Multiscale Progressive Damage Modelling in Continuously ...

AuthorEmil Jacob Pitz, BSc

SubmissionInstitute of PolymerProduct Engineering

Thesis SupervisorUniv.-Prof. Dipl.-Ing.Dr. mont. Zoltan Major

Assistant Thesis SupervisorDipl.-Ing. Dr. Matei-Constantin Miron

10, 2017

JOHANNES KEPLERUNIVERSITAT LINZAltenbergerstraße 694040 Linz, Osterreichwww.jku.atDVR 0093696

Multiscale ProgressiveDamage Modelling inContinuously ReinforcedComposite Structures

Master’s thesis

to confer the academic degree of

Diplom-Ingenieur

in the Master’s Program

Polymer Technologies and Science

Eidesstattliche Erklärung

Ich erkläre an Eides statt, dass ich die vorliegende Masterarbeit selbstständig und ohnefremde Hilfe verfasst, andere als die angegebenen Quellen und Hilfsmittel nicht benutztbzw. die wörtlich oder sinngemäß entnommenen Stellen als solche kenntlich gemacht ha-be. Die vorliegende Masterarbeit ist mit dem elektronisch übermittelten Textdokumentidentisch.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Emil Pitz

ii

Acknowledgement

First of all, I wish to thank Univ.-Prof. Dipl.-Ing. Dr. mont. Zoltán Major, head of theInstitute of Polymer Product Engineering at Johannes Kepler University in Linz, Austria,for giving me the opportunity to write this thesis at his institute, employing me, therewithincluding me in a wonderful group of people, and for granting me vast freedom on the topicof my research, letting me develop own ideas in the field of my interest. As a large part ofthe research for this thesis was performed during a three months stay at Stevens Instituteof Technology in Hoboken, New Jersey, having a breathtaking view on the skyline ofManhattan, I also wish to thank him for the support during the stay, and for making thistrip possible in the first place.

I also owe particular thanks to Prof. Dr. Kishore Pochiraju at Stevens Institute ofTechnology, for all the help before and during my stay, supporting me in my research, andfor including me in his research group, thereby giving me the opportunity to meet andexchange with other young researcher and to gain invaluable experience.

I am especially grateful towards my colleague and friend Dipl.-Ing. Dr. Matei-ConstantinMiron, for teaching me, for challenging, and always encouraging, trusting and believing inme, for all our discussions and all the stories told.

For performing the majority of the experimental testing required for this thesis andfor always having best advice regarding the applied world, I wish to thank my friend andcolleague Michael Lackner. And finally, it is of great importance to me, to thank all hithertounmentioned colleagues and friends at the Institute of Polymer Product Engineering, moreparticularly my friend Dipl.-Ing. Anna Kalteis for countless exhilarating coffee breaks,Florian Kiehas, BSc. for always being astoundingly positive, and many others.

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Abstract

The current thesis deals with the failure modelling in continuously reinforced compositestructures taking into account the material’s inherent microstructure.

For that, first the implementation of a constitutive model for transversely isotropicdamage in fibre reinforced composites as a User Material Subroutine for Abaqus Stan-dard (UMAT) is presented. Damage initiation in the transversely isotropic linear elas-tically modelled composite is governed by the Strain Invariant Failure Theory (SIFT),utilising the first strain invariant and the second deviatoric strain invariant for dam-age initiation. To phenomenologically account for the material’s microstructure, the ho-mogenised macro strain is related to the micro strain by means of strain amplificationfactors determined from simulations of Representative Volume Elements (RVEs) with dif-ferent fibre arrangements utilising Periodic Boundary Conditions (PBCs). For damageevolution, linear softening behaviour is assumed. The implemented constitutive model issubsequently expanded to account for nonlinearity within the matrix phase of the compos-ite, by means of an additional damage variable evolving as a function of the distortionalstrain in the matrix phase. This is implemented as a User Material Subroutine for AbaqusExplicit (VUMAT).

Additionally, the Micromechanics Analysis Code based on the Generalized Method ofCells (MAC/GMC), developed by NASA at Glenn Research Centre in Cleveland, Ohio,is examined and composite Repeating Unit Cells (RUCs) are developed, having as inputparameters purely the constituents’ material behaviour. Within the implemented RUCs,plastic deformation and failure of the composite material’s matrix phase is taken into ac-count and modelling of fibre failure is based on statistical distribution of the fibre strengthby means of the Curtin fibre failure model.

For validation of the numerically implemented models, experimental tests of compositelaminates with varying layups are conducted. These experiments are simulated utilisingthe implemented models and the obtained results compared to the experimental data.

iv

Kurzfassung

Fokus der vorliegende Masterarbeit ist die Fehlervorhersage in faserverstärkten Verbund-werkstoffstrukturen unter Berücksichtigung der, dieser Materialklasse inhärenten, Mikrostruk-tur.

Dafür wird zunächst die Implementierung eines Materialgesetzes zur Modellierung vontransversal isotroper Schädigung in Verbundwerkstoffen als benutzerdefiniertes Materialin Abaqus Standard (UMAT) präsentiert. Schadensinitiierung in dem transversal isotrop,linear elastisch modellierten Werkstoff ist implementiert mittels der Strain Invariant Fai-lure Theory (SIFT), wobei erreichen eines kritischen Wertes der ersten Invarianten desDehnungstensors bzw. der zweiten Invarianten des deviatorischen Dehnungstensors zurSchadensinitiierung führt. Die Materialmikrostruktur wird phänomenologisch mittels Deh-nungsverstärkungsfaktoren miteinbezogen, wobei diese den makroskopischen Dehnungszu-stand auf den mikroskopischen zurückführen. Die Dehnungsverstärkungsfaktoren werdendabei durch Simulation von mikromechanischen Blöcken mit unterschiedlichen Faseranord-nungen ermittelt, wobei dafür periodische Randbedingungen verwendet werden müssen.Nach Schadensinitiierung im Material wird lineares Erweichungsverhalten modelliert. Dassoeben beschriebene Materialgesetz wird im weiteren Verlauf der Arbeit erweitert, umnicht-lineares Verhalten in der Matrixphase des Verbundwerkstoffes zu berücksichtigen.Dies ist mittels einer zusätzlichen Schadensvariable im Modell, die als Funktion der Ver-zerrungen in der Matrixphase wächst, als benutzerdefiniertes Material in Abaqus Explicit(VUMAT) implementiert.

Zusätzlich wird der Micromechanics Analysis Code based on the Generalized Methodof Cells (MAC/GMC), entwickelt am NASA Glenn Research Centre in Cleveland, Ohio,untersucht und repräsentative Einheitszellen (RUC’s) zur Verbundwerkstoffmodellierungwerden entwickelt, wobei nur das Verhalten von Matrixphase und Faserphase als Einga-beparameter bekannt sein müssen. Dabei wird plastische Deformation und Versagen derMatrixphase bei der Modellierung berücksichtigt, genauso wie die statistische Verteilungder Faserfestigkeiten mittels Verwendung eines Modells für Faserschädigung entwickelt vonCurtin.

Um die entwickelten numerischen Modelle zu verifizieren wurden experimentelle Zug-versuche von unterschiedlichen Verbundwerkstofflaminaten durchgeführt. Die Ergebnissevon Simulationen dieser Zugversuche, unter Verwendung der implementierten Materialm-odelle, werden mit den experimentellen Ergebnissen verglichen.

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1

Contents

Eidesstattliche Erklärung ii

Acknowledgement iii

Abstract iv

Kurzfassung v

Acronyms 3

Vector and Tensor Notation 4

1. Introduction and Objective 5

2. Strain Invariant Failure Theory 82.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2. Continuum Mechanics Framework . . . . . . . . . . . . . . . . . . . . . . . 9

2.2.1. Deformation Gradient Tensor . . . . . . . . . . . . . . . . . . . . . . 92.2.2. Strain Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2.3. Invariants of the Strain Tensor . . . . . . . . . . . . . . . . . . . . . 12

2.3. Determination of Strain Amplification Factors . . . . . . . . . . . . . . . . . 132.3.1. Periodic Boundary Conditions . . . . . . . . . . . . . . . . . . . . . 132.3.2. Micromechanical Blocks . . . . . . . . . . . . . . . . . . . . . . . . . 152.3.3. Amplification of the Strain Invariants . . . . . . . . . . . . . . . . . 192.3.4. Failure Bodies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.4. Implementation as a User Material in Abaqus Standard . . . . . . . . . . . 212.4.1. Introduction and Variables to be defined . . . . . . . . . . . . . . . . 212.4.2. Damage Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.4.2.1. Damage Evolution Law . . . . . . . . . . . . . . . . . . . . 242.4.2.2. Viscous Regularisation . . . . . . . . . . . . . . . . . . . . 26

2.4.3. Stress Calculation in incremental Formulation . . . . . . . . . . . . . 272.4.4. Determination of damaged Jacobian Matrix . . . . . . . . . . . . . . 282.4.5. Determination of Strain Energy Density and dissipated Energy . . . 292.4.6. Known Restrictions . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.5. Matrix Nonlinearity and development of a User Material in Abaqus Explicit 302.5.1. Introduction and Variables to be defined . . . . . . . . . . . . . . . . 30

2

2.5.2. Considering Matrix Nonlinearity in the Model . . . . . . . . . . . . . 322.5.3. Stress Calculation in incremental Formulation . . . . . . . . . . . . . 33

3. Micromechanics Analysis Code based on the Generalized Method of Cells 353.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.2. Derivation of the GMC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.2.1. Differentiation to the FEA . . . . . . . . . . . . . . . . . . . . . . . 413.2.2. Shear Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.3. Comparison with other Homogenisation Approaches . . . . . . . . . . . . . 443.3.1. Voigt and Reuss Model . . . . . . . . . . . . . . . . . . . . . . . . . 443.3.2. Mori Tanaka Homogenisation . . . . . . . . . . . . . . . . . . . . . . 453.3.3. Method Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.4. Modelling of Composite Structures . . . . . . . . . . . . . . . . . . . . . . . 473.4.1. Incremental Plasticity . . . . . . . . . . . . . . . . . . . . . . . . . . 473.4.2. Tsai-Hill Failure Criterion . . . . . . . . . . . . . . . . . . . . . . . . 503.4.3. Curtin Fibre Failure . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4. Experimental Results and Material Characterisation 554.1. Epoxy Resin Tensile Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554.2. Laminate Tensile Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

5. Results and Analysis 595.1. Critical Strain Invariants and Damage Evolution . . . . . . . . . . . . . . . 595.2. Micromechanical Composite Model . . . . . . . . . . . . . . . . . . . . . . . 615.3. Comparison with Experimental Results . . . . . . . . . . . . . . . . . . . . 63

5.3.1. 0°-tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635.3.2. 90°-tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 655.3.3. ±45°-tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665.3.4. ±15°-tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

6. Conclusion and Outlook 70

A. Appendix 73A.1. UMAT Readme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73A.2. VUMAT Readme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75A.3. Amplification Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

List of Figures 79

List of Tables 82

Bibliography 83

3

Acronyms

CTE Coefficient of Thermal Expansion

FEA Finite Element Analysis

FVR Fibre Volume Ratio

GMC Generalised Method of Cells

HFGMC High Fidelity Generalized Method of Cells

MAC/GMC Micromechanics Analysis Code based on the Generalized Method of Cells

MOC Method of Cells

PBC Periodic Boundary Condition

RT Room Remperature

RUC Repeated Unit Cell

RVE Representative Volume Element

SDV Solution Dependent Variable

SIFT Strain Invariant Failure Theory

UMAT User Material Subroutine for Abaqus Standard

VUMAT User Material Subroutine for Abaqus Explicit

4

Vector and Tensor Notation

Nota bene: The vector and tensor notation utilised throughout this thesis is based on thenotation introduced in the lecture ‘Tensorial Rheology’ held by Mag. Dr. Gerhard Ederat the Johannes Kepler University, Linz in 2016 and was chosen by the author because,even though sometimes seeming intricate, unambiguous differentiation between scalar andtensorial quantities is given while a comprehensible structure is maintained.

Generally the order of a tensor is denoted by a respective number of underlines, so that:

v denotes a vector (first order tensor),

M denotes a second-order tensor,

T(n)

denotes a nth-order tensor.

Inner product (also dot or scalar product) between two second order tensors S, T ∈ V ⊗ V

is defined as:S • T := siktik.

Tensor product between two second order tensors S, T ∈ V ⊗ V is defined as:(T S

): V → V

where a ↦→(T S

)(a) := T

(S (a)

).

Inverse of a Tensor T is denoted as:

T−1

where T T−1 = 1,

with 1 being the unit tensor. The transpose of a tensor T is denoted as:

T T .

The trace of a tensor T is defined by:

tr(T)

= tii.

Without establishing a definition at this place, the determinant of a tensor T is denotedas:

det(T T).

5

1. Introduction and Objective

Starting with the introduction of glass fibre reinforced polymer matrix composites in the1940s, the development of high performance composite materials continues to date and thegrowth of the composite market is expected to continuously grow [1]. As an example ofrecent projects in industry utilising composite materials on a large scale, the developmentof commercial aircraft such as Airbus A350 and Boeing 787 may be mentioned, with morethan 50 % of the aircraft’s material by weight being composite materials. In fields suchas aerospace, composites are of major interest as a result of their attractive weight-savingproperties, directly translating to fuel and with that cost saving during operation.

Being multi-phase materials, generally three structural levels can be distinguished incomposites: The micro-level, defining the distribution of inclusions in a fibrous ply, themeso-level, defining the internal structure of the reinforcement and variations of fibre ori-entation within a laminate layup, and the macro-level, describing the overall geometryof the composite structure [1], [2] (schematically depicted in Fig. 1.1). For modelling ofcomposites, that is in this context the simulation or analysis of the behaviour of a fullyconsolidated composite material, two basic approaches are commonly referred to: themacromechanical approach and the micromechanical approach [1]. While the macrome-chanical approach strictly models the composite on the macro-level utilising effective prop-erties usually obtained from experimental data, the micromechanical approach explicitlyconsiders the constituents’ behaviour and their arrangement on the micro-level in orderto determine the effective composite properties.

Figure 1.1.: Schematic representation of the structural levels observed in composite mate-rials.

6

When trying to account for damage and failure in composite materials, the macrome-chanical approach can become somewhat problematic, as widely varying behaviour anddiffering damage modes are observed depending on loading direction and history, arisingthe necessity to perform extensive composite testing to account for this highly anisotropicfailure behaviour [1]. By taking into account the composite micromechanics though, thephysics of damage on the micro-level can be captured, giving the opportunity to predictdamage based on the constituents behaviour or at least reducing the required extend ofexperimental data.

In the current thesis, two micromechanics based though inherently different modellingapproaches are examined with the objective to model the behaviour and damage of con-tinuously reinforced carbon fibre composites under monotonic tensile loading conditions.For this purpose, a constitutive model describing the behaviour of composites with failureof the material governed by the Strain Invariant Failure Theory (SIFT) and subsequentstrain softening is developed by the author and a micromechanical composite model basedon the constituents’ behaviour within the commercially available Micromechanics AnalysisCode based on the Generalized Method of Cells (MAC/GMC) is employed. An overviewof the general methodology followed is depicted in Fig. 1.2.

Figure 1.2.: Depiction of the general methodology followed in this thesis.

The SIFT, a composite failure theory, phenomenologically accounts for the material’s

7

micromechanics by means of strain amplification factors relating the macro strain within ahomogenised model to the composite micro strain. For implementation within the FiniteElement Analysis (FEA) software Abaqus, a User Material Subroutine for Abaqus Standard(UMAT) and a User Material Subroutine for Abaqus Explicit (VUMAT) are developed,providing the composite material’s constitutive response and utilising damage initiationgoverned by the SIFT and subsequent damage evolution. The implemented model isfurther expanded to account for matrix nonlinearity by evaluating distortional deformationwithin the matrix phase.

Additionally, MAC/GMC developed at NASA’s Glenn Research Centre in Cleveland,Ohio, is utilised to build a composite model purely from the constituents’ behaviour, takinginto account plasticity within the matrix phase and modelling failure of the individualphases. Compared to many common analytical micromechanics codes, MAC/GMC givesthe opportunity to actually determine local gradients (e.g. displacement, stress, strain,etc.) on the microscale due to the heterogeneous material composition and with thatemploy local failure models, while remaining computationally very efficient. Using theadditional subroutine FEAMAC, MAC/GMC can be coupled to the FEA software Abaqus,in order to evaluated the material response in the integration points of the finite elementsby means of the constitutive law provided in MAC/GMC.

The focus of this thesis is set on the numerical implementation of failure models ratherthan on the experimental characterisation of composites, though a number of compositetests were conducted in order to obtain necessary material parameters for the differentmodelling approaches and to validate the computational models.

The structure of the thesis is as follows. In Chapter 2, a description of the SIFT,deriving the continuum mechanical background, explaining the determination of strainamplification factors utilising Periodic Boundary Conditions (PBCs) in Abaqus and theirapplication, and giving an overview of the obtained failure envelopes, is provided, followedby the implementation of the SIFT as user subroutines for Abaqus, where the material’sconstitutive law is provided and progressive damage is employed.

Chapter 3 gives an overview of the theoretical background for MAC/GMC and placesthe theory in the context of other micromechanical modelling approaches. Furthermore,the material models for modelling fibre and matrix phase within MAC/GMC are derivedin detail.

Experimental material tests conducted in the course of this thesis and their evaluationis presented in Chapter 4. The obtained material properties are then employed in Chap-ter 5 in order to apply the composite models implemented as user subroutines in Abaqusdeveloped by the author and in MAC/GMC. Additionally, the obtained numerical resultsare compared to the experimental data to validate the implemented models.

Finally, in Chapter 6, an overview of the findings and the conducted work is given.

8

2. Strain Invariant Failure Theory

2.1. Introduction

The SIFT is a failure criterion initially proposed by Gosse and Christensen in [3] foranalysing damage initiation within the polymer of a composite material and then by Tay,Tan, Tan, et al. in [4] applied in a more general framework for determining failure incontinuously reinforced composites. Being based, as the name suggests, on the straininvariants, failure initiation in this physically based theory occurs once the first straininvariant, describing dilatational deformation within the composite, or the second de-viatoric strain invariant, governing the distortional deformation, exceed a critical value[3]. The composite micromechanics are taken into account phenomenologically by meansof strain amplification factors obtained from modelling idealised composite RepeatingUnit Cells (RUCs), examining three different fibre arrangements (square, diamond, andhexagonal, see Section 2.3.2) and extracting the local micromechanical strain from variouspositions within these RUCs.

Due to the inhomogeneity at micro level in a composite from the randomly distributedfibres embedded in matrix, strain and stress distributions within fibre and matrix at themicro level must be non-uniform even under uniform external loading [5]. Utilising thestrain amplification factors, the local micro strain state can be related to the applieduniform macro strain, presuming linear dependence of micro and macro strain state.

Within the SIFT, the first strain invariant is amplified in the matrix phase accounting fordilatational matrix failure while the second deviatoric strain invariant is amplified in thematrix as well as the fibre phase, taking into account failure due to distorional deformationwithin matrix or fibre phase. The critical invariants are obtained from laminate testing,though it should be emphasised that these critical values are effective properties of thelaminate and not criteria for pure matrix or fibre [4].

It has to be stressed that within the SIFT, the composite behaviour in tension andcompression is modelled similarly (except in the current implementation there is no failurein hydrostatic pressure from the first strain invariant, see Sections 2.3.3 and 2.3.4) andeffects such as fibre kinking or pressure dependent matrix behaviour are not taken intoaccount [6].

