ILSB Report 206 hjb/ILSB, 080407

(preliminary version)

ILSB Report / ILSB-Arbeitsbericht 206

(supersedes CDL–FMD report 3–1998)

A SHORT INTRODUCTION TO BASIC ASPECTS

OF CONTINUUM MICROMECHANICS

H.J. Bohm

Institute of Lightweight Design and Structural Biomechanics (ILSB)

Vienna University of Technology

updated: April 7, 2008

c©2007, 2008 Helmut J. Bohm

CONTENTS

Notes on this Document

List of Variables

1. INTRODUCTION

1.1 Inhomogeneous Materials

1.2 Homogenization and Localization

1.3 Overall Behavior, Material Symmetries

1.4 Basic Modeling Strategies in Continuum Micromechanics of Materials

2. MEAN FIELD APPROACHES

2.1 General Relations

2.2 Eshelby Tensor and Dilute Matrix–Inclusion Composites

2.3 Mean Field Methods for Thermoelastic Composites with Aligned Reinforce-

ments

2.4 Hashin–Shtrikman Estimates

2.5 Other Estimates for Elastic Composites with Aligned Reinforcements

2.6 Mean Field Methods for Inelastic Composites

2.7 Mean Field Methods for Composites with Nonaligned Reinforcements

2.8 Mean Field Methods for Non-Ellipsoidal Reinforcements

2.9 Mean Field Methods for Diffusion-Type Problems

3. VARIATIONAL BOUNDING METHODS

3.1 Classical Bounds

3.2 Improved Bounds

3.3 Bounds for the Nonlinear Behavior

3.4 Some Comparisons of Mean Field and Bounding Predictions

4. GENERAL REMARKS ON APPROACHES BASED ON DISCRETE

MICROGEOMETRIES

4.1 Microgeometries

4.2 Numerical Engineering Methods

5. PERIODIC MICROFIELD APPROACHES

5.1 Basic Concepts of Unit Cell Models

5.2 Boundary Conditions, Application of Loads, Evaluation of Microfields

5.3 Unit Cell Models for Continuously Reinforced Composites

5.4 Unit Cell Models for Discontinuously Reinforced Composites

5.5 PMA Models for Some Other Inhomogeneous Materials

5.6 PMA Models for Diffusion-Type Problems

i

6. EMBEDDING APPROACHES AND RELATED METHODS

7. WINDOWING APPROACHES

8. MULTI-SCALE MODELS

9. CLOSING REMARKS

APPENDIX: SOME ASPECTS OF MODELING DAMAGE AT THE MICROLEVEL

REFERENCES

ii

Notes on this Document

The present document is an expanded and modified version of the CDL–FMD report

3–1998, “A Short Introduction to Basic Aspects of Continuum Micromechanics”, which

is based on lecture notes prepared for the European Advanced Summer Schools “Fron-

tiers of Computational Micromechanics in Industrial and Engineering Materials” held

in Galway, Ireland, in July 1998 and in August 2000. Similar lecture notes were used

for graduate courses in a Summer School held at Ameland, the Netherlands, in October

2000 and during the COMMAS Summer School held in September 2002 at Stuttgart,

Germany. All of the above documents, in turn, are based on the micromechanics section

of the lecture notes for the course “Composite Engineering” (317.003) offered regularly

at the Vienna University of Technology.

The course notes “A Short Introduction to Continuum Micromechanics” (Bohm,2004)

for the CISM Course on Mechanics of Microstructured Materials held in July 2003 in

Udine, Italy, may be viewed as an abridged version of the present report that employs a

somewhat different notation.

The present document is continuously updated to reflect current developments in con-

tinuum micromechanics as seen by the author.

For information on the research work in the field of micromechanics of materials currently

carried out at the Institut fur Leichtbau und Struktur-Biomechanik (formerly the Institut

fur Leichtbau und Flugzeugbau) of Vienna University of technology see the web pages

http://www.ilsb.tuwien.ac.at/ilfb/ilfb ra3e.html

iii

List of Variables

〈 〉 Volume average

a: Aspect ratio of inclusions, reinforcements

`: Characteristic length of lower length scale (microscale)

L: Characteristic length of higher length scale (macroscale)

Ωs: Volume of sample or region

Γs: Surface of sample or region

ΩUC: Volume of unit cell

ΓUC: Surface of unit cell

Ω(p): Volume occupied by phase (p)

V (p) Volume fraction of phase (p)

ξ: Inclusion, reinforcement volume fraction

η,ζ: Microstructural parameters for higher order bounds

ε: Strain tensor (rank 2; vector in Voigt notation)

〈ε〉: Tensor of (volume) averaged strains

ε′: Tensor of (local) fluctuating strains

εa: Tensor of the applied mechanical (far field) strains

〈ε〉(p): Tensor of averaged strains in phase (p)

εt: Tensor of unconstrained eigenstrains in inclusion

ετ : Tensor of equivalent eigenstrains in equivalent homogeneous inclusion

εc: Tensor of constrained strains in inclusion

σ: Stress tensor (rank 2; vector in Voigt notation)

〈σ〉: Tensor of (volume) averaged stresses

σ′: Tensor of (local) fluctuating stresses

σa: Tensor of applied (far field) stresses

〈σ〉(p): Tensor of averaged stresses in phase (p)

A: Phase strain concentration tensor (rank 4; matrix in Voigt notation)

B: Phase stress concentration tensor (rank 4; matrix in Voigt notation)

a: Phase thermal strain concentration tensor (rank 2; vector in Voigt notation)

b: Phase thermal stress concentration tensor (rank 2; vector in Voigt notation)

C: Compliance tensor (rank 4; matrix in Voigt notation)

E: Elasticity tensor (rank 4; matrix in Voigt notation)

t: Specific thermal stress tensor (rank 2; vector in Voigt notation)

iv

α: Thermal expansion coefficient tensor (rank 2; vector in Voigt notation)

I: Identity tensor (rank 4; matrix in Voigt notation)

S: Eshelby tensor (rank 4; matrix in Voigt notation)

t: Traction vector

ta: Vector of applied tractions

nΓ: Surface normal vector

E: Young’s modulus

G: Shear modulus

K: Bulk modulus

ν: Poisson’s ratio

KT: Transverse bulk modulus

T : Temperature

σ: Stress component

ε: Strain component

γ: Shear angle

α: Coefficient of thermal expansion

Constituents (or “phases”, i.e. matrix, inclusions, inhomogeneities, reinforcements, . . .)

are denoted by superscripts in brackets, e.g. (p). Axial and transverse properties of trans-

versely isotropic materials are marked by subscripts A and T , respectively, and effective

properties are denoted by a superscript asterisk ∗.

Voigt notation (also known as Nye or Voigt–Mandel notation) is used in chapters 1 to 3

of these notes, i.e. the stress and strain tensors are formally denoted as pseudo-vectors

with the components

σ =

σ(1)σ(2)σ(3)σ(4)σ(5)σ(6)

=

σ11

σ22

σ33

σ12

σ13

σ23

ε =

ε(1)ε(2)ε(3)ε(4)ε(5)ε(6)

=

ε11

ε22

ε33

γ12

γ13

γ23

,

where γij = 2εij are the shear angles1). Tensors of rank 4, such as elasticity, compliance,

concentration and Eshelby tensors, take the form of 6×6 matrices.

1) This notation allows strain energy densities to be obtained as U = 12σε. Furthermore,

the stress and strain “rotation tensors” TΘσ and TΘ

ε follow the relationship TΘε = [(TΘ

σ )−1]T .

v

vi

1. INTRODUCTION

In the present document some basic issues and some important approaches in the field

of continuum micromechanics of materials are discussed. The main emphasis lies on

application related (or “engineering”) aspects, and neither a comprehensive theoretical

treatment nor a review of the pertinent literature are attempted. For more formal treat-

ments of many of the concepts and methods discussed in what follows see e.g. (Mura,1987;

Aboudi,1991; Nemat-Nasser/Hori,1993; Suquet,1997; Markov,2000; Bornert et al.,2001;

Torquato,2002; Milton,2002; Qu/Cherkaoui,2006). Short overviews were given e.g. by

Hashin(1983) and Zaoui (2002). Discussions of the history of the development of the

field can be found in (Markov,2000; Zaoui,2002).

Due to the author’s research interests, more room is given to the thermomechanical

behavior of two-phase materials showing a matrix–inclusion topology and especially to

metal matrix composites (MMCs), than to materials with other phase topologies or phase

geometries. Extending the methods presented here to other types of inhomogeneous

materials, however, in general does not cause principal difficulties.

1.1 Inhomogeneous Materials

Many industrial and engineering materials as well as the majority of “natural” materials

are inhomogeneous, i.e. they consist of dissimilar constituents (or “phases”) that are

distinguishable at some (small) length scale. Each constituent shows different material

properties and/or material orientations and may itself be inhomogeneous at some even

smaller length scale(s). Typical examples of inhomogeneous materials are composites,

concrete, polycrystalline materials, porous and cellular materials, functionally graded

materials, wood, and bone.

An important aim of theoretical studies of multiphase materials lies in homogenization,

i.e. in deducing their overall (“effective” or “apparent”) behavior2) (e.g. stiffness, thermal

expansion and strength properties, heat conduction and related transport properties,

electrical and magnetic properties, electromechanical properties, . . .) from the corre-

sponding material behavior of the constituents (and of the interfaces between them) and

from the geometrical arrangement of the phases. In what follows, the main focus will lie

on describing the thermomechanical behavior of inhomogeneous two-phase materials by

methods of continuum mechanics. It should be kept in mind, however, that there is a large

body of literature applying analogous or related continuum approaches to other physi-

cal properties of inhomogeneous materials, see e.g. (Hashin,1983; Markov/Preziosi,2000;

Milton,2002) as well as sections 2.9 and 5.6 of the present report.

2) A number of authors restrict the term “effective material properties” to the response ofproper Representative Volume Elements, compare section 1.2, of an infinitely large sample, orof periodic (model) composites, whereas the designation “apparent material properties” is usedfor results obtained from smaller volume elements such as windows (Huet,1990).

– 1 –

In micromechanical approaches the stress and strain fields in an inhomogeneous material

are split into contributions corresponding to different length scales3). It is assumed that

these length scales are sufficiently different so that for each pair of them

• fluctuations of the stress and strain fields at the smaller length scale (microfields,

“fast variables”) influence the macroscopic behavior at the larger length scale only

via their volume averages and

• gradients of the stress and strain fields as well as compositional gradients at the

larger length scale (macrofields, “slow variables”) are not significant at the smaller

length scale, where these fields appear to be locally constant and can be described

in terms of uniform “applied” or “far field” stresses and strains.

Formally, this splitting of the strain and stress fields into slow and fast contributions can

be written as

ε(x) = 〈ε〉+ ε′(x) and σ(x) = 〈σ〉+ σ′(x) , (1)

where 〈ε〉 and 〈σ〉 are the (slow) macroscopic fields, whereas ε′ and σ′ stand for the

microscopic fluctuations.

For example, in layered composites (“laminates”) there are at least three length scales

that can be described by continuum mechanics, which are often called the macro scale

(component or sample), the mesoscale (individual plies), and the microscale (fiber–matrix

structure of the plies)4). Smaller length scales in such material systems may or may

not be amenable to continuum mechanical descriptions; for an overview of methods

applicable below the continuum range see e.g. (Raabe,1998). Viewed from some given

length scale the behavior of a material that is inhomogeneous at some smaller length scale

can be described by the behavior of an energetically equivalent homogenized continuum,

provided the separation between the two length scales is sufficiently large and the above

conditions are met.

If the above relations between “fast” and “slow” variables are not fulfilled to a sufficient

degree (e.g. due to insufficiently separated length scales, near free surfaces of inhomo-

geneous materials, at macroscopic interfaces adjoined by at least one inhomogeneous

material, or in the presence of marked compositional or load gradients, an example for

the latter being inhomogeneous beams under bending loads), special homogenization

methods must be used, such as second order schemes explicitly accounting for deforma-

tion gradients (Kouznetsova et al.,2002), and the homogenized continuum is nonlocal,

see e.g. (Feyel,2003).

An important classification scheme for inhomogeneous materials is based on the micro-

scopic phase topology. In matrix–inclusion arrangements (as found in particulate and

3) Micromechanics of materials deals with the bridging of relative length scales in inhomo-geneous materials (as described by the ratio between the characteristic lengths of the upper andlower scales). Continuum micromechanics in most cases cannot per se resolve effects of absolutelength scales, such as the absolute size of inhomogeneities, compare chapter 9.

4) This nomenclature is far from universal, the naming of the length scales of inhomogeneousmaterials being notoriously inconsistent in the literature.

– 2 –

fibrous materials, such as most composite materials, porous materials, and closed cell

foams5)) only the matrix shows a connected topology and the constituents play clearly

distinct roles. In interwoven phase arrangements (as found e.g. in functionally graded

materials or in open cell foams) and in many polycrystals (“granular materials”), in

contrast, the phases do not show clearly different topologies.

