2II Mechanics and Acoustics of Viscoelastic Inclusion Reinforced Composites-micro--macro Modeling of Effective

Universite catholique de Louvain

Faculte des sciences appliquees

Mechanics and Acoustics of viscoelastic

inclusion reinforced composites:

micro-macro modeling of effective

properties.

Christophe Friebel

Les membres du jury

Pr. C. Bailly Universite catholique de Louvain, BelgiquePr. I. Doghri Universite catholique de Louvain, Belgique (Promoteur)Pr. M. Geers Technische Universiteit Eindhoven, Pays-BasPr. P. Geubelle University of Illinois at Urbana-Champaign, USAPr. V. Legat Universite catholique de Louvain, Belgique (Promoteur)Pr. T. Pardoen Universite catholique de Louvain, BelgiquePr. J.-C. Samin Universite catholique de Louvain, Belgique (President)

These presentee en vue de l’obtention du gradede docteur en sciences appliquees

November 2007

Remerciements

Les travaux de recherche realises lors de ces annees de these n’auraient puetre menes a bien sans le soutien et la comprehension de nombre de personnesauxquelles je tiens a exprimer mes remerciements les plus sinceres.

Mes promoteurs tout d’abord, les Professeurs Issam Doghri et Vincent Legat.Le Professeur Issam Doghri etait deja le promoteur de mon travail de find’etudes, lequel portait egalement sur la micro-mecanique des composites. Jele remercie chaleureusement de m’avoir propose une these sous sa direction afinde poursuivre ces recherches. Je voudrais aussi lui exprimer ma profonde grati-tude pour la confiance qu’il m’a toujours accordee. Je souhaite aussi remercierle Professeur Vincent Legat pour son soutien et sa disponibilite. Tout deux onteu des roles complementaires, a la fois sur le plan scientifique et sur le plan hu-main. Je tiens aussi a les remercier conjointement pour leurs encouragementstout au long de ces annees de recherche.

Je souhaite aussi remercier chaleureusement les professeurs Christian Bailly etThomas Pardoen, membres de mon comite d’encadrement, ainsi que les autresmembres de mon jury de these, les professeurs Marc Geers, Philippe Geubelle etJean-Claude Samin, pour les appreciations et commentaires constructifs qu’ilsont portes sur cette dissertation

Il y aurait tant de choses a dire au sujet d’Olivier, avec qui j’ai partage lebureau pendant quatre ans. Ce fut un reel plaisir de travailler avec lui. Notrecollaboration ne se limitait cependant pas aux seuls aspects scientifiques. Nouspartagions d’autres centres d’interet. Je n’oublierai jamais ces formidablesannees passees en sa companie. J’ai la chance a present de travailler a nouveauavec lui.

Je n’oublierai pas non plus tous les bons moments passes avec Amine. J’ai nonseulement beaucoup appris grace a lui sur le plan scientifique, mais nous avonspris egalement beaucoup de plaisir a organiser ensemble les activites didactiquespour les etudiants.

Je pense par ailleurs a tous les autres membres de l’unite MEMA et a toutes

iv

les personnes du batiment Euler que j’ai eu la chance de connaitre tout au longde ces annees: Laurent Delannay, Maxime Melchior, Laurent White, OlivierLietaer, Brieux Delsaute, Olivier Gourgue, Yamen Othmani, Abdellatif Selmi,Richard Comblen, Slim Kammoun, et tous les autres.

Mes remerciements vont aussi a Michele Sergant, pour sa bonne humeur etpour toute l’aide qu’elle m’a apportee, en particulier dans le dedale des tachesadministratives.

Je souhaite aussi remercier ma famille, mes parents, ma soeur Stephanie etson fiance Daniel, pour leur soutien depuis que j’ai commence cette these etmeme bien avant, lorsque j’ai decide de quitter la Belgique pour passer quelquesannees en France, ou encore, lorsque je redigeais mon travail de fin d’etudes.

Pour finir, je tiens a remercier tous mes amis. Qu’ils veuillent bien me pardon-ner d’avoir disparu sans donner de nouvelles pendant des mois d’affilee, occupea mes recherches qui aboutissent a la presente publication. Je vais essayer defaire autrement . . .

Christophe

Contents

Introduction 1

1 Static elastic homogenization models 5

1.1 Two-phase composites . . . . . . . . . . . . . . . . . . . . . . . 6

1.1.1 Voigt and Reuss models . . . . . . . . . . . . . . . . . . 6

1.1.2 Dilute inclusion model . . . . . . . . . . . . . . . . . . . 7

1.1.3 Mori-Tanaka scheme . . . . . . . . . . . . . . . . . . . . 8

1.1.4 Interpolative scheme . . . . . . . . . . . . . . . . . . . . 9

1.1.5 Self-consistent scheme . . . . . . . . . . . . . . . . . . . 9

1.2 Multi-phase composites . . . . . . . . . . . . . . . . . . . . . . 9

1.3 Composites with coated inclusions . . . . . . . . . . . . . . . . 12

1.3.1 A two-level recursive scheme . . . . . . . . . . . . . . . 12

1.3.2 Other generic procedures . . . . . . . . . . . . . . . . . 13

1.3.3 A two-step method . . . . . . . . . . . . . . . . . . . . . 14

1.3.4 A comparative study . . . . . . . . . . . . . . . . . . . . 14

1.4 Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . 16

1.4.1 Composites with aligned reinforcements . . . . . . . . . 17

1.4.2 Composites with misaligned reinforcements . . . . . . . 23

1.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

A Expressions for Eshelby’s tensor . . . . . . . . . . . . . . . . . . 28

B Micro finite elements . . . . . . . . . . . . . . . . . . . . . . . . 32

B.1 Unit-cell modeling of particulate composites . . . . . . . 32

B.2 Unit-cell modeling of long fiber composites . . . . . . . 33

vi CONTENTS

B.3 RVE modeling of composites . . . . . . . . . . . . . . . 36

C On the Mori-Tanaka estimates of the transverse tensile modulus 37

D Notations and units . . . . . . . . . . . . . . . . . . . . . . . . 39

2 Quasi-static viscoelastic homogenization models 41

2.1 Homogeneous material response . . . . . . . . . . . . . . . . . . 42

2.1.1 1D illustrative example . . . . . . . . . . . . . . . . . . 42

2.1.2 3D tensorial formalism . . . . . . . . . . . . . . . . . . . 43

2.2 Homogenization procedures . . . . . . . . . . . . . . . . . . . . 44

2.2.1 Effective harmonic properties . . . . . . . . . . . . . . . 45

2.2.2 Effective time properties . . . . . . . . . . . . . . . . . . 45

2.3 Numerical simulations . . . . . . . . . . . . . . . . . . . . . . . 47

2.3.1 Frequency behavior of two-phase composites . . . . . . 47

2.3.2 Frequency behavior of coated-inclusion reinforced materials 57

2.3.3 Time behavior of two-phase composites . . . . . . . . . 62

2.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

A Notations and units . . . . . . . . . . . . . . . . . . . . . . . . 68

3 Dynamic viscoelastic homogenization models 69

3.1 A generic modeling framework . . . . . . . . . . . . . . . . . . 70

3.1.1 Steady-state dynamics . . . . . . . . . . . . . . . . . . . 70

3.1.2 Integral equation of the displacement field . . . . . . . . 71

3.1.3 Effective displacement field . . . . . . . . . . . . . . . . 72

3.1.4 Dispersion equation for effective plane waves . . . . . . 73

3.2 Models derived from this framework . . . . . . . . . . . . . . . 75

3.2.1 One-particle approximation and conditional means . . 75

3.2.2 Self-consistent schemes . . . . . . . . . . . . . . . . . . . 78

3.2.3 A class of explicit schemes . . . . . . . . . . . . . . . . . 78

3.2.4 Static limit of the models . . . . . . . . . . . . . . . . . 79

3.3 An additive model for wave attenuation . . . . . . . . . . . . . 80

3.3.1 Scattering and absorption cross-sections . . . . . . . . . 81

3.3.2 The additive attenuation model . . . . . . . . . . . . . . 85

CONTENTS vii

3.4 Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . 86

3.4.1 Glass-particle reinforced epoxy composite . . . . . . . . 87

3.4.2 Glass/Epoxy and Rubber/PMMA systems . . . . . . . . 89

3.4.3 Hard and heavy particles in a soft and light matrix . . . 91

3.4.4 Steel-particle reinforced PMMA composite . . . . . . . 92

3.4.5 Lead-particle reinforced epoxy composite . . . . . . . . 96

3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

A Solution of the one-particle scattering problem . . . . . . . . . 99

B Expressions for tensors λp and Λp . . . . . . . . . . . . . . . . 102

C Notations and units . . . . . . . . . . . . . . . . . . . . . . . . 106

4 Finite element acoustic simulations 109

4.1 Equation for the acoustic pressure . . . . . . . . . . . . . . . . 110

4.2 Acoustic impedance . . . . . . . . . . . . . . . . . . . . . . . . 110

4.2.1 Characteristic impedance of acoustic media . . . . . . . 110

4.2.2 Specific impedance of acoustic components . . . . . . . 111

4.2.3 Absorption coefficient . . . . . . . . . . . . . . . . . . . 111

4.3 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . 112

4.4 Discontinuous Galerkin formulation . . . . . . . . . . . . . . . . 113

4.4.1 Local weak form . . . . . . . . . . . . . . . . . . . . . . 114

4.4.2 Numerical fluxes . . . . . . . . . . . . . . . . . . . . . . 114

4.4.3 Internal Penalty method . . . . . . . . . . . . . . . . . . 114

4.4.4 Numerical implementation . . . . . . . . . . . . . . . . . 115

4.4.5 Convergence analysis . . . . . . . . . . . . . . . . . . . . 115

4.5 Numerical simulations . . . . . . . . . . . . . . . . . . . . . . . 121

4.5.1 Application to silencers . . . . . . . . . . . . . . . . . . 121

4.5.2 A coupled problem or how to hide a submarine . . . . . 124

4.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

A Specific impedance of an acoustic layer . . . . . . . . . . . . . . 130

B Notations and units . . . . . . . . . . . . . . . . . . . . . . . . 132

Conclusions 133

viii CONTENTS

List of publications 137

References 139

Introduction

In many applications, there is a growing interest in the use of composites. Thecombination of materials with different characteristics gives rise to propertiesthat couldn’t have been reached by either of the components alone. Compos-ites may be of several kinds (e.g. laminated, woven) depending on how theconstituents are put together. Inclusion reinforced materials are considered inthis work. These are composites made up with particles or fibers embedded ina matrix.

The development of new technologies allows to produce more and more inno-vative composites. Numerical approaches are essential to design their char-acteristics in order to obtain specific properties for given applications. Theyprovide a cheaper solution as compared to experiments. Efficient models forthe prediction of composites’ effective properties are however required.

There are at least two ways of addressing numerically the behavior of inclu-sion reinforced materials. A direct approach (e.g. the finite element method)is accurate and provides localized results in the microstructure. Such predic-tions are useful but require high computational costs. This approach evenbecomes unrealistic when composites are involved in simulations of real struc-tures. Mean-field homogenization schemes provide a cost-effective alternative.Based on simplifying assumptions, these methods predict the overall behaviorin a much cheaper way, which renders feasible simulations involving compositestructures.

The development of homogenization schemes strongly depends on the consti-tutive laws of the constituents (e.g. linear, non-linear, rate-dependent). In thiswork, we are concerned with linear viscoelastic inclusion reinforced compositesin the scope of acoustic attenuation applications.

When an acoustic wave hits an interface, part of its energy is reflected to themedium while the other is transmitted and propagates on the other side. If theinterface material is elastic and homogeneous, the amplitude of the transmittedwave remains constant over the thickness of the boundary. Viscoelastic mate-rials possess the ability to convert sound energy to other forms of energy (e.g.

2 Introduction

into heat). This is called absorption. As a result, the intensity of the transmit-ted wave diminishes with the distance. Another form of attenuation is due toscattering. Scattering is the reflection of the sound in directions other than itsoriginal direction of propagation. In composite materials, scattering occurs be-cause of the reinforcements. The combined effect of scattering and absorptionin viscoelastic particulate composites may be highly profitable. The energythat is scattered by the particles is absorbed by the viscoelastic matrix, leadingto an increase of attenuation in the material and a decrease of reflected power.Applications related to attenuative walls or anechoic coatings take advantageof this combined effect.

It is therefore of major interest to develop, verify and validate homogenizationmodels that take both viscoelasticity and scattering into account. Throughoutthe chapters, we will make homogenization schemes evolve from linear elasticity(Chap. 1) to linear viscoelasticity (Chap. 2), accounting for scattering inchapter 3. Chapter 4 links up these approaches to the modeling of the acousticfield in fluids interacting with composite walls.

There exists a large body of literature on mean-field homogenization schemes(see for instance the review article of Tucker III and Liang, 1999). They werefirst developed in linear elasticity. Most of them (e.g. the Mori and Tanaka(1973) model) are based on the well known result of Eshelby (1957). Chapter1 recalls some efficient schemes for two-phase composites (e.g. the interpola-tive double-inclusion scheme (Nemat-Nasser and Hori, 1999; Lielens, 1999))as well as a generic two-step approach (Camacho et al., 1990; Lielens, 1999;Friebel, 2002; Pierard et al., 2004) for multiphase inclusion reinforced mate-rials. A special attention is drawn to composites with coatings for which anoriginal two-level recursive scheme is proposed (Friebel et al., 2006). An an-alytical comparative study is performed with the two-step approach and themulti-inclusion method by Nemat-Nasser and Hori (1999). Several numericalsimulations of polymer reinforced composites are shown. Finite element calcu-lations are used for verification purposes.

Pioneering contributions to the homogenization of linear viscoelastic compositeswere made by Hashin (1965, 1970) and Christensen (1969). In 1965, Hashin pre-dicted the effective bulk modulus of two-phase viscoelastic composites. Chris-tensen followed in 1969 with formulae for both bulk and shear complex mod-uli. Hashin also gave results for the shear modulus in 1970. All these modelsare based on the composite spheres assemblage developed by Hashin (1962)for elastic composites and make use of the well-known correspondence princi-ple between linear elasticity and linear viscoelasticity. The major restrictionin those models is that they consider macroscopically isotropic composites,therefore they cannot handle fibrous composites for instance. Afterwards, thecorrespondence principle has been heavily used in order to extend homogeniza-tion models from linear elasticity to linear viscoelasticity (see for instance a

3

comparative study on fibrous composites by Chandra et al., 2002). All theaforementioned models are for two-phase composites. Our proposal in chapter2 is to generalize the interpolative double-inclusion scheme (two phase), thetwo-step (multiphase) and the two-level (composites with coatings) proceduresfor linear viscoelastic behaviors. An extensive verification and validation effortis carried out for numerous viscoelastic composite systems. The predictions– harmonic and time dependent – of our proposed methods as well as thoseof other authors are compared against experimental data and finite elementsimulations.

A large body of literature also exists on numerical schemes that account forscattering by inclusions in composite materials. Moon and Mow (1970) pro-posed a model for a dilute suspension of rigid spheres in an elastic matrix. Datta(1976) obtained longitudinal wave velocities through the scattering P-waves bya distribution of rigid spheroids. With a similar probabilistic approach, Datta(1977) extended the model to longitudinal and shear wave speeds in the caseof elastic ellipsoids, yet in the long-wave region. Brauner and Beltzer (1988)put forward a formulation that accounts for energy losses in a viscoelastic ma-trix (see also Beltzer and Brauner, 1987; Beltzer et al., 1983). Their theorywas re-examined with some modifications and applied to particle or fiber re-inforced polymers by Biwa (2001); Biwa et al. (2002, 2003). Variational tech-niques and self-consistent estimates have been proposed by Sabina and Willis(1988); Sabina et al. (1993); Smyshlyaev et al. (1993) for elastic composites.Assuming constant fields in the inclusions, the models by Sabina et al. (1993)and Smyshlyaev et al. (1993) are able to handle multiple phases of spheroidalreinforcements, aligned or oriented. These works were followed by those ofKanaun et al. (2004) and Kanaun and Levin (2005) who lifted the constantfields approximation for two-phase elastic composites with spherical particles.In chapter 3, we propose a generic formulation that brings together the modelsby Sabina et al. (1993), Smyshlyaev et al. (1993), Kanaun et al. (2004) andKanaun and Levin (2005). The latter is developed for viscoelastic behaviorsand may thus be considered as an extension of the aforementioned schemes.A dilute-inclusion model for two-phase particulate composites is derived fromthe formulation. It is based on the same single-scattering approximation asthe additive attenuation model by Biwa et al. (2002) which is also presentedin chapter 3. Numerical simulations are performed to compare the predictionsof all these approaches. A special emphasis is put on the comparison betweenour dilute-inclusion scheme and the model by Biwa et al. (2002).

Chapter 4 proposes to incorporate the homogenization schemes developed inthe previous chapters within the simulation of coupled fluid/solid acoustic prob-lems. To this end, a discontinuous Galerkin (DG) formulation for the Helmholtzequation, which describes time-harmonic acoustic and (visco)elastic waves, isproposed. DG methods provide a higher rate of convergence and require lower

4 Introduction

mesh resolution than standard finite element methods. A large review on thedevelopment of discontinuous Galerkin methods can be found in Cockburn et al.(2000). Our formulation is inspired by the works of Arnold et al. (2002) whoclassified numerous DG approaches. A specific method, commonly referred toas the internal penalty method, is implemented to perform numerical simula-tions on 2D and axisymmetric problems. The effective behavior of a particlereinforced viscoelastic rubber, designed for anechoic coating applications, ispredicted with help of homogenization schemes. The simulations show theinfluence of taking particle scattering into account on the predicted reflectedsound power.

Chapter 1

Static elastichomogenization models

Linear elastic composites made up with inclusions embedded in a matrix areconsidered. The generic term inclusions means fibers (short or long), plateletsor spherical particles. The reinforcements are approximated by spheroids, i.e.ellipsoids with an axis of revolution. Eshelby-based mean-field homogenizationschemes provide a very cost-effective way of predicting the overall elastic proper-ties of those composites. Section 1.1 deals with two-phase materials for whichthe spheroids are aligned and identically shaped. Common (e.g. Voigt andReuss bounds, the model of Mori and Tanaka (1973)) and less classical (theinterpolative model of Lielens (1999)) homogenization schemes are recalled.These models are extended to composites with multiple phases of reinforce-ments (different shapes, materials and orientations) in Sec. 1.2 with a two-stepprocedure which was first advocated by Camacho et al. (1990). Section 1.3handles the particular case of composites with coated inclusions. An originaland competitive (see Sec. 1.4) two-level approach is proposed. A compara-tive study with other existing models is shown. The numerical simulations ofSec. 1.4 focus on the effective elastic properties of polymer matrix composites.Finite element calculations on composites’ micro-structures are performed forverification purposes. The method is briefly commented in Appendix B. Acomplete overview can be found in Pierard (2006).

6 Static elastic homogenization models

1.1 Two-phase composites

Consider a representative volume element (RVE) Ω of a two-phase compositewith randomly distributed and identically shaped aligned inclusions. The ma-trix – which occupies a region Ω0 – and the inclusions – with domain Ω1 = Ω\Ω0

– have uniform stiffnesses, respectively given by C0 and C1. The volume frac-tion of inclusions is noted v1.

Let this RVE be subjected to linear boundary displacements corresponding to amacroscopic strain ε. Define fourth-order strain concentration tensors Bε andAε that relate the average strain in the inclusion phase to the average strainin the matrix phase and to the macro strain respectively:

〈ε〉Ω1= Bε : 〈ε〉Ω0

, 〈ε〉Ω1= Aε : ε . (1.1)

Tensors Bε and Aε are actually linked to each-other:

Aε = Bε : ((1 − v1)I + v1Bε)

−1, Bε = (1 − v1) (I − v1A

ε)−1

: Aε . (1.2)

From the Hill-mandel condition, defining the macro stress as σ = 〈σ〉Ω, theoverall stiffness tensor reads

C = [(1 − v1)C0 + v1C1 : Bε] : [(1 − v1)I + v1Bε]

−1, (1.3)

= C0 + v1 (C1 − C0) : Aε . (1.4)

Different homogenization schemes are developed for different expressions of con-centration tensors, i.e. from different assumptions on the interaction law be-tween the phases. Basic models, such as Voigt or Reuss, use simple – simplisticindeed – interaction laws. They are not very powerful in general. More elabo-rated schemes (the dilute inclusion model, the Mori-tanaka, interpolative andself-consistent schemes) take into account the geometry of the micro-structure.Assuming the fillers can be approximated by ellipsoidal inclusions, all of themare based on the fundamental solution of Eshelby (1957).

1.1.1 Voigt and Reuss models

Voigt and Reuss models are not restricted to two-phase composites, even notto inclusion – of any shape they are – reinforced materials. These two simplemodels only involve a single micro-structural parameter, the volume fraction.We however introduce them here for clarity and comparison with other two-phase schemes. Their extension to multi-phase composites is obvious.

Voigt model assumes uniform strain in the RVE, i.e. Bε = I. This constitutesan upper (resp. lower) bound for the strain concentration tensor in the case ofstiff (resp. soft) reinforcements. The effective stiffness tensor then reads

CVoigt = (1 − v1)C0 + v1C1 . (1.5)

Two-phase composites 7

Reuss model assumes uniform stress in the RVE, leading to Bε = C−11 : C0.

This constitutes a lower (resp. upper) bound for the strain concentration tensorin the case of stiff (resp. soft) reinforcements. The overall stiffness tensor thenreads

CReuss =((1 − v1)C

−10 + v1C

−11

)−1. (1.6)

Both assumptions are too simplistic to model with enough accuracy the com-plete three dimensional elastic behavior of composites in general. Fibrous com-posites with long aligned fibers are in some extent an exception. Together, Voigtand Reuss schemes provide acceptable estimates for the effective longitudinal,using Eq. (1.5), and transverse, with Eq. (1.6), tensile moduli.

In the case of isotropic components, Voigt and Reuss models respectively pro-vide upper and lower bounds for the effective shear and bulk moduli.

1.1.2 Dilute inclusion model

For very low concentration of fillers, their distribution in the matrix is verysparse and interactions between the inclusions are negligeable. The RVE andits loading are idealized by a single inclusion embedded in an infinite matrixsubjected to some far-field strain ε∞.

Consider that the ellipsoidal inclusion is made of the matrix material but under-goes a stress-free deformation ε∗ in the surrounding medium. Eshelby (1957)proved that the strain field that is due to ε∗ is uniform in the inclusion andgiven by

ε1 = S : ε∗ , (1.7)

where S is the fourth-order Eshebly’s tensor (see Appendix A). Consequently,by superposition, the total stress field in the inclusion reads

σ1 = C0 : (ε∞ + S : ε∗ − ε∗) . (1.8)

Inside the original inhomogeneity – undergoing no eigenstrain – the stress fieldequals

σ1 = C1 : (ε∞ + S : ε∗) , (1.9)

for the same strains ε∞ + S : ε∗. Expressing that the stresses are identical inboth cases, one finds the value of the eigenstrain

ε∗ = −((

C−10 : C1 − I

)−1+ S

)−1

: ε∞ (1.10)

that asserts the mechanical equivalence between both problems. The strain inthe single inclusion is thus linked to the far-field strain through

ε1 =

[

I − S :((

C−10 : C1 − I

)−1+ S

)−1]

: ε∞ . (1.11)


The fourth-order tensor in the right-hand side of the above relation can berewritten

H (C0,C1) =(I + S :

(C−1

0 : C1 − I))−1

. (1.12)

As the strain field in the inclusion is uniform, 〈ε〉Ω1= ε1. The strains in the

matrix have two contributions, the far-field ε∞ and the non-uniform perturba-tion due to the inhomogeneity. Because the matrix is of infinite extent and theperturbation field decreases with the distance we have

〈ε〉Ω0' 〈ε〉Ω ' ε∞ , (1.13)

leading to

Bε ' Aε ' H (C0,C1) (1.14)

for the dilute inclusion model.

1.1.3 Mori-Tanaka scheme

The Mori-Tanaka scheme was proposed by Mori and Tanaka (1973) to incor-porate interactions between the inclusions. It is also based on the solution ofEshelby’s problem. It’s development (see Mori and Tanaka, 1973; Lielens, 1999;Friebel, 2002; Pierard, 2006) is similar to that of the dilute inclusion model.The main difference is that the mechanical equivalence condition cannot besatisfied exactly but only in an average sense.

The strain concentration tensor for the M-T scheme is found to be

BεMT = H (C0,C1) , (1.15)

leading to the following interpretation proposed by Benveniste (1987) : eachinclusion behaves as isolated seeing as far-field the the average strain in thematrix. Unlike the dilute case we have Aε 6= Bε.

The M-T scheme provides good estimates of the stiffness tensor for two-phasecomposites with low to moderate – namely up to 25 − 30% – volume fractionof aligned inclusions.

For high volume fractions of fillers the composite behaves as if inclusions madeof the matrix material were embedded in a matrix made of the inclusion mate-rial. Switching the properties of both phases, this gives the inverse Mori-Tanakascheme for which the strain concentration tensor reads

BεIMT = (H (C1,C0))

−1 . (1.16)

Mori-Tanaka and Inverse Mori-Tanaka models correspond in general to Hashin-Shtrikman upper and lower bounds, respectively.

Multi-phase composites 9

1.1.4 Interpolative scheme

The interpolative D-I model is based on the Nemat-Nasser and Hori (1999)double-inclusion model. For intermediate concentration of inclusions, Lielens(1999) proposed to interpolate the Hashin-Shtrikman strain concentration ten-sors,

BεDI =

[

(1 − f(v1)) (BεMT)

−1+ f(v1) (Bε

IMT)−1]−1

, (1.17)

with the non-linear interpolation function f(v1) = (v1 + v21)/2. This combina-

tion of BεMT and Bε

IMT ensures physical results in both cases of very hard orvery soft – with respect to the stiffness of the matrix – inclusions.

1.1.5 Self-consistent scheme

The self-consistent approach assumes that each inclusion is isolated in an in-finite homogeneous medium with properties C – the unknown homogenizedstiffness – and subjected to ε at infinity. From the single inclusion problem onefinds

AεSC = H

(C,C1

). (1.18)

The equation for the overall stiffness thus becomes implicit and requires aniterative procedure to be solved. This kind of approach is in general well suitedto polycrystals but less satisfying for two-phase inclusion reinforced materials.

1.2 Multi-phase composites

Let the representative volume element Ω of a multi-phase composite be a matrixwith N families of spheroidal inclusions. Each family Ωi with volume fraction vi

inside the RVE is characterized by an aspect ratio Ariand a stiffness tensor Ci.

Let also ψi(p) be the orientation distribution function describing the orientationof the inclusions belonging to family i. The matrix Ω0 has concentration v0 anda stiffness tensor C0. The term phase will be used to denote a family (phase ifor 1 ≤ i ≤ N) or the matrix (phase 0). Each phase’s material is linear elasticand homogeneous.

A general two-step homogenization procedure was originally proposed by Ca-macho et al. (1990) and afterwards studied by Lielens (1999) and Pierard et al.(2004). The starting point of the method is the decomposition of the RVE intoa set of pseudo-grains (see figure 1.1). Each pseudo-grain Ωi,p is a two-phasecomposite containing the matrix material and all the inclusions of family (i)aligned in direction p. The relative volume fraction of the reinforcements ina pseudo-grain is set to (1 − v0). The idea is the following. As illustrated on


DECOMPOSITION

FIRST STEP SECOND STEP

Figure 1.1: The 2-step homogenization procedure. The composite is decomposed intopseudo-grains. STEP 1: homogenization of each pseudo-grain. STEP 2: homogeniza-tion of the set of homogenized pseudo-grains.

figure 1.1, each pseudo-grain is first homogenized individually with a suitablescheme for this kind of composite. Afterwards the set of homogeneous pseudo-grains is itself homogenized. At the end, a volume average over the entire RVEis obtained as an average over families and orientations of the volume averagesover the pseudo-grains:

〈•〉Ω =

N∑

i=1

vi

(1 − v0)

∮

〈•〉Ωi,pdψi(p)=

⟨

〈•〉Ωi,p

⟩

i,Ψi

(1.19)

The derivation of the above formula is done as follows,

〈•〉Ω =1

V (Ω)

N∑

i=1

V

(

∪p

Ωi,p

)

〈•〉∪p

Ωi,p=

N∑

i=1

∮

〈•〉Ωi,p

dV (Ωi,p)

V (Ω), (1.20)

withdV (Ωi,p)

V (Ω)=

vi

(1 − v0)dψi(p) , (1.21)

expressing the volume conservation of the inclusions of a given family alignedin a given direction inside the corresponding pseudo-grain and inside the entireRVE.

Homogenization of each pseudo-grain. Let Ω1i,p and Ω0

i,p respectively bethe inclusion and matrix phases of the pseudo-grain Ωi,p and Bε

i,p the related

Multi-phase composites 11

strain concentration tensor i.e.

〈ε(x)〉Ω1i,p

= Bεi,p : 〈ε(x)〉Ω0

i,p. (1.22)

Once defined, a strain concentration tensor gives rise to an effective stiffnesstensor

Ci,p = [v0C0 + (1 − v0)Ci : Bεi,p] : [v0I + (1 − v0)B

εi,p]−1 (1.23)

that links average stress and strain at pseudo-grain level

〈σ(x)〉Ωi,p= Ci,p : 〈ε(x)〉Ωi,p

. (1.24)

Each pseudo-grain is a two-phase composite with aligned inclusions. Any ho-mogenization scheme of Sec. 1.1 can be used to estimate Ci,p. The simpleVoigt (Bε

i,p = I) and Reuss (Bεi,p = C−1

i : C0) models will overestimateand underestimate the stiffness of the pseudo-grain, respectively. For the M-Tscheme, the concentration tensor reads

Bεi,p =

(I + SI,C0 :

(C−1

0 : Ci − I))−1

, (1.25)

where SI,C0 is Eshelby’s tensor with I corresponding to the inclusions withaspect ratio Ari

aligned in direction p. There is no restriction upon using theinterpolative scheme. The switch of material properties is performed at pseudo-grain level. Both M-T and D-I models are used in the numerical simulations ofSec. 1.4.

Homogenization of the set of homogenized pseudo-grain. To performthe second step we will assume that each homogenized pseudo-grain undergoesthe same deformation. The macroscopic stiffness tensor that links average stressand strain over the RVE is then given by

C =⟨Ci,p

⟩

i,Ψi. (1.26)

Using this Voigt-like hypothesis leads in general to better results compared tothose obtained by assuming that the pseudo-grains are sharing the same stressor by using M-T (same deformation in the matrix phase of all pseudo-grains)to perform the step. In linear elasticity (see e.g. Camacho et al., 1990; Pierardet al., 2004) as well as for elasto-plastic composites (see e.g. Doghri and Tinel,2005), very good predictions are obtained in many situations. Indeed the Voigt-like assumption can be intuitively understood (see Christensen, 1992; Lielens,1999) by thinking of a two-phase composite with misaligned long fibers. TheRVE for this composite may consist of mingled fibers crossing the boundaries.For the average strain in each fiber to be compatible with the average strain im-posed on the RVE, the displacement of each fiber must follow the displacement


of the surface it crosses. The fibers are thus acting in parallel. To achieve thatin the two-step model, we must suppose the same deformation in all pseudo-grains. The Voigt-like assumption seems therefore to be more appealing thanthe others. The M-T assumption should even be rejected because it might leadto physically unacceptable results (see Benveniste et al., 1991; Pierard et al.,2004). The case of a three-phase composite involving two sets of aspect ratiosconstitutes a classical example (see Sec. 1.4). Note that the two-step M-T/M-Tmodel turns out to be equivalent to the generalization of the M-T scheme tomultiple phases of reinforcements.

1.3 Composites with coated inclusions

Methods for coated inclusion-reinforced materials are discussed. We focus ongeneral approaches and propose two new ones: two-level and two-step schemes.The composites studied here are made up of 3 phases: matrix, inclusions andcoaxial coatings. All inclusions have the same aspect ratio and orientation.There are three linear elastic materials, one for each phase. It is also assumedthat both reinforcing phases have the same shape (identical aspect ratios). Thiscorresponds to most-frequently encountered situations (e.g., spherical particlesor long fibers coated with constant thickness layers).

