Mathematical Methods and Models in Composites

Mathematical Methods and Models in Composites

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Computational and Experimental Methods in Structures

Series Editor: Ferri M. H. Aliabadi (Imperial College London, UK)

Vol. 1 Buckling and Postbuckling Structures: Experimental, Analytical and Numerical Studies edited by B. G. Falzon and M. H. Aliabadi (Imperial College London, UK)

Vol. 2 Advances in Multiphysics Simulation and Experimental Testing of MEMS edited by A. Frangi, C. Cercignani (Politecnico di Milano, Italy), S. Mukherjee (Cornell University, USA) and N. Aluru (University of Illinois at Urbana Champaign, USA)

Vol. 3 Multiscale Modeling in Solid Mechanics: Computational Approaches edited by U. Galvanetto and M. H. Aliabadi (Imperial College London, UK)

Vol. 4 Boundary Element Methods in Engineering and Sciences by M. H. Aliabadi (Imperial College, UK) and P. Wen (Queen Mary University of London, UK)

Vol. 5 Mathematical Methods and Models in Composites edited by V. Mantic (University of Seville, Spain)

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ICP Imperial College Press

Computational and Experimental Methods in Structures – Vol. 5

Mathematical Methods and Models in Composites

Vladislav ManticUniversity of Seville, Spain

Editor

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Computational and Experimental Methods in Structures — Vol. 5MATHEMATICAL METHODS AND MODELS IN COMPOSITES

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September 24, 2013 9:25 9in x 6in Mathematical Methods and Models in Composites b1507-fm

PREFACE

The range of engineering applications of composite materials and structureshas grown substantially in recent years and this has led to an increasingdemand for their modelling. The purpose of this volume is to presenta variety of mathematical methods and models, which may be useful incomposite design, certification, analysis and characterization, simulation ofcomposite manufacturing and in the prediction of damage and failureof composites in service. The chapters, written by well-known scholarsalongside promising young researchers — engineers, mathematicians andphysicists — review recent advances in such methods and models andtheir applications to the solution of challenging engineering problems incomposites.

This volume contains 11 chapters, organized and grouped according totopic. Some homogenization methods in composite materials and structuresare introduced and analysed in Chapters 1–2. Several equivalent single-layerand layer-wise models for multilayered plates and shells are reviewed inChapters 3–5, also including an analysis of instabilities. Stroh formalismis applied to wave-propagation analysis in anisotropic homogenizedcomposites in Chapter 6. Promising model reduction techniques, whichcan be employed in highly efficient simulations of composite manufacturingprocesses, are introduced in Chapter 7. The aim of the various approachesand computational tools presented in Chapters 8–11 is the prediction ofdamage and fracture initiation and propagation in composites on themicro-, meso- and macroscales. Comprehensive explanations and referencesmake all the chapters self-contained.

Chapter 1 by A. Kalamkarov reviews the multiscale asymptotichomogenization method and some of its applications to composite materialsand three-dimensional thin-walled composite reinforced structures.Analytical formulae for the effective elastic properties of an equivalent

v


vi Preface

homogeneous material are derived through the analytical solution of thecorresponding unit-cell problems. Examples of the application of themethod to generally orthotropic grid-reinforced composite shells, carbonnanotubes and sandwich composite shells with cellular cores are presented.

Chapter 2 by M. Ostoja-Starzewski and S.I. Ranganathan considersthe scale-dependent properties of random microstructures employing theframework of stochastic (micro)mechanics consistent with the mathematicalstatement of the Hill–Mandel condition. The scaling from a statisticaltowards a representative volume element is analysed and the scale-dependent bounds and scaling laws in planar conductivity, linear and non-linear (thermo)elasticity, plasticity and Darcy permeability are studied andillustrated by examples.

Chapter 3 by C. Hwu presents the Stroh-like formalism for thinanisotropic plates with stretching-bending coupling. This formalism is apowerful complex-variable method for the analysis of general symmetricand unsymmetric composite laminates. Extensions of this formalism tohygrothermal and electro-elastic problems in composite laminates are alsointroduced. Analytical solutions of several problems for laminates with holesand cracks are shown.

Chapter 4 by E. Carrera and M. Cinefra provides, first, a comprehensivereview of available classical, refined, ‘zig-zag’ and layer-wise models formultilayered plates and shells, paying particular attention to their historicalorigins and evolution and to the key relations between them. Then, a refinedfinite element formulation for multilayered shell structures based on the firstauthor’s unified formulation is developed and numerically tested on classicaldiscriminating problems showing good convergence and robustness.

Chapter 5 by D. Bigoni, M. Gei and S. Roccabianca applies an incre-mental bifurcation theory of prestressed elastic solids to analyse instabilitiesthat often lead to delamination in multilayers, for different deformationpaths including finite tension/compression and finite bending. It is shownthat several instabilities such as Euler buckling, necking, surface instabilityand a number of wavelike modes may occur in a multilayer.

Chapter 6 by K. Tanuma and C.-S. Man introduces the Stroh formalismfor anisotropic elastodynamics and shows some of its applications.A perturbation formula for the phase velocities of Rayleigh waves thatpropagate along the free surface of a prestressed orthotropic half-space isderived, and the possibility of determination of the prestress in homogenizedorthotropic or transversely-isotropic composites by boundary measurementof phase velocities of Rayleigh waves is examined.


Preface vii

Chapter 7 by F. Chinesta, A. Leygue and A. Poitou reviews modelreduction techniques that make a significant speed up of computationpossible for different engineering problems. In particular, proper generalizeddecomposition (PGD) is revisited and applied to solving problemsencountered in composite structure manufacturing such as processoptimization and inverse analysis, coupling between global transport modelsand local kinetics in a multiscale framework and, finally, in a fully three-dimensional solution of models defined for plate geometries with two-dimensional computational complexity.

Chapter 8 by C.G. Davila, C.A. Rose and E.V. Iarve re-examinesthe most common modelling strategies for the prediction of localized anddistributed damage in laminated composites with an emphasis on the scaleof the damage idealization and size effects. The capabilities of cohesive lawsto represent crack initiation and propagation and the intrinsic limitationsof continuum damage models for modelling laminated composites areanalysed. An x-FEM technique modelling damage propagation by insertingcohesive cracks in arbitrary directions is introduced and its capabilities foravoiding some of the limitations of continuum damage models are shown.

Chapter 9 by T. Roubıcek, M. Kruzık and J. Zeman describes a generaland rigorous mathematical framework based on the so-called energeticsolutions for quasistatic and rate-independent processes of delaminationand debonding in composites. The process evolution is assumed to begoverned by the minimum-energy and minimum dissipation-potentialprinciples written in terms of a time-dependent Gibbs-type stored energyfunctional and a dissipation potential. This eventually leads to the solutionof a time-incremental variational problem. The methodology developedis applied to macroscopic delamination and microscopic fibre-matrixdebonding problems.

Chapter 10 by D. Leguillon and E. Martin presents a coupled criterion,in the framework of finite fracture mechanics, allowing prediction of thenucleation of a crack of finite length at stress concentration points. Thiscriterion uses conditions for energy and tensile stress and involves boththe fracture toughness and tensile strength of the material. It is appliedto a semi-analytical study of a transverse crack in a layer approachingthe interface with an adjacent layer, where this crack can stop, deflectoriginating a delamination crack or cross the interface.

Chapter 11 by V. Mantic, A. Barroso and F. Parıs introduces, first,a general semi-analytic procedure for the evaluation of singular stressesat anisotropic elastic multimaterial corners, also covering the case of


viii Preface

sliding frictional contact surfaces. Then, a least-squares fitting techniquefor extracting generalized stress intensity factors at such corners from FEMand BEM results is presented and used for the singularity analysis of a crackterminating at a ply interface in a laminate, and a bimaterial corner in adouble-lap joint. Finally, a criterion for failure initiation at the corner tipis proposed and a new experimental procedure with a modified Braziliandisc specimen for the determination of the corresponding failure envelopeis introduced and tested.

I would like to thank all the reviewers who recommended numerousimprovements to the original manuscripts. I am greatly indebted to all thecontributors to this book and I want to thank them for their efforts inproducing the final excellent versions of the chapters and for their patiencein waiting for the book submission and presentation. Special thanks go tothe Series Editor, Prof Ferri Aliabadi, for inviting me to edit this book andfor his professional advice.

V. ManticJanuary 2012


CONTRIBUTORS

Dr Alberto BarrosoSchool of EngineeringUniversity of SevilleSeville, Spain

Prof Davide BigoniDepartment of Mechanical and Structural EngineeringUniversity of TrentoTrento, Italy

Prof Erasmo CarreraAerospace DepartmentPolitecnico di TorinoTorino, Italy

Prof Francisco ChinestaEADS Corporate Foundation International ChairGEM, UMR CNRS-ECNNantes, France

Dr Maria CinefraAerospace DepartmentPolitecnico di TorinoTorino, Italy

Dr Carlos G. DavilaNASA Langley Research CenterHampton, Virginia, USA

ix


x Contributors

Dr Massimiliano GeiDepartment of Mechanical and Structural EngineeringUniversity of TrentoTrento, Italy

Prof Chyanbin HwuInstitute of Aeronautics and AstronauticsNational Cheng Kung UniversityTainan, Taiwan, Republic of China

Dr Endel V. IarveAir Force Research LaboratoryWright Patterson AFBOhio, USAUniversity of Dayton Research InstituteDayton, Ohio, USA

Prof Alexander L. KalamkarovDepartment of Mechanical EngineeringDalhousie UniversityHalifax, Nova Scotia, Canada

Dr Martin KruzıkDepartment of Physics, Faculty of Civil EngineeringCzech Technical UniversityPrague, Czech Republic

Prof Dominique LeguillonInstitut Jean Le Rond d’Alembert CNRS UMR 7190Universite Pierre et Marie CurieParis, France

Dr Adrien LeygueEADS Corporate Foundation International ChairGEM, UMR CNRS-ECNNantes, France

Prof Chi-Sing ManDepartment of MathematicsUniversity of KentuckyLexington, Kentucky, USA


Contributors xi

Prof Vladislav ManticSchool of EngineeringUniversity of SevilleSeville, Spain

Prof Eric MartinLaboratoire des Composites Thermo-Structuraux CNRS UMR 5801Universite BordeauxPessac, France

Prof Martin Ostoja-StarzewskiDepartment of Mechanical Science and EngineeringBeckman Institute, and Institute for Condensed Matter TheoryUniversity of Illinois at Urbana-ChampaignUrbana, Illinois, USA

Prof Federico ParısSchool of EngineeringUniversity of SevilleSeville, Spain

Prof Arnaud PoitouEADS Corporate Foundation International ChairGEM, UMR CNRS-ECNNantes, France

Dr Shivakumar I. RanganathanDepartment of Mechanical EngineeringAmerican University of SharjahSharjah, United Arabic Emirates

Dr Sara RoccabiancaDepartment of Mechanical and Structural EngineeringUniversity of TrentoTrento, Italy

Dr Cheryl A. RoseNASA Langley Research CenterHampton, Virginia, USA


xii Contributors

Prof Tomas RoubıcekMathematical Institute, Charles UniversityInstitute of Thermomechanics and Institute of Information Theory

and AutomationAcademy of Sciences of the Czech RepublicPrague, Czech Republic

Prof Kazumi TanumaDepartment of Mathematics, Graduate School of EngineeringGunma UniversityKiryu, Japan

Dr Jan ZemanDepartment of Mechanics, Faculty of Civil EngineeringCzech Technical UniversityPrague, Czech Republic


CONTENTS

Preface vContributors ix

1. Asymptotic Homogenization Methodand Micromechanical Models for CompositeMaterials and Thin-Walled Composite Structures 1

Alexander L. Kalamkarov

2. Scaling and Homogenization in Spatially Random Composites 61

Martin Ostoja-Starzewski and Shivakumar I. Ranganathan

3. Stroh-Like Formalism for General Thin LaminatedPlates and its Applications 103

Chyanbin Hwu

4. Classical, Refined, Zig-Zag and Layer-Wise Modelsfor Laminated Structures 135

Erasmo Carrera and Maria Cinefra

5. Bifurcation of Elastic Multilayers 173

Davide Bigoni, Massimiliano Gei and Sara Roccabianca

6. Propagation of Rayleigh Waves in Anisotropic Mediaand an Inverse Problem in the Characterization of Initial Stress 209

Kazumi Tanuma and Chi-Sing Man

xiii


xiv Contents

7. Advanced Model Order Reduction for SimulatingComposite-Forming Processes 247

Francisco Chinesta, Adrien Leygue and Arnaud Poitou

8. Modeling Fracture and Complex Crack Networksin Laminated Composites 297

Carlos G. Davila, Cheryl A. Rose and Endel V. Iarve

9. Delamination and Adhesive Contact Modelsand their Mathematical Analysis and Numerical Treatment 349

Tomas Roubıcek, Martin Kruzık and Jan Zeman

10. Crack Nucleation at Stress Concentration Pointsin Composite Materials — Application to CrackDeflection by an Interface 401

Dominique Leguillon and Eric Martin

11. Singular Elastic Solutions in Anisotropic MultimaterialCorners. Applications to Composites 425

Vladislav Mantic, Alberto Barroso and Federico Parıs

Index 497

September 24, 2013 9:25 9in x 6in Mathematical Methods and Models in Composites b1507-ch01

Chapter 1

ASYMPTOTIC HOMOGENIZATION METHODAND MICROMECHANICAL MODELS FOR

COMPOSITE MATERIALS AND THIN-WALLEDCOMPOSITE STRUCTURES

Alexander L. Kalamkarov

Department of Mechanical Engineering, Dalhousie UniversityHalifax, Nova Scotia, B3H 4R2, Canada

Abstract

The basics of the multiscale asymptotic homogenization method and itsapplication to the analysis of composite materials and thin-walled compositestructures are presented. The pertinent micromechanical models are developedand analytical formulae for the effective properties are obtained. Theasymptotic homogenization technique is applied to the analysis of three-dimensional (3D) grid-reinforced composites with generally orthotropicreinforcement materials. An analytical solution of the corresponding unit-cellproblem is obtained and explicit analytical formulae for the effective elasticproperties of 3D grid-reinforced composites of various structures are derived.The asymptotic homogenization of 3D thin-walled composite reinforcedstructures is presented, and the general homogenization model for a compositeshell is introduced. It is applied to the analysis of composite reinforcedshells and plates of practical importance, including rib- and wafer-reinforcedshells, orthotropic grid-reinforced composite shells and plates, and sandwichcomposite shells with cellular cores of different geometrical configurations.In particular, one of the considered examples demonstrates micromechanicalmodelling of carbon nanotubes. Analytical expressions for the effective stiffnessmoduli of these composite reinforced shells and plates are presented.

1.1 Introduction

The rapidly increasing popularity of composite materials and structuresin recent years can been seen through their incorporation in wide-rangingengineering applications. In particular, advanced composites are used to

1


2 A.L. Kalamkarov

reinforce and monitor components in civil and structural engineering(see, e.g., Kalamkarov et al. [1, 2]), in aerospace, automotive and marineengineering components of all sizes, medical prosthetic devices, sportsand recreational goods and others. Success in the practical applicationof composites largely depends on the ability to predict their mechanicalproperties and behaviour through the development of appropriatemechanical models. The micromechanical modelling of compositestructures, however, can be rather complicated as a result of the distributionand orientation of the multiple inclusions and reinforcements within thematrix, and their mechanical interactions on a local (micro)level. Therefore,it is important to establish micromechanical models that are neither toocomplicated to be developed and applied, nor so simple that they cannotreflect the real mechanical properties and behaviour of the compositematerials and structures.

At present, asymptotic techniques are applied in many cases tothe micromechanics of composites. Various asymptotic approaches tothe analysis of composite materials have apparently reached theirconclusion within the framework of the mathematical theory of asymptotichomogenization. Indeed, the proof of the possibility of homogenizing acomposite material with a regular structure, i.e. of examining an equivalenthomogeneous solid instead of the original inhomogeneous composite solid,is one of the principal results of this theory. The theory of homogenizationhas also indicated a method of transition from the original problem(which contains in its formulation a small parameter related to the smalldimensions of the constituents of the composite) to a problem for ahomogeneous solid. The effective properties of this equivalent homogeneousmaterial are determined through the solution of so-called local problemsformulated for the unit cell of the composite material. These solutions alsoenable the calculation of local stresses and strains in the composite material.

In the present chapter we will review the basics of asymptotichomogenization in Section 1.2 and consider the formulation of unit-cellproblems in Section 1.3. In Section 1.4, the asymptotic homogenizationtechnique is applied to the analysis of 3D grid-reinforced composites withgenerally orthotropic reinforcement materials. An analytical solution ofthe corresponding unit-cell problems is obtained and explicit analyticalformulae for the effective elastic properties of 3D grid-reinforced compositesof various structures are derived. The general homogenization compositeshell model is presented in Section 1.5. It is applied to the analysisof thin-walled composite structures, including rib- and wafer-reinforced


Asymptotic Homogenization Method and Micromechanical Models 3

shells, and sandwich composite shells with honeycomb fillers. Analyticalexpressions for the effective stiffness moduli of these composite reinforcedshells are obtained. Section 1.6 discusses the application of the generalhomogenization composite shell model to the analysis of generallyorthotropic grid-reinforced composite shells. Analytical solutions of thepertinent unit-cell problems are obtained, and used in Section 1.7 toderive formulae for the effective stiffness moduli of practically importanttypes of grid-reinforced composite shells with orthotropic reinforcements.In particular, one of the examples considered in Section 1.7 demonstratesanalytical modelling of the mechanical behaviour of carbon nanotubes.Finally, Section 1.8 presents the application of the general homogenizationcomposite shell model to the analysis of sandwich composite shells withcellular cores of different geometrical configurations.

The present chapter is largely based on research results obtained bythe author and his graduate students.

1.2 Asymptotic Homogenization Method

For the past 25 years, asymptotic homogenization methods have provento be powerful techniques for the study of heterogeneous media. Someof these classical tools today include multiscale expansions [3–8], G- andΓ-convergence [9, 10] and energy methods [11, 12].

An approach based on Fourier analysis has been proposed in [13, 14].This method works in the following way. First, the original operator istransformed into an equivalent operator in the Fourier space. StandardFourier series are used to expand the coefficients of the operator and aFourier transform is used to decompose the integrals. Next, the Fouriertransforms of the integrals are expanded using a suitable two-scaleexpansion and the homogenized problem is finally derived by merelyneglecting high-order terms in the above expansions when moving to thelimit as the period tends to zero.

The method of orientational averaging was proposed in [15]. It is basedon the following assumptions: a characteristic volume (repeated throughoutthe bulk of the composite) is isolated from the composite medium. Theproperties of the composite as a whole are assumed to be the same as thoseof this characteristic volume. In the case of ideally straight fibres, the setof fibres is represented in the form of an array of unidirectional reinforcedcylinders. The papers on homogenization using wavelet approximations [16]and non-smooth transformations [17] should also be mentioned.


4 A.L. Kalamkarov

In this section we describe a variant of the asymptotic homogenizationapproach that will be used later. For simplicity, we will start with a two-dimensional (2D) heat conduction problem. However, these results willremain valid for other kinds of transport coefficients such as electricalconductivity, diffusion, magnetic permeability, etc. Due to the well-knownlongitudinal shear–transverse conduction analogy, see [18], the elasticantiplane shear deformation can also be evaluated in a similar mathematicalway. This will be followed by a summary of asymptotic homogenizationapplied to an elasticity problem for a 3D composite solid. An analogousasymptotic homogenization technique has been developed for a number ofmore complicated non-linear models, see [3, 5, 11].

Let us consider a transverse transport process through a periodiccomposite structure, when the fibres are arranged in a periodic squarelattice, as shown in Fig. 1.1.

The characteristic size l of the inhomogeneities is assumed to be muchsmaller than the global size L of the whole structure: l L. Assumingperfect bonding conditions on the interface ∂Ω between the constituents,the governing boundary-value problem can be written as follows:

ka(∂2ua

∂x21

+∂2ua

∂x22

)= −fa in Ωa, um = uf ,

km∂um

∂n= kf

∂uf

∂non ∂Ω. (1.1)

Here and in the following, variables indexed by m correspond to the matrix,and those indexed by f correspond to the fibres, index a takes both of thesereferences: a = m or a = f . Generally, the boundary-value problem (1.1)has a number of different physical interpretations, but here it is discussed

l

fibre

matrix

x2

x1

Fig. 1.1. Composite material with hexagonal array of cylindrical fibres.



with reference to heat conduction. Then, in the above expressions, ka are theheat conductivities of the constituents, ua is a temperature distribution, fa

is a density of heat sources and ∂/∂n is a derivative in the normal directionto the interface ∂Ω. Let us now consider the governing boundary-valueproblem (1.1) using the asymptotic homogenization method [3–8]. We willintroduce a dimensionless small parameter ε = l/L, ε 1, characterizingthe rate of heterogeneity of the composite structure.

In order to separate the micro- and macroscale components of thesolution we introduce the so-called slow (x ) and fast (y) coordinates

y1 =x1

ε, y2 =

x2

ε(1.2)

and we express the temperature field in the form of an asymptoticexpansion:

ua = u0(x) + εua1(x,y) + ε2ua2(x,y) + · · · , (1.3)

where x = x1e1+x2e2, y = y1e1+y2e2, and e1 and e2 are the Cartesian unitvectors. The first term u0(x) of expansion (1.3) represents the homogeneouspart of the solution; it changes slowly within the whole domain of thematerial and does not depend on the fast coordinates. All the further termsuai (x,y), i = 1, 2, 3, . . . , describe the local variation of the temperaturefield on the scale of the heterogeneities. In the perfectly regular case theperiodicity of the medium induces the same periodicity for uai (x,y) withrespect to the fast variables:

uak(x,y) = uak(x,y + Lp), (1.4)

where Lp = ε−1lp, lp = p1l1 + p2l2, ps = 0,±1,±2, . . . , and l1 and l2 arethe fundamental translation vectors of the square lattice.

The spatial derivatives are defined as follows:

∂

∂x1→ ∂

∂x1+ ε−1 ∂

∂y1,

∂

∂x2→ ∂

∂x2+ ε−1 ∂

∂y2. (1.5)

Substituting expressions (1.2), (1.3) and (1.5) into the governing boundary-value problem (1.1) and splitting it with respect to equal powers of ε onecomes to a recurrent sequence of problems:

∂2ua1∂y2

1

+∂2ua1∂y2

2

= 0 in Ω, [um1 = uf1 ]|∂Ω,

[km

∂um1∂m

− kf∂uf1∂m

= (kf − km)∂u0

∂n

]∣∣∣∣∣∂Ω

;

(1.6)


6 A.L. Kalamkarov

ka(∂2u0

∂x21

+∂2u0

∂x22

+ 2∂2ua1∂x1∂y1

+ 2∂2ua1∂x2∂y2

+∂2ua2∂y2

1

+∂2ua2∂y2

2

)

= −fa in Ω, (1.7)

[um2 = uf2 ]|∂Ω,

[km

∂um2∂m

− kf∂uf2∂m

= kf∂uf1∂n

− km∂um1∂n

]∣∣∣∣∣∂Ω

;

and so on.

Here ∂/∂m is a derivative in the normal direction to the interface ∂Ω inthe fast coordinates y1, y2.

The boundary-value problem (1.6) allows evaluation of the higher-order component uai (x,y) of the temperature field; owing to the periodicitycondition (1.4) it can be considered within only one periodically repeatedunit cell. It follows from the boundary-value problem (1.6) that the variablesx and y can be separated in u1(x,y) by assuming

u1(x,y) =∂u0(x)∂xl

U1(y) +∂u0(x)∂x2

U2(y), (1.8)

where U1(y) and U2(y) are local functions for which problem (1.6) yieldsthe following unit-cell problems:

∂2U1(y)∂y2

1

+∂2U1(y)∂y2

2

= 0,∂2U2(y)∂y2

1

+∂2U2(y)∂y2

2

= 0 in Ω,

Um1 (y) = Uf1 (y), Um2 (y) = Uf2 (y) on ∂Ω,

(1.9)

km∂Um1 (y)∂m

− kf∂Uf1 (y)∂m

= (kf − km)n1,

km∂Um2 (y)∂m

− kf∂Uf2 (y)∂m

= (kf − km)n2 on ∂Ω.

The effective heat conductivity can be determined from the boundary-value problem (1.7). The following homogenization operator over the unit-cell area Ω0 will be applied to Eq. (1.7):[∫∫

Ωm0

(· · ·)dy1dy2 +∫∫

Ωin0

(· · ·)dy1dy2]L−2,

where Ωm0 and Ωin0 denote unit-cell areas occupied by the matrix andinclusion, respectively.



Terms containing ua2 will be eliminated by means of Green’s theoremand taking into account the boundary conditions (1.7) and the periodicitycondition (1.7), which yields:

[(1 − c)km + ckf ](∂2u0

∂x21

+∂2u0

∂x22

)

+km

L2

∫∫Ωm

0

(∂2um1∂x1∂y1

+∂2um1∂x2∂y2

)dy1dy2

+kf

L2

∫∫Ωin

0

(∂2uf1∂x1∂y1

+∂2uf1∂x2∂y2

)dy1dy2

= −[(1 − c)fm + cff ],

(1.10)

where c is the fibre volume fraction.The homogenized heat conduction equation can be obtained by

substituting expression (1.8) for u1(x,y) into Eq. (1.10), which yields

〈kij〉∂u20(x)

∂xi∂xj= −〈f〉 (1.11)

〈kij〉 = [(1 − c)km + ckf ]δij +km

L2

∫∫Ωm

0

∂Umj∂yi

dy1dy2

+kf

L2

∫∫Ωin

0

∂Ufj∂yi

dy1dy2, (1.12)

where 〈f〉 = (1 − c)fm + cff is the effective density of heat sources; δijis Kronecker’s delta; indexes i, j, l = 1, 2; and the summation over therepeated indexes is implied.

Note that in general the homogenized material will be anisotropic, and〈kij〉 in Eq. (1.11) is a tensor of effective coefficients of heat conductivity.Tensor 〈kij〉 is defined by the expression (1.12), and it can be readilycalculated as soon as the unit-cell problems (1.9) are solved and the localfunctions U1(y) and U2(y) are found. The unit-cell problems (1.9) canbe solved analytically or numerically. The approximate methods of theiranalytical solution will be presented below in a number of practicallyimportant cases.

Let us now consider the asymptotic homogenization of an elasticityproblem for a 3D periodic composite material occupying region Ω with aboundary S; see Fig. 1.2.


8 A.L. Kalamkarov

(b)

y3

Matrix

y2

Reinforcement

y1

x1

(a)

x2

Reinforcement

Unit cell, Y

Boundary, S

x3

Domain,

ε

Ω

Fig. 1.2. (a) Three-dimensional periodic composite structure, (b) unit cell, Y .

We assume that the region Ω is made up by the periodic repetitionof the unit cell Y in the form of a parallelepiped with dimensions εYi, i =1, 2, 3. The elastic deformation of this composite solid is described by thefollowing boundary-value problem:

∂σεij∂xj

= fi in Ω, uε(x) = 0 on S, (1.13)

σεij = Cijkleεkl, eεij =

12

(∂uεi∂xj

+∂uεj∂xi

)in Ω, (1.14)

where Cijkl, is a tensor of elastic coefficients. The coefficients Cijkl areassumed to be periodic functions with a unit cell Y . Here and in thefollowing all Latin indexes assume values 1, 2, 3, and repeated indexesare summed.

The introduction of the fast variables yi = xi

ε, i = 1, 2, 3, similar to

Eq. (1.2), into Eqs. (1.13) and (1.14) and the rule of differentiation (1.5)leads to the following boundary-value problem:

∂σεij∂xj

+1ε

∂σεij∂yj

= fi in Ω, uε(x,y) = 0 on S, (1.15)

σεij(x,y) = Cijkl(y)∂uεk∂xl

(x,y) in Ω. (1.16)

The next step is to expand the displacements and as a result the stressesinto the asymptotic expansions in powers of the small parameter ε, as forexpansion (1.3):

uε(x,y) = u(0)(x,y) + εu(1)(x,y) + ε2u(2)(x,y) + · · · (1.17)

σεij(x,y) = σ(0)ij (x,y) + εσ

(1)ij (x,y) + ε2σ

(2)ij (x,y) + · · · , (1.18)



where all the above functions are periodic in y with the unit cell Y .Substituting Eqs. (1.17) and (1.18) into Eqs. (1.15) and (1.16), whileconsidering at the same time the periodicity of u(i) in y , reveals that u(0) isindependent of the fast variable y ; see [5] for details. Subsequently, equatingterms with similar powers of ε results in the following set of equations:

∂σ(0)ij (x,y)∂yj

= 0, (1.19)

∂σ(1)ij (x,y)∂yj

+∂σ

(0)ij (x,y)∂xj

= fi, (1.20)

where

σ(0)ij = Cijkl

(∂u(0)

k

∂xl+∂u(1)

k

∂yl

), (1.21)

σ(1)ij = Cijkl

(∂u(1)

k

∂xl+∂u(2)

k

∂yl

). (1.22)

Substitution of Eq. (1.21) into Eq. (1.19) yields:

∂

∂yj

(Cijkl

∂u(1)k (x,y)∂yl

)= −∂Cijkl(y)

∂yj

∂u(0)k (x)∂xl

. (1.23)

Due to the separation of variables in the right-hand side of Eq. (1.23) thesolution of Eq. (1.23) can be written as follows, as with Eq. (1.8):

u(1)n (x,y) =

∂u(0)k (x)∂xl

Nkln (y), (1.24)

where Nkln (y)(n, k, l = 1, 2, 3) are periodic functions with a unit cell Y

satisfying the following equation:

∂

∂yj

(Cijmn(y)

∂Nklm(y)∂yn

)= −∂Cijkl

∂yj. (1.25)

It is observed that Eq. (1.25) depends only on the fast variable y and itis entirely formulated within the unit cell Y . Thus, the problem (1.25)is appropriately called an elastic unit-cell problem. Note that insteadof boundary conditions, this problem has a condition of a periodiccontinuation of functions Nkl

m(y).


10 A.L. Kalamkarov

If inclusions are perfectly bonded to the matrix on the interfacesof the composite material, then the functions Nkl

m(y) together with theexpressions [(Cijkl + Cijmn(y)∂Nkl

m(y)∂yn

)n(c)j ], i = 1, 2, 3, must be continuous

on the interfaces. Here, n(c)j are the components of the unit normal to the

interface.The next important step in the homogenization process is achieved

by substituting Eq. (1.24) into Eqs. (1.21) and (1.22), and the resultingexpression into Eq. (1.20). The result is then integrated over the domain Yof the unit cell (with volume |Y |), remembering to treat x as a parameteras far as integration with respect to y is concerned. After cancelling outterms that vanish due to the periodicity, we obtain the homogenized globalproblem

Cijkl∂2u(0)

k (x)∂xj∂xl

= fi in Ω, u(0)(x) = 0 on S, (1.26)

where the following notation is introduced:

Cijkl =1|Y|∫

Y

(Cijkl(y) + Cijmn(y)

∂Nklm

∂yn

)dv. (1.27)

Similarly, substitution of Eq. (1.24) into Eq. (1.21) and then integratingthe resulting expression over the domain of the unit cell Y yields:

⟨σ

(0)ij

⟩=

1|Y|∫

Y

σ(0)ij (y)dv = Cijkl

∂u(0)k

∂xl. (1.28)

Equations (1.26) and (1.28) represent the homogenized elasticityboundary-value problem. The coefficients Cijkl given by Eq. (1.27) arethe effective elastic coefficients of the homogenized material. They arereadily determined as soon as the unit-cell problem (1.25) is solved and thefunctions Nkl

m(y) are found. It is observed that these effective coefficients arefree from the complications that characterize the original rapidly varyingelastic coefficients Cijkl(y). They are universal for a composite materialunder study, and can be used to solve a wide variety of boundary-valueproblems associated with the given composite material.

It should be noted that while asymptotic homogenization leads to amuch simpler problem for an equivalent homogeneous material with certaineffective properties, the construction of a solution in the vicinity of theboundary S of the composite solid, i.e. at the distances of the order ofε, remains beyond the capabilities of classical homogenization. In order



to determine the stresses and strains near the boundary, a boundary-layer problem should be considered as an extension to the asymptotichomogenization. A boundary-layer method in asymptotic homogenizationwas developed by Kalamkarov [5]. This approach was further developedby Kalamkarov and Georgiades [19] in the asymptotic homogenization ofsmart periodic composites. The exponential decaying of boundary layerswas proved in Oleynik et al. [10] for problems with a simple geometry.

New generalized integral transforms for the analytical solution of theboundary-value problems for composite materials have been developed byKalamkarov [5, 20], and Kalamkarov et al. [21].

The properties of boundary layers in periodic homogenization inrectangular domains that are either fixed or have an oscillating boundaryare investigated in [22]. Such boundary layers are highly oscillating nearthe boundary and decay exponentially fast in the interior to a non-zerolimit, which the authors called a boundary-layer tail. It is shown that theseboundary-layer tails can be incorporated into the homogenized equation byadding dispersive terms and a Fourier boundary condition. Although findingthe explicit analytical solutions of boundary-layer problems in the theoryof homogenization still remains an open problem, the effective numericalprocedures have been proposed in [23, 24].

1.3 Unit-Cell Problems

As we have seen in Section 1.2, the derivation of the homogenized equationsfor the periodic composites includes solution of the unit-cell problems (1.9)or (1.25). In some particular cases these problems can be solved analyticallyproducing exact solutions, for example for laminated composites and grid-reinforced structures; see [5, 25–27]. The explicit formulae for the effectivemoduli are very useful, especially for the design and optimization ofcomposite materials and structures [27, 28]. But in the general case, theunit-cell problems cannot be solved analytically and therefore numericalmethods should be used. In some cases, approximate analytical solutionsof the unit-cell problems can be found, and explicit formulae for theeffective coefficients can be obtained due to the presence of additional smallparameters within the unit cell, not to be confused with the small parameterof inhomogeneity.

For a small volume fraction of inclusions, c cmax, one can use thethree-phase model [29–31]. It is based on the following assumption: theperiodically heterogeneous composite structure is approximately replaced


12 A.L. Kalamkarov

by a three-phase medium consisting of a single inclusion, a matrix layerand an infinite effective medium with homogenized mechanical properties.An asymptotic justification of the three-phase composite model is givenin [29].

For laminated composite materials, the unit-cell problems (1.9) and(1.25) are one dimensional and they can be solved analytically. Using thisanalytical solution, the effective properties of laminated composites can beobtained in an explicit analytical form from Eqs. (1.12) and (1.27); see[5, 27]. In the more complicated case of generally anisotropic constituentmaterials, explicit formulae for the effective elastic, actuation, thermalconductivity and hygroscopic absorption properties of laminated smartcomposites have been derived by Kalamkarov and Georgiades [32]. Inparticular the following explicit formula for the effective elastic coefficientsof a laminated composite in the case of generally anisotropic constituentmaterials is derived in [32]:

Cijkl = 〈Cijkl〉 − 〈Cijm3C−1m3q3Cq3kl〉

+ 〈Cijm3C−1m3q3〉〈C−1

q3p3〉−1〈C−1p3n3Cn3kl〉, (1.29)

where the angle brackets denote a rule of mixture, and as earlier indicatedall Latin indexes assume values 1,2,3, and repeated indexes are summed.

For fibre-reinforced periodic composites the unit-cell problem (1.25)becomes two dimensional, and it can be solved analytically for some simplegeometries, or numerically; see [5, 33, 34].

1.4 Three-Dimensional Grid-Reinforced Composites

In this section, we will apply the above described asymptotichomogenization technique to the analysis of a 3D composite structurereinforced with N families of reinforcements. An example of such astructure with three families of mutually perpendicular reinforcementsis shown in Fig. 1.3 [35]. We assume that the members of each familyare made of individual, generally orthotropic materials and have relativeorientation angles θn1 , θ

n2 , θ

n3 (where n = 1, 2, . . . , N) with the y1, y2, y3 axes

respectively.It is further assumed that the orthotropic reinforcements have

significantly higher elasticity moduli than the matrix material, so weare justified in neglecting the contribution of the matrix phase in theanalytical treatment. Clearly, for the particular case of framework or lattice



Fig. 1.3. (a) Cubic grid-reinforced structure and (b) its unit cell.

(a) (b)

2

1

3

y3

y2

y1

ηη

η

Fig. 1.4. Unit cell for a single reinforcement family in original (a) and rotated(b) microscopic coordinates.

network structures the surrounding matrix is absent and this is modelledby assuming zero matrix rigidity.

The nature of the network structure of Fig. 1.3 is such that it wouldbe more efficient if we first considered a simpler type of unit cell madeof only a single reinforcement as shown in Fig. 1.4. Having solved this,the effective elastic coefficients of more general structures with severalfamilies of reinforcements can be determined by the superposition of thefound solution for each of them separately. In following this procedure,one must naturally accept the error incurred at the regions of intersectionbetween the reinforcements. However, our approximation will be quiteaccurate because these regions of intersection are highly localized and do


14 A.L. Kalamkarov

not contribute significantly to the integral over the entire volume of theunit cell. A mathematical justification for this argument in the form of theso-called principle of the split homogenized operator can be found in [4].

In order to calculate the effective coefficients of the simpler structureof Fig. 1.4, the unit-cell problem given by Eq. (1.25) must be solved andsubsequently Eq. (1.27) must be applied. The problem formulation for thestructure shown in Fig. 1.4 begins with the introduction of the followingnotation [5, 35]:

bklij = Cijmn(y)∂Nkl

m (y)∂yn

+ Cijkl . (1.30)

With this definition in mind, the unit cell of the problem given by Eq. (1.25)can be solved as:

bmmmm = Cmmmm +

λmm1 Cm1q21 + Cm6q22 + Cm5q23+λmm2 Cm1q31 + Cm6q32 + Cm5q33+λmm3 Cm6q21 + Cm2q22 + Cm4q23+λmm4 Cm6q31 + Cm2q32 + Cm4q33+λmm5 Cm5q21 + Cm4q22 + Cm3q23+λmm6 Cm5q31 + Cm4q32 + Cm3q33

(1.31)

bmnmn = Cmnmn +

λmm1 Cmn11q21 + Cmn12q22 + Cmn13q23+λmn2 Cmn11q31 + Cmn12q32 + Cmn13q33+λmm3 Cmn12q21 + Cmn22q22 + Cmn23q23+λmn4 Cmn12q31 + Cmn22q32 + Cmn23q33+λmm5 Cmn13q21 + Cmn23q22 + Cmn33q23+λmm6 Cmn13q31 + Cmn23q32 + Cmn33q33

,

(1.32)

where there is no summation on either index m or n. The CIJ

(I, J = 1, 2, 3, . . . , 6) in Eq. (1.31) are the elastic coefficients of theorthotropic reinforcements in the contracted notation; see, e.g., [5, 36].These components are obtained from Cijkl by the following replacementof subscripts: 11 → 1, 22 → 2, 33 → 3, 23 → 4, 13 → 5, 12 → 6.The resulting CIJ are symmetric, i.e., CIJ = CJI . The coefficients qij inEqs. (1.31) and (1.32) represent the components of the matrix of direction



cosines characterizing the axes of rotation in Fig. 1.4. The constants λkli inEqs. (1.31) and (1.32) satisfy the following linear algebraic equations:

A1λkl1 +A2λ

kl2 +A3λ

kl3 +A4λ

kl4 +A5λ

kl5 +A6λ

kl6 +Akl7 = 0

A8λkl1 +A9λ

kl2 +A10λ

kl3 +A11λ

kl4 +A12λ

kl5 +A13λ

kl6 +Akl14 = 0

A15λkl1 +A16λ

kl2 +A17λ

kl3 +A18λ

kl4 +A19λ

kl5 +A20λ

kl6 +Akl21 = 0

A22λkl1 +A23λ

kl2 +A24λ

kl3 +A25λ

kl4 +A26λ

kl5 +A27λ

kl6 +Akl28 = 0

A29λkl1 +A30λ

kl2 +A31λ

kl3 +A32λ

kl4 +A33λ

kl5 +A34λ

kl6 +Akl35 = 0

A36λkl1 +A37λ

kl2 +A38λ

kl3 +A39λ

kl4 +A40λ

kl5 +A41λ

kl6 +Akl42 = 0,

(1.33)

where Akli are constants that depend on the geometric parameters of theunit cell and the material properties of the reinforcement. The explicitexpressions for these constants can be found in [35]. Once the system inEq. (1.33) is solved, the obtained coefficients λkli are substituted back intoEqs. (1.31) and (1.32) to determine the bklij coefficients. In turn these areused to calculate the effective elastic coefficients of the 3D grid-reinforcedcomposite structures by integrating over the volume of the unit cell.

The effective elastic moduli of the 3D grid-reinforced composite withgenerally orthotropic reinforcements with a unit cell shown in Fig. 1.4 areobtained on the basis of Eq. (1.27), which, on account of notation (1.30),becomes:

Cijkl =1|Y |∫Y

bklijdv. (1.34)

Noting that the bklij are constants in the considered case, and denotingthe length and cross-sectional area of the reinforcement (in coordinatesy1, y2, y3) by L and A respectively, and the volume of the unit cell by V ,the effective elastic coefficients become

Cijkl =ALVbklij = Vf b

klij , (1.35)

where Vf is the volume fraction of the reinforcement within the unit cell.For structures with more than one family of reinforcements (a particularcase of which is shown in Fig. 1.3) the effective moduli can be obtained bysuperposition. The influence of a fibre coating on the mechanical propertiesof fibre-reinforced composites was analysed in [37].


16 A.L. Kalamkarov

1.4.1 Examples of 3D grid-reinforced composite structures

Let us now apply the above asymptotic homogenization model to calculatethe effective elastic coefficients for three different examples of 3D grid-reinforced composite structures. We will also compare the analytical(asymptotic homogenization) results with numerical (finite element)calculations. We will consider consequently a grid-reinforced structure withtwo families of mutually perpendicular reinforcements (structure A1 shownin Fig. 1.5); a 3D model with three mutually perpendicular reinforcementsoriented along the three coordinate axes (structure A2 shown in Fig. 1.3);and, finally, a 3D structure with a rhombic arrangement of orthotropicreinforcements: two reinforcements are oriented in the y1−y2 plane at 45 toone another with a third reinforcement oriented along the y3 axis (structureA3 shown in Fig. 1.6).

The properties of the orthotropic reinforcement and isotropic matrixmaterials are listed in Table 1.1.

The results are shown below in Figs. 1.7–1.9. Figure 1.7 shows thevariation of the effective elastic coefficient E1 = E3 for the structure A1 vs.the total reinforcement volume fraction. Three different lines are shownin this figure. The first line represents the asymptotic homogenization(AHM) results, the second line represents the finite element (FEM)results considering the reinforcement contribution only (i.e. neglectingthe matrix), and the third line represents the FEM results with boththe reinforcement and matrix contributions; see [38] for details. Certaininteresting observations are apparent from Fig. 1.7. First of all the highdegree of conformity between the first and second lines validates the

Fig. 1.5. (a) Grid-reinforced composite structure A1, with reinforcements oriented alongthe y1 and y2 directions. (b) Unit cell of structure A1.



Fig. 1.6. (a) 3D grid-reinforced composite structure A3, with reinforcements arrangedin a rhombic fashion. (b) Unit cell of structure A3.

Table 1.1. Material properties.

Material properties of carbon reinforcement

E1 E2 E3 G12 G13 G23 ν12 ν13 ν23

173.0 GPa 33.1 GPa 5.2 GPa 9.4 GPa 8.3 GPa 3.2 GPa 0.036 0.25 0.171

Material properties of epoxy matrix E ν3.6 GPa 0.35

Eff

ecti

ve e

last

ic c

oeff

icie

nt (

MP

a)

1 3E E= (FEM)

Reinforcements only

1 3E E= (FEM)

Reinforcements and matrix

1 3E E= (AHM)

Total reinforcement volume fraction

Fig. 1.7. Variation of the effective stiffness moduli E1 (or E3) for structure A1 (shownin Fig. 1.5).


18 A.L. Kalamkarov

Eff

ectiv

e el

astic

coe

ffic

ient

(M

Pa)

1 2 3E E E= = (FEM)

Reinforcements only

1 2 3E E E= = (FEM)

Reinforcements and matrix

1 2 3E E E= = (AHM)


Fig. 1.8. Variation of the effective stiffness moduli E1 = E2 = E3 for structure A2

(shown in Fig. 1.3).

3E (FEM) Reinforcements only

3E (FEM) Reinforcements

and matrix

3E (AHM)


Eff

ectiv

e el

astic

coe

ffic

ient

(M

Pa)

Fig. 1.9. Variation of the effective stiffness modulus E3 for structure A3 (shown inFig. 1.6).

accuracy of the asymptotic homogenization model in the case when thematrix contribution is neglected. We recall that we have assumed thatthe reinforcements were much stiffer than the matrix and we consequentlyneglected the contribution of the latter. The discrepancy between the firstand third lines is due to the contribution of the matrix. Figure 1.7 alsovalidates another assumption of the asymptotic homogenization model. Inusing superposition to determine the effective properties of structures withtwo or more families of reinforcements, an error will be incurred at theregion of overlap between the reinforcements. However, we assume that forthe practical purposes this error will not contribute significantly to the



integral in Eq. (1.34) and thus will not appreciably affect the effectivecoefficients. This assumption is confirmed by the excellent agreementbetween the first and second lines in Fig. 1.7. Of course we expect that inmore complex unit-cell structures with a larger extent of overlap betweenreinforcements this error could be more pronounced. This will be illustratedin subsequent examples.

We now turn our attention to structure A2 shown in Fig. 1.3 for whichthe variation of the effective elastic moduli vs. the reinforcement volumefraction is shown in Fig. 1.8. Again, three lines are plotted correspondingto the AHM results, the FEM (contribution of reinforcements only) andFEM (contribution of reinforcements and matrix) results. It should beexpected that the discrepancy between the AHM results and the FEMresults (considering only the reinforcements) is higher for structure A2 thanfor structure A1. This is attributed to the larger volume of overlap betweenthe various reinforcements in the unit cell of structure A2.

The final structure to be considered is structure A3 shown in Fig. 1.6,for which the variation of E3 vs. the reinforcement volume fraction is shownin Fig. 1.9. As with the previous example, the discrepancy between thelower two lines is attributed to the regions of overlap between the differentreinforcement families. The difference between the upper two lines is thecontribution of the matrix on the effective elastic coefficients.

1.5 Asymptotic Homogenization of Thin-Walled CompositeReinforced Structures

In numerous engineering applications, composite materials are used in theform of thin-walled structural members such as shells and plates. Theirstiffness and strength combined with the reduced weight and associatedmaterial savings offer very impressive possibilities. It is very commonthat the reinforcing elements such as embedded fibres or surface ribsform a regular array with a period much smaller than the characteristicdimensions of the whole composite structure. Consequently the asymptotichomogenization analysis becomes applicable.

An asymptotic homogenized model for plates with periodicinhomogeneities in tangential directions was developed for the first time byDuvaut [39, 40]. In these works the asymptotic homogenization procedurewas applied directly to a 2D plate problem. Evidently, the asymptotichomogenization method cannot be applied directly to 3D thin compositelayers if their small thickness (in the direction in which there is no


20 A.L. Kalamkarov

periodicity) is comparable with the small dimensions of the periodicitycell (in the two tangential directions). To deal with the 3D problem fora thin composite layer, a modified asymptotic homogenization approachwas proposed by Caillerie [41, 42] in heat conduction studies. It consistsof applying a two-scale asymptotic formalism directly to the 3D problemfor a thin inhomogeneous layer with the following modification. Twosets of ‘rapid’ coordinates are introduced. Two tangential coordinates areassociated with the rapid periodic variation in the composite properties.The third one is in the transverse direction and it is associated with thesmall thickness of the layer. It takes into account that there is no periodicityin this transverse direction. There are two small parameters: one is ameasure of the periodic variation in the two tangential directions and theother is a measure of the small thickness. Generally these two parametersmay or may not be of the same order of magnitude, although in practicalapplications, they are commonly small values of the same order. Kohn andVogelius [43–45] adopted this approach in their study of the pure bendingof a thin, linearly elastic homogeneous plate with wavy surfaces.

A generalization of this approach to the most comprehensive case ofa thin 3D composite layer with wavy surfaces (which model the surfacereinforcements) was offered by Kalamkarov [5, 25, 26]; see also [27]. In theseworks the general asymptotic homogenization model for a composite shellwas developed by applying the modified two-scale asymptotic techniquedirectly to 3D elastic and thermoelastic problems for a thin curvilinearcomposite layer with wavy surfaces; see Fig. 1.10. Homogenization modelswere also developed for non-linear problems for composite shells; see [46, 47].

Fig. 1.10. Thin 3D curvilinear composite layer (a) with a periodicity cell Ωδ (b).



The homogenization models developed for a composite shell were appliedin the design and optimization of composite and reinforced shells [27, 28].Most recently, this technique was adopted in modelling smart compositeshells and plates [19, 48–57]. The general homogenization model for acomposite shell has found numerous applications in the analysis of variouspractically important composite structures. Grid-reinforced and network-thin generally orthotropic composite shells, as well as 3D network-reinforced composite structures are studied in [58–61]. Sandwich compositeshells and in particular honeycomb sandwich composite shells madeof generally orthotropic materials are analysed in [62–64]. Asymptotichomogenization was also applied by Kalamkarov et al. [65–67] to calculatethe effective properties of carbon nanotubes and carbon nanotube-reinforced structures.

Let us now summarize the above general homogenization model for acomposite shell; see [5, 27] for details. Consider a general thin 3D compositelayer of a periodic structure with the unit cell Ωδ shown in Fig. 1.10. Inthis figure, α1, α2 and γ are the orthogonal curvilinear coordinates, suchthat the coordinate lines α1 and α2 coincide with the main curvature linesof the mid-surface of the carrier layer, and coordinate line γ is normal toits mid-surface (at which γ = 0).

The thickness of the layer and the dimensions of the unit cell of thecomposite material (which define the scale of the composite material’sinhomogeneity) are assumed to be small compared with the dimensionsof the whole structure. These small dimensions of the periodicity cell arecharacterized by a small parameter δ.

The unit cell Ωδ shown in Fig. 1.10(b) is defined by the followingrelations:

−δh1

2< α1 <

δh1

2, −δh2

2< α2 <

δh2

2, γ− < γ < γ+,

γ± = ± δ2± δF±

(α1

δh1,α2

δh2

)(1.36)

Here, δ is the thickness of the layer, δh1 and δh2 are the tangentialdimensions of the periodicity cell Ωδ. The functions F± in Eq. (1.36) definethe geometry of the upper (S+) and lower (S−) reinforcing elements, forexample, the ribs or stiffeners; see Figs. 1.10 and 1.11. If there are noreinforcing elements then F+ = F− = 0 and the composite layer has auniform thickness of the order of δ as for example in the case shown inFig. 1.12.


22 A.L. Kalamkarov

(a) (b)

Fig. 1.11. (a) Wafer-reinforced shell and (b) its unit cell.

Fig. 1.12. Sandwich composite shell with honeycomb filler.

The periodic inhomogeneity of the composite material is modelled bythe assumption that the elastic coefficients Cijkl (α1, α2, γ) are periodicfunctions of the variables α1 and α2 with a unit cell Ωδ.

The elasticity problem for the above 3D thin composite layer isformulated as follows:

∂σij∂αj

= fi

σij = Cijkl(α1, α2, γ)ekl, ekl =12

(∂uk∂αl

+∂ul∂αk

), (1.37)

σijn±j = p±i .

Here fi, p±i and uk represent body forces, surface tractions and the

displacement field, respectively; n±j is the unit normal to the upper and



lower wavy surfaces γ±(α1, α2) and is given by

n± =−∂γ

±

∂α1,−∂γ

±

∂α2, 1(

1H2

1

(∂γ±

∂α1

)2

+1H2

2

(∂γ±

∂α2

)2

+ 1

)−1/2

,

(1.38)where H1 and H2 are the Lame coefficients defined by

H1 = A1(1 + κ1γ); H2 = A2(1 + κ2γ), (1.39)

where A1(α1, α2) and A2(α1, α2) are the coefficients of the first quadraticform and κ1 and κ2 are the main curvatures of the mid-surface of the carrierlayer (γ = 0).

We introduce the following fast variables, ξ = (ξ1, ξ2), and z:

ξ1 =α1A1

δh1, ξ2 =

α2A2

δh2, z =

γ

δ. (1.40)

The displacements and stresses are expressed in the form of thefollowing two-scale asymptotic expansions:

ui(α, ξ, z) = u(0)i (α) + δu

(1)i (α, ξ, z) + δ2u

(2)i (α, ξ, z) + · · · ,

σij(α, ξ, z) = σ(0)ij (α, ξ, z) + δσ

(1)ij (α, ξ, z) + δ2σ

(2)ij (α, ξ, z) + · · · .

(1.41)

As a result of the asymptotic homogenization procedure, see [5, 27]for details, the following relations for the displacements and stresses arederived:

u1 = v1(α) − δz

A1

∂w(α)∂α1

+ δUµν1 eµν + δ2V µν1 τµν + O(δ3)

u2 = v2(α) − δz

A2

∂w(α)∂α2

+ δUµν2 eµν + δ2V µν2 τµν + O(δ3)

u3 = w(α) + δUµν3 eµν + δ2V µν3 τµν + O(δ3)

(1.42)

σij = bµνij eµν + δb∗µνij τµν . (1.43)

Here and in the following Latin indexes assume values 1,2,3; Greek indexes1,2; and repeated indexes are summed; the mid-surface strains are denotedas follows: e11 = e1, e22 = e2 (elongations), e12 = e21 = ω/2 (shear),τ11 = k1, τ22 = k2 (bending) and τ12 = τ21 = τ (twisting).


24 A.L. Kalamkarov

The following notation is used in Eq. (1.43):

blmij =1hβCijnβ

∂U lmn∂ξβ

+ Cijn3∂U lmn∂z

+ Cijlm, (1.44)

b∗lmij =1hβCijnβ

∂V lmn∂ξβ

+ Cijn3∂V lmn∂z

+ zCijlm. (1.45)

The functions U lmn (ξ1, ξ2, z) and V lmn (ξ1, ξ2, z) in Eqs. (1.42), (1.44)and (1.45) are solutions of the unit-cell problems. Note that all the abovefunctions are periodic in the variables ξ1 and ξ2 with periods A1 and A2,respectively. The above unit-cell problems are formulated as follows:

1hβ

∂bλµiβ∂ξβ

+∂bλµi3∂z

= 0

1hβn±β b

λµiβ + n±3 b

λµi3 = 0 at z = z±

(1.46)

1hβ

∂b∗λµiβ

∂ξβ+∂b∗λµi3

∂z= 0

1hβn±β b

∗λµiβ + n±3 b

∗λµi3 = 0 at z = z±,

(1.47)

where n+i and n−

i are components of the normal unit vector to the upper(z = z+) and lower (z = z−) surfaces of the unit cell, respectively, definedin the coordinate system ξ1, ξ2, z.

If inclusions are perfectly bonded to the matrix on the interfaces ofthe composite material then the functions U lmn and V lmn together withthe expressions [ 1

hβn

(c)β bλµiβ + n

(c)3 bλµi3 ] and [ 1

hβn

(c)β b∗λµiβ + n

(c)3 b∗λµi3 ] must be

continuous on the interfaces. Here n(c)i are the components of the unit

normal to the interface.It should be noted that unlike the unit-cell problems of ‘classical’

homogenization models, e.g., Eqs. (1.9) and (1.25), those set by Eqs. (1.46)and (1.47) depend on the boundary conditions at z = z± in the z directionrather than on the periodicity.

After the local functions U lmn (ξ1, ξ2, z) and V lmn (ξ1, ξ2, z) are foundfrom the unit-cell problems given by Eqs. (1.44)–(1.47), the functionsblmij (ξ1, ξ2, z) and b∗lmij (ξ1, ξ2, z) given by Eqs. (1.44) and (1.45) can becalculated. These local functions define the stress σij as in from Eq. (1.43).They also define the effective stiffness moduli of the homogenized shell.Indeed the constitutive relations of the equivalent anisotropic homogeneous



shell, that is between the stress resultants N11, N22 (normal), N12 (shear)and moment resultants M11, M22 (bending), M12 (twisting) on the onehand, and the mid-surface strains e11 = e1, e22 = e2 (elongations), e12 =e21 = ω/2 (shear), τ11 = k1, τ22 = k2 (bending), τ12 = τ21 = τ (twisting)on the other, can be represented as follows (see [5, 27] for details):

Nαβ = δ〈bλµαβ〉eλµ + δ2〈b∗λµαβ 〉τλµMαβ = δ2〈zbλµαβ〉eλµ + δ3〈zb∗λµαβ 〉τλµ

. (1.48)

The angle brackets in Eq. (1.48) denote averaging by the integration overthe volume of the 3D unit cell:

〈f (ξ1, ξ2, z)〉 =1|Ω|∫

Ω

f(ξ1, ξ2, z) dξ1dξ2dz.

The coefficients in the constitutive relations Eq. (1.48) 〈bλµαβ〉, 〈b∗λµαβ 〉,〈zbλµαβ〉 and 〈zb∗λµαβ 〉 are the effective stiffness moduli of the homogenizedshell.

The following symmetry relationships for the effective stiffness modulihave been proved in [5]:⟨

bmnij⟩

=⟨bijmn⟩,⟨zbmnij

⟩=⟨b∗ijmn⟩,⟨zb∗mnij

⟩=⟨zb∗ijmn

⟩. (1.49)

The mid-surface strains eλµ(α1, α2) and τλµ(α1, α2) can be determinedby solving a global boundary-value problem for the homogenized anisotropicshell with the constitutive relations (1.48); see [5, 27] for details. It shouldbe noted that, as can be observed from Eq. (1.48), there is a followingone-to-one correspondence between the effective stiffness moduli and theextensional [A] coupling [B] and bending [D] stiffnesses familiar fromclassical composite laminate theory (see e.g. [36]):[A B

B D

]

=

δ⟨b1111⟩

δ⟨b2211⟩

δ⟨b1211⟩

δ2⟨zb1111

⟩δ2⟨zb2211

⟩δ2⟨zb1211

⟩δ⟨b2211⟩

δ⟨b2222⟩

δ⟨b1222⟩

δ2⟨zb2211

⟩δ2⟨zb2222

⟩δ2⟨zb1222

⟩δ⟨b1211⟩

δ⟨b1222⟩

δ⟨b1212⟩

δ2⟨zb1211

⟩δ2⟨zb1222

⟩δ2⟨zb1212

⟩δ2⟨b∗1111

⟩δ2⟨b∗2211

⟩δ2⟨b∗1211

⟩δ3⟨zb∗1111

⟩δ3⟨zb∗2211

⟩δ3⟨zb∗1211

⟩δ2⟨b∗2211

⟩δ2⟨b∗2222

⟩δ2⟨b∗1222

⟩δ3⟨zb∗2211

⟩δ3⟨zb∗2222

⟩δ3⟨zb∗1222

⟩δ2⟨b∗1211

⟩δ2⟨b∗1222

⟩δ2⟨b∗1212

⟩δ3⟨zb∗1211

⟩δ3⟨zb∗1222

⟩δ3⟨zb∗1212

⟩

.

(1.50)


26 A.L. Kalamkarov

It is worth mentioning at this point that the coordinates ξ1 and ξ2,defined by Eq. (1.40) in terms of the functions A1(α1, α2) and A2(α1, α2),are involved in the local problems. Functions A1(α1, α2) and A2(α1, α2) arethe coefficients of the first quadratic form of the mid-surface of the carrierlayer. That means that if the mid-surface is not a developing surface, so thatthese functions are not constant, the effective stiffness coefficients will alsodepend on the macroscopic coordinates α1 and α2 through these functions.Therefore, even in the case of the originally homogeneous material, we mayfind structural inhomogeneity after the homogenization process.

The unit-cell problems given by Eqs. (1.44), (1.46) and (1.45), (1.47)have been solved analytically for a number of structures of practical interest,and the explicit analytical formulae for the effective stiffness moduli havebeen obtained for the following types of composite and reinforced shells andplates: angle-ply fibre-reinforced shells, grid-reinforced and network shells[5, 25, 27, 55, 56, 58, 59]; rib- and wafer-reinforced shells [5, 26, 27, 49, 50, 54];sandwich composite shells, in particular, honeycomb sandwich compositeshells made of generally orthotropic materials [5, 27, 62−64] and carbonnanotubes [65–67].

As examples of these results, we will present here the analyticalformulae for the effective stiffness moduli of a wafer-reinforced shell(Fig. 1.11) and a sandwich composite shell with a honeycomb filler(Fig. 1.12).

All the non-zero effective stiffness moduli of the wafer-reinforced shellshown in Fig. 1.11 are obtained as follows, (see [5, 26, 27, 50] for details):

⟨b1111⟩

=E

(3)1

1 − ν(3)12 ν

(3)21

+ E(2)1 F

(w)2 ,

⟨b2222⟩

=E

(3)2

1 − ν(3)12 ν

(3)21

+ E(1)2 F

(w)1 ,

⟨b1122⟩

=⟨b2211⟩

=ν

(3)12 E

(3)1

1 − ν(3)12 ν

(3)21

,⟨b1212⟩

= G(3)12 ,

⟨zb1111

⟩=⟨b∗1111

⟩= E

(2)1 S

(w)2 ,

⟨zb2222

⟩=⟨b∗2222

⟩= E

(1)2 S

(w)1 ,

⟨zb∗1111

⟩=

E(3)1

12(1 − ν

(3)12 ν

(3)21

) + E(2)1 J

(w)2 ,

⟨zb∗2222

⟩=

E(3)2

12(1 − ν

(3)12 ν

(3)21

) + E(1)2 J

(w)1 ,

⟨zb∗1122

⟩=⟨zb∗2211

⟩=

ν(3)12 E

(3)1

12(1 − ν(3)12 ν

(3)21 )

,



⟨zb∗1212

⟩=G

(3)12

12+G

(1)12

12

(H3t1h1

−K1

)+G

(2)12

12

(H3t2h2

−K2

),

(1.51)

where

K1 =96H4

π5A1h1

√√√√G(1)12

G(1)23

∞∑n=1

[1 − (−1)n]n5

tanh

√√√√G

(1)23

G(1)12

nπA1t12H

,

K2 =96H4

π5A2h2

√√√√G(2)12

G(2)13

∞∑n=1

[1 − (−1)n]n5

tanh

√√√√G

(2)13

G(2)12

nπA2t22H

.

(1.52)

Here the superscripts indicate the elements of the unit cell Ω1, Ω2 and Ω3;see Fig. 1.11(b); A1 and A2 are the coefficients of the first quadratic formof the mid-surface of a carrier layer; F (w)

1 , F(w)2 , S(w)

1 , S(w)2 and J

(w)1 , J

(w)2

are defined as follows:

F(w)1 =

Ht1h1

, F(w)2 =

Ht2h2

, S(w)1 =

(H2 +H)t12h1

,

S(w)2 =

(H2 +H)t22h2

, J(w)1 =

(4H3 + 6H2 + 3H)t112h1

,

J(w)2 =

(4H3 + 6H2 + 3H)t212h2

.

(1.53)

All the non-zero effective stiffness moduli of the sandwich compositeshell with a honeycomb filler shown in Fig. 1.12 are obtained as follows (see[5, 27] for details):

⟨b1111⟩

=⟨b2222⟩

=2E0t01 − ν2

0

+√

34EHt

a,⟨b1212⟩

=E0t0

(1 + ν0)+

√3

12EHt

a,

⟨b1122⟩

=⟨b2211⟩

=2ν0E0t01 − ν2

0

+√

312

EHt

a,

⟨zb∗1111

⟩=⟨zb∗2222

⟩=

E0

1 − ν20

(H2t0

2+Ht20 +

2t303

)+

√3

48EH3t

a,

⟨zb∗1122

⟩=⟨zb∗2211

⟩=

ν0E0

1 − ν20

(H2t0

2+Ht20 +

2t303

)+

√3

144EH3t

a,


28 A.L. Kalamkarov

⟨zb∗1212

⟩=

E0

2(1 + ν0)

(H2t0

2+Ht20 +

2t303

)+

EH3t

12(1 + ν)a

×[

3 + ν

4√

3− 128H

(√

3π5At)

∞∑n=1

tanh(π(2n− 1)At/(2H))(2n− 1)5

].

(1.54)

The first terms in Eq. (1.54) define the contribution from the top andbottom carrier layers of the sandwich shell while the latter terms representthe contribution from the honeycomb filler. E0 and ν0 are the propertiesof the material of the carrier layers, and E and ν of the honeycombfoil material. We have confined our attention here to the case of equalcoefficients of the first quadratic form of the mid-surface of the shell,i.e. A1 = A2 = A. The details of the derivation of Eq. (1.54) and morecomplicated cases of composite sandwich shells with generally orthotropicconstituent materials will be presented below in Section 1.8.

1.6 Generally Orthotropic Grid-ReinforcedComposite Shell

In this section, we will apply the asymptotic homogenization modelto a composite shell reinforced with a number of families of parallelreinforcing elements. An example of a shell with two mutually perpendicularreinforcement families is shown in Fig. 1.13. We assume that thereinforcements are made of generally orthotropic materials and that theyare much stiffer than the surrounding matrix material. As such, we mayneglect the contribution of the matrix in the ensuing analysis.

As in the approach we used above in Section 1.4, we will first considera simpler type of shell with only one family of reinforcements. The effective

Orthotropic reinforcements

Fig. 1.13. Composite shell with two families of orthotropic reinforcements.



γ

α2

α1

δh1

δh2δ

ϕ

z

ξ2

ξ1

A1

A21

ϕ′

zγξξα

2

11

α2

→→→

Fig. 1.14. Unit cell in microscopic (ξ1, ξ2, z) and macroscopic (α1, α2, γ) variables.

elastic coefficients of more general structures with several reinforcementfamilies will be determined by superposition.

Consider the unit cell of Fig. 1.14 shown both before and after theintroduction of the microscopic variables ξ1, ξ2 and z, defined in Eq. (1.40).Note that the matrix [C] of elastic coefficients of an orthotropic materialreferenced to a coordinate system that has been rotated by the angle ϕ ofthe reinforcing grid orientation (in the ξ1−ξ2 plane) with respect to theprincipal material coordinate system coincides with that of a monoclinicmaterial and has the following form:

[C] =

C11 C12 C13 0 0 C16

C12 C22 C23 0 0 C26

C13 C23 C33 0 0 C36

0 0 0 C44 C45 00 0 0 C45 C55 0

C16 C26 C36 0 0 C66

(1.55)

After this coordinate transformation, the shape of the unit cell changes andthe angle between the reinforcement and the ξ1 axis changes to angle ϕ′

(see Fig. 1.14) through the following relationship ϕ′ = arctan(A2h1A1h2

tanϕ).


30 A.L. Kalamkarov

We begin by first solving for bλµij from Eq. (1.44). These local functionsare given as follows for an orthotropic material of reinforcement:

bλµ11 =1h1C11

∂Uλµ1

∂ξ1+

1h2C12

∂Uλµ2

∂ξ2+ C13

∂Uλµ3

∂z

+C16

[1h1

∂Uλµ2

∂ξ1+

1h2

∂Uλµ1

∂ξ2

]+ C11λµ (1.56)

bλµ22 =1h1C12

∂Uλµ1

∂ξ1+

1h2C22

∂Uλµ2

∂ξ2+ C23

∂Uλµ3

∂z

+C26

[1h1

∂Uλµ2

∂ξ1+

1h2

∂Uλµ1

∂ξ2

]+ C22λµ (1.57)

bλµ33 =1h1C13

∂Uλµ1

∂ξ1+

1h2C23

∂Uλµ2

∂ξ2+ C33

∂Uλµ3

∂z

+C36

[1h1

∂Uλµ2

∂ξ1+

1h2

∂Uλµ1

∂ξ2

]+ C33λµ (1.58)

bλµ12 =1h1C16

∂Uλµ1

∂ξ1+

1h2C26

∂Uλµ2

∂ξ2+ C36

∂Uλµ3

∂z

+C66

[1h1

∂Uλµ2

∂ξ1+

1h2

∂Uλµ1

∂ξ2

]+ C12λµ (1.59)

bλµ13 = C55

[1h1

∂Uλµ3

∂ξ1+∂Uλµ1

∂z

]+ C45

[1h2

∂Uλµ3

∂ξ2+∂Uλµ2

∂z

]+ C13λµ

(1.60)

bλµ23 = C45

[1h1

∂Uλµ3

∂ξ1+∂Uλµ1

∂z

]+ C44

[1h2

∂Uλµ3

∂ξ2+∂Uλµ2

∂z

]+ C23λµ.

(1.61)

In order to reduce the complexity of the associated problems, weintroduce a new coordinate system η1, η2, z obtained via rotation throughan angle ϕ′ around the z axis, such that the η1-coordinate axis coincideswith the longitudinal axis of the reinforcing element and the η2-coordinateaxis is perpendicular to it; see Fig. 1.15. With this transformation the



η2

η1

z

Fig. 1.15. Coordinate transformation to the microscopic coordinates (η1, η2, z).

problem at hand is now independent of the η1 coordinate and will onlydepend on η2 and z. Consequently the order of the differential equations isreduced by one and the analysis of the problem is simplified. Thus the bλµijfunctions from Eqs. (1.56)–(1.61) can be written as follows:

bλµ11 = − 1h1C11 sinϕ′ ∂U

λµ1

∂η2+

1h2C12 cosϕ′ ∂U

λµ2

∂η2+ C13

∂Uλµ3

∂z

+C16

[− 1h1

sinϕ′ ∂Uλµ2

∂η2+

1h2

cosϕ′ ∂Uλµ1

∂η2

]+ C11λµ (1.62)


λµ1

∂η2+


λµ2

∂η2+ C23

∂Uλµ3

∂z

+C26

[− 1h1

sinϕ′ ∂Uλµ2

∂η2+

1h2

cosϕ′ ∂Uλµ1

∂η2

]+ C22λµ (1.63)


λµ1

∂η2+


λµ2

∂η2+ C33

∂Uλµ3

∂z

+C36

[− 1h1

sinϕ′ ∂Uλµ2

∂η2+

1h2

cosϕ′ ∂Uλµ1

∂η2

]+ C33λµ (1.64)


λµ1

∂η2+


λµ2

∂η2+ C36

∂Uλµ3

∂z

+C66

[− 1h1

sinϕ′ ∂Uλµ2

∂η2+

1h2

cosϕ′ ∂Uλµ1

∂η2

]+ C12λµ (1.65)


32 A.L. Kalamkarov

bλµ13 = C55

[− 1h1

sinϕ′ ∂Uλµ3

∂η2+∂Uλµ1

∂z

]

+C45

[1h2

cosϕ′ ∂Uλµ3

∂η2+∂Uλµ2

∂z

]+ C13λµ (1.66)

bλµ23 = C45

[− 1h1

sinϕ′ ∂Uλµ3

∂η2+∂Uλµ1

∂z

]

+C44

[1h2

cosϕ′ ∂Uλµ3

∂η2+∂Uλµ2

∂z

]+ C23λµ (1.67)

and the unit-cell problem and associated boundary condition (1.46) can berewritten in terms of the coordinates η2 and z as follows:

− sinϕ′

h1

∂

∂η2bλµi1 +

cosϕ′

h2

∂

∂η2bλµi2 +

∂

∂zbλµi3 = 0 (1.68)

[n′

2

(− sinϕ′

h1bλµi1 +

cosϕ′

h2bλµi2

)+ n′3b

λµi3

]∣∣∣∣

= 0, (1.69)

where n′2 and n′

3 are the components of the unit vector normal to thelateral surface of the reinforcement with respect to the η1, η2, z coordinatesystem, and the suffix stands for the matrix/reinforcement interface.We will now solve the system defined by Eqs. (1.62)–(1.68) and associatedboundary condition (1.69) by assuming that the local functions Uλµ1 andUλµ2 are linear in η2 and are independent of z, whereas Uλµ3 is linear in z

and independent of η2. That is the solution can be found as follows:

Uλµ1 = Aλµη2, Uλµ2 = Bλµη2, Uλµ3 = Cλµz, (1.70)

where Aλµ, Bλµ and Cλµ are constants to be determined. Equation (1.70) issubstituted into the expressions (1.62)–(1.67), which allows the calculationof the aforementioned constants in conjunction with Eq. (1.69). Aftersolving the pertinent system of algebraic equations the results are thenback-substituted into Eqs. (1.62)–(1.67) to yield the following formulae forall the non-zero local functions bklij :

bλµ11 =

C12λµ[Λ4Λ7 + Λ8Λ3] + C11λµ[Λ5Λ7 − Λ9Λ3] + C22λµ[Λ6Λ7] + C33λµΛ3

Λ7

hA2 tan φ

A1Λ4 + Λ5 +

A22 tan2 φ

A21

Λ6

i+ A2 tan φ

A1[Λ8Λ3] − Λ9Λ3

(1.71)



bλµ22 =


Λ7

hA1

A2 tan φΛ4 +

A21

A22 tan2 φ

Λ5 + Λ6

i+ A1

A2 tan φ[Λ8Λ3] − A2

1A2

2 tan2 φΛ9Λ3

(1.72)

bλµ12 =


Λ7

hΛ4 + A1

A2 tan φΛ5 + A2 tan φ

A1Λ6

i+ [Λ8Λ3] − A1

A2 tan φΛ9Λ3

(1.73)

where the quantities Λ1,Λ2, . . . ,Λ9 can be found in [60].We now turn our attention to the local functions b∗klij . We begin

by expanding Eq. (1.45) keeping Eq. (1.55) in mind as well as thecoordinate transformation defined by Figs. 1.14 and 1.15. The resultingexpressions are:

b∗λµ11 = − 1h1C11 sinϕ′ ∂V

λµ1

∂η2+

1h2C12 cosϕ′ ∂V

λµ2

∂η2+ C13

∂V λµ3

∂z

+C16

[− 1h1

sinϕ′ ∂Vλµ2

∂η2+

1h2

cosϕ′ ∂Vλµ1

∂η2

]+ zC11λµ (1.74)

b∗λµ22 = − 1h1C12 sinϕ′ ∂V

λµ1

∂η2+


λµ2

∂η2+ C23

∂V λµ3

∂z

+C26

[− 1h1

sinϕ′ ∂Vλµ2

∂η2+

1h2

cosϕ′ ∂Vλµ1

∂η2

]+ zC22λµ (1.75)

b∗λµ33 = − 1h1C13 sinϕ′ ∂V

λµ1

∂η2+


λµ2

∂η2+ C33

∂V λµ3

∂z

+C36

[− 1h1

sinϕ′ ∂Vλµ2

∂η2+

1h2

cosϕ′ ∂Vλµ1

∂η2

]+ zC33λµ (1.76)

b∗λµ12 = − 1h1C16 sinϕ′ ∂V

λµ1

∂η2+


λµ2

∂η2+ C36

∂V λµ3

∂z

+C66

[− 1h1

sinϕ′ ∂Vλµ2

∂η2+

1h2

cosϕ′ ∂Vλµ1

∂η2

]+ zC12λµ (1.77)

b∗λµ13 = C55

[− 1h1

sinϕ′ ∂Vλµ3

∂η2+∂V λµ1

∂z

]

+C45

[1h2

cosϕ′ ∂Vλµ3

∂η2+∂V λµ2

∂z

]+ zC13λµ (1.78)


34 A.L. Kalamkarov

b∗λµ23 = C45

[− 1h1

sinϕ′ ∂Vλµ3

∂η2+∂V λµ1

∂z

]

+C44

[1h2

cosϕ′ ∂Vλµ3

∂η2+∂V λµ2

∂z

]+ zC23λµ. (1.79)

Similarly, the unit-cell problem (1.47) becomes:

− sinϕ′

h1

∂

∂η2b∗λµi1 +

cosϕ′

h2

∂

∂η2b∗λµi2 +

∂

∂zb∗λµi3 = 0 (1.80)

[n′2

(− sinϕ′

h1b∗λµi1 +

cosϕ′

h2b∗λµi2

)+ n′3b

∗λµi3

]∣∣∣∣

= 0. (1.81)

Since the local functions b∗klij are related to the bending deformations itis expected that the pertinent solution will depend on the shape of thereinforcing elements (unlike the corresponding bklij coefficients). Indeed thepresence of the z coordinates in Eqs. (1.74)–(1.79) implies exactly that.From the practical viewpoint, let us assume that the reinforcing elementshave a circular cross-section. From the coordinate transformation fromα1, α2 to ξ1, ξ2, defined by Eq. (1.40), we note that the cross-sectionwill change from circular to elliptical with the eccentricity e′ given by thefollowing formula:

e′ =(

1 − A21A

22

A22h

21 sin2 ϕ+A2

1h22 cos2 ϕ

)1/2

. (1.82)

Additionally the components n′2 and n′3 (clearly n′1 = 0) of the unit vector

normal to the surface of the reinforcing element are:

n′2 = η2[1 − (e′)2]−1 and n′3 = z. (1.83)

It is possible to satisfy the differential equation (1.80) and boundaryconditions (1.81) by assuming that the functions V λµi have the followinggeneral functional form:

V λµi = W λµi1 η2z +W λµ

i2

η22

2+Wλµ

i3

z2

2, (1.84)

where W λµij are constants to be determined. The determination of the local

functions b∗λµij follows in a straightforward, albeit algebraically tediousmanner. Keeping Eqs. (1.82) and (1.83) in mind, we first substitute



Eq. (1.84) into Eqs. (1.80) and (1.81) and calculate the constants Wλµij by

comparing terms with like powers of η2 and z. Once the Wλµij functions

are determined they are substituted into Eq. (1.84) and the resultingexpressions are back-substituted into Eqs. (1.74)–(1.79) to obtain thedesired local functions b∗λµij . As a result, the following expressions for thefunctions b∗λµ11 , b∗λµ22 and b∗λµ12 are found:

b∗λµ11 = zBλµ11 ; b∗λµ22 = zBλµ22 ; b∗λµ12 = zBλµ12 , (1.85)

where

Bλµ11 =∑

5

∑6 −∑

2

∑3∑

1

∑2 −∑

4

∑5

, Bλµ22 =∑

3

∑4 −∑

1

∑6∑

1

∑2 −∑

4

∑5

,

Bλµ12 =A2 tanϕ

2A1Bλµ11 +

A1

2A2 tanϕBλµ22 . (1.86)

Explicit expressions for∑

1,∑

2, . . . ,∑

6, which depend on thegeometric parameters of the unit cell and the material properties of thereinforcements, can be found in [60].

1.6.1 Calculation of the effective elastic coefficients

The effective elastic coefficients for the reinforced generally orthotropiccomposite shell of Fig. 1.14 can be calculated by means ofexpressions (1.71)–(1.73), (1.85) and (1.86). Let us denote the volume ofone reinforcing element within the unit cell of Fig. 1.14 by δ3V . Then, theeffective elastic coefficients are given by:

⟨bλµij⟩

=1|Ω|∫

Ω

bλµij dv =V

h1h2bλµij ,

⟨zbλµij

⟩=

1|Ω|∫

Ω

zbλµij dv = 0,

⟨b∗λµij

⟩= 0,

⟨zb∗λµij

⟩=

V

16h1h2Bλµij . (1.87)

The corresponding results for composite shells reinforced by more thanone family of orthotropic reinforcements can be obtained from Eq. (1.87)by superposition. In doing so, we accept an error incurred due to stressvariations at the regions of overlap of the reinforcements. However, thiserror is small and will not contribute significantly to the integral over thevolume of the unit cell as discussed earlier in Section 1.4.


36 A.L. Kalamkarov

1.7 Examples of Grid-Reinforced Composite Shells withOrthotropic Reinforcements

The mathematical model and methodology presented in Section 1.6 canbe used in analysis and design to tailor the effective elastic properties ofthe above reinforced composite shells to meet the criteria of a particularapplication, by selecting the appropriate shape of the shells as wellas the type, number, orientation and geometric characteristics of thereinforcements. In this section, we will apply our general solution to differentcomposite shells and plates. In the first example we will consider a generalcomposite shell reinforced with isotropic reinforcements. In the secondexample the special case of a cylindrical shell will be considered. In thethird example, we will obtain the closed-form expressions for the effectiveelastic properties of single-walled carbon nanotubes. In the fourth examplewe will consider general multilayered composite shells and illustrate ourresults with a typical three-layer cylindrical shell. Finally, we will observehow our model can be used to derive the effective elastic coefficients ofgrid-reinforced composite plates. Without loss of generality, we will assumethat in the considered grid-reinforced structures all reinforcements havesimilar cross-sectional areas and are made of the same material. If desired,however, the model allows for each family of reinforcements to have specificgeometric and material properties.

Example 1.1: Composite shell reinforced with isotropic

reinforcements

In the case of isotropic reinforcements all the non-zero effective stiffnessmoduli are [5, 25, 27]:

⟨b1111⟩

=VA4

1

h1h2Θ4E cos4 φ;

⟨b2222⟩

=VA4

2

h1h2Θ4E sin4 φ;

⟨b1211⟩

=VA3

1A2

h1h2Θ4E cos3 φ sinφ;

⟨b1222⟩

=VA1A

32

h1h2Θ4E cosφ sin3 φ;

⟨b2211⟩

=⟨b1212⟩

=VA2

1A22

h1h2Θ4E cos2 φ sin2 φ;

(1.88)

⟨zb∗1111

⟩=

V

16h1h2

EA41

(1 + ν)Θ4cos2 φ[2A4

2Ψ sin2 φ+ cos2 φ(1 + ν)]



⟨zb∗2211

⟩=

V

16h1h2

EA21A

22

(1 + ν)Θ4cos2 φ sin2 φ[−2A2

1A22Ψ + 1 + ν]

⟨zb∗1211

⟩=

V

16h1h2

EA31A2

(1 + ν)Θ4cosφ sinφ

× [A22Ψ(A2

2 sin2 φ−A21 cos2 φ) + cos2 φ(1 + ν)]

⟨zb∗1212

⟩=

V

16h1h2

EA21A

22

2(1 + ν)Θ4

× [(A21 cos2 φ−A2

2 sin2 φ)Ψ + 2 cos2 φ sin2 φ(1 + ν)]

⟨zb∗1222

⟩=

V

16h1h2

EA1A32

(1 + ν)Θ4cosφ sinφ

× [A21Ψ(A2

1 cos2 φ−A22 sin2 φ) + sin2 φ(1 + ν)]

⟨zb∗2222

⟩=

V

16h1h2

EA42

(1 + ν)Θ4sin2 φ[2A4

1Ψ cos2 φ+ sin2 φ(1 + ν)].

(1.89)

In Eqs. (1.88) and (1.89), E and ν are Young’s modulus and Poisson’s ratioof the reinforcement and

Θ = A21 cos2 ϕ+A2

2 sin2 ϕ, Ψ =√

Θ2 +A21A

22. (1.90)

Example 1.2: Thin cylindrical shell

The second example represents a cylindrical composite shell (i.e. we canassume that A1 = A2 = 1) reinforced with a single family of reinforcementsparallel to the longitudinal axis of the shell (ϕ = 00) as shown in Fig. 1.16.The effective elastic coefficients of this structure can readily be determinedfrom Eq. (1.87) with the use of the solutions (1.71)–(1.73) and (1.86).

Fig. 1.16. Cylindrical composite shell with a single family of orthotropic reinforcements.


38 A.L. Kalamkarov

Table 1.2. Reinforcement material properties.

Property E1 E2 = E3 G12 = G13 G23 ν12 = ν13 = ν23

Value 152.0 GPa 4.1 GPa 2.9 GPa 1.5 GPa 0.35

b aPM1111 bz aPM11

11∗

1111zb∗

1111b

1 2R V h h=

Fig. 1.17. Plot of˙b1111

¸and

˙zb∗11

11

¸vs. R, volume fraction of reinforcement for a

composite shell reinforced with a single family of orthotropic reinforcements (shown inFig. 1.16).

Although the resulting expressions are too lengthy to be reproduced here,typical coefficients will be presented graphically for the reinforcementmaterial properties given in Table 1.2.

Figure 1.17 shows a typical plot for the variation of⟨b1111⟩

and⟨zb∗1212

⟩vs. R for the reinforced shell of Fig. 1.16, where R is the ratio of the volumeof one reinforcing element within the unit cell to the volume of the entireunit cell. In other words, R is the volume fraction of the reinforcements andcan be expressed as:

R = V/(h1h2). (1.91)

As expected, both the bending and extensional stiffnesses in thedirection of the reinforcements increase with an increase in the volumefraction. Clearly, all the effective coefficients can be modified to fit differentrequirements by changing either the geometrical characteristics of the shelland reinforcements or by changing the type and number of reinforcementfamilies.



Example 1.3: Single-walled carbon nanotube

Of particular interest in the context of cylindrical grid-reinforced shells isthe case of a single-walled carbon nanotube (SWCNT). Carbon nanotubesare a recently discovered allotrope of carbon comprising long-chainedmolecules of carbon with carbon atoms arranged in a hexagonal networkto form a tubular structure. They are classified as single- or multi-walleddepending on the number of walls. Typically, the nanotubes are about20 to 150 A in diameter and about 1000 to 2000 A in length. And theydemonstrate remarkable strength and stiffness properties: Young’s modulusof 1.3 ± 0.5TPa and tensile strength of 150 GPa for an SWCNT. Carbonnanotubes can be made by rolling up a carbon-sheet in various ways.Figrue 1.18(b) shows the unit cell of an SWCNT in the so-called ‘arm-chair’ configuration.

As Fig. 1.18 demonstrates, the periodic nature of SWCNTs makes themparticularly amenable to study by asymptotic homogenization techniquesand the micromechanical model developed in Section 1.6; see Kalamkarovet al. [65, 66] for details. In this micromechanical model the C–C bondsare modelled by bars as shown in Fig. 1.18(b). It is assumed that thesebars are of circular cross-section with the material properties E and ν.Equations (1.88)–(1.90) can be applied to the geometry of the unit cell ofthe SWCNT shown in Fig. 1.18(b), and all the effective stiffnesses moduli

2α

1α

2δ

δ

2l

l

(a) (b)

Fig. 1.18. (a) Schematic representation of an SWCNT; (b) unit cell of a SWCNT.


40 A.L. Kalamkarov

entering the constitutive relations (1.48) as coefficients can be calculated.As a result, the following constitutive relations of the homogenized SWCNTwere obtained [65, 66]:

N11 = δ2E

l

π

16√

3(3ε11 + ε22)

N22 = δ2E

l

π

16√

3(ε11 + 3ε22) (1.92)

N12 = δ2E

l

π

16√

3ε12

M11 = δ3E

(1 + ν)lπ√

3768

[(4 + 3ν)k11 + νk22]

M22 = δ3E

(1 + ν)lπ√

3768

[νk11 + (4 + 3ν)k22] (1.93)

M12 = δ3E

l

π√

3768

k12.

The constitutive relations given by Eqs. (1.92) and (1.93) can be furtherapplied to derive the analytical formulae for the engineering constants ofSWCNTs. In particular, Eq. (1.92) yields the following formula for theeffective Young’s moduli E11 and E22 of SWCNTs:

E11 = E22 = ESWCNT =π

6√

3δE

l. (1.94)

Using Eq. (1.92) we also obtain ν12 = 0.33 and the following formula forthe effective shear modulus G12 of a SWCNT:

G12 =π

32√

3δE

l. (1.95)

It can be observed from Eq. (1.94) that the Young’s modulus of a carbonnanotube increases with increasing bar diameter δ and with decreasing C–Clink length l; see Fig. 1.18(b). In other words, this can be interpreted bysaying that the Young’s modulus of a SWCNT increases with decreasingtube diameter d. This dependency of Young’s modulus on the tube diameterof an SWCNT is consistent with experimental observations. Using typicalvalues of E = 5.488 × 10−6 N/nm2, δ = 0.147nm and l = 0.142nm, the



j = 1

j = 2

.

.

.

j = N

δaN

. . .

Fig. 1.19. Composite N-layered reinforced shell with each layer reinforced with a familyof orthotropic reinforcements.

effective Young’s and shear moduli of SWCNTs were determined fromEqs. (1.94) and (1.95) to be 1.71TPa and 0.32TPa, respectively [66]. Theseresults compare favourably with the results of other researchers who usedexperimental or numerical techniques in their analyses; see e.g. [68, 69].

Example 1.4: Laminated grid-reinforced composite shell

with generally orthotropic reinforcements

In this example we will analyse a laminated composite shell formedby N layers, each layer reinforced with a single family of orthotropicreinforcements; see Fig. 1.19. We assume that the family of reinforcementsin the jth layer of the shell makes an angle ϕj with the coordinate line α1.The distance between the axis of the jth reinforcement from the shell’s mid-surface is denoted by δaj (in the α1, α2, γ coordinate system) as shownin Fig. 1.19.

One may derive expressions for the effective properties in a similar wayas demonstrated in Section 1.6 after modifying the unit-cell problems inEqs. (1.44) and (1.45) by replacing z with (z′ + aj). The procedure, thoughalgebraically tedious, is straightforward. The final results show that thelocal functions bklij remain as in Eqs. (1.71)–(1.73) while the local functionsb∗klij become

b∗λµij = z′Bλµij + ajbλµij , (1.96)

where bλµij and Bλµij are given by Eqs. (1.71)–(1.73) and (1.86) after repla-cing tanϕ with tanϕj and Cλµmn with Cλµmn(j). Finally, the effective properties


42 A.L. Kalamkarov

as calculated by summation over all N layers are:

⟨bλµij⟩

=N∑j=1

bλµij γj ,⟨zbλµij

⟩=⟨b∗λµij

⟩=

N∑j=1

bλµij γjaj,

⟨zb∗λµij

⟩=

N∑j=1

(Bλµij γj

16+ a2

jbλµij γj

), (1.97)

where γj is the volume fraction of reinforcements in the jth layer and isgiven by:

γj =Vjh1h2

. (1.98)

We will now illustrate Example 1.4 by considering a three-layercomposite shell with orthotropic reinforcements oriented at ϕ = 60,ϕ = 90 and ϕ = 120 as shown in Fig. 1.20. The effective stiffness moduliare readily obtained from Eq. (1.97) and although the resulting expressionsare too lengthy to be reproduced here, some of the effective stiffnesses willbe presented graphically. We will assume that the shell layers are cylindricaland that the reinforcements have the properties given in Table 1.2.

Figure 1.21 shows the variation of effective stiffnesses⟨b1111⟩and

⟨b2222⟩

vs. γj . Without loss of generality we assume that the reinforcement volumefraction is the same in each of the three layers. As expected, the extensionalstiffness for the shell is larger in the α2 direction than in the α1 directionbecause there are more reinforcements either entirely (middle layer) orpartially (top and bottom layers) oriented in the α2 direction. For the samereason the bending stiffness in the α2 direction is larger than its counterpartin the α1 direction as shown in Fig. 1.22.

δh1 δ

ϕ = 60o

ϕ = 90o

ϕ = 120o

Fig. 1.20. Unit cell for three-layer composite shell.



2222b

1111b

Reinforcement volume fraction γ j

MPa

Fig. 1.21. Plot of typical extensional effective stiffness moduli vs. reinforcement volumefraction per layer γj .

MPa

Reinforcement volume fraction j

2222zb∗

1111zb∗

Fig. 1.22. Plot of typical bending effective elastic stiffness vs. reinforcement volumefraction per layer γj .


44 A.L. Kalamkarov

Example 1.5: Composite plates with generally orthotropic

reinforcements

As a final example, we will now apply the obtained general results forthe case of a thin plate reinforced with a grid of generally orthotropicreinforcements, shown in Fig. 1.23.

The effective stiffness moduli for the grid-reinforced plate can readilybe obtained from Eqs. (1.71)–(1.73), (1.85) and (1.86) by letting A1 =A2 = 1. The results are too lengthy to reproduce here, and for purposesof illustration we will compare graphically some of the effective stiffnessespertaining to the two structures shown in Figs. 1.24 and 1.25. The structureof Fig. 1.24 consists of two mutually perpendicular families of orthotropicreinforcements (ϕ = 0 and ϕ = 90) forming a rectangular reinforcinggrid. This structure will be referred to in the following as S1.

Matrix

Orthotropicreinforcements

Fig. 1.23. Composite plate reinforced with a network of orthotropic bars.

Fig. 1.24. Grid-reinforced plate (structure S1) with reinforcements arranged at anglesϕ = 0 and ϕ = 90.



Fig. 1.25. Grid-reinforced plate (structure S2) with reinforcements arranged at anglesϕ = 45, ϕ = 90 and ϕ = 135.

1111b MPa

h1h2

VR =

S1

S2

Fig. 1.26. Plot of elastic stiffness˙b1111

¸vs. volume fraction of reinforcements R for

structures S1 and S2.

The structure of Fig. 1.25 has three families of orthotropic reinforce-ments oriented at ϕ = 45, ϕ = 90 and ϕ = 135 forming a triangularreinforcing grid. This structure will be referred to as S2. The unit cellsof S1 and S2 are also shown in Figs. 1.24 and 1.25. In the ensuing plotswe will assume that the reinforcements have the elastic properties givenin Table 1.2. The different effective stiffnesses are plotted vs. the volumefraction of the reinforcements R defined in Eq. (1.91).

Figure 1.26 shows the variation of⟨b1111⟩

vs. R for the two structures S1

and S2. It can be observed that the stiffness in the ξ1 direction is larger for S1


46 A.L. Kalamkarov

2222b MPa

S1

S2

h1h2

VR =

Fig. 1.27. Plot of elastic stiffness˙b2222

¸vs. volume fraction of reinforcements R for

structures S1 and S2.

than for S2 because S1 has more reinforcements oriented in the ξ1 direction.S2 has an overall larger number of reinforcements but one of them is orientedentirely in the ξ2 direction and therefore makes no contribution to thestiffness in the ξ1 direction while the other two are oriented at an angle tothe ξ1 axis and therefore only partially contribute to the value of

⟨b1111⟩. For

the same reason we expect that the trend in the ξ2 direction will be reversedand that S2 should be stiffer. Indeed Fig. 1.27 shows precisely that.

Similar considerations hold for the remaining effective stiffness moduli.Figure 1.28 shows the variation of

⟨zb∗1111

⟩vs. R for S1 and S2. We note that

this coefficient characterizes the bending stiffness of the composite plate inthe ξ1−z plane. On the basis of the arguments given above the value ofthe

⟨zb∗1111

⟩coefficient is higher for S1 than S2. It is important to note,

however, that all of these trends and characteristics can be easily modifiedby changing the size, type, angular orientation, etc. so that the desirableelastic coefficients are obtained to conform to a particular application.

1.8 Sandwich Composite Shells with Cellular Cores

Composite sandwich structures with cellular cores have found numerousengineering applications; see, e.g., [70, 71]. In many cases, the methodsof synthesis and fabrication of these structures are controlled by thephase separation processes with surface tension being the controllingphysical factor; see Christensen [72]. Therefore, the hexagonal cell that



1111zb∗ ⟩ MPa⟨ S1 S2

h1h2

VR =

Fig. 1.28. Plot of˙zb∗1111

¸vs. volume fraction of reinforcements R for structures S1

and S2.

has the minimal surface area will have a big advantage in relationto the thermoelastic performance of sandwich structures in engineeringapplications.

We considered earlier in Section 1.5 an example of a three-layeredsandwich shell with a hexagonal honeycomb filler; see Fig. 1.12. Now wewill present the details of the derivation of the effective stiffness moduligiven earlier in Eq. (1.54) in the case of isotropic constituent materials. Wewill apply the above-introduced asymptotic homogenization technique toexamine the composite sandwich shell with hexagonal honeycomb filler witha periodicity cell consisting of ten individual elements as shown in Fig. 1.29.We assume that A1 = A2 = 1 and that all elements of the sandwichstructure shown in Fig. 1.29 are made of different generally orthotropicmaterials.

As noted above, the matrix [C] of elastic coefficients of an orthotropicmaterial referenced to a coordinate system that has been rotated by anangle ϕ with respect to its principal material coordinate system coincideswith that of a monoclinic material and has the form given in Eq. (1.55).The analytical solution of local problems (1.44), (1.46), and (1.45), (1.47)in the considered case can be found with the assumption that the thicknessof each of the cell elements is small in comparison to the other dimensions,i.e. under the conditions t h1, h2, H , t0 H and H ∼ h1, h2. Thisassumption is very appropriate for sandwich structures with cellular cores


48 A.L. Kalamkarov

Fig. 1.29. Unit cell of a three-layered composite sandwich with honeycomb filler.

used in engineering applications. On the basis of this analytical solution forall non-zero effective stiffness moduli we obtain (see [63, 64] for details):

⟨b1111⟩

=E

(1)2 F1

1 − ν(1)12 ν

(1)21

+E

(2)2 F2

1 − ν(2)12 ν

(2)21

+10∑i=3

E(i)FiS4(i)h

−42

(C2

(i)h−21 + S2

(i)h−22

)−2

⟨b2222⟩

=E

(1)1 F1

1 − ν(1)12 ν

(1)21

+E

(2)1 F2

1 − ν(2)12 ν

(2)21

+10∑i=3

E(i)FiC4(i)h

−42

(C2

(i)h−21 + S2

(i)h−22

)−2

⟨b1122⟩

=⟨b2211⟩

=ν

(1)12 E

(1)1 F1

1 − ν(1)12 ν

(1)21

+ν

(2)12 E

(2)1 F2

1 − ν(2)12 ν

(2)21

+10∑i=3

E(i)FiC2(i)S

2(i)h

−21 h−2

2

(C2

(i)h−21 + S2

(i)h−22

)−2



⟨b1212⟩

= G(1)12 F1 +G

(2)12 F2

+10∑i=3

E(i)FiC2(i)S

2(i)h

−21 h−2

2

(C2

(i)h−21 + S2

(i)h−22

)−2

⟨zb∗1111

⟩=

E(1)1 J1

1 − ν(1)12 ν

(1)21

+E

(2)1 J2

1 − ν(1)12 ν

(1)21

+10∑i=3

E(i)JiS4(i)h

−42

(C2

(i)h−21 + S2

(i)h−22

)−2

⟨zb∗2222

⟩=

E(1)2 J1

1 − ν(1)12 ν

(1)21

+E

(2)2 J2

1 − ν(1)12 ν

(1)21

+10∑i=3

E(i)JiC4(i)h

−41

(C2

(i)h−21 + S2

(i)h−22

)−2

⟨zb∗1122

⟩=⟨zb∗2211

⟩=

ν(1)12 E

(1)1 J1

1 − ν(1)12 ν

(1)21

+ν

(2)12 E

(2)1 J2

1 − ν(2)12 ν

(2)21

+10∑i=3

E(i)JiC2(i)S

2(i)h

−21 h−2

2

(C2

(i)h−21 + S2

(i)h−22

)−2

⟨zb∗1212

⟩= G

(1)12 J1 +G

(2)12 J2 +

10∑i=3

G(i)12 Ji

C2

(i)S2(i)h

−21 h−2

2(C2

(i)h−21 + S2

(i)h−22

)2− 8H

3π5h1

∞∑n=1

tanh[π(2n− 1)t/(2H)](2n− 1)5

).

(1.99)

In the above formulae, the superscripts (1), (2) and (i), i = 3, 4, . . . , 10,refer to the corresponding elements of the unit cell Ω1,Ω2,Ω3, . . . ,Ω10,shown in Fig. 1.29, and indicate the material properties of the correspondingstructural element; C(i) and S(i) stand for cosϕi and sin ϕi, respectively,where ϕi is an angle that element Ωi makes with the α2-axis, i = 3, 4, . . . , 10;and the quantities F1, F2, . . . , F10 and J1, J2, . . . , J10 are defined asfollows:

Fi = 〈1〉Ωi = t0; Ji = 〈z2〉Ωi =t03

(3H2

4+

3H t02

+ t20

), for i = 1, 2


50 A.L. Kalamkarov

Fi = 〈1〉Ωi =√

39Ht

a; Ji = 〈z2〉Ωi =

√3

108H3 t

a, for i = 3, 4, 5, 6

Fi = 〈1〉Ωi =√

318

Ht

a; Ji = 〈z2〉Ωi =

√3

216H3 t

a, for i = 7, 8, 9, 10.

(1.100)

Here F1, F2, . . . , F10 are the cross-sectional areas and J1, J2, . . . , J10 arethe moments of inertia of the cross-sections of the corresponding elementsΩ1,Ω2, . . . ,Ω10 relative to the middle surface of the shell, and are calculatedin the coordinate system ξ1, ξ2, z.

It is seen in Eq. (1.99) that the first terms represent the contributionfrom the top and bottom face carriers and the latter terms describe thecontribution of the sandwich core. If we now consider a particular casewhen the face carriers and honeycomb core are made of similar orthotropicmaterial then Eq. (1.99) reduces to the following formulae [64]:⟨

b1111⟩

= 2E1t0/(1 − ν12ν21) + 1.1732E1ν21Ht/(ν12a)⟨b2222⟩

= 2E2t0/(1 − ν12ν21) + 0.5152E2Ht/(a)⟨b1122⟩

=⟨b2211⟩

= 2ν12E1t0/(1 − ν12ν21) + 0.3908E1ν21Ht/(ν12a)⟨b1212⟩

= 2G12t0 + 0.3908E2H t/a⟨zb∗1111

⟩= E1t0(0.5H2 +Ht0 + 0.6667t20)/(1 − ν12ν21)

+ 0.0976E1ν21H3t/(ν12a)⟨

zb∗2222

⟩= E2t0(0.5H2 +Ht0 + 0.6667t20)/(1 − ν12ν21)

+ 0.0429336E2H3t/a⟨

zb∗1122

⟩=⟨zb∗2211

⟩= ν12E1t0(0.5H2 +H t0 + 0.6667t20)/(1 − ν12ν21)

+ 0.03256E1ν21H3t/(ν12a)⟨

zb∗1212

⟩= G12t0(0.5H2 +Ht0 + 0.6667t20) + 0.03256E2H

3t/a.

(1.101)

Note that Eqs. (1.99)–(1.101) generalize the earlier given Eq. (1.54) formuch more complicated cases of generally orthotropic constituent materials.

1.8.1 Examples of sandwich shells

The effective properties of sandwich composite structures can be tailored tomeet the requirements of a particular application by changing the geometric



Fig. 1.30. (a) Hexagonal-triangular and (b) star-hexagonal cored composite sandwichstructures.

parameters including the thickness of the face carriers, the width and cross-sectional areas of the core elements or the relative height of the core, theangular orientation of the elements and by changing the materials of thecore and face carriers. This is demonstrated below where a comparisonof the effective elastic properties of two different cellular cores is made;see Fig. 1.30. The unit cells of the analysed structures are shown inFigs. 1.31 and 1.32. Both structures have hexagonal symmetry beginningwith the triangular cell and maintaining the basic 60, 120 angulararchitecture.

The analytical formulae for all the effective stiffness moduli of thesesandwich structures have been derived in the case when the structuralelements are made of generally orthotropic materials. All the non-zeroeffective stiffness moduli are given in Table 1.3 [64].


52 A.L. Kalamkarov

Fig. 1.31. Unit cell of hexagonal-triangular core in global (α1, α2, γ) and local (ξ1, ξ2, z)coordinates.

Figures 1.33–1.37 show the variation of the effective stiffness moduliof the hexagonal-triangular and star-hexagonal cores vs. core height H .A graphite/epoxy core material is considered with the following properties:C11 = 183.443GPa, C12 = C13 = 4.363GPa, C22 = C33 = 11.662GPa,C23 = 3.918GPa, C44 = 2.87GPa, C55 = C66 = 7.17GPa, ν12 = ν13 =0.01593, ν21 = ν31 = 0.28003 and ν23 = ν32 = 0.33.