1.2 Homogenization and Localization

For any region of an inhomogeneous material the microscopic strain and stress fields, ε(x)

and σ(x), and the corresponding macroscopic responses, 〈ε〉 and 〈σ〉, can be formally

linked by localization (or projection) relations of the type

ε(x) = A(x)〈ε〉

σ(x) = B(x)〈σ〉 . (2)

If, in addition, the region is sufficiently large and contains no significant macroscopic

gradients of stress, strain or composition, homogenization relations can be defined as

〈ε〉 =1

Ωs

∫

Ωs

ε(x) dΩ =1

2Ωs

∫

Γs

(u(x)⊗ nΓ(x) + nΓ(x)⊗ u(x)) dΓ

〈σ〉 =1

Ωs

∫

Ωs

σ(x) dΩ =1

Ωs

∫

Γs

t(x)⊗ x dΓ , (3)

where Ωs and Γs stand for the volume and the surface of the region under considera-

tion, u(x) is the deformation vector, t(x) = σ(x)nΓ(x) are the surface traction vectors,

nΓ(x) are surface normal vectors, and ⊗ is the dyadic product of vectors defined as

[a⊗b]ij=aibj. A(x) and B(x) are called mechanical strain and stress concentration ten-

sors (or influence functions; Hill,1963), respectively. The surface integral formulation

for 〈ε〉 given above holds only for perfect interfaces between the constituents, in which

case the mean strains and stresses in a control volume, 〈ε〉 and 〈σ〉, are fully deter-

mined by the surface displacements and tractions; otherwise correction terms involving

the displacement jumps across imperfect interfaces or cracks must be accounted for, see

e.g. (Nemat-Nasser/Hori,1993). In the absence of body forces the microstresses σ(x) are

self-equilibrated (but not necessarily zero). In the above form, eqn.(2) applies only to

linear elastic behavior, but it can be modified to cover thermoelastic behavior, compare

eqn.(16), and extended to the nonlinear range (e.g. to elastoplastic materials described

by secant or incremental plasticity). For a discussion of homogenization at finite defor-

mations see e.g. (Nemat-Nasser,1999).

Equations (1) and (3) immediately imply that volume averages of fluctuations vanish for

sufficiently large integration volumes,

1

Ωs

∫

Ωs

ε′(x) dΩ = 0 =1

Ωs

∫

Ωs

σ′(x) dΩ . (4)

5) With respect to their thermomechanical behavior, porous and cellular materials canusually be treated as inhomogeneous materials in which one constituent has vanishing stiffness,thermal expansion, conductivity etc.

– 3 –

Similarly, surface integrals over the microscopic fluctuations of appropriate variables

typically vanish6).

For inhomogeneous materials that show sufficient separation between the length scales

the relation

∫

Ωs

σ(x) ε(x) dΩ =

∫

Ωs

σ(x) dΩ

∫

Ωs

ε(x) dΩ (5)

can be shown to hold for general statically admissible stress fields σ and kinematically

admissible strain fields ε, see (Hill,1952). This equation is known as Hill’s macrohomo-

geneity condition, the Mandel–Hill condition or the energy equivalence condition, com-

pare (Bornert,2001; Zaoui,2001). For the two special cases of homogeneous stress and

homogeneous strain boundary conditions (for which the above relation can be shown to

hold by transforming the volume integrals into surface integrals, to which the fluctua-

tions σ′ and ε′ give zero contributions) it is referred to as Hill’s lemma. Equation (5)

implies that the volume averaged strain energy density of an inhomogeneous material can

be obtained from the volume averages of the stresses and strains, provided the micro-

and macroscales are sufficiently well separated7). Accordingly, homogenization proce-

dures can be interpreted as finding a homogeneous comparison material (or “reference

material”) that is energetically equivalent to a given microstructured material.

The microgeometries of real inhomogeneous materials are at least to some extent random

and, in the majority of cases of practically relevance, their detailed microgeometries are

highly complex. As a consequence, exact expressions for A(x), B(x), ε(x), σ(x) etc.

in general cannot be provided with reasonable effort and approximations have to be in-

troduced. Typically, these approximations are based on the ergodic hypothesis, i.e., the

heterogeneous material is assumed to be statistically homogeneous. This implies that

sufficiently large volume elements (“homogenization volumes”) selected at random po-

sitions within the sample have statistically equivalent phase arrangements and give rise

to the same averaged material properties8). As mentioned above, these material proper-

ties are referred to as the overall or effective material properties of the inhomogeneous

material.

Ideally, the homogenization volume should be chosen to be a proper representative volume

element (or reference volume element, RVE), i.e., a subvolume of Ωs that is sufficiently

6) Whereas integrals over products of slow and fast variables vanish, integrals over productsof fluctuating components do not vanish in general, compare section 2.4.

7) For a discussion of the Hill condition for finite deformations and related issues see e.g.(Khisaeva/Ostoja-Starzewski,2006).

8) Some inhomogeneous materials are not statistically homogeneous by design (e.g. func-tionally graded materials in the direction(s) of the gradient(s) and, consequently, may requirenonstandard treatment. For such materials it is not possible to define effective material prop-erties in the sense of eqn.(6) and representative volume elements may not exist. Statisticalinhomogeneity may also be introduced by manufacturing processes.

– 4 –

large to be statistically representative of the microgeometry of the material9). Such a

volume element must be sufficiently large to allow a meaningful sampling of the mi-

crofields and sufficiently small for the influence of macroscopic gradients to be negligible

and for an analysis of the microfields to be possible10). For discussions of homogenization

volumes and RVEs see e.g. (Hashin,1983; Nemat-Nasser/Hori,1993; Drugan/Willis,1996;

Ostoja-Starzewski,1998; Zaoui,2001; Bornert et al.,2001; Kanit et al.,2003; Stroeven et

al.,2004; Gitman et al.,2007) as well as section 4.1. Comparisons of RVEs conforming to

different definitions can be found in (Trias et al.,2006; Swaminathan et al.,2006).

In practice it tends to be rather difficult to study (or, in fact, to identify) proper RVEs

of actual inhomogeneous materials (as opposed to simplified phase arrangements) and,

accordingly, in models based on discrete microgeometries typically volume elements are

used that are only approximations to RVEs.

1.3 Overall Behavior, Material Symmetries

The homogenized strain and stress fields of an elastic inhomogeneous material as obtained

by eqn.(3), 〈ε〉 and 〈σ〉, can be linked by effective elastic tensors E∗ and C∗ as

〈σ〉 = E∗〈ε〉

〈ε〉 = C∗〈σ〉 , (6)

which may be viewed as the elastic tensors of an appropriate equivalent homogeneous

material. Using eqn.(3) these effective elastic tensors can be obtained from the local

elastic tensors E(x) and C(x) and from the concentration tensors A(x) and B(x) as

volume averages

E∗ =1

Ωs

∫

Ωs

E(x)A(x) dΩ

C∗ =1

Ωs

∫

Ωs

C(x)B(x) dΩ . (7)

9) A verbatim interpretation of the above definition leads to the concept of “geometricalRVEs”, which is based directly on the phase geometry of an inhomogeneous material, typ-ically characterized by statistical descriptors (Pyrz,1994; Zeman/Sejnoha,2001; Jeulin,2001).More generally, a representative volume element may be defined as a volume that shows thesame overall response with respect to some given physical property independently of its po-sition and orientation within the sample and irrespective of the applied boundary conditions(Hill,1963; Sab,1992; Huet,1999) with the exception of free surfaces. The sizes of such “physicalRVEs” depend on the physical property considered, compare e.g. (Kanit et al.,2003; Swami-nathan/Ghosh,2006) and it has been argued that no such RVE exists for softening materialbehavior (Gitman et al.,2007). The volume averages and higher statistical moments of stresses,strains, and other variables are the same for all physical RVEs and are equal to the correspondingensemble averages.

10) This requirement was symbolically denoted as MICROMESOMACRO by Hashin(1983), where MICRO and MACRO have their “usual” meanings and MESO stands for thelength scale of the homogenization volume. As noted by Nemat-Nasser (1999) it is the dimensionrelative to the microstructure relevant for a given problem that is important for the size of anRVE.

– 5 –

Other effective properties of inhomogeneous materials, e.g. tensors describing their ther-

mophysical behavior, can be obtained by analogous operations.

The resulting homogenized behavior of many multiphase materials can be idealized as

being statistically isotropic or quasi-isotropic (e.g. for composites reinforced with spher-

ical particles, randomly oriented particles of general shape or randomly oriented fibers,

many polycrystals, many porous and cellular materials, random mixtures of two phases)

or statistically transversely isotropic (e.g. for composites reinforced with aligned fibers

or platelets, composites reinforced with nonaligned reinforcements showing an axisym-

metric orientation distribution function, etc.), compare (Hashin,1983). Of course, lower

material symmetries of the homogenized response may also be found (e.g. in textured

polycrystals).

Statistically isotropic multiphase materials show the same overall behavior in all direc-

tions, and their effective elasticity tensors (or “material tensors”) and thermal expansion

tensors take the form (in Voigt notation)

E∗ =

E∗

11 E∗

12 E∗

12 0 0 0E∗

12 E∗

11 E∗

12 0 0 0E∗

12 E∗

12 E∗

11 0 0 00 0 0 E∗

44 0 00 0 0 0 E∗

44 00 0 0 0 0 E∗

44 = 12 (E∗

11 − E∗

12)

α∗ =

α∗

11

α∗

11

α∗

11

000

. (8)

Two independent parameters are sufficient for describing such an overall linear elastic

behavior (e.g. the effective Young’s modulus E∗=E∗

11 −2E∗

122

E∗

11+E∗

12, the effective Poisson

number ν∗=E∗

12

E∗

11+E∗

12, the effective shear modulus G∗=E∗

44=E∗/2(1 + ν∗), the effective

bulk modulus K∗=E∗/3(1− 2ν∗), or the effective Lame constants) and one is required

for the effective thermal expansion behavior in the linear range (the effective coefficient

of thermal expansion α∗=α∗

11).

The effective elasticity and thermal expansion tensors for statistically transversely

isotropic materials have the structure

E∗ =

E∗

11 E∗

12 E∗

12 0 0 0E∗

12 E∗

22 E∗

23 0 0 0E∗

12 E∗

23 E∗

22 0 0 00 0 0 E∗

44 0 00 0 0 0 E∗

44 00 0 0 0 0 E∗

66 = 12 (E∗

22 − E∗

23)

α∗ =

α∗

11

α∗

22

α∗

22

000

, (9)

where 1 is the axial direction and 2–3 is the transverse plane of isotropy. Generally, the

thermoelastic behavior of transversely isotropic materials is described by five independent

elastic constants and two independent coefficients of thermal expansion. Appropriate

elastic parameters in this context are e.g. the axial and transverse effective Young’s mod-

uli, E∗

A=E∗

11 −2E∗2

12

E∗

22+E∗

23and E∗

T=E∗

22 −E∗

11E∗223+E∗

22E∗212−2E∗

23E∗212

E∗

11E∗

22−E∗212

, the axial and transverse

effective shear moduli, G∗

A=E∗

44 and G∗

T=E∗

66, the axial and transverse effective Poisson

– 6 –

numbers, ν∗

A=E∗

12

E∗

22+E∗

23and ν∗

T=E∗

11E∗

23−E∗212

E∗

11E∗

22−E∗212

, as well as the effective transverse bulk mod-

ulus K∗

T=E∗

A/2[(1 − ν∗

T)(E∗

A/E∗

T) − 2ν∗

A2]. The transverse (“in-plane”) properties are

related via G∗

T=E∗

T/2(1+ν∗

T), but for general transversely isotropic materials there is no

linkage between the axial properties E∗

A, G∗

A and ν∗

A beyond the above definition of K∗

T.

Both an axial and a transverse effective coefficient of thermal expansion, α∗

A=α∗

11 and

α∗

T=α∗

22, are required. For the special case of materials reinforced by aligned continuous

fibers Hill’s (1964) relations

E∗

A = ξE(f)A + (1− ξ)E(m) +

4(ν(f)A − ν(m))2

(1/K(f)T − 1/K

(m)T )2

(

ξ

K(f)T

+1− ξ

K(m)T

−1

K∗

T

)

ν∗

A = ξν(f)A + (1− ξ)ν(m) +

ν(f)A − ν(m)

1/K(f)T − 1/K

(m)T

(

ξ

K(f)T

+1− ξ

K(m)T

−1

K∗

T

)

(10)

allow the effective moduli E∗

A and ν∗

A to be expressed by K∗

T, some constituent properties,

and the fiber volume fraction ξ. Equations (10) can be used to reduce the number of in-

dependent effective elastic parameters required for unidirectional continuously reinforced

composites to three.

The the overall material symmetries of inhomogeneous materials and their effect on var-

ious physical properties can be treated in full analogy to the symmetries of crystals as

discussed e.g. in (Nye,1957). The influence of the overall symmetry of the phase arrange-

ment on the overall mechanical behavior of inhomogeneous materials can be marked11),

especially on the post-yield and other nonlinear responses to mechanical loads, compare

e.g. (Nakamura/Suresh,1993; Weissenbek,1994). Accordingly, it is important to aim at

approaching the symmetry of the actual material as closely as possible in any modeling

effort.

1.4 Basic Modeling Strategies in Continuum Micromechanics of Materials

Homogenization procedures aim at finding a volume element’s responses to prescribed

mechanical loads (typically far field stresses or far field strains) or temperature excursions

and to deduce from them the corresponding overall properties. The most straightforward

application of such studies is materials characterization, i.e. simulating the overall mate-

rial response under simple loading conditions such as uniaxial tensile tests. Analysis of

this kind can be performed by all approaches described here (with the possible exception

of bounding methods).

Many homogenization procedures can also be employed directly as micromechanically

based constitutive material models at higher length scales, i.e. they can link the full ho-

mogenized stress tensor to the full homogenized strain tensor or provide the corresponding

11) Overall properties described by lower order tensors, e.g. the thermal expansion response,are less sensitive to symmetry effects, compare (Nye,1957).

– 7 –

instantaneous stiffnesses for nonlinear materials12). This, of course, requires the capabil-

ity of evaluating the overall response of an inhomogeneous material under any loading

condition and for any loading history, see also chapter 8. Compared to semiempirical

constitutive laws, as proposed e.g. in (Davis,1996), micromechanically based constitutive

models have both a clear physical basis and an inherent capability for “zooming in” on

the local phase stresses and strains by using localization procedures. The latter allow the

local response of the phases to be evaluated when the macroscopic response or state of a

sample or structure is known, e.g. for predicting the onset of plastic yielding or damage

of the constituents.