All the restrictions above are introduced for clarity only as none of them is re-ally a limitation. Dealing with much more general multi-phase composites withcoated inclusions, the virtual decomposition step of the two-step homogeniza-tion procedure (Fig. 1.1) leads to three-phase pseudo-grains of the type studiedhere. The following methods should then be used to achieve the first step (ho-mogenization of each pseudo-grain), the second step remaining unchanged.

1.3.1 A two-level recursive scheme

We propose a two-level procedure based on the idea that the matrix sees re-inforcements that are themselves composites. We thus propose a two-levelrecursive application of homogenization schemes. As illustrated on figure 1.2(top), each coated inclusion is seen (deepest level) as a two-phase composite (asingle inclusion inside a matrix made of the interphase material) which, oncehomogenized, plays the role of a homogeneous reinforcement for the matrixmaterial (highest level). At each level a homogenization scheme suitable fortwo-phase materials is needed. Using Eshelby-based methods supposes thatthe effective properties at the deepest level are the same as those of a matrixbody made of interphase material and reinforced with a large number of smalland randomly positioned inclusions having the same aspect ratio and volumefraction as the real ones.

Composites with coated inclusions 13

HIGH LEVEL

STEP 2

DECOMPOSITION STEP 1

DEEP LEVEL

Figure 1.2: Schematic view of the two-level (top) and two-step (bottom) homogeniza-tion procedures for the effective properties of composites with coated inclusions. Foreach step/level a two-phase homogenization model is required.

Several remarks must be made about this recursive scheme. Suppose that eachmaterial is isotropic. At the highest level we have to deal with homogeneoustransversely isotropic inclusions – the outcome of the deepest level. One mustthen care about the calculation of Eshelby’s tensor when using Lielens (1999)interpolative model: the transversely isotropic material will also play the role ofthe matrix. Nevertheless, Eshelby’s result is still valid and analytical formulae(see Withers, 1989) still exist provided the revolution axis of the spheroid isaligned with the matrix’ principal direction – which is the case here. At thedeepest level both materials are isotropic but the volume fraction of fillers isin general high. The interpolative D-I scheme or even the inverse M-T modelmay thus be more adapted for this level than the M-T model.

1.3.2 Other generic procedures

Another generic way to handle coatings is through the multi-inclusion methodby Nemat-Nasser and Hori (1999) which is an extension of their double-inclusionmodel. For a three-phase composite with coatings the behavior is approximatedby that of an inclusion coated with a layer of interphase, itself surrounded by alayer of matrix and embedded inside a reference material (Fig. 1.3(a)). Nemat-Nasser and Hori (1999) showed that the overall stiffness tensor obtained withthis multi-inclusion method is identical to the one predicted by their multi-phase composite model. In other words, the coatings behave as if they were


reference

interphase

inclusion

matrix

(a) (b)

Figure 1.3: Schematic view of the Multi-Inclusion Method (a) and the Multi-PhaseComposite Model (b) by Nemat-Nasser and Hori (1999).

a separate phase in matrix: inside the reference material stays an inclusion ofmatrix embedding separate inclusions of the other components (Fig. 1.3(b)).To make things simple, choosing as reference material the one of the matrixwill lead in both cases to the well known generalized M-T scheme where thecoatings are considered as a distinct reinforcing phase.

1.3.3 A two-step method

We propose a two-step scheme which treats coatings as separate reinforcementsbut instead of using a generalized M-T for the three-phase coated composite,we virtually decompose the composite into an aggregate of two pseudo-grains,each containing the matrix material either reinforced with real inclusions orwith inclusions made of coating material (Fig. 1.2 (bottom)). Each two-phasepseudo grain is homogenized (first step) and then the effective properties of theaggregate are computed (second step). We suggest using interpolative D-I orM-T in the first step and Voigt in the second.

Examples in Sec. 1.4 show that under severe conditions, our proposed two-leveland two-step schemes give remarkable predictions while a direct M-T methodleads to erroneous results.

1.3.4 A comparative study

In order to exhibit the differences between those three procedures, the effectivestiffness tensors are hereafter developed for generalized M-T, the two-step (M-T,Voigt) and the two-level (M-T,M-T) schemes. The three phases are denotedΩ0 (matrix), Ω1 (real inclusions) and Ω2 (real coatings) while Ω stands for thecomposite. Volume fractions of components inside the composite are denoted vi

Composites with coated inclusions 15

and elastic tensors Ci for i = 0, 1, 2. We also introduce the following notation:

BεCi,Cj

=(I + SI,Ci

:(C−1

i : Cj − I))−1

. (1.27)

For a matrix material Ci reinforced with aligned inclusions of material Cj andshape I, this is the expression of the strain concentration tensor relating strainaverages between inclusion and matrix phases that corresponds to the M-Tscheme.

The direct extension of the M-T scheme handles the coated inclusions asif they were not coated. The overall stiffness tensor reads

C = C0 + v1 (C1 − C0) : Aε1 + v2 (C2 − C0) : Aε

2 , (1.28)

where Aε1 and Aε

2 are the strain concentration tensors linking average strainsin each reinforcing phase to the average strains in the composite:

〈ε〉Ωi = Aεi : 〈ε〉Ω ;

Aεi = Bε

C0,Ci: (v0I + v1B

εC0,C1 + v2B

εC0,C2)

−1; i = 1, 2 . (1.29)

The two-step (M-T,Voigt) model leads to the definition of two pseudo-grains Ω01 (matrix + real inclusions) and Ω02 (matrix + inclusions made ofinterphase material). The macroscopic stiffness

C = C0 + v1 (C1 − C0) : Aε1,01 + v2 (C2 − C0) : Aε

2,02 (1.30)

has a similar expression (compare (1.30) to (1.28)) except that the concentra-tion tensors do not relate the same quantities as before. Each makes the linkbetween average strains in the inclusion phase of a pseudo-grain (Ω1

01 or Ω102)

and the macro strains in that pseudo-grain:

〈ε〉Ω10i

= Aεi,0i : 〈ε〉Ω0i

;

Aεi,0i = Bε

C0,Ci: (v0I + (1 − v0)B

εC0,Ci

)−1

; i = 1, 2 . (1.31)

In the two-level (M-T,M-T) scheme, we first homogenize the coated in-clusions with M-T. We thus obtain an effective stiffness

C21 = C2 +v1

v1 + v2(C1 − C2) : Aε

1,21 , (1.32)

where the concentration tensor Aε1,21 considers the inclusions as the reinforcing

phase Ω121 of the aggregate Ω21 (inclusions + coatings) and relates the average

strains as follows:

〈ε〉Ω121

= Aε1,21 : 〈ε〉Ω21

;

Aε1,21 = (v1 + v2)B

εC2,C1 : (v2I + v1B

εC2,C1)

−1. (1.33)


Next, we homogenize the matrix reinforced with those homogenized inclusions.As expected, this leads to an expression which is nested with the one of Eq.(1.32):

C = C0 + (v1 + v2)(C21 − C0

): Aε

21 . (1.34)

Moreover, the strain concentration tensor Aε21 depends also on the deepest

level overall stiffness C21 and reads for the M-T scheme:

〈ε〉Ω21= Aε

21 : 〈ε〉Ω ;

Aε21 = Bε

C0,C21:(

v0I + (v1 + v2)Bε

C0,C21

)−1

. (1.35)

For the differences to be better underlined, the average strains in thecoated phase (i.e., the real inclusions) are written with respect to the macrostrains 〈ε〉Ω for each one of the 3 methods:

〈ε〉Ω1 = Aε1 : 〈ε〉Ω (1.36)

〈ε〉Ω1 = Aε1,01 : 〈ε〉Ω (1.37)

〈ε〉Ω1 = Aε1,21 : Aε

21 : 〈ε〉Ω (1.38)

In the two-step scheme (eq. (1.37)) the average deformation in the real inclu-sions does not depend on what happens in the coatings, because of the Voigtassumption in the second step (under applied strain). For the direct extensionof the M-T model (eq. (1.36)) the deformations in the coated and the coat-ing phases are linked (Aε

1 also involves BεC0,C2). This is also true for the

two-level approach (eq. (1.38)). This latter scheme exhibits a multiplicativedecomposition of the strain concentration tensor. Note that a similar decompo-sition was found by Aboutajeddine and Neale (2005) in their new formulationof the double-inclusion model.

1.4 Numerical Simulations

Homogenization schemes based on the two-step (Sec. 1.2) and two-level (Sec.1.3) approaches are applied to predict the elastic behavior of multi-phase com-posites. Aligned and misaligned reinforcements are considered. A compositewith coatings is shown. These examples illustrate the results we obtained, re-garding the elastic properties of polymer based composites. Many more caseshave been considered. They can be found in Friebel (2002); Pierard et al.(2004); Selmi et al. (2007).

Numerical Simulations 17

0

2

4

6

8

10

12

14

16

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

Normalized tensile modulus E3/E0 [-]

Total volume fraction of inclusions (v1=v2) [-]

3-phase composite with short fibers and spherical particles of the same material. Uniaxial tensile loading in the fiber direction

E0= 1 GPa, ν0=0.35

E1=50 GPa, ν1=0.3 , Ar1=50

E2=50 GPa, ν2=0.3 , Ar2=1

Two-step (M-T/Voigt)Two-step (M-T/M-T)Two-step (M-T/Reuss)M-T 3-phase (Taya and Chou, 1981)

Figure 1.4: Effective longitudinal tensile modulus versus total volume fraction offillers. Reinforcements differ only by the shape. Comparison between various two-step schemes and results after Taya and Chou (1981).

1.4.1 Composites with aligned reinforcements

Academic hybrid composites after Taya and Chou (1981) are considered first inorder to underline two aspects of multi-phase homogenization schemes. First,the two-step approach with M-T for both steps and the generalized M-T modelprovide the same predictions. Second, using M-T for the second step leads, insome cases, to estimates that may be considered as out of range. A polymermatrix reinforced with long and aligned single-walled nanotubes (SWNT) isstudied afterwards. This composite is modeled as a three-phase material withcoated fibers. The two-level predictions are in very good agreement with unit-cell FE calculations, for all elastic constants.

Hybrid composites. We study three-phase composites with aligned sphero-idal inclusions. Two cases are considered. In the first one (Fig. 1.4) thereinforcements differ only by the shape – i.e. same material but two sets ofaspect ratios – while for the second (Fig. 1.5) only the stiffnesses are different– i.e. same aspect ratio but distinct materials. Material parameters for thesecomposites are taken from Taya and Chou (1981) who studied them with thegeneralized M-T scheme.

Figure 1.4 illustrates two things. First, as claimed in Sec. 1.2, the generalized


1

2

3

4

5

6

7

8

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

Normalized tensile modulus E3/E0 [-]

Total volume fraction of inclusions (v1=v2) [-]

3-phase composite with identically shaped short fibers of different materials. Uniaxial tensile loading in the fiber direction

E0= 1 GPa, ν0=0.35

E1=10 GPa, ν1=0.3 , Ar1=10

E2=50 GPa, ν2=0.3 , Ar2=10

Two-step (M-T/Voigt)Two-step (M-T/M-T)Two-step (M-T/Reuss)

Figure 1.5: Effective longitudinal tensile modulus versus total volume fraction offillers. Reinforcements differ only by the material. Comparison between various two-step schemes.

M-T model is equivalent to the two-step (M-T/M-T) scheme. A proof of thisresult can be found in Friebel (2002). Second, the (M-T/M-T) predictions arenot comprised between the (M-T/Reuss) and (M-T/Voigt) estimates. From ourexperience, this is always the case for hybrid composites with aligned inclusionsof different aspect ratios. The conditions formulated by Benveniste et al. (1991)that ensure the required symmetries of the macro stiffness tensor are indeed notfulfilled in this case. The latter are satisfied for the second composite for whichall spheroids are similarly shaped. Figure 1.5 reports our two-step predictions.As one could expect the (M-T/M-T) scheme is in the range. More simulationsshowing the same behavior of the M-T scheme for hybrid composites can befound in Friebel (2002) and Pierard et al. (2004).

Young’s modulus [GPa] Poisson’s ratio [−]

LaRC-SI 3.8 0.4Continuum graphene 2520 0.25

Table 1.1: Elastic constants of LaRC-SI (after Odegard et al., 2003) and continuumgraphene (after Selmi et al., 2007).


Figure 1.6: 3D periodic unit cell for a 25%long SWNT reinforced polymer. Unlikethe 2D FE model, the inner parts (i.e. thevoids) of the nanotubes are not meshed.

Long SWNT/Polyimide composite. Selmi et al. (2007) have studied ef-fective elastic properties of various carbon nanotube reinforced polymers. Wecontributed to this study with our coated inclusion micro-mechanical modelsdescribed in section 1.3 as well as with FE simulations.

The first part of the work of Selmi et al. (2007) consists in the modeling ofthe nanotubes themselves. Each nanotube is considered as a sheet of graphenerolled on itself. The chemical structure (i.e. inter-atomic distance, angle, chem-ical bond) of the graphene sheet is known. Using a method developed by Ode-gard et al. (2002) the chemical model is first replaced by a truss model andnext by a continuum model. All these models are equivalent regarding defor-mation energy. The procedure ends-up with two in-plane elastic constants forthe graphene sheet (see table 1.1) which are assumed to be the two parametersof an isotropic elastic solid. More details on this homogenization of graphenesheets can be found Selmi et al. (2007).

The second part is dedicated to the homogenized properties of single wallednanotube (SWNT) reinforced polymers. As a typical application we consider apolyimide (LaRC − SI) reinforced with aligned continuous SWNT. In order toapply our two-level and two-step homogenization methods we model the SWNTas two-phase materials. Each hollow tube is approximated by a continuumgraphene phase of thickness Rc − Rv coating a fiber shaped void phase withradius Rv. Data for the internal (Rv = 0.53 nm) and external (Rc = 0.87 nm)radii come from Odegard et al. (2002). The material properties of the matrixand the coatings are summarized in table 1.1. The Poisson’s ratio of the thirdphase equals the one of the continuum graphene while its elastic modulus is setto a small value to mimic a very soft material.

We also made unit cell FE simulations using the procedure described in Ap-pendix B. Two kinds of unit cells were considered: 2D unit-cells with kinematic


boundary conditions and 3D unit cells with periodic boundary conditions. Forthe 2D simulations, an elastic material with very low stiffness (the same as forthe Eshelby based homogenization models) is used to mimic the void phase.The third phase (voids) is not meshed in the 3D unit-cells (Fig. 1.6).

Selmi et al. (2007) proposed another method – a sequential method – to addressthe effective elastic properties aligned SWNT reinforced polymers. It combinesa generic iterative procedure (Ben Hamida and Dumontet, 2003) and the solu-tion of a boundary value problem (BVP) specific to the microsrtucture of thesecomposites. The main idea of the iterative approach is that many homogeniza-tion schemes are efficient only for low concentration of reinforcements. Addingthe fillers little by little – followed each time by a homogenization process –until the final concentration is reached provides a way to improve those models.Selmi et al. (2007) propose to solve the following two dimensional BVP at eachiteration: a corona of continuum graphene embedded in a circular matrix –the polymer at the first iteration, the outcome of the previous step otherwise– subjected to elementary loadings. The solution is analytical.

We now focus on the dependence of the overall elastic constants on the nan-otubes volume fraction. The results of this parametric study are reported onfigures 1.7 to 1.11. Starting with longitudinal tension, all models predict thesame effective tensile modulus (Fig. 1.7) which indeed obeys the rule of mix-ture. All models show approximately the same trend for the Poisson’s ratio(Fig. 1.8) but only the two-level estimates fit the FE results. Under transversetension (Fig. 1.9) or shear (Figs. 1.10 and 1.11) the two-step approach – gener-alized M-T included – fails in a spectacular way. On the contrary, the two-leveland sequential methods provide pretty good predictions in those cases. Bothmodels are indeed always close to each-other except for the longitudinal Pois-son’s ratio (Fig. 1.8). We don’t have any explanation for this issue yet; thiswould require further investigations.

It is rather surprising that the effective transverse tensile modulus (Fig. 1.9)exhibit a kind of ”knee” at approximately 1% SWNT volume fraction. This”knee” is observed for all Eshelby-based methods and confirmed by both FEmodels. One reason for this behavior is the high contrast between the moduli ofthe phases. The dependence of the transverse modulus on the volume fractionis typical of a Mori-Tanaka scheme for two-phase fibrous composites when thestiffness contrast becomes important. An analytical proof of this result is givenin Appendix C. The fact that the reinforcements are hollow tubes can alsoexplain this drop in the modulus. This explanation does however not holdas it is, especially for the two-step schemes which do not see this particulargeometry. The relative concentrations of the three phases must also come intoplay. The latter are indeed linked to the geometry of the reinforcements. Thereason is therefore related to the combined effect of all these aspects: stiffnesscontrast, geometry of the reinforcements and relative volume fractions.


0

20

40

60

80

100

120

140

160

180

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

normalized tensile modulus (E3/Em) [-]

SWNT volume fraction [-]

SWNT/LaRC-SI composite under longitudinal unaxial tension

all curves/points superimposed

Abaqus (2D unit-cell)Abaqus (3D unit-cell)Two-level (M-T/M-T)SequentialTwo-step (M-T/M-T)Two-step (M-T/Voigt)

Figure 1.7: Effective longitudinal tensile modulus versus volume fraction of SWNT.Comparison between various homogenization schemes and FE unit-cell models.

0.33

0.34

0.35

0.36

0.37

0.38

0.39

0.4

0.41

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

Poisson’s ratio

ν 31 [-]


SWNT/LaRC-SI composite under longitudinal uniaxial tension

Abaqus (2D unit-cell)Two-level (M-T/M-T)SequentialTwo-step (M-T/M-T)Two-step (M-T/Voigt)

Figure 1.8: Effective Poisson’s ratio (longitudinal loading) versus SWNT volume frac-tion. Comparison between various homogenization schemes and FE unit-cell models.


0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

normalized tensile modulus (E1/Em) [-]


SWNT/LaRC-SI composite under transverse unaxial tension


Figure 1.9: Effective transverse tensile modulus versus SWNT volume fraction. Com-parison between various homogenization schemes and FE unit-cell models.

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

normalized shear modulus (G12/Gm) [-]


SWNT/LaRC-SI composite under transverse shear


Figure 1.10: Effective transverse shear modulus versus SWNT volume fraction. Com-parison between various homogenization schemes and FE unit-cell models.


0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

normalized shear modulus (G31/Gm) [-]


SWNT/LaRC-SI composite under longitudinal shear

Abaqus (3D unit-cell)Two-level (M-T/M-T)SequentialTwo-step (M-T/M-T)Two-step (M-T/Voigt)

Figure 1.11: Effective longitudinal shear modulus versus SWNT volume fraction.Comparison between various homogenization schemes and FE unit-cell models.

1.4.2 Composites with misaligned reinforcements

Composites made of two materials may be regarded as multi-phased when theinclusions, even identically shaped, do not share the same orientation. Suchmaterials fall into the same category as the hybrid composites discussed above.The ill behavior of the generalized M-T scheme is observed for fibers or plateletsrandomly oriented (2D and/or 3D) in a polymer matrix.

Short SWNT/Epoxy composites SWNT reinforced polymers were alsonumerically investigated by Lusti and Gusev (2004). They considered short orlong SWNT, aligned or randomly oriented, in an Epoxy matrix. The authorsmodeled the nanotubes as massive cylinders with hemispherical caps at bothends. We thus prefer thinking about two-phase fibrous composites rather thanthree-phase materials with coatings. Consequently, we avoid two-level based

Young’s modulus [GPa] Poisson’s ratio [−]

Epoxy 3.0 0.35Carbon 1220.0 0.275

Table 1.2: Elastic constants of epoxy and carbon after Lusti and Gusev (2004).


0

2

4

6

8

10

12

14

0 0.02 0.04 0.06 0.08 0.1 0.12

normalized tensile modulus (E/Em) [-]

volume fraction of fibers [-]

Epoxy matrix with randomly oriented carbon fibers Uniaxial tension test

3D random orientation

2D random orientation(in-plane tension)

Ar = 100

Two-step (M-T/Voigt)Two-step (M-T/M-T)FE (Lusti and Gusev, 2004)

Figure 1.12: Epoxy matrix with 3D and 2D randomly oriented carbon short fibers.Dependence of the effective tensile modulus on the volume fraction of fillers. Com-parison between homogenization schemes predictions and FE results.

homogenization schemes for this study. The elastic properties for the fibers andthe matrix are reported in table 1.2.

Lusti and Gusev (2004) designed orthorhombic cells containing 50 equally sizedand fully aligned fibers as well as cubic cells with 350 randomly oriented (3Dor 2D planar) fibers distributed by employing a Monte Carlo algorithm. Allgeometries were meshed with tetrahedral elements – up to several million ele-ments were necessary in some cases – and periodic boundary conditions wereapplied. They conducted both direct calculations on the cubic cells with mis-aligned reinforcements and orientation averaging of the moduli coming fromorthorhombic cells. It is not clear which method was chosen to obtain their re-sults given on Fig. 1.12. But the authors claim that a perfect match is obtainedbetween both approaches for the 3D random orientation while deviations of upto 8% can be observed in the 2D case.

We made simulations with the two-step (M-T/Voigt) and (M-T/M-T) schemes.Figure 1.12 reports our predictions of the effective tensile moduli. A perfectmatch is obtained with the (M-T/Voigt) scheme for both orientation distribu-tions. The (M-T/M-T) model is also satisfying regarding the FE results. Ithowever overestimates the (M-T/Voigt) response. This well know behavior (seee.g. Friebel, 2002; Pierard et al., 2004) of the generalized M-T scheme is dueto the fact that each pseudo-grain of the virtual decomposition has a different


1

1.5

2

2.5

3

0 5 10 15 20 25

Normalized tensile modulus E/E0 [-]

Volume fraction of platelets [%]

Polyester matrix with randomly oriented mica plateletsUniaxial tensile loading

E0 = 4.5 GPa, ν0 = 0.38E1 = 172 GPa, ν1 = 0.2, Ar = 0.04

Two-step (M-T/Voigt)Two-step (M-T/M-T)Two-step (M-T/Reuss)corrected Halpin-Tsai (van Es, 2001)experimental (van Es, 2001)

Figure 1.13: Randomly oriented mica platelets reinforcing a polyester matrix. Depen-dence of the effective tensile modulus on the volume fraction of fillers. Comparisonbetween homogenization schemes predictions, randomized corrected Halpin-Tsai es-timates and experimental data reported in van Es (2001).

orientation.

Selmi et al. (2007) performed the same simulations with the two-level approach.The coating phase was assigned the material properties of the massive fibers.Its volume fraction was determined from the chirality of the nanotubes. Theagreement with the FE results is, without any surprise, less satisfying. Moredetails and additional comparisons with composites studied by Lusti and Gusev(2004) can be found in Selmi et al. (2007).

Mica platelets/Polyester composites Polyester with randomly orientedmica platelets (Ar < 1) has been studied numerically by van Es (2001) withhelp of a yet simple but phenomenological – at least to some extent – model.The latter is based on laminate theory, 3D randomizing of planes with ran-domly oriented inclusions and corrected Halpin-Tsai equations. The proposedformula for the overall tensile modulus reads E = 0.49E// +0.51E⊥, where E//

and E⊥ are the effective axial and transverse tensile moduli of the associatedunidirectional composite. They are calculated from the Halpin-Tsai equationsusing scaled aspect ratios – the scaling may change from a modulus to another– obtained by fitting the model on other numerical schemes (e.g. M-T) orexperiments. More details can be found in van Es (2001).


Predictions obtained with the above described model are compared to our two-step homogenization results and experimental data on figure 1.13. We observethat M-T/Voigt is in good agreement with both the results of van Es (2001) andthe experimental data, at least for small volume fractions of platelets. Again,the M-T/M-T predictions are out of range.

1.5 Conclusions

In this chapter, common (e.g. the Mori and Tanaka (1973) scheme) and lessclassical (the interpolative model of Lielens (1999)) homogenization schemesfor two-phase elastic composites with aligned inclusions have been recalled.

These models have been generalized to multiphase composites with a two-stepapproach (Camacho et al., 1990; Lielens, 1999; Friebel, 2002; Pierard et al.,2004). This generic procedure allows to handle spheroidal reinforcements withdifferent aspect ratios and discrete or distributed orientations.

Attention has been drawn to composites with coated inclusions for which anoriginal two-level recursive method is proposed. The latter may be used stand-alone (for three-phase materials with coatings) or to achieve the first step of thetwo-step approach (e.g. for composites with misaligned coated reinforcements).An analytical comparative study has been done, showing differences and sim-ilarities between the two-level procedure and other homogenization schemesfrom the literature.

Numerical simulations have been performed to predict the overall elastic prop-erties of polymer reinforced composites. They show two things. First, thetwo-step (M-T/M-T) model, which is indeed equivalent to the generalized M-Tscheme (see Friebel, 2002; Pierard et al., 2004), might provide effective mod-uli which are outside (M-T/Reuss) and (M-T/Voigt) estimates. This happenswhen the inclusions have different shapes and/or orientations. Benveniste et al.(1991) showed that the generalized M-T scheme might lead to a non-symmetricmacroscopic stiffness tensor in the case of reinforcements with different aspectratios. Second, thinking of the coatings as a separate reinforcing phase (i.e.inclusions made of the coating material embedded in the matrix), thus usingthe two-step approach, leads in general to dramatically poor predictions. Onthe contrary, the two-level method is in remarkable agreement with FE simu-lations. The same conclusions have been drawn by Friebel et al. (2006) whoextended both the two-step and two-level procedures to multiphase viscoelasticcomposites (see Chap. 2).

Many other extensions have been proposed recently. Regarding two-phase com-posites, Doghri and Ouaar (2003), followed by Doghri and Friebel (2005), haveproposed an incremental formulation of the M-T and interpolative schemes for

Conclusions 27

elastoplastic materials. Both latter models were also developed in Pierard et al.(2004) to predict effective thermo-elastic properties. For elasto-viscoplastic be-haviors, Pierard and Doghri (2006) (see also Pierard et al., 2007) have ex-tended the M-T scheme in the framework of the affine formulation of Masson(1998). For multiple phases of reinforcements, the two-step approach was gen-eralized to inelastic materials by Doghri and Tinel (2005) and Doghri and Tinel(2006). Thermo-elastic properties of multiphase composites were also addressedin Doghri and Tinel (2006). Ouaar et al. (2007) developed the two-step (M-T/Voigt) scheme in the scope of three-phase brittle composites: elastoplasticfibers and cavities in a matrix obeying a non-linear damage model.

Considering the remarkable predictions obtained with the two-level approachfor elastic coated inclusion reinforced composites, the method needs to be gen-eralized to other material behaviors∗ (e.g. thermo-elasticity, elastoplasticity).

∗The viscoelastic extension is developed in Chap. 2


Appendix A Expressions for Eshelby’s tensor

Eshelby (1957) addressed the problem of an ellipsoidal inclusion undergoing astress-free strain ε∗ in a infinite homogeneous isotropic elastic medium. Theauthor proved that the resulting strain field in the inclusion is uniform. Thelatter is linearly linked to the eigenstrain through ε = S : ε∗, where S isthe fourth-order Eshelby’s tensor. S depends only on the geometry of theinclusion and the Poisson’s ratio ν of the medium. It has minor symmetries.Major symmetries hold only if the inclusion is of spherical shape. Eshelby’stensor has an analytical expression for a spheroidal inclusion. An infinitely longcylinder of elliptic cross-section also gives rise to analytical formulae (see Mura,1987). Analytic developments were found by Withers (1989) for spheroids intransverse isotropic media provided the axis of revolution and the anisotropyaxis are aligned. In more general cases S has to be evaluated numerically(see Gavazzi and Lagoudas, 1990). We hereafter report the expressions of S

calculated by Friebel (2002) for ellipsoids and spheroids (including 3 limit cases)in isotropic media.

Ellipsoidal inclusion: Consider an ellipsoid with half-axes a1, a2, a3 re-spectively aligned with axes 1, 2, 3 of a Cartesian coordinate system. Thenon-vanishing components of Eshelby’s tensor S read

8π(1 − ν)S1111 =a21I1 − a2

2I2a21 − a2

2

+a21I1 − a2

3I3a21 − a2

3

+ 2(1 − ν)I1 ,

8π(1 − ν)S1122 = −a21I1 − a2

2I2a21 − a2

2

+ 2νI1 ,

8π(1 − ν)S1133 = −a21I1 − a2

3I3a21 − a2

3

+ 2νI1 ,

8π(1 − ν)S2222 =a22I2 − a2

3I3a22 − a2

3

+a22I2 − a2

1I1a22 − a2

1

+ 2(1 − ν)I2 ,

8π(1 − ν)S2233 = −a22I2 − a2

3I3a22 − a2

3

+ 2νI2 ,

8π(1 − ν)S2211 = −a22I2 − a2

1I1a22 − a2

1

+ 2νI2 , (A.1)

8π(1 − ν)S3333 =a23I3 − a2

1I1a23 − a2

1

+a23I3 − a2

2I2a23 − a2

2

+ 2(1 − ν)I3 ,

8π(1 − ν)S3311 = −a23I3 − a2

1I1a23 − a2

1

+ 2νI3 ,

8π(1 − ν)S3322 = −a23I3 − a2

2I2a23 − a2

2

+ 2νI3 ,

Expressions for Eshelby’s tensor 29

8π(1 − ν)S1212 = (1 − ν)(I1 + I2) − a21I1 − a2

2I2a21 − a2

2

,

8π(1 − ν)S2323 = (1 − ν)(I2 + I3) − a22I2 − a2

3I3a22 − a2

3

,

8π(1 − ν)S3131 = (1 − ν)(I3 + I1) −a23I3 − a2

1I1a23 − a2

1

.

Terms I1, I2, I3 – note that I1 + I2 + I3 = 4π – involved in (A.1) are constantsthat depend on the ratios between the half-axes of the ellipsoid. They can becomputed numerically with help of

I1 =4πa1a2a3

(a21 − a2

2) (a21 − a2

3)1/2

(F − E) , (A.2)

I2 =4πa1a2a3

(a22 − a2

3) (a21 − a2

3)1/2

(

`

a21 − a2

3

´

E −`

a22 − a2

3

´

F

a21 − a2

2

− a3

`

a21 − a2

3

´1/2

a1a2

)

,

(A.3)

I3 =4πa1a2a3

(a22 − a2

3) (a21 − a2

3)1/2

(

a2

`

a21 − a2

3

´1/2

a1a3− E

)

, (A.4)

where F et E are incomplete elliptic integrals of the first and second kind,

F = F (θ, k) =

Z θ

0

dw`

1 − k2 sin2 w´1/2

, (A.5)

E = E(θ, k) =

Z θ

0

`

1 − k2 sin2 w´1/2

dw , (A.6)

with amplitude and module respectively given by

θ = arcsin

„

1 − a23

a21

«1/2

, (A.7)

k2 =a21 − a2

2

a21 − a2

3

, (A.8)

assuming a1 > a2 > a3.