Figure 1.33 shows that the stiffness⟨b1111⟩

in the ξ1 direction issignificantly larger for the hexagonal-triangular core than its star-hexagonalcounterpart because the former has more reinforcements oriented in the ξ1direction. Although the latter structure has an overall similar number ofreinforcements to those of the hexagonal-triangular core, the horizontal



Fig. 1.32. Unit cell of star-hexagonal core in global (α1, α2, γ) and local (ξ1, ξ2, z)coordinates.

component of the reinforcements of that structure is much shorter inlength in the ξ1 direction and therefore makes a significant difference inthe homogenized stiffness in the ξ1 direction, while making almost nocontribution in the elastic properties in the ξ2 direction. In other words, itis expected that the stiffness properties in the ξ2 direction for the two coresremain almost unchanged. Indeed, Fig. 1.34 proves this trend precisely.

Figure 1.35 shows the variation of the effective stiffness⟨zb∗1111

⟩vs.

core height H . It is obvious that the presence of additional members in thehexagonal-triangular core has contributed to making the structure stifferthan its star-hexagonal counterpart. In Fig. 1.36, the effective stiffnesses ofthe two cores are very close due to the same reason as explained for Fig. 1.34.


54 A.L. Kalamkarov

Table 1.3. Effective stiffness moduli of hexagonal-triangular and star-hexagonal cores(A1 = A2 = 1, a = 2.5).

Effective stiffness Hexagonal-triangular core Star-hexagonal coremoduli shown in Figs. 1.30(a) and 1.31 shown in Figs. 1.30(b) and 1.32

˙b1111

¸16.6276E1ν21Ht/(ν12a) 3.3366E1ν21Ht/(ν12a)

˙b2222

¸2.4250 E2Ht/(a) 2.3287 E2Ht/(a)

˙b1122

¸=

˙b2211

¸5.5424E1ν21Ht/(ν12a) 1.1122E1ν21Ht/(ν12a)

˙b1212

¸5.5424E2Ht/a 1.1122E2Ht/a

˙zb∗1111

¸1.3856 E1ν21H3t/(ν12a) 0.2780 E1ν21H3t/(ν12a)

˙zb∗2222

¸0.2022E2H3t/a 0.1941E2H3t/a

˙zb∗1122

¸=

˙zb∗2211

¸0.4620 E1ν21H3t/(ν12a) 0.0927 E1ν21H3t/(ν12a)

˙zb∗1212

¸0.4620 E2H3t/a 0.0927 E2H3t/a

0

50

100

150

200

250

300

350

5 10 15 20 25

H

<b11

11>/

E1

Hexagonal-triangular

Star-hexagonal

Fig. 1.33. Effective stiffness˙b1111

¸/E1 vs. core height H of hexagonal-triangular and

star-hexagonal sandwich cores.

0

0.5

1

1.5

2

2.5

3

5 10 15 20 25

H

<b22

22>/

E2

Hexagonal-triangular

Star-hexagonal

Fig. 1.34. Effective stiffness˙b2222





0

2000

4000

6000

8000

10000

12000

14000

16000

5 10 15 20 25

H

<zb

11*1

1>/

E1 Hexagonal-

triangular

Star-hexagonal

Fig. 1.35. Effective stiffness˙zb∗1111



0

20

40

60

80

100

120

140

5 10 15 20 25

H

<zb

22*2

2>/

E2 Hexagonal-

triangularStar-

hexagonal

Fig. 1.36. Effective stiffness˙zb∗2222



0

50

100

150

200

250

300

350

5 10 15 20 25

H

<b12

12>/

E2 4

3

2

1

0

5

6

<zb

12*1

2>/

E2

<b1212>/E2,

hex.-trian.

<b1212>/E2,

star-hex.

<zb12*12>/E2,

hex.-trian.

<zb12*12>/E2,

star-hex.

Fig. 1.37. Effective stiffness moduli˙b1212

¸/E2 and

˙zb∗1212

¸/E2 vs. core height H of

hexagonal-triangular and star-hexagonal sandwich cores.


56 A.L. Kalamkarov

Finally, Fig. 1.37 shows the variation of effective stiffnesses⟨b1212⟩

and⟨zb∗1212

⟩vs. core height H . As expected both the extensional and torsional

stiffnesses in the direction of the reinforcements increase with the height ofthe core.

1.9 Conclusion

Asymptotic homogenization is a mathematically rigorous and powerfultool for analysing composite materials and structures. The proof of thepossibility of homogenizing a composite material of a regular structure,i.e. of examining an equivalent homogeneous solid instead of the originalinhomogeneous composite solid, is one of the principal results of thistheory. The method of asymptotic homogenization also gives a procedureof transition from the original problem (which contains in its formulation asmall parameter related to the small dimensions of the constituents of thecomposite) to a problem for a homogeneous solid. The effective propertiesof this equivalent homogeneous material are determined through thesolution of unit-cell problems. An important advantage of the asymptotichomogenization is that in addition to the effective properties it allows thehighly accurate determination of the local stress and strain distributionsdefined by the microstructure of the composite materials.

This chapter reviews the basics of the asymptotic homogenization ofcomposite materials and thin-walled composite structures. The asymptotichomogenization of 3D thin-walled composite reinforced structures isconsidered and the general homogenization composite shell model isintroduced. Micromechanical models are derived and applied to obtainanalytical formulae for the effective stiffnesses of generally orthotropic grid-reinforced composite materials, composite shells and plates, rib- and wafer-reinforced shells, as well as sandwich composite shells with cellular cores ofdifferent geometrical configuration.

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Chapter 2

SCALING AND HOMOGENIZATIONIN SPATIALLY RANDOM COMPOSITES

Martin Ostoja-Starzewski∗ and Shivakumar I. Ranganathan†∗Department of Mechanical Science and Engineering

Beckman Institute, and Institute for Condensed Matter TheoryUniversity of Illinois at Urbana-Champaign, Urbana, IL 61801, USA

†Department of Mechanical Engineering, American University of Sharjah,Sharjah, 26666, UAE

Abstract

We review the key issues involved in scaling and homogenization of randomcomposite materials. In the first place, this involves a Hill–Mandel condition inthe setting of stochastic micromechanics. Within this framework, we introducethe concept of a scaling function that describes “finite-size scaling” of thermallyconducting or elastic crystalline aggregates. While the finite size is representedby the mesoscale, the scaling function depends on an appropriate measurequantifying the single-crystal anisotropy. Based on the scaling function, weconstruct a material scaling diagram, from which one can assess the approachfrom a statistical volume element (SVE) to a representative volume element(RVE) for many different materials. We demonstrate these concepts with thescaling of the fourth-rank elasticity and the second-rank thermal conductivitytensors. We discuss the trends in approaching the RVE for planar conductivity,linear/non-linear (thermo)elasticity, plasticity, and Darcy permeability.

2.1 Introduction

In this chapter, we review the progress on the scale-dependent properties ofrandom microstructures. Consider a simple tension test of a polycrystallineaggregate such as a bar of copper with randomly oriented grains. Thisis perhaps among the earliest concepts taught in an introductory solidmechanics or materials science class. One of the important outcomes of such

61


62 M. Ostoja-Starzewski and S.I. Ranganathan

a test would be to construct the stress-strain curve (true or engineering)and extract material properties such as the Young’s modulus, yield stress,ultimate stress, etc. At that stage, it is seldom mentioned that suchproperties are in fact scale-dependent and it is inherently assumed thatthe sample size is “significantly larger” than the grain size.

A possible approach to address the preceding question would beto introduce the notion of a representative volume element (RVE) bydetermining its size. Following Hill [1]:

A Representative Volume is a sample that (a) is structurallyentirely typical of the whole mixture on average and (b) containsa sufficient number of inclusions for the apparent overall modulito be effectively independent of the surface values of traction anddisplacement, so long as these values are macroscopically uniform.

The same idea was independently developed by Mandel and Dantu [2].Strictly speaking, the apparent overall modulus becomes independent ofthe applied boundary conditions only at an infinite length scale, but forall practical purposes, one could approximate the size of the RVE atfinite scales within a few percent error. Below the RVE, the responseinvolves statistical scatter and such a volume element is therefore calledthe statistical volume element (SVE). Besides this situation of a domaincontaining a very large (mathematically infinite) set of microscale elements(e.g. grains or inhomogeneities), the RVE is very clearly defined for a unitin a periodic microstructure. We shall not pursue the latter situation here.

With reference to the example of the simple tension test on a copperbar, the above statement implies that the response of the specimen willbe realization dependent if the sample size is not representative. It is ofinterest here to analyze: (i) the approach of the SVE towards the RVE,and (ii) establish scaling laws in a variety of random microstructures suchas those depicted in Fig. 2.1. Indeed, these form but a small subset of amyriad of different microstructures (e.g. [3, 4]), but the methodology weoutline on simple models of random composites can readily be applied tomore complex material systems.

We employ the framework of stochastic (micro)mechanics consistentwith the mathematical statement of the Hill–Mandel condition statedabove. Such an approach is extremely versatile and can be used not onlyin linear elastic problems but also in bounding the response of hyperelasticmaterials, elasto-plastic materials, heat conduction, porous media; e.g. see


Scaling and Homogenization in Spatially Random Composites 63

(a) (b)

(d)(c)

Fig. 2.1. (a) Random polycrystal in 2D. (b) Random polycrystal in 3D where the colorscale represents the different orientation (random) of each grain. (c) Circular-inclusioncomposite, showing a mesoscale window. (d) Microstructure of trabecular bone obtained

by micro-CT imaging.

[5, 6]. In the following sections we discuss the scale-dependent bounds andscaling laws on the various constitutive behaviors of random materials.

2.2 Scaling in Heat Conduction

Single crystals exhibit an anisotropic behavior in heat conduction. Ingeneral, thermal conductivity is a second-rank tensor that is completelydefined by three independent components in the principal direction. Forexample, trigonal, hexagonal, and tetragonal single crystals demonstratea uniaxial thermal character with two of the principal components ofthe conductivity tensor being equal. On the other hand, a randomlyoriented polycrystal exhibits isotropic behavior at the RVE scale. Thus, thefollowing are the key aspects at the intermediate scale of the SVE: (i) the



sample response is in general anisotropic and realization dependent; (ii) theisotropic response may be recovered by averaging over a sufficient numberof ensembles; (iii) the apparent conductivity is very much dependent onthe applied boundary conditions; and (iv) with increasing length scales,the SVE approaches the RVE. It is apparent from these observations thatin going from the level of a single crystal to that of a polycrystallineaggregate, the conductivity is scale dependent and isotropy is approachedindicating the reduction in the number of independent constants necessaryto define the conductivity tensor completely. Another related question iswhether one could establish scaling laws unifying a class of materials. Forinstance, hematite and quartz possess a trigonal crystal structure at thesingle-crystal level and thereby one is tempted to seek similarities in theirscale-dependent properties. In the subsequent discussion, we attempt toanswer some of the above questions using stochastic (micro)mechanics asthe tool and the Hill–Mandel condition as the basis.

2.2.1 The Hill–Mandel condition

Consider a specific realization Bδ(ω) of a random medium Bδ. Here, ω(∈ Ω)indicates a specific realization of the microstructure taken from the samplespace Ω and the subscript δ, defined as

δ = (NG)13 , (2.1)

is the dimensionless parameter specifying the mesoscale of thepolycrystalline aggregate comprising NG grains. In the following, δ willsometimes be referred to as window size.

Now, adapting the idea of the Hill–Mandel condition, one can establishthe following relationship [6–8]:

q · ∇T = q · ∇T ⇔∫∂Bδ

(q · n− q · n)(T −∇T · x)dS = 0, ∀x ∈ ∂Bδ.

(2.2)

Here q is the heat flux, ∇T is the temperature gradient, and x is the positionvector. The overbar operator (•) indicates volume averaging. Equation (2.2)suggests three types of boundary condition:

(i) uniform essential (EBC): T = ∇T 0 · x, (2.3a)

(ii) uniform natural (NBC): q · n = q0 · n, (2.3b)

(iii) mixed-orthogonal: (q · n − q0 · n)(T −∇T 0 · x) = 0. (2.3c)



Fig. 2.2. Methodology for obtaining scale-dependent bounds in heat conduction.

By increasing the mesoscale δ (effectively, the number of grains in Bδ)and by setting up stochastic boundary-value problems with the aboveboundary conditions and upon ensemble averaging, one obtains boundson the constitutive response of the aggregate. Now, the condition (2.3a)results in a mesoscale (i.e. δ-dependent) conductivity tensor Ce

δ , (2.3b) is amesoscale resistivity tensor Snδ , while (2.3c) yields a mesoscale conductivityor resistivity tensor (depending on the interpretation). The superscripts eand n denote quantities obtained under essential and natural boundaryconditions, respectively. The condition (2.3c) is understood with thestipulation that one must not simultaneously specify both the heat fluxand the temperature gradient in any given direction on any portion of theboundary.

The methodology outlined here works provided the hypotheses ofspatial homogeneity and ergodicity hold for the random field Θ(x, ω) ofmaterial parameters involved. In particular, we assume Θ(x, ω) to be awide-sense stationary (WSS) random field with a constant mean and finite-valued autocorrelation [6]:

〈Θ(x1)〉 = µ,

〈Θ(x1)〈Θ(x1)Θ(x1 + h)〉〉 = RΘ(h) <∞.(2.4a)

The operator 〈•〉 indicates ensemble averaging. The random field Θ(x, ω)is mean ergodic providing its spatial average equals the ensemble average:

1V

∫V

Θ(x, ω)dV = Θ(x) = 〈Θ(x)〉 =∫

Ω

Θ(x, ω)dP. (2.4b)

The proposed methodology is illustrated in Fig. 2.2. The single crystal hasa reference conductivity tensor Cref

pq with three independent constants in



its principal directions c1, c2, and c3. By using a set of uniformly generatedrotation tensors (Fig. 2.2), the reference tensor is rotated to assign thematerial property for each individual crystal in the polycrystal. Using(2.3a,b), the boundary-value problems are solved and mesoscale tensorsestablished so that upon ensemble averaging one obtains bounds on theaggregate conductivity.

2.2.2 Bounds on the conductivity

At this point we recall the ergodicity and WSS properties of themicrostructure and obtain the hierarchy of scale-dependent bounds asfollows:

〈Sn1 〉−1 ≤ · · · ≤ 〈Snδ′〉−1 ≤ 〈Snδ 〉−1 ≤ · · · ≤ Ceff∞ · · · ≤ 〈Ce

δ〉≤ 〈Ce

δ′〉 · · · ≤ 〈Ce1〉, ∀ δ′ ≤ δ. (2.5)

Such bounds date back to Ostoja-Starzewski and Schulte [7] and Jianget al. [9]; see also [10]. Using (2.5) along with the definition of the isotropicconductivity tensor, we obtain the following hierarchy of bounds on theisotropic conductivity measure:

cH ≤ · · · ≤ 〈cnδ′〉 ≤ 〈cnδ 〉 ≤ · · · ≤ ceff∞≤ · · · 〈ceδ〉 ≤ 〈ceδ′〉 ≤ cA, ∀ δ′ ≤ δ, (2.6)

where 1/cH = (1/c1 + 1/c2 + 1/c3)/3 and cA = (c1 + c2 + c3)/3 are theharmonic (Reuss type) and the arithmetic mean (Voigt type) estimatesof the conductivity. While the hierarchy type of behavior of mesoscaletensors appears evident, it has been found in Ostoja-Starzewski [11] that thecoefficient of variation of the second invariant of Ce

δ as well as Seδ for severaldifferent planar random microstructures (all generated by homogeneousPoisson point fields) equals ∼0.55.

2.2.3 Scaling function in heat conduction

For a given realization Bδ(ω) of a random medium Bδ on some mesoscaleδ, (2.3a) yields a mesoscale random conductivity tensor Ce

δ(ω) such that

qδ(ω) = Ceδ(ω) · ∇T 0. (2.7)

Similarly, (2.3b) yields a mesoscale random resistivity tensor Snδ (ω) suchthat

∇Tδ(ω) = Snδ (ω) · q0. (2.8)



In general, for any given realization ω(∈ Ω), Ceδ(ω), and Snδ (ω) are

anisotropic. We obtain an isotropic response only by assigning the crystalorientations uniformly (distributed uniformly on a unit sphere; also see [12])and upon ensemble averaging over the realization space. Thus the ensemble-averaged isotropic conductivity and resistivity tensors can be expressed asfollows

〈Ceδ〉 = 〈ceδ〉I, (2.9)

〈Snδ 〉 =1

〈cnδ 〉I. (2.10)

In the above, I represents the second-rank identity tensor, while 〈ceδ〉and 〈cnδ 〉 are ensemble-averaged isotropic conductivity measures under theessential and natural boundary conditions, respectively. By contracting(2.9) and (2.10) we obtain the following scalar equation:

〈Ceδ〉 : 〈Snδ 〉 = 3

〈ceδ〉〈cnδ 〉

. (2.11)

In the limit δ → ∞ the conductivity tensor must be the exact inverse ofthe resistivity tensor, and so we obtain

limδ→∞

〈Ceδ〉 : 〈Snδ 〉 = 3. (2.12)

Now, we postulate the following relationship between the left-hand sideof (2.11) and (2.12), that is

〈Ceδ〉 : 〈Snδ 〉 = lim

δ→∞〈Ce

δ〉 : 〈Snδ 〉 + g(c1, c2, c3, δ)

= limδ→∞

〈Ceδ〉 : 〈Snδ 〉 + g(k1, k2, c3, δ). (2.13)

In the above, g(c1, c2, c3, δ) [“or” g(k1, k2, c3, δ)] defines the scaling functionand k1 = c1c

−13 and k2 = c2c

−13 are two non-dimensional parameters.

Substituting (2.13) and (2.12) into (2.11), we obtain

g(k1, k2, c3, δ) = 3( 〈ceδ〉〈cnδ 〉

− 1). (2.14)

Notice that the right-hand side of (2.14) is dimensionless. Thus, the scalingfunction is dependent only on the non-dimensional parameters and takesthe form g(k1, k2, δ) and can be determined numerically by the solutionof boundary-value problems subject to the uniform essential and naturalboundary conditions.



2.2.4 Some properties of and bounds on the scaling

function

The scaling function g(k1, k2, δ) introduced in (2.14) has the followingproperties (see also [8, 13]):

g(k1, k2, δ = ∞) = 0. (2.15)

Again, the scaling function becomes null if the crystals are locallyisotropic:

g(k1 = k2 = 1, δ) = 0. (2.16)

One can further establish the following bounds on the scaling functionfor aggregates made up of single crystals with uniaxial thermal character(k1 = k2 = k for trigonal, hexagonal, and tetragonal single crystals. k isalso a measure of a single crystal’s anisotropy)

0 ≤ g(k, δ) ≤ 23

(√k − 1√

k

)2

. (2.17)

The lower bound in (2.17) is easily established using Eq. (2.6)and (2.14). Also at any given scale the following inequality holds(using Eq. (2.6)) ( 〈ceδ〉

〈cnδ 〉− 1)

≤(cA

cH− 1). (2.18)

Using the definitions of cA and cH in Eq. (2.6), we obtain the upperbound.

2.2.5 Numerical simulations

We proved several properties of the scaling function in Section 2.2.4. We nowproceed to derive a suitable form for the scaling function based on numericalresults. Since it is not practical here to perform numerical simulations onall known crystals, we restrict our study to the materials listed in Table 2.1.

We perform numerical simulations with the number of grains, NG =20, 23, 26, and 29 (i.e. δ = 1, 2, 4, and 8). The following loading conditionshave been imposed to run the stochastic boundary-value problems:

Dirichlet problem: ∇T 0 = 4k (2.19a)

Neumann problem: q0 = −18k (2.19b)



Table 2.1. Material parameters.

Single crystal propertyThermal (W/m.C)

Crystal Thermal axes and Scalingsystem character orientation Materials c1 = c2 c3 k = c1/c3 k−1 function

Cubic Isotropic c1 = c2 = c3 aluminum 208 208 1 1 0copper 410 410 1 1 0

Trigonal Uniaxial c1 = c2 = c3 calcite 4.18 4.98 0.84 1.19 g1(k, δ)hematite 14.6 12.17 1.20 0.83 ∼= g1(k, δ)quartz 6.5 11.3 0.58 1.74 g2(k, δ)

Hexagonal Uniaxial c1 = c2 = c3 graphite 355 89 3.99 0.25 g3(k, δ)

Based on the numerical results for these materials, one can postulate asuitable form for the scaling function and develop the material scalingdiagram.

We begin the discussion by studying the scale-dependent response ofcalcite and hematite. As seen from Table 2.1, these materials have verydifferent single-crystal conductivities. Notice that, although the anisotropyindex of calcite is greater than one and that of hematite is less thanone, their product is almost equal to one. In other words the anisotropymeasure of calcite is almost the reciprocal of that of hematite. The scale-dependent bounds for calcite and hematite are shown in Figs. 2.3 and 2.4,respectively. Notice that since the anisotropy index for these crystals isclose to one, the Voigt and Reuss bounds are very close to one another.As observed in these plots, we obtain the upper bound on applicationof (2.19a) and the lower bound using (2.19b). Also, as we increase thenumber of grains (or as the mesoscale size δ increases), we obtain tighterbounds on the aggregate conductivity. The scaling functions for the calciteand hematite aggregates have similar trends and are quite close to oneanother (Fig. 2.5). We attribute the discrepancies to the finite number ofrealizations used to obtain the ensemble average and the slight differencesin the anisotropy index. Based on these observations, we conclude that twomaterials with single-crystal anisotropy indices k and k−1 represent thesame scaling function. Mathematically,

g(k, δ) = g

(1k, δ

). (2.20)

Let us now consider the scale-dependent bounds on quartz and graphiteas illustrated in Figs. 2.6 and 2.7. Owing to the higher single-crystalanisotropy index (or its reciprocal), the Voigt and Reuss bounds for quartz



Window size

Con

duct

ivit

y (W

/m.K

)

1 2 3 4 5 6 7 84.41

4.42

4.43

4.44

4.45

Calcite-VoigtCalcite-ReussCalcite-EBCCalcite-NBC

Fig. 2.3. Bounds on the aggregate response (calcite).

Window size

Con

duct

ivit

y (W

/m.K

)

1 2 3 4 5 6 7 813.66

13.68

13.7

13.72

13.74

13.76

13.78

13.8

Hematite-VoigtHematite-ReussHematite-EBCHematite-NBC

Fig. 2.4. Bounds on the aggregate response (hematite).



Window size

Scal

ing

func

tion

1 2 3 4 5 6 7 80

0.01

0.02

0.03HematiteCalcite

Fig. 2.5. Scaling function (calcite and hematite).

Window size

Con

duct

ivity

(W/m

.K)

2 4 6 87.5

7.6

7.7

7.8

7.9

8

8.1

8.2

Quartz-VoigtQuartz-ReussQuartz-EBCQuartz-NBC

Fig. 2.6. Bounds on the aggregate response (quartz).



Window size

Con

duct

ivit

y (W

/m.K

)

1 2 3 4 5 6 7 8160

180

200

220

240

260

280

Graphite-VoigtGraphite-ReussGraphite-EBCGraphite-NBC

Fig. 2.7. Bounds on the aggregate response (graphite).

and graphite are much farther apart compared to calcite or hematite. Again,the application of (2.19a) and (2.19b) bounds the aggregate conductivityfrom above and below, respectively. The scaling functions for quartz andgraphite are plotted in Figs. 2.8 and 2.9, respectively. We immediatelynotice that for any given scale, the scaling functions for graphite and quartzhave much larger values than hematite or graphite. Since the anisotropyindex (or its reciprocal) for graphite is much larger than that of quartz,its scaling function takes markedly higher values than that of quartz at allscales.

2.2.6 Constructing the scaling function

In Fig. 2.10, we compile the scaling function for all the different crystalsconsidered in the previous section. It is useful to rewrite (2.17) as follows

0 ≤ 3

2(√

k − 1√k

)2 g(k, δ) ≤ 1. (2.21)



Window size

Sca

ling

func

tion

1 2 3 4 5 6 7 8-0.05

0

0.05

0.1

0.15

0.2

0.25

Quartz

Fig. 2.8. Scaling function (quartz).

Window size

Scal

ing

func

tion

1 2 3 4 5 6 7 80

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Graphite

Fig. 2.9. Scaling function (graphite).



Window size

Sca

ling

func

tion

1 2 3 4 5 6 7 8

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

GraphiteQuartzHematiteCalcite

Fig. 2.10. Scaling function (compiled).

We now proceed to plot the rescaled scaling function defined in(2.21). Interestingly, from Fig. 2.11 it appears that rescaling boundsall the materials closely. We attribute the discrepancies (especially foraggregates with few grains (small mesoscale δ)) to the finite numberof realizations employed to construct the scaling function. The aboveessentially means that we could approximate the mean rescaled functiong∗(δ) to be independent of the single-crystal anisotropy. The scalingfunction can now be redefined as follows

g(k, δ) =23

(√k − 1√

k

)2

g∗(δ), (2.22)

where g∗(δ) represents the material-independent rescaled function. Basedon the mean values of g∗(δ) in Fig. 2.11, we construct the effective averagedrescaled function and fit it using an exponential function. The effectivefunction and its fit are illustrated in Fig. 2.12. Based on this fit, g∗(δ) takesthe following form

g∗(δ) = exp[−0.9135(δ− 1)0.5]. (2.23)



Window size

Res

cale

d fu

ncti

on

1 2 3 4 5 6 7 80

0.5

1

1.5

QuartzGraphiteHematiteCalcite

Fig. 2.11. Rescaled scaling function.

Window size

Eff

ecti

vere

sca

led

func

tion

1 2 3 4 5 6 7 8-0.2

0

0.2

0.4

0.6

0.8

1

1.2

ActualFit

Fig. 2.12. Effective rescaled scaling function and fit.



In view of (2.22), the scaling function takes the following form

g(k, δ) =23

(√k − 1√

k

)2

exp[−0.9135(δ− 1)0.5], δ = N13g . (2.24)

This particular form of the scaling function satisfies all the propertiesdefined in Section 2.2.4. Consistent with (2.20), it does not distinguishbetween k and k−1. Equation (2.24) also suggests that the scaling functionis exact in the single-crystal anisotropy index. The empirical form comesfrom its dependence on the window size. The form of (2.24) has been used toreconstruct the scaling function for the different polycrystals and is plottedin Fig. 2.13. It is evident from the plot that (2.24) captures the scalingfunction reasonably well.

Window size

Sca

ling

func

tion

1 2 3 4 5 6 7 8-0.005

0

0.005

0.01

0.015

0.02

0.025

Hematite-actualCalcite-actualHematite-reconstructedCalcite-reconstructed

Window size

Scal

ing

func

tion

1 2 3 4 5 6 7 8-0.05

0

0.05

0.1

0.15

0.2

0.25

Quartz-actualQuartz-reconstructed

Window size

Scal

ing

func

tion

1 2 3 4 5 6 7 80

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Graphite-actualGraphite-reconstructed

(a) (b)

(c)

Fig. 2.13. Goodness of fit: (a) calcite and hematite, (b) quartz, (c) graphite.



Anisotropy index

Win

do

w s

ize

1 2 3 4100

101

102

f=0.0001f=0.001f=0.01f=0.1f=0.3f=0.6f=1.0f=10.0

Fig. 2.14. Contours of scaling function (for 0.0001 ≤ g ≤ 10).

2.2.7 Constructing the scaling function

It is now possible to construct the contours of the scaling function in thek − δ space based on Eq. (2.24) as illustrated in Fig. 2.14. Notice that, asthe scaling function decreases, the curves shift towards a higher mesoscaleand vice versa.

Theoretically the scaling function becomes zero only when the numberof grains is infinite (i.e. δ ≈ ∞) or when the crystal is locally isotropic withk = 1 (such as cubic crystals). For all practical purposes, one defines theRVE based on some finite value of the scaling function. We chose a valueof 0.01 for the scaling function to construct Fig. 2.15 since the discrepancyin the conductivity predictions for most crystals based on the essentialand natural boundary conditions lie within 0.5%. The plots give an ideaof the number of grains necessary to homogenize the aggregate responsein the anisotropic heat conduction for hexagonal (Fig. 2.15(a)), trigonal(Fig. 2.15(b)), and tetragonal crystals (Fig. 2.15(c)). Since cubic crystalsare locally isotropic, the number of grains necessary is trivial and equalsone. We notice from these plots that two aggregates made up of singlecrystals with anisotropy indices k and k−1 scale identically. Also, we need



C

Anisotropy index

Num

ber

of g

rain

s

0.5 1 1.5 2 2.5 3 3.5100

101

102

103

104

Emerald

Nephelite

Anglesite

Pyromorphite

Pyrrhotite

Apatite

Parisite

Graphite

f=0.01 (Mesoscale)

Macroscale

Microscale

Mic

rosc

ale

Hexagonal single crystals

D

Anisotropy index

Num

ber

of g

rain

s

0.6 0.8 1 1.2 1.4 1.6 1.8100

101

102

103Quartz

Troostite

Tourmaline

Penninite

Chabazite

Phenacite

Eudialyte

Cinnabarite

Argyrodite

Hematite

Ilmenite

Corundum

Mesitite

Calcite

Dolomite

f=0.01 (Mesoscale)

Microscale

Macroscale

Microscale

Trigonal single crystals

Anisotropy index

Num

ber

of g

rain

s

0.6 0.8 1 1.2 1.4 1.6 1.8100

101

102

103

Urea

Erythritol

Zircon

Scapolite

Vesuvianite

Cassiterite

Phosgenite

Wulfenite

Calomel

Scheelite

Rutile

Octahedrite

f=0.01 (Mesoscale)

Tetragonal single crystals

Macroscale

Microscale

Microscale

(a) (b)

(c)

D

C

Fig. 2.15. Material scaling diagram at g = 0.01: (a) hexagonal single crystals,(b) trigonal single crystals, (c) tetragonal single crystals.

more grains (or crystals) to homogenize the aggregate response as k or k−1

depart from unity.

2.3 Scaling in Elasticity

In this section, we consider the elastic properties of polycrystals atthe microscale, mesoscale, and macroscale. Single crystals are typicallyanisotropic elastically and the extent of anisotropy can be quantifiedby using various measures of anisotropy [14–16]. At the microscale,a polycrystalline sample consists of relatively few grains exhibiting arealization-dependent anisotropic response. As the length scale increasesand as the RVE is approached, the aggregate sample typically consistsof many grains and the response becomes realization independent andisotropic. Much like our observations in the heat conduction problem, the



elastic properties are indeed scale dependent and our attempt is to establishunifying scaling laws for a class of elastic crystals. For instance, copper andtantalum have a cubic crystal system at the single-crystal level and therebyone is tempted to seek similarities in their scale-dependent properties. Withthese objectives in mind, we re-state the Hill–Mandel condition for elasticproperties and then employ stochastic (micro)mechanics to determine thescale-dependent elastic properties and to establish scaling laws.

2.3.1 The Hill–Mandel condition

The Hill–Mandel condition in elasticity stems from the consideration of theequivalence of energetic and mechanical interpretations of stored energy[2, 17−21]:

σij : εij = σij : εij ⇒ 1V

∫∂V

σ′ij : ε′ijdV = 0

⇔∫∂Bδ

(ti − σijnj)(ui − εijxj)dS = 0 (2.25)

The three boundary conditions that satisfy (2.25) are:

(i) uniform displacement (Dirichlet): ui = ε0ijxj , (2.26a)

(ii) uniform traction (Neumann): ti = σ0ijnj , (2.26b)

(iii) mixed-orthogonal: (ti − σ0ijnj)(ui − ε0ijxj) = 0. (2.26c)

With increasing mesoscale δ, we obtain scale-dependent bounds (Fig. 2.16)on the elastic response of the aggregate by setting up and solvingboundary-value problems consistent with Eq. (2.26). The condition (2.26c)is understood with the stipulation that one must not simultaneously specifyboth the traction and the displacement in any given direction on any portionof the boundary.

2.3.2 Bounds on the elastic response

At this point we recall that the ergodicity and WSS properties of themicrostructure, together with the variational principles of elasticity theory,imply a hierarchy of scale-dependent bounds on the elastic response

〈St1〉−1 ≤ · · · ≤ 〈Stδ′〉−1 ≤ 〈Stδ〉−1 ≤ · · · ≤ Ceff∞ · · · ≤ 〈Cd

δ〉≤ 〈Cd

δ′〉 · · · ≤ 〈Cd1〉, ∀ δ′ ≤ δ. (2.27)

The superscripts t and d denote quantities obtained under the applicationof traction and displacement boundary conditions, respectively. Such



Fig. 2.16. Methodology for obtaining scale-dependent bounds for elastic polycrystals.

bounds date back to Huet [19] and Sab [22] and have been applied tomicrostructures with various geometries [6, 23−28]. Using (2.27) along withthe definition of the isotropic elasticity tensor, we obtain the followinghierarchy of bounds on the shear and bulk moduli:

GR ≤ · · · ≤ 〈Gtδ′〉 ≤ 〈Gtδ〉 ≤ · · · ≤ Geff∞

≤ · · · 〈Gdδ〉 ≤ 〈Gdδ′〉 · · · ≤ GV , (2.28a)

KR ≤ · · · ≤ 〈Ktδ′〉 ≤ 〈Kt

δ〉 ≤ · · · ≤ Keff∞

≤ · · · 〈Kdδ 〉 ≤ 〈Kd

δ′〉 · · · ≤ KV , ∀ δ′ ≤ δ, (2.28b)

where (GR,KR) and (GV ,KV ) represent, respectively, the Reuss and Voigtestimates of the shear and bulk moduli.

2.3.3 Elastic scaling function

For a given realization Bδ(ω) of a random medium Bδ on some mesoscaleδ, (2.26a) yields a mesoscale random stiffness tensor Cd

δ(ω) such that

σδ(ω) = Cdδ(ω) : ε0. (2.29)

Similarly, (2.26b) yields a mesoscale random compliance tensor Stδ(ω) suchthat

εδ(ω) = Stδ(ω) : σ0. (2.30)



By uniformly distributing the crystal orientations and upon ensembleaveraging, we recover the isotropic aggregate response. In such a case, theaveraged stiffness and compliance tensors can be expressed in terms of theshear modulus G and the bulk modulus K as follows:

〈Cdδ〉 = 2〈Gdδ〉K + 3〈Kd

δ 〉J, (2.31)

〈Stδ〉 =1

2〈Gtδ〉K +

13〈Kt

δ〉J. (2.32)

In the above, J and K represent the spherical and the deviatoric parts ofthe unit fourth-order tensor I. By contracting (2.31) and (2.32), we obtainthe following scalar equation

〈Cdδ〉 : 〈Stδ〉 = 5

〈Gdδ〉〈Gtδ〉

+〈Kd

δ 〉〈Kt

δ〉. (2.33)

In the limit δ → ∞ the stiffness tensor must be the exact inverse of thecompliance tensor, and so we obtain

limδ→∞

〈Cdδ〉 : 〈Stδ〉 = 6. (2.34)

Now, we postulate the following relationship between the left-hand sideof (2.33) and (2.34), that is

〈Cdδ〉 : 〈Stδ〉 = lim

δ→∞〈Cd

δ〉 : 〈Stδ〉 + f(Cij , δ), (2.35)

where f(Cij , δ) defines the elastic scaling function. The parameter Cijrepresents all the independent single-crystal elastic constants dependingon the crystal type. For aggregates made up of cubic single crystals,Cij ≡ (C11, C12, C44) and for triclinic systems, Cij will include all the21 independent single-crystal constants. Substituting (2.35) and (2.34) in(2.33), we obtain

f(Cij , δ) = 5〈Gdδ〉〈Gtδ〉

+〈Kd

δ 〉〈Kt

δ〉− 6. (2.36)

For the specific case of cubic crystals, the bulk modulus is scaleindependent [29, 30] and (2.36) can be rewritten as

f(C11, C12, C44,δ) = 5( 〈Gdδ〉〈Gtδ〉

− 1). (2.37)



The scaling function f(Cij , δ) introduced in (2.36) has the followingproperties

f (Cij , δ = ∞) = 0. (2.38)

Equation (2.38) states that the scaling function is identically zero at infinitemesoscale. Again, the scaling function becomes null if the crystals are locallyisotropic, that is

f(iso(Cij), δ) = 0. (2.39)

The term iso(Cij) accounts for all the possible combinations of the single-crystal elastic constants that will ensure an isotropic single-crystal response.One can further establish the following bounds on the scaling function

f(Cij ,∞) ≤ f(Cij , δ) ≤ f(Cij , 1), ∀ 1 ≤ δ ≤ ∞. (2.40)

Using (2.38) and (2.36) in (2.40), we obtain

0 ≤ f(Cij , δ) ≤ AU4 (1) = 5GV

GR+KV

KR− 6, ∀ 1 ≤ δ ≤ ∞, (2.41)

where the quantity AU4 (1) = 5GV /GR + KV /KR − 6 represents the so-called universal anisotropy index quantifying the single-crystal anisotropy(see also [14]) of the fourth-rank elasticity tensor. That index is increasinglyproving to be of use to many researchers in solid state physics andgeophysics; e.g. see [101] for the mapping of the entire Earth’s surface interms of AU4 (1) = 5GV /GR +KV /KR − 6.

Based on (2.41) one can interpret the elastic scaling function in (2.36)as the evolution of the equivalent anisotropy in the mesoscale domain,thus

f(Cij , δ) = AU4 (δ). (2.42)

The simplest form for (2.42) having a separable structure:

f(Cij , δ) = AU4 (δ) = AU4 (1)h4(δ), (2.43)

is a very good approximation for single-phase aggregates made up of singlecrystals of cubic type. The scaling function depends on the single-crystal



anisotropy and the mesoscale, and takes the following form based onnumerical simulations (see also [29]):

f(Cij , δ) = AU4 (δ) = AU4 (1)h4(δ)

=65

(√A− 1√

A

)2

exp[−0.767(δ − 1)0.5]. (2.44)

Again, as discussed previously in Section 2.2.3, one may determine thesize of the RVE by choosing a convenient value for the scaling function.Based on (2.44), in Fig. 2.17 we plot the contours of the scaling functionin the (A, δ) space. It is evident that for a fixed value of the scalingfunction, the mesoscale size increases with an increase in the single-crystal anisotropy. In other words the higher the single-crystal anisotropy,the greater the number of grains necessary to homogenize the aggregateresponse. This fact is again confirmed in Fig. 2.18 (plotted for f = 0.23) fora variety of aggregates made up of cubic single crystals. Again, notice thedistinct regions (microscale, limiting mesoscale, and macroscale) in theseplots.

Fig. 2.17. Contours of the scaling function in the (A, δ) space.



Fig. 2.18. Material scaling diagram for polycrystals made up of cubic single crystals(f = 0.23).

2.4 Scaling in Inelastic and Non-Linear Materials

2.4.1 Thermoelasticity

First recall that the thermal expansion phenomenon reflects a couplingbetween mechanical and thermal fields. It is grasped by the thermal straincoefficient αkl (a second-rank tensor), a material property. Alternatively,one can also employ the thermal expansion stress coefficient Γkl given bythe constitutive relation Γij = −Cijklαkl, with Cijkl being the stiffnesstensor. For heterogeneous materials, that relation does not hold unless thedV element is a homogeneous domain within any constituent phase or theRVE for a homogenized composite. To study the RVE size in the lattercase, one again uses the Hill–Mandel condition as the starting point, andfollows the scaling concepts introduced elsewhere. Thus, besides Eqs. (2.27)and (2.28a,b), one also needs to set up bounds on the effective thermalexpansion and the specific heat [31, 32]. In particular, depending on therelations between the material constants of a composite’s constituents, forthe thermal expansion coefficients we have two cases of hierarchies:

(i) α1 > α2 ≥ 0 and k1 > k2:

α∗ ≥ · · · ≥ 〈αnδ 〉 ≥ 〈αnδ′〉 ≥ · · · ≥ 〈αn1 〉, ∀ δ′ ≤ δ (2.45)



(ii) α1 > α2 ≥ 0 and k1 < k2:

α∗ ≤ · · · ≤ 〈αnδ 〉 ≤ 〈αnδ′〉 ≤ · · · ≤ 〈αn1 〉, ∀ δ′ ≤ δ. (2.46)

Similarly, there are two cases of hierarchies for the thermal stresscoefficients:

(i) 0 ≥ Γ1 > Γ2 and k1 > k2:

Γ∗ ≤ · · · ≤ 〈Γeδ〉 ≤ 〈Γeδ′〉 ≤ · · · ≤ 〈Γe1〉, ∀ δ′ ≤ δ, (2.47)

(ii) 0 ≥ Γ1 > Γ2 and k1 < k2:

Γ∗ ≥ · · · ≥ 〈Γeδ〉 ≥ 〈Γeδ′〉 ≥ · · · ≥ 〈Γe1〉, ∀ δ′ ≤ δ. (2.48)

2.4.2 Elasto-plasticity

Turning to a two-phase elastic-hardening plastic composite, the constitutiveresponses of both phases p(= 1, 2), are taken in the following form [17]:

dε′ij =dσ′

ij

2Gp+ h · dfp · ∂fp

∂σijwhenever fp = cp and dfp ≥ 0

dε′ij = dσ′ij/2Gp whenever fp < cp

dε = dσ · (1 − 2νp)2Gp(1 + νp)

everywhere(dε =

dεii3, dσ =

dσii3

).

(2.49)

Here primes indicate deviatoric tensor components; Gp is a shear modulus,vp is Poisson’s ratio, fp is a yield function, cp is a material constant, and his an isotropic hardening parameter.

Under monotonically increasing loading, elasto-plastic hardeningcomposites can be treated as physically non-linear elastic materials.Since the stiffness and compliance tensors are not constant any more,we consider the tangent stiffness and compliance moduli (CTdδ or STtδ ),defined as

dσ = CTdδ : dε = CTd

δ : dε0; dε = STtδ : dσ = STtδ : dσ0. (2.50)

Here the superscript d (or t) indicates the response obtained underthe displacement or traction boundary conditions. As noticed by Jiang



et al. [33], there is a hierarchy of upper and lower bounds on the effectivetangent modulus:

〈STS1 〉−1 ≡ 〈STt1 〉−1 ≤ · · · ≤ 〈STtδ′ 〉−1 ≤ 〈STtδ 〉−1 ≤ · · · ≤ (ST∞)−1

≡ CT∞ ≤ · · · ≤ 〈CTd

δ 〉 ≤ 〈CTdδ′ 〉 ≤ · · · ≤ 〈CTd

1 〉≡ 〈CTT

1 〉, ∀ δ′ ≤ δ, (2.51)

where 〈STS1 〉−1 and 〈CTT1 〉 are recognized as the tangential Sachs and Taylor

bounds, respectively. Sample results are shown in Figs. 2.19 and 2.20 [34].Note that the flow rule on the mesoscale (under both kinds of boundarycondition) for a composite made of phases with a normal flow rule is notnecessarily normal.

An application of the above concepts to imperfect masonry structures isreported in Zeman and Sejnoha [35] and Cavalagli et al. [36], while there isa computation of RVE size using the mixed-orthogonal boundary conditionin Salahouelhadj and Haddadi [37]; see also [38]. A perspective of trends inmultiscale plasticity is given in McDowell [39].

A recent experimental study investigating the RVE size is reportedin Efstahiou et al. [40]; see also [41–43]. A thermomechanics frameworkallowing an effective physical interpretation of the micromechanical internalvariables and parameters in the RVE is developed in Einav and Collins[44]; see also [45, 46]. The geodesic character of the strain field patterns,apparently almost self-evident in Fig. 2.19, has been investigated in Jeulinet al. [47].

2.4.3 Finite elasticity

The assumption inherent in the finite hyperelasticity theory is the exist-ence of a strain energy function ψ per unit volume of an undeformed body,which depends on the deformation of the object and its material properties.The equation of state of the material in the reference configurationthen takes the form Pik = ∂ψ/∂Fik, where Pik is the first Piola–Kirchhoff stress tensor and Fik is the deformation gradient tensor.These two tensors form a conjugate pair, which satisfies the Hill–Mandelcondition:

PijFij = P ijF ij . (2.52)



Fig. 2.19. Equivalent plastic strain patterns in matrix-inclusion composites (atmesoscale δ = 6 and 20) under traction (middle row) and displacement (bottom row)boundary conditions, from Jiang et al. [33].

Following the same procedure as in Section 2.2.2, we consider thefunctional:

PUi =∫V0

ψ(Ui,k)dV −∫ST

t0iUidS, (2.53)

where Ui is an admissible displacement field and t0i is the specified boundarytraction. The relation (2.53) represents the finite elasticity counterpart



(a) (b)

Fig. 2.20. Ensemble-averaged elasto-plastic stress-strain curves (a) and yield surfaces(b), depending on the boundary conditions imposed.

of the principle of the minimum potential energy for infinitesimal elasticdeformation. The functional PUi assumes a local minimum for the actualsolution ui providing

∫V0

∂2ψ∂ui,k∂up,q

di,kdp,qdV > 0, for all non-zero di suchthat di = 0 [48]. Under the uniform displacement boundary condition,which in non-linear elasticity has the form ui = (F 0

ik − δik)xk , ∀xk ∈ S0,the upper bound on the effective response is [49, 50]:

〈Ψ(F0,∆)〉 ≤ 〈Ψ(F0, δ)〉 ≤ 〈Ψ(F0, δ′)〉 ≤ 〈Ψ(F0, 1)〉, for ∀ δ′ < δ < ∆,

(2.54)

where Ψ(ω,F0) =∫V0ψ(ω,X,F)dV , and ∆ and 1 denote the RVE size and

inhomogeneity size, respectively.Considering the complementary energy principle [48], under the

uniform traction boundary condition ti = P 0iknk, ∀xj ∈ S0, we have another

scale-dependent hierarchy of bounds on the effective properties:

〈Ψ∗(P0)〉∆ ≤ 〈Ψ∗(P0)〉δ ≤ 〈Ψ∗(P0)〉δ′ ≤ 〈Ψ∗(P0)〉1, for ∀ δ′ < δ < ∆,

(2.55)



where Ψ∗(ω,P0) =∫V0

∂ψ∂Uik

Uik − ψdV and we assume∫V0

∂2ψ∂ui,k∂up,q

dik

dpqdV > 0 for all non-zero dik satisfying ∂∂xk

( ∂2ψ∂ui,k∂up,q

dpq) = 0 in V0 and∂2ψ

∂ui,k∂up,qdpqnk = 0 on ST . Ψ∗(ω,P0) is a complementary strain energy

function, which in non-linear elasticity is generally unknown.For the RVE-sized composite material, the application of different

types of boundary conditions leads to a similar response and thereforeΨ∗(ω,P0) = Ψ∗(ω,F0) and Ψ(ω,F0) = Ψ(ω,P0). Then, the lower bound(2.55) can be written as

〈Ψ(P0)〉1 − V0〈P : F〉1 + V0〈P : F〉∆≤ 〈Ψ(P0)〉δ′ − V0〈P : F〉δ′ + V0〈P : F〉∆≤ 〈Ψ(P0)〉δ − V0〈P : F〉δ + V0〈P : F〉∆≤ 〈Ψ(P0)〉∆, for ∀ δ′ < δ < ∆, (2.56)

where we used the following relation between the complementary and strainenergy functions:

Ψ∗(ω,P0) = V0(P : F) − Ψ(ω,P0). (2.57)

Bounds (2.54) and (2.56) allow us to estimate the convergence rateand consequently the RVE size for any non-linear composite satisfyingthe convexity requirement on the strain energy function. Sample resultsobtained by finite element analysis are shown in Figs. 2.21, 2.22, and2.23. A comparison of scaling in linear versus non-linear thermoelasticmaterials is reported in [31]. See [51] for a related development in elastomersand Ghysels et al. [52] for an application of similar concepts in biologicalstructures. A different approach, not involving the convexity assumption,has just been reported in Temizer and Wriggers [53].

2.4.4 Permeability of porous media

Darcy’s law, which describes the permeability of a fluid-saturated material,has the form

U = −Kµ

· ∇p, (2.58)

where U is the Darcy velocity, ∇p is the applied pressure gradient, µ is thefluid viscosity, and K is the permeability, is considered to be the governingpartial differential equation for fluid flow in porous media at small Reynolds



Fig. 2.21. Finite element mesh of a composite in (a) undeformed and (b) deformed(under uniform traction boundary conditions) configurations.

Fig. 2.22. Probability densities for the stored strain energy density of a non-linearcomposite such as in Fig. 2.21 under (a) uniform displacement and (b) uniform tractionboundary conditions.

numbers. A flow distribution for one realization of particles is shown inFig. 2.24.

Note that the relation (2.58) is a purely phenomenological equation(it is not derivable from the Navier–Stokes equation), and its validityneeds assessment through a corresponding Hill–Mandel condition for flowin porous media. The latter is

p,iUi = p,iUi ⇔∫∂B

(p− p,jxj)(Uini − Uini)dS = 0. (2.59)



Fig. 2.23. Finite elasticity stress-strain responses of a random composite under uniaxialloading, KUBC (and SUBC) stand for kinematic (and static) uniform boundaryconditions.

It follows that there are three possible boundary conditions: (i) p = p,ixi,(ii) Uini = U0ini, and (iii) (p − p,ixi)(Uini − U0in) = 0 on ∂B satisfying(2.59). Following a derivation analogous to what was carried out in theabove sections, one obtains the scale-dependent hierarchy of mesoscalebounds [54]:

〈Kesδ 〉 ≥ 〈Ke

2δ〉 ≥ · · · ≥ Keff

= (Reff )−1 ≥ · · · ≥ 〈Rn2δ〉−1 ≥ 〈Rn

δ 〉−1, ∀ δ′ < δ, (2.60)

where R is the resistance of porous media that satisfies K = R−1 forhomogenized media. Sample results are shown in Fig. 2.25.

2.4.5 Comparative numerical results

In the preceding sections we outlined basic theoretical results on scalingin linear elasticity, finite elasticity, elasto-plasticity, thermoelasticity, and



Fig. 2.24. Schematic of a flow distribution in one realization of particles in a randomfluid-saturated material, leading to a question: On what scale is Darcy’s law valid? Amesoscale window is also shown.

permeability. In this section we compare the relative convergence rates tothe RVE of these different physical systems. For common reference, weconsider a two-phase composite, consisting of circular inclusions randomlydistributed in a matrix according to a Poisson point process with exclusion;see Fig. 2.1(c). Finite element analyses of such composites were conductedusing ABAQUS 6.5. In order to compare the scaling from the SVE to theRVE between the physical systems, a so-called discrepancy is employed:

D =Reδ −Rnδ

(Reδ +Rnδ )/2. (2.61)

Here Reδ is the response under the essential boundary conditions and Rnδis the response under the natural boundary conditions, both interpretedaccording to a given physical situation. For linear elasticity Reδ =tr〈Ce〉δ , Rnδ = tr〈Sn〉−1

δ , for linear thermoelasticity Reδ = 〈Γeii〉δ , Rnδ =−〈Cnijkl〉δ〈αnkl〉δ (see Fig. 2.1), for plasticity Reδ = 〈he〉δ, Rnδ = 〈hn〉δ, andfor non-linear elasticity Reδ = 〈Ψe〉δ, Rnδ = 〈Pn : Fn − Ψn〉δ.

The material parameters used for modeling and the computationalresults are summarized in Table 2.2. Here κ is the bulk modulus, µis the shear modulus, κ0 is the initial bulk modulus, µ0 is the initialshear modulus, and superscripts (i) and (m) denote inclusion and matrix



20

25

30

35

40

45

0 20 40 60 80 100

δ (L/d)

Tr(

Kij)

/2 (x

10-9

m2 )

Dirichlet B.C.

Neumann B.C.

2D Round Disks60%porosity

1

1.2

1.4

1.6

1.8

2

2.2

2.4

0 20 40 60 80 100

δ (L/d)

Kije R

jkn(I

=k

=1, o

r 2)

80%

70%

60%

50%

Round Disk Model, porosity:

(a)

(b)

Fig. 2.25. Effect of increasing window scale on the convergence of the hierarchy (2.60)of mesoscale bounds, obtained by computational fluid mechanics. (b) Effect of increasingwindow scale on the convergence of (2.60) as a function of the mesoscale δ obtained bycomputational fluid mechanics.

accordingly. In order to compare physically different problems, linear-typeconstitutive relations were assumed, i.e. linear isotropic hardening inplasticity and neo-Hookean strain energy function in non-linear elasticity.In linear thermoelasticity we need to consider the mismatch not only ofthe thermal expansion coefficients but also of the elastic properties of bothphases, otherwise α becomes a scale-dependent constant parameter.

As can be seen from the table, the discrepancy increases in the followingorder: linear elasticity, plasticity, linear thermoelasticity, and non-linear



Table 2.2. Mismatch and discrepancy values for the mesoscale δ = 16.

Mismatch D[%]

Linear elasticityµ(i)

µ(m)= 10,

κ(i)

κ(m)= 1 2.28

Linear thermoelasticityµ(i)

µ(m)= 10,

κ(i)

κ(m)= 10,

α(i)

α(m)=

1

105.51

Plasticityh(i)

h(m)= 10,

E(i)

E(m)= 1 2.29

Non-linear elasticityµ

(i)0

µ(m)0

= 10,κ(i)0

κ(m)0

= 1 5.86

Flow in porous mediatr(K(i))

tr(K(m))= ∞ 27

elasticity. By nature, the mismatch between the solid and fluid phases inporous media is infinite, so the discrepancy value for permeability cannot becompared to the results in elasticity. The numerical results for this case showD = 27% for the sample size δ = 16, which obviously cannot be consideredhomogeneous in the sense of Hill. While in linear elasticity the discrepancyD is not a function of strain, in non-linear elasticity it strongly depends ondeformation. In general, the value shown in Table 2.2 is the average overthe considered deformation range.

2.5 Conclusions

Microstructural randomness is present in just about all solid materials.In this chapter, we reviewed the scaling from a statistical volume element(SVE) to a representative volume element (RVE). Using micromechanics,the RVE is approached in terms of two hierarchies of bounds stemming,respectively, from Dirichlet and Neumann boundary-value problems set upon the SVE. In Zohdi and Wriggers [55] one can find a comprehensiveintroduction to basic homogenization theory, microstructural optimization,and the multifield analysis of heterogeneous materials; see also [22, 56].A separate area of research and application is the extensive methodology,based on mathematical morphology, used to model a wide range of randomcomposite materials, typically using a few parameters calibrated by theimage analysis of real microstructures [57–60].

Besides the settings of conductivity, (thermo)elasticity, elasto-plasticity, and Darcy permeability discussed in this review, the scaling fromSVE to RVE has been (and continues to be) examined in various other



fields of mechanics and physics, for instance: composite materials [61, 62],random plates [63], strength and damage [64], functionally graded materials[65], Fickian diffusion [66], solidification [67], thermodynamics and heattransfer [68, 69], granular media [70–72], and the morphogenesis of fractals[73]. Related efforts compare scale-dependent bounds, those resulting fromperiodic three-dimensional polycrystal models have been pursued in [74, 75],or determine the effects of the clustering of inclusions [76].

In the late 90s the seminal paper by Forest and Sab [77] launched theeffort to find a homogeneous (Cosserat) micropolar continuum smoothinga spatially heterogeneous Cauchy material. The objective of this line ofresearch is to trade highly detailed information about a Cauchy-typemicrostructure for a less detailed description via an equivalent Cosseratcontinuum; see also [78–87]. A broad overview of multiscale models ofmicrostructured materials — indeed, dating back to works by Navier,Cauchy, and Poisson — can be found in [88].

Another line of studies initiated more recently by Li and Liu [89] seeksto replace an inhomogeneous Cosserat microstructure by an equivalent,spatially homogeneous Cosserat continuum [90]. This is motivated bythe fact that almost every micropolar-type material (granular mass, fiberframework, stone masonry, . . . ) is not perfectly periodic, but always displayssome disorder. The Hill–Mandel condition generalized in that contextalso provides the basis for homogenization in the presence of dynamiceffects.

We end with a note that the methodology outlined in this reviewforms a microstructure-mechanics basis for setting up mesoscale continuumrandom fields and stochastic finite element methods [91–94]. Such fieldsare essential for solving macroscopic boundary-value problems lacking aseparation of the scales, i.e. where heterogeneous components are presentbut a finite element is not larger than the RVE. In that case, the finiteelement is analogous to the SVE. In view of the boundary conditions(2.26), the mesoscale random field is not only scale dependent, but also non-unique, its properties being anisotropic even if the RVE-level properties areisotropic. The stiffness matrix is dictated by the boundary condition (2.26a)while the flexibility matrix is dictated by (2.26b). These matrices enter,respectively, into minimum potential and complementary formulations offinite element methods.

While this chapter has been concerned with scaling and homogeni-zation in random materials where a separation of scales can be attained,the subject matter meets even more formidable challenges in fractal



media. A theoretical formulation in this direction, based on dimensionalregularization, is under development [95–100].

Acknowledgements

This work was made possible by the NSF support under the grant CMMI-1030940.

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Chapter 3

STROH-LIKE FORMALISM FOR GENERAL THINLAMINATED PLATES AND ITS APPLICATIONS

Chyanbin Hwu

Institute of Aeronautics and AstronauticsNational Cheng Kung University

Tainan, Taiwan, R.O.C.

Abstract

If laminates are unsymmetric they will be stretched as well as bent evenunder pure in-plane forces or pure bending moments. The coupled stretching-bending theory of laminates was developed to study the mechanical behaviorof thin laminated plates. Since this theory considers the linear variationof displacements across the thickness direction, by separating the thicknessdependence it is easy to get general solutions through the complex-variableapproach. An elegant and powerful complex-variable method called Strohformalism is well known for problems in two-dimensional linear anisotropicelasticity. In this chapter, its counterpart, generally called Stroh-like formalism,will be introduced to deal with the coupled stretching-bending theory oflaminates. Moreover, its extension to hygrothermal problems and electro-elasticcomposite laminates, and its applications to the problems of holes and cracksin laminates will all be presented. Some representative numerical examples arethen shown to illustrate the advantage and necessity of the analytical closed-form solutions obtained by the Stroh-like formalism.

3.1 Introduction

The virtually limitless combinations of ply materials, ply orientations, andply stacking sequences offered by laminated plates considerably enhance thedesign flexibility inherent in composite structures. In practical applications,to take advantage of the designable characteristics of composite laminates,there is always the option of designing an unsymmetric laminated plate.

103


104 C. Hwu

In that case, stretching-bending coupling may occur no matter what kindof loading is applied on the laminated plates.

In most practical applications, the plates are usually designed withmuch smaller thickness than the other two dimensions; such configurationsare commonly called thin plates. According to observations of the actualmechanical behavior of thin laminated plates, Kirchhoff’s assumptions areusually made in classical lamination theory. If the effects of transverse sheardeformation are considered, shear deformation theory may be considered forrelatively thick laminated plates. Both classical lamination theory and sheardeformation theory are based upon the assumption of the linear variationof displacement fields in the thickness coordinate, and are usually called thefirst-order plate theory. Sometimes it is difficult to describe the deformationof thick laminates by first-order plate theory. By considering the warping ofthe cross-section that probably occurs in thick laminates, high-order platetheory was developed by assuming the displacement fields of higher-ordervariation in the thickness coordinate. In this chapter, the main concernis the applicability of the complex-variable method to general laminatedplates; to avoid the complexity involved in the thickness direction onlyclassical lamination theory for thin plates is considered.

It is well known that the complex-variable method is a powerfulmethod for two-dimensional elasticity problems as well as plate bendingproblems. For two-dimensional linear anisotropic elasticity, there are twomajor complex-variable formalisms presented in the literature. One isLekhnitskii formalism [1–2], which starts with equilibrated stresses givenin terms of stress functions followed by the compatibility equations,and the other is Stroh formalism [3–6], which starts with compatibledisplacements followed by the equilibrium equations. For plate bendinganalysis, Lekhnitskii bending formalism and Stroh-like bending formalismswap over. That is, Lekhnitskii bending formalism starts with compatibledeflections whereas Stroh-like bending formalism starts with equilibratedbending moments. This interesting connection provides a hint to mixedformalism for the coupled stretching-bending analysis of general laminatedplates [7], which is different from the displacement formalism initiatedby Lu and Mahrenholtz [8], and improved by Cheng and Reddy [9] andHwu [7]. Further discussion and interpretation of Stroh-like formalism werepresented in [10–13]. In addition to the above mentioned works, variousLekhnitskii-oriented complex-variable formulations and their applicationshave also been proposed in the literature, for example [14–19] withdisplacement formulation, [20–22] with mixed formulation, and [23–26],


Stroh-Like Formalism for General Thin Laminated Plates and its Applications 105

which considers all types of non-degenerate and degenerate anisotropicplates.

Through an appropriate combination of displacement formalism andmixed formalism, Stroh-like formalism was developed for the coupledstretching-bending analysis of general laminated plates [7]. Since there ismuch detail in the development of Stroh-like formalism, to save space inthis chapter most of the results are presented without detailed derivation.The reader may refer to the book, Anisotropic Elastic Plates [6], or theoriginal papers cited in the following sections for the detailed derivation ofStroh-like formalism and its applications.

This chapter is divided into seven sections. Following the introductionin this section, Stroh-like formalism will be introduced in Section 3.2. Theextension to hygrothermal problems and electro-elastic composite laminateswill then be presented in Sections 3.3 and 3.4, respectively.

Since Stroh-like formalism has been purposely arranged into the formof Stroh formalism for two-dimensional linear anisotropic elasticity, almostall the mathematical techniques developed for two-dimensional problemscan be transferred to the coupled stretching-bending problems. Thus, bysimple analogy many unsolved lamination problems can be solved if theircorresponding two-dimensional problems have been solved successfully.With this advantage, several applications of Stroh-like formalism canbe found in the literature, for example [27–36]. In order to show theapplications of Stroh-like formalism, problems with holes and cracks inlaminates are considered in Section 3.5. These problems are: (i) holesin laminates subjected to uniform stretching and bending moments, (ii)holes in laminates subjected to uniform heat flow and moisture transfer,(iii) holes in electro-elastic laminates, and (iv) cracks in laminates. Somerepresentative numerical examples are then shown in Section 3.6 toillustrate the advantage and necessity of the analytical explicit closed-formsolutions presented in this chapter.

3.2 Stroh-Like Formalism

A commonly used orthotropic material in engineering applications isunidirectional fiber-reinforced composite. Laminated composites are madeby laying various unidirectional fiber-reinforced composites. A single layerof a laminated composite is generally referred to as a ply or lamina. A singlelamina is generally too thin to be directly used in engineering applications.Several laminae are bonded together to form a structure termed a laminate.


106 C. Hwu

The overall properties of laminates can be designed by changing the fiberorientation and the stacking sequence of the laminae. The most popularway to describe the overall properties and macromechanical behavior ofa laminate is classical lamination theory [37]. According to observationsof the actual mechanical behavior of laminates, Kirchhoff’s assumptionsare usually made in this theory: (a) The laminate consists of perfectlybonded laminae and the bonds are infinitesimally thin as well as non-shear-deformable. Thus, the displacements are continuous across laminaboundaries so that no lamina can slip relative to another. (b) A lineoriginally straight and perpendicular to the middle surface of the laminateremains straight and perpendicular to the middle surface of the laminatewhen the laminate is deformed. In other words, transverse shear strains areignored, i.e., γ13 = γ23 = 0. (c) The normals have constant length so thatthe strain perpendicular to the middle surface is ignored, i.e. ε3 = 0.

Based upon Kirchhoff’s assumptions, the displacement fields, thestrain–displacement relations, the constitutive laws, and the equilibriumequations can be written as follows:

Ui = ui + x3βi, β1 = −w,1, β2 = −w,2,ξij = εij + x3κij , εij = (ui,j + uj,i)/2, κij = (βi,j + βj,i)/2,

Nij = Aijklεkl +Bijklκkl Mij = Bijklεkl +Dijklκkl,

Nij,j = 0, Mij,ij + q = 0, Qi = Mij,j , i, j, k, l = 1, 2,

(3.1)

where U1 and U2 are the displacements in the x1 and x2 directions; u1

and u2 are their associated middle surface displacements; βi, i = 1, 2, arethe negative slopes; w is the displacement in the x3 direction; ξij , εij ,and κij are, respectively, the strains, the midplane strains, and the platecurvatures; Nij and Mij are the stress resultants and bending moments;Aijkl, Bijkl, and Dijkl are, respectively, the tensors of extensional, coupling,and bending stiffnesses; Qi is the transverse shear force; and q is the lateraldistributed load applied on the laminates. The subscript comma stands fordifferentiation, e.g., w,1 = ∂w/∂x1, and repeated indices imply summationthrough 1 to 2.

In order to find a solution satisfying all the basic equations in (3.1)in the coupled stretching-bending analysis of general laminates, severaldifferent complex-variable methods have been proposed in the literature.The main differences between these methods are: (1) the choice ofthe primary and secondary functions, (2) the organization of the finalexpressions of the general solutions, (3) the establishment of a material



eigen-relation, and (4) the consideration of degenerate anisotropic plates.These differences are discussed in the following.

(1) The choice of the primary and secondary functions starts from theconsideration of the constitutive laws given in (3.1)3, which can be rewrittenin matrix form as

N

M

=[A B

B D

]ε0

κ

. (3.2)

By inversion or semi-inversion, we have

ε0

κ

=[A∗ B∗

B∗ D∗

]N

M

,

N

κ

=

[A B

−BTD

]ε0

M

, (3.3)

or

ε0

M

=

[A B

−BTD

]−1N

κ

=

[A

∗B

∗

−B∗T D∗

]N

κ

, (3.4)

where

A∗ = A−1 +A−1BD∗BA−1, B∗ = −A−1BD∗,

D∗ = (D −BA−1B)−1,

A = A−BD−1B, B = BD−1, D =D−1,

A∗ = A−1, B∗ = −A−1B, D∗

= D −BA−1B.

(3.5)

If the formulation takes the compatible displacements ui, w as theprimary functions, we have a single-valued midplane strain ε0 andcurvature κ. The stress resultants N and bending moments M can beobtained from (3.2). The equilibrium equations can then be expressed interms of the displacements. The problems remaining are the system ofpartial differential equations of u1, u2, and w in which the plate propertiesare expressed byA,B, andD. This is the usual step taken in the literature,for example, [7−9, 13, 14], and is generally called displacement formalism.

If the formulation takes the Airy stress function F and the platedeflection w as the primary functions, the stress resultants N are obtaineddirectly from the Airy stress function, and the equilibrium equationsfor the in-plane problem will be satisfied automatically. A single-valuedcurvature κ is obtained through the second-order differentiation of theplate deflection w. With the equilibrated N and the compatible κ, themidplane strain ε0 and bending moment M can be obtained from (3.4).


108 C. Hwu

The compatibility equation for the midplane strain ε0 and the equilibriumequations for plate bending moments M can then be expressed in termsof F and w. The problems remaining are the system of partial differentialequations of F and w in which the plate properties are expressed by A∗, B∗,and D∗. This is called mixed formalism and has been used in [20, 23].

An alternative choice for the above mixed primary functions is thecompatible displacements u1, u2 and the equilibrated bending momentpotentials ψ1 and ψ2. With this choice, we get the compatible midplanestrain ε0 and the equilibrated bending moment M . We get N and κ

from the second equation of (3.3). The equilibrium equation of N andthe compatibility equation of κ will then lead to the system of partialdifferential equations in terms of u1, u2, ψ1, and ψ2 in which the plateproperties are expressed by A, B, and D. This approach was taken byHwu [7] and Lu [10].

(2) The organization of the final expressions of the general solutionscan roughly be distinguished as being in component form or matrix form.The former expresses the general solutions for each component of thedisplacements, stress resultants, and bending moments, whereas the latterorganizes the solutions in terms of vectors and matrices. Examples ofcomponent form expressions proposed in the literature are [14, 20] andexamples of matrix form expressions are [7−9, 12, 23, 24].

(3) The establishment of a material eigen-relation is important in thecomplex-variable method since the material eigenvalues µα, which havebeen proved to be complex numbers, are the key parameters of the complexvariables zα(= x+µαy). In Lekhnitskii formalism this relation is representedby the characteristic polynomial equation, whereas in Stroh formalismit is represented by a standard eigen-relation Nξ = µξ in which N iscalled the fundamental elasticity matrix and ξ is the material eigenvector.Using Lekhnitskii formalism, the analytical expressions of the materialeigenvectors can be obtained explicitly in terms of the material eigenvalues.Using the standard eigen-relation of Stroh formalism, many propertiesand identities that are useful for the derivation of explicit analyticalsolutions can be obtained for the material eigenvalues and eigenvectors,which is fundamental to the success of Stroh formalism in solving alarge variety of problems in general anisotropic elasticity. In practicalapplications, these two different formalisms can benefit each other. Onemay refer to [14, 20, 23, 24] for Lekhnitskii formalism and [6−9, 12] for Strohformalism.

(4) Most of the complex-variable methods proposed in the literatureconsider only non-degenerate anisotropic plates. That is, the formulation is



based upon the assumption that the material eigenvalues are distinct. Fordegenerate anisotropic plates, some or all of the material eigenvalues arerepeated and cannot yield a complete set of independent eigensolutions.Thus, the solutions constructed through the eigensolutions of distincteigenvalues are not general enough and should be modified for degenerateanisotropic plates. To get a complete set of independent eigensolutions,one can add the generalized eigensolutions obtained by differentiatingthe eigensolutions with respect to the eigenvalue. Full consideration ofdegenerate anisotropic plates can be found in [23–26]. It should benoted that if a solution is obtained by the complex-variable method fornon-degenerate anisotropic plates, it is applicable to degenerate platesanalytically if its final expression does not contain the material eigenvectormatrices, and it is also applicable to degenerate plates numerically if a smallperturbation of the material eigenvalues is made [38].

As stated above, several different complex-variable methods have beenproposed in the literature for the coupled stretching-bending analysis ofgeneral laminated plates. Their main differences have been described bythe names: displacement formalism or mixed formalism, component formor matrix form, Lekhnitskii formalism or Stroh formalism, non-degenerateanisotropic plates or degenerate anisotropic plates. Since each methodpossesses its own merits, the best approach may be the combination ofall their merits. From this viewpoint, the Stroh-like formalism proposed byHwu [7] is the most appropriate one to be introduced in this chapter. InHwu’s Stroh-like formalism, the final expression of the general solution iswritten in matrix form and is derived from displacement formalism, whereasthe associated material eigen-relation is derived from mixed formalismand the explicit expression of the material eigenvector is obtained byLekhnitskii formalism. Although analytically it is valid only for non-degenerate anisotropic plates, numerically by perturbation it is applicablefor all different types of anisotropic plates including degenerate anisotropicplates. Moreover, through the identities established using the materialeigen-relation, many complex form solutions can be converted into realform. In this case, the final analytical solutions can still be applied tothe degenerate anisotropic plates even if they have been derived for non-degenerate anisotropic plates.

By selecting u1, u2, β1, and β2 as the primary functions, generalsolutions satisfying all the basic equations in (3.1) with q = 0 have beenobtained as [7]

ud = 2ReAf(z), φd = 2ReBf(z), (3.6a)


110 C. Hwu

where

ud =uβ

, φd =

φ

ψ

,

A = [a1 a2 a3 a4], B = [b1 b2 b3 b4],

f(z) =

f1(z1)f2(z2)f3(z3)f4(z4)

, zk = x1 + µkx2, k = 1, 2, 3, 4,

(3.6b)

and

u =u1

u2

, β =

β1

β2

=−w,1−w,2

, φ =

φ1

φ2

, ψ =

ψ1

ψ2

. (3.6c)

In Eq. (3.6c), φi, i = 1, 2, are the stress functions related to the in-planeforces Nij , and ψi, i = 1, 2, are the stress functions related to the bendingmoments Mij , transverse shear forces Qi, and effective transverse shearforces Vi(= Qi +Mij,j). Their relations are

N11 = −φ1,2, N22 = φ2,1, N12 = φ1,1 = −φ2,2 = N21,

M11 = −ψ1,2, M22 = ψ2,1, M12 = ψ1,1 − η = −ψ2,2 + η = M21,

Q1 = −η,2, Q2 = η,1,

V1 = −ψ2,22, V2 = ψ1,11,(3.7a)

where

η =12(ψ1,1 + ψ2,2). (3.7b)

The material eigenvalues µk and their associated eigenvectors ak andbk can be determined from the following eigen-relation

Nξ = µξ, (3.8a)

where

N = ItNmIt, ξ =ab

, (3.8b)



and

N =

[N1 N2

N3 NT1

], Nm =

[(Nm)1 (Nm)2(Nm)3 (Nm)T1

], It =

[I1 I2

I2 I1

],

(Nm)1 = −T−1m RT

m, (Nm)2 = T−1m = (Nm)T2 ,

(Nm)3 = RmT−1m RT

m − Qm = (Nm)T3 , I1 =

[I 0

0 0

], I2 =

[0 0

0 I

],

(3.8c)

and I is a 2 × 2 unit matrix. Note that the material eigenvalues µk havebeen assumed to be distinct in the general solution (3.6). Moreover, thefour pairs of material eigenvectors (ak,bk), k = 1, 2, 3, 4, are assumed to bethose corresponding to the eigenvalues with positive imaginary parts. Formaterials whose eigenvalues are repeated, a small perturbation in their valuemay be introduced to avoid degeneracy problems [38] or a modification ofthe general solution can be made [39].

In (3.8b), Nm is the 8 × 8 fundamental elasticity matrix of mixedformalism whose explicit expressions have been obtained in [40]. In (3.8c),the three 4 × 4 real matrices Qm,Rm, and Tm are related to the matricesA, B, and D by

Qm =

A11 A16 B16/2 B12

A16 A66 B66/2 B62

B16/2 B66/2 −D66/4 −D26/2

B12 B62 −D26/2 −D22

, (3.9a)

Rm =

A16 A12 −B11 −B16/2

A66 A26 −B61 −B66/2

B66/2 B26/2 D16/2 D66/4

B62 B22 D12 D26/2

, (3.9b)

Tm =

A66 A26 −B61 −B66/2

A26 A22 −B21 −B26/2

−B61 −B21 −D11 −D16/2

−B66/2 −B26/2 −D16/2 −D66/4

. (3.9c)


112 C. Hwu

Note that unlike the two-dimensional elastic case, Qm and Tm definedin (3.9a) and (3.9c) are not positive definite. However, the existence of theinverse of Tm used in (3.8c) for the calculation of (Nm)i is guaranteed,since it has been proved through the derivation of the explicit expressionsof (Nm)i [40] or the relation between displacement formalism and mixedformalism [7, 11].

With the general solutions presented in (3.6), the next important task isconstructing an environment to help researchers find solutions satisfying theboundary conditions for a given problem. It is helpful if this environmentcontains the following features: (1) the relations for the physical quantitieson the rotated coordinate; (2) the identities for the conversion of a complexform solution into a real form solution; and (3) the explicit expressions forthe matrices used in the formulation.

(1) To be suitable for general problems, which may have arbitrarygeometrical shape, it is always desirable to have the relations for thephysical quantities expressed in other coordinates. If a quantity is identifiedas a tensor, the change to the components of the quantity can be made byfollowing the transformation laws of tensors. From this viewpoint, withthe conventional approach one usually applies the transformation law ofsecond-order tensors to obtain the bending moments (Mn,Ms,Mns) andin-plane forces (Nn, Ns, Nns), and applies the transformation laws of first-order tensors to obtain the transverse shear force (Qn, Qs), and then oneuses the definition of effective transverse shear force to obtain (Vn, Vs).Here, the subscripts n and s denote, respectively, the normal and tangentialdirections of the normal-tangent (n−s) coordinate system. In the Stroh-likeformalism the secondary functions are the stress functions φ1, φ2, ψ1, andψ2 instead of the stresses. Thus, it is more convenient if we express therelations calculating the stress resultants and bending moments in the n−scoordinates using the stress functions instead of their quantities in thex1 − x2 coordinates. The relations obtained in [34] are as follows,

Nn = nT tn = nTφ,s, Nns = sT tn = sTφ,s,

Ns = sT ts = −sTφ,n, Nsn = nT ts = −nTφ,n = Nns,

Mn = nTmn = nTψ,s, Mns = sTmn = sTψ,s − η,

Ms = sTms = −sTψ,n, Msn = nTms = −nTψ,n + η = Mns,

Qn = η,s, Qs = −η,n, Vn = (sTψ,s),s,

Vs = −(nTψ,n),n, η = (sTψ,s + nTψ,n)/2,

(3.10)



where tn and mn are the surface traction and surface moment alongthe surface with normal n, and ts and ms are the surface traction andsurface moment along the surface with normal s, which is perpendicular tothe direction n; n and s are the directions normal and tangential to theboundary and can be written as

nT = (− sin θ, cos θ), sT = (cos θ, sin θ), (3.11)

where θ denotes the angle from the positive x1-axis to the direction s in aclockwise direction.

In addition to the relations shown in (3.10), another important relationthat cannot be obtained directly through simple tensor transformation isthe generalized material eigen-relation. This relation is established througha consideration of the dual coordinate system in which the displacementsand stress functions are referred to the x1 − x2 coordinate system, whereasother terms such as the elastic constants are referred to a rotated coordinatesystem x∗1 − x∗2. The generalized material eigen-relation presented in [41] is

N(ω)ξ = µ(ω)ξ, (3.12a)

where

N(ω) = ItNm(ω)It, ξ =ab

, (3.12b)

N(ω) =

[N1(ω) N2(ω)

N3(ω) NT1 (ω)

], Nm(ω) =

[(Nm(ω))1 (Nm(ω))2(Nm(ω))3 (Nm(ω))T1

],

(Nm(ω))1 = −T−1m (ω)RT

m(ω), (Nm(ω))2 = T−1m (ω), (3.12c)

(Nm(ω))3 = Rm(ω)T−1m (ω)RT

m(ω) − Qm(ω),

and

µ(ω) =− sinω + µ cosωcosω + µ sinω

. (3.12d)

In (3.12), Qm(ω),Rm(ω), and Tm(ω) are related to the matrices Qm,Rm,and Tm defined in (3.9) by

Qm(ω) = Qm cos2 ω + (Rm + RTm) sinω cosω + Tm sin2 ω,

Rm(ω) = Rm cos2 ω + (Tm − Qm) sinω cosω − RTm sin2 ω,

Tm(ω) = Tm cos2 ω − (Rm + RTm) sinω cosω + Qm sin2 ω,

(3.12e)


114 C. Hwu

in which ω denotes the angle between the transformed and originalcoordinates.

(2) Since all physical quantities are real, it is desirable to have real formsolutions instead of complex form solutions. Moreover, if the solutions canbe written in a real form that does not involve complex numbers — materialeigenvalues µα and their associated material eigenvector matrices A andB — they will not have the degeneracy problems raised by the repetitionof material eigenvalues. Therefore, if the solutions can be converted intoreal form, they are not only convenient for practical applications but arealso applicable for all the different types of laminated plates includingdegenerate plates. Thus, the identities converting the complex form intoreal form are important for the analytical derivation using the Stroh-likeformalism.

In the eigen-relation Nξ = µξ, N is real, whereas µ and ξ are complex.In other words, this eigen-relation is the foundation for converting theidentities connecting the complex µ, A, and B to real N. Several identitiesderiving from this relation have been obtained for the Stroh formalismof two-dimensional anisotropic elasticity [5,6]. Since the material eigen-relations (3.8) and (3.12) of the Stroh-like formalism have been purposelyorganized into the same forms as those of the Stroh formalism, all theidentities derived from this eigen-relation can be applied to the Stroh-likeformalism without further proof. Therefore, in this chapter no identitieswill be presented. For those interested in these identities, please refer tothe books [5,6] for anisotropic elasticity.

(3) The explicit expressions for the matrices used in the formulation areuseful when one tries to find the analytical solutions for each componentof the physical quantities such as displacements, strains, and stresses. Theadvantage of the matrix form is that the solution can be expressed in anelegant and easily programmed way. Moreover, through proper arrangementof the matrices, it is possible that one solution form will be suitable forseveral different kinds of problem. The typical example is the generalsolution shown in (3.6a), which is a matrix form solution suitable for two-dimensional elasticity problems, plate bending problems as well as coupledstretching-bending problems, and the materials considered can be any kindsof anisotropic or piezoelectric materials. In other words, this simple matrixform solution (3.6a) can be applied to a wide range of problems andmaterials, and hence, it is important to know the explicit expressions ofthe material eigenvector matrices A and B in this solution.



As stated in (2), a key feature of Stroh formalism is its eigen-relationrelating the complex material eigenvectors to real material properties.Several real matrices connected through this eigen-relation, such as thefundamental elasticity matrix N, the generalized fundamental elasticitymatrix N(θ), and the Barnett–Lothe tensors L, S, and H, are crucial whenthe analytical solutions lead to real form expressions. Because these realmatrices can be obtained directly from the material properties, they existfor all types of materials — degenerate or non-degenerate. In other words,if the final solution of a particular elasticity problem can be expressed interms of these real matrices, even if they are derived using an assumptionof non-degenerate materials, they are still valid for degenerate materials.

Therefore, no matter whether complex or real, getting the explicitexpressions for A, B, N, N(θ), L, S, and H is an important stagein understanding the effects of the material properties in problems ofanisotropic elasticity. For plane problems with anisotropic elastic materials,most of the explicit expressions have been well documented in [5, 6].Their corresponding explicit expressions for general piezoelectric materialscovering all the possible two-dimensional states such as generalized planestrain or plane stress and short circuit or open circuit, were provided in [42].In my recent study [41], one can find the explicit expressions of the Stroh-like formalism for the coupled stretching-bending analysis.

3.3 Extended Stroh-Like Formalism — HygrothermalStresses

In the previous section, the constitutive relations pertain only to anenvironment with constant temperature and constant moisture, i.e., notemperature or moisture changes are allowed. However, in laminatedstructures hygrothermal changes occur frequently during fabrication andstructural usage. When a homogeneous body is completely free to deform,the hygrothermal changes may produce only hygrothermal strains for whichthe body is free of stresses. However, for a composite laminate eachindividual lamina is not completely free to deform. Lamina stresses aretherefore induced by the constraints placed on deformation by adjacentlamina. Thus, the study of hygrothermal stresses in laminates is importantfor practical engineering design. Like the extension of Stroh formalismto anisotropic thermoelasticity, in this section we extend the Stroh-likeformalism to the hygrothermal stress analysis of laminates.


116 C. Hwu

When hygrothermal effects are considered, the basic equations shownin (3.1) should be modified to

Ui = ui + x3βi, T = T 0 + x3T∗, H = H0 + x3H

∗,

β1 = −w,1, β2 = −w,2,qi = −Kt

ijT0,j −K∗t

ij T∗,j −Kt

i3T∗, mi = −Kh

ijH0,j −K∗h

ij H∗,j −Kh

i3H∗,

εij =12(ui,j + uj,i), κij =

12(βi,j + βj,i),

Nij = Aijklεkl +Bijklκkl −AtijT0 −AhijH

0 −BtijT∗ −BhijH

∗,

Mij = Bijklεkl +Dijklκkl −BtijT0 −BhijH

0 −DtijT

∗ −DhijH

∗,

Nij,j = 0, Mij,ij + p = 0, Qi = Mij,j , qi,i + q = 0,

mi,i +m = 0, i, j, k, l = 1, 2,

(3.13)

where Ui, T , and H are the displacements, temperature, and moisturecontent of the plates, ui, T 0, and H0 are the middle surface displacements,temperature, and moisture content, and βi, T ∗, and H∗ are the negativeslopes, and the rates of change of temperature and moisture content acrossthe thickness. qi and mi denote the heat flux resultant and moisture transferresultant; Atij , B

tij , D

tij and Ahij , B

hij , D

hij are the corresponding tensors for

the thermal and moisture expansion coefficients; Ktij ,K

hij and K∗t

ij ,K∗hij

are the coefficients related to the heat conduction and moisture diffusioncoefficients; p, q, and m are the lateral distributed load, heat flux, andmoisture concentration transfer applied to the laminates.

Since the basic Eqs. (3.13) are quite general, it is not easy to find asolution satisfying all these basic equations. Here, two special cases thatoccur frequently in engineering applications are considered. One is the casewhen the temperature and moisture distributions depend on x1 and x2

only, i.e., T ∗ = H∗ = 0, and the other is the case when the temperatureand moisture distributions depend on x3 only, i.e., T = T 0 + x3T

∗ andH = H0 + x3H

∗ in which T 0, T ∗, H0, and H∗ are constants independentof x1 and x2. The general solutions for these two special cases have beenobtained in [28] and are called the extended Stroh-like formalism since theyare the extensions of Stroh-like formalism.

Case 1: Temperature and moisture content depend on x1 and x2 only

In this case, the temperature and moisture content are assumed to varyin the x1−x2 plane and distribute uniformly in the thickness direction.



The lateral distributed load, heat flux, and moisture concentrationtransfer applied on the laminates are neglected. With this consideration,T ∗ =H∗ = p= q=m=0, and the basic equations in (3.13) can be simplified.By following the steps for the Stroh formalism of two-dimensionalthermoelasticity [43], the general solutions satisfying all the basic equationsin (3.13) can be found as [28]

T = 2Reg′t(zt), H = 2Reg′h(zh),qi = −2Re(Kt

i1 + τtKti2)g

′′t (zt),

mi = −2Re(Khi1 + τhK

hi2)g

′′h(zh),

ud = 2ReAf(z) + ctgt(zt) + chgh(zh),φd = 2ReBf(z) + dtgt(zt) + dhgh(zh),

(3.14)

in which zt = x1 + τtx2, zh = x1 + τhx2, and τt, τh and (ct,dt), (ch,dh)are thermal and moisture eigenvalues and eigenvectors, which can bedetermined by the following eigen-relations:

Kt11 + 2τtKt

12 + τ2t K

t22 = 0, Kh

11 + 2τhKh12 + τ2

hKh22 = 0,

Nηt = τtηt + γt, Nηh = τhηh + γh,(3.15a)

where

N =

[N1 N2

N3 NT1

], ηt =

ctdt

, ηh =

chdh

, (3.15b)

and N is the fundamental elasticity matrix defined in (3.8c), γt and γh aretwo 8×1 complex vectors related to the elastic constants and the coefficientsof thermal and moisture expansion [6].

Case 2: Temperature and moisture content depend on x3 only

If the temperature and moisture content depend on x3 only, the generalsolutions satisfying all the basic equations in (3.13) can be expressed as [28]

ud = 2ReAf(z),φd = 2ReBf(z) − x1ϑ2 + x2ϑ1,

(3.16a)

where

ϑi = αtiT0 +αhiH

0 +α∗ti T

∗ +α∗hi H

∗, i = 1, 2. (3.16b)


118 C. Hwu

αti and α∗ti are defined by

αt1 =αtA1

αtB1

, αt2 =

αtA2

αtB2

, αtAi =

At1iAt2i

, αtBi =

Bt1iBt2i

,

(3.16c)

and

α∗ti =

αtBiαtDi

, αtBi =

Bt1iBt2i

, αtDi =

Dt

1i

Dt2i

, i = 1, 2. (3.16d)

The same expressions as (3.16c) and (3.16d) are defined for αhi and α∗hi by

replacing superscript t with h.With the general solution shown in (3.14) or (3.16) for the generalized

displacement and stress function vectors, the middle surface displacementsui and slopes βi can be obtained directly from the components of thegeneralized displacement vector ud. For the stress resultants Nij , shearforcesQi, effective shear forces Vi, and bending momentsMij , we can utilizethe relations shown in (3.7).

3.4 Expanded Stroh-Like Formalism — Electro-ElasticLaminates

Consider an electro-elastic laminate made of fiber-reinforced compositesand piezoelectric materials. Although it may exhibit electric-elastic couplingeffects that are more complicated than those of single-phase piezoelectricmaterials, a similar extension from pure elastic materials to piezoelectricmaterials for a two-dimensional analysis can still be applied to the coupledstretching-bending analysis of electro-elastic laminates. For piezoelectricanisotropic elasticity, to include the piezoelectric effects the constitutivelaws, the strain–displacement relations, and the equilibrium equations canbe written as follows [44]

σij = CEijklεkl − ekijEk,

Dj = ejklεkl + ωεjkEk ,εij =

12(ui,j + uj,i),

σij,j = 0,Di,i = 0,

i, j, k, s = 1, 2, 3, (3.17)

in which ui, σij , εij , Dj , and Ek are, respectively, the displacement, stress,strain, electric displacement (also called induction), and electric field;CEijkl, ekij , and ωεjk are, respectively, the elastic stiffness tensor at constant



electric field, piezoelectric stress tensor, and dielectric permittivity tensorat constant strain. By letting

Dj = σ4j , −Ej = u4,j = 2ε4j , j = 1, 2, 3,

Cijkl = CEijkl, i, j, k, l = 1, 2, 3,

Cij4l = elij , i, j, l = 1, 2, 3,

C4jkl = ejkl, j, k, l = 1, 2, 3,

C4j4l = −ωεjl, j, l = 1, 2, 3,

(3.18)

the basic equations (3.17) can be rewritten in an expanded notation as

σIj = CIjKlεKl, εIj =12(uI,j + uj,I), σIj,j = 0,

I,K = 1, 2, 3, 4, j, l = 1, 2, 3,(3.19)

where uj,4 = 0. Through the use of the expanded notation for piezoelectricmaterials, the basic equations for the coupled mechanical-electrical analysisof electro-elastic composite laminates can be rewritten in the form of (3.1)by expanding the range of certain subscripts from 2 to 4 [29]. And, hencethe general solutions satisfying the basic equations can also be writtenin the same matrix form as shown in (3.6) for the Stroh-like formalism [6].The only difference is the dimension and content of the matrices. Thus, thisformalism is called expanded Stroh-like formalism due to the expansion ofthe matrix dimension.

3.5 Holes and Cracks

Due to the stress concentration induced by the existence of holes and cracks,anisotropic plates containing holes or cracks have been studied extensivelyin two-dimensional problems. Owing to the mathematical complexity, notmany analytical results have been presented for coupled stretching-bendingproblems of holes and cracks in general composite laminates. In this sectionwe will show some results for laminates containing holes or cracks presentedin the literature using the Stroh-like formalism.

3.5.1 Holes in laminates under uniform stretching

and bending moments

Consider an unbounded composite laminate with an elliptical hole subjectedto in-plane forces N11 = N∞

11 , N22 = N∞22 , N12 = N∞

12 , and out-of-plane


120 C. Hwu

a2

b2

∞11M

∞22M

∞12N

∞11N

∞12M

∞12N

∞12M

∞22N

∞11M

∞12N

∞11N

∞12M

∞22M

∞12N

∞12M

∞22N

Fig. 3.1. A composite laminate weakened by an elliptical hole subjected to in-planeforces and out-of-plane bending moments.

bending moments M11 = M∞11 , M22 = M∞

22 , M12 = M∞12 , at infinity

(Fig. 3.1). The contour of the elliptical hole is represented by

x1 = a cosϕ, x2 = b sinϕ, (3.20)

where 2a and 2b are the major and minor axes of the ellipse and ϕ is a realparameter. There is no load around the edge of the elliptical hole, i.e.,

Nn = Nns = 0, Mn = Vn = 0, (3.21)

along the hole boundary.A solution satisfying all the basic equations (3.1) and the boundary

condition (3.21) has been found as [27]

ud = x1d∞1 + x2d∞

2 − ReA < ζ−1α > B−1(am∞

2 − ibm∞1 ),

φd = x1m∞2 − x2m∞

1 − ReB < ζ−1α > B−1(am∞

2 − ibm∞1 ),

(3.22)

in which the angular bracket <> stands for a diagonal matrix in whicheach component is varied according to the subscript α, and

ζα =zα +

√z2α − a2 − µ2

αb2

a− iµαb. (3.23)



m∞1 = (N∞

11 , N∞12 ,M

∞11 ,M

∞12 )T ,m∞

2 = (N∞12 , N

∞22 ,M

∞12 ,M

∞22 )T , and d∞

1

and d∞2 are the vectors containing the quantities for the shear strains and

curvatures at infinity, which are related to m∞1 and m∞

2 byd∞

1

d∞2

=

[Q∗ R∗

R∗T T∗

]m∞

1

m∞2

, (3.24a)

where

Q∗ =

A∗11 A∗

16/2 B∗11 B∗

16/2

A∗16/2 A∗

66/4 B∗61/2 B∗

66/4

B∗11 B∗

61/2 D∗11 D∗

16/2

B∗16/2 B∗

66/4 D∗16/2 D∗

66/4

, (3.24b)

R∗ =

A∗16/2 A∗

12 B∗16/2 B∗

12

A∗66/4 A∗

26/2 B∗66/4 B∗

62/2

B∗61/2 B∗

21 D∗16/2 D∗

12

B∗66/4 B∗

26/2 D∗66/4 D∗

26/2

, (3.24c)

T∗ =

A∗66/4 A∗

26/2 B∗66/4 B∗

62/2

A∗26/2 A∗

22 B∗26/2 B∗

22

B∗66/4 B∗

26/2 D∗66/4 D∗

26/2

B∗62/2 B∗

22 D∗26/2 D∗

22

, (3.24d)

and A∗ij , B

∗ij , D

∗ij are the components of A∗,B∗,D∗ defined in (3.5).

According to the relations given in (3.10) we know that the calculationof the stress resultants and bending moments relies upon the calculationof the differentials φd,s and φd,n. Along the hole boundary, the real formexpressions have been obtained as φd,s = 0 and

φd,n = sin θG1(θ)m∞

1 −[I +

a

bG3(θ)

]m∞

2

− cos θ[

I +b

aG3(θ)

]m∞

1 + G1(θ)m∞2

, (3.25a)

where

G1(θ) = NT1 (θ) − N3(θ)SL−1, G3(θ) = −N3(θ)L−1. (3.25b)


122 C. Hwu

L, S, and H are three real matrices which are called the Barnett–Lothetensors defined by

H = 2iAAT , L = −2iBBT , S = i(2ABT − I). (3.26)

3.5.2 Holes in laminates under uniform heat flow

and moisture transfer

Using the extended Stroh-like formalism presented in Section 3.3, thehygrothermal stresses in composite laminates disturbed by an elliptical holesubjected to uniform heat flow and moisture transfer in the x1–x2 plane orx3 direction were solved in [28]. To save space only the solutions for theproblem under uniform heat flow and moisture transfer in the x3 directionare presented in this section.

If the temperature and moisture content are assumed to vary linearlyin the thickness direction, and T = Tu and H = Hu on the top surface andT = Tl and H = Hl on the bottom surface, we have

T 0 =Tl + Tu

2, H0 =

Hl +Hu

2, T ∗ =

Tl − Tuh

, H∗ =Hl −Hu

h,

(3.27)

where h is the thickness of the laminated plate.The field solution of this problem has been obtained as

ud = ReA < ζ−1α > B−1(aϑ2 − ibϑ1),

φd = ReB < ζ−1α > B−1(aϑ2 − ibϑ1) − x1ϑ2 + x2ϑ1,

(3.28)

where ϑ1 and ϑ2 are related to T 0, H0, T ∗, and H∗ by the equation shownin (3.16b). Along the hole boundary, the real form expressions of φd,n havebeen obtained as

φd,n = cos θ[ϑ1 + G1(θ)ϑ2 +

b

aG3(θ)ϑ1

]

+ sin θ[ϑ2 − G1(θ)ϑ1 +

a

bG3(θ)ϑ2

]. (3.29)

3.5.3 Holes in electro-elastic laminates under uniform

loads and charges

Consider an unbounded electro-elastic composite laminate with an ellipticalhole subjected to the uniform generalized forces N∞

11 , N∞22 , N

∞12 , N

∞41 , N

∞42 ,

and uniform generalized moments M∞11 ,M

∞22 ,M

∞12 , M∞

41 ,M∞42 at infinity.



The generalized forces and momentsN4i andM4i, i = 1, 2, are related to theelectric displacement [29]. The contour of the elliptical hole is representedby (3.20). If the hole edge and the upper and lower surfaces of the laminateare free of traction and electric charge, the boundary conditions of thisproblem can be expressed by the same matrix form equation as that forthe laminates discussed in Section 3.5.1. The only difference is that thedimension of the vectors φd, m∞

1 , and m∞2 is now 6 × 1 instead of 4 × 1.

Due to the equivalence of the mathematical formulation, the solutions ofthis problem can also be expressed by the matrix form expressions shownin (3.22)–(3.25).

3.5.4 Cracks in laminates

Consider an unbounded composite laminate containing a through-thicknesscrack loaded at infinity and the crack is assumed to lie on the x1-axiswith its center located at the origin. Since an elliptical opening can beconsidered to be a crack of length 2a by letting the minor axis b be zero,the field solutions for cracks can therefore be obtained from the solutions forproblems for elliptical holes with b = 0. After deriving the field solutions,the stress intensity factors k can be calculated directly through the followingdefinition:

k =

KII

KI

KIIB

KIB

= limr→0

√2πr

N12

N22

M12

M22

= limr→0

√2πrm2, (3.30)

where r is the distance ahead of the crack tip; KII ,KI ,KIIB ,KIB are,respectively, the stress intensity factors of the shearing mode, openingmode, twisting mode, and bending mode. From the relations given in (3.7)and the definition (3.30), we get

k = limr→0

√2πr(φd,1 − ηi3), (3.31a)

where

iT3 = (0 0 1 0). (3.31b)


124 C. Hwu

Using the relation (3.31) and the solutions for the problems of elliptical holes(3.22)2 with b = 0, the explicit solutions for the stress intensity factors canbe obtained as [45]

k =√πam∞

2 − η0i3, (3.32a)

where

η0 = M∞12 + (G1m∞

2 )4/2, (3.32b)

and G1 = G1(0) which is defined in (3.25b).

3.6 Numerical Examples

For the analytical closed-form solutions presented in the previous section,several numerical examples are illustrated in this section. All the examplesconsider an unbounded laminate composed of different combinations ofgraphite/epoxy fiber-reinforced composite laminae. Each lamina thicknessis 1 mm, and the material properties of the graphite/epoxy are

E1 = 181 GPa, E2 = 10.3 GPa, G12 = 7.17 GPa, ν12 = 0.28,

αt11 = 0.02 × 10−6/C, αt22 = 22.5 × 10−6/C,

kt11 = 1.5 W/mC, kt22 = 0.5 W/mC,

(3.33)

where E1 and E2 are the Young’s moduli in the x1 and x2 directions,respectively; G12 is the shear modulus in the x1–x2 plane; ν12 is the majorPoisson’s ratio and is related by ν21E1 = ν12E2 to the minor Poisson’s ratioν21; αt11 and αt22 are the coefficients of thermal expansion in the fiber andits transverse directions, respectively; and kt11 and kt22 are the coefficientsof heat conduction. All the other values of αtks, k

tij , i = j, are zero. Note

that the properties given in (3.33) are for the plane-stress condition oforthotropic materials, whose properties in the thickness direction are notrequired. With the material properties (3.33) and layup sequence, the fielddeformations and stresses as well as the resultant forces and momentsaround the hole boundary can be calculated by following the procedureshown in the flowchart Fig. 3.2.

3.6.1 Holes

Consider a [+45/0/+45/−45] unsymmetric laminate containing a circularhole. Figure 3.3 shows the resultant forces and bending moments around the



Laminate construction

nn hhh ,...,,,,...,, 2121 θθθ

Calculate the eigen-relation (3.8)

Generalized matrices (3.12e))(),(),( θθθ mmm TRQ

In-plane forces and out-of-plane bending moments at infinity Temperature changes:

∞∞21 ,mm

Hole geometry, a and b

Lamina properties

Thermal problem: 121221 ,,, νGEE

Generalized fundamental matrices (3.12c))(),(),( 321 θθθ NNN

Matrices (3.5)*** D,B,A

General solutions for the whole field (3.22) Thermal problem: (3.28)

dd φ,u

Laminate stiffness [6]

(3.5)

(3.9)

Thermal problem:

[6]; (3.16c,d)

, ,m m mQ R T

DB,A,

, * * *A,B, D A ,B ,D

Material eigenvalues (3.8)Material eigenvector matrices A, B (3.6b&3.8) Barnett–Lothe tensors S, H, L (3.26);

kµ

Fundamental matrices N1, N2, N3 (3.8)

, t tij ijkα

, ,t t tij ij ijA B D *,t t

i i

0 *, T T

(3.24a)1 2, ∞ ∞d d

Resultant forces and moments around the hole boundary (3.25)Thermal problem: (3.29)Stress intensity factors for crack problem: (3.32)

Fig. 3.2. Flowchart for the calculation of deformations and stresses, and resultantforces and moments, around the hole boundary, and the stress intensity factors for crackproblems.

hole boundary under different loading conditions. From this figure we seethat if an unsymmetric composite laminate is subjected to in-plane forcesonly or out-of-plane bending moments only, both bending moments and in-plane forces will be induced around the hole boundary, which is reasonableand expected due to the existence of the coupling stiffnesses.

To show the effect of hole shape, five different ratios, b/a = 5, 3, 1,1/3, 1/5, were considered. Figure 3.4 shows that the maximum values ofNs increase as the ratio b/a increases. Moreover, the maximum values ofNs are located at ϕ = 90 and 270 when the ratio b/a ≥ 1. On the otherhand, when the ratio b/a < 1 the location of the maximum values has a


126 C. Hwu

x1

nsM

sM

sN

p

nsM

sM

sN

m

(a) (b)

x1

Fig. 3.3. Forces and moments around the circular hole in an unsymmetric compositelaminate under different loading conditions. (a) N∞

11 = p; (b) M∞11 = m [27].

-2

-1

0

0 30 60 90 120 150 180 210

(degree)

240 270 300 330 360

1

2

3

4

5

6

7

8

9

10

11

pNs

ˆb / a = 5

b / a = 1/5

b / a = 1b / a = 3

b / a = 1/3

ϕ

Fig. 3.4. Force around an elliptical hole in an unsymmetric laminate subjected to N∞11 =

p, M∞11 = m (m = p × 1) [27].



tendency, although not quite clear, to shift to near ϕ = 180 and ϕ = 360.A similar situation occurs for the variation of Ms and Mns [27].

3.6.2 Thermal environment

To show the necessity of the analytical solutions, here we considerthe simplest case where only in-plane stretching occurs under uniformtemperature changes in a unidirectional laminate. Consider a [0]4unidirectional laminate containing a through-thickness circular holesubjected to a temperature change from 0C to 100C over the entirelaminate, i.e., T 0 = 100C and T ∗ = 0 in (3.13). Figure 3.5 comparesthe resultant hoop stress Ns around the hole boundary obtained by thesolution (3.14) with three different element types, Plane42, Shell99, andSolid45, from the commercial finite element software ANSYS. In the ANSYSsimulation, 72 nodes around the hole boundary and a 1:100 hole/plate ratiowere used to approximate the unbounded laminate. Figure 3.5 shows thatthe results for Plane42 agree well with the present solution while a largerdiscrepancy occurs for the other element types. To illustrate the reason for

-0.50 30 60 90 120

(degree)

150 180

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3Present

Plane42

Shell99

Solid45

sN (GPa ⋅ mm)

ϕ Fig. 3.5. Resultant force Ns around the circular hole boundary when the laminate issubjected to a uniform temperature change [28].