In the following section, the fundamentals of the SIFT will be discussed. For that,the extraction of the amplification factors from simulating micromechanical blocks, andthe implementation of the failure theory with progressive damage in the finite elementsoftware Abaqus as a UMAT with purely linear elastic matrix behaviour upon damage ini-

9

tiation is described. The presented constitutive model is subsequently expanded to takeinto account matrix nonlinearity in order to achieve better agreement with experimentalresults, which is implemented as a VUMAT for Abaqus Explicit. Within the developedUMAT and VUMAT, respectively, damage initiation is governed by the SIFT, while thematerials constitutive response prior to damage initiation is provided by a transverselyisotropic (non)linear elastic material law exhibiting linear softening behaviour after dam-age initiation. In addition to developing a UMAT and VUMAT, for determining the strainamplification factors a macro was developed that automatically, upon input of matrix andfibre material properties, fibre diameter, and Fibre Volume Ratio (FVR), simulates themicromechanical blocks utilising PBCs and writes the extracted amplification factors tofiles that are subsequently utilised within the UMAT and VUMAT subroutines.

2.2. Continuum Mechanics Framework

2.2.1. Deformation Gradient Tensor

For developing the SIFT at this place the required continuum mechanics framework shallbe established, being restricted to introducing the displacement gradient tensor, from thatdeveloping the strain tensor and its invariants.

Let the motion of a body, being imagined as an assemblage of particles with the positionof each of the particles described by a vector X in the orthonormal basis Ei in the initialconfiguration and by the vector x in the orthonormal basis ei in the deformed configuration,be described by a mathematical mapping function Φ relating the initial and the currentparticle positions [7]:

x = Φ (X, t) . (2.1)

It should be noted that commonly this mapping function is expressed as:

x = x(X, t) (2.2)

though in this context for better distinction to the vector x the notation Φ shall be utilised.In the remainder of this text it is also presumed that the bases Ei and ei coincide.

Considering now two material particles Q1 and Q2 in the vicinity of a third materialparticle P in the undeformed material configuration as depicted in Fig. 2.1, the positionsof Q1 and Q2 relative to P are given by the elemental vectors dXi:

dXi = XQi−XP i = 1, 2. (2.3)

With the mapping function Φ, the positions of the material particles can be determinedin the deformed configuration according to:

xp = Φ(XP , t), (2.4)

10

here only depicted for the particle P but working in a similar manner for the particles Q1

and Q2.

X3, x3

X2, x2X1, x1

undeformedconfiguration

deformedconfiguration

P

p

X

x(t)Q1

q1

dX1

dx1(t)

Q2

q2

dX2

dx2(t)

Figure 2.1.: General motion and deformation of an arbitrary deformable body where P ,Q1, and Q2 are three arbitrary material points in the initial configuration andp, q1, and q2 the same points in the deformed configuration (adapted from[7]).

With that the elemental vectors in the current configuration become:

dxi = xqi− xp = Φ(XP + dXi, t) − Φ(XP , t) i = 1, 2, (2.5)

which leads to the introduction of the deformation gradient tensor F :

F = ∇ Φ = ∂Φ

∂X. (2.6)

F transforms any elemental vector dX in the initial configuration to the correspondingvector dx in the current, deformed configuration:

dx = F dX. (2.7)

2.2.2. Strain Measures

Having introduced the deformation gradient tensor F , the right Cauchy-Green deformationtensor C is given according to:

C = F TF , (2.8)

and with that the Lagrangian or Green strain tensor E is defined as:

E = 12(C − 1

)(2.9)

11

where 1 is the second order unit tensor. With E, the difference in scalar product of theelemental vectors in initial and deformed configuration can be found by:

12 (dx1 • dx2 − dX1 • dX2) = dX1 • E dX2. (2.10)

It is at this place important to note that the Green strain tensor is merely one ofmany possible measures of strain, in particular applicable for problems involving largemotions of the solid body [8], in fact in [8] in the context of FEA it is said that ‘a largenumber of different strain measures exist. [. . . ] For the same physical deformation differentstrain measures will report different values in large-strain analysis. The optimal choice ofstrain measure depends on analysis type, material behavior, and (to some degree) personalpreference.’

Additionally to the Green strain tensor, the definition of the small strain tensor εs,commonly used in elementary elasticity textbooks, shall be given here [8]. Consideringcases where both the strains and the rotations are small, it is appropriate to take intoaccount only linear terms of the Green strain tensor, which gives the small strain tensorby [9]:

εs = 12(F T + F

)− 1. (2.11)

As the determination of the strain amplification factors for the SIFT (see Section 2.3)and all further simulations are conducted in Abaqus, an overview of the strain measureutilised within Abaqus in nonlinear, incremental framework shall be given here as well. Bydefault the strain increment is computed in Abaqus by integrating the rate of deformationtensor D over the time increment [8]:

∆ε =∫ tn+1

tn

D dt, (2.12)

where D is obtained from the velocity gradient tensor L by:

D = 12(L+ LT

), (2.13)

withL = F F−1 (2.14)

whereF =

∂F

∂t. (2.15)

With the strain increment, the total strain εn+1 at increment n+1 is determined from the

total strain εn

at increment n, being rotated into the current material frame of referenceby means of an incremental rotation tensor ∆R according to:

εn+1 = ∆R ε

n∆RT + ∆ε

n. (2.16)

12

If the principal directions remain fixed in the material axes the integrated strain is equiv-alent to the logarithmic strain εL given by:

εL = lnV =∫ tn+1

0D dt, (2.17)

where V is the left stretch tensor:

V =√F F T , (2.18)

otherwise the integrated strain is an approximation of the same.

2.2.3. Invariants of the Strain Tensor

For any second order tensor the three scalar quantities (here exemplary written for thestrain tensor ε):

I1 = tr ε

I2 = 12

[(tr ε

)2− tr

(ε2)]

I3 = det ε

(2.19)

are invariant with respect to change of vector base and are therefore called the invariantsof ε [10]. In particular for the strain tensor, the volumetric part of the strain εvol withina material can be obtained with the three invariants of the strain tensor by:

εvol = I1 + I2 + I3, (2.20)

though in practice the most significant component of the volumetric strain is given by thefirst strain invariant [11], therefore:

εvol ≈ I1. (2.21)

Having introduced the invariants of the strain tensor, the deviatoric strain tensor εdev

can be established, being a measure of distortion independent of volume change [10]:

εdev

= ε− 13I11. (2.22)

In the framework of the SIFT, the first invariant of the strain tensor I1 and the secondinvariant of the deviatoric strain tensor, denominated J2, are utilised as failure initia-tion criteria, representing the volumetric or dilatational and distortional effective strain,respectively.

Evaluating Eq. (2.19), the first invariant is given by [4]:

I1 = εx + εy + εz (2.23)

13

and the second deviatoric strain invariant utilising tensorial strain components accordingto:

J2 = 13(ε2

x + ε2y + ε2

z − εxεy − εxεz − εyεz

)+(ε2

xy + ε2xz + ε2

yz

)(2.24)

which can be simplified to:

J2 = 16[(εx − εy)2 + (εx − εz)2 + (εy − εz)2

]+(ε2

xy + ε2xz + ε2

yz

). (2.25)

Written in engineering strains with γxy = 2εxy, γxz = 2εxz, and γyz = 2εyz, J2 is given by:

J2 = 16[(εx − εy)2 + (εx − εz)2 + (εy − εz)2

]+ 1

4(γ2

xy + γ2xz + γ2

yz

). (2.26)

The SIFT utilises the equivalent von Mises strain, which can be obtained from the seconddeviatoric strain invariant by:

εvm =√

3J2. (2.27)

2.3. Determination of Strain Amplification Factors

For determining the strain amplification factors a python script was developed that, uponentering the geometrical (fibre diameter and fibre volume ratio) and material properties(transversely isotropic linear elastic fibre properties, isotropic linear elastic matrix prop-erties), automatically creates three Representative Volume Elements (RVEs) with thethree fibre arrangements proposed in [3] (see Section 2.3.2), assigns the material prop-erties, meshes the parts, applies PBCs, runs simulations for each fibre arrangement ineach loading scenario, and extracts the amplification factors at the respective positionsin the micromechanical blocks (see Section 2.3.2) writing them into files for use with theUMAT or VUMAT subroutine. The modelling assumptions and theoretical foundation inmodelling the RVEs for extracting the amplification factors and the actual amplificationof the strain invariants shall be described in detail on the following pages.

2.3.1. Periodic Boundary Conditions

Presuming an arbitrary composite material can be represented by a single periodic RVE,the modelling of such a RVE enables the examination of the behaviour of the compositeon the micro level and the prediction of the bulk material’s homogenised properties fromthe constituents’ properties [12]. In order to model a RVE within the framework of dis-placement based FEA in such a way that there is no influence of edge effects, that is theRVE has to be modelled as if being part of the continuous bulk material, commonly socalled PBCs are utilised.

For a cubical RVE of length L and with the edges of the cube defined by the orthogonalbasis system e1, e2, e3 the periodic boundary conditions (PBC) link the local displacementvector u of the nodes on opposite faces of the RVE to the far field macroscopic displacement

14

gradient F according to [12]–[15]:

u (x1, x2, 0) − u (x1, x2, L) =(F − 1

)Le3

u (x1, 0, x3) − u (x1, L, x3) =(F − 1

)Le2

u (0, x2, x3) − u (L, x2, x3) =(F − 1

)Le1.

(2.28)

In Abaqus, PBCs are applied utilising ‘linear constraint equations’ [8] on opposite nodesof the RVE, specifying the displacements of the nodes and that of an arbitrary referencepoint to impose the far field displacement gradient, as linear combination according toEq. (2.28). For this node based approach of applying PBCs, the mesh has to be conforming,that is the surface mesh is required to be identical on opposite sides of the modelled RVE,which can be achieved utilising adequate mesh constraints in Abaqus or by means of athird party mesh generator allowing the generation of symmetric meshes (e.g. meshingwithin Siemens NX using the function ‘2D dependent’).

To apply PBCs, a reference point has to be in the Abaqus assembly for every pairof opposite faces of the RVE and an equation constraint has to be created for everytranslational degree of freedom, respectively, for every individual node on the respectivesurface and its opposite node, coupling the node pairs to the respective reference point.

Exemplary imposing of PBCs is shown in Fig. 2.2 for an arbitrary 2D unit cell consistingof 3 × 3 elements with regular arrangement of the nodes on the edges so that node basedequation constraints can be applied. There, for nodes on the xmin and xmax (left and

NodeNode constrained to RPx

Node constrained to RPy

RPx

RPy

x1, y0 x2, y0 x3, y0

x1, y3 x2, y3 x3, y3

x0, y0

x0, y1

x0, y2

x0, y3

x3, y1

x3, y2

y

x

Figure 2.2.: Application of equation constrains for PBCs to an arbitrary 2D-RVE consist-ing of 3 × 3 elements.

right) faces of the RVE, the following equation constraints have to be applied:

ux (x0, yi) − ux (x3, yi) − ux (RPx) = 0

uy (x0, yi) − uy (x3, yi) − uy (RPx) = 0 where i = 0, 1, 2, 3,(2.29)

where ux and uy are the translational degrees of freedom in x and y-direction, respectively,for the respective node or reference point as depicted in Fig. 2.2. It has to be noted that

15

to impose the constraint Abaqus eliminates the first nodal variable specified [8], that is:

ux (x0, yi) = 0

uy (x0, yi) = 0 where i = 0, 1, 2, 3,(2.30)

therefore eliminating the degrees of freedom of the respective nodes. The remaining equa-tion constraints for the nodes on the ymin and ymax (lower and upper) faces are hencegiven by:

ux (xj , y0) − ux (xj , y3) − ux (RPy) = 0

uy (xj , y0) − uy (xj , y3) − uy (RPy) = 0 where j = 1, 2, 3,(2.31)

while x0, y0 must not be explicitly coupled to x0, y3 and RPy as this would lead to over-constraining of the node x0, y0, as its degrees of freedom are already constrained fromEq. (2.30). Sufficient coupling of x0, y0 to x0, y3 though is given as x0, y0 is already cou-pled to x3, y0 and RPx, which in turn is coupled to x3, y3 and RPy, which again in turnis coupled to x3, y0 and RPx, therefore all degrees of freedom of x0, y0 are automaticallycoupled to x0, y3 from existing couplings.

For a 3D-RVE this can be extended in a similar manner, where instead of corners nowedges must not be overconstrained. By applying displacements to the reference points inthe model, deformation is imposed on the RVE.

2.3.2. Micromechanical Blocks

The three fibre arrangements (square, hexagonal, diamond) proposed in [4] for determiningstrain amplification factors were adapted in such a way that an actual periodic RUC isobtained (Fig. 2.3), assuming that the three different fibre arrangements well represent theoccurring spatial fibre formations within an actual composite with statistically distributedfibres. As mentioned in the introduction of this section, the three micromechanical blocksare automatically created from user input of the desired FVR and fibre diameter by a selfdeveloped python macro in Abaqus. The interface between fibre and matrix phase withinthe RVEs, which generally represents an unknown in micromechanical models [5], is heremodelled as perfect, that is displacement continuity is imposed at the interface.

The developed micromechanical blocks thereby do not necessarily represent the smallestpossible RUC but could be chosen smaller [5]. Because of the applied PBCs in a validperiodic microcell, the actual size of the RUC does not influence the results but only af-fects the computation time to solve the problem. As computation time is not critical inthis given case, the RUCs were arbitrarily chosen to be as similar to the blocks proposedby Tay, Tan, Tan, et al. in [4] as possible, while obtaining actual periodic microcells.For extracting the amplification factors in every fibre arrangement three specific positionswithin the matrix (depicted M1 - M3), eight positions at the fibre matrix interface butwithin the fibre phase (F1 - F8), and one position at the middle of the fibre were chosen(F9), respectively (the notation of amplification factors can be seen in Fig. 2.4 for the

16

z

y

Smallest unit cellFibre phaseMatrix phase

(a) Square.

z

y

Smallest unit cellFibre phaseMatrix phase

(b) Hexagonal.

z

y

Smallest unit cellFibre phaseMatrix phase

(c) Diamond.

Figure 2.3.: Micromechanical blocks utilised for determining strain amplification factors.The locations for extracting the amplification factors are shown together withthe outline of the smallest unit cell required for modelling the respective fibrearrangement.

square fibre arrangement). These positions for extracting the amplification factors (po-sitions for all fibre arrangements are depicted in Fig. 2.3, for square fibre arrangementsee also Fig. 2.4) were chosen similarly to the positions proposed in [4] without furtherinvestigation, assuming the positions are capturing the most critical locations within theRUCs, that is the positions exhibiting the highest strain amplification. It should alsobe mentioned that generally any arbitrary number of amplification factors at any posi-tion within the micromechancial block can be extracted and utilised to amplify the straininvariants.

F 9

M1

M3

M2

F 1

F 5

F 7 F 3

F 8 F 2

F 6 F 4z

y

Figure 2.4.: Identification of the locations for extracting strain amplification factors de-picted for the square fibre arrangement (adapted from [4]).

For every fibre arrangement six different loadings (εxx, εyy, εzz, εxy, εxz, εyz) are exam-ined one after the other by applying a respective displacement to the RVE (see Fig. 2.5), inorder to determine the strain amplification for every respective loading. To approximatethe loading state within the bulk material, except for the applied displacement all degreesof freedom of the RVE are constrained, thereby modelling a plane strain state.

17

εxx y

z

x

εyy

y

z

x

εzz

y

z

x

εxy y

z

x

εxz

y

z

x

εyz

y

z

x

Figure 2.5.: Symbolic depiction of the loading scenarios imposed onto the micromechanicalblocks for determining the strain amplification factors (adapted from [4]).

In Fig. 2.6 the deformed configuration of the micromechanical blocks with hexagonalfibre arrangement is depicted for all loadings indicating the different strain distributionsfrom the fibre inclusion. For a strain applied in longitudinal (fibre) direction (εxx, seeFig. 2.6a) no strain concentration can be identified as fibre and matrix are loaded inparallel, which also indicates the applicability of the Voigt model (Rule of mixture) fordetermining the longitudinal composite stiffness (see Section 3.3.1). In contrast, for allremaining loadings, strain amplifications are exhibited due to the differing stiffness of fibreand matrix.

To obtain the strain amplification factors, element sets are created within the Pythonmacro, containing the elements of the respective phase (fibre or matrix) adjacent to theposition at which the amplification factor should be determined (Fig. 2.3). For the re-spective loading the corresponding strain component values at the elements’ integrationpoints (reduced integration linear hexahedral elements (C3D8R) are utilised, hence oneintegration point per element) in the respective set are extracted and averaged and theobtained value is normalised with respect to the average applied strain, determined byaveraging the respective strain component from all integration points within the micro-cell. With that the amplification factor Apos

ij for the globally applied (macro)strain εij at

18

εxx, εyy, εzz (-)2.2 · 10−2

2.0 · 10−2

1.9 · 10−2

1.7 · 10−2

1.6 · 10−2

1.4 · 10−2

1.3 · 10−2

1.1 · 10−2

1.0 · 10−2

8.5 · 10−3

7.0 · 10−3

5.5 · 10−3

4.0 · 10−3 (a) εxx. (b) εyy. (c) εzz.

εxy, εxz, εyz (-)2.9 · 10−2

2.7 · 10−2

2.4 · 10−2

2.2 · 10−2

2.0 · 10−2

1.8 · 10−2

1.5 · 10−2

1.3 · 10−2

1.1 · 10−2

8.7 · 10−3

6.5 · 10−3

4.2 · 10−3

2.0 · 10−3 (d) εxy. (e) εxz. (f) εyz.

Figure 2.6.: Deformed configuration of the micromechanical blocks with hexagonal fibrearrangements at a respective applied strain of εij = 0.01.

position pos in the respective micromechanical block is determined by:

Aposij =

εposij

εij. (2.32)

All extracted amplification factors in the different loadings are subsequently exported astext files, each containing the factors for one fibre arrangement, respectively. Havingthree micromechanical blocks, each in six loading conditions and with twelve positions forextracting amplification factors a total of 216 (3 × 6 × 12) factors are obtained. It shouldbe noted that the amplification factors have to be obtained only once for every set of fibreand matrix properties and FVR, where for every FVR the size of the respective RUCs isadapted such that the desired FVR is obtained.

Exemplary in Table 2.1 the amplification factors obtained at the M1 and F1 position(see Fig. 2.4) with input data according to Table 5.3 are depicted, while an entire listof all determined amplification factors is given in Appendix A.3. As an example in thesquare fibre arrangement with a globally applied strain of εyy = 0.01, at the M1 positiona local strain of 2.073 × 10−2 is extracted, which results in an amplification factor ofAM1,square

yy = 2.073, which can be found in Table 2.1. As mentioned before, for loading infibre direction (εxx) no strain amplification is observed (amplification factor equal to one)in any arrangement or location.