Spheroidal inclusion: Eshelby’s tensor is analytic because the incompleteelliptic integrals (A.5) and (A.6) are. For a spheroid with aspect ratio Ar = a3

a1

– a1 = a2 and the axis of revolution is x3 – the non-vanishing components of


S read

S1111 = S2222 =1

4(1 − ν)

2(2 − ν)g − 1

2−

„

A2r − 1

4

«

3g − 2

A2r − 1

ff

,

S3333 =1

2(1 − ν)

2(1 − ν)(1 − g) + g − A2r3g − 2

A2r − 1

ff

,

SP(11,22) =1

4(1 − ν)

−(1 − 2ν)g +1

2− 1

4

3g − 2

A2r − 1

ff

,

S1133 = S2233 =1

4(1 − ν)

−(1 − 2ν)g + A2r

3g − 2

A2r − 1

ff

, (A.9)

S3311 = S3322 =1

4(1 − ν)

4ν(1 − g) − g + A2r3g − 2

A2r − 1

ff

,

SP(1,2)P(1,2) =1

4(1 − ν)

(1 − 2ν)g +1

2− 1

4

3g − 2

A2r − 1

ff

,

SP(2,3)P(2,3) =1

4(1 − ν)

(1 − ν)(2 − g) − g + A2r

3g − 2

A2r − 1

ff

,

SP(3,1)P(3,1) = SP(2,3)P(2,3) ,

with

g =

8

>

>

<

>

>

:

Ar

A2r − 1

„

Ar − cos−1 Ar√1 − A2

r

«

: 0 < Ar < 1 oblate spheroid

Ar

A2r − 1

„

Ar − cosh−1 Ar√A2

r − 1

«

: 1 < Ar <∞ prolate spheroid

and where P(a, b, c, . . .) are all the permutations of a, b, c, . . . For exampleSP(1,2)P(1,2) represents S1212, S1221, S2112 and S2121, while SP(11,22) standsfor S1122 and S2211. Note that numerical errors may occur for Ar close to 1 ifterm 3g−2

A2r−1 is not properly handled.

Limit cases: Expressions (A.9) are only valid for spheroids with aspect ratioAr > 0, Ar 6= 1. Limit cases are found using

limAr→0

g = 0 , (A.10)

lim|Ar−1|→0

g =2

3, lim

|Ar−1|→0

3g − 2

A2r − 1

=2

5, (A.11)

limAr→∞

g = 1 . (A.12)

The non vanishing components of Eshelby’s tensor for flat disks (Ar → 0),spheres (Ar = 1) and long fibers with circular cross-section (Ar → ∞) arereported hereafter.

Expressions for Eshelby’s tensor 31

Flat disks

S3333 = 1 ,

S3311 = S3322 =ν

1 − ν, (A.13)

SP(2,3)P(2,3) = SP(3,1)P(3,1) =1

2.

Spheres

S1111 = S2222 = S3333 =7 − 5ν

15(1 − ν),

SP(11,22) = SP(22,33) = SP(33,11) =5ν − 1

15(1 − ν), (A.14)

SP(1,2)P(1,2) = SP(2,3)P(2,3) = SP(3,1)P(3,1) =4 − 5ν

15(1 − ν).

Long fibers with circular cross-section

S1111 = S2222 =5 − 4ν

8(1 − ν),

SP(11,22) =4ν − 1

8(1 − ν),

S1133 = S2233 =ν

2(1 − ν), (A.15)

SP(1,2)P(1,2) =3 − 4ν

8(1 − ν),

SP(2,3)P(2,3) = SP(3,1)P(3,1) =1

4.


u=cs

te

u

Figure B.1: Periodic array arrangement of spherical particles in a matrix. The unitpattern – a hexagonal-based prism (left) – is approximated by a cylinder (middle),itself reduced to an axisymmetric unit cell (right). The latter is loaded in tension.

Appendix B Micro finite elements

The finite element method provides a powerful tool to simulate the responseof composite materials. It is precise and can be used to verify the simplify-ing assumptions inherent to Eshelby-based mean-field homogenization models.The drawback of the FE approach is its computational cost. The latter ismuch higher than the one of homogenization schemes. When the microstruc-ture is periodic, FE simulations can be performed on a so-called unit cell, whichconsists in the unit patern. The cell can even be simplified by symmetry consid-erations. For complex (non-periodic) microstructures, the unit cell approachis not applicable. A whole representative element has to be discretized andloaded. Boundary conditions are of kinematic, static, mixed or periodic type.They must be specified with care. They have a great influence on the depen-dence of the convergence rate of the method on the filler-size to RVE-sive ratio.Periodic boundary conditions are advised. In addition, they enable to obtainaverage fields, such as stresses and strains, in an easy way. The use of periodicboundary conditions is detailed in Kouznetsova (2004) and Pierard (2006). Ourfinite element simulations are performed with ABAQUSr (2005).

B.1 Unit-cell modeling of particulate composites

Two-phase composites with a random distribution of spherical particles areisotropic. A way to approximate the behavior – actually in tension only –of such materials is to consider a pileup of matrix layers containing sphericalparticles arranged in a hexagonal array. The unit pattern of this periodicmicrostructure is a hexagon-based prism with a sphere at its center (Fig. B.1).Approximating the prism with a cylinder, it is reduced to an axisymmetric unitcell. The symmetry plane can also be taken into account.

This unit cell model can indeed be extended to aligned spheroidal or cylindrical

Micro finite elements 33

u

.u

u=cs

te

u=cs

te

u=cste

Figure B.2: Two-dimensional periodic (dotted lines) and reduced (dashed lines) unitcells for composites with hexagonal array arrangement of long fibers. Kinematicboundary conditions are applied on the reduced unit cell to achieve transverse (mid-dle) and axial (right) tension tests.

shaped particles. It however provides only a means to estimate the behavior ofsuch materials under longitudinal tension.

B.2 Unit-cell modeling of long fiber composites

Two phase composites made of aligned long fibers randomly distributed in amatrix exhibit in most cases (e.g. if all materials are isotropic) a transverseisotropic behavior. The periodic microstructure which is more likely to reflecttransverse isotropy consists in a hexagonal array arrangement of the fibers.

Assuming plane strain and classical general plane strain conditions, a two di-mensional unit cell can be meshed to perform FE analyzes. Periodic boundaryconditions are required. Taking advantage of the two symmetry planes – usingtherefore kinematic or mixed boundary conditions – only a quarter of the latterunit cell may be considered (Fig. B.2).

In both cases, two elementary loadings are enough to compute all but oneengineering moduli of this transverse isotropic material. The out-of-plane (orlongitudinal) shear modulus is left unknown. The latter can be obtained usingmodified general plane strain conditions or from classical FE calculations onthe three dimensional (full) unit cell with periodic boundary conditions.

Description of the method. The elastic stress-strain relation for homoge-neous transverse isotropic media with e3 as anisotropy axis is written in the


orthonormal basis (e1, e2, e3) under the engineering matrix form:

σ11

σ22

σ33

σ12

σ23

σ31

=

C1111 C1122 C1133

C1122 C1111 C1133

C1133 C1133 C3333

C1212

C3131

C3131

ε11ε22ε332ε122ε232ε31

(B.1)

The engineering moduli – longitudinal tensile modulus E3, transverse tensilemodulus E1 = E2, in-plane Poisson’s ratio ν12 = ν21, out-of-plane Poisson’sratios ν31 = ν32 and ν13 = ν23, in-plane shear modulus G12 = G21, bulkmodulus K – are obtained as

E1 =1

S1111, E3 =

1

S3333, (B.2)

ν12 = −S1122

S1111, ν13 = −S1133

S1111, ν31 = −S1133

S3333, (B.3)

G12 =E1

2(1 + ν12), (B.4)

K =1

2S1111 + S3333 + 2 (S1122 + 2S1133), (B.5)

where S = C−1 is the compliance matrix.

Under plain strain conditions – ε33 = 0, ε23 = 0, and ε31 = 0 – the stress strainrelation (B.1) is simplified and one gets

C1133 =σ33

ε11 + ε22, (B.6)

C1111 =σ22ε22 − σ11ε11

ε222 − ε211, (B.7)

C1122 =σ11ε22 − σ22ε11

ε222 − ε211. (B.8)

For general plane strain conditions, the assumptions are ε23 = 0, ε31 = 0 andε33 = ε 6= 0. C3333 can thus be computed with

C3333 =σ33 − C1133 (ε11 + ε22)

ε. (B.9)

Appropriate boundary conditions are imposed to achieve (plane strain) trans-verse and (general plane strain) axial tensions on the periodic or the reduced

Micro finite elements 35

E1 E2 G12 G13 ν12 ν31 ν13Experiment 150 225 55 58 0.31 0.28 0.18FE∗ 148 220 54.9 57.3 0.336 0.281 0.189FE† 146.79 220.53 54.93 - 0.336 0.284 0.189D-I 154.48 220.61 57.89 62.36 0.334 0.281 0.197M-T 144.52 220.56 53.68 57.24 0.346 0.284 0.186

Table B.1: Aluminum matrix with long stiff alumina fibers aligned in the 3-direction.Comparison of our M-T and D-I predictions against experimental data (Jansson,1992) and 2D unit cell – assuming hexagonal array arrangement of the fibers – FE(∗Jansson (1992),†this work) results. Moduli are given in GPa.

unit cell. From the average stresses and strains, the overall stiffness matrixis (partially) filled-in using (B.6) to (B.9). In the case of kinematic or mixedboundary conditions, stresses and strains must be explicitly averaged over theunit cell. For periodic boundary conditions, these averages are inferred fromthe displacements and reaction forces at master and slave nodes.

Illustrative example. The procedure is applied to compute the engineeringmoduli of a ductile aluminum alloy matrix reinforced with aligned long stiffalumina fibers. The mechanical properties are Ef = 344.5 GPa, νf = 0.26 forthe fibers and Em = 68.9 GPa, νm = 0.32 for the matrix. They were all takenfrom Jansson (1991). Note that the volume fraction of fibers given in Jansson(1992) is wrong; the correct value is vf = 55% and can be found in Jansson(1991).

The predictions are compared to experimental data and FE estimates afterJansson (1992) on Tab. B.1. Both FE models are conducted on 2D unitcells, assuming hexagonal array arrangement of the fibers. They however differfrom the definition of the unit cell and the boundary conditions. We usedthe rectangular unit cell with kinematic boundary conditions (Fig. B.2). Thisexplains why we don’t report any prediction for the effective longitudinal shearmodulus G13. Jansson (1992) reduced this cell to half of its size thanks to itscenter of symmetry. The latter is thus of triangular shape. Mixed boundaryconditions are applied. The author – because he had access to the FE sourcecode – was able to perform a longitudinal shear test under modified generalplane strain conditions. M-T and D-I estimates are also reported on Tab.B.1. All numerical models provide very good predictions, which is especiallyremarkable for the Eshelby-based homogenization schemes.


B.3 RVE modeling of composites

More realistic FE simulations are performed on representative volume elements(Fig. B.3), not unit cells. This is the case when the composite is of complexgeometry (e.g. misaligned fibers, multiple phases of reinforcements, aggregatesof inclusions, . . . ) and cannot be approximated by a periodic microstructure,or, even for simple geometries (e.g. aligned fibers), when the materials enter thenon-linear regime. For non-linear behaviors the FE overall response stronglydepends on the unit cell, i.e. on the arrangement of the fibers in the matrix (seee.g. Hom, 1992; Jansson, 1992; Levy and Papazian, 1990). This effect is not soimportant or even negligeable in the linear regime. To take an example, theeffective elastic properties of a fibrous composite are almost identical assumingeither a square or a hexagonal array arrangement of the long fibers (see e.g.Jansson, 1992).

Figure B.3: Three-dimensional representative volume element of a spherical particlereinforced composite. There are 30 identically sized spheres. The volume fraction is30%. When an inclusion crosses a plane of the cube, the missing part is repeated onthe other side. Periodic boundary conditions hold.

On the Mori-Tanaka estimates of the transverse tensile modulus 37

Appendix C On the Mori-Tanaka estimates of

the transverse tensile modulus

In the following, it is proved that the Mori-Tanaka estimates of the transverseYoung’s modulus (ET ) for two-phase fibrous composites can be a non-convexfunction of the volume fraction (v ∈ [0, 1]) of fibers. This is illustrated on Fig.C.1. To simplify the analysis we assume long fibers (Ar → ∞), materials withidentical Poisson’s ratio (ν0 = ν1 , ν ∈]0, 1/2[) and a matrix softer than thefibers (x , E1/E0 > 1). The proof is based on the sign of the second partial

derivative of ET with respect to (w.r.t) v. It is shown that ∂2ET

∂v2 (v = 1) > 0

while the sign of ∂2ET

∂v2 (v = 0) depends on the stiffness contrast (x). We believethat the result can be generalized.

For v = 1, the second partial derivative of ET w.r.t v is positive. Thesecond partial derivative of ET w.r.t v has the following expression:

∂2ET

∂v2(v = 1) =

(1 + ν)(1 − x)2E0

32x(1 − ν)2g1

T (ν, x) (C.1)

0.9

1

1.1

1.2

1.3

1.4

1.5

0 5 10 15 20

normalized effective modulus (ET/E0) [-]

volume fraction of fibers [%]

2-phase fibrous composites with different stiffness contrastsDependence of the transverse Young’s modulus on the volume fraction of fibers

E1/E0=1

E1/E0=2

E1/E0=10

E1/E0=100

E1/E0=100

ν0=ν1=0.3 Ar=∞

E1/E0=2

Mori-TanakaVoigtReuss

Figure C.1: Two-phase fibrous composites with different stiffness contrasts. TheMori-Tanaka estimates of the effective transverse Young’s modulus becomes a non-convex function of the volume fraction of fibers when the stiffness contrast increases(see the curve corresponding to E1/E0 = 100).


where

g1T (ν, x) = (64ν3 − 16ν2 − 55ν + 25)x2+

(−128ν3 + 128ν2 − 58ν + 16)x+ 64ν3 − 48ν2 + 9ν − 1 (C.2)

To show that ∂2ET

∂v2 (v = 1) > 0 it suffices to prove that g1T (ν, x) > 0. This

can be achieved by showing that g1T (ν, 1) is positive and that g1

T (ν, x) is anincreasing function of x.

g1T (ν, 1) is positive because ∀ν ∈]0, 1/2[:

g1T (ν, 1) = 64 (1 − ν)

︸︷︷︸

>0

(5/8 − ν)︸︷︷︸

>0

(C.3)

g1T (ν, x) is an increasing function of x because ∀ν ∈]0, 1/2[:

∂g1T

∂x(ν, x) = 128 (ν + 1)

︸︷︷︸

>0

(5/8 − ν)2︸︷︷︸

>0

x− 128ν3 + 128ν2 − 58ν + 16 (C.4)

∂g1T

∂x(ν, 1) = 96 [ν − (7 +

√5)/8]

︸︷︷︸

<0

[ν − (7 −√

5)/8]︸︷︷︸

<0

(C.5)

For v = 0, the second partial derivative of ET w.r.t v becomes negativewhen the contrast between the moduli increases. The second partialderivative of ET has the following expression:

∂2ET

∂v2(v = 0) =

−2(1 − ν)(1 + ν)(1 − x)2E0

[1 + (3 − 4ν)x]2[(1 − 2ν) + x]

2 g0T (ν, x) (C.6)

where

g0T (ν, x) = (16ν4 − 24ν3 + 9ν2)x4 + (8ν3 − 6ν2)x3+

(−32ν4 + 40ν3 − 43ν2 + 40ν − 14)x2 + (56ν3 − 108ν2 + 80ν − 20)x+

16ν4 − 16ν3 − 20ν2 + 24ν − 6 (C.7)

To show that ∂2ET

∂v2 (v = 0) changes its sign w.r.t x, it suffices to prove the resulton g0

T (ν, x).

g0T (ν, 1) is negative because ∀ν ∈]0, 1/2[:

g0T (ν, 1) = 8 (8ν − 5)

︸︷︷︸

<0

(1 − ν)2︸︷︷︸

>0

(C.8)

g0T (ν, x) is positive when x→ ∞ because ∀ν ∈]0, 1/2[:

sg(g0T (ν, x)) = sg(ν2(4ν − 3)2) for x ∼ ∞ (C.9)

Notations and units 39

Appendix D Notations and units

Notation Description Units

Ω representative volume elementΩi region occupied by phase ip orientation vectorψi(p) orientation distribution function for inclusions of

phase iAri aspect ratio of inclusions of phase ivi volume fraction of phase i

Ωji,p region occupied by phase j in the pseudo-grain of

type i and orientation p〈〉Ω average over the RVE〈〉Ωi

average over phase i

〈〉Ωi,paverage over pseudo-grain Ωi,p

〈〉i,Ψiaverage over inclusions and orientations

ε strain tensorε∗ eigenstrain tensorε∞ far-field strain tensorσ stress tensor Pa

C i stiffness tensor of material i Pa

C i,p overall stiffness tensor of pseudo-grain Ωi,p Pa

C overall stiffness tensor PaS Eshelby’s tensorI fourth order identity tensor1 second order identity tensorBε, Aε strain concentration tensorsBε

i,p , Aεi,p strain concentration tensors at pseudo-grain level

Chapter 2

Quasi-static viscoelastichomogenization models

When inertial effects are neglected, the equation of motion is formsimilar tothe static equilibrium equation for elastic solids. This is known as the quasi-static approximation. In harmonic regime, the stress-strain relation for homo-geneous linear viscoelastic materials may be written in an elastic-like form. Thetransition from time domain to (complex) frequency domain is achieved withhelp of the Laplace-Carson transform (LCT). This is commonly referred to asthe elastic-viscoelastic correspondence principle. Dealing with inclusion rein-forced composites (two-phase, multiphase, with coatings), this principle allowsto generalize any elastic homogenization scheme (two-phase models, two-stepapproach, two-level method) to linear viscoelastic behaviors. Only a changeof variable is required to address the effective harmonic properties of suchheterogeneous materials. The inverse LCT is required in order to obtain thedependence of the overall behavior on time. Section 2.1 recalls some generalpoints (e.g. storage and loss moduli, relaxation form of the stess-strain relation,LCT) about linear viscoelastic solids. The framework for the development ofviscoelastic homogenization schemes is specified in Sec. 2.2. We refer to Chap.1 for details on the models. Section 2.2 also reports a method for the inver-sion of the LCT. The numerical simulations (steady-state and dynamic) of Sec.2.3 involve several polymer based viscoelastic composites. Various numericalapproaches (homogenization schemes and finite element models) are comparedtogether and/or against experimental data from the literature.

42 Quasi-static viscoelastic homogenization models

2.1 Homogeneous material response

We briefly recall the mechanical response of linear viscoelastic materials toharmonic oscillations. A one-dimensional example is given to illustrate that thecomplex moduli – which completely characterize the material subjected to thiskind of loadings – can be viewed as the LCT of the time moduli. Afterwards, thetensorial formalism will help us to stay as generic as possible in the developmentof homogenization schemes (Sec. 2.2).

2.1.1 1D illustrative example

Consider a homogeneous viscoelastic specimen subjected to a uniaxial strainhistory ε(t) = ε0 sin(ωt) with amplitude ε0 small enough for the material toremain in the linear regime. The corresponding stress response, detailed inWineman and Rajagopal (2000), is given by:

σ(t) = ε0 [E′(ω) sinωt+ E′′(ω) cosωt] ,

= ε0[E′(ω)2 + E′′(ω)2

]1/2sin(ωt+ δ(ω)) ,

(2.1)

where tan δ(ω) = E′′(ω)E′(ω) and E′ and E′′ are respectively the storage and loss

moduli in tension. The terms storage and loss can be interpreted by consideringthe mechanical work per loading cycle:

W =

∮

σdε =

2π/ω∫

0

σ∂ε

∂tdt

=ωε20E

′(ω)

2

2π/ω∫

0

sin(2ωt)dt

︸︷︷︸

in-phase

+ωε20E

′′(ω)

2

2π/ω∫

0

[1 + cos(2ωt)] dt

︸︷︷︸

out-of-phase

.

(2.2)

The in-phase part vanishes. The out-of-phase results in a dissipation per cycleand is equal to:

Wdis = πε20E′′ . (2.3)

In other words, the strain energy associated with the in-phase stress and strainstored during loading can be recovered during unloading without loss, while theenergy supplied to the material by the out-of-phase components is convertedirreversibly to heat. The maximum in-phase stored energy occurs at quarter-cycle and is given by:

Wsto =ε20E

′

2. (2.4)

Homogeneous material response 43

The ratio between dissipated and stored energies can be computed as:

Wdis

Wsto= 2π tan δ(ω) , (2.5)

and the loss factor tan δ(ω) clearly appears as a measure of the material damp-ing capacity. Storage and loss moduli can also be viewed as real and imaginaryparts of the so-called complex tensile modulus E∗(ω). Defined asE∗ = E′+iE′′,with i2 = −1, it links, for a sinusoidal history of deformation, the stress to thestrain through the relation

σ(t) = =[E∗(ω)ε0e

iωt]. (2.6)

Moreover, if we define the LCT f(s) of a function f(t) by s times its Laplacetransform,

f(s) = Lc [f(t)] (s) = sL [f(t)] (s) = s

∞∫

0

f(t)e−stdt , (2.7)

the complex modulus in tension appears to be the LCT of the correspondingtime modulus for s = iω. To the complex strain history ε(t) = ε0e

iωt corre-sponds the complex stress:

σ(t) =

t∫

−∞

E(t− τ)ε(τ)dτ = iωε0

t∫

−∞

E(t− τ)eiωτdτ

=

iω

∞∫

0

E(τ)e−iωτdτ

︸︷︷︸

E∗(ω)

ε(t) by a change of variable. (2.8)

2.1.2 3D tensorial formalism

The three dimensional constitutive equation for a homogeneous linear viscoelas-tic material is often written under the integral relaxation form

σ(t) = G(t) : ε(0) +

t∫

0

G(t− τ) : ε(τ) dτ with ε(0) = limt→0t>0

ε(t) (2.9)

where σ(t), ε(t) and ε(t) are the second-order stress, strain and strain ratetensors respectively and G(t) is the fourth-order relaxation tensor. This stan-dard solid model can in most cases be inverted into an equivalent integral creep


form. The fourth-order creep tensor J relates the strain to the stress history.

In the Laplace-Carson domain (see hereafter) this is reflected by J = G−1

. Foran isotropic material, the relaxation tensor has the following expression

G(t) = 2G(t)I +

(

K(t) − 2

3G(t)

)

1⊗ 1 , (2.10)

where I and 1 are the symmetric fourth- and second-order identity tensors,respectively, and K(t) and G(t) the bulk and shear moduli.

If ε(t) is a continuous and piece-wise differentiable function of time, the linearviscoelastic stress-strain relation (2.9) can be transformed by applying LCT(2.7) on both of its sides, leaving us after some simple manipulations with:

σ(s) = G(s) : ε(s) , i.e. σij(s) = Gijnm(s)εmn(s) . (2.11)

Each component Gijkl(s) of the fourth-order tensor G(s) is the LCT of Gijkl(t).It is a complex function of the complex variable s. As already illustrated inSec. 2.1, taking the Gijkl(s) for s = iω provides us with a means to completelycharacterize the frequency response of the material. Examples in connectionwith the numerical simulations of Sec. 2.3 are given hereafter. For isotropicmaterials, the complex tensile modulus is computed with help of the fourth-order creep tensor in the complex plane:

E∗(ω) =1

J1111(iω). (2.12)

For transversely isotropic materials with anisotropy axis along the third direc-tion, the complex transverse plane strain tensile modulus is given by:

E∗12,2(ω) = G1111(iω) − G1122(iω)G2211(iω)

G2222(iω). (2.13)

2.2 Homogenization procedures

Taking advantage of the correspondence principle, the prediction of the fre-quency behavior of heterogeneous viscoelastic solids is not more difficult thanthe prediction of the overall mechanical properties of multi-phase elastic com-posites. The problem is brought from time to frequency domain where thehomogenization procedures of Chap. 1 (e.g. two-step and two-level methods)are applicable. In order to obtain storage and loss moduli characterizing the re-sponse of the material, only a change of variable is needed as shown in Sec. 2.1.Effective time properties require the inverse LCT. Except for simple homog-enization schemes (e.g. Voigt), the inversion has to be achieved numerically.The collocation method (Schapery, 1962) is advised.

Homogenization procedures 45

2.2.1 Effective harmonic properties

Let the representative volume element (RVE) Ω of a multi-phase compositebe a matrix with N families of spheroidal inclusions. Each family Ωi withvolume fraction vi inside the RVE is characterized by an aspect ratio Ari

and arelaxation tensor Gi(t). Let also ψi(p) be the orientation distribution functiondescribing the orientation of the inclusions belonging to family i. The matrixΩ0 has concentration v0 and a relaxation tensor G0(t). The term phase will beused to denote a family (phase i for 1 ≤ i ≤ N) or the matrix (phase 0). Eachphase’s material is linear viscoelastic and homogeneous with a constitutive lawgiven by Eq. (2.9).

Let the RVE be subjected to linear boundary displacement u(x, t) and applythe LCT (Eq. (2.7)) on the time variable t for all involved equations (i.e. con-stitutive equations, boundary conditions, ...). What we get is a fictitious RVEin the Laplace-Carson domain (variable s) with boundary conditions u(x, s)and for which the material behavior of each phase 1 ≤ i ≤ N is given by:

σ(x, s) = Gi(s) : ε(x, s) ; ∀x ∈ Ωi . (2.14)

This situation is identical to that of a multi-phase composite with homogeneouslinear elastic reinforcements (see Camacho et al., 1990; Lielens, 1999; Pierardet al., 2004). Any homogenization procedure for this kind of materials (Chap.1) can thus be extended to predict the frequency behavior of multi-phase vis-coelastic composites.

2.2.2 Effective time properties

Homogenization of viscoelastic multiphase composites is not achieved in timedomain. The LCT brings the whole problem into the complex domain. Takingadvantage of similarities with elastic composites, homogenization schemes areextended in a straightforward manner, except that everything depends on thecomplex variable s. The frequency behavior comes out by a simple changeof variable s = iω. The overall time properties (e.g. how do the effectivemoduli evolve w.r.t. time?) cannot be obtained in such an easy way. Anumerical inversion of the LCT is required. The collocation method proposedby Schapery (1962) is hereafter presented. It is easy to implement and theapproximate function can be evaluated at any time once some coefficients aredetermined. The method was advocated by Masson (1998) and also successfullytested by Pierard and Doghri (2006) in the framework of the affine formulationfor elasto-viscoplastic composites.

Principle of the collocation method. Let us consider an unknown timefunction f(t) which we are able to evaluate at any point of the transformed


domain, i.e. f(s) is known. The approximation f(t) is developed into a n-terms Dirichlet series with an additional affine term,

f(t) = A+Bt+k=n∑

k=1

bk (1 − e−t/θk)︸︷︷︸

basis functions

, (2.15)

and its transform is fitted with the transform of the series over a number m ofdistinct points, i.e.

f(sl) = A+B

sl+

k=n∑

k=1

bk1 + slθk

, 1 ≤ l ≤ m. (2.16)

The above relations constitute a linear system with n+2 unknowns: A, B andbk for 1 ≤ k ≤ n. The number of variables is reduced to n since

A = lims→+∞

f(s) = limt→0

f(t) (2.17)

and B = lims→0

sf(s) = limt→+∞

f(t)

t, (2.18)

giving the sufficient condition m ≥ n to ensure uniqueness of the solution ofsystem (2.16) in a least square sense. Once the unknowns fixed, the func-tion f can be evaluated at any time t with help of Eq. (2.15). FollowingSchapery (1962) the total square error between the function and its approxima-tion (

∫∞

0 (f(t)− f(t))2dt) is minimized by collocating the LCT of the Dirichlet

series and f at n points s = 1/θ (i.e. m = n and sk = 1/θk ; 1 ≤ k ≤ n).

Application to viscoelastic composites. The unknown time function is inthis case a tensor: the effective fourth-order relaxation tensor G. As each com-ponent Gijkl is only a function of time, the generalization of the aforementionedprocedure is obvious.

The shape of the basis functions is similar to the terms of a Prony series (com-pare Eqs. (2.15) and (2.19)). The set of collocation points θk should then atleast contain all the relaxation times of all the materials involved in the com-posite. This is however not enough and most authors advise to choose abouttwenty equispaced points on a logarithmic scale (see e.g. Pierard and Doghri,2006).

The last issue is the limit tensors, A and B, which have to be known whateverthe homogenization procedure. This is actually not a problem: A correspondsto the initial elastic response and B vanishes since G(t) remains finite for eacht. Indeed, for standard linear viscoelastic materials, the moduli (e.g. shearmodulus) are expressed in terms of Prony series, leadind to A > 0, B = 0 andbk 6 0 (1 6 k 6 n) in Eq. 2.15.

Numerical simulations 47

Materials Matrix InclusionsPVC DOP Stabilizers Paraloid

weight fraction [%] 61.5 24.5 1 13relative density1 (water=1) [-] 0.9 0.9861 0.9volume fraction [%] 86.71 13.29

Table 2.1: Components’ weight and volume fractions of the Paraloid particle re-inforced (PVC+DOP) matrix composite studied by Redaelli (2002). 1Sources:http://www.inchem.org, http://www.chemicalland21.com

2.3 Numerical simulations

All the theoretical aspects exposed up to here, namely the viscoelastic extensionof the two-step and two-level approaches described in Chap. 1 and the colloca-tion method, are hereafter applied to predict the frequency and time behaviorof viscoelastic composites with two or three phases, including coatings. Twentydistinct composites are presented: sixteen two-phase materials and four withthree phases. Shear or (plane strain) tensile complex moduli are estimated withrespect to frequency (6 cases), time (10 cases) or volume fraction of fillers (4cases) using many different homogenization schemes. The latter are analyzedas often as possible and compared with respect to each other. Each time, thepredictions are validated against experimental data from the literature and/orverified with FE results.

2.3.1 Frequency behavior of two-phase composites

The numerical materials involved here are made of two phases for which at leastone is viscoelastic. In all cases the inclusions are assumed aligned, reducing thetwo-step procedure to classical homogenization methods. M-T and Lielens’interpolative schemes are used to predict the shear or tensile complex modulus,either with respect to frequency or, for a given one, in function of the volumefraction of fillers. The latter are spherical particles or long fibers. FE unit cellor RVE results and experimental data are used for verification and validation.

PVC matrix with spherical Paraloid inclusions. The composite wasprepared by Redaelli (2002). It involves two incompatible polymers, Paraloid –commercial name for polybutyl acrylate/methyl methacrylate – and polyvinylchloride (PVC). The PVC had to be plasticized by adding dioctyl phtalate(DOP) and small quantities of stabilizers entered also in the preparation. Thematerial data are summarized in Tab. 2.1. The original composition was givenin weight fractions of components. With help of the densities we computed


6.75

7

7.25

7.5

7.75

8

8.25

8.5

8.75

9

9.25

-0.4 0 0.4 0.8 1.2 1.6

log storage shear modulus (G´) [Pa]

log frequency [Hz]

PVC+DOP(24%w) matrix reinforced with 13%w Paraloid particles under shear.

Paraloid+24%wDOP

PVC+24%wDOP

experimental (Redaelli, 2002)virgin materialsMori-Tanaka

Figure 2.1: Storage shear modulus as a function of frequency of a 2-phase viscoelasticcomposite at 0C. Comparison between experimental measurements and estimatesobtained with the Mori-Tanaka scheme.

the volume fraction of each phase (the 1% stabilizers were shared half andhalf between PVC and DOP). Storage and loss shear moduli were determinedexperimentally by Redaelli (2002) for the matrix, the inclusions and the blendover a frequency range from 0.5 to 50Hz. With regard to the geometry of thereinforcements, Scanning Electron Analysis (SEM) images also coming fromRedaelli (2002) show the Paraloid inclusions to be of spherical shape (i.e. Ar =1).

The experimental measurements provided only the complex shear moduli. Wecalculated the bulk modulus of each phase with help of the Poisson’s ratioswhich we assumed constant. The matrix (PVC+DOP) Poisson’s ratio wasfixed to 0.49 according to Chazeau et al. (1999) who studied DOP plasticizedPVC – in proportions similar to the material of Redaelli (2002) – and reporteda value of 0.5 at 280 K. The knowledge of the Paraloid Poisson’s ratio is lessimportant since it has no effect – at least for the M-T scheme – on the effectiveshear modulus. The value of 0.4 was used.

We made numerical simulations with the M-T model. Our estimates of thecomplex shear modulus are confronted to the experimental results on Figs. 2.1and 2.2. The loss modulus (Fig. 2.2) is pretty well predicted while the storageone (Fig. 2.1) is overestimated. The relative error is of about 15%. This mightbe related to the assumptions that we made on the composition of the blend:


5.5

6

6.5

7

7.5

8

8.5

-0.4 0 0.4 0.8 1.2 1.6

log loss shear modulus (G˝) [Pa]

log frequency [Hz]

PVC+DOP(24%w) matrix reinforced with 13%w Paraloid particles under shear.