128 C. Hwu

-0.30 30 60 90

(degree)

120 150 180

-0.2

-0.1

0

0.1

0.2

0.3Present

Plane42

Shell99

Solid45Nn / Ns Nns / Ns

ϕ0 30 60 90

(degree)

120 150 180

ϕ

-0.3

-0.2

-0.1

0

0.1

0.2

0.3Present

Plane42

Shell99

Solid45

(a) (b)

Fig. 3.6. Resultant forces around the hole boundary under uniform temperaturechanges. (a) Nn/Ns; (b) Nns/Ns [28].

this, numerical values of the resultant tractions Nn/Ns and Nns/Ns aroundthe hole boundary, which should vanish for a traction-free hole, are plottedin Figs. 3.6(a) and (b). These two figures provide strong evidence for why alarger discrepancy occurs for Shell99 and Solid45 since they did not satisfythe traction-free hole boundary condition, which is satisfied by Plane42approximately and by the present solution exactly. Because the elementtype Plane42 can only be used for in-plane problems, for general stretching-bending coupling problems one must choose a shell or solid element. It isthen expected that a large discrepancy between the analytical solutions andthose of ANSYS will occur for general unsymmetric laminates. In otherwords, due to the approximate nature of any finite element software, toavoid the calculation of inaccurate solutions it is important to have a goodreference such as the exact solutions for unbounded plates shown in thischapter. Several numerical examples for unsymmetric laminates under auniform heat flow in the x1–x2 plane or in the x3 direction can be foundin [28].

3.6.3 Electro-elastic coupling

Due to the complexity of electro-elastic coupling, not many numericalexamples can be found in the literature. To learn how to calculatethe physical responses for electro-elastic laminates subjected to in-planeforces, out-of-plane bending moments, or electric displacements, one may



refer to [29] for [E/0/45/E] composed of two layers of a graphite/epoxyfiber-reinforced composite in the middle and two piezoelectric layers ofleft-hand quartz at the top and bottom. From the discussions therein,we see that to avoid numerical ill-conditioning before employing theexplicit solutions shown in this chapter a proper dimensional adjustment isnecessary, since the elastic stiffness constants are usually of the order above9, whereas the dielectric permittivity is usually of the order below −12, if SIunits are used. For example, the constitutive relation shown in (3.17) can berewritten as

σij/E0 = (CEijkl/E0)εkl − ekij(Ek/E0),

Dj = ejklεkl + (E0ωεjk)(Ek/E0),

(3.34)

in which E0 is a reference number, such as 109 N/m2, used for scaleadjustment. In numerical calculations, Eq. (3.34) allows the replacementof the material properties CEijkl and ωεjk by CEijkl/E0 and E0ω

εjk. With this

replacement the output values of the stresses and electric fields are σij/E0

and Ek/E0, which should be multiplied by E0 to return them to theiroriginal units.

3.6.4 Cracks

From the solution shown in (3.32) we see that the stress intensity factorKIIB depends on the material properties, and all the other stress intensityfactors are independent of the material properties. To illustrate the effectof the material, two additional materials were considered in crack problemsbesides the graphite/epoxy laminates. One was glass/epoxy and the otherwas boron/epoxy. Their material properties are

Glass/epoxy:

E1 = 38.6 GPa, E2 = 8.27 GPa, G12 = 4.14 GPa, ν12 = 0.26.

Boron/epoxy:

E1 = 209 GPa, E2 = 19 GPa, G12 = 6.6 GPa, ν12 = 0.21.

Figure 3.7 shows that the larger the crack length 2a the higher the stressintensity factors KIB and KIIB , and the Mode II stress intensity factorKIIB depends on both the crack geometry and the material properties.


130 C. Hwu

0 2 4 6 8 100

1

2

3

4

5

6

7

8

)(

/

21

22

mm

MKi∞

∞22/ MKIB

graphite/epoxyIIB MK ∞ )/( 22

glass/epoxyIIB MK ∞ )/( 22

boron/epoxyIIB MK ∞ )/( 22

ha /2

Fig. 3.7. Stress intensity factors of composite laminates.

3.7 Conclusions

From the presentation in this chapter, it can be seen that most of thefeatures of the Stroh formalism for two-dimensional anisotropic elasticityhave been preserved in the Stroh-like formalism for coupled stretching-bending analysis. Using this advantage, most of the problems in coupledstretching-bending analysis, which could not be solved previously have nowbeen solved by referring to the solutions obtained for the two-dimensionalelasticity problems.

Acknowledgements

The author would like to thank the National Science Council, Taiwan,R.O.C. for support through Grant NSC 96-2221-E-006-174.

References

[1] Lekhnitskii S.G., 1963. Theory of Elasticity of an Anisotropic Body, MIR,Moscow.

[2] Lekhnitskii S.G., 1968. Anisotropic Plates, Gordon and Breach SciencePublishers, New York.

[3] Stroh A.N., 1958. Dislocations and cracks in anisotropic elasticity, Phil. Mag.,7, 625–646.



[4] Stroh A.N., 1962. Steady state problems in anisotropic elasticity, J. Math.Phys., 41, 77–103.

[5] Ting T.C.T., 1996. Anisotropic Elasticity: Theory and Applications, OxfordScience Publications, New York.

[6] Hwu C., 2010. Anisotropic Elastic Plates, Springer, New York.[7] Hwu C., 2003. Stroh-like formalism for the coupled stretching-bending

analysis of composite laminates, Int. J. Solids Struct., 40(13–14), 3681–3705.[8] Lu P., Mahrenholtz O., 1994. Extension of the Stroh formalism to an analysis

of bending of anisotropic elastic plates, J. Mech. Phys. Solids, 42(11), 1725–1741.

[9] Cheng Z.Q., Reddy J.N., 2002. Octet formalism for Kirchhoff anisotropicplates, Proceedings of the Royal Society of London, Series A, 458, 1499–1517.

[10] Lu P., 2004. A Stroh-type formalism for anisotropic thin plates with bending-extension coupling, Arch. Appl. Mech., 73, 690–710.

[11] Cheng Z.Q., Reddy J.N., 2005. Structure and properties of the fundamentalelastic plate matrix, ZAMM Z. Angew. Math. Mech., 85(10), 721–739.

[12] Fu Y.B., Brookes D.W., 2006. Edge waves in asymmetrically laminatedplates, J. Mech. Phys. Solids, 54, 1–21.

[13] Fu Y.B., 2007. Hamiltonian interpretation of the Stroh formalism inanisotropic elasticity, Proceedings of the Royal Society of London, Series A,463, 3073–3087.

[14] Becker W., 1991. A complex potential method for plate problems withbending extension coupling, Arch. Appl. Mech., 61, 318–326.

[15] Becker W., 1992. Closed-form analytical solutions for a Griffith crack in anon-symmetric laminate plate, Compos. Struct., 21, 49–55.

[16] Becker W., 1993. Complex method for the elliptical hole in an unsymmetriclaminate, Arch. Appl. Mech., 63, 159–169.

[17] Becker W., 1995. Concentrated forces and moments on laminates withbending extension coupling, Compos. Struct., 30, 1–11.

[18] Zakharov D.D., Becker W., 2000. Boundary value problems for unsymmetriclaminates, occupying a region with elliptic contour, Compos. Struct., 49,275–284.

[19] Zakharov D.D., Becker W., 2000. Singular potentials and double-forcesolutions for anisotropic laminates with coupled bending and stretching,Arch. Appl. Mech., 70, 659–669.

[20] Chen P., Shen Z., 2001. Extension of Lekhnitskii’s complex potentialapproach to unsymmetric composite laminates, Mech. Res. Commun., 28(4),423–428.

[21] Chen P., Shen Z., 2001. Green’s functions for an unsymmetric laminate withan elliptic hole, Mech. Res. Commun., 28(5), 519–524.

[22] Chen P., Nie H., 2004. Green’s function for bending problem of anunsymmetrical laminated plate with bending-extension coupling containingan elliptic hole, Arch. Appl. Mech., 73, 846–856.

[23] Yin W.L., 2003. General solutions of anisotropic laminated plates, ASME J.Appl. Mech., 70(4), 496–504.


132 C. Hwu

[24] Yin W.L., 2003. Structures and properties of the solution space of generalanisotropic laminates, Int. J. Solids Struct., 40, 1825–1852.

[25] Yin W.L., 2005. Green’s function of anisotropic plates with unrestrictedcoupling and degeneracy, Part 1: the Infinite Plate, Compos. Struct., 69,360–375.

[26] Yin W.L., 2005. Green’s function of anisotropic plates with unrestrictedcoupling and degeneracy, Part 2: other domains and special laminates,Compos. Struct., 69, 376–386.

[27] Hsieh M.C., Hwu C., 2003. Explicit solutions for the coupled stretching-bending problems of holes in composite laminates, Int. J. Solids Struct.,40(15), 3913–3933.

[28] Hsieh M.C., Hwu C., 2006. Hygrothermal stresses in unsymmetriclaminates disturbed by elliptical holes, ASME J. Appl. Mech., 73,228–239.

[29] Hwu C., Hsieh M.C., 2005. Extended Stroh-like formalism for the electro-elastic composite laminates and its applications to hole problems, SmartMater. Struct., 14, 56–68.

[30] Cheng Z.Q., Reddy J.N., 2003. Green’s functions for infinite and semi-infiniteanisotropic thin plates, J. Appl. Mech., 70, 260–267.

[31] Cheng Z.Q., Reddy J.N., 2004. Laminated anisotropic thin plate with anelliptic inhomogeneity, Mech. Mater., 36, 647–657.

[32] Hwu C., Tan C.Z., 2006. In-plane/out-of-plane concentrated forces andmoments on composite laminates with elliptical elastic inclusions, Int. J.Solids Struct., 44, 6584–6606.

[33] Hwu C., 2003. Stroh-Like complex variable formalism for bending theory ofanisotropic plates, ASME J. Appl. Mech., 70, 696–707.

[34] Hwu C., 2004. Green’s function for the composite laminates with bendingextension coupling, Compos. Struct., 63, 283–292.

[35] Hwu C., 2005. Green’s functions for holes/cracks in laminates with stretchingbending coupling, ASME J. Appl. Mech., 72, 282–289.

[36] Hwu C., 2010. Boundary integral equations for general laminated plates withcoupled stretching-bending deformation, Proceedings of the Royal Society,Series A, 466, 1027–1054.

[37] Jones R.M., 1974. Mech. Compos. Mater., Scripta, Washington, DC.[38] Hwu C., Yen W.J., 1991. Green’s functions of two-dimensional anisotropic

plates containing an elliptic hole. Int. J. Solids Struct., 27, No. 13,1705–1719.

[39] Ting T.C.T., Hwu C., 1988. Sextic formalism in anisotropic elasticity foralmost non-semisimple matrix N, Int. J. Solids Struct., 24(1), 65–76.

[40] Hsieh M.C., Hwu C., 2002. Explicit expressions of the fundamental elasticitymatrices of Stroh-like formalism for symmetric/unsymmetric laminates,Chin. J. Mech.-Ser. A, 18(3), 109–118.

[41] Hwu C., 2010. Some explicit expressions of Stroh-like formalism for coupledstretching-bending analysis, Int. J. Solids Struct., 47, 526–536.



[42] Hwu C., 2008. Some explicit expressions of extended Stroh formalism fortwo-dimensional piezoelectric anisotropic elasticity, Int. J. Solids Struct., 45,4460–4473.

[43] Hwu C., 1990. Thermal stresses in an anisotropic plate disturbed by aninsulated elliptic hole or crack, ASME J. Appl. Mech., 57(4), 916–922.

[44] Rogacheva N.N., 1994. The Theory of Piezoelectric Shells and Plates, CRCPress, London.

[45] Hsieh M.C., 2005. Various coupling analyses of composite laminates withholes/cracks, PhD thesis, Institute of Aeronautics and Astronautics, NationalCheng Kung University.




Chapter 4

CLASSICAL, REFINED, ZIG-ZAG AND LAYER-WISEMODELS FOR LAMINATED STRUCTURES

Erasmo Carrera and Maria Cinefra

Aerospace Department, Politecnico di Torino,Corso Duca degli Abruzzi 24, 10129 Torino, Italy

Abstract

This chapter overviews classical and advanced theories for laminated platesand shell structures. Findings from existing historical reviews are used toconfirm that the advanced theories can be grouped and referred to as:Lekhnitskii multilayered theories, Ambartsumian multilayered theories, andReissner multilayered theories. The unified formulation proposed by the firstauthor, which is known as CUF (Carrera’s unified formulation), is used to makenumerical assessments of various laminated plate/shell theories. The chapterends by giving details of a recent reliable finite element formulation forlaminated shell analysis. It is embedded in the CUF framework and it leadsto the classical, zig-zag, and layer-wise models as particular cases. Numericalmechanisms such as shear and membrane locking are contrasted by developingan appropriate choice of shape functions and mixed assumed shear straintechniques.

4.1 Introduction

Two-dimensional (2D) modeling of multilayered plates and shellsrequires appropriate theories. The discontinuity of physical/mechanicalproperties in the thickness direction makes theories that were origi-nally developed for one-layered structures inadequate, such as theCauchy–Poisson–Kirchhoff–Love thin plate/shell theories [1–4], or theReissner and Mindlin [5, 6] first-order shear deformation theory (FSDT)as well as higher-order models such as that by Hildebrand, Reissner,and Thomas [7]. These theories are in fact not able to reproducepiecewise continuous displacement and transverse stress fields in the

135


136 E. Carrera and M. Cinefra

Fig. 4.1. C0z -requirements. Single-layered and three-layered structures.

thickness direction, which are usually experienced by multilayeredstructures. These two fields are often described in the literature as zig-zag effects and interlaminar continuity, respectively (see also the three-dimensional solutions reported by Pagano [8]). In [9] these two effects havebeen summarized using the acronymC0

z -requirements, that is displacementsand transverse stresses must be C0-continuous functions in the z-thicknessdirections. A qualitative comparison of displacement and stress fields ina one-layered and a multilayered structure is depicted in Fig. 4.1. Thispicture clearly shows that theories designed for one-layer structures areinappropriate to analyze multilayered ones.

A number of refinements of classical models as well as theories designedfor multilayered structures have been proposed in the literature over thelast four decades. Due to the form of displacement fields (see Fig. 4.1),the latter are often referred to as “zig-zag” theories. For a complete reviewof this topic, readers who are interested can refer to the many availablesurvey articles on beams, plates, and shells. Among these, excellent reviewsare quoted in the articles by Ambartsumian [10], Librescu and Reddy[11], Grigolyuk and Kulikov [12], Kapania and Raciti [13], Kapania [14],Noor and co-authors [15–17], Reddy and Robbins [18], Carrera [19], as wellas in the books by Librescu [20] and Reddy [21]. These articles reviewtheories that deal with layer-wise models (LWMs) and equivalent single-layer models (ESLMs). Following Reddy [21], it is intended that the number


Classical, Refined, Zig-Zag and Layer-Wise Models for Laminated Structures 137

of displacement variables is kept independent of the number of constitutivelayers in the ESLM, while the same variables are independent in each layerfor LWM cases.

Although these review works are excellent, in the authors’ opinion,there still exists the need for a historical review with the aim of giving clearanswers to the following questions:

(1) Who first presented a zig-zag theory for a multilayered structure?(2) How many different and independent ESL zig-zag theories have been

proposed in open literature?(3) Who first proposed the theories for question 2?(4) Are the original works well recognized and mentioned correctly in

subsequent articles?(5) What are the main differences among the available approaches to

multilayered structures?

The answers to these five points could be extremely useful to analystsof layered structures. Furthermore, it will give an insight into early and,equally very interesting, ideas and methods such as those by Lekhnitskii[22], which could be extended and applied to further problems. In particular,answers to questions 1 and 3 could establish a sort of historical justice,therefore permitting us to give to “Caesar what belongs to Caesar and toGod what belongs to God.”

This chapter is therefore a historical review of “zig-zag” theories, whichcan describe what have previously been called C0

z -requirements, in view ofquestions 1–5. These topics have already been documented in the historicalnote by the first author [23]. The findings in that paper are reconsidered inthe first part of this chapter.

The present chapter considers mostly ESLMs. For the sake ofcompleteness, a few comments on layer-wise cases are given in a separatesection. A further limitation of the present chapter is that it is restrictedto axiomatic-type approaches. The three multilayered theories discussedintroduce initial assumptions: stress function forms were assumed byLekhnitskii, transverse shear stress fields were assumed by Ambartsumian,while both displacements and transverse shear stresses were assumed in theframework of the mixed theorem proposed by Reissner. Therefore, thoseworks which are based on asymptotic expansion such as those in [24–26]have not been discussed in the present chapter.



The latter part of this chapter considers the development of a refinedshell finite element formulation, which is based on Carrera’s unifiedformulation (CUF) [27, 28].

The most common mathematical models used to describe shellstructures may be classified into two classes according to their differentphysical assumptions. The Koiter model [29] is based on the Kirchhoffhypothesis. The Naghdi model [30] is based on the Reissner–Mindlinassumptions, which take into account transverse shear deformation. It isknown that when a finite element method is used to discretize a physicalmodel, numerical locking may arise from hidden constraints that arenot well represented in the finite element approximation. In the Naghdimodel both transverse shear and membrane constraints appear as the shellthickness becomes very small, thus locking may arise. The most commonapproaches proposed to overcome the locking phenomenon are the standarddisplacement formulation with higher-order elements [31, 32] or techniquesof reduced-selective integration [33, 34]. But these introduce other numericalproblems.

With reference to works by Bathe and others [35–37], the presentauthors have employed the mixed interpolation of tensorial components(MITC) method, coupled to CUF, to overcome the locking phenomenon.This method has been applied to both ESL and LW variable kinematicmodels contained in CUF in order to analyze multilayered structures. Nine-node cylindrical shell elements have been considered. The performance ofthe new element has been tested by solving benchmark problems involvingvery thin shells as well as multilayered shells. The results show that theelement has good convergence and robustness when the thicknesses becomevery small. In particular, the study of multilayered structures demonstratesthat the zig-zag and LW models provide more accurate solutions than thesimple ESL models.

4.2 Who First Proposed a Zig-Zag Theory?

To the best of the authors’ knowledge, Lekhnitskii should be consideredas the first contributor to the theory for multilayered structures. In [22],in fact, Lekhnitskii proposed a splendid method able to describe zig-zageffects (for both in-plane and through-the-thickness displacements) andinterlaminar continuous transverse stresses. This is proved by Fig. 4.2,taken from Lekhnitskii’s pioneering work [22], which shows an interlaminarcontinuous transverse shear stress field (τ1 and τ2 are shear stresses in



Fig. 4.2. C0z -form of a transverse shear stress in a two-layered structure.

the layers 1 and 2, respectively) with discontinuous derivatives at the layerinterface (the first author thanks Prof Shifrin, who provided the originalarticle in Russian, and D. Carrera for providing an Italian translation ofthe same article). In other words, the C0

z–requirements of Fig. 4.1 wereentirely accounted for by Lekhnitskii [22].

The authors believe it would be of relevant interest to quote the originalderivations made by Lekhnitskii. It is, in fact, difficult to obtain the originalarticle by Lekhnitskii, which has no English translation. Furthermore, thetheory proposed by Lekhnitskii is very interesting and the method usedcould be a starting point for future developments. The following detailedderivation is therefore taken directly from Lekhnitskii’s original paper,written in Russian. A few changes in notation are made. A briefer treatmentcan be found in the English translation of the book ([38]; Section 18 ofChapter 3, p. 74).

This section closes with a few remarks on the theory proposed byLekhnitskii:

(1) Lekhnitskii’s theory describes the zig-zag form of both longitudinal andthrough-the-thickness displacements, in particular:

(a) The longitudinal displacements uk have a cubic order in thez–thickness direction.

(b) The through-the-thickness displacement wk varies according to aparabolic order in z.

(2) Lekhnitskii’s theory gives the interlaminar continuous transversestresses σzz and σxz.



(3) The stresses obtained by Lekhnitskii fulfill the 3D indefinite equilibriumequations (this fundamental property is intrinsic in the used stressfunction formulation).

(4) Stresses and displacements were obtained by employing:

(a) Compatibility conditions for stress functions.(b) Strain–displacement relations.(c) Compatibility conditions for displacements at the interface:

uk−1 = uk, wk−1 = wk, k = 2, Nl. (4.1)

(d) Homogeneous conditions at the bottom and top surfaces for thetransverse stresses:

σ1zz = σNl

zz = 0, σ1xz = σNl

xz = 0, for z = 0, h. (4.2)

(e) Interlaminar equilibrium for the transverse stresses:

σk−1zz = σkzz, σk−1

xz = σkxz, k = 2, Nl. (4.3)

(5) No post-processing is used to recover transverse stresses.(6) The thickness stress σzz are neglected. Nevertheless, the Poisson effects

on the thickness displacement wk are fully retained.(7) Full retention of Koiter’s recommendation would require a different

assumption for the stress functions (the authors do not know any workthat does so).

Although Lekhnitskii’s theory was published in the mid-1930s and reportedin a short paragraph of the English edition of his book [38], it has beensystematically forgotten in the recent literature. An exception is the workby Ren [39–41], which is documented in the next paragraph.

4.3 The Lekhnitskii–Ren Theory

This is the first of the three discussed theories. It is named after the authorof the original work, Lekhnitskii and the author who first extended the workto plates, Ren. Due to the original stress function formulation, the presentapproach could also be referred to as a “stress approach.”

To the best of the authors’ knowledge, Ren is the only scientist who hasused Lekhnitskii’s work as described in the previous section. In two papers[39, 40], Ren has, in fact, extended Lekhnitskii’s theory to orthotropic and



anisotropic plates. Further applications to vibration and buckling weremade in a third paper written in collaboration with Owen [41]. These threepapers are the unique contributions known to the authors that have beenmade under the framework of Lekhnitskii’s theory. As these three papershave been published in journals that are easily available worldwide, a fulldescription of Ren’s extension of Lekhnitskii’s theory to plates has been,therefore, omitted. Nevertheless, it is of interest to make a few commentson Ren’s work in order to make explicit the stress and displacementfields that were introduced by Ren to analyze the response of anisotropicplates.

On the basis of the form of τkxz obtained by Lekhnitskii, it appearedreasonable to Ren, see [39], to assume the following distribution oftransverse shear stresses in a laminated plate, composed of Nl orthotropiclayers (x, y, and z are the coordinates of the reference system depicted inFig. 4.3):

σkxz(x, y, z) = ξx(x, y)ak(z) + ηx(x, y)ck(z)

σkyz(x, y, z) = ξy(x, y)bk(z) + ηy(x, y)gk(z).(4.4)

z

z

h

h k

x yk k

z 0k

k=1

k=3

k=2

k

k=N l

Ω

Ωk

k k

x,y

Fig. 4.3. Multilayered plate.



Four independent function of x, y were introduced to describe thetransverse shear stresses. The layer constants are parabolic functions ofthe thickness coordinate z. As in Lekhnitskii, the displacement fields areobtained by integrating the strain–displacement relations.

In contrast to the work by Lekhnitskii, it is underlined that thetransverse strain εzz was discarded by Ren. This assumption contrastswith Koiter’s recommendation already mentioned. The constants ofintegration are determined by imposing compatibility conditions for thedisplacements at the interface. The displacement field assumes the followingform:

uk(x, y, z) = u0(x, y) − w,x + ξx(x, y)Ak(z) + ηx(x, y)Ck(z)

vk(x, y, z) = v0(x, y) − w,y + ξy(x, y)Bk(z) + ηy(x, y)Gk(z)

w(x, y, z) = w0(x, y),

(4.5)

where Ak(z), Bk(z), Ck(z), and Gk(z) are obtained by integrating thecorresponding ak(z), bk(z), ck(z), and gk(z). That is, Eqs. (4.5) represent apiecewise continuous displacement field in the thickness direction z, whichis cubic in each layer. An extension to generally anisotropic layers has beenprovided by the same author in the article already mentioned [40].

The displacement model of Eqs. (4.5) can be used in the framework ofknown variational statements, such as the principle of virtual displacements(PVD) to formulate the governing equations for anisotropic plates as wellas finite element models. This was done in [39–41]. No shell applications ofthe Lekhnitskii–Ren theory are known to the authors.

4.4 The Ambartsumian–Whitney–Rath–Das Theory

This is the second of the three discussed theories. Ambartsumian wasthe author of the original work [42–45]; Whitney [46] both extended thetheory to anisotropic plates and introduced the theory to the scientificcommunity in the West; Rath and Das [47], extended Whitney’s work toshell geometries.

The Ambartsumian–Whitney–Rath–Das (AWRD) approach has thepeculiarity of having the same number of unknown variables as first-order shear deformation theory, i.e. three displacements and two rotations(or shear strains). It was originated by Ambartsumian [42, 43] whorestricted the formulation to orthotropic layers. Here attention will focuson the work by Whitney [46] who first applied and extended it to generallyanisotropic and symmetrical and asymmetrical plates. For simplicity, only



symmetrical laminated plates are outlined. Details can be read in the above-mentioned articles and books. The transverse shear stresses are assumedto be:

σkxz(x, y, z) = [Qk55f(z) + ak55]φx(x, y) + [Qk45f(z) + ak45]φy(x, y)

σkyz(x, y, z) = [Qk45f(z) + ak55]φx(x, y) + [Qk44f(z) + ak44]φy(x, y).(4.6)

The Ambartsumian case can be obtained by putting Qk45 = ak45 =0.f(z) is a function of the thickness coordinate which is assumed to bedifferent in the symmetrical and unsymmetrical laminate cases. A parabolicform for f(z) has mostly been considered (an explicit formula forunsymmetrical cases was given by Whitney). The layer constants ak44, a

k45,

and ak55 are determined by imposing the continuity conditions of transverseshear stresses at the interface while top-bottom homogeneous conditionsare used to determine the form of f(z). Notice that the top-bottominhomogeneous conditions for transverse shear stresses were addressedby Ambartsumian [42, 43], along with a method to compute transversenormal stresses. These two facts have not been addressed in subsequentwork.

The transverse shear strains related to the assumed transverse shearstress fields are:

γkxz(x, y, z) = [f(z) + Sk55ak55 + Sk45a

k45]φx(x, y) + [Sk55a

k45 + Sk45a

k44]φy(x, y)

γkyz(x, y, z) = [Sk44ak44 + Sk45a

k45]φx(x, y) + [f(z) + Sk44a

k44 + Sk45a

k55]φy(x, y),

(4.7)in which the following compliances have been introduced:

Sk55 =Qk55D

, Sk45 = −Qk45

D, Sk44 =

Qk44D

, D = Qk44Qk55 − (Qk45)

2.

By assuming the transverse displacement is constant in the thicknessdirection, i.e. εzz=0, on integrating the shear strains the displacement fieldhas the following form:

uk(x, y, z) = −zw,x + [J(z) + gk1 (z)]φx(x, y) + gk2 (z)φy(x, y)

vk(x, y, z) = −zw,y + [J(z) + gk3 (z)]φy(x, y) + gk4 (z)φx(x, y)

w(x, y, x) = w0(x, y, z),

(4.8)



where

J(z) =∫f(z)dz

gk1 (z) = [Sk55ak55 + Sk45a

k45]z + dk1

gk2 (z) = [Sk55ak55 + Sk45a

k45]z + dk2

gk3 (z) = [Sk55ak55 + Sk45a

k45]z + dk3

gk4 (z) = [Sk55ak55 + Sk45a

k45]z + dk4 .

(4.9)

dk1 , dk2 , dk3 , and dk4 are calculated by imposing the compatibility of the in-plane displacement at each interface. Equations (4.8) are the starting pointfor any analytical or computational study of multilayered plates.

An extension to doubly curved shells and a dynamic case of Whitney’swork was made by Rath and Das [47].

Dozens of papers have been presented over recent decades that dealwith zig-zag effects and interlaminar continuous transverse shear stresses,and which have stated that new theories were being proposed. The authorsbelieve that these articles should be considered as simplified cases of theAWRD theory or the AWRD theory itself. Unfortunately, the original workand authors (Ambartsumian, Whitney, Rath, and Das) are not mentioned,or rarely cited, in the literature lists of this large number of articles. Thishistorical unfairness has been corrected in [23].

4.5 The Reissner–Murakami–Carrera Theory

A third approach to laminated structures originated in two papers byReissner [48, 49] in which a mixed variational equation, namely the Reissnermixed variational theorem (RMVT) was proposed. The displacement andtransverse stress variables are independently assumed in RMVT. This thirdapproach is the only one that was entirely developed in the West. Reissner[48] proposed a mixed theorem and traced the manner in which it couldbe developed; Murakami [50, 51], a student under Prof Reissner in SanDiego, was the first to develop a plate theory on the basis of RMVTand introduced fundamental ideas on the application of RMVT in theframework of ESLM; Carrera [9, 52] presented a systematic way to useRMVT to develop plate and shell theories and introduced a weak formof Hooke’s law (WFHL), which reduces mixed theories to classical modelswith only displacement variables.

RMVT fulfills completely and a priori the C0z -requirements by

assuming two independent fields for displacements u = u, v, w, and



transverse stresses σn = σxz, σyz , σzz (bold letters denote arrays). Briefly,RMVT puts 3D indefinite equilibrium equations (and related equilibriumconditions at the boundary surfaces, which for brevity are not written here)and compatibility equations for transverse strains in a variational form. The3D equilibrium equations in the dynamic case are:

σij,j − ρ ui = pi i, j = 1, 2, 3, (4.10)

where ρ is the mass density and double dots denote acceleration while(p1, p2, p3)=p are volume loadings. The compatibility conditions fortransverse stresses can be written by evaluating transverse strains in twoways: using Hooke’s law εnH = εxzH

, εyzH, εzzH

and using a geometricalrelation εnG= εxzG , εyzG

, εzzG; the subscript n denotes transverse/normalcomponents. Hence

εnH − εnG = 0. (4.11)

RMVT therefore states:∫V

(δεpTGσpH + δεnTGσnM + δσn

TM (εnG − εnH))dV

=∫V

ρδu u dV + δLe. (4.12)

The superscript T signifies an array transposition and V denotes the3D multilayered body volume while the subscript p denotes in-planecomponents, respectively. Therefore, σp = σxx, σyy, σxy and εp =εxx, εyy, εxy. The subscript H underlines that stresses are computed viaHooke’s law. The variation of the internal work has been split into in-plane and out-of-plane parts and involves the stress from Hooke’s law andthe strain from the geometrical relations (subscript G). δLe is the virtualvariation of the work done by the external layer-force p. Subscript M

underlines that transverse stresses are those of the assumed model.The first application of RMVT was due to Murakami [50, 51], who

developed a refinement of Reissner–Mindlin type theories. First a zig-zagform of the displacement field was introduced by means of two “zig-zag”functions (Dx, Dy):

uk(x, y, z) = u0(x, y) + zφx(x, y) + ξk(−1)kDx(x, y)

vk(x, y, z) = v0(x, y) + zφy(x, y) + ξk(−1)kDy(x, y) (4.13)

w(x, y, z) = w0(x, y).



z

x,y

ED1

Zig-Zag

Fig. 4.4. Geometrical meaning of Murakami’s zig-zag function. Linear case.

ξk=2zk/hk is a dimensionless layer coordinate (zk is the physicalcoordinate of the kth layer whose thickness is hk). The exponent k changesthe sign of the zig-zag term in each layer. This trick reproduces thediscontinuity of the first derivative of the displacement variables in thez-direction. The geometrical meaning of the zig-zag function is explainedin Figs. 4.4 and 4.5.

The transverse shear stresses fields were assumed to be parabolic byMurakami [50] in each layer and interlaminar continuous according to thefollowing formula:

σkxz(x, y, z) = σktxz(x, y)F0(zk) + F1(zk)kx(x, y) + σkbxz(x, y)F2(zk)

σkyz(x, y, z) = σktyz(x, y)F0(zk) + F1(zk)Rky(x, y) + σkbyz(x, y)F2(zk),

(4.14)

where σktxz(x, y), σktyz(x, y), σkbxz(x, y), and σkbyz(x, y) are the top and bottomvalues of the transverse shear stresses, while Rkx(x, y), and Rky(x, y) arethe layer stress resultants. The introduced layer thickness coordinatepolynomials hold:

F0 = −1/4 + ξk + 3ξ2k, F1 =3 − 12ξ2k

2hk, F2 = −1/4− ξk + 3ξ2k.

The homogeneous and inhomogeneous boundary conditions at the top-bottom plate surfaces can be linked to the introduced stress field.



z

x,y

Zig-Zag

ED3

Fig. 4.5. Geometrical meaning of Murakami’s zig-zag function. Higher-degree case.

Toledano and Murakami [53] introduced transverse normal strain andstress effects by using a third-order displacement field for both in-plane andout-of-plane components and a fourth-order transverse stress field for bothshear and normal components. This paper is the first paper in the ESLMframework in which Koiter’s recommendation is retained.

A generalization of RMVT to plate/shell theories has been providedby Carrera [28,54–60]. The displacements and transverse stress componentswere assumed as follows:

uk = Ftukt + Fbu

kb + Fru

kr = Fτu

kτ τ = t, b, r

r = 2, 3, . . . , NσknM = Ftσ

knt + Fbσ

knb + Frσ

knr = Fτσ

knτ k = 1, 2, . . . , Nl.

(4.15)

The subscripts t and b denote values for the top and bottom surface layer,respectively. These two terms consist of the linear part of the expansion.The thickness functions Fτ (ξk) can now be defined at the kth layer level:

Ft =P0 + P1

2, Fb =

P0 − P1

2, Fr = Pr − Pr−2,

r = 2, 3, . . . , N (4.16)

in which Pj = Pj(ξk) is the jth order Legendre polynomial defined in theξk domain: −1 ≤ ξk ≤ 1. For instance, the first five Legendre polynomials



are:

P0 = 1, P1 = ξk, P2 = (3ξ2k − 1)/2, P3 =5ξ3k2

− 3ξk2,

P4 =35ξ4k

8− 15ξ2k

4+

38.

The chosen functions have the following properties:

ξk =

1 when Ft = 1; Fb = 0; Fr = 0

−1 when Ft = 0; Fb = 1; Fr = 0.(4.17)

The top and bottom values have been used as unknown variables. Such achoice makes the model particularly suitable, in view of the fulfillment ofthe C0

z -requirements. The interlaminar transverse shear and normal stresscontinuity can therefore be linked by simply writing:

σknt = σk+1nb , k = 1, Nl − 1. (4.18)

In those cases in which the top/bottom plate/shell stress values areprescribed (zero or imposed values), the following additional equilibriumconditions must be accounted for:

σ1nb = σnb, σNl

nt = σnt, (4.19)

where the over-bar denotes the imposed values for the plate boundarysurfaces.

Examples of the application of RMVT to laminated plates in theequivalent single-layer model were presented in the already mentionedarticles [50, 51, 53]. The results obtained for the cylindrical bending ofcross-ply symmetrically laminated plates showed an improvement indescribing the in-plane response with respect to the first-order sheardeformation theory [51]. Applications to unsymmetrically laminated plateswere presented in [53]. Shell applications based on [51] were developed byBhaskar and Varadan [61] and Jing and Tzeng [62]. Bhaskar and Varadan[61] underlined the severe limitation of the transverse shear stress a priorievaluated by the assumed model. Finite element applications of this modelhave been developed. The linear analysis of thick plates was discussed byRao and Meyer-Piening [63]. Linear and geometrically non-linear static anddynamic analyses were considered by Carrera [54, 64] and co-authors [65].Partial implementations to shell elements were proposed by Bhaskar andVaradan [66]. A full shell implementation has recently been given by Brankand Carrera [67].



The limitations of the equivalent single-layer analysis were known toToledano and Murakami [53] who applied RMVT in a multilayered model.A linear in-plane displacement expansion was expressed in terms of theinterface values in each layer while the transverse shear stresses wereassumed parabolic. It was shown that the accuracy of the resulting theorieswas independent of layout. Transverse normal stress and related effects werediscarded and the analysis showed severe limitations when analyzing thickplates. A more comprehensive evaluation of layer-wise theories for linear andparabolic expansions was made by the first author in [55] where applicationsto the static analysis of plates were presented. Subsequent work extendedthe analysis to dynamic cases [28, 56, 59, 60] and shell geometry [52, 57, 58].A more exhaustive overview work of based on the Reissner theorem hasbeen provided in [19].

4.6 Remarks on the Theories

In the authors’ opinion the work by Lekhnitskii is the most relevantcontribution to multilayered structure modeling:

• L1. This is the first work to account for the C0z -requirements.

• L2. Even though Lekhnitskii restricted his analysis to a cantileveredmultilayered beam, he quoted explicit formulas for transverse stresses anddisplacement fields (Eqs. (4.4) and (4.5)), which are valid at all pointsof the considered beam. This could be extremely useful in assessing newanalytical and numerical models.

• L3. The work by Lekhnitskii shows how multilayered structures problemscan be handled. For instance, it is clear in [22] that the inclusion ofa transverse normal stress would require a different choice of stressfunctions.

• L4. The stress function formulation leads to in-plane and transversestress fields which fulfill “by definition” the 3D equilibrium equations.Stresses were calculated by Lekhnitskii by solving a boundary-valueproblem for the compatibility equations written in terms of a stressfunction. In particular, the evaluation of transverse stresses does notrequire any post-processing procedure such as Hooke’s law or integrationof 3D equilibrium equations.

• L5. Although transverse normal stresses are neglected, the transversedisplacement varies in the beam thickness according to a piecewise-parabolic distribution. A direct attempt to include the transverse



normal stress effect would require an appropriate choice for the stressfunction.

Concerning Lekhnitskii–Ren plate theory observe that:

• LR1. The transverse shear stresses are continuous at the interfaces andparabolic in each layer. Furthermore, homogeneous conditions are fulfilledat the top-bottom plate surfaces.

• LR2. Four independent functions defined on Ω are used to expresstransverse shear stresses. Layer constants, which are parabolic in eachlayer, are used to describe the transverse shear stresses.

• LR3. Expressions for the layer constants were given by Ren. In otherwords their calculation does not require any imposition of transverseshear stresses.

• LR4. The in-plane displacements are continuous at each interface and arecubic in each layer.

• LR5. Seven independent variables defined on Ω were used to describethe displacement and stress fields in the laminated plates. Four areused for the transverse shear stresses and three for the displacementscorresponding to the chosen reference surface Ω.

• LR6. According to Lekhnitskii, Ren neglects the transverse normal stressσzz . In contrast to Lekhnitskii, the transverse normal strain εzz isdiscarded by Ren.

• LR7. The transverse shear stresses are calculated by Ren directly usingEqs. (4.4). Hooke’s Law is not used and integration of the 3D equilibriumequations is not required.

Regarding the Ambartsumian–Whitney–Rath–Das theory notice that:

• AWRD1. As LR1.• AWRD2. Two independent functions defined on Ω are used to express

transverse shear stresses (Eqs. (4.6)).• AWRD3. Layer constants, parabolic in each layer, must be computed

by imposing transverse shear stress continuity at each interface whilethe form of the f(z) function is found by imposing top-bottom layerhomogeneous conditions.

• AWRD4. As LR4.• AWRD5. Five independent variables defined on Ω are used to describe

the displacement and stress fields in a laminated plate/shell, which is twoless than LR.

• AWRD6. As LR6.



• AWRD7. The literature shows that much better evaluations fortransverse shear stresses are obtained via integration of the 3D equili-brium equations, with respect to Eqs. (4.6).

• AWRD8. The extension to a shell requires a reformulation of thedisplacement models and related layer constants.

For the Reissner–Murakami–Carrera theory observe that:

• RMC1. As LR1. In this case, homogeneous as well as inhomogeneousconditions for transverse stresses can be included.

• RMC2. At least 2Nl + 1 independent variables must be used for eachtransverse stress component. However, these variables can be expressedin terms of the displacement variables using a weak form of Hooke’slaw.

• RMC3. The in-plane displacements are continuous at each interface andcan be chosen linear or of higher order in each layer.

• RMC4. The number of independent variables can be chosen arbitrarilyaccording to RMC3.

• RMC5. Interlaminar continuous transverse normal stresses/strains can beeasily described by the RMC theory. These effects were, in fact, includedin the early development of the RMC theory, fulfilling the fundamentalKoiter’s recommendation.

• RMC6. As for the AWRD theory much better evaluations for transversestresses are obtained via integration of the 3D equilibrium equations,with respect to assumed forms, e.g. Eqs. (4.14).

• RMC7. The extension to a shell does not require any changes in eitherdisplacement or stress fields.

4.7 A Brief Discussion on Layer-Wise Theories

The previous discussion has been restricted to ESLM. In this class oftheories the number of unknown variables doesn’t depend on the numberof layers (it is intended that for the RMC theory this restriction is only fordisplacement variables). The use of independent variables in each layer,as in the layer-wise description, increases computational costs. On theother hand, such a choice permits one to include “naturally” the zig-zag form of displacements in the thickness direction and in general cansignificantly improve for the response of very thick structures. In thisrespect, the authors’ experience suggests that the layer-wise descriptionis mandatory for thick plate/shell analyses and in any other problems in



which the response is essentially a layer response. In particular, in [60]the first author showed that the use of a sufficiently high order for thedisplacement fields in the layers could lead to a description with acceptableaccuracy of the transverse stresses directly computed by Hooke’s law. Manylayer-wise theories have been proposed. So called global/local approacheshave also been proposed, see Reddy [21]. Excellent overviews can be foundin the review articles and books mentioned in the Introduction.

To the best of the authors’ knowledge, there is no layer-wise theorybased on the LR approach. Works with a layer-wise description in theframework of the AWRD theories have recently been discussed by Choand Averill [68]. Studies on the use of the RMC theory have been madefor plates by Toledano and Murakami [53] and extended to higher-ordercases (including normal stress effects), dynamics, and shells in the Carrera’sarticles [55, 60].

4.8 CUF Shell Finite Elements

The efficient load-carrying capabilities of shell structures make them veryuseful in a variety of engineering applications. The continuous developmentof new structural materials leads to ever more complex structural designsthat require careful analysis. Although analytical techniques are veryimportant, the use of numerical methods to solve mathematical shell modelsof complex structures has become an essential ingredient in the designprocess. The finite element method (FEM) has been the fundamentalnumerical procedure in the analysis of shells.

In this section, a new shell finite element approach based on variablekinematic models within Carrera’s unified formulation [27, 28] is presented.Elements with nine nodes and cylindrical geometry are considered.Referring to Bathe and others [35–37], the MITC method is used toovercome the locking phenomenon. The governing equations are derived inthe framework of the CUF in order to apply FEM. Some numerical resultsare provided to show the efficiency of the new element.

4.8.1 Geometry of cylindrical shells

Let us consider a cylindrical shell. In a system of Cartesian coordinates(O, x, y, z), the region occupied by the mid-surface of the shell is:

S = (x, y, z) ∈ R3 : −L/2 < x < L/2, y2 + z2 = R2 (4.20)



where L and R are the length and the radius of the shell, respectively. Letus consider a curvilinear coordinate system (α, β, z) placed at the center ofthe upper part of the mid-surface. The 3D medium corresponding to theshell is defined by the 3D chart given by:

Φ(α, β, z) = φ(α, β) + za3(α, β), (4.21)

where a3 is the unit vector normal to the tangent plane. Then, the mid-surface S of the cylindrical shell is described by the following 2D chart:

φ1(α, β) = α

φ2(α, β) = R sin(β/R)

φ3(α, β) = R cos(β/R).

(4.22)

With this choice, the region Ω ⊂ R2 corresponding to the mid-surface S isthe rectangle:

Ω = (α, β) : −L/2 < α < L/2, −Rπ < β < Rπ. (4.23)

Using these geometrical assumptions, the strain–displacement relations canbe obtained by considering the linear part of the 3D Green–Lagrange straintensor. Remembering that in the unified formulation the unknowns arethe components of the displacement uτ (α, β), vτ (α, β), and wτ (α, β), forτ = 0, 1, . . . , N , the geometrical relations for the kth layer of a multilayercylindrical shell can be written as follow:

εkαα = Fτukτ,α

εkββ = Fτ

[(1 +

zkRk

)wkτRk

+(

1 +zkRk

)vkτ,β

]

εkαβ = Fτ

[ukτ,β +

(1 +

zkRk

)vkτ,α

]εkαz = wkτ,αFτ + ukτFτ,z

εkβz = Fτ

[wkτ,β − vkτ

Rk

]+ Fτ,z

[(1 +

zkRk

)vkτ

]εkzz = wkτFτ,z,

(4.24)

where Rk is the radius of the mid-surface of the layer k. The thicknessfunctions Fτ are Taylor functions (1, z, z2, . . .) if the approach used isESL or combinations of Legendre polynomials if the approach is LW(see Eqs. (4.16)). For more details of the geometrical description and the



procedure to obtain the strain–displacement relations, the reader can referto [69].

The previous geometrical relations can be expressed in matrix form as:

εkp =(Dkp +Ak

p)uk

εkn =(DknΩ +Dk

nz −Akn)u

k,(4.25)

where subscripts (p) and (n) indicate in-plane and normal components,respectively, and the differential operators are defined as follows:

Dkp =

∂α 0 0

0 Hk∂β 0∂β Hk∂α 0

, Dk

nΩ =

0 0 ∂α

0 0 ∂β0 0 0

,

Dknz = ∂z ·Ak

nz = ∂z

1 0 0

0 Hk 00 0 1

, (4.26)

Akp =

0 0 0

0 0 1RkHk

0 0 0

, Ak

n =

0 0 0

0 1Rk

00 0 0

, (4.27)

where Hk = (1 + zk/Rk).

4.8.2 MITC method

Considering the components of the strain tensor in the local coordinatesystem (ξ, η, z), the MITC shell elements are formulated using — insteadof the strain components directly computed from the displacements —an interpolation of these strain components within each element usinga specific interpolation strategy for each component. The correspondinginterpolation points — called the tying points — are shown in Fig. 4.6 fora nine-node shell element (MITC9 shell element). For more details see [69].

The interpolating functions are arranged in the following arrays:

Nm1 = [NA1, NB1, NC1, ND1, NE1, NF1]

Nm2 = [NA2, NB2, NC2, ND2, NE2, NF2] (4.28)

Nm3 = [NP , NQ, NR, NS ].

For convenience, we will indicate with the subscripts m1, m2, andm3 the quantities calculated for the points (A1, B1, C1, D1, E1, F1),(A2, B2, C2, D2, E2, F2), and (P,Q,R, S), respectively.



ξ

η

A1

C1

E1 F1

D1

B1

εε dnastnenopmoC αα αz εε dnastnenopmoC ββ βz Component εαβ

ξ

η

A2

B2

C2

D2 F2

E2ξ

η

P

R S

Q

Fig. 4.6. Tying points for MITC9 shell finite element

According to the MITC method, the strain components are interpolatedon the tying points as follows:

εkp =

εk11

εk22

εk12

=

Nm1 0 0

0 Nm2 00 0 Nm3

εk11m1

εk22m2

εk12m3

= [N1]

εk11m1

εk22m2

εk12m3

εkn =

εk13

εk23

εk33

=

Nm1 0 0

0 Nm2 00 0 1

εk13m1

εk23m2

εk33

= [N2]

εk13m1

εk23m2

εk33

,

(4.29)

which defines the matrixes N1 and N2.Applying the finite element method, the unknown displacements are

interpolated on the nodes of the element by means of the Lagrangian shapefunctions Ni (for i = 1, . . . , 9):

uk = FτNiqkτi, (4.30)

where qkτiare the nodal displacements and the unified formulation is

applied. Substituting in Eqs. (4.25) the geometrical relations become:

εkp =Fτ (Dkp +Ak

p)(NiI)qkτi

εkn =Fτ (DknΩ −Ak

n)(NiI)qkτi

+ Fτ,zAknz(NiI)q

kτi,

(4.31)

where I is a 3 × 3 identity matrix.



If the MITC technique is applied, the geometrical relations arerewritten as follows:

εkτpim=Fτ [Ck

3im]qkτi

εkτnim=Fτ [Ck

1im]qkτi

+ Fτ,z [Ck2im

]qkτi,

(4.32)

where the introduced matrixes are:

[Ck1im

] = [N2]

[(Dk

nΩ −Akn)(NiI)]m1(1, :)

[(DknΩ −Ak

n)(NiI)]m2(2, :)[(Dk

nΩ −Akn)(NiI)](3, :)

[Ck2im

] = [N2]

[Ak

nz(NiI)]m1(1, :)[Ak

nz(NiI)]m2(2, :)[Ak

nz(NiI)](3, :)

(4.33)

[Ck3im

] = [N1]

[(Dk

p +Akp)(NiI)]m1(1, :)

[(Dkp +Ak

p)(NiI)]m2(2, :)[(Dk

p +Akp)(NiI)]m3(3, :)

.

(1,:), (2,:), and (3,:) respectively indicate that the first, second, or thirdline of the relevant matrix is considered.

4.8.3 Governing equations

This section presents the derivation of the governing equations based on theprinciple of virtual displacements (PVD) for a multilayered shell subjectedto mechanical loads. CUF can be used to obtain the so-called fundamentalnuclei, which are simple matrices representing the basic elements fromwhich the stiffness matrix of the whole structure can be computed.

The PVD for a multilayered shell with Nl layers is:

Nl∑k=1

∫Ωk

∫Ak

δεkpG

TσkpC + δεknG

TσknC

dΩkdzk =

Nl∑k=1

δLke , (4.34)

where Ωk and Ak are the integration domains in the plane (α,β) andthe z-direction, respectively, and T indicates the transpose of a vector.The first member of the equation represents the variation of internal workδLkint and δLke is the external work. G means geometrical relations and C

constitutive relations. The first step in deriving the fundamental nucleiis the substitution of the constitutive equations (C) in the variational



statement of PVD, which are:

σkpC = σkspjn= Ck

pp εkspjn

+Ckpn ε

ksnjn

σknC = σksnjn= Ck

np εkspjn

+Cknn ε

ksnjn

(4.35)

with

Ckpp =

Ck11 Ck12 Ck16

Ck12 Ck22 Ck26

Ck16 Ck26 Ck66

Ck

pn =

0 0 Ck13

0 0 Ck23

0 0 Ck36

Cknp =

0 0 00 0 0Ck13 Ck23 Ck36

Ck

nn =

Ck55 Ck45 0Ck45 Ck44 00 0 Ck33

,

(4.36)

and C are the material coefficients.Then, one substitutes the geometrical relations (4.32) and the

constitutive equations (4.35) into the variational statement (4.34) to obtainthe governing equation system:

δukτiT

: Kkτsijuu uksj = P k

uτi, (4.37)

where Kkτsijuu is the fundamental nucleus of the stiffness array, which is

expanded according to the indexes τ ,s and i,j in order to obtain thematrix for the whole shell. P k

uτ is the fundamental nucleus of the externalmechanical load. The explicit form of the stiffness fundamental nucleus isthe following:

Kkτsij11 = Ck55Nim1 Nm1Nn1 Ωk

Njn1 Fτ,zFs,z Ak

+Ck11Ni,αm1 Nm1Nn1 ΩkNj,αn1 FτFs Ak

+Ck16Ni,βm3 Nm3Nn1 ΩkNj,αn1 FτFs Ak

+Ck16Ni,αm1 Nm1Nn3 ΩkNj,βn3 FτFs Ak

+Ck66Ni,βm3 Nm3Nn3 ΩkNj,βn3 FτFs Ak

Kkτsij12 = −Ck45

1Rk

Nim1 Nm1Nn2 ΩkNjn2 Fτ,zFs Ak

+Ck45Nim1 Nm1Nn2 ΩkNjn2 HkFτ,zFs,z Ak



+Ck12Ni,αm1 Nm1Nn2 ΩkNj,βn2 HkFτFs Ak

+Ck16Ni,αm1 Nm1Nn3 ΩkNj,αn3 HkFτFs Ak

+Ck26Ni,βm3 Nm3Nn2 ΩkNj,βn2 HkFτFs Ak

(4.38)

+Ck66Ni,βm3 Nm3Nn3 ΩkNj,αn3 HkFτFs Ak

Kkτsij13 = Ck13Ni,αm1 Nm1Nj Ωk

FτFs,z Ak

+Ck36Ni,βm3 Nm3Nj ΩkFτFs,z Ak

+Ck121Rk

Ni,αm1 Nm1Nn2 ΩkNjn2 HkFτFs Ak

+Ck261Rk

Ni,βm3 Nm3Nn2 ΩkNjn2 HkFτFs Ak

+Ck55Nim1 Nm1Nn1 ΩkNj,αn1 Fτ,zFs Ak

+Ck45Nim1 Nm1Nn2 ΩkNj,βn2 Fτ,zFs Ak

Kkτsij21 = −Ck45

1Rk

Nim2 Nm2Nn1 ΩkNjn1 FτFs,z Ak

+Ck45Nim2 Nm2Nn1 ΩkNjn1 HkFτ,zFs,z Ak

+Ck12Ni,βm2 Nm2Nn1 ΩkNj,αn1 HkFτFs Ak

+Ck16Ni,αm3 Nm3Nn1 ΩkNj,αn1 HkFτFs Ak

+Ck26Ni,βm2 Nm2Nn3 ΩkNj,βn3 HkFτFs Ak

+Ck66Ni,αm3 Nm3Nn3 ΩkNj,βn3 HkFτFs Ak

Kkτsij22 = Ck22Ni,βm2 Nm2Nn2 Ωk

Nj,βn2 H2kFτFs Ak

+Ck26Ni,βm2 Nm2Nn3 ΩkNj,αn3 H2

kFτFs Ak

+Ck26Ni,αm3 Nm3Nn2 ΩkNj,βn2 H2

kFτFs Ak

+Ck66Ni,αm3 Nm3Nn3 ΩkNj,αn3 H2

kFτFs Ak(4.39)

+Ck441R2k

Nim2 Nm2Nn2 ΩkNjn2 FτFs Ak

−Ck441Rk

Nim2 Nm2Nn2 ΩkNjn2 HkFτFs,z Ak

−Ck441Rk

Nim2 Nm2Nn2 ΩkNjn2 HkFτ,zFs Ak

+Ck44Nim2 Nm2Nn2 ΩkNjn2 H2

kFτ,zFs,zAk



Kkτsij23 = Ck22

1Rk

Ni,βm2 Nm2Nn2 ΩkNjn2 H2

kFτFs Ak

+Ck23Ni,βm2 Nm2Nj ΩkHkFτFs,z Ak

+Ck261Rk

Ni,αm3 Nm3Nn2 ΩkNjn2 H2

kFτFs Ak

+Ck36Ni,αm3 Nm3Nj ΩkHkFτFs,z Ak

−Ck451Rk

Nim2 Nm2Nn1 ΩkNj,αn1 FτFs Ak

−Ck441Rk

Nim2 Nm2Nn2 ΩkNj,βn2 FτFs Ak

+Ck45Nim2 Nm2Nn1 ΩkNj,αn1 HkFτ,zFs Ak

+Ck44Nim2 Nm2Nn2 ΩkNj,βn2 HkFτ,zFs Ak

Kkτsij31 = Ck55Ni,αm1 Nm1Nn1 Ωk

Njn1 FτFs,z Ak

+Ck45Ni,βm2 Nm2Nn1 ΩkNjn1 FτFs,z Ak

+Ck121Rk

Nim2 Nm2Nn1 ΩkNj,αn1 HkFτFs Ak

+Ck13 NiNn1 ΩkNj,αn1 Fτ,zFs Ak

+Ck261Rk

Nim2 Nm2Nn3 ΩkNj,βn3 HkFτFs Ak

+Ck36 NiNn3 ΩkNj,βn3 Fτ,zFs Ak

Kkτsij32 = Ck22

1Rk

Nim2 Nm2Nn2 ΩkNj,βn2 H2

kFτFs Ak

+Ck23 NiNn2 ΩkNj,βn2 HkFτ,zFs Ak

+Ck261Rk

Nim2 Nm2Nn3 ΩkNj,αn3 H2

kFτFs Ak

+Ck36 NiNn3 ΩkNj,αn3 HkFτ,zFs Ak

−Ck451Rk

Ni,αm1 Nm1Nn2 ΩkNjn2 FτFs Ak

(4.40)

−Ck441Rk

Ni,βm2 Nm2Nn2 ΩkNjn2 FτFs Ak

+Ck45Ni,αm1 Nm1Nn2 ΩkNjn2 HkFτFs,z Ak

+Ck44Ni,βm2 Nm2Nn2 ΩkNjn2 HkFτFsAk



Kkτsij33 = Ck22

1R2k

Nim2 Nm2Nn2 ΩkNjn2 H2

kFτFs Ak

+Ck231Rk

Nim2 Nm2Nj ΩkHkFτFs,z Ak

+Ck231Rk

NiNn2 ΩkNjn2 HkFτ,zFs Ak

+Ck33 NiNj ΩkFτ,zFs,z Ak

+Ck55Ni,αm1 Nm1Nn1 ΩkNj,αn1 FτFs Ak

+Ck45Ni,βm2 Nm2Nn1 ΩkNj,αn1 FτFs Ak

+Ck45Ni,αm1 Nm1Nn2 ΩkNj,βn2 FτFs Ak

+Ck44Ni,βm2 Nm2Nn2 ΩkNj,βn2 FτFsAk

where · · ·Ωkindicates

∫Ωk. . . dΩk and · · ·Ak indicates

∫Ak. . . dzk.

4.9 Numerical Examples

The model introduced does not involve an approximation of the geometryof the shell and it accurately describes the curvature of the shell. However,the locking phenomenon is still present. In this work, the model is combinedwith a simple displacement formulation. The CUF, coupled with the MITCmethod, allows us to increase the degree of approximation by increasingthe order of expansion of the displacements in the thickness directionand the number of elements used. Firstly, the reliability of the model isanalyzed. Two classical discriminating test problems are considered: thepinched cylinder with a diaphragm [62], which is the most severe test forboth inextensional bending modes and complex membrane states; and theScordelis–Lo problem [63], which is extremely useful for determining theability of a finite element to accurately solve complex states of a membranestrain.

The pinched shell has been analyzed in [32] and the essential shape isshown in Fig. 4.7. It is simply supported at each end by a rigid diaphragmand singularly loaded by two opposing forces acting at the midpoint ofthe shell. Due to the symmetry of the structure the computations havebeen performed, using a uniform decomposition, on an octave of the shell.The physical data given in Table 4.1 have been assumed. The following



Fig. 4.7. Pinched shell.

Table 4.1. Physical data for pinched shell.

Young’s modulus E 3 × 106 psi = 20.684 × 109 N/m2

Poisson’s ratio ν 0.3Load P 1 lb = 0.154 KgLength L 600 in = 15.24 mRadius R 300 in = 7.62 mThickness h 3 in = 0.0762 m

symmetry conditions and boundary conditions are applied:

vs(α, 0) = 0

us(0, β) = 0

vs(α,Rπ/2) = 0

vs(L/2, β) = ws(L/2, β) = 0

(4.41)

with s = 0, 1, . . . , N .In Table 4.2 the transversal displacement at the loaded point C is

presented for several decompositions [n × n] and different theories. Thehigh-order equivalent single-layer theories in the CUF are indicated withthe acronym ESLN , where N is the order of expansion. The exact solutionis given by Flugge in [70] 1.8248×10−5 in. The table shows that the MITC9



Table 4.2. Pinched shell. Transversaldisplacement w in ×105 .

Theory [4 × 4] [10 × 10] [13 × 13]

Koiter 1.7891 1.8231 1.8253Naghdi 1.7984 1.8364 1.8398ESL1 1.9212 1.9583 1.9617ESL2 1.7805 1.8361 1.8408

ESL3 1.7818 1.8380 1.8428ESL4 1.7818 1.8380 1.8428

Fig. 4.8. Scordelis–Lo roof.

element has good convergence and robustness on increasing the mesh size.According to Reddy [21], the results obtained with high-order theories aregreater than the reference value because Flugge uses a classical shell theory.Indeed, the solution calculated with the Koiter model for mesh [13 × 13]is very close to the exact solution, while the Naghdi model, which takesinto account the shear energy, gives a higher value, as one would expect.The ESL theory with linear expansion (ESL1) produces such a high valuebecause a correction for Poisson locking has been applied (for details ofPoisson locking one can refer to [72]), but for cylindrical shell structuresthis correction causes problems. The remaining theories give almost thesame results and they converge to the same value (1.842 × 10−5 in) byincreasing the order of expansion and the number of elements used.

The second problem (the Scordelis–Lo problem [71]) concerns acylindrical shell known in the literature as a barrel vault, see Fig. 4.8. The



Table 4.3. Physical data for barrel vault.

Young’s modulus E 4.32 × 106 lb/h2 = 20.684 × 109 N/m2

Poisson’s ratio ν 0.0

Load P 90 lb/ft2 = 4309.224 N/m2

Length L 50 ft = 15.24 mRadius R 25 ft = 7.62 mThickness h 0.25 ft = 0.0762 mAngle θ0 2π/9 rad

0.275

0.28

0.285

0.29

0.295

0.3

0.305

0.31

4 5 6 7 8 9 10 11 12 13

w[ft

]

n

exactESL4(s)

ESL4(m+)

Fig. 4.9. Scordelis–Lo problem. Transversal displacement w [ft] at the point B of themid-surface S.

shell is simply supported on diaphragms, is free on its other sides, and isloaded by its own weight P . The physical data given in Table 4.3 have beenassumed. The computations were performed only on a quarter of the shell,using a uniform decomposition.

The exact solution for this problem is given by McNeal and Harderin [73] in terms of the transversal displacement at the point B: 0.3024 ft.In Fig. 4.9, the solution is given for several decompositions [n × n]. Theperformance of the MITC9 element in which a correction for both shearand membrane locking has been applied (m+) is compared with an elementin which only shear locking has been corrected (s). The figure confirms theconclusions for the pinched shell: the results converge to the exact solutionon increasing the number of elements used. Moreover, the figure shows thatfor thin shells (h/R = 0.01) the correction for membrane locking is essentialbecause for coarse meshes the solution (m+) is much higher than the (s)



solution. One can conclude that the MITC9 element is completely lockingfree. The theory used for this analysis is ESL4 but the behavior is the sameas for the other models.

Finally, a multilayered shell was analyzed in order to show thesuperiority of LW and the zig-zag models compared to ESL models. Theorthotropic cylindrical shell studied by Varadan and Bhaskar in [74] isconsidered. The ends are simply supported. The loading is an internalsinusoidal pressure, applied normal to the shell surface, given by:

p+z = p+

z sin(mπα

a

)sin(nπβ

b

), (4.42)

where the wave numbers are m = 1 and n = 8. The amplitude of theload p+

z is assumed to be 1. a is the length and b the circumference of thecylinder.

The cylinder is made up of three equal layers with lamination(90/0/90). Each layer is assumed to be made of a square symmetricunidirectional fibrous composite material with the following properties:

EL/ET = 25

GLT /ET = 0.5

GTT /ET = 0.2

νLT = νTT = 0.25,

(4.43)

where L is the direction parallel to the fibers and T is the transversedirection. The length a of the cylinder is assumed to be 4Rβ, and theradius Rβ , referred to the mid-surface of the whole shell, is 10. Sincethe cylinder is a symmetric structure and it is symmetrically loaded, thecomputations were performed only on an octave of the shell, using a uniformdecomposition.

The solution is given in terms of the transversal displacement w fordifferent values of the thickness ratio Rβ/h, where h is the global thicknessof the cylinder. According to [74], the following dimensionless parameter isused:

w = w10ELh3

p+z R4

β

. (4.44)

The results are presented in Table 4.4 and are compared with the 3D-elasticity solution given by Varadan and Bhaskar in [74]. The transversal



Table 4.4. Varadan and Bhaskar. Dimensionless transversaldisplacement at the max-loading point in z = 0.

Rβ/h 2 4 10 50 100 500

3D 10.1 4.009 1.223 0.5495 0.4715 0.1027ESL4 9.682 3.782 1.1438 0.5456 0.4707 0.1029ESL3 9.664 3.785 1.1439 0.5456 0.4707 0.1029ESL2 8.280 2.971 0.9540 0.5378 0.4692 0.1029

ESL1 8.925 3.015 0.9559 0.5380 0.4696 0.1034Naghdi 8.421 2.872 0.9382 0.5370 0.4688 0.1029Koiter 0.4094 0.4796 0.5205 0.5209 0.4656 0.1029

ESLZ3 9.791 3.987 1.224 0.5493 0.4715 0.1029

ESLZ2 9.596 3.866 1.191 0.5479 0.4712 0.1029ESLZ1 10.228 3.901 1.191 0.5457 0.4694 0.1028

LW4 10.267 4.032 1.225 0.5493 0.4715 0.1029LW3 10.256 4.031 1.225 0.5493 0.4715 0.1029

LW2 9.789 3.971 1.223 0.5493 0.4715 0.1029LW1 9.689 3.874 1.191 0.5477 0.4710 0.1029

displacement is calculated on the mid-surface of the multilayered shell(z = 0), at the max-loading point. An [8 × 8] mesh was used, which issufficient to ensure numerical convergence. Equivalent single-layer (ESLN),zig-zag (ESLZN), and layer-wise (LWN) theories in the CUF are employedfor the analysis. The classical Koiter’s and Naghdi’s models were alsoused for comparison. One can note that the solution obtained with theclassical models is completely wrong, while the ESL theories give a moreaccurate solution by increasing the order of expansion N , especially forhigh-thickness ratios. If one also takes into account the zig-zag effects inthe displacements using Murakami’s zig-zag function (ESLZ models), theresults improve again and the ESLZ3 theory provides approximately the3D solution even for very thick shells. Finally, the table shows that the LWtheories give the best results even when the order of expansion is nothigh (N = 2,3), according to the assertions made in the introduction ofthis chapter about C0

z -requirements. This behavior is particularly visiblefor thick shells (Rβ/h = 2, 4). For very thin shells (Rβ/h = 500) all thetheories converge to the 3D solution and this fact demonstrates once againthe numerical efficiency of the new approach. Note that the LW3 and LW4models give a solution slightly higher than the 3D solution for very thickshells. This is due to a curvature approximation along the thickness, whichcan be easily eliminated by considering the shell to be composed of thinnerfictitious layers with the same properties.



4.10 Conclusions

In this chapter, it has been shown that there are three independent waysof introducing “zig-zag” theories for the analysis of multilayered plates andshells. In particular, it has been established that:

• Lekhnitskii [22] was the first to propose a theory for multilayeredstructures that describes the zig–zag form of a displacement field inthe thickness direction and the interlaminar equilibrium of transversestresses.

• Three different and independent theories are proposed in the literature.Apart from the one by Lekhnitskii [22], the other two approaches werebased on work by Ambartsumian [44, 55] and Reissner [48], respectively.

• Based on the authors’ historical considerations, which are documentedin this chapter, it is suggested that these three approaches are calledthe Lekhnitskii–Ren, Ambartsumian–Whitney–Rath–Das, and Reissner–Murakami–Carrera theories, respectively.

• As far as the Ambartsumian–Whitney–Rath–Das theory is concerned,it should be underlined that other developments, even though derivedindependently by other authors (such as those originated by Yu [75],Chou and Carleone [76], Disciuva [77], Bhaskar and Varadan [78], Choand Parmerter [79], among others), are applications of the AWRD theory.

Even though most of this discussion has been about the so-called ESLMs,these being more relevant for the subject of this chapter, a brief outline ofLWMs was given in Section 4.7.

The author would encourage scientists who are working on theanalysis of multilayer structures to return to the fundamental work byLekhnitskii [22], Ambartsumian [44, 45], and Reissner [48]. There is, infact, a significant amount to learn from these works and probably morecould be done, on the basis of these fundamental works, to obtain a betterunderstanding of the mechanics of multilayered structures. In particular,future developments could be to extend Lekhnitskii’s theory as well as theReissner theorem. This latter, in the authors’ opinion, is the natural toolfor the analysis of multilayered structures.

As a final remarks the authors are clearly aware that this historicalreview may be not complete. The authors are aware that other significantarticles and papers could exist on this subject that have not beenconsidered. However, what has been quoted in this chapter will help to



assign the right credit concerning the contributions and contributors tomultilayered theory.

The final part of this chapter discussed the development of a refinedshell finite element approach based on Carrera’s unified formulation. TheCUF has been coupled to the MITC method to overcome the lockingphenomenon that affects finite element analysis. The reliability of theapproach has been tested by considering classical discriminating problems,such as the pinched cylinder studied in [70] and the Scordelis–Lo problemanalyzed in [71], and the approach has shown good convergence androbustness on growing the mesh size. Moreover, the accuracy of thesolution has been demonstrated to improve by increasing the order ofexpansion of the displacements in the thickness direction. Finally, theorthotropic multilayered cylinder studied by Varadan and Bhaskar in [74]was considered. From this analysis one can conclude that for the studyof multilayered structures it is mandatory to consider zig-zag effects inthe displacements in order to obtain the 3D solution. This is possible byintroducing Murakami’s zig-zag function in the ESL models or by usingthe LW models briefly discussed in this chapter, which allow us to useindependent variables in each layer. This gives the best results.

For clarity, Table 4.5 summarizes the features of the theories cited inthis chapter for the analysis of laminated structures:

• Classical = classical models such as Kirchoff–Love, Reissner–Mindlin, andso on;

• CUF-ESL = equivalent single-layer theories contained in the CUF, inwhich a high order of expansion in the thickness direction is used forboth the in-plane and ransversal displacements;

• L = Lekhnitskii theory;• LR = Lekhnitskii–Ren theory;

Table 4.5. Available theories for laminated structures.

Theory σnM ZZ IC εzz σzz

Classical [1–6]CUF-ESL [9, 27] • •L [22] • • •LR [38–40] • •AWRD [41–46] • •RMC [47–53] • • • • •CUF-LW-D [60] • • •CUF-LW-M [54–60] • • • • •



• AWRD = Ambartsumian–Whitney–Rath–Das theory;• RMC = Reissner–Murakami–Carrera theory, based on ESL approach for

displacement variables;• CUF-LW-D = layer-wise models contained in the CUF, based on the

principle of virtual displacements (PVD);• CUF-LW-M = layer-wise models contained in the CUF, based on the

Reissner mixed variational theorem (RMVT).

The features, considered in the table, are the following:

• σnM = the transverse shear and normal stresses are unknown variableswith the displacements;

• ZZ = zig-zag effects are considered in the displacements;• IC = interlaminar continuity of the transverse stresses is fulfilled;• εzz = thickness stretching effects are considered, εzz = 0;• σzz = Koiter’s recommendation is fulfilled, σzz = 0.

The symbol • indicates that the theory satisfies the corresponding feature.

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Chapter 5

BIFURCATION OF ELASTIC MULTILAYERS

Davide Bigoni, Massimiliano Gei and Sara Roccabianca

Department of Mechanical and Structural EngineeringUniversity of Trento, Via Mesiano 77, I-38123 Trento, Italy

Abstract

The occurrence of a bifurcation during loading of a multilayer sets a limit onstructural deformability, and therefore represents an important factor in thedesign of composites. Since bifurcation is strongly influenced by the contactconditions at the interfaces between the layers, mechanical modelling of these iscrucial. The theory of incremental bifurcation is reviewed for elastic multilayers,when these are subject to a finite strain before bifurcation, correspondingto uniform tension/compression and finite bending. The interlaminar contactis described by introducing linear imperfect interfaces. Results are criticallydiscussed in view of applications and available experiments.

5.1 Introduction

Natural (geological formations, biological materials) and man-made(sandwich panels, submarine coatings, microelectronic devices, ceramiccapacitors) structures are often made up of layers of different materialsglued together, the so-called ‘multilayers’. Large strain in these structuresis achieved (i) as an industrial need (for instance when forming metallicmultilayers [1], ‘wrapping’ of an engineered tissue around a tubular supportto create an artificial blood vessel [2] or bending of multilayer flexible solarcells [3]), (ii) under working conditions (for instance when multilayer filmsare employed for flexible packaging) or (iii) as a natural process (for instanceduring morphogenesis of arteries or geological formations or when the leaf ofa plant bends to trap an insect, to disperse seeds or to resist dehydration).In all these cases, the occurrence of various forms of bifurcation sets limitsto deformation performance. For instance compressive strain is limited

173


174 D. Bigoni, M. Gei and S. Roccabianca

Fig. 5.1. Left: A stiff (30 mm thick, neoprene) layer bonded by two compliant (100 mmthick foam) layers in a rigid-wall, confined compression apparatus (note that separationbetween sample and wall has occurred on the right upper edge of the sample).Centre: Creases on the compressive side of a rubber strip, coated on the tensile side witha 0.4mm thick polyester transparent film, subject to flexure. Right: Bifurcation of a two-layer rubber block under finite bending, evidencing long-wavelength bifurcation modes(the stiff layer, made of natural rubber, is on the compressive side of a neoprene block).

by buckling and subsequent folding (see the example on the left-handside of Fig. 5.1), uniform tensile strain may terminate with shear bandformation and growth, while uniform flexure may lead to the formationof bifurcation modes such as creases and undulations (see the example inthe centre and on the right-hand side of Fig. 5.1). Bifurcation is thereforean important factor in the design of multilayered materials, and so ithas been the focus of a thorough research effort, which was initiated byMaurice A. Biot [4] and continued by many others. In particular, elasticlayered structures deformed in plane strain and subject to a uniform stateof stress have been analysed by Dorris and Nemat-Nasser [5], Steif [6–9],Papamichos, Vardoulakis and Muhlhaus [10], Dowaikh and Ogden [11],Benallal, Billardon and Geymonat [12], Triantafyllidis and Lehner [13],Triantafyllidis and Leroy [14], Shield, Kim and Shield [15], Ogden andSotiropoulos [16] and Steigmann and Ogden [17] as a bifurcation problem ofan isolated layer subject to uniform tension or compression [18–20]. Layeredstructures subject to finite bending have been considered by Roccabianca,Bigoni and Gei [21,22], who found solutions both for the non-uniform stateof stress that develops during flexurea and for the related incremental

aThe solution of finite flexure of an elastic multilayered structure is interesting fromdifferent points of view, since the stress state induced by bending is complex (it mayinvolve for instance the presence of more than one neutral axis) and strongly influencesbifurcation.


Bifurcation of Elastic Multilayers 175

Fig. 5.2. Bifurcation through compression of a finely layered metamorphic rock hasinduced severe folding. This is an example of a so-called ‘accommodation structure’(Trearddur Bay, Holyhead, N. Wales, UK; the coin in the photo is a pound).

bifurcation problem. These findings relied on a generalization of previousresults for plane-strain bending of an elastic block given by Rivlin [23] andon analyses of incremental bifurcations [24–30].

The bifurcation loads and modes are strongly sensitive to the bondingconditions between the layers, which may be perfect (as in the case of therock shown in Fig. 5.2), but often they may involve the possibility of slipand detachments, the so-called ‘delaminations’ (as in the cases shown inFig. 5.3). A simple way to account for this crucial behaviour is to introduceinterfacial laws at the contact between layers. The simplest model of theselaws is linear and can be formulated by assuming that the interface hasnull [31–33] or finite [34–36] thickness. We will limit our attention to zero-thickness linear interfaces, across which the nominal traction incrementremains continuous, but linearly related to the jump in incrementaldisplacement, which is unrestricted. For simplicity, the materials formingthe multilayer are assumed hyperelastic and incompressible, according tothe general framework laid by Biot [4], in which Mooney–Rivlin and Ogdenmaterials [37], as well as the J2-deformation theory of plasticity materials,are particular cases. Therefore, the constitutive laws are broad enough toembrace the behaviour of rubber, plastics, geological materials, but alsoductile metals subject to proportional loading, as they can be representedin terms of the J2-deformation theory.



Fig. 5.3. Bifurcation through compression with detachment of layers: Left: A stiff (1 mmthick) plastic coating has detached from the foam substrate to which it was initially glued.Right: Three layers of foam subject to compression show folding with detachment, clearlyvisible near the edges of the sample.

After the introduction of the constitutive laws for the material andthe interfaces (Section 5.2), we start with the problem of an elasticincompressible structure made of straight layers connected through linearinterfaces and deformed in a state of uniform biaxial stress, for whichincremental bifurcations are sought (Section 5.3). We conclude with thecase of finite bending of a layered elastic block, deformed under plane strain(Section 5.4).

5.2 Notations and Governing Equations

The notations employed in this chapter and the main equations governingequilibrium in finite and incremental elasticity are now briefly reviewed.Let x 0 denote the position of a material point in some stress-free referenceconfiguration B0 of an elastic body. A deformation ξ is applied, mappingpoints of B0 to those of the current configuration B indicated by x =ξ(x 0). We identify its deformation gradient by F , i.e. F = gradξ, and wedefine the right C and the left B Cauchy–Green tensors as C = FTF andB = FFT .

For isotropic incompressible elasticity, the constitutive equations canbe written as a relationship between the Cauchy stress T and B as

T = −πI + α1B + α−1B−1, detB = 1, (5.1)



where π is an arbitrary Lagrangian multiplier representing hydrostaticpressure and α1 and α−1 are coefficients (such that α1 > 0 and α−1 ≤ 0),which may depend on the deformation.

Alternatively, the principal stresses Ti (i = 1, 2, 3), which are alignedwith the Eulerian principal axes, can be obtained in terms of a strain-energyfunction W , which can be viewed as a function of the principal stretchesλi (i = 1, 2, 3). For an incompressible material, these relationships take theform (index i not summed)

Ti = −π + λi∂W (λ1, λ2, λ3)

∂λi, λ1λ2λ3 = 1. (5.2)

Equations (5.1) and (5.2) are linked through the following equations [33]

α1 =1

λ21 − λ2

2

[(T1 − T3)λ2

1

λ21 − λ2

3

− (T2 − T3)λ22

λ22 − λ2

3

],

α−1 =1

λ21 − λ2

2

[T1 − T3

λ21 − λ2

3

− T2 − T3

λ22 − λ2

3

],

(5.3)

which express the coefficients α1 and α−1 in terms of the strain-energyfunction of the material.

In the absence of body forces, equilibrium is expressed in terms of thefirst Piola–Kirchhoff stress tensor S = TF−T (note that for incompressiblematerials detF = 1) as divS = 0, an equation defined on B0.

The loss of uniqueness of plane-strain incremental boundary-valueproblems is investigated, so that the incremental displacements are givenby

u(x ) = ξ(x 0), (5.4)

where, as in the following, a superposed dot is used to denote a first-orderincrement and an updated Lagrangian formulation (where the governingequations are defined in the current configuration B) is adopted. Theincremental counterpart of equilibrium is expressed by divΣ = 0, wherethe updated incremental first Piola–Kirchhoff stress is given by

Σ = SFT , S = TF−T −TLTF−T . (5.5)

The linearized constitutive equation is

Σ = CL − πI , (5.6)



where L = gradu and C is the fourth-order tensor of instantaneous elasticmoduli (possessing the major symmetries). Incompressibility requires thattrL = 0. Since Σ = T − TLT (see Eq. (5.5)), the balance of rotationalmomentum yields Σ12 − Σ21 = T2L12 − T1L21, and a comparison withEq. (5.6) shows that (no sum on indices i and j)

Cijji + Ti = Cjiji (i = j). (5.7)

For a hyperelastic material, the components of C can be defined in termsof the strain-energy function W .

For the plane problem addressed here they depend on two incrementalmoduli [4], namely

µ =λ

2

(λ4 + 1λ4 − 1

dW

dλ

), µ∗ =

λ

4

(dW

dλ+ λ

d2W

dλ2

), (5.8)

where W =W (λ, 1/λ, 1), due to incompressibility. In the following,examples are given for two specific materials both of which are initiallyisotropic elastic solids. One is the Mooney–Rivlin material, for which

W =µ0

2(λ2

1 + λ22 − 2), (5.9)

where λ1 and λ2 are the principal in-plane stretches and µ0 is the shearmodulus in the undeformed configuration. Due to incompressibility λ = λ1

and λ2 = 1/λ, so that

T1 = µ0(λ2 − λ−2) and µ = µ∗ =µ0

2(λ2 + λ−2), (5.10)

where the former is the uniaxial tension law (along axis x1). Notice thatthe ratio between T1 and µ is:

T1

µ=

2(λ2 − λ−2)λ2 + λ−2

, (5.11)

and its value always ranges between −2 (infinite compression) and 2 (infinitetension). The other material analysed in this section is the J2-deformationtheory solid introduced by Hutchinson and Neale [38], for which

W =K

N + 1εN+1, µ =

KεN coth(2ε)2

, µ∗ =KNεN−1

4, (5.12)



where K is a material parameter, N ∈ ]0, 1] is the hardening exponentand ε is the maximum principal logarithmic strain (ε = lnλ). The uniaxialstress-strain law turns out to be T1 = KεN . For this material, the governingequilibrium equations become hyperbolic when [39]

εsb =√N [2εsb coth(2εsb) −N ], (5.13)

a threshold which corresponds to the emergence of shear bands in thedeformed solid.

At the interfaces between layerswe employ the compliant interface modelof Suo, Ortiz and Needleman [31] and Bigoni, Ortiz and Needleman [32] forwhich the jump between incremental stress and incremental displacement canbe written, in components (in a reference system with the axis 1 orthogonal tothe interface), as

Σ11 = S1m(u+m − u−m), Σ21 = S2m(u+

m − u−m); (5.14)

here Sij , the instantaneous stiffness of the interface, is a 2 × 2 constantmatrix whose components have dimension [stress/length]. It is importantto notice that the model depends on the situation, as in the present case, inwhich the stress vector at the interface is null for the fundamental path. Thelimiting cases of a traction-free and perfectly bonded interface correspondto Sij ≡ 0 and to Sij → ∞, respectively. S11 represents the normal stiffnessand S22 the shear stiffness of the interface. S12 and S21 are the couplingbetween the normal and shear responses and, in the applications, will bechosen equal to zero. In (5.14), the terms ()+ and ()− indicate quantities forthe two sides of the interface. In addition to (5.14), continuity of tractionacross the interface has to be imposed, namely

Σ+n = Σ−n . (5.15)

5.3 Uniaxial Tension/Compression of an Elastic Multilayer

In this section bifurcation is analysed for a multilayered elastic structurewith straight interfaces separating orthotropic, incompressible layersdeformed in plane-strain tension and compression. The fundamental pathis characterized by finite, uniform deformations, and the loss of uniquenessin the form of waves of vanishing velocity is considered. The materialsin the layers obey a general hyperelastic incompressible constitutive lawand specific results are presented for Mooney–Rivlin and J2-deformationtheory materials. Different boundary conditions are imposed at the external



surfaces of the multilayered structure, namely, traction free, and bondingto an elastic or undeformable substrate. The possibility of shear-bandinstability, due to the loss of ellipticity as seen in the equilibrium equations,is also analysed.

5.3.1 Equations for a layer

A laminated structure composed of n-layers is considered, subjectto homogeneous large deformation in the fundamental path, so thatequilibrium and compatibility are trivially satisfied. Plane-strain conditionsare assumed with the principal directions of deformation aligned normal andparallel to the layers (Fig. 5.4), with the additional assumption that eachlayer, along the fundamental path, is subjected to a uniaxial stress alongdirection x2. The possibility of bifurcation from the homogeneous state isinvestigated by adopting an updated Lagrangian formulation of the fieldequations where the current configuration is taken as a reference.

The material is a non-linear, orthotropic, incompressible elastic solidand obeys the incremental constitutive equation (5.6). In the absence ofbody forces, incremental equilibrium requires divΣ = 0. In each layer, non-homogeneous incremental solutions are considered in the form

uj = wj(x1)eikx2(j = 1, 2), p = q(x1)eikx2 . (5.16)

The functions wj(x1) and q(x1) will, in general, differ from layer to layer,but the wave number k is taken to be the same for all layers. A chainsubstitution of Eq. (5.16) into the constitutive law (5.6) and, finally, intothe incremental equilibrium equations yields a system of three constant-coefficient ordinary differential equations for the three unknown functions

layer 1

layer n

x1

x2

layer p-1

layer p

layer p+1

Fig. 5.4. Sketch of the laminated structure. Note that a linear interface is present ateach junction between layers.



wj(x1) and q(x1). The solution is

w1(x1) = b1eτ1x1 + b2e

τ2x1 + b3eτ3x1 + b4e

τ4x1 ,

w2(x1) =i

k[τ1b1eτ1x1 + τ2b2e

τ2x1 + τ3b3eτ3x1 + τ4b4e

τ4x1 ],

q(x1) =12[(C2222 − C1111 +M)(τ1b1eτ1x1 + τ2b2e

τ2x1)

+ (C2222 − C1111 −M)(τ3b3eτ3x1 + τ4b4eτ4x1)],

(5.17)

in which M =√L2 − 4C1212C2121 and L = 2C1221+2C1122−C1111−C2222.

Coefficients τs (s = 1, . . . , 4, τ2 = −τ1, τ4 = −τ3) are the eigenvaluesof the equilibrium equations and depend on k, µ, µ∗ and T2. Using thestandard classification of regimes, coefficients τs may be: (i) real numbersin the elliptic imaginary regime, (ii) two complex conjugate pairs in theelliptic complex regime, (iii) purely imaginary numbers in the hyperbolicregime and (iv) two purely imaginary and two real numbers in the parabolicregime. Departure from the elliptic range corresponds to the occurrenceof shear bands. In the following, examples are given for two previouslyintroduced materials, namely, the Mooney–Rivlin and the J2-deformationtheory constitutive models.

Focusing now on the conditions at the interface between layers p andp + 1 in Fig. 5.4, a substitution of wj(x1) and q(x1) into Eqs. (5.14) and(5.15) yields the interfacial conditions in terms of coefficients bps and b(p+1)

s .In matrix form these are

H p−bp = H (p+1)+b(p+1), (5.18)

where vectors bp and b(p+1) collect coefficients bs for the two layers sharingthe interface, while H p− and H (p+1)+ are the interfacial matrices for layer− and +, respectively

H p−=

2

66666664

(eτx1τΓ)p−1 (eτx1τΓ)

p−2 (eτx1τΓ)

p−3 (eτx1τΓ)

p−4

(eτx1∆)p−1 (eτx1∆)p−

2 (eτx1∆)p−3 (eτx1∆)p−

4

(eτx1 [τΓ + Θ])p−1 (eτx1 [τΓ + Θ])

p−2 (eτx1 [τΓ + Θ])

p−3 (eτx1 [τΓ + Θ])

p−4

(eτx1 [ik∆ + Ξ])p−1 (eτx1 [ik∆ + Ξ])p−

2 (eτx1 [ik∆ + Ξ])p−3 (eτx1 [ik∆ + Ξ])p−

4

3

77777775

,



H (p+1)+ =

2

6666666664

(eτx1τΓ)(p+1)+

1 (eτx1τΓ)(p+1)+

2 (eτx1τΓ)(p+1)+

3 (eτx1τΓ)(p+1)+

4

(eτx1∆)(p+1)+

1 (eτx1∆)(p+1)+

2 (eτx1∆)(p+1)+

3 (eτx1∆)(p+1)+

4

(eτx1Θ)(p+1)+

1 (eτx1Θ)(p+1)+

2 (eτx1Θ)(p+1)+

3 (eτx1Θ)(p+1)+

4

(eτx1Ξ)(p+1)+

1 (eτx1Ξ)(p+1)+

2 (eτx1Ξ)(p+1)+

3 (eτx1Ξ)(p+1)+

4

3

7777777775

,

(5.19)

where the entries in the matrices are

(eτx1τΓ)p−s = eτ

ps x

p−1 τps Γps , (eτx1∆)p

−s = eτ

ps x

p−1 ∆p

s ,

(eτx1[τΓ + Θ])p−s = eτ

ps x

p−1 [τps Γps + Θp−

s ],

(eτx1 [ik∆ + Ξ])p−s = eτ

ps x

p−1 [ik∆p

s + Ξp−s ],

(5.20)

and the expressions for Γps, ∆ps, Θp−

s and Ξp−s are, respectively,

Γp1 = Γp2 =[Cp1111

2+Cp2222

2− Cp1122 +

Mp

2

],

Γp3 = Γp4 =[Cp1111

2+Cp2222

2− Cp1122 −

Mp

2

],

∆ps =

[Cp1221 + Cp1212

(τpsk

)2],

Θp−s = Sp−11 + iSp−12

τpsk, Ξp

−s = Sp−21 + iSp−22

τpsk.

(5.21)

Relation (5.18) holds at every interface. To complete the analysis, theboundary conditions at the external surfaces 1+ and n− need to be set.

5.3.1.1 Traction free at the external surface of the multilayer

With reference to the external surface 1+ of the multilayer, vanishing ofthe nominal tractions requires

Σ1+

11 = 0, Σ1+

21 = 0, (5.22)



which can be written in matrix form as

C 1+b1 = 0,

C 1+=

[(eτx1τΓ)1

+

1 (eτx1τΓ)1+

2 (eτx1τΓ)1+

3 (eτx1τΓ)1+

4

(eτx1∆)1+

1 (eτx1∆)1+

2 (eτx1∆)1+

3 (eτx1∆)1+

4

].

(5.23)

A similar result can be obtained for the free boundary n−.

5.3.1.2 Bonding to an elastic half-space at the external surface of themultilayer

When an elastic half-space is coated with a multilayer, the elastic solutionhas to decay within it with depth x1 → +∞ (or x1 → −∞), acondition implying vanishing of the two coefficients bs corresponding to theeigenvalues τs with positive (or negative) real part. Therefore, the interfacialmatrices for half-spaces at the upper (label 1) and lower (label n) externalsurfaces of the multilayer are

H 1−=

(eτx1τΓ)1−

1 (eτx1τΓ)1−

3

(eτx1∆)1−

1 (eτx1∆)1−

3

(eτx1[τΓ + Θ])1−

1 (eτx1 [τΓ + Θ])1−

3

(eτx1 [ik∆ + Ξ])1−

1 (eτx1 [ik∆ + Ξ])1−

3

,

H (n+1)+ =

(eτx1τΓ)(n+1)+

2 (eτx1τΓ)(n+1)+

4

(eτx1∆)(n+1)+

2 (eτx1∆)(n+1)+

4

(eτx1Θ)(n+1)+

2 (eτx1Θ)(n+1)+

4

(eτx1Ξ)(n+1)+

2 (eτx1Ξ)(n+1)+

4

.

(5.24)

5.3.1.3 Bonding to an undeformable substrate at the external surfaceof the multilayer

In the case when the external surface of the multilayer is jointed to a smoothundeformable constraint, the normal component of the velocity and the



tangential nominal traction have to vanish. With reference to the surface1+ these conditions are

v1+

1 = 0, Σ1+

21 = 0, (5.25)

which in matrix form become

C 1+b1 = 0,

C 1+=

eτ

11x

1+1 eτ

12x

1+1 eτ

13x

1+1 eτ

14x

1+1

(eτx1∆)1+

1 (eτx1∆)1+

2 (eτx1∆)1+

3 (eτx1∆)1+

4

. (5.26)

5.3.1.4 Bonding to an undeformable substrate with a compliantinterface at the external surface of the multilayer

In this case, the interfacial constitutive law, Eq. (5.14), is used between theexternal elastic layer of the multilayer and an undeforming substrate, whichbehaves as a rigid constraint. At the external surface 1+, this boundarycondition is

C 1+b1 = 0,

C 1+ =2

64

(eτx1 [τΓ − Θ])1+

1 (eτx1 [τΓ − Θ])1+

2 (eτx1 [τΓ − Θ])1+

3 (eτx1 [τΓ − Θ])1+

4

(eτx1 [ik∆ − Ξ])1+

1 (eτx1 [ik∆ − Ξ])1+

2 (eτx1 [ik∆ − Ξ])1+

3 (eτx1 [ik∆ − Ξ])1+

4

3

75,

(5.27)whilst when imposed at the external surface n−, it becomes

Cn−bn = 0,

Cn−=

2

64

(eτx1 [τΓ + Θ])n−1 (eτx1 [τΓ + Θ])n−

2 (eτx1 [τΓ + Θ])n−3 (eτx1 [τΓ + Θ])n−

4

(eτx1 [ik∆ + Ξ])n−1 (eτx1 [ik∆ + Ξ])n−

2 (eτx1 [ik∆ + Ξ])n−3 (eτx1 [ik∆ + Ξ])n−

4

3

75.

(5.28)

5.3.2 Bifurcation criterion

The set of equations for interfacial and boundary conditions forms a linearsystem, where the coefficients bs of all layers are the unknowns. When elastichalf-spaces are not present, the dimension of the linear system is 4n× 4n.When one elastic half-space or two half-spaces are considered as external



boundary conditions, the order of the linear system becomes 2(2n− 1) or4(n− 1), respectively.

A critical bifurcation condition is attained when a non-trivial solutionis possible. This occurs when the system is singular. In terms of interfacialmatrices the system can be written as:

C 1+: 0

H 1− −H 2+:

.. .. .. .. ..

: H (n−1)− −H n+

0 : −Cn−

b1

b2

..

bn−1

bn

= 0 ⇒ Yb = 0,

(5.29)

and the bifurcation criterion becomes det(Y ) = 0. The system can bereduced using the transfer matrix method [40,41]. In particular, using theinterfacial condition (5.18), the vector bp can be expressed in terms of bp+1

as

bp = (H p−)−1H (p+1)+bp+1, (5.30)

so that, as a consequence, b1 can be given as a function of bn

b1 = Ωbn, Ω = (H 1−)−1H 2+

(H 2−)−1H 3+

. . . (H (n−1)−)−1H n+,

(5.31)

where Ω is the transfer matrix. The linear system is therefore reduced tofour equations in four unknowns:

C 1+b1 = 0

C n−bn = 0

⇒C 1+

Ωbn = 0

Cn−bn = 0

⇒C 1+

Ω

C n−

bn = 0 ⇒ Xbn = 0,

(5.32)

and the bifurcation criterion becomes det(X ) = 0. Finally notice that,when the number of the layers increases, numerical difficulties due toill-conditioning may be encountered using the transfer matrix method. Aninvestigation of these problems, well known in the case of infinitesimalelasticity [42–44], would be interesting, but falls beyond the scope of thischapter.



5.3.3 Results and discussion

The above-described general formulation is applied now to a few simplebifurcation problems. As already remarked, S12 = S21 = 0 is assumed.Moreover, the analysis is limited for simplicity to S11 = S22. Results fordifferent interfacial compliances are calculated in terms of the ratio c/h,where h is the thickness of a representative layer and c is given by

c =µ∗1S11

, (5.33)

where µ∗1 is for layer 1. Parameter c/h is zero for perfect bonding and infinite

when the interface becomes a separation surface between two disjointedlayers.

In Figs. 5.5 and 5.8, the logarithmic strain ε versus kh (where k = k/2πis the inverse of the wavelength of the bifurcation mode) is shown for a J2-deformation theory material. Cauchy stress replaces ε in Fig. 5.6 for theMooney–Rivlin material. A null transversal stress T1 = 0 has been imposedin the fundamental path for all analysed cases. For the J2-material, loss ofellipticity may occur before bifurcation into a diffuse mode. In particular,condition (5.13) gives εsb = 0.3216 for N = 0.1.

0

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0 0.2 0.4 0.6 0.8 1 1.2 1.4

ε

kh

c/h = 0

c/h = 0.1

c/h = 1

c/h = 10Nlay = 0.1, Nsub = 0.4Ksub/Klay = 2

0.3216, loss of ellipticity in the layer

single layer

Fig. 5.5. Bifurcation logarithmic strain for a layer bonded to a half-space and loadedunder plane-strain uniaxial tension for a J2-deformation theory material.



0

0.20

0.40

0.60

0.80

1

1.20

1.40

1.60

1.80

2.00

T/

c/h = 0

c/h = 1

c/h = 0.1

c/h = 100

c/h = 10

a/ b = 3single layer

0 0.2 0.4 0.6 0.8 1 1.2 1.4kh

Fig. 5.6. Bifurcation stress for a periodic multilayer in which the representative cell ismade of three layers jointed through an imperfect interface and externally bonded toa smooth undeformable substrate. The structure, made of Mooney–Rivlin material, isloaded under plane-strain uniaxial compression.

5.3.3.1 Layer bonded to a half-space

The compression case was analysed by Bigoni, Ortiz and Needleman [32]and therefore only the behaviour under tension is investigated here, for aJ2-deformation theory material (Fig. 5.5). The substrate is stiffer than thelayer: Ksub/Klay = 2, Nsub = 0.4, Nlay = 0.1. The effect of the interfacialcompliance gives a strong reduction in the bifurcation critical strain. In theshort wavelength limit (kh → ∞) all curves tend to the surface instabilityvalue for the layer (ε = 0.2524). For sufficiently large wavelength modesshear bands occur in the layer.

5.3.3.2 Periodic multilayered structures

Following [7,13], a periodic multilayered structure can be analysed undercertain restrictions as a bifurcation problem of a representative cell(Fig. 5.7), subject to the boundary conditions of contact with smoothundeformable substrates. We consider the layers joined with the imperfectinterface defined by Eq. (5.14).

In the example, a ratio µa/µb = 3 is assumed for the Mooney–Rivlinmaterial (Fig. 5.6, where T replaces the uniaxial stress T2). In the case of



h

2h

h

2h

h

h

h

h

Representative cell

a

b

a

b

a

b

a

b

Fig. 5.7. Periodic multilayer structure and representative cell.

0

0.10

0.20

0.30

0.40

0.50

0.60

c/h = 0

c/h = 0.1

c/h = 1

c/h = 10

Na = 0.4, Nb = 0.1Ka/Kb = 2

0.3216 loss of ellipticity in b

single layer

0 0.2 0.4 0.6 0.8 1 1.2 1.4kh

ε

Fig. 5.8. Bifurcation logarithmic strain for a periodic multilayer in which therepresentative cell is made of three layers jointed through an imperfect interface andexternally bonded to a smooth undeformable substrate. The representative cell analysedto model a periodic multilayer is made of J2-deformation theory material and is loadedunder plane-strain uniaxial compression.

perfect bonding, bifurcation is impossible when kh < 1.1 and the interfaceinstability (T/µ = −1.9216) is approached when kh → ∞. When theinterfacial compliance increases, the bifurcation load reduces, and the singlelayer solution [18] is recovered in the limit case of complete separation. Thecase of compression for J2-deformation theory material is analysed for thevalues Ka/Kb = 2, Na = 0.4, Nb = 0.1 (Fig. 5.8). In all cases, a portionof the curves falls beyond the loss of ellipticity threshold. This portion, in



which homogeneous deformation is terminated by strain localization in theweaker layer, becomes larger as the interface becomes stiffer.

5.4 Bending of Elastic Multilayers with ImperfectInterfaces

In this section we consider elastic multilayers subject to finite flexure, inwhich the different layers are jointed with imperfect interfaces allowing forfull transmission of normal traction and imperfect transmission of sheartraction, which are linearly related to a possible jump in the tangentialincremental displacement. These conditions are again given by Eqs. (5.14),written now in a cylindrical coordinate system, but with normal stiffnessSr → ∞. Note that such an interface is not ‘activated’ during finitebending of a multilayer (since shear tractions are not present at theinterfaces between different layers), so that the solution for finite flexureis identical both for perfect and imperfect bonding when Sr → ∞, but thebifurcation thresholds are strongly affected by the tangential stiffness of theinterface Sθ.

The solution for pure bending of an elastic layered thick plate (of initial‘global’ dimensions l0×h0, see Fig. 5.9) made up ofN layers jointed throughinterfaces, which allow complete transmission of normal tractions, followsfrom an ‘appropriate assembling’ of solutions relative to the bending of alllayers taken separately, a problem analysed by Rivlin [23]. This solution

h(1

)

h(3)0

l 0

θθ

e r

r

eθ

x1x01

x2

h(2

) h(3

)

ri

h

x02

h(2)0h(1)

0

h0

Referenceconfiguration

Deformedconfiguration

xx0

Fig. 5.9. Sketch of a generic layered elastic thick plate subject to finite bending.



is now briefly explained, with reference to a generic layer (the s-th) of theconsidered multilayer (see [21] for more details).

5.4.1 Kinematics

With reference to Fig. 5.9, the generic layer, denoted by the superscript‘(s)’ (s = 1, . . . , N), is considered in the reference stress-free configurationof a Cartesian coordinate system O

(s)0 x

0(s)1 x

0(s)2 x

0(s)3 , centred at its centroid,

with basis vectors e0i (i = 1, 2, 3), x0(s)

1 ∈ [−h(s)0 /2, h(s)

0 /2], x0(s)2 ∈

[−l0/2, l0/2], and with x0(s)3 denoting the out-of-plane coordinate.

The deformed configuration of each layer is a sector of a cylindrical tubeof semi-angle θ, which can be referred to as a cylindrical coordinate systemO(s)r(s)θ(s)z(s), with basis vectors er, eθ and ez, r(s) ∈ [r(s)i , r

(s)i + h(s)],

θ(s) ∈ [−θ,+θ], and with out-of-plane coordinate z(s) (Fig. 5.9).The deformation is prescribed so that a line at constant x0(s)

1 transformsto a circular arc at constant r(s), while a line at constant x0(s)

2 remainsstraight but inclined at constant θ(s). The out-of-plane deformation is null,so that x0(s)

3 = z(s). The incompressibility constraint means that

r(s)i =

l0h(s)0

2θh(s)− h(s)

2, (5.34)

where h(s) is the current thickness of the circular sector, to be determined.The deformation is described by the functions

r(s) = r(s)(x0(s)1 ), θ(s) = θ(s)(x0(s)

2 ), z(s) = x0(s)3 , (5.35)

so that the deformation gradient takes the form

F (s) =dr(s)

dx0(s)1

er ⊗ e01 + r(s)

dθ(s)

dx0(s)2

eθ ⊗ e02 + ez ⊗ e0

3, (5.36)

and we can therefore identify the principal stretches as

λ(s)r =

dr(s)

dx0(s)1

, λ(s)θ = r(s)

dθ(s)

dx0(s)2

and λ(s)z = 1. (5.37)

Imposition of the incompressibility constraint with Eq. (5.35) yields

r(s) =

√2α(s)

x0(s)1 + β(s), θ(s) = α(s)x

0(s)2 , (5.38)



so that, using Eq. (5.37), the principal stretches can be evaluated as

λ(s)r =

1α(s)r(s)

, λ(s)θ = α(s)r(s) and λ(s)

z = 1, (5.39)

where α(s) and β(s) (Eq. (5.38)) are constants which can be determinedby imposing the boundary conditions, which for the s-th layer, are thefollowing:

• At x0(s)2 = ±l0/2, θ(s) = ±θ, which from Eq. (5.38)2, θ(s) = ±α(s)l0/2,

yield

α(s) =2θl0, (5.40)

where it is worth noting that α(s) is now independent of the index s;• At x0(s)

1 = −h(s)0 /2, r(s) = r

(s)i , which from Eqs. (5.34) and (5.38)1,

r(s)i = r(s)(−h(s)

0 /2), yield

β(s) = r(s)2

i +l0h

(s)0

2θ. (5.41)

The N layers are assumed to be imperfectly bonded to each otheras previously explained, so that continuity of the radial displacements ispreserved, and therefore the interfaces do not affect the bending solution.Therefore, we have

r(s)i = r

(s−1)i + h(s−1) (s = 2, . . . , N), (5.42)

with r(1)i given by r(1)i = l0h(1)0 /(2θh(1))−h(1)/2 (see Eq. (5.34)). Repeated

use of Eqs. (5.34) and (5.42) can be employed to express all thicknessesh(s) (s = 2, . . . , N) in terms of the thickness of the first layer h(1), whichremains the sole kinematical unknown of the problem. In particular, sinceEq. (5.42) is imposed at each of the N − 1 interfaces between layers,all radial coordinates r(s) share the same origin O of a new cylindricalcoordinate system Orθz, common to all deformed layers (Fig. 5.9 on theright); therefore, the index s on the local current coordinates will be omittedin the following, so that the deformed configuration of the multilayer will bedescribed in terms of the global system Orθz. From the kinematic analysis,all the stretches are obtained in the multilayer and represented as

λr =l0

2θr, λθ =

2θrl0

and λz = 1; (5.43)



moreover, the current thickness of the s-th layer h(s) becomes a function ofh(s−1), namely

h(s) = − l0h(s−1)0

2θh(s−1)− h(s−1)

2+

√√√√( l0h(s−1)0

2θh(s−1)+h(s−1)

2

)2

+l0h

(s)0

θ

(s = 2, . . . , N). (5.44)

We may conclude that all current thicknesses are known once the thicknessof the first layer h(1) is known (and this will be determined from the solutionof the boundary-value problem described in the following section).