19

Table 2.1.: Amplification factors determined from the varying fibre arrangements at theM1 and F1 location within the micromechanical blocks.Position εxx (-) εyy (-) εzz (-) εxy (-) εxz (-) εyz (-)

M1, square 1.000 2.073 0.934 2.888 0.492 1.379M1, diamond 1.000 1.578 1.583 1.470 1.485 0.654M1, hexagonal 1.000 1.400 0.941 1.894 0.928 1.480F1, square 1.000 0.613 0.543 0.352 0.287 0.438F1, diamond 1.000 0.468 0.527 0.274 0.332 0.551F1, hexagonal 1.000 0.476 0.481 0.279 0.293 0.515

2.3.3. Amplification of the Strain Invariants

For determining the critical invariants, the strain components extracted from a homogenisedmaterial model (εxx, . . . ) of the composite are amplified with the respective amplificationfactors (Apos

i ), where pos indicates the position within the micromechancial blocks and i

the corresponding loading.The amplified invariants are hence obtained as follows, the superscript a indicating the

amplified invariant at position pos:

Ia,pos1 = Apos

xx εxx +Aposyy εyy +Apos

zz εzz (2.33)

Ja,pos2 = 1

6

[(Apos

xx εxx −Aposyy εyy

)2+ (Apos

xx εxx −Aposzz εzz)2 +

(Apos

yy εyy −Aposzz εzz

)2]

+

14

[(Apos

xy εxy

)2+ (Apos

xz εxz)2 +(Apos

yz εyz

)2],

(2.34)

where the first strain invariant is amplified with only amplification factors in the matrixphase and the second deviatoric invariant is amplified in the matrix and fibre phase,respectively.

After amplification with the respective factors from all three micromechancial blocks,for failure initiation the maximum obtained values are determined:

Ia1 = max

pos(Ia,pos

1 ) where pos = M1,M2,M3 (2.35)

Ja2,matrix = max

pos(Ja,pos

2 ) where pos = M1,M2,M3 (2.36)

Ja2,fibre = max

pos(Ja,pos

2 ) where pos = F1, . . . , F9. (2.37)

Where for J2-driven failure the equivalent or von Mises strain is utilised for failure initia-tion:

εmatrixvm =

√3Ja

2,matrix

εfibrevm =

√3Ja

2,fibre.(2.38)

Comparing these amplified invariants with the respective critical values, three failure

20

criteria are obtained while four failure modes can be distinguished:

• Matrix failure due to dilational deformation within the matrix phase (Ia1 ),

• Matrix failure due to distortional deformation (Ja2,matrix) within the matrix phase,

• Fibre failure, where the critical invariant in the fibre phase is obtained at the F9-position, that is at the centre of the fibre (Ja,F 9

2,fibre),

• Interface failure when Ja2,fibre is critical at the F1-F8 position.

It is hence hypothesised that the second deviatoric strain invariant amplified in the fibrephase incorporates not only the fibre but also the interface behaviour, as a number ofamplification factors are obtained very near to the fibre-matrix interface [4]. Tay, Tan,Tan, et al. in [4] though mention that ‘it is very difficult to test this hypothesis directly’.For the implementation described within this thesis this hypothesis though was adaptedand good results were obtained.

Furthermore, in implementing the SIFT it is assumed that there is no failure initiationfrom the first strain invariant amplified within the matrix phase (Ia

1 ) in hydrostatic pres-sure, that is the critical invariant is positive (see also Section 2.3.4). In the development ofthe SIFT in [3], [4], [16] the influence of hydrostatic pressure on the composite behaviour isnot examined though Pinho, Darvizeh, Robinson, et al. in [6] establish open failure slopesin hydrostatic pressure within the LaRC05 composite failure criterion.

2.3.4. Failure Bodies

From amplification of the strain invariants three failure criteria are obtained governingdamage initiation within the composite material. These failure criteria create three partlyintersecting failure bodies within the general stress space defining the borders of materialintegrity. In order to illustrate the shape of the failure bodies and to examine the influenceof strain amplification on the shape, in Fig. 2.7 the failure bodies are depicted in the stressspace (nota bene!) for σ11 and σ22, while all remaining stresses are kept at zero, utilisingthe amplified strains (Fig. 2.7a) and the homogenised strains (Fig. 2.7b), respectively, andhaving as input parameters the properties determined in Chapter 5.

In Fig. 2.7a the three failure bodies resulting from the three amplified invariants canbe distinguished with the ‘zone of no failure’ marked, though, as the second deviatoricstrain invariant amplified within the matrix phase accounts for distortional deformationof the matrix, in the depicted stress state with all stresses except σ11 and σ22 being zero,initial failure in any failure mode is governed by either the first strain invariant withinthe matrix phase, or the second deviatoric strain invariant in the fibre phase (for betterrepresentation therefore the εmatrix

vm failure body is not depicted in its entirety). As notedearlier, it is presumed that only positive values of the first invariant lead to failure, that is,in matrix compression (negative σ22) matrix failure occurs only once the second deviatoricmatrix strain invariant reaches a critical value.

21

−2,000 2,000

−500

500

no failure

σ11

σ22I1εmatrix

vm

εfibrevm

(a) Amplified strains.

−2,000 2,000

−500

500

no failure

σ11

σ22I1εmatrix

vm

εfibrevm

(b) Homogenised strains.

Figure 2.7.: SIFT failure bodies resulting from the three critical invariants. The failurebodies obtained with amplified (2.7a) and homogenised strains (2.7b), respec-tively, are compare utilising the same critical invariants for both cases.

Comparing the failure bodies obtained from the amplified strain values (Fig. 2.7a) andobtained from the homogenised strain values (Fig. 2.7b) it is obvious that by amplifyingthe strain values, the shape and the orientation of the failure body within the stress spacechanges.

2.4. Implementation as a User Material in Abaqus Standard

2.4.1. Introduction and Variables to be defined

For implementing user defined constitutive models in Abaqus Standard the Abaqus usersubroutine UMAT in the programming language Fortran has to be utilised [8]. Withina UMAT subroutine, the new stress state at the end of an increment according to therespective constitutive law has to be provided and for the Newton algorithm utilisedby Abaqus in predicting the next increment, the Jacobian matrix (also tangent stiffnessmatrix) of the constitutive equation:

∂∆σ∂∆ε , (2.39)

where Voigt notation is utilised here and ∂∆σ is the stress and ∂∆ε the strain increment,has to be given. Correct determination of the Jacobian matrix thereby is required forquadratic convergence of the Newton scheme, while an incorrect definition slows conver-gence considerably (linear convergence), if not impedes convergence entirely, though theresults are not affected.

Additionally the specific elastic strain energy and, in order to assess the influence ofviscous dissipation (see Section 2.4.2.2) utilised to improve convergence in the currentmodel, the creep dissipation energy are to be defined within the UMAT subroutine, thoughthese quantities do not affect the results but are provided in the output of a simulation

22

purely for information purposes. To save material point data, such as damage variables,so called Solution Dependent Variables (SDVs) are utilised.

To obtain the required quantities within the UMAT the simulated composite materialis modelled transversely isotropic linear elastic prior to damage initiation, where damageinitiation is governed by the SIFT. For that it is necessary to read-in the files containingthe strain amplification factors determined with an additional Python macro, which isobtain by means of the user subroutine UEXTERNALDB in Abaqus, where the factorsare stored as common block, that is a variable that is saved between calls to the subroutineand therefore available at every increment, and provided to the UMAT subroutine. Ascommon blocks can restrict thread safety (availability of multi CPU usage for the analysis)when there are multiple write accesses from different threads to the same common blockat the same time, the amplification factors are only read-in once at the beginning of theanalysis (or beginning of a restart analysis) and subsequently read-only access occurs toensure thread safety.

After damage initiation, a damage evolution law is defined and stress and Jacobianmatrix have to be modified accordingly. An overview depicting the implementation ofthe UMAT is given in Fig. 2.8, while the determination of these required quantities fora user material utilised in modelling composites with the SIFT for failure initiation andprogressive damage shall be discussed in the following sections.

2.4.2. Damage Evolution

For modelling failure of the material subsequent to damage initiation in the current im-plementation of the SIFT as a UMAT progressive damage is introduced in the model,which shall be discussed in this section. The present damage formulation is based onthe ‘Hypothesis of Strain Equivalence’, where a constitutive equation of a damaged mate-rial is determined from the corresponding constitutive equation of a fictitious undamagedmaterial merely by replacing the stress tensor σ within the constitutive equation by thecorresponding effective stress tensor σ, that is the stress acting on the undamaged mate-rial, while the same strain state is given in the damaged and undamaged material [18],[19]. Therefore, in case of a linear elastic response prior to damage initiation with theinitial or undamaged compliance tensor S0

(4), it can be found that:

ε = S0(4)σ =

[S0

(4)M

(4)(d)]σ = S

(4)(d)σ, (2.40)

where M(4)

(d) is the fourth order damage effect tensor, d the damage variable describingthe current state of damage, and S

(4)(d) the current (damaged) compliance tensor.

Knowing the initial and damaged compliance tensor, the damage effect tensor can there-fore be obtained by:

M(4)

(d) =(S0

(4)

)−1S

(4)(d) , (2.41)

23

Apply strain increment ∆ε

Read-in amplification factors

Start of Analysis

εn+1 = εn + ∆ε

Calculate undamagedstress and Jacobian:σun

n+1 = σunn + C ∆ε

∂∆σ∂∆ε = C

Failure criteria

Damageonset?

Calculate damage variablesand damage effect tensor M

Calculate damaged stress:σn+1 = M−1 σun

n+1,

Update ∂∆σ∂∆ε

Calculate strain energy and‘creep’ dissipation energy

Abaqus solver

Converged?

yes

no

For

each

inte

grat

ion

poin

t

Nex

tin

crem

ent

until

end

ofan

alys

is

yes

Iter

ate

until

conv

erge

d

no

UMAT

UEXTERNALDB

Figure 2.8.: Schematic describing the implementation of the SIFT as a UMAT in AbaqusStandard (adapted from [17]).

relating the effective (undamaged) stress σ to the current stress σ by:

σ = M−1(4)

(d) σ, (2.42)

while the damaged (current) stiffness tensor C(4)

(d) can be obtained from the undamagedstiffness tensor C0

(4)by:

C(4)

(d) = M−1(4)

(d) C0(4). (2.43)

In Voigt (contracted) notation the damage effect tensor can be written as a 6x6 matrix,

24

relating the effective and acting stress vector:

σ = M−1 (d) σ. (2.44)

For reasons of simplicity, the contracted notation of stresses and strains will be utilisedthroughout the following sections.

In the damaged material state, the current compliance matrix for an orthotropic material(S (d)) is determined according to Matzenmiller, Lubliner, and Taylor in [20] and extendedfor the 3D-case according to Wang, Pineda, Ranatunga, et al. in [21]:

S (d) =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

1(1−d1)E11

− ν12E11

− ν13E11

0 0 0− ν21

E221

(1−d2)E22− ν23

E220 0 0

− ν31E33

− ν32E33

1(1−d3)E33

0 0 00 0 0 1

(1−d4)G230 0

0 0 0 0 1(1−d5)G13

00 0 0 0 0 1

(1−d6)G12

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦, (2.45)

with the six scalar damage variables d1,. . . ,d6. These damage variables are set to zerofor the undamaged material (in that case the damaged compliance matrix in Eq. (2.45) issimilar to the compliance matrix for an arbitrary linear elastic orthotropic material) andupon damage initiation and further straining evolve to one, for the fully damaged stateaccording to the damage evolution law (see Section 2.4.2.1) [8].

Having a transversely isotropic material, it is assumed that only the damage variablesfor fibre failure (d1, longitudinal direction) and matrix failure (d2, transversal direction)are independent while the remaining damage variables are calculated from the followingrelationships [22]:

d2 =d3 (2.46)

d4 =1 − (1 − d2)(1 − d3) = 1 − (1 − d2)2 (2.47)

d5 =1 − (1 − d1)(1 − d2) (2.48)

d6 =1 − (1 − d1)(1 − d3) = 1 − (1 − d1)(1 − d2), (2.49)

where the damage variables in shear are obtained from the respective contributing normalcomponents.

2.4.2.1. Damage Evolution Law

For the evolution of the damage variables subsequent to damage initiation a linear damageevolution law was employed in the presented implementation [8]:

d (δeq) =δfinal

eq

(δeq − δinit

eq

)δeq

(δfinal

eq − δiniteq

) (2.50)

25

with δeq being the equivalent displacement and the superscript init and final denotingequivalent displacement at failure initiation and at maximum degradation (zero remainingstiffness), respectively. The given evolution law results in linear softening of the material(see Fig. 2.9). To prevent ‘healing’ of the material upon unloading, the damage variableat time t is defined as:

di (t) = maxt∗≤t

{di (t∗)} , (2.51)

which guarantees that unloading and subsequent reloading always follow the same pathindependent of the damage state (see also Fig. 2.9).

δiniteq δfinal

eq

σiniteq d = 0

d = 0.25

d = 0.5

Equivalent displacment δeq

Equi

vale

ntst

ressσ

eq

Figure 2.9.: Linear damage evolution law.

In the current implementation for every initiation criterion an independent evolutionlaw is utilised resulting in three different laws evolving damage variable d2, that is ma-trix failure due to I1, εmatrix

vm , and εfibrevm at the interface and a single law in fibre failure

(d1, εfibrevm -failure), though the laws for fibre damage evolution and interface failure are

modelled similarly as both are determined from the same initiation criterion. The equiv-alent displacement at failure initiation and final failure is obtained from the respectivestrain invariant at initiation (critical strain invariant) and final failure multiplied withthe characteristic element length (here designated as lc) at the beginning of the analysis,which is provided to the subroutine by Abaqus as variable, while the current equivalentdisplacement is obtained from the respective invariant and the critical element length inthe current increment, exemplary here given for I1:

δiniteq,I1 = I1l

t=0c

δfinaleq,I1

= cI1lt=0c

δeq,I1 = I1ltc,

(2.52)

where the factor c is a fitting factor input by the user in order to define the damage

26

evolution law and given according to:

c =δfinal

eq

δiniteq

. (2.53)

In order to establish a valid damage evolution law it has to be true that c > 1, wherehigher c-values lead to more ductile damage evolution while a value close to one representsfaster stiffness degradation. Therefore c relates to the strain energy release rate Gc, whichcan be obtained by integrating the stress-displacement curve (Fig. 2.9) defined from thedamage evolution law [8]. A total of three fitting factors (cI1 , c

εmatrix,cvm

, cεfibre,c

vm) have to

be defined by the user of the UMAT in order to define all damage evolution laws, that isone factor for every initiation criterion.

Defining the evolution laws in terms of a fitting factor instead of the physically basedstrain energy release rate in this case leads to simplified implementation of the progressivedamage as first, the strain energy release rate had to be adjusted for mesh size utilisingfor example the ‘crack band theory’ as proposed in [22] and second, because by definingthe damage evolution law purely in displacements, there is no necessity to formulate anequivalent stress, which proved to be difficult for the current failure theory as a resultof the amplification of the macro-strains with strain amplification factors. Hence thedamage evolution law is fully defined by having the material stiffness, the damage initiationcriterion, given by the critical strain invariants, and a fitting factor c.

It should be mentioned that the damage variables d1 and d2 actually must not evolve toone, as there has to be some remaining stiffness in order for the Newton scheme to convergeand element deletion is not available within a UMAT subroutine [16]. In the currentimplementation, the user has to specify the maximum value for the damage variables (e.g.0.99, that is 1 % stiffness remaining once the material is ‘fully damaged’). If no progressivedamage is desired in the model, this value has simply be set to zero, otherwise a valueclose to but less than one is recommended.

2.4.2.2. Viscous Regularisation

Materials exhibiting stiffness degradation often exhibit convergence difficulties when util-ising implicit analysis algorithms such as implemented in Abaqus Standard as the tangentstiffness matrix can become non positive definite [22]. In order to sustain a positive definiteJacobian matrix for sufficiently small time increments, a viscous regularisation scheme isutilised, where the regularised damage variable dv is defined by the following evolutionlaw [22]:

ddv

dt = 1η

(d− dv) , (2.54)

with the viscosity coefficient η and d denoting the non-regularised damage variable.Utilising numerical integration the regularised damage variable can be discretised in

27

time t as follows [8]:

dv |tn+∆t=∆t

η + ∆td |tn+∆t + η

η + ∆tdv |tn . (2.55)

Choosing a small value for the viscosity parameter η usually results in better convergence ofthe Newton scheme, due to slower evolution of the damage variable as the time increment∆t is reduced, without significantly influencing the results, as the solution of the ‘viscoussystem’ relaxes towards the inviscid case for t

η → ∞ [22]. To estimate the influence ofviscous regularisation, the dissipated energy as a result of regularisation is integrated andoutputted to the history output (see Section 2.4.5). This energy should be small comparedto the strain energy in the model [8].

2.4.3. Stress Calculation in incremental Formulation

In the undamaged state the total stress at the end of the current increment can be deter-mined in incremental formulation from the previous stress state σn, which is automaticallyrotated into the current material frame of reference by Abaqus, and the strain increment∆ε by [8]:

σn+1 = σn + C ∆ε. (2.56)

Once damage occurs in the model, the effective stress σ, that is the stress acting in afictitious undamaged material subjected to a similar strain (see Section 2.4.2), which isobtained with the undamaged material stiffness matrix according to Eq. (2.56), can berelated to the actual stress acting on the damaged material, utilising the inverse of thedamage effect tensor M :

σn+1 = M−1 (dn) σn+1 = M−1 (dn)[σn + C ∆ε

]. (2.57)

The above relation is implemented within the UMAT-subroutine, where the effective orundamaged stress, being equal to the stress in the undamaged configuration, is determinedfor every iteration and upon damage initiation the damage effect tensor is additionallyobtained to calculate the stress within the damaged material from the effective stress. Itshould be added that by having a material orientation defined in the model, Abaqus takesinto account the rotation of the material such that the strain vector is always definedin terms of the material orientation [8], hence this has not to be accounted for in thesubroutine.

28

2.4.4. Determination of damaged Jacobian Matrix

In the user subroutine the stress increment can be obtained from:

∆σ = Cd

∆ε, (2.58)

where Cd

is the current stiffness matrix obtained from the initial stiffness matrix C0 by:

Cd

= M−1 C0. (2.59)

When there is no damage present in the model, the damaged stiffness matrix is equal tothe undamaged.

By differentiating Eq. (2.58) the Jacobian matrix can be obtained [8]:

∂∆σ∂∆ε = C

d+∂C

d

∂εε

= Cd

+(∂C

d

∂d1ε

)(∂d1∂ff

∂ff

∂ε

)+(∂C

d

∂d2ε

)(∂d2∂fm

∂fm

∂ε

),

(2.60)

where ff is the fibre failure criterion (equivalent strain in fibre phase, εfibrevm ) and fm the

respective active (whichever is higher) matrix failure criterion (I1, εmatrixvm , or εfibre

vm in thecase of interface failure).

In order to improve convergence in the model, viscous regularisation is utilised withinthe subroutine for both damage variables (see Section 2.4.2.2):

ddv1

dt = 1η

(d1 − dv1) (2.61)

ddv2

dt = 1η

(d2 − dv2) . (2.62)

The time discretised regularised damage variables (Eq. (2.55)) differentiated with respectto the respective non-regularised damage variable produce:

∂dv1

∂d1= ∂dv

2∂d2

= ∆tη + ∆t . (2.63)

Therefore, the Jacobian matrix with viscous regularisation has to be expanded with theabove term and can determined as [8]:

∂∆σ∂∆ε = C

d+[(

∂Cd

∂dv1ε

)(∂d1∂ff

∂ff

∂ε

)+(∂C

d

∂dv2ε

)(∂d2∂fm

∂fm

∂ε

)] ∆tη + ∆t . (2.64)

The obtained damaged Jacobian matrix is non symmetric, hence the non symmetric equa-tion solver in Abaqus has to be utilised in order to retain the quadratic convergence rateof the Newton scheme.