Paraloid+24%wDOP

PVC+24%wDOP

experimental (Redaelli, 2002)virgin materialsMori-Tanaka

Figure 2.2: Loss shear modulus as a function of frequency of a 2-phase viscoelasticcomposite at 0C. Comparison between experimental measurements and estimatesobtained with the Mori-Tanaka scheme.

the material used for the inclusion phase in our model is not pure Paraloid– no experimental data were reported in Redaelli (2002) for pure Paraloidat 0C – but plasticized Paraloid (Paraloid+DOP); the volume fractions ofthe components were computed from the densities of the materials; constantPoisson’s ratios were assumed; other sources (see Tab. 2.1 and above) wereused to fix the unknown material parameters.

Epoxy and copolymer matrix materials reinforced with ceramic par-ticles. Marra et al. (1999) studied two composites. Both consist of a poly-meric matrix reinforced with Ca-modified PbTiO3 – a ceramic – spherical in-clusions. The materials used for the matrix are Epon 828 and P(VDF-TrFE).For each of them, the authors made experimental measurements of the tensilestorage and loss moduli. Next, they assumed constant (and thus real) Pois-son’s ratios – values were taken from the literature (see Marra et al., 1999,and references therein) – to compute the real and imaginary parts of the com-plex shear and bulk moduli. The reinforcing material is supposed elastic andits parameters were obtained from previous works of other authors (see Marraet al., 1999, and references therein). The (visco)elastic parameters of thesethree components are reported in Table 2.2 for the angular frequency of in-terest ω = 10 rad s−1. Marra et al. (1999) collected experimental data onthe complex tensile modulus for the heterogeneous materials. In addition, they


G′ [GPa] G′′ [GPa] K′ [GPa] K′′ [GPa]

Epon 828 1.2851 1.2524 10−2 4.1560 4.0501 10−2

P(VDF-TrFE) 5.6789 10−1 2.0221 10−2 2.4386 8.6832 10−2

E [GPa] ν [−]

Epon 828 0.36P(VDF-TrFE) 0.392Ca-modified PbTi03 127.6 0.2046

Table 2.2: Mechanical properties of Ca-modified PbTi03 (elastic), Epon 828 (vis-coelastic) and P(VDF-TrFE) (viscoelastic) at angular frequency ω = 10 rad s−1 (afterMarra et al., 1999).

made axisymmetric unit cell FE calculations and used a simple analytical modelin order to predict the frequency behavior in tension. We made numerical sim-ulations on these composites with the M-T scheme and the interpolative D-Imodel (D-I). The predictions we obtained are confronted to Marra et al. (1999)results on Figs. 2.3 to 2.6.

Looking at these four figures all together, one can observe that none of the nu-merical methods gives results very close to the experimental data. The misfitis much more pronounced for the tensile loss modulus than for the storage one(compare Figs. 2.4 and 2.6 to Figs. 2.3 and 2.5, resp.). The worst predictionsare obtained for the loss modulus of the Ca-modified PbTi03/Epon 828 com-posite (Fig. 2.4). One reason for this bad behavior is that all these numericalsimulations have been conducted with input data computed by assuming con-stant Poisson’s ratios (see above). A real Poisson’s ratio implies indeed thesame time dependency for the shear, bulk and tensile moduli. The impact ofsuch a restriction might be not negligible and may not be suitable for thesematerials, especially for Epon 828.


0

5

10

15

20

25

30

35

40

45

50

55

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

tensile storage modulus (E´) [GPa]

ceramic volume fraction [-]

Ca-modified PbTiO3 spherical inclusions reinforcing aEpon 828 matrix under harmonic uniaxial tension (ω=10 rads-1)

D-I

M-T

Marra et al. 1999experimentalanalytical modelFE model

Figure 2.3: Storage tensile modulus versus volume fraction of inclusions of a ceramicelastic particle reinforced Epon 828 viscoelastic matrix. Comparison between experi-mental results, FE calculations and various homogenization scheme predictions.

0

50

100

150

200

250

300

350

400

450

500

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

tensile loss modulus (E˝) [MPa]


Ca-modified PbTiO3 spherical inclusions reinforcing aEpon 828 matrix under harmonic uniaxial tension (ω=10 rads-1)

D-I

M-T


Figure 2.4: Loss tensile modulus versus volume fraction of inclusions of a ceramicelastic particle reinforced Epon 828 viscoelastic matrix. Comparison between experi-mental results, FE calculations and various homogenization scheme predictions.


0

5

10

15

20

25

30

35

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8



Ca-modified PbTiO3 spherical inclusions reinforcing aP(VDF-TrFE) matrix under harmonic uniaxial tension (ω=10 rads-1)

D-I

M-T


Figure 2.5: Storage tensile modulus versus volume fraction of inclusions of a ceramicelastic particle reinforced P(VDF-TrFE) viscoelastic matrix. Comparison betweenexperimental results, FE calculations and various homogenization scheme predictions.

0

100

200

300

400

500

600

700

800

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8



Ca-modified PbTiO3 spherical inclusions reinforcing aP(VDF-TrFE) matrix under harmonic uniaxial tension (ω=10 rads-1)

D-I

M-T


Figure 2.6: Loss tensile modulus versus volume fraction of inclusions of a ceramicelastic particle reinforced P(VDF-TrFE) viscoelastic matrix. Comparison betweenexperimental results, FE calculations and various homogenization scheme predictions.


stiff material soft materialshear bulk shear bulkτi Gi τi Ki τi Gi τi Ki

3 3.162 10000 40000 0.032 2.512 100.000 300010 17.783 0.100 10.000 316.228 10032 100.000 0.316 56.234

100 316.228 1.000 316.228316 1000.000 3.162 1000.000

1000 5623.413 10.000 199.526316210000.000 31.623 50.119

10000 562.341 100.000 19.95331623 141.254 316.228 12.589

100000 56.234 1000.000 2.512316228 17.783 3162.278 1.698

1000000 5.623 10000.000 1.2023162278 3.162 31622.777 1.148

10000000 1.778 100000.000 1.096

G0 = 17948.761 K0 = 48000 G0 = 1677.979 K0 = 3300

Table 2.3: Mechanical properties of the idealized viscoelastic materials used by Brin-son and Lin (1998) and Fisher and Brinson (2001). Relaxation times (in sec) andweights (Gi = G0wi, Ki = K0wi, in bar) involved in the Prony series.

Viscoelastic matrix with viscoelastic long fibers. The components in-volved in the composite explored by Brinson and Lin (1998) are two idealizedisotropic viscoelastic materials. Prony series are used to describe the timeevolution of their shear and bulk moduli:

Y (t) = Y0

[

1 −n∑

i=1

wi

(

1 − e−t/τi

)]

, Y0 = Y (t = 0) . (2.19)

The relaxation times (τi) and weights (wiY0) are listed on Table 2.3 for bothmaterials’ moduli. The sets of values were chosen so that one of the materials isat all times stiffer (the stiff material) than the other (the soft material) and thattheir loss peaks do not coincide. Notice also that in both cases shear and bulkmoduli do not have the same time dependence. The complex Poisson’s ratioshave therefore a non-zero imaginary part. Both materials will alternativelyplay the role of the matrix and the inclusions with a volume fraction of stiffmaterial always equal to 36%.

This time the reinforcements are no longer of spherical shape but consist of longfibers (Ar → ∞), resulting in transversely isotropic heterogeneous materials.Brinson and Lin (1998) were interested in the complex transverse plane straintensile modulus of these two composites. To this end, they made FE calcula-


1

1.5

2

2.5

3

3.5

4

4.5

5

-10 -8 -6 -4 -2 0 2

log transverse storage tensile modulus (E´) [bar]

log ω [rad/s]

Long fiber composite made of 36% stiff and 64% soft viscoelastic materialsunder plane-strain transverse uniaxial tension.

stiff material only

soft material only64% soft fibers

36% stiff fibers

FE-SQR (Brinson and Lin 1998)FE-HEX (Brinson and Lin 1998)Mori-TanakaLielens’ Interpolation

Figure 2.7: Plane strain transverse storage tensile modulus as a function of frequencyof two long fiber composites. Both phases bulk and shear moduli are described withProny series. The interpolative and Mori-Tanaka schemes are confronted to two FEunit cells calculations.

tions with two sorts of unit cells. The models differed by the fiber arrangementin the matrix: square (FE-SQR) and hexagonal (FE-HEX) array (see Brinsonand Lin, 1998, for details). The authors also made simulations with the M-Tscheme.

We compare our interpolative D-I and M-T predictions of the composites’ stor-age and loss moduli to those FE results on Figs. 2.7 and 2.8. In all cases, the FEand the homogenization models predict similar frequency behaviors. However,the estimates obtained with the interpolative D-I match always the FE-HEXones. This is very satisfying because the hexagonal array fiber arrangement ismore likely to reflect the transverse isotropy of the material (Jansson, 1992).The M-T predictions – identical to those obtained by Brinson and Lin (1998)– are less satisfying for a fiber volume fraction of 64%. As in the linear elasticcase (see Pierard et al., 2004) the analytical prediction is improved with theD-I model.

Viscoelastic matrix softened by viscoelastic particles. With the samematerials as Brinson and Lin (1998) we design another composite, changingonly the microstructure. The latter consists in soft spherical particles withvolume fraction 30% embedded in the stiff material matrix. The loading is also


-4

-3

-2

-1

0

1

2

3

4

5

-10 -8 -6 -4 -2 0 2

log transverse loss tensile modulus (E˝) [bar]

log ω [rad/s]

Long fiber composite made of 36% stiff and 64% soft viscoelastic materialsunder plane-strain transverse uniaxial tension.

stiff material only

soft material only

64% soft fibers

36% stiff fibers

FE-SQR (Brinson and Lin 1998)FE-HEX (Brinson and Lin 1998)Mori-TanakaLielens’ Interpolation

Figure 2.8: Plane strain transverse loss tensile modulus as a function of frequencyof two long fiber composites. Both phases bulk and shear moduli are described withProny series. The interpolative and Mori-Tanaka schemes are confronted to two FEunit cells calculations.

changed. The composite is now subjected to simple shear.

We made FE simulations to verify the predictions of the M-T scheme. Theywere done on a 3D RVE containing 30 randomly distributed and identicallysized spheres. The latter has been kindly lent by Pierard (2006) who builtit up. Periodic boundary conditions hold. A snapshot of the RVE and itsFE mesh is reported on Fig. B.3 of Chap. 1. The FE results are stemmingfrom computations on this single numerical RVE. No averaging on predictionsobtained for different random distributions of particles or numbers of sphereswas performed. However, the number of inclusions (30) and the algorithmthat was used to calculate their locations ensure that the results are closeto those that would have been obtained for a RVE of infinite size. This hasbeen previously checked (see Pierard, 2006; Pierard et al., 2007, and referencestherein) for non-linear materials, for which there is much more scattering inthe FE calculations than for linear elastic or viscoelastic behaviors.

An excellent agreement is obtained between the M-T scheme and this veryrealistic FE model for both storage (Fig. 2.9) and loss (Fig. 2.10) shear moduliover the whole frequency range.


1e+00

1e+01

1e+02

1e+03

1e+04

1e+05

1e-06 1e-04 1e-02 1e+00 1e+02

Storage modulus [bar]

Angular frequency ω [rad/s]

Spherical viscoelastic particles softening a viscoelastic matrixShear loading

stiff matrix

soft particles

Mori-TanakaFE (3D RVE, periodic BCs)

Figure 2.9: Storage modulus versus angular frequency of a particle softened viscoelas-tic composite subjected to shear loading. Comparison between M-T predictions andFE calculations on a 3D RVE with periodic boundary conditions.

1e-02

1e-01

1e+00

1e+01

1e+02

1e+03

1e+04

1e-06 1e-04 1e-02 1e+00 1e+02

Loss modulus [bar]

Angular frequency ω [rad/s]

Spherical viscoelastic particles softening a viscoelastic matrixShear loading

stiff matrix

soft particles

Mori-TanakaFE (3D RVE, periodic BCs)

Figure 2.10: Loss modulus versus angular frequency of a particle softened viscoelasticcomposite subjected to shear loading. Comparison between M-T predictions and FEcalculations on a 3D RVE with periodic boundary conditions.


2.3.2 Frequency behavior of coated-inclusion reinforcedmaterials

The following composites are made of elastic fibers or particles coated with aviscoelastic material and embedded in a viscoelastic matrix. Various homog-enization methods able to handle coatings are compared together: the two-step procedure, our original two-level method, the direct extension of the M-Tscheme – which is a particular case of the Multi-Inclusion method of Nemat-Nasser and Hori – and Benveniste’s model. The predicted complex tensile mod-uli (with respect to frequency or volume fraction of inclusions) are confrontedto experimental data and/or FE calculations.

Viscoelastic matrix with elastic long fibers and viscoelastic coatings.Keeping both idealized viscoelastic materials of Table 2.3 and taking as thirdcomponent an elastic one, Fisher and Brinson (2001) considered the following3-phase composites: 30% elastic long fibers reinforcing a soft (resp. stiff) vis-coelastic matrix with 10% stiff (resp. soft) viscoelastic interphase. The fiberswith elastic shear and bulk moduli G = 40000 Pa and K = 100000 Pa arealways the stiffest of the three phases.

Assuming a hexagonal array arrangement of the coated fibers, Fisher and Brin-son (2001) calculated – with a unit cell similar to the FE-HEX one used byBrinson and Lin (1998) – the complex transverse tensile modulus E∗

2 (ω) ofboth composites. To this end, two sets of boundary conditions had to be con-sidered: one set in order to compute the shear modulus G∗

12(ω) and another forthe transverse plane strain tensile modulus E∗

12,2(ω). The complex transversetensile modulus is finally obtained as:

E∗2 (ω) = 4G∗

12(ω)

(

1 − G∗12(ω)

E∗12,2(ω)

)

. (2.20)

Fisher and Brinson (2001) also made numerical simulations with the generalizedM-T scheme. The corresponding results are labeled Mori-Tanaka (3 phases) inFigs. 2.11 to 2.14. As briefly exposed in Sec. 1.3 of Chap. 1, the same schemecould be applicable if the interface material were not coating the inclusions.Therefore we used a two-step M-T/Voigt procedure labeled 2-step (M-T,Voigt)in Figs. 2.11 to 2.14. Taking the FE predictions as reference, the simulationsshow that in case of soft matrix and a stiff interphase the predictions of bothhomogenization methods (direct M-T and 2-step M-T/Voigt) match perfectlythe FE-HEX ones (Figs. 2.11 and 2.12). On the contrary, it is clearly nottrue if the phases’ materials are switched (Fig. 2.13 and 2.14): neither thedirect extension of M-T nor the two-step scheme (M-T,Voigt) shows a behaviorcomparable to the one of the unit cell.


1

1.5

2

2.5

3

3.5

4

4.5

-7 -6 -5 -4 -3 -2 -1 0 1 2


log ω [rad/s]

Stiff elastic long fibers reinforcing a soft viscoelastic matrixwith stiff viscoelastic interphase under transverse uniaxial tension

2 curves (M-T,M-T) and (D-I,M-T)superimposed upon M-T (3 phases)and 2-step (M-T,Voigt)

Fisher and Brinson 2001Mori-Tanaka (3 phases)FE-HEXBenveniste

1

1.5

2

2.5

3

3.5

4

4.5

-7 -6 -5 -4 -3 -2 -1 0 1 2


log ω [rad/s]



2-step and 2-level homogenization schemes2-step (M-T,Voigt)2-level (D-I,M-T) and (M-T,M-T)

Figure 2.11: Transverse storage tensile modulus as a function of frequency of a coatedelastic long fiber composite. The bulk and shear moduli of the coatings and matrixphases are expanded in Prony series. Comparison between various predictive methods.

-1

0

1

2

3

4

-7 -6 -5 -4 -3 -2 -1 0 1 2


log ω [rad/s]




-1

0

1

2

3

4

-7 -6 -5 -4 -3 -2 -1 0 1 2


log ω [rad/s]




Figure 2.12: Transverse loss tensile modulus as a function of frequency of a coatedelastic long fiber composite. The bulk and shear moduli of the coatings and matrixphases are expanded in Prony series. Comparison between various predictive methods.


1.5

2

2.5

3

3.5

4

4.5

5

5.5

-7 -6 -5 -4 -3 -2 -1 0 1 2


log ω [rad/s]

Stiff elastic long fibers reinforcing a stiff viscoelastic matrixwith soft viscoelastic interphase under transverse uniaxial tension

(D-I,M-T) (M-T,M-T)


1.5

2

2.5

3

3.5

4

4.5

5

5.5

-7 -6 -5 -4 -3 -2 -1 0 1 2


log ω [rad/s]


(D-I,M-T) (M-T,M-T)


Figure 2.13: Transverse storage tensile modulus as a function of frequency of a coatedelastic long fiber composite. The bulk and shear moduli of the coatings and matrixphases are expanded in Prony series. Comparison between various predictive methods.

-1

0

1

2

3

4

5

-7 -6 -5 -4 -3 -2 -1 0 1 2


log ω [rad/s]


(M-T,M-T)

(D-I,M-T)


-1

0

1

2

3

4

5

-7 -6 -5 -4 -3 -2 -1 0 1 2


log ω [rad/s]


(M-T,M-T)

(D-I,M-T)


Figure 2.14: Transverse loss tensile modulus as a function of frequency of a coatedelastic long fiber composite. The bulk and shear moduli of the coatings and matrixphases are expanded in Prony series. Comparison between various predictive methods.


Another method to evaluate the effective behavior of three phase compositeswith coated inclusions is the one by Benveniste et al. (1989). The model as-sumes that “(...) the strain field in each part of the reinforcement phases f org (...) are assumed to be equal to the fields in a single inclusion of phase for g which is embedded in an unbounded matrix medium (...) and subjectedto remotely applied strains (...) which are equal to the yet unknown averagestrain in the matrix”. Although this assumption is nothing else than thinkingthe coatings as a separate phase combined to a M-T interpretation, this is notthe generalized M-T scheme. The difference lies in the computation of the con-centration tensors which link average strains or stresses in the reinforcementphases to the corresponding averages in the matrix. Without going into details,the model by Benveniste et al. (1989) approximates these tensors “(...) by thosefound when the coated inclusion is embedded in an unbounded matrix mediumsubjected to the average matrix stresses (or strains) at infinity”. By solvingaverage stresses and strains on a set of auxiliary problems – each problem isdefined by the (simple) loading applied at infinity – expressions for the concen-tration tensors are found. Compared to the generalized M-T scheme – for whichthe tensors are computed with help of Eshelby’s result – this model is stronglylimited by the geometry. Actually, Benveniste et al. (1989) presented theirmodel for aligned coated long fibers only. With help of the elastic-viscoelasticcorrespondence principle, Fisher and Brinson (2001) applied the method ofBenveniste et al. (1989) to predict the complex transverse tensile modulus oftheir two materials. Again, remarkable results are obtained for the stiff in-terphase composite (Figs. 2.11 and 2.12) while the overall behavior is missedcompletely in the case of soft coatings (Figs. 2.13 and 2.14).

We made numerical simulations with our two-level homogenization approachtaking M-T or interpolative D-I for the deepest level and M-T for the highest(Chap. 1, Fig. 1.2). The corresponding labels are 2-level (M-T,M-T) and 2-level (D-I,M-T) respectively. Our predictions are confronted to all other resultson Figs. 2.11 to 2.14. The complex transverse tensile modulus is still very wellpredicted in the case of a stiff coating (Figs. 2.11 and 2.12) and, unlike theother homogenization schemes, this new recursive method gives results thatshow good agreement with the FE-HEX ones when the interphase is madeof the soft material (Figs. 2.13 and 2.14). One also observes that using theinterpolative model for the deepest level leads to slightly better estimates.


0

10

20

30

40

50

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8



Coated Ca-modified PbTiO3 spherical inclusions reinforcing aEpon 828 matrix under harmonic uniaxial tension (ω=10 rads-1)

(D-I,Voigt)

(M-T,Voigt)

2-step and 2-level homogenization schemes2-step (M-T,Voigt) and (D-I,Voigt)2-level (D-I,M-T)

0

10

20

30

40

50

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8




(D-I,Voigt)

(M-T,Voigt)

Marra et al. 1999experimentalFE model with interphase

Figure 2.15: Storage tensile modulus versus volume fraction of inclusions of a coatedceramic elastic particle reinforced Epon 828 viscoelastic matrix. Comparison betweenexperiments and three-phase numerical models: FE calculations and various homog-enization schemes.

Epoxy and copolymer matrix materials reinforced with coated ce-ramic particles. Marra et al. (1999) modified their original FE model byadding a viscoelastic layer that surrounds each inclusion. The idea they fol-lowed was that “energy losses likely occur at or near the interface of the ceramicand the matrix”. The inclusion and matrix materials are those of Table 2.2.The interphase’s properties are those of the matrix except its Young’s loss mod-ulus which is α times higher. The volume fraction of the layer vi is linked tothe one of the particles vp through the relation vi = vpf/(1−f) for a given con-stant f . The results they obtained with this model are labeled FE model withinterphase in Figs. 2.15 to 2.18. The values of the two additional unknowns, αand f , were determined by fitting the predictions on the experimental points,which explains the remarkable agreement.

Taking the same values as Marra et al. (1999) for the new parameters, we usedthree-phase homogenization models to predict the complex tensile moduli ofboth composites. In each case (Figs. 2.15 to 2.18), our two-level scheme (D-I,M-T) underestimates the FE response and its predictions are always close tothe two-step (M-T,Voigt) ones. The high volume fraction of ceramic particles(coated or not) the M-T model has to deal with explains these trends. Thetwo-step (D-I,Voigt) works on the contrary pretty well for the storage moduliof both composites (Figs. 2.15 and 2.17). The loss moduli however are not


0

50

100

150

200

250

300

350

400

450

500

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8




(D-I,Voigt)

(M-T,Voigt)


0

50

100

150

200

250

300

350

400

450

500

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8




(D-I,Voigt)

(M-T,Voigt)


Figure 2.16: Loss tensile modulus versus volume fraction of inclusions of a coatedceramic elastic particle reinforced Epon 828 viscoelastic matrix. Comparison betweenexperiments and three-phase numerical models: FE calculations and various homog-enization schemes.

as well estimated, especially for the reinforced epoxy matrix (Fig. 2.16). Incomparison with the case of the copolymer matrix (f = 0.1, α = 2) the contrastbetween the loss moduli is much more pronounced (α = 33) and the layer ismuch thinner (f = 0.02). This is a severe situation. The two-level model mightovercome this issue with the interpolative D-I scheme at both levels. Thisrequires the extension to the complex plane of Eshelby’s tensor formulae foran inclusion in a transversely isotropic matrix. This is not as straightforwardas it seems to be. The expressions developed by Withers (1989) in the elasticcase already involve complex numbers. Care must be taken to ensure they donot conflict with the complex moduli of viscoelastic materials. This will beinvestigated in a future work.

2.3.3 Time behavior of two-phase composites

The materials hereafter should be qualified as academic since neither the com-posites nor the homogeneous components attempt to represent any existingmaterial. They were actually picked up by Yi et al. (1998) who chose them toillustrate their “(...) systematic way of obtaining the effective viscoelastic mod-uli in time and frequency domain (...) for viscoelastic composites with periodicmicrostructure”. The latter methodology is in form not much different from


0

5

10

15

20

25

30

35

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8



Coated Ca-modified PbTiO3 spherical inclusions reinforcing aP(VDF-TrFE) matrix under harmonic uniaxial tension (ω=10 rads-1)

(D-I,Voigt)

(M-T,Voigt)


0

5

10

15

20

25

30

35

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8




(D-I,Voigt)

(M-T,Voigt)


Figure 2.17: Storage tensile modulus versus volume fraction of inclusions of a coatedceramic elastic particle reinforced P(VDF-TrFE) viscoelastic matrix. Comparisonbetween experiments and three-phase numerical models: FE calculations and varioushomogenization schemes.

0

100

200

300

400

500

600

700

800

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8




(D-I,Voigt)

(M-T,Voigt)


0

100

200

300

400

500

600

700

800

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8




(D-I,Voigt)

(M-T,Voigt)


Figure 2.18: Loss tensile modulus versus volume fraction of inclusions of a coatedceramic elastic particle reinforced P(VDF-TrFE) viscoelastic matrix. Comparisonbetween experiments and three-phase numerical models: FE calculations and varioushomogenization schemes.


0

5

10

15

20

25

-3 -2 -1 0 1 2

log transverse tensile modulus E [-]

log time [s]

Elastic long fibers in a viscoelastic matrixunder plane strain uniaxial tension

matrix: E=3+17e-t , ν=0.38

fibers: E=20 , ν=0.21

Interpolative Scheme (D-I)vf = 10%vf = 30%vf = 50%vf = 70%

0

5

10

15

20

25

-3 -2 -1 0 1 2


log time [s]


matrix: E=3+17e-t , ν=0.38

fibers: E=20 , ν=0.21

FE-SQR (Yeong-Moo Yi et al. 1998)vf = 10%vf = 30%vf = 50%vf = 70%

Figure 2.19: Elastic long fiber reinforced viscoelastic matrix (one relaxation time).Time evolution of the plane strain transverse tensile modulus. Comparison betweenhomogenization and finite element predictions for various volume fractions of fibers.

what we do: compute the effective complex moduli in a transformed domain(Laplace or Laplace-Carson) and invert them numerically into time domain.However, while both numerical inversion tools are almost identical, the micro-macro transitions are drastically different. Assuming a periodic microstructureYi et al. (1998) use FE unit cell calculations (in the transformed domain) inorder to predict the effective moduli, while our estimates are computed byEshelby-based mean field homogenization schemes.

Various two-phase composites are examined here. Each of them consists ofelastic or viscoelastic long fibers embedded in a viscoelastic matrix (detailedmaterial compositions are to be found on Figs. 2.19 to 2.21). Transverseuniaxial loading under plane strain conditions is assumed.

Our numerical simulations have been conducted with the interpolative D-Imodel due to the high volume fraction of fibers. As explained in section 2.2.2 theset of collocation points should at least include the involved relaxation times,namely 1s and 10s. For these simple academic examples, they were chosen asθk = k for 1 ≤ k ≤ 10.

The predicted transverse plane strain elastic moduli are compared to those ob-tained by Yi et al. (1998) on Figs. 2.19 to 2.21. These three figures representno less than ten distinct composites – dissimilarities lie in the concentration orin the material of the phases – and in each case both FE and D-I estimates


0

5

10

15

20

25

-3 -2 -1 0 1 2 3


log time [s]


matrix: E=3+7e-t+10e-t/10 , ν=0.38

fibers: E=20 , ν=0.21

Interpolative Scheme (D-I)vf = 10%vf = 30%vf = 50%vf = 70%

0

5

10

15

20

25

-3 -2 -1 0 1 2 3


log time [s]


matrix: E=3+7e-t+10e-t/10 , ν=0.38

fibers: E=20 , ν=0.21

FE-SQR (Yeong-Moo Yi et al. 1998)vf = 10%vf = 30%vf = 50%vf = 70%

Figure 2.20: Elastic long fiber reinforced viscoelastic matrix (two relaxation times).Time evolution of the plane strain transverse tensile modulus. Comparison betweenhomogenization and finite element predictions for various volume fractions of fibers.

0

5

10

15

20

25

-3 -2 -1 0 1 2 3


log time [s]

50% viscoelastic long fibers in a viscoelastic matrixunder plane strain uniaxial tension

material 1

material 2

composite 1(2): material 1(2) for the fibers

E=3+17e-t/10

ν=0.38E=3+17e-t

ν=0.38

Interpolative Scheme (D-I)composite 1composite 2

0

5

10

15

20

25

-3 -2 -1 0 1 2 3


log time [s]

50% viscoelastic long fibers in a viscoelastic matrixunder plane strain uniaxial tension

material 1

material 2

composite 1(2): material 1(2) for the fibers

E=3+17e-t/10

ν=0.38E=3+17e-t

ν=0.38

FE-SQR (Yeong-Moo Yi et al. 1998)composite 1composite 2

Figure 2.21: Long fiber composite made of two viscoelastic materials. Time evolutionof the plane strain transverse tensile modulus. Comparison between homogenizationand finite element predictions for both phase configurations.


are almost identical. This is very satisfying although these two models do notrefer to the same microstructure. A periodic unit cell used in FE simulationscan indeed be a good approximation of randomly distributed inclusions (as-sumed by mean-field homogenization) for linear elastic composites, but notalways for nonlinear composites (see Adams, 1970; Jansson, 1992; Weissenbeket al., 1994; Bohm and Han, 2001; Segurado et al., 2002; Ji and Wang, 2003;Saraev and Schmauder, 2003; Carrere et al., 2004; Iwamoto, 2004). Due to theelastic-viscoelastic correspondence principle, the good agreement between FEsimulations and D-I estimates can be considered in our case as a verification.However, in order to have a complete verification, comparisons should be madewith FE calculations conducted on RVEs (not unit cells) containing a randomdistribution of the fibers.

These more realistic FE simulations are very expensive in regard to the CPUand user times which are both already high in the case of periodic microstruc-tures. Remember that in the FE analyses for each fiber volume fraction (herefive in total) a new FE mesh has to be built. On the contrary, only a change inthe value of a single parameter is needed for the interpolative scheme. Moreover,homogenization models provide a three dimensional response: the output is atensor from which the moduli are extracted. With the FE based method, newanalyzes, numerical inversions and sometimes new meshes would be necessaryif we were interested in other moduli.

2.4 Conclusions

In this chapter, we presented mean-field homogenization schemes for matrixmaterials reinforced with multiple phases of ellipsoidal inclusions, either coatedor not, in the linear viscoelastic regime. The general two-step and two-levelhomogenization procedures of Chap. 1 have been extended from linear elas-ticity to linear viscoelasticity thanks to the elastic-viscoelastic correspondenceprinciple. The overall harmonic properties are obtained in a straightforwardmanner by a change of variable in the Laplace-Carson domain. A numerical in-version procedure is required to predict the dependence of the effective modulion time.

A special emphasis was put on an extensive verification and validation of theproposed methods against FE results and available experimental data . Severecases (high volume fractions of inclusions, high contrasts between materialsproperties, soft or stiff matrices) were simulated.

For two-phase composites, several simulations were presented in frequency do-main (Figs. 2.1 to 2.8). As compared to reference unit-cell FE results, thepredictions of the interpolative D-I model were always excellent and better

Conclusions 67

than those of M-T for high volume fractions of inclusions (see Figs. 2.3 to 2.6,2.7 and 2.8). The same remarkable predictions of interpolative D-I were alsoobserved in the time domain (Figs. 2.19 to 2.21).

For composites with coated inclusions, the conclusions that can be drawn fromthe numerical simulations (Figs. 2.11 to 2.18) are less clear. A general commentwould be that the two-level or two-step schemes, or both of them give excellentpredictions, except in one instance. Indeed, two-step and two-level approachesmay provide almost identical predictions, yet in very good agreement withFE calculations (Figs. 2.11 and 2.12). But with the same microstructure,switching only material properties, the two-step method fails while the two-level procedure succeeds (Figs. 2.13 and 2.14) . Three cases (Figs. 2.15, 2.17and 2.18) for which the two-step (D-I/Voigt) is the most performing have beenshown. Both approaches fail dramatically in one case (Fig. 2.16). For thesefour last cases, the comparison is not really fair. A complete study wouldinclude the two-level (D-I/D-I) scheme for which additional developments arerequired.

There are at least a few directions for future work. The first one is relatedto the aforementioned comment. The development of the two-level (D-I/D-I)model requires the expression of Eshelby’s tensor in a transversely isotropicviscoelastic matrix. The results of Withers (1989) in elasticity can help. Moregenerally, one issue is to better assess the predictive capabilities and limitationsof the proposed two-step and two-level schemes for coated composites.