5.4.2 Stress

Let us analyse now the stress state within the multilayer and consider thatthe Cauchy stress tensor in a generic layer s can be written as

T (s) = T (s)r er ⊗ er + T

(s)θ eθ ⊗ eθ + T (s)

z ez ⊗ ez, (5.45)

where, from the constitutive equations (5.2),

T (s)r = −π(s) + λr

∂W (s)

∂λr, T

(s)θ = −π(s) + λθ

∂W (s)

∂λθ, (5.46)

T (s)z = −π(s) +

∂W (s)

∂λz

∣∣∣∣λz=1

.

Since the stretches only depend on r, the chain rule of differentiation

d ·dr

=∂ ·∂λr

dλrdr

+∂ ·∂λθ

dλθdr

, (5.47)

together with Eqs. (5.46) and the derivatives of stretches with respect to r(calculated from Eq. (5.39)), can be used in the equilibrium equations

∂T(s)r

∂r+T

(s)r − T

(s)θ

r= 0,

∂T(s)θ

∂θ= 0, (5.48)

to obtain the identities

dW (s)

dr= −T

(s)r − T

(s)θ

r=dT

(s)r

dr. (5.49)



Therefore, identifying λθ with λ, for a Mooney–Rivlin material(Eq. (5.9)), we arrive at the expressions

T (s)r = W (s) + γ(s) =

µ(s)0

2

(λ2 +

1λ2

)+ γ(s),

T(s)θ =

(λW (s)

)′+ γ(s) =

µ(s)0

2

(3λ2 − 1

λ2

)+ γ(s),

(5.50)

where W (s)(λ) = W (s)(1/λ, λ, 1), γ(s) is an integration constant and ()′

denotes differentiation with respect to the stretch λ. The component T (s)z

can be inferred from Eq. (5.46).Constants γ(s) (s = 1, . . . , N) and thickness h(1) can be calculated

by imposing: (i) continuity of tractions at interfaces between layers (N −1 equations) and (ii) traction-free boundary conditions at the externalboundaries of the multilayer (two equations). Considering N layers, thetraction continuity at the interfaces is

T (s−1)r (r(s−1)

i + h(s−1)) = T (s)r (r(s)i ) (s = 2, . . . , N), (5.51)

while null traction at the external surfaces of the multilayer yields

T (1)r (r(1)i ) = 0, T (N)

r (r(N)i + h(N)) = 0. (5.52)

Therefore, γ(N) can be calculated from Eq. (5.52)2 and specified for aMooney–Rivlin strain-energy function as

γ(N) = −µ(N)0

2

[(αr(N)

e )2 +1

(αr(N)e )2

], (5.53)

while employing Eq. (5.51), the following recursive formulae are obtained

γ(s−1) =µ

(s)0 − µ

(s−1)0

2

[(αr(s−1)

e )2 +1

(αr(s−1)e )2

]+ γ(s) (s = 2, . . . , N),

(5.54)

where r(s)i = r(s−1)e (see Eq. (5.42)).



Considering now Eq. (5.52)1 and evaluating γ(1) from Eq. (5.54) writtenfor s = 2, we obtain an implicit expression to be solved for h(1)

µ(1)0

2

[(αr(1)i )2 +

1

(αr(1)i )2

]+µ

(2)0 − µ

(1)0

2

[(αr(1)e )2 +

1

(αr(1)e )2

]+ γ(2) = 0,

(5.55)

in which r(1)i , r(1)e and γ(2) are all functions of h(1), through Eqs. (5.44) and(5.54).

The obtained solution allows determination of the complex stress andstrain fields within a thick, multilayered plate, when subject to finitebending. For instance, we show in Fig. 5.10 the deformed geometriesfor a four-layer structure (with l0/h0 = 1, thickness ratios: h(b)

0 /h(a)0 = 2,

h(c)0 /h

(a)0 =3 and h

(d)0 /h

(a)0 =4 and stiffness ratios: µ(a)/µ(d) =27,

µ(b)/µ(d) = 9 and µ(c)/µ(d) = 3), together with graphs of the dimensionlessCauchy principal stresses Tr(r)/µ(a) (the transverse component) andTθ(r)/µ(a) (the circumferential component).

Note that the transverse stress is always compressive, while thedistribution of Tθ(r) strongly depends on the stiffness of the layer underconsideration and always has a null resultant, so that it is equivalent to thebending moment loading the plate. For all cases, the neutral axis (the linecorresponding to vanishing circumferential stress) is drawn. Note that in thesketch on the right two neutral axes are visible, an interesting feature which

l /h =1h /h =20 0

0 0(b) (a)

h /h =3h /h =4

0 0

0 0

(c) (a)

(d) (a)

T / , T /a) a)r

1T / (a)

T /r(a)

material (a)material (b)material (c)material (d)

(a) (d)/ =27(b) (d)

(c) (d)/ =9/ =3

T / (a)

T / (a) T /r(a)

+ -

-

/4= rad = /2 rad = rad

T /r(a)

neutral axesneutral axes

Fig. 5.10. Undeformed and deformed shapes and internal stress states for finite bending

of a Mooney–Rivlin four-layer structure with l0/h0 = 1, thickness ratios: h(b)0 /h

(a)0 = 2,

h(c)0 /h

(a)0 = 3 and h

(d)0 /h

(a)0 = 4 and stiffness ratios: µ(a)/µ(d) = 27, µ(b)/µ(d) = 9 and

µ(c)/µ(d) = 3. Dashed lines represent the neutral axes. Note that two neutral axes arevisible in the figure on the right.



may occur, depending on the geometry and on the properties of layers, fora multilayered plate under finite bending (see [22] for details).

5.4.3 Incremental bifurcations superimposed on finite

bending of an elastic multilayered structure

We address in this section the plane-strain incremental bifurcation problemof the multilayered thick plate subject to the previously solved (Section 5.4)finite bending deformation. For simplicity we consider the problem of abilayered structure made of Mooney–Rivlin material, but considerationof additional layers or different constitutive equations is straightforward.The incremental equilibrium is again expressed in terms of the updatedincremental first Piola–Kirchhoff stress Σ by

divΣ = 0, (5.56)

where Σ is given by Eq. (5.5) in terms of the gradient of incrementaldisplacements L, which in cylindrical components can be written as

L = ur,rer ⊗ er +ur,θ − uθ

rer ⊗ eθ + uθ,reθ ⊗ er +

ur + uθ,θr

eθ ⊗ eθ,

(5.57)

and is subject to the constraint trL = 0 (incremental incompressibility),namely,

rur,r + ur + uθ,θ = 0. (5.58)

The linearized constitutive equation is given by Eq. (5.6) and for aMooney–Rivlin material, the components of C can be written as functions oftwo incremental moduli, denoted by µ and µ∗, Eq. (5.10), and depending onthe value of the current strain. In cylindrical coordinates, the non-vanishingcomponents of C are [18,45]

Crrrr = Cθθθθ = 2µ∗ + p, Cθrθr = µ− Γ,

Crθrθ = µ+ Γ, Crθθr = Cθrrθ = µ+ p,(5.59)

where Γ and p are given by

Γ =Tθ − Tr

2, and p = −Tθ + Tr

2, (5.60)



describing the state of prestress. Therefore, the incremental constitutiveequations (5.6) take, for each layer, the explicit form

Σrr = −π + (2µ∗ + p)ur,r,

Σθθ = −π + (2µ∗ + p)ur + uθ,θ

r,

Σrθ = (µ+ Γ)ur,θ − uθ

r+ (µ+ p)uθ,r,

Σθr = (µ+ p)ur,θ − uθ

r+ (µ− Γ)uθ,r.

(5.61)

We seek bifurcations represented by an incremental displacement fieldin the form

ur(r, θ) = f(r) cosnθ,

uθ(r, θ) = g(r) sinnθ,

π(r, θ) = k(r) cosnθ,

(5.62)

so that Eq. (5.58) can be reformulated as

g = − (f + rf ′)n

, (5.63)

and the incremental equilibrium equations as

k′ = Df ′′ +(C,r +D,r +

C + 2Dr

)f ′ +

E(1 − n2)r2

f,

k =r2C

n2f ′′′ +

F + 3Cn2

rf ′′ +(F

n2−D

)f ′ − 1 − n2

n2

F

rf,

(5.64)

where coefficients C, D, E and F can be expressed (for a Mooney–Rivlinmaterial) as

C = µ− Γ =µ0

λ2, D = 2µ∗ − µ =

µ0

2λ4 + 1λ2

,

E = µ+ Γ = µ0λ2, F = rC,r + C = −µ0

λ2.

(5.65)

By differentiating Eq. (5.64)2 with respect to r and substituting theresult into Eq. (5.64)1, a single differential equation in terms of f(r) is



obtained

r4f ′′′′ + 2r3f ′′′ − (3 + n2(λ4 + 1))r2f ′′

+(3 + n2(1 − 3λ4))rf ′ + (n2 − 1)(3 + n2λ4)f = 0,(5.66)

defining the function f(r) within a generic layer. Once f(r) is knownfor each layer, the other functions, g(r) and k(r), can be calculated byemploying Eqs. (5.63) and (5.64)2, respectively, so that function f(r)becomes the primary unknown.

The differential Eq. (5.66) for the functions f (s)(r) (s = 1, . . . , N) iscomplemented by the following boundary conditions:

• Continuity of incremental tractions at interfaces:

Σ(s)rr

∣∣∣r=r

(s)e

= Σ(s+1)rr

∣∣∣r=r

(s+1)i

, Σ(s)θr

∣∣∣r=r

(s)e

= Σ(s+1)θr

∣∣∣r=r

(s+1)i

; (5.67)

• Continuity of the radial component of the incremental displacement atthe interfaces:

u(s)r

∣∣∣r=r

(s)e

= u(s+1)r

∣∣∣r=r

(s+1)i

; (5.68)

• Imperfect ‘shear-type’ interface (obtained from Eq. (5.14) taking Sr →∞)

Σ(s)θr

∣∣∣r=r

(s)e

= Sθ(u

(s+1)+

θ − u(s)−

θ

), (5.69)

where Sθ is a positive shear stiffness coefficient, so that perfect bondingis recovered in the limit Sθ → ∞;

• For dead-load tractions on the external surfaces, the boundary conditionsat r = r

(1)i and r = r

(N)e are

Σ(1),(N)rr

∣∣∣r=r

(1)i ,r

(N)e

= 0, Σ(1),(N)θr

∣∣∣r=r

(1)i ,r

(N)e

= 0. (5.70)

On the boundaries θ = ±θ we require that shear stresses andincremental normal displacements vanish

Σ(s)rθ

∣∣∣θ=±θ

= 0, u(s)θ

∣∣∣θ=±θ

= 0, (5.71)



a condition which is achieved if sinnθ = 0 (see Eq. (5.62)) or equivalentlyusing Eq. (5.40), if

n =2mπαl0

(m ∈ N). (5.72)

5.4.4 An example: bifurcation of a bilayer

The critical angle θcr and the critical stretch λcr (on the compressive sideof the specimen) for a bilayer at bifurcation are shown in Fig. 5.11 asfunctions of the aspect ratio l0/h0 (the unloaded height of the specimen isl0 and global thickness is h0, see Fig. 5.9), for the thickness and stiffnessratios h(lay)

0 /h(coat)0 = 10 and µ(coat)/µ(lay) = 20, respectively. In the figure,

bifurcation curves are shown for different values of the integer parameterm which, through Eq. (5.72), defines the circumferential wave number n.Obviously, for a given value of l0/h0 the bifurcation threshold is set by thevalue of m providing the minimum (or maximum) value of the critical angle(or stretch).

In the same figure, the threshold for surface instability of the ‘soft’layer material (λsurf ≈ 0.545 [4]) is also shown. It can be deduced fromthe figure that a diffuse mode, which set the bifurcation thresholds, alwaysexists before surface instability for each aspect ratio l0/h0. It is importantto observe that the occurrence of the critical diffuse mode is very closeto the surface instability when the coating is located on the tensile sideof the specimen (Fig. 5.11). The critical angle at bifurcation is given inFig. 5.12 as a function of the aspect ratio l0/h0, for two values of thecoating thickness, h(lay)

0 /h(coat)0 = 10, 20, and when the coating layer is

on the tensile side. In the same figure, results for the uncoated layer arealso shown for comparison.

It is evident from the figures that the bifurcation solution for asingle layer is approximated by a straight line, so that we can define anapproximate solution

θcr = 0.712 l0/h0, (5.73)

which is very useful for applications. We may also notice that a linearrelation between θcr and l0/h0 is found for the bilayer (Figs. 5.11, 5.12and 5.16); however, the inclination of such lines depends on the elastic andthickness contrasts between the layer and coating, so that it is difficult toobtain a simple formula like Eq. (5.73) in this case.

The effects of an imperfect interface on bifurcations of a layered blockunder bending have never been analysed, so we limit the discussion to a



0 1 2 3 4 5 6 7 8 9 100

0.5

1

1.5

2

2.5

3

θ

3.5

0 1 2 3 4 5 6 7 8 9 100.54

0.545

0.55

0.555

0.56

0.565

0.57

cr

m=1

2

3

4

5

6

7

8

9

10

m=1 2 3 4 5 6 7 8 9 10

l /h0 0

l /h0 0

cr

annular configuration

surface instability

surfa

ceinsta

bility

h / =100(lay) h0

coated layer:(coat) (lay)/ =20

(coat)

θ

Fig. 5.11. Critical angle θcr and critical stretch λcr (evaluated at the internal boundary,

r = r(1)i ) versus aspect ratio l0/h0 of a Mooney–Rivlin coated bilayer subject to bending

with h(lay)0 /h

(coat)0 = 10 and µ(coat)/µ(lay) = 20. The coating is located on the tensile

side of the structure. In both plots, a small circle denotes a transition between twodifferent integer values of m (the parameter which sets the circumferential wave number).



(coat) (lay)/ =20coated layer:

h / =200(lay) (coat)h0

h / =100(lay) (coat)h0

0 2 4 6 8 10


0

0.5

1

1.5

2

2.5

3

θ

θ

3.5

1 3 5 7 9

uncoatedlayer

m=9 m=10m=10

8

9

7

6

5

3

2

4

1

cr

l /h0 0

Fig. 5.12. Comparison between the critical angle θcr at bifurcation versus aspect ratiol0/h0 of two Mooney–Rivlin coated bilayers subject to bending with the coating on

the tensile side of the structure, with µ(coat)/µ(lay) = 20 and h(lay)0 /h

(coat)0 = 10

and 20, respectively. On each curve, a small symbol denotes a transition between twodifferent integer values of m (the parameter which sets the circumferential wave number).Bifurcation angles for a single, uncoated layer are also shown.

simple situation, while a more detailed presentation will be the subjectof future research. The simple example analysed in Figs. 5.13 and 5.14pertains to a uniform elastic block divided into two identical layers throughan imperfect interface of stiffness Sθ and Sr → ∞. Results presentedin Figs. 5.13 and 5.14 are in terms of the critical bending angle forbifurcation θcr versus the initial ‘global’ aspect ratio l0/h0, as a functionof the dimensionless interfacial stiffness parameter Sθh0/µ0. Results forseveral values of this parameter (ranging between 0 and 1000) are shownin Fig. 5.13, while those results for only two (namely, 0 and 10) are shownin Fig. 5.14.

Only the smallest circumferential number m = 1 was considered forFig. 5.13, so that θ is not always ‘critical’, since for low values of the aspectratio l0/h0 the onset of instability is associated with higher values of m.

A general conclusion that can be drawn from the results shown inFigs. 5.13 and 5.14 is that the bifurcation threshold strongly depends on thedimensionless parameter Sθh0/µ0, which yields an important decrease in



0 5 10 150

0.5

1

1.5

2

2.5

3

θ3.5

cr

l /h0 0

annular configurationm=1

S h /0 0

r = (u -u )S + -

perf

ect b

ondi

ng

0

5

10

25501000

θ

θ θ θ θ

θ

Fig. 5.13. Critical angle θcr at bifurcation (m = 1) versus the ‘global’ aspect ratiol0/h0 for two Mooney–Rivlin identical layers subjected to flexure and jointed througha ‘shear-type’ imperfect interface of dimensionless stiffness Sθh0/µ0. Perfect bonding

corresponds to Sθh0/µ0 → ∞.

the bifurcation angles with respect to the perfectly bonded case, approachedwhen Sθh0/µ0 → ∞.

5.4.5 Experiments on coated and uncoated rubber blocks

under bending

To substantiate the theoretical results for the bifurcation of layeredstructures subject to finite bending, Roccabianca, Bigoni and Gei [21,22]designed and performed experiments, similar to those initiated by Gentand Cho [46,47]. In these experiments, a finite flexure was imposed on



m=1

1

2

3

l /h0 0

annular configurationsu

rface

inst

abili

ty

cr

0 5 100

0.5

1

1.5

2

2.5

3

3.5

r = (u -u )S + -


surfa

cein

stab

ility

cr

l /h0 00 5 100

0.5

1

1.5

2

2.5

3

3.5

m=1

1

2

3

4

5

2

3 2

S h / =100 0S h / =00 0θ

θ

θ θ θ θ

θ θθ

Fig. 5.14. Bifurcation angles θcr versus the ‘global’ aspect ratio l0/h0 for two Mooney–Rivlin identical layers subjected to flexure and jointed through a ‘shear-type’ imperfectinterface as in Fig. 5.13. Left: Sθh0/µ0 = 0; right: Sθh0/µ0 = 10. The small numbersnear a curve denote the value of the circumferential number m. The lower boundary ofthe grey region is the bifurcation threshold for perfect bonding.

screw loading devicerubber block

scale for bending angle measure

post-bifurcation pattern

onset of bifurcation

Fig. 5.15. Left: Device used to impose a finite bending (of semi-angle θ equal to 35in the photo). Right: Bifurcation of a 20 × 4 × 100 mm3 rubber block, coated with twopolyester 0.2mm thick films on the tensile side. Top: Onset of bifurcation (θ = 40,creases become visible). Lower: Post-bifurcation pattern (θ = 50, creases invade thewhole specimen).



(coat) (lay)/ =500

0 2 4 6 8 10 12 14 16 180

0.5

1

1.5

2

2.5

3

θ3.5

cr annular configuration

l /h0 0

uncoatedlayer

experiment:experiment: uncoated

h /h =20(lay) (coat)0 0

experiment: h /h =10(lay) (coat)0 0

1

2

3

5

coated layer:

h /h =20(lay) (coat)0 0

h /h =10(lay) (coat)0 0

1

m=12m=11m=9

8

7

5

4

3

2

1

4

2

3

4

5

Fig. 5.16. Experimental results versus theoretical predictions for the bifurcationopening semi-angle θcr of uncoated and coated rubber strips subject to finite bending,versus the aspect ratio l0/h0 of the undeformed configuration. The shear moduli ratioµ(coat)/µ(lay) of the coated layers has been taken equal to 500, while two thickness ratios

h(lay)0 /h

(coat)0 equal to 20 and 10 were considered. The critical theoretical configurations

(for h(lay)0 /h

(coat)0 = 20) corresponding to bifurcation points Ωi (i = 1, . . . , 5) are

sketched the right on the figure.

uncoated and coated elastic blocks (made of natural rubber), glued to twometallic platelets, which were forced to bend by a simple screw loadingdevice (Fig. 5.15 left; see also [21]).

Different coatings and blocks were tested. Bending results for threeuncoated rubber strips (made of natural rubber with a ground-state shearmodulus µ(lay) ∼= 1 N/mm2) and ten coated strips of the same dimensionswith two types of coating (both made of a polyester transparent filmhaving µ(coat) ∼= 500N/mm2 but with different thicknesses), all situatedon the tensile side of the structure, are shown in Fig. 5.16. At a certainstage of finite bending, namely at a certain bending semi-angle θcr,creases can be detected on the surface of the sample, as in Fig. 5.15



on the right (and in Fig. 5.1 in the centre). This has been identifiedwith the appearance of small wavelength bifurcations and compared withtheoretical predictions for uncoated layers and for a layer with a stiffcoating on the tensile side of the specimen, in terms of the critical bendingsemi-angle (θcr) at bifurcation versus the aspect ratio of the samples,Fig. 5.16. Experiments demonstrate that the trend predicted by thetheory is qualitatively very well followed, while quantitatively experimentalvalues for bifurcation angles are often a bit lower than the theoreticalpredictions, a result consistent with observations by Gent and Cho [46]. Thefact that experimental results substantiate theoretical predictions allowsus to conclude that bifurcation theory can be successfully employed topredict the deformational capabilities of a composite plate subject to finitebending.

In the case when the coating is applied to the compressed side,long wavelengths become visible in the experiment, as qualitativelydemonstrated in Fig. 5.1 on the right (see also [22]), while quantitativeevaluation still requires further investigation.

5.5 Conclusions

The load-carrying capacity of laminated structures is often limited by theoccurrence of various instabilities at different structural levels. Amongthese, delamination is the best known. Accordingly, there is a largeliterature where bifurcations and instabilities of multilayers are analysedfrom a variety of perspectives.

We have shown that the theory of incremental bifurcation of prestressedelastic solids, in which each layer is treated as an elastic non-linearcontinuum and plate-like approximations are not introduced, can beeffectively used to find threshold loads for delamination involving complexbifurcation modes. The presented framework is broad enough to includeseveral constitutive laws modelling the mechanical response of (i) theinterfaces (for instance spring-like or shear-type junctions) and (ii) thelayers (for instance Mooney–Rivlin and J2-deformation theory of plasticitymaterials).

The bifurcation analysis, carried out for different deformation pathsincluding finite tension/compression of straight layers and finite bending,reveals a number of different instabilities that may occur in a multilayer,including Euler buckling, necking, surface instability and various wave-like



modes. The occurrence of one or another form of instability is stronglyrelated to the interfacial conditions between the layers.

Acknowledgements

D.B. gratefully acknowledges the support from the European UnionSeventh Framework Programme under contract no. PIAP-GA-2011-286110-INTERCER2. M.G. and S.R. gratefully acknowledge the support of ItalianMinistry of Education, University and Research under PRIN grant no.2009XWLFKW, ‘Multi-scale modelling of materials and structures’.

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Chapter 6

PROPAGATION OF RAYLEIGH WAVESIN ANISOTROPIC MEDIA AND AN INVERSE

PROBLEM IN THE CHARACTERIZATIONOF INITIAL STRESS

Kazumi Tanuma∗ and Chi-Sing Man†∗Department of Mathematics, Faculty of Science and Technology

Gunma University, Kiryu 376-8515, Japan

†Department of Mathematics, University of Kentucky,Lexington, KY 40506-0027, USA

Abstract

Composite materials behave in the long wavelength limit as if they arehomogeneous but, more often than not, carry strong anisotropy. We review theStroh formalism for dynamic elasticity and apply this formalism to derive aperturbation formula for the phase velocities of Rayleigh waves that propagatealong the free surface of a prestressed half-space where, when unstressed, theprincipal part of the elasticity tensor is orthotropic or transversely isotropic.An objective of the present study is to examine the possibility of andthe problems and issues regarding the determination of the prestress in anotherwise orthotropic or transversely isotropic composite material by boundarymeasurements of the phase velocities of Rayleigh waves.

6.1 Introduction

In the long wavelength limit composite materials behave as if they arehomogeneous but, more often than not, carry strong anisotropy (see forexample Chapter 17 of [1], [2], Chapter 11 of [3], and [4]).a The Stroh

aHere we cite one concrete example. Consider wave propagation in a composite thatconsists of an elastic homogeneous matrix reinforced by a random distribution of alignedcontinuous cylindrical elastic fibers. It is well established that such a composite can bemodeled as a transversely isotropic homogeneous elastic medium when the wavelength

209


210 K. Tanuma and C.-S. Man

formalism is a powerful and elegant mathematical method developed forthe analysis of problems of two-dimensional deformations in anisotropicelasticity [6–12]. It reveals simple structures hidden in the equations ofmotion (or equilibrium) for such problems and provides a systematicapproach for tackling these equations. In particular the Stroh formalismis an effective tool in the study of Rayleigh waves, long since a topic ofutmost importance in non-destructive evaluation, seismology, and materialsscience.

In this chapter we will use the Stroh formalism to derive a perturbationformula for the phase velocities of Rayleigh waves that propagate alongvarious directions on the free surface of a prestressed half-space where,when unstressed, the principal part of the elasticity tensor is orthotropicor transversely isotropic. The effects of the initial stress are grouped underthe perturbative part of the constitutive equation. In our previous study[13] we derived a perturbation formula for the phase velocity of Rayleigh

waves that propagate along the free surface of a prestressed half-space;that formula expresses the shift of the phase velocity of the Rayleigh wavesfrom its value for the comparative unstressed and isotropic medium. Unlikethat work, here we allow the base material to be orthotropic or transverselyisotropic (and the anisotropy may be strong), which is typical of directionalfiber-reinforced composite materials (see for example [1,2,4,14], Chapter 15of [15], and [16]). An objective of the present study is to examine thepossibility of and the problems and issues regarding the determinationof the prestress in an otherwise orthotropic or transversely isotropiccomposite material by boundary measurements of the phase velocities ofRayleigh waves.

The present chapter reviews the essence of the Stroh formalism fordynamic elasticity, summarizes an approach based on this formalismto derive a first-order perturbation formula for the phase velocity ofRayleigh waves, and outlines what information on the prestress of abody of a composite material can, in principle, be delivered by boundary

of the propagating wave is much longer than the fiber diameter. See for example thetheoretical and experimental study of Datta and Ledbetter [5], where they calculated

and measured the macroscopic elastic constants of a composite that consisted of analuminum 6061 alloy matrix reinforced by 0.14-mm-diameter boron fibers. They usedtwo methods in their measurements of the elastic constants. In the resonance methodthey used oscillations of frequencies that ranged from 30 to 50 kHz. In the ultrasonic-velocity method they used waves at frequencies near 10 MHz.


Propagation of Rayleigh Waves in Anisotropic Media 211

measurements of Rayleigh-wave phase velocities. One objective of thischapter is to introduce the reader to the full paper [17].

6.2 Basic Elasticity in Anisotropic Materialswith Initial Stress

Here we consider a macroscopically homogeneous, anisotropic, prestressedelastic medium and small elastic motions superimposed on it. Let R denotethe set of real numbers. Suppose that the medium occupies a region B inR

3 with a smooth boundary. We take the given initial configuration of Bas the reference configuration for the description of the elastic motions.

Let E = E(u) =(εij)i,j=1,2,3

be the infinitesimal strain tensor

εij =12

(∂ui∂xj

+∂uj∂xi

), (6.1)

where u = u(x) = (u1, u2, u3) is the displacement at the place x pertainingto the superimposed small elastic motion and (x1, x2, x3) are the Cartesiancoordinates of x.

In the theoretical context of linear elasticity with initial stress (see [18]and the references therein), the constitutive equation can be written as

S = T

+H T

+ L[E], (6.2)

where S = (Sij)i,j=1,2,3 is the first Piola–Kirchhoff stress, T

=(Tij

)i,j=1,2,3

the initial stress, H = (∂ui/∂xj)i,j=1,2,3 the displacement gradient, and L

the incremental elasticity tensor. The initial stress T

is symmetric

Tij = T

ji, i, j = 1, 2, 3 (6.3)

and the incremental elasticity tensor L, regarded as a fourth-order tensorthat maps symmetric tensors E onto symmetric tensors, has major andminor symmetries

Lijkl = Lklij = Ljikl, i, j, k, l = 1, 2, 3. (6.4)

Then we can rewrite (6.2) componentwise as

Sij = Tij +

3∑k,l=1

(Tjl δik + Lijkl)

∂uk∂xl

, (6.5)



where δik is Kronecker’s delta. Both L and T

depend implicitly on theformative history of the material body in the given initial configuration,but we will keep this dependence implicit and simply take L and T

as given

fourth-order and second-order tensors, respectively.Note that when T

= 0 the constitutive equation (6.2) is reduced to the

generalized Hooke’s law

S = C[E] or Sij =3∑

k,l=1

Cijkl εkl,

where L now collapses to the classical elasticity tensor C. Furthermore,we say that the elastic material is isotropic if the elasticity tensor C =(Cijkl

)1≤i,j,k,≤3

satisfies

Cijkl =3∑

p,q,r,s=1

QipQjq Qkr QlsCpqrs, i, j, k, l = 1, 2, 3 (6.6)

for any orthogonal tensor Q = (Qij)i,j=1,2,3. For isotropic materials, thecomponents Cijkl(i, j, k, l = 1, 2, 3) can be written as

Cijkl = λ δijδkl + µ(δikδjl + δilδkj)

with the Lame constants λ and µ. We say that the elastic material isanisotropic if it is not isotropic.

We will study dynamic deformations of the elastic medium in B. Lett denote the time and ρ the uniform mass density of the material in thegiven initial (prestressed) configuration. We assume that the initial stress

T

satisfies the equation of equilibrium with the body force (per unit mass)b = b(x) = (b1, b2, b3):

3∑j=1

∂

∂xjTij + ρ bi = 0, i = 1, 2, 3.

Substitution of (6.5) into the equation of motion with the body force b,namely

3∑j=1

∂

∂xjSij + ρ bi = ρ

∂2

∂t2ui, i = 1, 2, 3,



leads to the elastic wave equation written in terms of the displacement

3∑j=1

∂

∂xj

(Bijkl

∂uk∂xl

)= ρ

∂2

∂t2ui, i = 1, 2, 3, (6.7)

where

Bijkl = δikTjl + Lijkl (6.8)

are the effective elastic coefficients which have the major symmetry

Bijkl = Bklij , i, j, k, l = 1, 2, 3 (6.9)

but do not have the minor symmetries because of the term δikTjl.

Hereafter in this chapter we assume the strong ellipticity condition:

The matrix( 3∑j,l=1

Bijkl ξjξl

)i↓k→1,2,3

is positive definite for

any non-zero vector ξ = (ξ1, ξ2, ξ3) ∈ R3.b (6.10)

This condition is guaranteed if the incremental elasticity tensor L

satisfies the strong convexity condition:

3∑i,j,k,l=1

Lijkl εij εkl > 0 for any non-zero 3 × 3

real symmetric matrix (εij)

and if the initial stress T

is sufficiently small.We have treated T

and L as if they are independent. In fact they are

generally related through their mutual dependence on the formative historyof the material body in question. In particular L generally depends on T

.

There are models for the dependency of L on T. The simplest case of which

is found when L is an isotropic bilinear function of T

and E relative to the

bThe subscript i ↓ k → 1, 2, 3 means that i and k are the row and column number,respectively, and both numbers run from 1 to 3.



initial configuration. In this case L is given by

L(T)[E] = λ(trE)I + 2µE + β1(trE)(trT

)I + β2(trT

)E

+ β3

((trE)T

+ (trET

)I)

+ β4(ET

+ TE), (6.11)

where λ and µ are the Lame constants and βi (i = 1, ..., 4) are materialconstants (see [19]). For instance classical acoustoelastic theory, where theinitial stress is caused by a deformation of an isotropic elastic materialfrom a stress-free natural configuration, in effect uses (see Section 9 of[20]) the constitutive equation (6.2) with L given by (6.11) and the fourβi coefficients expressed in terms of three third-order elastic constants andthe Lame constants of the isotropic elastic material.

We take as the sole reference configuration the initial configuration,which already has the initial stress and material anisotropy, and we makeno presumption of their origin. Thus our study begins with the constitutiverelation (6.2) and our theory is applicable so long as infinitesimal motionssuperimposed on this initial configuration are elastic. We assume thesymmetries (6.3) for T

and (6.4) for L but do not assume any other material

symmetry.In Section 6.3 we give a brief review of the Stroh formalism for elastic

wave equations. This formalism, which reveals simple structures hidden inthe equations of anisotropic elasticity, is powerful and elegant. One of itsapplications is a systematic study of Rayleigh waves in anisotropic materialspresented in Sections 6.4 and 6.5.

6.3 The Stroh Formalism

In this section we review the Stroh formalism for elastic wave equations ina manner as self-contained, readable, and brief as possible. Rayleigh wavespropagate along free surfaces, i.e. these waves produce no traction at theboundary. However, the formalism in this section applies to surface wavesthat do not necessarily satisfy a zero-traction condition at the boundary.In the subsequent sections we will restrict our attention to Rayleigh waves.

All the theorems, lemmas, etc. in Sections 6.3 and 6.4 can be foundin [8–12]. In the following, except for several important results, we shallrefrain from referring to the literature at each point.

We assume that the elastic medium and the initial stress arehomogeneous, and hence the effective elastic coefficients

(Bijkl

)i,j,k,l=1,2,3



are independent of x. In this case (6.7) becomes

3∑j,k,l=1

Bijkl∂2uk∂xj∂xl

= ρ∂2

∂t2ui, i = 1, 2, 3. (6.12)

The only assumptions we make for (Bijkl)i,j,k,l=1,2,3 are their majorsymmetry (6.9) and the strong ellipticity condition (6.10).

Let x = (x1, x2, x3) be the position vector and let m = (m1,m2,m3)and n = (n1, n2, n3) be orthogonal unit vectors in R3. Let C denote theset of complex numbers.c We consider the motion of a homogeneous elasticmedium that occupies the half-space n · x = n1x1 + n2x2 + n3x3 ≤ 0, andseek solutions to (6.12) of the form

u = (u1, u2, u3) = a e−√−1 k(m·x+pn·x−v t) ∈ C

3 (6.13)

in n · x ≤ 0 (cf., for example, Section 2 of [10]).When Im p, the imaginary part of p ∈ C, is positive, a solution of the

preceding form describes a surface wave in n · x ≤ 0; it propagates alongthe surface n · x = 0 in the direction of m with the wave number k andthe phase velocity v > 0, has the polarization defined by a constant vectora, and decays exponentially as n · x −→ −∞. When Im p < 0, the solutionblows up as n · x −→ −∞. We exclude such a solution.

A surface wave described by the form (6.13) depends on the projectionof x on the plane spanned by the two orthogonal unit vectors m and n.

Let us determine the equations that a ∈ C3 and p ∈ C in (6.13) mustsatisfy. Substituting (6.13) into (6.12) and noting that

∂u

∂xj= −√−1 k(mj + pnj)a e−

√−1 k(m·x+pn·x−v t)

and

∂u

∂t=

√−1 k v a e−√−1 k(m·x+pn·x−v t),

cWe also use the symbol C to denote the classical elasticity tensor. It should be clearfrom the context what the symbol means when it appears.



we get 3∑j,l=1

Bijkl (mj + pnj)(ml + pnl) − ρ v2δik

i↓k→1,2,3

a

=

3∑j,l=1

Bijklmjml − ρ v2δik

+ p

3∑j,l=1

Bijklmjnl +3∑

j,l=1

Bijkl njml

+ p23∑

j,l=1

Bijkl njnl

i↓k→1,2,3

a = 0, (6.14)

where δik is the Kronecker delta. We introduce the 3 × 3 real matrices

Q =

3∑j,l=1

Bijklmjml − ρ v2δik

i↓k→1,2,3

, (6.15)

R =

3∑j,l=1

Bijklmjnl

i↓k→1,2,3

, T =

3∑j,l=1

Bijkl njnl

i↓k→1,2,3

.

Taking the major symmetry (6.9) into account, we rewrite equation(6.14) as

[Q + p(R + RT ) + p2T]a = 0, (6.16)

where the superscript T denotes transposition. For the existence of a non-trivial vector a = 0, we observe that p satisfies the sextic equation

det[Q + p(R + RT ) + p2T] = 0. (6.17)

Lemma 6.1.

(1) Matrices Q and T are symmetric. Moreover, matrix T is positivedefinite.

(2) When v = 0, the characteristic roots pα (1 ≤ α ≤ 6), i.e. the solutionsto the sextic equation (6.17), are not real and they occur in complexconjugate pairs.



Proof. The symmetries of Q and T follow immediately from the majorsymmetry (6.9) and the positive definiteness of T from the strong ellipticitycondition (6.10). Assertion (2) also follows easily from (6.10). For details,refer, for example, to (2) of Lemma 1.1 of [12].

Next we examine the traction on the surface n ·x = 0 produced by thesurface-wave solution (6.13). Since the outward unit normal of this surfaceis the vector n, the traction t on n · x = 0 is given by

t =

3∑j=1

Sij nj

i↓1,2,3

∣∣∣∣∣∣n·x=0

which, by (6.5), (6.8), and (6.13) and (6.15), becomesd

t =( 3∑j,k,l=1

Bijkl∂uk∂xl

nj

)i↓1,2,3

∣∣∣∣n·x=0

= −√−1k( 3∑j,l=1

Bijkl (ml + pnl)nj)i↓k→1,2,3

a e−√−1 k(m·x−v t)

= −√−1k [RT + pT] a e−√−1 k(m·x−v t).

Hence we define a vector l ∈ C3 as

l = [RT + pT] a. (6.18)

Then

t = −√−1 k l e−√−1 k(m·x−v t) (6.19)

is the traction on the surface n · x = 0 produced by (6.13).By (1) of Lemma 6.1, T−1 exists. Hence from (6.18) we get

p a = −T−1RT a + T−1 l, p l = [pRT + p2T]a. (6.20)

The last equation becomes, by (6.16) and the first equation of (6.20),

p l = −[Q + pR]a = −Qa− R(−T−1RT a + T−1 l)

= [−Q + RT−1RT ]a − RT−1l. (6.21)

dHere we restrict our attention to the case where the traction produced by T

on n ·x = 0

vanishes, i.e.P3

j=1 T

ijnj = 0 (i = 1, 2, 3) on n · x = 0.



Thus, from (6.20) and (6.21) we obtain:

Theorem 6.1. Let[al

]be a column vector in C

6 whose first three

components consist of a vector a ∈ C3 that satisfies (6.16) and whose

last three components consist of the vector l∈C3 given by (6.18). Thenthe following six-dimensional eigen-relation holds:

N[al

]= p

[al

], (6.22)

where N is the 6 × 6 real matrix defined by

N =

[ −T−1RT T−1

−Q + RT−1RT −RT−1

]. (6.23)

We call the eigenvalue problem (6.22) Stroh’s eigenvalue problem.

When v = 0, it follows from (2) of Lemma 6.1 that the solutions pα (1 ≤α ≤ 6) to (6.17), i.e. the eigenvalues of N, are not real. As v increases fromv = 0, at some point Eq. (6.17) ceases to have only complex roots. Sincewe are concerned with the surface-wave solution (6.13), where Im p > 0, weshall restrict our attention to the range of v for which all the solutions to(6.17) are complex.

Let m = (m1, m2, m3) and n = (n1, n2, n3) be orthogonal unit vectorsin R

3, which are obtained by rotating the orthogonal unit vectors m and naround their vector product m × n by an angle φ (−π ≤ φ < π) so that

m = m(φ) = m cosφ+ n sinφ, n = n(φ) = −m sinφ+ n cosφ. (6.24)

Let Q(φ),R(φ), and T(φ) be the 3 × 3 real matrices given by

Q(φ) =

3∑j,l=1

Bijkl mjml

i↓k→1,2,3

− ρ v2 cos2 φ I,

R(φ) =

3∑j,l=1

Bijkl mj nl

i↓k→1,2,3

+ ρ v2 cosφ sinφ I,

T(φ) =

3∑j,l=1

Bijkl nj nl

i↓k→1,2,3

− ρ v2 sin2 φ I, (6.25)



where I is the 3 × 3 identity matrix. Then Q(0),R(0), and T(0) are equalto Q,R, and T in (6.15), respectively.

As we shall see in Proposition 6.2, Definition 6.3, and Theorem 6.4,we have introduced the matrices Q(φ),R(φ), and T(φ) for the ultimatepurpose of finding properties of the vectors a and l in (6.22) and of thesurface impedance matrix Z(v), which is defined later through these vectors.

Definition 6.1. The limiting velocity vL = vL(m,n) is the lowest velocityfor which the matrices Q(φ) and T(φ) become singular for some angle φ:

vL = infv > 0|∃φ; detQ(φ) = 0= infv > 0|∃φ; detT(φ) = 0. (6.26)

We will give a characterization of vL = vL(m,n) in terms of bodywaves whose direction of propagation is on the m–n plane.

A solution to (6.12) of the form

u = (u1, u2, u3) = a e−√−1 k

(em·x−c(φ) t

)(6.27)

represents a body wave with direction of propagation m, wave number k,velocity c(φ), and polarization a = a(φ). Substituting this into (6.12), weeasily observe that ρ c(φ)2 and a(φ) are an eigenvalue and an eigenvectorof the positive definite matrix called the acoustic tensor

3∑j,l=1

Bijklmjml

i↓k→1,2,3

, (6.28)

respectively.Let

λi(φ) (i = 1, 2, 3), 0 < λ1(φ) ≤ λ2(φ) ≤ λ3(φ)

be the eigenvalues of the acoustic tensor (6.28). Corresponding to theseeigenvalues, there exist three body waves, which have direction of propa-gation m, phase velocity

ci(φ) =

√λi(φ)ρ

(i = 1, 2, 3), 0 < c1(φ) ≤ c2(φ) ≤ c3(φ), (6.29)

and polarizations ai =ai(φ)∈R3 (i=1, 2, 3), which are mutually ortho-

gonal.



Proposition 6.1.

vL = vL(m,n) = min−π

2<φ<π2

c1(φ)cosφ

. (6.30)

Proof. We see from the first equation in (6.25) that the eigenvalues ofQ(φ) are

λi(φ) − ρ v2 cos2 φ, i = 1, 2, 3.

Hence the assertion follows from the first equality in (6.26).

A useful construction of vL(m,n) is in terms of the slowness sectionin the m–n plane. The slowness section in the m–n plane consists of thethree closed curves generated by the radius vectors

1ci(φ)

m, i = 1, 2, 3 (−π < φ ≤ π).

The curve corresponding to the slowest velocity c1(φ) defines the silhouetteof the slowness section and is called the outer profile.

It follows from (6.30) that

v−1L = max

−π2<φ<

π2

1c1(φ)

cosφ. (6.31)

Since φ is the angle of rotation about m × n between the m-axis and thevector m,

1c1(φ)

cosφ

is the projection on the m-axis of the point on the outer profile. Then by(6.31), the limiting slowness vL(m,n)−1 is the absolute maximum of theset of such projections. Thus, we obtain:

Corollary 6.1. In the m–n plane, let L be a line parallel to the n-axis approaching the slowness section from the right and making the firsttangential contact with the outer profile at some point T . The limitingslowness vL(m,n)−1 is the projection of T on the m-axis. Let φ be theangle between

−→OT and the m-axis (−π

2< φ < π

2). Then

v−1L = max

−π2<φ<

π2

1c1(φ)

cosφ =1

c1(φ)cos φ (6.32)



m

n

T

v (m,n)-1L

L

φ

Fig. 6.1. Slowness section and limiting velocity.

(see Fig. 6.1). The corresponding body wave represented by (6.27) propagatesin the direction of the radius vector

−→OT with the velocity c1(φ) and

its polarization is an eigenvector of the acoustic tensor (6.28) at φ= φ

pertaining to the smallest eigenvalue ρ c1(φ)2.

Example 6.1. The effective elastic coefficients of a material in anunstressed and isotropic state are given by

Bijkl = λ δijδkl + µ(δikδjl + δilδkj), (6.33)

where λ and µ are the Lame constants. Then it follows thate

Q(φ) = (λ+ µ)m ⊗ m + (µ− ρv2 cos2 φ) I,

detQ(φ) = (µ− ρv2 cos2 φ)2(λ+ 2µ− ρv2 cos2 φ).

Taking into account that µ < λ + 2µ, which is guaranteed by the strongconvexity condition for (6.33) (cf. the paragraph which follows (6.10)), weimmediately see that vL for the material in question is given by vIso

L =√

µρ

eHenceforth we will use the notations ⊗ and × to denote the tensor product and thevector product, respectively.



for any m and n. The acoustic tensor (6.28) of this material is given by

(λ+ 2µ)m ⊗ m + µ(n ⊗ n + ⊗ ),where = m×n = m×n; its eigenvalues are λ+2µ (simple) and µ (double).Hence the slowness sections of this material are three circles centered atthe origin, two of which have the same radius

√ρµ

and the other has radius√ρ

λ+2µ.

The interval 0 < v < vL is called the subsonic range.f

We present below several fundamental properties of the matricesQ(φ),T(φ), and N and the sextic equation (6.17) in the subsonic range.The proof of the following lemma is given, for example, in Lemma 3.2 of [12].

Lemma 6.2.

(1) The symmetric matrices Q(φ) and T(φ) are positive definite for all φif and only if 0 ≤ v < vL.

(2) When 0 ≤ v < vL, the solutions pα (1 ≤ α ≤ 6) to the sextic equation(6.17), i.e. the eigenvalues of N, are not real and they occur in complexconjugate pairs.

Henceforth we will restrict our attention to the subsonic range 0 ≤ v <

vL and take

Im pα > 0, α = 1, 2, 3. (6.34)

Now let 0 ≤ v < vL, and let[aαlα

](α = 1, 2, 3) be linearly independent

eigenvectors of the eigenvalue problem (6.22) pertaining to the eigenvaluespα (α = 1, 2, 3, Im pα > 0), respectively. Motivated by (6.13) and (6.19),we observe that the general formg of the surface-wave solution is given by

u =3∑

α=1

cα aα e−√−1 k(m·x+pαn·x−v t) (6.35)

fThe reader should not consider “subsonic” and “ultrasonic” to be related adjectives.“Ultrasonic” refers to stress waves with frequencies above 2 × 104 Hz, the upper limit ofhuman hearing. Rayleigh waves, which have a phase velocity in the subsonic range, canhave ultrasonic frequencies.gWe have used the term “general” because aα (α = 1, 2, 3) are linearly independent inC

3, which we shall see soon.



and the corresponding traction on the surface n · x = 0 produced by thesolution above is given by

t = −√−1k

3∑α=1

cαlαe−√−1k(m·x−vt), (6.36)

where cα (1 ≤ α ≤ 3) are arbitrary complex constants. The arrangement(6.34) guarantees that the surface-wave solution (6.35) in the subsonic rangedecays exponentially as n · x −→ −∞.

Remark 6.1. When the six-dimensional eigenvalue problem (6.22) doesnot have six linearly independent eigenvectors, generalized eigenvector(s)

must be introduced. Let[aαlα

]∈ C

6 (α = 1, 2, 3) be linearly independent

eigenvector(s) and generalized eigenvector(s) of N associated with theeigenvalues pα(α = 1, 2, 3). Then the form of the surface-wave solution(6.35) must be modified according to the degeneracy of the eigenvalueproblem (6.22). However, the corresponding displacements on the surfacen · x = 0 have the same form

u =3∑

α=1

cαaαe−√−1k(m·x−v t), (6.37)

and the corresponding tractions on n · x = 0 all have the same form

t = −√−1 k3∑

α=1

cαlαe−√−1k(m·x−v t). (6.38)

Let us turn to the rotated orthogonal unit vectors (6.24) in the m–nplane and to the matrices (6.25) defined through these rotated vectors. For0 ≤ v < vL, let N(φ) be the 6 × 6 real matrix defined by

N(φ) =

[−T(φ)−1R(φ)T T(φ)−1

−Q(φ) + R(φ)T(φ)−1R(φ)T −R(φ)T(φ)−1

]. (6.39)

Note that N(0) is equal to N defined by (6.23).The following property of Stroh’s eigenvector problem is fundamental

to the Stroh formalism. It serves as a basis for the derivation of the Barnett–Lothe integral formalism and is also elegant in itself.



Theorem 6.2. For 0 ≤ v < vL, let[al

]be an eigenvector of N(0)

associated with the eigenvalue p = p0. Then

N(φ)[al

]= p(φ)

[al

](6.40)

for all φ, and the eigenvalue p(φ) of N(φ) satisfies the Riccati equation

d

dφp = −1 − p2 (6.41)

with p(0) = p0.

Various proofs of this theorem can be found in several papers listedat the beginning of this section. Here, by avoiding the introduction of“dynamical elastic coefficients,” we present a slight improvement of theproof given in [12].

Proof. We denote the differentiation ddφ by ′. From (6.24) we get

m′ = −m sinφ+ n cosφ = n, n′ = −m cosφ− n sinφ = −m.

Then from (6.25) it follows that

Q(φ)′ =

3∑j,l=1

Bijklnjml +3∑

j,l=1

Bijklmjnl

i↓k→1,2,3

+ 2ρv2 cosφ sinφI = R(φ) + R(φ)T

R(φ)′ =

3∑j,l=1

Bijklnj nl −3∑

j,l=1

Bijklmjml

i↓k→1,2,3

+ ρv2(cos2 φ− sin2 φ)I = T(φ) − Q(φ)

T(φ)′ = − 3∑j,l=1

Bijklmjnl +3∑

j,l=1

Bijklnjml

i↓k→1,2,3

− 2ρ v2 cosφ sinφI = −R(φ) − R(φ)T . (6.42)

Put

H(φ) = [Q + p(R + RT ) + p2T](φ).



From (6.42) we have

H′(φ) = [R + RT + p′(R + RT ) + 2p (T − Q)

+ 2p p′ T − p2(R + RT )](φ)

= [−2pQ + (1 + p′ − p2)(R + RT ) + 2p(1 + p′)T](φ).

Suppose that p(φ) satisfies the Riccati equation (6.41). Then the equationabove becomes

H′(φ) = −2p[Q + p(R + RT ) + p2T](φ) = −2p(φ)H(φ). (6.43)

Now put

h(φ) = [Q + p(R + RT ) + p2T](φ)a = H(φ)a,

where a satisfies (6.16). Then we get

h(0) = 0. (6.44)

It follows from (6.43) that

h′(φ) = −2p(φ)H(φ)a = −2p(φ)h(φ). (6.45)

The solution to (6.41) is p(φ) = tan(φ0−φ) with tan(φ0) = p0 and Im p0 = 0or p(φ) ≡ ±√−1, both of which are bounded smooth functions of φ. Thusfrom (6.44) and (6.45) we obtain

h(φ) = [Q + p(R + RT ) + p2T](φ)a = 0 (6.46)

for all φ.Now put

l(φ) = [RT + pT](φ)a.

Then, by the assumption of the theorem, we have

l(0) = l.

It follows from (6.41) and (6.42) that

l′(φ) = [T− Q + p′T − p(R + RT )](φ)a

= −[Q + p(R + RT ) + p2T](φ)a = −h(φ),

which is equal to zero by (6.46). Hence we obtain

l(φ) = [RT + pT](φ)a = l(0) = l (6.47)



for all φ. In the same manner that (6.16) and (6.18) lead to the eigen-relation (6.22), from (6.46) and (6.47) we obtain (6.40).h

When N(0) has generalized eigenvectors, they do not have the simpleinvariance as in Theorem 6.2, but there is a rule that describes theirdependence on φ, for which we refer to the literature given at the beginningof this section.

We now proceed to the Barnett–Lothe integral formalism.

Definition 6.2. For 0 ≤ v < vL, we define the 6× 6 real matrix S = S(v)to be the angular average of the 6 × 6 matrix N(φ) over [−π, π]:

S =

[S1 S2

S3 ST1

]=

12π

∫ π

−πN(φ)dφ, (6.48)

where S1 = S1(v),S2 = S2(v), and S3 = S3(v) are 3 × 3 real matricesdefined by

S1 =12π

∫ π

−π−T(φ)−1R(φ)T dφ, S2 =

12π

∫ π

−πT(φ)−1dφ,

S3 =12π

∫ π

−π−Q(φ) + R(φ)T(φ)−1R(φ)T dφ, (6.49)

and Q(φ), R(φ), and T(φ) are given by (6.25).

By (1) of Lemma 6.2, the matrices S2 and S3 are symmetric and S2 ispositive definite for 0 ≤ v < vL.

Now we take the angular average of Stroh’s eigenvalue problem.

Theorem 6.3. For 0 ≤ v < vL, let[aαlα

]be an eigenvector or generalized

eigenvector of N(0) corresponding to the eigenvalues pα (α = 1, 2, 3) withIm pα > 0. Then for 0 ≤ v < vL,

S[aαlα

]=

√−1[aαlα

]. (6.50)

hIn the proof, we have assumed that p(φ) satisfies the Riccati equation (6.41). However,it follows that p(φ), being the eigenvalue of N(φ) in (6.40), must be the solution to (6.41),because an eigenvalue that corresponds to the same eigenvector is unique.



Proof. When[aαlα

]is an eigenvector of N(0) corresponding to the

eigenvalues pα (α = 1, 2, 3) with Im pα > 0, we take the angular average ofboth sides of (6.40). For p(φ) = tan(φ0 − φ) with tan(φ0) = p0, Im p0 > 0and for p(φ) ≡ ±√−1 it follows that 1

2π

∫ π−π p(φ)dφ =

√−1, which,combined with (6.48), leads to (6.50).

When[aαlα

]is a generalized eigenvector of N(0), the proof of (6.50)

is a little complicated: refer to the literature given at the beginning of thissection.

Proposition 6.2. For 0≤ v <vL, let[aαlα

](α=1, 2, 3) be linearly

independent eigenvector(s) or generalized eigenvector(s) of N(0) corres-ponding to the eigenvalues pα (α = 1, 2, 3) with Im pα > 0. Then theirdisplacement parts aα (α = 1, 2, 3) are linearly independent.

Proof. The first three rows of the system (6.50) are written, by using thenotation in (6.49), as

S1aα + S2lα =√−1aα, α = 1, 2, 3.

Then we get

S2lα = (√−1I − S1)aα, α = 1, 2, 3.

Since S2 is invertible, multiplying both sides by S−12 , we obtain

lα = (√−1S−1

2 − S−12 S1)aα, α = 1, 2, 3. (6.51)

Suppose that aα(α = 1, 2, 3) are linearly dependent. Then there exists a setof complex numbers (c1, c2, c3) = (0, 0, 0) such that

3∑α=1

cαaα = 0.

Then from (6.51) it follows that

3∑α=1

cαlα = 0,



and therefore3∑

α=1

cα

[aαlα

]= 0.

This contradicts the assumption that[aαlα

](α = 1, 2, 3) are linearly

independent.

Now we can define the surface impedance matrix, which was firstintroduced by Ingebrigtsen and Tonning [21] and later given by Lotheand Barnett [8] in the framework of the Stroh formalism.

Definition 6.3. For 0≤ v <vL, let[aαlα

](α=1, 2, 3) be linearly indepen-

dent eigenvector(s) or generalized eigenvector(s) of N(0) corresponding tothe eigenvalues pα (α = 1, 2, 3) with Im pα > 0. The surface impedancematrix Z(v) is the 3 × 3 matrix given by

Z(v) = −√−1[l1, l2, l3][a1, a2,a3]−1, (6.52)

where [l1, l2, l3] and [a1,a2,a3] denote 3 × 3 matrices, which consist of thecolumn vectors lα and aα, respectively.

Therefore, Z(v) expresses a linear relationship between (i) thedisplacements given at the surface n · x = 0 of the form (6.37), whichpertain to the surface-wave solution of the form (6.35)i propagating in thedirection of m with the phase velocity v, and (ii) the tractions needed tosustain them at that surface.

Remark 6.2. There is an arbitrariness in the choice of the linearlyindependent eigenvectors and generalized eigenvectors. However, Theorem6.4, below, implies that the arbitrariness is canceled out in the product ofthe two matrices in (6.52), and hence Z is well defined.

Theorem 6.4. For 0 ≤ v < vL,

Z(v) = S−12 +

√−1S−12 S1, (6.53)

where the matrices S1 and S2 are given by (6.49).

Proof. It is obvious from (6.51) and (6.52).

iWhen N = N(0) has generalized eigenvectors, the form of (6.35) will have to be modifiedslightly.



Let[aαlα

](α= 1, 2, 3) be linearly independent eigenvectors or

generalized eigenvectors of N(0) corresponding to the eigenvalues pα(α= 1, 2, 3) with Im pα > 0. Then from Theorem 6.3 it follows that

S2

[aαlα

]= −

[aαlα

](α = 1, 2, 3). (6.54)

We see from (6.22) that the complex conjugates of[aαlα

](α = 1, 2, 3)

are linearly independent eigenvectors or generalized eigenvectors of N(0)corresponding to the complex conjugates of the eigenvalues pα (α = 1, 2, 3).Hence we take the complex conjugate of both sides of (6.54) to seethat (6.54) holds for six linearly independent eigenvectors or generalizedeigenvectors of N(0). Therefore we get

S2 = −I (0 ≤ v < vL), (6.55)

where I denotes the 6 × 6 identity matrix. Then the blockwise expressionof (6.55) obtained from (6.48) gives

S21 + S2S3 = −I, S1S2 + S2ST1 = 0 (0 ≤ v < vL). (6.56)

Since S2 is symmetric and invertible for 0 ≤ v < vL, it follows from thesecond equality in (6.56) that

S−12 S1 = −(S−1

2 S1)T (0 ≤ v < vL), (6.57)

which implies that S−12 S1 is antisymmetric. Hence from (6.53) we obtain:

Corollary 6.2. The surface impedance matrix Z(v) is Hermitian for 0 ≤v < vL.

6.4 Rayleigh Waves in Anisotropic Materials

Rayleigh waves are elastic surface waves, which propagate along thetraction-free surface n · x = 0 with a phase velocity in the subsonic range0 < v < vL, and whose amplitude decays exponentially with depth belowthat surface.

Let m and n be orthogonal unit vectors in R3. Following the setting of

Section 6.3, we consider Rayleigh waves that propagate along the surfacen · x = 0 in the direction of m with the phase velocity vR satisfying0 < vR < vL, and whose amplitude decays exponentially as n · x −→ −∞,



and which produce no tractions on n · x = 0. Here vL = vL(m,n) is thelimiting velocity in Definition 6.1.

We take the C3-vectors aα and lα (α = 1, 2, 3) so that

[aαlα

](α = 1, 2, 3)

are linearly independent eigenvector(s) or generalized eigenvector(s) of N =N(0) at v = vR associated with the eigenvalues pα (α = 1, 2, 3, Im pα > 0).Then the existence of Rayleigh waves implies that the correspondingtraction on n · x = 0 given by (6.38) vanishes for v = vR. In other words,there exists a set of complex numbers (c1, c2, c3) = (0, 0, 0) such that

3∑α=1

cαlα = 0 at v = vR, (6.58)

which is equivalent to

det[l1, l2, l3] = 0 at v = vR. (6.59)

We will give a characterization of the Rayleigh-wave velocity vRin terms of the surface impedance matrix Z(v) defined by (6.52), orequivalently by (6.53), which we shall use in investigating the perturbationof vR in the next section.

Theorem 6.5. A necessary and sufficient condition for the existence ofRayleigh waves in the half-space n ·x ≤ 0, which propagate along the surfacen · x = 0 in the direction of m with the phase velocity vR in the subsonicrange 0 < v < vL, is

det Z(v) = 0 at v = vR (0 < vR < vL). (6.60)

Proof. Recall that Rayleigh waves produce no tractions on n ·x = 0. Theproof is obvious from (6.59), Proposition 6.2, and (6.52).

The surface impedance matrix Z(v) (0 ≤ v < vL) has the followingfundamental properties:

(1) Z(v) is Hermitian for 0 ≤ v < vL (Corollary 6.2).(2) Z(0) is positive definitej (cf. [8], Section 7.D of [9], Section 6.6 of [11],

and [22,23]).

jThis assertion holds when (Bijkl)i,j,k,l=1,2,3 satisfies the strong convexity condition.However, since we have assumed that a deviation of the medium from its unstressedstate caused by the initial stress is small (see the comment just after (6.10)), the assertionremains valid.



(3) The Hermitian matrix ddv

Z(v) is negative definite for 0 < v < vL (cf.[8,24,25]).

Proofs of assertions (2) and (3) are long: refer to the literature mentionedabove.

Using assertion (3), we obtain (cf. Theorem 7 of [10], [12]):

Lemma 6.3. For 0 ≤ v < vL, the eigenvalues of Z(v) are monotonicdecreasing functions of v.

Therefore, the eigenvalues of Z(v) decrease monotonically with v

in the interval 0≤ v < vL from their positive values at v= 0, and oneof them becomes zero when v equals the Rayleigh-wave velocity vR(Theorem 6.5).

Using the integral expression (6.53) for Z(v), we get (cf. the commentsafter Theorem 7 of [10], [12]):

Lemma 6.4. At most one eigenvalue of Z(v) can be negative at v = vL.

The last two lemmas, combined with Theorem 6.5, imply:

Corollary 6.3. For given orthogonal unit vectors m and n, the phasevelocity of a Rayleigh wave is uniquely determined if the Rayleigh waveexists.k

Example 6.2. The effective elastic coefficients of a material in anunstressed and isotropic state are given by (6.33). Algebraic manipulationsof (6.52) give

Z(v) = −√−1[µ p1⊗ +

V p3

1 + p1p3m ⊗ m +

V p1

1 + p1p3n ⊗ n

+(

2µ− V

1 + p1p3

)(m ⊗ n − n⊗ m)

], (6.61)

where

V = ρ v2, p1 = p2 =√−1

√µ− V

µ, p3 =

√−1

√λ+ 2µ− V

λ+ 2µ

kThe polarization vectorP3

α=1 cαaα in (6.37) of the Rayleigh wave at the surfacen · x = 0 is also uniquely determined up to a constant multiplicative factor.



and = m × n is the vector product and ⊗ denotes the tensor product.Here we have used the explicit expressions of aα and lα (α = 1, 2, 3) givenin (3.36) and (3.158) of [12].

When v = vIsoL =

√µρ,

p1 = 0, p3 =√−1

√λ+ µ

λ+ 2µ.

Hence

Z(vIsoL ) = µ

√λ+ µ

λ+ 2µm ⊗ m −√−1µ(m ⊗ n− n ⊗ m),

whose eigenvalues are 0 and the two real roots of the quadratic equationof p:

p2 − µ

√λ+ µ

λ+ 2µp− µ2 = 0.

Obviously, this has two roots of opposite sign, which implies that one ofthe eigenvalues of Z(v) is negative at v = vIso

L . Therefore a Rayleigh waveexists for any m and n. The phase velocity vIso

R (< vIsoL ) of the Rayleigh

wave is obtained through (6.60) and (6.61) from the equation

V p1

1 + p1p3

V p3

1 + p1p3+(

2µ− V

1 + p1p3

)2

= 0,

which is equivalent to the cubic equation

V 3 − 8µV 2 +8µ2(3λ+ 4µ)

λ+ 2µV − 16µ3(λ+ µ)

λ+ 2µ= 0.

In the next section, using Theorem 6.5 we derive an equation for thephase velocity of Rayleigh waves (i.e. a secular equation) that propagateon the free surface of a material in an unstressed and orthotropic ortransversely isotropic state and study the perturbation of the phase velocitycaused by the presence of the initial stress and/or caused by the deviationof the incremental elasticity tensor from its comparative orthotropic ortransversely isotropic state.



6.5 Perturbation of the Phase Velocity of Rayleigh Wavesin Prestressed Anisotropic Media when the BaseMaterial is Orthotropic

Let us turn to the constitutive equation (6.2). As a base material we takean orthotropic mediuml with elasticity tensor C

Orth. Suppose the basematerial can be such that it occupies the half-space x3 ≤ 0 and its planesof reflectional symmetry coincide with the three coordinate planes. Thenthe elasticity tensor C

Orth is expressed with the Voigt notation as

COrth = (COrth

rs ) =

C11 C12 C13 0 0 0C22 C23 0 0 0

C33 0 0 0C44 0 0

C55 0Sym. C66

. (6.62)

Suppose that the incremental elasticity tensor L is composed of anorthotropic part C

Orth and a perturbative part A, the latter of whichexpresses a deviation of the material from its orthotropic unperturbed state.For the perturbative part A, we assume the major and minor symmetries

aijkl = aklij = ajikl, i, j, k, l = 1, 2, 3

but do not assume any material symmetry. In the Voigt notation, A can bewritten as

A = (ars) =

a11 a12 a13 a14 a15 a16

a22 a23 a24 a25 a26

a33 a34 a35 a36

a44 a45 a46

a55 a56

Sym. a66

(6.63)

and the 21 components in the upper triangular part of matrix (6.63) aregenerally all independent. In this setting, L can be written as a fourth-order

lWe say that an elastic material is orthotropic if there exist three mutually orthogonalplanes such that (6.6) holds for any orthogonal tensor Q = (Qij)i,j=1,2,3 pertaining toa reflection with respect to one of these planes.



tensor on symmetric tensors E in the form

L[E] = COrth[E] + A[E]; (6.64)

here the comparative orthotropic part COrth and the perturbative part A are

also treated as fourth-order tensors onE. In this section we shall investigatehow A affects the phase velocity of Rayleigh waves.

We also consider the change of the phase velocity of Rayleigh wavesdue to the presence of the initial stress. We assume that the surface x3 = 0of the half-space is free of traction. Since the initial stress is here takento be homogeneous, zero traction at the boundary x3 = 0 implies that thecomponents T

i3 (i = 1, 2, 3) of T

must vanish. Then the second-order tensor

T

is represented by a 3 × 3 matrix as

T

=

T11 T

12 0

T12 T

22 0

0 0 0

. (6.65)

Thus, in our constitutive equation (6.2) with (6.64), A and T

expressthe deviation of the medium in question from its comparative orthotropicand unstressed state and then we consider what influence A and T

exert

upon the phase velocity of Rayleigh waves that propagate along the surfaceof the material half-space.

As we mentioned in Section 6.2, L generally depends on T. Hence A

also may depend on T. In this section we keep the dependence of L and

A on T

implicit. In the theorem and remark below, when we refer to theeffect of T

on the perturbation formulas, we mean only the contribution of

the initial stress through the HT

term in (6.2).For an orthotropic base material whose elasticity tensor is given by

(6.62), when there exist Rayleigh waves propagating along the surface ofthe half-space x3 ≤ 0 in the direction of the 2-axis, the phase velocity vOrth

R

satisfies the secular equation

ROrth(v) = 0, (6.66)

where

ROrth(v) = C33C44(C22 − V )V 2 − (C44 − V )(C33(C22 − V ) − C2

23

)2(6.67)

and V = ρv2. Equations (6.66) and (6.67) follow from (6.59) (cf. [26]).



As mentioned in the introduction of this chapter, our purpose is toderive a perturbation formula that shows how A and T

affect the phase

velocity of Rayleigh waves from its comparative orthotropic and unstressedvalue vOrth

R . Now we state a theorem that gives the perturbation formula.

Theorem 6.6. In a prestressed medium whose incremental elasticitytensor L and initial stress T

are given by (6.64) and (6.65) respectively,

the phase velocity of Rayleigh waves which propagate along the surface ofthe half-space x3 ≤ 0 in the direction of the 2-axis can be written, up toterms linear in the perturbative part A of L and the initial stress T

, as

vR = vOrthR − 1

2ρ vOrthR

[γ22(vOrth

R )a22 (6.68)

+ γ23(vOrthR )a23 + γ33(vOrth

R )a33 + γ44(vOrthR )a44 − T

22

],

where

γij(v) =Nij(v)D(v)

(ij = 22, 23, 33, 44), (6.69)

N22(v) = C33[−2C44(C22C33 − C223) + 2(C22C33 − C2

23 + C33C44)V

+ (C44 − 2C33)V 2],

N23(v) = 4C23(C44 − V )(C22C33 − C223 − C33V ),

N33(v) = (C22 − V )[−2C44(C22C33 − C223) + 2(C22C33 − C2

23 + C33C44)V

+ (C44 − 2C33)V 2] =C22 − V

C33N22(v),

N44(v) =−VC44

(C22C33 − C223 − C33V )2,

D(v) = (C22C33 − C223)(C22C33 − C2

23 + 2C33C44)

+ 2C33[C22C44 − 2(C22C33 − C223) − C33C44]V

+ 3C33(C33 − C44)V 2,

V = ρv2.

Remark 6.3. Only four components a22, a23, a33, and a44 of theperturbative part A of L and one component T

22 of the initial stress T

affect

the first-order perturbation of the phase velocity vR. This is also true for



the case where the base material is unstressed and isotropic [13]. When thebase material is generally anisotropic, Song and Fu [27] obtained a formulaon the first-order perturbation of the phase velocity of Rayleigh waves; thatformula involves the eigenvalues and the engenvectors of Stroh’s eigenvalueproblem for the base material. Also, they applied their formula to the casewhere the base material is monoclinic. There they asserted that for Rayleighwaves polarized in a symmetry plane of the monoclinic material, which wetake to be the 2–3 plane in this instance, the first-order perturbation ofvR will not involve any components of A in which suffix 1 appears at leastonce, i.e., no components of A other than a22, a23, a24, a33, a34, and a44 willaffect the first-order perturbation of vR. For more discussion on work [27]refer to [17].

When the base material is transversely isotropicm and the axis ofsymmetry coincides with the 3-axis, its elasticity tensor C

Trans is expressedin the Voigt notation as

CTrans = (CTrans

rs ) =

C11 C12 C13 0 0 0C11 C13 0 0 0

C33 0 0 0C44 0 0

C44 0Sym . (C11 − C12)/2

.

(6.70)

Corollary 6.4. In a prestressed medium whose incremental elasticitytensor L and initial stress T

are given by L = C

Trans + A and (6.65)respectively, the first-order perturbation formula for the phase velocity ofRayleigh waves that propagate along the surface of the half-space x3 ≤ 0in the direction of the 2-axis is given by (6.68) and (6.69), where vOrth

R isreplaced by the phase velocity vTrans

R of Rayleigh waves propagating on thesurface of the half-space x3 ≤ 0 of the comparative transversely isotropicand unstressed medium, which solves (6.66).

For a base material with T

= 0, A = O, and L = COrth, the limiting

velocity vOrthL of the surface waves that propagate along the surface of the

mWe say that an elastic material is transversely isotropic if there exists a unit vectorn such that (6.6) holds for any orthogonal transformation Q = (Qij)i,j=1,2,3 whichsatisfies Qn = n or Qn = −n.



half-space x3 ≤ 0 in the direction of the 2-axis is given by

ρ(vOrthL )2 = min(C66, V

∗), (6.71)

where

V ∗ = sup0 < V ≤ min(C22, C44)|√C33(C22 − V )

+√C44(C44 − V ) ≥ |C23 + C44|,

and the corresponding surface impedance matrix is given by

ZOrth(v) = (Zij)i↓j→1,2,3 = ZOrth(v)T, (6.72)

Z11 =√C55(C66 − V ), Z12 = Z13 = 0,

Z22 =

√C44(C22 − V )√

C33(C22 − V ) +√C44(C44 − V )

×√

(√C33(C22 − V ) +

√C44(C44 − V ))2 − (C23 + C44)2,

Z33 =

√C33(C44 − V )√

C33(C22 − V ) +√C44(C44 − V )

×√

(√C33(C22 − V ) +

√C44(C44 − V ))2 − (C23 + C44)

2,

Z23 =−√−1

√C44√

C33(C22 − V ) +√C44(C44 − V )

×(√

C33C44(C22 − V ) − C23

√C44 − V

),

where V = ρ v2. (Cf. Section 12-10 of [11] and Section 3.8 of [12].)From Theorem 6.5 it can be proved (cf. [11,12]) that Rayleigh waves

that propagate along the surface of the half-space x3 ≤ 0 in the directionof the 2-axis exist if V ∗ ≤ C66 and that the secular equation for the phasevelocity of the Rayleigh waves in the base material is given by Z22Z33 +Z2

23 = 0, which is found to be equivalent to (6.66).

Sketch of Proof of Theorem 6.6. Below we shall briefly sketch aderivation of the perturbation formula (6.68) in Theorem 6.6. We willinvestigate the effects of A and T

on the surface impedance matrix Z(v)

pertaining to the surface waves that propagate in the direction of the 2-axisalong the surface of the half-space x3 ≤ 0 of an elastic medium whose



incremental elasticity tensor and the initial stress have the forms (6.64)and (6.65).

Since we are concerned with the terms in vR up to those linear in theperturbative part A of L and the initial stress T

, suppose that we can write

the surface impedance matrix Z(v), up to terms linear in A and T, as

Z(v) ≈ ZOrth(v) + ZPtb(v).

We will use the notation ≈ to indicate that we are retaining terms up tothose linear in the perturbative part A of L and the initial stress T

and are

neglecting the higher-order terms. Note that each component of ZPtb(v) is

a linear function of A and T. From (6.72) we can write

Z(v) ≈

Z11 + ζ11 ζ12 ζ13

ζ12 Z22 + ζ22 Z23 + ζ23

ζ13 −Z23 + ζ23 Z33 + ζ33

,

where

ZPtb(v) = ZPtb(v)T

= (ζij)i↓j→1,2,3.

Hence it follows that

det Z(v) ≈ (Z11 + ζ11)(Z22 + ζ22)(Z33 + ζ33)

− (Z11 + ζ11)(Z23 + ζ23)(−Z23 + ζ23)

≈ (Z11 + ζ11)(Z22Z33 + Z223 + Z33ζ22 + Z22ζ33

+Z23(ζ23 − ζ23)).

We see from Z11 =√C55(C66 − V ) and (6.71) that Z11 > 0 in the subsonic

range 0 < v < vOrthL of surface waves that propagate along the surface of

the half-space x3 ≤ 0 in the direction of the 2-axis of the base material.Since T

and A are sufficiently small, Z11 + ζ11 > 0 in the subsonic range

of the surface waves in question. Thus we obtain from Theorem 6.5 anapproximate secular equation for vR:

∆(v) = 0, (6.73)

where

∆(v) = Z22Z33 + Z223 + Z33ζ22 + Z22ζ33 + Z23(ζ23 − ζ23).



Lemma 6.5. The effects of the initial stress T

and the perturbativepart A of the incremental elasticity tensor L on the approximate secularequation (6.73), more precisely, on the components ζ22 and ζ33 of ZPtb(v)and ζ23 − ζ23, to first order of T

and A, come only from a22, a23, a33, a44,

and T22.

To prove this lemma we use the integral representation (6.53) of Z(v).The matrix S1 can be written as

S1 ≈ SOrth1 + SPtb

1 .

Here

SOrth1 = SOrth

1 (v) =12π

∫ π

−π(−TOrth

v (φ)−1ROrthv (φ)T )dφ

is of zeroth order in T

and A, where

ROrthv (φ) =

3∑j,l=1

COrthijkl mjnl

i↓k→1,2,3

+ ρ v2 cosφ sinφ I,

TOrthv (φ) =

3∑j,l=1

COrthijkl njnl

i↓k→1,2,3

− ρv2 sin2 φI,

and

SPtb1 = SPtb

1 (v) =12π

∫ π

−π(−TOrth

v (φ)−1RPtb(φ)T

+TOrthv (φ)−1TPtb(φ)TOrth

v (φ)−1ROrthv (φ)T )dφ

is of first order in T

and A, where

RPtb(φ) =3∑

j,l=1

TjlmjnlI +

3∑j,l=1

aijklmjnl

i↓k→1,2,3

,

TPtb(φ) =3∑

j,l=1

Tjlnj nl I +

3∑j,l=1

aijklnjnl

i↓k→1,2,3

.

We also have

S2 ≈ SOrth2 + SPtb

2 ,



where

SOrth2 = SOrth

2 (v) =12π

∫ π

−πTOrthv (φ)−1dφ

is of zeroth order in T

and A and

SPtb2 = SPtb

2 (v) =−12π

∫ π

−πTOrthv (φ)−1TPtb(φ)TOrth

v (φ)−1dφ

is of first order in T

and A. Hence we see from

Z(v) = S−12 +

√−1S−12 S1 ≈ ZOrth(v) + ZPtb(v)

that

ZPtb(v) = −(SOrth2 )−1SPtb

2 (SOrth2 )−1 (6.74)

+√−1[−(SOrth

2 )−1SPtb2 (SOrth

2 )−1SOrth1 + (SOrth

2 )−1SPtb1 ]

= − (SOrth2 )−1SPtb

2 ZOrth(v) +√−1(SOrth

2 )−1SPtb1 .

Looking carefully at the components ζ22 and ζ33 of (6.74) and ζ23 − ζ23, wecan confirm the lemma. For details refer to [17].

From the lemma we immediately obtain

Proposition 6.3. The effects of the initial stress T

and the perturbativepart A of the incremental elasticity tensor L on the phase velocity vR, tofirst order of T

and A, come only from a22, a23, a33, a44, and T

22.

This proposition allows us to reduce the case where the perturbativepart A is generally anisotropic and the initial stress T

is generally given

by (6.65) to the orthotropic case with uniaxial stress in the propagationdirection, which provides a highly simplified derivation of the perturbationformula (6.68). For details refer to [17].

6.6 An Inverse Problem on Recovery of Initial Stress

Consider a body of a composite material, which occupies the half-spacex3 ≤ 0 and, after homogenization, is transversely isotropic (with the 3-axisdefined by the unit vector e3 as the ∞-axis) except for the presence

of a prestress T

given by (6.65). In this section we will investigate

what information about the prestress T

could be inferred from boundary



measurements of the phase velocities of Rayleigh waves that propagatealong the free surface.

In the following we will choose the 1- and 2-axes of the Cartesiancoordinate system in question arbitrarily. We assume that the constitutiveequation of the composite material is of the form (cf. [28] for a different butsimilar setting)

S = T

+HT

+ L(T)[E] = T

+HT

+ C[E] + D[T

,E], (6.75)

where the elasticity tensor C and the acoustoelastic tensor D satisfy

QC[E]QT = C[QEQT ], QD[T,E]QT = D[QT

QT ,QEQT ], (6.76)

respectively, for all orthogonal transformations Q that obey Qe3 = e3 orQe3 = − e3. Restriction (6.76)1 dictates that C is given by C

Trans or by(6.62) with the additional conditions

C11 = C22, C13 = C23, C44 = C55, C66 =12(C11 − C12). (6.77)

In the following, for simplicity we will adopt the special assumption that D

is an isotropic bilinear function of E and T. With the understanding that

E and T

are written as Ekl and Tmn, respectively, we then have [19]

Dijklmn = β1δijδklδmn +12β2(δikδjl + δilδjk)δmn

+12β3((δimδjn + δinδjm)δkl + (δkmδln + δknδlm)δij)

+14β4(δikδlmδjn + δilδkmδjn + δikδlnδjm

+ δilδknδjm + δimδjlδkn + δimδjkδln + δinδjlδkm + δinδjkδlm),

(6.78)

where β1, β2, β3, and β4 are material constants. Assumption (6.78), whilesimplistic in the sense that it could not be expected to be adequatefor a transversely isotropic composite that is strongly anisotropic, servesthe purpose of illustrating all the main issues pertaining to the inverseproblem in question. We refer the reader to Remark 6.4 for furthercomments and to [17] for a full account of the case where D is transverselyisotropic.



Here the acoustoelastic tensor D plays the role of the perturbative partA in (6.64). It follows from (6.78) that the parameters a22, a23, a33, anda44 in (6.68) are given by the following formulas:

a22 = D2222mnTmn = (β1 + β2)(T

11 + T

22) + 2β3T

22 + 2β4T

22, (6.79)

a23 = D2233mnTmn = β1(T

11 + T

22) + β3T

22, (6.80)

a33 = D3333mnTmn = (β1 + β2)(T

11 + T

22), (6.81)

a44 = D2323mnTmn =

12β2(T

11 + T

22) +

12β4T

22. (6.82)

Note that T12 does not appear in these equations.

Let σ1 and σ2 be the principal stresses in the 1–2 plane, and let the1′- and 2′-axes define the corresponding principal directions of the initialstress. Let ψ be the angle between the 1′-axis and the 2-axis. We choosethe labels 1′ and 2′ such that 0 < ψ ≤ π/2. The non-trivial components ofthe initial stress are then given by

T11 =

12(σ1 + σ2) − 1

2(σ1 − σ2) cos 2ψ, T

12 =

12(σ1 − σ2) sin 2ψ,

T22 =

12(σ1 + σ2) +

12(σ1 − σ2) cos 2ψ. (6.83)

Substituting the expressions for T11 and T

22 into Eqs. (6.79)–(6.82),

we have

a22 = (β1 + β2 + β3 + β4)(σ1 + σ2) + (β3 + β4)(σ1 − σ2) cos 2ψ, (6.84)

a23 =(β1 +

12β3

)(σ1 + σ2) +

12β3(σ1 − σ2) cos 2ψ, (6.85)

a33 = (β1 + β2)(σ1 + σ2), (6.86)

a44 =(

12β2 +

14β4

)(σ1 + σ2) +

14β4(σ1 − σ2) cos 2ψ. (6.87)

Substituting the preceding expressions and that for T22 into (6.68), we

obtain for Rayleigh waves propagating in the 2-direction the formula

vR = vTransR +A(σ1 + σ2) +B(σ1 − σ2) cos 2ψ, (6.88)



where vTransR is the solution of the secular equation (6.66) for the unstressed

medium, and

A = − 12ρvTrans

R

(γ22(β1 + β2 + β3 + β4) + γ23

(β1 +

12β3

)

+ γ33(β1 + β2) + γ44

(12β2 +

14β4

)− 1

2

), (6.89)

B = − 12ρvTrans

R

(γ22(β3 + β4) +

12β3γ23 +

14β4γ44 − 1

2

). (6.90)

Now consider Rayleigh waves with propagation direction m = (cos θ,sin θ, 0) and let vR(θ) be the phase velocity. Let ϕ be the angle of rotationabout the 3-axis that will bring the 1- and 2-axes to the 1′- and 2′-directions,respectively. A moment’s reflection will convince the reader that vR(θ) isgiven by (6.88) with the angle ψ = θ − ϕ, namely:

vR(θ) = vTransR +A(σ1 + σ2) +B(σ1 − σ2) cos 2(θ − ϕ). (6.91)

The recovery of ϕ (i.e. the principal directions of T), of σ1 − σ2, and

of σ1 + σ2 from boundary measurements of Rayleigh-wave phase velocitieshave different degrees of difficulty. We will discuss these problems in turnbelow:

(1) Recovery of the principal directions of T. By measuring vR(θ) for

various θ, we can determine ϕ without any knowledge of vTransR , A,

B, σ1, σ2.(2) Recovery of σ1 − σ2. Measurement of vR(θ) alone will not suffice for

determination of σ1 − σ2. We will also have to ascertain the value ofthe acoustoelastic constant B. If it is feasible to conduct experimentsin which vR(θ) is measured for various θ while the sample is subjectedto additional (known) applied stresses, then B can be determined.Another possibility is to estimate C22, C23, C33, C44, β3, and β4 throughmicromechanical modeling or some homogenization scheme and thenuse (6.90) to calculate B.

(3) Recovery of σ1 + σ2. Even if we can perform experiments such asthose described in item (2) above, we still cannot recover σ1 + σ2

from measurement of vR(θ) unless vTransR has already been found.

If the initial stress T

is a result of the manufacturing process of thecomposite material, then it may not be possible to prepare unstressedsamples with the same C as that of the composite material. In that



case, getting theoretical estimates of C22, C23, C33, and C44 throughmicromechanical modeling or homogenization would be the only waythat might lead us out of this difficulty. Since we have been treatingthe effects of the initial stress T

as a first-order perturbation, we have

assumed a priori that the size of the term A(σ1 + σ2) is at least anorder of magnitude smaller than vTrans

R . For the approach outlined hereto work, the measured value of vR(θ) and the theoretical estimate ofvTransR should be sufficiently accurate and the size of the acoustoelastic

constant A sufficiently large so that the errors involved do not maskthe contribution of σ1 + σ2 when this sum is at significant levels.

Remark 6.4. As mentioned above, for transversely isotropic compositesthat are strongly anisotropic, the isotropic assumption (6.78) for D will beinadequate. It should be replaced by its transversely isotropic counterpart,which has 16 material constants (instead of four for the isotropic case).The procedure that leads to formula (6.88) for vR and the formula itself,however, remain valid, although the parameters A and B are now givenby somewhat more complicated formulas that involve, besides elasticcoefficients from C, seven material constants from D (see [17] for details).

The discussion above on the recovery of the principal directions of T, of

σ1 − σ2, and of σ1 + σ2 from boundary measurements of Rayleigh-wavephase velocities requires only minor modifications that result from thefact that the formulas for A and B now involve more material constants.In fact, the sheer number of such constants might make their experimentaldetermination arduous or even unfeasible. A potentially more practicalapproach is to design experiments based on formula (6.88) to measure theacoustoelastic constants A and B directly. Alternatively, one may considerdeveloping micromechanical models or homogenization schemes by whichthe material parameters of the homogenized material that appear in C andD can be estimated from the structure of the composite and the mechanicalproperties of its constituents.

Acknowledgements

We are grateful to Prof Vladislav Mantic, who kindly invited us tosubmit this chapter and who gave us many helpful suggestions aboutthe literature on composite materials. We also thank the reviewers whoread our manuscript carefully and gave us many comments. The work ofTanuma was partly supported by Grant-in-Aid for Scientific Research (C)



(No. 22540111), Society for the Promotion of Science, Japan. The researchefforts of Man were supported in part by a grant (No. DMS-0807543) fromthe US National Science Foundation.

References

[1] Rose J.L., 1999. Ultrasonic Waves in Solid Media, Cambridge UniversityPress, Cambridge.

[2] Datta S.K., 2000. Wave propagation in composite plates and shells, inComprehensive Composite Materials, Vol. 1 (A. Kelly, C. Zweben, eds),Elsevier, Amsterdam, pp. 511–558.

[3] Ostoja-Starzewski M., 2008. Microstructural Randomness and Scaling inMechanics of Materials, Chapman & Hall/CRC, Taylor & Francis Group.

[4] Datta S.K., Shah A.H., 2009. Elastic Waves in Composite Media andStructures: With Applications to Ultrasonic Nondestructive Evaluation, CRCPress, Taylor & Francis Group.

[5] Datta S.K., Ledbetter H.M., 1983. Elastic constants of fiber-reinforced boron-aluminum: Observation and theory, Int. J. Solids Struct., 19, 885–894.

[6] Stroh A.N., 1958. Dislocations and cracks in anisotropic elasticity, Phil. Mag.,3, 625–646.


[8] Lothe J., Barnett D.M., 1976. On the existence of surface-wave solutions foranisotropic elastic half-spaces with free surface, J. Appl. Phys., 47, 428–433.

[9] Chadwick P., Smith G.D., 1977. Foundations of the theory of surface wavesin anisotropic elastic materials, Adv. Appl. Mech., 17, 303–376.

[10] Barnett D.M., Lothe J., 1985. Free surface (Rayleigh) waves in anisotropicelastic half-spaces: the surface impedance method, Proc. R. Soc. Lond. A,402, 135–152.

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[12] Tanuma K., 2007. Stroh formalism and Rayleigh waves, J. Elasticity, 89,5–154.

[13] Tanuma K., Man, C.-S., 2006. Perturbation formula for phase velocity ofRayleigh waves in prestressed anisotropic media, J. Elasticity, 85, 21–37.

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[15] Nayfeh A.H., 1995. Wave Propagation in Layered Anisotropic Media withApplications to Composites, North-Holland, Amsterdam.

[16] Wu T.-T., Liu Y.-H., 1999. On the measurement of anisotropic elasticconstants of fiber-reinforced composite plate using ultrasonic bulk wave andlaser generated Lamb wave, Ultrasonics, 37, 405–412.

[17] Tanuma K., Man C.-S., Du W., 2013. Perturbation of phase velocity ofRayleigh waves in pre-stressed anisotropic media with orthorhombic principalpart, Math. Mech. Solids, 18, 301–322.



[18] Man C.-S., Lu W.Y., 1987. Towards an acoustoelastic theory formeasurement of residual stress, J. Elasticity, 17, 159–182.

[19] Man C.-S., 1998. Hartig’s law and linear elasticity with initial stress, InverseProblems, 14, 313–319.

[20] Tanuma K., Man C.-S., 2008. Perturbation formulas for polarizationratio and phase shift of Rayleigh waves in prestressed anisotropic media,J. Elasticity, 92, 1–33.

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Chapter 7

ADVANCED MODEL ORDER REDUCTIONFOR SIMULATING COMPOSITE-FORMING

PROCESSES

Francisco Chinesta, Adrien Leygue and Arnaud Poitou

EADS Corporate Foundation International Chair“Advanced Modeling of Composite-Manufacturing Processes”

GEM, UMR CNRS-ECN,1 rue de la Noe, BP 92101, F-44321 Nantes cedex 3, France

Abstract

Efficient simulation of composite-manufacturing processes remains a challeng-ing issue today despite the impressive progress made in mechanical modeling,numerical analysis, discretization techniques, and computer science during thelast decade. However, composite manufacturing involves multiscale models inspace and time, highly non-linear and anisotropic behavior, strongly coupledmultiphysics, and complex geometries. Moreover, optimization (shape and pro-cess optimization), inverse analysis (parameter identification, non-destructivetesting, . . . ), and process control all need solutions from many direct problems,as fast and accurately as possible. In this context, model reduction techniquesconstitute an appealing simulation choice, making it possible to speed upcomputations by several orders of magnitude and even to solve previouslyunsolved models.

7.1 Introduction

Composite-manufacturing processes involve many different physics. Impreg-nation of reinforcements needs the efficient solution of models of flowsthrough porous media that can deform due to the flow action. Severalissues exist when one considers such flows: their three-dimensional (3D)nature in the presence of anisotropic plies with different orientations, theprediction of the permeability for each layer but also for the laminate, themodeling of impregnation that is at the origin of the residual porosity,

247


248 F. Chinesta, A. Leygue and A. Poitou

among many others. Another aspect concerns resin-curing kinetics and itsthermomechanical couplings for thermosets or the cooling process of ther-moplastics, which for semicrystalline polymers also involves crystallization.The microstructural morphologies affect the macroscopic thermomechanicalbehavior of the formed parts. In some cases an amount of thermoplasticis added to thermoset resins in order to improve resistance to impacts.In that case another physics is involved: namely, that concerning thedemixing that results in a discrete thermoplastic phase transforming intoa continuous thermoset phase. Again, the microstructural morphology isa key point in understanding the resulting macroscopic behavior includingdamage mechanics. Residual stresses are the main manifestation of all thesephysics and its prediction is a key point when optimizing and/or controllingprocesses in order to limit the magnitude of the residual stress-induceddeformations.

All these coupled physics coexist and exhibit multiscale and localizedbehaviors in space and time. Due to the multiscale description, micro-macro modeling is mandatory, and appropriate inter-scale bridges, otherthan classical homogenization, must be defined. Another important issueconcerns the nature of the macroscopic models defined in plate or shelldomains characterized by having a dimension (the thickness) several ordersof magnitude lower than the other representative in-plane dimensions.This fact, even if not a major conceptual issue, is a real nightmare forsimulation purposes. This situation is not new: plate and shell theorieswere successfully developed many years ago and they have been intensivelyused in structural mechanics. These theories make use of kinematic andstatic hypotheses to reduce the 3D nature of mechanical models to two-dimensional (2D) reduced models defined at the shell or plate middlesurface. In the case of elastic behaviors the derivation of such reducedmodels is quite simple and it constitutes the foundations of classical plateand shell theories. Today, most commercial codes for structural mechanicsapplications have different types of plate and shell finite elements, even inthe case of multilayered composites plates or shells.

However, in composite-manufacturing processes the physics encoun-tered in such multilayered plate or shell domains is much more rich, because,as previously indicated, it involves chemical reactions, crystallization,and strongly coupled thermomechanical behaviors. The complexity of theinvolved physics makes it impossible to introduce pertinent hypotheses forreducing the dimensionality of the model from 3D to 2D. In that case fully3D modeling is compulsory, and because of the richness of the thickness


Advanced Model Order Reduction for Simulating Composite-Forming Processes 249

description (many coupled physics and many plies differently oriented)the approximation of the fields involved in the models needs thousands ofnodes distributed along the thickness direction. Thus, fully 3D descriptionsinvolve thousands of millions of degrees of freedom that should be solvedmany times because of the history-dependent thermomechanical behavior.Moreover, when we are considering optimization or inverse identification,many direct problems have to be solved in order to minimize the costfunction. In the case of inverse analysis the cost function is the differencebetween the predicted and measured fields.

Today, the solution of such fully 3D models involved in composite-manufacturing processes remains intractable despite the impressive progressmade in mechanical modeling, numerical analysis, discretization techniques,and computer science during the last decade. New numerical techniquesare needed for such complex scenarios, able to proceed to the solution offully 3D multiphysics models in geometrically complex parts (e.g. wholeaircraft). The well-used mesh-based discretizations techniques fail becausethe excessive number of degrees of freedom involved in the fully 3Ddiscretizations where very fine meshes are required in the thickness direction(despite its reduced dimension) and also in the in-plane directions toavoid meshes that are too distorted and also because some processes (e.g.tape placement) imply thermomechanical loads moving on the plate orshell requiring sufficiently fine meshes for the plate or shell surfaces. Weare at a real impasse. The only getaway is to explore new discretizationstrategies able to circumvent or at least alleviate the drawbacks of mesh-based discretizations of fully 3D models defined in plate or shell domains,as well as the complex plate and shell assemblages usually encountered incomposite structures.

Another important issue encountered in the simulation of compositemanufacturing is related to process control and optimization. In general,optimization implies the definition of a cost function and the search for theoptimum process parameters defining the minimum of that cost function.The procedure starts by choosing a tentative set of process parameters.Then the process is simulated by discretizing it. Finding the solution ofthe model is the most costly step of the optimization procedure. As soonas the solution is available, the cost function can be evaluated and itsoptimality checked. If the chosen parameters do not define a minimum(at least local) of the cost function, the process parameters should beupdated and the solution recomputed. The procedure continues until theminimum of the cost function is reached. Obviously, nothing ensures that



the minimum is global, so more sophisticated procedures exist in orderto explore the domain defined by the parameters to escape from localminimum traps. Parameter updating is carried out to ensure the highestvariation of the cost function, i.e. in order to move along the directionof the cost function gradient. However, to identify the direction of thegradient one should compute not only the fields involved in the processmodel but also the derivatives of fields with respect to the different processparameters. The evaluation of these derivatives is not in general an easytask. Conceptually, one could imagine that by perturbing slightly only oneof the parameters involved in the process optimization and then solvingthe resulting model, one could estimate using a finite difference formula thederivative of the cost function with respect to the perturbed parameter. Byperturbing sequentially all the parameters we could find the derivatives ofthe cost function with respect to all the process parameters (also knownas sensibilities) defining the cost function gradient direction on which thenew trial set of parameters should be chosen. There are many strategiesfor updating the set of process parameters and the interested reader canfind most of them in the books focusing on optimization procedures.Our interest here is not the analysis of optimization strategies, but onlypointing out the necessity of reducing the number of direct solutions ofthe model. As we discussed in the previous paragraphs the solution ofthe process model is a tricky task that demands significant computationalresources and usually implies extremely large computing times. The usualoptimization procedures are inapplicable because they need numeroussolutions of the problem defining the model of the process, one for eachtrial set of process parameters. The same issues are encountered whendealing with inverse analysis in which material or process parameters areexpected to be identified from numerical simulation, by looking for theunknown parameters such that computed fields agree with those measuredexperimentally.

Until now the getaway has consisted of using more and more powerfulcomputing platforms (parallel and distributed computing architectures,GPUs, acceleration techniques based on the preconditioning of the alge-braic systems obtained after discretization, multigrid and multidomaintechniques, among many others). But the verdict is implacable: all thesetechniques are not enough, neither today nor will they be in the next decade,when the complexity of the models is sufficient for industrial interest. Thebrute force approach is no longer a valuable alternative and consequentlynew proposals are urgently needed.



One alternative is the use of model reduction strategies. Model reduc-tion is based on the fact that the solution of many models contains muchless information than the one assumed a priori when the discrete modelwas built. Below we illustrate that the solution of some models usuallyencountered in computational engineering and science only involves areduced amount of information. For this purpose we will revisit the so-calledproper orthogonal decomposition (POD, also known as the Karhunen–Loeve decomposition), which will be illustrated by the analysis of anacademic problem: the solution of the transient heat transfer equation.The main drawback in using such a strategy lies in the fact that somea priori knowledge is compulsory in order to define the reduced model thatcould then be applied for solving ‘similar’ problems. However, because PODleads to a separated representation of the problem solution, this separateddescription could be postulated a priori and then the functions involved inthe decomposition calculated. This idea is the heart of the so-called propergeneralized decomposition (PGD), which will be revisited in Section 7.3and then applied for solving different problems encountered in manufac-turing composite structures. Process optimization and inverse analysis areaddressed in Section 7.4. The coupling between global transport modelsand local kinetics in a multiscale framework is revisited in Section 7.5.Finally, Section 7.6 focuses on the fully 3D solution of models defined inplate geometries.

7.2 Model Reduction: Information Versus RelevantInformation

Consider a mesh having M nodes, and associate with each node anapproximation function (e.g. a shape function in the framework of finiteelements); we implicitly define an approximation space wherein a discretesolution of the problem is sought. For a transient problem, one must thuscompute M values (the nodal values in the finite element framework) ateach time step. For non-linear problems, this implies the solution of at leastone linear algebraic system of size M at each time step, which becomescomputationally expensive when M increases.

In many cases, however, the problem solution lives in a subspace ofdimension much smaller than M , and it makes sense to look for a reduced-order model whose solution is computationally much cheaper to obtain.This constitutes the main idea behind the POD reduced modeling approach,which we revisit below.



7.2.1 Extracting relevant information: proper orthogonal

decomposition

We assume that the field of interest u(x, t) is known at the nodes xi ofa spatial mesh for discrete times tm = m · ∆t, with i ∈ [1, · · · ,M ] andm ∈ [0, · · · , P ]. We use the notation u(xi, tm) ≡ um(xi) ≡ umi and defineum as the vector of nodal values umi at time tm. The main objective ofPOD is to obtain the most typical or characteristic structure φ(x) from theum(x), ∀m [1]. For this purpose, we maximize the scalar quantity

α =

∑Pm=1

[∑Mi=1 φ(xi)um(xi)

]2∑M

i=1(φ(xi))2, (7.1)

which amounts to solving the following eigenvalue problem:

cφ = αφ. (7.2)

Here, the vector φ has i-component φ(xi) and c is the two-pointcorrelation matrix

cij =P∑

m=1

um(xi)um(xj) =P∑

m=1

um · (um)T , (7.3)

which is symmetric and positive definite. With the matrix Q defined as

Q =

u11 u2

1 · · · uP1

u12 u2

2 · · · uP2

......

. . ....

u1M u2

M · · · uPM

, (7.4)

we have

c = Q ·QT . (7.5)

7.2.2 Building the POD reduced-order model

In order to obtain a reduced model, we first solve the eigenvalue problemEq. (7.2) and select the N eigenvectors φi associated with the eigenvaluesbelonging to the interval defined by the highest eigenvalue α1 and α1 dividedby a large enough number (e.g. 108). In practice, N is found to be much



lower than M . These N eigenfunctions φi are then used to approximatethe solution um(x), ∀m. Thus, let us define the matrix B = [φ1 · · ·φN ], i.e.

B =

φ1(x1) φ2(x1) · · · φN (x1)

φ1(x2) φ2(x2) · · · φN (x2)

......

. . ....

φ1(xM ) φ2(xM ) · · · φN (xM )

. (7.6)

Now, let us assume for illustrative purposes that an explicit time-stepping scheme is used to compute the discrete solution um+1 at timetm+1. One must thus solve a linear algebraic system of the form

Gm um+1 = Hm. (7.7)

A reduced-order model is then obtained by approximating um+1 in thesubspace defined by the N eigenvectors φi, i.e.

um+1 ≈N∑i=1

φiζm+1i = B ζm+1. (7.8)

Equation (7.7) then reads

Gm B ζm+1 = Hm, (7.9)

and multiplying both terms by BT

BTGm B ζm+1 = BTHm. (7.10)

The coefficients ζm+1 defining the solution of the reduced-order modelare thus obtained by solving an algebraic system of size N instead of M .When N M , as is the case in numerous applications, the solution ofEq. (7.10) is thus preferred because of its much reduced size.

Remark 7.2.1. The reduced-order model Eq. (7.10) is built a posterioriby means of the already computed discrete field evolution. Thus, onecould wonder about the point of the whole exercise. In fact, two beneficialapproaches are widely considered (see e.g. [1–8]). The first approachconsists in solving the large original model over a short time interval, thusallowing for the extraction of the characteristic structure that defines thereduced model. The latter is then solved over larger time intervals, with



the associated computing time savings. The other approach consists insolving the original model over the entire time interval, and then usingthe corresponding reduced model to solve similar problems very efficiently,for example, with slight variations in material parameters or boundaryconditions.

7.2.3 Illustrating the construction of a reduced-order model

We consider the following 1D heat transfer problem, written in dimension-less form:

∂u

∂t= λ

∂2u

∂x2, (7.11)

with constant thermal diffusivity λ = 0.01, t ∈ (0, 30], and x ∈ (0, 1). Theinitial condition is u(x, t = 0) = 1 and the boundary conditions are givenby ∂u

∂x|x=0,t = q(t) and ∂u

∂x|x=1,t = 0.

Equation (7.11) is discretized by using an implicit finite element methodon a mesh with M = 100 nodes, where a linear approximation is definedon each of the Me = 99 elements. The time step is set to ∆t = 0.1. Theresulting discrete system can be written as:

K um+1 = M um + qm+1, (7.12)

where the vector qm+1 accounts for the boundary heat flux source at tm+1.First, we consider the following boundary heat source:

q(t) =

1 t ≤ 10

0 t > 10. (7.13)

The computed temperature profiles are depicted in Fig. 7.1 at dis-crete times tm =m, for m=1, 2, . . . , 30. The red curves correspond tothe heating stage up to t=10, while the blue curves for t> 10 illustratethe heat transfer by conduction from the warmest zones towards thecoldest ones.

From these 30 discrete temperature profiles, we compute the matricesQ and c in order to build the eigenvalue problem (7.2). The three largesteigenvalues are found to be α1 = 1790, α2 = 1.1, α3 = 0.1, while theremaining eigenvalues are such that αj < α1 · 10−8, 4 ≤ j ≤ 100. A reducedmodel involving a linear combination of the three eigenvectors for the firstthree largest eigenvalues should thus be able to approximate the solutionwith great accuracy. In order to account for the initial condition, it is



0 0.2 0.4 0.6 0.8 11

1.05

1.1

1.15

1.2

1.25

1.3

1.35

1.4

X

Tem

pera

ture

Fig. 7.1. Temperature profiles corresponding to the source term Eq. (7.13) at discretetimes tm = m, for m = 1, 2, . . . , 30. The red curves correspond to the heating stage upto t = 10, while the blue curves for t > 10 illustrate the heat transfer by conduction

from the warmest zones towards the coldest ones.

convenient to include it in the approximation basis (even though it isthen no longer orthogonal). Figure 7.2 shows the resulting approximationfunctions in normalized form, i.e. φj

‖φj‖ (j = 1, 2, 3) and u0

‖u0‖ . Defining thematrix B as follows

B =[

u0

‖u0‖φ1

‖φ1‖φ2

‖φ2‖φ3

‖φ3‖], (7.14)

we obtain the reduced model for Eq. (7.12),

BTKB ζm+1 = BTMB ζm + BTqm+1, (7.15)

which involves four degrees of freedom only. The initial condition in thereduced basis is (ζ0)T = (1, 0, 0, 0).

Equation (7.15) and the relationship um+1 =Bζm+1 then yield approx-imate solution profiles at a very low cost indeed. The results are shown inFig. 7.3 and they cannot be distinguished at the scale of the drawing fromthose of the complete problem (7.12).

In order to illustrate our Remark 1.2.1, let us now use the reducedmodel (7.15) as is to solve a problem different from the one that was used



0 0.2 0.4 0.6 0.8 1−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

X

Nor

mal

ized

eig

enfu

nctio

nsInitial temperatureFirst eigenfunctionSecond eigenfunctionThird eigenfunction

Fig. 7.2. Reduced-order approximation basis involving the initial condition and theeigenvectors corresponding to the three largest eigenvalues.