29

2.4.5. Determination of Strain Energy Density and dissipated Energy

The strain energy density W in non-incremental framework can be obtained from [23]:

W =∫σij dεij , (2.65)

where the indices imply summation. In incremental formulation the strain energy densityat the end of an increment can be determined according to:

Wn+1 = Wn +∫ tn+1

tn

σij dεij . (2.66)

Utilising numerical integration for evaluating the integral in the above expression, thefollowing formulation is established [8]:

Wn+1 = Wn + 12 [(σn+1 + σn) · ∆ε] . (2.67)

The dissipated energy from viscous regularisation is calculated from the difference of thestrain energy density with and without regularisation, which additionally requires thedetermination of the stress obtained with non-regularised damage variables in the subrou-tine. Evaluating the difference in strain energy density the dissipated energy from viscousregularisation can be found by [8]:

Wdiss,n+1 = Wdiss,n + 12[{

(σn+1 + σn) −(σd

n+1 + σdn

)}· ∆ε

]. (2.68)

with σd being the stress without viscous regularisation.

2.4.6. Known Restrictions

While the subroutine implemented as described in the previous sections was found tobe generally working well, minor restrictions especially in element availability arise asthe UMAT was implemented covering only certain conditions. The subroutine generallywas implemented only for 3D-elements, that is elements having six stress components, ascovering also planar elements would have required reformulation of a large part of thesubroutine to account for the reduced matrices.

Additionally, functionality of progressive damage is only given for elements having asingle integration point (e.g. reduced integration hexahedron - C3D8R, the subroutinewas not tested for tetrahedral elements though functionality should be given) as otherwisegradients of the damage variables between the integration point within a single elementarise, which lead to large and unrealistic deformations of the element and localisation ofthe damage at this integration point because of reduced stiffness at a single integrationpoint. To prevent hourglassing in the reduced integration elements, an hourglass stiffness(Ghour) has to be specified. The hourglass stiffness can be estimated from the shear moduli

30

according to [8]:Ghour ≈ 1

3 (G12 +G13 +G23) . (2.69)

As there is some remaining stiffness even at a state of fully damaged material (see Sec-tion 2.4.2.1) and because of the unavailability of element deletion in a UMAT subroutine,large deformations after damage can lead to unrealistic results. As for composites thoughgenerally small deformations are examined, this should commonly not be of concern. Tocircumvent this issue, a python script was written that stops an analysis a specified num-ber of times, every time checks for fully damaged elements, deletes these elements fromthe model, and resumes the simulation. This though shall not be discussed in greaterdetail in this thesis.

It should also be mentioned that despite having implemented viscous regularisation,for small values of η, slow convergence rates can be observed, which is to be expectedthough for constitutive laws exhibiting softening behaviour, while large viscosity valuescan influence the results.

2.5. Matrix Nonlinearity and development of a User Material inAbaqus Explicit

2.5.1. Introduction and Variables to be defined

Modelling composites by means of a linear elastic transversely isotropic material modelupon damage initiation, as has been done in implementing the SIFT as a UMAT withprogressive damage described in the previous sections, good agreement with experimen-tal data is found in most cases (see Chapter 5), though in certain circumstances (e.g.±45° laminate in tension) a highly nonlinear material response is encountered, which isnot captured by the linear elastic material model (see Fig. 2.10). According to Pinho,Darvizeh, Robinson, et al. in [6], the behaviour of pure matrix is mainly influenced by thesecond deviatoric strain invariant. As within the SIFT the second deviatoric invariant inthe matrix phase is readily available from amplification of the homogenised solution, thematrix stiffness can easily be modelled as a function of the distortional matrix strain, asperformed for the LaRC05 failure criterion by Pinho, Darvizeh, Robinson, et al. in [6].It has to be taken into account though that the strain amplification factors are obtainedutilising a purely linear elastic formulation of the matrix material, hence the amplifiedstrain invariants only validly relate the macro strain state to the micro strain state solong as linear elastic behaviour is encountered in both cases. Despite this in the currentimplementation it was assumed that the deviatoric second strain invariant amplified bymeans of ‘elastic‘ amplification factors represent an accurate enough approximation alsowithin the nonlinear homogenised response.

The implementation of nonlinear matrix response as an addition to damage initiationwith the SIFT and progressive damage will be described in the following section. For

31

reasons of reduced complexity this was performed utilising the VUMAT subroutine inAbaqus Explicit for defining the constitutive material law. A VUMAT thereby was chosenbecause in Abaqus Explicit, due to the explicit integration algorithm, no Newton schemeis utilised, which omits the necessity to define the Jacobian matrix of the constitutiveequation that proved to be complex for the nonlinear case [8]. Instead within a VUMATsubroutine, only the stress state at the end of the increment has to be defined. A furtheradvantage of the VUMAT subroutine is the possibility to define an element deletion flagin case of a fully damaged element, that is once this flag is active, the element is deletedfrom the model and there is no remaining stiffness. In the current implementation, thisflag is activated once complete failure in fibre direction is obtained, as it is assumed thatfibre failure leads to the loss of any load carrying capacity of the material, while matrixfailure merely leads to (almost) zero stiffness in matrix direction.

Having expanded the constitutive model to consider matrix nonlinearity, experimentallyobtained results exhibiting large deviations from purely linear behaviour prior to finalfailure are well approximated by the developed model, as is exemplary shown for tensiletests of ±45° laminate specimens in Fig. 2.10. More detailed validation of the model usingexperimental results can be found in Chapter 5.

0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.0350

20

40

60

80

100

120

Strain (-)

Stre

ss(M

Pa)

ExperimentalSIFTSIFT nonlinear

Figure 2.10.: Highly nonlinear response observed when testing ±45° laminate specimensand modelling of the tests utilising the developed constitutive laws.

The general schematic of the VUMAT implementation is given in Fig. 2.11 and thedetails differing from the development of the SIFT-UMAT, as described in the previoussections, will be developed in the next sections.

32

Apply strain increment ∆ε

Read-in amplification factors

Start of Analysis

εn+1 = εn + ∆ε

Calculate undamaged stress:σun

n+1 = σunn + C ∆ε

Failure criteria

Damageonset?

Calculate damage variables

Calculate damaged stress:σi,n+1 = (1 − di)σi,n+1

Calculate matrix nonlinearityparameter and adapt stress

Calculate strain energy

Abaqus solver

yes

no

For

each

inte

grat

ion

poin

t

Nex

tin

crem

ent

until

end

ofan

alys

is

VUMAT

VUEXTERNALDB

Figure 2.11.: Schematic describing the implementation of the SIFT as a VUMAT in AbaqusExplicit (adapted from [17]).

2.5.2. Considering Matrix Nonlinearity in the Model

In the current modelling approach matrix nonlinearity is numerically implemented byintroducing an additional damage variable, denoted as r in the following, having an ex-ponential damage evolution law based on the second deviatoric strain invariant amplifiedwithin the matrix phase. The evolution law was arbitrarily chosen to fit the experimen-tal data as good as possible while having a function of low complexity and few fitting

33

parameters, and is given by:

r = A+ (1 −A) exp(

−εmatrixvm

C

), (2.70)

where A and C are fitting parameters with 0 < A < 1 and 0 < C. The damage variabler purely evolves as a function of the second deviatoric strain invariant within the matrixphase, though, in order to obtain a linear damage evolution law, upon damage initiationaccording to the SIFT within the matrix phase and evolution of the damage variable d2,r is held constant. Again, to prevent ‘healing’ of the material, within the ‘time history’ ris defined as:

r (t) = maxt∗≤t

{r (t∗)} . (2.71)

It should be stressed that the current modelling approach is not a plastic materialresponse but merely a nonlinear elastic constitutive model, representing a reduction inmatrix stiffness as a function of the equivalent strain within the matrix phase, that wasdeveloped to show the capabilities of the current modelling approach. To include plasticeffects, that is residual deformation upon unloading additionally to a nonlinear stiffness,the model has to be further refined.

In Fig. 2.12 exemplary the material response with purely elastic and with nonlinear elas-tic matrix behaviour, respectively, is depicted while both material responses exhibit equalstrain at damage initiation and strain at final failure and both having a linear damageevolution law upon damage initiation. The dashed lines within Fig. 2.12 for the nonlin-ear material response represent unloading and subsequent reloading paths, showing thestiffness reduction within the nonlinear regime and subsequently after damage initiation.

2.5.3. Stress Calculation in incremental Formulation

As the formulation of a Jacobian matrix is not necessary with an explicit integrationalgorithm, the formulation of a damaged stiffness matrix is not required and therefore,the damaged stress is readily available (here shown in scalar formulation) from the damagevariables di and the nonlinearity parameter r. The damage variables thereby are obtainedin a similar manner as described in Section 2.4.2.1, though no viscous regularisation isimplemented as, due to lack of the Newton scheme in the explicit solver, convergence isnot an issue. Therefore, the stress in the damaged, nonlinear configuration is obtainedfrom the effective stress σ, that is again the stress in a fictitious undamaged material, hereadditionally also without any nonlinear effects, objected to the same strain, according to:

σ1,n+1 = (1 − di)σ1,n+1 and

σi,n+1 = r(1 − di)σi,n+1 where i = 2, . . . , 6.(2.72)

It has to be emphasized that in order to account for nonlinearity within the matrix phase,the stress in fibre direction σ1 is not adapted by the nonlinearity factor r, but only stress

34

εinit εfinal

Strain

Stre

ssNonlinear matrixLinear elastic matrixUnloading and reloading path

Figure 2.12.: Comparison of linear and nonlinear material response and subsequent lineardamage evolution. The strain at damage initiation and final damage is equalin both cases. Dashed lines represent unloading and subsequent reloadingpaths.

components in matrix direction (σ2 and σ3) and the shear stresses, as it is presumed thatthe shear stresses are influenced by matrix nonlinearity as well.

Within a VUMAT subroutine in Abaqus Explicit, the strain energy per unit mass shallbe defined [8]. From the obtained stress the strain energy per unit volume is determinedin a similar way as described in Section 2.4.5 and subsequently scaled with the massdensity in order to obtain the strain energy per unit mass. The dissipated energy fromviscous regularisation is not determined as regularisation is not implemented due to lackof necessity for the explicit solver.

35

3. Micromechanics Analysis Code based onthe Generalized Method of Cells

3.1. Introduction

In predicting effective properties of multiphase materials, such as composites, homogeni-sation methods like the Mori-Tanaka mean field homogenisation [24] (presented in Sec-tion 3.3.2) can provide accurate estimates [1] while remaining computationally very effi-cient. These closed-form analytical models though provide merely mean fields (e.g. stress,strain, displacement, etc.) for the fibre and matrix phase, respectively, while actually localfields can vary significantly in the constituents. To overcome the limitation of neglectedlocal field gradients Aboudi, Arnold, and Bednarcyk therefore developed, relying on theassumption that the multiphase material structure of a composite can be represented bya periodic RUC, the Method of Cells (MOC) in a first step. The MOC subdivides thecomposite’s matrix phase in a RUC into three regions (see Fig. 3.1), each having distinctlocal fields.

h

l

x2

x3

(a) Composite microstructure.

h2

h1

l1 l2

x(2)2

x(1)3γ = 1

β = 2

x(1)2

x(1)3

γ = 1β = 1

x(2)2

x(2)3γ = 2

β = 2

x(1)2

x(2)3

γ = 2β = 1

(b) Doubly periodic RUC modelling thecomposite structure.

Figure 3.1.: Modelling of a composite by means of a 2×2 doubly periodic RUC. In 3.1b theRUC is depicted with the local subcell coordinate system, having its origin inthe centre of the subcell, and the subcell numbering being illustrated (adaptedfrom [1]).

Addressing the limitations of having merely four subcells (one fibre and three matrix)in a RUC, the MOC was further extended into the Generalised Method of Cells (GMC),subdividing the composite into an arbitrary number of subcells (derivation see Section 3.2).This allows to capture more variations in the local field gradients and enables betterapproximation of the inclusion shape (see Fig. 3.2), while the GMC remains extremelycomputationally efficient.

36

Matrix

Inclusion

(a) MOC microcell. (b) GMC or HFGMCmicrocell.

Figure 3.2.: Approximation of a continuous fibre inclusion utilising a doubly periodic 2 × 2MOC microcell (3.2a) and a 26 × 26 GMC or HFGMC microcell (3.2b), re-spectively (adapted from [1]).

A shortcoming of the GMC and MOC, and in fact of most analytical micromechanicsmodels, is the lack of shear coupling (arising of shear stresses from applied normal stressesor strains as is observed in reality or also captured by FEA, see Fig. 3.3), resulting fromemploying a first-order displacement field. This lack of shear coupling is not much of aproblem in many applications, as for example in modelling of continuously reinforced com-posites, while under certain conditions (e.g. short fibre composites, staggered structures[11]) lacking shear coupling can lead to highly ineffective properties prediction.

To account for shear coupling, the High Fidelity Generalized Method of Cells (HFGMC)was developed by employing a second-order displacement field in contrast to the GMC’sfirst-order displacement field, resulting in stresses and strains varying linearly within theHFGMC subcells. This extension introduces more unknown variables in the model and ne-cessitates the use of integration points in the subcells in order to handle the now arbitrarilyvarying inelastic strain fields in the subcells. All of this leads to highly increased compu-tational cost in order to be able to incorporate shear coupling in the model. Furthermorethough, also a better representation of the stress concentrations around the inclusions ofa composite are found with the HFGMC compared to the GMC due to the existence ofstress and strain gradients within a subcell (see Fig. 3.3). These gradients on the otherhand introduce a dependency of the solution on the size of the subcells and with thatthe degree of geometrical refinement in the RUC. Further elaboration on the influence ofshear coupling on the overall predicted composite stiffness can be found in Section 3.2.2.

MOC, GMC, and HFGMC are implemented in the software MAC/GMC developed atNASA’s Glenn research centre for application in modelling composite microcells [25]. InMAC/GMC the MOC and GMC are implemented for doubly periodic (material is mod-elled in plane periodic and infinitely extending in transversal direction (see Fig. 3.1b),e.g. continuously reinforced composite) as well as triply periodic (material modelled pe-riodically in all three directions of space, e.g. short fibre reinforced composites) RUCswhile with the HFGMC only doubly periodic unit cells can be examined. For modelling ofstructures using one of the presented micromechanics approaches, MAC/GMC can be cou-pled with the commercial finite element software Abaqus utilising the code FEAMAC [26].

37

σ11 (MPa)2,3002,1081,9171,7251,5331,3421,1509587675753831920 (a) σ11, GMC. (b) σ11, HFGMC. (c) σ11, FEA.

σ22 (MPa)4.03.52.92.41.81.30.70.2

−0.3−0.9−1.4−2.0−2.5 (d) σ22, GMC. (e) σ22, HFGMC. (f) σ22, FEA.

σ23 (MPa)2.21.81.51.10.70.40.0

−0.4−0.7−1.1−1.5−1.8−2.2 (g) σ23, GMC. (h) σ23, HFGMC. (i) σ23, FEA.

Figure 3.3.: Modelling of a continuously reinforced composite utilising the GMC, HFGMC,and FEA, respectively, by means of a single fibre RUC. A strain of ε11 = 0.01is applied, the 11-direction being the fibre direction and the resulting stressesare depicted. Comparing 3.3g and 3.3h, lack of shear coupling in the GMCis obvious from the fact, that there are no shear stresses arising in the GMC-microcell. The stress concentrations along the fibre inclusion are better rep-resented with the HFGMC compared to the GMC, while very similar resultsare obtained with the HFGMC and FEA.

FEAMAC, implemented as a UMAT [8], calls MAC/GMC at runtime in every integrationpoint of the simulated structure to evaluate the current material response under the givenloading conditions.

38

3.2. Derivation of the GMC

At this point an overview of the thermomechanical formulation with the triply periodicGMC shall be given according to Aboudi, Arnold, and Bednarcyk in [1], determiningthe GMC constitutive law for an arbitrary RUC consisting of Nα ×Nβ ×Nγ subcells (seeFig. 3.4) with α, β and γ being the running subcell indices with α = 1, 2, . . . , Nα; β = 1, 2,. . . , Nβ and γ = 1, 2, . . . , Nγ . The RUC is modelled in the global coordinates x1, x2 andx3, and within the center of each subcell (αβγ) the local coordinates x(α)

1 , x(β)2 and x

(γ)3

are introduced (see Fig. 3.1b). Each subcell (αβγ) has a size of dα in x1-direction, hβ inx2-direction and lγ in x3-direction and can consist of any arbitrary inelastic material. Inthis deviation also strains due to thermal expansion, resulting from non-zero Coefficientsof Thermal Expansion (CTEs), are considered in order to point out the most generalcase possible, even though thermal expansion will not be taken into account in the laterexamples of application, meaning constant temperature is assumed in all examples.

α = 1

α = 2α = 3

β = 1 β = 2γ = 1γ = 2

γ = 3

l

h

d

x2

x1

x3

Figure 3.4.: Geometry of a triply periodic RUC consisting of 3 × 2 × 3 subcells (adaptedfrom [1]).

For subcell (αβγ) the neighbouring subcell in x1 direction is labelled (αβγ) with α beingdefined as

α =

⎧⎨⎩α+ 1 α < Nα

1 α = Nα.(3.1)

This notation guarantees for α < nα that the neighbouring subcell in x1 direction is thesubcell with indices α+ 1, β and γ in the same RUC, while for α = Nα the neighbouringsubcell is subcell (1βγ) within the adjoining RUC. Furthermore, this ensures continuityat the interfaces between neighbouring RUCs. The indices β and γ are similarly definedfor the x2 and x3 directions, respectively.

For the deviation of the GMC a first-order displacement field is utilised with the displace-ment in each subcell u(αβγ)

i being linearly expanded in terms of distance from the subcellcentre. Introducing the displacement components at the subcell centre ω(αβγ)

i (x) with thevector x = (x1, x2, x3) denoting the position of the subcell centre in the global coordinatesystem (this vector is constant at subcell level), and three microvariables χ(αβγ)

i , ϕ(αβγ)i ,

39

and ψ(αβγ)i , denoting the linear dependence of the displacement u(αβγ)

i on the local subcellcoordinates ( x(α)

1 , x(β)2 , x

(γ)3 ). With this the first-order displacement expansion in subcell

(αβγ) is given by:

u(αβγ)i = ω

(αβγ)i (x) + x

(α)1 χ

(αβγ)i + x

(β)2 ϕ

(αβγ)i + x

(γ)3 ψ

(αβγ)i i = 1, 2, 3, (3.2)

while repeated Greek letters do not imply summation.Between neighbouring subcells of the RUC, the displacement components have to be

continuous, which implies that for all subcells

u(αβγ)i

⏐⏐⏐x

(α)1 = dα

2= u

(αβγ)i

⏐⏐⏐x

(α)1 = −dα

2(3.3)

has to be true in x1-direction. Equal conditions have to hold in x2 and x3-direction. Thesecontinuity conditions are imposed at the interfaces in an average (integral) sense.

It can now be shown that the subcell total strain components ε(αβγ)ij and total stress

components σ(αβγ)ij have to be constant within a given subcell and hence are independent

of the local coordinates x(α)1 , x

(β)2 and x

(γ)3 . From this has to follow that the average

and pointwise stress and strain fields within a subcell are identical, exemplary writtenhere for the stress components with an overlined field variable representing the averagecomponents:

σ(αβγ)ij = σ

(αβγ)ij . (3.4)

The elastic-viscoplastic and temperature-dependent material occupying subcell (αβγ)can be modelled with the elastic stiffness tensor of the material C(αβγ)

ijkl by the followingconstitutive equation:

σ(αβγ)ij = C

(αβγ)ijkl

(ε

(αβγ)kl − ε

I(αβγ)kl − ε

T (αβγ)kl

), (3.5)

with the thermal strain components εT (αβγ)ij and the inelastic strain components εI(αβγ)

ij .It shall be show, that a set of continuum equations independent of the microvariables

and modelling the overall behaviour of the triply periodic multiphase composite can beobtained, by employing the displacement and traction continuity conditions at the subcellinterfaces.