The two-step and two-level homogenization procedures have to be generalizedfor the prediction of thermo-viscoelastic properties, which is of major interestfor viscoelastic materials. One could be inspired by the works of Pierard et al.(2004); Doghri and Tinel (2006).

Another subject for future work is to extend the proposed methods to nonlinearviscoelasticity, perhaps by using the so-called affine formulation, which wasdeveloped successfully for elasto-viscoplasticity (e.g. Masson, 1998; Pierard andDoghri, 2006).


Appendix A Notations and units


Ω representative volume elementΩi region occupied by phase ip orientation vectorψi(p) orientation distribution function for inclusions of

phase iAri aspect ratio of inclusions of phase ivi volume fraction of phase i

t time sω angular frequency rad s−1

i imaginary unit number i2 = −1s complex variable in the Laplace-Carson domain rad s−1

Lc [ ], ˆ Laplace-Carson transform Lc [f ] (s) = f(s) = s

∞Z

0

f(t)e−stdt

<, = real and imaginary parts

x position mu displacement m

ε strain tensorσ stress tensor Pa

Gi relaxation tensor of material i PaG relaxation tensor PaJ creep tensor Pa−1

G overall relaxation tensor PaI fourth order identity tensor1 second order identity tensor

E∗, E′, E′′complex, storage and loss tensile moduli PaG∗, G′, G′′complex, storage and loss tensile moduli Pa

Chapter 3

Dynamic viscoelastichomogenization models

This chapter deals with propagation of plane waves in elastic and viscoelas-tic composites. Linear elastic composites and static loadings were consideredin Chap. 1. In Chap. 2, with the quasi-static approximation, homogeniza-tion methods were transposed from elasticity to viscoelasticity thanks to theelastic-viscoelastic correspondence principle. A generic formulation that han-dles both elastic and viscoelastic multiphase (including coated reinforcements)composites unifies a wide range of Eshelby-based mean-field homogenizationschemes. When inertial effects are not ignored, the correspondence principleis still valid but Eshelby’s result does not hold anymore. It however has anequivalent within the scope of plane wave propagation in inclusion reinforcedmaterials: the one-particle scattering problem. Many models of the literature(e.g. Sabina and Willis, 1988; Sabina et al., 1993; Smyshlyaev et al., 1993;Kanaun et al., 2004; Kanaun and Levin, 2005) are based on the solution of thiselementary problem. A generic formulation that unifies most of these models,extends them to viscoelastic behaviors and proposes new schemes is detailed inSecs. 3.1 and 3.2. A different approach (Biwa, 2001; Biwa et al., 2002, 2003),however also related to the one-particle scattering problem, is introduced inSec. 3.3. Section 3.4 aims to compare the predictions of all these models re-garding the effective acoustic properties (phase velocities, attenuation factors)of two-phase particulate composites. Experimental data from the literatureare taken as reference. Unfortunately, finite element results (or stemming fromsimilar approaches) are lacking.

70 Dynamic viscoelastic homogenization models

3.1 A generic modeling framework

A generic framework is built up to address plane wave propagation in multi-phase inclusion reinforced viscoelastic composites. It attempts to group withina unique formulation various models of the literature (e.g. Sabina and Willis,1988; Sabina et al., 1993; Smyshlyaev et al., 1993; Kanaun et al., 2004; Kanaunand Levin, 2005). The latter are developed for elastic composites. The fol-lowing approach encompasses also an extension to viscoelastic behaviors. Thesteady-state equations of motion exhibit the analogy between elasticity and vis-coelasticity. Their solution is written in integral form. Concentration tensorsare introduced and the solution is averaged over the heterogeneous medium.Assuming plane waves, the effective dispersion equation is obtained.

3.1.1 Steady-state dynamics

Consider a region Ω of a heterogeneous medium which is free of body forces.The stress field σ within Ω satisfies the equation of motion

∇ · σ(x, t) − ρ(x)∂2

∂t2u(x, t) = 0 , ∀x ∈ Ω , ∀ t > 0 (3.1)

where ρ is the density and u the displacement field. Let the Laplace-Carsontransform Lc defined for any time function f by∗

Lc [f ] (s) , s

∞∫

0

f(t)est dt , f(s) (3.2)

operate on Eq. (3.1). It results in

∇ · σ(x, s) − s2ρ(x)u(x, s) = 0 , ∀x ∈ Ω , ∀ s (3.3)

provided u(x, 0) = 0 and ∂∂tu(x, 0) = 0.

For standard linear viscoelastic materials, the stress in the heterogeneous bodydepends on the history of deformation through the convolution (relaxationform)

σ(x, t) =

t∫

−∞

G(x, t− τ) :∂

∂τε(x, τ) dτ , ∀x ∈ Ω , ∀ t > 0 (3.4)

∗This definition differs from Eq. (2.7) of Chap. 2 by the minus sign. The reason is thata time dependency of e

−iωt is assumed in this chapter.

A generic modeling framework 71

where the kernel G is the fourth order relaxation tensor – which is space andtime dependent – and ε is the strain field linked to the displacement by

ε(x, t) =1

2

(∇u(x, t) + ∇T u(x, t)

). (3.5)

The Laplace-Carson transform puts Eq. (3.4) under the elastic-like form

σ(x, s) = G(x, s) : ε(x, s) , ∀x ∈ Ω , ∀ s > 0 (3.6)

while the kinematics in the time-transformed domain, linking ε to u, is keptidentical to Eq. (3.5). This is known as the elastic-viscoelastic correspondenceprinciple.

The transformed problem (Eqs. (3.3) and (3.6)) is equivalent to the initialone (Eqs. (3.1) and (3.4)) if the displacement field in the medium is sought inharmonic regime. For a time dependency of e−iωt – ω is the angular frequency– the time independent part derives from u by the change of variable s = iωand verifies

∂

∂xj

[

Gijkl(x, iω)∂

∂xluk(x, iω)

]

+ ω2ρ(x)ui(x, iω) = 0 , ∀x ∈ Ω , (3.7)

once the kinematics is introduced in Eq. (3.3)

As it is a steady-state solution, the initial conditions are not relevant, whichjustifies the assumption made in Eq. (3.3) on the displacement and its firsttime derivative for t = 0.

Thanks to the elastic-viscoelastic correspondence principle, Eq. (3.7) can beregarded with identical approaches as for a heterogeneous body made of elasticconstituents. The only difference is the dependence on the angular frequencydue to the viscoelasticity. Such a parallel has already been exhibited in Chap. 2for viscoelastic composites subjected to harmonic loadings when inertial effectsare not taken into account.

3.1.2 Integral equation of the displacement field

Let Ω now be a representative volume element (RVE) of a composite madeof inclusions embedded inside a matrix. All materials are homogeneous andviscoelastic. Material p – index 0 stands for the matrix – has relaxation tensorGp and density ρp. It occupies a region Ωp in concentration vp in the RVE.Define the following variations for each reinforcing phase p > 0

∆Gp(x, t) , ∆Gp(t)UΩp(x) ,

[Gp(t) − G0(t)

]UΩp

(x) (3.8)

∆ρp(x) , ∆ρpUΩp(x) , [ρp − ρ0]UΩp

(x) (3.9)


with UΩpthe characteristic function of Ωp and rewrite Eq. (3.7) as

−

∂

∂xj

[

G0ijkl(iω)

∂

∂xluk(x, iω)

]

+ ω2ρ0ui(x, iω)

=

∑

p>0

∂

∂xj

[

∆Gpijkl(x, iω)

∂

∂xluk(x, iω)

]

+ ω2∆ρp(x)ui(x, iω)

, ∀x ∈ Ω .

(3.10)

Using the Green function G0ik of the operator −

(∂

∂xj

[

G0ijkl(iω) ∂

∂xl

]

+ ω2ρ0δik

)

one obtains the following integral equation for the time-independent part of thedisplacement field in the composite,

ui(x, iω) = u0i (x, iω)+∑

p>0

∫

Ω

G0ik(x − y)

∂

∂yj

[

∆Gpkjmn(y, iω)εnm(y, iω)

]

+ ω2G0ik(x − y)∆ρp(y)uk(y, iω) dy , (3.11)

where the strains ε have been reintroduced. The field u0 is the solution of theassociated homogeneous equation. It is the field that would have existed inthe medium without the presence of inclusions. If the RVE is such that thereis no inclusion across its boundaries, integration by part combined with thedivergence theorem gives:

ui(x, iω) = u0i (x, iω)+∑

p>0

∫

Ω

∂

∂xjG0

ik(x − y)∆Gpkjmn(y, iω)εnm(y, iω)

+ ω2G0ik(x − y)∆ρp(y)uk(y, iω) dy . (3.12)

The latter assumption will hold in the sequel.

3.1.3 Effective displacement field

For any field q in the composite, it’s expectation value at point x is given by

E [q(x)] =∑

p

P[x ∈ Ωp]E [q(x)|x ∈ Ωp] (3.13)

where E [q(x)|x ∈ Ωp] is the expectation value of q at x conditional upon findingmaterial p at this point and P[x ∈ Ωp] is the probability of finding material p

A generic modeling framework 73

at x. These probabilities are uniform and equal to the volume fractions if theinclusions are randomly dispersed in the matrix, as assumed in the following.

Taking the expectation values of both sides of Eq. (3.12), the average displace-ment field in the composite reads

E [ui(x, iω)] = u0i (x, iω)+

∑

p>0

∫

Ω

∂

∂xjG0

ik(x − y)E[

∆Gpkjmn(y, iω)εnm(y, iω)

]

+ ω2G0ik(x − y)E [∆ρp(y)uk(y, iω)] dy (3.14)

because u0 and G0 do not depend on the reinforcing phases. Moreover, using

Eq. (3.13), it simplifies into

E [ui(x, iω)] = u0i (x, iω)+

∑

p>0

vp

∫

Ω

∂

∂xjG0

ik(x − y)∆Gpkjmn(iω)E [εnm(y, iω)|y ∈ Ωp]

+ ω2G0ik(x − y)∆ρpE [uk(y, iω)|y ∈ Ωp] dy , (3.15)

as ∆Gp(y, iω) and ∆ρp(y) vanish for y /∈ Ωp when p > 0.

Let the conditional means of both displacement and deformation fields in eachreinforcing phase p > 0 be linked to the corresponding average fields in thecomposite through

E [ε(y, iω)|y ∈ Ωp] = Ap(iω) : E [ε(y, iω)] , (3.16)

E [u(y, iω)|y ∈ Ωp] = ap(iω) : E [u(y, iω)] , (3.17)

where Ap and ap are constant tensors – they are constant with respect toposition, they still depend at least on frequency and material properties – ofthe fourth and second order respectively. The effective displacement field takesthe following form

E [ui(x, iω)] = u0i (x, iω)+

∑

p>0

vp

∫

Ω

∂

∂xjG0

ik(x − y)∆Gpkjmn(iω)Ap

nmrs(iω)E [εsr(y, iω)]

+ ω2G0ik(x − y)∆ρpa

pkr(iω)E [ur(y, iω)] dy , (3.18)

once these tensors are introduced in Eq. (3.15).

3.1.4 Dispersion equation for effective plane waves

In the sequel, attention is drawn on plane waves propagating in the composite.To this end, the RVE is supposed to be wide enough to support such waves


(i.e. Ω is of infinite extent) and a Fourier transform (over space), defined forany space function f by

F [f ] (q) =

∫

f(x)e−iq·x dx , (3.19)

is applied on Eq. (3.18). The effective displacement field, in the Fourier domain,then reads

E [ui(q, iω)] = u0i (q, iω)+

∑

p>0

vp

iqjG0ik(q)∆Gp

kjmn(iω)Apnmrs(iω)E [εsr(q, iω)]

+ω2G0ik(q)∆ρpa

pkr(iω)E [ur(q, iω)]

, (3.20)

where the same notation is used for functions (with argument x or y in Eq.(3.18)) and their Fourier transforms (with argument q). Because tensors Ap

have minor symmetry – from their definition (Eq. (3.16)) – and from thekinematics in the Fourier domain,

εsr(q, iω) =i

2(qsur(q, iω) + qrus(q, iω)) , (3.21)

the overall field is rewritten:

E [ui(q, iω)] = u0i (q, iω)−

G0ik(q)

∑

p>0

vp

qjqs∆Gpkjmn(iω)Ap

nmrs(iω) − ω2∆ρpapkr(iω)

E [ur(q, iω)] .

(3.22)

Note that Eqs. (3.20) and (3.21) are obtained provided the boundary integralterms that appear in the Fourier transforms of the derivatives vanish. Thisassumption holds because Ω is of infinite extent.

The operator(

qjqlG0ijkl(iω) − ω2ρ0δik

)

is the Fourier transform of the differ-

ential operator −(

∂∂xj

[

G0ijkl(iω) ∂

∂xl

]

+ ω2ρ0δik

)

for which G0ik is the Green’s

function. Therefore, as(


)

u0i (q, iω) = 0, Eq. (3.22) is

equivalent to

[ (


)

+

∑

p>0

vp

(

qjql∆Gpijmn(iω)Ap

nmkl(iω) − ω2∆ρpapik(iω)

) ]

E [uk(q, iω)] = 0 .

(3.23)

Models derived from this framework 75

If the mean wave is a plane wave with wave number q , its Fourier transformis a Dirac function δ(q − q) and the above relation simplifies into

(

kj klG0ijkl(iω) − ω2ρ0δik

)

+

∑

p>0

vp

(

kj kl∆Gpijmn(iω)Ap

nmkl(iω) − ω2∆ρpapik(iω)

)

= 0 . (3.24)

The tensorial form of this so-called dispersion equation reads

q · G · q − ω2ρ = 0 , (3.25)

where the overall density ρ and relaxation G tensors are given by

ρ(iω) = ρ01 +∑

p>0

vp (ρp − ρ0)ap(iω) , (3.26)

G(iω) = G0(iω) +∑

p>0

vp

(Gp(iω) − G0(iω)

): Ap(iω) , (3.27)

with 1 the second order identity tensor.

Up to now, no specific scheme has been defined. The generic framework isbased on relations between conditional means regarding displacement and de-formation fields, i.e. on the existence of Ap and ap. The way these tensors arecomputed is the key point in the development of a model.

3.2 Models derived from this framework

For each reinforcing phase a one-particle problem is designed. A single inclusionof type p > 0 is embedded in a infinite reference medium and subjected to anincident wave. Each problem gives rise to two tensors λp and Λp. Explicitforms are found provided the wave in the reference material is a plane waveand the inclusion is of spherical shape. Tensors ap and Ap are related to λp andΛp, respectively. The link depends on the choice of the reference media, theincident waves propagating inside them and assumptions on how these incidentfields and the effective fields in the composite are related to each other.

3.2.1 One-particle approximation and conditional means

Let a representative inclusion of phase p > 0 be embedded in some referencemedium and subjected to some incident wave. Both the surrounding medium


x0

Up

pI

0x

x

O

(x)

Figure 3.1: The mean value at point x ofphase p is obtained as a sum of solutions atx obtained for different particle locations.The integration domain Up(x) is the regionof particles’ center (x0) for which x still liesinside the particles.

and the wave may depend on the type of the reinforcement. Assume thatthis simplified problem can be solved. Denote by f(x,x0) a solution fieldcorresponding to the single inclusion centered at x0. The link with the averagesolution field in phase p of the original composite is achieved as follows (Sabinaet al., 1993):

E [f(x)|x ∈ Ωp] ,1

|Up(x)|

∫

Up(x)

f(x,x0) dx0 . (3.28)

As illustrated on Fig. 3.1, Up(x) is the region of particles’ center for which x stilllies inside the particles. A formal definition reads Up(x) = x0 : x − x0 ∈ Ipwith Ip the inclusion of type p centered at the origin (x0 = 0). |Up(x)| is thevolume of Up(x).

Expectation value of the displacement field. Consider a representativeinclusion of phase p centered at the origin and a plane wave ur in the referencematerial. Let the displacement field in the inclusion be represented in the form

ui(x) = λik [urk(x)] = λik

[Uke

iq·x], (3.29)

where λ is a linear operator which depends only on the properties of the inclu-sion and the reference medium. If the inclusion is now centered in x0, a changeof variable gives

ui(x) = λik

[

eiq·(x−x0)]

e−iq·(x−x0)urk(x) , λu

ik(x − x0)urk(x) . (3.30)

The expectation value of the displacement field in phase p then reads

E [u(x)|x ∈ Ωp] = λp · ur(x) , (3.31)

where the second order tensor λp is defined by

λp ,1

|Up(x)|

∫

Up(x)

λu(x − x0) dx0 =1

|Ip|

∫

Ip

λu(y) dy . (3.32)


Expectation value of the deformation field. A relation similar to Eq.(3.31) holds for the associated deformation field. Its expectation value in thereinforcing phase verifies

E [ε(x)|x ∈ Ωp] = Λp : εr(x) , (3.33)

where Λp is the fourth order tensor defined by

Λp ,1

|Up(x)|

∫

Up(x)

Λε(x − x0) dx0 =1

|Ip|

∫

Ip

Λε(y) dy . (3.34)

Tensor Λε links the strains in the inclusion to the incident strains:

εij(x) , Λεijkl(x − x0)ε

rlk(x) . (3.35)

Its components are obtained by differentiating Eq. (3.30),

Λεijkl(y) = −1

2

(∂

∂yjλu

ik(y) +∂

∂yiλu

jk(y)

)iql

q · q +1

2

(λu

ik(y)δjl + λujk(y)δil

),

(3.36)using the fact that ur is a plane wave and minor symmetries.

On the computation of Λp and λp. Because the wave in the referencemedium is a plane wave ur(x) = Ueiq·x, one has

λpikur

k(x)e−iq·x =1

|Ip|

∫

Ip

λuik(y)Uk dy =

1

|Ip|

∫

Ip

λik

[Uke

iq·y]e−iq·y dy =

1

|Ip|

∫

Ip

ui(y)e−iq·y dy , (3.37)

using successively Eqs. (3.32) and (3.30), assuming the representative inclu-sion is located at the origin (x0 = 0). This means that λp can be computedby integration of the so-called transmitted displacement field over the regionoccupied by the inclusion, once the one-particle scattering problem has beensolved for ur. The same holds for Λp with the strains,

Λpijklε

rlk(x)e−iq·x =

1

|Ip|

∫

Ip

Λεijkl(y)iqlUk dy =

1

|Ip|

∫

Ip

Λεijkl(y)iqlUke

iq·ye−iq·y dy =

1

|Ip|

∫

Ip

Λεijkl(y)εr

lk(y)e−iq·y dy =1

|Ip|

∫

Ip

εij(y)e−iq·y dy , (3.38)

making use of Eqs. (3.34) and (3.35). Expressions for the displacement anddeformation fields in the inclusion are required in order to compute both tensors


with Eqs. (3.37) and (3.38). The latter exist when the inclusion is of sphericalshape. The one-particle scattering problem can be solved explicitly in this case.Its solution is reported in Appendix A. An approximate solution was proposedby Sabina and Willis (1988); Sabina et al. (1993) for a spheroidal inclusion. Inthis approximation the fields do not depend on the position inside the inclusion.

Tensors Λp and λp do not define any model yet. The reference medium and theacting plane wave still have to be specified. Expressions of Ap and ap dependon these choices which determine a specific scheme.

3.2.2 Self-consistent schemes

Self-consistent schemes are obtained when the reference medium (the same forall types of inclusions) has the effective properties of the composite. The actingplane wave propagating in the reference medium is chosen as the mean wavefield in the composite:

ur(x) , Ueiq·x = E [u(x)] . (3.39)

For a single population of spherical elastic particles, this is the effective mediummethod (EMM) of Kanaun et al. (2004) in its first version. Dealing with multi-ple phases of aligned spheroidal elastic reinforcements, this approach is equiv-alent to the self-consistent scheme of Sabina et al. (1993). The latter has beengeneralized to misoriented inclusions by Smyshlyaev et al. (1993). Recall how-ever that both latter methods are based on an approximate solution of theone-particle scattering problem.

The self-consistent approach leads to ap = λp and Ap = Λp. Both thesetensors depend on the (unknown) effective properties. The dispersion equation(Eqs. (3.25), (3.26), (3.27)) is implicit. It has to be solved with an iterativeprocedure.

3.2.3 A class of explicit schemes

We propose to build a class of models for which the reference medium is thematrix of the composite. Tensors Λp and λp depend on the (known) propertiesof the constituents. The dispersion equation (Eqs. (3.25), (3.26), (3.27)) issolved in an explicit way. The drawback is that the acting plane wave has tobe determined explicitly. The latter should represent the average field in thematrix phase:

ur(x) , U0eiq0·x = E [u(x)|x ∈ Ω0] . (3.40)


As a consequence, the definitions of Ap, Λp, ap and λp give

Ap = Λp :(v0I +

∑

p>0

vpΛp)−1

, (3.41)

ap = λp ·(v01 +

∑

p>0

vpλp)−1

, (3.42)

where 1 and I are the second and fourth order identity tensors, respectively.

A dilute inclusion scheme. For a very low concentration of inclusionsE [u(x)|x ∈ Ω0] can be approximated by E [u(x)] in Eq. (3.40). Each inclusionis then subjected to a plane wave propagating in the matrix, the latter beingthe wave that would have existed without the presence of the surrounding in-clusions. This is known as the single- or independent-scattering approximation.For this so-called dilute inclusion scheme Ap = Λp and ap = λp. Unlike theself-consistent schemes, these tensors are explicit. They are computed withhelp of the matrix properties instead of the effective ones.

A slight improvement of the dilute inclusion scheme. The above modelmay be slightly improved as follows: the incident field of the single-scatteringapproximation is used to compute Λp and λp, tensors Ap and ap are thenobtained from Eqs. (3.41) and (3.42). This method does not take multiple-scattering into account. It should therefore lead to improved predictions mainlyin the long wave region for which scattering is weak or even non-existent(static/quasi-static limit). Compared to the dilute inclusion model, no ad-ditional development or computational effort are required.

Multiple-scattering models. A way to handle multiple-scattering is to addthe fields scattered on the surrounding inclusions as part of the incident wavein the one-particle scattering problem. How to determine these scattered fieldsis still an open issue. Recall that for the one-particle problem to be solvablethe incident field has to be a plane wave, or a superposition of plane waves.Anyway, this requires further developments. Connections may be found withthe Effective Field Method (EFM) of Kanaun and Levin (2005). This model ishowever implicit: the single inclusion is embedded in the matrix and subjectedto a plane wave propagating in the matrix but with effective (unknown) wavenumber. This results in a modified one-particle scattering problem.

3.2.4 Static limit of the models

Tensors λp and Λp are the building blocks of the schemes developed in Secs.3.2.2 and 3.2.3. The static versions of these models are determined by taking


One-particle based models Classical elastic modelsDilute-Inclusion Generalized Dilute-InclusionDilute-Inclusion (slightly improved) Generalized Mori-TanakaEMM (Kanaun et al., 2004) Self-ConsistentS-C (Sabina et al., 1993) Generalized Self-ConsistentEFM (Kanaun and Levin, 2005) Mori-Tanaka

Table 3.1: Equivalence relations between models based on the one-particle scatteringproblem in the static regime and classical mean-field Eshelby-based elastic homoge-nization schemes (generalized means extension to multi-phase composites).

the limit of λp and Λp for ω → 0.

Assuming both materials of the one-particle scattering problem are purely elas-tic (no attenuation), the expressions reported in Appendix B give

limω→0

λp = 1 (3.43)

andlimω→0

Λp =(I + SI,Cr

:(C−1

r : Cp − I))−1

, (3.44)

where Cr and Cp are the stiffnesses of the reference medium and the particle,respectively. SI,Cr

is Eshelby’s tensor, computed with the Poisson’s ratio ofthe reference material for a spherical inclusion. These limit values also holdfor elastic materials with attenuation (because the attenuation factors vanishwhen ω → 0) and for standard viscoelastic materials provided Cr and Cp arethe long term stiffness tensors.

As a result, ap → 1, whatever the model. This means (see Eq. (3.26)) nothingelse than that the effective density is the volume fraction weighted sum of theconstituents’ densities.

The right hand side of Eq. (3.44) relates the (constant) strains in the particle tothe remote (constant) deformation field. It is actually H (Cr,Cp), as definedby Eq. (1.12) of Chap. 1. Table 3.1 summarizes the equivalence between themodels of Secs. 3.2.2 and 3.2.3 in the static regime and mean-field Eshelby-based elastic homogenization schemes.

3.3 An additive model for wave attenuation

This section describes another micro-mechanical model able to estimate waveattenuation in viscoelastic inclusion reinforced materials. It is an extension ofthe method proposed by Beltzer and Brauner (1987) for elastic composites. The

An additive model for wave attenuation 81

original model was modified by Biwa (2001) and Biwa et al. (2002) to accountfor absorption losses in the constituents, i.e. to handle viscoelastic components.The approach is based on the assumption of single scattering/absorption. Themodel is thus only valid for composites with low concentration of reinforce-ments. A differential (incremental) scheme (not presented here) is proposed byBiwa et al. (2003) for higher volume fraction of inclusions. It follows the sameidea – at each step, an infinitesimal amount of fillers is added to the equivalentmatrix medium of the previous step – similarly to the sequential method ex-posed in Sec. 1.4 of Chap. 1. The dilute inclusion model of Biwa (2001); Biwaet al. (2002) is based on the evaluation of so-called scattering and absorptioncross-sections. It models wave attenuation in inclusion reinforced materials ina rather simple way. However, effective phase velocities cannot be predicted.

3.3.1 Scattering and absorption cross-sections

The average energy flux vector – the analogous of the time averaged Poyntingvector for electromagnetic waves – of a time harmonic wave is a real-valued andtime-independent vector. It is obtained by averaging over one cycle in time theinstantaneous energy flux and reads

S =1

T

∫ T

0

−1

2< [σ · u∗] dt (3.45)

for a time dependence of e−iωt. In the above expression σ is the stress tensor,u is the time derivative of the displacement field and T = 2π/ω is the period.In the case of plane waves with initial amplitude U0 propagating along x3 in ahomogeneous viscoelastic medium its expression simplifies to

S =ω

2U0U

∗0 e

−2=[k]x3< [k(λ+ 2µ)]e3 , (3.46)

for P-waves, and

S =ω

2U0U

∗0 e

−2=[κ]x3< [κµ] e1 , (3.47)

for S-waves with polarization vector e1. The imaginary part of the wave num-bers vanishes for purely elastic materials . There is no decay of energy and thePoynting vectors are constant in that case.

For a plane wave incident to a particle, part of it is scattered while the otheris refracted. The ability of the particle to scatter and absorb energy from theincident wave is characterized by its scattering and absorption cross-sections.The scattering (resp. absorption) cross-section γsca (resp. γabs) is defined asthe ratio between the average scattered (resp. refracted) energy on the particleboundary surface and a reference average energy flux. Their expressions are


given by

γsca =1

Sref

1

T

∫ T

0

∫

Γ

−1

2<[

σsca · usca∗

]

· ndΓdt (3.48)

γabs =1

Sref

1

T

∫ T

0

∫

Γ

−1

2<[

σabs · uabs∗

]

· (−n) dΓdt (3.49)

where Γ is the particle boundary surface and n its outgoing normal. Thereference energy flux Sref is the one of the matrix, i.e. the medium embeddingthe particle. It is defined as the module of (3.46) or (3.47) depending on thetype of wave. For viscoelastic media the value at the center of the particle(x3 = 0) is considered.

Scattering and absorption cross-sections have been calculated for both P-wavesand S-waves acting on a spherical particle with radius r. (3.48) and (3.49)require integrating the product of two associated Legendre functions. Formulaecan be found in mathematical handbooks. The expressions for γsca and γabs,normalized by the particle surface, are given by

γsca

4πr2=

X

n>0

2

2n+ 1=

(

f25 (k0r)f

21∗(k0r) + f2

7 (k0r)f22∗(k0r)

(κ0r)2|k0an|2

+f26 (κ0r)f

22∗(κ0r) + f2

8 (κ0r)ˆ

f26∗(κ0r) + 2f2

2∗(κ0r)

˜

(κ0r)2|k0bn|2

+f25 (k0r)f

22∗(κ0r) + f2

6 (k0r)f24∗(κ0r)

(κ0r)2(k0an)(k0bn)∗

+f26 (κ0r)f

21∗(k0r) + f2

8 (κ0r)f22∗(k0r)

(κ0r)2(k0bn)(k0an)∗

),

<[k0r] (3.50)

and

γabs

4πr2= −ρ1

ρ0

X

n>0

2

2n+ 1=

(

f15 (k1r)f

11∗(k1r) + f1

7 (k1r)f12∗(k1r)

(κ1r)2˛

˛k0a′n

˛

˛

2

+f16 (κ1r)f

12∗(κ1r) + f1

8 (κ1r)ˆ

f16∗(κ1r) + 2f1

2∗(κ1r)

˜

(κ1r)2˛

˛k0b′n

˛

˛

2

+f15 (k1r)f

12∗(κ1r) + f1

6 (k1r)f14∗(κ1r)

(κ1r)2(k0a

′n)(k0b

′n)

∗

+f16 (κ1r)f

11∗(k1r) + f1

8 (κ1r)f12∗(k1r)

(κ1r)2(k0b

′n)(k0a

′n)

∗

),

<[k0r] (3.51)


for P-waves, and,

γsca

4πr2=

X

n>1

n(n+ 1)

2n+ 1=

(

n(n+ 1)f27 (κ0r)f

23∗(κ0r)

2|dn|2

+f25 (k0r)f

21∗(k0r) + n(n+ 1)f2

7 (k0r)f23∗(k0r)

(κ0r)2|κ0cn|2

+f26 (κ0r)f

22∗(κ0r) + n(n+ 1)f2

8 (κ0r)f24∗(κ0r)

(κ0r)2|κ0en|2

+f25 (k0r)f

22∗(κ0r) + n(n+ 1)f2

7 (k0r)f24∗(κ0r)

(κ0r)2(κ0cn)(κ0en)∗

+f26 (κ0r)f

21∗(k0r) + n(n+ 1)f2

8 (κ0r)f23∗(k0r)

(κ0r)2(κ0en)(κ0cn)∗

),

<[κ0r] (3.52)

and

γabs

4πr2= −ρ1

ρ0

X

n>1

n(n+ 1)

2n+ 1=

(

n(n+ 1)f17 (κ1r)f

13∗(κ1r)

2

˛

˛d′n˛

˛

2

+f15 (k1r)f

11∗(k1r) + n(n+ 1)f1

7 (k1r)f13∗(k1r)

(κ1r)2˛

˛κ0c′n

˛

˛

2

+f16 (κ1r)f

12∗(κ1r) + n(n+ 1)f1

8 (κ1r)f14∗(κ1r)

(κ1r)2˛

˛κ0e′n

˛

˛

2

+f15 (k1r)f

12∗(κ1r) + n(n+ 1)f1

7 (k1r)f14∗(κ1r)

(κ1r)2(κ0c

′n)(κ0e

′n)

∗

+f16 (κ1r)f

11∗(k1r) + n(n+ 1)f1

8 (κ1r)f13∗(k1r)

(κ1r)2(κ0e

′n)(κ0c

′n)

∗

),

<[κ0r] (3.53)

for S-waves. Scalars an, bn, cn, dn and en (resp. a′n, b′n, c′n, d′n and e′n) are thecoefficients involved in the spherical function expansion of the scattered (resp.refracted) displacement field. They are solutions of linear systems obtainedfrom the perfect bound condition at particle/matrix interface. The f i

m (m =1, . . . , 8) functions involve spherical Bessel (i = 1) and Hankel (i = 2) functionsof the first kind. See Appendix A for the details.