0 0.2 0.4 0.6 0.8 10.9

0.95

1

1.05

1.1

1.15

1.2

1.25

1.3

1.35

1.4

X

Tem

pera

ture

Fig. 7.3. Global (continuous line) versus reduced-order (symbols) model solutions.



0 0.2 0.4 0.6 0.8 1

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

X

Tem

pera

ture

Fig. 7.4. Global (continuous line) versus reduced-order (symbols) model solutions forthe source term Eq. (7.16). The reduced-order approximation basis is that obtained fromthe solution of a different thermal problem, with the source term Eq. (7.13).

to derive it. While keeping all other specifications identical, we now imposeinstead of (7.13) a substantially different boundary heat source:

q(t) =

t

20t ≤ 20

t− 305

t > 20. (7.16)

The solution of the reduced model is compared to that of the completeproblem in Fig. 7.4. Even though the reduced approximation basis functionsare those obtained from the thermal model related to the boundarycondition (7.13), the reduced model yields a very accurate representationof the solution of this rather different problem.

7.2.4 Discussion

The above example illustrates the significant value of model reduction. Ofcourse, one would ideally want to be able to build a reduced-order approx-imation a priori, i.e. without relying on a knowledge of the (approximate)solution of the complete problem. One would then want to be able to assess



the accuracy of the reduced-order solution and, if necessary, to enrich thereduced approximation basis in order to improve accuracy (see e.g. ourearlier studies [9] and [1]). The proper generalized decomposition (PGD),which we describe in general terms in the next section, is an efficient answerto these questions.

The above POD results also tell us that an accurate approximatesolution can often be written as a separated representation involving fewterms. Indeed, when the field evolves smoothly, the magnitude of the(ordered) eigenvalues αi decreases very fast with increasing index i, andthe evolution of the field can be approximated from a reduced number ofmodes. Thus, if we define a cutoff value ε (e.g. ε = 10−8 · α1, α1 being thehighest eigenvalue), only a small numberN of modes are retained (N M)such that αi ≥ ε, for i ≤ N , and αi < ε, for i > N . Thus, one canwrite:

u(x, t) ≈N∑i=1

φi(x) · Ti(t) ≡N∑i=1

Xi(x) · Ti(t). (7.17)

For the sake of clarity, the space modes φi(x) will be denoted inthe following as Xi(x). Equation (7.17) represents a natural separatedrepresentation, also known as a finite sum decomposition. The solution,which depends on space and time, can be approximated as a sum of asmall number of functional products, with one of the functions dependingon the space coordinates and the other on time. The use of separatedrepresentations like (7.17) is at the heart of the PGD.

Thus, we expect that the transient solution of numerous problemsof interest can be expressed using a very reduced number of functionalproducts each involving a function of time and a function of space. Ideally,the functions involved in these functional products should be determinedsimultaneously by applying an appropriate algorithm to guarantee robust-ness and optimality; in view of the non-linear nature of the separatedrepresentation, this will require a suitable iterative process.

To our knowledge, the unique precedent to the PGD algorithm forbuilding a separated space-time representation is the so-called radialapproximation introduced by Ladeveze [10–12] in the context of computa-tional solid mechanics.

In terms of performance, the verdict is simply impressive. Consider atypical transient problem defined in 3D physical space. Use of a standardincremental strategy with P time steps (P is of the order of millions in



industrial applications) requires the solution of P 3D problems. By contrast,using the space-time separated representation (7.17), we must solve N ·m3D problems for computing the space functions Xi(x), and N · m 1Dproblems for computing the time functions Ti(t). Here, m is the numberof non-linear iterations needed for computing each term of the finite sum.For many problems of practical interest, we find that N ·m is of order 100.The computational time savings afforded by the separated representationcan thus reach many orders of magnitude.

7.3 Proper Generalized Decomposition at a Glance

Consider a problem defined in a space of dimension d for the unknown fieldu(x1, · · · , xd). Here, the coordinates xi denote any usual coordinate (scalaror vectorial) related to physical space, time, or conformation space, forexample, but they could also include problem parameters such as boundaryconditions or material parameters. We seek a solution for (x1, · · · , xd) ∈Ω1 × · · · × Ωd.

PGD yields an approximate solution in the separated form:

u(x1, · · · , xd) ≈N∑i=1

F 1i (x1) × · · · × F di (xd). (7.18)

The PGD approximation is thus a sum of N functional products eachinvolving a number d of functions F ji (xj) that are unknown a priori. It isconstructed by successive enrichment, whereby each functional product isdetermined in sequence. At a particular enrichment step n+1, the functionsF ji (xj) are known for i ≤ n from the previous steps, and one must computethe new product involving the d unknown functions F jn+1(xj). This isachieved by invoking the weak form of the problem under consideration.The resulting discrete system is non-linear, which implies that iterationsare needed at each enrichment step. A low-dimensional problem can thusbe defined in Ωj for each of the d functions F jn+1(xj).

If M nodes are used to discretize each coordinate, the total numberof PGD unknowns is N × M × d instead of the Md degrees of freedominvolved in standard mesh-based discretizations. Moreover, all numericalexperiments carried out to date with PGD show that the number of termsN required to obtain an accurate solution is not a function of the problemdimension d, but instead depends on the regularity of the exact solution.PGD thus avoids the exponential complexity with respect to the problemdimension.



In many applications studied to date, N is found to be as small as afew tens, and in all cases the approximation converges towards the solutionassociated with the complete tensor product of the approximation basesconsidered in each Ωj . Thus, we can be confident about the generality of theseparated representation (7.18), but its optimality depends on the solutionregularity and the nature of the differential operator. When an exactsolution of a particular problem can be represented with enough accuracyby a reduced number of functional products, the PGD approximation isoptimal. If the solution is a strictly non-separable function, the PGD solverproceeds to enrich the approximation until it includes all the elements of thefunctional space, i.e. the Md functions involved in the full tensor productof the approximation bases in each Ωj .

Let us now consider in more detail a specific example.

7.3.1 Proper generalized decomposition of a generic

parametric model

In this section, we illustrate PGD by considering the following parametricheat transfer equation:

∂u

∂t− k∆u− f = 0. (7.19)

Here (x, t, k) ∈ Ω× I ×, and the source term f is assumed constant.The conductivity k is viewed as a new coordinate defined in the interval .Thus, instead of solving the thermal model for different discrete values ofthe conductivity parameter, we will solve a more general problem, the priceto pay being an increase of the problem dimensionality. However, as thecomplexity of PGD scales only linearly (and not exponentially) with thespace dimension, consideration of the conductivity as a new coordinate stillallows one to efficiently obtain an accurate solution.

The weak form of Eq. (7.19) is:

∫Ω×I×

u∗ ·(∂u

∂t− k∆u− f

)dx dt dk = 0, (7.20)

for all test functions u∗ selected in an appropriate functional space.The PGD solution is sought in the form:

u (x, t, k) ≈N∑i=1

Xi(x) · Ti(t) ·Ki(k). (7.21)



At enrichment step n of the PGD algorithm, the following approxima-tion is already known:

un(x, t, k) =n∑i=1

Xi(x) · Ti(t) ·Ki(k). (7.22)

We wish to compute the next functional product Xn+1(x) · Tn+1(t) ·Kn+1(k), which we write as R(x) · S(t) ·W (k) for notational simplicity.

Thus, the solution at enrichment step n+ 1 is

un+1 = un +R(x) · S(t) ·W (k). (7.23)

Inspired by the calculus of variations, we propose the simplest choicefor the test functions u∗ in Eq. (7.20):

u∗ = R∗(x) · S(t) ·W (k) +R(x) · S∗(t) ·W (k) +R(x) · S(t) ·W ∗(k).

(7.24)

With the trial and test functions given by Eqs. (7.23) and (7.24)respectively, Eq. (7.20) is a non-linear problem that must be solved bymeans of a suitable iterative scheme. In our earlier papers [13] and [14], weused Newton’s method. Simpler linearization strategies can also be applied,however. The simplest one is an alternating direction, fixed-point algorithm,which was found remarkably robust in the present context. Each iterationconsists of three steps that are repeated until reaching convergence, thatis, until reaching the fixed point. The first step assumes S(t) and W (k)are known from the previous iteration and computes an update for R(x)(in this case the test function reduces to R∗(x) · S(t) · W (k)). From thejust-updated R(x) and the previously-used W (k), we can update S(t)(with u∗ = R(x) · S∗(t) · W (k)). Finally, from the just-computed R(x)and S(t), we update W (k) (with u∗ = R(x) · S(t) ·W ∗(k)). This iterativeprocedure continues until convergence. The converged functions R(x), S(t),and W (k) yield the new functional product of the current enrichment step:Xn+1(x) = R(x), Tn+1(t) = S(t), and Kn+1(k) = W (k). The explicit formof these operations is described below.

Computing R(x) from S(t) and W (k):

We consider the weak form of equation (7.19):∫Ω×I×

u∗(∂u

∂t− k∆u− f

)dx dt dk = 0. (7.25)



Here, the trial function is given by

u(x, t, k) =n∑i=1

Xi(x) · Ti(t) ·Ki(k) +R(x) · S(t) ·W (k). (7.26)

Since S and W are known from the previous iteration, the test functionreads

u∗(x, t, k) = R∗(x) · S(t) ·W (k). (7.27)

Introducing (7.26) and (7.27) into (7.25) yields

∫Ω×I×

R∗ · S ·W ·(R · ∂S

∂t·W − k · ∆R · S ·W

)dx dt dk

= −∫

Ω×I×R∗ · S ·W · Rn dx dt dk, (7.28)

where Rn is the residual at enrichment step n:

Rn =n∑i=1

Xi·∂Ti∂t

·Ki −n∑i=1

k · ∆Xi · Ti ·Ki − f. (7.29)

Since all functions involving time and conductivity have been deter-mined, we can integrate Eq. (7.28) over I × . With the followingnotation,

w1 =∫W 2dk s1 =

∫I

S2dt r1 =∫

Ω

R2dx

w2 =∫kW 2dk s2 =

∫I

S · dSdtdt r2 =

∫Ω

R · ∆R dx

w3 =∫W dk s3 =

∫I

S dt r3 =∫

Ω

R dx

wi4 =∫W ·Ki dk si4 =

∫I

S · dTidtdt ri4 =

∫Ω

R · ∆Xi dx

wi5 =∫kW ·Ki dk si5 =

∫I

S · Ti dt ri5 =∫

Ω

R ·Xi dx

(7.30)



Equation (7.28) reduces to∫Ω

R∗ · (w1 · s2 ·R − w2 · s1 · ∆R) dx

= −∫Ω

R∗ ·(

n∑i=1

wi4 · si4 ·Xi −n∑i=1

wi5 · si5 · ∆Xi − w3 · s3 · f)dx.

(7.31)

Equation (7.31) defines in weak form an elliptic steady-state boundary-value problem for the unknown function R, which can be solved by usingany suitable discretization technique (finite elements, finite volumes, . . . ).Another possibility consists of returning to the strong form of Eq. (7.31):

w1 · s2 · R− w2 · s1 · ∆R

= −(

n∑i=1

wi4 · si4 ·Xi −n∑i=1

wi5 · si5 · ∆Xi − w3 · s3 · f), (7.32)

which can be solved by using any classical collocation technique (finitedifferences, SPH, . . . ).

Computing S(t) from R(x) and W (k):

In the present case, the test function is written as

u∗(x, t, k) = S∗(t) ·R(x) ·W (k), (7.33)

and the weak form becomes∫Ω×I×

S∗ ·R ·W ·(R · ∂S

∂t·W − k · ∆R · S ·W

)dx dt dk

= −∫

Ω×I×S∗ ·R ·W · Rn dx dt dk. (7.34)

Integrating over Ω ×, one obtains∫I

S∗ ·(w1 · r1 · dS

dt− w2 · r2 · S

)dt

= −∫I

S∗ ·(

n∑i=1

wi4 · ri5 ·dTidt

−n∑i=1

wi5 · ri4 · Ti − w3 · r3 · f)dt.

(7.35)



Equation (7.35) represents the weak form of the ordinary differentialequation (ODE) defining the time evolution of the field S, which can besolved by using any stabilized discretization technique (streamline upwind,discontinuous Galerkin, . . . ). The strong form of Eq. (7.35) is

w1 · r1 · dSdt

− w2 · r2 · S

= −(

n∑i=1

wi4 · ri5 ·dTidt

−n∑i=1

wi5 · ri4 · Ti − w3 · r3 · f). (7.36)

Equation (7.36) can be solved by using backward finite differences orhigher-order Runge–Kutta schemes, among many other possibilities.

Computing W (k) from R(x) and S(t):

The test function is now given by

u∗(x, t, k) = W ∗(k) · R(x) · S(t), (7.37)

and the weak form becomes∫Ω×I×

W ∗ ·R · S ·(R · ∂S

∂t·W − k · ∆R · S ·W

)dx dt dk

= −∫

Ω×I×W ∗ ·R · S · Rn dx dt dk. (7.38)

Integration over Ω × I yields∫W ∗ · (r1 · s2 ·W − r2 · s1 · k ·W ) dk

= −∫W ∗ ·

(n∑i=1

ri5 · si4 ·Ki −n∑i=1

ri4 · si5 · k ·Ki − r3 · s3 · f)dk.

(7.39)

Equation (7.39) does not involve any differential operator. The corre-sponding strong form reads

(r1 · s2 − r2 · s1 · k) ·W = −(

n∑i=1

(ri5 · si4 − ri4 · si5 · k) ·Ki − r3 · s3 · f).

(7.40)



This is an algebraic problem, which is hardly a surprise since theoriginal equation (7.19) does not contain derivatives with respect to theparameter k. Introduction of the parameter k as an additional modelcoordinate does not increase the cost of a particular enrichment step.It does, however, necessitate more enrichment steps, i.e. more terms(higher N) in the decomposition (7.21).

7.3.2 Discussion

The construction of each term in Eq. (7.21) needs a certain numberof iterations because of the non-linearity of the problem related to theapproximation given by Eq. (7.21). Denoting bymi the number of iterationsneeded for computing the i-th sum in Eq. (7.21), let m =

∑i=Ni=1 mi be the

total number of iterations involved in the construction of the separatedapproximation, Eq. (7.21). It is easy to note that the solution procedureinvolves the solution of m 3D problems in the construction of the spacefunctions Xi(x), i = 1, · · · , N ; m 1D ordinary differential equations inthe construction of functions Ti(t), and finally m diagonal linear systemsrelated to the definition of functions Ki(k). In general m rarely exceedsten. On the other hand, the number N of sums needed to approximate thesolution of a given problem depends on the solution regularity itself butall experiments carried out so far reveal that this number ranges from afew tens to slightly more than one hundred. Thus, we can conclude thatthe complexity of the solution procedure is of some tens of 3D solutions(the cost for the 1D problems being negligible with respect to the one forthe 3D problems). In contrast, if we follow a classical approach, we wouldsolve a 3D problem at each time step and for each value of the parameter k.In common applications the complexity can easily reach millions of 3Dsolutions. The CPU time savings by applying PGD can be several ordersof magnitude.

Note also that another possibility exists consisting of the separation ofthe 3D physical space into a sequence of 1D ones:

u(x, t, k) ≈i=N∑i=1

Xi(x) · Yi(y) · Zi(z) · Ti(t) ·Ki(k). (7.41)

This possibility reduces drastically the complexity mentioned before,allowing the solution of models involving hundreds of dimensions and theequivalent of 10300 degrees of freedom [14] in a few minutes using a standard



laptop. Techniques in this framework for non-parallelepipedic domains inthree or more dimensions have been analyzed in [15].

7.4 Applying PGD in Optimization and Inverse Analysis

In this section we analyze the applicability of PGD to the field ofoptimization and inverse analysis, by introducing all design parameters, inthe case of optimization, or the sources of uncertainty, in the case of inverseanalysis, as new extra coordinates, and then employing the techniquedescribed in the previous section for solving the resulting multidimensionalmodel. By solving the resulting multidimensional model only once, we candetermine the value of the field of interest for any values of the designparameters in optimization problems or the uncertain parameters in inverseproblems.

7.4.1 Process optimization through the example

of pultrusion processes

We begin by briefly describing the pultrusion process in order to introducethe difficulties usually encountered while optimizing such a process. Pultru-sion is a continuous process for producing constant cross-sectional profilecomposites. During this process, the fiber reinforcements are saturated withresin, and are then pulled through a heated die. The resin gradually curesinside the die while generating heat. At the exit, pullers draw the compositeout and a traveling saw cuts it to the designed length. This process issketched in Fig. 7.5.

Fig. 7.5. Pultrusion process. FBG denotes fiber Bragg grating, and CFRP denotescarbon-fiber-reinforced plastic.



For decades, engineers relied on experience to define the optimalparameters for pultrusion processes. The parameters in the thermal modelfor pultrusion processes are the temperatures at the different heaters locatedon the die wall (see Fig. 7.5), the inlet temperature, and the profile speed.However, with the fast growing use of this technique [16,17], more effectivemethods should be used in the design of this process. Many numericalmodels have been proposed, each with its own advantages and drawbacksbut most of the simulation procedures solve the thermal model many timesgenerating different values of the process parameters to find the optimum.The resulting 2D steady-state heat transfer equation based on Fourier’slaw was usually solved by using an appropriate discretization technique(finite elements, finite differences, or finite volumes among many otherpossibilities) and the optimum is searched using common deterministicoptimization procedures, genetic algorithms, or neural networks.

Thus, optimizing the process implies defining a cost function thatshould be minimized for the optimal set of parameters (heater temper-atures, inlet temperature, profile velocity, . . . ). The usual optimizationtechniques proceed by choosing a trial set of parameters and then solvingthe resulting model. From that solution an appropriate physically derivedcost function is evaluated and if the stopping criterion is not satisfied (opti-mality criteria), the parameters are updated according to an appropriateoptimization algorithm and then the solution is recomputed. The algorithmproceeds until it reaches a minimum, in general a local one. This procedureimplies numerous solutions with the consequent impact on computing time.More recent strategies based on neural networks and genetic algorithms alsoneed the solution of many direct problems [18] being therefore confrontedwith the same challenging issues as standard deterministic optimizationprocedures.

The strategy proposed here is based upon using the unknown processparameters as new coordinates of the model. In fact, the coordinates, orspace dimensions, represent the (not necessarily physical) location of thesolution. Thus, we can compute the solution of the problem for any valueof the unknown parameters (in a fixed, bounded, interval, of course). Thisconverts the values of the unknown parameters into values of the newdimensions of the space in which the model is established.

This strategy faces a challenging problem if the number of parameters ofthe model increases. It is well known that the number of degrees of freedomfor a mesh-based method (say, finite element, finite difference, . . . ) increasesexponentially with the number of dimensions. Thus, in a squared domain



the number of degrees of freedom grows as the number of degrees of freedomalong each spatial direction to the power of the number of dimensions. Forinstance, in 2D, 100 nodes along each direction give rise to 1002. In 3D,the number of degrees of freedom rises to 1003 and so on. This exponentialincrease of the number of degrees of freedom can be literally out of reachfor today’s computers even if the number of dimensions increases onlymoderately. This phenomenon is known as the curse of dimensionality.

Of course, to deal efficiently with this problem a strategy different frommesh-based methods should be employed. Although efficient techniquesexist for a moderate number of spatial dimensions, such as sparse grid meth-ods [19], these techniques fail when the number of dimensions increases.Here, a different approach has been employed, based on the use of PGD,previously described and deeply analyzed in [20] and the references therein.PGD techniques construct an approximation of the solution by means of asequence of products of separable functions. These functions are determinedon the fly, as the method proceeds, with no initial assumption on their form.This work is not focused on the optimization procedure itself, that is a workin progress, but only on the solution of the parametric multidimensionalmodel using PGD, which allows us to compute the temperature field forany temperature applied by the heaters.

In modeling the pultrusion process, as sketched in Fig. 7.5, we considerthe process within the oven as modeled by the following 2D convection-diffusion equation:

ρc(∂u

∂t+ v

∂u

∂x

)= k∆u+ q, (7.42)

where k is the thermal conductivity, q is the internal heat generated by theresin-curing reaction, ρ is the density, c is the specific heat, and v is theprofile speed.

The die is equipped with three heaters as depicted in Fig. 7.5 whosetemperatures constitute the process parameters to be optimized. Weconsider a constant internal heat generation value q. In the numericaltest presented here we considered the profile velocity v and the inlettemperature as given. Because both these parameters are assumed to beknown they are therefore not involved in the optimization processes, so noextra coordinates associated with them will be introduced in the pultrusionparametric thermal model.

In the present case the temperature field u is a function of five differentcoordinates, where three of them are the three temperatures prescribed in



three regions on the die wall. In fact u = u(x, y, θ1, θ2, θ3), where θ1, θ2,and θ3 are the temperatures of the heaters I, II, and III, shown in Fig. 7.5,respectively. The separated representation of u is:

u(x, y, θ1, θ2, θ3) =i=N∑i=1

Fi(x, y) · Θ1i(θ1) · Θ2i(θ2) · Θ3i(θ3). (7.43)

The boundary conditions of this problem can be written as:

u(x = 0, y, θ1, θ2, θ3) = u0

u(x ∈ L1, y = 0 or y = h, θ1, θ2, θ3) = θ1

u(x ∈ L2, y = 0 or y = h, θ1, θ2, θ3) = θ2

u(x ∈ L3, y = 0 or y = h, θ1, θ2, θ3) = θ3

∇u(L, y, θ1, θ2, θ3) · (1, 0) = 0

∇u(x /∈ L1 ∪ L2 ∪ L3, y = 0, θ1, θ1, θ2, θ3) · (0,−1) = 0

∇u(x /∈ L1 ∪ L2 ∪ L3, y = h, θ1, θ1, θ2, θ3) · (0, 1) = 0,

where L1, L2, and L3 are the intervals of the x-coordinates for heaters I, II,and III respectively, L is the length of the die, and h its width.

Proceeding as indicated in the previous section we obtain the solutionfor all possible boundary conditions that could be applied to this pultrusionprocess. One of the possible solutions for a particular choice of the boundaryconditions (heaters temperatures) is depicted in Fig. 7.6.

If we compare the solution in Fig. 7.6 to the one obtained by thefinite element method for the same values of the heater temperatures, thedifference (using the L2-norm) is found to be less than 10−5 for N up to30 modes.

This procedure can be extended to shape optimization by introducingthe design shape parameters as extra coordinates and then solving thephysical model for any choice of such parameters, that is, solving themodel for any geometry [21]. From this general solution one could try tocompute the best design by minimizing an appropriate cost function. It isimportant to recall that because the design parameters are included as extracoordinates, one could compute the analytical expression of the derivativesof the unknown field with respect to these extra coordinates, derivativesthat constitute the sensitivities employed in most optimization strategies.It is well known that the calculation of these sensitivities is a tricky taskwhen the usual solution strategies are employed.



−0.5 0 0.5 1 1.5 2 2.5 30

0.5

1

1.5100

150

200

250

300

350

400

450

Fig. 7.6. Solution for θ1 = 400K, θ2 = 300K, and θ3 = 200 K.

7.4.2 Inverse analysis

From the previous examples we can see that any model can be enrichedby introducing as extra coordinates any kind of parameters: boundary con-ditions, initial conditions, source terms, applied loads, parameters definingthe geometry, material parameters like the material conductivity, . . .

We have verified that one could solve a thermal model by assumingthat the temperature over the whole boundary is given by a number ofparameters introduced in the model as extra coordinates. Thus, once thesolution is available, we can particularize it for any choice of boundaryconditions without the necessity of recomputing the problem solution.

The advantages of this approach are evident, not only for optimizationpurposes but also in inverse analysis, in which for example extra informationis available at some part of the boundary, whereas nothing is known about acomplementary part of the boundary. This scenario is usually encounteredin manufacturing processes, where the acquisition of the temperature orheat fluxes at some part of the boundary could be technically difficult,whereas in other places it is easy to measure both temperatures and heatfluxes.

Another exciting application of this approach lies in its use in non-destructive testing. This topic is extremely important for industry and it iswell known that the computational approach is a challenging issue. Imagine



a structure containing a defect whose size and position are unknown.There are different experimental techniques that proceed by measuring aparticular field and then numerically identify the size and position of thedefect. For this purpose the physical model is solved for a given size andposition of the defect and the solution is compared to the field measuredexperimentally. If the fields, that is the experimentally measured one andthe one obtained numerically, differ, the size and/or position of the defect isupdated by applying an appropriate optimization procedure, and the modelis solved again. This procedure is repeated until it converges. Obviously, thenumerical procedure is quite expensive because for each tentative size andposition of the defect a direct problem must be solved.

In our approach, all parameters defining the hypothetical defect(size and position in the previous discussion) could be introduced asextra coordinates in the model, and then the solution of the resultingmultidimensional problem allows us to evaluate the field of interest forany position and size of such a defect.

To illustrate the procedure we consider the 1D steady-state heatequation

∂

∂x

(k(x)

∂u

∂x

)= −q, x ∈ Ω = (0, L), (7.44)

with homogenous boundary conditions, i.e. u(0) = u(L) = 0.We assume that the conductivity takes a value k1 everywhere except

within an inclusion of size d whose location is unknown, where the materialconductivity is k1 − k2, k2 < k1. Let Ξ(x) be the characteristic function forthe inclusion:

Ξ(x) =

1 if x ∈ [s, s+ d]0 elsewhere

, (7.45)

where s ∈ [0, L− d]. The material conductivity can be written as:

k(x) = k1 − k2 · Ξ(x). (7.46)

Now, the solution is searched in the separated form:

u(x, s) ≈i=N∑i=1

Xi(x) · Si(s), (7.47)

where functions Xi(x) and Si(s) are computed by applying the strategydescribed in Section 7.3. Figure 7.7 depicts the temperature field for anyposition of the inclusion, i.e. u(x, s).



Fig. 7.7. Temperature field for all possible positions of a material inclusion.

We have successfully tested this procedure in the case of more complexthermal models in 2D geometrically complex domains. The analysis ofthe application of the strategy to more complex scenarios involving morecomplex physics, inclusions parameterized from several parameters, or thecoexistence of many defects, is a work in progress.

7.5 Coupling Transport and Kinetics

In this section we analyze some different possibilities for coupling local andglobal models in a multiscale framework. Global models are described bypartial differential equations (PDEs). The heat transfer equation is thesimplest example. Thermal source terms can be associated with manychemical reactions taking place in the whole physical domain, each onemodeled by a local kinetic equation, i.e. an ordinary differential equationwhose time integration is performed at each position and involves localquantities only.

Until recently it was widely accepted that the most difficult task in thesolution of such coupled models was the solution of the PDE. However, byusing separated representations within the context of PGD, PDEs can besolved extremely fast, and then, unexpectedly, the most costly step is theintegration of the systems of ODEs governing the different kinetics, at eachlocation of interest (nodes or integration points).



In this section we reconsider the problem described above and proposesome alternatives that could significantly reduce computations when manyreactions are involved, as is usually encountered in modeling polymerizationprocesses.

For the sake of simplicity, we consider again the heat equation definedin a 1D physical space Ω:

∂u

∂t−K∆u = f(x, t) in Ω × (0, Tmax], (7.48)

with the following initial and boundary conditionsu(x, 0) = u0, x ∈ Ω,u(x, t) = ug, (x, t) ∈ ∂Ω × (0, Tmax].

(7.49)

We assume that the source term depends on the local value of the rfields Ci(x, t), i = 1, . . . , r:

f(x, t) =i=r∑i=1

γi · Ci(x, t) (7.50)

where the time evolution of the r fields Ci(x, t) is governed by r simulta-neous ODEs (the so-called kinetic model). For the sake of simplicity weconsider the linear case, the non-linear one reduces to a sequence of linearproblems by applying an appropriate linearization strategy [22]. The systemof linear ODEs is at each point x ∈ Ω:

dCi(x, t)dt

=j=r∑j=1

αij(x) Cj(x, t). (7.51)

We assume that the kinetic coefficients αij evolve smoothly in Ω,because in practical applications these coefficients depend on the solutionof the diffusion problem, u(x, t). For the sake of simplicity and without lossof generality, from now on we assume those coefficients are constant (theywere assumed to be evolving linearly in the physical space in [23]).

Now, we describe three possible coupled solution of Eqs. (7.48) and(7.51):

(1) The simplest strategy consists of using a separated representationof the global problem solution (7.48) whereas the local problemsare integrated in the whole time interval at each nodal position (orintegration point). Obviously, this strategy implies the solution of rordinary differential equations at each node (or integration point).



Moreover, the resulting fields Ci(x, t), i = 1, . . . , r, do not have aseparated structure and for this reason before injecting these fields intothe global problem (7.48) we should separate them by invoking, forexample, the singular value decomposition (SVD) leading to:

Ci(x, t) ≈q=m∑q=1

XC,iq (x) · TC,iq (t). (7.52)

As soon as the source term has a separated structure, the procedureillustrated in the previous sections can be applied to compute the newtrial solution of the global problem:

u(x, t) ≈N∑i=1

Xui (x) · T ui (t). (7.53)

Thus, this coupling strategy requires the solution of many localproblems (for all species) and at all nodal positions (or integrationpoints). Moreover, after these solutions (which could be performed inparallel) the SVD must be applied in order to separate them prior toinserting them into the PGD solver of the global problem (7.48).

(2) The second coupling strategy lies in globalizing the solution of the localproblems. Thus, we assume that the field related to each species canbe written in a separated form:

Ci(x, t) ≈q=m∑q=1

XC,iq (x) · TC,iq (t), i = 1, . . . , r (7.54)

and now, we apply the procedure described in the previous sectionsto build up the reduced separated approximation, i.e. to constructall the functions involved in (7.54). Thus, instead of solving the r

ODEs in Eq. (7.52) (that define r coupled 1D problems) at eachnodal position (or integration point), we should solve only r higherdimensional coupled models defined in the physical space and time.Obviously, if the number of nodes (or integration points) is important(mainly when 3D physical spaces are considered) the present couplingstrategy could offer significant CPU time savings.

This strategy allows us to compute directly a separated represen-tation, and then, with respect to the previous one, the applicationof SVD is avoided. However, if the number of species is high, thecomputational efforts can become important, because the space-time



separated representation must be applied to each one of the consideredspecies.

(3) The third alternative, which in our opinion is the most promisingfor solving models involving many species and can be as large as isnecessary, implies the definition of a new variable C(x, t, c), whichas we can see contains an extra coordinate c, with discrete nature,and which takes integer values: c = 1, . . . , r, so that C(x, t, i) ≡Ci(x, t), i = 1, . . . , r. Thus, we have increased the dimensionality ofthe problem but now only a single problem has to be solved insteadof the r problems involved in the previous strategy. This increasein the model dimensionality is not dramatic because as argued inSection 7.3, the separated representation allows us to circumvent thecurse of dimensionality, giving fast and accurate solutions of highlymultidimensional models. Now, the issue is the derivation of thegoverning equation for this new variable C(x, t, c) and the separatedrepresentation constructor able to define the approximation:

C(x, t, c) ≈q=S∑q=1

XCq (x) · TCq (t) · Aq(c). (7.55)

As this strategy was retained in our simulations in [23] we will focuson its associated computational aspects in the next section.

7.5.1 Fully globalized local models

The third strategy above implies the solution of a single multidimensionalmodel involving the field C(x, t, c). This deserves some additional com-ments. The first concerns the discrete nature of the kinetic equations:

dCi(x, t)dt

=j=r∑j=1

αij(x) · Cj(x, t), i = 1, . . . , r. (7.56)

Now, by introducing C(x, t, c), such that C(x, t, i) ≡ Ci(x, t), thekinetic equations can be written as:

dC

dt= Lc(C), (7.57)

where Lc is an operator in the c-coordinate.



If we try to discretize Eq. (7.57) by finite differences, we would writeat each node (xk , tp, i):

C(xk , tp, i) − C(xk, tp−1, i)∆t

= Lc(C)|i, (7.58)

where

Lc(C)|i =j=r∑j=1

αij · C(xk, tp, j) (7.59)

represents the discrete form of the c-operator at point i.Now, we return to the separated representation constructor for defining

the approximation:

C(x, t, c) ≈q=S∑q=1

XCq (x) · TCq (t) ·Aq(c). (7.60)

To define such an approximation one should repeat the procedurefully illustrated in Section 7.3. As the operator here is less standard wesummarize the main steps.

We assume that the first n iterations allow us to compute the first nsums of Eq. (7.60):

C(x, t, c) ≈q=n∑q=1

XCq (x) · TCq (t) ·Aq(c) (7.61)

and now, we look for the enrichment R(x) · S(t) ·W (c), such that

C(x, t, c) ≈q=n∑q=1

XCq (x) · TCq (t) ·Aq(c) +R(x) · S(t) ·W (c)

= Cn(x, t, c) +R(x) · S(t) ·W (c) (7.62)

satisfies∫Ω

∫ Tmax

0

∫ r

0

C∗(x, t, c) ·(dC

dt− Lc(C)

)dc dt dx = 0. (7.63)

Obviously, due to the discrete character of the third coordinate, anintegration quadrature consisting of r points, c1 = 1, . . . , cr = r will beconsidered later.



Now, for computing the three enrichment functions we consider again(as in Section 7.3) the alternating-direction strategy, which proceeds inthree steps (and are repeated until they reach convergence):

(1) Assuming functions S(t) and W (c) are known, the trial function isC∗(x, t, c) = R∗(x) · S(t) ·W (c). Thus the weak form (7.63) is:

∫Ω

∫ Tmax

0

∫ r

0

R∗ · S ·W · (R · S′ ·W −R · S · Lc(W )) dc dt dx

= −∫

Ω

∫ Tmax

0

∫ r

0

R∗ · S ·W ·q=n∑q=1

(XCq · (TCq )′ ·Aq

)dc dt dx

+∫

Ω

∫ Tmax

0

∫ r

0

R∗ · S ·W ·q=n∑q=1

(XCq · TCq · Lc(Aq)

)dc dt dx,

(7.64)

where S′ = dSdt and (TCq )′ =

dTCq

dt .Now, the time integrals and those involving the c-coordinate can be

evaluated. The ones involving the coordinate c are:

∫ r

0

W ·W dc =i=r∑i=1

W (ci)2, (7.65)

where as above ci = i, ∀i,∫ r

0

W · Lc(W ) dc =i=r∑i=1

W (ci) ·

j=r∑j=1

αijW (cj)

(7.66)

and similar expressions can be derived for the integrals in the right-hand member.

Thus, finally:

ξx∫

Ω

R∗ ·R dx =∫

Ω

R∗F x(x) dx, (7.67)

where the coefficient ξx contains all integrals in the time andc-coordinates for the left-hand member of Eq. (7.64) and F x(x) allintegrals appearing in the right-hand member. The strong form relatedto Eq. (7.67) is:

ξxR(x) = F x(x), (7.68)



whose algebraic nature describes the fact that the kinetic model is localand does not involve space derivatives.

(2) Assuming functions R(x) and W (c) are known, the trial function isC∗(x, t, c) = R(x) · S∗(t) ·W (c). Thus the weak form (7.63) is:

∫Ω

∫ Tmax

0

∫ r

0

R · S∗ ·W · (R · S′ ·W −R · S · Lc(W )) dc dt dx

= −∫

Ω

∫ Tmax

0

∫ r

0

R · S∗ ·W ·q=n∑q=1

(XCq · (TCq )′ ·Aq) dc dt dx

+∫

Ω

∫ Tmax

0

∫ r

0

R · S∗ ·W ·q=n∑q=1

(XCq · TCq · Lc(Aq)) dc dt dx.

(7.69)

Now, the integrals defined in the physical space Ω must be computedbut this task does not involve additional difficulties.Finally, this gives:

∫ Tmax

0

S∗ ·(ξtS + υt

dS

dt

)dt =

∫ Tmax

0

S∗F t(t) dt, (7.70)

where the coefficients ξt and υt contain all integrals in the space andthe c-coordinate for the left-hand member of Eq. (7.69) and F t(t) theassociated integrals appearing in the right-hand member. The strongform for Eq. (7.70) is:

υtdS

dt+ ξtS(t) = F t(t), (7.71)

whose first-order differential nature results from the first-order timederivatives in the kinetic model.

(3) Assuming functions R(x) and S(t) are known, the trial function isC∗(x, t, c) = R(x) · S(t) ·W ∗(c). Thus the weak form (7.63) is:

∫Ω

∫ Tmax

0

∫ r

0

R · S ·W ∗ · (R · S′ ·W −R · S · Lc(W )) dc dt dx

= −∫

Ω

∫ Tmax

0

∫ r

0

R · S ·W ∗ ·q=n∑q=1

(XCq · (TCq )′ ·Aq) dc dt dx



+∫

Ω

∫ Tmax

0

∫ r

0

R · S ·W ∗ ·q=n∑q=1

(XCq · TCq · Lc(Aq)) dc dt dx.

(7.72)

After integrating over space and time:∫ r

0

W ∗ · (ξc W − υc Lc(W )) dc =∫ r

0

W ∗F c(c) dc, (7.73)

where the coefficients ξc and υc contain all integrals in space andtime for the left-hand member of Eq. (7.72) and F c(c) the associatedintegrals appearing in the right-hand member. The strong form forEq. (7.73) is:

−υc Lc(W ) + ξcW (c) = F c(c), (7.74)

which gives the algebraic system:

−υcj=r∑j=1

αij W (cj) + ξcW (ci) = F c(ci), i = 1, . . . , r. (7.75)

Again, its algebraic nature comes from the nature of the kinetic model.

Remark 7.5.1. For the sake of simplicity we have illustrated the solutionprocedure within an alternating-direction fixed-point framework, but thesolutions in [23] were computed by enforcing the residual minimization fullydescribed in [20].

The interested reader can refer to [20] for more details of the model cou-pling and numerical results showing the attraction and potentiality of thisglobalization of local models and the coupling strategy.

7.6 Advanced Simulation of Models Defined for PlateGeometries: 3D Solutions with 2D ComputationalComplexity

In this section we describe the challenging issues related to the fully 3Dsolution of models defined for plate geometries. We consider two scenarios,the first one related to the flow of a viscous Newtonian fluid in a porousmedium occupying a plate domain, and the second one associated with thefully 3D elastic solution of multilayered composites.



7.6.1 Fully 3D simulation of a flow through a porous

preform in a plate mold

In the following we illustrate the construction of the PGD of the flow modelwithin a porous medium for a plate domain Ξ = Ω × I with Ω ⊂ R2 andI = [0, H ] ⊂ R, as often encountered in models of resin transfer moldingprocesses. This model is obtained by combining Darcy’s law, which relatesthe fluid velocity with the pressure gradient

v = −K · ∇p (7.76)

and the flow incompressibility

∇ · v = 0. (7.77)

Introducing Eq. (7.76) into Eq. (7.77) gives:

∇ · (K · ∇p) = 0. (7.78)

We consider that the mold contains a laminate preform composedof P different anisotropic plies, each one characterized by a well-definedpermeability tensor Ki(x, y) that is assumed constant in the ply thickness.Moreover, without loss of generality, we assume the same thickness h for thedifferent layers composing the laminate. Thus, we can define a characteristicfunction representing the position of each layer i = 1, . . . , P as:

ξi(z) =

1, zi ≤ z ≤ zi+1

0, otherwise,(7.79)

where zi = (i−1)·h defines the location of ply i in the plate thickness. Now,the laminate preform permeability can be given in the following separatedform:

K(x, y, z) =i=P∑i=1

Ki(x) · ξi(z), (7.80)

where x denotes the in-plane coordinates, i.e. x = (x, y) ∈ Ω.The weak form of Eq. (7.78) is:

∫Ξ

∇p∗ · (K · ∇p) dΞ = 0, (7.81)



with the test function p∗ defined in an appropriate functional space. Thesolution p(x, y, z) is required in the separated form:

p(x, z) ≈j=N∑j=1

Xj(x) · Zj(z). (7.82)

In the following we illustrate the construction of such a decomposition.For this purpose we assume that at iteration n < N the solution is alreadyknown:

pn(x, z) =j=n∑j=1

Xj(x) · Zj(z) (7.83)

and that at the present iteration n + 1 we are looking for the solutionenrichment:

pn+1(x, z) = pn(x, z) +R(x) · S(z). (7.84)

The test function involved in the weak form is required as:

p∗(x, z) = R∗(x) · S(z) +R(x) · S∗(z). (7.85)

By introducing Eqs. (7.84) and (7.85) into Eq. (7.81) we get:

∫Ξ

∇R∗ · S

R∗ · dSdz

+

∇R · S∗

R · dS∗

dz

·

K ·

∇R · S

R · dSdz

dΞ

= −∫

Ξ

∇R∗ · S

R∗ · dSdz

+

∇R · S∗

R · dS∗

dz

·Qn dΞ, (7.86)

where ∇ denotes the plane component of the gradient operator, i.e. ∇T =( ∂∂x ,

∂∂y

) and Qn denotes the flux at iteration n:

Qn = K ·j=n∑j=1

∇Xj(x) · Zj(z)

Xj(x) · dZj(z)dz

. (7.87)

Now, as the enrichment process is non-linear, we propose looking forthe couple of functions R(x) and S(z) by applying an alternating-directionfixed-point algorithm. Thus, assuming R(x) known, we compute S(z) andthen we update R(x). The process continues until it reaches convergence.



The converged solutions allow us to define the next term in the finite sumdecomposition: R(x) → Xn+1(x) and S(z) → Zn+1(z).

7.6.1.1 Building the separated representation

In this section we illustrate each step of the separated representation solver,in particular the construction of the functions R(x) and S(z).

(1) Computing R(x) from S(z).When S(z) is known, the test function reduces to:

p∗(x, z) = R∗(x) · S(z) (7.88)

and the weak form (7.86) reduces to:

∫Ξ

∇R∗ · S

R∗ · dSdz

·

K ·

∇R · S

R · dSdz

dΞ = −

∫Ξ

∇R∗ · S

R∗ · dSdz

· Qn dΞ.

(7.89)

Now, as all the functions involving the coordinate z are known, theycan be integrated in I = [0, H ]. Thus, if we consider:

K =

(K k

kT κ

)(7.90)

with

k =

(Kxz

Kyz

)(7.91)

and κ = Kzz, then we can define:

Kx =

∫I

K · S2 dz∫I k · dS

dz· S dz∫

IkT · dS

dz· S dz

∫I κ ·

(dS

dz

)2

dz

(7.92)

and

(Qx)n =j=n∑j=1

∫I

K · S · Zj dz∫Ik · dZj

dz· S dz∫

IkT · dS

dz· Zj dz

∫Iκ · dS

dz· Zjdz

dz

·(∇Xj(x)Xj(x)

),(7.93)



which allows us to write Eq. (7.89) in the form∫Ω

(∇R∗

R∗

)·(Kx ·

(∇RR

))dΩ = −

∫Ω

(∇R∗

R∗

)· (Qx)n dΩ, (7.94)

which defines an elliptic 2D problem defined in Ω and represents themiddle plane of the plate.

(2) Computing S(z) from R(x).When R(x) is known the test function is:

p∗(x, z) = R(x) · S∗(z) (7.95)

and the weak form (7.86) reduces to:

∫Ξ

∇R · S∗

R · dS∗

dz

·

K ·

∇R · SR · dS

dz

dΞ = −

∫Ξ

∇R · S∗

R · dS∗

dz

· Qn dΞ.

(7.96)

Now, as all functions involving the in-plane coordinates x = (x, y) areknown, they can be integrated in Ω. Thus, using the previous notation,we can define:

Kz =

∫

Ω

(∇R) · (K · ∇R) dΩ∫

Ω

(∇R) · k ·R dΩ∫Ω

(∇R) · k ·R dΩ∫

Ω

κ · R2 dΩ

(7.97)

and

(Qz)n =j=n∑j=1

∫

Ω

(∇R) · (K · ∇Xj) dΩ∫

Ω

(∇R) · k ·Xj dΩ∫Ω

(∇Xj) · k ·R dΩ∫

Ω

κ ·Xj ·R dΩ

· Zj(z)dZjdz

(z)

, (7.98)

which allows us to write Eq. (7.96) in the form

∫I

S∗

dS∗

dz

·

Kz ·

S

dS

dz

dz = −

∫I

S∗

dS∗

dz

· (Qz)n dz, (7.99)

which defines a 1D boundary-value problem.



7.6.1.2 Numerical example

Traditionally, the flow of a viscous fluid through a porous mediumoccupying a plate mold was assumed to be 2D to make possible realisticsimulations of industrial interest.

When the plate mold consists of a laminate preform composed of severalanisotropic plies with different principal directions of anisotropy, the mainissue lies in the definition of an equivalent permeability tensor representingthe whole laminate preform.

One could expect that an appropriate equivalent permeability tensorcould be defined by averaging the permeability of the different plies thatcompose the laminate preform. To analyze the validity of this approach, westart by considering a laminate preform composed of two identical plies ofa unidirectional reinforcement placed in the rectangular mold depicted inFig. 7.8.

The in-plane permeability tensor of each ply can be written using theprincipal anisotropy directions:

K =

(K1 0

0 K2

). (7.100)

From now on we will consider a coordinate system such that thex-coordinate is aligned in the direction of the longest plate edge, they-coordinate defines the plate width, and the z-coordinate its thickness.Now, we assume the principal anisotropy directions of the first ply turnedan angle θ with respect to the coordinate system, whereas the second plyis placed symmetrically, that is, by forming an angle of −θ.

In this case, the in-plane permeability of both plies, K1 and K2

respectively, is:

K1 = QT(θ) K Q(θ) (7.101)

and

K2 = QT(−θ) K Q(−θ), (7.102)

Fig. 7.8. Laminate preform composed of two symmetrically oriented anisotropic plies.



where Q(θ) and Q(−θ) are the rotation tensors for the angles θ and −θ,respectively.

If we define the equivalent in-plane permeability of the laminate pre-form K from the simple average of the in-plane permeability of both plies,it is easy to verify that the resulting equivalent in-plane permeability is:

K =

(K1 · cos2(θ) +K2 · sin2(θ) 0

0 K2 · cos2(θ) +K1 · sin2(θ)

). (7.103)

Now, if we apply a pressure drop ∆P between the inlet plate facelocated at x = 0 and the outlet located in the opposite face x = L, thepressure distribution will be strictly linear and the associated velocity fieldwill be uniform and unidirectional according to

p(x, y, z) = P0 +x

L· ∆P, (7.104)

where P0 denotes the inlet pressure, from which the velocity field v is:

v(x, y, z) =

−K · ∆P

L00

. (7.105)

One would expect fluid trajectories parallel to the x-axis and a residencetime tR given by L = ‖v‖ · tR.

In order to check the validity of the assumed hypotheses, we solvethe fully 3D model using the strategy presented in the previous section.Figure 7.9 depicts the resulting pressure field as well as some flow pathlines,from which we can see the complex 3D behavior of flows in plates composedof many plies differently oriented.

Obviously, more complex scenarios could be solved. Figure 7.10 depictsthe pressure fields and some flow pathlines in a complex plate involving aseries of cylindrical obstacles. The consideration of hundreds of plies does

Fig. 7.9. Resulting pressure field and associated flow pathlines.



Fig. 7.10. Laminate preform occupying a more complex plate geometry.

not have a significant impact on the computing time, because by using theseparated representation solver (PGD) the introduction of plies only affectsthe solution of functions defined in the thickness coordinate, which is a 1Dproblem whose solution is very cheap from a computational point of view.

In these examples we can see quite complex behavior of the flowtrajectories, which could indicate the possible existence of mixing becausea hypothetical chaotic behavior is being analyzed.

7.6.2 Fully 3D simulation of thermomechanical models

defined in plate domains

In this section, we apply the PGD method to the simulation of the linearelastic behavior of plate-shaped domains. The PGD method allows us toseparately search for the in-plane and the out-of-plane contributions to thefully 3D solution, allowing significant savings in computational time andmemory resources. The method is validated with a simple case and its fullpotential is then presented in the simulation of the behavior of multilayeredcomposite plates and honeycomb plates.

When computing the elastic response of plates, 2D plate theories areusually preferred to the numerically expensive fully 3D elastic problems.Going from a 3D elastic problem to a 2D plate theory usually involvessome kinematic hypothesis [24] on the evolution of the solution throughthe thickness of the plate.

Despite the quality of existing plate theories, their solution close tothe plate edge is usually wrong because the displacement field is truly 3D



in those regions and does not satisfy the kinematic hypothesis. Indeed,the kinematic hypothesis is a good approximation when Saint-Venant’sprinciple holds. Moreover, as discussed in Section 7.1, complex physics needsthe solution of fully 3D models because no appropriate hypothesis can beintroduced to give the desired dimensionality reduction.

In an attempt at solving fully 3D models but keeping 2D complexityconcerning the computing cost, we use the PGD widely described in pre-vious sections, by assuming a separated representation of the displacementfield

u(x, y, z) =

u(x, y, z)v(x, y, z)w(x, y, z)

≈

N∑i=1

uixy(x, y) · uiz(z)vixy(x, y) · viz(z)wixy(x, y) · wiz(z)

, (7.106)

where uxy(x, y), vxy(x, y), and wxy(x, y) are functions of the in-planecoordinates whereas uz(z), vz(z), and wz(z) are functions involving thethickness coordinate.

Because neither the number of terms in the separated representationof the displacement field nor the dependence on z of the functions uiz(z),viz(z), and wiz(z) is assumed a priori, the approximation is flexible enough torepresent the fully 3D solution, being obviously more general than classicalplate theories that assume particular a priori behaviors in the thicknessdirection.

The separated representation solution converges towards the solutionobtained using a fully tensor product of the approximation bases definedby the in-plane and thickness approximation basis, as with a fully 3Dfinite element discretization, making possible fast and accurate solutionsof problems unapproachable in practice using the more widely used finiteelement method.

7.6.2.1 Problem formulation and solution strategy

Let us consider a linear elasticity problem in a domain Ω. The weakformulation associated with such a problem is:∫

Ω

ε(u∗) · K · ε(u) dΩ =∫

Ω

u∗ · fd dΩ +∫

ΓN

u∗ · Fd dΓ, (7.107)

where K is the generalized 6×6 Hooke tensor, fd represents the volumetricbody forces, and Fd represents the surface forces applied on the Neumannboundary ΓN . The separation of variables introduced in Eq. (7.106) yields



the following expression for the components of the strain tensor:

ε(u(x, y, z)) =∑

uxy,x uz

vxy,y vz

wxy wz,z

uxy,y uz + vxy,x vz

uxy uz,z + wxy,xwz

vxy vz,z + wxy,y wz

. (7.108)

Depending on the number of non-zero elements in the K matrix, thedevelopment of ε(u∗) · K · ε(u) involves many terms, 21 in the case of anisotropic material and 41 in the case of highly anisotropic behavior.

Assuming that the first n modes of the solution have already beencomputed, we focus on the solution enrichment for the computation of thenext functional couple, according to:

un+1(x, y, z) = un(x, y, z) +R(x, y) · S(z). (7.109)

As previously, we consider that the test function is given by:

u∗(x, y, z) = R(x, y) · S(z) +R(x, y) · S(z). (7.110)

Introducing the trial and test functions given by Eqs. (7.109) and(7.110) respectively, into the weak form (7.107), we obtain:∫

x,y

∫z

[ε(u∗(x, y, z)

)·K · ε

(un+1(x, y, z)

)]dxdy dz

=∫x,y

∫z

u∗ · fd dxdy dz +∫

ΓN

u∗ · Fd dΓ. (7.111)

Because the simultaneous calculation of R(x, y) and S(z) is a non-linearproblem, a linearization strategy is compulsory, the simplest one being thealternating-direction fixed-point method previously described and used.

Given an initial approximation S(0)(z) of S(z) that is arbitrarily chosen,all z dependent functions are known (in this case S(z) = 0) and Eq. (7.111)therefore reduces to a 2D (x and y dependent) problem, where R(x, y) isthe unknown field. Its solution yields R(1)(x, y), the first approximation ofR(x, y). Then by approximating R(x, y) by R(1)(x, y) in Eq. (7.111), wesimilarly obtain a 1D problem (z dependent), which allows us to compute



S(1)(z)–the next approximation of S(z). This fixed-point loop keeps runninguntil it reaches convergence, i.e.:∫

Ω

(R(j)(x, y) · S(j)(z) −R(j−1)(x, y) · S(j−1)(z)

)2

dxdy dz ≤ ε, (7.112)

where ε is a small enough parameter.The solution is enriched with new modes until it reaches a normalized

residual norm small enough. The converged separated representationrepresents the fully 3D solution of the linear elasticity problem definedin the plate domain, whose solution only involves the solution of some 2Dproblems involving the in-plane coordinates and some 1D problems relatedto the functions involving the thickness coordinate. Thus, 3D solutions canbe computed while keeping a 2D complexity.

To validate the proposed technique, we consider a square plate andwe compare the classical 3D linear elastic finite element solution and theone obtained using PGD with an equivalent discretization, that is, the 2Dfunctions involving the in-plane coordinates in PGD are approximated usingthe same mesh that the finite element method considered on the platesurface, and the 1D functions involving the thickness coordinate when usingPGD are approximated using the same number of nodes that was consideredin the thickness finite element approximation.

Let us consider the square plate depicted in Fig. 7.11. The appliedload consists of a uniform pressure applied on the upper face. The finiteelement solution was calculated by considering a uniform mesh composedof 100×100×50 eight-node elements. The PGD solution was found using auniform mesh composed of 100×100 four-node elements to approximate thefunctions involving the in-plane coordinates, whereas a uniform 1D meshcomposed of 50 two-node elements was used to approximate the functionsinvolving the thickness coordinate.

Fig. 7.11. Problem geometry and computed solution by applying PGD — only thesolution in half of the domain is depicted for the sake of clarity.



Fig. 7.12. Error with respect to the FEM solution considering the usual energy norm.

The solution computed using PGD is depicted in Fig. 7.11. Nine termswere needed to describe the solution, most of them to describe the 3D effectsthat appear in the neighborhood of the boundaries where the displacementwas prescribed. In order to compute these nine terms in the separatedrepresentation 165 2D and 1D problems were solved. Figure 7.12 shows theenergy density error, considering as reference solution the one computedusing the finite element method (FEM). This error is everywhere lowerthan 0.3% except in the vicinity of the plate corners where it reaches avalue of 0.57%. This error can be reduced by considering more terms in theseparated representation, i.e. higher N .

Because of the different methods used for the 2D and 1D problemsby the PGD solver, one could consider very different resolutions in bothapproximations, that is, the characteristic size of the mesh used for solvingthe 2D problems does not affect the characteristic mesh considered inthe approximation of functions involving the thickness coordinate, andvice versa. Moreover, one could use different discretization techniques forsolving both problems.

7.6.2.2 Discussion

We will compare the complexity of PGD-based solvers to the standard finiteelement method. For the sake of simplicity, we will consider a hexahedraldomain discretized using a regular structured grid with Nx, Ny, and Nznodes in the x, y, and z directions, respectively. Even if the domain



thickness is much smaller than the other characteristic in-plane dimensions,the physics in the thickness direction could be quite rich, especially whenwe consider composite plates composed of hundreds of plies in which thecomplex physics requires fully 3D descriptions. In this case, thousands ofnodes in the thickness direction could be required to represent accuratelythe solution behavior in that direction. In usual mesh-based discretizationstrategies, this fact induces a challenging issue because the number ofnodes involved in the model scales with Nx × Ny × Nz; however, if oneapplies a PGD-based discretization, and the separated representation of thesolution involvesN modes, one would solveN 2D problems for the functionsinvolving the in-plane coordinates and N 1D problems for the functionsinvolving the thickness coordinate. The computational time for the solutionof the 1D problems can be neglected when compared to the time requiredfor solving the 2D ones. Thus, the PGD complexity scales as N ×Nx×Ny.

By comparing both complexities, we can see that as soon as Nz N

the PGD-based discretization leads to impressive computing time savings,making possible the solution of models previously unsolved, even using low-performance computing platforms.

Figure 7.13 compares the CPU time of both PGD and FEMdiscretizations for solving the linear elasticity problem previously described,

Fig. 7.13. Comparison of PGD and FEM 3D discretizations. DOF = degrees of freedom.



as a function of the number of in-plane degrees of freedom Nx × Ny, andof the number of degrees of freedom in the thickness Nz.

In the previous sections we have implicitly assumed that the generalizedHooke tensor K does not depend on the in-plane coordinates but this is nota restriction on applying the PGD-based solver. An efficient implementationof PGD requires a separated representation of the elasticity tensor only:

Kij(x, y, z) =∑l

K l,xyij (x, y)K l,z

ij (z). (7.113)

For example, by making Kij z-dependent, only one mode is necessaryto represent the material parameters of a laminate preform composite platewhere each ply could have different principal directions of anisotropy.

In the following, we consider a quite complex structure consisting of twoplates with a honeycomb in between, as depicted in Fig. 7.14. It is possibleto define a function χ(x, y, z), taking a unit value inside the honeycombcells and vanishing in the cell walls as well as in the upper and lower plates,having the following separated representation: χ(x, y, z) = χxy(x, y) ·χz(z).

Assuming that the honeycomb cells are filled with a material describedby the generalized Hooke tensor K1, while the cell walls and the upper andlower plates can be described with a tensor K2, we define the generalizedHooke tensor for the whole structure as:

K(x, y, z) = K1 · χ(x, y, z) + K2 · (1 − χ(x, y, z)). (7.114)

Remark 7.6.1. Even if the honeycomb cells are empty, we must considera virtual material with a very small rigidity because the assumption of a

Fig. 7.14. Honeycomb structure between two thin plates.



Fig. 7.15. Multilayered composite laminate and honeycomb structures.

null elasticity tensor produces numerical instabilities whose origin is beinganalyzed.

We can therefore see that only two modes are needed for a separatedrepresentation of the elasticity tensor in this complex geometry.

Figure 7.15 depicts the solutions obtained by applying the PGDsolver to both structures, the multilayered composite laminate and thehoneycomb. Both computations were carried out on a simple laptopcomputer, at a computational cost that is negligible with respect to the3D finite element solution, which is unapproachable even using powerfulcomputing platforms.

7.7 Conclusions

The simulation of composite-manufacturing processes remains today achallenging issue despite the recent impressive progress in computationalresources. The simulation of processes of industrial interest needs advancednumerical strategies, radically different to the ones now widely used, usuallybased on finite element, finite difference, or finite volume discretizations.

In this chapter we explored the potential of techniques based onmodel reduction, and in particular those making use of the so-called PGD.This technique allows us to circumvent the curse of dimensionality whenaddressing models defined in high-dimensional spaces. We proved thatit could constitute a real breakthrough for addressing shape or processoptimization as well as inverse analysis.

On the other hand, thanks to the separated representation that PGDuses, fully 3D solutions in plate domains could be achieved by separating thein-plane and out-of-plane (thickness) coordinates. Thus fully 3D solutionscould be computed while keeping 2D computational complexity.



Thanks to this novel approach we have solved models previouslyunsolved, without using powerful computing platforms. All the resultsshown here were computed using Matlab on a standard laptop.

Model reduction strategies open new possibilities in simulatingcomposite-manufacturing processes. It is too early to define the possibilitiesand the limitations of this approach. In any case, the first results seem verypositive and are encouraging for further development.

Acknowledgements

The authors acknowledge the contribution of many colleagues to thischapter: Brice Bognet and Chady Ghnatios from GEM at Ecole Centrale ofNantes, Elias Cueto from the University of Zaragoza in Spain, and AmineAmmar from ENSAM Angers.

References

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[9] Ammar A., Ryckelynck D., Chinesta F., Keunings R., 2006. On the reductionof kinetic theory models related to finitely extensible dumbbells, J. Non-Newton Fluid, 134, 136–147.

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[14] Ammar A., Mokdad B., Chinesta F., Keunings R., 2007. A new family ofsolvers for some classes of multidimensional partial differential equationsencountered in kinetic theory modeling of complex fluids. Part II: transientsimulation using space-time separated representations, J. Non-Newton.Fluid, 144, 98–121.

[15] Gonzalez D., Ammar A., Chinesta F., Cueto E., 2010. Recent advances in theuse of separated representations, Int. J. Numer. Meth. Eng., 81(5), 637–659.

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[17] Martin J., 2006. Pultruded composites compete with traditional constructionmaterials, Reinforced Plastics, 50(5), 20–27.

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Chapter 8

MODELING FRACTURE AND COMPLEX CRACKNETWORKS IN LAMINATED COMPOSITES

Carlos G. Davila∗, Cheryl A. Rose∗ and Endel V. Iarve†,‡∗NASA Langley Research Center

Hampton, VA 23681, USA

†Air Force Research Laboratory, Wright Patterson AFB, OH 45433

‡University of Dayton Research InstituteDayton, OH 45469, USA

Abstract

Recent advances in computational methods and numerous published demons-trations of successful representations of the propagation of composite damagemechanisms indicate that the day is imminent when reliable tools for thevirtual testing of composite structures will replace some mechanical testingin the design and certification process. Given these rapid developments andthe apparent diversity of the proposed approaches, it is necessary to formulatethe conditions under which a given model can be expected to work and whenit will cease to be adequate. In this chapter, we examine the fundamentalconcepts that are required for predicting damage in composites with the intentof providing a basis to help select the idealizations that are necessary, physicallyreasonable, and computationally tractable. Issues of the objectivity of fracturepropagation with continuum damage models are discussed and the applicationof the extended finite element method to avoid these difficulties is explored.

8.1 Introduction: Damage Idealization and Scale

Fracture in a composite structure is the result of the evolution of discretedamage events, such as fiber/matrix debonding, matrix cracking,delamination between plies, and fiber failure. These damage modes evolvein various combinations that depend on the stacking sequence and plythicknesses and cause redistributions of stresses in the failing composite.Some combinations of damage may reduce local stress concentrations, whileothers may precipitate a structural collapse. Therefore, a methodology

297


298 C.G. Davila, C.A. Rose and E.V. Iarve

(a) Fiber fracture (b) Matrix transverse crack (c) Delamination

Fig. 8.1. Damage mechanisms in laminated composites.

capable of predicting structural strength must take into account damageinitiation and propagation.

However, the details of the mechanisms that lead to failure arenot fully understood due to the complexity of the idealization of theindividual constituent responses and their interactions. The presence oftwo constituents, the fiber and the matrix, and the extreme anisotropyin both stiffness and strength properties result in damage mechanisms atdifferent levels. The damage mechanisms can be divided into intralaminarand interlaminar damage. As shown in Fig. 8.1, intralaminar damagemechanisms correspond to fiber fracture and matrix cracking, whereasinterlaminar damage mechanisms correspond to the interfacial separationof the plies (delamination).

The formulation of the governing physical principles of damageevolution depends on the scale of the idealization of the damage process,which may span from molecular dynamics to structural mechanics, andincludes the intermediate scales of micro- and mesomechanics. The damagemodels shown in Fig. 8.2 illustrate four typical scales of damage idealization.The micromechanical scale model shown in Fig. 8.2(a) represents what isnormally the smallest scale of composite damage idealization, in whichdetailed matrix energy-dissipating mechanisms such as matrix plasticityand damage and fiber/matrix interface cracking are represented [1].The representation of damage at this level is typically based on areduction of the material stiffness. Hence, a fracture is representedas a band of localized volumetric stiffness reduction, referred to as a“weak discontinuity,” as opposed to a “strong discontinuity” in whichvoids are represented by displacement discontinuities in the model. Dueto computational constraints, micromechanical models are typically twodimensional (2D) and represent domains much smaller than a ply thickness.


Modeling Fracture and Complex Crack Networks in Laminated Composites 299

Structural scale(Strong discontinuity) • Fracture mechanics and

modifications • Strain softening, cohesive laws

a.

Mesoscale(Weak discontinuity) • Continuum damage

mechanics (CDM)

b. Mesoscale(Strong discontinuity) • x-FEM + Cohesive laws

c.

Microscale(Weak discontinuity) • Representation volume element

(RVE) models (unit cell)

d.

Scale of idealization Damage type Typical approaches

Through-the-thickness crack or delamination

Intralaminar damage

Discrete damage

Fiber/matrix interface, matrix plasticity and

damage

Fig. 8.2. Levels of damage idealization from microscale to structural scale.

Consequently, they are useful for representing a composite hardeningresponse before cracks localize at either the ply level or on a larger scale.

The discrete damage mechanics (DDM) model shown in Fig. 8.2(b)represents a new class of analysis methods in which the plies are representedat the mesoscale level, i.e. the material is assumed to be homogeneouswith orthotropic properties, and where the extended finite element method(x-FEM) is used to insert cracks and delaminations in locations that areindependent of the mesh orientation. DDM models can be used to representcomplex networks of transverse matrix cracks and delaminations using asingle cohesive law [2].

The most common idealization for composite damage is the continuumdamage mechanics (CDM) model at the mesoscale level, which assumes thatplies are composed of a homogeneous material with orthotropic propertiesand damage modes, such as fiber fracture, fiber kinking, matrix cracking,and delamination, are represented as a reduction in the correspondingstiffnesses [3,4]. In CDM models, localized damage is therefore represented



as a weak discontinuity, as opposed to the strong discontinuities used inDDM models. An example of a mesoscale model of a notched wing skinsubjected to compression loads is shown in Fig. 8.2(c).

Mesoscale approaches such as DDM and CDM have severalcomputational advantages compared to micromechanical-level models.These models use material properties that can be determined fromlamina-level characterization tests, and imperfections and variabilities thatmust be considered at lower scales can be disregarded. To account formicromechanical effects such as fiber/matrix debonding, some mesoscalemethods perform concurrent analyses on idealized unit-cell models thatrepresent typical distribution patterns of fibers in the matrix. Othermechanisms, such as transverse matrix cracking or fiber kinking, whichdepend on mesoscale data such as ply thickness, are usually taken intoaccount with failure criteria and in situ strengths. These failure criteriaare used in conjunction with material degradation schedules to softenthe material properties associated with a particular mode of failure. Thedelamination failure mode is represented at the mesoscale level as a strongdiscontinuity between plies, which is generally modeled using either cohesiveelements or non-linear springs [5].

The mesoscale CDM approach is one of the most widely usedapproaches to calculate the damage tolerance of structural components.However, the diffused representation of damage in the CDM approach maylack sufficient resolution to capture some important damage interactionsat the micromechanical scale. In particular, mesoscale CDM models havedifficulty predicting the correct propagation of matrix cracks parallel to thefibers. Furthermore, the damage models for intralaminar damage may notinteract correctly with the delamination models, e.g. [6,7].

Finally, the model of a through-crack in a fuselage panel illustratedin Fig. 8.2(d) illustrates an example of a structural-level damage model.The crack is represented as a strong discontinuity, and prediction of thepropagation of the crack could be based on a strain-softening law ora criterion based on the critical energy release rate [8]. However, anystructural-level crack propagation criterion is strongly dependent on thematerial system and laminate configuration and consequently must bedetermined for each new material system and laminate stacking sequence. Inaddition, structural-level semi-empirical fracture models cannot address thecharacteristics of the crack-tip damage zone nor the complex interactions



between micro- and macro-failures associated with the crack-extensionprocess. Instead, the crack-tip damage zone is simulated as some “effective”notch-tip damage zone that is assumed to grow in a self-similar manner.In many cases, self-similar crack growth is not observed, and the lack ofresolution in the damage mechanisms often renders structural-level damagemodels inaccurate after a short propagation of damage.

It is clear from the preceding overview of typical damage modelingstrategies that the conceptual idealization of damage, i.e. the identification,characterization, and formulation of the governing physical mechanismsthat constitute damage evolution, are different at each scale of idealization.Damage idealizations at lower structural scales have higher resolution andkinematic freedom and can capture multiple damage mechanisms with aseparate damage law for each mechanism. These damage laws are likely tobe simpler and require a lower fracture toughness than the damage lawsused in coarser, higher-level models to represent the same global response.For instance, mesoscale DDM models may only need one simple cohesivelaw to represent a variety of matrix damage patterns, while CDM modelsneed multiple empirical stiffness degradation laws and interacting activationfunctions to represent the softening of the material.

The choice of a modeling approach is further complicated by thefact that there is a vast number of published demonstrations of differentsuccessful representations of the propagation of damage. A variety ofdamage models have been championed and, given the complexity of thesemethods, their differences and shortcomings are unclear. The objectiveof the present chapter is to examine the common features and basicprinciples of well-established approaches for predicting damage propagationin composites to establish which damage mechanisms are important andhow to select the idealizations that are necessary, physically reasonable,and computationally tractable.

In the following sections, we first examine the capabilities of cohesivelaws to represent crack initiation and propagation. Then, the concepts ofthe continuum representation of composite material response are discussedand the intrinsic limitations of continuum damage models for laminatedcomposites are outlined. Finally, an emerging modeling technique based onx-FEM, which overcomes many of the limitations of continuum damagemodels is presented and the capabilities of the x-FEM methodology areillustrated with some examples.



8.2 Crack Initiation and Propagation

8.2.1 Linear elastic fracture of composites

The ability to predict crack propagation in composites emerged four decadesago with the development of computational methods based on the theoryof linear elastic fracture mechanics (LEFM). In particular, the virtualcrack closure technique [9] is a computationally simple procedure withwhich to calculate the energy release rate (ERR) for each mode of fracturealong a crack front. According to the Griffith criterion [10], when theseERR values exceed the corresponding critical fracture toughness, the crackshould propagate. However, LEFM is limited to applications in which thefracture process zone is confined to the immediate neighborhood of thecrack tip itself, so it cannot be applied to a number of important crackingproblems involving some of the tougher, more ductile structural materialsand adhesives, which are sometimes described as quasi-brittle [11], thatfracture after extensive non-linear deformation.

For many fracture processes in composite materials and structures,the fracture process zone may be relatively large compared to otherstructural dimensions. The fracture process zone is a non-linear zonecharacterized by plastic deformations and progressive material softeningdue to non-linear material deformations, such as microcracking, voidformation, and fiber/matrix pullout. The size of the fracture processzone is dependent on the type of material softening and it must beconsidered in many situations of crack growth in composite structures. Forexample, the development of a process zone gives rise to stable growthand crack growth resistance. The apparent fracture toughness increaseswith crack growth — an effect called the R-curve — until the processzone is fully developed. In addition, the effects of material strength, theR-curve, and structural size on the residual strength of structures made ofthese materials may be misunderstood, disregarded, or perceived as solelystatistical if the fracture process zone is not taken into account in fracturecalculations.

Reduced-singularity criteria such as the Mar–Lin criterion [8,12] orSun’s two-parameter fracture criterion [13] were devised for extending theclassical linear theory to situations where the stress distribution aheadof the crack tip does not follow a square-root singularity. However, thesecriteria can easily be supplanted by non-linear fracture mechanics (NLFM).NLFM provides a framework for characterizing crack growth resistance andfor analyzing initial amounts of stable crack growth.



The cohesive crack model is a NLFM methodology, which wasdeveloped to simulate the non-linear fracture response near the crack tip.Cohesive crack models have the ability to describe the process of voidnucleation from inclusions. Therefore, they can be applied to an initiallyun-cracked structure and can describe the entire fracture process, fromno crack to complete structural failure. In the following sections, theformulation of cohesive laws is presented. We examine the differences inthe failure/fracture of a quasi-brittle material subjected to either a uniformstress field or a “singular” stress field caused by a crack tip. Next, weexamine the ability of cohesive laws to account for the size effect onstructural strength and different aspects of composite failure related tomaterial characteristics. Finally, the effect of material softening on the sizeof the fracture process zone, and the R-curve effect is demonstrated.

8.2.2 Cohesive laws

Cohesive crack models are based on kinematic descriptions that use strongdiscontinuities in the displacement field. Cohesive interfacial laws arephenomenological mechanical relations between the traction σ and theinterfacial separation δ such that with increasing interfacial separation,the tractions across the interface reach a maximum and then decrease andvanish when complete decohesion occurs. It can be shown by performing aJ-integral calculation along a contour surrounding a notch tip that theresulting work of interfacial separation is related to Griffith’s fracturecriterion [10]:

Gc =∫ δf

0

σ(δ)dδ, (8.1)

where Gc is the critical energy release rate (ERR) and δf is the criticalinterfacial separation.

The complete representation of a fracture in a continuum requiresan ability to model both the initiation and growth of the fracture. Theprinciples of the representation of a fracture in a continuum can beillustrated by considering a quasi-brittle bar of length L and cross-sectionA,as shown in Fig. 8.3(a). To model the fracture of the bar, a cohesive interfaceis introduced. The crack is assumed to open according to the softening lawshown in Fig. 8.3(b). Initially, the crack is assumed to be elastic and thecrack closing forces are related to the interfacial displacement jump δcoh bya high penalty stiffness K. If the displacement jump exceeds a critical value



L+ +

K

F,

Gc

c

(1-d)K

(a) (b)

Fig. 8.3. (a) Elastic bar with a cohesive crack. (b) Bilinear cohesive law.

δi the crack closing forces are assumed to soften linearly such that the areaunder the traction-displacement curve is equal to the fracture toughnessGc. Complete separation is achieved when the displacement jump exceedsδf . Therefore the bilinear cohesive law can be expressed in two parts:

σ =

Kδ, δ < δi

(1 − d)Kδ, δi ≤ δ < δf, (8.2)

where the damage variable d is a function of the displacement jump andaccounts for the reduction in the load-carrying ability of the material as aresult of damage.

The response of the bar can be obtained from the compatibility andequilibrium equations:

∆ = δbar + δcoh and σ = σbar = σcoh. (8.3)

The deformation of the bulk material is δbar = σL/E where E is Young’smodulus. If the bar is undamaged (σ ≤ σc) the opening of the cohesivecrack is δcoh = σ/K. Therefore the force-displacement response of the barbefore damage is

F = Aσ = EA∆

L+ EK

. (8.4)

It can be observed that as long as K E/L the compliance introducedby the cohesive law can be neglected. It is also clear from Eq. (8.4) thatthe maximum load that the bar can withstand Fc = Aσc and the responseof the bar before failure depend on strength alone and not on a fracturecriterion.

For a bilinear cohesive law, the relationship between the damagevariable d and the displacement jump δcoh has the form:

d =δf (δcoh − δi)δcoh(δf − δi)

, (8.5)



where δi = σc

Kand δf = 2Gc

σc. Therefore, the response of the bar after

damage initiation is obtained by substituting d, ∆, δi, and δf into Eq. (8.4),which results in:

F = Aσ = EA∆ − 2Gc

σc

L− 2EGc

σ2c

+ EK

. (8.6)

The response of the bar after damage initiation is stable under displacementcontrol only if ∂F

∂∆≤ 0, which assuming that K E/L gives

L ≤ 2EGcσ2c

. (8.7)

The right-hand side of Eq. (8.7) is a characteristic dimension associatedwith material properties that is typically much smaller than the structuraldimensions. Consequently the failure of a uniformly stressed problem, suchas the present bar, is typically unstable, i.e. no quasistatic equilibriumsolution exists, and the failure load is independent of the fracture toughnessGc. As a result of the unstable fracture, the shape of the softening responseis of no consequence. On the other hand, when cohesive models are usedto predict the propagation of crack fronts, the fracture toughness Gc is thedominant material property and the effect of the strength σc may be minor.

In the following section, the fundamental concepts in the analysis ofcrack propagation using cohesive laws and LEFM are compared and someprinciples related to the effect of size on structural strength are outlined.

8.2.2.1 Length of the fracture process zone

LEFM assumes that the material is elastic and that the mechanisms thatconsume the fracture energy act at the crack tip. NLFM was initiated byIrwin [14] with a model for ductile solids based on an elastic/perfectlyplastic material response to describe the effect of plastic material behaviorin the vicinity of the crack tip on fracture propagation. By assuming thatplasticity affects the stress field only in the vicinity of the crack tip, Irwinestimated the size of the plastic region by equating the yield strength tothe stress of the elastic field ahead of the leading crack tip (see Fig. 8.4).A generalization of Irwin’s model for material softening due to damagewas proposed by Bazant and Planas [15]. In the Irwin–Bazant model, thetraction profile in the inelastic zone ahead of the trailing crack tip followsa general expression:

σ = σc

(x

lp

)β, (8.8)



where lp is the size of the cohesive zone (or the fracture process zone), x isthe distance from the trailing crack tip, and β is a parameter that describesthe stress field in the process zone.

In the elastic zone, the traction profile follows the expression given bythe LEFM solution. Accounting for an offset r1 to be determined, the stresssingularity at the crack tip given by the linear elastic solution is:

σ =

√EGc

2π(x− r1). (8.9)

The length lp and the offset r1 are obtained by assuming that the tractiongiven by Eqs. (8.8) and (8.9) are equal at x = lp, and that the areas A1 andA2 shown in Fig. 8.4 are also equal. The crack propagates when the energyrelease rate G is equal to the critical value Gc. Therefore, the size of thecohesive zone when the crack is propagating in a self-similar, steady-statefashion can be solved using the previous equations, resulting in:

l∞p =β + 1π

EGcσ2c

. (8.10)

The notation l∞p indicates that Eq. (8.10) is valid only when the structuraldimensions are infinite compared to the crack length, in which case theelastic stress field is represented accurately by Eq. (8.9). Finite elementanalysis results indicate that the steady-state length of the process zone for

Traction

x

Trailing crack tip

Cohesive traction profile (Bažant)

LEFM stress singularity

r1

lp

A1

A2

Leading crack tip

Plastic traction profile (Irwin)

Fig. 8.4. Stress profile ahead of the crack tip [15].



a bilinear law, such as the one represented in Fig. 8.3(b), is approximatedwell using β = 1 in Eq. (8.10) (see Turon et al. [16]).

In many situations of crack growth in composite structures, the processzone length lp may be relatively large compared to other structuraldimensions. The fracture process zone length is generally considered torepresent an intrinsic characteristic of a material response and the factthat it can range from about 10 nm for a silicon wafer to several metersfor concrete underlines the diversity in the response of different materials[17]. However, when the structural dimensions are small, the boundaryconditions can influence the stress distribution ahead of the crack tip andempirical corrections must be applied to Eq. (8.10). This can be the case, forinstance, in delamination, where the process zone length is shorter than thel∞p estimated by Eq. (8.10), especially for thin adherends [16]. Converselyin a notched specimen whose height is less than the crack length, the non-singular stresses become significant and induce a process zone that is longerthan l∞p [13]. The development of analytical expressions for predictingthe length of the process zone accurately for general configurations is thesubject of ongoing research.

For many fracture processes in composite materials and structures,the length of the process zone must be considered because its formationis responsible for an increase in fracture toughness with crack growth, aresponse denoted as the resistance curve or the R-curve. In the presenceof an R-curve, the toughness measured during crack propagation typicallyincreases monotonically until reaching a steady-state value. In the case ofdelamination, the increase in toughness with crack growth is attributedmostly to fiber bridging across the delamination plane. Since it is generallyassumed that fiber bridging only occurs in unidirectional test specimens andnot in general laminates, the toughness of the material is taken as the initialtoughness and the toughness for steady-state propagation is ignored [18].However, recent experimental work on the delamination between generalinterfaces indicates that R-curve effects are always present in delaminationsuch that the steady-state values of the critical ERR is typically betweenfour and five times greater than the initiation values [19–21]. In a through-the-thickness fracture of composite laminates (Fig. 8.2(d)), the R-curveeffect is caused by a combination of damage mechanisms leading the processzone, and fiber bridging in the trailing region of the process zone. ThisR-curve response makes it difficult to predict the effect of structural sizeon strength using LEFM, even in the case of self-similar propagation, asdescribed in the following section. Test results for notched laminated panels



of different sizes indicate that their strength cannot normally be predictedusing a constant fracture toughness [8].

Finally, a knowledge of the length of the process zone lp is also usefulfor determining the finite element mesh requirements for a given materialsince it is necessary to use more than three elements in the process zone [22].

8.2.2.2 Size effects

The goal of any damage theory is to predict the effect of size, i.e. thechange in strength when the spatial dimensions are scaled up from thecoupon to the structure. No viable physical theory exists without a clearunderstanding of the effects of scaling. The LEFM theory, in which allthe fracture processes are assumed to occur at the crack tip, exhibitsthe strongest possible size effect. LEFM scaling predicts that the nominalstrength is inversely proportional to the square root of structure size, asshown in the following example. Consider the notched specimen shownin Fig. 8.5(a), with an initial crack length a0 and subjected to a tensilestress σu. Let D, b, and t be the width, the length, and the thickness ofthe specimen, respectively. The elastic solution can be approximated as astress field consisting of two regions: one which is loaded elastically Vel andanother which is unloaded Vul. The elastic energy in the specimen for acrack length a is then

Ue =σ2u

2EVel =

σ2u

2E(Db− ka2)t, (8.11)

k Vel

Vul

σu

k Vel

Vul

log D

log σu

LEFM

Yield or Strength Criteria

21

lp

log Dc

log

a0

a0

( )

c

cu

D

DD

+=

1

σσ

σ

σ

u

D

b

(a) LEFM (b) NLFM (c) Scale effect

Fig. 8.5. Brittle and quasi-brittle crack propagation in a uniformly-loaded specimen.



where E is the elastic modulus and the slope k of the boundary between Veland Vul is a constant that is independent of the specimen dimension D andcrack length a. Griffith’s fracture criterion states that crack propagationoccurs when the change in elastic energy per new surface area created isequal to the fracture toughness Gc. Therefore,

−1t

[∂Ue∂a

]σu

=σ2u

Eka = Gc. (8.12)

The failure strength σu is obtained by solving Eq. (8.12), which givesσu =

√EGc/ka. Since the failure strength decreases with the crack length

a, the maximum strength for the specimen corresponds to the initial cracklength a0:

σMAXu =

√EGcka0

. (8.13)

Consider a series of experiments conducted on specimens with the samematerial and the same geometric proportions but with different dimensions.On a logarithmic plot, the strength in Eq. (8.13) scales as a line withslope −1/2, as shown in Fig. 8.5(c).

In contrast with the LEFM assumptions, a crack in a quasi-brittlematerial can propagate in a stable manner until a damage zone is formedbehind the crack tip. The effect of the damage zone is equivalent to anincrement lp to the crack length, as shown in Fig. 8.5(b). Therefore thefailure strength can be expressed as:

σu =

√EGc

k(a0 + lp). (8.14)

Since a0 is proportional to D, the strength represented by Eq. (8.14)results in the scaling law

σu(D) =σc√

1 + DDc

, (8.15)

which can be plotted on a logarithmic scale as shown in Fig. 8.5(c).Equation (8.15) is known as Bazant’s scaling law [11]. It can be observedthat the failure of small specimens is governed by strength considerations,while that of large specimens is governed by LEFM. Furthermore the degreeto which size effects on structural strength can be predicted by LEFMdepends on the laminate notch sensitivity, which is a function of both



the laminate material and the notch length. The notch sensitivity can beexpressed by the dimensionless ratio η of the notch length a over lp:

η =a

lp

η < 5, ductile damage, plasticity5 ≤ η ≤ 100, quasi-brittle fracture mechanics

η > 100, brittle.(8.16)

The load-carrying capability of notch-ductile components is dictated bystrength; that of notch-brittle parts is dictated by fracture toughness; andfor the range in between these extremes, both strength and toughness playa role. The process zone length for polymeric composites is of the orderof a few millimeters, so notched panels with notch lengths greater thanapproximately 1–10 cm are notch-brittle [3].

Specimens that do not exhibit localized fracture planes, such as inthe case of brooming, splitting, etc., may exhibit much enhanced notch-ductility. For instance, notch-ductility is enhanced by multiple crackingand matrix splitting along the fiber direction. It has been shown that thefracture toughness of a composite laminate increases with fiber strengthand decreases with fiber/matrix shear strength [23].

Since cohesive laws are defined in terms of strength and fracturetoughness, they can represent equally well the propagation of cracks asthe softening of a material, as in the example of the tension-loaded quasi-brittle bar, which gives them the ability to produce the entire range ofstructural scaling represented by Eq. (8.15) and shown in Fig. 8.5(c).

8.2.2.3 Softening law and the R-curve effect

The relationship between the functional form of the material softening law,the length of the process zone, and the shape of the R-curve has not receivedmuch attention from the computational mechanics community. As describedin the previous section, the physics of stable crack growth should be viewedas the gradual development of a fracture process zone behind the cracktip that produces a stabilizing influence on crack growth, characterized bya rising R-curve. For a bilinear softening curve such as the one shown inFig. 8.3(b), the R-curve can be approximated by the expression [24]:

GR(∆a) =

Gc

∆alp

(2 − ∆a

lp

)for ∆a < lp

Gc for ∆a ≥ lp

, (8.17)

where lp can be estimated from Eq. (8.10) with β = 1.



Fig. 8.6. Trilinear cohesive law obtained by the superposition of two bilinear laws.

In composite materials more than one physical phenomenon is ofteninvolved in the fracture process. Some phenomena act at small openingdisplacements, which are confined to correspondingly small distances fromthe crack tip, and others act at higher displacements, which extend furtherinto the crack wake. In these situations more complicated softening lawsthan the bilinear softening law may be necessary to capture the correctcrack growth response. For example, when fiber bridging or friction effectsare present, a softening law may be required that has a peak at low crackdisplacements to represent the tip process zone, and a long tail at highcrack displacements to represent the bridging in the wake of the crack.

In this case, multilinear softening laws can be obtained by combiningtwo or more bilinear cohesive laws, as illustrated in Fig. 8.6. Thetwo underlying linear responses may be seen as representing differentphenomena, such as a quasi-brittle delamination fracture characterized bya short critical opening displacement δc1, and fiber bridging characterizedby a lower peak stress and a longer critical opening displacement δc2.A multilinear cohesive law can provide a more accurate approximationof the process zone length and a more accurate approximation of anexperimentally determined R-curve.

Consider a trilinear cohesive law such as that shown in Fig. 8.6. Todescribe such a trilinear law, it is convenient to consider the superpositionof two bilinear cohesive laws that peak at the same displacement jump.Two bilinear softening responses are used for convenience and do notnecessarily correspond to two distinct failure modes, which could peakat different displacement jumps. In fact, the bridging strength does nottypically contribute to the peak strength, which is associated with theintrinsic fracture process prior to the bridging process. Consequently, a



trilinear cohesive law can be described by the proportions: σc1 = nσc,σc2 = (1 − n)σc, G1 = mGc, and G2 = (1 − m)Gc with 0 ≤ n, m ≤ 1,so that

Gc = G1 +G2 and σc = σc1 + σc2. (8.18)

A procedure for determining the strength ratio n and the toughnessratio m that approximate an experimentally determined R-curve ispresented below. On the basis of Eq. (8.17), an expression for an R-curvethat results from the sum of two bilinear cohesive laws [24] is defined as:

GR(∆a) = nGc∆alp1

(2 − n

m

∆alp1

)︸︷︷︸

=G1 if ∆a≥mn lp1

+ (1 − n)Gc∆alp1

(2 − (1 − n)

(1 −m)∆alp1

)︸︷︷︸

=G2 if ∆a≥ 1−m1−n lp1

,

(8.19)

where lp1 = 2πEGc

σ2c

is the length of the process zone for a single bilinearcohesive law. If the two superposed bilinear cohesive laws are ordered suchthat m/n ≤ (1−m)/(1−n), then the process zone length for the resultingtrilinear law is

lp2 =1 −m

1 − nlp1. Otherwise, lp2 =

m

nlp1. (8.20)

Consequently lp2 ≥ lp1, i.e. the process zone length for a trilinear cohesivelaw is longer than that of the corresponding bilinear cohesive law. Forexample, consider the problem examined in [24] of a crack with an initiallength a0 = 25mm in a material with modulus E = 70GPa, fracturetoughness Gc = 180kJ/m2, and strength σc = 2000MPa. The length ofthe process zone is equal to lp = 2.01mm, and the associated R-curveobtained from Eq. (8.17) is shown in Fig. 8.7. The R-curve for a trilinearcohesive law defined by m = 0.556 and n = 0.944 obtained from Eq. (8.20)is also shown in Fig. 8.7.

8.2.2.4 Mixed-mode cohesive laws

In structural applications of composites, crack growth is likely to occurunder mixed-mode loading. Therefore, a general formulation for cohesivelaws must address mixed-mode fracture. A mixed-mode cohesive law canbe illustrated in a single three-dimensional (3D) map by representing ModeI on the 2–3 plane, and Mode II in the 1–3 plane, as shown in Fig. 8.8. Thetriangle O − Y − δFI is the bilinear material response in pure Mode I and



0

40

80

120

160

200

20 25 30 35 40 45

GR ,

kJ/m

2

Crack Length, a, mm.

=17.8 mm

G

G

1=100 kJ/m2

2=90 kJ/m2

=2.01 mm1

2

Bilinear

Trilinear

Fig. 8.7. R-curves for bilinear and trilinear cohesive laws.

O − S − δFII is the bilinear material response in pure Mode II. It can beobserved that the tensile strength Y is lower than the shear strength S, andthe ultimate displacement in shear can be larger than in tension. In this3D map, any point on the 0–1–2 plane represents a mixed-mode relativedisplacement. Under a mixed mode, damage initiates at δ0 and completefracture is reached at δF . Consequently the tractions for Mode I and Mode IIunder mixed-mode loading follow the reduced curves O − Y M − δFMI andO−SM−δFMII , respectively. The areas under these two curves represent thefracture energies under the mixed mode. In the model proposed by Turonet al. [25], the initial strength and the critical value of the ERR in themixed-mode cohesive interface damage are functions of the mode-mixityparameter β:

β = 1 −(δnδ

)2

, (8.21)

where δn is the displacement jump in the direction normal to the cracksurface. The critical strength and fracture toughness values for mixed-modeloading are defined as

σ2c = Y 2 + (S2 − Y 2)β,Gc = GI + (GII −GI)β,

(8.22)



F

F

F

Y

Traction

Y

0FM

FM

S

S0

3

1

2

M

M

Fig. 8.8. Mixed-mode cohesive law.

where Y and S are the transverse tensile strength and shear strength,respectively, of a composite ply and GI and GII are the Mode I and IIcritical ERR values, respectively. This mixed-mode model has been adoptedby a number of authors [26–30] and has been extended to trilinear cohesivelaws by Hansen et al. [31].

Despite the maturity of cohesive laws, some issues regarding theprediction of crack propagation under mixed-mode conditions remainunresolved. According to any of the mode-mixity measures defined in thereferences cited above, the mode ratio is rarely if ever constant duringfracture. Even in specimens such as the mixed-mode bending (MMB)specimen, where from the LEFM point of view the mode ratio is constantduring propagation, it can be observed that the opening displacementsat damage initiation are dominated by Mode II, and that immediatelybefore complete separation the displacement jumps are mostly in Mode I.Sørensen et al. [32] and Turon et al. [33] have observed that the ratio ofinterlaminar strengths affects the prediction of delamination propagation,even when the crack length is long and the propagations should respondaccording to LEFM. By enforcing the condition of a non-negativedamage rate under variable mode mixity, Turon obtained a relationshipbetween interlaminar strengths and fracture toughnesses that has theform:

τ0 = σ0

√GIIcGIc

, (8.23)



where τ0 represents the interlaminar shear strength and σ0 represents thepeel strength. When the shear strength is set according to Eq. (8.23)and the LEFM assumptions are valid then the load-displacement curvefor propagation predicted from a mixed-mode cohesive formulation isequivalent to that predicted using LEFM. However, additional research isneeded for a full understanding of mixed-mode crack propagation.

8.3 Continuum Representation of Material Response

8.3.1 Distributed damage vs. localization of fracture

All materials exhibit irreversible non-linearities, which can be due toplasticity, damage, and fracture. When a material’s constitutive tangentrelationship between the strain increments and the stresses is positivedefinite, as shown in Fig. 8.9, the response is said to be of a hardeningtype. Hardening is a macroscopic, distributed, and irreversible materialresponse that smears the stress concentrations and eliminates anystress singularities. Examples of hardening damage include plasticity anddistributed damage mechanisms at the microstructural scale. Differentmaterial non-linearities can be identified by comparing the unloading pathsto the loading paths. Unloading path A indicates that the material islinear elastic. Path B indicates that the material has undergone plasticdeformation. Path C indicates the presence of additional non-linearitiessuch as cracking. The constitutive tangential stiffness for loading path D isnon-positive definite and the response is said to be of a softening type, whichcorresponds to the development and coalescence of voids and microcracks.

Hardening

Softening

τ

γ

G12

A B C D

Fig. 8.9. Typical shear stress-strain response exhibiting multiple non-linearities.



As a consequence of softening, damage localizes along a fracture surfacewhile material adjacent to the fracture surface unloads elastically.

A number of models have been proposed to represent material non-linearities. The hardening of composites can be modeled with localconstitutive models, i.e. models in which the stresses at a given point dependuniquely on the history of the strains up to that point. These models rely onthe implicit assumption that the material can be treated as a continuumat any arbitrarily small scale. These models are described by differentialequations and lack the notion of characteristic length.

Alternatively, the constitutive response of a material can be approxi-mated using a spatially periodic representative volume element (RVE)to represent the micromechanical response of individual constituents andtheir interactions [34,35]. The degree of complexity of the system is oftenreduced by considering a 2D approximation of the 3D continuum, assumingplane-strain conditions for a 2D model of the material. It is assumedthat each RVE deforms in a repetitive way, identical to its neighbors.Periodic boundary conditions are imposed on each RVE in order to ensurecompatibility of the deformation field along the boundaries. RVEs mustbe large enough to reflect the stochastic fluctuations of material propertieson the pertinent scale while the computational requirements call for anRVE to be as small as possible. However, even when the RVE is large,extending the analysis to a highly non-linear material response that leads tolocalization of damage renders the periodicity in the boundary conditions,and consequently the RVE approach, unsatisfactory [36].

Until Bazant and others developed the concept of crack bands [37],the idea of using strain softening to represent cracking in a continuum wascontroversial [38]. It was often argued that materials with a non-positivedefinite tangential moduli tensor do not exist. The point in the deformationhistory where the tangential stiffness of the constitutive model loses itspositive definite properties indicates the formation of discontinuities. Fromthe mathematical point of view, the so-called loss of ellipticity of thegoverning differential equation induces numerical difficulties related to theill-posedness of the boundary-value problem [39,40]. Since an infinitesimalchange in the data can cause a finite change in the solution, ill-posednessis manifested by the pathological sensitivity of numerical results to finiteelement discretization.

The problem can be easily illustrated by considering a simpleexample of a quasi-brittle bar loaded in monotonic tension, as shown inFig. 8.10.



L+

EF,

(a) (b)

Fig. 8.10. (a) Bar under tensile load. (b) Constitutive response with linear softening.

The constitutive damage model is a function of the strain given by thefollowing expression:

σ(ε) =

Eε, ε ≤ εi

(1 − d)Eε, εi < ε < εf , d =εf (ε− εi)ε(εf − εi)

0, ε > εf

, (8.24)

where d is a scalar damage variable and E is the elastic modulus ofthe material. While the strains do not exceed εi, the force-displacementrelationship of the bar is

F = Aσ =EA

Lδ. (8.25)

The material properties and the geometry of a real bar cannot beexactly uniform. Assuming that the strength of a small region of thebar ΩB is lower than the strength of the remaining portion of the barΩA, damage localizes in the region ΩB . Consequently the material in ΩBsoftens and in order to satisfy the equilibrium conditions the materialin ΩA unloads elastically. Equilibrium dictates that the stress along thebar is

σ = σA = σB ⇔ σ

E= εA = εi

εf − εBεf − εi

. (8.26)

Compatibility of the displacements gives

(L− LB)εA + LBεB = δ. (8.27)

Using Eqs. (8.26) and (8.27) to solve for εA and εB gives the force-displacement relationship during the damage process:

F = Aσ = EAδ − LBεfL− LB

εf

εi

. (8.28)



Fig. 8.11. Force-displacement response of a bar loaded in tension for different numericaldiscretizations.

As Eq. (8.28) indicates, the force-displacement relationship for damagedepends on the length of damage localization LB , which can take anyvalue between zero and L. Consequently the problem has infinitely manysolutions, as the post-peak solutions illustrated in Fig. 8.11 indicate, and itis not clear which of these solutions is correct. In addition, some post-peakresponses such as that for LB/L = 0.01 are meaningless because they haveno solution once the critical displacement is exceeded.

In finite element analyses, the length LB is related to the element lengthLe. Consider a model of the bar composed of Ne linear beam elements ofequal length. The length of the localized zone is LB = Le = L/Ne. Thepost-peak solution given by Eq. (8.28) is therefore strongly dependent onthe number of elements Ne. As the number of elements tends to infinity,the post-peak response approaches the initial (linear elastic) solution.Furthermore, the energy dissipated in the localized zone is calculated as∫V

∫ εf

0σdεdV = 1/2σcεfALe, which depends on the element length Le.

At the limit, when the element size tends to zero, the computationalmodel predicts failure without any energy being dissipated, a physicallyunacceptable result. If the crack-band model does not permit reducing theelement size to zero, convergence cannot be defined and the boundary-value



problem becomes ill-posed. Somehow, the boundary-value problem has tobe regularized.

In the context of continuum mechanics, so-called non-local techniquesare available to resolve spurious mesh sensitivity and to retain theobjectivity of the numerical response. Most of these techniques introducespatial interaction terms that have a smearing effect on the deformationfields, and thus preclude localization in a plane [41,42]. A similar smearingcan be obtained in a computationally simpler technique using rate-dependent (viscous) properties [43]. The material properties necessary forcrack smearing, or strain softening, are chosen such that the width of alocalizing diffuse crack band in a continuum is equal to the characteristiclength, which is associated with the material response.

Alternatively, the objectivity of the numerical solution can be simplyachieved by adjusting the post-peak material response using a characteristicelement length Le. This technique, proposed by Bazant and Oh [37], consistsof ensuring that the computed dissipated energy due to the fracture processis constant and equal to the product between the fracture energy Gc andthe crack surface A. Solving the resulting equation for the ultimate straingives:∫

V

∫ εf

0

σdεdV = GcA⇒ ALeEεiεf

2= GcA⇒ εf =

2GcLeEεi

. (8.29)

When Eq. (8.29) is substituted into (8.28), the response of the bar becomes

F = EAδ − 2Gc

σc

L− 2EGc

σ2c

. (8.30)

The response of the bar represented by Eq. (8.30) is independent of thecrack band or discretization length Le. In addition, the response given byEq. (8.30) is identical to the response obtained from (8.6) using a cohesivecrack model, provided that the penalty stiffness of the cohesive law is avery large number: K E/L.

It can be observed that the adjusted constitutive model takes intoaccount a size effect, since the response of the bar depends on the length,L: a longer bar has a more brittle post-peak response. It can be shownthat under displacement control, the post-failure response described byEq. (8.30) is stable only if L ≤ 2EGc/σ2

c .Another important aspect in the simulation of fracture using crack-

band models is that there exists a maximum size of the finite elements that



E

c

E

c

E

c

Softening Limit Snap-back

Fig. 8.12. Snap-back at constitutive level.

can be used in the simulation of a crack band. In order to avoid snap-back inthe constitutive model illustrated in Fig. 8.12, the ultimate strain cannot beless than εi = σc/E. Under this circumstance, the maximum characteristicsize of the finite element is:

ALMaxe

σ2c

2E= GcA⇒ LMax

e =2EGcσ2c

. (8.31)

The implications of Eq. (8.31) are important: the use of an element sizelarger than LMax

e , which for some composite damage modes is a fraction ofa millimeter, results in the overestimation of the critical ERR. When it iscomputationally impractical to use a sufficiently fine mesh, a workaroundconsists of artificially reducing the strength in order to preserve the correctvalue of Gc [22,44].

8.3.2 Idealization of damage modes in composite materials

The complex damage mechanisms occurring in advanced compositematerials result in additional difficulties in the numerical simulation offailure. While some intralaminar damage mechanisms such as transversematrix cracking (Fig. 8.1(b)) and delamination (Fig. 8.1(c)) occur in easilyidentifiable fracture planes that are parallel to the fiber direction, otherssuch as fiber failure are more difficult to idealize. These complexities makethe representation of intralaminar damage using a kinematic descriptionbased on strong discontinuities a formidable task. A more computationallytractable approach consists of using failure criteria based on thehomogenized stress or strain state to idealize the mechanisms of failure.

8.3.3 Failure criteria and strength

Many failure criteria have been proposed to predict the onset of matrixcracking and fiber fracture, some of which are described and compared



in the worldwide failure exercise (WWFE) [45,46]. However, few criteriacan represent several relevant aspects of the failure process of laminatedcomposites, e.g. the increase of apparent shear strength when applyingmoderate values of transverse compression, or the detrimental effect ofthe in-plane shear stresses in failure by fiber kinking. The LaRC03 failurecriteria [47] and subsequent evolutions [48] address some of the limitationsof other failure criteria as identified from the WWFE. For example, theLaRC criteria account for the effect of ply thickness, fiber misalignmentin compression, and the effect of shear non-linearity on fiber kinking andin situ strength.

8.3.4 Crack tunneling and in situ strength

Transverse matrix cracking is often considered a benign mode of failurebecause it normally causes such a small reduction in the overall stiffness ofa structure that it is difficult to detect during a test. However, transversematrix cracks can have a strong effect on the development of damage. Workby Green et al. [49] and others indicates that scaling effects in the failure ofthe matrix produce different modes of failure. Thicker plies were found tocrack and cause delaminations at relatively low loads, while thinner pliesresulted in more brittle failure mechanisms with cleaner through-thicknessfracture surfaces.

To predict matrix cracking in a laminate subjected to in-plane shearand transverse tensile stresses, a failure criterion must account for the in situstrengths. The in situ effect, originally detected in Parvizi et al.’s [50] tensiletests of cross-ply glass-fiber-reinforced plastics, is characterized by highertransverse tensile and shear strengths of a ply when it is constrained byplies with different fiber orientations in a laminate, compared with thestrength of the same ply in a unidirectional laminate. The in situ strengthalso depends on the number of plies clustered together and on the fiberorientations of the constraining plies. The results of Wang’s [51] tests of[0/90n/0] carbon/epoxy laminates indicate that thinner plies exhibit ahigher transverse tensile strength.

Both experimental and analytical methods [52] have been proposed todetermine the in situ strengths. The in situ strengths are calculated usingfracture mechanics solutions for the propagation of cracks in a constrainedply. For typical ply thicknesses, it can be assumed that defects exist in aply and that these material defects span the thickness of the ply, as shownin Fig. 8.13. Cracking of the ply can be assumed to occur when the ply



Fig. 8.13. In situ strength determined from the propagation of a slit crack in a ply.

Fig. 8.14. Transverse tensile strength as a function of ply thickness [51].

is loaded above the load required to propagate the slit crack in the fiberdirection (tunneling). It can be shown that the stress required for tunneling(or the in situ strength) can be approximated as [52]

Y Tis =

√8GIcπtΛ0

22

, (8.32)

where GIc is the Mode I fracture toughness of the matrix, t is the plythickness, and Λ0

22 is an elastic modulus of the material. Predicted in situstrengths as a function of the ply thickness and some experimental valuesof in situ strength are shown in Fig. 8.14 for a T300/944 graphite/epoxy.A similar approach is followed to calculate the in situ shear strength.

In multidirectional composite laminates subjected to uniform stressstates, cracks accumulate during the loading process. As the loading on



an individual ply is increased, new cracks suddenly appear, initially atrather random locations, and then with a progressively more uniform crackspacing. Eventually new sets of cracks appear deterministically equallyspaced between the original cracks. At some load the crack density reachesa saturation value at which a different event occurs, such as delaminationor fiber failure in an adjacent ply.

As the density of cracks in each ply of the laminate increases, networksof cracks are formed, which can link up through the thickness of thelaminate by inducing delaminations. A simplified damage progressionsequence of coupled transverse matrix cracking and interlaminardelamination is shown in Figs. 8.15(b)–(d) for the case of a laminated platesubjected to a tensile load. Initially, the laminate is undamaged, as shownin Fig. 8.15(a). As a result of the load application, transverse matrix cracksform in different plies of the laminate, as shown in Fig. 8.15(b). In theabsence of a stress concentration, the locations of the initial matrix cracks

Fig. 8.15. Idealized damage progression sequence in a laminated composite platesubjected to tensile loading: (a) initial stage without damage, (b) matrix cracking stage,(c) delamination stage, linking up matrix cracks in various plies, and (d) specimenfracture.



are random, and cannot be known a priori. At some value of the appliedload, delaminations initiate from the matrix cracks, Fig. 8.15(c). Thesedelaminations can connect matrix cracks in adjacent plies, which can causethe disintegration of the laminate.