Besides materials characterization and constitutive modeling, there are a number of other

important applications of continuum micromechanics, among them studying local phe-

nomena in inhomogeneous materials, e.g. the initiation and evolution of damage on the

microscale, the nucleation and growth of cracks, the stresses at intersections between

macroscopic interfaces and free surfaces, the effects of local instabilities, and the inter-

actions between phase transformations and microstresses. For these behaviors details

of the microstructure tend to be of major importance and may determine the macro-

scopic response, an extreme case being the mechanical strength of brittle inhomogeneous

materials.

Because for realistic phase distributions the analysis of the spatial variations of the

microfields in sufficiently large volume elements tends to be beyond present capabilities,

approximations have to be used. For convenience, the majority of the resulting modeling

approaches may be treated as falling into two groups. The first of them comprises

methods that describe the microgeometries of inhomogeneous materials on the basis of

(limited) statistical information13):

• Mean Field Approaches (MFAs) and related methods (see chapter 2): The mi-

crofields within each constituent are approximated by their phase averages 〈ε〉(p)

and 〈σ〉(p), i.e. piecewise (phase-wise) uniform stress and strain fields are employed.

Such descriptions typically use information on the microscopic topology, the re-

inforcements’ shape and orientation, and, to some extent, on the statistics of the

12) The overall thermomechanical behavior of homogenized materials is typically richer thanthat of the constituents, i.e. the effects of the interaction of the constituents in many cases can-not be satisfactorily described by simply adapting material parameters without changing thefunctional relationships in the constitutive laws (for example, a composite consisting of a matrixthat follows J2 plasticity and elastic reinforcements has pressure dependent macroscopic plasticbehavior, and two constituents showing viscoplastic power law behavior with different expo-nents in general do not give rise to a power-law type overall response). Accordingly, predicteduniaxial stress–strain curves are not sufficient for calibrating macroscopic constitutive modelsfor describing multiaxial behavior.

13) General phase arrangements can be described by n-point correlation functions that givethe probability that a some geometrical configuration of n points lies within prescribed phases.For statistically isotropic materials 2-point correlations correspond to the volume fractions. Forlower overall symmetries they also provide information on the phases’ shapes.

– 8 –

phase distribution. The localization relations then take the form

〈ε〉(p) = A(p)〈ε〉

〈σ〉(p) = B(p)〈σ〉 (11)

and the homogenization relations become

〈ε〉(p) =1

Ω(p)

∫

Ω(p)

ε(x) dΩ with 〈ε〉 =∑

p

V (p)〈ε〉(p)

〈σ〉(p) =1

Ω(p)

∫

Ω(p)

σ(x) dΩ with 〈σ〉 =∑

p

V (p)〈σ〉(p) (12)

where (p) denotes a given phase of the material, Ω(p) is the corresponding phase

volume, and V (p)=Ω(p)/∑

k Ω(k) is the volume fraction of the phase. In contrast to

eqn.(2), for MFAs the phase concentration tensors A and B are not functions of

the spatial coordinates.

Mean field approaches tend to be formulated in terms of the phase concentra-

tion tensors, they pose relatively low computational requirements, they have been

highly successful in describing the thermoelastic response of inhomogeneous ma-

terials, and their use for modeling nonlinear inhomogeneous materials is a subject

of active research.

• Variational Bounding Methods (see chapter 3): Variational principles are used

to obtain upper and (in many cases) lower bounds on the overall elastic tensors,

elastic moduli, secant moduli, and other physical properties of inhomogeneous ma-

terials. Many analytical bounds are obtained on the basis of phase-wise constant

stress (polarization) fields. Bounds — in addition to their intrinsic value — are

important tools for assessing other models of inhomogeneous materials. Further-

more, in many cases one of the bounds provides good estimates for the physical

property under consideration, even if the bounds are rather slack (Torquato,1991).

Many bounding methods are closely related to MFAs.

The second group of approximations is based on studying discrete microstructures and

includes

• Periodic Microfield Approaches (PMAs), also referred to as periodic homogeniza-

tion schemes or unit cell methods, see chapter 5: The inhomogeneous material is

approximated by an infinitely extended model material with a periodic phase ar-

rangement14). The resulting periodic microfields are usually evaluated by analyz-

ing unit cells (which may describe microgeometries ranging from rather simplistic

to highly complex ones) via analytical or numerical methods. Unit cell methods

14) Whereas mean field methods make the implicit assumption that there is no long rangeorder in the inhomogeneous material, periodic phase arrangements imply the existence of justsuch an order.

– 9 –

are often used for performing materials characterization of inhomogeneous mate-

rials in the nonlinear range, but they can also be employed as micromechanically

based constitutive models. The high resolution of the microfields provided by

PMAs can be very useful for studying the initiation of damage at the microscale.

However, because they inherently give rise to periodic configurations of damage,

PMAs are not suited to investigating phenomena such as the interaction of the

microstructure with macroscopic cracks.

Periodic microfield approaches can give detailed information on the local stress

and strain fields within a given unit cell, but they tend to be computationally

expensive.

• Embedded Cell or Embedding Approaches (ECAs; see chapter 6): The inhomo-

geneous material is approximated by a model material consisting of a “core” con-

taining a discrete phase arrangement that is embedded within some outer region

to which far field loads or displacements are applied. The material properties of

this outer region may be described by some macroscopic constitutive law, they can

be determined self-consistently or quasi-self-consistently from the behavior of the

core, or the embedding region may take the form of a coarse description and/or

discretization of the phase arrangement. ECAs can be used for materials char-

acterization, and they are usually the best choice for studying regions of special

interest, such as crack tips and their surroundings, in inhomogeneous materials.

Like PMAs, embedded cell approaches can resolve local stress and strain fields in

the core region at high detail, but tend to be computationally expensive.

• Windowing Approaches (see chapter 7): Subregions (“windows”) — typically of

rectangular or hexahedral shape — are chosen (“cut out”) from a given phase

arrangement and subjected to boundary conditions that guarantee energy equiv-

alence between the micro- and macroscales. For the special cases of macroho-

mogeneous stress and strain boundary conditions, respectively, lower and upper

estimates for and bounds on the overall behavior of the inhomogeneous material

can be obtained.

• Other homogenization approaches employing discrete microgeometries, such as the

statistics-based non-periodic homogenization scheme of Cui et al. (Li/Cui,2005).

Because this group of methods explicitly study mesodomains (in the sense of section 1.2)

they are sometimes referred to as “mesoscale approaches”. Figure 1 shows a sketch of a

microstructure as well as PMA, ECA and windowing approaches applied to it.

For small inhomogeneous samples (i.e. samples the size of which exceeds the microscale by

not more than, say, an order of magnitude) the full microstructure can be modeled and

studied, see e.g. (Guidoum,1994; Papka/Kyriakides,1994; Silberschmidt/Werner,2001;

Luxner et al.,2005). In such cases boundary effects (and thus the boundary conditions

applied to the sample as well as the sample’s size in terms of the characteristic length of

the inhomogeneities) typically play a prominent role in the mechanical behavior. Note

that, even though such problems certainly pertain to continuum micromechanics, they

– 10 –

ORIGINAL CONFIGURATION PERIODIC APPROXIMATION

WINDOWEMBEDDED CONFIGURATION

FIG.1: Schematic sketch of a random matrix–inclusion microstructure and of the mod-ified microgeometries used by periodic microfield, embedded cell and windowing ap-proaches for modeling it.

do not involve a scale transition and describe somewhat different physical problems com-

pared to mesoscale approaches.

Further descriptions of inhomogeneous materials, such as rules of mixture (isos-

train and isostress models) and semiempirical formulae like the Halpin–Tsai equations

(Halpin/Kardos,1976), are not discussed here due to their typically rather weak physi-

cal background and their rather limited predictive capabilities. For brevity, a number

of models with solid physical backgrounds, such as expressions for self-similar compos-

ite sphere assemblages (CSA, Hashin,1962) and composite cylinder assemblages (CCA,

Hashin/Rosen,1964), information theoretical approaches (Kreher,1990), and the semiem-

pirical methods proposed for describing the elastoplastic behavior of composites rein-

forced by aligned continuous fibers (Dvorak/Bahei-el-Din,1987), are not covered within

the present discussions, either.

For studying materials that are inhomogeneous at a number of (sufficiently widely spread)

length scales (e.g. materials in which well defined clusters of inhomogeneities are present),

hierarchical procedures that use homogenization at more than one level are a natural

extension of the above concepts. Such multi-scale models are the subject of a short

discussion in chapter 8.

– 11 –

2. MEAN FIELD APPROACHES

In this section relations are given for two-phase materials only, extensions to multiphase

materials being rather straightforward in most cases. Special emphasis is put on effective

field methods of the Mori–Tanaka type, which may be viewed as the simplest mean

field approaches for modeling inhomogeneous materials that encompass the full physical

range of phase volume fractions15). Unless specifically stated otherwise, the material

behavior of both reinforcements and matrix is taken to be linear (thermo)elastic. Perfect

bonding between the constituents is assumed in all cases. There is an extensive body

of literature covering mean field approaches, most of which can be classified as effective

field or effective medium theories, and related methods, so that the following treatment

is far from complete.

2.1 General Relations

For linear thermoelastic materials the overall stress–strain relations can be denoted in

the form

〈σ〉 = E∗〈ε〉+ t∗∆T

〈ε〉 = C∗〈σ〉+ α∗∆T , (13)

where t∗ = −E∗α∗ is the overall specific thermal stress tensor (i.e. the overall stress

response of the fully constrained material to a purely thermal unit load) and ∆T is

the (spatially uniform) temperature difference with respect to some stress-free reference

temperature. The constituents, here a matrix (m) and inhomogeneities (i), are also taken

to behave thermoelastically, i.e.

〈σ〉(m) = E(m)〈ε〉(m) + t(m)∆T 〈σ〉(i) = E(i)〈ε〉(i) + t(i)∆T

〈ε〉(m) = C(m)〈σ〉(m) + α(m)∆T 〈ε〉(i) = C(i)〈σ〉(i) + α(i)∆T , (14)

where the relations t(m) = −E(m)α(m) and t(i) = −E(i)α(i) hold16).

From the definition of volume averaging, eqn.(3), and from eqn.(12) one obtains the

following relations for the phase averaged fields

〈ε〉 = ξ〈ε〉(i) + (1− ξ)〈ε〉(m) = εa

〈σ〉 = ξ〈σ〉(i) + (1− ξ)〈σ〉(m) = σa , (15)

15) Because Eshelby and Mori–Tanaka methods are specifically suited for matrix–inclusion-type microtopologies, the expression “composite” is often used in the present chapter instead ofthe more general designation “inhomogeneous material”.

16) In mean field theories the constitutive relations such as eqns.(13) and (14) are typicallyformulated in terms of total averaged strains and thus of the averaged fields defined in eqns.(3)and (12). Contributions of inelastic strains in Hooke’s law are accounted for by appropriateeigenstresses s, which in the case of thermoelasticity take the form s = t∆T = −Eα∆T .

– 12 –

provided no displacement jumps are present. Here ξ=V (i)=Ω(i)/(Ω(i) + Ω(m)) stands for

the volume fraction of the inhomogeneities, εa is the applied (far field) strain and σa is

the applied (far field) stress. In the isothermal case, the equalities 〈ε〉=εa and 〈σ〉=σa

correspond to homogeneous strain and homogeneous stress boundary conditions, respec-

tively17), perfectly bonded interfaces being also assumed in the former case.

The phase averaged strains and stresses can be related to the overall strains, stresses,

and temperature changes by the elastic and thermal phase strain and stress concentration

tensors A, a, B, and b (Hill,1963), respectively, which for thermoelastic inhomogeneous

materials are defined by the expressions

〈ε〉(m) = A(m)〈ε〉+ a(m)∆T 〈ε〉(i) = A(i)〈ε〉+ a(i)∆T

〈σ〉(m) = B(m)〈σ〉+ b(m)∆T 〈σ〉(i) = B(i)〈σ〉+ b(i)∆T (16)

(compare eqn.(11) for the isothermal elastic case). By using eqns.(15) and (16), the

phase strain and stress concentration tensors (or strain and stress concentration tensors

for short) can be shown to fulfill the relations

ξA(i) + (1− ξ)A(m) = I ξa(i) + (1− ξ)a(m) = 0

ξB(i) + (1− ξ)B(m) = I ξb(i) + (1− ξ)b(m) = 0 . (17)

From eqn.(7) the effective elasticity and compliance tensors of the composite can be

obtained from the properties of the phases and from the concentration tensors as18)

E∗ = ξE(i)A(i) + (1− ξ)E(m)A(m)

C∗ = ξC(i)B(i) + (1− ξ)C(m)B(m) . (18)

The effective specific thermal stress tensor and the effective thermal expansion coefficient

tensor can be written as

t∗ = ξ[E(i)a(i) + t(i)] + (1− ξ)[E(m)a(m) + t(m)]

α∗ = ξ[C(i)b(i) + α(i)] + (1− ξ)[C(m)b(m) + α(m)] , (19)

respectively. Alternatively, the overall coefficients of thermal expansion can be obtained

from the Mandel–Levin formula (Mandel,1965; Levin,1967) as

α∗ = (1− ξ)[B(m)]T α(m) + ξ[B(i)]T α(i) , (20)

17) By using the Gauss theorem it can be shown theorem that the average stresses in a volumeelement of an inhomogeneous material with perfect phase boundaries are fully determined bythe tractions on its boundaries and the average strains are determined by the displacements atthe boundaries also for more general boundary conditions, compare eqn.(3).

18) Because E∗ and its inverse C∗ depend not only on the properties of the constituents,but also on the concentration tensors and thus on the microscopic phase arrangement, it can beargued that overall properties of inhomogeneous materials, e.g. their effective Young’s modulus,are not material parameters in the strict sense.