Focusing on P-waves only, Biwa et al. (2002) report the scattering and absorp-tion cross-sections for an elastic glass particle in a viscoelastic epoxy mediumand for a rubber particle embedded in a PMMA matrix, both materials beingviscoelastic. Cross-sections computed with (3.50) and (3.51) are compared tothe data of Biwa et al. (2002) on Figs. 3.2 to 3.4. A perfect match is ob-tained for the glass/epoxy system (Fig. 3.2). The absorption cross-section (notshown) vanishes because the inclusion is purely elastic. Figure 3.3 shows somemismatch between both calculations of γsca for the rubber/PMMA composite


1e-05

1e-04

1e-03

1e-02

1e-01

1e+00

1e+01

1e-03 1e-02 1e-01 1e+00

Normalized scattering cross-section

γsca/4

πa2 [-]

Normalized frequency af/cL0 [-]

P-wave acting on a spherical glass particleembedded in an epoxy medium

particle radius a=150µmEpoxy P-wave velocity cL

0=2.54mm/µs

Biwa et al. (2002)viscoelastic epoxyelastic epoxy

1e-05

1e-04

1e-03

1e-02

1e-01

1e+00

1e+01

1e-03 1e-02 1e-01 1e+00


γsca/4

πa2 [-]


P-wave acting on a spherical glass particleembedded in an epoxy medium

particle radius a=150µmEpoxy P-wave velocity cL

0=2.54mm/µs

This workviscoelastic epoxyelastic epoxy

1e-05

1e-04

1e-03

1e-02

1e-01

1e+00

1e+01

1e-02 1e-01 1e+00


γsca/4

πa2 [-]

Normalized frequency af/cT0 [-]

S-wave acting on a spherical glass particle embedded in an epoxy medium

particle radius a=150µmEpoxy S-wave velocity cT

0=1.16mm/µs

This workviscoelastic epoxyelastic epoxy

Figure 3.2: Scattering cross-section for P-waves (left) and S-waves (right) propagatingin an epoxy medium and acting on a spherical glass particle (see Tab. 3.2 for materialproperties).

1e-05

1e-04

1e-03

1e-02

1e-01

1e+00

1e+01

1e-03 1e-02 1e-01 1e+00


γsca/4

πa2 [-]


P-wave acting on a spherical rubber particleembedded in a PMMA medium

particle radius a=50µmPMMA P-wave velocity cL

0=2.74mm/µs

Biwa et al. (2002)viscoelastic PMMAelastic PMMA

1e-05

1e-04

1e-03

1e-02

1e-01

1e+00

1e+01

1e-03 1e-02 1e-01 1e+00


γsca/4

πa2 [-]



particle radius a=50µmPMMA P-wave velocity cL

0=2.74mm/µs

This workviscoelastic PMMAelastic PMMA

1e-05

1e-04

1e-03

1e-02

1e-01

1e+00

1e+01

1e-02 1e-01 1e+00


γsca/4

πa2 [-]

Normalized frequency af/cT0 [-]

S-wave acting on a spherical rubber particleembedded in a PMMA medium

particle radius a=50µmPMMA S-wave velocity cT

0=1.38mm/µs

This workviscoelastic PMMAelastic PMMA

Figure 3.3: Scattering cross-section for P-waves (left) and S-waves (right) propagatingin a PMMA medium and acting on a spherical rubber particle (see Tab. 3.3 formaterial properties).


0

1

2

3

4

5

6

7

8

0 1 2 3 4 5 6

Normalized absorption cross-section

γabs/4

πa2 [-]

Frequency f [MHz]


Biwa et al. (2002)particle radius a=50 µmparticle radius a=20 µm

0

1

2

3

4

5

6

7

8

0 1 2 3 4 5 6


γabs/4

πa2 [-]

Frequency f [MHz]


This workparticle radius a=50 µmparticle radius a=20 µm

0

2

4

6

8

10

12

14

16

0 1 2 3 4 5 6


γabs/4

πa2 [-]

Frequency f [MHz]

S-wave acting on a spherical rubber particleembedded in a PMMA medium

This workparticle radius a=50 µmparticle radius a=20 µm

Figure 3.4: Absorption cross-section for P-waves (left) and S-waves (right) propagat-ing in an PMMA medium and acting on a spherical rubber particle (see Tab. 3.3 formaterial properties).

when af/c0L > 2 10−1. These differences might be linked to numerical approx-imations or to the truncation of the spherical function expansion. It may alsobe related to the radius of the particles used for the numerical simulations.The latter was not reported in Biwa et al. (2002). We used a = 50µm. Theabsorption cross-sections (Fig. 3.4) are on the contrary identical. Note howeverthat the maximal value of af/c0L is reached with a = 50µm and f = 5.5 MHzand equals approximately 1e−1.

S-wave cross-sections obtained with (3.52) and (3.53) are also plotted on Fig-ures 3.2 to 3.4. As aforementioned, Biwa et al. (2002) only studied P-waves.Unfortunately no other data were available for comparison.

3.3.2 The additive attenuation model

To model wave attenuation in inclusion reinforced materials Biwa et al. (2002)assume that the spatial decay of the average energy flux 〈e〉 of plane wavesobeys an additive decomposition. In the case of viscoelastic composites theauthors propose

d 〈e〉dx3

= −〈Isca〉 − 〈Iabs〉 − 〈Imat〉 (3.54)

for a plane wave propagating in the x3 direction. The three contributions – perunit volume and time averaged – to the energy decay are the scattering loss rate〈Isca〉, the absorption loss rate in the particles 〈Iabs〉 and the absorption loss


rate in the matrix 〈Imat〉. Elastic materials do not contribute the absorptionloss, i.e. 〈Iabs〉 and/or 〈Imat〉 vanishes.

The assumption of low concentration of inclusions is introduced to evaluate thecontributions (scattering and absorption) of the particles. Mutual interactionsare neglected and the loss rates are estimated additively from the single-particlescattering. For ns inclusions per unit volume, they are given by

〈Isca〉 = nsγsca 〈e〉0 , 〈Iabs〉 = nsγ

abs 〈e〉0 , (3.55)

where γsca and γabs are the scattering and absorption cross-sections, and 〈e〉0stands for the reference energy flux† (denoted Sref in Sec. 3.3.1). The spatialdecay of the average energy flux for a plane wave propagating in a homoge-neous medium with attenuation factor α0 is obtained from (3.46) or (3.47) andequals −2α0 〈e〉0 at first-order. The absorption loss rate in the matrix is thenestimated with

〈Imat〉 = (1 − v1)2α0 〈e〉0 , (3.56)

where v1 is the volume fraction of inclusions. This rather intuitive result hasbeen demonstrated by Biwa (2001). It holds for inclusions with arbitrary shapefor which the product of α0 and their size along the propagation direction issmall.

Let α be the effective attenuation factor of the composite. The average energyflux decay is approximated at first order by −2α 〈e〉0. It is assumed that thereference energy flux can be approximated by the one of the matrix. The lowconcentration of particles justifies this assumption. Putting all together, theeffective attenuation coefficient reads

α = (1 − v1)α0 +1

2ns (γsca + γabs) = α0 + v1

3

8πr3(γsca + γabs) − α0

,

(3.57)

with r the radius of the spherical particles.

The overall attenuation for P-waves (resp. S-waves) is estimated using (3.57)with α0 the longitudinal (resp. transverse) attenuation coefficient of the matrixand γsca and γabs obtained for a P-wave (resp. S-wave) propagating in thematrix and acting on a single inclusion. The model of Biwa et al. (2002) issimple but provides estimates for wave attenuation only.

3.4 Numerical Simulations

The hereafter simulations cover a rather wide range of two-phase particulatecomposites. The latter are made up with either two elastic phases (1 case),

†Note that there is a typo in the definition of 〈e〉0 in Eq. (Biwa et al., 2002, 15).


λ+ 2µ [GPa] µ [GPa] ρ [kg/m3]

storage loss storage lossEpoxy 7.61 0.283 1.58 0.128 1180Glass 77.8 - 26.0 - 2470

Table 3.2: Material properties for the glass/epoxy composite after Biwa et al. (2002).

elastic spheres in a viscoelastic matrix (3 cases) or two viscoelastic materials(2 cases). Attenuation factors and phase velocities are shown for longitudinalwaves only. There are two reasons for that. First, the experimental datathat can be found in the literature are most of the time devoted to P-wavesonly. Second, the numerical predictions reported in Kanaun et al. (2004) andKanaun and Levin (2005) regarding S-waves remain equivocal‡. Attention isdrawn on the comparison between the dilute-inclusion scheme of Sec. 3.2.3 andthe additive attenuation scheme of Biwa et al. (2002). Both of them are basedon the independent scattering assumption. Confrontation with other numericalapproaches and/or experimental data from the literature are shown wheneveravailable. Unfortunately there are no results stemming from direct calculations(e.g. finite elements). This issue is discussed in the conclusions.

3.4.1 Glass-particle reinforced epoxy composite

Biwa et al. (2002) considered the following composite: an epoxy (TRA-CAST3012) matrix reinforced with spherical glass particles. The low volume frac-tion of fillers (8.6%) allows to assume a dilute suspension of inclusions. It thusmakes sense to compare our dilute inclusion scheme with the additive attenu-ation model on this system. The material properties of both components aresummarized in table 3.2. The radius of the particles equals 150µm. This com-posite was first studied experimentally by Kinra et al. (1980). The authorsmeasured the acoustic properties of epoxy over the frequency range f = 0.3to 5 MHz. They found that both cL and cT are independent of the frequency:cL = 2.54 mm/µs and cT = 1.16 mm/µs. The longitudinal attenuation factorαL was found to increase linearly with f . A linear regression gave αL = α+mfwith α = 0.001 nepers/mm and m = 0.0456 nepers/mm− MHz. No data werereported for the transverse attenuation factor. Measurements coming from an-other experimental study (see Biwa et al., 2002, and references therein) areconsidered instead for the lacking property. Computing phase velocities andthe longitudinal attenuation factor from the complex moduli of table 3.2, onechecks that all data are consistent. The elastic properties for glass – Kinraet al. (1980) report E = 64.89 GPa and ν = 0.249 – are consistent as well.

‡The authors recently confirmed that this is only due to misprints in their papers.


0

0.05

0.1

0.15

0.2

0.25

0.3

0 0.2 0.4 0.6 0.8 1

Attenuation factor

α L [mm-1]

Non-dimensional longitudinal wave number of the matrix k0a [-]

Epoxy matrix reinforced with 8.6% glass particles.Frequency dependence of the longitudinal attenuation factor.

epoxy matrix

particle radius: a=150µm

experimental (Kinra et al., 1980)v0α0+nsγsca/2Dilute-Inclusion

Figure 3.5: Effective attenuation factor for P-waves in a glass/epoxy composite. Com-parison between experimental data, the dilute inclusion scheme of Sec. 3.2.3 and theadditive attenuation model of Biwa et al. (2002).

0.9

0.95

1

1.05

1.1

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

Normalized phase velocity cL/cL0 [-]


Epoxy matrix reinforced with 8.6% glass particles.Frequency dependence of the longitudinal phase velocity.

particle radius: a=150µmmatrix phase velocity: c0L=2.54mm/µs

experimental (Kinra et al., 1980)Dilute-Inclusion

Figure 3.6: Effective phase velocity for P-waves in a glass/epoxy composite. Com-parison between experimental data and the dilute inclusion scheme of Sec. 3.2.3.


λ+ 2µ [GPa] µ [GPa] ρ [kg/m3]

storage loss storage lossPMMA 8.88 0.07 2.25 0.09 1180Rubber 3.30 0.14+0.035f† 0.55 0.10 1100†f : frequency [MHz]

Table 3.3: Material properties for the rubber/PMMA blend after Biwa et al. (2002).

Figure 3.5 reports the predictions and the experimental data for the effectivelongitudinal attenuation factor. Both models provide similar estimates of αL.They are even identical for k0a < 0.4 and show a very good agreement with themeasurements over the latter frequency range. For higher frequencies (k0a >0.4) the schemes diverge, slowly however. The agreement with the experimentbecomes less satisfactory for both of them but remains, at least qualitatively,acceptable. Similar conclusions can be drawn for the effective longitudinalphase velocity. Figure 3.6 shows that the predictions are accurate for k0a < 0.4.Beyond this value only qualitative results should be considered. The error-barsin Fig. 3.6 are based on the accuracy of the measurements for cL reported inKinra et al. (1980): ±0.3% for the matrix and ±1% for the composite. Recallthat the model after Biwa et al. (2002) cannot predict phase velocities.

3.4.2 Glass/Epoxy and Rubber/PMMA systems

The results exposed in the previous section may lead to conclude that ourdilute inclusion model and the scheme of Biwa et al. (2002) are more or lessequivalent regarding the prediction of attenuation factors. Except in the long-wave region, they actually differ. Roughly, the higher is the value of k0a, themore the differences are significant.

Three other glass/epoxy composites were designed by Biwa et al. (2002) withthe same material properties as before. Only the radius of the spherical par-ticles is changed: a = 1, a = 10, and a = 100µm. The authors also studiedrubber-particle reinforced PMMA systems with the same set of radii for the in-clusions. The material properties for rubber and PMMA are reported in table3.3. The dilute-inclusion scheme and the additive attenuation model are com-pared together on Fig.3.7 for the glass/epoxy composites and Fig. 3.8 for therubber/PMMA blends. Although 20% volume fraction of fillers is far too highfor the independent scattering assumption to remain valid, these comparisonsunderline similarities and differences between both methods. First, both modelspredict similarly shaped frequency dependence of the longitudinal attenuationfactor. Second, the differences regarding the amplitude of αL grow with k0a.Third, our dilute scheme seems to predict sometimes higher sometimes lower


0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10

Attenuation factor [mm-1]

Frequency [MHz]

Epoxy matrix reinforced with 20% glass particles. Frequency dependenceof the longitudinal attenuation factor for different particles’ radii.

100µm

10µm1µm

reduced attenuationof the matrix (v0α0)

Dilute-Inclusion

v0α0+nsγsca/2

Figure 3.7: Effective attenuation factor for P-waves in three glass/epoxy compos-ites. Comparison between the dilute inclusion scheme of Sec. 3.2.3 and the additiveattenuation model of Biwa et al. (2002).

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10

Attenuation factor [mm-1]

Frequency [MHz]

PMMA matrix reinforced with 20% rubber particles. Frequency dependenceof the longitudinal attenuation factor for different particles’ radii.

100µm

10µm

1µm

reduced attenuationof the matrix (v0α0)

v0α0+ns(γsca+γabs)/2

Dilute-Inclusion

Figure 3.8: Effective attenuation factor for P-waves in three rubber/PMMA compos-ites. Comparison between the dilute inclusion scheme of Sec. 3.2.3 and the additiveattenuation model of Biwa et al. (2002).


0.7

0.8

0.9

1

1.1

-2 -1 0 1 2

Normalized phase velocity cL/cL0 [-]

Non-dimensional longitudinal wave number of the matrix log(k0a) [-]

10% hard and heavy particles reinforcing a soft and light matrix.P-wave phase velocity

SC (Kanaun et al., 2004)EMM (Kanaun et al., 2004)Dilute-InclusionDilute-Inclusion (improved)

-2 -1

D-I

M-T

S-C

Figure 3.9: Effective longitudinal phase velocity for a highly contrasted elastic com-posite. Comparison between various numerical models. The arrows are the values ob-tained with the static self-consistent (S-C), Mori-Tanaka (M-T) and dilute-inclusion(D-I) schemes.

attenuation than the model of Biwa et al. (2002). It is not just related to theviscosity of the reinforcing phase, i.e. whether γabs vanishes or not. Performingthe same numerical simulations on the rubber/PMMA composites with the realparts of the complex moduli for the particles gives a figure (not shown here)which is similar to 3.8.

3.4.3 Hard and heavy particles in a soft and light matrix

This virtual composite was designed by Kanaun et al. (2004). It consists in asoft and light elastic matrix reinforced with hard and heavy spherical elasticparticles: E1/E0 = 50, ν1 = 0.3, ν0 = 0.4, ρ1/ρ0 = 10. The authors consideredtwo volume fractions of inclusions (10 and 30%). They analyzed the compositewith the EMM and another self-consistent model. The latter is based on anapproximate solution of the one-particle scattering problem proposed by Sabinaand Willis (1988) and Sabina et al. (1993) and for which the field inside theinclusion is assumed to be constant. Both P-waves and S-waves are studied inKanaun et al. (2004). The same authors (Kanaun and Levin, 2005) developedan Effective Field Method (EFM) to analyze shear waves in this composite.Surprisingly no comparisons were made with their first model in the secondstudy. Uncertainties remain regarding the results reported in Kanaun et al.


-7

-6

-5

-4

-3

-2

-1

0

-2 -1 0 1 2

Non-dimensional attenuation factor log(

α La) [-]

Non-dimensional longitudinal wave number of the matrix log(k0a) [-]

10% hard and heavy particles reinforcing a soft and light matrix.P-wave attenuation factor

SC (Kanaun et al., 2004)EMM (Kanaun et al., 2004)Dilute-Inclusionv0α0+nsγsca/2

Figure 3.10: Effective longitudinal attenuation factor for a highly contrasted elasticcomposite. Comparison between various numerical models.

(2004) and Kanaun and Levin (2005) for S-waves. The present comparisonsbetween numerical models will therefore be restricted to longitudinal acous-tic properties. Only the 10% composite is considered because of the singlescattering assumption of some models.

The predicted phase velocities (Fig. 3.9) are almost the same. Both self-consistent models are very close to each-other. The dilute-inclusion schemediffers from the others mostly in the mid-wave region. The improvement (notshown on the whole figure) proposed in Sec. 3.2.3 is useful in the long-waveregion only. The model becomes equivalent to the static M-T scheme.

Except for the self-consistent method in the long-wave region, all predictions ofthe effective attenuation factor (Fig. 3.10) are more or less equivalent in log-logscale. The dilute-inclusion scheme and its improved version (not shown) areidentical. They both predict a decrease of the attenuation factor for log(k0a) >1.75. This is due to the number of terms in the spherical series representationswhich is too low.

3.4.4 Steel-particle reinforced PMMA composite

This composite material has been studied by many authors, either experimen-tally (Kinra and Ker, 1983; Kinra, 1985) or numerically (Kanaun et al., 2004).It consists in spherical steel particles randomly distributed in a PMMA matrix.


cL αL cT αT ρ vf

[mm/µs] [nepers/mm] [mm/µs] [nepers/mm] [g/cm3] [%]

PMMA 2.63 0.02 @ 1MHz 1.32 not avail. 1.16 84.8Steel 5.94 negligible 3.22 not avail. 7.80 15.2

The radius of the steel particles is 0.55 mm.

Table 3.4: PMMA matrix reinforced with spherical steel particles. Acoustic propertiesof the materials and composite constitution after Kinra (1985).

Kanaun et al. (2004) compare the EMM predictions to the experimental dataof Kinra (1985) and report a volume fraction of 11.5% for the inclusions. Thelatter value is wrong. The correct concentration of 15.2% is given in Kinraand Ker (1983) and Kinra (1985). Their numerical predictions are howeverright: if the value of 11.5% were used the EMM wouldn’t have given the ex-pected static self-consistent estimates for ω → 0 (see Sec. 3.2.4). The otherproperties of the composite are summarized in Tab. 3.4. The transverse at-tenuation factors are missing for both components. As αL is negligeable forsteel, one can reasonably set αT = 0. Kinra and Ker (1983) assumed thatthe longitudinal attenuation factor for PMMA is a linear function of the fre-quency: αL = mLf , with mL = 0.02 nepers/mm-MHz. The material propertiesof PMMA used in Biwa et al. (2002) (see Tab. 3.3) give attenuation factors forwhich the ratio αT /αL equals approximately 10. We thus set αT = mT f , withmT = 0.2 nepers/mm-MHz. Kanaun et al. (2004) had no option – the EMMwas developed in elasticity only – but to assume that PMMA is purely elastic(i.e. αL = αT = 0).

Kinra (1985) measured the overall longitudinal phase velocity and attenuationfactor. Detailed information on the experimental set-up can be found in Kinraand Ker (1983); Kinra (1985) and references therein. The experimental data arereported on Figs. 3.11 and 3.12, respectively. Numerical predictions obtainedwith the EMM (Kanaun et al., 2004), the dilute-inclusion scheme of Sec. 3.2.3,its improved version and the additive attenuation model of Biwa et al. (2002)(Fig. 3.12 only) are also plotted on these figures. The EMM provides thebetter estimates for both properties. The three remaining schemes give similarpredictions which are far away from the measurements in the mid-wave region.This is not really surprising. The volume fraction of particles is rather hight.The independent scattering assumption – all models but the EMM are basedon this assumption – does not hold anymore. Multiple scattering is to a certainextent implicitly taken into account by the EMM. The improved version ofthe dilute-inclusion scheme is the more capable of all explicit model. This ispromising for further developments.

The experimental data show positive jumps in the phase velocity (Fig. 3.11)


at k0a ' 0.6, 1.2 and 1.5. The first one is more or less well captured by all nu-merical schemes. The others are not predicted, whatever the model. FollowingKinra (1985), these rapid changes in the phase velocity are related to resonanceeffects. The first one is the rigid-body-translation (or zeropole) resonance of therelatively heavy inclusions. The cutoff frequency at which it occurs separatesthe so-called low-frequency acoustical and high-frequency optical branches ofthe dispersion curve. The elementary model of Moon and Mow (1970) pro-vides an approximate value of the cutoff frequency Ωc for random particulatecomposites:

Ω2c =

9(1 − 2ν0)

5 − 6ν0

[ρ0

ρ1+v1v0

]

. (3.58)

In the above formula, νi, ρi and vi are the Poisson’s ratios, densities and vol-ume fractions of the matrix (i = 0) and the particles (i = 1), respectively.This model relies on rather restrictive assumptions: the inclusions are heavy(ρ1/ρ0 → ∞), perfectly rigid, homogeneously distributed, in dilute suspension;the frequency is small (ka 1). Kinra (1985) shows that the predicted cut-off frequency is however in good agreement with experimental data in manysituations. The theory of Moon and Mow (1970) is based on the equation ofmotion of a rigid inclusion. Kinra (1985) pointed out that ”(. . . ) Ωc is roughlythe same as the resonance frequency of a single sphere in an unbounded ma-trix (. . . ), slightly modified by modified by the concentration” of the particles.This explains why all these models, which are based on the one-particle scat-tering problem, predict the rapid change in the phase velocity at a frequencyapproximately equal to Ωc obtained with Eq. (3.58). The cutoff frequencyΩc = 0.575 calculated with Eq. (3.58) is shown on Fig. 3.11. A high increaseof the attenuation can also be observed at the cutoff frequency on Fig. 3.12.

None of the models predicts subsequent jumps in the phase velocity (Fig. 3.11)and oscillations of the attenuation factor (Fig. 3.12). Surprisingly, these jumpsoccur roughly at the same frequencies as for the periodic steel/PMMA compos-ite studied by Kinra and Ker (1983). All parameters are unchanged except thedistribution of the particles. Periodic composites exhibit additional resonancesat frequencies above Ωc. They are due to the periodicity of the lattice (Kinraet al., 1998). Indeed, some periodicity remains in the random composite dueto the preparation of the specimens. The samples actually consist in a pile upof layers. Each layer contains a planar random distribution of particles withone sphere in the thickness. Kinra (1985) advances other reasons for these ve-locity jumps. They are referred to as monopole, dipole, . . . resonance effects.The numerical schemes presented in this chapter cannot handle these othertypes of resonance. Models for periodic composites exist (see e.g. Toledanoand Murakami, 1987). The multipole resonance effects may be addressed withthe resonant scattering theory and its S-matrix – S stands for Strahlung orscattering – formalism, introduced in acoustics by Flax et al. (1978).


0.75

0.8

0.85

0.9

0.95

1

1.05

1.1

0 0.5 1 1.5 2 2.5 3

Relative phase velocity vL/vL0 [-]


Steel/PMMA composite: P-wave phase velocity

steel particles radius a=0.55mm

steel volume fraction v1=15.2%

cutoff frequency(Moon and Mow, 1970)

Experiment (Kinra, 1985)Dilute-InclusionDilute-Inclusion (improved)EMM (Kanaun et. al., 2004)

0 0.5

D-I

M-T

S-C

Figure 3.11: Effective phase velocity for P-waves in a Steel/PMMA composite. Com-parison between various numerical models and experimental data. The arrows are thevalues obtained with the static self-consistent (S-C), Mori-Tanaka (M-T) and dilute-inclusion (D-I) schemes. The cutoff frequency obtained with Eq. (3.58) is shown.

0

0.05

0.1

0.15

0.2

0.25

0 0.5 1 1.5 2 2.5 3

Non-dimensional attenuation factor

α La [-]


Steel/PMMA composite: P-wave attenuation factor

steel particles radius a=0.55mm

steel volume fraction v1=15.2%

matrix

Exp. Toneburst (Kinra, 1985)Exp. Spectroscopy (Kinra, 1985)Dilute-InclusionDilute-Inclusion (improved)v0α0+nsγsca/2EMM (Kanaun et. al., 2004)

Figure 3.12: Effective attenuation factor for P-waves in a Steel/PMMA composite.Comparison between various numerical models and experimental data.


cL αL cT αT ρ vf

[mm/µs] [nepers/mm] [mm/µs] [nepers/mm] [g/cm3] [%]

Epoxy 2.64 0.043 @ 1MHz 1.20 not avail. 1.202 94.6Lead 2.21 0.026 @ 2MHz 0.86 not avail. 11.3 5.4

The radius of the lead particles is 0.66 mm.

Table 3.5: Epon 828Z epoxy matrix reinforced with spherical lead particles. Acousticproperties of the materials and composite constitution after Kinra (1985).

3.4.5 Lead-particle reinforced epoxy composite

A low concentration lead/epoxy composite is considered. It has been experi-mentally studied by Kinra (1985). The properties for lead and epoxy are re-ported in Tab. 3.5. A linear dependence of the attenuation factors on thefrequency is assumed: α = mf . This gives mL = 0.043 nepers/mm-MHzand mL = 0.013 nepers/mm-MHz for epoxy and lead, respectively. A ratioαL/αT of about 5 is obtained for the matrix of the glass/epoxy compositestudied by Biwa et al. (2002). For lead, assuming a constant Poisson’s ra-tio, αL/αT = cL/cT . One finds mT = 0.215 nepers/mm-MHz for epoxy andmT = 0.0334 nepers/mm-MHz for lead.

The effective longitudinal phase velocity is reported on Fig. 3.13. The agree-ment between the experimental data of Kinra (1985) and the dilute-inclusionestimates is good. This is due to the low volume fraction (5.4%) of particles.The authors were not able to measure the phase velocity in the long-wave re-gion. This is related to the apparatus used for the measurements and radiusof the lead balls (0.66 mm). The rapid change in the phase velocity at cutofffrequency is well predicted. The elementary model of Moon and Mow (1970)gives Ωc = 0.371. No additional jumps at higher frequencies are observed. Ac-cording to Kinra (1985), there are two reasons for that. First, lead is heavyand the zeropole resonance is the dominant effect. Second, the epoxy matrixis highly attenuative and mask other types of resonance.

3.5 Conclusions

In this chapter, homogenization schemes for the propagation of plane wavesin viscoelastic particulate composites have been studied. A formulation thatunifies and generalizes various models of the literature (e.g. Sabina and Willis,1988; Sabina et al., 1993; Smyshlyaev et al., 1993; Kanaun et al., 2004; Kanaunand Levin, 2005) was proposed. New schemes have been developed. All thosemodels are based on the elementary problem of the scattering of a planemonochromatic wave by a single particle embedded in an infinite reference

Conclusions 97

0.75

0.8

0.85

0.9

0.95

1

1.05

1.1

0 0.5 1.0 1.5 2 3 4 5 6

Relative phase velocity vL/vL0 [-]


Lead/Epoxy composite: P-wave phase velocity

lead particles radius a=0.66mm

lead volume fraction v1=5.4%

cutoff frequency(Moon and Mow, 1970)

Experiment (Kinra, 1985)Dilute-Inclusion

0 0.35

D-I

Figure 3.13: Effective phase velocity for P-waves in a Lead/Epoxy composite. Com-parison between the dilute-inclusion model and experimental data. Note the changeof scale at k0a = 2. The arrow is the value obtained with the static dilute-inclusion(D-I) scheme. The cutoff frequency obtained with Eq. (3.58) is shown.

medium. They differ by the choice of the reference material and the actingwave. Connections with homogenization schemes in the static or quasi-staticregime has been shown. The model of Biwa et al. (2002) was introduced (seealso Biwa, 2001; Biwa et al., 2003). The latter is also based on the one-particlescattering problem. It is however more simple and allows to predict effectiveattenuation but not the speed of waves.

Special emphasis was put on the comparison of these homogenization modelswith each other. A large number of numerical simulations have been performed.Only experimental data were available for validation purposes. It is of particu-lar interest to compare our dilute-inclusion scheme with the approach of Biwaet al. (2002). Although both are based in part on the same assumptions (one-particle scattering problem, independent scattering), they lead to predictionswhich are significantly different in some cases. The experimental case studieswere not well-tailored to decide between them.

This opens the way for further research. A direct approach is required to verifyhomogenization schemes in the framework of wave propagation in viscoelasticparticulate composites (e.g. to better assess the validity of the independentscattering assumption). A finite element model would be very expensive unlessthe composite is periodic. Semi-analytical methods could be a workaround.


Attempts were initiated with the CIVA (2007) platform which is a simulationsoftware for non-destructive testing developed by the commissariat a l’energieatomique (CEA). This software however implements models based on the Bornapproximation which is indeed equivalent to the independent scattering as-sumption. Another solution could be found in the works of Dr. P. Laugier§

who develop a finite difference approach in the scope of non-destructive testingfor medical applications (e.g. to study the aging of bones). According to Dr.R. Marklein¶, the elastodynamic finite integration technique (EFIT) may beuseful address this issue of verification.

§Directeur de Recherche CNRS, Laboratoire d’imagerie parametrique, Universite Pierre

et Marie Curie, Paris, France¶Universitat Kassel, Kassel, Hessen, Germany

Solution of the one-particle scattering problem 99

Appendix A Solution of the one-particle scat-

tering problem

The one-particle scattering problem is the diffraction of a plane monochromaticwave on a single inclusion embedded in a infinite medium. An explicit solutionexists when the particle is of spherical shape and both media are made ofhomogeneous isotropic elastic materials. This solution is hereafter detailedfor longitudinal and transverse waves. The latter is also valid for viscoelasticmaterials thanks to the elastic-viscoelastic correspondence principle. In thefollowing, k and κ are longitudinal and transverse wave numbers, respectively,r is the radius of the particle, index 0 (resp. 1) refers to the surrounding media(resp. the inclusion).

Expressions for the displacement fields are given in a spherical coordinate sys-tem (r, φ, θ) with polar axis e3 (i.e. θ ∈ [0, π] and φ ∈ [0, 2π]) and origin thecenter of the particle. It is made use of spherical Bessel and Hankel functionsof the first kind and integer order. The former are defined as

jn(z) =

√π

2

Jn+ 12(z)

√z

, (A.1)

where Jn+ 12

are the Bessel functions of the first kind and half integer order.Spherical Hankel functions of the first kind are defined as hn = jn + iyn, wherethe spherical Bessel functions of the second kind yn are related to Yn+ 1

2, the

Bessel functions of the second kind and half integer order, in the same way asEq. (A.1). One finds:

hn(z) = jn(z) + i (−1)n+1j−(n+1)(z) . (A.2)

The solution also involves associated Legendre functions Pmn defined as

Pmn (x) = (−1)m(1 − x2)m/2 dm

dxmPn(x) , (A.3)

with Pn the Legendre polynomials. All these functions obey recurrence anddifferentiation rules which can be found in all mathematical handbooks.