8.3.5 Continuum damage models for composite materials

Continuum damage mechanics models for composite materials werepioneered by Ladeveze and LeDantec [53], Matzenmiller et al. [54], andothers based on previous work by Kachanov [55], Lemaıtre et al. [56], andothers. In these composite damage models, a distinction is made betweenthe different failure modes, especially between fiber and matrix failure.These models include a progressive softening of the material response,with internal damage variables describing the softening response. However,a softening response causes a localization of the strains along a surfaceknown as the failure surface and these strain-softening models typicallydo not include a characteristic length. Therefore they exhibit pathologicaldependences on element size. Upon reducing the mesh size to zero, suchanalyses predict that failure would occur with zero energy dissipation.

To resolve this lack of objectivity with respect to element size,a characteristic length must be inserted into the constitutive model.The evolution of intralaminar damage in laminated composites can berepresented by softening laws that define the evolution of damage in terms ofthe fracture energy dissipated in each damage mode. Most damage models,such as the progressive damage model for composites provided in Abaqus R©

[28] and typical cohesive elements [25,57], represent the evolution of damagewith linear softening laws that are described by a maximum traction and acritical energy release rate.

Using the LaRC04 failure criteria [47] as damage activation functionsFM it is possible to formulate a continuum damage model to predict thepropagation of M damage mechanisms occurring at the intralaminar level.Each damage activation function predicts one type of damage mechanismusing the following equations:

FM := φM (εt) − rtM ≤ 0, (8.33)

where rtM are internal variables (equal to 1 at time t = 0), and the functionsφM (εt) of the strains εt correspond to the LaRC04 failure criteria. Whena damage activation function is satisfied, FM ≥ 0, the associated damagevariable dM takes on a positive, non-zero value less than or equal to 1, and



the ply compliance tensor is affected by the presence of damage. Using themodel proposed by Maimı [4], the compliance matrix of a damaged ply isdefined as:

H =

1(1 − d1)E1

−υ12

E20

−υ12

E2

1(1 − d2)E2

0

0 01

(1 − d6)G12

, (8.34)

where d1 is the damage variable associated with fiber fracture, d2 is thedamage state variable associated with matrix cracking, and d6 is a damagevariable associated with both damage mechanisms.

In addition to the damage activation functions and damagedcompliance tensor, it is necessary to define the evolution laws for the damagevariables dM . The damage evolution laws need to ensure that the computedenergy dissipated is independent of the refinement of the mesh.

A complete definition of a continuum damage model for the simulationof intralaminar damage can be found in [4,44]. The algorithm for theintegration of the damage constitutive model was implemented inan Abaqus R© UMAT subroutine. The CDM model simulates localizedintralaminar damage using strain softening constitutive models. In orderto avoid mesh-dependent solutions, energy dissipation is regularized foreach damage mechanism using a modification of the crack-band model.To avoid physically unacceptable snap-backs of the material response, themaximum allowable element size is determined using closed-form equations[44]. If the element size exceeds this maximum for any damage mode, thecorresponding strength is automatically reduced to preserve the correctERR.

8.4 CDM: Limitations

To predict the ultimate strength of composite structures, it is necessaryto have an accurate numerical representation of all damage modes andtheir interactions. Some of the most complex damage models availablerely on CDM to represent the intralaminar damage modes (e.g. transversematrix cracking and fiber failure) and use cohesive zone models to capturedelamination between ply interfaces. Some of the combined CDM/cohesivemodels, such as the impact models of Lopes et al. [58], rely on extremely



fine meshes with one or more elements through the thickness of a ply,while others use stacks of shell elements to represent sublaminates withinlarger structures [59,60]. However, despite advances in progressive damagemodeling, recent studies (e.g., van der Meer and Sluys [61]) indicate thatCDM models coupled with cohesive zone models may not always representlaminate failure sequences properly. These deficiencies are particularlyevident when the observed fracture mode exhibits matrix splitting andpullouts [49] or when the fracture is characterized by a strong couplingbetween transverse matrix cracking and delamination [7,61].

The deficiencies of the predictive capabilities consist of several issues,including the incorrect prediction of the damage zone size normal to thefracture direction when using crack-band models and the inability of localCDM models to reliably predict matrix crack paths. These limitationsare mostly due to the fact that CDM models are usually implemented as“local” rather than “non-local” models [36], i.e. the evolution of damage ina local CDM model is evaluated at individual integration points withoutconsideration of the state of damage at neighboring locations. The followingdiscussion pertains mostly to such local implementations, since non-localdamage models are less widely used due to the difficulty in implementingthem within the finite element method.

The premise of the crack-band approach for regularizing CDM modelsis that damage localizes into a band with a width equivalent to theelement dimension. If the element size is smaller than the damage processzone, the crack-band approach may not correctly predict the width of thedamage zone nor the local stress field. Consequently the stress redistributionresulting from damage development may be inaccurately predicted and canpotentially result in inaccurate representation of damage mode interactionsand failure sequences.

As a result of homogenization and damage localization, CDM modelshave difficulties predicting crack paths. Since homogenization eliminatesthe distinction between the fibers and the matrix, a CDM model cannotdistinguish between cracks that propagate along fiber directions fromthose that cross fibers [61]. In CDM models implemented with damagelocalization, the damage state at any integration point in the model dependsonly on the stress field at that point rather than the damage state ofneighboring points. Therefore, the direction of damage evolution is drivenonly by the instantaneous local stress distribution. In other words, the localdirection of cracking may be predicted correctly by the failure criteria, but ifthe morphology of the material is not properly accounted for in the damage



ττ

τ τ

21

Damage band

21

a

12

12

Propagation of shear damage along fiber. Propagation across fiber direction.

(a) (b)

Fig. 8.16. Idealized propagation of shear damage (adapted from [61]).

model, the sequence of failures that eventually defines the path of a crackat a macroscopic level may be predicted incorrectly.

The potential inability of CDM models to determine the correctdirection of propagation is particularly evident when the stress field isdominated by shear. Consider two different plies in a laminate with a notchand subjected to shear, as shown in Fig. 8.16. In both situations, the stresslevel required to initiate matrix microcracks is correctly predicted by thefailure criterion. Furthermore, both situations would result in an identicalsequence of failures, since the stress field is identical. However, it is clearlynot the same to propagate a crack in a sequence of linked microcracks(Fig. 8.16(a)) as it is to propagate a crack band across fibers (Fig. 8.16(b)).Matrix cracking in a shear band running parallel to the fibers is a relativelybrittle failure mechanism, whereas matrix cracking normal to the fibersproduces a damage band that requires much more work to propagate.

The sensitivity of CDM predictions on the finite element meshorientation also contributes to the difficulty in predicting the crack path.Although the objectivity of the solution with respect to element size isaddressed with the crack-band approach described in the previous section,the predicted damage may be dependent on mesh orientation and elementshape. When strain-softening constitutive models are used in a finiteelement simulation, damage tends to propagate along preferred directions,consisting of either element edges or element diagonals. A demonstrationof the sensitivity of simulation results to mesh orientation is provided inFig. 8.17 for a unidirectional compact tension (CT) specimen with fibersoriented at 90 to the load direction. Results are presented for simulations



Mesh aligned with crack direction. Mesh inclined to crack direction.

(a) (b)

Fig. 8.17. Effect of mesh orientation on crack path in a unidirectional CT specimen.

obtained with a mesh oriented parallel to the fiber direction (Fig. 8.17(a))and with an inclined mesh in front of the crack tip (Fig. 8.17(b)). Thecrack should propagate along the fiber direction. However, the results showdirectional bias, and the simulated crack band propagates in the directionof the element alignment.

The tendency for damage to localize along mesh lines can be partiallyattributed to shear locking [62]. In the CDM methodology, a crackor displacement discontinuity is represented by a degradation of thecorresponding terms in the constitutive stiffness. As the crack opens, thestiffness degradation is such that stress should not be transferred acrossthe crack faces. However, such unloading may not occur due to in-planeshear locking. Shear locking here refers to inappropriate shear stress transferacross a widely open smeared crack, which occurs when an element cannotshear without inducing tensile strains. In a ply with orthotropic properties,a matrix crack should be represented by setting the transverse shearmodulus G12 and the transverse Young’s modulus E22 to zero. However,quadrilateral elements have been shown to exhibit coupling between γ12

and ε11 unless the element edges are aligned with the softening band or areoriented at 45 to the band [63]. Furthermore the tendency of the element tolock is dependent on the order of integration of the element: fully integratedelements are more susceptible to pathological in-plane shear locking thanreduced-integration elements.

The shear stress transfer across an open smeared cracked caused byshear locking can result in inaccurate prediction of stress redistributionafter damage development. Iarve et al. [64] demonstrated shear locking



for a simple case of a unidirectional [08] graphite/epoxy open-hole tensionspecimen. Experiments show that this specimen exhibits splitting cracksparallel to the load and tangential to the hole. Splitting near holes andnotches reduces the stress concentration, so predicting their effect isessential in obtaining the ultimate strength with any accuracy. Iarve’sprogressive damage analyses were conducted using a radial-type meshpattern with different levels of mesh refinement, where a radial meshtypically consists of a pattern of elements radiating from a circular holeand ending at a rectangular boundary. It was demonstrated that it isnot possible to predict the stress relaxation with a radial mesh due tolongitudinal splitting unless the fiber-direction modulus E11 is also set tozero.

When damage localizes and a fracture path is known a priori, arelatively simple approach to circumvent some of the limitations of CDMnoted above consists of aligning the mesh with the direction of fracture [65].Mesh alignment can be used to force a matrix crack to localize along thefiber direction. The benefit of mesh alignment is demonstrated below for anopen-hole tension specimen. Predictions were obtained for an [08] IM7-8552laminate using the continuum damage model provided within the Abaqus

finite element code [28]. Each ply was modeled using quadrilateral reducedintegration shell elements, S4R, to minimize the tendency for locking.Analyses were obtained using a radial mesh and an aligned mesh, as shownin Fig. 8.18. The loading direction and fiber direction are parallel to theX-axis.

Radial mesh

(a)

Aligned mesh

(b)

Fig. 8.18. Splitting damage predicted using a radial mesh and an aligned mesh.



Load versus end displacement.

(a)

Normalized axial stress distribution.(b)

Fig. 8.19. Effect of mesh type on predicted ultimate failure and fiber-stress relaxationdue to splitting in a [08] laminate.

The damage zones predicted in the region of the hole using the radialand aligned meshes are shown in Fig. 8.18. The results indicate thatboth models predict longitudinal splitting tangential to the hole. The loadversus end-shortening response obtained with the two models is shown inFig. 8.19(a). The radial mesh model predicts failure of the specimen at 65%of the load predicted by the model with the aligned mesh. The failure load isseverely underestimated by the radial mesh due to the inability of rotatedelements to represent shearing along the axial split. The tensile stressesalong the width (Y direction) of the specimen are shown in Fig. 8.19(b).

The results obtained with the aligned mesh indicate that the stressconcentration at the hole is nearly eliminated as a consequence of the splitas expected. The results obtained using the radial mesh show insufficientstress relaxation at the edge of the hole, indicating that damaged elementstransfer shear load across the damaged band even though the matrix-dependent moduli G12 and E22 have been set to zero.

8.5 Regularized x-FEM (Rx-FEM) Framework

The extended finite element method (x-FEM) is a technique that can beused to predict the location and evolution of matrix cracks in compositeswhile avoiding the aforementioned limitations associated with CDM models.x-FEM is a mesh enrichment technique based on a pioneering concept



by Moes et al. [66], which facilitates the introduction of displacementdiscontinuities such as cracks at locations and along directions that areindependent of the underlying finite element mesh. Although most of theresearch on x-FEM is devoted to arbitrary crack propagation in isotropicmaterials, recent applications to composite materials include delaminationmodeling and textile composite architecture representation [67]. Huynhet al. [68] provide a review of contemporary developments in x-FEM as wellas novel applications to interfacial crack analysis in 2D and 3D problems.

Modeling a matrix crack that propagates parallel to the fiber directionin a ply is conceptually straightforward using x-FEM. However, it is moredifficult to model networks of matrix cracks in a laminate where thefracture planes of matrix cracks in individual plies intersect at commoninterfaces and can cause delaminations that link the cracks throughthe thickness. Within the traditional x-FEM approach, the difficulty inmodeling networks of linked matrix cracks could be addressed by developinga special enrichment for multiple crack situations or by connecting twoenriched/cracked elements. Such connections were recently accomplishedin a quasi-2D formulation, for example, by van der Meer and Sluys [69] andLing et al. [70].

Another direction in which x-FEM is being developed is the regularizedextended finite element method (Rx-FEM) proposed by Iarve [71–74]. InRx-FEM, the step function approaches to describe the crack surface, whichare used in x-FEM, are replaced by a continuous function. Displacementshape functions are used to approximate the step function, and theGauss integration can be retained for element stiffness matrix computationregardless of the orientation of the crack. The cohesive connection betweentwo plies in which cracks have been introduced can be established bycomputing integrals of the products of the shape functions at the plyinterface. Therefore, with the Rx-FEM technique a kinematically powerfulmodel of crack networks can be constructed in which transverse matrixcracks are inserted parallel to the fiber direction at locations determinedusing a failure criterion.

A simulation begins without any initial matrix cracks. As the loadingis increased, matrix cracks are inserted according to the LaRC03 failurecriterion [47]. The criterion is evaluated at each integration point and ifthe criterion is exceeded a matrix crack oriented in the fiber direction isadded. The crack is inserted using the displacement enrichment necessaryto model the displacement jump. The magnitude of the jump is initially zeroand is controlled by an interface cohesive law (Turon et al. [25]). The same



cohesive law is used at the ply interfaces to represent potential delaminationsurfaces. A Newton–Raphson procedure is applied to find the equilibriumsolution at each load step of the implicit incremental solution.

The following section describes the formulation of Rx-FEM. Our goalhere is to highlight the concepts. More detailed discussion of the formulationcan be found in [71] and [73]. Then, a few examples that illustrate theapplication of Rx-FEM to predict matrix cracking-induced delaminationfailure in unnotched laminated composites are considered. The ability ofthe model to predict the effects of ply thickness and ply orientation onmatrix cracking and delamination are discussed as well.

8.5.1 Matrix crack modeling using Rx-FEM

A discontinuous displacement field over a crack surface Γα can be re-presented using two continuous displacement fields u1 and u2 and theHeaviside step function H as follows [66]:

u = H(fα)u1 + (1 −H(fα))u2, (8.35)

where fα is a signed distance function of the crack surface Γα. This functionis defined for an arbitrary point as the distance from the point to the cracksurface. The signed distance function is positive if the point is located inthe direction of the normal to the crack surface and negative if it is locatedin the direction opposite to the normal. The strain field and subsequentlythe stress field are computed similarly on each side of the crack from thecontinuous displacement fields u1 and u2 as

ε = H(fα)ε1 + (1 −H(fα))ε2, (8.36)

σ = H(fα)σ1 + (1 −H(fα))σ2. (8.37)

The calculation of the strain energy of a volume, i.e. an elementcontaining a crack Γα, is more difficult because the approximations givenin Eqs. (8.35)–(8.37) are discontinuous across the surface Γα. Therefore,separate computations of the stiffness matrix are required on each side ofthe crack using complex element subdivisions and the associated integrationpoints. Nevertheless, the strain energy of the volume of interest V can becast in the following form:

W =∫V

(H(fa)W1 + (1 −H(fa))W2)dV, (8.38)



where W1 and W2 are computed from the strain and stress on each sideof the crack and the aforementioned integration detail is hidden by thepresence of the step function. The cohesive energy over the crack surface Sαcan be written using the cohesive law provided in Eq. (8.2) as

M =∫Sα

(∫ δ

0

σ(δ)dδ

)ds, (8.39)

where the displacement jump δ = ‖u1 − u2‖ and the normal vector to thecrack surface can be readily computed. Performing the surface integrationis not straightforward because the crack surface inside the volume V hasto be discretized in order to perform the surface integration. On the otherhand, if there were a practical way of dealing with the step function andits derivatives, the surface integral in Eq. (8.39) would be calculated as avolume integral using

M =∫V

|∇H |(∫ δ

0

σ(δ)dδ

)dV. (8.40)

Equation (8.40) can be readily understood since

∇H(fα) = D(fα)∇fα, (8.41)

where D(x) is the Dirac delta function, and the gradient of the signeddistance function is by definition the unit normal vector to the cracksurface so that |∇H(fα)| = D(fα). However, the transition from thevolume integral to a surface integral in Eq. (8.40) is intuitive, especiallyconsidering the 1D case where the “volume” is a line and the “crack surface”is a point. Clearly evaluation of Eq. (8.40) within the standard x-FEMframework has no practical application and needs to be evaluated by surfacediscretization.

In the regularized x-FEM formulation, the step function H(fα) isreplaced with a continuous function H(x). In the formulation proposedby Iarve [71], the function H(x) is expressed as a superposition of thedisplacement approximation functions. This function is equal to 0 or 1everywhere except in the vicinity of the crack surface. If the displacementapproximation functions are Xi(x), then

H(x) =∑

hiXi(x) (8.42)



and the coefficients hi are obtained as

hi =

∫VX

〈fα〉i fdVα∫

VXidV

or hi = 0.5 + 0.5

∫VXifαdV∫

VXi|fα|dV . (8.43)

As indicated in Eq. (8.43), the coefficient hi is equal to 0 or 1 if the signeddistance function does not change sign in the support domain of the shapefunction Xi. On the other hand, when the crack crosses the support domainof the shape function, 0 < hi < 1, the evaluation of Eqs. (8.43) requiresdetermination of the signed distance function from the crack surface, whichis a trivial task in the case of a straight crack that is normal to the plyinterfaces and has a prescribed orientation.

The strain energy, Eq. (8.38), and the cohesive energy, Eq. (8.40), canbe evaluated in the Rx-FEM formulation, whereH(fα) is replaced by H(x),using standard Gauss quadratures for a crack of arbitrary direction andlocation, which is fully defined by the hi coefficients. The price for suchsimplicity and robustness is the non-zero width of the crack surface, whichis associated with the width of the gradient zone |∇H | > 0. The width ofthis zone is a function of the mesh size. In other words, standard x-FEMencloses the crack inside the element and Rx-FEM smears the crack surfaceover possibly more than one element. Energetically, however, the cohesiveenergy of Rx-FEM will approach that of the regular crack with meshrefinement. Reducing the mesh size, i.e. reducing the size of the supportdomain of the shape functions, reduces the size of the band where Xi is notconstant. Therefore, the difference between the step function representationof the crack surface H(fα) and the continuous representation H(x) isreduced.

The ability to maintain the Gauss quadratures in “cracked” elementsalso allows for simple calculation of the cross products of shape functionson the surfaces between plies with arbitrary fiber and matrix crackingdirections. This is illustrated in Fig. 8.20, where two plies with differentcrack directions are shown, and the regularized step functions H associatedwith the crack in each ply are shown by shading. Enriched functions ineach ply are obtained by multiplying the shape functions by the respectivestep functions. Since the step functions are smooth functions and do notalter the support domains for the shape functions, the cross integrals of theenriched functions in the two plies can be easily computed by the Gaussquadrature.



Fig. 8.20. Two plies with different cracking directions. The cracking direction in eachply and the shaded level plot of the regularized step function H are shown.

8.6 Rx-FEM Simulations

Numerical results are presented in the following sections to verify andillustrate the proposed methodology. First, we consider in detail the ini-tiation of delamination from a given matrix crack in a transversecrack tension (TCT) specimen. Next, we simulate failure in multilayeredcomposites and evaluate the ply thickness and lamination effects on matrixcracking and delamination initiation and propagation.

8.6.1 Transverse crack tension test

The TCT specimen was designed to measure Mode II interlaminar fracturetoughness [75]. It consists of three unidirectional (θ = 0) plies withthicknesses t, 2t, and t and where the middle ply is cut at the specimen mid-length prior to curing. The specimen is subjected to axial tensile loading,and when a critical load is exceeded, delaminations between the middleply and the top and bottom plies develop suddenly. These delaminationscontinue to propagate in a stable manner with increasing load until thedelaminations reach the grips, at which point the load is carried by the twoundamaged outer plies.



Conventional FE model

(a)

Rx-FEM model

(b)

Fig. 8.21. Conventional finite element (FE) model and Rx-FEM model of the TCTspecimen.

A TCT specimen was analyzed using a conventional finite element (FE)model, shown in Fig. 8.21(a), and an Rx-FEM model, shown in Fig. 8.21(b),to illustrate the ability of the Rx-FEM method to insert transverse matrixcracks at arbitrary locations and orientations, and to demonstrate theability to represent the correct interactions between transverse matrixcracks and the corresponding delaminations. Although Fig. 8.21 shows theentire laminate thickness, only half of the laminate thickness was modeledand symmetry conditions were applied to the mid-surface of the laminate.The shaded red regions in the figures show the initial middle ply crack.In the case of the conventional FE model, the crack in the middle plyis aligned with a mesh line and is simply modeled by using unconnecteddouble nodes.

For the Rx-FEM model, a curved non-uniform mesh was used todemonstrate the mesh independence of the approach. In the Rx-FEMmodel, the middle ply crack was inserted at the start of the analysis and isnot aligned with the mesh cell boundaries. The total number of elementsin the longitudinal direction is 120 for both models. However, the localdensity of the Rx-FEM mesh near the delamination crack tip is not uniformdue to the irregularity of the mesh. In both models, the delaminationsbetween the plies were modeled using the cohesive technique describedabove and one element was used through the thickness of each ply (dueto symmetry half of the middle ply was represented by 1 element). Thematerial properties for T300/914C from [75] are summarized in Table 8.1.The thermal prestress is not considered since all plies have the same



Table 8.1. Material properties used in the analyses.

T300/914C [12] T300/976 [14]

E11 (GPa) 139.9 138E22, E33 (GPa) 10.1 10.3G23 (GPa) 3.7 3.1G12,13 (GPa) 4.6 5.5ν23 0.436 0.66

ν12, ν13 0.3 0.3α11 (1/0C) — 0.4 × 10−6

α22, α33 (1/0C) — 2.54 × 10−5

T − T0 (0C) — –125Yt (MPa) 80 37.9Yc (MPa) 300 200S (MPa) 100 100GIC (J/m2) 120 157GIIC (J/m2) 500 315

20

25

30

35

40

45

50

55

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Regular FE Rx-FEM

Axial displacement (mm)

Axi

al s

tres

s (G

Pa)

Fig. 8.22. Load versus displacement curves of the TCT specimen predicted using theconventional FE model and the Rx-FEM model.

orientation and there is no mismatch of thermal expansion propertiesbetween plies. The load versus applied displacement curves predicted bythe two models are shown in Fig. 8.22. The two responses are nearlyidentical.



Fig. 8.23. Fracture sequence in (25/−25/90n)s laminates: (a) just prior to delamination;(b) subsequent to delamination; (c) just prior to final failure. Reprinted with permissionfrom [76].

8.6.2 Effect of ply thickness

One of the key factors affecting the matrix cracking and delaminationfailure modes in laminated composites is the ply thickness. A systematicexperimental study of delamination failure as a function of ply thicknesswas conducted by Crossman and Wang [76]. A T300/934 [±25/90n]slaminate family with n = 1, . . . , 8 was subjected to uniaxial tensile loading,perpendicular to the 90 ply, and failure loads and patterns were carefullydocumented. The delamination patterns (hatched lines) and crack densities(spacing between horizontal lines) can be observed in Fig. 8.23 for three loadlevels and for two thicknesses of the 90 ply, namely n = 3 and 8.

The differences observed in the shapes of the delaminations in the twolaminates are evident: in the n = 8 case, the delamination is funneling offthe individual matrix cracks, whereas for the thinner plies, the delaminationspreads over multiple transverse matrix cracks, and is referred to as “oystershaped” in [76]. A significant difference between the two cases is also seenin the 90 ply transverse crack densities.

The results of simulating the tensile loading of these two laminatesare presented in Fig. 8.24. The material properties for T300/936 used inthe model are shown in Table 8.1. To illustrate the damage developmentprocess, damage variable contours for both transverse matrix cracks anddelaminations are plotted on the undeformed geometry. The areas ofdelamination correspond to interfaces where the value of the damagevariable d exceeds 0.995. The Rx-FEM transverse matrix cracks in each



[±25/908]s

(a)

[±25/903]s Step 1

(b)

[±25/903]s Step 2

(c)

Fig. 8.24. Predicted cracking and delamination patterns in [±25/908]s and [±25/903 ]slaminates. Blue areas correspond to predicted delaminations at the 90/−25 interface andgreen areas correspond to predicted delaminations at the 25/−25 interface.

ply correspond to surfaces where the discontinuity function H is equalto 0.5.

Predicted matrix cracking and delamination patterns for the laminateswith n = 8 and n = 3 are shown in Fig. 8.24. Blue areas correspondto predicted delaminations at the 90/−25 interface, and green areascorrespond to predicted delaminations at the 25/−25 interface. The stateof cracking and delamination immediately before the complete failure ofthe [±25/908]s laminate is shown in Fig. 8.24(a). The delaminations in the[±25/903]s laminate evolve extremely rapidly before failure. A sequence oftwo states of delamination at nearly identical loads is shown in Figs. 8.24(b)and (c). It can be observed that the predicted density of matrix crackingfor n = 8 is significantly lower than predicted for the thinner n = 3 case.In addition, the shape of the delamination in Fig. 8.24(a) is very similarto the experimental funnel-type delamination shown in Fig. 8.23. In boththe experimental observations and the predictions, thin delamination areasaccompany all matrix cracks.

The delaminations predicted for n = 3 (Figs. 8.24(b) and (c)) covermultiple transverse cracks and have shapes consistent with the experimental



results shown in Fig 8.23. The extent of the delamination in Fig. 8.24(b)is very similar to that in Fig. 8.23. It is likely that the larger predicteddelamination in Fig. 8.24(c) corresponds to an unstable equilibrium state,which is unlikely to be caught in the experiment. Since the predicted extentof the delaminations and the crack density is in good agreement with theexperimental observations for both ply thicknesses, it is concluded that theeffect of ply thickness on matrix cracking and delamination evolution is wellrepresented by the Rx-FEM model.

8.6.3 Internal delamination versus edge delamination

In the case of the [±25/908]s laminates considered above, the delaminationevolution process initiates from the intersection of matrix cracks and thefree edges, leading to the eventual disintegration of the laminate. It isof interest to evaluate the present methodology for characterizing theprocess of matrix crack-induced damage accumulation in laminates withdifferent ply orientations, where the delamination and matrix crackingevolution and interaction patterns may vary. A number of angle-plylaminate configurations were experimentally and analytically investigatedby Johnson and Chang [77]. The T300/976 graphite fiber material system(see Table 8.1 for ply level properties) was used. Tensile failure of a[±45/90]s laminate and a [±602]s laminate, both considered in [77], witha ply thickness of 0.127mm are considered below. These laminates do notcontain any 0 plies and completely lose their load-carrying capacity as aresult of matrix cracking and delamination.

Predicted matrix crack and delamination damage evolution patternsfor a [±45/90]s laminate and for a [±602]s laminate are shown in Fig. 8.25.Damage patterns are shown at three load levels, including the load levelimmediately preceding the simulated final failure. For both laminatesconsidered, a few cracks develop in the very early stages of loading (notshown). All of the matrix cracks then quickly grow though the width of thespecimen.

The general damage evolution process of the [±45/90]s specimen issimilar to the edge delamination initiated process seen before in the[±25/908]s laminates. Triangular-shaped delaminations initiate in multiplelocations on the +45/−45 and −45/90 interfaces at the matrix crack andfree edge intersections, as shown in Figs. 8.25(a), (b), and (c). As the loadingincreases, the delaminations grow inwardly and expand in size until theyconnect the two edges and the interfaces via matrix cracks, at which pointthe specimen fails.



[±45/90]s [602/−602]s

Transverse Crack

Delaminations

45/-45 Interface-45/90 Interface

60/-60 Interface

(a)

(b)

(c)

(d)

(e)

(f)

Initiation of delamination

Intermediate load

Imminent final failure

Fig. 8.25. Predicted damage at three increasing load levels in a [45/−45/90]s laminate(a,b,c) and in a [602/−602 ]s laminate (d,e,f).

The failure process in the [602/−602]s specimen is starkly differentcompared to the previous laminate. Delamination initiation andpropagation are not anchored around the outer edges of the specimenas in the [±45/90]s specimen. Delamination in this case initiates in theinterior of the specimen at the matrix crack intersections, as shown inFig. 8.25(d). As the load is increased, the delamination grows in theinterior of the specimen. Figure 8.25(e) shows a delamination band ofalmost uniform length through the entire width of the specimen, which thenextends and allows the matrix cracks to separate the plies (Fig. 8.25(f)).This difference in failure mechanisms between the two laminates has beenobserved experimentally [77].

The ability to address various failure mechanisms arising in non-traditional composite laminates without modifying the analysis frameworkand/or mesh is a critical advantage of x-FEM technology. Such capabilityis becoming increasingly important with aerospace companies focused onincreasing the structural efficiency of composites and breaking away fromtraditional laminate design.



8.7 Conclusions

Vast numbers of new computational models capable of predicting thedamage processes in composites are continuously advocated. Giventhe increasing complexity of these numerical methods, the task ofdistilling technological breakthroughs by sorting through vastly differentdemonstrations of successful representations of the propagation of damagemechanisms is daunting. In the present chapter, an overview of thefundamentals of the idealization of damage in composites was presentedwith an emphasis on identifying the issues associated with the scale ofdamage idealization and size effects. It was shown that the ability ofdamage models to predict the initiation and propagation of damage isrelated to the shape of the softening law and its corresponding characteristiclength, which depends on the material properties and on the scale(resolution) of the idealization selected. The capabilities of advancedcontinuum damage mechanics (CDM) models were reviewed and theirpathological deficiencies were discussed. In particular, the conditions weredemonstrated under which cohesive laws and continuum damage modelscan achieve objectivity with respect to the mesh size, and how crackpropagation using CDM models is constrained by mesh orientation anddamage localization. Finally the use of an extended finite element techniqueto model damage propagation by inserting cohesive cracks in arbitrarydirections was presented as an emerging technology that avoids some ofthe limitations of CDM models.

Acknowledgements

The authors are grateful to Prof Pedro P. Camanho for helpful dis-cussions and some unpublished material that was used in this chapter.The authors thank Dr. David Mollenhauer, AFRL, for his technicalcontributions. Funding under NASA NRA NNX08AB05A and the AFRLcontract FA8650-05-D-5052 with the University of Dayton ResearchInstitute are gratefully acknowledged by Endel Iarve.

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[72] Patzak B., Jirasek M., 2003. Process zone resolution by extended finiteelements, Eng. Fract. Mech., 70, 957–977.



[73] Benvenuti E., 2008. A regularized XFEM framework for embedded cohesiveinterfaces, Comp. Meth. Appl. Mech. Eng., 197, 4367–4378.

[74] Oliver J., Huespe A.E., Sanchez P.J., 2006. A comparative study on finiteelements for capturing strong discontinuities: E-FEM vs X-FEM, Comp.Meth. Appl. Mech. Eng., 195, 4732–4752.

[75] Konig M., Kruger R., Kussmaul K., von Alberti M., Gadke M., 1997.Characterizing static and fatigue interlaminar fracture behaviour of a firstgeneration graphite/epoxy composite, in Composite Materials: Testing andDesign (S.J. Hooper, ed), ASTM, West Conshohocken, PA, pp. 60–81.

[76] Crossman S.W., Wang A.S.D., 1982. The dependence of transverse crackingand delamination on ply thickness in graphite/epoxy laminates, in Damagein Composite Materials, ASTM International, pp. 118–139.

[77] Johnson P., Chang F.-K., 2001. Characterization of matrix crack-inducedlaminate failure. Part I: Experiments, J. Compos. Mater., 35, 2009–2035.




Chapter 9

DELAMINATION AND ADHESIVE CONTACTMODELS AND THEIR MATHEMATICAL

ANALYSIS AND NUMERICAL TREATMENT

Tomas Roubıcek∗,†,‡, Martin Kruzık‡,§ and Jan Zeman¶∗Mathematical Inst., Charles University, Sokolovska 83

CZ-186 75 Praha 8, Czech Republic

†Inst. of Thermomechanics of the ASCR, Dolejskova 5CZ-182 00 Praha 8, Czech Republic

‡Institute of Information Theory and Automation of the ASCRPod vodarenskou vezı 4, CZ-182 08 Praha 8, Czech Republic

§Department of Physics, Faculty of Civil EngineeringCzech Technical University, Thakurova 7


¶Department of Mechanics, Faculty of Civil EngineeringCzech Technical University, Thakurova 7


Abstract

This chapter reviews mathematical approaches to inelastic processes on

the surfaces of elastic bodies. We mostly consider a quasistatic and rate-independent evolution at small strains and the so-called energetic solution.This concept is applied e.g. to brittle/elastic delamination, cohesive contactproblems, and to delamination in various modes. Besides the theoreticaltreatment, numerical experiments are also presented. Finally, generalizationsto dynamical and thermodynamical processes are outlined, together with anextension to the homogenization of composite materials with debonding phases.

9.1 Introduction

This chapter is devoted to formulation, mathematical, and numericaltreatments of inelastic processes on surfaces. Through this chapter, withthe exception of Sections 9.6.5 and 9.6.6, these inelastic processes areconsidered to be quasistatic (no inertia is taken into account) and

349


350 T. Roubıcek, M. Kruzık and J. Zeman

rate-independent (no internal time scale is considered). We will addressonly small-strain models. Our general framework will solely be basedon the hypothesis that the evolution is governed by a time-dependentGibbs-type stored energy functional E (involving external loading) anda dissipation potential R, which, being degree-1 positively homogeneous,reflects the rate-independence of the process (i.e. invariance under anymonotone rescaling of time). Both functionals are defined on a suitablestate space considered here in the form U ×Z . With only a small lossof generality, we assume that R involves just the z-component of a stateq = (u, z) ∈ U ×Z . This distinguishes z as a “slow” variable, while u is a“fast” variable because its velocity is not controlled by any dissipation.

To introduce the concept of the energetic approach informally into sucha class of rate-independent systems, we consider a uniform discretization ofa fixed finite time interval [0, T ] with a fixed time step τ . Given the initialdata (u0, z0) := (u(0), z(0)) the values of the state variables at the k-thtime level (ukτ , z

kτ ) := (uτ (kτ), zτ (kτ)) with k = 1, 2, . . . , T/τ follow from

the solution of the incremental variational problem

minimize (u, z) → E (kτ, u, z) + R(z − zk−1τ )

subject to (u, z) ∈ U ×Z

. (9.1)

This energy minimization provides a unified approach to both the continuumthermodynamics of inelastic solids, e.g. [1,2], as well as to computationalinelasticity, [3–5]. In addition, rigorous mathematical theory has beenestablished to study the time-continuous behavior of the problem (9.1)(corresponding to the limit τ → 0), see [6–9] and Section 9.3 below.

As a prototypical application of the energetic framework in modelingcomposite materials and structures, we now briefly introduce a simplifiedelastic delamination model (sometimes also called debonding) treated inmore detail later in Section 9.5. Here, we restrict our attention to two elasticdomains Ω1 and Ω2, sharing an interface ΓC. The structure is subjected toa time-dependent hard-device loading with a Dirichlet boundary conditionimposed by displacements wD(t) acting at part of the external boundaryΓD (with uD(t) denoting an extension of wD(t) to bodies Ω1 and Ω2).

In this context, u+uD is used to denote the displacement field . Moreover,we will work with an internal variable z (possibly vectorial) describinginelastic delamination processes on the boundary ΓC. Various possibilitiesare presented in this chapter. The simplest scenario works with a scalar-valued delamination parameter considered as a function ofx ∈ ΓC, with z= 1and z=0 corresponding to coherent and fully damaged interfacial points


Delamination and Adhesive Contact Models 351

x, respectively. The time-independent set U consists of the kinematicallyadmissible displacements, satisfying the homogeneous Dirichlet boundaryconditions on ΓD and frictionless contact conditions on ΓC:

u = 0 on ΓD and[[u]]n≥ 0 on ΓC, (9.2)

where [[u]]n denotes the normal component of the displacement jump [[u]].Analogously, the set of admissible internal variables Z is defined as

0 ≤ z ≤ 1 on ΓC. (9.3)

Now, given admissible u and z, the stored energy functional reads

E (t, u, z) :=2∑i=1

∫Ωi

C(i)e(u):e

(12u+uD(t)

)dx+

∫ΓC

z

2K

[[u]]·[[u]] dS,

(9.4)where e(u) denotes the small-strain tensor associated with displacementu, C

(i) stores the stiffness tensor of the i-th domain, and K is the elasticinterfacial stiffness. The dissipation rate is defined as

R(.z) :=

∫ΓC

a|.z| dS if.z ≤ 0 on ΓC,

∞ otherwise,(9.5)

with a denoting the fracture energy, representing the energy dissipatedby complete delamination. The value ∞ in (9.5) is used to ensureunidirectionality of the delamination phenomena, i.e. no healing of theinterface is admissible during the loading process. Sometimes only a finitelyvalued R is considered, which means that healing (also called rebonding)is allowed, cf. e.g. [10–12]. Such models are, however, mathematically lessdifficult in some aspects than (9.5) and their interpretation is rather limited(as the configuration, possibly shifted after complete delamination, has atendency to remember its initial state after healing if not combined with aninterfacial plasticity like in [105]), and will not be particularly addressed inthis chapter.

After the energy functionals (9.4) and (9.5) have been specified,the tools and techniques presented in the remainder of this chapterwill allow us to study the delamination evolution rigorously, includingtheoretically supported numerical simulations. Note that the generalityof the energetic framework makes it easy to incorporate more realisticinterfacial constitutive laws naturally and to couple the delaminationphenomena with other inelastic properties such as plasticity, damage, or



phase transformations. In addition, the energy functional (9.4) can easily beadapted to the periodic homogenization theory, which makes the frameworkdirectly applicable to the analysis of e.g. fiber/matrix debonding in fibrouscomposites, see also Section 9.7 for a concrete example.

9.2 Concepts in Quasistatic Rate-Independent Evolution

We will consider a Banach spacea X ⊃ Z on which R is defined ashaving a domain with a nonempty interior and being coercive, and degree-1homogeneous in the sense R(λz) = λR(z) for any λ ≥ 0. The set X mustbe a Banach space (to define the degree-1 homogeneity), but here also U

and Z will always be Banach spaces.Formally, for E : [0, T ]×U ×Z →R∪∞ and R : Z → [0,∞], the rate-

independent evolution we have in mind is governed by the following systemof doubly non-linear degenerate parabolic/elliptic variational inclusions:

∂uE (t, u, z) 0 and ∂R(.z) + ∂zE (t, u, z) 0, (9.6)

where.z := dz

dt and the symbol “∂” refers to a (partial) subdifferential,b andR(·), E (t, ·, z), and E (t, u, ·) are convex functionals in all specific modelsconsidered in this chapter.

In mechanics, internal variables and the inclusion (9.6), sometimesalso called Biot’s equation [13], are used for so-called generalized standardmaterials [14], cf. also [15] specifically for adhesive contact. There are soundvariational principles supporting the abstract model (9.6), although theirapplicability should not be overestimated.

The first inclusion in (9.6) expresses the minimum-energy principle,asserting that at any time t the displacement u minimizes u → E (t, u, z(t)).

Assuming for simplicity E smooth and denoting the partial differentialsby E ′

t , E ′u, and E ′

z, one can postulate a so-called Lagrangian in the form

L (t, u, z,.z) :=

ddt

E + R

= E ′t (t, u, z) + 〈E ′

u(t, u, z),.u〉 + 〈E ′

z(t, u, z),.z〉 + R(

.z)

= E ′t (t, u, z) + 〈E ′

z(t, u, z),.z〉 + R(

.z), (9.7)

aA normed linear space which is complete is called a Banach space. Recall that a normedspace X is complete if every Cauchy sequence in X converges to a limit in X.bRecall that the subdifferential ∂f(x) of a convex function f : X → R∪∞ at a point x isdefined as the convex closed subset ∂f(x) := x∗ ∈X∗; ∀ v ∈X: f(x)+〈x∗, v−x〉 ≤ f(v)of the dual space X∗; conventionally, 〈·, ·〉 : X∗ ×X → R denotes the duality pairingbetween the Banach space X and its dual X∗ := x∗: X → R, linear and continuous.



where we also used the first inclusion in (9.6). Then the second inclusion in(9.6) can be derived as the first-order optimality condition for

.z → L (t, u, z,

.z) is minimal for any time t. (9.8)

One refers to (9.8) as a minimum dissipation-potential principle, cf. [1–3,5].The degree-1 homogeneity of R still allows for further interpretation of

the flow rule, i.e. the second inclusion in (9.6). Defining the convex “elasticdomain” K := ∂R(0), the second inclusion in (9.6) only means 〈ω− z, w−.z〉 ≥ 0 for any w and any ω ∈ ∂R(w), where we introduced the so-calledthermodynamical driving force z ∈ −∂zE (t, u, z). In particular, for w = 0one obtains

〈z, .z〉 = maxω∈K

〈ω, .z〉. (9.9)

To derive (9.9), we used that z∈ ∂R(.z)⊂ ∂R(0)=K thanks to the degree-1

homogeneity of R(·), so that always 〈z, .z〉 ≤ maxω∈K〈ω, .z〉. The identity(9.9) means that the dissipation due to the driving force z is maximalprovided that the order-parameter rate

.z is kept fixed, while the vector

of possible driving stresses ω varies freely over all admissible drivingstresses from K. This resembles the so-called Hill’s maximum-dissipationprinciple [16], cf. also [17–20].

Let us also observe that the set K = ∂R(0) determines R becauseR = δ∗K with δK denoting the so-called indicator functionc of the set Kand δ∗K is its Legendre–Fenchel conjugate.d In terms of K, by standardconvex-analysis calculus [21], (9.9) can also be written as

.z ∈ [∂δ∗K ]−1(z) = ∂[[δ∗K ]∗](z) = ∂δK(z) = NK(z), (9.10)

where NK(z) denotes the normal conee to K at z. This is the well-known principle from plasticity theory, stating that the rate of the internalparameter z (representing, as a special case in plasticity theory, the plastic-deformation rate) belongs to the cone of outward normals to the elasticitydomain, see also [17,22] or [23, Section 3.2], [24, Section 2.4.4] or [25,Section 2.6].

cThis means that δK(v) = 0 if v ∈ K and δK(v) = ∞ if v ∈ K.dThe Legendre–Fenchel conjugate f∗ : X∗ → R ∪ ∞ of a function f : X → R ∪ ∞is defined as f∗(x∗) := supx∈X〈x∗, x〉 − f(x).eRecall that the normal cone NK(x) to a convex set K ⊂ X at x ∈ X is defined asNK(x) := x∗ ∈X∗; ∀ v ∈K : 〈x∗, v − x〉 ≤ 0.



9.3 Mathematical Concepts to Solve the System (9.6)

To design the concept of a weak solution to (9.6), we use z as in Section 9.2,and use the definition to write the three subdifferentials in (9.6) as thesystem of three inequalities

∀(t, u, z) : R(z)−〈z(t), z − .z(t)〉 ≥ R(

.z(t)), (9.11a)

E (t, u, z(t)) ≥ E (t, u(t), z(t)), (9.11b)

E (t, u(t), z) + 〈z(t), z − z(t)〉 ≥ E (t, u(t), z(t)), (9.11c)

where the first and third inequalities mean that z ∈ ∂R(.z) and −z ∈

∂zE (t, u(t), z(t)), respectively, while (9.11b) states that u(t) minimizes theenergy E (t, ·, z(t)). As R is homogeneous only of degree 1,

.z can be expected

to be bounded only in the Bochner–Lebesgue space L1(0, T ; X ),f or ratherin the corresponding space of measures. Thus the expression 〈z, .z〉 in (9.11a)might not be well defined or not allow for a mathematical treatmentregarding various limit procedures. Thus it is desirable to convert it toa more suitable form. For this, we use the formal chain rule

E (T, u(T ), z(T ))− E (0, u0, z0) =∫ T

0

ddt

E (t, u, z) dt

=∫ T

0

(E ′t (t, u, z)−〈z, .z〉) dt; (9.12)

cf. (9.7). We integrate (9.11a) over [0, T ] and substitute for∫ T0

R(.z) dt the

total variation

DissR(z, [0, T ]) := supN∑j=1

R(z(tj) − z(tj−1)), (9.13)

where the supremum is taken over all partitions 0 ≤ t0 < t1 < ... <

tN−1 ≤ tN ≤ T of [0, T ]; indeed, if z is absolutely continuous, then

fThe notation Lp(·) stands for the Banach space of measurable functions whose p-power isintegrable on the indicated domain, here [0, T ]. Such spaces are called Lebesgue spaces.If the functions take values in a general Banach space X then one applies a naturalgeneralization of measurability due to Bochner.



DissR(z, [0, T ]) =∫ T0

R(.z) dt. This turns (9.11a) into

E (T, u(T ), z(T )) + DissR(z, [0, T ])−∫ T

0

E ′t (t, u, z) dt− E (0, u0, z0)

≤∫ T

0

(R(z) − 〈z, z〉) dt. (9.14)

Note that (9.14) implies the energy inequality, more specifically for z = 0we get

E (T, u(T ), z(T ))︸︷︷︸stored energyat time t = T

+ DissR(z, [0, T ])︸︷︷︸energy dissipated

during [0, T ]

≤∫ T

0

E ′t (t, u, z) dt︸︷︷︸

work done bymechanical load

+ E (0, u0, z0)︸︷︷︸stored energyat time t = 0

.

(9.15)

We will use the notation B([0, T ]; U ) for the Banach space of boundedmeasurable functions [0, T ] → U defined everywhere, and BV([0, T ]; X )for functions [0, T ] → X with bounded variation;g recall that X ⊃ Z is aBanach space on which R is coercive, i.e. R(z) ≥ ε‖z‖ for some ε > 0.

Definition 9.1. (Weak solutions.) The process (u, z, z) : [0, T ] → U ×Z ×Z ∗ is called a weak solution of the initial-value problem given by(U ×Z ,E ,R) and the initial condition (u0, z0) if u ∈ B([0, T ]; U ), z ∈B([0, T ]; Z ) ∩ BV([0, T ]; X ), z ∈ B([0, T ]; Z ∗) and

(i) the inequality (9.14) with (9.13) holds for any z smooth enough, sothat (9.14) is well defined,

(ii) (9.11b) holds for any u ∈ U and any t ∈ [0, T ],(iii) (9.11c) holds for any z ∈ Z and any t ∈ [0, T ],(iv) the initial conditions u(0) = u0 and z(0) = z0 hold.

The advantage of the above definition is that it is completely derivativefree, i.e. no time derivative

.z and no (sub)differentials of E or R occur

explicitly in Definition 9.1. In fact, if E (t, ·, z) or E (t, u, ·) are not convexand thus (9.6) loses meaning, Definition 9.1 still yields a certain generalizedsolution. If a weak solution (u, z, z) is such that

.z is absolutely continuous

(i.e..z is not a measure but an L1-function) and if E (t, ·, z) and E (t, u, ·) are

gRecall that the variation of z : [0, T ] → X is defined as supPN

j=1 ‖z(tj ) − z(tj−1)‖,where ‖ · ‖ is the norm on X and the supremum is taken over all partitions of [0, T ].



convex, then (u, z) solves the original problem (9.6) for a.e. time t ∈ [0, T ].This justifies the above definition.

A drawback of the above definition is its rather low selectivity.h Aninconvenience is also the involvement of the driving-force field z valued inZ ∗, which obviously does not bear any generalization for spaces Z lackinga linear structure, like the Griffith-delamination problem (9.39)–(9.40),below. Thus one may be tempted to make further generalizations. As R isdegree-1 homogeneous and convex, ∂R(

.z) ⊂ ∂R(0) and thus −E ′

z(t, u, z) ∈∂R(

.z) implies R(z) + 〈E ′

z(t, u, z), z〉 ≥ R(0) = 0. The convexity of E (t, ·, ·)and 〈E ′

u(t, u, z), u〉 then further imply R(z)+E (t, u+ u, z+ z)−E (t, u, z) ≥0, which is obviously just (9.16), below. Thus we arrive at the following:

Definition 9.2. (Energetic solutions, [8,26,27].) The process (u, z):[0, T ]→U ×Z is called an energetic solution to the initial-value problemgiven by (U ×Z ,E ,R) and the initial condition (u0, z0) if u ∈ B([0, T ];U ), z ∈ B([0, T ]; Z ) ∩ BV([0, T ]; X ), and

(i) the energy inequality (9.15) holds,(ii) the following stability inequality holds for any t ∈ [0, T ]:

∀(u, z) ∈ U ×Z : E (t, u, z) ≤ E (t, u, z) + R(z − z), (9.16)

(iii) the initial conditions u(0) = u0 and z(0) = z0 hold.

In fact, any energetic solution satisfies (9.15) as an equality and thus,more conventionally, the definition of energetic solutions instead employsthe energy equality. If an energetic solution exists it is also a weak solution.Indeed, if (9.15) holds then (9.14) follows because R ≥ 0 and

⟨z, z

⟩ ≥ 0due to the fact that 0 ∈ ∂R(0). Taking z := z in (9.16) gives (9.11b).Finally, setting u := u in (9.16) and exploiting the convexity of E (t, u, ·)proves (9.11c).

An important step is to generalize Definition 9.2 to make E (t, ·, ·)nonconvex, which is just a typical situation in the modeling of quasistaticdelamination processes. Roughly speaking, energetic solutions evolve assoon as it is energetically not disadvantageous. It should be noted that thismay, however, not be exactly always in full agreement with the responseof real systems where some other rate-dependent phenomena may come

hFor example, weak solutions to the brittle delamination problem in the formulation(9.45) do not necessarily have the so-called Griffith property and do not recover theoriginal problem (9.39)–(9.40), through the formula (9.46), in contrast to the energeticsolutions (see Definition 9.2), which enjoy this property.



into play on some occasions. For this reason, there are also some othersolutions that are sometimes applicable and which successfully competewith energetic solutions, cf. also [7] for a comparison with other conceptsin general and [28–32] in the context of crack propagation. In particular, awell-motivated class of concepts is based on adding a small viscosity into oneor both inclusions in (9.6). Passing such viscosity to zero leads to so-calledvanishing-viscosity solutions to the rate-independent problem (9.6). In thecontext of delamination, the vanishing-viscosity in the second inclusion in(9.6) was exploited in [106, 107], leading to a definition which involves adefect measure balancing the energetics.

An efficient theoretical tool to prove the existence of energeticsolutions is the implicit time discretization of (9.6). Being constructive, itsimultaneously suggests a conceptual numerical algorithm; cf. Remark 9.3below. Considering, for simplicity, an equidistant partition of [0, T ] with atime step τ > 0, it formally leads to the recursive problem

∂uE (kτ, ukτ , zkτ ) 0 and ∂R

(zkτ − zk−1

τ

τ

)+ ∂zE (kτ, ukτ , z

kτ ) 0

(9.17)

for k = 1, 2, . . . , T/τ , starting from u0τ = u0 and z0

τ = z0. Note that, in fact,∂R is homogeneous of degree 0 so that the factor 1/τ can be omitted. Thepotential structure of the problem allows for a conceptually constructiveway to obtain a solution to (9.17), namely by solving the incrementalglobal-minimization problem, defined earlier by (9.1). Note that (9.17), thetime-discretized analog of (9.6), is a first-order optimality condition for anysolution to (9.1). This is also called the direct method for solving (9.17),i.e. no approximate problem is in principle needed to ensure the existenceof a solution to (9.17), see e.g. [33]. Of course, if E (t, ·, ·) is not convex,as is usual in delamination problems, (9.17) might admit other solutions.Equation (9.1) encompasses the energetic-solution concept, which in turnis easily amenable to mathematical and numerical treatment. The followingassertion uses quite minimal hypotheses:

Proposition 9.1. (Existence of time-discrete solutions.) If E (t, ·, ·)is lower semicontinuous and coercivei on U ×Z and also R ≥ 0 is lowersemicontinuous, then the incremental problem (9.1) possesses a solution.

iThis essentially means that the sub level sets of E (t, ·, ·), i.e. (u, z) ∈U ×Z ; E (t, u, z) ≤ c, are, for any c ∈ R that makes them nonempty, compact insome topology of U ×Z which makes E (t, ·, ·) and R(·) lower semicontinuous.



Let us consider a solution (ukτ , zkτ ) ∈ U ×Z of the incremental problem(9.1) at the level k. Comparing the energy value of (9.1) for a solution in thetime step k with energy at arbitrary (u, z), we obtain the discrete stability:

E (kτ, ukτ , zkτ ) ≤ E (kτ, u, z) + R(z − zk−1

τ ) − R(zkτ − zk−1τ )

≤ E (kτ, u, z) + R(z − zkτ ), (9.18)

where we also used the degree-1 homogeneity and the convexity of R, whichyields the triangle inequality R(z − zk−1

τ ) ≤ R(zkτ − zk−1τ ) + R(z − zkτ ).

Comparing the energy value of a solution at the level k with that fora solution (uk−1

τ , zk−1τ ) of the incremental problem (9.1) at the level k − 1

gives E (kτ, ukτ , zkτ )+R(zkτ −zk−1τ ) ≤ E (kτ, uk−1

τ , zk−1τ )+R(zk−1

τ −zk−1τ ) =

E (kτ, uk−1τ , zk−1

τ ), which yields an upper estimate of the energy balance inthe k-th step:

E (kτ, ukτ , zkτ ) + R(zkτ − zk−1

τ ) − E ((k − 1)τ, uk−1τ , zk−1

τ )

≤ E (kτ, uk−1τ , zk−1

τ ) − E ((k − 1)τ, uk−1τ , zk−1

τ )

=∫ kτ

(k−1)τ

E ′t (t, u

k−1τ , zk−1

τ )dt. (9.19)

Eventually, writing the stability (9.18) at the level k − 1 and testing it by(u, z) = (ukτ , zkτ ) gives a lower estimate of the energy balance in the k-thstep:

E (kτ, ukτ , zkτ ) + R(zkτ − zk−1

τ ) − E ((k − 1)τ, uk−1τ , zk−1

τ )

= E ((k − 1)τ, ukτ , zkτ ) +

∫ kτ

(k−1)τ

E ′t (t, u

kτ , z

kτ )dt+ R(zkτ − zk−1

τ )

−E ((k − 1)τ, uk−1τ , zk−1

τ ) ≥∫ kτ

(k−1)τ

E ′t (t, u

kτ , z

kτ )dt. (9.20)

It is convenient to introduce the notation for the piecewise constantinterpolants uτ and uτ , defined by

uτ (t) := ukτ for t ∈ ((k − 1)τ, kτ ], (9.21a)

uτ (t) := uk−1τ for t ∈ [(k − 1)τ, kτ). (9.21b)

The notation zτ and zτ has an analogous meaning. Moreover, we define

E τ (t, u, z) := E (kτ, u, z) for t ∈ ((k − 1)τ, kτ ]. (9.21c)



In terms of these interpolants, one can write (9.18), (9.19), and (9.20)summed over k in a compact form (9.22)–(9.23):

Proposition 9.2. (Stability and two-sided energy estimate.) LetR be degree-1 positively homogeneous and let E ′

t (·, u, z) ∈ L1(0, T ) for any(u, z). Then the discrete stability

∀(u, z) ∈ U ×Z : E τ (t, uτ (t), zτ (t)) ≤ E τ (t, u, z) + R(z − zτ (t))

(9.22)

holds for any t ∈ [0, T ], and, for any s = kτ ∈ [0, T ], k ∈ N, the followingtwo-sided energy inequality holds:∫ s

0

E ′t (t, uτ (t), zτ (t)) dt

≤ E τ (s, uτ (s), zτ (s)) + DissR(zτ , [0, s]) − E τ (0, u0, z0)

≤∫ s

0

E ′t (t, uτ (t), zτ (t)) dt. (9.23)

The two-sided energy estimate may facilitate the numerical solution ofthe global-optimization problem (9.1). It should be emphasized that in allthe applications considered in Sections 9.4–9.6, the incremental problem(9.1) involves a nonconvex functional, whose minimization is therefore verydelicate, and iterative procedures need good starting points. Various back-tracking strategies based only on the two-sided energy estimate (9.23) havebeen designed and tested in [34–36] for similar kinds of problems, see alsoSection 9.5.2 for additional details.

Let us assume, with some restriction of generality but still covering allproblems presented here, that:

∃ ε > 0 ∀ t, u, z : E (t, u, z) ≥ ε(‖u‖2U + ‖z‖2

Z ) − 1/ε, (9.24a)

∃ γ ∈ L1(0, T ) ∀ t, u, z : |E ′t (t, u, z)| ≤ γ(t)(1 + ‖u‖U ), (9.24b)

∃ ε > 0 ∀ z : R(z) ≥ ε‖z‖X . (9.24c)

Proposition 9.3. (Convergence of discrete solutions.) Let (9.24)hold, u0 be stable, and E (t, ·, z) be strictly convex. For τ → 0, there is asubsequence of the sequence of approximate solutions (uτ , zτ )τ>0, which



converges to some (u, z) in the sense

uτ (t) → u(t) in U for any t ∈ [0, T ], (9.25a)

zτ (t) → z(t) in Z for any t ∈ [0, T ], (9.25b)

DissR(zτ , [0, t]) → DissR(z, [0, t]) for any t ∈ [0, T ], (9.25c)

E ′t (·, uτ (·), zτ (·)) → E ′

t (·, u(·), z(·)) in L1(0, T ). (9.25d)

Moreover, every (u, z) obtained by such a limit process is an energeticsolution to the problem (E ,R, z0).

The proof of Proposition 9.3 conventionally relies on the following steps:

(1) A priori estimates, derived from the upper energy estimate in (9.23)by using the coercivity/growth assumption (9.24) and Gronwall’sinequality: namely one gets

‖uτ‖B([0,T ];U ) ≤ C, (9.26a)

‖zτ‖B([0,T ];Z )∩BV([0,T ];X ) ≤ C. (9.26b)

(2) The selection of convergent subsequences using Banach’s and Helly’sprinciples; the latter is used for the z-component, which has a boundedvariation. Using in addition the strict convexity of E (t, ·, z), one canshow that the u-component converges at each time t.

(3) Passage to the limit of the discrete stability (9.22) by finding a so-calledmutual-recovery sequence [37], i.e.∀ stable sequencej (tk, uk, zk) → (t, u, z) ∀ (u, z) ∃(uk, zk) :

lim supk→∞

(E (tk, uk, zk)+ R(zk − zk) − E (tk, uk, zk))

≤ E (t, u, z)+ R(z − z) − E (t, u, z). (9.27)

(4) Passage to the limit by weak lower-semicontinuity in the upper energyestimate, i.e. in the second inequality in (9.23).

Merging Propositions 9.1 and 9.3, one obtains:

Corollary 9.1. (Existence of energetic solutions.) Under the assum-ptions of Propositions 9.1 and 9.3, energetic solutions in the sense ofDefinition 9.2 do exist.

jA sequence (tk , uk, zk)k∈N is called stable if supk∈N E (tk , uk, zk) <∞ and if E(tk , uk, zk) ≤ E (tk , u, z) + R(z − zk) for all (u, z) ∈ U ×Z .



Remark 9.1. (Special unidirectional processes.) In many delami-nation models R has the special form

R(.z) = δK(

.z) + 〈a, .z〉 (9.28)

with some cone K ⊂ Z and some a ∈ Z ∗ non-negative in the sense that〈a, z〉 ≥ 0 for any z ∈ K. Then one can evaluate DissR(z, [0, T ]) in (9.14)very explicitly. Indeed, any solution satisfying (9.14) must have

.z ∈ K,

hence z(tj) − z(tj−1) ∈ K for any tj ≥ tj−1, so that (9.13) gives

DissR(z, [0, T ]) = supN∑j=1

〈a, z(tj) − z(tj−1)〉 = sup 〈a, z(tN ) − z(t0)〉

= 〈a, z(T ) − z(0)〉 = R(z(T )− z(0)). (9.29)

In this particular case, one can equivalently consider the dissipationpotential R0 : Z → 0,∞ as R0(

.z) = δK(

.z), if one augments the stored

energy by the term 〈a, z〉. In view of (9.6), this is obvious when writing

∂R(.z) + ∂zE (t, u, z) = ∂[R0(

.z) + 〈a, .z〉] + ∂zE (t, u, z)

= ∂R0(.z) + a+ ∂zE (t, u, z)

= ∂R0(.z) + ∂zE0(t, u, z)

for E0(t, u, z) := E (t, u, z) + 〈a, z〉. The philosophy behind this formula isthat the contribution to the stored energy via a unidirectional process cannever be gained back, and it is thus stored forever, which means that it isdissipated. This alternative setting has been considered, e.g., in [38–41].It should be emphasized that this purely mechanical alternative is nolonger equivalent in the full thermodynamical context when the dissipatedenergy contributes to heat production, in contrast to the stored energy,cf. Section 9.6.6.

Remark 9.2. (More general dissipation.) Sometimes it is useful toconsider R = R(u, z,

.z), and then the inclusion (9.6) modifies to

∂uE (t, u, z) 0, ∂.zR(u, z,.z) + ∂zE (t, u, z) 0. (9.30)

A priori estimates based on the test by.z are the same. Now in general,

DissR(z; [0, T ]) in (9.15) depends also on the u-component and in terms ofa placeholder q = (u, z) is defined by

DissD(q; [0, T ]) := supN∑i=1

D(q(ti−1, ·), q(ti, ·)

), (9.31)



where the supremum is taken over all partitions of the type 0 ≤ t0 < t1< · · · < tN ≤ T , N ∈ N; here D denotes a so-called dissipation distancedefined in [42], reflecting the minimum dissipation-potential principle(9.8), by

D(q1, q2) := inf

∫ 1

0

R(u, z,.z)dt;

q = (u, z) ∈ C1([0, 1]; X ), q(0) = q1, q(1) = q2

. (9.32)

As before, one assumes the positive one-homogeneity of R(u, z, ·). Thisimplies the triangle inequality

∀ q1, q2, q3 ∈ U ×Z : D(q1, q3) ≤ D(q1, q2) + D(q2, q3). (9.33)

In terms of the dissipation distance, the incremental problem (9.1) takesthe form

minimize (u, z) → E (kτ, u, z) + D((uk−1τ , zk−1

τ ), (u, z))subject to (u, z) ∈ U ×Z .

(9.34)

An important step in the conceptual generalization is to consider thedissipation distance D ≥ 0 satisfying (9.33) without any reference to R,and even without requiring any linear structure on U ×Z . The mutual-recovery-sequence condition (9.27) then modifies to

lim supk→∞

(E (tk , qk)+ D(qk, qk) − E (tk, qk)) ≤ E (t, q)+ D(q, q) − E (t, q).

(9.35)

Note that, if R = R(.z) and Z is a Banach space, formula (9.32) yields

D(q1, q2) = R(z2 − z1) with qi = (ui, zi), and one obtains the former case.

Remark 9.3. (Numerics.) One can further approximate U and Z in(9.34) by some finite-dimensional Banach subspaces Uh and Zh. Thus, inconcrete situations, we obtain computationally implementable numericalstrategies, as also demonstrated in Section 9.5.2. Besides, one has aconvergence analysis for h→ 0 at one’s disposal in specific cases; essentially,the proofs reduce to finding a suitable mutual-recovery sequence forconditions similar to (9.27) or (9.35), but involving also a sequence offunctionals Eh, which coincides with E on [0, T ]×Uh×Zh while being= ∞ elsewhere, cf. [43]. For an example see (9.51) below.



Fig. 9.1. Illustration of the geometry and the notation.

9.4 Quasistatic Brittle Delamination, The Griffith Concept

We now present models of quasistatic delamination that can be covered bythe general abstract ansatz (9.6) or (9.30). We start, in this section, withmodels of brittle delamination.

Let Ω ⊂ Rd (d = 2, 3) be a bounded Lipschitz domain,k and let

us consider its decomposition into a finite number of mutually disjointLipschitz subdomains Ωi, i = 1, . . . , N . Further denote Γij = ∂Ωi ∩ ∂Ωjthe (possibly empty) boundary between Ωi and Ωj. Thus, Γij represents aprescribed (d − 1)-dimensional surface, which may undergo delamination.We assume that the boundary ∂Ω is the union of two disjoint subsets ΓD

and ΓN, with

L d−1(∂Ωi ∩ ΓD) > 0, i = 1, . . . , N, (9.36)

where L d−1 denotes the (d − 1)-dimensional Lebesgue measure. On theDirichlet part of the boundary ΓD we impose a time-dependent boundarydisplacement wD(t), while the remaining part ΓN is assumed to be free.Therefore, any admissible displacement u : ∪Ni=1Ωi → Rd has to be equalto a prescribed “hard-device” loading wD(t) on ΓD.

We consider here the case of linear elasticity determined, on eachsubdomain Ωi, by the elastic-moduli tensor C(i). Moreover, we take intoaccount the local non-interpenetration of matter by requiring, for thedisplacement u, that [[u]]ij ·νij ≥ 0 on Γij , where νij denotes the unit normalto Γij oriented from Ωj to Ωi. Here, [[u]]ij denotes the jump u|Ωi − u|Ωj ,

kRecall that a domain is called Lipschitz if its boundary can be covered by a finitenumber of graphs of Lipschitz functions. Roughly speaking, it excludes corners of 0 or360, otherwise most domains considered in engineering are Lipschitz.



with u|Ωi being the trace on ∂Ωi of the restriction of u to Ωi. For

A ⊂ ΓC :=⋃i<j

Γij , (9.37)

we consider the stored-energy functional E in the form

E (t, u, A) :=

N∑i=1

12

∫Ωi

C(i)e(u):e(u) dx if u = wD(t) on ΓD,

[[u]]n ≥ 0 on ΓC,

[[u(x)]]= 0 for x ∈ A,

∞ elsewhere,

(9.38)

where e(u)= 12 (∇u) + 1

2∇u is the small-strain tensor, and where [[u]]stands for the particular jump [[u]]ij on the whole union ΓC of Γijs,and where we write for brevity [[u]]n ≥ 0 on ΓC instead of [[u]]ij · νij≥ 0 on Γij , i < j. This means, in particular, that the part A of thecontact boundary ΓC is perfectly glued while the rest ΓC\A is completelydelaminated and a frictionless unilateral, so-called Signorini contact takesplace there. This unilateral condition is important for preventing anunphysical delamination by a mere compression of the surface.

During the time-dependent loading wD, the glued part A=A(t)possibly evolves. In the simplest model, this process is consideredunidirectional, i.e. healing is not allowed so that t → A(t) is non-increasing,and for activation of delamination one needs (and thus dissipates) a specificenergy a : ΓC → R+ (in joules per unit area). The dissipated energy(understood also as the so-called dissipation distance) is then

D(A1, A2) :=

∫A2\A1

a(x)dS if A1 ⊂ A2 ⊂ ΓC,

∞ otherwise.(9.39)

In particular, the dissipated energy does not depend on particular fracturemodes; cf. Section 9.6.2 below for a refinement of this model. The philosophyof such a quasistatic evolution is related to the Griffith fracture criterion[44], which states that a crack grows as soon as the energy release is

more than the fracture toughness, here determined by a in (9.39). This“geometrical” framework was used in the small-strain setting in [29,32,45]



and also large strains in [31] for polyconvex materials,l and in [46] forquasiconvex materials.m

The definition of the energetic solutions (9.15) involves the timederivative of the stored energy, which is hardly defined for (9.38) unlesswD is constant in time. Therefore, using the additive shift u− uD(t) (whereuD is a suitable extension of the formerly defined wD) we resort to time-dependent Dirichlet boundary conditions. Thus, up to an irrelevant time-dependent constant, (9.38) transforms to

E (t, u, A) :=

N∑i=1

∫Ωi

C(i)e(u):e

(u2

+uD(t))dx if u = 0 on ΓD,

[[u]]n ≥ 0 on ΓC,

[[u(x)]] = 0 for x∈A,∞ elsewhere,

(9.40)

if one assumes that the Dirichlet loading allows for an extension such that

uD|ΓC = wD. (9.41)

Note that, with a slight abuse of notation, the symbol u in (9.40) andthereafter denotes the shifted displacement field satisfying u = 0 on ΓD.After such a shift of the Dirichlet boundary conditions, E ′

t does exist andone has the following simple formula for it:

E ′t (t, u, A) =

N∑i=1

∫Ωi

C(i)e(u):e

(.uD(t)

)dx. (9.42)

Definingn

U := u ∈ W 1,2(Ω\ΓC; Rd); u|ΓD = 0, (9.43a)

Z := A ⊂ ΓC; A measurable (9.43b)

lMaterials whose stored energy density is a convex function of the deformation gradient,its cofactor, and determinant.mMaterials whose stored energy density f is quasiconvex, i.e., f(F )|Ω| ≤ R

Ωf(F +

∇u(x)) dx for all smooth mappings u : Ω → R3 vanishing at ∂Ω.nNotation W k,p(Ω) stands for the Banach space of functions on Ω whose k-th derivativesbelong to Lp-space.



and adopting a generalization from Remark 9.2, one can claim:

Proposition 9.4. Let wD ∈W 1,1(I;W 1/2,2(ΓD; Rd)) with W 1/2,2

denoting a Sobolev–Slobodetskiı spaceo allow for an extensionuD ∈W 1,1(I;W 1,2 (Ω; Rd)) satisfying (9.41) and let A0 ⊂ ΓC be measurable.Then the problem (U ×Z ,E ,D , A0) defined by (9.39), (9.40), and (9.43),possesses at least one energetic solution in the sense of Definition 9.2.

Having such an energetic solution (u,A), it is natural to consider theshifted (u+uD, A) as a solution to the original problem with E from (9.38)even if E ′

t is not well defined. Also, let us note that there is no explicit linearstructure for the As that would allow us to write first-order optimalityconditions like (9.6) but, nevertheless, the concept of energetic solutionsstill works.

Further, it is convenient to reformulate this problem in a way thatZ is a subset of a Banach space. We introduce a so-called delaminationparameter z : ΓC → [0, 1], meaning a fraction of fixed adhesive: z(x) = 0means complete delamination and z(x) = 1 means 100% perfect bonding,while z(x) = 1

2means that 50% of the adhesive is debonded at x ∈ ΓC.

Here it is appropriate to consider the model[[u(x)

]]= 0 for a.e. x ∈ ΓC such that z(x) > 0, (9.44)

expressing that delamination can occur only if the adhesive is completelydebonded, i.e. only if z(x) = 0. Thus, instead of (9.40), we now consider

E (t, u, z) :=

N∑i=1

∫Ωi

C(i)e(u):e

(u2

+uD(t))

dx if u = 0 on ΓD,

[[u]]n ≥ 0 on ΓC

z[[u]] = 0 on ΓC,

0 ≤ z ≤ 1 on ΓC,

∞ elsewhere,

(9.45a)

the dissipation potential

R(.z) :=

∫ΓC

a|.z| dS if.z ≤ 0 on ΓC,

∞ otherwise,(9.45b)

oThe space W 1/2,2(ΓD), involving fractional derivatives, is just the space of traces onΓD of all functions from W 1,2(Ω).



and, instead of (9.43b),

Z := L∞(ΓC). (9.45c)

Now we can use Definition 9.2:

Proposition 9.5. Let wD ∈ W 1,1(I;W 1/2,2(ΓD)) and 0 ≤ z0 ≤ 1 bemeasurable. The problem (U ×Z ,E ,R, z0) defined by (9.43a) and (9.45)possesses energetic solutions in the sense of Definition 9.2.

The relation between the previous “geometrical” concept used inProposition 9.4 and this “functional” concept is that, if z takes only values0 or 1, i.e. always z = χA for some A ⊂ ΓC,p then

D(A1, A2) = R(z2 − z1) with z1 = χA1 , z2 = χA2 . (9.46)

It has been proved in [47] that any energetic solution (u, z) to the brittledelamination problem, whose existence was stated in Proposition 9.5, isof the Griffith type in the sense that z takes only values 0 or 1. Thus,in particular, Proposition 9.4 is proved if Proposition 9.5 is proved. Thelatter essentially relies on the explicit construction of the mutual-recoverysequence for condition (9.27) from [48], namely

uk := uk and zk :=zkz/z where z > 00 where z = 0.

(9.47)

Note that 0 ≤ zk ≤ zk always and, if one considers uk → u weakly inW 1,2(Ω; Rd) and zk → z weakly* in L∞(ΓC; Rd),q then also uk → u weaklyand zk → z weakly*.

As the adhesive does not exhibit any elastic response in model (9.38),we refer to it as a brittle delamination.

The classical formulation corresponding to (9.6) with E from (9.45a)and R from (9.45b) consists in the equilibrium of forces on each subdomainΩi and several complementarity problems. Using a simplified notation

pHere χA denotes the characteristic function of a set A, i.e. χA(x) = 1 if x ∈ A whileχA(x) = 0 otherwise.qThe adjective “weak*” refers to testing by functions from a so-called pre-dualspace. Here, as L∞(ΓC; Rd)= L1(ΓC; Rd)∗, weak* L∞(ΓC; Rd)-convergence means thatlimk→∞

RΓC

zk ·ϕ dS =RΓC

z·ϕdS for every ϕ ∈ L1(ΓC; Rd).



C = C(x) = C(i) if x ∈ Ωi, at a current time t, we can write it as:

div σ = 0, σ = C(i)e(u), in Ωi, i = 1, . . . , N, (9.48a)

u = wD(t, ·) on ΓD, (9.48b)

σν = 0 on ΓN, (9.48c)

[[σ]]ν = 0,[[u]]n ≥ 0, σn(u)[[u]]n = 0σn(u) ≤ 0 wherever z(t, ·) = 0z[[u]] = 0.z ≤ 0, ξ ≤ a,

.z(ξ − a) = 0,

ξ ∈ N[0,1](z) + ∂zI([[u]], z)

on ΓC, (9.48d)

where N[0,1](·) : R ⇒ R is the normal-cone mapping, I denotes the indicatorfunction of the constraint z[[u]] = 0, ξ is the driving “force” for thedelamination, and σν := C

(i)e(u)|Γν is the traction stress on Γ =ΓC orΓ = ΓN adjacent to Ωi. Moreover, its normal and tangential components onΓC are denoted σn(u) = (σν) · ν and σt(u) = σν − ((σν) · ν)ν, respectively,so that we have the decomposition σν = σnν + σt. Notice that, since byour choice ν turns out to be the inner unit normal on Ω1, σn in (9.48d) isnon-positive.

9.5 Elastic-Brittle Delamination

In contrast with Section 9.4, we will consider that the adhesive has anelastic response, which is called elastic-brittle delamination.

9.5.1 The model and its asymptotics to brittle delamination

Assuming a linear response of the adhesive, the possible modification of(9.45a) is

EK(t, u, z) :=

N∑i=1

∫Ωi

C(i)e(u):e

(u2

+uD(t))

dx

+∫

ΓC

12zK

[[u]] · [[u]] dS if u = 0 on ΓD,

[[u]]n ≥ 0 on ΓC,

0 ≤ z on ΓC,

∞ elsewhere,

(9.49)



with K being a positive-definite matrix representing the elastic response ofthe adhesive. Note that the constraint z ≤ 1 cannot be active if the initialcondition z0 satisfies it, therefore it is not explicitly involved in EK.

The classical formulation corresponding to (9.6) with E = EK from(9.49) and R from (9.45b) consists of an equilibrium of forces on eachsubdomain Ωi and three complementarity problems on ΓC, correspondingto the three subdifferentials in functionals EK and R. Before shifting theDirichlet conditions, it is (ρ is the Lagrange multiplier to the constraintz ≥ 0):

div σ = 0, σ = C(i)e(u), in Ωi, i = 1, . . . , N,

(9.50a)

u = wD(t, ·) on ΓD, (9.50b)

σν = 0 on ΓN, (9.50c)

[[σ]]ν = 0σν + zK[[u]] = 0

[[u]]n ≥ 0, σn(u) ≤ 0, σn(u)[[u]]n = 0.z ≤ 0,

12

K[[u]]·[[u]] + ρ ≤ a

.z

(12

K[[u]] · [[u]] + ρ− a

)= 0

z ≥ 0, ρ ≤ 0, ρz = 0.

on ΓC, (9.50d)

Mathematically speaking, elastic-brittle delamination is a regulari-zation of brittle delamination. In fact, the condition z[[u]] = 0 in (9.45a)can obviously be modified to

√z[[u]] = 0 with entirely the same effect,

and then, after a penalization using a quadratic penalty with an L2-type(possibly anisotropic) norm, (

∫ΓC

Ku ·u dS)1/2 yields exactly (9.49). Thus,one can expect convergencer for K → ∞ to brittle delamination. This hasbeen proved in [48].

For a computer implementation, one also needs a spatial discretization.The simplest choice is P1-finite elements for u and P0-finite elements forz, assuming that all Ωi are polyhedral and triangulated consistently on thejoint boundary ΓC. The mutual-recovery sequence, cf. Remark 9.3 above,

rThe shorthand notation K → ∞ means that the minimal eigenvalue of K goes to ∞.



can be taken as:

uh := ΠU ,hu, zh := zhΠZ ,h(z/z), (9.51)

where z(x)/z(x) is defined 0 if z(x) = 0, h = 1/k denotes the mesh size,and ΠU ,h and ΠZ ,h are standard projectors on the finite-element subspace,the latter making element-wise constant averages.s Merging it with a timediscretization, the general result follows [37]. If we consider P1-elements forz with ΠZ,h the corresponding projector, we can take zh = ΠZ,h(z − ‖zh −z‖L∞(ΓC))+, where + denotes the positive part. Merging the convergenceto the brittle limit K → ∞ with numerical approximation (τ, h) → (0, 0)seems possible only in a rather implicit way, cf. also [47,49] for this type ofresult.t

Altogether, we can summarize:

Proposition 9.6. Assume (9.43a) and (9.45c). Let (uK, zK) denote theenergetic solution to the problem (EK,R) from (9.49) and (9.45b). Let(uK,τ , zK,τ ) stand for the approximate solutions obtained by the implicitsemidiscretization in time (with a time step τ), and (uK,τ,h, zK,τ,h) for itsnumerical approximation constructed by this implicit time discretizationand the above finite element method (FEM) discretization in space (with ha mesh parameter). Also let the above qualification of wD and z0 be satisfied.Then:

(i) If K → ∞, then (uK, zK) converges (in terms of subsequences) toenergetic solutions to the brittle problem (E ,R) from (9.45a) and(9.45b) in the sense (9.25). The same holds also for (uK,τ , zK,τ ) forK → ∞ and τ → 0.

sNote that the product of element-wise constant functions zh and ΠZ ,h(z/z) is againelement-wise constant, hence zh ∈ Zh. As 0 ≤ ΠZ ,h(z/z) ≤ 1, we have also 0 ≤ zh ≤ zh,hence R(zh − zh) < ∞. As ΠZ ,h(z/z) → z/z in any Lp(ΓC), p < +∞, and zh → zweakly, from (9.51) we have zh → z(z/z) = z weakly* in fact in L∞(ΓC) due to thea priori bound of values in [0,1].tMore explicitly H occurring in Proposition 9.6(iii) might be supported by a localLipschitz continuity of (z, u) → z[[u]]2:L2(ΓC)×W 1/2,2(Ω)→L1(ΓC) and by a rateof approximation by finite element (FE) discretization in these norms, cf. [43,Proposition 3.3]. If d = 3, this continuity is due to ‖z1[[u1]]2 − z2[[u2]]2‖L1(ΓC) ≤‖z1 − z2‖L2(ΓC)‖[[u1]]‖2

L4(ΓC)+ 2‖z2‖L∞(ΓC) (‖[[u1]]‖L4(ΓC) + ‖[[u2]]‖L4(ΓC)) ‖[[u1 −

u2]]‖L4/3(ΓC), and then due to the continuity of the trace operator W 1/2,2(Ω) → L2(ΓC).To get the rate of convergence, it seems inevitable to use a gradient theory for z. ThenH(K) ∼ o(|K|−1/2) is expected.



(ii) For K fixed and for (τ, h) → (0, 0), (uK,τ,h, zK,τ,h) converges (in termsof subsequences) to energetic solutions of (EK,R).

(iii) There H : Rd×d → R+ converging to 0 sufficiently fast for K → ∞such that the “stability criterion” h ≤ H(K) ensures the convergenceof (uK,τ,h, zK,τ,h) (in terms of subsequences) to energetic solutions ofthe brittle problem (E ,R).

9.5.2 Numerical implementation

The theoretical developments presented up to this point providea convenient framework for an implementable numerical scheme bydiscretizing the time-incremental formulation (9.1) in the space variablesby standard finite element methods, recall Remark 9.3. Hence, each domainΩi is triangulated using elements with a mesh size h, cf. Remark 9.3. Recallthat we assume that the discretization conforms, i.e. that two interfacialnodes belonging to the adjacent domains Ωi and Ωj are geometricallyidentical, and that the same mesh is used to approximate variables u

and z. Note that, in the following, we denote by boldface letters nodaldiscretized variables and omit the subscripts K and h. Now, the finiteelement discretization with a suitable numbering of nodes yields a discreteincremental problem in the form

minimize (u, z) → Eh(kτ,u, z) + Rh(z − zk−1τ )

subject to BEu = 0, BIu ≥ 0, zk−1τ ≥ z ≥ 0.

(9.52)

Here, u ∈ Rnu stores the nodal displacements for individual subdomainsand z ∈ R

nz designates the delamination parameters associated withinterfacial element edges. The discretized stored energy functional has theform, related to (9.49),

Eh(t,u, z) = uTK

(12u + wD(t)

)+

12[[u

]]Tk(z)

[[u

]], (9.53)

where K = diag (K1,K2, . . . ,KN ) is a symmetric positive semi-definiteblock-diagonal stiffness matrix of order nu (derived from C(i)), [[u]] ∈ Rnk

stores the displacement jumps at interfacial nodes, and k is a symmetricpositive-definite interfacial stiffness matrix of order nk, which is derivedfrom K and depends linearly on z. The discrete dissipation potential isexpressed as

Rh(z) = −aTz, (9.54)



Table 9.1. Conceptual implementation of the alternating minimization algorithm.

(1) Set j = 0 and z(0) = zk−1τ

(2) Repeat

(a) Set j = j + 1(b) Solve for u(j):

minimize u → Eh(kτ, u, z(j−1))subject to BEu = wD(kτ) BIu ≥ 0

ff(9.55)

(c) Solve for z(j):

minimize z → Eh(kτ, u(j), z) + Rh(z − zk−1τ )

subject to zk−1τ ≥ z ≥ 0

)(9.56)

(d) Until ‖z(j) − z(j−1)‖ < η

(3) Set ukτ = u(j) and zk

τ = z(j)

where the entries of a ∈ Rm store the amount of energy dissipated bythe complete delamination of an interfacial element; see [50] for additionaldetails. The constraints in problem (9.52) consist of the homogeneousDirichlet boundary conditions prescribed at nodes specified by a full-rankmE × nu Boolean matrix BE, nodal non-penetration conditions specifiedby a full-rank matrix BI ∈ RmI×nu storing the corresponding componentsof the normal vector, and the box constraints on the internal variable.

The discrete incremental problem (9.52) represents a large-scale non-convex program (due to the k(z) term), which is very difficult to solveusing a monolithic approach. Nevertheless, it can be observed that theproblem is separately convex with respect to the variables u and z. Thisdirectly suggests using the alternating minimization algorithm, proposedby Bourdin et al. [51] for variational models of fracture. In the currentcontext, the algorithm is summarized in Table 9.1.

The individual sub problems of the alternating minimization algorithmcan efficiently be resolved using specialized solvers. In particular, step (9.55)now becomes a quadratic programming problem, for which optimal duality-based solvers have been recently developed [52,53]. Owing to the piecewiseconstant approximation of the delamination parameters, problem (9.56) canbe solved locally element-by-element in a closed form, see also [54,55] foradditional details.



Table 9.2. Conceptual implementation of energy-based back-tracking-in-timeprocedure.

(1) Set k = 1, z0 = z(0) = 1

(2) Repeat

(a) Determine zkτ using the alternating minimization algorithm for time tk

and initial value z(0)

(b) Set z(0) = zkτ

(c) If

Z tkτ

tk−1τ

∂tEh(t, ukτ , zk

τ )dt ≤ Eh(tk , ukτ , zk

τ ) + Rh(zk−1τ − zk

τ )

−Eh(tk−1, uk−1τ , zk−1

τ ) ≤Z tk

τ

tk−1τ

∂tEh(t, uk−1τ , zk−1

τ )dt,

set k = k + 1(d) Else set k = k − 1(e) Until k > T/τ

Note that this method allows for a non-constant C(i) and works equallywell for non-homogeneous materials. Let us, however, mention that if allC(i) are independent of x, one can alternatively (and more efficiently)apply boundary element methods (BEM) thus combining recent advancesin BEM-based solvers for the Signorini problem [56] with developments incomputational materials science [57–60].

Even though the alternating minimization algorithm performs wellfor a wide range of computational examples, it only converges to a localminimizer of the objective function (9.52), cf. [61,62], whereas the energeticsolution concept relies on global energy minimization. To overcome thisdiscrepancy, Mielke et al. [36] proposed a heuristic back-tracking algorithmbased on the two-sided energy inequality (9.23). The resulting algorithmproceeds as shown in Table 9.2.

It should be noted that there is generally no guarantee that thealgorithm will locate the global optimum of the objective function(9.52); nevertheless computational experiments suggest that it deliverssolutions with lower energies than the basic alternating minimizationscheme [36,61]. An alternative approach is offered by stochastic optimiza-tion techniques [34,35], but this comes at the expense of a substantialincrease of computational cost.



Fig. 9.2. Setup of the flexure test.

9.5.3 Illustrative example

The basic features of the proposed numerical treatment will bedemonstrated by means of the response of a two-layer beam in bending,imposed by a vertical displacement at the mid-span. The geometrical detailsof the experiment, adapted from [63], are shown in Fig. 9.2; the thicknessrefers to a plane-stress model used in the calculations. The elastic propertiesof the bulk material are characterizedu by Young’s modulus of 75 GPa andPoisson’s ratio of 0.3 (corresponding to aluminum), the interfacial fracturetoughness is set to a = 25 Jm−2 and the maximum vertical displacementamounts to 1 mm at T = 1. The problem is discretized by identical isoscelesright triangles with side length h and uniform time step τ .

The energetics of the delamination process is shown in Fig. 9.3(a),highlighting the difference between local energy minimization and the timeback-tracking scheme. In particular, the local scheme predicts initiallyelastic behavior, followed by almost complete delamination of the twolayers at t ≈ 0.56, accompanied by interfacial energy dissipation. However,exactly at this step the two-sided inequality is violated, as detected by theback-tracking algorithm. Inductively using this solution as the initial guessof the alternating iterative scheme, the algorithm returns to the originalelastic path, thereby predicting a response leading to a lower value ofthe total energy for t ∈ [0.46, 0.56]. During the whole time interval, thecontribution of the stored interfacial energy remains relatively small, owingto the large value of the interfacial stiffness. Notice that a small part of theinterface remains intact even at T = 1, see Fig. 9.3(b), due to the presenceof compressive traction at the mid-span. This explains why the dissipated

uThis means we use an isotropic material with C determined by Ce:e = λ|trace e|2 +2µ|e(u)|2 with the so-called Lame constants λ = νE/((1 + ν)(1−2ν)) and µ = E/(2+ 2ν),when E denotes Young’s modulus and ν Poisson’s ratio.



(a) (b)

Fig. 9.3. (a) Illustration of the back-tracking procedure, (b) distribution of delaminationparameter at T = 1; h = 1mm, τ = 0.025 and K = 105I GPa/m; EΩ =energy stored inbulk, EΓ = the interfacial contribution.

(a) (b)

Fig. 9.4. Convergence for h → 0: (a) energetics and (b) force-displacement diagram;τ = 0.025 and K = 105

I GPa/m; EΩ = energy stored in bulk, EΓ = the interfacialcontribution.

energy in Fig. 9.3(a) saturates at a slightly smaller value than 60 Nmm,which corresponds to complete delamination.

Figure 9.4(a) demonstrates the convergence of the approximatesolutions for h → 0. The results confirm that the overall energetic pictureis almost independent of the spatial discretization, and that no spuriousnumerical oscillations are observed. The same conclusion holds for the force-displacement diagrams, shown in Fig. 9.4(b).

Finally, the convergence of the debonding process as K → ∞ isillustrated in Fig. 9.5. Notice that, as with h → 0, the energetics appearsto be only mildly dependent on the interfacial stiffness and that, already



Fig. 9.5. Convergence for K → ∞ towards the brittle model devised in [48]; τ = 0.025and h = 0.5 mm; EΩ = energy stored in bulk, EΓ = the interfacial contribution.

for K ≈ 104I GPa/m, the FE-based solution accurately approximates theGriffith-type behavior discussed in Section 9.4.

9.6 Various Refinements and Enhancements

The models introduced so far represent a very basic scenario, which wasintentionally simplified to make the explanation of the underlying conceptseasier. Engineers, however, deal with various advanced ideas not so fardiscussed. The goal of this section is to demonstrate that they too can beinvolved in this theory.

9.6.1 Cohesive contacts

The adhesive model presented in Section 9.5 yields a discontinuous responseof the mechanical stress σ := ∂[[u]](z2K[[u]] · [[u]]) = zK[[u]] within a displace-ment-controlled experiment. Namely, starting from an unstressedconfiguration, the stress linearly increases with a prescribed [[u]] until thedriving force z := ∂z(z2K[[u]] · [[u]]) = 1

2K[[u]] · [[u]] reaches the activation

threshold a used in (9.45b), then z jumps to zero and the mechanical stressjumps to zero, too; Fig. 9.6 depicts the isotropic case K = κI with someκ > 0. The engineering literature often considers instead the continuousresponse of the mechanical stress, however. It is referred to as a cohesive-type contact and urges some modification of the above model. One simple



Fig. 9.6. Schematic illustration of the response of the driving force z, the delaminationz, and the mechanical stress σ in model (9.49) with K = κI and (9.45b) during thedisplacement-controlled experiment.

option is to modify EK from (9.49) as follows

EK1,K2(t, u, z) :=

8>>>>>>>>>>>>>><>>>>>>>>>>>>>>:

NXi=1

Z

Ωi

C(i)e(u):e

“u

2+ uD(t)

”dx

+

Z

ΓC

zK1[[u]]+ z2K2[[u]]

2· ˆ

u˜

+κ0

r|∇Sz|r dS if u = 0 on ΓD,

[[u]]n ≥ 0 on ΓC

0 ≤ z ≤ 1 on ΓC,∞ elsewhere,

(9.57)

where we used a (d − 1)-dimensional “surface” gradientv ∇S and assumer > d−1 and for mathematical reasons κ0 > 0. This last term has a similar“non-local” effect as in the frequently used gradient theory in damage; alsothe analysis and especially the constructions of a mutual-recovery sequencein the sense of [37] are the same as in damage models. In particular, for1 < r ≤ d − 1 one has to use a sophisticated construction from [64,65],otherwise a simpler construction from [66] works, too. In the context ofdelamination, gradient theory has been used, e.g., in [38, Chapter 14]or [67,68].

To demonstrate the response of this model, consider the isotropicadhesive response K1 =κ1I and K2 = κ2I. Then, the mechanical stress σ:= ∂[[u]](1

2(κ1z+ κ2z

2)|[[u]]|2)= (κ1z+κ2z2)[[u]] within a pulling experiment

vThe notation ∇Sz of the surface gradient stands for ∇z − ν(ν · ∇z).



Fig. 9.7. Schematic illustration of the response of the refined model (9.57) with K1 =κ1I and K2 = κ2I and (9.45b) under the pulling experiment.

again linearly increases with [[u]] until the driving force z := ∂z(12(κ1

z+ κ2z2)|[[u]]|2) = ( 1

2κ1 +κ2z)|[[u]]|2 reaches a, which happens for |[[u]]| =√

2a/(κ1 + 2κ2), and then z starts evolving while holding z = a, i.e. z =aκ2|[[u]]|−2 − κ1

2κ2, until it arrives at 0, which happens for |[[u]]| =

√2a/κ1;

thus the mechanical stress decays as σ = (κ1z+κ2z2)[[u]] = (a

2

κ2|[[u]]|−4 −

κ21

4κ2)[[u]] to zero; see Fig. 9.7. The continuous response of σ (Fig. 9.7, right)

is addressed as a cohesive-zone model, cf. e.g. [69,70]. As (9.49), a featureof (9.57) is that it is separately quadratic both in the u- and the z-variable,so one can advantageously use alternating minimization algorithms tosolve the incremental minimization problems of the type (9.1) as in[36,51,62].

More generally, one can consider a continuous, increasing functionφ: [0, 1] → R

+ with φ(0)= 0, and replace the z-term under the surfaceintegral in (9.49) by φ(z)|[[u]]|2. Repeating the previous arguments, theresponse in a tensile experiment starting from z=1 exhibits a quadraticdependence on the driving force z=φ′(z)|[[u]]|2 until it reaches the activationthreshold a, which happens for |[[u]]|=√

a/φ′(1). Then z starts evolvingwhile holding z= a, which yields z= [φ′]−1(|[[u]]|2/a) and the actual stressσ= 2φ([φ′]−1(|[[u]]|2/a))[[u]], until it arrives at 0.

Up to the gradient term, the equivalent effect can be obtained by asubstitution of φ(z) by a new delamination variable, say ζ. This leads to thestored-energy term ζ|[[u]]|2 and the dissipation distance a|φ−1(ζ)− φ−1(ζ)|if ζ ≤ ζ, which corresponds to the dissipation metric a|

.ζ|/φ′(φ−1(ζ)). Thus,

in terms of this new variable, we obtain the situation

effective activation =a

φ′(φ−1(ζ)), stress = 2ζ

[[u]], (9.58)



while the driving force is |[[u]]|2. In fact, having an “optically” non-associativew model like (9.58), one can conversely explicitly construct thedissipation metric, which is instead an exceptional situation due to theone-dimensionality and uni-directionality of the considered delaminationprocess.

9.6.2 Delamination in Modes I, II and mixed modes

Dissipation in the so-called Mode I (delamination by opening) is lessthan in the so-called Mode II (delamination by shearing); sometimes thedifference may be tens or even hundreds of percent and under generalloading it depends on the so-called fracture-mode-mixity angle, cf. [71–74].Microscopically, the additional dissipation in Mode II may be explainedby a plastic process both in the adhesive itself and in a narrow bulkvicinity of the delamination surface before the actual delamination starts,cf. [72,75], or by some rough structure of the interface, cf. [76]. Theseplastic processes do not manifest in Mode I if the plastic strain is valuedin R

d×ddev = the “incompressible” (=trace free) symmetric strain, as usually

considered. Modeling the narrow plastic strip around ΓC is computationallydifficult, and thus the various simplified phenomenological models are worthconsidering.

An immediate reflection of the standard engineering approach as e.g. in[77–79] is to modify (9.45b) by an activation threshold a = a(ψ) dependingon the so-called fracture-mode-mixity angle ψ. For K = diag(κn, κt, κt), thelatter is defined as

ψ = ψ([[u]]) := arc tan

√κt|[[u]]t|2κn|[[u]]n|2 , (9.59)

where [[u]]t and [[u]]n stand for the tangential and the normal displacementjump, arising in the decomposition [[u]] = [[u]]nν + [[u]]t with [[u]]n = [[u]] · νwith ν a unit normal to ΓC. Typical phenomenology is that κt < κn (usuallyreaching no more than 80% of κn). A typical, phenomenological form of a(·)used in engineering [77] is, e.g.,

a(ψ) := aI

(1 + tan2((1 − λ)ψ)

), (9.60)

wHere “non-associative” means that there is no unique activation threshold associatedwith the dissipation mechanism. Sometimes the adjective “non-associative” insteadmeans that the dissipative forces do not have any potential.



where aI = a(0) is the activation threshold for fracture Mode I and λ is theso-called fracture-mode-sensitivity parameter. E.g., for moderately strongfracture-mode sensitivity, which means the ratio aII/aI is about 5–10 (withaII = a(90) being the activation threshold for the pure fracture Mode II),one has λ about 0.2–0.3; cf. [79]. Therefore, this model uses the dissipationrate from Remark 9.2 in a general form, namely

R(u,.z) :=

∫ΓC

a(ψ([[u]]))|.z| dS if

.z ≤ 0 on ΓC,


An immediate idea is to use a semi-implicit time discretization, leading toa modification of the incremental minimization problem (9.1) as follows

minimize (u, z) → EK(kτ, u, z) + R(uk−1τ , z − zk−1

τ )subject to (u, z) ∈ U ×Z .

(9.62)

The convergence of this method is indeed guaranteed in some cases [80],in particular when the stored energy is uniformly convex. This, however, isnot the case for EK. The discrete stability inequality (9.18) is modified to

E (kτ, ukτ , zkτ ) ≤ E (kτ, u, z) + R(uk−1

τ , z − zkτ ) (9.63)

and the following energy inequalities hold:

E (kτ, ukτ , zkτ ) + R(uk−1

τ , zkτ − zk−1τ ) − E ((k − 1)τ, uk−1

τ , zk−1τ )

≤∫ kτ

(k−1)τ

E ′t (t, u

k−1τ , zk−1

τ ) dt (9.64)

and

E (kτ, ukτ , zkτ ) + R(uk−2

τ , zkτ − zk−1τ ) − E ((k − 1)τ, uk−1

τ , zk−1τ )

≥∫ kτ

(k−1)τ

E ′t (t, u

kτ , z

kτ ) dt. (9.65)

The first inequality follows from the minimality of (ukτ , zkτ ) when compared

with (uk−1τ , zk−1

τ ) while the second is implied by the discrete stability (9.63)of (uk−1

τ , zk−1τ ) with (u, z) = (ukτ , zkτ ). We then get only W 1,1 bounds

on piecewise affine interpolants of zkτ . Hence, concentrations of.z can

appear in the limit z, which is thus a function of the bounded variationonly. To pass to the limit in the dissipation term one would need toenhance the sophisticated techniques developed in [81,82] and then onlyget the energy inequality. We refer to [54,55] for more partial results. There



is, however, an obvious peculiarity in direct application of the previousconcepts from Remark 9.2 because the dissipation distance D = DR

defined implicitly by (9.32) here can be evaluated explicitly as D(q1, q2) =∫ΓC

min0≤ψ≤π/2 a(ψ)|z1 − z2|dS if z2 ≤ z1 a.e. on ΓC, otherwise it isinfinite. The existence of an energetic solution of the model determined by(EK,DR) defined by (9.49) and (9.61) can be conventionally shown, but suchsolutions do not distinguish particular modes at all. This is a quite well-known effect in non-associative models, indicating that sometimes otherconcepts for solutions are more relevant, cf. [7,83]. Altogether, the analysisof (9.62) (or e.g. a fully implicit modification of it) is not entirely clear.Moreover, a question remains as to whether one can indeed model thedesired mode-mixity-sensitive effect in all situations in such a way. It islikely that the higher gradients in u are needed to control [[u]] in C(Γ; Rd)to give a good meaning to

∫ T0

R(u,.z) dt with R(u,

.z) from (9.61), i.e. to∫

ΓCa(ψ([[u]]))| .z|(dS).x

To overcome these drawbacks, but still considering an additionaldissipation in Mode II, one can alternatively consider an additional inelasticprocess on ΓC. For this, we may introduce a dissipative variable representingthe “plastic” tangential slip sp on ΓC, and devise a plastic-like model withkinematic-type hardening for it, namely

Z := L∞(ΓC)×L2(ΓC; Rd−1), X := L1(ΓC)×L1(ΓC; Rd−1), (9.66a)

E (t, u, z, sp) :=

N∑i=1

∫Ωi

C(i)e(u):e

(u2

+ uD(t))

dx

+∫

ΓC

z(κn

2|[[u]]

n|2 +

κt

2|[[u]]

t− sp|2

)

+κH

2|sp|2 +

κ0

r|∇Sz|r dS if u = 0 on ΓD,

[[u]]n ≥ 0 on ΓC,

0 ≤ z ≤ 1 on ΓC,

∞ elsewhere,(9.66b)

R(.z,

.sp) :=

∫ΓC

a1|.z| + a2|.sp|dS if.z ≤ 0 a.e. on ΓC,

∞ otherwise,(9.66c)

xIn fact, this scenario indeed works in the viscous Kelvin–Voigt rheology fromSection 9.6.5 below, as demonstrated recently in [84].



Fig. 9.8. Schematic illustration of the response of the mechanical stress in model (9.66)under pulling and shearing experiments; the left-hand side (Mode I) corresponds toFig. 9.6 (right); 2κtaI ≥ a2

2 is assumed so that aII ≥ aI .

with [[u]] = [[u]]nν + [[u]]t, [[u]]n = [[u]] · ν and ν a unit normal to ΓC, whileU is again from (9.43a). Rigorously, sp ∈ L2(ΓC; Rd−1) is considered as a(d−1)-dimensional vector field embedded in Rd space to give meaning to theexpression [[u]]t− sp. As in (9.57), we again use a gradient theory for z withr > d − 1 and κ0 > 0 to facilitate the construction of the mutual-recoverysequence.y When d = 3, the physical dimensions are: [aI ] = J/m2

, [a2] = J/m3, and [κt] = [κn] = [κH ] =J/m4. The activation criterion to triggerdelamination is now

12(κn|

[[u]]n|2 + κt

∣∣[[u]]t− sp|2) ≤ aI . (9.67)

Starting from the initial conditions sp,0 = 0 and z0 = 1, the responsefor pure Mode I is essentially the same as in Fig. 9.6 (right), because noevolution of sp is triggered by for opening. To analyze the response pureMode II, realize that the tangential stress σt is a derivative of E with respectto [[u]]t, and thus σt(u, sp) = κt([[u]]t − sp) if z = 1. In analogy with theplasticity, the slope of evolution of sp under hardening is κt/(κt+κH). From(9.67), one can see that delamination is triggered if 1

2κt|[[u]]t−sp|2 = 1

2σ2

t /κt

reaches the threshold aI , i.e. if the tangential stress σt achieves the threshold√2aIκt, as depicted in Fig. 9.8 (right). The delamination in Mode II is thus

triggered under the tangential displacement

sII =√

2κ3taI − a2κt +

√2κtκ2

HaI

κtκH(9.68a)

yWe can use the damage-type construction for z, i.e. zk = (z − ‖z − zk‖L∞(ΓC))+ and

the binomial trick [37] for sp; cf. [85] for details.



and, after some algebra, one can see that the overall dissipated energy is

aII = aI + a2κH

(√

2κtaI − a2), (9.68b)

provided 2κtaI ≥ a22. The fracture-mode sensitivity aII/aI is then indeed

more than 1, namely 1 + a2(√

2κtaI − a2)/(κHaI). The surface plastic slipstops evolving after delamination and, as used for (9.68b), only if afterdelamination the driving stress κHsII has a magnitude less than a2. In viewof (9.68a), it needs κtaI < 2a2

2. Thus, to produce the desired effects, ourmodel should work with parameters satisfying

12κtaI < a2

2 ≤ 2κtaI . (9.69)

The validity of this model has been tested numerically in [85,86].An interesting open problem is the limit passage of this model under a

suitable scaling to a brittle model as in Proposition 9.6.Let us note that in both of the mode-mixity-sensitive models considered

in this section the stored energy involves z linearly. However, it is notdifficult to combine cohesive-zone-type models from Section 9.6.1 with thesemode-mixity-sensitive models. Thus, e.g., (9.61) can be generalized to

R(u, z,

.z)

:=

∫ΓC

a(ψ([[u]]), z)|.z| dS if

.z ≤ 0 on ΓC,


To facilitate the mathematical analysis, one again needs the stored energyto be augmented by the delamination gradient.