– 13 –

where BT denotes the transpose of B.

The stress and strain concentration tensors for a given phase are linked to each other by

expressions such as

A(m) = C(m)B(m)E(i)[I + (1− ξ)(C(m) −C(i))B(m)E(i)]−1 = C(m)B(m)E∗

B(m) = E(m)A(m)C(i)[I + (1− ξ)(E(m) − E(i))A(m)C(i)]−1 = E(m)A(m)C∗ . (21)

In addition, relations can be derived that connect the thermal strain and stress concen-

tration tensors of a given phase with the elastic strain and stress concentration tensors,

respectively (see e.g. Benveniste/Dvorak,1990; Benveniste et al.,1991), typical expres-

sions being19)

a(m) = [I− A(m)][E(i) −E(m)]−1[t(m) − t(i)]

b(m) = [I− B(m)][C(i) −C(m)]−1[α(m) − α(i)] . (22)

From eqns.(18) to (22) it is evident that the knowledge of one elastic phase concentration

tensor is sufficient for describing the full thermoelastic behavior of a two-phase inhomo-

geneous material within the mean field framework20). There are a number of additional

relations that allow e.g. the phase concentration tensors to be obtained from the overall

and phase elastic tensors, see e.g. (Bohm,1998).

The macrohomogeneity condition and Hill’s lemma, eqn.(5), can be denoted for elastic

inhomogeneous materials as

〈σT ε〉 =∑

p

1

Ω(p)

∫

Ω(p)

εT (x)E(p) ε(x) dΩ = 〈ε〉T〈σ〉 = 〈ε〉

TE∗〈ε〉 . (23)

For detailed discussions of Hill’s lemma for the case of periodicity boundary conditions

see e.g. (Suquet,1987; Michel et al.,1999).

2.2 Eshelby Tensor and Dilute Matrix–Inclusion Composites

A large proportion of the mean field descriptions used in continuum micromechanics of

materials are based on the work of Eshelby (1957), who studied the stress and strain

distributions in homogeneous media containing a subregion that spontaneously changes

its shape and/or size (undergoes a “transformation”) so that it no longer fits into its pre-

vious space in the parent medium. Eshelby’s results show that if an elastic homogeneous

ellipsoidal inclusion in an infinite linear elastic matrix is subjected to a uniform strain εt

(called the “stress-free strain”, “unconstrained strain”, “eigenstrain”, or “transformation

19) Relations for the inclusion concentration tensors corresponding to eqns.(21) and (22) takeanalogous forms.

20) Similarly, n−1 elastic phase concentration tensors must be known for describing theoverall thermoelastic behavior of an n-phase material.

– 14 –

strain”), uniform stress and strain states are induced in the constrained inclusion21). The

uniform strain in the constrained inclusion (the “constrained strain”), εc, is related to

the stress-free strain εt by the expression

εc = Sεt , (24)

where S is called the (interior point) Eshelby tensor. For eqn.(24) to hold, εt may be any

kind of eigenstrain which is uniform over the inclusion (e.g. a thermal strain, or a strain

due to some phase transformation which involves no changes in the elastic constants of

the inclusion).

For mean field descriptions of dilute matrix–inclusion composites, of course, it is the

stress and strain fields in inhomogeneous inclusions (“inhomogeneities”) embedded in

a matrix that are of interest. Such cases can be handled on the basis of Eshelby’s

theory for homogeneous inclusions, eqn.(24), by introducing the concept of equivalent

homogeneous inclusions. This strategy involves replacing an actual perfectly bonded

inhomogeneous inclusion, which has different material properties than the matrix and

which is subjected to a given unconstrained eigenstrain εt, with a (fictitious) “equivalent”

homogeneous inclusion on which an appropriate (fictitious) “equivalent” eigenstrain ετ

is made to act. This equivalent eigenstrain must be chosen in such a way that the same

stress and strain fields, 〈σ〉(i) and 〈ε〉(i)=εc, are obtained in the constrained state for the

actual inhomogeneous inclusion and the equivalent homogeneous inclusion. Following

(Withers et al.,1989) this process can be visualized as consisting of “cutting and welding”

operations as shown in fig. 2.

The concept of the equivalent homogeneous inclusion can be extended to cases where

a uniform mechanical strain εa or external stress σa is applied to a perfectly bonded

inhomogeneous elastic inclusion in an infinite matrix. Here, the strain in the inclusion,

〈ε〉(i), will be a superposition of the applied strain and of an additional term εc due to the

constraint effects of the surrounding matrix. A fair number of different expressions for

concentration tensors obtained by such procedures have been reported in the literature,

among the simplest being those proposed by Hill (1965) and elaborated by Benveniste

(1987). For deriving them, the condition of equal stresses and strains in the actual

inclusion (elasticity tensor E(i)) and the equivalent inclusion (elasticity tensor E(m)) under

an applied far field strain εa is denoted as

〈σ〉(i) = E(i)〈ε〉(i) = E(m)(〈ε〉(i) − ετ ) ,

〈ε〉(i) = εa + εc = εa + Sετ (25)

where eqn.(24) is used to describe the constrained strain of the equivalent homogeneous

inclusion. From these expressions and from eqn.(16) the inclusion strain concentration

tensor can be obtained as

A(i)dil = [I + SC(m)(E(i) −E(m))]−1 (26)

21) For dilute inclusions this “Eshelby property” is limited to ellipsoidal shapes(Lubarda/Markenscoff,1998). For nondilute periodic arrangements of inclusions, however, non-ellipsoidal shapes give rise to homogeneous fields (Liu et al.,2007).

– 15 –

FIG.2: Sketch depicting an equivalent inclusion procedure for a matrix–inclusion systemloaded by a transformation strain εt (Withers et al.,1989).

and the inclusion stress concentration tensor is found as

B(i)dil = E(i)[I + SC(m)(E(i) − E(m))]−1C(m)

= [I + E(m)(I− S)(C(i) −C(m))]−1 . (27)

compare (Hill,1965; Benveniste,1987). Alternative expressions for dilute mechanical and

thermal inclusion concentration tensors were given e.g. by Mura (1987), Wakashima et

al. (1988), and Clyne/Withers (1993). All of the above relations were derived under the

assumption that the inhomogeneities are dilutely dispersed in the matrix and thus do

not “feel” any effects due to their neighbors (i.e. they are loaded by the unperturbed

applied stress σa or applied strain εa, the so called dilute case). Accordingly, they are

independent of the reinforcement volume fraction ξ.

The stress and strain fields outside a transformed homogeneous or inhomogeneous inclu-

sion in an infinite matrix are not uniform on the microscale22) (Eshelby,1959). Within

the framework of mean field theories, which aim to link the average fields in matrix and

inhomogeneities with the overall response of inhomogeneous materials, however, it is only

the average matrix stresses and strains that are of interest. For dilute composites, such

expressions follow directly by combining eqns.(26) and (27) with eqn.(17). Estimates for

22) The fields outside a single inclusion can be described via the (exterior point) Eshelbytensor, see e.g. (Ju/Sun,1999). From the (constant) interior point fields and the (position)dependent exterior point fields the stress and strain jumps at the interface between inclusionand matrix can be evaluated.

– 16 –

the overall elastic and thermal expansion tensors can, of course, be obtained in a straight-

forward way from the concentration tensors by using eqns.(19) and (22). It must be kept

in mind, however, that the dilute expressions are strictly valid only for vanishingly small

inhomogeneity volume fractions and give dependable results only for ξ 0.1.

The Eshelby tensor S depends only on the material properties of the matrix and on the as-

pect ratio a of the inclusions, i.e. expressions for the Eshelby tensor of ellipsoidal inhomo-

geneities are independent of the material symmetry and properties of the inhomogeneities.

Expressions for the Eshelby tensor of spheroidal inclusions in an isotropic matrix are

given e.g. in (Pedersen,1983; Tandon/Weng,1984; Mura,1987; Clyne/Withers,1993)23),

the formulae being very simple for continuous fibers of circular cross-section (a → ∞),

spherical inclusions (a = 1), and thin circular disks (a → 0). Closed form expressions

for the Eshelby tensor have also been reported for spheroidal inclusions in a matrix of

transversely isotropic (Withers,1989) or cubic material symmetry (Mura,1987), provided

the material axes of the matrix constituents are aligned with the orientations of non-

spherical inclusions. In cases where no analytical solutions are available, the Eshelby

tensor can be evaluated numerically, compare (Gavazzi/Lagoudas,1990).

2.3 Mean Field Methods for Thermoelastic Composites with Aligned

Reinforcements

Theoretical descriptions of the overall thermoelastic behavior of composites with re-

inforcement volume fractions of more than a few percent must explicitly account for

collective interactions between inhomogeneities, i.e. for the effects of all surrounding

reinforcements on the stress and strain fields experienced by a given fiber or particle.

Within the mean field framework such reinforcement interaction effects as well as the

concomitant perturbations of the stress and strain fields in the matrix are accounted for

via suitable phase-wise constant approximations.

Beyond these “background effects”, there are interactions between individual reinforce-

ments which, on the one hand, give rise to inhomogeneous stress and strain fields within

each inhomogeneity (“intra-particle fluctuations”). On the other hand, they cause the

levels of the average stresses and strains in individual inhomogeneities to differ (“inter-

particle fluctuations”). These interactions are not accounted for by most mean field

methods24).

Mori–Tanaka-Type Estimates

One way for achieving this consists of approximating the stresses acting on an inhomo-

geneity, which may be viewed as the perturbation stresses caused by the presence of other

23) Instead of evaluating the Eshelby tensor for a given configuration, the so calledmean polarization factor tensor Q=E(m)(I − S) may be evaluated instead, see e.g. (PonteCastaneda,1996).

24) In the literature the designator “interacting models” is typically used for approaches thatexplicitly account for interactions between pairs or higher numbers of inhomogeneities, i.e., notfor Mori-Tanaka and self-consistent methods.

– 17 –

inhomogeneities (“image stresses”, “background stresses”, “mean field stresses”, “effec-

tive stresses”) superimposed on the applied far field stress, by an appropriate average

matrix stress 〈σ〉(m). The idea of combining this concept of an average matrix stress with

Eshelby-type equivalent inclusion approaches goes back to Brown and Stobbs (1971) as

well as Mori and Tanaka (1973). Effective field theories of this type are generically called

Mori–Tanaka methods.

It was recognized by Benveniste (1987) that in the isothermal case the central assumption

involved in Mori–Tanaka approaches can be expressed as

〈ε〉(i) = A(i)dil〈ε〉

(m) = A(i)dilA

(m)M 〈ε〉

〈σ〉(i) = B(i)dil〈σ〉

(m) = B(i)dilB

(m)M 〈σ〉 . (28)

These relations can be viewed as modifications of eqn.(11) in which the applied strain

and stress, εa and σa, are replaced by the (unknown) matrix strain or stress, 〈ε〉(m) and

〈σ〉(m), respectively, according to the effective field strategy.

Based on eqn.(28) simple and straightforward Mori–Tanaka-type expressions for the ma-

trix strain and stress concentrations tensors were proposed by (Benveniste,1987) as

A(m)M = [(1− ξ)I + ξA

(i)dil]

−1

B(m)M = [(1− ξ)I + ξB

(i)dil]

−1 , (29)

the corresponding relations for the inclusion strain and stress concentration tensors fol-

lowing from eqns.(28) as

A(i)M = A

(i)dil[(1− ξ)I + ξA

(i)dil]

−1

B(i)M = B

(i)dil[(1− ξ)I + ξB

(i)dil]

−1 , (30)

Equations (29) and (30) may be evaluated with any strain and stress concentration

tensors A(i)dil and B

(i)dil pertaining to dilute inhomogeneities in a matrix. If, for example,

eqns.(26) and (27) are employed, the Mori–Tanaka matrix strain and stress concentration

tensors take the form

A(m)M = (1− ξ)I + ξ[I + SC(m)(E(i) −E(m))]−1−1

B(m)M = (1− ξ)I + ξE(i)[I + SC(m)(E(i) − E(m))]−1C(m)−1 , (31)

compare (Benveniste,1987; Benveniste/Dvorak,1990; Benveniste et al.,1991).

A number of authors gave different but essentially equivalent Mori–Tanaka-type expres-

sions for the phase concentration tensors and effective thermoelastic tensors of inhomo-

geneous materials, among them (Pedersen,1983; Wakashima et al.,1988; Taya et al.,1991;

Pedersen/Withers,1992; Clyne/Withers,1993). Alternatively, the Mori–Tanaka method

can be formulated (Tandon/Weng,1984) to directly give the overall elasticity tensor as

E∗

M = E(m)

I− ξ[(E(i) −E(m))(

S− ξ(S− I))

+ E(m)]−1[E(i) −E(m)]

−1, (32)

– 18 –

FIG.3: Sketch of an aligned ellipsoidally distributed spatial arrangement of aligned el-lipsoidal inhomogeneities that is closely related to Mori–Tanaka-type approaches (a=2.0).

which can be easily modified for porous materials by letting E(i) → 0 to give

E∗

M,por = E(m)[

I +ξ

1− ξ(I− S)−1

]

−1. (33)

Equation (33) should not, however, be used for void volume fractions that are much in

excess of, say, ξ = 0.2525).

In accordance with their derivation, Mori–Tanaka-type theories at all volume fractions

describe composites consisting of aligned ellipsoidal inhomogeneities embedded in a ma-

trix, i.e. inhomogeneous materials with a distinct matrix–inclusion microtopology. More

precisely, it can be shown (Ponte Castaneda/Willis,1995) that Mori–Tanaka methods are

a special case of Hashin–Shtrikman variational estimates (compare section 2.4) in which

the reference material is the matrix and the spatial arrangement of the (aligned) inhomo-

geneities follows an “ellipsoidal distribution” with the same aspect ratio and orientation

as the inhomogeneities themselves, compare fig. 326).