Longitudinal wave. Consider an incident P-wave of unit amplitude prop-agating along e3 in the infinite media. It can be represented by a series ofspherical vector functions (see e.g. Kanaun et al., 2004, and references therein):

uI

=

∞X

n=0

−in+1(2n + 1)

k0

8

>

>

>

<

>

>

>

:

n d

drjn(k0r)Pn(cos θ)

o

er

+n jn(k0r)

r

d

dθPn(cos θ)

o

eθ

9

>

>

>

=

>

>

>

;

. (A.4)


The scattered and transmitted fields can also be presented in the form of seriesof spherical vector functions. They are given by

uS

=

∞X

n=0

8

>

>

>

>

<

>

>

>

>

:

n

"

and

drhn(k0r) + bn

n(n + 1)

rhn(κ0r)

#

Pn(cos θ)o

er

+n

"

anhn(k0r)

r+ bn

1

r

d

dr[rhn(κ0r)]

#

d

dθPn(cos θ)

o

eθ

9

>

>

>

>

=

>

>

>

>

;

(A.5)

and

uT

=

∞X

n=0

8

>

>

>

>

<

>

>

>

>

:

n

"

a′

n

d

drjn(k1r) + b

′

n

n(n + 1)

rjn(κ1r)

#

Pn(cos θ)o

er

+n

"

a′

n

jn(k1r)

r+ b

′

n

1

r

d

dr[rjn(κ1r)]

#

d

dθPn(cos θ)

o

eθ

9

>

>

>

>

=

>

>

>

>

;

, (A.6)

respectively. Scalars an, bn, a′n and b′n in Eqs. (A.5) and (A.6) are to befound from the conditions at inclusion/matrix interface. If a perfect bound isassumed, they are solution of the following linear system:

2

6

6

6

6

6

4

f21 (k0r) f2

2 (κ0r) −f11 (k1r) −f1

2 (κ1r)

f23 (k0r) f2

4 (κ0r) −f13 (k1r) −f1

4 (κ1r)

f25 (k0r) f2

6 (κ0r) −µ1µ0

f15 (k1r) −

µ1µ0

f16 (κ1r)

f27 (k0r) f2

8 (κ0r) −µ1µ0

f17 (k1r) −

µ1µ0

f18 (κ1r)

3

7

7

7

7

7

5

2

6

6

4

k0ank0bnk0a′

nk0b′n

3

7

7

5

= in+1

(2n + 1)

2

6

6

6

4

f11 (k0r)

f13 (k0r)

f15 (k0r)

f17 (k0r)

3

7

7

7

5

, (A.7)

where µ stands for the shear modulus. The f1m (m = 1, . . . , 8) are radial

functions:

f11 (kr) = njn(kr) − krjn+1(kr) , f1

2 (κr) = n(n+ 1)jn(κr) , f13 (kr) = jn(kr) ,

f14 (κr) = (n+ 1)jn(κr) − yjn+1(κr) ,

f15 (kr) = (n2 − n− (κr)2

2)jn(kr) + 2krjn+1(kr) ,

f16 (κr) = n(n+ 1) [(n− 1)jn(κr) − κrjn+1(κr)] ,

f17 (kr) = (n− 1)jn(kr) − krjn+1(kr) ,

f18 (κr) = (n2 − 1 − (κr)2

2)jn(κr) + κrjn+1(κr) . (A.8)

The same definitions hold for functions f2m (m = 1, . . . , 8) using hn instead of

jn. The dependence of f im (i = 1, 2; m = 1, . . . , 8) on order n is omitted for

clarity.

Transverse wave. Let an incident S-wave of unit amplitude and polarizatione1 propagate along e3 in the infinite media. It’s spherical vector functions seriesreads

uI

=

∞X

n=1

in(2n + 1)

n(n + 1)

8

>

>

>

>

>

>

>

>

>

<

>

>

>

>

>

>

>

>

>

:

cos ϕn

−

i

κ0rn(n + 1)jn(κ0r)P

1n(cos θ)

o

er

cos ϕn

jn(κ0r)P1

n(cos θ)

sin θ−

i

κ0r

d

dr[rjn(κ0r)]

d

dθP

1n(cos θ)

o

eθ

− sin ϕn

jn(κ0r)d

dθP

1n(cos θ) −

i

κ0r

d

dr[rjn(κ0r)]

P1n(cos θ)

sin θ

o

eϕ

9

>

>

>

>

>

>

>

>

>

=

>

>

>

>

>

>

>

>

>

;

.

(A.9)

Solution of the one-particle scattering problem 101

The scattered,

uS

=

∞X

n=1

8

>

>

>

>

>

>

>

>

>

<

>

>

>

>

>

>

>

>

>

:

cos ϕn

+

"

cnd

drhn(k0r) + en

n(n + 1)

rhn(κ0r)

#

P1n(cos θ)

o

er

cos ϕn

dnhn(κ0r)P1

n(cos θ)

sin θ+

"

cnhn(k0r)

r+ en

1

r

d

dr[rhn(κ0r)]

#

d

dθP

1n(cos θ)

o

eθ

− sin ϕn

dnhn(κ0r)d

dθP

1n(cos θ) +

"

cnhn(k0r)

r+ en

1

r

d

dr[rhn(κ0r)]

#

P1n(cos θ)

sin θ

o

eϕ

9

>

>

>

>

>

>

>

>

>

=

>

>

>

>

>

>

>

>

>

;

,

(A.10)

and transmitted,

uT

=

∞X

n=1

8

>

>

>

>

>

>

>

>

>

<

>

>

>

>

>

>

>

>

>

:

cos ϕn

+

"

c′

n

d

drjn(k1r) + e

′

n

n(n + 1)

rjn(κ1r)

#

P1n(cos θ)

o

er

cos ϕn

d′

njn(κ1r)P1

n(cos θ)

sin θ+

"

c′

n

jn(k1r)

r+ e

′

n

1

r

d

dr[rjn(κ1r)]

#

d

dθP

1n(cos θ)

o

eθ

− sin ϕn

d′

njn(κ1r)d

dθP

1n(cos θ) +

"

c′

n

jn(k1r)

r+ e

′

n

1

r

d

dr[rjn(κ1r)]

#

P1n(cos θ)

sin θ

o

eϕ

9

>

>

>

>

>

>

>

>

>

=

>

>

>

>

>

>

>

>

>

;

,

(A.11)

fields involve coefficients cn, dn, en, c′n, d′n and e′n which have to be determinedat particle boundary. Under the same assumptions as for P-waves, two linearsystems are obtained:

2

6

6

6

6

6

4

f21 (k0r) f2

2 (κ0r) −f11 (k1r) −f1

2 (κ1r)

f23 (k0r) f2

4 (κ0r) −f13 (k1r) −f1

4 (κ1r)

f25 (k0r) f2

6 (κ0r) −µ1µ0

f15 (k1r) −

µ1µ0

f16 (κ1r)

f27 (k0r) f2

8 (κ0r) −µ1µ0

f17 (k1r) −

µ1µ0

f18 (κ1r)

3

7

7

7

7

7

5

2

6

6

4

κ0cnκ0enκ0c′nκ0e′n

3

7

7

5

=in+1(2n + 1)

n(n + 1)

2

6

6

6

4

f12 (κ0r)

f14 (κ0r)

f16 (κ0r)

f18 (κ0r)

3

7

7

7

5

(A.12)

and2

4

f23 (κ0r) −f1

3 (κ1r)

f27 (κ0r) −

µ1µ0

f17 (κ1r)

3

5

»

dnd′

n

–

= −

in(2n + 1)

n(n + 1)

"

f13 (κ0r)

f17 (κ0r)

#

. (A.13)


Appendix B Expressions for tensors λp and Λp

Recall that the behavior of the composite is idealized as follows: each inclusionbehaves as isolated in some reference medium and subjected to some incidentwave propagating in the reference medium. For each reinforcing phase p, tensorλp (resp. Λp) relates the expectation value of the displacement (resp. deforma-tion) field in a representative inclusion to the displacement (resp. deformation)field in the reference medium. If the inclusion is of spherical shape and theacting wave is a plane wave, the solution of one-particle scattering problem(Appendix A) allows to find explicit forms of λp and Λp. Their expressions aregiven hereafter. In order to not overload the notations, a single inclusion phaseis assumed and the upper-script p is omitted. Without any restrictions thematrix is taken for the reference medium. With r the radius of the particles,variables xi = kir and yi = κir are defined for i = 0, 1, where index 0 (resp.1) refers to the matrix (resp. the inclusion). They correspond to longitudinaland transverse non-dimensional wave numbers, respectively.

In the Cartesian coordinate system, tensor λ has the following representation:

λij = hδij ; i, j = 1, 2, 3 . (B.1)

Scalar h depends on the type of wave. For P-waves, h = hL takes the form

hL = −3∞

X

n=0

(−i)n+1

(

»

f13 (x1)f

11 (x0)

x20

+ gn(x1, x0)

–

(k0a′n) +

f12 (y1)f

13 (x0)

x20

(k0b′n)

)

,

(B.2)where gn is defined as

gn(x1, x0) =x1jn+1(x1)jn(x0) − x0jn+1(x0)jn(x1)

x21 − x2

0

. (B.3)

In the case of S-waves, h = hT reads

hT = −3

2

∞X

n=1

(−i)n+1n(n+ 1)

(

f13 (x1)f

13 (y0)

y20

(κ0c′n) − ign(y1, y0)d

′n

+

»

f13 (y1)f

14 (y0)

y20

+ gn(y1, y0)

–

(κ0e′n)

)

. (B.4)

The components of tensor Λ are given by

Λpijkl =

[

H1L +H2

L(δil −1

3)

]

δijδkl +HT

2(1 − δijδkl)δikδjl , (B.5)

where

H1L = −

∞X

n=0

(−i)n+1 x21

x20

gn(x1, x0)(k0a′n) (B.6)

Expressions for tensors λp and Λp 103

and

H2L = −9

2

∞X

n=0

(−i)n+1

( "

f11 (x1)

`

f12 (x0) − 2f1

1 (x0)´

x40

+f13 (x1)f

16 (x0)

x40

+

2

3

x21

x20

gn(x1, x0)

–

(k0a′n) +

"

f14 (y1)f

16 (x0)

x40

+f12 (y1)

`

f12 (x0) − 2f1

1 (x0)´

x40

#

(k0b′n)

)

(B.7)

and

HT = −3

2

∞X

n=1

(−i)n+1n(n+ 1)

(

»

2f11 (x1)f

17 (y0) + f1

3 (x1)f18 (y0)

y40

+f13 (x1)f

13 (y0)

y20

–

(κ0c′n) − i

»

f13 (y1)f

17 (y0)

y20

+ gn(y1, y0)

–

d′n

+

»

2f12 (y1)f

17 (y0) + f1

4 (y1)f18 (y0)

y40

+f13 (y1)f

14 (y0)

y20

+ gn(y1, y0)

–

(κ0e′n)

)

. (B.8)

The expressions for hL, hT , H1L, H2

L and HT turn out to be equivalent to theone reported in Kanaun et al. (2004) to what we believe are typos:

hL : brackets are missing in front of a′n in Eq. (Kanaun et al., 2004, 4.13).

hT : Eq. (B.4) is equivalent to Eq. (Kanaun et al., 2004, 4.17)

H1L : minus sign is missing in Eq. (Kanaun et al., 2004, 4.15).

H2L : Eqs. (Kanaun et al., 2004, 4.15 and 4.16) are equivalent to Eq. (B.7)

provided the observation regarding the sign of H1L is taken into account.

HT : Eqs. (B.8) and (Kanaun et al., 2004, 4.18) are equivalent.

Computing these tensors is a tedious task. It requires the integration of threeassociated Legendre functions:

1∫

−1

Pm1

j1(z)Pm2

j2(z)Pm3

j3(z)dz . (B.9)

Formulae can be found in Mavromatis and Alassar (1999).

Expressions in the static regime. The static fields are obtained for n 6 2in the series representations. Higher order terms are negligeable for ω → 0.


The limit values of the solutions – only the transmitted fields are required toevaluate λ and Λ – of systems (A.7), (A.12) and (A.13) read

k0a′0 ∼ −3iy2

0

3y21 + (1 − µ1

µ0) (4x2

1 − 3y21)

k0b′0 ∼ −k0a

′0

for n = 0,

k0a′1 ∼

15y21 + 10(y2

0 − y21) + (1 − µ1

µ0)y2

1

5y21 + (1 − µ1

µ0) (4x2

1 − 3y21)

x0

x1

k0b′1 ∼

−5(y20 − y2

1) + (1 − µ1

µ0)(6x2

1 − 5y21

)

5y21 + (1 − µ1

µ0) (4x2

1 − 3y21)

x0

y1

κ0c′1 ∼

−5(y20 − y2

1) + (1 − µ1

µ0)y2

1

5y21 + (1 − µ1

µ0)(4x2

1 − 3y21)

y0x1

d′1 ∼ 3

2iy0y1

κ0e′1 ∼ 1

2

15y21 + 5(y2

0 − y21) + (1 − µ1

µ0)(12x2

1 − 10y21)

5y21 + (1 − µ1

µ0)(4x2

1 − 3y21)

y0y1

for n = 1, and

k0a′2 ∼

75i[

7(2y21 + 3y2

0) + (1 − µ1

µ0)(2y2

1 − 14y20)]

y20

[

15 − 2(1 − µ1

µ0)(

2x20

y20

+ 3)] [

35 − (1 − µ1

µ0)(

19y21

y20− 24

x21

y20

)]x2

0

x21

k0b′2 ∼

25i[

7(3y21 − 3y2

0) + (1 − µ1

µ0)(24x2

1 + 14y20 − 21y2

1)]

y20

[

15 − 2(1 − µ1

µ0)(

2x20

y20

+ 3)] [

35 − (1 − µ1

µ0)(

19y21

y20− 24

x21

y20

)]x2

0

y21

κ0c′2 ∼

25i[

7(3y21 − 3y2

0) + (1 − µ1

µ0)(3y2

1 + 14x20)]

y20

[

15 − 2(1 − µ1

µ0)(

2x20

y20

+ 3)] [

35 − (1 − µ1

µ0)(

19y21

y20− 24

x21

y20

)]y20

y21

d′2 ∼ 1

6

−25

5 − (1 − µ1

µ0)

y20

y21

κ0e′2 ∼ 1

6

25i[

21(3y21 + 2y2

0) − (1 − µ1

µ0)(28x2

0 − 72x21 + 63y2

1)]

y20

[

15 − 2(1 − µ1

µ0)(

2x20

y20

+ 3)] [

35 − (1 − µ1

µ0)(

19y21

y20− 24

x21

y20

)]y20

y21

for n = 2. They are obtained using rational function approximations of spher-

Expressions for tensors λp and Λp 105

ical Bessel and Hankel functions. Using the same technique, one finds

hL ∼ 1

3

x1

x0k0a

′1 + 2

y1x0k0b

′1

∼ 1

hT ∼ 1

3

x1

y0κ0c

′1 + 2

y1y0κ0e

′1

∼ 1

and conclude (Eq. (B.1)) that λ ∼ 1 for either longitudinal or transverse waves.Rearranging the terms in Eq. (B.5) once the limit values

H1L ∼ i

3

x21

x20

k0a′0 ∼ x2

1

3y21 + (1 − µ1

µ0) (4x2

1 − 3y21)

y20

x20

H2L ∼ − i

5

x2

1

x20

k0a′2 + 3

y21

x20

k0b′2

∼ 15x20

15y20 − 2(1 − µ1

µ0) (2x2

0 + 3y20)

y20

x20

HT ∼ −2i

5

x2

1

y20

(κ0c′2) + 3

y21

y20

(κ0e′2)

∼ 15x20

15y20 − 2(1 − µ1

µ0) (2x2

0 + 3y20)

y20

x20

have been introduced leads to Λ ∼ H (C0,C1), with Ci (i = 0, 1) the (long-term) elastic stiffness tensors of the materials.


Appendix C Notations and units


Ω representative volume elementΩp region occupied by phase p

UΩp characteristic function of Ωp: UΩp(x) =

1 if x ∈ Ωp

0 if x 6∈ Ωp

vp volume fraction of phase pρp density of material p kg m−3

t time sω angular frequency rad s−1

i imaginary unit number i2 = −1s complex variable in the Laplace-Carson domain rad s−1

Lc [ ], ˆ Laplace-Carson transform Lc [f ] (s) = f(s) = s

∞Z

0

f(t)estdt

F [ ] Fourier transform F [f ] (q) =

Z

f(x)e−iq·x

<, = real and imaginary partsE [ ], P[ ] expectation value, probabilityd differential∂ partial derivative∇ gradient∇· divergence· scalar product

x position mu displacement m

ε strain tensorσ stress tensor Pa

Notations and units 107


Gp relaxation tensor of material p PaG relaxation tensor Pa

G overall relaxation tensor PaI fourth order identity tensor1 second order identity tensorρ overall density tensor kgm−3

Ap, ap concentration tensorsΛp, λp solution tensors for the one-particle scattering problem

k, κ longitudinal and transverse wave numbers m−1

αL, αT longitudinal and transverse attenuation factors nepers/mvL, vT longitudinal and transverse phase velocities m s−1

γsca, γabs scattering and absorption cross-sections m2

Chapter 4

Finite element acousticsimulations

This chapter addresses the issue of modeling the acoustic field in fluids in-teracting with inclusion reinforced composite walls. The latter may be eithersubstituted by appropriate boundary conditions – the transmitted field is how-ever lost – or modeled as bulk materials. In both cases, the homogenizationschemes of Chaps. 1, 2 and 3 are of major interest. They provide a very cost-effective way of specifying the acoustic properties of an equivalent homogeneouswall. Choosing the model is critical. If viscosity is neglected (Chap. 1) and/orif scattering is ignored (Chap. 2), the ability for the wall to attenuate soundwaves may be underestimated. Linear acoustics and non dissipative fluids ini-tially at rest are assumed. Section 4.1 recalls how the Helmholtz equation isobtained from the fundamental laws of fluid dynamics. Section 4.2 introducesmajor concepts of acoustics such as acoustic impedance. Boundary conditionsfor locally reacting walls are derived from these notions in Sec. 4.3. A discon-tinuous Galerkin (DG) formulation of the Helmholtz equation is proposed inSec. 4.4. A specific method, known as the internal penalty (IP) method, isimplemented in 2D and for axisymmetric problems. The numerical simulationsof Sec. 4.5 involve interfaces between fluids (air or water) and solids (porousmaterials, particulate composites). Attention is drawn on the modeling thewalls.

110 Finite element acoustic simulations

4.1 Equation for the acoustic pressure

With the assumptions of non-dissipative fluids, small perturbations and nomean flow, the equation for the propagation of acoustic pressure p in homoge-neous fluids reads (see e.g. Morse and Ingard (1986) and Bruneau (1998))

[1

c20

∂2

∂t2− ∆

]

p = −f , (4.1)

where f represents the source term and c0 is the speed of sound in the fluid.

A Fourrier transform of Eq. (4.1) leads to the Helmholtz equation in thefrequency domain (the same notation is used for p and f and their Fourriertransform)

[∆ + k2

0

]p = f , (4.2)

where k0 = ωc0

is the wave number in the fluid, with ω the angular frequency.

Further developments (e.g. for dissipative fluids) can be found in Morse andIngard (1986) and Bruneau (1998).

4.2 Acoustic impedance

Acoustic impedance is a frequency dependent material property. Distinctionhas to be observed between the characteristic acoustic impedance of a medium(e.g. air) and the specific impedance of an acoustic component (e.g. a wall).The former represents the resistance to the propagation of sound in the medium.Together with the characteristic impedance, the latter determine the reactionof the wall to an incident acoustic wave (e.g. reflection/transmission of acousticpressure, sound intensity absorption). The specific acoustic impedance modelsthe behavior of a boundary, getting rid of modeling wave propagation on theother side of it. This approach is a good approximation when the normaldisplacement of a point on the interface depends only on the acoustic pressureat this point. Walls of this kind are referred to as locally reacting. For wallswith non-local (or extended) reaction (e.g. membranes, laminated structures)the acoustic field is fully coupled with the vibration state of the interface.

4.2.1 Characteristic impedance of acoustic media

The characteristic impedance Z0 of a medium (e.g. air, water) is a materialproperty defined as

Z0 = ρ0c0 , (4.3)

Acoustic impedance 111

Z0 ρc00=

rigid surface

Z

V

vn

pfluid

Figure 4.1: Interface between a fluidmedium with characteristic acousticimpedance Z0 and a wall of specificacoustic Z.

where ρ0 is the density of the medium and c0 is the sound speed. It repre-sents the resistance of the medium to the propagation of waves. For a planewave propagating in a medium with characteristic impedance Z0, the acousticpressure p is proportional to the particle velocity v:

p = Z0v . (4.4)

The characteristic impedance of air at room temperature – c0 = 343.2 m/s andρ0 = 1.204 kg/m3 at 20 C and 1 atm – is Z0 = 413.2 Ns/m3. Sound speedand density are much higher in water. Its characteristic acoustic impedance isabout 3400 times that of air.

4.2.2 Specific impedance of acoustic components

The specific acoustic impedance of an acoustic component (e.g. a locally react-ing wall) is the ratio of acoustic pressure to particle velocity at it’s connectionpoint. Consider (Fig. 4.1) a fluid domain bounded by a rigid surface coveredwith a material of specific impedance Z. The acoustic pressure p and normalparticle velocity v at fluid/material interface are related through

Z =p

(v − V ) · n , (4.5)

where n is the outgoing normal (w.r.t. to the fluid domain) and V is anarbitrary velocity imposed to the rigid surface. The specific acoustic impedancerepresents the opposition to the flow of sound through the surface. It is ingeneral a complex number Z = R+ iX with a resistive part, R, and a reactivepart, X .

4.2.3 Absorption coefficient

Sound waves are reflected/transmitted at interface between two media withdifferent acoustic impedances. The ratios of reflected (pr) and transmitted (pt)pressures to the incident pressure (pi) are frequently expressed as the pressure


reflection coefficient, R , and pressure transmission coefficient, T :

R =pr

pi, T =

pt

pi. (4.6)

For a sound wave propagating in a media with acoustic impedance Z0, thepressure reflection coefficient at boundary with a material of acoustic impedanceZs reads

R =Zs − Z0

Zs + Z0. (4.7)

The pressure transmission coefficient is T = 1 −R.

The ratio of absorbed energy to the incident energy at interface is called theabsorption coefficient. It is given by

α = 1 − |R|2 , (4.8)

where R is the pressure reflection coefficient. The fraction of the incident waveintensity that is reflected is 1 − α.

Consider the interface between a fluid with characteristic impedance Z0 = ρ0c0and an acoustic component of specific impedance Zs = Rs + iXs. If the latteris purely reactive, α = 0 and no energy is absorbed. If it is purely resistive, theabsorption coefficient is less than 1:

α = 1 −(Rs − ρ0c0Rs + ρ0c0

)2

. (4.9)

4.3 Boundary conditions

At interface, the specific acoustic impedance links the acoustic pressure andnormal particle velocity in a linear way. This provides a means to specifyboundary conditions for locally reacting walls. They are easier to handle in thefrequency domain than in the time domain.

Frequency dependent boundary conditions. At the fluid/wall boundary(away from any sources) the Euler equation reads

v · n = − 1

iωρ

∂p

∂n. (4.10)

Using this expression of v ·n in the definition of the specific acoustic impedance(Eq. (4.5)) leads to a generic form of boundary conditions for locally reactingwalls:

∂p

∂n+ ik0βp = −iωρV · n . (4.11)

Discontinuous Galerkin formulation 113

In the above formula, β = Z0

Z is the normalized (w.r.t. the characteristic

admittance of the fluid Z−10 ) admittance of the wall, k0 = ω

c0is the wave

number in the fluid. Boundary conditions of this kind are commonly calledRobin or mixed boundary conditions. The specific impedance of a perfectlyrigid material is infinite (i.e. β = 0) and Eq. (4.11) becomes a Neumannboundary condition:

∂p

∂n= −iωρV · n . (4.12)

These boundary conditions are homogeneous when no velocity is applied to thewall (V = 0). Homogeneous Dirichlet boundary conditions,

p = 0 , (4.13)

hold at interface between a dense fluid and a gaseous medium, its reaction beingin general negligible.

Time dependent boundary conditions. In the time domain Eqs. (4.13),(4.12) and (4.11) become:

- Homogeneous Dirichlet condition

p = 0 (4.14)

- Neumann condition∂p

∂n= −ρ∂V

∂t· n (4.15)

- Robin condition∂p

∂n+

1

c

∂β

∂t∗ p = −ρ∂V

∂t· n (4.16)

A convolution appears in the boundary condition of mixed type. Analogybetween Fourier and Z transforms allows to circumvent this issue. The convo-lution is substituted by a recurrence relation. More details on this workaroundcan be found in Ozyoruk and Long (1997) and Ozyoruk et al. (2001).

4.4 Discontinuous Galerkin formulation

The Helmholtz equation is rewritten to obtain a pressure-velocity formulation.The boundary integral terms that appear in the local weak form are approxi-mated using so-called numerical fluxes. Attention is drawn on a specific choicefor these fluxes. It leads to the Internal Penalty (IP) method. A convergenceanalysis is performed on 2D and axisymmetric (2.5D) test cases.


4.4.1 Local weak form

Let the Helmholtz equation be rewritten as

σ = ∇uk20u+ ∇ · σ = −f , (4.17)

where u is the acoustic pressure. Define test functions τ and v for σ and u,respectively. For each element K with boundary ∂K, the local weak form,

< σ · τ >K= − < u∇ · τ >K + un · τ ∂K , (4.18)

< k2uv >K − < σ · ∇v >K=< fv >K − σ · nv ∂K , (4.19)

is obtained after integration, using the divergence theorem.

4.4.2 Numerical fluxes

Numerical fluxes are introduced to handle discontinuity of the finite elementsolution across the elements’ boundaries. Arnold et al. (2002) propose familiesof numerical fluxes putting various discontinuous Galerkin methods together.

Consider an elementK of the triangulation. On each edge e ∈ ∂K the numericalsolutions uh and σh are approximated by he

u and heσ, respectively. A family of

numerical fluxes is given by

heu = uh + γe · [[uh]] , (4.20)

heσ = ∇uh − ηe[[uh]] + βe[[∇uh]] , (4.21)

where and [[]] are the average and jump operators, respectively. None ofthese numerical fluxes depend on σh. As a consequence, variable σh can belocally eliminated and the system is sparser.

4.4.3 Internal Penalty method

The parameters involved in heu and he

σ determine the DG method. Settingγe = βe = 0 generates the Internal Penalty method for which Eqs. (4.18) and(4.19) become

< σh · τh >K= − < uh∇ · τ h >K + uhn · τh ∂K (4.22)

and

< k2uhvh >K − < σh · ∇vh >K=< fvh >K − (∇uh − ηe[[uh]]) · nvh ∂K , (4.23)

respectively. We will focus on the latter formulation in the sequel.


Boundary conditions. The above expressions are written for an interiorelement K. A neighbor element exists along each edge e ∈ ∂K. The averagevalues on e and jumps across e are determined from the solutions on each sideof e. For boundary edges, uh, [[uh]] and ∇uh have to be specified accordingto the type of boundary condition (Sec. 4.3):

- Dirichlet boundary condition u = u

uh = u ; [[uh]] = 0 ; ∇uh = ∇uh . (4.24)

- Neumann boundary condition ∂u∂n

= g

uh = uh ; [[uh]] = 0 ; ∇uh · n = g . (4.25)

- Robin boundary condition ∂u∂n

+ λu = g

uh = uh ; [[uh]] = 0 ; ∇uh · n = g − λuh . (4.26)

4.4.4 Numerical implementation

The IP method is implemented in 2D and 2.5D. The axisymmetric formulationin the (r, z) plane,

< rσ · τ >K= − < ru∇ · τ >K − < ruτ · ~er

r>K

︸︷︷︸

additional term

+ run · τ ∂K , (4.27)

< rk2uv >K − < rσ · ∇v >K=< rfv >K − rσ · nv ∂K , (4.28)

differs from the 2D case by an additional term and the presence of the vari-able r in all integrands. The computational domain is meshed with triangularelements. Linear or quadratic shape functions are used to interpolate boththe acoustic pressure uh and its gradient σh at the nodes of the triangulation.These approaches are commonly referred to as conform P1-P1 (linear shapefunctions) and conform P2-P2 (quadratic shape functions).

4.4.5 Convergence analysis

The rate of convergence of the IP method is analyzed on two one-dimensionalproblems. The first consists in a strip of plane of infinite length and finitewidth. The acoustic pressure is imposed (Dirichlet BC) on one side of the bandA complete reflection (homogeneous Neumann BC) is assumed on the otherside. The second can be viewed as an infinitely long cylinder embedded in anunbounded (Robin BC) medium. The cylinder is perfectly rigid and vibrates(Neumann BC) in the direction of its radius.


1e-11

1e-10

1e-09

1e-08

1e-07

1e-06

1e-05

1e-04

1e-03

1e-02

1e-01

1 0.1 0.01 0.001

Error

h

Convergence rate for the pressure (L2 norm of u)

1

2.5

1

3.5

linearquadratic

Figure 4.2: Convergence rate of the IP method (2D) for the acoustic pressure. Linearand quadratic shape functions on triangular elements.

Problem #1. The following boundary value problem is solved with the 2Dformulation on a structured mesh of triangular elements:

[∂2

∂x2+ (k0L)2

]

p(x) = 0 ∀x ∈]0, 1[ (4.29)

p(x) = 1 for x = 0 (4.30)

∂p

∂x(x) = 0 for x = 1 (4.31)

The non-dimensional parameter k0L – L is the width of the strip – is set to 2π.This corresponds to one wave-length in the computational domain.

The mesh has one element of size hmin in the y-direction. Homogeneous Neu-mann BCs are imposed on y = 0 and y = hmin. Mesh refinement is performedalong the x-direction from h = 1 to h = hmin.

The parameter ηe which weakly enforces continuity has dimension of 1/length-unit. The numerical simulations are performed with a constant value of ηe.Following Arnold et al. (2002), the value of 3/hmin – 3 is for the number ofedges of the triangular elements – is chosen.

Linear and quadratic shape functions are successively used to interpolate theacoustic pressure and its gradient on the triangulation. The rate of convergence(L2 norm) is reported on Fig. 4.2, for the acoustic pressure, and on Figs. 4.3and 4.4, for both components of the acoustic velocity (variable σ indeed, but


1e-08

1e-07

1e-06

1e-05

1e-04

1e-03

1e-02

1e-01

1e-00

1 0.1 0.01 0.001

Error

h

Convergence rate for the velocity (L2 norm of σx)

1

1.5

1

2.5

linearquadratic

Figure 4.3: Convergence rate of the IP method (2D) for the acoustic velocity in thedirection of sound flow. Linear and quadratic shape functions on triangular elements.

1e-08

1e-07

1e-06

1e-05

1e-04

1e-03

1 0.1 0.01 0.001

Error

h

Convergence rate for the velocity (L2 norm of σy)

1

1.5

1

2.5

linearquadratic

Figure 4.4: Convergence rate of the IP method (2D) for the acoustic velocity in thetransverse direction of sound flow. Linear and quadratic shape functions on triangularelements.


1e-11

1e-10

1e-09

1e-08

1e-07

1e-06

1e-05

1e-04

1e-03

1e-02

1e-01

1 0.1 0.01 0.001

Error

h

Convergence rate for the pressure (L2 norm of u)

1

2.5

1

3.5

linearquadratic

Figure 4.5: Convergence rate of the IP method (2.5D) for the acoustic pressure.Linear and quadratic shape functions on triangular elements.

they are proportional). An asymptotic rate of 2.5 is reached for the pressurewith linear shape functions. It falls down to 1.5 for the velocity. We believethat this is due to the numerical flux he

σ which is independent of σh. A rateof convergence of 2.5 should be reached with he

σ = σh − ηe[[uh]] instead ofhe

σ = ∇uh − ηe[[uh]]. This will however decrease the sparsity of the matrix.These rates of convergence are increased by 1 with quadratic shape functions.