The influence of the mixed-mode behavior will be illustrated with anexample of the mixed-mode flexure test [63, Section 3.2] shown in Fig. 9.9.The material properties of the bulk material are the same as in Section 9.5.3,the elastic-brittle interface is now characterized by the stiffnesses κn =810 GPa/m and κt = 760 GPa/m. The Mode I and Mode II fractureenergies are set to aI = 200 Jm−2 and aII = 900 Jm−2, in order to achieve

Fig. 9.9. Setup of the mixed-mode flexure test.



(a) (b)

Fig. 9.10. (a) Energetics of mixed mode flexure test (EΩ = energy stored in bulk, EΓ =interfacial contribution) and convergence for h → 0 , and (b) evolution of mode-mixityangles for h = 0.5 mm; τ = 0.025.

a more ductile structural response. The prescribed mid-span displacementequals 2.5 mm at T = 1. The non-associative model (9.61) with (9.49) isused.

Figure 9.10(a) summarizes energetics of the delamination evolution.After the initial elastic regime, delamination initiates in a combined normaland shear mode, see Fig. 9.10(b). This is accompanied by a high increase ofthe dissipated energy. With the increasing load, however, the mode mixitygradually changes towards the opening mode. The production of dissipatedenergy decreases and the interfacial stored energy almost vanishes; seeFig. 9.11 for an illustration. The response remains almost independent ofthe mesh size h. Moreover, the back-tracking algorithm remained inactivefor the whole loading range, which confirms the energy stability of thedelamination evolution. Note that the peaks in the mode-mixity angles inFig. 10(b) are related to the changes of the sign of the tangential slip [[u]]t,recall Eq. (9.59).

Remark 9.4. (Mode III.) Delamination by twisting (i.e. Mode III)exhibits specific behavior and is often also considered, though we notconsider this sort of model here. In fact, it would be possible to model suchregimes by making the activation threshold dependent on the angle between∇Sz and the tangential stress. Obviously, there needs to be compactness interms of ∇Sz, which would have to occur “non-linearly” in the model, sothat an even higher gradient of z is involved in E .



Fig. 9.11. Ten snapshots of delamination evolution during the flexure test of thespecimen from Fig. 9.9; displacements are magnified five times.

9.6.3 Multi-threshold delamination

Some of the engineering literature incorporates parallel breakable springswith different elastic and inelastic properties, cf. [78,79]. On the continuum-mechanical level, this idea can be reflected by a generalization of theprevious model by considering J different adhesives acting simultaneouslyon ΓC:

E (t, u, z1, . . . , zJ) :=

N∑i=1

∫Ωi

C(i)e(u):e

(u2

+ uD(t))

dx

+∫ΓC

∑Jj=1 zjKj [[u]]

2· [[u]]dS

if u = 0 on ΓD,

[[u]]n ≥ 0 on ΓC,

0 ≤ zj ≤ 1 on ΓC,

∞ elsewhere,(9.71a)



R(.z1, . . . ,

.zI) :=

J∑j=1

∫ΓC

aj∣∣.zj∣∣dS if max

j=1,...,J

.zj ≤ 0 on ΓC,

∞ otherwise.

(9.71b)

Again, the advantage of the energetic formulation is that there is no problemin combining multi-threshold models with cohesive-zone models fromSection 9.6.1 and/or the mode-mixity-sensitive models from Section 9.6.2.Also, instead of parameterizing the various adhesives by a discreteparameter j = 1, . . . , J , one could use a “continuous” parameter.

9.6.4 Combinations with other inelastic processes

The definite advantage of the energetic formulation is that one can easilycombine the above delamination models with other inelastic processes, likedamage or plasticity in the bulk.

Let us illustrate this by a simple example, augmenting the modelfrom Section 9.5 by linearized, single-threshold plasticity with kinematichardening. The additional variable is then the plastic strain π valued inRd×ddev := π ∈Rd×d, π = π, trπ= 0, and the plastic response is determined

by the convex “elasticity” domain S(i) ⊂ Rd×ddev and by a hardening tensor

H(i) on each Ωi. After shifting the Dirichlet conditions, the functionals are:

EK(t, u, z, π) :=

N∑i=1

∫Ωi

C(i)(e(u) − π):

(e(u

2+ uD(t)

)− π

2

)+ H

(i)π:π dx

+∫

ΓC

12zK

[[u]] · [[u]] dS if u = 0 on ΓD,

[[u]]n ≥ 0 on ΓC,

0 ≤ z on ΓC,

∞ elsewhere,

(9.72a)

R(.z,

.π) :=

∫ΓC

a|.z| dS +N∑i=1

∫Ωi

δ∗S(i)(.π) dx if

.z ≤ 0 on ΓC,

∞ otherwise.

(9.72b)

and, instead of (9.43b),

Z := L∞(ΓC) × L2(Ω; Rd×ddev ). (9.72c)



The classical formulation corresponding to (9.6) with E and R from(9.45b) is an equilibrium of forces on each subdomain Ωi and fourcomplementarity problems on ΓC, corresponding to the four subdifferentialsoccurring in the involved functionals EK and R. Before shifting the Dirichletconditions, this formulation is:

div σ = 0, σ = C(i)(e(u) − π).π ∈ NS(i)(H(i)π − σ)

in Ωi, i = 1, . . . , N,

(9.73a)

u = wD(t, ·) on ΓD, (9.73b)

σν = 0 on ΓN, (9.73c)

[[σ]]ν = 0

σν + zK[[u]] = 0[[u]]n≥ 0, σn(u) ≤ 0, σn(u)

[[u]]n

= 0.z ≤ 0,

12

K[[u]] · [[u]] + ρ ≤ a

.z

(12

K[[u]] · [[u]] + ρ− a

)= 0

z ≥ 0, ρ ≤ 0, ρz = 0,

on ΓC, (9.73d)

where NS(i) denotes the normal cone to S(i), as in (9.10).There are both mathematical and engineering studies combining

elasto-plasticity with cracks, cf. e.g. [87,88]; this is a highly non-trivialproblem because the crack path is not a priori prescribed, in contrast todelamination and perfect (the so-called Prandtl–Reuss) plasticity, which hasbeen considered in [88]. A combination with damage models in the bulk isalso easily possible. Recent studies [47,64] reveal that the delaminationmodel from Section 9.4 can be obtained as the limit when, instead ofa surface ΓC undergoing delamination, one considers a narrow strip ofmaterial undergoing damage and the thickness of that strip goes to zero.A challenging conjecture is whether the refined models from Section 9.6can be justified in this way, e.g. whether considering a narrow strip of amaterial undergoing damage and plasticity with kinematic hardening mightrecover the mode-mixity-dependent model (9.66) under a suitable scaling.This could support former engineering studies as, e.g., in [75,89].



9.6.5 Dynamical adhesive contact in visco-elastic materials

So far, we have considered only quasistatic models, which have relativelybroad applicability. In some situations, additional effects must be taken intoaccount, however. In particular, even under very slow loading, spontaneousrupture of weak surfaces ΓC may lead to the emission of elastic waves in thebulk, which may backward interact with the rate-independent delaminationhosted on ΓC. Thus inertial effects must be considered. It is natural to takeinto account also attenuation in the bulk. Considering the Kelvin–Voigtrheology, the simplest model from Section 9.5 thus modifies to:

(i) ..u − div σ = 0, σ = C(i)e(u) + D(i)e(.u) in Ωi, i = 1, . . . , N,

(9.74a)

u = wD(t, ·) on ΓD, (9.74b)

σν = 0 on ΓN, (9.74c)

[[σ]]ν = 0

σν + zK[[u]] = 0[[u]]n≥ 0, σn(u) ≤ 0, σn(u)

[[u]]n

= 0

.z ≤ 0,

12

K[[u]] · [[u]] + ρ ≤ a

.z

(12

K[[u]] · [[u]] + ρ− a

)= 0

z ≥ 0, ρ ≤ 0, ρz = 0

on ΓC, (9.74d)

where (i) > 0 is a mass density and D(i) is a fourth-order symmetricpositive definite tensor determining the attenuation of the materialoccupying the domain Ωi. This model is analyzed by a semidiscretizationin time leading to a recursive increment formula. We refer to [90] for moredetails.

For an analogous model but with viscous (instead of activated rate-independent) adhesion we refer to [11, Chapter 5] or e.g. [10], and, withplasticity in the bulk, to [91].

The typical application of dynamic adhesive contact is in the modelingof spontaneous rupture on lithospheric faults (i.e., weak surfaces in thelanguage of mechanical engineers) with the emission of elastic waves havingthe capability to trigger, e.g., another possible rupture on an adjacentfault and inelastic damage, examples of which occur on the Earth’s surfaceas earthquakes. Typical rupture processes run in pure Mode II becausethe enormous gravity pressures on the faults deep under the Earth’s



surface exclude Mode I. Thus, Signorini contact might even be a priorisimplified to the condition [[u]]n = 0 on ΓC. An important phenomenonthat facilitates spontaneous rupture within earthquake modeling is the so-called slip weakening; cf. models and discussion e.g. in [92–98]. There areseveral options for how to describe the weakening phenomena within themodels presented above. One can, e.g., modify model (9.66) by consideringthe activation threshold aI in (9.66c) dependent on the plastic slip sp or,perhaps more physically, on

∫ t0| .sp|dt, which then makes the dissipation non-

associative. This option is capable of modeling one particular rupturing. Forrepeated rupturing (combined with healing), a dependence of both a1, anda2 in (9.66c) on delamination parameter z can be considered instead, cf.also [105].

Another possibility arises if E (t, u, ·) is concave. In model (9.57) inSection 9.6.1 with K1 = κ1I and K2 = κ2I, this occurs if κ2 is small,namely 0>κ2>−κ1/2. Then Fig. 9.6 is relevant but the rupturehappens under the mechanical stress σ = (κ1 +κ2)

√2a/(κ1 +2κ2) when

|[[u]]| =√

2a/(κ1 + 2κ2) as in Fig. 9.7. Thus, one can model delaminationweakening. Note that the initial stress σ leading to delamination may bemade very large by sending κ2 ↓ −κ1/2, even if the total dissipated energymay be independently moderate. One can see the weakening effect also from(9.58): if κ2 < 0 then φ is strictly concave hence φ′ is decreasing and thusthe effective activation threshold (9.58) gets smaller if η decreases (rangingover a monotone branch of φ of course).

The weakening phenomenon may also occur in the multi-thresholdmodels from Section 9.6.3: consider K1 > K2 > · · · (= ordering of positivedefinite matrices) and a1 > a2 > · · · ; initially the first adhesive layer canwithstand high stress but when debonded the next adhesive layers canwithstand smaller and smaller stresses.

9.6.6 Thermodynamics of adhesive contact

Interesting features might be triggered in non-isothermal situations.Mechanical stresses in thermally expanding materials can be createdby spatially varying temperature profiles. Also, merging materials withdifferent thermal expansion coefficients (as is typical in laminatedcomposites) creates mechanical stresses even with a spatially equilibratedtemperature. Such a thermomechanical load may lead to delamination.This may naturally influence heat transfer through the delaminatedsurfaces. Hence, besides the usual thermomechanical coupling due to viscousdissipation and thermal expansion in the bulk, coupling by delaminationalso occurs.



Focusing again on the Kelvin–Voigt rheology as in Section 9.6.5, thethermodynamically consistent model, naturally involving the additionalvariable θ as temperature, is

..u − div σ = 0σ = C(i)(e(u) − θE(i)) + D(i)e(

.u)

c(i)(θ).θ − div(L(i)∇θ) = (D(i)e(

.u) − θC(i)E(i)):e(

.u)

in Ωi, (9.75a)

u = wD(t, ·) on ΓD, (9.75b)

σν = 0 on ΓN, (9.75c)

[[σ]]ν = 0

σν + zK[[u]] = 0

[[u]]n ≥ 0, σn(u) ≤ 0, σn(u)[[u]]n = 0.z ≤ 0,

12

K[[u]] · [[u]] + ρ ≤ a = a0 + a1

.z

(12

K[[u]] · [[u]] + ρ− a

)= 0

z ≥ 0, ρ ≤ 0, ρz = 0

[[L∇θ]] · ν = −a1.z,

L∇θ · ν +η([[u]], z)[[θ]] = 0

on ΓC, (9.75d)

where · denotes the average of traces from both sides of ΓC, and whereE(i) is the matrix of thermal expansion coefficient, which may depend onΩi as C

(i) and D(i). Further c(i) = c(i)(θ) is the heat capacity, and L

(i)

is the positive-definite heat-conductivity tensor, and η = η([[u]], z

)is the

heat-transfer coefficient through the delaminating boundary ΓC.Note that only a part a1/a = a1/(a1 + a0) of the mechanical energy

dissipated during delamination contributes to heat production on ΓC, whilethe rest a0/a is irreversibly stored (=dissipated) in the debonded adhesivewithout contributing to the heat balance.

Mode II causes considerably more heating than Mode I, as experi-mentally documented in [99]. E.g., bearing in mind (9.76), one may considerthe splitting

a0(ψ) := aI , a1(ψ) := aI tan2((1 − λ)ψ), (9.76)

which indicates that plain delamination does not contribute to heatproduction at all, and only the additional dissipation for Mode IIcontributes to heat production on the delaminating surface.

For model (9.66), it would be natural to involve the dissipation viaa2| .sd| as a measure-valued heat source acting on ΓC while the dissipation



by pure opening along the surface, i.e. aI |.z|, would contribute instead to the

stored energy. Mathematical analysis of such a problem, as well as numericalexperiments, seem challenging. For mode-mixity-independent dissipation,this model has been analyzed in [100]; for Griffith-type rate-dependentadhesion see also [101]. Recently, mode-mixity-sensitive dissipation was alsoscrutinized in [84] by implementing the concept of non-simple materials.

9.7 Conclusion and Further Application

In this chapter, we surveyed some existing models and proposed a menagerieof new ones for delamination under small strains. The main purpose hasbeen to cover them using the unified concept of quasistatic evolutionof the form (9.6) or (9.30) — also to pursue their energetics — andto outline the rigorous mathematical and numerical analysis based onthe concept of the so-called energetic solutions. This approach suggestsefficient computational algorithms and allows for mathematically supportedsimulations. It also allows us relatively easily and routinely to combinevarious mutually competing inelastic processes both on the delaminatingsurface or in the bulk, and to devise advanced mathematically supportedmodels and launch numerical simulations of such non-trivial processes. Inaddition, under certain conditions the approach may also be combined withrate-dependent processes in the bulk, such as viscosity in a sufficientlydissipating rheology (e.g. Kelvin–Voigt), inertia, and even thermal processesas, e.g., thermo-visco-plasticity.

Although the focus of this chapter was on the modeling of macroscopicdelamination problems, the presented framework can easily be adapted tothe homogenization of composites with debonding interfaces. In the contextof two-scale homogenization, the stored energy associated with the unit cellproblem is [102,103]:

EK(t, u, z) :=

N∑i=1

12

∫Ωi

C(i)(e(u)+E(t)):(e(u) + E(t)) dx

+∫

ΓC

12φ(z)K

[[u]] · [[u]] +

κ0

r|∇Sz|r dS if u is Ω-periodic,∫

Ω

udx = 0,

[[u]]n ≥ 0 on ΓC,

0 ≤ z on ΓC,

∞ elsewhere,



where E(t) designates the macroscopic strain tensor, u now denotes aperiodic microscopic displacement, and the function φ is used to modelthe cohesive contact with a piecewise linear traction-separation law, recallSection 9.6. The dissipative potential remains unchanged, as well as thenumerical treatment of the incremental problem. The periodic boundaryconditions and the macroscopic strain are incorporated by the Lagrange

Table 9.3. Material data.

Matrix Young’s modulus 1 MPaMatrix Poisson’s ratio 0.4Fiber Young’s modulus 150 MPaFiber Poisson’s ratio 0.3Interfacial fracture energy, a 0.02 J/m2

Interfacial elastic stiffnesses, κn = κt 0.5 GPa/mInterfacial cohesive contact function φ(z) = z/(1 − z + 10−4)

Fig. 9.12. Energetics of single-fiber debonding and three selected snapshots ofdisplacement (magnified 20×) showing the spatial distribution of stress.



multipliers technique introduced, e.g., in [104]. As an example, we considera cross-section of a unit cell of a fiber-reinforced composite, subject to abi-axial macroscopic stretching E11 = E22 = 1.5% at T = 1. The geometryof the problem is defined by the fiber volume fraction equal to 50% and thediameter of the fiber is taken as 10 µm. The material data for individualphases and interfaces appear in Table 9.3; the gradient term was neglectedby setting κ0 = 0.

The energetics of the progressive debonding is shown in Fig. 9.12,together with representative snapshots of the debonding evolution. Dueto the prescribed cohesive law, we capture the gradual transition from astiff elastic interface, i.e., when the highest values of stress exists in thefiber, to the completely debonded configuration. In this situation the wholeload is carried by the matrix phase and the stored interfacial energy dropsto zero.

This simple study is complemented with a numerical simulationof debonding evolution in a complex 20-particle unit cell subject tomacroscopic shear E12 = 1%. The results in Fig. 9.13 confirm that

Fig. 9.13. Debonding in a fiber-reinforced composite: selected snapshots of a graduallyloaded representative cell containing 20 fibers with depicted displacements (magnified20×) showing the spatial distribution of stress.



the energetic approach, combined with robust duality-based solvers,captures the complex mechanisms of multiple contact, sliding, and gradualdebonding between fibers and matrix in geometrically complicated real-world material samples.

We entirely omitted models at large strains. Let us, however, pointout that the advantage of the energetic formulation is that the quasistaticdelamination models can be relatively easily formulated for in large strains,in contrast to dynamical viscous models of the type (9.74).

Acknowledgements

The authors warmly thank Vladislav Mantic, Ctirad Matyska, AlexanderMielke, Riccarda Rossi, and Marita Thomas for many fruitful discussionsand comments, and Pavel Gruber for performing numerical simulations. Weextend our thanks to two anonymous referees whose valuable remarks andsuggestions helped us to improve the clarity of the original manuscript.The authors also acknowledge partial support from the grants A 100750802(GA AV CR), 106/08/1379, 201/09/0917, 106/09/1573, and 201/10/0357(GA CR). T.R. also acknowledges the hospitality of Universidad de Sevilla,where this work was partly undertaken, covered by Junta de Andaluciathrough the project IAC 09-III-6321.

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[99] Rittel D., 1999. Thermomechanical aspects of dynamic crack initiation, Int.J. Fracture, 99, 199–209.



[100] Rossi R., Roubıcek T., 2011. Thermodynamics and analysis of rate-independent adhesive contact at small strains, Nonlinear Anal., Theor., 74,3159–3190.

[101] Bonetti E., Bonfanti G., Rossi R., 2009. Thermal effects in adhesive contact:Modelling and analysis, Nonlinearity, 22, 2697–2731.

[102] Miehe C., 2002. Strain-driven homogenization of inelastic microstructuresand composites based on an incremental variational formulation, Int. J.Numer. Meth. Engrg., 55, 1285–1322.

[103] Mielke A., Timofte A., 2007. Two-scale homogenization for evolutionaryvariational inequalities via the energetic formulation, SIAM J. Math.Analysis, 39, 642–668.

[104] Michel J., Moulinec H., Suquet P., 1999. Effective properties of compositematerials with periodic microstructure: A computational approach,Comput. Methods Appl. Mech. Engrg., 172, 109–143.

[105] Roubıcek T., Soucek O., Vodicka R. 2013. A model of rupturing lithosphericfaults with reoccurring earthquakes, SIAM J. Appl. Math., 73, 1460–1488.

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[107] Roubıcek T., Panagiotopoulos C.G., Mantic V. Quasistatic adhesive contactof visco-elastic bodies and its numerical treatment for very small viscosity,Zeit. f. Angew. Math. Mech., online (DOI:10.1002/zamm.201200239).


Chapter 10

CRACK NUCLEATION AT STRESS CONCENTRATIONPOINTS IN COMPOSITE MATERIALS — APPLICATION

TO CRACK DEFLECTION BY AN INTERFACE

Dominique Leguillon∗ and Eric Martin†∗Institut Jean Le Rond d’Alembert CNRS UMR 7190

Universite Pierre et Marie Curie, 4 place Jussieu, 75005 PARIS, France

†Laboratoire des Composites Thermo-Structuraux CNRS UMR 5801Universite Bordeaux 1, 3 rue La Boetie, 33600 PESSAC, France

Abstract

The coupled criterion predicts crack nucleation at a stress concentration point.It is a twofold criterion that uses conditions for energy and tensile stress andinvolves both the toughness and tensile strength of the material. In general,the crack jumps a finite length and then either stops or goes on growing.The criterion has proven its effectiveness in many situations encounteredin homogeneous materials like V- and U-notches and predictions agreereasonably with experimental measurements. It can also be used to studyspecific mechanisms of degradation of composites such as delamination or fibre

debonding. It has recently been used successfully to predict the initiation ofdelamination from a stress-free edge; the application discussed in this chapteris the deflection of transverse cracks by an interface. However, it is valid forboth delamination between layers and fibre debonding.

10.1 Introduction

Delamination is the main cause of failure of multimaterials and especiallyof composite laminates: the components separate leading to total failureor at least to a weakened structure. This topic is still the subject ofnumerous contemporary works and a detailed and recent list of referencescan be found in Martin et al. [1], which will give a better overview of theproblem. There are at least two important origins of delamination under

401


402 D. Leguillon and E. Martin

static load [2]: the first is the classic initiation process, which occurs at theintersection of a free edge with the interface between two layers. It is a zoneof stress concentration described in elasticity by a singular displacementfield [1, 3]. The other is less obvious because it is an internal process andtherefore not directly detectable: the deflection of transverse cracks presentin the most disoriented layers relative to the tensile direction [4–6]. Theclassical situation in a laminate concerns the presence of plies oriented90 with respect to the loading direction. Transverse microcracks are thengenerated, which coalesce to form a transverse crack that is deflected whenit reaches the interface with a 0 ply. This deflection mechanism creates adelamination crack [7]. Crack deflection at the fibre/matrix interface is alsoa prerequisite for the activation of toughening mechanisms like multiplematrix cracking in ceramic matrix composites (CMCs) [8, 9].

We propose in this chapter to present the coupled criterion, whichpredicts crack nucleation at stress concentration points [10]. It combinesstress and energy conditions that do not require the definition of acharacteristic fracture length selected more or less arbitrarily. Thiscriterion has been applied successfully to several situations in compositematerials: delamination originating from a stress-free edge within ageneralized plane-strain elasticity framework [1, 11] and crack kinking outof an interface within a plane-strain framework [12, 13]. It is illustrated hereby the second mechanism mentioned above, transverse crack deflection byan interface. The analysis will be developed for plane strain and could betheoretically extended to 3D although a number of technical difficultiesremain [14].

10.2 The Coupled Criterion

To establish this criterion, a good generic model is the three-point bend-ing test on a V-notched specimen made of a homogeneous material(Fig. 10.1).

In composite materials, the criterion applies at the intersection of theinterface between two layers and a free edge for instance [1], or at the endof a transverse crack impinging on an interface as illustrated in the nextsections.

The coupled criterion uses two conditions to predict the nucleation ofcracks in areas of stress concentrations in brittle materials: the maximumtensile or shear stress that the structure can sustain and an energy balancebetween the stored energy and the energy required to induce fracture [10].


Crack Nucleation at Stress Concentration Points in Composite Materials 403

Fig. 10.1. The three-point bending test on a V-notched homogeneous specimen.

The first condition refers to the tensile strength σc (or shear strength τc)while the other relies on the toughness Gc of the material (or interface).These two conditions must be satisfied simultaneously.

The form taken by these two conditions is from the theory ofsingularities and the asymptotic expansions of the displacement field U

and the stress field σ in the vicinity of the origin, the so-called Williams’expansion (in polar coordinates with origin at the singular point formed bythe notch root):

U(r, θ) = R + krλu(θ) + · · ·σ(r, θ) = krλ−1s(θ) + · · ·

(10.1)

λ is the singularity exponent (1/2 ≤ λ ≤ 1 for a V-notch), k (MPa·m1−λ)is the generalized stress intensity factor (GSIF), u(θ) and s(θ) are twoangular functions and R is a constant (the rigid translation of the origin).Coefficient k depends on the whole geometry of the structure and on theremote applied load. The exponent and the angular functions depend onlyon the local geometry and elastic properties; they are solutions to aneigenvalue problem and are either known analytically in some simple casesor can anyway be determined numerically using a simple algorithm. Clearlythe stress components tend to infinity as r → 0; this is why it is called asingular point.

Note that here and in the following, we only address the tensile stressbut extension to the shear component is straightforward.

The GSIF k can be computed using a path-independent integral Ψ[15, 16], valid for any elastic fields satisfying the equilibrium to 0 (i.e.vanishing boundary conditions and the balance equation within the domain



surrounded by the integration path):

k =Ψ(U(r, θ), r−λu−(θ))Ψ(rλu(θ), r−λu−(θ))

with

Ψ(U, V ) =12

∫Γ

(σ(U)·n·V − σ(V )·n·U)ds, (10.2)

where Γ is a contour encompassing the notch root and starting and finishingon the stress-free edges of the notch and n is its normal pointing towardthe origin. Relation (10.2) is based on two properties:

(1) If λ is an eigenvalue then so is −λ. The so-called dual mode [15] or‘super singular’ function [17] r−λu−(θ) is a mathematical solution tothe previous eigenvalue problem, which has presently no special physicalmeaning (in particular, the elastic energy associated with this functionis unbounded in the vicinity of the origin).

(2) For any pair of eigensolutions rαuα(θ) and rβuβ(θ), β = − α⇒Ψ(rαuα(θ), rβuα(θ))=0. This is a kind of bi-orthogonality property(note that Ψ is not a scalar product), which allows extraction of thecoefficient k. This result is a consequence of the path independenceof Ψ.

Here, the only role of the dual mode is to be a mathematical extractionfunction; however, these modes will play a greater role in the matchedasymptotic procedures both for the inner and outer expansions (see (10.14)and (10.15) in Section 10.3).

The stress condition (e.g. the maximum tensile stress criterion) involvesthe tensile component σ of the stress tensor σ acting on the presupposedcrack path defined by the direction θ0 prior to its onset; it provides anupper bound to the admissible crack extension lengths a (λ− 1 < 0):

σ = krλ−1s(θ0) + · · · ≥ σc for 0 ≤ r ≤ a⇒ kaλ−1s(θ0) ≥ σc. (10.3)

The coefficient s(θ0) is a dimensionless constant derived from s, which canbe normalized to s(θ0) = 1 if the failure direction (i.e. θ0) is known [15].Relation (10.3)2 is enough to imply σ ≥ σc along the whole presupposedcrack path since σ is a decreasing function of the distance to the singularpoint.

As will be shown in Section 10.3, expansions (10.1) can be used todefine an expansion of the potential energy variation when a small crack



extension appears in direction θ0. Its leading term provides a lower boundof the crack extension length (2λ− 1 > 0):

Ginc(θ0) = −W (a) −W (0)a

= A(θ0)k2a2λ−1 + · · · ≥ Gc, (10.4)

where W (x) is the potential energy of the structure embedding a crackextension with length x. Ginc(θ0) is the so-called incremental energy releaserate, because it depends on the increment a; emphasis is put on the factthat we do not consider the limit as a→ 0 as for the Griffith criterion [18].The incremental and differential criterion are identical if λ = 1/2. Ginc(θ0)is the rate of potential energy change prior to and following the onset of anew crack with length a. The scaling coefficient A(θ0) (MPa−1) is anotherconstant depending on the local properties and on the direction θ0 of theshort crack but not on the remote applied load which occurs in (10.4)through the only coefficient k. A complete definition of A is given in thefollowing section.

The compatibility between these two inequalities provides acharacteristic length ac at initiation (Fig. 10.2):

ac =GcA(θ0)

(s(θ0)σc

)2

. (10.5)

Initiation is in general (i.e. if λ > 1/2) an unstable mechanism. The crackjumps the length ac and then continues to grow or not, but ac is not definedas a crack arrest length. Essentially, below this length the balance betweenthe stored energy and the energy consumed during failure does not hold: nocrack smaller than ac can be observed. This jump length is still a functionof θ0.

Then we deduce an Irwin-like [19] condition on the GSIF k, which playsthe classical role of the stress intensity factor (SIF) KI

k ≥ kc =(

GcA(θ0)

)1−λ(σcs(θ0)

)2λ−1

. (10.6)

For a crack embedded in a homogeneous body, then λ = 1/2 andk = KI , relation (10.6) coincides with the well-known Irwin criterion. Astraight edge in a homogeneous material is a limit case where there is nostress concentration, then λ = 1 and inequality (10.6) coincides with themaximum tensile stress criterion.



Fig. 10.2. Schematic view of the determination of ac (10.5). (1) For a small remote load,crack extension lengths fulfilling the stress and the energy conditions are incompatible.(2) When the remote load increases, the two bounds are closer to each other. (3) Failureoccurs when the two bounds merge giving ac. Arrows indicate the motion of the curvesand points when the remote load increases.

The direction θ0 was assumed to be known; if not, one has to checkall the possible directions and maximize the denominator in (10.6), i.e.minimize the value of kc.

A single mode is involved in (10.1); for a V-notch this correspondsto the symmetric case as shown in Fig. 10.1. Generalizations can bemade to account for more complex loadings, where it is then necessary todetermine both the load causing failure and the direction of the nucleatingcrack [20].

The computation of the scaling coefficient A(θ0), using matchedasymptotic expansions, will be the topic of the next section. For simplicitythe dependency on θ0 will be omitted.

Remark : Martin [1, 21, 22] and Hebel [23] and their co-workers apply thecoupled criterion numerically without going through the semi-analyticalasymptotic expansion procedure. The tensile stress σ along the presupposedcrack path and the incremental energy release rate Ginc (10.4) are extractedfrom a direct finite element (FE) computation, which requires taking special



care with the mesh refinement in the vicinity of the region where thenew crack initiates. Inequalities (10.3) and (10.4) are employed withoutcalculating λ, k, s and A, and reduce to

σ = σFE(r) ≥ σc for 0 ≤ r ≤ a and

Ginc = −WFE(a) −WFE(0)a

≥ Gc. (10.7)

a is the smaller length fulfilling the two inequalities. Indeed (10.7) is themost general definition of the coupled criterion and can be used in all cases.

This approach allows situations to studied, which cannot be taken intoaccount in the asymptotic approach, like crack arrest after a short initiationfor instance. However, it does not reveal directly (analytically) the roleplayed by the different geometric parameters of the structure, such as thelayer thickness for an adhesive layer or an interphase for instance [24].

10.3 Matched Asymptotic Expansions

Numerically solving an elasticity problem in a domain Ωa embedding ashort crack of length a at the root of a V-notch (Fig. 10.3) presents somedifficulties because of the small size of the perturbation. Drastic meshrefinements are needed for the very small details.

It is better to represent the solution in the form of an outer expansionor far field:

Ua(x1, x2) = U0(x1, x2) + small correction, (10.8)

where U0 is the solution to the same elasticity problem but now posed onthe unperturbed domain Ω0 (Fig. 10.1), which can be considered as thelimit of Ωa as a→ 0 (the short crack is not visible).

Fig. 10.3. Onset of a short crack of length a at the root of the V-notch.



It is clear that this solution U0 is a satisfying approximation of Ua

away from the perturbation, i.e. outside a neighbourhood of it, hence itsdesignation as the outer field (or far field or remote field).

Evidently, this information is incomplete, particularly when we areinterested in fracture mechanisms. We therefore dilate the space variablesby introducing the change of variables yi = xi/a. In the limit when a→ 0,we obtain an unbounded domain Ωin (looking like the enlarged frame inFig. 10.3) in which the length of the crack is now equal to 1.

We then look for a different representation of the solution in the formof an expansion known as the inner expansion or near field:

Ua(x1, x2) = Ua(ay1, ay2) = F0(a)V 0(y1, y2)

+F1(a)V 1(y1, y2) + · · · , (10.9)

where F1(a)/F0(a) → 0 as a → 0. Since there are no conditions at infinityto give well-posed problems for V 0 and V 1, matching rules are used. Theremust be an intermediate zone (close to the perturbation for the far field andfar from it for the near field) where both the inner and outer expansionsare valid.

The behaviour of the far field near the origin is described by theexpansion in powers of r as previously encountered in Eq. (10.1):

U0(x1, x2) = R + krλu(θ) + · · · (10.10)

Then the matching conditions can be written as follows:

F0(a)V 0(y1, y2) ≈ R, F 1(a)V 1(y1, y2) ≈ kaλρλu(θ), (10.11)

when ρ = r/a =√y21 + y2

2 → ∞ (the symbol ≈ means here ‘behaves as atinfinity’), thus

F0(a) = 1; V 0(y1, y2) = R; F1(a) = kaλ; V 1(y1, y2) ≈ ρλu(θ).

(10.12)

This matching statement is nothing else than the so-called remote load atinfinity. Using superposition, it becomes:

V 1(y1, y2) = ρλu(θ) + V1(y1, y2) with V

1(y1, y2) ≈ 0. (10.13)



More precisely, the behaviour of V1(y1, y2) at infinity can be described by

the dual mode ρ−λu−(θ) to ρλu(θ) (see Section 10.2):

V1(y1, y2) = κρ−λu−(θ) + · · · (10.14)

This expansion is analogous to (10.10) but is at infinity; κ is the GSIF andmissing terms tend to 0 faster than ρ−λ at infinity. This detail is generallyuseless for our purpose but it may play a role elsewhere. It has been usedrecently to determine the length of a crack using full field measurementsand digital image correlation (DIC) [25]. It allows the small correctionmentioned in (10.8) to be specified:

Ua(x1, x2) = U0(x1, x2) + kκa2λ(r−λu−(θ) + U1(x1, x2)) + · · · (10.15)

Finally Eq. (10.9) becomes

Ua(x1, x2) = Ua(ay1, ay2) = R + kaλV 1(y1, y2) + · · · (10.16)

The function V 1(y1, y2) is computed using FEs in an artificially bounded(at a large distance from the perturbation) domain with either prescribeddisplacements or forces along the new fictitious boundary.

We have now at our disposal a description of the elastic solution priorto and following the onset of a short crack and we are able to calculate thechange in potential energy W (a)−W (0), which can be expressed using thepath independent integral Ψ encountered in (10.2)

−(W (a) −W (0)) = Ψ(Ua, U0). (10.17)

Then, substituting the above expansions, once for a = 0 and once for a = 0,into (10.17) leads to the expression (10.4) with Ginc = −(W (a)−W (0))/aand

A = Ψ(V 1(y1, y2), ρλu(θ)). (10.18)

Figure 10.4 shows the dimensionless function A∗ = E∗A (where E∗ = E

for plane stress and E∗ = E/(1−ν2) for plane strain, with E being Young’smodulus and ν Poisson’s ratio for the homogeneous isotropic material)for different V-notch openings ω (Fig. 10.1) and for a crack located alongthe bisector (symmetric case). It can be used as a master curve valid forany elastic isotropic material; the role of Poisson’s ratio in A has beenverified numerically. Note that A∗ = 2π for ω = 0 as a consequence of thenormalization of the eigenmode (10.1). The tensile stress σ = k/rλ−1 along



0

2

4

6

8

0 30 60 90 120 150 180

ω (°)

A*

Fig. 10.4. The dimensionless coefficient A∗ vs. the V-notch opening ω in the symmetriccase.

Fig. 10.5. PMMA Compact Tension V-notched Specimen (CTS).

the bisector, leading to σ = k/√r for ω = 0 (a crack) whereas it is usually

σ = k/√

2πr.

10.4 Application to the Crack Onset at a V-Notchin a Homogeneous Material

Tensile tests have been carried out on poly(methyl methacrylate) (PMMA)V-notched specimens (E = 3250MPa, ν = 0.3, Gc = 0.325MPa.mm, σc =75MPa) for different V-notch openings from 30 to 160 (Fig. 10.5) [26].

The tensile test was then numerically simulated by finite elements for anarbitrary prescribed load F0 (note here that special care must be taken given



0

1000

2000

3000

4000

5000

0 30 60 90 120 150 180

F (N)

ω

Fig. 10.6. Applied force F at failure of V-notched specimens of PMMA as a function ofthe notch opening ω. Comparison between experiments (diamonds) and prediction (solidline) using the coupled criterion [26].

Fig. 10.7. Schematic view of a transverse crack impinging on an interface.

the lack of symmetry of the specimen) and the GSIF k0 was extracted using(10.2). A scaling with the critical value kc (10.6) provides the correspondingforce F = F0 × kc/k0 at failure. A comparison between predicted andmeasured failure forces is illustrated in Fig. 10.6, which exhibits a fairagreement.

10.5 Application to the Deflection of Transverse Cracks

We now consider a transverse crack as depicted schematically in Fig. 10.7.Despite there being a pre-existing crack, the singular exponent at itstip, which impinges the interface, is not 1/2 as usual and the situationdiffers from that of a crack in a homogeneous material (Fig. 10.8). For



0

0.2

0.4

0.6

0.8

-3 -2 -1 0 1 2 3

λ

Ln(E2/E1)

Fig. 10.8. Singular exponent vs. ratio of Young’s moduli for two adjacent materialswith the same Poisson’s ratio ν1 = ν2 = 0.3.

homogeneous isotropic components, if E2 > E1 (the first case) then λ > 1/2(a weak singularity, which is less harmful than a crack) and vice versa ifE2 < E1 (the second case) then λ < 1/2 (a strong singularity, more harmfulthan a crack). Here Ei is Young’s modulus for ply number i (it is assumedthat the two Poisson’s ratios ν1 and ν2 are equal, otherwise the rule is closeto the above but slightly altered by the contrast in ν). This obviously leadsto substantially different results in terms of rupture. We immediately noticeaccording to (10.4) that if a → 0 then Ginc → 0 in the first case whereasGinc → ∞ in the second.

This property also affects the (differential) energy release rate G ofa crack approaching and crossing the interface. As the crack approachesthe interface, there remains a ligament of length l between the crack tipand the interface (Fig. 10.13) and G → 0 (respectively, G → ∞) for aweak singularity (respectively, strong singularity) as l → 0. Symmetrically,after crossing the interface the crack tip is at a distance a from it (Fig. 10.7)and G increases from 0 (weak singularity) or decreases from infinity (strongsingularity) as a increases. This behaviour is shown in Fig. 10.9 for differentratios of Young’s moduli: E2/E1 = 0.1, 0.2, 0.5, 1, 2, 5 and 10. The resultswere obtained using FEs and a variable crack tip location, counted negativeif the crack is growing toward the interface and positive after the crossing(Fig. 10.9). Even if the energy is globally calculated at the structure level,it requires strong mesh refinement in the area of interest to give a goodgeometrical description.



0

-4 -3 -2 -1 0 1 2 3

G

l a

Fig. 10.9. The behaviour of the energy release rate when a crack approaches an interface(left) and then crosses it (right) for different ratios of Young’s moduli: E2/E1 = 0.1(dashed line and diamonds), 0.2 (dashed line and squares), 0.5 (dashed line and triangles),

1 (solid line and circles), 2 (solid line and triangles), 5 (solid line and squares) and 10(solid line and diamonds). The units are not important, the emphasis is on the generaltrends — whether they are increasing or decreasing functions when approaching theinterface.

Fig. 10.10. The mechanisms of crack penetration and deflection.

The question that arises now is: does such a transverse crack stop,penetrate Material 2 or deflect along the interface to give a delaminationcrack (Fig. 10.10)?

Let us consider again inequality (10.4) for two cases: penetration ofMaterial 2 (index p) and deflection along the interface (index d). G(I)

c andG

(2)c are the interface and Material 2 toughness, respectively:

Gincd = Adk

2a2λ−1d + · · · ≥ G(I)

c and Gincp = Apk

2a2λ−1p + · · · ≥ G(2)

c .

(10.19)

Two cases can be considered: a doubly symmetric deflection (Fig. 10.10) ora single asymmetric one; the only change is in the coefficient Ad but this



0

0.5

1

-3 -2 -1 0 1 2 3Ln(E2/E1)

Ad/Ap

Fig. 10.11. Dimensionless ratio Ad/Ap vs. ratio of Young’s moduli for two adjacentmaterials for a doubly symmetric deflection.

does not lead to a big difference [27]. The following analysis will be carriedout for the first case.

Deflection is promoted if the first inequality in (10.19) is fulfilledwhereas the second one is not, then:

G(I)c

G(2)c

= R ≤ AdAp

(adap

)2λ−1

. (10.20)

The dimensionless ratio Ad/Ap is plotted in Fig. 10.11 for various values ofE2/E1 (Ei is the Young’s modulus of material i and ν1 = ν2 = 0.3).

He and Hutchinson [28] obtained a similar result (although differently)but simplified thanks to a dubious assumption. They considered the(differential) energy release rates Gp and Gd respectively at the tip of apenetrated crack and a deflected one, the two crack extensions being equal,i.e. ad = ap, and obtained a condition for the toughness ratio of the interfaceand Material 2, which is clearly equivalent to the ratio Ad/Ap according to(10.19) if ad = ap:

R ≤ GdGp

. (10.21)

A discussion of this specific point can be found in [29, 30].Clearly it is possible to determine the two characteristic lengths ad and

ap in (10.20) using the stress condition, provided λ > 1/2 (otherwise both



G and σ are decreasing functions of the distance to the singular point andthe coupled criterion can no longer be used).

10.5.1 λ > 1/2

If λ > 1/2, according to (10.5) (σ(I)c and σ

(2)c are the tensile strengths of

the interface and Material 2, respectively):

ad =G

(I)c

Ad

(sd

σ(I)c

)2

; ap =G

(2)c

Ap

(sp

σ(2)c

)2

. (10.22)

Thus deflection is promoted if

R ≤ AdAp

(sdsp

σ(2)c

σ(I)c

) 2λ−11−λ

. (10.23)

The special case λ = 1 cannot be met; it would correspond to an infinitelycompliant Material 1 compared to Material 2. Knowing that the ratio sd/spremains of the same order of magnitude as 1, it is clear from (10.23) thatthe tensile strength ratio plays a crucial role, which can significantly alterthe criterion proposed by He and Hutchinson.

As illustrated in Fig. 10.12, deflection will be favoured even more asMaterial 2 becomes increasingly resistant (i.e. σ(2)

c > σ(I)c ). Note that

0

0.5

1

1.5

2

0 0.5 1 1.5 2 2.5Ln(E2/E1)

0.25

1

4R

Fig. 10.12. Failure map of the criterion (10.23) for different values of the ratio of

strengths σ(2)c /σ

(I)c = 0.25, 1, 4 as a function of the material contrast E2/E1. The

dashed line corresponds to Ad/Ap (Fig. 10.11). Below the continuous line conditions arefavourable for deflection along the interface, and above to penetration of Material 2.



Parmigiani and Thouless [31] derived the same tendency with cohesive zonemodels (CZMs).

Reference is made in both cases to the tensile stress; this is clearfor penetration but less obvious for deflection. However, considering theeigenmode governing the stress field before crack propagation, one can checkthat the tensile component σ is equal to or larger than the shear one τ : theratio σ/τ grows from 1 to 2.8 as E2/E1 varies from 1 to 10. In addition,knowing that shear failure is generally more difficult than tension, it seemsreasonable to consider only the tensile component.

This analysis could be included in a homogenization process wherethe domain shown in Fig. 10.7 would be a representative volume element(RVE). The slenderness of this cell can be used to take into account thedifferent densities of transverse cracks [5].

10.5.2 λ < 1/2

If λ < 1/2 then this coupled criterion approach is not valid becausethe energy release rate is now a decreasing function of the distanceto the singular point (see (10.4)) and the energy condition no longergives any lower bound for the crack extension length [27, 32]. Under amonotonic loading the crack grows continuously: there is no crack jump.Moreover, according to (10.19), Gd and Gp tend to infinity as ad and apdecrease to 0, which prevents the direct use of the energy release rate atthe very start of the crack growth process. No rigorous conclusion canbe derived in this situation. He and Hutchinson [28] still proposed touse (10.21) or equivalently the ratio Ad/Ap. Another approach based onthe maximum dissipated energy is proposed in Leguillon et al. [27]; however,this corresponds better to the geometrical situation analysed in the nextsection (Fig. 10.13).

10.6 The Cook and Gordon Mechanism

Due to the decay to 0 or the unbounded growth of the energy release rate(Fig. 10.9), it should be pointed out that the geometric situation shownin Fig. 10.7 cannot be achieved by a crack growing in Material 1 andapproaching the interface. It can only be obtained by a mechanical actionlike a saw cut.

Otherwise, we have to consider a crack in Material 1 with its tip a smalldistance l from the interface as shown in Fig. 10.13 [32, 33].



Fig. 10.13. A crack growing in Material 1 and approaching the interface, at a distance l.

Assuming a small increment δl l at the tip of this crack and accordingto (10.4), the (differential) energy release rate is

G1 = − limδl→0

W (l + δl) −W (l)δl

= k2A1 limδl→0

(l + δl)2λ − l2λ

δl+ · · ·

= 2λk2A1l2λ−1 + · · · (10.24)

A1 is given by (10.18) where the perturbation is the small ligament withwidth l instead of a crack extension. Moreover, since the crack is growingin Material 1:

G1 = G(1)c ⇒ k2l2λ−1 =

G(1)c

2λA1. (10.25)

This relation means that in these conditions knowing l or the applied loadis somewhat equivalent.

10.6.1 λ > 1/2

If λ > 1/2, G1 decreases to 0 as l → 0 and thus drops below G(1)c (Material

1’s toughness). An overload must occur for the situation to evolve. Thereis a three-way conflict: the crack still grows in Material 1, the interfacedebonds ahead of the crack tip or the crack jumps and penetrates Material 2(Fig. 10.14) [34]. The latter mechanism will not be discussed here. Anothermechanism called step-over, where the crack reinitiates in the secondmaterial, leaving a ligament in its wake, was discussed in [22, 32, 35].

There are now two small parameters l and ad (respectively, ap) fordebonding (respectively, penetration), which is an additional difficulty.If one is very small compared to the other, it can be neglected in a firststep. If they are of the same order of magnitude the expansions can becarried out with either of them. For technical reasons it is easier to use l.



Fig. 10.14. The conflict between the crack growing in Material 1, the crack jumping inMaterial 2 and the interface debonding.

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5

µ/2

Bd

Fig. 10.15. Bd(µ) (MPa−1) for material contrast E2/E1 = 10.

By analogy with the single parameter case, the stress and energy conditionsare now (see [36] for details of the proof):

σ = klλ−1σ(µd) ≥ σ

(I)c

Gincd = k2Bd(µd)l2λ−1 ≥ G

(I)c ,

(10.26)

where µd = ad/l. The function Bd (Fig. 10.15) is an increasing functionof µ and replaces Ad. It is derived from the calculation of A (10.18).There are four cases: the ‘unperturbed’ one (Fig. 10.7) and the successivecases illustrated in Fig. 10.13 and the middle and right of Fig. 14. σ isa decreasing function (Fig. 10.16), which replaces s and is the tensilestress associated with V 1 along the presupposed crack path prior to anycrack extension (i.e., with the inner term calculated for the geometry ofFig. 10.13).



0

0.2

0.4

0.6

0.8

0 1 2 3 4 5

µ /2

σ

Fig. 10.16. σ(µ) for material contrast E2/E1 = 10.

The equation for the dimensionless characteristic length µd is derivedfrom (10.25) and (10.26)2:

Bd(µd) = 2λA1G

(I)c

G(1)c

. (10.27)

The dimensionless debonding length µd is small if the interface toughnessG

(I)c is small.

For material contrast E2/E1 = 10, λ = 0.667 (Fig. 10.8), A1 =0.479MPa−1, thus ifG(I)

c = G(1)c , then from (10.25)Bd(µd) = 0.639MPa−1,

µd/2 = 1.9 (Fig. 10.15) and σ(µd) = 0.345 (Fig. 10.16). Thus the conditionfor an interface debonding ahead of the primary crack is

k ≥ kc =

(G

(I)c

Bd(µd)

)1−λ(σ

(I)c

σ(µd)

)2λ−1

. (10.28)

Note that (10.25) and (10.26)1 give l, which is not useful for criterion (10.28)which requires only µd:

l =G

(1)c

2λA1

(σ(µd)

σ(I)c

)2

. (10.29)

The ligament width is small if the tensile strength σ(I)c is high. Since ad =

µdl and according to (10.27) and (10.29) the debond length is large for ahigh toughness and a small tensile strength of the interface, which is oftenthe case for polymer adhesives for instance.



0

0.5

1

1.5

0 1 2 3 4 5

µ/2

γ

Fig. 10.17. The evolution of the energy release rate (normalized by G(1)c ) at the tip of

the primary crack after the onset of interface debonding for material contrastE2/E1 = 10.

The primary crack may stop at a distance l: it depends on how theenergy release rate G1 evolves after the onset of the debonding. Figure 10.17shows the ratio γ = G1/G

(1)c as a function of the dimensionless debonding

length µd for E2/E1 = 10 and with ν = 0.3 in both materials.Obviously, on the one hand, if the characteristic debonding length µd is

smaller than a given value (roughly µd/2 = 4 in the present case, Fig. 10.17)the energy release rate increases as debonding occurs and the primary crackrestarts and definitely breaks the ligament. On the other hand, this ligamentdoes not disappear and can only be observed if the debonding length is large(µd/2 = 4 in the present case).

10.6.2 λ < 1/2

If λ < 1/2, G1 increases as l → 0. For a given l, if the load (i.e. a givenGSIF k) is such that (10.25) holds then the crack accelerates toward theinterface. As it impinges on the interface, there is an excess of energy inthe balance:

∆W1 = kA1l2λ −G(1)

c l = G(1)c

(12λ

− 1)l. (10.30)

Hence, the crack will deflect and the (differential) energy release rate Gdwill decrease as the debond length increases (which is calculated using asmall increment δad ad at the tip of the deflected crack and passing tothe limit δad → 0 as for G1, see (10.24)). Following (10.24) and (10.25), it



drops below the interface toughness at a distance ad such that

Gd = 2λk2Ada2λ−1d = G(I)

c ⇒(adl

)2λ−1

=G

(I)c

G(1)c

A1

Ad. (10.31)

At this point, the excess energy is now

∆W = ∆W1 + ∆Wd =(G(1)c l +G(I)

c ad

)( 12λ

− 1). (10.32)

Thus the crack will continue to grow with a characteristic length δad untilit consumes this excess energy:

δad =∆W

G(I)c

=

(G

(1)c

G(I)c

l + ad

)(12λ

− 1). (10.33)

Of course ad + δad is an upper bound of the delamination length. Someof the excess energy will be dissipated by dynamic effects, such as elasticwaves producing noise for example.

As already mentioned, comparison with a crack advancing in a straightline and penetrating Material 2 is not considered here. This mechanism ismore difficult to describe and is the subject of a work in progress.

The modelling described in Sections 10.5 and 10.6 can be extended tothe anisotropic case provided it can still be split into plane and antiplaneproblems. This is the case for cross-ply laminates of carbon-fibre-reinforcedpolymers, where each layer is orthotropic in the appropriate basis, but notfor angle-ply laminates [1].

10.7 Conclusion

Plane-strain elasticity is the main framework of this chapter. The coupledcriterion can be extended without major difficulty to a generalized plane-strain assumption in the analysis of delamination of angle-ply laminates[1, 11]. It is also possible to describe in the same way crack kinking outof an interface [12, 13], although this makes use of complex exponents andSIFs [37, 38]. Together with the analysis of crack deflection by the interfacedeveloped here, this covers a wide range of problems of failure in compositelaminates; especially since anisotropy, which has not been mentioned in thischapter, does not complicate things too much as long as the assumption ofplane or generalized plane elasticity still holds.

Obviously the extension to 3D raises the most difficulties. There areno major conceptual changes, but everything becomes technically much



more complicated. The crack extension is no longer simply described bytwo parameters, e.g. direction and length: the complete geometry must betaken into account. An attempt was made to predict the nucleation of smalllens-shaped cracks along a straight crack front subject to Mode III remoteloading [14]. Nevertheless it is clear that much remains to be done in thisdomain.

References

[1] Martin E., Leguillon D., Carrere N., 2010. A twofold strength and toughnesscriterion for the onset of free-edge shear delamination in angle-ply laminates,Int. J. Solids Struct., 47, 1297–1305.

[2] Sridharan S., 2008. Delamination behaviour of composites, in Materials,Woodhead Publishing Ltd, Cambridge, UK.

[3] Brewer J.C., Lagace P.A., 1998. Quadratic stress criterion for initiation ofdelamination, J. Comp. Mat., 122, 1141–1155.

[4] Nairn J.A., Hu S., 1992. The initiation and growth of delaminations inducedby matrix microcracks in laminated composites, Int. J. Fracture, 57, 1–24.

[5] Delisee I., 1996. Etude de mecanismes de delaminage des composites croisescarbone/epoxy, PhD thesis, Universite Pierre et Marie Curie, Paris.

[6] Ladeveze P., Lubineau G., Marsal D., 2006. Towards a bridge between themicro- and mesomechanics of delamination for laminated composites, Comp.Sci. and Technology, 66, 698–712.

[7] Blazquez A., Mantic V., Parıs F., McCartney L.N., 2008. Stress statecharacterization of delamination cracks in [0/90] symmetric laminates byBEM, Int. J. Solids Struct., 45, 1632–1662.

[8] Martin E., Peters P.W.M., Leguillon D., Quenisset J.M., 1998. Conditionsfor crack deflection at an interface in ceramic matrix composites, Mat. Sci.Eng. A, 250, 291–302.

[9] Carrere N., Martin E., Lamon J., 2000. The influence of the interphase andassociated interfaces on the deflection of matrix cracks in ceramic matrixcomposites, Comp. Part A, 31, 1179–1190.

[10] Leguillon D., 2002. Strength or toughness? A criterion for crack onset at anotch, Eur. J. Mech. — A–Solid, 21, 61–72.

[11] Pipes B.R., Pagano N.J., 1970. Interlaminar stresses in composite laminatesunder uniform axial extension, J. Comp. Mat., 4, 538–548.

[12] Leguillon D., Murer S., 2008. Crack deflection in a biaxial stress state, Int. J.Fracture, 150, 75–90.

[13] Leguillon D., Murer S., 2008. A criterion for crack kinking out of an interface,Key Eng. Mat., 385–387, 9–12.

[14] Leguillon D., 2003. Computation of 3D singular elastic fields for the pre-diction of failure at corners, Key Eng. Mat., 251–252, 147–152.

[15] Leguillon D., Sanchez-Palencia E., 1987. Computation of Singular Solutionsin Elliptic Problems and Elasticity, John Wiley & Son, New York, andMasson, Paris.



[16] Labossiere P.E.W., Dunn M.L., 1999. Stress intensities at interface cornersin anisotropic bimaterials, Eng. Fract. Mech., 62, 555–575.

[17] Henninger C., Roux S., Hild F., 2010. Enriched kinematic fields of crackedstructures, Int. J. Solids Struct., 47, 3305–3316.

[18] Griffith A.A., 1920. The phenomenon of rupture and flow in solids, Phil.Trans. Roy. Soc. London A, 221, 163–198.

[19] Irwin G., 1958. Fracture, in Hand. der Physic, Springer, Berlin, Vol. VI,pp. 551–590.

[20] Yosibash Z., Priel E., Leguillon D., 2006. A failure criterion for brittle elasticmaterials under mixed-mode loading, Int. J. Fracture, 141, 289–310.

[21] Martin E., Leguillon D., 2004. Energetic conditions for interfacial failure inthe vicinity of a matrix crack in brittle matrix composites, Int. J. SolidsStruct., 41, 6937–6948.

[22] Martin E., Poitou B., Leguillon D., Gatt J.M., 2008. Competition betweendeflection and penetration at an interface in the vicinity of a main crack,Int. J. Fracture, 151, 247–268.

[23] Hebel J., Dieringer R., Becker W., 2010. Modelling brittle crack formationat geometrical and material discontinuities using a finite fracture mechanicsapproach, Eng. Fract. Mech., 77, 3558–3572.

[24] Leguillon D., Laurencin J., Dupeux M., 2003. Failure of an epoxy jointbetween two steel plates, Eur. J. Mech. A–Solid., 22, 509–524.

[25] Leguillon D., 2011. Determination of the length of a short crack at a V-notchfrom a full filed measurement. Int. J. Solids Struct. 48, 884–892.

[26] Leguillon D., Murer S., N. Recho J. Li, 2009. Crack initiation at a V-notchunder complex loadings — Statistical scattering, International Conferenceon Fracture, ICF12, Ottawa, Canada, 12–17 July 2009.

[27] Leguillon D., Lacroix C., Martin E., 2000. Matrix crack deflection at aninterface between a stiff matrix and a soft inclusion, C. R. Acad. Sci. Paris,328, serie IIb, 9–24.

[28] He M.Y., Hutchinson J.W., 1989. Crack deflection at an interface betweendissimilar elastic materials, Int. J. Solids Struct., 25, 1053–1067.

[29] Martinez D., Gupta V., 1994. Energy criterion for crack deflection atan interface between two orthotropic media, J. Mech. Phys. Solids, 42,1247–1271.

[30] He M.Y., Evans A.G., Hutchinson J.W., 1994. Crack deflection at aninterface between dissimilar elastic materials: Role of residual stresses, Int.J. Solids Struct., 31, 3443–3455.

[31] Parmigiani J.P., Thouless M.D., 2006. The roles of toughness and cohesivestrength on crack deflection at interfaces, J. Mech. Phys. Solids, 54,266–287.

[32] Leguillon D., Lacroix C., Martin E., 2000. Interface debonding ahead of aprimary crack, J. Mech. Phys. Solids, 48, 2137–2161.

[33] Martin E., Leguillon D., Lacroix C., 2002. An energy criterion for theinitiation of interface failure ahead of a matrix crack in brittle matrixcomposites, Compos. Interf., 9, 143–156.



[34] Cook J., Gordon J.E., 1964. A mechanism for the control of crack propagationin all brittle systems, Proc. Roy. Soc A, 282, 508–520.

[35] Quesada D., Picard D., Putot C., Leguillon D., 2009. The role of the interbedthickness on the step-over fracture under overburden pressure, Int. J. RockMech. Min., 46, 281–288.

[36] Leguillon D., Quesada D., Putot C., Martin E., 2007. Size effects for crackinitiation at blunt notches or cavities, Eng. Fract. Mech., 74, 2420–2436.

[37] Rice J.R., 1988. Elastic fracture mechanics concept for interfacial cracks,J. Appl. Mech., 55, 98–103.

[38] He M.Y., Hutchinson J.W., 1989. Kinking of a crack out of an interface,J. Appl. Mech., 56, 270–278.


Chapter 11

SINGULAR ELASTIC SOLUTIONS INANISOTROPIC MULTIMATERIAL CORNERS.

APPLICATIONS TO COMPOSITES

Vladislav Mantic, Alberto Barroso and Federico Parıs

School of Engineering, University of SevilleCamino de los Descubrimientos s/n, Seville, 41092 Spain

Abstract

An approach to the evaluation of linear elastic solutions in anisotropicmultimaterial corners under generalized plane-strain or plane-stress conditionsis developed. This approach works for quite general configurations of piecewisehomogeneous multimaterial corners covering discontinuities in geometry,materials and boundary conditions. Open and closed (periodic) cornersincluding any finite number of single-material wedges converging at the cornertip are considered. General homogeneous boundary conditions and slidingfriction contact can be imposed at corner boundaries and perfect adhesion orsliding friction contact can be imposed at corner interfaces. An anisotropic dryfriction model is generally assumed, representing contact between surfaces witha strongly oriented surface topography or texture. A semi-analytic approachto the corner singularity analysis based on the Lekhnitskii–Stroh formalismof anisotropic elasticity, a transfer matrix concept for single-material wedgesand a matrix formalism for boundary and interface conditions, is developedand implemented in a symbolic computation tool. A least-squares fittingtechnique for extracting generalized stress intensity factors (GSIFs) fromfinite element and boundary element results is proposed and implemented.Singularity analysis of a crack terminating at a ply interface in a laminateand of a bimaterial corner in a double-lap joint between a composite laminateand a metal layer is carried out as an application of the developed theory.Finally, a criterion for failure initiation at a closed corner tip based on GSIFsand the associated generalized fracture toughnesses is proposed, and a novelexperimental procedure for the determination of the corresponding failureenvelope is introduced and accomplished.

425


426 V. Mantic, A. Barroso and F. Parıs

11.1 Introduction

Composites are heterogeneous materials, which on microscales andmesoscales are usually considered as piecewise homogeneous materials.Therefore models of composites and their adhesive joints with othercomposites or metals on micro-, meso- or macroscale often include interfacesbetween dissimilar materials, sometimes with potential cracks either alonginterfaces or terminating at interfaces [1]. Singular points, where there arediscontinuities in the idealized geometry, material, interface or boundaryconditions, can easily be identified in these models. We refer to a neigh-bourhood of a singular point as a corner, and to the singular point itselfas the corner tip (having in mind a 2D model). In particular, we referto a neighbourhood of a singular point where several materials meet as amultimaterial corner, see Figs. 11.1 and 11.2, and definitions in Section 11.3.

Linear elastic solutions for models of composites and their adhesivejoints under mechanical and thermal loads may involve unbounded stressesat a corner tip, called singular stresses [2–8]. After the first fundamentalcontributions analysing singular stresses at homogeneous isotropic elasticcorners in [9,10], see also [11], considerable effort has been made by

M1

M2

M3

Mk

MmMM

...

...

θ0 θ1

θ2

θ3

θk-1

θk

θmθM-1

θM

...

...

bound.cond.

bound.cond.

θ0,...,θM materials

wedges

frictionless or frictionalcontact interface

x1

x2

θr

(W≤M)

θm-1

W0 ,...,ϑϑ

0ϑ1ϑ

2ϑ

1-wϑ

wϑ

Wϑ

1-Wϑ

Wϑ

0ϑ

Fig. 11.1. A multimaterial corner (2D view).


Singular Elastic Solutions in Anisotropic Multimaterial Corners 427

3s

)( Wr ϑs

x3

)( Wϑn

1

W

3s

)( 0ϑrs

)( 0ϑn0ϑ

1ϑ

1−Wϑ

Wϑ

)(ϑrs

3s

),( ωϑk

),( ωϑm

ω

ω

wedge face plane at ϑ

),( ωϑk

µarctan

normal plane to wedge face at ϑ

)(ϑn

),,( µωϑµn

µarctan

),,( µωϑµs

Fig. 11.2. Local Cartesian coordinate systems at a multimaterial corner (3D view).

many authors to analyse singular stresses at multimaterial isotropic andanisotropic elastic corners [12–25]. A rigorous mathematical analysis ofthese solutions can be found in [26–27].

The present work is focused on linear elastic anisotropic materialssubjected to a generalized plane-strain state [28–29] where three-dimensional (3D) displacements depend on only two Cartesian coordinates(or on the corresponding polar coordinates), namely ui =ui(x1, x2)=ui(r, θ) with i=1, 2, 3 or i= r, θ, 3. The results obtained can easily beadapted to generalized plane-stress states [29] if required. The majorityof practical problems found in composites can be studied under thesehypotheses.

In general, a singular elastic solution in a neighbourhood of thecorner tip can be represented by the following asymptotic series expansion(a rigorous mathematical justification can be found in the fundamentalworks [30,31], see also [2–5,26–27]), given by products of power-logarithmicterms in radial coordinate and angular functions:

ui(r, θ) ∼=N∑n=1

Q∑q=0

L∑l=0

Knqlrλn+q logl rg(nql)

i (θ) + · · · , (11.1)

where Knql are generalized stress intensity factors (GSIFs), λn arecharacteristic or singular exponents (eigenvalues), and g

(nql)i (θ) are

characteristic or singular angular shape functions (eigenfunctions), N ≥ 1



and Q ≥ 0 can be finite or infinite numbers whereas L ≥ 0 is a finitenumber.

The terms with the exponent q= 0 are referred to as principal terms,because the terms with q≥ 1 are due to the curvature of the cornerboundaries and interfaces, the material non-homogeneity in the radialdirection, non-homogeneous boundary or interface conditions or bodyforces in the neighbourhood of the corner tip. In the present work, forthe sake of simplicity, only straight boundaries and interfaces, piecewisehomogeneous materials (in particular, those which are homogeneous inthe radial direction), homogeneous boundary and interface conditions andvanishing body forces are considered in the neighbourhood of the cornertip. Thus, Q=0, the index q being omitted hereinafter. The so-calledcharacteristic exponents λn are defined by the roots of a complex analytic(holomorphic) function given as the determinant of a matrix, referred to asa characteristic matrix, whose elements are also complex analytic functions.The null space of this characteristic matrix determines the characteristicangular shape functions g

(nql)i (θ). The vanishing determinant condition

defines a transcendental equation referred to as a characteristic equation(eigenequation) of the corner, which depends on the problem configurationin the neighbourhood of the corner tip: geometry, material properties,and boundary and interface conditions. Under the above assumptions,L ≥ 1 is associated only with special cases with repeated roots of thecharacteristic equation of the corner, whose algebraic multiplicity is largerthan the geometric multiplicity [7,28,32,33]. For the sake of simplicity,L = 0 is assumed, and the index l is omitted as well. In view of the aboveassumptions, the displacements and stresses in the neighbourhood of thecorner tip can be represented by the following asymptotic series expansions:

ui(r, θ) ∼=N∑n=1

Knrλng

(n)i (θ) + · · · , (11.2)

σij(r, θ) ∼=N∑n=1

Knrλn−1f

(n)ij (θ) + · · · (11.3)

If the number of terms in the series N is finite, a regular remainderterm vanishing at the corner tip should be added to these series to obtainthe complete solution in the corner. The terms in these series, referredto as power-type singularities, solve the elliptic system of three partialdifferential equations of generalized plane strain and satisfy the boundaryand interface conditions in the neighbourhood of the corner tip. All power



type singularities are defined by three elements: the characteristic exponentλn, the characteristic angular shape functions g

(n)i (θ) for displacements

and f (n)ij (θ) for stresses, and the generalized stress intensity factor (GSIF)

Kn. The characteristic angular functions are smooth functions inside eachhomogeneous material in the corner tip neighbourhood, but may be non-smooth or even discontinuous at material interfaces. In an elastic boundary-value problem with a domain including one or more corners, the associatedcharacteristic exponents and functions depend only on the local problemconfiguration (geometry, material properties, and boundary and interfaceconditions) in a neighbourhood of the corner tip, whereas the GSIFs dependon the global problem configuration. If the boundary-value problem withvanishing body forces is linear, the GSIFs are linear functionals of theboundary conditions.

The rigid body motions are included in (11.2) and (11.3) for λn = 0(translations) and λn = 1 (small rotations) with appropriate definitions ofg(n)i (θ) and the corresponding f

(n)ij (θ) = 0. The terms with 0 < λn < 1

give rise to unbounded (singular) stresses with a finite elastic strain energy.It is assumed that λn are naturally ordered satisfying Reλn ≤ Reλn+1 (Redenotes the real part of a complex number).

Many different approaches to the corner singularity analysis, theevaluation of the characteristic exponents and functions, have beenproposed in the past. In particular, with reference to the singularity analysisof a linear elastic anisotropic multimaterial corner in a generalized plane-strain state, several analytical, semi-analytical and numerical approachesare available at present. A numerical approach [2,13–15] is expected tobe more general and capable of analysing corner singularity problemsnot tractable by analytical or semi-analytical approaches. The advantagesof an analytical or semi-analytical approach, which provides an explicitclosed-form expression of the characteristic equation of the corner, areits essentially arbitrarily high precision, fast computation and excellentpossibilities for parametric studies and for understanding the influence ofdifferent parameters of the local corner configuration on the values andnature of the characteristic exponents and the behaviour of characteristicfunctions.

Following the original proposal by Ting [16], several authors [18–21]have shown that for the development of a general semi-analytical approachto the singularity analysis of linear elastic anisotropic multimaterial cornersin a generalized plane-strain state (assuming perfect adhesion betweenmaterials) it is very advantageous to employ the powerful Lekhnitskii–Stroh



formalism of anisotropic elasticity [34–36], see also [28–29,37], togetherwith a transfer matrix for a single-material angular sector (single-materialwedge) in the corner. In fact, this methodology is essentially fully analyticalexcept for the numerical evaluation of the complex roots of the Lekhnitskii–Stroh sextic polynomial for each anisotropic material in the corner. Inthe case of homogenized unidirectional fibre-reinforced composite laminas,represented by transversely isotropic materials, the roots of this sexticpolynomial can be evaluated analytically leading to a fully analyticalapproach for corner singularity analysis [25]. It is worth pointing outthat the application of the transfer matrix concept for all the single-material wedges in the corner allows the size of the matrix in the vanishingdeterminant condition to be made as small as possible. It should also bementioned that the expressions introduced in [19,21] can be applied directlyto the corner singularity analysis for any kind of anisotropic material,namely non-degenerate, degenerate or extraordinary degenerate with alower geometric than algebraic multiplicity of the repeated roots of thesextic polynomial, according to the classification in [38].

The above described methodology was suitably adapted by Mantic et al.[22] to the simpler case of the singularity analysis of linear elastic anisotropicmultimaterial corners in antiplane strain, governed by a scalar linear second-order elliptic equation (a generalization of the Laplace equation).

A general semi-analytical approach to the corner singularity analysisfor linear elliptic systems of second-order partial differential equations inthe plane was developed by Costabel and Dauge [17,39], employing theirprevious fundamental mathematical results [31]. This general mathematicalframework covers, as a particular case, linear elastic anisotropic materialsunder generalized plane-strain conditions. Their approach was implementedin a computational tool capable of carrying out the singularity analysisof multimaterial corners including anisotropic non-degenerate materials[17]. It is interesting to observe that, although not apparent at first sight,the approach due to Costabel and Dauge is in essence closely related tothe above-described approaches based on the Lekhnitskii–Stroh formalism.In their approach, first, the complex roots of the symbol determinant of thepartial differential system of linear elasticity in the plane are found, andthen, the solution basis in the form of terms in the series expansion (11.2)is constructed analytically. This symbol determinant is, in fact, given bythe Lekhnitskii–Stroh sextic polynomial.

Whereas, as discussed above, an arbitrarily high accuracy canbe achieved in the evaluation of characteristic exponents and functions,the accuracy in the evaluation of GSIFs is, in general, substantially



worse because a numerical solution of the global elastic problem (typicallyby means of finite or boundary element methods, FEM or BEM), orexperimental tests (e.g. using photoelasticity), and usually also somepost-processing of the results are needed [6,8,40,41], see Section 11.6 forother references. Nevertheless, a few fast and highly accurate methods forthe evaluation of stresses in the presence of crack and corner singularitiesin isotropic elastic materials are already available [41,42].

In real composite structures, only high values of stresses (high stressconcentrations), instead of singular stresses, are expected at these corners;first because strict discontinuities are hardly present in a real structure (e.g.a sharp corner tip in the model is usually rounded in a real structure), andsecond because a zone of non-linear behaviour (due to large strains, plasticity,damage, etc.) usually appears in the neighbourhood of the corner tip.

The failure of a composite specimen, structure or joint may be initiatedat a corner due to these high stress concentration there. Nevertheless,if the size of the non-linear zone is sufficiently small with respect to ageometrical characteristic length (adjacent layer thickness, crack length,corner side length, etc.) the linear elastic solution for the idealized modelmay satisfactorily represent the solution in almost the whole volume ofthe structure and it will essentially determine the solution behaviour in thenon-linear zone. In this case, failure initiation at a corner is governed by thelinear elastic solution, and in particular by its asymptotic series expansionin the neighbourhood of the corner tip, given by GSIFs, which could thenbe used in predictions of strength. A typical form of failure initiation ata corner tip, also considered in the present study, is the onset of a crack[9,43–54]. In fact, the ultimate aim of the present study is to contributeto improving the accuracy and reliability of the strength predictions ofcomposites in cases where a failure initiates at a multimaterial corner inthe form of a crack.

The overall objective of the present work is the development of:(i) a general semi-analytical procedure for the singularity analysis of linearelastic anisotropic multimaterial corners in generalized plane strain (i.e.the evaluation of characteristic exponents and functions) with frictionlessor friction contact surfaces in the corner, (ii) a general and sufficientlyaccurate and reliable numerical procedure for the extraction of GSIFs inthese corners from FEM or BEM results and (iii) a procedure for failureassessment of corners in composites and their joints.

For the first objective, the methodology developed in [19] is generalizedto include boundary surfaces and interfaces with frictionless or frictioncontact in the neighbourhood of the corner tip. Although several relevant



studies of singular elastic solutions at interface cracks and corners withfrictionless contact or sliding friction contact have been published forisotropic [55–61] and also for anisotropic materials [62–75], it seems thata fully general approach to the singularity analysis of elastic anisotropicmultimaterial corners in generalized plane strain including sliding frictionalsurfaces has not been developed yet. One of the open issues is relatedto the angle between the friction shear stress vector and the relativetangential displacement vector in the case where the in-plane and antiplanedisplacements are coupled. This fact is the reason for including in thepresent work a general and powerful matrix formalism for imposing rathergeneral boundary and interface conditions, in particular frictionless andfrictional sliding contact, by generalizing the methodology introducedin [19,71]. This matrix formalism, which is especially suitable for astraightforward computational implementation, allows the characteristiccorner matrix to be assembled in a fully automatic way for any finite numberof materials and contact surfaces in the corner.

With reference to the second objective, a least-squares procedure forthe extraction of multiple GSIFs at this kind of corner is developed andimplemented [40] as a computational tool for post-processing results ofa BEM code [75–79] solving boundary-value problems with linear elasticanisotropic materials in generalized plane strain.

The above described computation tools are used to study a crackterminating at the interface between two plies in a [0/90]S laminatesubjected to longitudinal tension and a critical bimaterial corner in anadhesively bonded double-lap joint. An altered configuration of the cornerin this double-lap joint including an interface crack with sliding frictioncontact is also studied.

With reference to the third objective, the results of the singularityanalysis for the critical corner in the double-lap joint are further usedwhen analysing experimental results obtained by testing a novel modifiedconfiguration of the Brazilian disc specimen including the same bimaterialcorner [80,81] with two stress singularities. These tests provide a roughapproximation of the failure envelope in the plane of GSIFs normalizedby the pertinent values of the generalized fracture toughnesses. A cornerfailure criterion based on this failure envelope is proposed. The results forreal adhesively bonded double-lap joints show a satisfactory agreement withthe proposed failure criterion. The procedure developed can be appliedas a general methodology for the failure assessment of these kind ofmultimaterial corners.



11.2 Lekhnitskii–Stroh Formalism for Linear ElasticAnisotropic Materials

The Lekhnitskii–Stroh complex-variable formalism [34–36,82], or simplyStroh formalism, is a powerful and efficient theoretical tool for the analysisof anisotropic elastic problems. This section summarizes the fundamentalsof this formalism employed in the analysis of singular stresses at multilateralcorners in generalized plane strain. A comprehensive explanation of theLekhnitskii–Stroh formalism and of its numerous applications can be foundin [28,29,37]. The application of the theoretical framework presented hereto generalized plane-stress problems is straightforward; see [29] for thepertinent conversions of material constants.

11.2.1 Basic equations

Let xi (i = 1, 2, 3) be a Cartesian coordinate system. The constitutive lawof a linear elastic anisotropic material relating the Cartesian componentsof stresses σij and displacements ui has the following form, at small strainsεij = 1

2(ui,j + uj,i):

σij = Cijklεkl = Cijkluk,l, (11.4)

where Cijkl is the positive definite and symmetric fourth-order tensor ofelastic stiffnesses, satisfying the symmetry relations Cijkl = Cjikl = Cklij

and the positivity condition for the strain energy density 12Cijklεijεkl =

12σijεij > 0 for any non-zero εij . Then, the equilibrium equations (in the

absence of body forces) can be written in terms of displacements as

Cijkluk,lj = 0. (11.5)

Under generalized plane-strain conditions the displacement fielddepends only on the plane coordinates x1 and x2, i.e. ui = ui(x1, x2)(i = 1,2,3). Hence, ε33 = 0. In this case (11.5) represents a linearelliptic system of three second-order partial differential equations in twodimensions for the 3D displacement vector field ui. Let a solution of (11.5)be written as a function of a variable z defined by a linear combination of x1

and x2,

ui = aif(z), with z = x1 + px2, (11.6)

where f(z) is an arbitrary analytic function of z, while p and ai areconstants to be determined.



By differentiating the displacements in (11.6) twice with respect to xland xj and substituting in (11.5), the following condition for the numberp and vector a is obtained, taking into account that f(z) is an arbitraryfunction:

[Ci1k1 + p(Ci1k2 + Ci2k1) + p2Ci2k2]ak = 0. (11.7)

Let the 3 × 3 matrices Q, R and T be defined as

Qik = Ci1k1, Rik = Ci1k2, Tik = Ci2k2, (11.8)

then, (11.7) can be written in the matrix form:

[Q + p(R + RT ) + p2T]a = 0. (11.9)

It can be shown that Q and T are symmetric and positive definite [28,29].The components of the stress tensor σi1 and σi2 can be obtained by

substituting the derivatives of the displacements uk,l = akf′(z)(δ1l + pδ2l)

and the definitions in (11.8) into (11.4),

σi1 = (Qik + pRik)akf ′(z), σi2 = (Rki + pTik)akf ′(z). (11.10)

σ33 is determined using the constitutive law and the condition ε33 = 0. Therelations (11.10) can be rewritten as

σi1 = −pbif ′(z), σi2 = bif′(z), (11.11)

by defining

b = (RT + pT)a = −1p(Q + pR)a. (11.12)

By introducing the stress function vector ϕ as

ϕi = bif(z), (11.13)

the stress tensor components in (11.10) can be expressed as

σi1 = −ϕi,2, σi2 = ϕi,1. (11.14)

The homogeneous linear system in (11.9) has a non-trivial solution ifand only if its determinant is zero,

|Q + p(R + RT ) + p2T| = 0. (11.15)

The determinant in (11.15) is a polynomial of six degrees with realcoefficients in a single variable p. The condition for a vanishing determinant



in (11.15) is referred to as the Lekhnitskii–Stroh sextic equation of theanisotropic material in generalized plane strain. This polynomial has sixcomplex roots (three pairs of complex conjugate values), called eigenvalues,pα (α = 1, . . . , 6). These eigenvalues are usually sorted by the sign of theimaginary part of pα as follows (Im denotes the imaginary part and theoverbar the complex conjugate value):

Im pα > 0, pα+3 = pα (α = 1, 2, 3). (11.16)

Let aα denote the eigenvector associated with pα (α = 1, 2, 3) in (11.9).bα is obtained from (11.12). Then,

aα+3 = aα and bα+3 = bα (α = 1, 2, 3). (11.17)

11.2.2 Sextic eigen-relation. Stroh orthogonality

and closure relations

The two equalities in (11.12) can be rewritten in the form:[−Q 03×3

−RT I3×3

] [aαbα

]= pα

[R I3×3

T 03×3

][aαbα

], (11.18)

where I3×3 and 03×3 are the 3× 3 identity and zero matrices, respectively.Multiplying (11.18) by the inverse of the 6× 6 matrix on the right-handside of (11.18) leads to the sextic eigen-relation:[

−T−1RT T−1

RT−1RT − Q −RT−1

][aαbα

]= pα

[aαbα

]⇒ Nξα = pαξα, (11.19)

where N is the 6 × 6 fundamental elasticity matrix [83], ξTα = (aTα ,bTα ) isthe right eigenvector of N (N is non-symmetric) and pα is the associatedeigenvalue.

Equation (11.19) is valid for all α when N has three linearly indepen-dent eigenvectors ξα (α = 1, 2, 3). Then, N is called simple if all the pα aredifferent and semisimple if there are repeated pα but with three independenteigenvectors ξα (α = 1, 2, 3). The associated materials are referred to asnon-degenerate materials.

If N has less than three linearly independent eigenvectors ξα associatedwith pα (α = 1, 2, 3), i.e. the algebraic multiplicity of a repeated eigenvalueis larger than its geometric multiplicity, some expressions of the Strohformalism have to be modified [28,84–86].



When there are two linearly independent eigenvectors ξα associatedwith pα (α=1, 2, 3), N is called non-semisimple and the associatedmaterials are known as degenerate materials. Then, for p1 = p2,

Nξ1 = p1ξ1, Nξ2 = p1ξ2 + ξ1, Nξ3 = p3ξ3, (11.20)

where ξ2 is a generalized eigenvector.When there is only one linearly independent eigenvector ξα associated

with pα (α=1, 2, 3), N is called extraordinary non-semisimple, and theassociated materials are known as extraordinary degenerate materials.Then, for p1 = p2 = p3 = p,

Nξ1 = pξ1, Nξ2 = pξ2 + ξ1, Nξ3 = pξ3 + ξ2, (11.21)

where ξ2 and ξ3 are generalized eigenvectors.For each case, the relations satisfied by the eigenvalues pα and the

associated eigenvectors ξTα = (aTα ,bTα ) can be presented in the followingexplicit form:

— Non-degenerate case (three linearly independent eigenvectors). Forα = 1, 2, 3,

[Q + (R + RT )pα + Tp2α]aα = 0, (11.22)

bα = (RT + pαT)aα = − 1pα

(Q + pαR)aα. (11.23)

— Degenerate case (two linearly independent eigenvectors). For α = 1,3Eqs. (11.22) hold, and for the generalized eigenvector ξT2 = (aT2 ,b

T2 )

with p2 = p1 = p:

— −Q + (R + RT )p+ Tp2a2 = −[2pT + R + RT ]a1,

b2 = Ta1 + [RT + pT]a2.(11.24)

— Extraordinary degenerate case (one linearly independent eigenvector).Denoting p1 = p2 = p3 = p, for (p1, ξ1) Eqs. (11.22) hold with α = 1,for (p2, ξ2) Eqs. (11.24) hold, and for the generalized eigenvector ξT3 =(aT3 ,b

T3 ):

−[Q + (R + RT )p+ Tp2]a3 = −[2pT + R + RT ]a2 − Ta1,

b3 = Ta2 + [RT + pT]a3.(11.25)

See [19,38] for further details of the classification of N.



The eigenvectors or generalized eigenvectors ξTα = (aTα ,bTα) define the

(3 × 3) complex matrices, A = [a1, a2, a3] and B = [b1,b2,b3], whichare employed in representations of the displacement and stress functionvectors.

Inasmuch as N is non-symmetric, the left eigenvector fulfils the relation:

NTη = pη, (11.26)

where η = (bT ,aT ). It is easy to show that the right and left eigenvectorsfor the different eigenvalues are orthogonal:

ηα · ξβ = 0, for pα = pβ . (11.27)

In general the following Stroh orthogonality and closure relations(written in a compact form) can be deduced, after a suitable normalizationof the eigenvectors (in particular, the eigenvectors are normalized accordingto ηα · ξβ = δαβ):

XX−1 = X−1X = I6×6, (11.28)

where

X =

[A A

B B

], X−1 =

[ΓBT ΓAT

ΓBT ΓAT

], (11.29)

I6×6 is the 6 × 6 identity matrix and Γ is expressed as

ΓND =

1 0 00 1 00 0 1

, ΓD =

0 1 01 0 00 0 1

, ΓED =

0 0 10 1 01 0 0

,

(11.30)

with superscripts ND, D and ED referring to the non-degenerate,degenerate and extraordinary degenerate cases, respectively.

Degenerate cases appear, strictly speaking, when a particularcombination of the elastic stiffnesses of the material leads to theLekhnitskii–Stroh sextic equation (11.15) with repeated roots (eigenvalues)whose algebraic multiplicity is larger than its geometric multiplicity, see(11.9). Thus, from an engineering point of view, the degenerate casesmight be considered as very particular cases, and any degenerate casecan be obtained as a limit of non-degenerate cases with respect to acontinuous variation of the elastic stiffnesses. However, it should be stressedthat in such a limit procedure the behaviour of the eigenvectors ξα,



and correspondingly of the matrices A and B, can be discontinuous, inparticular their magnitudes can become infinite.

For composite materials reinforced with long fibres, which typicallybehave as transversely isotropic materials, the particular spatial orienta-tions of these materials can lead to a non-semisimple N, cf. [25,37,87].The fact that a spatial orientation of some materials, irrespective of thevalues of the elastic stiffnesses, can lead to a non-semisimple N, makes theanalysis of degenerate cases relevant for applications of the Lekhnitskii–Stroh formalism to composite materials.

11.2.3 Representation of displacement and stress

function vectors

The displacement and stress function vector solution of an anisotropicelastic problem under generalized plane-strain conditions can be expressedas a linear combination of terms written in compact (sextic) form as follows[19,28,84–86]:

w(x1, x2) = X

[F(x1, x2) 03×3

03×3 F(x1, x2)

]v, with

w =

[u

ϕ

]and v =

[q

q

], (11.31)

where q= (q1, q2, q3)T and q=(q1, q2, q3)T are in general 3 × 1 constantvectors with real or complex components, and the elements of the 3 × 3matrices F and F are defined by a complex analytic function f of complexvariables zα = x1 + pαx2 and zα = x1 + pαx2, and by its first and secondderivatives f ′ and f ′′. The structure of F and F depends on the number oflinearly independent eigenvectors ξα, as follows,

— Non-degenerate case:

F =

f(z1) 0 0

0 f(z2) 0

0 0 f(z3)

, F =

f(z1) 0 0

0 f(z2) 0

0 0 f(z3)

. (11.32)



— Degenerate case (z1 = z2):

F =

f(z1) x2f

′(z1) 0

0 f(z1) 0

0 0 f(z3)

, F =

f(z1) x2f

′(z1) 0

0 f(z1) 0

0 0 f(z3)

.(11.33)

— Extraordinary degenerate case (z1 = z2 = z3 = z):

F =

f(z) x2f

′(z)12x2

2f′′(z)

0 f(z) x2f′(z)

0 0 f(z)

, F =

f(z) x2f

′(z)12x2

2f′′(z)

0 f(z) x2f′(z)

0 0 f(z)

.

(11.34)

Finally, in view of (11.14), the traction vector t at a point (x1,x2) ona contour, with the unit normal vector n = (n1,n2) to this contour at(x1,x2), can be computed by the tangential derivative of ϕ with respect tothis contour,

t(x1, x2) = −∂ϕ∂s

(x1, x2), (11.35)

where s = (−n2, n1) is the unit tangential vector to this contour. Thus, azero traction t along a contour corresponds to a constant (possibly zero)stress function vector ϕ along this contour.

11.3 Elastic Multimaterial Corner

11.3.1 Corner configuration

Consider an elastic anisotropic multimaterial corner composed of a finitenumber of single-material wedges with plane faces intersecting at astraight corner edge coincident with the x3-axis of the Cartesian andcylindrical coordinate systems (x1, x2, x3) and (r, θ, x3), respectively. Letthe corner be subjected to a generalized plane-strain state with zero bodyforces and constant interface conditions at the plane interfaces betweenthe single-material wedges. Perfect bonding (traction equilibrium anddisplacement continuity), friction and frictionless sliding are considered



at these interfaces. We distinguish between an open corner with twoouter plane boundary faces, intersecting at the corner edge, where eitherconstant homogeneous orthogonal or sliding friction boundary conditionsare prescribed, and a closed corner (also called a periodic corner) withno such outer boundary faces. A homogeneous orthogonal boundarycondition represents a general case of either displacement, traction or mixedhomogeneous boundary conditions where the displacement and tractionvectors are perpendicular to each other (including the case where one ofthem is zero), cf. [12,19,28,71,88].

Typical 2D and 3D representations of an open corner are shown inFigs. 11.1 and 11.2. Notice that the corner tip in Fig. 11.1 is a 2D view of thecorner edge in Fig. 11.2. Let the corner contain M (M ≥ 1) single-materialwedges (in the following also referred to as materials), where material Mm

(m = 1, . . . ,M) is defined by angles θ in the angular sector θm−1 < θ < θm.In an open corner, the boundary conditions are defined at angles θ0 andθM , where 0 < θM − θ0 ≤ 360 and the interface conditions at θm(m =1, . . . ,M − 1). In a closed corner, the interface conditions are prescribed atθm(m = 0, . . . ,M), where θM − θ0 = 360.

Let the sequence of all materials in the corner be partitioned tosubsequences of the maximum length of consecutive materials with perfectbonding at the common interfaces. The materials in such a subsequenceare grouped together and referred to as a wedge. The number of wedges ina corner is W ≥ 1 (W ≤ M), and the sequence of wedges is indexed bysubscript w (w = 1, . . . ,W ), see Fig. 11.1. Wedge w is defined by angles θin the angular sector ϑw−1 < θ < ϑw. Thus, ϑ0 = θ0 and ϑW = θM . Asshown in Section 11.5.1, the fact that materials in a wedge are perfectlybonded together allows us to work advantageously with a wedge as if it werea single entity, by defining a transfer matrix for the whole wedge. It followsfrom the wedge definition that friction or frictionless sliding is prescribed atthe interface between two consecutive wedges. If there is no such interfacein the corner the whole multimaterial corner is considered as a wedge, i.e.W = 1.

The present friction model can be described as a rate-independent dryCoulomb friction model with a linear variation of the limit shear tractionwith respect to the normal traction. Surface topography and texture canbe isotropic or anisotropic. This sometimes requires a generalization ofthe standard isotropic friction model. Suitable anisotropic friction modelswith either an associated sliding rule (given by a version of the maximumdissipation principle) or a non-associated sliding rule have been studied in



[89–92], see also [93]. Without loss of generality, we assume that suitable,physically based or experimentally determined, functions defining the angleof the sliding velocity ωu and kinetic (or dynamic) friction coefficient µ interms of the angle of the frictional shear traction ω are given at each frictionsurface by:

ωu = ωu(ω) − sliding rule,

µ = µ(ω) > 0 − friction rule, (11.36)

where both angles, ωu and ω, are measured with respect to the coordinatesystem in the wedge face.

Functions ωu(ω) and µ(ω) are periodic with period 2π. In the caseof an associated sliding rule, the polar diagram of the directionallydependent friction coefficient µ(ω) is typically given by an ellipsewith the following relationship for the sliding angle tan(ωu − α) =(µ(0 + α)/µ(90 + α))2 tan(ω−α), where µ(0 +α) and µ(90 +α) definethe major and minor semi-axes of this ellipse rotated by an angle α withrespect to the coordinate system. In the usual isotropic friction modelωu = ω and µ is a constant independent of ω, the ellipse being replaced bya circle.

11.3.2 Boundary and interface conditions.

Matrix formalism

In this section, a powerful matrix formalism for a compact representationof different boundary and interface conditions in the multimaterial corner,suitable for an efficient computer implementation, is introduced.

11.3.2.1 Coordinate systems

The following orthonormal vector basis attached to the wedge faces, cf.[19,71], is employed to define the homogeneous orthogonal boundary andinterface conditions:

(sr(ϑ), s3,n(ϑ)), (11.37)

with the Cartesian components of these vectors defined as

sr(ϑ) =

− cosϑ

− sinϑ0

, s3 =

0

01

, n(ϑ) =

− sinϑ

cosϑ0

, (11.38)



where ϑ is the wedge face angle. Additionally, when defining the slidingfriction condition, the following three orthonormal vector bases attached tothe wedge faces are employed:

(k(ϑ, ω),m(ϑ, ω),n(ϑ)), (11.39)

(k(ϑ, ωu),m(ϑ, ωu),n(ϑ)), (11.40)

(nµ(ϑ, ω, µ), sµ(ϑ, ω, µ),m(ϑ, ω)), (11.41)

with

k(ϑ, ω) = cosωsr(ϑ) + sinωs3, (11.42)

m(ϑ, ω) = − sinωsr(ϑ) + cosωs3, (11.43)

nµ(ϑ, ω, µ) =n(ϑ) + µk(ϑ, ω)√

1 + µ2, (11.44)

sµ(ϑ, ω, µ) =k(ϑ, ω) − µn(ϑ)√

1 + µ2, (11.45)

where ω and ωu, respectively, are the angles measured from the vector sr(ϑ)and are the directions of the friction shear and sliding in the wedge face,and µ is the corresponding kinetic (or dynamic) friction coefficient at thewedge face. Notice that, in the isotropic friction model the vector bases in(11.39) and (11.40) coincide. Subscript w has been omitted in the abovevector definitions for the sake of simplicity.

11.3.2.2 Boundary condition matrices

The usual homogeneous orthogonal boundary conditions for the first andlast corner faces, w = 0 andW , can be expressed formally, in view of (11.35)and assuming a zero stress function vector ϕ at the corner edge (r = 0), bya linear relation for r >0:

Du(ϑw)u(r, ϑw) + Dϕ(ϑw)ϕ(r, ϑw) = 0, (11.46)

where Du(ϑw) and Dϕ(ϑw) are 3 × 3 real matrices defined in Table 11.1,fulfilling the following orthogonality relations, cf. [19, 71]:

Du(ϑw)DTϕ(ϑw) = Dϕ(ϑw)DT

u (ϑw) = 0, (11.47)

with superscript T denoting the transpose.



Table 11.1. Boundary condition matrices Du and Dϕ for homogeneousorthogonal boundary conditions, with ϑ = ϑw for w = 0 and W .

Matrix definition

Boundary condition Du(ϑ) Dϕ(ϑ)

Free 03×3 I3×3

Fixed I3×3 03×3

Symmetry (only uθ restricted) [n(ϑ), 0,0]T [0, sr(ϑ), s3]T

Antisymmetry (only uθ allowed) [sr(ϑ), s3,0]T [0,0,n(ϑ)]T

Only ur restricted [sr(ϑ), 0,0]T [0,n(ϑ), s3]T

Only ur allowed [n(ϑ), s3,0]T [0,0, sr(ϑ)]T

Only u3 restricted [s3, 0,0]T [0, sr(ϑ), n(ϑ)]T

Only u3 allowed [sr(ϑ), n(ϑ), 0]T [0,0, s3]T

Table 11.2. Boundary condition matrices Du, Dϕ, Du and Dϕ for Coulomb slidingfriction, with ϑ = ϑw, ω = ωw and ωu = ωu

w(ωw) and µ = µw(ωw) for w =0 and W .

Du(ϑ, ω, ωu) = [n(ϑ), 0,m(ϑ, ωu),0]T Dϕ(ϑ, ω, µ) = [0, sµ(ϑ, ω, µ), 0,m(ϑ, ω)]T

Du(ϑ, ω, ωu) = [0,k(ϑ, ωu)]T Dϕ(ϑ, ω, µ) = [nµ(ϑ, ω, µ), 0]T

In general, the boundary conditions of friction contact are non-linear,because of the unilateral Signorini conditions of impenetrability and non-adhesion, and the Amontons–Coulomb law for dry friction. Nevertheless,under the present hypothesis that the whole face is sliding in the samedirection, i.e. ω, ωu and µ are constant at the whole face, and assuming amonotonic loading from the unloaded state, the sliding friction boundarycondition at the first and last corner faces, w = 0 and W , can also beexpressed formally by a linear relation, cf. [71]:

Du(ϑw, ωw, ωuw(ωw))u(r, ϑw) + Dϕ(ϑw, ωw, µw(ωw))ϕ(r, ϑw) = 0,

(11.48)

where Du(ϑw, ωw, ωuw(ωw)) and Dϕ(ϑw, ωw, µw(ωw)) are 4×3 real matricesdefined in Table 11.2. These matrices, however, do not fulfil orthogonalityrelations similar to (11.47). The fact that these matrices are rectangular,instead of the square matrices used in (11.46), implying four boundaryconditions instead of three as would be expected, is associated with thefact that a certain direction of shear traction (or equivalently the slidingdirection) is assumed here, although in general it is unknown. Recall, that



the friction coefficient µ has only positive values. The fulfilment of thecompression condition (the normal stresses are negative or vanish) and thedissipative character of friction (the friction shear stress is exerted in adirection that opposes sliding) should be checked at the wedge face afterthe problem is solved.

Notice that the boundary condition of frictionless sliding is referred toin Table 11.1 as a symmetry boundary condition, which requires a checkafter the problem is solved to ensure that the compression condition (thenormal stresses are negative or vanish) is fulfilled at the wedge face.

In the following, a 6 × 6 real matrix DBC including all the previouslydefined boundary condition matrices, is introduced and applied to partitionthe 6 × 1 vector w defined in (11.31) into the prescribed and unknownsubvectors. We refer to DBC as the main boundary-condition-matrix. DBC

is defined for homogeneous orthogonal boundary conditions and a slidingfriction boundary condition, for a wedge face of angle ϑw, for w = 0 andW , respectively, as

DBC(ϑw) =

[Du(ϑw) Dϕ(ϑw)

Du(ϑw) Dϕ(ϑw)

]=

[Du(ϑw) Dϕ(ϑw)

Dϕ(ϑw) Du(ϑw)

](11.49)

and

DBC(ϑw, ωw, µw(ωw), ωuw(ωw)) =

[Du(ϑw, ωw, ωuw) Dϕ(ϑw, ωw, µw)

Du(ϑw, ωw, ωuw) Dϕ(ϑw, ωw, µw)

],

(11.50)

where the 2×3 real matrices Du(ϑw, ωw, ωuw) and Dϕ(ϑw, ωw, µw) are givenin Table 11.2. It is straightforward to check that DBC is an orthogonalmatrix, i.e.

DBCDTBC = DT

BCDBC = I6×6, (11.51)

where I6×6 is the 6×6 identity matrix. Evidently ωw, µw and ωuwcan be different for w = 0 and W . As shown in Section 11.5.2, thegeneral orthogonality relation (11.51) is very useful in the application ofdifferent boundary conditions in the corner singularity analysis. Althoughan orthogonality relation analogous to (11.51) obtained for orthogonalboundary conditions can be found in [28] (Section 14.1) and [88] in a slightlydifferent context, the orthogonality relation (11.51) appears to be new forCoulomb sliding friction for both isotropic and anisotropic friction models.



It is easy to show that if the vector w(r, ϑw) in (11.31) is multipliedfrom the left by the matrix DBC , given either by (11.49) or (11.50), theprescribed and the unknown components of w(r, ϑw) appear grouped intwo separate blocks, wP (r, ϑw) and wU (r, ϑw), respectively,

DBCw(r, ϑw) =

[wP (r, ϑw)

wU (r, ϑw)

]. (11.52)

As follows from the above relations, wP (r, ϑw) = 0 and wU (r, ϑw) are3 × 1 vectors for orthogonal boundary conditions, whereas they are 4 × 1and 2 × 1 vectors, respectively, for friction sliding. From the orthogonalityrelation (11.51) and the fact that wP (r, ϑw) is a zero vector, it is obtained,for w = 0 and W , that

w(r, ϑw) = DTBC

[wP (r, ϑw)

wU (r, ϑw)

]=

[DTu

DTϕ

]wU (r, ϑw) = DT

BCwU (r, ϑw).

(11.53)Inasmuch as the order of the prescribed and unknown variables is not

relevant, the definition of the main boundary-condition matrix DBC is notunique, e.g. equivalent definitions can be obtained by permutations of rowscorresponding to wP (r, ϑw) and wU (r, ϑw) separately.

11.3.2.3 Interface condition matrices

Similarly as in (11.46) and (11.48), the interface conditions between thewedges, for frictionless or frictional sliding, respectively, can be expressed,using (11.35) and assuming a zero stress function vector ϕ at the corneredge (r = 0), formally by linear relations for r > 0 and 1 ≤ w ≤W − 1:

D1(ϑw)ww(r, ϑw) + D2(ϑw)ww+1(r, ϑw) = 0, (11.54)

and

D1(ϑw, ωw, µw(ωw), ωuw(ωw))ww(r, ϑw)

+D2(ϑw, ωw, µw(ωw), ωuw(ωw))ww+1(r, ϑw) = 0, (11.55)

where Di(ϑw) and Di(ϑw, ωw, µw, ωuw) (i = 1,2), respectively, are real6 × 6 and 7 × 6 matrices defined in Tables 11.3 and 11.4, ww(r, ϑw) andww+1(r, ϑw), respectively, are 6 × 1 vectors of displacements and stressfunctions (11.31) associated with the wedges of number w and w + 1 anddefined at the wedge interface given by the angle ϑw.



Table 11.3. Interface condition matrices D1, D2, D1 and D2

for frictionless sliding, with ϑ = ϑw and 1 ≤ w ≤ W − 1.

D1(ϑ) = 1√2

"−n(ϑ) 03×1 03×3 03×1

03×1 sr(ϑ) −I3×3 s3

#T

D2(ϑ) = 1√2

"n(ϑ) 03×1 03×3 03×1

03×1 sr(ϑ) I3×3 s3

#T

D1(ϑ) = 1√2

"n(ϑ) 03×1

√2sr(ϑ) 03×1

√2s3 03×1

03×1 n(ϑ) 03×1 03×1 03×1 03×1

#T

D2(ϑ) = 1√2

"n(ϑ) 03×1 03×1

√2sr(ϑ) 03×1

√2s3

03×1 n(ϑ) 03×1 03×1 03×1 03×1

#T

Table 11.4. Interface condition matrices D1, D2, D1 and D2 for slidingfriction, with ϑ = ϑw, ω = ωw, ωu = ωu

w(ωw), µ = µw(ωw) and 1 ≤ w ≤ W−1.

D1(ϑ, ω, µ, ωu) = 1√2

"−n(ϑ) 03×1 03×3 −m(ϑ, ωu) 03×1

03×1 sµ(ϑ, ω, µ) −I3×3 03×1 m(ϑ, ω)

#T

D2(ϑ, ω, µ, ωu) = 1√2

"n(ϑ) 03×1 03×3 m(ϑ, ωu) 03×1

03×1 sµ(ϑ, ω, µ) I3×3 03×1 m(ϑ, ω)

#T

D1(ϑ, ω, µ, ωu) = 1√2

"n(ϑ) 03×1

√2k(ϑ, ωu) 03×1 m(ϑ, ωu)

03×1 nµ(ϑ, ω, µ) 03×1 03×1 03×1

#T

D2(ϑ, ω, µ, ωu) = 1√2

"n(ϑ) 03×1 03×1

√2k(ϑ, ωu) m(ϑ, ωu)

03×1 nµ(ϑ, ω, µ) 03×1 03×1 03×1

#T

A matrix analogous to the matrix DBC in the case of boundaryconditions is defined and applied for interface conditions as well. A 12× 12real matrix DI is defined for the frictionless and frictional sliding interfaceconditions, respectively, as

DI(ϑw) =

[D1(ϑw) D2(ϑw)

D1(ϑw) D2(ϑw)

](11.56)

and

DI(ϑw, ωw, µw(ωw), ωuw(ωw))

=

[D1(ϑw, ωw, µw, ωuw) D2(ϑw , ωw, µw, ωuw)

D1(ϑw, ωw, µw, ωuw) D2(ϑw , ωw, µw, ωuw)

], (11.57)



where Di(ϑw) and Di(ϑw, ωw, µw(ωw), ωuw(ωw)) (i = 1, 2), respectively, are6×6 and 5×6 real matrices, defined in Tables 11.3 and 11.4. We refer to DI

as the main interface-condition matrix. We can check by direct evaluationthat DI is an orthogonal matrix, i.e.

DIDTI = DT

I DI = I12×12, (11.58)

where I12×12 is the 12 × 12 identity matrix. This general orthogonalityrelation will be useful in Section 11.5.2 when applying interface conditionsin the corner singularity analysis.