For two-phase composites Mori–Tanaka estimates coincide with Hashin–Shtrikman-type

bounds (compare section 3.1) and, accordingly, their predictions for the overall Young’s

and shear moduli are always on the low side for composites reinforced by stiff aligned

or spherical reinforcements (see the comparisons in section 3.4 as well as tables 2 to

6) and on the high side for materials containing compliant reinforcements in a stiffer

matrix. In high contrast situations they tend to under- or overestimate the effective

25) Mori–Tanaka expressions for two-phase materials are based on the assumption that theshape of the inhomogeneities can be described by ellipsoids of a fixed aspect ratio throughout thedeformation history. In the case of porous materials with high void volume fractions, e.g. closed-cell cellular materials and foams, deformation at the microscale takes place mainly by bendingand buckling of the matrix “walls” between voids, see e.g. (Gibson/Ashby,1988). Such largestrain effects cannot be properly described by Mori–Tanaka approaches, which, consequently,tend to markedly overestimate the effective stiffness of cellular materials.

26) Geometries with ellipsoidal symmetry as shown in fig. 3 can be thought of as affinelydeformed versions of the composite sphere assemblage (CSA) geometries that were proposed byHashin (1962).

– 19 –

elastic properties by a considerable margin (compare table 6). Similar tendencies hold

for multi-phase composites. For discussions of further issues with respect to the range

of validity of Mori–Tanaka theories for elastic inhomogeneous two-phase materials see

(Christensen et al.,1992).

Mori–Tanaka-type theories can be implemented into computer programs in a straightfor-

ward way. Because they are explicit algorithms, all that is required are matrix additions,

multiplications, and inversions plus expressions for the Eshelby tensor. Despite their

limitations, Mori–Tanaka approaches provide useful accuracy for the elastic contrasts

pertaining to most practically relevant composites. This combination of features makes

them important tools for evaluating the stiffness and thermal expansion properties of

inhomogeneous materials that show a matrix–inclusion topology with aligned inhomo-

geneities or voids. Mori–Tanaka-type approaches for thermoelastoplastic materials are

discussed in section 2.4 and “extended” Mori–Tanaka approaches for nonaligned rein-

forcements in section 2.7.

Classical Self-Consistent Estimates

Another group of estimates for the overall thermomechanical behavior of inhomogeneous

materials are self-consistent schemes (or effective medium theories), in which an inhomo-

geneity or some phase arrangement is embedded in the effective material (the properties

of which are not known a priori)27). Figure 4 shows a schematic comparison of the mate-

rial and loading configurations underlying Eshelby, Mori–Tanaka, classical self-consistent,

and generalized self-consistent (compare section 3.4) approaches.

effeffmmm

i i i i

σ σ σσa tot a a(m)

GSCSCSCSMTMdiluteFIG.4: Schematic comparison of the Eshelby method for dilute composites, Mori–Tanaka approaches, and classical as well as generalized self-consistent schemes.

If the kernel consists of an inhomogeneity only, classical (or two-phase) self-consistent

schemes are obtained (CSCS, see e.g. Hill,1965), which formally can be described by

relations such as

E∗

SCS = E(m) + ξ[E(i) −E(m)]Ai,∗dil . (34)

Because Ai,∗dil, the strain concentration tensor of an inhomogeneity embedded in the

effective medium in analogy to eqn.(26), is a function of E∗

SCS, this expression is implicit

27) Within the classification of micromechanical methods given in section 1.4, self-consistentmean field methods can also be viewed as as analytical ECAs.

– 20 –

and E∗

SCS has to be solved for by self-consistent iteration using a scheme of the type

En+1 = E(m) + ξ[E(i) − E(m)][I + SnCn(E(i) −En)]−1

Cn+1 =[

En+1

]

−1. (35)

The Eshelby tensor used in this expression, Sn, pertains to an inclusion embedded in

the n-th iteration for the effective medium and has to be recomputed for each iteration.

Generally, classical self-consistent schemes are best suited to describing the overall elastic

properties of two-phase materials that do not show a matrix–inclusion microtopology in

at least part of their range of volume fractions28), but they have also been employed for

studying particle reinforced MMCs in which the grain size of the matrix and the size of

the particulates are similar (Corvasce et al.,1991). When microstructures described by

the CSCS show a geometrical anisotropy of the transversely isotropic type (e.g. in terms

of the free lengths in the phases in metallographical sections), this anisotropy determines

the “inhomogeneity” aspect ratio to be used.

For porous materials classical self-consistent schemes predict a breakdown of stiffness

due to the percolation of the pores at ξ = 12 for spherical voids and at ξ = 1

3 for aligned

cylindrical voids (Torquato,2002).

Because self-consistent schemes are by definition implicit methods, their computational

requirements are in general higher than those of Mori–Tanaka-type approaches. Like

Mori–Tanaka approaches, they can form the basis for describing the behavior of nonlinear

inhomogeneous materials.

Differential Schemes

A further group of mean field approaches besides effective field and self-consistent meth-

ods are differential schemes (McLaughlin,1977; Norris,1985). They can be envisaged as

involving repeated cycles of adding small concentrations of inhomogeneities to a mate-

rial followed by homogenizing. According to (Hashin,1988) the resulting overall elastic

tensors can be described by the differential equations

dE∗

D

dξ=

1

1− ξ(E(i) − E∗

D)Ai,∗dil

dC∗

D

dξ=

1

1− ξ(C(i) −C∗

D)Bi,∗dil (36)

with the initial conditions E∗

D=E(m) and C∗

D=C(m), respectively, at ξ=0. In analogy to

eqn.(34) Ai,∗dil and B

i,∗dil depend on E∗

D. Differential schemes describe aligned matrix–

28) Classical self consistent schemes can be shown to correspond to perfectly disorderedmaterials (Kroner,1978) or self-similar hierarchical materials (Torquato,2002).In the context of two-phase materials typical targets are interwoven composites and functionallygraded materials. Multi-phase expressions analogous to eqn.(34) are often used to describe themechanical behavior of polycrystals.

– 21 –

inclusion microgeometries with markedly polydisperse distributions of inhomogeneity

sizes (corresponding to the repeated homogenization steps) 29).

Because of their association with specialized microgeometries (that are not typical of

“classical composites”, in which reinforcements tend to be monodispersely sized) and due

to their higher mathematical complexity (as compared e.g. to Mori–Tanaka-approaches)

Differential Schemes have not been used extensively for studying the mechanical behavior

of composite materials.

2.4 Hashin–Shtrikman Estimates

The stress field in an inhomogeneous material can be expressed in terms of the one in a

homogeneous comparison (or reference) medium as

σ(x) = E0ε(x) + τ(x) = E0(ε0 + ε 0(x)) + τ(x) (37)

where τ(x) is known as the polarization tensor, E0 is the elastic tensor of the comparison

material, ε0 is the homogeneous strain in the reference material, and ε 0(x) is the local

strain fluctuation tensor associated with τ(x). Equation (37) can be interpreted as using

the polarizations for describing the “fast” contributions to the actual stress field, σ(x),

while the “slow” contributions are covered by the behavior of the reference material.

Phase-wise constant polarization tensors

τ (p) = (E(p) − E0)〈ε〉(p) (38)

can be obtained by phase averaging τ over phase (p).

Equation (38) can be used as the starting point for deriving general mean field strain

concentration tensors of the type

A(p)HS = (L0 + E(p))−1

[

ξ(L0 + E(i))−1 + (1− ξ)(L0 + E(m))−1]

−1. (39)

This expression leads to estimates of the overall elastic stiffness of microstructured two-

phase materials in the form

E∗

HS = E(m) + ξ(L0 − E(m))[

I + (1− ξ)(L0 + E(m))−1(E(i) − E(m))]

−1, (40)

which are referred to as Hashin–Shtrikman elastic tensors. The tensor L0 appearing in

eqns.(39) and (40) is known as the overall constraint tensor (Hill,1965) and is defined as

L0 = E0[(S0)−1 − I] . (41)

29) With the exception of the CSCS for two-dimensional conduction (Torquato/Hyun,2001),the mean field methods discussed in section 2.3 do not exactly correspond to monodisperseinclusion–matrix microstructures, i.e., they are not realizable as single-scale dispersions.

– 22 –

Here the Eshelby tensor S0 must be evaluated with respect to the comparison mate-

rial. L0 can be shown to depend on the two-point statistics of the microgeometrical

arrangement, see (Bornert,2001).

Standard mean field models can be obtained as special cases of Hashin–Shtrikman meth-

ods by appropriate choices of the comparison material. For example, using the matrix as

the comparison material results in Mori–Tanaka methods, whereas the selection of the

effective material as comparison material leads to classical self-consistent schemes.

Generalized Self-Consistent Estimates

The above Hashin–Shtrikman formalism can be extended in a number of ways, one of

which consists in not only studying inhomogeneities but more complex geometrical enti-

ties, called patterns or motifs, embedded in a matrix, see (Bornert,1996; Bornert,2001).

Typically the stress and strain fields in such patterns are inhomogeneous even in the di-

lute case and numerical methods have to be resorted to in evaluating the corresponding

overall constraint tensors. A simple example of such a pattern is a spherical particle

or cylindrical fiber surrounded by a layer of matrix at an appropriate volume fraction,

which, when embedded in the effective material, allows to recover the three-phase or gen-

eralized self-consistent scheme (GSCS) of (Christensen/Lo,1979; Christensen/Lo,1986),

compare fig. 4. GSCS-expressions for the overall elastic moduli are obtained by consider-

ing the differential equations describing the elastic response of such three-phase regions

under appropriate boundary and loading conditions. The GSCS in its original version

is limited to inhomogeneities of spherical or cylindrical shape, where it describes mi-

crogeometries of the CSA and CCA types, respectively, and gives rise to third order

equations for the shear modulus G and the transverse shear modulus GT (for the other

moduli, CSA and CCA results are available). Extensions of the GSCS to multi-layered

spherical inhomogeneities (Herve/Zaoui,1993) and to aligned ellipsoidal inhomogeneities

(Huang/Hu,1995), were reported in the literature.

Generalized self-consistent schemes give excellent results for inhomogeneous materials

with matrix–inclusion topologies and are accordingly highly suited for obtaining estimates

for the thermoelastic moduli of composite materials reinforced by (approximately) spher-

ical particles or aligned continuous fibers. They are, however, not mean-field schemes30)

in the sense of section 2.3.

2.5 Other Estimates for Elastic Composites with Aligned Reinforcements

A number of interpolative schemes have been proposed, which generate estimates between

mean-field and related estimates describing, on the one hand, some composite and, on the

other hand, a material in which the roles of matrix and reinforcements are interchanged

(referred to, e.g., as Mori–Tanaka and Inverse Mori–Tanka schemes), see e.g. (Lielens

30) Even though GSCS schemes are not a mean field methods in the strict sense, pertinentphase concentration tensors can easily be obtained from the overall elastic tensors, so that meanfield formalisms may be used to generate e.g. effective coefficients of thermal expansion.

– 23 –

et al.,1997). Such approaches, while being capable of providing improved estimates

for given composites, suffer from the drawback of not being associated with realizable

microgeometries.

Recently three-point estimates for the thermoelastic properties of two-phase materials

were developed by Torquato (1998), which correspond to the same phase arrangements

and use the same three-point microstructural parameters η and ζ as discussed for three-

point bounds in section 3.2. These estimates lie between the corresponding bounds and

give excellent analytical predictions for the overall thermoelastic response of inhomoge-

neous materials that show the appropriate microstructures and are free of flaws.

2.6 Mean Field Methods for Inelastic Composites

Models for microstructured materials with viscoelastic constituents are closely related to

those for elastic composites. Relaxation moduli and creep compliances can be obtained

by applying micromechanical approaches in the Laplace transformed domain, where the

problems are analogous to elastic ones for the same microgeometries. Standard mi-

cromechanical methods can be used in conjunction with complex moduli for steady state

vibration analysis of viscoelastic composites. For correspondence principles between de-

scriptions pertaining to elastic and viscoelastic inhomogeneous materials, respectively,

see e.g. (Hashin,1983). The extension of mean field estimates and bounding methods to

elastoplastic, viscoelastoplastic, and damaging inhomogeneous materials, however, has

proven to be challenging.

(Thermo)Elastoplastic Composites

The aim of mean field models of microstructured materials with at least one elastoplastic

constituent consists in providing accurate estimates for the material response for any

load state and load history at reasonable computational cost. The main difficulties in

attaining this goal lie in the typically strong intra-phase fluctuations of the microstress

and microstrain fields in elastoplastic inhomogeneous materials and in the path depen-

dence of plastic behavior. The responses of the constituents can thus vary markedly at

the microscale, with each material point essentially following a different trajectory in

stress space. Accordingly, even a 2-phase elastoplastic composite effectively behaves as

a multiphase material and phase averages over matrix and reinforcements are less useful

descriptors than in the linear elastic case.

Mean field models of elastoplastic inhomogeneous materials31) typically are based on

solving sequences or sets of linearized problems in terms of linear inhomogeneous ref-

erence media. Accordingly, choices have to be made with respect to the linearization

procedure, the linear homogenization model, and the phase-wise equivalent stresses and

31) The present chapter is restricted to continuum mechanical descriptions of thermoelasto-plastic composites. In a separate line of development, phase averaged stress fields obtainedby Eshelby-type methods have been used in dislocation-based descriptions of nonlinear ma-trix behavior, e.g. for studying the response of MMCs to temperature excursions, see e.g.(Taya/Mori,1987).

– 24 –

equivalent strains to be used in evaluating the elastoplastic constituent material behavior,

see (Zaoui,2001).