Problem #2. The acoustic pressure outside the pulsing cylinder obeys thefollowing equation and conditions:

[

r2∂2

∂r2+ r

∂

∂r+ (k0r1)

2r2]

p(r) = 0 ∀r ∈]1,∞[ (4.32)

∂p

∂r(r) = 1 for r = 1 (4.33)

Sommerfeld conditions for r → ∞ (4.34)

The change of variable x = (k0r1)r, where r1 is the radius of the cylinder,transforms Eq. (4.32) into the Bessel differential equation of order n = 0. It’ssolution is given by p(x) = AJ0(x) +BY0(x), where J0 (resp. Y0) is the Besselfunction of the first (resp. second) kind and order n = 0. The infinite domainwith Sommerfeld conditions at infinity is replaced with a computational interval


1e-08

1e-07

1e-06

1e-05

1e-04

1e-03

1e-02

1e-01

1 0.1 0.01 0.001

Error

h

Convergence rate for the velocity (L2 norm of σr)

1

1.5

1

2.5

linearquadratic

Figure 4.6: Convergence rate of the IP method (2.5D) for the acoustic velocity in thedirection of sound flow. Linear and quadratic shape functions on triangular elements.

1e-08

1e-07

1e-06

1e-05

1e-04

1 0.1 0.01 0.001

Error

h

Convergence rate for the velocity (L2 norm of σz)

1

1.5

1

2.5

linearquadratic

Figure 4.7: Convergence rate of the IP method (2.5D) for the acoustic velocity in thetransverse direction of sound flow. Linear and quadratic shape functions on triangularelements.


of finite extent ]1, r2

r1[ with a mixed condition,

[∂

∂r+ ik0β

]

p(r) = 0 , (4.35)

of unit normalized admittance (β = 1) at the artificial boundary. With r2

r1= 4

and k0r1 = 2π, the size of the computational domain equals 3 wave-lengths.

The approach is the same as for the previous case regarding the interpolation,the mesh refinement and the DG parameter ηe. The axisymmetric formulation(Eqs. (4.27) and (4.28)) is used instead of the 2D one.

The rate of convergence is reported on Figs. 4.5, 4.6 and 4.7 for the acousticpressure, the radial velocity and the axial velocity, respectively. The sameconclusions as for the 2D formulation hold for the axisymmetric case.


4.5 Numerical simulations

The transmission loss (TL) in cylindrical silencers is analyzed with the IPmethod. Three examples are shown. First, we consider rigid walls. Next, theexpansion chamber is lined with an absorbing layer. The lining material iseither modeled as an equivalent fluid medium or substituted by a boundarycondition. The predicted TL are in good agreement with experimental dataand other numerical results from the literature. A coupled fluid/solid problemis considered afterwards. The reflection loss (RL) of a viscoelastic compositelayer coating and elastic wall is studied. The numerical simulations link up allthe modeling aspects exposed along the chapters of this work.

4.5.1 Application to silencers

The transmission loss is one of the characteristic design parameter for silencers.It is defined as the ratio of the incident sound power to the transmitted soundpower. TL values are useful when comparing the performance of one silencergeometry to the next. Transmission loss can be measured or calculated inseveral ways (see e.g. Bilawchuk and Fyfe, 2003, and references therein). Aconvenient method is the 3-Pole method. The latter is illustrated on Fig. 4.8.It requires the knowledge (measured or computed) of the acoustic pressure –other methods may need velocity data – at 3 locations along the silencer. TheTL is given by

TL = 20 log

∣∣∣∣

pi

pe

∣∣∣∣, (4.36)

where pi and pe are the incoming and exiting portions of rms∗ acoustic pressurewave, respectively. The incoming rms sound pressure is computed with help ofthe data recorded at the two first locations,

pi = rmsp1 − p2e

−ik0x12

1 − e−ik0x12, (4.37)

∗root mean square

p1

p2

p3

x12

source anechoicend

Figure 4.8: 3-Pole method : the transmission loss is computed with help of the acousticpressure at 3 locations along the silencer.


where x12 = x2 − x1 is the distance between the recording positions. Theprocedure requires an anechoic chamber at the termination of the silencer. Theoutgoing rms pressure is simply

pe = rms p3 , (4.38)

with p3 the acoustic pressure near the end of the silencer. It is worth notingthat this procedure is based on plane wave propagation.

A silencer with rigid walls. Bilawchuk and Fyfe (2003) modeled the trans-mission loss in a cylindrical single expansion chamber with the vibro-acousticfinite element code SYSNOISEr (2003). The same silencer is studied with theIP method.

The dimensions of the silencer can be found on Fig. 4.9. The walls are perfectlyrigid. At the outlet, the anechoic chamber is replaced by a boundary with unitnormalized admittance (homogeneous Robin BC). A unit velocity is imposedat the inlet. The frequency ranges from 10 Hz to 1200 Hz. The silencer is filledwith air of characteristic acoustic impedance Z0 = 416.5 rayls (c0 = 340.0 m s−1,ρ0 = 1.225 kgm−3).

The unstructured mesh consists in 1230 second-order triangular elements. Theirdiameter corresponds to approximately 28 elements per wave-length for thehighest frequency. The simulations are performed with ηe = 600.

The SYSNOISE and IP predictions are compared together on Fig. 4.9. Ex-perimental data after Bilawchuk and Fyfe (2003) are also reported. Both finiteelement estimates of the TL are almost identical and in very good agreementwith the measurements.

A silencer with polyester lining. Cylindrical single expansion chamberswith polyester lining have been studied numerically and experimentally bymany authors (see e.g. Wu et al., 2002; Lou et al., 2003; Mehdizadeh andParaschivoiu, 2005). One of the objectives followed by these authors is to showthe influence of modeling the liner either as a bulk material or with an appro-priate boundary condition. The numerical approaches are of two kinds: three-dimensional finite elements (Mehdizadeh and Paraschivoiu, 2005) and boundaryelement methods (Wu et al., 2002; Lou et al., 2003). A representative silenceris analyzed with the IP method.

The polyester liner is considered as a porous material. The model of Delanyand Bazley (1970) allows to model a porous material as an equivalent fluid witha complex characteristic acoustic impedance. The empirical equations for the


0

2

4

6

8

10

12

14

16

0 100 200 300 400 500 600 700 800 900 1000 1100 1200

Transmission Loss [dB]

Frequency [Hz]

Experiment (Bilawchuck and Fyfe, 2003)FE 3D (Bilawchuck and Fyfe, 2003)DG (IP method)

Transmission Loss in a cylindrical single expansion chamber (rigid walls)

0.453m

0.5m0.5m

0.0745m 0.02625m

Figure 4.9: Transmission loss in a cylindrical silencer with rigid walls. Compari-son between measured data, 3D finite element (SYSNOISE) results and predictionsobtained with the axisymmetric implementation of the IP method.

impedance Zc and the propagation constant Γc are

Zs = ρ0c0[1 + 0.0571X−0.754 − i 0.087X−0.732

], (4.39)

Γc = iω

c0

[1 + 0.0978X−0.700 − i 0.189X−0.595

], (4.40)

where X = ρ0fσ . The flow resistivity σ is in Pa s/m2, f is the frequency in Hz

and ω = 2πf is the angular frequency. Air in the pores has a characteristicimpedance Z0 = ρ0c0. The speed of sound in the equivalent fluid and its densityread

cc =iω

Γcand ρc = −i

ZcΓc

ω, (4.41)

respectively. The model involves a single parameter for the porous material:its flow resistivity. For polyester, Mehdizadeh and Paraschivoiu (2005) reporta value of 16000 Pa s/m2.

In order to avoid to model wave propagation in the liner, the air/polyesterinterface is substituted for a boundary condition. A question arises: how tomodel the specific acoustic impedance of the wall? Neither Mehdizadeh andParaschivoiu (2005) nor Lou et al. (2003) give any information on the proce-dure they followed. A simple model is reported in Appendix A. The specific


Figure 4.10: Snapshot of the finite element meshes (half parts only, same scaling) forthe cylindrical silencers with polyester lining. Left: the liner is meshed (1300 second-order triangles, 2906 nodes). Right: the liner is replaced by the Robin boundarycondition (734 second-order triangles, 1631 nodes).

impedance is determined from the characteristic impedance, the wave numberand the thickness of the medium layer. The geometrical parameters requiredby Eq. (A.8) are r1 = 3 in and r2 = 4 in. Characteristic impedance and wavenumber for the polyester are those obtained with the Delany and Bazley (1970)model.

The dimensions of the silencer are reported on Figs. 4.11 and 4.12 for thebulk-reacting and locally-reacting approaches, respectively. The radius of thechamber is decreased by the thickness of the liner in the second case. Themeshes are faced to one another on Fig. 4.10. They are almost identical in theregion occupied by air. The size of the elements corresponds to approximately 6elements per wave-length along the axis of the cylinder for the high frequencies.This number is scaled by a factor 3 at the air/polyester interface. A unitacoustic pressure is applied at the inlet of the silencer.

The TL predictions obtained with the IP method are plotted against frequencyon Figs. 4.11 and 4.12. All other data come from Mehdizadeh and Paraschivoiu(2005) who reported the experimental and BEM results of Wu et al. (2002).When wave propagation in the liner is taken into account (Fig. 4.11) thenumerical models show a good agreement with the experiments. The locally-reacting approaches (Fig. 4.12) are inefficient, especially in the mid-frequencyregion. The BEM and IP methods even differ drastically for these frequencies.This might be related to differences in the modeling of the specific acousticimpedance.

4.5.2 A coupled problem or how to hide a submarine

Not being visible to other naval vessels is one of the main issues for submarines.During the second world war, rubber coatings with air-filled cavities – such coat-ings are said to be of Alberich type – were developed to prevent submarines


0

10

20

30

40

50

60

70

0 500 1000 1500 2000 2500 3000 3500


Frequency [Hz]

Experiment (Wu et al., 2002)FEM 3D (Mehdizadeh and Paraschivoiu, 2005)BEM (Wu et al., 2002)DG (IP method)

Transmission Loss in a cylindrical single expansion chamber (1in liner)

18in

9in9in4in 1in

Figure 4.11: Transmission loss in a cylindrical silencer with polyester lining. Theporous liner is modeled by an equivalent fluid medium. Comparison between experi-ment, 3D FEM, BEM and IP method.

0

20

40

60

80

100

120

0 500 1000 1500 2000 2500 3000 3500


Frequency [Hz]

Experiment (Wu et al., 2002)BEM (Wu et al., 2002)DG (IP method)

Transmission Loss in a cylindrical single expansion chamber (1in liner)

18in

9in9in3in 1in

Figure 4.12: Transmission loss in a cylindrical silencer with polyester lining. Theliner is substituted for a Robin boundary condition. Comparison between experiment,BEM and IP method.


from being located by sonars. Rubber was chosen because of two reasons. Itscharacteristic impedance is not so far away from that of the water. Wave at-tenuation is relatively high in rubber. When sound from an active sonar entersthe coating, energy is scattered by the cavities (Fig. 4.13) and absorbed by thesurrounding rubber matrix. The reflected power is reduced. It is more difficultfor the sonar to catch the weakened echo. Nowadays, composite materials withanechoic properties are investigated in many fields.

This last section is not intended to design a new type of coating, but ratherto show how the homogenization schemes of Chaps. 1, 2 and 3 can be usedin coupled fluid/solid acoustic problems. To this aim, wave propagation insolids has to be modeled. A simple, albeit restrictive, approach is proposed. Itshows how one can take advantage of the DG method in order to have a unifiedformulation for both media and their coupling. All together, these numericalmodels are used to predict the reflection loss of a particle reinforced rubbercoating on the wall of a submarine.

Fluid-Solid coupling. Consider the interface between a fluid and a solid.In the fluid, the acoustic pressure p satisfies the Helmholtz equation

[∆ + k2

f

]p = 0 , (4.42)

where kf = ωcf

is the propagation constant, with cf the speed of sound in the

medium. If longitudinal waves only are taken into account in the solid, theequation of motion becomes

[∆ + k2

s

]u = 0 , (4.43)

where u is the (scalar) displacement field. The propagation constant depends

on the material moduli and density: ks = ωcs

, cs =√

λs+2µs

ρs. The differential

operator ∆ + k2 holds for both fluid and solid domains. It is applied on theacoustic pressure in the fluid and on the longitudinal displacement in the solid.The DG approach allows to handle both Eqs. (4.42) and (4.43) within a uniqueformulation. The change of variable is achieved on the interface edges of thetriangulation. The conditions at fluid/solid interface,

p = − (λs + 2µs)∂u

∂n, (4.44)

∂p

∂n= ρfω

2u , (4.45)

are incorporated in the numerical fluxes.


Composite coating

Water Steel

Figure 4.13: The steel wall of a submarine is coated with a layer of rubber composite.The viscous behavior of rubber and the scattering by the particles are responsible forthe attenuation of reflection waves.

Reflection loss. The efficiency of the coating is determined by its reflectionloss. The latter is defined as the ratio of the reflected sound power to theincident sound power. The RL is computed in a similar way as the TL forsilencers, with help of the acoustic pressure at two locations in the water. It isgiven by

TL = 20 log

∣∣∣∣

pr

pi

∣∣∣∣, (4.46)

where pi and pr are the incoming and reflected portions of rms acoustic pres-sure wave, respectively. The incoming and reflected rms sound pressures areobtained as

pi = rmsp1 − p2e

−ik0x12

1 − e−ik0x12(4.47)

pr = rmsp1 − p2e

ik0x12

1 − eik0x12(4.48)

where x12 = x2 − x1 is the distance between the positions where the data arerecorded. Again, plane waves are assumed.

Illustrative example. Consider the situation depicted on Fig. 4.13. It isinspired by the work of Ivansson (2005). The steel wall of a submarine is coveredwith a 7 mm thick rubber composite layer. The wall is assumed to be of infiniteextent in the propagation direction. This allows to focus on the behavior ofthe coating for waves coming from the water. Reflection waves that would havetaken place in the steel wall due to impedance mismatch at the interface withanother medium are neglected.

Ivansson (2005) assumes the following regarding the reinforcements: a singlelayer of spherical particles; they are arranged in square array at mid-thickness


of the coating; the grid size 4.7 mm; the radius of the spheres is 1.5 mm; theyare made of a hard and incompressible material with specific gravity 8. Thiscorresponds to approximately 10% volume fraction of inclusions. No additionalinformation on the filling material is given. A Young’s modulus of 210 GPa anda Poisson’s ratio of 0.49 are chosen. The properties of the rubber matrix usedby Ivansson (2005) are determined as follows. Its density is that of water(ρ0 = 1000 kg/m3). Longitudinal waves travel at the speed of sound in water(c0 = 1480 m/s). Viscous effects are modeled with an attenuation† of 0.1 dBper wave-length. The transverse properties are the outcome of an optimizationprocedure on the composite coating designed to provide a zero reflection ata given frequency. This leads to low phase speed and high attenuation. Therubber material of Biwa et al. (2002) is considered to be more realistic. Itsproperties (Chap. 3, Tab. 3.3) are used instead.

The viscoelastic version of the Mori-tanaka model (Chaps. 1, 2) and the diluteinclusion scheme of Chap. 3 are used to predict the effective acoustic propertiesof the coating. The acoustic field in the water is obtained with the IP method.Three homogeneous media are involved in the simulation: the water, the coatingand the steel wall. The fluid/solid coupling is handled as explained above.

The scattering by the particles leads to an increase of the attenuation factorof the material as predicted by the dilute inclusion model (Fig. 4.14, right).Consequently, the coating absorbs more energy and the reflected power is re-duced (Fig. 4.14, left). The quasi-static MT scheme also predicts an increasedattenuation of the material. It is less pronounced because scattering is nottaken into account (Fig. 4.14, right). The absorbed energy is underestimated.More power is reflected by the coating (Fig. 4.14, left).

4.6 Conclusions

This chapter was devoted to sound propagation in fluids interacting with com-posite walls. Dissipative effects were neglected for the fluids. Small perturba-tions and absence of mean flow were assumed. The acoustic pressure satisfiesthe Helmholtz equation for which a discontinuous Galerkin formulation wasproposed. The latter is based on a family of numerical fluxes after Arnoldet al. (2002).

The internal penalty method was derived and implemented in 2D and for ax-isymmetric problems. Linear and quadratic shape functions were used to in-terpolate both the acoustic pressure and its gradient on the elements of thetriangulation. A convergence analysis has been performed. It shows a asymp-totic rate of convergence of order k + 3/2 (resp. k + 1/2) – k is the order of

†1 dB = ln 1020

nepers ' 0.1151 nepers

Conclusions 129

-5

-4

-3

-2

-1

0 10 20 30 40

Reflection loss [dB]

Frequency [kHz]

Underwater reflection loss of a steel half-planecoated with a 7mm thick layer of rubber composite

Rubber onlyComposite (MT,no scattering)Composite (Dilute Inclusion)

0

0.005

0.01

0.015

0.02

0 10 20 30 40

P-wave attenuation factor [nepers/mm]

Frequency [kHz]

Rubber matrix with 10% hard sphericalparticles of radius 1.5mm

Rubber onlyComposite (MT,no scattering)Composite (Dilute Inclusion)

Figure 4.14: Left: underwater reflection loss of a rubber composite layer on a steelwall. Right: longitudinal attenuation factor of the coating material.

interpolation – for the L2 norm of the acoustic pressure (resp. the particlevelocity). A higher rate of convergence (k + 3/2) should be reached for thevelocity by choosing a corresponding velocity-dependent numerical flux.

A first set of numerical simulations was about the transmission loss in silencers.The effect of polyester as lining material for the expansion chamber has beenstudied. The porous liner was modeled as an equivalent fluid with help of theDelany and Bazley (1970) formulae. The homogenization schemes of Chaps. 1to 3 are not well adapted to this kind of materials in this case. The predictedTL were in very good agreement with other numerical results and experimentaldata from the literature.

The reflection loss of a particulate rubber composite coating has been consid-ered afterwards. The latter was modeled as an equivalent homogeneous mate-rial with effective properties obtained from homogenization schemes of Chaps.2 and 3. When sound enters the coating, energy is scattered by the particlesand absorbed by the viscoelastic matrix. The simulations show the influenceof accounting for scattering in the homogenization model on the prediction ofthe RL.

More complex (e.g. geometry) and realistic (e.g. materials) simulations shouldbe performed and compared to experimental measurements. This would re-quire the 3D extension of the model (IP method, fluid/solid interface) or theintegration of homogenization schemes in existing softwares for acoustics.


Appendix A Specific impedance of an acoustic

layer

A boundary condition intended to replace the interface between two acousticmedia is developed. One can then go without the modeling of a region where theacoustic fields are not especially sought. The computational costs are narroweddown at the same time.

Consider a plane wave propagating along the x-direction of a Cartesian coordi-nate system in a medium with characteristic acoustic impedance Z1. Assumethat this wave is in normal incidence to a second acoustic medium with charac-teristic impedance Z2. Let the sound pressure waves in both media be written

pi(x) = Aieikix +Bie

−ikix , i = 1, 2 (A.1)

where ki are the wave numbers. The interface is located at x = a. Thesecond medium is of finite length and terminates on a rigid wall at x = b > a.Continuity of acoustic pressure and normal velocity at interface and completereflection at boundary give the following set of relations:

A1eik1a +B1e

−ik1a = A2eik2a +B2e

−ik2a (pressure) (A.2)

A1eik1a −B1e

−ik1a =Z1

Z2

(A2e

ik2a −B2e−ik2a

)(velocity) (A.3)

A2 = B2e−2ik2b (reflection) (A.4)

The interface with the second medium is replaced by a locally reacting wall.The Robin boundary condition at x = a reads

A1eik1a −B1e

−ik1a +Z1

Z

(A1e

ik1a +B1e−ik1a

)= 0 , (A.5)

where Z is the specific acoustic impedance of the wall. Its value is determinedafter simple algebra on the above four equations. One finds

Z = −iZ2 cot (k2d) , (A.6)

where d = b− a is the thickness of the layer.

Consider now a wave that propagates along the radius of a cylindrical coordi-nate system. The interface is located at r = r1 > 0 and the second mediumterminates on a rigid material at r = r2 > r1. The pressure fields are solutionof the Bessel equation of order n = 0:

pi(r) = AiJ0(kir) +BiY0(kir) , i = 1, 2 . (A.7)

Specific impedance of an acoustic layer 131

Continuity conditions at r = r1 and perfect reflection at r = r2 allows to findthe specific acoustic impedance of the substituting boundary:

Z = iZ2Y1(k2r2)J0(k2r1) − J1(k2r2)Y0(k2r1)

Y1(k2r2)J1(k2r1) − J1(k2r2)Y1(k2r1). (A.8)

Unlike the plane wave case, Z depends explicitly on the position of the boundarylayer.


Appendix B Notations and units


t, t0 time sω angular frequency rad s−1

P , P0 pressure Pap acoustic pressure PaT , T0 temperature KS, S0 specific entropy J K−1 kg−1

ρ, ρ0 density kg m−3

v, v0 particle velocity m s−1

d differential∂ partial derivative∇ gradient∇· divergence∆ laplacian· scalar product

F [ ], ˆ Fourier transform f(ω) =

∞Z

−∞

f(t)e−iωtdt

c, c0 speed of sound m s−1

k, k0 wave number rad m−1

Z0, Z characteristic, specific acoustic impedance N sm−3

β normalized specific admittance

K element of the triangulation∂K boundary of Ke edge of the triangulation

uh FE solution for p Paσh FE solution for ∇p Pam−1

heu numerical flux for uh across e Pa

heσ numerical flux for σh across e Pa m−1

[[]] jump operator averaging operatorηe DG parameter for the IP method m−1

Conclusions

Throughout the chapters of this thesis, we developed homogenization schemesfor the effective mechanical and acoustical properties of linear viscoelastic in-clusion reinforced composites. Generic procedures were studied and proposed,starting from the statics and elastic materials, followed by viscoelastic behaviorsand the quasi-static approximation, to end up with dynamic models. Quasi-static and dynamic schemes were used in the simulation of coupled fluid/solidacoustic problems, showing the influence of accounting for particle scattering.

For elastic composites, our main contributions regard homogenization schemesfor coated inclusion reinforced materials. An original two-level recursive pro-cedure has been proposed. At each level, a homogenization model suitable fortwo-phase composites with aligned reinforcements is required. That makes theprocedure very generic. The latter is used as stand-alone (i.e. for three-phasecomposites) or to achieve the first step of the two-step approach (Camachoet al., 1990) for multiphase materials (e.g. for composites with misalignedcoated fibers). The two-level approach has been successfully applied to singlewalled carbon nanotube (SWNT) reinforced polymers for which 3D periodicunit-cell finite element calculations were performed to verify the predicted effec-tive elastic constants. Based on the interpretation of the multi-inclusion modelby Nemat-Nasser and Hori (1999) – the coating phase behaves as a separate in-clusion phase in the matrix – the two-step approach has also been proposed forcomposites with coatings. Its application to SWNT reinforced polymers showeddramatically poor estimates of the effective moduli. An analytical comparativestudy has been performed between the two-step (M-T,Voigt) scheme, the two-step (M-T,M-T) model – indeed equivalent to the generalized M-T scheme –and the two-level (M-T,M-T) method. It showed that, unlike the other models,the two-level approach leads to a multiplicative decomposition of the strainconcentration tensor. A similar decomposition was found by Aboutajeddineand Neale (2005) in their new formulation of the double-inclusion model.

With the quasi-static approximation, i.e. when inertial effects are ignored,we generalized the generic two-step and two-level procedures to predict both

134 Conclusions

time dependent and harmonic effective mechanical properties (i.e. storage andloss moduli) of linear viscoelastic multiphase composites. Thanks to the elastic-viscoelastic correspondence principle, any linear elastic homogenization schemecan indeed be extended to linear viscoelastic behaviors. This principle has beenwidely used in the literature, but was never applied to the interpolative double-inclusion model by Lielens (1999), neither to the two-step approach, nor to thetwo-level procedure. In addition, time dependent predictions which require thenumerical inversion of the Laplace-Carson transform, are not so common in theliterature. We put a special emphasis on an extensive verification and validationof the proposed extensions against FE calculations and available experimentaldata. Twenty distinct composites were studied. They were made up with twoor three phases and subjected to various loadings (shear, uniaxial tension, planestrain uniaxial tension). The predictions were compared against available ex-perimental data, unit-cell finite element (FE) results and other homogenizationestimates from the literature. FE calculations with periodic boundary condi-tions have also been performed on 3D representative volume elements. Fortwo-phase composites, several simulations were presented in both frequencyand time domain. As compared to reference unit-cell FE results, the predic-tions of the interpolative D-I model were always excellent and better than thoseof M-T for high volume fractions of inclusions. For composites with coated in-clusions, unlike the elastic case of SWNT reinforced polymers (see above), theconclusions that can be drawn from the numerical simulations remain open.Two-step and two-level approaches may both success, both fail, or one failswhile the other successes. All four cases have been observed. However, for stiffelastic fibers coated with a soft viscoelastic layer and embedded in a stiff vis-coelastic matrix, two-level schemes performed remarkably well while two othermethods, namely direct M-T and Benveniste et al. (1989) procedure gave verywrong estimates.

Dealing with effective acoustic properties (plane wave phase velocities, atten-uation factors) of viscoelastic inclusion reinforced composites, we proposed ageneric framework for the development of homogenization schemes. It unifieswithin a single formulation several models of the literature (e.g. Sabina andWillis, 1988; Sabina et al., 1993; Smyshlyaev et al., 1993; Kanaun et al., 2004;Kanaun and Levin, 2005). All these methods were developed in linear elastic-ity. The proposed formulation may therefore be considered as a generalizationto linear viscoelastic behaviors. All these models have in common the solu-tion of an elementary problem: the one-particle scattering problem. It is theequivalent of the single inhomogeneity problem (Eshelby, 1957) when inertialeffects are not neglected. The link between the long-wave limit (ω → 0) of thedynamic schemes and the static models has been shown. With the assump-tion of independent-scattering, a dilute-inclusion scheme was derived from theproposed formulation and further slightly improved, mainly in the long-wave

135

region. We also studied the additive attenuation model by Biwa et al. (2002)which is based on the same assumption. Several numerical simulations wereperformed on two-phase composites with spherical particles. Various homoge-nization schemes were compared together and against experimental data fromthe literature. A special attention was drawn on the comparison between bothaforementioned models. The results showed that they are not equivalent andmay indeed provide estimates that are far away from each other. We were notable to decide between them, mainly because no predictions stemming fromdirect approaches (e.g. finite elements) were available.

As an application, we considered coupled fluid/solid acoustic problems forwhich the walls are made of composite materials. To this end, a discontinuousGalerkin (DG) formulation was proposed for the Helmholtz equation which de-scribes time-harmonic acoustic and (visco)elastic waves. The choice of numer-ical fluxes that weakly enforce continuity across the edges of the triangulationwas inspired by the works of Arnold et al. (2002). The so-called internal penalty(IP) method was implemented in 2D and for axisymmetric problems. Piece-wise discontinuous linear and quadratic shape functions were used to interpolateboth the acoustic pressure and its gradient on nodes of triangular elements. Aconvergence analysis showed the expected higher rate of convergence of the IPmethod. A first set of numerical simulations involved a porous polyester as linermaterial for cylindrical silencers with a single-expansion chamber. The homog-enization schemes we developed are not well suited to this kind of materialsfor that application (high porosity, the inclusions are air-filled). Their overallbehavior is better addressed with the equivalent fluid model by Delany andBazley (1970). Several results from the literature were successfully reproduced.Finally, we considered a particle reinforced viscoelastic rubber composite andit’s application to anechoic coating for submarines. Quasi-static and dynamichomogenization schemes were used to predict the effective properties of thecoating. Although several restrictive assumptions were made – the geometryis simplified, the modeling does not account for shear waves in the solids, theinfluence of external pressure is neglected as well as a potential mean flow – thenumerical simulations showed the influence of accounting for particle scatteringon the reduction of the reflected acoustic power.

There are at least a few directions for future work. One issue is to better as-sess the predictive capabilities and limitations of the proposed two-step andtwo-level schemes for coated composites. This cannot be completely achievedwithout the development of the two-level (D-I,D-I) model which requires theexpression of Eshelby’s tensor for a transverse isotropic viscoelastic matrix.The works of Withers (1989) in linear elasticity would be the starting point.As temperature has a great influence on the properties of viscoelastic mate-rials, the generic two-step and two-level approaches have to be extended tolinear thermo-viscoelastic behaviors. This research could be based on the de-

136 Conclusions

velopments of Pierard et al. (2004) and Doghri and Tinel (2006). Anothersubject for future work is to extend these methods to nonlinear viscoelasticity,perhaps by using the so-called affine formulation, which was developed success-fully for elasto-viscoplasticity (e.g. Masson, 1998; Pierard and Doghri, 2006;Pierard, 2006). Regarding effective acoustic properties of inclusion reinforcedmaterials, more comparisons between existing models, verifications with directapproaches and validations are required. Many models have been developedbut very few were compared together (for instance the schemes by Kanaunet al., 2004; Kanaun and Levin, 2005). A parametric study would be of greatinterest. A direct approach would be very helpful for verification purposes. Al-though some methods are written for multiphase composites, none of them, toour knowledge, has been validated. Finally, dynamic homogenization schemesshould be involved in more complex (e.g. geometry) and realistic (e.g. materi-als) coupled fluid/solid acoustic simulations, with comparisons to experimentalmeasurements.

List of publications

Refereed papers in international journals

Pierard, O., Friebel, C. and Doghri, I. (2004). Mean-field homogenization ofmulti-phase thermo-elastic composites: a general framework and its validation.Composites Science and Technology, 64(10-11):1587-1603.

Doghri, I. and Friebel, C. (2005). Effective elasto-plastic properties of inclusion-reinforced composites. Study of shape, orientation and cyclic response. Me-chanics of Materials, 37(1):45-68.

Friebel, C., Doghri, I. and Legat, V. (2006). General mean-field homogeniza-tion schemes for viscoelastic composites containing multiple phases of coatedinclusions. International Journal of Solids and Structures, 43(9):2513-2541.

Selmi, A., Friebel, C., Doghri, I. and Hassis, H. (2007). Prediction of the elasticproperties of single walled carbon nanotube reinforced polymers: a comparativestudy of several micromechanical models. Composites Science and Technology,67:2071-2084.

Conference proceedings

Doghri, I., Friebel, C. and Ouaar, A. (2003). Two-scale mechanics of inclusion-reinforced elasto-plastic composites: modeling and computation. In Khan, A.S., editor, Dislocations, plasticity and metal forming, pages 427-429, Fulton,Maryland. NEAT Press.

Friebel, C., Doghri, I. and Legat, V. (2006). Mechanics and Acoustics of Vis-coelastic Composites by a Micro-Macro Mean-Field Approach. In proc. of IIIEuropean Conference on Computational Mechanics, Lisbon, Portugal.

Conference presentations

Doghri, I., Friebel, C. and Ouaar, A. (2003). Two-scale mechanics of inclusion-reinforced elasto-plastic composites: modeling and computation. In DEPOS18, Colloque francophone sur la deformation des polymeres solides, CentreCPCV de Saint-Prix (Val dOise), France.

138 List of publications

Friebel, C. and Doghri, I. (2003). Homogenization of two-phase elasto-plasticcomposites. Effet of the reinforcements shape and orientation for monotonicand cyclic loadings. In 6th National Congress on Theorical and Applied Me-chanics, Gent, Belgium.

Friebel, C., Doghri, I. and Ouaar, A. (2003). Micro-Macro mechanics of inclu-sion-reinforced elasto-plastic composites: modeling and computation. In LUX-FEM03, Luxembourg city, Luxembourg.

Friebel, C., Doghri, I. and Legat, V. (2006). Mean-Field HomogenizationSchemes for Effective Mechanical and Acoustical Properties of ViscoelasticComposites. In 7th National Congress on Theoretical and Applied Mechanics,Mons, Belgium.

Conference and colloquium posters

Friebel, C., Doghri, I. and Legat, V. (2006). Mechanics and Acoustics of Vis-coelastic Composites by a Micro-Macro Mean-Field Approach. In Approchesmulti-echelles en mecanique des materiaux, Colloque National MECAMAT,Aussois, France.

Friebel, C. Doghri, I. and Legat, V. (2007). Mechanics and Acoustics of Poly-mer Composites. Micro-Macro Modeling of Effective Properties. In MecaTechenterprise-research lab. meeting, Louvain-la-Neuve, Belgium.

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2II Mechanics and Acoustics of Viscoelastic Inclusion Reinforced Composites-micro--macro Modeling of Effective

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