The equations relating the subcell strain vector εs to the average composite strain vector(macrostrain) ε can be written in matrix form according to:

AGε

s= J ε (3.6)

withε = (ε11, ε22, ε33, ε23, ε13, ε12) , (3.7)

εs

=(ε(αβγ), . . . , ε(NαNβNγ)

), (3.8)

40

and the matrices J and AG

containing geometrical properties of the RUC.Evaluating continuity of the tractions between neighbouring subcell interfaces and RUC

interfaces and using Eq. (3.5) it can be obtained that:

AM

(ε

s− εI

s− εT

s

)= 0 (3.9)

with the matrix AM

involving the elastic properties C(αβγ) of each subcell and the inelasticεI

sand thermal εT

sstrain matrices being similarly composed as Eq. (3.8). Combining and

solving Eqs. (3.6) and (3.9) leads to:

εs

= A ε+D(εI

s+ εT

s

)(3.10)

with

A =[A

M

AG

]−1 [0J

]=

⎡⎢⎢⎢⎣A(111)

...A(NαNβNγ)

⎤⎥⎥⎥⎦

D =[A

M

AG

]−1 [A

M

0

]=

⎡⎢⎢⎢⎣D(111)

...D(NαNβNγ)

⎤⎥⎥⎥⎦ .(3.11)

The concentration tensors A(αβγ) and D(αβγ) depend only on the local elastic moduli,shape, and volume fraction of the local phases and therefore are constant.

With the concentration tensors and the effective elastic subcell stiffness tensor thesubcell stress tensor can be determined from the uniform overall strain tensor (appliedmacrostrain) ε by:

σ(αβγ) = C(αβγ)[A(αβγ)ε+D(αβγ)

(εI

s+ εT

s

)−(εI(αβγ)

s+ εT (αβγ)

s

)]. (3.12)

Furthermore, the effective elastic composite (homogenised macromechanical) stiffness ten-sor C∗ is determined from:

C∗ = 1dhl

Nα∑α=1

Nβ∑β=1

Nγ∑γ=1

dαhβlγC(αβγ)A(αβγ) (3.13)

and with that the effective elastoplastic thermomecanical of the composite is given by:

σ = C∗(ε− εI − εT

)(3.14)

with the composite inelastic strain tensor εI and the average thermal strain tensor εT .The established GMC constitutive law, derived from evaluating the displacement and

traction continuity conditions at the subcell and RUC interfaces, relates the average subcellstress and strain to the overall uniform RUC strain and the subcell inelastic and thermal

41

strains by means of the concentration tensors A(αβγ) and D(αβγ). The effective RUCstiffness tensor C∗ is determined, modelling the homogenised composite macromechanicallaw.

For computational implementation of the GMC the constitutive law can be reformulated,reducing the number of equations to solve for better computational efficiency. Furtherelaboration can be found in [1].

3.2.1. Differentiation to the FEA

Although both the FEA as well as the GMC utilise subvolume discretisation, both methodsare inherently different, therefore a short overview of the underlying differences shall begiven at this place [1]. It should be noted that all details given here for the GMC areapplicable to the HFGMC as well.

While the GMC was specifically developed by Aboudi, Arnold, and Bednarcyk to modelcomposite RUCs amongst others for the determination of effective properties, the FEA,through satisfying the mechanical equilibrium by means of the weak formulation of theenergy/virtual work principle and from that determining the displacements at the elementnodes, is inherently a far more general approach. To emphasise the comprehensiveness ofthe FEA as an example just the multitude of different element types available for differentmodelling purposes in order to capture the desired phenomenon while achieving optimumcomputational efficiency and at the same time preventing undesired artefacts such as forexample locking effects, shall be pointed out at this place. Equilibrium conditions in theGMC in contrast are imposed in a volume average sense within the subcells utilising thestrong form.

In order to model periodic RUCs within the FEA though, it is necessary to addition-ally impose the periodicity conditions by means of displacement based periodic boundaryconditions being applied to the unit cell’s nodes (see Section 2.3.1). In contrast, the GMCby definition imposes continuity of displacements and tractions in an average sense at theRUC’s surfaces without the need of additional boundary conditions.

Satisfying displacement continuity at the subcell interfaces in the GMC compared to thepointwise satisfaction in the FEA implies that a square approximation of an inclusion inthe GMC does not necessarily represent an actually square inclusion but may be utilisedto model a circular inclusion.

In conclusion, the FEA and GMC are based on inherently different theories and the FEAgenerally is a far more universal approach. The GMC is specifically designed to efficientlyand easily model composite RUCs and in that both theories can achieve similar results,in particular similar results are obtained with the HFGMC and FEA (see Section 3.2.2),

42

3.2.2. Shear Coupling

As mentioned in Section 3.1 the GMC does not account for shear coupling, that is noshear stresses arise from an applied normal strain or stress, due to the utilised first orderdisplacement field (see Fig. 3.3). Therefore, the HFGMC was developed utilising a secondorder displacement field to account for shear coupling in the microstructure though at afar higher computational cost. To illustrate the influence of shear coupling on the deter-mined composite stiffness at this point the predicted effective properties of a continuouslyreinforced composite and of a staggered structure determined by means of the GMC andHFGMC, respectively, are compared. Furthermore, in the case of the staggered structurethe local stress fields will be examined for both methods.

Using the input parameters for fibre and matrix as specified in Table 5.3 a doublyperiodic 26 × 26 RUC (see Fig. 3.2b) was modelled to determine the effective propertiesof a continuously reinforced composite utilising the GMC and HFGMC, respectively. Theobtained results can be found in Table 3.1 with index ‘|’ representing longitudinal or fibredirection and index ‘⊥’ representing transverse direction. The longitudinal and transversalYoung’s moduli show only negligible deviations for both methods while the shear modulishow deviations of almost 8 % in the case of G|⊥. The higher shear moduli are therebypredicted with the HFGMC because the shear coupling within the microcell is taken intoaccount. Overall though, similar results are obtained with both theories and the neglect ofshear coupling does not render the GMC ineffective in the case of continuously reinforcedcomposites as the deviations from considering shear coupling are minor in all cases.

Table 3.1.: Comparison of the GMC and HFGMC for the determination of the effectivemoduli of a continuously reinforced composite.

E|| (MPa) E⊥⊥ (MPa) G|⊥ (MPa) G⊥⊥ (MPa) ν|⊥ (-)

GMC 95880 4785 1846 1456 0.2573HFGMC 95880 4856 1990 1534 0.2554Relative deviation 0 % 1.5 % 7.8 % 5.4 % -0.7 %

In order to give an example where shear coupling becomes a first-order effect and there-fore approximations utilising the GMC are inadequate, a staggered structure is examinedin the following. Biocomposites, such as e.g. nacre, enamel, or dentin bone, exhibit atthe most elementary level a generic ‘brick and mortar’ microstructure, that is staggeredmineral bricks embedded in a soft (protein) matrix [11]. This microstructure exhibits highfracture toughness due to stress-redistribution and crack-stopping mechanisms. Upon anapplied tensile stress the mineral bricks carry the tensile load while the embedding matrixtransfers the load between the bricks by means of shear stresses in the matrix.

Following the microstructure as observed in nacre, a RUC was built having high mod-ulus platelets embedded in a soft matrix with an arbitrary difference in stiffness betweeninclusion and matrix of four magnitudes (input parameters see Table 3.2). In the doublyperiodic RUC consisting of 30 × 38 subcells, the platelets were arranged as a staggered

43

structure (see Fig. 3.5), extending infinitely in transverse direction.

Table 3.2.: Material and geometry input parameters for modelling of a staggered structure.Ematrix 1 MPaEinclusion 10000 MPaInclusion length 5µmInclusion aspect ratio 10Inclusion volume fraction 0.54

Matrix

Inclusion2

1

Figure 3.5.: Microcell of a staggered structure consisting of platelets of high stiffness em-bedded in a soft matrix. The platelets are extending infinitely in out-of-planedirection (doubly periodic RUC). The microcell width is scaled for betterrepresentation.

The stiffness in direction of platelet orientation (1-direction) determined by means ofthe GMC and HFGMC, respectively is given in Table 3.3. As can be seen, the modu-lus determined utilising the HFGMC is more than twice the GMC-modulus. This largedeviation results from the load carrying mechanism in staggered structures, that is thenormal load being carried by the stiff platelets and transferred between the platelets byshear stresses in the matrix. These shear stresses do not arise utilising the GMC due tothe lack of shear coupling.

Table 3.3.: Longitudinal stiffness of a staggered structure determined by means of theGMC and HFGMC, respectively.

E11 (MPa)

GMC 17.9HFGMC 47.2Relative deviation 163.2 %

Examining the stress distribution within the microcells as depicted in Fig. 3.6, the lack ofshear stresses in the GMC-microcell becomes apparent (Fig. 3.6c). As the stresses betweenthe platelets in the GMC-microcell are solely transmitted by normal stresses in the lowstiffness matrix, constant normal stresses arise along the platelet direction (Fig. 3.6a).In contrast, in the HFGMC-microcell higher normal stresses arise showing a non-uniformdistribution along the platelets (Fig. 3.6b) due to stress transmission by shear stresses inthe matrix.

44

σ11 (MPa)1.000.920.830.750.670.580.500.420.330.250.170.080.00

(a) GMC, σ11. (b) HFGMC,σ11.

σ12 (MPa)0.250.210.170.120.080.040.00

−0.04−0.08−0.13−0.17−0.21−0.25

(c) GMC, σ12. (d) HFGMC,σ12.

Figure 3.6.: Microcell of a staggered structure with an applied strain in direction of inclu-sion orientation of ε11 = 0.01 utilising the GMC and HFGMC, respectively.The lack of shear coupling in the GMC becomes apparent.

In the case of staggered structures therefore the GMC becomes highly inefficient in pre-dicting the effective properties as shear stresses arising from an applied normal stress orstrain largely contribute to the overall load carrying capacity. For more accurate predic-tions here the HFGMC has to be utilised in order to capture shear coupling.

3.3. Comparison with other Homogenisation Approaches

In order to show the differences between stiffness predictions utilising the GMC and otherhomogenisation approaches, exemplary the GMC results will be compared to approxima-tions utilising the Mori-Tanaka homogenisation approach and the Voigt and Reuss model(upper and lower bound model). A short introduction to the mentioned homogenisationprocedures will be given first.

3.3.1. Voigt and Reuss Model

The simplest relationship predicting the longitudinal properties of a continuously rein-forced composite from those of its constituents is given by the Voigt model (or Rule ofMixture)[27]. Assuming parallel loading of fibres and matrix and hence equal strains inboth phases the composite longitudinal modulus E|| is given by a weighted average of theconstituents moduli Ef and Em [28]:

E|| = vfEf + (1 − vf )Em (3.15)

with vf being the fibre volume fraction. The Voigt model provides the upper bound forthe longitudinal stiffness.

Assuming equal stress in matrix and fibre phase the Reuss model (Inverse Rule of Mix-ture) [29] can be obtained, predicting the transversal composite stiffness E⊥⊥ according

45

to:

E⊥⊥ =(vf

Ef+ 1 − vf

Em

)−1

. (3.16)

The Reuss model provides the lower bound for the transverse composite stiffness.

3.3.2. Mori Tanaka Homogenisation

The semi-analytical mean field homogenisation approach based on the Mori-Tanaka [24]theory has shown to be a very efficient method in predicting the effective properties ofa composite [1], [30]. The Mori-Tanaka approach originally addressed the determinationof the average internal stress in a matrix containing inclusions by means of eigenstrains.The term eigenstrain as utilised here was adapted from the German ‘Eigenspannungenund Eigenspannungsquellen’ by Mura in [31] meaning nonelastic strains such as strainsdue to thermal expansion or misfit strains causing eigenstresses in a body free from anyexternal force or surface constraint. In the original paper of Eshelby [32] on which theMori-Tanaka theory is based, eigenstrains were referred to as ‘stress-free transformations’.

Eshelby in [32] determined a function for the problem of a single ellipsoidal inclusioninside an infinite body being cut out, undergoing an eigenstrain and then being weldedback into the cavity it formerly occupied (see Fig. 3.7), relating the strain inside theellipsoidal volume to the eigenstrain [30].

Figure 3.7.: Depiction of Eshelby’s problem: An ellipsoidal inclusion inside an infinite ma-trix being cut out, undergoing an eigenstrain and being welded back (reprintedfrom [30]).

Starting from the effective stiffness tensor C∗ for a two-phase composite given by:

C∗ = C(1) + v2(C(2) − C(1)

)A(2) (3.17)

with C(1) and C(2) being the stiffness tensor for matrix and inclusion phase, respectivelyand v2 being the inclusion volume fraction, a concentration tensor A(2) is given such that:

ε(2) = A(2)ε0, (3.18)

where ε(2) is the average inclusion strain and ε0 is the externally applied homogeneousstrain at the surface. In the problem of a single inclusion in an infinite matrix with strain

46

ε(1) it is assumed that:ε(1) ≈ ε0. (3.19)

Mori and Tanaka assume that the problem of a two-phase composite with an arbitrarynumber of inclusions with an average strain ε(2) can be approximated by the problem ofa single inclusion embedded in an infinite matrix subjected to a strain ε(1) (Fig. 3.8) andtherefore approximating the multi-inclusion problem with Eshelby’s solution.

Figure 3.8.: Exemplification of the Mori-Tanaka approach: The multi-inclusion problemis approximated by a single-inclusion problem and subsequently solved basedan Eshelby’s problem (reprinted from [30]).

With that a tensor B is found relating the volume average strain over all inclusions tothe average matrix strain:

B = H(I, C(1), C(2)

)(3.20)

and being a function of the inclusion I and the matrix and inclusion stiffness tensors.Because of Eq. (3.19) this approach is theoretically restricted to rather small inclusion

volume fractions (approximately maximum 25 %), though good approximations can stillbe achieved for higher volume fractions [30].

3.3.3. Method Comparison

The moduli obtained by means of the upper and lower bound models, Mori-Tanaka meanfield homogenisation, and MAC/GMC for a continuously reinforced composite with linearelastic isotropic matrix and inclusion are shown in Fig. 3.9. In MAC/GMC a doublyperiodic 2 × 2 GMC-RUC (as depicted in Fig. 3.2a) was utilised.

The results for the longitudinal moduli determined with Mori-Tanaka homogenisationand MAC/GMC were found to be similar to the longitudinal modulus determined with

47

the upper bound model (Voigt-model) and hence are not shown in Fig. 3.9.Comparing the transversal moduli determined by means of mean field homogenisation

and with MAC/GMC, both theories predict, as is expected, a higher stiffness than theReuss model (lower bound model). Both theories as well as the Reuss model though exhibithighly matrix dominated behaviour showing a rapid increase in predicted modulus onlyat volume fractions higher than approximately 70 %. The highest transversal modulus isgenerally predicted with the GMC.

0 0.2 0.4 0.6 0.8 1Fibre Volume Fraction (-)

Stiff

ness

Upper BoundLower BoundMori Tanaka transversalMAC/GMC transversalPure MatrixPure Fibre

Figure 3.9.: Comparison of the transversal moduli of a continuously reinforced compositedetermined utilising the Mori-Tanaka homogenisation approach and the GMCwith the results of upper and lower bound models. For better reference purematrix and inclusion stiffness are depicted as well.

3.4. Modelling of Composite Structures

In the following sections an overview of the theoretical foundations of material modelsutilised in modelling composite RUCs in MAC/GMC is given. Implementation of theRUCs and obtained results are then described in Chapter 5.

3.4.1. Incremental Plasticity

Being one of the simplest plasticity models the classical incremental plasticity based onthe von Mises yield criterion (see Fig. 3.10), also referred to as classical metal plasticity,is commonly used in FEA-codes. Though it was initially developed for metals, restrictedapplicability is also given for polymers with limitations being for example the missingpressure dependence [33]. In MAC/GMC rate-independent incremental plasticity is im-plemented with linear isotropic hardening, that is uniform expansion of the yield surfacein all directions in stress space (see Fig. 3.11) with a (piecewise defined from up to eight

48

points) linear hardening law (as depicted in Fig. 3.10) [25], [34]. An overview of the ba-sic concepts involved in the implementation of rate-independent incremental plasticity asavailable in MAC/GMC shall be given in the following.

εplastic εtotal

σy

irreversible reversible

E

ET

Strain

Stre

ss

Figure 3.10.: Schematic stress-strain response of an elastic-plastic material with linearhardening (adapted from [1]).

σ22

σ11

σ33

σy

σy

σy

initial yield surface

expanded yield sur-faces after plasticdeformation

Figure 3.11.: Evolving of the yield surface during plastic deformation with isotropic hard-ening of the material depicted in the deviatoric plane (adapted from [34],[35]).

A basic assumption of elastic-plastic constitutive models is that the deformation can bedecomposed in an elastic and a plastic or inelastic part, which can be written in its mostbasic form with the displacement gradient tensor [8]:

F = F elF pl. (3.21)

49

Instead of this decomposition an ‘additive strain rate decomposition’:

ε = εel + εpl (3.22)

with the elastic εel and the plastic strain rate εpl, respectively, can be utilised to formulatethe plasticity model in the case of infinitesimal strains (negligible compared to unity).This strain rate decomposition can be integrated and one obtains for the total strain:

ε = εel + εpl. (3.23)

The elastic part of the material response is thereby presumed to be given by a strainenergy density potential U , so that the stress is given according to:

σ = ∂U

∂εel. (3.24)

The region of purely elastic response in the material is defined by a yield function f sothat:

f < 0 (3.25)

in the case of purely elastic behaviour and during plastic flow:

f = 0. (3.26)

When inelastic flow is occurring, the plastic strain increment is given by [1]:

∆εpl = ∆λ∂f∂σ

(3.27)

where λ is a measure for the amount of plastic flow in the system [8].The Mises yield condition, which is valid for an isotropic material with linear elasticity

and incompressibility of the material once yielding occurs is given by [8], [36]:

q =√

32S • S, (3.28)

where S is the deviatoric stress tensor:

S = σ − 13tr

(σ)

1 (3.29)

with the unit tensor 1, and says that yielding occurs once q reaches the material’s yieldstress σy for a bar in tension:

q = σy. (3.30)

With this yield criterion Eq. (3.27) can be integrated utilising for example the ‘radialreturn method’ according to Simo and Hughes in [37] to obtain the plastic strain at the

50

end of an increment while assuring that the stress state is kept on the evolving yield surfacewhile plastic strain is being accumulated [1]. For further details regarding integration ofEq. (3.27) the reader is referred to the literature as this would exceed the scope of thisthesis.

3.4.2. Tsai-Hill Failure Criterion

The Tsai-Hill composite failure criterion is based on Hill’s yield criterion, which was ini-tially developed as a theory of yielding for anisotropic metals by generalising the Misesyield criterion to orthotropic materials [38]–[40].

Let for such an anisotropic metal, where the anisotropy has three mutually orthogonalplanes of symmetry everywhere in the material and presuming the same behaviour intension and compression, the yield stresses in principal anisotropic direction be given byX, Y , and Z, respectively and the yield stresses in shear with respect to the principalaxes of anisotropy be given by R, S, and T [39]. With this, the Mises yield criterion(Eq. (3.28)) can be expanded in such a way, that yielding initiates once:

F (σyy − σzz)2 +G(σzz − σxx)2 +H(σxx − σyy)2 + 2Lσyz + 2Mσxz + 2Nσxy = 1 (3.31)

where F , G, H, L, M , and N are constants describing the state of anisotropy at the currentmaterial point. These constants can be obtained from the engineering yield strengths by:

F = 12

( 1Y 2 + 1

Z2 − 1X2

)G = 1

2

( 1Z2 + 1

X2 − 1Y 2

)H = 1

2

( 1X2 + 1

Y 2 − 1Z2

)L = 1

2R2

M = 12S2

N = 12T 2 .