In the literature several lines of development of mean field and Hashin–Shtrikman-type

approaches for elastoplastic inhomogeneous materials can be found. Historically, the

most important of them have been secant plasticity concepts based on deformation

theory, see e.g. (Tandon/Weng,1988; Dunn/Ledbetter,1997), and incremental plastic-

ity models, compare e.g. (Hill,1965; Lagoudas et al.,1991; Karayaka/Sehitoglu,1993;

Pettermann,1997). Other relevant methods are the “tangent” concept (Molinari et

al.,1987) and mean field applications of the Transformation Field Analysis (TFA; Dvo-

rak/Benveniste,1992; Dvorak,1992). The latter describes inelastic strains via transfor-

mation strains and uses elastic accommodation for stresses and strains in the elasto-

plastic regime, resulting in computationally efficient algorithms. A further recent de-

velopment are affine formulations (Masson et al.,2000), in which the behavior of vis-

coplastic composites is studied on the basis of thermoelastic problems. A fairly recent

overview of mean field methods for elastoplastic composites can be found in (Ponte

Castaneda/Suquet,1998).

Secant models aim at directly arriving at solutions in terms of the overall response for

a given load state. They are based on the deformation theory of plasticity and can be

formulated in terms of potentials, which allows for a concise mathematical presenta-

tion (Bornert/Suquet,2001). Secant models treat elastoplastic composites as nonlinearly

elastic materials and are, accordingly, limited to monotonic loading and radial (or ap-

proximately radial) trajectories of the constituents in stress space during loading32),

which precludes their use as micromechanically based constitutive models in multi-scale

analysis. Advanced secant formulations, however, are highly suitable for materials char-

acterization, where they give excellent results.

Incremental mean field models, affine approaches and the TFA are not subject to lim-

itations with respect to loading paths. However, in their standard forms incremen-

tal MFAs and the TFA tend to overestimate the overall strain hardening in the post-

yield regime to such an extent that they are of limited practical applicability, especially

for matrix dominated deformation modes, compare e.g. (Gilormini,1995; Suquet,1997;

Chaboche/Kanoute,2003). Recent developments involve the use of tangent operators

that reflect the macroscopic symmetry of the composite, e.g. “isotropized” operators33)

for statistically isotropic materials, see e.g. (Bornert,2001), and algorithmic modifica-

tions, compare (Doghri/Ouaar,2003; Doghri/Friebel,2005). These improvements have

succeeded in markedly reducing the above weaknesses of incremental mean field mod-

32) The condition of radial loading paths in stress space at the constituent level is generallyviolated by a considerable margin in the phases of elastoplastic inhomogeneous materials, evenfor macroscopic loading paths that are are perfectly radial, see e.g. (Pettermann,1997). Thiseffect is due to changes in the accommodation of the phase stresses and strains in inhomogeneousmaterials upon the changes in tangential stiffness in any constituent upon yielding and, to alesser extent, in the strain hardening regime.

33) For a discussion of a number of issues pertaining to the use of “isotropic” versus“anisotropic” tangent operators see e.g. (Chaboche/Kanoute,2003).

– 25 –

els such as incremental Mori–Tanaka methods for particle reinforced composites, making

them suitable for practical applications. Modifications to the Transformation Field Anal-

ysis have also been proposed, see e.g. Chaboche et al.(2001). Major improvements to the

TFA can be achieved at limited computational cost by describing the inelastic strains

at the microscale via nonuniform transformation fields, the plastic strain field being ap-

proximated by suitable generalized strains referred to as plastic modes that are extracted

from unit cell models (Michel/Suquet,2004).

A problem common to all mean field approaches for elastoplastic composites lies in

handling the intraphase variations of the stress and strain fields within the assump-

tion of phase-wise uniformity34). Improvements in this respect have been obtained by

evaluating the phase averages of the von Mises equivalent stresses from energy considera-

tions (Qiu/Weng,1992; Hu,1996) or by using statistics-based theories (Buryachenko,1996;

Ju/Sun,2001). Within mean field methods that use secant plasticity, accounting for fluc-

tuations in the effective equivalent stress gives rise to “modified secant models”, see e.g.

(Ponte Castaneda/Suquet,1998), which have shown excellent agreement with predictions

from multi-particle unit cell models in a materials characterization context (Segurado et

al.,2002). This approach, however, is not suitable for incremental methods, for which an

empirical correction was proposed by Delannay et al.(2007).

In what follows a short discussion is given of the Incremental Mori–Tanaka (IMT) method

developed by Pettermann (1997). Incremental mean field methods can be formulated in

terms of phase averaged strain and stress rate tensors, d〈ε〉(p) and d〈σ〉(p), which can be

expressed in analogy to eqn.(16) as

d〈ε〉(p) = A(p)t d〈ε〉+ a

(p)t dT

d〈σ〉(p) = B(p)t d〈σ〉+ b

(p)t dT , (42)

where d〈ε〉 stands for the far field mechanical strain rate tensor, d〈σ〉 for the far field

stress rate tensor, and dT for a spatially uniform temperature rate. A(p)t , a

(p)t , B

(p)t , and

b(p)t are instantaneous strain and stress concentration tensors, respectively. Using the

assumption that the inhomogeneities show elastic and the matrix elastoplastic material

behavior35), the overall instantaneous (tangent) stiffness tensor can be written in terms

of the phase properties and the instantaneous concentration tensors as

E∗

t = E(i) + (1− ξ)[E(m)t − E(i)]A

(m)t

= [C(i) + (1− ξ)[C(m)t −C(i)]B

(m)t ]−1 , (43)

34) In general, the phase average of the equivalent stress in a given phase is greater thanthe equivalent stress computed from the phase averages of the stress components, which leadsto a tendency of simple mean field methods to overestimate overall yield and flow stresses. Asan extreme case, evaluating the von Mises equivalent stresses from the phase averaged stresscomponents in such approaches leads to predictions that materials with spherical reinforcementswill not yield under overall hydrostatic loads and homogeneous temperature loads.

35) Analogous expressions can be derived for elastoplastic inhomogeneities in an elastic ma-trix or, in general, for composites containing any required number of elastoplastic phases.

– 26 –

in analogy to eqn.(18). Note that eqn.(43) is formulated such that d〈σ〉 = E∗

t d〈ε〉, i.e.

E∗

t is a “continuum” tangent operator, compare (Doghri/Ouaar,2003).

Using the Mori–Tanaka formalism of Benveniste (1987), compare eqn.(31), the instanta-

neous matrix concentration tensors take the form

A(m)t = (1− ξ)I + ξ[I + StC

(m)t (E(i) − E

(m)t )]−1−1

B(m)t = (1− ξ)I + ξE(i)[I + StC

(m)t (E(i) − E

(m)t )]−1C

(m)t

−1 . (44)

The instantaneous Eshelby tensor St depends on the current state of the matrix material

and in general has to be evaluated numerically due to the anisotropic structure of the

instantaneous tangent operator E(m)t . Expressions for the instantaneous coefficients of

thermal expansion and for the instantaneous thermal concentration tensors can be derived

in analogy to the corresponding thermoelastic relations, e.g. eqns.(19) and (22).

Formulations of eqns.(42) to (44) that are directly suitable for implementation as mi-

cromechanically based constitutive models at the integration point level within Finite

Element codes can be obtained by replacing rates such as d〈σ〉(p) with finite increments

such as ∆〈σ〉(p). It is worth noting that in the resulting incremental Mori–Tanaka meth-

ods no assumptions on the overall yield surface and the overall flow potential are made,

the effective material behavior being entirely determined by the incremental mean field

equations and the constitutive behavior of the phases. However, mapping of the stresses

onto the yield surface cannot be handled at the level of the homogenized material and

stress return mapping must be applied to the matrix at the microscale instead. Accord-

ingly, the constitutive equations describing the overall behavior cannot be integrated

directly (as is the case for homogeneous elastoplastic materials), and iterative algorithms

are required. For example, Pettermann (1997) used an implicit Euler scheme in an im-

plementation of an incremental Mori–Tanaka method as a user supplied material routine

(UMAT) for the commercial Finite Element code ABAQUS (Simulia, Pawtucket, RI).

Algorithms of this type can also handle thermal expansion effects36) and temperature

dependent material parameters.

Incremental Mori–Tanaka methods, especially when they incorporate improvements as

proposed by Doghri and Friebel (2005), offer a combination of useful accuracy, flexibility

in terms of inhomogeneity geometries, and relatively low computational requirements.

Accordingly, they are attractive candidates for use at the lower length scale in multi-scale

models, compare chapter 8, of ductile matrix composites. Similar methods have also been

employed to study the nonlinear response of inhomogeneous materials due to microscopic

damage or to combinations of damage and plasticity, see e.g. (Tohgo/Chou,1996; Guo et

al.,1997).

36) Elastoplastic inhomogeneous materials such as metal matrix composites typically showa hysteretic thermal expansion response, i.e. the “coefficients of thermal expansion” are notmaterial properties in the strict sense. This dependence of the thermal expansion behavior onthe instantaneous mechanical response requires special treatment within the IMT framework,compare (Pettermann,1997).

– 27 –

Self-consistent schemes can also be combined with secant, incremental and affine

formulations to describe elastoplastic inhomogeneous materials, see e.g. (Hill,1965;

Hutchinson,1970; Berveiller/Zaoui,1979; Bornert,1996). They have also been extended

to account for microscopic damage in addition to plasticity (Estevez et al.,1999;

Gonzalez/LLorca,2000). Such procedures (as well as the weaknesses and strengths of the

approaches) are closely related to those discussed above for the Mori–Tanaka method37).

Analogous statements hold for Hashin–Shtrikman estimates (and bounds, compare sec-

tion 3.3) in general, which have e.g. been developed to account for geometrical nonlinear-

ities due to the evolution of of pore shapes and orientations in porous materials, compare

(Kailasam et al.,2000).

Finally, it is worth mentioning that in the elastoplastic range the “Eshelby property”

(i.e. uniformity) of the stress and strain fields within the inhomogeneities is lost even at

low reinforcement volume fractions, both intra-inhomogeneity and inter-inhomogeneity

fluctuations being caused by interactions with neighboring fibers or particles, see e.g.

(Bohm/Han,2001).

2.7 Mean Field Methods for Composites with Nonaligned Reinforcements

The overall symmetry of composites reinforced by nonaligned short fibers in many cases

is isotropic (random fiber orientations) or transversely isotropic (fiber arrangements

with axisymmetric orientation distributions, e.g., fibers with planar random orienta-

tion). However, processing conditions can give rise to a wide range of fiber orientation

distributions and, consequently, lower overall symmetries, compare e.g. (Allen/Lee,1990).

The statistics of the microgeometries of nonaligned matrix–inclusion composites can be

described on the basis of orientation distribution functions (ODFs) and aspect ratio or

length distribution functions (LDFs) of the reinforcements, which can be determined

experimentally. In addition, information is required on the type of phase geometry to

be described. At elevated fiber volume fractions local domains of (nearly) aligned fibers

are typically observed, which give rise to a “grain-type” mesostructure, referred to as an

aggregate system (Eduljee/McCullough,1993). Composites of this type are appropriately

modelled via two-step homogenization schemes that involve meso-level averages at the

grain level, see e.g. (Camacho et al.,1990; Pierard et al.,2004). At low to moderate fiber

volume fractions, in contrast, the orientations of neighboring fibers are essentially inde-

pendent within the geometrical constraints of non-penetration. Such dispersed systems

(Eduljee/McCullough,1993) can be studied via single-step mean field schemes involving

configurational averaging procedures, which may encompass aspect ratio averaging in

addition to orientational averaging.

Orientational averaging of tensor valued variables can be done by direct (numerical)

integration, see e.g. (Pettermann et al.,1987), or on the basis of expansions of the

37) In analogy to the elastic case, Mori–Tanaka based methods are more suitable for describ-ing elastoplastic matrix–inclusion composites, whereas two-phase self-consistent schemes aremore suited for modeling inhomogeneous materials with topologically indistinguishable phases,e.g. polycrystals or interwoven materials.

– 28 –

ODF in terms of generalized spherical harmonics (Viglin expansions), compare e.g. (Ad-

vani/Tucker,1987). The latter approach can be formulated in a number of ways, e.g. via

texture coefficients or texture matrices, compare (Siegmund et al.,2004). For a discus-

sion of a number of issues relevant to configurational averaging see Eduljee/McCullough

(1993).

Mori–Tanaka methods have been adapted to describing the elastic behavior of rather

general microstructures consisting of nonaligned inhomogeneities embedded in a matrix.

A typical starting point for such “extended” Mori–Tanaka algorithms are dilute stress

concentration tensors for reinforcements that have some given orientation with respect

to the global coordinate system, B(i),Θdil , which can be expressed as

B(i),Θdil = TΘ

σ

[

I + E(m)(I− S)(C(i) −C(m))]

−1TΘ

σ

−1. (45)

on the basis of eqn.(27). Here TΘσ stands for the stress transformation tensor from the

local (reinforcement based) to the global coordinate system. A phase averaged dilute

fiber stress concentration tensor can then be obtained by orientational averaging as

B(i)

dil =

∫

Θ

ρ (Θ)B(i),Θdil dΘ =

⟨

ρ B(i),Θdil

⟩

(46)

where Θ stands for the orientation angle and the orientation distribution function ρ(Θ)

is assumed to be normalized to give 〈ρ〉 = 1. The core statement of the Mori–Tanaka

approach according to Benveniste (1987), eqn.(28), can then be written in the form

〈σ〉(i) = B(i)

M〈σ〉 = B(i)

dil B(m)

M 〈σ〉 = B(i)

dil〈σ〉(m) (47)

This allows the phase averaged Mori–Tanaka concentration tensors to be expressed as

B(m)

M = [(1− ξ)I + ξB(i)

dil]−1

B(i)

M = B(i)

dil [(1− ξ)I + ξB(i)

dil]−1 (48)

in analogy to standard Mori–Tanaka methods, compare eqns.(29) and (30).