Similarly, as for the boundary conditions, the fact that the matricesin (11.55) are rectangular, instead of the square matrices used in (11.54),in particular implying seven interface conditions instead of six as would beexpected, is associated with the fact that some direction of friction sheartraction (or equivalently sliding direction) is assumed here, although ingeneral it is unknown. The fulfilment of the compression condition and thedissipative character of friction should be checked at the interface after theproblem is solved.

The frictionless sliding condition also requires a check after the problemis solved to ensure the compression condition is fulfilled at the interface.

By multiplying the 12 × 1 vector (wTw(r, ϑw),wT

w+1(r, ϑw))T from theleft by the matrix DI , given either by (11.56) or (11.57), the prescribedand the unknown components of w(r, ϑw) appear grouped in two separateblocks, wP (r, ϑw) and wU(r, ϑw), respectively,

DI

[ww(r, ϑw)

ww+1(r, ϑw)

]=[wP (r, ϑw)wU (r, ϑw)

]. (11.59)

As follows from the above relations, wP (r, ϑw) = 0 and wU (r, ϑw) are 6×1vectors for frictionless sliding, whereas they are 7 × 1 and 5 × 1 vectors,respectively, for sliding friction. From the orthogonality relation (11.58) andthe fact that wP (r, ϑw) is a zero vector, we obtain, for 1 ≤ w ≤W −1, that

[ww(r, ϑw)

ww+1(r, ϑw)

]= DT

I

[wP (r, ϑw)wU (r, ϑw)

]=

[DT

1

DT2

]wU (r, ϑw). (11.60)

Similarly, as for the boundary conditions, the definition of the maininterface-condition matrix DI is not unique, e.g. equivalent definitionscan be obtained by permutations of rows corresponding to wP (r, ϑw) andwU (r, ϑw) separately.



In the case of a closed corner, ϑW = ϑ0 and wW+1(r, ϑW ) shouldbe replaced by w1(r, ϑ0) in the relations (11.59) and (11.60), written forw = W . This is why a closed corner is sometimes referred to as periodiccorner.

Finally, it should be mentioned that for a genuine 2D problem, wherethe in-plane-strain state can be uncoupled from the antiplane-strain state,the above vectors and matrices are correspondingly reduced. In particular,a significant reduction takes place in the sliding friction case, where asomewhat unpleasant feature with more boundary or interface conditionsthan usual, and consequently with rectangular boundary or interfacecondition matrices, is removed, as we know a priori that the sliding takesplace in the x1−x2 plane, thus ω = ωu and ω equals either 0 or ±180.Subsequently the vectors s3 and m and the angles ω and ωu essentiallydisappear from the formulation, cf. [71].

11.4 Singular Elastic Solution in a Single-Material Wedge.Transfer Matrix

The particular geometrical configuration of the multimaterial corner shownin Fig. 11.1 allows us to take advantage of the transfer matrix conceptsimilarly as in [16, 19]. The idea is explained briefly in the following. Letus assume a particular kind of elastic state in a single-material wedge Mm

(defined by the angular sector between θm−1 and θm), allowing separationof the variables in polar coordinates with the same radial dependence inall components of the displacement and stress function vectors. Then, wecan relate the displacement and stress function vectors at both outer radialfaces of the wedge, wm(r, θm−1) and wm(r, θm), by a transfer matrix Em.It should be stressed that such a transfer matrix depends on the kind of theelastic state considered. If we can enforce continuity of the displacement andstress function vectors across interfaces between several perfectly bondedsingle-material wedges we arrive at a transfer matrix for the whole sequenceof bonded wedges simply by sequentially multiplying the transfer matricesof all the single-material wedges in the sequence.

With reference to the representation of the displacement and stressfunction vector in (11.31) for the analysis of problems with stresssingularities, in this section we will assume the following simple form forthe complex analytic functions, in view of (11.2) and (11.3):

f(zα) = zλα and f(zα) = zλα (α = 1, 2, 3), (11.61)



where λ is a real or complex characteristic exponent. The complex variableszα and zα in (11.61) can be expressed as

zα = x1 + pαx2 = r(cos θ + pα sin θ) = rζα(θ),zα = x1 + pαx2 = r(cos θ + pα sin θ) = rζα(θ),

(11.62)

considering the polar coordinate system (r, θ) centred at the corner tip(r=0).

11.4.1 Non-degenerate materials

Let us assume that the material in a single-material wedge Mm is non-degenerate, see Section 11.2. Then, by substituting (11.61) into (11.31) andtaking into account (11.32), the displacement and stress function vectorfields in this wedge can be written in the following compact form:

w(r, θ) = rλXZλ(θ)v, w(r, θ) =[u(r, θ)ϕ(r, θ)

], v =

[qq

], (11.63)

where X is defined in (11.29), and Zλ(θ) is a diagonal matrix

Zλ(θ) =[ 〈ζλ∗ (θ)〉 03×3

03×3 〈ζλ∗ (θ)〉], (11.64)

and

〈ζλ∗ (θ)〉 = diag[ζλ1 (θ), ζλ2 (θ), ζλ3 (θ)],

〈ζλ∗ (θ)〉 = diag[ζλ1 (θ), ζλ2 (θ), ζλ3 (θ)],

with

ζλα(θ) = (cos θ + pα sin θ)λ and ζλα(θ) = (cos θ + pα sin θ)λ.

Thus, F and F in (11.32) are expressed as

F(x1, x2) = rλ⟨ζλ∗ (θ)

⟩, F(x1, x2) = rλ

⟨ζλ∗ (θ)

⟩. (11.65)

According to (11.63), if 0<λ< 1 (or 0<Re(λ)< 1, if λ is a complexnumber), the associated stresses become singular at the origin ofcoordinates, i.e. they may become unbounded for r → 0+. If λ is a realnumber, then q is the complex conjugate of q, and u and ϕ are also realfunctions. If λ is a complex number, then q is not necessarily the complexconjugate of q, and u and ϕ are also complex functions. It can be deducedthat if λ is a solution, then λ is also a solution, and the superposition of thesolutions (11.63) for λ and λ leads to real-valued expressions of u and ϕ.



If (11.63) is evaluated for the single-material wedge Mm at θ = θm−1

and θ = θm, and v is eliminated, we obtain:

wm(r, θm−1) =rλXZλ(θm−1)vwm(r, θm) =rλXZλ(θm)v

⇒wm(r, θm)=Em(λ, θm, θm−1)wm(r, θm−1),

(11.66)where

Em(λ, θm, θm−1) = XZλ(θm)[Zλ(θm−1)]−1X−1. (11.67)

The 3 × 3 complex matrix Em(λ, θm, θm−1), referred to as the transfermatrix for the single-material wedge Mm, depends on the wedge materialproperties, through the matrix X and its inverse X−1 defined in (11.29)and the eigenvalues pα, on the wedge geometry, given by the angles θm−1

and θm, and on the characteristic exponent λ. An explicit expression forZλ(θm)[Zλ(θm−1)]−1 is obtained following [16, 94],

Zλ(θm)[Zλ(θm−1)]−1 = Zλ(θm, θm−1)

=

[〈ζλ∗ (θm, θm−1)〉 03×3

03×3 〈ζλ∗ (θm, θm−1)〉

], (11.68)

where

〈ζλ∗ (θm, θm−1)〉 = diag[ζλ1 (θm, θm−1), ζλ2 (θm, θm−1), ζλ3 (θm, θm−1)],

(11.69)

with

ζα(θm, θm−1) =ζα(θm)ζα(θm−1)

= cos(θm − θm−1) + pα(θm−1) sin(θm − θm−1),

(11.70)

and

pα(θm−1) =pα cos(θm−1) − sin(θm−1)pα sin(θm−1) + cos(θm−1)

. (11.71)

11.4.2 Degenerate materials

Let us assume that the material in a single-material wedge Mm isdegenerate, see Section 11.2. Then, by substituting (11.61) for (α = 1, 3)into (11.31) and taking into account (11.33), the displacement and stress



function vector fields in this wedge can be written in the following compactform:

w(r, θ) = rλXZ(θ, λ)v, w(r, θ) =[u(r, θ)ϕ(r, θ)

], v =

[qq

], (11.72)

where X is defined in (11.29), and Z(θ, λ) is defined as

Z(θ, λ) =[Ψ(p∗, θ, λ) 03×3

03×3 Ψ(p∗, θ, λ)

], (11.73)

with

Ψ(p∗, θ, λ) =

ζλ1 (θ) K(p1, θ, λ)ζλ1 (θ) 0

0 ζλ1 (θ) 00 0 ζλ3 (θ)

,

K(p1, θ, λ) =λ sin(θ)ζ1(θ)

. (11.74)

Now, by applying a procedure similar to that of the previous section,first evaluating (11.72) at θ = θm−1 and θ = θm, and then eliminating v,we again arrive at

wm(r, θm) = Em (λ, θm, θm−1)w(r, θm−1), (11.75)

where

Em(λ, θm, θm−1) = XZ(θm, λ)[Z(θm−1, λ)]−1X−1 (11.76)

is the transfer matrix for the single-material wedge Mm with a degeneratematerial. According to [19]

Z(θm, λ)[Z(θm−1, λ)]−1 =

[Ψ(p∗, θm, θm−1, λ) 03×3

03×3 Ψ(p∗, θm, θm−1, λ)

],

(11.77)

where

Ψ(p∗, θm, θm−1, λ)

=

ζλ1 (θm, θm−1) K(p1, θm, θm−1, λ)ζλ1 (θm, θm−1) 0

0 ζλ1 (θm, θm−1) 0

0 0 ζλ3 (θm, θm−1)

,

(11.78)



with ζα(θm, θm−1) defined in (11.70) and

K(p1, θm, θm−1, λ) =λ sin(θm − θm−1)ζ1(θm)ζ1(θm−1)

. (11.79)

Isotropic materials are a typical example of degenerate anisotropicmaterials with a triple eigenvalue p = i =

√−1 and two linearlyindependent eigenvectors. All the above expressions for these materialssimplify due to the fact that ζα(θ) = cos θ + i sin θ = eiθ. In particular,ζα(θm, θm−1) in (11.70) and K in (11.79) can be rewritten as

ζα(θm, θm−1) = ei (θm−θm−1), and K(i, θm, θm−1, λ)

=λ sin(θm − θm−1)ei(θm+θm−1)

. (11.80)

Expressions for the complex matrices A and B for an isotropic elasticmaterial in [19, 28] define the complex matrix X in (11.29).

11.4.3 Extraordinary degenerate materials

Let us assume that the material in a single-material wedge Mm isextraordinary degenerate, see Section 11.2. Then, by substituting (11.61)with pα = p, zα = z and ζα(θ) = ζ(θ) = cos θ + p sin θ, into (11.31) andtaking into account (11.34), the displacement and stress function vectorfields in this wedge can be written in the form (11.72), where,

Z(θ, λ) =[Ψ(p, θ, λ) 03×3

03×3 Ψ(p, θ, λ)

], (11.81)

with

Ψ(p, θ, λ) = ζλ(θ)

1 K(p, θ, λ)12(1 − λ−1)K2(p, θ, λ)

0 1 K(p, θ, λ)0 0 1

,

K(p, θ, λ) =λ sin(θ)ζ(θ)

. (11.82)

By applying a procedure similar to that of the previous sections, wearrive at transfer expressions in the same form as in (11.75) and (11.76),



where, according to [19],

Z(θm, λ)[Z(θm−1, λ)]−1 =

[Ψ(p, θm, θm−1, λ) 03×3

03×3 Ψ(p, θm, θm−1, λ)

],

(11.83)

Ψ(p, θm, θm−1, λ) = ζλ(θm, θm−1)

×

1 K(p, θm, θm−1, λ) K(p, θm, θm−1, λ)Z(p, θm, θm−1, λ)0 1 K(p, θm, θm−1, λ)0 0 1

,(11.84)

and ζ(θm, θm−1) is defined similarly as in (11.70), K(p, θm, θm−1, λ) isdefined similarly as in (11.79), and Z(p, θm, θm−1, λ) is defined as

Z(p, θm, θm−1, λ) =12

(K(p, θm, θm−1, λ) − sin θm−1

ζ(θm−1)− sin θmζ(θm)

). (11.85)

11.5 Characteristic System for the Singularity Analysisof an Elastic Multimaterial Corner

Closed-form expressions of the transfer matrix for a single-material wedgemade of any anisotropic linear elastic material were obtained in Section 11.4by assuming a simple power law form f(zα) = zλα for the complex analyticfunctions appearing in a general representation of any elastic solution inthe Lekhnitskii–Stroh formalism (11.31). In the present section this transfermatrix is used first to generate the transfer matrix of a sequence of perfectlybonded single-material wedges and then to assemble the correspondingcharacteristic system (also called an eigensystem) of the multimaterialcorner with some boundary and/or interface conditions. The solution ofthis characteristic system gives the characteristic exponents of the cornerproblem and characteristic angular functions defining the singular elasticstate in the corner. The formulation is quite general considering any finitenumber of single-material wedges bonded or with friction or frictionlesscontact between them, and any homogeneous orthogonal or contactboundary conditions for an open corner. Isotropic and also anisotropicfriction contact conditions may be considered at the contact faces. Thepowerful matrix formalism introduced in the previous sections allows us towrite general expressions in a compact form suitable for a straightforwardcomputer implementation.



11.5.1 Transfer matrix for a multimaterial wedge

Let a multimaterial wedge w (w = 1, . . . ,W ), see Fig. 11.1, be definedby a sequence of perfectly bonded single-material wedges with indices m =iw, iw+1, . . . , jw−1, jw, where iw and jw are the indices of the first and lastmaterial in the wedge w. Notice that θiw−1 = ϑw−1 and θjw = ϑw. Then,a wedge transfer matrix can be defined as follows. Using the continuityconditions corresponding to the hypothesis of perfect bonding between thematerials wm(r, θm) = wm+1(r, θm) (iw ≤ m < m+ 1 ≤ jw), and the 6× 6transfer matrix Em(λ) for each single-material wedge, it is easy to arrive atthe transfer relation for the wedge w, relating the elastic variables betweenthe wedge external faces at angles ϑw−1 and ϑw:

ww(r, ϑw) = Kw(λ)ww(r, ϑw−1)

or [uw(r, ϑw)

ϕw(r, ϑw)

]=

[K(1)w (λ) K(2)

w (λ)

K(3)w (λ) K(4)

w (λ)

][uw(r, ϑw−1)

ϕw(r, ϑw−1)

], (11.86)

where the expression for the 6× 6 wedge transfer matrix Kw is obtainedby the sequential product of the transfer matrices Em(λ) of all the single-material wedges in the multimaterial wedge w,

Kw(λ) = Ejw (λ) · Ejw−1(λ) · · · Eiw+1(λ) ·Eiw (λ). (11.87)

The transfer relation (11.86) can be rewritten in the following matrixform suitable for the easy assembly of the characteristic system of amultimaterial corner:

[Kw(λ) − I6×6]

[ww(r, ϑw−1)

ww(r, ϑw)

]= 06×1. (11.88)

11.5.2 Characteristic system assembly

The following linear system collects all the wedge transfer relations (11.88)for the corner,

Kcorner ext.(λ)wcorner ext. = 06W×1, (11.89)



where the 6W × 12W extended complex matrix of transfer relations of themultimaterial corner is defined as

Kcorner ext.(λ)

=

K1(λ) −I6×6 06×6 06×6 · · · · · · 06×6

06×6 06×6 K2(λ) −I6×6 · · · · · · 06×6

......

......

. . ....

...06×6 06×6 06×6 06×6 · · · KW (λ) −I6×6

(11.90)

and the 12W × 1 vector of elastic variables at wedge faces in the corner isdefined as

wcorner ext. =

w1(r, ϑ0)w1(r, ϑ1)w2(r, ϑ1)w2(r, ϑ2)

...wW (r, ϑW−1)wW (r, ϑW )

. (11.91)

Let ϑ = (ϑ0, ϑ1, . . . , ϑW−1, ϑW ) define the vector of polar angles ofwedge faces in the whole multimaterial corner. Let the sliding frictioncondition be prescribed at F (0 ≤ F ≤W +1) boundary faces or interfaces(between wedges) whose polar angles are given by the sequence ϑki (i =1, . . . , F ) and 0 ≤ k1 < k2 < · · · < kF−1 < kF ≤ W . The correspondingfunctions defining the friction coefficients µki(ωki) and the angles of slidingdirection ωuki

(ωki) in terms of a priori unknown angles of frictional sheartraction ωki are gathered in the following vectors of the assumed functionsµ = (µk1 , µk2 , . . . , µkF−1 , µkF ) and ωu = (ωuk1 , ω

uk2, . . . , ωukF−1

, ωukF).

Additionally, we define the vector of unknown values of the angles offrictional shear traction ω = (ωk1 , ωk2 , . . . , ωkF−1 , ωkF ). In the followingexpressions, for the simple case of isotropic friction, the vector ωu canbe omitted as the angles of friction shear stress and sliding anglescoincide and the vector µ represents just friction coefficient valuesat each contact surface, independent of the angles of friction shearstress.



11.5.2.1 Open multimaterial corner

The following 12W × 12W extended matrix of boundary and interfaceconditions for an open multimaterial corner collects all the matrices forthe boundary and interface conditions of the multimaterial corner in a waythat it is compatible with the definition of the vector wcorner ext. in (11.91):

Dcorner ext.(ϑϑϑ,ωωω,µµµ,ωωωu) = blocked diag[DBC(ϑ0),DI (ϑ1), . . . ,DBC(ϑW )]

=

DBC(ϑ0) 06×12 06×12 · · · 06×6

012×6 DI (ϑ1) 012×12 · · · 012×6

......

.... . .

...06×6 06×12 06×12 · · · DBC(ϑW )

. (11.92)

Notice that some or all of the boundary or interface condition matrices,DBC or DI , included in the definition of Dcorner ext.(ϑϑϑ,ωωω,µµµ,ωωωu) mayadditionally depend on the values of the angle of friction shear stress, andon the assumed functions for the friction coefficient and for the angle ofsliding at the pertinent boundary surface or interface, which have not beenexplicitly indicated in the right-hand side of (11.92) for the sake of notationsimplicity. When there are no sliding friction conditions in the corner, ω,µµµ and ωωωu are omitted in (11.92). The matrix Dcorner ext.(ϑϑϑ,ωωω,µµµ,ωωωu) isorthogonal because of the orthogonality relations (11.51) and (11.58) of thediagonal submatrices.

By reordering the vector of elastic variables at the wedge faces in (11.91)into subvectors of prescribed and unknown variables according to theboundary and interface condition relations (11.52) and (11.59), respectively,the vector denoted as wcorner PU is obtained. This 12W×1 vector is reducedto a (6W − F ) × 1 vector wcorner U by omitting the prescribed zero valuesof these variables,

wcorner PU =

wP (r, ϑ0)wU (r, ϑ0)wP (r, ϑ1)wU (r, ϑ1)

...wP (r, ϑW )wU (r, ϑW )

, wcorner U =

wU (r, ϑ0)wU (r, ϑ1)

...wU (r, ϑW )

. (11.93)



By collecting all the boundary and interface condition relations in thecorner, (11.52) and (11.59), respectively, we can write, first,

wcorner PU = Dcorner ext.(ϑϑϑ,ωωω,µµµ,ωωωu)wcorner ext. (11.94)

and, then, taking into account that Dcorner ext.(ϑϑϑ,ωωω,µµµ,ωωωu) is an orthogonalmatrix,

wcorner ext. = DTcorner ext.(ϑϑϑ,ωωω,µµµ,ωωω

u)wcorner PU. (11.95)

In fact, the last relation collects all the relations (11.53) and (11.60)for the boundary surfaces and interfaces of an open multimaterial corner.By substituting this relation into (11.89) we obtain

Kcorner ext.(λ)DTcorner ext.(ϑϑϑ,ωωω,µµµ,ωωω

u)wcorner PU = 06W×1. (11.96)

Finally, by removing the columns of the matrix Kcorner ext.(λ)DT

corner ext.(ϑ,ω,m,ωu) multiplied by the prescribed zero values of w(r, ϑw)(w = 0,W ) the final form of the characteristic system for the singularityanalysis of an open multimaterial corner (also called a corner eigensystem)is achieved:

Kcorner(λ,ωωω)wcorner U = 06W×1, (11.97)

where only the unknown values of λ and ωωω remain as arguments of thecharacteristic matrix of an open multimaterial corner

Kcorner(λ,ωωω)

=

266664

K1DTBC(ϑ0) −DT

1 (ϑ1) 06×n2 · · · 06×nW−1 06×nW

06×n0 K2DT2 (ϑ1) −DT

1 (ϑ2) · · · 06×nW−1 06×nW

......

.... . .

......

06×n0 06×n1 06×n2 · · · KW DT2 (ϑW−1) −DT

BC(ϑW )

377775

,

(11.98)

nw is the number of rows in matrices DBC(ϑw), for w = 0, W , and (defindin (11.53)) in the matrices Di(ϑw), for w = 1, . . . ,W − 1 and i = 1, 2.Thus, according to Section 11.3.2, nw = 3, for w = 0, W , and nw =6, for w = 1, . . . ,W − 1, except for the faces where the sliding frictioncondition is prescribed at ϑw, then nw is smaller by 1, i.e. nw = 2 andnw = 5, respectively. Therefore, the matrix Kcorner(λ) is a 6W×6W squarematrix if the sliding friction condition is not prescribed at any boundarysurface or interface of the corner (i.e. F = 0), whereas with a sliding friction



condition at one or more corner boundary surfaces or interfaces (i.e. F >

0) Kcorner(λ,ωωω) is a 6W × (6W − F ) rectangular matrix. In general, theelements of the matrix Kcorner(λ,ωωω) are transcendental complex analyticfunctions (holomorphic functions) of λ, including also real parameters ωωω forsliding friction conditions.

It is instructive to see how the characteristic matrix of the cornerKcorner(λ,ωωω) simplifies when there is only one wedge (i.e. W = 1) to a6 × 6, 6 × 5 or 6 × 4 matrix, respectively, depending on whether there arenone, or one or two boundary surfaces with a sliding friction condition:

Kcorner(λ,ω) = [K1(λ)DTBC(ϑ0) −DT

BC(ϑ1)]. (11.99)

When there are only orthogonal boundary conditions prescribed at bothboundary surfaces of the wedge (W = 1), the characteristic system canfurther be reduced, as in [19] (see also [71]), to the form:

Kcorner reduced(λ)wU (r, ϑ0) = 0, (11.100)

where

Kcorner reduced(λ) = Du(ϑ1)K(1)1 (λ)DT

u (ϑ0) + Du(ϑ1)K(2)1 (λ)DT

cϕ(ϑ0)

+Dϕ(ϑ1)K(3)1 (λ)DT

u (ϑ0) + Dϕ(ϑ1)K(4)1 (λ)DT

cϕ(ϑ0)

(11.101)

is a 3 × 3 matrix defined by partitioning the wedge transfer matrix K1(λ)to four 3 × 3 matrices in (11.86).

11.5.2.2 Closed multimaterial corner (periodic corner)

The following 12W × 12W rectangular matrix of boundary and interfaceconditions for a closed multimaterial corner collects all the matrices ofboundary and interface conditions of the corner in such a way that it iscompatible with the definition of vector wcorner ext. in (11.91):

Dcorner ext.(ϑϑϑ,ωωω,µµµ,ωωωu)

=

D2(ϑ0) 06×12 06×12 · · · 06×12 D1(ϑW )D2(ϑ0) 06×12 06×12 · · · 06×12 D1(ϑW )012×6 DI(ϑ1) 012×12 · · · 012×12 012×6

......

.... . .

......

012×6 012×12 012×12 · · · DI(ϑW−1) 012×6

.

(11.102)



Comments similar to those given after Eq. (11.92) are valid here as well.In particular the matrix Dcorner ext.(ϑϑϑ,ωωω,µµµ,ωωωu) is orthogonal. Recall thatϑW = ϑ0 + 360.

By reordering the vector of elastic variables for the wedge faces in(11.91) into subvectors of prescribed and unknown variables according tothe interface condition relation (11.59), the vector denoted as wcorner PU isobtained. This 12W×1 vector is reduced to a (6W − F )×1 vector wcorner U

by omitting the prescribed zero values of these variables,

wcorner PU =

wP (r, ϑ0)wU (r, ϑ0)wP (r, ϑ1)wU (r, ϑ1)

...

wP (r, ϑW−1)wU (r, ϑW−1)

, wcorner U =

wU (r, ϑ0)wU (r, ϑ1)

...wU (r, ϑW−1)

. (11.103)

Several differences with respect to the similar vectors in (11.93) should benoted. In particular, while wP (r, ϑ0) and wU (r, ϑ0) are 3 × 1 vectors (or4 × 1 and 2 × 1, respectively, for a sliding friction condition) in (11.93),representing the prescribed boundary conditions and unknowns at thecorner boundary, they are 6×1 vectors (or 7×1 and 5×1, respectively, for asliding friction condition) in (11.103), representing the prescribed interfaceconditions and interface unknowns.

By collecting all the interface condition relations in the corner (11.59),we can write, first,

wcorner PU = Dcorner ext.(ϑϑϑ,ωωω,µµµ,ωωωu)wcorner ext. (11.104)

and, then, taking into account that Dcorner ext.(ϑϑϑ,ωωω,µµµ,ωωωu) is an orthogonalmatrix,

wcorner ext. = DTcorner ext.(ϑϑϑ,ωωω,µµµ,ωωω

u)wcorner PU. (11.105)

In fact, the last relation collects all the relations (11.60) for the interfacesof a closed multimaterial corner. By substituting this relation into (11.89)we obtain

Kcorner ext.(λ)DTcorner ext.(ϑϑϑ,ωωω,µµµ,ωωω

u)wcorner PU = 06W×1. (11.106)



Finally, by removing the columns of the matrix Kcorner ext.(λ)DT

corner ext.(ϑϑϑ,ωωω,µµµ,ωωωu) multiplied by the prescribed zero values of

wP (r, ϑw) (w = 0,W −1) the final form of the characteristic system for thesingularity analysis of a closed multimaterial corner (also called a cornereigensystem) is achieved:

Kcorner(λ,ωωω)wcorner U = 06W×1, (11.107)

where only the unknown values of λ and ω remain as arguments of thecharacteristic matrix of a closed multimaterial corner

Kcorner(λ,ωωω)

=

266666664

K1DT2 (ϑ0) −DT

1 (ϑ1) 06×n2 · · · 06×nW−2 06×nW−1

06×n0 K2DT2 (ϑ1) −DT

1 (ϑ2) · · · 06×nW−2 06×nW−1

......

.... . .

......

06×n0 06×n1 06×n2 · · · KW−1DT2 (ϑW−2) −DT

1 (ϑW−1)

−DT1 (ϑ0) 06×n1 06×n2 · · · 06×nW−2 KW DT

2 (ϑW−1)

377777775

,

(11.108)

nw is the number of rows in the matrices Di(ϑw), for w = 0, . . . ,W −1 andi = 1, 2. Thus, according to Section 11.3.2, nw = 6 for w = 0, . . . ,W − 1,except for the interfaces where the sliding friction condition is prescribedat ϑw, nw then being smaller by 1, i.e. nw = 5. Therefore, the matrixKcorner(λ) is a 6W × 6W square matrix if the sliding friction condition isnot prescribed at any corner interface (i.e. F = 0), whereas with the slidingfriction condition at one or more corner interfaces (i.e. F > 0) Kcorner(λ,ωωω)is a 6W × (6W − F ) rectangular matrix.

11.5.3 Solution of the characteristic system. Singular

elastic solution

We are looking for non-trivial solutions wcorner U = 0 of the homogeneouslinear system (11.97) or (11.107), respectively, for an open or closed corner,which define the characteristic (singular) elastic solutions of the cornerverifying the prescribed boundary and/or interface conditions. Any non-trivial solution wcorner U = 0 of (11.97) or (11.107) is a (right) null vectorof the matrix Kcorner(λ,ωωω). Thus, first we need to find the characteristic(singular) values of λ and ωωω, which will provide a rank deficient matrix



Kcorner(λ,ωωω) (rank Kcorner(λ,ωωω) < 6W −F ) with a non-trivial (right) nullspace.

In the most usual case without friction contact conditions, i.e. whenF = 0, the linear system (11.97) or (11.107), with a 6W×6W square matrix,is a kind of non-linear eigenvalue problem of the multimaterial corner

Kcorner(λ)wcorner U = 06W×1. (11.109)

The usual procedure for solving this non-linear eigenvalue problembegins by finding the roots λ of the matrix determinant, which are thesolutions of the characteristic equation of the corner (also called the cornereigenequation),

detKcorner(λ) = 0. (11.110)

These roots, referred to as the characteristic (singular) exponents (alsocalled eigenvalues), are fundamental in the corner singularity analysis. Fromthe above, in general detKcorner(λ) is a transcendental complex analyticfunction (a holomorphic function) of λ. Characteristic exponents usuallyform an infinite discrete set in the complex plane defining an infinite setof characteristic elastic solutions for the considered corner configuration.It can be shown that if a complex λ is root of (11.110), its complex conjugateλ is a root of (11.110) as well. While characteristic exponents λ with Reλ >0 lead to elastic solutions in the corner with a finite elastic strain energy,Reλ < 0 correspond to elastic solutions with infinite strain energy, due toa non-integrable singularity at the corner tip. In particular, characteristicexponents with 0 < Reλ < 1 correspond to singular elastic solutions in thecorner with unbounded stresses and strains at the corner tip but a finiteelastic strain energy.

With friction contact conditions, i.e. when F ≥ 1, linear system (11.97)or (11.107), with a 6W×(6W−F ) rectangular matrix, is an overdeterminedhomogeneous system. An efficient and reliable way to determine if it has anon-trivial solution for some particular values of λ and ωωω, and to evaluatesuch a non-trivial solution, is to compute the singular value decomposition(SVD) of the matrix Kcorner(λ,ωωω) [95, 96], by evaluating singular valuesσi ≥ 0 (i = 1, . . . , 6W − F ). Let σmin(Kcorner(λ,ωωω)) denote the smallestsingular value. Then, the characteristic exponents λ and characteristicfriction-angles ωωω (which in the isotropic friction case coincide with thecharacteristic sliding angles ωωωu) are determined by solving the following



characteristic equation of the corner with sliding friction contact:

σmin(Kcorner(λ,ωωω)) = 0. (11.111)

Due to the periodicity of the functions of ωki (i = 1, . . . F ) involved inthe definition of Kcorner(λ,ωωω), see (11.36) and (11.42)–(11.45), we typicallysearch for characteristic friction angles in the interval −180 ≤ ωki < 180.

It may be useful to write the characteristic equation of the corner asa condition for a vanishing determinant, which is (at least theoretically)equivalent to (11.111), by taking into account that the squares of thesingular values σ2

i are eigenvalues of the matrix of the least-squares systemfor (11.97) or (11.107),

det(KTcorner(λ,ωωω)Kcorner(λ,ωωω)) = 0. (11.112)

An advantage of this explicit form of the characteristic equation of thecorner, in comparison with (11.111), is that we can search for roots ofa transcendental complex analytic function of λ and ωki (providing thefunctions in (11.36) are also analytic functions) similarly as in (11.110).However, a disadvantage of (11.112) may be that the algebraic multiplicityof the roots λ is doubled.

Previous formulations for a corner singularity with sliding frictioncontact are usually limited to corner configurations with uncoupled plane-strain and antiplane-strain states [67, 70, 71, 73]. The novelty of the presentformulation is that in a corner singularity problem with friction undergeneralized plane-strain conditions we find not only the characteristicexponents λ as usual, but also the characteristic friction angles ωωω, whichsolve the characteristic equation of the corner, (11.111) or (11.112).

It should be stressed that the present procedure leads to a closed-formanalytic expression of the matrix Kcorner(λ,ωωω), where the only numericallycomputed values are the roots pα of the Lekhnitskii–Stroh sextic equationfor anisotropic materials (11.15), if analytic expressions of these roots arenot available. With the exception of very simple corner configurations, acomputational tool for symbolic computations should be used to evaluatea closed-form expression of Kcorner(λ,ωωω). Furthermore, such a tool forsymbolic computation can be used to evaluate a closed-form analyticexpression of the corner eigenequation (11.110) or (11.112). In the presentwork, Mathematica [97] has successfully been used for these purposes.

Finding the real or complex roots λ of non-linear eigenequations inthe form of a vanishing determinant condition in (11.110) (and similarly



in (11.112)), which are in general given by complex analytic functions(holomorphic functions), can be efficiently carried out by Muller’s method[98,99] or by more sophisticated algorithms [100]. Nevertheless, special careshould be taken in finding all the roots in a region of interest, usually definedby an interval of the real part of λ, e.g. 0 < Reλ ≤ 1. For this purpose,the argument principle (e.g. [101]) is an excellent tool for identifying thenumber of roots of a holomorphic function, e.g. h(λ) = detKcorner(λ), in aparticular region of the complex plane. According to the argument principle,the following integral along a closed contour C (without self-intersections)in the complex plane:

J =1

2πi

∮C

h′(λ)h(λ)

dλ, (11.113)

with h′(λ) denoting the complex derivative of h(λ), gives the change in theargument of h(λ) around this contour (the difference between the final andinitial values of the continuously varying polar angle of the complex numberh(λ)) divided by 2π,

J =12π

[arg(h(λ))]C . (11.114)

It can be shown that J equals the number of zeros of h(λ) (which has nopoles), including their multiplicity, inside the domain defined by C. Thisstatement is valid if there are no zeros of h(λ) on the contour C itself. Theintegral in (11.113) can be evaluated by a numerical quadrature, usuallygiving a number very close to an integer representing the number of zerosof h(λ). Nevertheless, in the present work we employ the representation(11.114). A couple of examples of applications of the argument principleare given in Sections 11.7.1 and 11.7.2.

With sliding friction conditions at some corner surfaces, and takinginto account that a singular value of a matrix is always a real non-negative number, it appears that an efficient way of finding the solutionsof the characteristic equation of the corner (11.111) is to apply a globalminimization procedure to σmin(Kcorner(λ,ωωω)) as an objective function ofλ and ωωω from a feasible region of interest, e.g. defined by 0 < Reλ < 1 and−180 ≤ ωki < 180. The region of interest should be thoroughly exploredby an automatic and reliable minimization procedure capable of finding allthe minimizer pairs λ and ωωω for which the objective function vanishes, i.e.σmin(Kcorner(λ,ωωω)) = 0. Note that the minimization procedure requires the



computation of the SVD of Kcorner(λ,ωωω) for all pairs λ and ωωω consideredin the minimization iterations.

Once a particular value of the characteristic exponent λ, and withfriction contact also of the vector of the characteristic friction angles ωωω,is obtained as a solution of the characteristic equation of the corner, thecorresponding behaviour of the displacements and stresses inside the cornercan easily be computed. The procedure starts by computing a correspondingnon-trivial solution wcorner U = 0 of the homogeneous linear systems (11.97)or (11.107) considering a fixed value of the radial coordinate r, e.g. r = 1.The values of wcorner U with the pertinent radial dependence are obtainedby multiplying it by rλ, wcorner U(r) = rλwcorner U(r = 1). Completingwcorner U with zero values of wP(r, ϑw), see (11.93) and (11.103), leads tothe corresponding wcorner PU = 0. Then, wcorner ext. in (11.91) is evaluatedby means of (11.95) or (11.105). Now, from ww(r, ϑw−1), known for anyw = 1, . . . ,W , we can compute wm(r, θm−1) for all single-material wedgesin a multimaterial wedge w, with indices m = iw, . . . , jw, by sequentiallyemploying the transfer relation (11.66). Finally, we compute the Cartesiancomponents of the displacement and stress function vectors, u(r, θ) andϕ(r, θ), inside a single-material wedge m by a transfer relation analogousto (11.66):

wm(r, θ) = Em (λ, θ, θm−1)wm(r, θm−1) for θm−1 ≤ θ ≤ θm. (11.115)

Then, the displacement vector and stress tensor in cylindricalcoordinates are obtained as follows ([28] Section 7.3):

ur = −sTr (θ)u(r, θ), uθ = nT (θ)u(r, θ), u3 = sT3 u(r, θ), (11.116)

and

σrr = sTr (θ)ϕ,θ(r, θ)/r, σθθ = nT (θ)ϕ,r(r, θ),

σrθ = −nT (θ)ϕ,θ(r, θ)/r = −sTr (θ)ϕ,r(r, θ),

σr3 = −sT3 (θ)ϕ,θ(r, θ)/r, σθ3 = sT3 (θ)ϕ,r(r, θ), (11.117)

and σ33 is evaluated from the condition ε33 = 0. Taking into account thatif wcorner U = 0 is a solution of the homogeneous linear system (11.97) or(11.107) then also cwcorner U, where c = 0, solves this system as well, it isuseful to standardize the corresponding singular elastic solution obtained inthe corner. A practical approach is to make a stress component at a suitablydefined position in the corner equal to a given constant, e.g. requiring



σθθ(r = 1, θ∗) = (2π)Reλ−1 [102] where θ∗ is a specific angle in the corner(the angle of a wedge face, angle of the corner symmetry or antisymmetryplane if it exists, etc.). In this way the standardized characteristic angularfunctions g(n)

i (θ) and f (n)ij (θ) introduced in (11.2) and (11.3) are defined for

a particular characteristic exponent λ = λn.With reference to the corner singularity analysis with friction contact

conditions, and as explained in Section 11.3.2, each solution of thecharacteristic equation of the corner (11.111) obtained should be checkedto see if it satisfies the compression condition (σθθ(r, ϑw) ≤ 0) and thefriction dissipation condition (although the relative tangential displacementobtained by the solution of the characteristic system is parallel to k(ϑw , ωu),we still should check its orientation with respect to the friction shear stressto guarantee the dissipative character of the friction) at all the frictionsurfaces of the corner. It appears that oscillatory solutions for a complexcharacteristic exponent λ are not compatible with these compressionand friction dissipation conditions; consequently only real characteristicexponents λ are essentially admissible. Nevertheless, it is well known thatcomplex characteristic exponents λ may appear in cracks at the straightinterface between two anisotropic bodies with a frictionless contact zoneat the crack tip [65–70]. Thus, assuming solution continuity with respectto a vanishing friction coefficient, it can be expected that similar complexcharacteristic exponents λ can also appear for sliding friction contact ina corner singularity analysis, a hypothesis that should still be checkednumerically. Although, strictly speaking, such solutions are not admissible,in the global problem solution the portion of the contact zone adjacent tothe corner tip, where these compression and friction dissipation conditionscan be violated, could be of a very small size with respect to othercharacteristic lengths of the whole problem. Then, such a solution mightbe accepted, in the same way that crack face overlapping in an oscillatorysolution of an open model of interface cracks is accepted with the hypothesisof small scale contact [103].

11.6 Evaluation of GSIFs

When the characteristic exponents λn and the characteristic angular shapefunctions gni (θ) and fnij(θ) in the asymptotic series expansions of the elasticsolution in a corner in (11.2) and (11.3) are known, the only unknowns inthese series to be determined are the generalized stress intensity factors Kn

(GSIFs), which are coefficients of power-type singularities. The evaluation



Table 11.5. Classification of procedures for the evaluation of GSIFs.

Local/global techniques Global techniques

Extraction of GSIFs fromFEM or BEM solution inthe post-processing stage

Least-squares fitting[40, 108–111]

Conservative integrals[2, 105, 107, 112–117]

Local techniques Global techniques

Incorporation of singularityshape functions inproblem discretization

Quarter point elements[118–120] and othersingularity elements [121]

Functions in the wholedomain or boundary[41,122]

of the GSIFs usually requires a numerical model of the whole problem.Techniques for the evaluation of GSIFs can roughly be divided into fourbasic groups, see Table 11.5, according to their local or global characterand whether they are implemented in the solution or post-processing stageof FEM or BEM analysis. Local techniques are usually sensitive to theaccuracy of the numerical solution for stresses or displacements close to thecorner tip, while global techniques, working also, or only, with the elasticsolution far from the corner tip, are less sensitive to the solution accuracy atthe corner tip. Techniques for extracting GSIFs from a numerical FEM orBEM solution in the post-processing stage do not need to be incorporatedinto the FEM or BEM codes, but do not typically have as good accuracyas methods which directly incorporate the singularity shape functions intothe problem discretization, usually requiring a modification of these codes.References for techniques in these groups are included in Table 11.5; furtherinformation can be found in [6, 8, 41]. Examples of the evaluation of GSIFsinvolving anisotropic materials can be found in [104, 105] by means of theH-integral, and in [106, 107] by means of the M -integral along with othertechniques.

11.6.1 Least-squares fitting technique

The technique presented in this section is based on least-squares fitting ofthe finite asymptotic series expansions in (11.2) and (11.3) to the numericalresults for the displacements and/or stresses in a multimaterial corner [40].This is a reliable, accurate and easy-to-use technique for the extractionof GSIFs with no need to modify the FEM or BEM code applied. It hasno limits for the number of power stress singularities considered in theanalysis and shows an acceptable robustness when employing numericalresults relatively far from the corner tip.



The technique minimizes an error function J

J(K1, . . . ,KN ) = aJu(K1, . . . ,KN ) + bJσ(K1, . . . ,KN) (a, b ≥ 0),(11.118)

with

Ju(K1, . . . ,KN ) =∑

α=r,θ,3

Nr∑i=1

Nθ∑j=1

aα

×[useriesα (ri, θj ,K1, . . . ,KN ) − uBEMα (ri, θj)]2 (aα ≥ 0),

(11.119)

Jσ(K1, . . . ,KN ) =∑

α=r,θ,3

Nr∑i=1

Nθ∑j=1

bα

×[tseriesα (ri, θj ,K1, . . . ,KN) − tBEMα (ri, θj)]2 (bα ≥ 0),

(11.120)

where Ju and Jσ compute the sums of squares of differences between thenumerical and analytical solution in terms of displacements and stresses,respectively, at a number of points (usually nodes of a mesh) given by polarcoordinates (ri, θj). The numerical and analytical solutions are denotedwith the superscripts ‘BEM ’ and ‘series ’, as the first is obtained from aBEM model in the present work and the second is given by the asymptoticseries expansion in (11.2) and (11.3). In (11.118), a and b are weightingfactors, which allow us to consider only displacements (a, b) = (1, 0), onlystresses (a, b) = (0, 1) or both (a, b) = (l−2, σ−2), l and σ being somecharacteristic length and stress values so that the terms given by thedisplacements and stresses have values of the same order of magnitude.Similarly, the dimensionless weighting factors in (11.119) and (11.120),aα and bα, allow the isolated components of the nodal displacements andtractions to be used. The points used for the evaluation of J in (11.118)are placed in the present work, without loss of generality, along radial linesdefined by the corner boundaries and interfaces. This is a natural optionwhen a BEM model is used as any BEM mesh has nodes at these locations.Thus,Nr andNθ in (11.119) and (11.120) denote the number of BEM nodesat each radial line and the number of radial lines employed, respectively,as schematically illustrated in Fig. 11.3. Nevertheless, when using an FEMmodel, J can be evaluated at an arbitrary set of nodes in the corner.



Nr

Nr

NrNr

Nrj=1

j=2

...

j=3

j=Nθ -1

j=Nθ

Fig. 11.3. BEM nodes for least-squares fitting.

The set of GSIFs Kn (n = 1, . . . , N) which minimizes J is obtained bysolving the following linear system of equations:

∂J(K1, . . . ,KN )∂Kj

= 0 (j = 1, . . . , N). (11.121)

The present technique admits solutions with complex values ofGSIFs. If a characteristic exponent λ is a complex number, as in theopen model of interface cracks, then its complex conjugate λ is also acharacteristic exponent. The associated GSIF is also a complex numberK = KR + iKI, where KR and KI are real numbers. In this case, tworeal terms can be included in (11.2) and (11.3) instead of two complexterms. In the representation of displacements, one term would be equalto KR Rerλ gi(θ) and the other to KI Imrλ gi(θ), while in therepresentation of stresses, one term would be equal to KR Rerλ−1 fij(θ)and the other to KI Imrλ−1 fij(θ).

11.6.2 Implementation, accuracy and robustness

We will now briefly discuss additional issues regarding the implementation,accuracy and robustness of the above technique. First, the linear systemarising from (11.121) is calculated explicitly. For the sake of simplicity andwithout loss of generality, let only the radial displacement component urbe included in the error function J , by choosing ar =1, aθ = a3 = 0, a = 1and b = 0. The displacement ur at a point p (usually a BEM mesh node)defined by the radius ri and polar angle θj is approximated by using N



terms of the asymptotic series expansion representation (11.2) as:

useriesr (ri, θj) ∼=

N∑n=1

Knrλn

i g(n)r (θj) =

N∑n=1

apnKn, (11.122)

where apn = rλn

i g(n)r (θj) is the coefficient for the Kn term of the series

expansion of ur in (11.2) evaluated at the point p with the polar coordinates(ri, θj). The derivative of the error function J with respect to Kj evaluatedat P points (p = 1, . . . , P ) is a linear function of Kn (n = 1, . . . , N), whichcan be expressed as follows:

∂J

∂Kj= 2

P∑p=1

N∑n=1

apnKn − uBEMr (p)

apj

= 2P∑p=1

N∑n=1

apnKnapj − apjuBEMr (p)

= 0, (11.123)

where uBEMr (p) = uBEM

r (ri, θj) denotes the displacement ur at the point p.Equation (11.123) is written in matrix notation as:

AT · A ·K = AT · b, (11.124)

where A is a P × N matrix, P and N being respectively the number ofpoints (nodes) used for building the error function J , and the number ofterms in the series (11.2) expansion (the number of GSIF values). As longas the number of points P is greater than the number of terms consideredfor the displacement representation N , A is a rectangular matrix with morerows than columns and thus expected to have full rank N. K is the N × 1vector of unknowns (GSIF values) and b is the P × 1 vector of numericalresults for ur at the chosen points. Hence,

A =

a11 · · · a1N

......

aP1 · · · aPN

, K =

K1

...KN

, b =

b1...bP

. (11.125)

It is clear that the N ×N square matrix AT · A can have rank=N ifand only if the number of points P is equal to or greater than the numberN of GSIFs to be calculated. Only in that case can the inverse of AT · Aexist and be computed.



Equation (11.124) is a typical matrix expression that appears whensolving an overdetermined linear system

A ·K = b, (11.126)

using the 2-norm minimization minK‖A ·K− b‖2, see [95, 96]. Notice thatthe fulfilment of (11.126) corresponds to the vanishing differences in theerror function J , i.e. J = 0.

The solution b of the full rank least-squares problem is theoreticallyunique. Nevertheless, due to the nature of the matrix components apn =rλni g

(n)r (θj), the evaluation of the matrix AT ·A using only points very close

to the corner tip has been shown to give rise to ill-conditioned matrices withnumerically computed rank (AT ·A) < N . This ill-conditioning includes thecases in which the number of points P exceeds the number of GSIF termsNand the matrix A has numerically computed full rank N . The conditioningnumber for the 2-matrix norm [95, 96] κ(A) = ‖A‖2‖A+‖2 = σmax/σmin,where A+ is the pseudoinverse of A, and σmax and σmin, respectively,are the maximum and minimum singular values of A, gives an indicationof the conditioning of the problem. It has numerically been verified [40]that κ(A) in these cases (in which the number of points P exceeds onlyslightly the number of GSIF terms N) may be around 103 times higher thanthe conditioning number obtained in those cases where P is much greaterthan N .

Also the relative proximity between the nodes chosen for the evaluationof the error function J or, equivalently, matrix A has been shown to affectthe numerical conditioning of AT · A. When consecutive nodes are chosenfor the evaluation of J , the number of points needed for AT · A to havenumerically computed full rank has been shown to be significantly greaterthan when non-consecutive nodes are chosen.

Taking all these considerations into account, the least-squares solutionhas been computed in the present work, solving the system in (11.126)by means of the QR decomposition of matrix A, which is known to bemore accurate than directly solving the system in (11.124) with a possiblyill-conditioned matrix AT · A.

11.7 Examples of Singularity Analysis

Two problems regarding multimaterial corners with composite materialsare studied applying the computational tools developed in the previoussections and implemented in Mathematica [97] and numerical BEM codes



[78, 79] with the aim of showing the capabilities of these tools and validatingthem. Both problems are of unquestionable engineering interest. The firstis related to failure initiation in cross-ply laminates [0/90]s under tension.The second deals with the failure analysis of an adhesively bonded double-lap joint (between a composite laminate [0n] and an aluminium plate usingan epoxy adhesive layer) subjected to tension load.

11.7.1 Transverse crack terminating at the interface

in a [0/90]S laminate

A transverse crack in the inner 90 ply of a cross-ply [0/90]S laminate undertensile loading is considered, see [79] and Fig. 11.4(a). The transverse crackterminates perpendicularly at the interface with the outer 0 plies and thecrack faces are assumed to be free. The neighbourhood of the crack tip canbe considered as a trimaterial corner, Fig. 11.4(b). The elastic propertiesof the unidirectional fibre-reinforced plastic ply (AS4/8552) considered asan orthotropic material are E11 = 141.3GPa, E22 = E33 = 9.58GPa,G12 = G13 = 5.0GPa, G23 = 3.5GPa, ν12 = ν13 = 0.3, ν23 = 0.32, wheresubscript 1 denotes the fibre direction. The material in both plies is the samebut they have a different spatial orientation. Due to the material symmetriesand corner configuration, the in-plane and antiplane displacements areuncoupled.

In this study, first a complete singularity analysis of the trimaterialcorner shown in Fig. 11.4(b) is carried out by solving (11.110) by findingcharacteristic exponents with 0 < Reλ < 2. The influence of the treatmentof material degeneracy in the Lekhnitskii–Stroh formalism on the accuracyof the singularity analysis results is discussed. Then, the whole problemshown in Fig. 11.4(a) is solved by means of a BEM model such as thatused in [79], a few GSIFs are evaluated by a least-squares fitting procedure(Section 11.6), and finally, to check the computational tools developed in thepresent work, the stresses computed by BEM at interior points of the cornerare compared with those derived from the series approximation (11.3).

The roots of the characteristic equation of the corner (11.110) are found,by means of the argument principle (11.114), inside a rectangular contourin the complex plane defined by its corners 0.1− i0.9, 1.95− i0.9, 1.95+i0.9and 0.1+i0.9, see Fig. 11.5; no roots exist along this contour. The argumentprinciple indicates that inside the contour, in the domain with 0.1 < Reλ <1.95 and −0.9 < Imλ < 0.9, there are 11 roots of (11.110), includingtheir multiplicities. The characteristic exponents found are: λ1 = 0.471654



90o ply

90o ply

0o ply

transversecrack

θr

0.55

mm

0.55

mm

1.10

mm

90o0o 0o

(a)

(b)

Fig. 11.4. (a) Cross-ply [0/90]S laminate under tensile loading with a transverse crackterminated at the interface. (b) Neighbourhood of the transverse crack tip.

-1,0

-0,5

0,0

0,5

1,0

0,0 0,5 1,0 1,5 2,0

Im(λ

)

Re(λ)

λ8=1.73425(symmetric)

Singular solutions

· · ·

(antisymmetric)λ11=1.89369(symmetric)S

earc

hing

are

a fo

r th

e A

rgum

ent P

rinc

iple λ1=0.471654 (antiplane)

λ2=0.521510 (antisymmetric)λ3 (symmetric)=0.669888

λ4,5,6=1 (triple root)

λ7=1.52834

rigid body rotationantiplane symmetric

λ9,10 ± 0.308109 i=1.84194

Non-singular solutions

Fig. 11.5. Characteristic exponents for the open trimaterial corner in Fig. 11.4(b).

(antiplane), λ2 = 0.521510 (antisymmetric), λ3 = 0.669888 (symmetric),λ4,5,6 = 1 (a root of multiplicity three, one for rigid body rotation, oneis an antiplane term and the third is a symmetric term), λ7 = 1.52834(antiplane), λ8 = 1.73425 (symmetric), λ9,10 = 1.84194 ± 0.308109i (both



antisymmetric) and λ11 = 1.89369 (symmetric). The labels symmetricand antisymmetric refer to plane-strain elastic solutions corresponding toparticular terms in (11.2) and (11.3), which are symmetric or antisymmetricwith respect to the transverse crack plane, whereas the label antiplane refersto an antiplane elastic solution corresponding to a term in (11.2) and (11.3).In the case of complex roots, special care should be taken to identify them,especially those having Reλ > 1, because of typically large variations in thedeterminant of the characteristic matrix there, making it difficult to findroots using standard algorithms such as Muller’s method [98, 99], and thusmore sophisticated algorithms may be required [100].

In many previous works on corner singularity analysis with anisotropicmaterials, only non-degenerate materials (in the Lekhnitskii–Strohformalism of anisotropic elasticity) are treated explicitly, so it seemsuseful to analyse briefly the possible influence of an approximation ofa degenerate material by a non-degenerate one on the results of thesingularity analysis. For instance, let the unidirectional fibre-reinforcedplastic ply (AS4/8552) be modelled as an elastic transversely isotropicmaterial (instead of an orthotropic one) defined by five independent elasticconstants (instead of nine): E = 9.58GPa and ν = 0.32 define its isotropicbehaviour in the 2–3 plane, and E11 = 141.3GPa, ν12 = ν13 = 0.3and G12 = G13 = 5.0GPa. Notice that the shear modulus in the 2–3plane is G = E/2(1 + ν) = 3.629GPa (instead of G23 = 3.5GPa asgiven by the manufacturer considering this ply as an orthotropic material).A change in the spatial orientation of a transversely isotropic materialcan lead to a mathematical degeneracy in the Lekhnitskii–Stroh formalism[25, 37, 87] and if explicit expressions of the A and B matrices for thecorresponding degenerate material are not available, only approximatesingularity analysis of a corner including this material can be carried outthrough using a small perturbation of the elastic constants or spatialorientation of the material leading to a non-degenerate case. However,the accuracy of the results obtained by such a procedure is not knowna priori. To illustrate these facts, Fig. 11.6 shows the numerical resultsfor the characteristic exponents λ1 (antiplane), λ2 (antisymmetric) andλ3 (symmetric), considering small perturbations of the spatial orientationof the transversely isotropic material. These characteristic exponents wereobtained using a specific Mathematica [97] code developed for this kindof material in [25], employing standard machine precision for floatingpoint computation, which is about 15–16 decimal digits. In particular,the influence of the fibre-angle perturbation φ of the 90 ply, which is a



1,E-09

1,E-08

1,E-07

1,E-06

1,E-05

1,E-04

1,E-03

1,E-02

1,E-01

1,E+00

1,E+01

1,E-031,E-021,E-011,E+001,E+01

Perturbation angle φ (deg)

λi λi(φ=0o)λi(φ=0o)

λ1

λ2

λ3 Rel

ativ

eer

ror

x3

x1

x2

90o

90o

0o

Perturbation angle φfor the 90º layer

(inside x1–x3 plane)

Fig. 11.6. Influence of the fibre-angle perturbation φ in the 90 ply on the characteristicexponents for the open trimaterial corner in Fig. 11.4(b).

degenerate case [25], is studied. The relative error of the actual value ofλi with respect to the solution for a vanishing fibre-angle perturbationφ = 0 is plotted as a function of φ. Numerical instabilities are seen forsmall fibre-angle perturbations. Consequently, the characteristic exponentsobtained by approximating a degenerate material (φ = 0) with a non-degenerate one (φ = 0) with a sufficiently small fibre-angle perturbationmay lead to significant numerical errors, as the threshold angle below whichnumerical instabilities appear is a priori unknown, and depends on thecorner configuration. Therefore, in the present work all the mathematicallydegenerate cases are dealt with using the corresponding expressionsintroduced in the previous sections, which provide high accuracy in theevaluation of characteristic exponents and functions.

The plane-strain solution of the elastic problem defined in Fig. 11.4(a)is symmetric with respect to the transverse crack plane, and the crack isexpected to open because of the tensile load applied perpendicularly tothis plane. Thus, only the characteristic exponents marked as symmetricin Fig. 11.5, namely λ3 = 0.669888 for singular stresses and λ6 = 1,λ8 =1.73425 and λ11 =1.89369 for finite stresses, are included in the 2Dasymptotic series expansions (11.2) and (11.3) in the neighbourhood ofthe transverse crack tip. The GSIFs extracted from the BEM resultsusing the least-squares fitting technique, standardized following [102]in such a way that the stress component σθθ|θ=0 =K/(2πr)1−λ, areKr(λ=1) = −0.0000203291, K3(λ=0.669888) = 0.561634MPa ·mm0.330112,K6(λ=1) = 0.694945MPa, K8(λ=1.73425) = 0.522112MPa ·mm−0.73425 and



-500

0

500

1000

1500

2000

0 30 60 90 120 150 180

MPa

.

angle (deg)θ

σrseries

σrBEM

θr

σrθseries

σθseries

σrθΒΕΜ

σθΒΕΜ

0º ply 90º ply

0º ply90º ply

Fig. 11.7. Stress components evaluated by BEM and by the asymptotic series expansionat a distance r = 0.1mm from the corner tip.

K11(λ=1.89369) =−0.288119MPa ·mm−0.89369, theKr term being associatedwith rigid body rotation. The BEM results used in the least-squaresfitting are the two displacement components (ur, uθ) along the radial linesemerging from the corner tip at θ = 90 and 180, computed at the nodes ofthe BEM mesh in the range 10−5 mm < r < 0.3mm. Figure 11.7 shows theresults for the stress components as functions of the angular coordinate θ,for a fixed radial coordinate r = 0.1mm, obtained by both the BEM modeland the asymptotic series approximation (11.3) using the extracted GSIFsvalues. The fitting of both approximations for all three stress componentsis excellent along the whole range of θ. As could be expected, in view of therelative stiffnesses of the plies in the load direction parallel to the interface(the 0 ply is much stiffer than 90 ply), much higher stresses are observedin the 0 than in 90 ply. Notice the discontinuity of the σr stress componentat the interface (θ = 90) between the 0 and 90 plies.

11.7.2 Bimaterial corner in an adhesively bonded

double-lap joint

11.7.2.1 Singularity analysis of a closed corner

In an adhesively bonded double-lap joint between an aluminium plate and acomposite laminate [0n] with an epoxy adhesive layer as shown in Fig. 11.8,



metal

adhesive

metal

adhesive0o

adhesive

0o

adhesiveadhesive

0o

adhesive

0o

metal

adhesive

metal

adhesive

CFRP [0]12

(2.2 mm thickness)Aluminium(3.2 mm thickness)

12.5 mm

A B C D

Adhesive layerthickness = 0.1 mm

Fig. 11.8. Multimaterial corners in an adhesively bonded double-lap joint between analuminium plate and two carbon-fibre-reinforced polymer (CFRP) laminates.

several multimaterial corners can be identified. We will look at corner B,a closed bimaterial corner (the two materials are perfectly bonded at bothinterfaces), because it is a critical point at which failure typically initiates inthis type of joint [123, 124]. The elastic properties of the unidirectional fibre-reinforced plastic ply are the same as in the first example in Section 11.7.1;the elastic properties of the adhesive are E = 3.0 GPa and ν = 0.35 and ofthe aluminium E = 68.67GPa and ν = 0.33. Thermal stresses, which couldarise in the curing process, are not considered [125].

In this study, first a comprehensive singularity analysis of the closedbimaterial corner B, Fig. 11.8, is carried out solving (11.110) by findingcharacteristic exponents with 0 < Reλ < 1.5. Then, the double-lap jointproblem shown in Fig. 11.8 is solved by means of a BEM model such as thatused in [124]. Three GSIFs for elastic plane-strain solutions are evaluated bya least-squares fitting procedure (Section 11.6), and finally, in order to checkthe computational tools developed in the present work, a displacementcomponent computed by BEM at interior points of the corner is comparedwith the series approximation (11.2), identifying the contribution of eachterm in the series expansion.

The roots of the characteristic equation of the corner (11.110) arefound, using the argument principle (11.114), inside a rectangular contourin the complex plane defined by its corners 0.1− i0.9, 1.5− i0.9, 1.5 + i0.9and 0.1 + i0.9, see Fig. 11.9; there are no roots along this contour. Theargument principle indicates that inside the contour, in the domain with0.1 < Reλ < 1.5 and −0.9 < Imλ < 0.9, there are six roots of (11.110),including their multiplicities. The characteristic exponents found are allreal numbers: λ1 = 0.763236, λ2 = 0.813696, λ3 = 0.889389, λ4 = 1,λ5 = 1.106980, λ6 = 1.185066, all having a multiplicity equal to 1. The roots



-1,0

-0,5

0,0

0,5

1,0

0,0 0,5 1,0 1,5

Im(λ

)

Re(λ)

λ1=0.763236λ2=0.813696 (antiplane)

λ3=0.889389λ4=1 (rigid body rotation)

λ5=1.10698λ6=1.185066 (antiplane)

Sea

rchi

ng a

rea

for

the

Arg

umen

t Pri

ncip

le

Singular solutions Non-singular solutions

Fig. 11.9. Characteristic exponents for the closed bimaterial corner B in Fig. 11.8.

λ2 and λ6 correspond to antiplane solutions, while λ4 is for rigid bodyrotation.

The elastic problem of the double-lap joint defined in Fig. 11.8 is solvedunder plane-strain conditions. Thus, only two characteristic exponents λ1 =0.763236 and λ3 = 0.889389 for singular stresses and one characteristicexponent λ5 = 1.106980 for finite stresses are considered in the followingseries approximations of the displacements (where the rigid body rotationterm is also included) and stresses in the neighbourhood of the corner tip:

ui(r, θ) ∼= K1r0.763236g

(1)i (θ) +K3r

0.889389g(3)i (θ)

+K4r1g

(4)i (θ) +K5r

1.106980g(5)i (θ), (11.127)

σij(r, θ) ∼= K1

r0.236764f

(1)ij (θ) +

K3

r0.110611f

(3)ij (θ) +K5r

0.106980f(5)ij (θ).

(11.128)

As an example, Fig. 11.10 shows the angular shape functions for thefirst singular term with λ1 = 0.763236. Recall that f (n)

ij are dimensionless

functions whereas the dimension of g(n)i (with the exception of the

dimensionless g(4)i for rigid body rotation) is F−1L2 where F and L denote

force and length respectively. Notice that all the characteristic angularshape functions shown in Fig. 11.10 are continuous except for f (1)

rr , whichsuffers jumps at the interfaces (θ = 0 and 90).



-9

-6

-3

0

3

6

0 30 60 90 120 150 180 210 240 270 300 330 360

angle θ (deg)

fθθ frr

frθ

gθ

gr

-31.1

gr

frθ

gθ

fθθ

frr

Fig. 11.10. Angular shape functions for λ1: g(1)r (θ), g

(1)θ (θ), f

(1)θθ (θ), f

(1)rθ (θ) and f

(1)rr (θ).

0.00

0.50

1.00

1.50

2.00

2.50

3.00

3.50

4.00

45.0 47.0 49.0 51.0 53.0 55.0 57.0 59.0 61.0 63.0 65.0

laminado de carbono [0º]12

Aluminio

Deformada(x20)

Deformedshape (x20)

CFRP laminate [0]12

Aluminium

Fig. 11.11. Deformed shape of the BEM mesh and undeformed boundaries, detail ofthe overlap zone (in mm).

The plane-strain solution of the elastic problem shown in Fig. 11.8 canbe obtained analysing only half of the aluminium plate and one CFRPlaminate because of the problem symmetry. The right end of the CFRPlaminate is fixed while the left end of the aluminium plate has a tensilestress of 125 MPa. Figure 11.11 shows a detail (for the overlap area) of thedeformed shape calculated by the BEM model employed.

The GSIFs extracted from the BEM results using the least-squares fitting technique [40], standardized as proposed by [102] insuch a way that the stress component σθθ|θ=0 =K/(2πr)1−λ, areKr =K4(λ=1) = − 0.00356242, K1(λ=0.763236) = −0.00275036MPa ·mm0.236764, K2(λ=0.889389) = 0.0273839MPa ·mm0.110611, K3(λ=1.106980) =−0.0114328MPa ·mm−0.10698. The BEM results used in the least-squares



-0,0003

-0,0002

-0,0001

0

0,0001

0,0002

0,0003

0,0004

0,0005

0,0006

0 30 60 90 120 150 180 210 240 270 300 330 360

u r(m

m)

angle θ (deg)

term1

term2

term3

term 1 + term 2 term 1 + term 2 + term 3BEM solution

Fig. 11.12. Displacement component ur evaluated by BEM and by the asymptotic seriesexpansion at r = 0.0194 mm from the corner B tip.

fitting are the two displacement components (ur, uθ) along the radial linesemerging from the corner tip at θ = 0 and 90. The BEM mesh hasa progressive mesh refinement towards the corner tip, with the smallestelement having a length of 10−8 mm; the selected range of the nodes isfrom 10−6 mm to 0.025mm. Figure 11.12 shows the results for the urdisplacement component as a function of the angular coordinate θ, for afixed radial coordinate r = 0.0194mm, obtained by the BEM model and bythe asymptotic series approximation (11.2) particularized in (11.127) usingthe extracted GSIFs values. The contribution of each term in (11.127) isalso indicated, showing that even the approximation given by the two-termseries (term 1 + term 2) is reasonably close to the BEM results while thefitting of BEM results by the three-term series approximation (term 1 +term 2 + term 3) is excellent along the whole range of θ.

11.7.2.2 Singularity analysis of a corner including an interface crackwith sliding friction contact

This section shows some results of a singularity analysis of the corner Bin Fig. 11.8 altered by the presence of a crack at the interface betweenthe composite lamina and adhesive at θ = 90. Such cracks were observedin experimental tests of double-lap joints [123, 124]. During the service lifeof this kind of joint and in testing under cyclic loading, the crack facescan make contact with each other and slide in any direction. To show



the capability of the computational tools developed for the singularityanalysis of sliding friction contact with coupled in-plane and antiplanedisplacements, the angle of the fibres in the unidirectional composite laminawas changed from φ = 0 (parallel to the load in the original configuration inFig. 11.8) to φ = 90 (perpendicular to the load), while the fibres keep theirhorizontal position indicated in Fig. 11.8. An isotropic Coulomb frictionmodel, with ω = ωu in (11.36), is considered with a relatively large valuefor the friction coefficient µ = 1, in order to have a noteworthy influence offriction on the solution behaviour.

Solutions of the characteristic equation of the corner (11.111) are givenby pairs of characteristic exponents λ and characteristic friction angles ω.We searched for 0 < λ < 1 and −180 ≤ ω < 180. Plots of the contours ofthe function σmin(Kcorner(λ, ω)) from (11.111) for φ = 0 and 40 are shownin Fig. 11.13 as examples. Some of the zeros (global minima) are indicatedby arrows. For the case φ = 0, Fig. 11.13(a), besides the expected values ofthe characteristic friction angles ω = 0, ±90, ±180, characteristic valuesat ω = ±79.7 are unexpectedly found, which represent singular elasticsolutions with coupled in-plane and antiplane displacements, in spite of thesymmetry of this corner configuration with respect to the plane (x1–x2).For the case φ = 40, Fig. 11.13(b), all the singular solutions found havecoupled in-plane and antiplane displacements as expected due to the non-symmetric corner configuration. For all the solutions of (11.111) the energy

ω

λ

ω

λ(a) (b)

Fig. 11.13. Contour plots of σmin(Kcorner(λ, ω)) from (11.111) for fibre angles(a) φ = 0 and (b) φ = 40.



0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1

0 15 30 45 60 75 90

λλ

fibre angle φ (deg)

-180

-150

-120

-90

-60

-30

0

30

60

90

120

150

180

0 15 30 45 60 75 90

ω

fibre angle φ (deg)

(a)

(b)

Fig. 11.14. Plots of (a) characteristic exponents λ and (b) characteristic friction anglesω for corner B, Fig. 11.8, for a crack in sliding friction contact with friction coefficientµ = 1. Only characteristic values with singular elastic solutions satisfying the energy

dissipation condition are shown.

dissipation condition under proportional loading (Section 11.5.3) is checkeda posteriori, the arrows in Fig. 11.13 showing only those solutions (globalminima) which satisfy this condition.

The solutions of (11.111) are plotted in Fig. 11.14 as functions of thefibre angle φ. Only solutions satisfying the energy dissipation condition are



shown and the same symbols are used in both plots for the correspondingvalues of λ and ω. Only slight variations in the values of λ and ω areobserved. Some of the series of characteristic values shown in Fig. 11.14 areshorter than others because, surprisingly, the singular elastic solutions withthe remaining characteristic values violate the energy dissipation condition.

11.8 Failure Criterion for a Multimaterial Closed CornerBased on Generalized Fracture Toughness

This section introduces a quite general criterion for failure initiation at amultimaterial corner based on generalized stress intensity factors (GSIFs)and associated generalized fracture toughnesses. It is necessary to havean experimental procedure capable of generating the corresponding failureenvelope covering all fracture mode mixities.

Consider a multimaterial corner under a plane strain with two stresssingularities, represented by two terms (typically referred to as singularmodes) in the asymptotic series expansion (11.3) defined by characteristicexponents λ1 and λ2, which are assumed to govern the failure initiation atthe corner tip through the associated GSIFs K1 and K2. Then, a generalcorner failure criterion can be expressed as:

K = κC(ψ), (11.129)

where

K =

√(K1

K1C

)2

+(K2

K2C

)2

(11.130)

is a normalized GSIF modulus (a dimensionless magnitude), ψ is anormalized fracture-mode-mixity angle,

tanψ =K2/K2C

K1/K1C, (11.131)

and κC(ψ) is a dimensionless function of ψ giving the critical value ofK. K1C and K2C are the generalized fracture-toughness values for puresingular modes, i.e. when either K2 or K1 equals zero, respectively. Inparticular, they can be chosen so that κC(0) = κC(90) = 1. ψ = 0

corresponds to K1 > 0 and K2 = 0 while ψ = 90 to K2 > 0 andK1 = 0. Unlike the traditional fracture-mode-mixity definition for a crack,tanψ = (K2/K1), the inclusion of the generalized fracture toughnesses in(11.131) is due to the different dimensions of K1 and K2 in the present case.



The parameterization (ψ, κC(ψ)) defines a hypothetical failure envelopecurve in the plane (K1/K1C ,K2/K2C).

The size-scale effect should be taken into account when evaluatingGSIFs in geometrically similar specimens. From dimensional analysis, aGSIF can be expressed as [80]:

Kk = σnomR1−λkAk, (11.132)

where σnom is the nominal stress in the specimen, R is a characteristiclength of the specimen and Ak is a shape factor that takes into account thegeometry and material properties of the specimen.

Whereas a numerical procedure for the evaluation of GSIFs wasproposed in Section 11.6 and applied in Section 11.7, the determination ofparticular values of K1C and K2C, and of the whole failure envelope curveκC(ψ), requires the experimental testing of samples including the cornerunder several loading conditions. For the particular case of the bimaterialclosed corner B in the adhesively bonded joint in Fig. 11.8, a suitableprocedure proposed in [80] can be used to experimentally determine K1C,K2C and κC(ψ). This procedure is based on a novel modified configurationof the Brazilian disc specimen, including the bimaterial corner tip at thecentre of the disc (Fig. 11.15(a)) and subjected to a diametrical compressionP at an angle α (Fig. 11.15(b)).

GSIFs K1 and K2 were extracted from FEM results in [80] byemploying the least-squares fitting technique presented in Section 11.6.The numerical results for a particular configuration (t=1mm, R=1mmand P =100N) are depicted in Fig. 11.16. The diametrical compressionorientations providing (approximately) pure singular modes are α ≈ 13

adhesive

x

0ºCFRP

y

θr

t

R

adhesive

x

αP

P

0ºCFRP

y

(a) (b)

Fig. 11.15. Brazilian disc specimen with the bimaterial corner. (a) Geometry.(b) Loading.



-0,015

-0,010

-0,005

0,000

0,005

0,010

0,015

-0,150

-0,100

-0,050

0,000

0,050

0,100

0,150

0 30 60 90 120 150 180

K1

(MPa

mm

0.23

67)

K2

(MPa

mm

0.11

06)

angle α

K2

K1K2K1

adhesive

0ºx

α≈13ºP

P=100 N

P

adhesive

0ºx

α≈ ºP

P=100 N

P

adhesive x

α ≈143º

P=100N

0º

P

P

Fig. 11.16. Standardized values of K1 and K2 in the Brazilian disc specimen underdiametrical compression.

and α ≈ 115 for K2 ≈ 0, and α ≈ 60 and α ≈ 143 for K1 ≈ 0. Anglesα ≈ 13 and α ≈ 143, giving positive values of the non-vanishing GSIFs,were chosen to determine the generalized fracture toughnesses K1C andK2C , respectively. Let P exp denote the experimental load for which failureinitiation in the corner is observed, texp the real specimen thickness andRexp the real specimen radius. Then the following equation, obtained from(11.132), is used to determine K1C and K2C :

KkC = KFEMk

tFEM

texp

RFEM

Rexp

P exp

PFEM

(Rexp

RFEM

)1−λk

= KFEMk

σexpnom

σFEMnom

(Rexp

RFEM

)1−λk

, (11.133)

where superscript ‘FEM ’ denotes values from the above defined FEMmodel, and σFEM

nom and σexpnom are nominal stresses in the FEM model and

the real specimen, respectively.Figure 11.17 shows a real sample after failure and a scheme for the

failure path for α ≈ 13, the angle used to determine K1C .Experimental tests corresponding to any orientation of the diametrical

compression define critical pairs of GSIFs for which failure initiates in thecorner. All the critical pairs of GSIFs obtained can be represented in theplane (K1/K1C,K2/K2C) as shown in Fig. 11.18; see [81] for the detailsof these experiments. In Fig. 11.18, light (blue) circles are experimentalresults of single tests for particular load orientations (the load orientation



26º

13o

Fig. 11.17. Brazilian disc specimen tested (left) and failure path scheme (right) for aloading angle α ≈ 13.

-2

-1,5

-1

-0,5

0

0,5

1

1,5

-1,5 -1 -0,5 0 0,5 1 1,5

K2/K2C

K1/K1C

Experimental values

Average experimental values

Barroso et al., Refs. [123,124]

First quadrant

Third quadrant60º < αα < 115º

Second quadrant115º < α < 143º

Fourth quadrant13º < α < 60º

Barroso et al.Refs. [123,124]

envelope for the average failure load

α = 13º

α = 30º

α = 60º

α = 90º

α = 115º

α = 143º α = 150º

α = 120º

α = 0º = 180º

SAFE

UNSAFE

143º < α < 193º

ψ

)(ψκ C

Fig. 11.18. Experimental results for the critical pairs of GSIFs in the Brazilian discspecimens (circle marks) and double-lap joint specimens (triangular marks) and anapproximation of the corner failure envelope.



α is indicated on the plot), and black circles are the average values at thesame load orientation (α = 150 has only one specimen). The continuousline interpolates the black circles and represents an approximation of thefailure envelope curve, which can be used to define the corner failurecriterion in (11.129). Representations of the loading angle range leading toresults in each quadrant in Fig. 11.18 are plotted schematically to visualizethe correspondence between the loading angle α and the critical pairs ofGSIFs.

In addition to the experimental results obtained by the Brazilian discspecimen, Fig. 11.18 shows another set of experimental results obtainedpreviously by Barroso and co-workers [123, 124]. These results, indicatedby black triangles in Fig. 11.18, represent the critical pairs of GSIFs forwhich complete failure was observed in real double-lap joints between theunidirectional CFRP laminate and aluminium, see Fig. 11.8, subjectedto tension. Notice that both specimens, the Brazilian disc specimen andthe double-lap joint, have the same corner configuration at the cornertip with reference to geometry and materials. The critical pairs of GSIFsshown in Fig. 11.18 for both sets of experiments for the same fracturemode mixity are very close. The observed agreement between both sets ofexperiments is quite significant as the double-lap joints, although having thesame local corner configuration, are completely different in size, geometryand manufacturing process from the Brazilian disc specimens. While theBrazilian disc specimens were manufactured in an autoclave and theircharacteristic length (the radius) is 17mm, the double-lap joint specimenswere manufactured in a hot plate press and their characteristic lengthis 0.1mm (the adhesive layer thickness). Additionally, the Brazilian discspecimens were tested only a few days after manufacture whereas thedouble-lap joints were tested a long time after manufacture (about halfa year); this delay might lead to strength degradation due to moistureabsorption.

It is important to notice that the novel experimental procedureemploying the Brazilian disc specimen can directly be applied to thegeneration of failure criteria based on generalized fracture toughness forother multimaterial closed corners with two stress singularities under aplane strain. In fact, placing a very thin strip of Teflon R© at the interfaceso that it emerges from the corner tip can imitate a crack, as in [126] fora straight interface, allowing the present procedure to be applied to openmultimaterial corners with a corner angle ϑW − ϑ0 = 360.



11.9 Conclusions

A novel procedure for the singularity analysis of linear elastic anisotropicmultimaterial (piecewise homogeneous) corners in generalized plane strainincluding sliding friction contact surfaces has been developed andimplemented. This semi-analytic approach, based on the Lekhnitskii–Strohformalism of anisotropic materials, handles any kind of linear elasticmaterial. The approach avoids numerical uncertainties and instabilities,which may appear when degenerate materials are treated as limit cases ofnon-degenerate materials. Specifically in the generation of the characteristicmatrix of the corner, only the complex roots of the Lekhnitskii–Stroh sextic polynomial for an anisotropic material must, in general,be evaluated numerically. Nevertheless, for specific kinds of material,for example, transversely isotropic materials representing homogenizedunidirectional fibre-reinforced composites, even these roots can be evaluatedanalytically for any orientation of the material axes. The application ofthe transfer matrix concept for homogeneous single-material wedges alongwith a powerful matrix formalism for prescribed boundary and interface(transmission) conditions leads to a fully automatic generation of thecharacteristic system for open and closed (periodic) corners including anyfinite number of wedges and frictionless or frictional contact surfaces andany homogeneous orthogonal boundary conditions.

A general anisotropic friction model was considered. It is usefulfor modelling contact between anisotropic materials, in particular forunidirectional composite laminas. If the angles of friction shear stressesand relative tangential displacements in a contact surface are not knowna priori, which is the case where the in-plane and antiplane displacementsare coupled, these angles should be determined when solving the cornercharacteristic equation. In this case, the corner characteristic equationincludes, in addition to the characteristic exponent, one or more extraunknowns — the angles of friction shear stresses, depending on the numberof friction contact surfaces in the corner. This fact appears to be a newfeature of a corner characteristic equation to the best knowledge of theauthors. The general procedure developed for the corner singularity analysisis very suitable for straightforward computational implementation.

A general post-processing procedure to extract multiple generalizedstress intensity factors for multimaterial corners from a numerical solutionobtained by FEM or BEM has also been presented, implemented and tested.



The procedure employs least-squares fitting of a finite asymptotic seriesexpansion of the singular solution at the corner to the numerical results indisplacements and/or stresses along the boundaries and interfaces in thecorner. The procedure has no special requirements regarding the accuracyof the numerical results close to the corner tip, where larger errors in thenumerical solution are expected.

Singularity corner analyses have been carried out for a couple ofexamples of multimaterial corners in composites specimens: a transversecrack terminating at the interface between plies in a [0/90]S laminate, anda closed corner in a double-lap joint between a composite and a metal. Thecomputational tool developed has been validated by comparing the BEMsolution with the computed finite series approximation at the corner at a setdistance to the corner tip, whereas their values along the corner boundarieswere used in the fitting procedure. Furthermore, the corner in the latterexample was altered to include a friction interface crack, an additionalparametric study having been carried out for this particular case.

A novel experimental procedure for the determination of generalizedfracture-toughness values and the subsequent generation of a failurecriterion for closed anisotropic multimaterial corners under a plane strainwith two singularities has been proposed and tested. The procedure is basedon a kind of Brazilian disc specimen with the corner tip at its centreand loaded in compression at any position along the external perimeter.A failure envelope curve in the plane of the generalized stress intensityfactors normalized by the generalized fracture-toughness values has beendetermined experimentally, and used in the formulation of a failure criterionfor corners of this kind. Satisfactory agreement has been observed betweenthe predictions of this failure criterion and the failure loads of a real double-lap joint between a composite and a metal including such a corner.

Acknowledgements

The authors are grateful to Dr Antonio Blazquez and Dr Enrique Gracianifor their BEM codes and to Mr Israel G. Garcıa for carrying out calculationsin some of the numerical examples. The authors also thank Dr EnriqueGraciani for a correction of formula (11.133) in the first version of thischapter. Dr Daniane Vicentini has carried out and analysed tests with thenew Brazilian disc specimen. V. Mantic thanks Prof Dominique Leguillonfor his support and encouragement in analysing singularities with a slidingfriction contact. This work was supported by Junta de Andalucıa and



the European Social Fund (Proyectos de Excelencia P08-TEP-4051 andP08-TEP-4071) and by Ministerio de Ciencia e Innovacion de Espana(Proyecto MAT2009-14022).

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September 24, 2013 9:25 9in x 6in Mathematical Methods and Models in Composites b1507-index

INDEX

acoustic tensor, 219, 222

acoustoelasticacoustoelasticity, 214constant, 243, 244formula, 243tensor, 241

activation criterion, 382adhesive contact, 349adhesively bonded double-lap joint,

432, 476alternating minimization, 372

Ambartsumian multilayered theories,135

Ambartsumian–Whitney–Rath–Dastheory, 142, 150, 166, 168

angle-ply fibre-reinforced shell, 26angular sector, 440anisotropic

degenerate materials, 115, 430,436, 437, 450, 473

degenerate plates, 105, 107, 109elastic material, 212, 433extraordinary degenerate

material, 436, 437, 439, 452

friction, see frictionmultimaterial corners, 425non-degenerate materials, 430,

435–437, 449, 473non-degenerate plates, 105, 108,

109plates, 109, 141, 142

anisotropy index, 69, 82antiplane-strain solutions, 430, 473,

477

argument principle, 463

asymptoticexpansion, 403, 404, 406, 427,

428, 467, 475, 477, 479

homogenizationof elasticity problem, 7of heat conduction problem,

4

asymptotic homogenization method(AHM), 3, 4, 18, 56

autoclave, 486

back-tracking algorithm, 373Barnett–Lothe

integral formalism, 226

tensors, 115, 122, 125bifurcations, 173–175, 179, 180,

184–189, 195, 196, 198, 200–204body wave, 219

bondingimperfect (imperfect interface),

173, 187–189, 191, 197, 198,200–202

perfect, 4, 186, 188, 197, 201,202, 366, 439, 440, 448, 454

boundary

condition, 79, 425, 428, 432, 441homogeneous orthogonal,

440, 442, 444

matrix, 442, 444mixed-orthogonal, 79non-homogeneous, 428

layer, 11

measurements, 209, 243, 244

497


498 Index

boundary element method (BEM),373, 431, 466, 478

Brazilian disc specimen, 432, 483, 485

bridging, 311

brittle delamination, 367

carbon-fibre-reinforced polymer(CFRP), 266, 421

laminates, 476

carbon nanotube: Young’s modulus,40

Carrera’s unified formulation (CUF)shell finite elements, 152

geometry of cylindrical shells,152

governing equations, 156

mixed interpolation of tensorialcomponents (MITC) method,154

numerical examples, 160

Cauchy–Green tensors, 176

ceramic matrix composite, 402

characteristic

angular shape functions, 427,428, 429, 478

see also singular angularshape functions

elastic solutions, 461

see also singular elasticsolution

equation of the corner, 428, 461,462, 464

exponent, 427, 428, 461, 472,474, 477, 480, 481

see also singular exponent

friction angles, 461, 480, 481

matrix, 428

matrix of a closed multimaterialcorner, 460

matrix of an open multimaterialcorner, 457

sliding angles, 461

system for the singularityanalysis, 453, 457, 460

classical lamination theory, 104, 106

coating, 15, 173, 176, 198–200, 203,204

cohesive

contact, 376, 392

laws, 299, 303, 311, 312, 393

zone model, 378, 383

complementarity problems, 367, 369,387

complex analytic (holomorphic)function, 428, 433, 448, 453, 458,461–463

complex-variable methods, 103, 106,108, 109

compliance tensor, 80, 81, 85

composite

fibre-reinforced compositematerial, 15, 105, 118, 124,129, 209, 210, 393, 487

forming processes, 247

laminate, 25, 103, 105, 115, 119,122, 123, 125, 293, 307, 310,322, 341, 401, 402, 421, 425,475

manufacturing, 247, 270, 293

material, 208, 210, 401, 402, 421

random, see random

conductivity, 6, 12, 61, 65, 66, 94,260, 268, 271, 390

conservative integrals, 466

constitutive equation, 156, 176, 177,195, 211, 214, 241

continuum damage

mechanics (CDM), 299

models, 297, 301, 324, 329

convexity, 89, 356, 358, 360

strong, 213, 230

corner, 426

closed (periodic) multimaterial,440, 448, 458, 482–486

edge, 439

elastic anisotropic multimaterial,425, 427–432, 439, 453

failure criterion, 432, 482, 486

multimaterial, 426, 427

open multimaterial, 440, 457


Index 499

singularity analysis, seesingularity

tip, 426–428, 440, 486

Coulomb friction, see friction

coupled

criterion, 402, 406, 407, 411, 415,416, 421

mechanical-electrical analysis,119

stretching-bending analysis, 104,106, 130

crack

band model, 316, 318–320,325–328

deflection, 402, 411, 413–416,421

growth resistance, 302

impinging on an interface, 411

jump, 401, 405, 416–418

kinking, 402, 421

networks, 297, 331

terminating at an interface, 432,471

see also interface crack

critical energy release rate (ERR),300, 303, 324

cross-ply, 148, 321, 421, 471, 472

curse of dimensionality, 268, 275, 293

damage, 297, 299, 315, 351, 377, 386,431

activation functions, 325

discrete damage mechanics(DDM), 299

evolution, 301

mechanisms, 298, 301, 320, 324

modes, 299, 320, 324, 325

propagation, 301, 342

Darcy’s law, 89, 92, 280

debonding, 297, 350, 375, 393, 417,420

deformation gradient, 86, 176, 190,365

delamination, 175, 297, 300, 320, 323,335, 338–340, 349–351, 401, 402,413, 421

brittle, 367–369

elastic-brittle, 368

in visco-elastic materials, 388

mode mixity, 379

parameter, 350

thermodynamics, 389

weakening, 389

dimensional analysis, 483

direct method, 357

Dirichlet problem, 68

discrete damage mechanics (DDM),see damage

displacement

and stress function vectorsolution, 438

field, 22, 87, 106, 136, 141–143,144, 147, 149, 196, 287, 303,332, 350

formalism, 104, 105, 107, 109,112

dissipation

distance, 362

potential, 350, 361, 366, 371

dissipative character of friction, seefriction

double-lap joint, 425, 432, 475–479,485, 486

doubly non-linear inclusion, 352

driving force, 353, 356, 376–379

dual mode, 404, 409

dynamic

adhesive contact, 388

elasticity, 145, 209

effective

activation, 378, 389

coefficients of heat conductivity,7

elastic coefficients, 10, 16, 35,214, 221

stiffness moduli, 17, 18, 25–27,36, 40, 42, 44, 46, 48, 54, 55


500 Index

eigenvalue, 108, 110, 114, 117, 125,181, 183, 218–220, 222, 227, 231,236, 252, 254, 256, 258, 427,435–437, 461, 462

problem of the multimaterialcorner, 461

eigenvector, 108, 110, 114, 117, 125,219, 221–224, 226–230, 252–254,256, 435–438, 452

generalized, 223, 226–230, 436,437

elastic

anisotropic multimaterial corner,see corner

brittle delamination, 368, 370

delamination model, 350

solution, 183, 279, 308

linear, 306, 318, 409, 425,426, 431

singular, see singular

wave equation, 213, 214

see also isotropic elastic solid

elasticity, 4, 10, 22, 61, 79, 86, 94,103, 114, 118, 164, 176, 185, 210,214

fundamental matrix, 108, 111,115, 117, 435

linear, 91–94, 211, 287, 289, 291,363, 430

non-linear, 88, 89, 92–94

tensor, 82, 212, 233, 236, 241,292

incremental, 211, 233, 236

electro-elastic laminates, 118, 122,128

elliptic

linear system, 428, 430, 433

scalar linear second-orderequation, 430

ellipticity

loss of, 180, 186, 188, 316

strong ellipticity, 213, 215

energetic solution, 356

to elastic-brittle delamination,370

energy release rate (ERR), 302, 306,405, 412–414, 416, 417, 420

incremental, 405, 406

ensemble averaging, 65

ergodicity, 79

extended finite element method, seefinite element method (FEM)

extracting (or extraction of)generalized stress intensity factors(GSIFs), see stress intensity factor(SIF)

failure

criteria, 300, 320, 321, 324, 432,482, 486

envelope, 425, 432, 482, 483, 485

fibre, see fibre

initiation, 425, 431, 482

fibre

diameter, 210

failure, 297, 320, 323, 325

fracture, 298, 325

kinking, 299, 300, 321

reinforced composites, 105, 118,391

finite

bending, 174, 176, 189, 194, 195,201–204

flexure, 174, 189, 201

finite element method (FEM), 16, 19,89, 92, 127, 135, 138, 142, 148, 152,155, 160, 167, 251, 254, 263, 267,269, 287, 289, 306, 308, 316, 318,320, 326, 327, 329, 336, 369–371,406, 410, 425, 431, 466, 483

extended (x-FEM), 297, 299,330, 331

stochastic, 95

flow in porous media, 89, 90, 94

fracture, 297, 298, 302, 303, 307, 311,315, 323, 329, 372

(Griffith) criterion, 302, 303,309, 364, 405

linear elastic fracture mechanics(LEFM), 302


Index 501

mechanics, 299, 302, 310, 321

mode mixity, 313, 314, 482, 486

mode-mixity angle, 379, 384,482

mode sensitivity, 380, 383

model, 300

non-linear fracture mechanics(NLFM), 302

process zone, 302, 305, 307

toughness (or energy), 301, 302,304, 305, 307–310, 312–314,319, 322, 324, 335, 351, 364,374, 380, 383, 392, 401–403,413, 414, 417, 419, 421

generalized, 425, 432, 482,484, 486, 488

free edge, 340, 402, 404

friction

anisotropic, 425, 440–444, 446

coefficient, 441, 442, 444, 455,456, 465, 480

contact, 425, 431, 432, 443, 453,461, 462, 464, 465, 479–481

dissipation condition, 465

dissipative character of, 444, 447

isotropic Coulomb, 440, 444, 480

kinetic (or dynamic) coefficient,441, 442

or frictionless sliding, 425, 432,439–448, 455–463, 465,479–480

rule, 441

frictionless contact, 351, 431, 432,453, 465

generalized

eigenvector, see eigenvector

integral transforms, 11

plane strain, see plane strain

plane stress, see plane stress

stress intensity factor, see stressintensity factor (SIF)

grains, 61, 62, 64, 65, 68, 69, 74, 77,78, 83

grid-reinforced composite

material, 12, 16, 21

plate, 36, 44

shell, 28, 36, 41

Griffith fracture criterion, see fracture

H-integral, 466

hardening, 85, 93, 179, 299, 315, 316,381, 382, 386, 387

healing, 351, 364

hierarchy of bounds, 66, 80, 88

Hill–Mandel condition, 61, 62, 64, 79,84, 86, 90, 95

homogeneous orthogonal boundarycondition, see boundary condition

homogenization, 1, 6, 10, 11, 24, 61,94, 240, 243, 244, 248, 326, 349,352, 391, 416

honeycomb, 3, 21, 22, 26–28, 292

plates, 286

sandwich composite shell, 21, 26,27, 47, 48, 50

structure, 292, 293

hot plate press, 486

hygrothermal stresses, 115, 122

ill-conditioning, 129, 185, 470

in situ strength, 300, 321, 322

incremental variational problem,350

initial stress, 209–214, 230, 232,234–236, 238–240, 242–244

recovery of, 240

inner expansion, 404, 408

interface, 173, 175, 176, 179–182, 184,187–189, 191, 193, 197, 198,200–202, 204, 402, 403, 411–421

condition, 425, 428, 432, 440,441

matrix, 445, 447

non-homogeneous, 428

crack, 298, 432, 465, 468, 479,488

toughness, 419, 421

internal variable, 86, 324, 350, 351,372


502 Index

intralaminar damage modes, 325

inverse

analysis, 247, 249–251, 266, 270,293

problem, 209, 240, 266

Irwin criterion, 405

isotropic elastic solid, 178, 212, 214,221, 426, 431, 452

J2-deformation theory, 175, 178, 179,181, 186, 187, 204

material, 188

J-integral, 303

kinetic friction coefficient, see friction

kinetics, 248, 251, 272

Kirchhoff’s assumptions, 104, 106,138

laminate

[0/90]S laminate, 471

laminated

composite shell, 41

composites, 11, 12, 105, 297, 298,301, 321, 324, 332, 338, 389

plates, 103–105, 109, 114, 135,141, 148, 150

shell, 135, 150

layer-wise theories, 151

least-squares, 425, 432, 462, 466, 468,470, 474, 483

Lekhnitskii

bending formalism, 104

formalism, 104, 108, 109

multilayered theories, 135

Lekhnitskii–Ren plate theory, 140,142, 150, 167

Lekhnitskii–Stroh

formalism, 425, 429, 430, 433,438, 453, 471, 473

sextic equation, 435, 437, 462

sextic polynomial, 430

ligament, 412, 417, 419, 420

limiting velocity, 219–221

in isotropic material, 221

in orthotropic material, 236

long wavelength limit, 209

loss of uniqueness, 177, 179

macroscale, 5, 78, 83

matched asymptotic

expansion, 404, 406, 407

procedure, 404, 406

material

anisotropic elastic, seeanisotropic

composite, see composite

degenerate anisotropic, seeanisotropic

eigen-relation, 107–110

extraordinary degenerateanisotropic, see anisotropic

fibre-reinforced composite, seecomposite

generalized eigen-relation, 113

grid-reinforced composite, seegrid-reinforced composite

J2-deformation theory, seeJ 2-deformation theory

Mooney–Rivlin, 175, 178, 179,181, 187, 193–196, 199–202,204

non-degenerate, see anisotropic

random, see random

scaling diagram, 61, 69, 78, 84

softening, 302, 305, 310, see alsosoftening

matrix cracking, 297–300, 320, 321,323, 325–327, 332, 334, 335,338–340, 402

matrix formalism, 425, 432, 441, 453

maximum-dissipation principle, 353

mesh sensitivity, 319

mesoscale, 61, 63–66, 69, 74, 77–80,82, 83, 86, 87, 91–95, 299–301, 426

micromechanical scale, 300

microscale, 62, 78, 83, 299, 426

minimum dissipation potentialprinciple, 353, 362

minimum-energy principle, 352


Index 503

M -integral, 466

mixed formalism, 104, 105, 108, 109,111, 112

mixed-mode

bending (MMB), 314, 383

fracture, 312–315, 379, 383, 384,386, 387

model order reduction, 247

Mooney–Rivlin material, see material

multilayer, 145, 149, 166, 167,173–175, 179, 182–184, 187–193,204, 279

multilayered

composite, 279, 286, 335

composite laminate, 293

plates, 135, 141, 144, 166, 194,195, 248, 286

shells, 135, 138, 153, 156,164–166, 248

structures, 136–138, 149, 166,167, 174, 179, 187, 188, 195

theories, 135, 137, 167

multiscale framework, 251, 272

mutual recovery sequence, 360, 362,367, 377, 382

for FEM for delamination, 369

network shell, 26

Neumann problem, 68

neutral axes, 174, 194

non-associative model, 379, 381, 384

non-linear behaviour, 431

non-semisimple matrix, 436

extraordinary, 436

notch

ductility, 310

sensitivity, 309, 310

nucleation, 303, 401, 402, 422

optimization, 11, 21, 94, 247,249–251, 266–271, 359, 373

orthogonality relation, 442–445, 447,456

orthotropic, 233

reinforcements, 1–3, 12, 14–16,28, 35–38, 41, 42, 44, 45

outer expansion, 404, 407, 408

overdetermined homogeneous system,461

parabolic variational inclusion, 352

path independent integral, 403, 409

penetration, 413, 415–417

permeability, 4, 61, 89, 92, 94, 247,280, 284, 285

perturbation formula, 209, 210,234–237, 240

phase velocity, 209, 210, 215

of body wave, 219

of Rayleigh wave, see secularequation, 222, 229–237, 240,243

piecewise homogeneous materials,426, 428

Piola–Kirchhoff stress, 86, 177, 195,211

plane

strain, 174–177, 179, 180,186–188, 195, 316, 402, 409,421, 448, 473, 474, 476–478,482, 486

generalized, 115, 402, 421,425, 427–435, 438, 439,462

stress, 115, 124, 374, 409, 425

generalized, 427, 433

plasticity, 61, 85, 86, 91–94, 175, 204,298, 299, 305, 310, 315, 351, 353,382, 386–388

plate, 1, 19, 36, 44, 95, 104, 140–143,146, 148–150, 152, 248, 285, 286,291, 292

geometries, 279, 286

theory, 104, 144, 147, 286

Poisson’s ratio, 37, 85, 124, 140, 161,163, 374, 392, 409, 412

polarization, 215, 219, 221, 231

polycrystal, 63, 76, 78, 84, 95


504 Index

porous medium, 62, 89–91, 94, 247,279, 280, 284

post-processing, 140, 149, 431, 466power-logarithmic terms, 427

prestressed

anisotropic media, 211, 233, 235,236

elastic solids, 204half-space, 209 , 210

principal stress, 177, 194, 242

process

control, 247, 249

zone length, 305–308, 310–312,326

proper generalized decomposition(PGD), 251, 258–260

pultrusion processes, 266–269

QR decomposition, 470

quarter point elements, 466

quasi-brittle, 302, 303, 308–311, 316

quasistatic process, 349, 356

R-curve, 302, 307, 310, 311–313

radial approximation, 258

random

composite, 61, 62, 91field, 65, 95

materials, 63, 95

medium, 64, 66, 80,

microstructures, 61, 62, 66

rate-independent process, 350

Rayleigh wave, 209, 214, 229

phase velocity, see phasevelocity, see also secularequation

rebonding, 351

Reissner

mixed variational theorem, 144,168

multilayered theories, 135theorem, 149, 166

Reissner–Murakami–Carrera theory,144, 151, 168

repeated roots, 428, 430, 437

representative volume element(RVE), 61, 62, 94, 316, 416

residual stresses, 248

resistivity, 65–67

Riccati equation, 224–226

sandwich composite shell, 1, 3, 21, 22,26, 27, 46, 47, 51

scale-dependent bounds, 65, 66, 69,79, 80, 88, 91, 95

scaling, 61

function, 61, 66, 68, 72, 77, 80,81

secular equation

approximate, 238

for Rayleigh-wave velocity, 232

in orthotropic material, 234

semisimple matrix, 435

sextic

equation, 216, 222, 435, 437, 462

polynomial, 430, 434

shear bands, 174, 179–181, 187

shear locking, 163, 328

shell, 1, 19, 24–28, 35–38, 135, 136,138, 142, 144, 148, 149, 151–157,160–167

finite element, 152, 155, 167

structures, 138, 152, 162

theory, 144, 147

Signorini contact conditions, 364,389, 443

simple matrix, 435

single-material wedge, 425, 430, 439,440, 448

single-walled carbon nanotube(SWCNT), 39, 40

singular

angular shape functions, 427

see also characteristicangular shape functions

elastic solution, 306, 425, 427,460

exponent, 403, 411, 412, 427, 461

see also characteristicexponent


Index 505

point, 403, 404, 415, 416, 426

stresses, 426, 429, 431, 474, 477

singular value decomposition (SVD),274, 461, 464

singularity, 302, 306, 403, 411

(corner) analysis, 425, 429–432,444, 447, 453, 457, 460, 461,465, 470, 476, 480

exponent, see singular exponent

power-type, 428, 465

strong, 412

weak, 412

size effect, 303, 308, 309, 319, 342

sliding

friction contact, 425, 432, 440,442–444, 446, 447, 455,457–460, 462, 463, 465,479–481

see also friction

rule, 441

associated, 440, 441

non-associated, 440

slip weakening, 389

slowness section, 220, 221

of isotropic material, 222

softening, 299–303, 305, 310, 311,315–317, 319, 320, 324, 325, 327,328, 342

sparse grid, 268

spatial average, 65

splitting, 310, 326, 329, 330

stability inequality, 356, 380

statistical volume element (SVE), 61,62, 94

step-over mechanism, 417

stiffness matrix, 95, 156, 331, 332, 371

stochastic, 61

boundary-value problem, 65, 68

micromechanics, 61, 62, 64, 79

stored energy, 79, 350, 351, 355, 364,365, 371, 380, 383, 390

strength predictions, 431

stress

concentration, 119, 297, 315,323, 329, 330, 402, 405, 431

function vector, 118, 434,437–439, 442, 445, 448

stress intensity factor (SIF), 123, 124,129, 403, 405

generalized (GSIF), 403, 405,427, 429, 465, 474, 478

extraction (or evaluation)of, 404, 425, 431, 466, 474

Stroh

eigen-relation of the Strohformalism, 109, 110, 113, 117,125, 218, 435

expanded Stroh-like formalism,118, 119

extended Stroh-like formalism,115, 116, 122

formalism, 103–105, 108, 109,114, 117, 130, 209, 214, 223,228, 433, 435

see also Lekhnitskii–Strohformalism

orthogonality and closurerelations, 435, 437

Stroh-like bending formalism,104

Stroh-like formalism, 103–105,109, 112, 114–116, 119,130

Stroh’s eigenvalue problem, 218

angular average of, 226

subsonic, 222, 223, 229, 230, 238

surface

impedance matrix, 228

integral expression of, 228

of isotropic material, 231

of orthotropic material, 237

properties of, 230

topography, 425, 440

wave, 215

solution, 222, 223

symbol determinant, 430

symmetry

major, 211, 213, 215, 233

minor, 211, 213, 233


506 Index

tangential slip, 381, 384tensile

strength, 39, 313, 314, 321, 322,401, 403, 415, 419

stress, 308, 321, 330, 401,403–406, 409, 416, 418

texture, 425, 440thermodynamics of delamination, 389thermoelasticity, 84, 91–94, 115,

117thin three-dimensional composite

layer, 20, 21thin-walled composite reinforced

structure, 1, 2, 19three-dimensional periodic composite

material, 7three-point bending, 402time discretization

implicit, 357, 370semi-implicit, 380

transfer matrix, 185, 425, 430, 440,448, 450, 451, 453, 454, 458

transport, 4, 251, 272transverse

crack, 298, 335, 338, 339, 341,401, 402, 411, 413, 416,471–474

crack deflection, 402matrix cracking, 300, 320, 321,

323, 325, 326matrix cracks, 299, 331, 336,

338microcrack, 402

transversely isotropic, 209, 210, 232,236, 240, 241, 244, 430, 438,473

ultrasonic, 210, 222unidirectional fibre-reinforced

composite, 105 164, 487

composite laminas, 127, 321,430, 480, 486, 487

plastic ply, 325, 471, 473, 476unified formulation, 135, 152, 153,

155, 167uniform

displacement, 79, 88, 90traction, 79, 88, 90

unit-cell problem, 1–3, 6, 9, 11, 24,32, 34, 41, 300

unsymmetric laminate, 103, 124,128

V-notch, 402, 403, 406, 407, 409–411vanishing determinant condition, 428,

430, 462velocity

limiting velocity, 219, 221, 230,236

see also phase velocityvirtual crack closure technique, 302

wafer-reinforced shell, 1, 22, 26wavelength, 209weak solution, 355wedge, 425–427, 430, 439–442, 445,

448–456, 458, 464wide-sense stationary (WSS), 65Williams’ expansion, 403

see also asymptotic expansion

Young’s modulus, 37, 39, 40, 62, 124,161, 163, 304, 328, 374, 392, 409,412–414

zig-zag theory, 136–139, 144–147, 164,165

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