(3.32)

Tsai and Wu in [38] postulate a tensorial strength criterion of the general scalar form:

Fiσi + Fijσiσj = 1 (3.33)

where the contracted notation is used and therefore the indices apply summation withi, j = 1, 2, . . . 6. It should at this place be noted that from this, the well known Tsai-Wufailure criterion is obtained. To receive the Tsai-Hill criterion though, Tsai and Wu utilisethe notation in Eq. (3.33) to rewrite the otherwise unchanged Hill criterion from Eq. (3.31)in tensorial notation, by setting

Fi = 0 (3.34)

51

and choosing Fij appropriately.The resulting interactive failure criterion is identical to the Hill criterion but applied to

the framework of composite failure analysis by Tsai and Wu.In the further modelling of composite structures as presented in this thesis, the Tsai-Hill

criterion will be utilised to determine failure in isotropic epoxy resin. For this the strengthwas chosen in such a way that:

X2 = Y 2 = Z2 = R2

3 = S2

3 = T 2

3 . (3.35)

Utilising these properties in Eq. (3.31) it can be easily shown with Eq. (3.32) that oneobtains:

(σyy − σzz)2 + (σzz − σxx)2 + (σxx − σyy)2 + 6(σ2xy + σ2

xz + σ2yz) = 2X2. (3.36)

This equation though is identical to the Mises criterion Eq. (3.28) implying that for anisotropic material the Tsai-Hill or Hill criterion simplifies to the Mises criterion [38], whichwould be assumed from the fact that the Hill criterion is an expansion of the Misescriterion for anisotropic materials. In the context of an isotropic elastic-plastic materialwith classical incremental plasticity the Tsai-Hill criterion therefore, being identical to theMises criterion, determines the yield surface where upon reaching final failure occurs.

3.4.3. Curtin Fibre Failure

To model the mechanical response and damage progression of fibres within a compositetaking into account stochastical strength variations of the fibres, Curtin introduced amodel based on fibre strength statistics combined with a shear lag model in [41].

It is assumed that the individual fibres in a composite fracture independently and uponfailure of a single fibre, global load redistribution occurs. During loading of a composite infibre direction every individual fibre effectively experiences a uniaxial tension test, whichis identical to a hypothetical tension test of a single-fibre composite in a matrix with alarge failure strain. Defects occurring along the fibre are distributed statistically and thecumulative number of defects that can fail at a given stress σ in a fibre of length L is givenby a Weibull distribution [42]:

Φ (L, σ) = L

L0

(σ

σ0

)m

, (3.37)

with σ0 the stress required to cause one failure in a fibre of length L0 on average or L0 beingthe gauge length in a fibre tension test and m the Weibull parameter. The probability offibre failure can therefore be determined with:

Pf (L, σ) = 1 − exp (−Φ (L, σ)) . (3.38)

52

Therefore, after substituting Eq. (3.37) into Eq. (3.38), the probability of failure in fibresof length 2lf at a stress level of σL is given by [1], [43]:

q(lf ) = P (2lf , σL) = 1 − exp(

−2lfL0

σmL

σm0

). (3.39)

With the frictional sliding resistance τ between fibre and matrix and the fibre radius r itcan be found utilising shear-lag analysis and a simple force balance:

2rπlfτ = r2πσL (3.40)

which yields the approximate slip zone length:

lf = rσL

2τ . (3.41)

With Eq. (3.41), Eq. (3.39) can be written as:

q(lf ) = 1 − exp(

−(σL

σc

)m+1)

(3.42)

where

σc =(σm

0 τL0r

) 1m+1

. (3.43)

By means of Taylor series and retaining only linear terms, q(lf ) can be approximated as:

q(lf ) ≈(σL

σc

)m+1. (3.44)

The average fibre stress at an applied longitudinal stress σ on the composite and a FVRof vf is given by:

σavgf = σ

vf(3.45)

or, with q(lf ) being the fraction of broken fibres:

σavgf = σL

(1 − 1

2q(lf )). (3.46)

Substituting for q(lf ) provides the average fibre stress by:

σavgf = σL

{1 − 1

2

(σL

σc

)m+1}, (3.47)

taking into account progressive fibre breakage in a fibre bundle. Based on fibre strengthstatistics the Curtin model predicts an elastic response of the effective fibre upon initialloading and eventually initiating fibre damage and therefore a softening behaviour withincreasing stress level σL.

By maximising Eq. (3.47) with respect to σL and with Eq. (3.45), the ultimate tensile

53

strength σu can be obtained by [41]:

σu = vfσc

( 2m+ 2

) 1m+1 m+ 1

m+ 2 . (3.48)

The longitudinal fibre stress in the unbroken fibres σL is determined from the elasticfibre modulus Ef and the applied mechanical fibre strain εf :

σL = Efεf . (3.49)

The effective elastic modulus of the effective fibre E∗f on the other hand is given by:

E∗f =

σavgf

εf. (3.50)

Substituting for the average fibre stress σavgf , E∗

f is obtained by:

E∗f = 1

2

{1 + exp

[−(Efεf

σc

)m+1]}Ef . (3.51)

It should be noted that for Eq. (3.51) the exponential form for q(lf ) according to Eq. (3.42)and not the approximated formulation is utilised.

Aboudi, Arnold, and Bednarcyk in [1] note that the value of τ strongly affects the failurebehaviour of the effective fibre (Fig. 3.12). To account for the stochastic nature of τ itshould be fitted to available data rather than demanding it to be a single experimentallydetermined value. It is to be considered that after reaching a maximum effective fibrestress and an initial drop in the stress level upon further loading a further stress increaseis observed (see dotted lines in Fig. 3.12), which in practice is not the case. For modellingpurposes in MAC/GMC the fibre subcell stiffness can be zeroed once maximum fibrestress is reached in order to model the loss of load carrying capacity after fibre failure (seeremaining lines in Fig. 3.12) [25].

54

0.00 0.01 0.02 0.03 0.04 0.050

1,000

2,000

3,000

4,000

5,000

6,000

7,000

Strain (-)

Stre

ss(M

Pa)

Composite stress, τ = 3.5Fibre stress, τ = 3.5no subcell zeroingComposite stress, τ = 35Fibre stress, τ = 35

Figure 3.12.: Comparison of Curtin fibre failure with different values for τ implementedwith a doubly periodic 2 × 2 RUC. The effective fibre stress as well as theaverage composite stress are depicted. Except for the dotted lines the fibresubcell stress is zeroed upon reaching the ultimate fibre stress.

55

4. Experimental Results and MaterialCharacterisation

For implementing a micromechanics based material model and validating the developedcomputational models, a number of material tests were performed to gather the necessaryexperimental data. In order to determine the critical strain invariants required for theSIFT, tensile tests of 0°, 90°, and ±45° laminates are utilised, while for developing amicromechanical model in MAC/GMC, pure matrix and again 0° laminate tensile testsare required. For additional validation of the implemented models, tensile tests of ±15°laminates were performed. A short overview of the obtained experimental data shall begiven here.

4.1. Epoxy Resin Tensile Tests

For modelling the matrix behaviour within MAC/GMC, data from previously and inde-pendently of this thesis performed tension tests of epoxy resin was utilised. The tensiontests were performed at Transfercenter für Kunststofftechnik GmbH, Wels in accordancewith ISO 527-2 at Room Remperature (RT) utilising five ISO 527 Type 1A specimensmilled from a cast epoxy sheet. Longitudinal strain was measured by means of an ex-tensometer, while transversal strain for determining the Poisson’s ratio was obtained bymeans of a strain gauge.

The obtained stress-strain response is depicted in Fig. 4.1 while an overview of thetesting results is given in Table 4.1. During testing initial linear elastic behaviour of theepoxy resin in tension shifts to a highly nonlinear response upon larger strains, followedby subsequent brittle failure of the material.

Table 4.1.: Epoxy resin properties obtained from experimental tensile tests.E (MPa) s (MPa) σmax (MPa) s (MPa) εmax (-) s (-) ν (-) s (-)

2654 79 72.35 0.15 0.072 4.0·10−3 0.32 0.02

56

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.080

15

30

45

60

75

Strain (-)

Stre

ss(M

Pa)

Epoxy resin

Figure 4.1.: Stress-strain response of epoxy resin specimens in tension.

4.2. Laminate Tensile Tests

Tension tests of carbon fibre composite laminates with varying layups were performedutilising specimens milled from 10-layer composite sheets of the respective layup beingmanufactured from unidirectional CYCOM® 977-2 carbon fibre epoxy prepregs. For de-termination of the longitudinal and transversal lamina properties, respectively, 0◦ and 90◦

unidirectional specimens were tested, while for fitting the second deviatoric strain invari-ant in the matrix phase within the framework of the SIFT, ±45◦ laminates were examined.Additionally, for validation of the determined properties, ±15◦ laminates were tested.

An overview of the specimens and the respective testing conditions is given in Table 4.2.The specimens’ FVR as well as the utilised fibre type was not provided by the manufac-turer and was subsequently fitted in the used micromechanical models in order to fit theexperimental data (see Chapter 5). All composite tests were performed in-house utilisinga servo-hydraulic testing machine MTS 852 Damper Test System of the company MTSSystems Corporation (Eden Prairie, USA) with a 50 kN MTS force transducer (model661.20F-02) and taking the deformation during testing for subsequent strain determina-tion from piston displacement recorded with the machine’s displacement transducer.

An overview of the obtained material properties is provided in Table 4.3. It has tobe stressed that especially with the 0◦ specimens problems with specimen quality wereencountered, that is visible variations in fibre orientation partly explaining the large stan-dard deviations obtained for the tested moduli. Furthermore, during testing of the 0◦ and±45◦ specimens on a number of specimens slipping of the glued tabs was encountered. Forboth layups only a single specimen could be tested to final failure.

The averaged experimental stress-strain curves are outlined in Fig. 4.2. To avoid mul-tiple depiction only the combined graphs are given here, while more detailed depictions

57

Table 4.2.: Specimen geometry and testing conditions for composite laminate tensile tests.Layup

0◦ 90◦ ±15◦ ±45◦

Specimen geometry

Number of Layers 10Width (mm) 15 25 25 25Length (mm) 200Thickness (mm) 1Tab material aluminiumTab length (mm) 30 50 50 50

Testing conditions

Number of specimens 4 3 3 3Displacement rate (mm/min) 1Ambient temperature RT

Table 4.3.: Composite properties obtained from experimental tensile tests of differentlayups.

Specimen E (MPa) s (MPa) σmax (MPa) s (MPa) εmax (-) s (-)

UD 0◦ 93517 5768.4 1535.1 — 0.019 —UD 90◦ 2918.2 289.4 32.6 1.61 0.009 9.1·10−4

MD ±45◦ 8681.6 — 100.2 — 0.033 —MD ±15◦ 69149 4194.0 844.7 25.9 0.012 3.6·10−4

of the individual results for each layup are provided in Chapter 5 being compared to thenumerical results. For all tested laminates brittle failure is encountered with only the±45◦ specimen exhibiting large deviations from the initial linear response and showingsoftening behaviour upon higher deformations. During testing of 0° and ±15° specimens,fibre failure was observed while for 90° and ±45° laminates, matrix failure was encounteredupon reaching the critical load.

58

0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.0350

250

500

750

1,000

1,250

1,500

1,750

Strain (-)

Stre

ss(M

Pa)

0◦

90◦

±45◦

±15◦

Figure 4.2.: Stress-strain response of composite laminate specimens in tension.

59

5. Results and Analysis

In the following sections, the modelling of performed experimental laminate tests withthe user subroutines developed by the author for Abaqus with damage initiation governedby the SIFT and utilising the micromechanics software MAC/GMC, respectively, will bediscussed more in detail. Having determined the respective required material parame-ters and developed the models, the numerically obtained results will be compared to theexperimental results provided in Section 4.2.

5.1. Critical Strain Invariants and Damage Evolution

In contrast to MAC/GMC, where the code provides the homogenised composite materialresponse from the constituents’ behaviour, the SIFT represents merely a failure crite-rion phenomenologically taking into account the material’s microstructure, while in thecurrent implementation the composite material’s constitutive response is provided withinthe UMAT or VUMAT, respectively, developed by the author. Within the respectivesubroutine the composite is modelled as a transversely isotropic linear or nonlinear elas-tic material with linear stiffness degradation upon damage initiation. The initial linearelastic composite response has to be determined by means of experimental testing or anarbitrary micromechanics model, though the constituents’ behaviour has to be known fordetermining the strain amplification factors. In the course of the thesis the effective com-posite properties are determining utilising the micromechanics analysis code Digimat FE,where a RVE is examined utilising FEA in order to obtain the effective properties. The-oretically the macro developed for determining the strain amplification factors could beeasily expanded to additionally output the average effective composite properties, as threecomposite RUCs are modelled for the obtaining factors anyway, rendering unnecessarythe utilisation of a third party micromechanics software. This feature though has not yetbeen implemented.

Having the input parameters depicted in Table 5.3, the initial elastic effective com-posite properties were determined for the examined laminates utilising Digimat FE (seeTable 5.1). Additionally, again using the properties from Table 5.3, with a macro devel-oped by the author for Abaqus (see Chapter 2), the strain amplification factors for thegiven composite were examined (amplification factors are depicted in Tables A.1 to A.3).

Having obtained the strain amplification factors subsequently, to determine critical in-variants (Ic

1, εmatrix,cvm , εfibre,c

vm ), damage evolution law (cI1 , cεmatrix,c

vm, c

εfibre,cvm

), and theparameters governing the nonlinear matrix response (A, C) when utilising the VUMAT,

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Table 5.1.: Effective composite properties obtained utilising the FEA of a microcell inDigimat FE.

E|| (MPa) E⊥⊥ (MPa) G|⊥ (MPa) G⊥⊥ (MPa) ν|⊥ (-)

94390 4730 1994 1688 0.3769

the performed laminate tests were modelled in Abaqus and the respective parameters werechosen to fit the experimental data. Thereby Ic

1 and cI1 were determined from the testsof 90◦ laminates, εfibre,c

vm and cεfibre,c

vmfrom 0◦ tests, and εmatrix,c

vm , cεmatrix,c

vm, A, and C from

±45◦ laminates. The critical invariants for the UMAT were determined in order to fit therespective experimentally determined maximum stress. All fitted parameters are depictedin Table 5.2. The data from ±15◦ laminate tests is used for verification of the obtainedparameters. It has to be emphasized that all parameters determined are effective prop-erties, that is these parameters have to be determined, as shown here, from experimentallaminate tests and are not pure fibre or matrix properties.

Table 5.2.: Critical invariants and parameters for defining damage evolution laws obtainedfrom fitting experimental data.

Linear matrix be-haviour (UMAT)

Nonlinear matrix be-haviour (VUMAT)

Critical invariants

Ic1 (-) 0.011 0.0134εmatrix,c

vm (-) 0.062 0.16εfibre,c

vm (-) 0.021 0.021

Damage evolution

cI1 (-) 1.1 1.1c

εmatrix,cvm

(-) 4 2c

εfibre,cvm

(-) 1.1 1.1

Nonlinearity parameters

A — 0.1B — 0.25

Damage evolution was found to be mesh dependent, which generally is expected whenusing strain softening material behaviour. With smaller mesh, strain concentrations aremore prevalent within the model, leading to earlier damage initiation and degradation,while a less refined mesh exhibits more averaged strains.

It should be noted that for all models using the VUMAT in order to avoid excessivesimulation times, the tests were modelled with a displacement rate of 100 mm/s, comparedto the experimental displacement rate of 1 mm/min. Because of the implemented materialmodel being rate independent and due to the material’s low density the effect of inertiaon the results was found to be negligible.

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5.2. Micromechanical Composite Model

For simulating the experimental tests (see Section 4.2) utilising a micromechanical modelin MAC/GMC, the unidirectionally reinforced carbon fibre epoxy composite is modelled bymeans of a 2×2 doubly periodic GMC-RUC (see Chapter 3) within MAC/GMC, assumingthat the composite, having statistically distributed fibres, can be represented with theutilised single fibre unit cell. The matrix phase within the unit cell is modelled utilisingonce a linear elastic isotropic material model and once by means of an incrementally plasticconstitutive law, having as input parameters the experimentally obtained epoxy stress-strain response, in order to capture the nonlinear matrix behaviour at higher deformations(see Section 4.1). In both modelling approaches matrix failure is determined utilising theTsai-Hill failure criterion, having as failure stress in tension for the isotropic matrix theexperimentally determined tensile strength and determining the stress in shear failureaccording to the Mises-criterion (τmax = σmax/

√3), so that the Tsai-Hill criterion is

reduced to the Mises criterion (see Section 3.4.2). Once failure in a subcell occurs accordingto the Tsai-Hill criterion, failure of the respective subcell is modelled by reducing thesubcell stiffness to 0.1 % of the initial value, effectively zeroing the subcell stiffness.

For modelling the fibre behaviour within the RUC a transversally isotropic materialmodel is used taking the fibre properties from literature ([44], [45]). An overview of theelastic fibre and matrix material properties is given in Table 5.3.

Table 5.3.: Matrix and fibre elastic input parameters.Matrix input Fibre input [44], [45]

E (MPa) 2654 E|| (GPa) 230ν (-) 0.32 E⊥⊥ (GPa) 11.5σmax (MPa) 72 G|⊥ (GPa) 9.0

ν|⊥ (-) 0.18ν⊥⊥ (-) 0.4Fibre diameter (µm) 7FVR (-) (fitted) 0.41

In order to account for failure in the fibre phase, the Curtin fibre failure model (Sec-tion 3.4.3), implemented in MAC/GMC, taking the required model parameters again fromliterature [46] or from fitting to the experimental tests of 0◦ laminates (frictional slidingresistance between fibre and matrix τ and Weibull parameter m were fitted), respectively.The utilised Curtin parameters are depicted in Table 5.4. Once the maximum fibre stressaccording to the Curtin model is obtained within the fibre subcell to account for failureof the fibre, the subcell stiffness again is zeroed.

The effective elastic composite properties determined by means of a 2×2 GMC-RUCin MAC/GMC with the above input parameters are provided in Table 5.5. It should benoted that minor differences in effective properties compared to Table 3.1 result from thedifferent microcells utilised, where for the properties in Table 3.1 a 26×26 RUC was used,

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Table 5.4.: Parameters utilised for modelling fibre damage by means of the Curtin fibrefailure model [46].

D (µm) L0 (m) σ0 (MPa) τ (MPa) m (-)

7.0 0.01 4500 3.5 4.0

while for better computational efficiency in FEAMAC here the composite is modelled bymeans of a 2×2 RUC.

Table 5.5.: Effective composite properties obtained utilising the GMC.E|| (MPa) E⊥⊥ (MPa) G|⊥ (MPa) G⊥⊥ (MPa) ν|⊥ (-)

95880 4834 1854 1456 0.2572

For comparison with experimental data, the performed laminate tests were modelled inAbaqus utilising FEAMAC, a UMAT provided in combination with MAC/GMC allowingthe evaluation of the material response in every integration point using the micromechan-ical models in MAC/GMC. Once failure occurs within a subcell, convergence issues areencountered in Abaqus due to sudden zeroing of the subcell stiffness and therefore suddendrops in material stiffness. To circumvent stopping of the simulation once failure occurs,the Abaqus solver parameters were adapted in such a way that a larger number of cutbacksin increment size and smaller overall increment timesteps are allowed. As this may lead tohighly increased simulation times depending on the increment size, the simulations werestopped once a considerable drop in overall stiffness was observed. Additionally, it wasfound that due to the sudden drop in stiffness once failure initiates, the solutions werehighly mesh size dependent, as with larger mesh size stress and strain concentrations aremore averaged and hence failure occurs at a higher overall loading. An overview of thesimulation results and comparison with experimental data and simulations of the sametests utilising the earlier presented UMAT and VUMAT, respectively, will be given inSection 5.3.