“Extended” Mori–Tanaka methods for modeling the elastic behavior of dispersed mi-

crostructures that contain nonaligned inhomogeneities were developed by a number of

authors, compare e.g. (Benveniste,1990; Dunn/Ledbetter,1997; Pettermann et al.,1997;

Mlekusch,1999). These models differ mainly in the algorithms employed for orientational

or configurational averaging.

In addition to providing estimates for the overall thermoelastic behavior of composites

containing nonaligned reinforcements, “extended” Mori–Tanaka theories can also be used

for describing the dependence of the microscopic stress and strain tensors in individual

reinforcements in such materials on the latter’s orientation. For this purpose eqn.(28) is

rewritten as

B(i),ΘM = B

(i),Θdil B

(m)

M , (49)

– 29 –

-1

0

1

2

3

4

0 10 20 30 40 50 60 70 80 90

σ 1 (

MPa

)

Θ ( ° )

σ1 - MTMσax - MTMσ1 - UC

FIG.5: “Extended” Mori–Tanaka and unit cell predictions for the angular dependenceof the maximum principal stresses σ1 (solid line) in the randomly oriented fibers (aspectratio a = 5) of a SiC/Al MMC subjected to uniaxial tension. In addition, the variationof the axial stresses σxx is given (dashed line; note that the maximum principal stressesshow little variation for fibers oriented at angles in excess of Θ≈65 to the loadingdirection.) Numerical results are presented in terms of the mean values (solid circles)and the corresponding standard deviations within individual fibers; from (Duschlbaueret al.,2003).

where B(i),ΘM and B

(m)

M are defined by eqns. (45) and (48). Results obtained with the above

relation are in good agreement with numerical predictions for moderate reinforcement

volume fractions and elastic contrasts, compare fig. 5.

Even though they tend to be highly useful for practical applications, “extended” Mori–

Tanaka methods38) are of an ad-hoc nature and can lead to nonsymmetric effective

“elastic tensors” in a number of scenarios, especially in multi-phase materials or when

anisotropic phases are present (Benveniste et al.,1991; Ferrari,1991)39).

“Extended” Mori–Tanaka methods based on orientational averaging have also been em-

ployed in studies of the nonlinear behavior of nonaligned composites. Secant plastic-

ity schemes of the above type were used for describing ductile matrix materials, see

e.g. (Bhattacharyya/Weng,1994; Dunn/Ledbetter,1997), and incremental approaches for

modeling composites consisting of an elastoplastic matrix reinforced by nonaligned or

random short fibers were proposed by Doghri/Tinel (2005) and Lee/Simunovic (2000),

respectively, with the latter work also accounting for debonding between reinforcements

and matrix.

38) Alternative modified Mori–Tanaka models for nonaligned composites can be obtainedby using spatially averaged Eshelby tensors (Johannesson/Pedersen,1998). Elastic compositescontaining woven, braided, and knitted reinforcements may be treated in analogy to “extended”Mori–Tanaka models, the the orientation of the fiber tows (“yarns”) being described by appro-priate “equivalent orientations” (Gommers et al.,1998).

39) The reason for this behavior lies in the fact, mentioned in section 2.3, that Mori–Tanakamethods inherently describe aligned ellipsoidal symmetries of the type shown in fig. 3.

– 30 –

A rigorous approach for studying the effects of orientation and aspect ratio distribu-

tions on the overall elastic behavior of inhomogeneous materials was proposed by Ponte

Castaneda and Willis (1995) on the basis of Hashin–Shtrikman estimates (compare sec-

tion 2.4). It is based on “ellipsoid-in-ellipsoid” phase arrangements in which different

Eshelby tensors are used for the ellipsoids describing the two-point correlations of the

phase arrangement and for the ellipsoids describing the shapes of the inhomogeneities.

The Hashin–Shtrikman estimates of Ponte Castaneda and Willis are limited, however,

to reinforcement volume fractions that follow the requirement that the inner ellipsoids

(shapes of the inhomogeneities) must stay subvolumes of the outer ellipsoids (“positions”

of the inhomogeneities). The resulting range of unrestricted applicability is rather small

for markedly prolate or oblate reinforcements that show a considerable degree of mis-

alignment. For a discussion of other micromechanical methods related to these estimates

see (Hu/Weng,2000).

For the special case of randomly oriented fibers or platelets of a given aspect ratio,

“symmetrized” dilute strain concentration tensors may be constructed, which are known

as Wu tensors (Wu,1966). They can be combined with Mori–Tanaka methods40), see

e.g. (Benveniste,1987), or classical self-consistent schemes (Berryman,1980) to describe

composites with randomly oriented phases of the matrix–inclusion and interwoven types.

Another mean field approach for such materials, the Kuster–Toksoz (1974) model, essen-

tially is a dilute description applicable to matrix–inclusion topologies and tends to give

unphysical results at high reinforcement volume fractions. In addition, the Hashin–

Shtrikman approach of Ponte Castaneda and Willis (1995) can be adapted to ran-

domly oriented reinforcements. For recent discussions on the relationships between

some of the above approaches see e.g. (Berryman/Berge,1996; Hu/Weng,2000). Due

to the overall isotropic behavior of composites reinforced by randomly oriented fibers

or platelets, their elastic response must comply with the “standard” Hashin–Shtrikman

bounds (Hashin/Shtrikman,1963), see also table 4 in section 5.4.

Besides mean field and unit cell models (compare section 5.4) a number of other meth-

ods have been proposed for studying composites with nonaligned reinforcements. Most

of them are based on the assumption that the contribution of a given fiber to the overall

stiffness and strength depends solely on its orientation with respect to the applied load

and on its length, interactions between neighboring fibers being neglected. The paper

physics approach (Cox,1952) and the Fukuda–Kawata theory (Fukuda/Kawata,1974)

are based on summing up stiffness contributions of fibers crossing an arbitrary nor-

mal section on the basis of fiber orientation and length distribution functions, com-

pare also (Jayaraman/Kortschot,1996). Such theories use shear lag models (Cox,1952;

Fukuda/Chou,1982) or improvements thereof (Fukuda/Kawata,1974) to describe the be-

havior of a single fiber embedded in matrix, the results being typically given as modified

rules of mixtures with fiber direction and fiber length corrections. Laminate analogy ap-

proaches, see e.g. (Fu/Lauke,1998), approximate nonaligned reinforcement arrangements

40) When used for describing randomly oriented reinforcements via a Wu tensor, Mori–Tanaka methods are not liable to giving nonsymmetric overall elastic tensors.

– 31 –

by a stack of layers each of which handles one fiber orientation and, where appropri-

ate, one fiber length. Like related models that are based on orientational averaging of

the elastic tensors corresponding to different fiber orientations, see e.g. (Huang,2001;

Siegmund et al.,2004), and the so-called aggregate models (Toll,1992), laminate analogy

approaches use Voigt (strain coupling) approximations.

2.8 Mean Field Methods for Non-Ellipsoidal Reinforcements

When non-ellipsoidal, inhomogeneous or coated inclusions are subjected to a homoge-

neous eigenstrain the resulting stress and strain fields are in general inhomogeneous. As a

consequence, the interior field Eshelby tensors and dilute inclusion concentration tensors

depend on the position wthin such inhomogeneities. Analytical solutions are available

only for certain inclusion shapes, see e.g. (Mura,1987).

A straightforward extension of the mean field concept for such cases consists in using Es-

helby or dilute inclusion concentration tensors that are averaged over the inhomogeneity

(Duschlbauer,2004). Averaged (or “replacement”) dilute inclusion concentration tensors,

A(i,r)dil and B

(i,r)dil , can be obtained by setting up models with a single dilute inhomogeneity,

generating solutions for six linearly independent load cases e.g. by the Finite Element

method, and using eqn.(75) to evaluate the average strains or stresses, respectively, in

the phases.

In order to achieve a workable formulation it is necessary to replace, in addition, the

elastic tensors of the inhomogeneities, E(i) and C(i), with “replacement elastic tensors”

E(i,r) = E(m) +1

ξdil(E∗

dil − E(m))[A(i,r)dil ]−1

C(i,r) = C(m) +1

ξdil(C∗

dil −C(m))[B(i,r)dil ]−1 , (50)

where E∗

dil and C∗

dil are the effective elastic tensors of the dilute single-inhomogeneity

configurations, and ξdil is the corresponding reinforcement volume fraction. The replace-

ment elastic tensors ensure consistency within the mean field framework by fulfilling

eqns.(18).

An alternative ( but related) method of handling non-ellipsoidal inhomogeneities in a

mean field framework is provided by the compliance contribution tensor formalism pro-

posed by Kachanov et al. (1994). It uses the difference between the compliance tensors

of the matrix and of the dilutely reinforced material,

H(i,m)dil = C∗

dil −C(m) , (51)

which is referred to as the compliance contribution tensor, to formulate mean field ex-

pressions for the overall responses of inhomogeneous materials containing reinforcements

or holes of general shape. The C∗

dil, like above, must be evaluated from appropriate

dilute single-inhomogeneity configurations.

Both the replacement inclusion approach and thee compliance contribution formalism

are approximations applicable to non-ellipsoidal inhomogeneities of general shapes, and

both inhomogeneities and matrix may have general elastic symmetries.

– 32 –

2.9 Mean Field Methods for Diffusion-Type Problems

There are strong conceptual and algorithmic analogies between inclusion problems de-

scribing the elastostatic behavior of and diffusive transport processes in inhomogeneous

materials, see e.g. (Hashin,1983) and Table 1. The ranks of the involved tensors, how-

ever, are lower in the diffusion problems and the underlying differential equations are

different.

elasticity thermal conduction electrical conduction species diffusion

σ q th j qc

stress heat flux current density species flux

[N m−2] [W m−2] [A m−2] [kg m−2s−1]

ε ∆T E ∆C

strain temperature gradient electrical field concentration gradient

[] [K m−1] [V m−1] [m−1]

u T V C

displacements temperature electrical potential concentration

[m] [K] [V] []

E Kth Kel ρD

“elasticity” thermal conductivity electrical conductivity D: diffusivity

[N m−2] [W m−1K−1] [Ω−1m−1] [m2s−1]

C=E−1 Rth=K−1th Rel=K

−1el —

compliance thermal resistivity electrical resistivity

[N−1m2] [W−1m K] [Ω m]

S S S S

elastic diffusion diffusion diffusion

Eshelby tensor Eshelby tensor Eshelby tensor Eshelby tensor

TABLE 1: Analogies between the mathematical descriptions of elastic and diffusionproblems (ρ stands for the mass density).

Specifically, in diffusion-type problems (such as heat conduction, electrical conduction, or

diffusion of moisture)41) the effects of dilute ellipsoidal inhomogeneous inclusions embed-

41) The differential equations describing antiplane shear in elastic solids, d’Arcy creep flowin porous media, and equilibrium properties such as overall dielectric constants and magneticpermeabilities are also of the Poisson type and thus mathematically equivalent to diffusivetransport problems.

– 33 –

ded in a matrix can be described in analogy to eqns.(24) to (27) via “diffusion Eshelby

tensors”42). The Eshelby property takes the form of constant fluxes and constant gradi-

ents within dilute inclusions, and dilute gradient and flux concentration tensors can be

obtained in analogy to eqns.(26) and (27) as

A(i)dil = [I + S(m)R(m)(K(i) −K(m))]−1

B(i)dil = K(i)[I + S(m)R(m)(K(i) − K(m))]−1R(m) . (52)

Similar descriptions can also be devised for a number of coupled problems, such as the

electromechanical behavior of inhomogeneous materials with at least one piezoelectric

constituent, see e.g. (Huang/Ju,1994).

For diffusive transport in nondilute materials direct analoga of the Mori–Tanaka and

self-consistent approaches introduced in sections 2.3 and 2.7 can be derived, see e.g.

(Hatta/Taya,1986; Miloh/Benveniste,1988; Dunn/Taya,1993; Chen,1997). For example,

the equivalents of eqns.(29) and (30) take the form

A(m)MT = [(1− ξ)I + ξA

(i)dil]

−1 A(i)MT = A

(i)dil[(1− ξ)I + ξA

(i)dil]

−1

B(m)MT = [(1− ξ)I + ξB

(i)dil]

−1 B(i)MT = B

(i)dil[(1− ξ)I + ξB

(i)dil]

−1 . (53)

The classical self-consistent scheme takes the form

Kn+1 = K(m) + ξ[K(i) − K(m)][In + SnRn(K(i) − Kn)]−1

Rn+1 =[

Kn+1

]

−1, (54)

compare eqns.(35).

Replacement inhomogeneity models in analogy to section 2.8 can be used to handle

non-ellipsoidal and inhomogeneous reinforcements as well as reinforcements with finite

interfacial conductances. In the latter case provision must be make for the “replacement

inhomogeneity” to account for the interfacial temperature jumps, see e.g. the discussion in

(Duschlbauer,2004). Replacement conductivities or resistitivities must be evaluated from

consistency conditions equivalent to eqns.(50). Alternatively, resistivity contribution

tensors may be used to handle non-ellipsoidal or inhomogeneous inclusions in analogy to

the compliance contribution tensor formalism, eqn.(51).

Due to the lower order of the tensors involved in diffusion-type problems the latter tend

to be more benign and easier to handle. Extensive discussions of diffusion-type problems

in inhomogeneous media can be found e.g. in (Markov/Preziosi,2000; Torquato,2002;

Milton,2002).

42) In contrast to the “mechanical Eshelby tensor” S, which (like the elasticity tensor E) isof rank 4, the “diffusion Eshelby tensor” S is of rank 2 (like the conductivity tensor K), and,for spheroidal inclusions, it depends only on their aspect ratios. Explicit expressions for theelements of “diffusion Eshelby tensors” are given e.g. in (Hatta/Taya,1986; Duan et al.,2006).