At this place instead a number of restrictions discovered while using FEAMAC in com-bination with Abaqus version 6.14 running on a 64-bit system, will be discussed more indetail. It should be stressed here that FEAMAC is still in its beta phase, hence it isexpected that a number of errors are encountered. While working with FEAMAC it wasfound that an Abaqus analysis in combination with FEAMAC must not be run on morethan one CPU as otherwise the solver process ‘standard.exe’ remains in idle and the sim-ulation does not advance. A similar problem was encountered but later avoided duringdevelopment of the UMAT for implementing the SIFT, when write access to a variablewithin the UMAT code defined as common block, that is a variable being saved betweencalls to the subroutine, occurs from several threads at the same time, so it is assumed thatin FEAMAC this is caused in a similar manner.

Furthermore, it was found that once a contact definition is present in the analysed

63

model, no output is written when using FEAMAC, which when utilising a user subroutineis usually caused by either an entry of the Jacobian matrix or of the stress vector becominginfinite. A similar error is encountered when using the Curtin fibre failure model in FEA-MAC and the Weibull parameter m not being an integer. In contrast, utilising MAC/GMConly, m can be set to any arbitrary real number without encountering problems.

Finally, when trying to use a HFGMC model with FEAMAC, the analysis crashes with‘LoadLibraryA error 193’ during pre-processing. A summary of the encountered problemsis given in Table 5.6.

Table 5.6.: Modelling restrictions and bugs encountered in FEAMAC.Cause of error Encountered error

Multi CPU usageAfter finishing of pre-processor, stan-dard.exe process is running in idle andsimulation does not advance

Contact definition present within themodel No output written

m-parameter within Curtin model nointeger No output written

Utilisation of HFGMC Analysis crashes with ‘LoadLibraryAerror 193’

5.3. Comparison with Experimental Results

Hereafter, the numerically determined results will be compared to the experimentally de-termined data. In that context, the inherent differences between both examined compu-tational models shall be mentioned again. Whereas MAC/GMC provides a homogenisedmacroscopic material response for an arbitrary multi-phase material purely from the con-stituents’ behaviour according to the theory presented in Chapter 3, while taking intoaccount local field gradients within the individual constituents phase, the SIFT representspurely a theory governing failure initiation within a continuously reinforced compositephenomenologically accounting for the material’s microstructure by means of strain am-plification factors. Utilising the current implementation of the SIFT, the materials consti-tutive response is provided within the UMAT or VUMAT developed by the author, wheretransversely isotropic (non)linear elastic material behaviour is assumed and a progressivedamage model is implemented. As will be shown in the following though, these inherentlydifferent modelling approaches both provide good predictions of the composite behaviourexamined.

5.3.1. 0°-tests

A comparison of the simulated and experimental material response for 0◦ specimens isdepicted in Fig. 5.1 and Table 5.7. The predicted stiffness values all are well within the

64

experimentally determined value and the stress at failure σmax could be well fitted to theexperimental data. The marginally lower stiffness determined from the UMAT comparedto the VUMAT is assumed to result from utilising an implicit compared to an explicitsolver and differences in the displacement rate. Both UMAT and VUMAT predict lowerstrain at failure compared to the experimental results, because softening behaviour infibre direction as observed during the experiments, is not implemented but a linear elasticresponse in fibre direction is utilised.

Having implemented the Curtin fibre failure model within MAC/GMC, the experimen-tally observed softening behaviour could be well fitted by adjusting the Weibull parameterm to match the experimental strain at failure, though a more gradual decrease in stiff-ness is obtained from experiments than is predicted with the Curtin model. Remainingstiffness after fibre failure observed for the MAC/GMC material response results from theremaining matrix subcells that have not failed yet and hence still exhibit load carryingcapacity.

0.000 0.005 0.010 0.015 0.020 0.0250

250

500

750

1,000

1,250

1,500

1,750

Strain (-)

Stre

ss(M

Pa)

ExperimentalSIFTSIFT nonlinearMAC/GMC plasticMAC/GMC elastic

Figure 5.1.: Comparison of numerical and experimental data for 0° specimens.

Table 5.7.: Comparison of numerical and experimental data for 0° specimens.E (MPa) σmax (MPa) εmax (-)

Experimental 93517 ±5768.4 1535.1 0.019SIFT UMAT 93558 1577.6 0.017SIFT VUMAT 93636 1586.1 0.017MAC/GMC plastic 95920 1546.1 0.020MAC/GMC elastic 95920 1553.2 0.020

65

5.3.2. 90°-tests

Comparing results from specimens with transverse fibre direction in tension (Fig. 5.2and Table 5.8) to the numerically determined results, the stiffness predicted utilisingMAC/GMC as well as the stiffness from FEA analysis of a composite microcell in DigimatFE used for the UMAT and VUMAT, are approximately 65 % higher compared to theexperimental stiffness. This large deviation is assumed to be a result of poor specimenquality. Using the SIFT, stress at failure could be well fitted to the experimental data,while the strain at failure is predicted too low, which though is to be expected consideringthe difference in material stiffness. Accounting for matrix nonlinearity within the VUMAT,softening behaviour can be observed, though deviation from the linear response is onlymarginal as for uniaxial tension transverse to the fibre direction predominantly volumetricdeformation is observed, while the material nonlinearity in the current implementation isgoverned by deviatoric deformation within the matrix phase.

0.00 0.01 0.02 0.03 0.040

20

40

60

80

100

120

Strain (-)

Stre

ss(M

Pa)

ExperimentalSIFTSIFT nonlinearMAC/GMC plasticMAC/GMC elastic

Figure 5.2.: Comparison of numerical and experimental data for 90° specimens.

Table 5.8.: Comparison of numerical and experimental data for 90° specimens.E (MPa) σmax (MPa) εmax (-)

Experimental 2918.2 ±289.4 32.6 ±1.61 0.009 ±9.05·10−4

SIFT UMAT 4730.0 32.7 0.007SIFT VUMAT 4600.4 33.2 0.008MAC/GMC plastic 4834 106.7 0.031MAC/GMC elastic 4834 87.1 0.017

The transversal strength predicted by MAC/GMC with both elastic and elastoplas-tic matrix modelling largely overestimates the experimental values, which is assumed tomainly be a result of utilising a too coarse approximation of the fibre structure by the 2×2

66

unit cell, and hence an underestimation of the stress concentrations along the fibre inclu-sion. Furthermore, only a single fibre arrangement is modelled by the unit cell thereforenot accounting for the statistical fibre distribution within a composite possibly leading toeven higher stress concentrations. Also, the fibre matrix interface is modelled as perfectlybonded, hence not taking into account interface failure that could initiate further crackgrowth. Finally, by utilising an elastic or incrementally plastic, respectively, matrix mate-rial model with failure initiation according to the Tsai-Hill criterion, stress-state dependentfailure is not accounted for. In the stress-strain response ‘steps’ observed upon initial fail-ure of the matrix are a result of failure of individual subcells within the RUC. The overallhigher strength predicted for the subcell with elastoplastic matrix model is presumed tooriginate from plastic deformation within the subcells decreasing stress concentrationsfrom the fibre inclusion.

5.3.3. ±45°-tests

Comparing the experimental results of ±45° laminate tests (Fig. 5.3 and Table 5.9) andcomputational methods, all models predict a lower initial stiffness. As due to slipping ofthe glued tabs only a single specimen could be utilised for stiffness and strength determina-tion, no standard deviation among the experimental results could be determined and thehigh stiffness was presumed to originate from variations among the specimens, especiallytaking into account, that for ±15° laminate test the stiffness is correctly predicted fromall methods (see Section 5.3.4).

0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.0350

20

40

60

80

100

120

Strain (-)

Stre

ss(M

Pa)

ExperimentalSIFTSIFT nonlinearMAC/GMC plasticMAC/GMC elastic

Figure 5.3.: Comparison of numerical and experimental data for ±45° specimens.

Utilising the SIFT implemented as a UMAT, the stress at failure could be fitted well tothe experimental results, while, due to highly nonlinear behaviour of the tested specimenwhich is not captured by the transversely isotropic linear elastic material utilised withinthe UMAT, deviations in the shape of the curve are observed and the strain at failure

67

Table 5.9.: Comparison of numerical and experimental data for ±45° specimens.E (MPa) σmax (MPa) εmax (-)

Experimental 8681.6 100.2 0.033SIFT UMAT 7389.8 99.4 0.014SIFT VUMAT 7196.3 98.2 0.033MAC/GMC plastic 6846.6 72.0 0.019MAC/GMC elastic 7712.5 57.6 0.008

is underestimated. For this reason a VUMAT was developed taking into account matrixnonlinearity as a function of the distortional strain within the matrix phase, where theshape of the stress-strain response as well as failure stress and strain could be well fitted,while initial deviations between experimental and numerical curves are traced back to thegenerally too low predicted initial stiffness. Utilising UMAT and VUMAT, both predictdamage initiation initially in the matrix phase due to distortional deformation, thoughas element deletion can be implemented utilising a VUMAT, different failure modes areobserved. With the UMAT (see the distribution of the matrix damage variable d2 inFig. 5.4a), after failure initiation in the matrix phase, the matrix stiffness is degradedaccording to the implemented damage evolution law, leading to softening of the materialresponse and subsequent fibre failure. Having element deletion in the VUMAT additionallyto a differently defined damage evolution law for matrix failure in distortional deformationwith faster damage progression, which was found to better fit the experimental data incombination with matrix nonlinearity, loss of load carrying capacity of the matrix due tofailure leads to fibre failure and with that element deletion, resulting in actual breakingof the specimen (see Fig. 5.4b). In both simulations influence of the boundary conditionscould be observed having as a result initial damage at the regions of displacement appli-cation. This could be avoided by modelling the glued tabs on the specimen utilising acohesive contact definition, though for better comparability to the results obtained usingFEAMAC, where contact definitions are not available, this was acquiesced as failure inthe tested specimen was observed close to the glued tabs as well.

Using a RUC in MAC/GMC with linear elastic matrix behaviour, premature failure ofthe matrix subcells results in too low prediction of the composite strength. In contrast,when using incremental plasticity for the matrix phase, stress concentrations along thefibre inclusion are decreased due to plastic matrix deformation, resulting in better approx-imation of the experimental material response though still, due to fibre failure within theunit cell according to the Curtin failure model, strength is predicted lower than experi-mentally observed. This again is assumed to result from badly approximating the stressdistribution within the unit cell as a result of modelling the composite with a total of only4 subcells. Furthermore, due to the perfectly modelled fibre-matrix interface, higher stresslevels in the matrix phase are presumed that would otherwise be lower due to slipping atthe interface or delamination.

68

d2 (-)0.3710.3630.3560.3490.3420.3340.3270.3200.3130.3050.2980.2910.284

(a) Linear matrix be-haviour.

d2 (-)0.9900.9070.8250.7420.6600.5770.4950.4120.3300.2470.1650.0820.000

(b) Nonlinear matrix be-haviour.

Figure 5.4.: Matrix damage variable after reaching maximum stress for ±45° lami-nate specimens in tension simulated using the developed UMAT (5.4a) andVUMAT (5.4b).

5.3.4. ±15°-tests

For validation of the SIFT-parameters fitted from 0°, 90°, and ±45° laminate tests, addi-tionally the results are compared to ±15° tests, as can be found in Fig. 5.5 and Table 5.10.With both UMAT and VUMAT, experimental stiffness and stress and strain at failureare accurately predicted within the respective standard deviation. Utilising the UMAT,initial failure is observed due to distortional deformation within the matrix followed byfailure of the fibre phase, due to loss of load carrying capacity of the matrix. In contrast,using the VUMAT and having nonlinear matrix behaviour, a change in failure mode isobserved, where damage now initiates due to fibre failure, as a larger value for the criticalsecond deviatoric strain invariant within the matrix phase (εmatrix,c

vm ) was determined. Amarginal deviation from the linear behaviour is observed upon loading during the simula-tion utilising the VUMAT.

Table 5.10.: Comparison of numerical and experimental data for ±15° specimens.E (MPa) σmax (MPa) εmax (-)

Experimental 69149 ±4194.0 844.7 ±25.9 0.012 ±3.61·10−4

SIFT UMAT 73084 838.9 0.012SIFT VUMAT 72035 821.1 0.012MAC/GMC plastic 74229 800.1 0.011MAC/GMC elastic 75007 466.7 0.006

Comparing the results obtained with FEAMAC to the experimental data, again betterpredictions can be found by means of an elastoplastic matrix material model. Having

69

0.000 0.003 0.005 0.008 0.010 0.013 0.0150

200

400

600

800

1,000

Strain (-)

Stre

ss(M

Pa)

ExperimentalSIFTSIFT nonlinearMAC/GMC plasticMAC/GMC elastic

Figure 5.5.: Comparison of numerical and experimental data for ±15° specimens.

purely elastic matrix response, failure initiation within the matrix phase occurs beforereaching the experimental failure stress, while with an elastoplastic constitutive matrixlaw due to plastic deformation within the matrix phase, again a shift in failure mode toinitially fibre failure can be observed and a better prediction of the experimental failurestress and strain is obtained.

70

6. Conclusion and Outlook

The overall objective of this thesis was the development and examination of microme-chanics based models for the simulation and failure modelling of continuously reinforcedcomposite materials (Fig. 6.1). For this purpose, first the SIFT, phenomenologically ac-counting for the inherent material microstructure by means of strain amplification factors,determined from simulation of micromechanical blocks with different fibre arrangements,was implemented by means of user defined material models in the FEA-software suiteAbaqus. For that the materials’ constitutive response was modelled transversely isotropic(non)-linear elastic prior to damage and upon damage initiation governed by the SIFTdamage evolution exhibiting linear softening behaviour was implemented.

The initially developed UMAT for Abaqus Standard was implemented exhibiting lin-ear elastic material response prior to damage initiation, which was found to badly fitthe observed stress-strain response of ±45° laminates in tension, where a highly nonlin-ear material response was observed upon loading. In order to account for the materialnonlinearity, a VUMAT in Abaqus Explicit was developed, extending the existing consti-tutive model by an additional damage variable evolving as a function of the distortionaldeformation within the matrix phase of the composite.

Fitting the critical invariants governing damage initiation, damage evolution laws, and,in case of the VUMAT, nonlinear material law from experimental data of 0°, 90°, and±45° laminates in tension and further validating the parameters with tension tests of±15° laminates, generally good agreement with experimental data could be achieved forboth subroutines, with the nonlinear material implementation of the VUMAT showingbetter agreement with the highly nonlinear response of the ±45° laminates.

Overall it was shown that the current implementations of the constitutive law for com-posites enable accurate modelling of the mechanical behaviour of composites includingprecise prediction of damage onset and damage evolution.

Secondly, a composite material model was developed by means of the micromechanicsanalysis code MAC/GMC, having as input parameters purely the constituents’ behaviour.Thereby, in modelling a composite RUC, a linear elastic and an elastoplastic matrix modelwere compared, both having failure of the material governed by the Tsai-Hill failure crite-rion. The fibre phase within the developed RUCs was assumed to be transversely isotropiclinear elastic, with fibre failure implemented utilising the Curtin fibre failure model andthereby accounting for the statistical distribution of the strength of individual fibres.

Both established MAC/GMC composite models were compared to experimental results

71

by modelling the performed tests in Abaqus and evaluating the material response at the in-tegration points using FEAMAC, thereby coupling the Abaqus solver to MAC/GMC. Thepredictions for composite stiffness and strength from MAC/GMC using an elastoplasticmatrix model generally well match experimental data, except for transversal stiffness andstrength, which both are predicted too high. The higher predicted strength is assumed toresult from using a too coarse approximation of the fibre inclusion in MAC/GMC, there-fore not correctly capturing the inclusion’s stress concentrations, and from not accountingfor the statistical fibre distribution within the actual composite.

Having a linear elastic matrix material model, generally the composites strength isunderestimated, showing the necessity to account for plastic deformation within the matrixphase during loading.

Generally though, MAC/GMC proved to be a puissant and efficient tool in predict-ing the behaviour of the examined composite, having as input parameters merely theconstituents’ constitutive response, which reduces the necessary experimental data formodelling a continuously reinforced composite structure to a minimum.

Figure 6.1.: Schematic overview of the implemented and examined numerical models.

Though the focus of this thesis was set on the numerical implementation of the examinedtheories, the implemented models could be verified for the examined loading conditions,where experimental data was available. Of interest for further research though would

72

be predictions from the implemented models for composites in compression, as effectsoccurring in compression such as fibre kinking are generraly not accounted for in theexamined models. This though was not performed for the current thesis due to lack ofexperimental data.

Furthermore, matrix nonlinearity in the current VUMAT is accounted for purely byintroduction of an additional damage variable following an exponential evolution func-tion. This simplified nonlinearity model was mostly implemented in order to show thecapabilities of knowing the distortional deformation within the matrix phase, and can befurther expanded to additionally account for pressure dependent matrix behaviour, or toaccount for actual matrix plasticity instead of merely nonlinear behaviour, opening newopportunities for further research projects.

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A. Appendix

A.1. UMAT Readme

**********************************************************UMAT for linear elastic orthotropic material behaviour inlaminated composites with implementation of progressivedamage . Damage initiation is based on the Strain InvariantFailure Theory (SIFT ). The linear damage evolution law isgoverned by the SIFT damage variables . To improve convergence ,viscous regularization is implemented in the damage evolutionlaw. Damage can occur in two independent directions : fibredamage and matrix damage . This results in an unsymmetricjacobian matrix , hence the unsymmetric solution techniqueshall be used.**********************************************************The UMAT is implemented only for 3D- stress state! It wastested for C3D8R elements (8-node , linear hexaeder element ,reduced integration ). An hourglass stiffness (G) has to bespecified ! The stiffness can be estimated byG =1/3*( G12+G13+G23 ).**********************************************************Material properties for input:Ex , Ey , Ez , NUxy , NUxz , NUyz , Gxy , Gxz , Gyz , ( orthotropicmaterial properties )

J1_crit , I2_f_crit , I2_m_crit ,( critical invariants : first invariant in matrix phase ,second invariant in fibre phase , second invariant in matrixphase),

CF (1<CF , multiple of damage variable in fibre failure atwhich total failure occurs , CF=1 results in perfectlybrittle failure which may result in convergence issues ),

CM1 and CM2 (1<CM1 ,CM2 , multiple of damage variable in matrixfailure for first and second invariant , respectively , atwhich total failure occurs , CM1 ,CM2 =1 results in perfectlybrittle failure which may result in convergence issues ),

MAXD (0<MAXD <1, maximum value for the damage evolutionvariables , if MAXD =1 there is no remaining stiffness aftertotal failure of the element , this may result in convergence

74

issues !), if MAXD =0, there is no damage evolution .

ETA (0<ETA , viscosity for viscous regularisation of thedamage variable , ETA =0 may result in convergence issues ).For determining the impact of viscous regularisation onthe final result , the creep dissipation energy (ALLCD)can be compared to the strain energy (ALLSE) of the modelin the history output . ALLCD should be small comparedto ALLSE.

There are 28 STATEV used in the model

A material orientation has to be specified in the model.

Three .txt -files with the amplification factors have to bein the working directory . These files should be createdwith the given Abaqus macro. Upon executing the macroamplification_factors .py in Abaqus , the matrix and fibrematerial properties , the fibre volume ratio and the fibrediameter have to be entered . This automatically createsthe required .txt -files with the amplification factors .The Analysis will not run without these files providedin the working directory .**********************************************************Outputted Solution Dependent State Variables ( STATEV ):STATEV (1) - first strain invariant matrix phaseSTATEV (2) - equivalent strain matrix phaseSTATEV (3) - equivalent strain fibre phaseSTATEV (4) - damage initiation first invariant in

matrix phaseSTATEV (5) - damage initiation second invariant in

matrix phaseSTATEV (6) - damage initiation second invariant in

fibre phaseSTATEV (7) - matrix damage variable with viscous dissipationSTATEV (8) - fibre damage variable with viscous dissipationSTATEV (9) - matrix damage variable without viscous

dissipationSTATEV (10) - fibre damage variable without viscous

dissipationSTATEV (11) to STATEV (16) - used for calculation of

damage variablesSTATEV (17) to STATEV (22) - stress components used

for calculationSTATEV (23) to STATEV (28) - stress components without

viscous dissipation**********************************************************By inserting the following lines into the input file belowthe * DEPVAR flag (e.g. by editing the model keywords ), adescription of the STATEVs shows up in the field output :

75

1 ,"01 I1_m "," First strain invariant amplified in matrix phase"2 ,"02 eps_eq_m "," Equivalent strain amplified in matrix phase"3 ,"03 eps_eq_f "," Equivalent strain amplified in fibre phase"4 ,"04 SIFT_1 "," Damage initiation , first strain invariant amplified

in matrix phase"5 ,"05 SIFT_2 "," Damage initiation , equivalent strain amplified in

matrix phase"6 ,"06 SIFT_3 "," Damage initiation , equivalent strain amplified in

fibre phase"7 ,"07 mf"," Matrix failure damage variable "8 ,"08 ff"," Fibre failure damage variable "9 ,"09 mf_d "," Matrix failure damage variable without viscous

dissipation "10 ,"10 ff_d "," Fibre failure damage variable without viscous

dissipation "11 ,"11 IV_1 "," Internal variable "12 ,"12 IV_2 "," Internal variable "13 ,"13 IV_3 "," Internal variable "14 ,"14 IV_4 "," Internal variable "15 ,"15 IV_5 "," Internal variable "16 ,"16 IV_6 "," Internal variable "17 ,"17 IV_7 "," Internal variable "18 ,"18 IV_8 "," Internal variable "19 ,"19 IV_9 "," Internal variable "20 ,"20 IV_10 "," Internal variable "21 ,"21 IV_11 "," Internal variable "22 ,"22 IV_12 "," Internal variable "23 ,"23 IV_13 "," Internal variable "24 ,"24 IV_14 "," Internal variable "25 ,"25 IV_15 "," Internal variable "26 ,"26 IV_16 "," Internal variable "27 ,"27 IV_17 "," Internal variable "28 ,"28 IV_18 "," Internal variable "

A.2. VUMAT Readme

********************************************************************VUMAT for orthotropic material behaviour in laminated compositeswith implementation of non - linear ( pseudo elastic ) matrix behaviour andprogressive damage . Matrix material behaviour is governed by theequivalent strain in the matrix phase. Damage initiation is based on theStrain Invariant Failure Theory (SIFT ). The linear damage evolution lawis governed by the SIFT damage variables .Damage can occur in two independent directions : fibre damage and matrixdamage .********************************************************************The VUMAT is implemented only for 3D- stress state! It was tested for

76

C3D8R elements (8-node , linear hexaeder element , reduced integration ).********************************************************************Material properties for input:Ex , Ey , Ez , NUxy , NUxz , NUyz , Gxy , Gxz , Gyz , ( orthotropic materialproperties )

J1_crit , I2_f_crit , I2_m_crit ,( critical invariants : first invariant in matrix phase , secondinvariant in fibre phase , second invariant in matrix phase),

CF (1<CF , multiple of damage variable in fibre failure at which totalfailure occurs , CF=1 results in perfectly brittle failure which mayresult in convergence issues ), CM1 and CM2 (1<CM1 ,CM2 , multiple ofresult in convergence issues ), second invariant , respectively , atwhich total failure occurs , CM1 ,CM2 =1 results in perfectly brittlefailure which may result in convergence issues ),

MAXD (0<MAXD <1, maximum value for the damage evolution variables ,if MAXD =1 there is no remaining stiffness after total failure ofthe element , which may lead to crashing of the solver !),

A (first parameter governing the law for nonlinear matrixbehaviour , 0<A<1),C ( second parameter governing the law for nonlinear matrixbehaviour , 0<C, the smaller C, the faster the matrix behaviourdeviates from elastic behaviour ).

There are 24 STATEV used in the model

For element deletion state variable 9 shall be used as Status variablein the material definition

A material orientation has to be specified in the model.

Three .txt -files with the amplification factors have to be in theworking directory . These files should be created with the givenAbaqus macro. Upon executing the macroamplification_factors .py in Abaqus , the matrix and fibrematerial properties , the fibre volume ratio and the fibrediameter have to beentered . This automatically createsthe required .txt -files with the amplification factors .The Analysis will not run without these files providedin the working directory .********************************************************************Outputted Solution Dependent State Variables ( STATEV ):STATEV (1) - first strain invariant matrix phaseSTATEV (2) - equivalent strain matrix phaseSTATEV (3) - equivalent strain fibre phaseSTATEV (4) - damage initiation first invariant in matrix phaseSTATEV (5) - damage initiation second invariant in matrix phase

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STATEV (6) - damage initiation second invariant in fibre phaseSTATEV (7) - matrix damage variable with viscous dissipationSTATEV (8) - fibre damage variable with viscous dissipationSTATEV (9) - Status variableSTATEV (10) to STATEV (15) - undamaged stress componentsSTATEV (16) to STATEV (21) - total strain componentsSTATEV (22) - internal variableSTATEV (23) - characteristic element length at beginning of theanalysisSTATEV (24) - Nonlinearity parameter********************************************************************By inserting the following lines into the input file below the * DEPVARflag (e.g. by editing the model keywords ), a description of theSTATEVs shows up in the field output :

1 ,"01 I1_m "," First strain invariant amplified in matrix phase"2 ,"02 eps_eq_m "," Equivalent strain amplified in matrix phase"3 ,"03 eps_eq_f "," Equivalent strain amplified in fibre phase"4 ,"04 SIFT_1 "," Damage initiation , first strain invariant amplified inmatrix phase"5 ,"05 SIFT_2 "," Damage initiation , equivalent strain amplified inmatrix phase"6 ,"06 SIFT_3 "," Damage initiation , equivalent strain amplified infibre phase"7 ,"07 mf"," Matrix failure damage variable "8 ,"08 ff"," Fibre failure damage variable "9 ,"09 STATUS "," Status variable "10 ,"10 IV_1 "," Internal variable "11 ,"11 IV_2 "," Internal variable "12 ,"12 IV_3 "," Internal variable "13 ,"13 IV_4 "," Internal variable "14 ,"14 IV_5 "," Internal variable "15 ,"15 IV_6 "," Internal variable "16 ,"16 IV_7 "," Internal variable "17 ,"17 IV_8 "," Internal variable "18 ,"18 IV_9 "," Internal variable "19 ,"19 IV_10 "," Internal variable "20 ,"20 IV_11 "," Internal variable "21 ,"21 IV_12 "," Internal variable "22 ,"22 IV_13 "," Internal variable "23 ,"23 IV_14 "," Internal variable "24 ,"24 IV_15 "," Nonlinearity parameter "

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A.3. Amplification Factors

Table A.1.: Amplification factors determined for the diamond fibre arrangement.Position ε11 ε22 ε33 ε12 ε13 ε23M1 1.00E+00 1.58E+00 1.58E+00 1.47E+00 1.49E+00 6.54E-01M2 1.00E+00 1.58E+00 1.58E+00 1.48E+00 1.48E+00 6.57E-01M3 1.00E+00 1.30E+00 1.33E+00 1.57E+00 1.64E+00 1.64E+00F1 1.00E+00 4.68E-01 5.27E-01 2.74E-01 3.32E-01 5.51E-01F2 1.00E+00 4.88E-01 4.88E-01 3.05E-01 3.03E-01 5.87E-01F3 1.00E+00 5.25E-01 4.70E-01 3.31E-01 2.76E-01 5.53E-01F4 1.00E+00 4.90E-01 4.89E-01 3.05E-01 3.03E-01 5.83E-01F5 1.00E+00 4.69E-01 5.27E-01 2.75E-01 3.32E-01 5.51E-01F6 1.00E+00 4.91E-01 4.88E-01 3.06E-01 3.02E-01 5.82E-01F7 1.00E+00 5.24E-01 4.70E-01 3.30E-01 2.76E-01 5.55E-01F8 1.00E+00 4.94E-01 4.87E-01 3.09E-01 3.00E-01 5.82E-01F9 1.00E+00 4.69E-01 4.69E-01 3.04E-01 3.03E-01 6.42E-01

Table A.2.: Amplification factors determined for the hexagonal fibre arrangement.Position ε11 ε22 ε33 ε12 ε13 ε23M1 1.00E+00 1.40E+00 9.41E-01 1.89E+00 9.28E-01 1.48E+00M2 1.00E+00 1.06E+00 2.09E+00 5.05E-01 2.56E+00 1.06E+00M3 1.00E+00 1.40E+00 9.42E-01 1.89E+00 9.25E-01 1.48E+00F1 1.00E+00 4.76E-01 4.81E-01 2.79E-01 2.93E-01 5.15E-01F2 1.00E+00 5.12E-01 5.23E-01 2.78E-01 2.95E-01 4.38E-01F3 1.00E+00 5.24E-01 5.95E-01 2.56E-01 3.22E-01 3.79E-01F4 1.00E+00 5.10E-01 5.26E-01 2.76E-01 2.97E-01 4.36E-01F5 1.00E+00 4.75E-01 4.87E-01 2.75E-01 2.97E-01 5.08E-01F6 1.00E+00 5.11E-01 5.27E-01 2.76E-01 2.97E-01 4.38E-01F7 1.00E+00 5.24E-01 5.94E-01 2.56E-01 3.22E-01 3.80E-01F8 1.00E+00 5.12E-01 5.24E-01 2.78E-01 2.96E-01 4.34E-01F9 1.00E+00 5.21E-01 5.50E-01 2.71E-01 3.03E-01 4.24E-01

Table A.3.: Amplification factors determined for the square fibre arrangement.Position ε11 ε22 ε33 ε12 ε13 ε23M1 1.00E+00 2.07E+00 9.34E-01 2.89E+00 4.92E-01 1.38E+00M2 1.00E+00 9.31E-01 2.06E+00 4.87E-01 2.86E+00 1.38E+00M3 1.00E+00 1.02E+00 1.01E+00 1.56E+00 1.53E+00 1.95E+00F1 1.00E+00 6.13E-01 5.43E-01 3.52E-01 2.87E-01 4.38E-01F2 1.00E+00 5.68E-01 5.64E-01 3.22E-01 3.17E-01 4.58E-01F3 1.00E+00 5.50E-01 6.07E-01 2.93E-01 3.44E-01 4.34E-01F4 1.00E+00 5.62E-01 5.68E-01 3.15E-01 3.19E-01 4.55E-01F5 1.00E+00 6.15E-01 5.43E-01 3.53E-01 2.85E-01 4.37E-01F6 1.00E+00 5.64E-01 5.69E-01 3.14E-01 3.19E-01 4.50E-01F7 1.00E+00 5.47E-01 6.14E-01 2.89E-01 3.51E-01 4.37E-01F8 1.00E+00 5.71E-01 5.56E-01 3.25E-01 3.11E-01 4.57E-01F9 1.00E+00 6.12E-01 6.08E-01 3.21E-01 3.17E-01 3.93E-01

79

List of Figures

1.1. Schematic representation of the structural levels observed in composite ma-terials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2. Depiction of the general methodology followed in this thesis. . . . . . . . . . 6

2.1. General motion and deformation of an arbitrary deformable body where P ,Q1, and Q2 are three arbitrary material points in the initial configurationand p, q1, and q2 the same points in the deformed configuration (adaptedfrom [7]). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2. Application of equation constrains for PBCs to an arbitrary 2D-RVE con-sisting of 3 × 3 elements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.3. Micromechanical blocks utilised for determining strain amplification fac-tors. The locations for extracting the amplification factors are shown to-gether with the outline of the smallest unit cell required for modelling therespective fibre arrangement. . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.4. Identification of the locations for extracting strain amplification factors de-picted for the square fibre arrangement (adapted from [4]). . . . . . . . . . 16

2.5. Symbolic depiction of the loading scenarios imposed onto the micromechan-ical blocks for determining the strain amplification factors (adapted from[4]). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.6. Deformed configuration of the micromechanical blocks with hexagonal fibrearrangements at a respective applied strain of εij = 0.01. . . . . . . . . . . . 18

2.7. SIFT failure bodies resulting from the three critical invariants. The fail-ure bodies obtained with amplified (2.7a) and homogenised strains (2.7b),respectively, are compare utilising the same critical invariants for both cases. 21

2.8. Schematic describing the implementation of the SIFT as a UMAT in AbaqusStandard (adapted from [17]). . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.9. Linear damage evolution law. . . . . . . . . . . . . . . . . . . . . . . . . . . 252.10. Highly nonlinear response observed when testing ±45° laminate specimens

and modelling of the tests utilising the developed constitutive laws. . . . . . 312.11. Schematic describing the implementation of the SIFT as a VUMAT in

Abaqus Explicit (adapted from [17]). . . . . . . . . . . . . . . . . . . . . . . 32

80

2.12. Comparison of linear and nonlinear material response and subsequent lineardamage evolution. The strain at damage initiation and final damage is equalin both cases. Dashed lines represent unloading and subsequent reloadingpaths. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.1. Modelling of a composite by means of a 2 × 2 doubly periodic RUC. In3.1b the RUC is depicted with the local subcell coordinate system, havingits origin in the centre of the subcell, and the subcell numbering beingillustrated (adapted from [1]). . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.2. Approximation of a continuous fibre inclusion utilising a doubly periodic2 × 2 MOC microcell (3.2a) and a 26 × 26 GMC or HFGMC microcell(3.2b), respectively (adapted from [1]). . . . . . . . . . . . . . . . . . . . . . 36

3.3. Modelling of a continuously reinforced composite utilising the GMC, HFGMC,and FEA, respectively, by means of a single fibre RUC. A strain of ε11 = 0.01is applied, the 11-direction being the fibre direction and the resulting stressesare depicted. Comparing 3.3g and 3.3h, lack of shear coupling in the GMC isobvious from the fact, that there are no shear stresses arising in the GMC-microcell. The stress concentrations along the fibre inclusion are betterrepresented with the HFGMC compared to the GMC, while very similarresults are obtained with the HFGMC and FEA. . . . . . . . . . . . . . . . 37

3.4. Geometry of a triply periodic RUC consisting of 3 × 2 × 3 subcells (adaptedfrom [1]). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.5. Microcell of a staggered structure consisting of platelets of high stiffnessembedded in a soft matrix. The platelets are extending infinitely in out-of-plane direction (doubly periodic RUC). The microcell width is scaled forbetter representation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.6. Microcell of a staggered structure with an applied strain in direction ofinclusion orientation of ε11 = 0.01 utilising the GMC and HFGMC, respec-tively. The lack of shear coupling in the GMC becomes apparent. . . . . . . 44

3.7. Depiction of Eshelby’s problem: An ellipsoidal inclusion inside an infinitematrix being cut out, undergoing an eigenstrain and being welded back(reprinted from [30]). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.8. Exemplification of the Mori-Tanaka approach: The multi-inclusion prob-lem is approximated by a single-inclusion problem and subsequently solvedbased an Eshelby’s problem (reprinted from [30]). . . . . . . . . . . . . . . . 46

3.9. Comparison of the transversal moduli of a continuously reinforced com-posite determined utilising the Mori-Tanaka homogenisation approach andthe GMC with the results of upper and lower bound models. For betterreference pure matrix and inclusion stiffness are depicted as well. . . . . . . 47

3.10. Schematic stress-strain response of an elastic-plastic material with linearhardening (adapted from [1]). . . . . . . . . . . . . . . . . . . . . . . . . . . 48

81

3.11. Evolving of the yield surface during plastic deformation with isotropic hard-ening of the material depicted in the deviatoric plane (adapted from [34],[35]). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.12. Comparison of Curtin fibre failure with different values for τ implementedwith a doubly periodic 2 × 2 RUC. The effective fibre stress as well as theaverage composite stress are depicted. Except for the dotted lines the fibresubcell stress is zeroed upon reaching the ultimate fibre stress. . . . . . . . 54

4.1. Stress-strain response of epoxy resin specimens in tension. . . . . . . . . . . 564.2. Stress-strain response of composite laminate specimens in tension. . . . . . 58

5.1. Comparison of numerical and experimental data for 0° specimens. . . . . . 645.2. Comparison of numerical and experimental data for 90° specimens. . . . . . 655.3. Comparison of numerical and experimental data for ±45° specimens. . . . . 665.4. Matrix damage variable after reaching maximum stress for ±45° lami-

nate specimens in tension simulated using the developed UMAT (5.4a) andVUMAT (5.4b). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

5.5. Comparison of numerical and experimental data for ±15° specimens. . . . . 69

6.1. Schematic overview of the implemented and examined numerical models. . . 71

82

List of Tables

2.1. Amplification factors determined from the varying fibre arrangements atthe M1 and F1 location within the micromechanical blocks. . . . . . . . . . 19

3.1. Comparison of the GMC and HFGMC for the determination of the effectivemoduli of a continuously reinforced composite. . . . . . . . . . . . . . . . . 42

3.2. Material and geometry input parameters for modelling of a staggered struc-ture. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.3. Longitudinal stiffness of a staggered structure determined by means of theGMC and HFGMC, respectively. . . . . . . . . . . . . . . . . . . . . . . . . 43

4.1. Epoxy resin properties obtained from experimental tensile tests. . . . . . . . 554.2. Specimen geometry and testing conditions for composite laminate tensile

tests. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574.3. Composite properties obtained from experimental tensile tests of different

layups. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5.1. Effective composite properties obtained utilising the FEA of a microcell inDigimat FE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

5.2. Critical invariants and parameters for defining damage evolution laws ob-tained from fitting experimental data. . . . . . . . . . . . . . . . . . . . . . 60

5.3. Matrix and fibre elastic input parameters. . . . . . . . . . . . . . . . . . . . 615.4. Parameters utilised for modelling fibre damage by means of the Curtin fibre

failure model [46]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625.5. Effective composite properties obtained utilising the GMC. . . . . . . . . . 625.6. Modelling restrictions and bugs encountered in FEAMAC. . . . . . . . . . . 635.7. Comparison of numerical and experimental data for 0° specimens. . . . . . 645.8. Comparison of numerical and experimental data for 90° specimens. . . . . . 655.9. Comparison of numerical and experimental data for ±45° specimens. . . . . 675.10. Comparison of numerical and experimental data for ±15° specimens. . . . . 68

A.1. Amplification factors determined for the diamond fibre arrangement. . . . . 78A.2. Amplification factors determined for the hexagonal fibre arrangement. . . . 78A.3. Amplification factors determined for the square fibre arrangement. . . . . . 78

83

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