– 34 –

3. VARIATIONAL BOUNDING METHODS

Whereas mean field methods, unit cell approaches and embedding strategies can typically

be used for both homogenization and localization tasks, bounding methods are limited to

homogenization. Discussions in this chapter again are restricted to materials consisting

of two perfectly bonded constituents.

Rigorous bounds for the overall elastic properties of inhomogeneous materials are typi-

cally obtained from appropriate variational (minimum energy) principles. In what fol-

lows only outlines of bounding methods are given; formal treatments can be found e.g. in

(Nemat-Nasser/Hori,1993; Bornert,1996; Ponte Castaneda/Suquet,1998; Markov,2000;

Bornert et al.,2001; Milton,2002).

3.1 Classical Bounds

Hill Bounds

Classical expressions for the minimum potential energy and the minimum complementary

energy in combination with uniform stress and strain trial functions lead to the simplest

variational bounding expressions, the upper bounds of Voigt (1889) and the lower bounds

of Reuss (1929). In their tensorial form (Hill,1952) they are known as the Hill bounds

and can be written as

E∗

H−=

[

∑

(p)

V (p)C(p)]

−1

≤ E∗ ≤∑

(p)

V (p)E(p) = E∗

H+ . (55)

These bounds, while universal and very simple, do not contain any information on the

microgeometry of an inhomogeneous material beyond the phase volume fractions, and

are typically too slack to be of much practical use43), but in contrast to the Hashin–

Shtrikman and higher order bounds they also hold for control volumes that are too small

to be proper RVEs.

Hashin–Shtrikman-Type Bounds

Considerably tighter bounds on the overall behavior of inhomogeneous materials can

be obtained from a variational formulation due to Hashin and Shtrikman (1962). It

is formulated in terms of a homogeneous reference material and of polarization fields

τ(x) that describe the difference between the microscopic stress fields in the inhomoge-

neous material and a homogeneous reference material, compare eqn.(37). In combination

with the macrohomogeneity condition (compare section 2.1) polarization fields allow the

43) The bounds on the Young’s and shear moduli obtained from eqn.(55) are equivalent toVoigt and Reuss expressions in terms of the corresponding phase moduli only if the constituentshave appropriate Poisson’s ratios. Due to the homogeneous stress and strain assumptions usedfor obtaining the Hill bounds, the corresponding phase strain and stress concentration tensors

are A(p)V =I and B

(p)R =I, respectively.

– 35 –

complementary energy of the composite to be formulated in such a way that the “fast”

and “slow” contributions are separated, giving rise to the Hashin–Shtrikman variational

principle. The latter can be written in the form

FHS(τ) =1

V

∫

V

τ(x)[E0 − E(x)]−1 τ(x) +(

τ(x)− 〈τ〉∗)

ε 0(τ(x)) + 2τε0 dΩ , (56)

compare e.g. (Gross/Seelig,2001). The stationary values of FHS take the form

FHS = ε0 (E∗ −E0) ε0

and are maxima when (E(x) − E0) is positive definite and minima if (E(x) − E0) is

negative definite.

In order to allow an analytical solution for the above variational problem, the highly

complex position dependent polarizations τ(x) are approximated by phase-wise constant

polarizations τ (p), compare eqn.(38). This has the consequence that the strain fluctuations

ε 0(τ (p)) can be evaluated by an Eshelby-type eigenstrain procedure. It can be shown that

for the case of isotropic constituents and isotropic overall behavior the proper Eshelby

tensor to be used is the one for spheres, i.e. for inclusion aspect ratio a = 1. By optimizing

the Hashin–Shtrikman variational principle with respect to the τ (p) the tightest possible

bounds within the Hashin–Shtrikman scheme are found.

In the case of two-phase materials in which the matrix is softer than the reinforcements,

i.e. for (K(i)−K(m)) > 0 and (G(i)−G(m)) > 0, the above procedure leads to lower bounds

for the elasticity tensors of the form

E∗

HS− = E(m) + ξ[

(E(i) − E(m))−1 + (1− ξ)S(m)C(m)]

−1

= E(m) + ξ(E(i) − E(m)) A(i)M , (57)

where A(i)M corresponds to the Mori–Tanaka strain concentration tensor given in eqn.(30).

The upper bound Hashin–Shtrikman elasticity tensors are obtained as

E∗

HS+ = E(i) + (1− ξ)[

(E(m) −E(i))−1 + ξS(i)C(i)]

−1

(58)

by exchanging the roles of matrix and inhomogeneities.

The Hashin–Shtrikman bounds proper, eqn.(57) and (58), were stated by Hashin and

Shtrikman (1963) in terms of bounds on the effective bulk modulus K∗ and the effective

shear modulus G44). They apply to essentially all practically relevant inhomogeneous

44) Whereas engineers tend to describe elastic material behavior in terms of Young’s moduliand Poisson’s ratios (which can be measured experimentally in a relatively straightforwardway), many homogenization expressions are best formulated in terms of bulk and shear moduli,which directly describe the physically relevant hydrostatic and deviatoric responses of materials.For obtaining bounds on the effective Young’s modulus E∗ from the results on K∗ and G∗ see(Hashin,1983), and for bounding expressions on the Poisson numbers ν∗ see (Zimmerman,1992).

– 36 –

materials with isotropic phases that show (statistically) isotropic overall symmetry45)

and fulfill the condition (K(i)−K(m))(G(i)−G(m)) > 0. The Hashin–Shtrikman formalism

also allows to recover the Voigt and Reuss bounds when the reference material is chosen

to be much stiffer or much softer than either of the constituents.

Another set of Hashin–Shtrikman bounds is applicable to (statistically) transverse

isotropic composites reinforced by aligned continuous fibers46). For a discussion of

evaluating these bounds for composites reinforced by transversely isotropic fibers see

(Hashin,1983).

Further developments led to the Walpole (1966) and Willis (1977) bounds, which pertain

to thermoelastically anisotropic overall behavior due to anisotropic constituents, statis-

tically anisotropic phase arrangements, and/or non-spherical reinforcements, a typical

example being the aligned ellipsoidal microgeometry sketched in fig.3. These bounds

include the Hashin–Shtrikman bounds discussed above as special cases and they can

also handle situations where a “softest” and “stiffest” constituent cannot be identified

unequivocally so that a fictive reference material must be used.

Hashin–Shtrikman-type bounds are not restricted to materials with matrix–inclusion

topologies, but hold for any phase arrangement of the appropriate symmetry and phase

volume fractions. In addition, it is worth noting that Hashin–Shtrikman-type bounds

are sharp, i.e. they are the tightest bounds that can be given for the type of geometrical

information used (volume fraction and overall symmetry, corresponding to two-point

correlations). From the practical point of view it is of interest that for two-phase materials

Mori–Tanaka estimates are equal to one of the Willis bounds, compare e.g. (Weng,1990),

while the other bound can be obtained after a “color inversion” (i.e. after exchanging the

roles of inhomogeneities and matrix)47). Accordingly, the bounds can be evaluated for

fairly general aligned phase geometries by simple matrix algebra such as eqn.(32). For

elevated phase contrasts the lower and upper Hashin–Shtrikman-type bounds tend to be

rather far apart, however.

45) Avellaneda and Milton (1989) constructed hierarchical laminates that are statisticallyisotropic and the overall conductivities of which do not fall within the pertinent Hashin–Shtrikman bounds. Accordingly, statistical isotropy by itself is not a sufficient condition forguaranteeing the validity of expressions such as eqns.(57) and (58).For a discussion of Hashin–Shtrikman-type bounds of macroscopically isotropic composites withanisotropic constituents see e.g. Nadeau/Ferrari (2001).

46) These bounds apply to general (transversely isotropic) two-phase materials, the (“in-plane”) phase geometries of which are translation invariant in the “out-of-plane” or “axial”direction.

47) Put more precisely, the lower bounds for two-phase materials are obtained from Mori–Tanaka methods when the more compliant material is used as matrix phase, and the upperbounds when it is used as inclusion phase. For matrix–inclusion materials with more than twoconstituents, Mori–Tanaka expressions are bounds if the matrix is the stiffest or the softestphase, and fall between the appropriate bounds otherwise (Weng,1990). Note, however, thatextended Mori–Tanaka methods for nonaligned composites as discussed in section 2.7 in generaldo not coincide with bounds.

– 37 –

Composites reinforced by randomly oriented fibers or platelets follow the Hashin–

Shtrikman bounds for statistically isotropic materials, eqns.(57) and (58). A discussion

of Hashin–Shtrikman-type bounds on the elastic responses of composites with more gen-

eral fiber orientation distributions can be found in (Eduljee/McCullough,1993). In the

case of porous materials (where the pores may be viewed as an ideally compliant phase)

only upper Hashin–Shtrikman bounds can be obtained, the lower bounds for the elastic

moduli being zero. The opposite situation holds in case one of the constituents is rigid.

Hashin–Shtrikman-type variational formulations can also be employed to generate

bounds for more general phase arrangements. Evaluating the polarizations for “compos-

ite regions” consisting of inhomogeneities embedded in matrix gives rise to Herve–Stolz–

Zaoui bounds (Herve et al.,1991). When complex phase patterns are to be considered

(Bornert,1996; Bornert et al.,1996) numerical methods must be employed for evaluating

the polarization fields.

Bounds for Discrete Microstructures

Hashin–Shtrikman-type bounds can also be derived for (simple) periodic microgeome-

tries, see e.g. (Nemat-Nasser/Hori,1993). Among other bounding approaches for such

microstructures are those of Bisegna and Luciano (1996), which uses approximate vari-

ational principles evaluated from periodic unit cells via Finite Element models, and of

Teply and Dvorak (1988), which evaluates bounds for the elastoplastic behavior of fiber

reinforced composites with periodic hexagonal phase arrangements. In addition, win-

dowing methods as discussed in chapter 7 may be used to generate sequences of bounds

on the overall behavior of given phase arrangements.

3.2 Improved Bounds

When more complex trial functions are used in variational bounding, their optimization

requires statistical information on the phase arrangement in the form of n-point corre-

lation functions48). This way improved bounds can be generated that are significantly

tighter than Hashin–Shtrikman-type expressions.

Three-point bounds for statistically isotropic two-phase materials can be formulated in

such a way that the information on the phase arrangement statistics is contained in

two three-point microstructural parameters, η(ξ) and ζ(ξ). The evaluation of these

parameters for given arrangement statistics is a considerable task. However, analyti-

cal expressions or tabulated data in terms of the reinforcement volume fraction ξ are

available for a number of generic microgeometries of practical importance, among them

statistically homogeneous isotropic materials containing identical, bidisperse and polydis-

perse impenetrable (“hard”) spheres (that describe matrix–inclusion composites) as well

as monodisperse interpenetrating spheres (“boolean models” that can describe many

interwoven phase arrangements), and statistically homogeneous transversely isotropic

materials reinforced by impenetrable or interpenetrating aligned cylinders.

48) For discussions on statistical descriptions of phase arrangements see e.g. (Torquato,1991;Ghosh et al.,1997; Pyrz/Bochenek,1998; Torquato,1998; Torquato,2002).

– 38 –

References to some three-point bounds and to a number of expressions for η and ζ

applicable to some two-phase composites can be found in section 3.4, where results from

mean field and bounding approaches are compared. For reviews of higher order bounds

for elastic (as well as other) properties of inhomogeneous materials see (Torquato,1991;

Torquato,2002).

Improved bounds typically provide highly useful information for low and moderate phase

contrasts (as typically found in technical composites), but even they tend to be rather

slack for elevated phase contrasts.

3.3 Bounds for the Nonlinear Behavior

Analoga to the Hill bounds for nonlinear inhomogeneous materials were introduced by

Bishop and Hill (1951). For polycrystals the nonlinear equivalents to Voigt and Reuss

expressions are usually referred to as Taylor and Sachs bounds, respectively.

In analogy to mean field estimates for elastoplastic material behavior, nonlinear bounds

are typically obtained by evaluating a sequence of linear bounds. Talbot and Willis

(1985) extended the Hashin–Shtrikman variational principles to obtain one-sided bounds

(i.e. upper or lower bounds, depending on the material combination) on the nonlinear

mechanical behavior of inhomogeneous materials.

An important development was the derivation of a variational principle by Ponte

Castaneda (1992) that allows upper bounds on the effective nonlinear behavior of in-

homogeneous materials to be generated on the basis of upper bounds for the elastic re-

sponse49). It uses a series of inhomogeneous reference materials, the properties of which

have to be obtained by optimization procedures for each strain level. Essentially, the vari-

ational principle guarantees the best choice for the comparison material at a given load.

The Ponte Castaneda bounds are closely related to mean field approaches using improved

secant plasticity methods (compare the remarks in section 2.4). For higher order bounds

on the nonlinear response of inhomogeneous materials see also (Talbot/Willis,1998).

The study of bounds — like the development of improved estimates — for the overall

nonlinear mechanical behavior of inhomogeneous materials has been an active field of

research during the last decade, see the recent reviews by Suquet(1997), Ponte Castaneda

and Suquet (1998), and Willis (2000).

3.4 Some Comparisons of Mean Field and Bounding Predictions

In order to give some idea of the type of predictions that can be obtained by different

mean field (and related) approaches and by bounding methods for the thermomechanical

responses of inhomogeneous thermoelastic materials, results for the overall elastic moduli

49) The Ponte Castaneda bounds are rigorous for nonlinear elastic inhomogeneous materialsand, on the basis of deformation theory, are very good approximations for materials with atleast one elastoplastic constituent. Applying the Ponte Castaneda variational procedure toelastic lower bounds does not necessarily lead to a lower bound for the inelastic behavior.

– 39 –

and coefficients of thermal expansion are presented as functions of the reinforcement

volume fraction ξ in this section. All comparisons are based on E-glass particles or fibers

embedded in an epoxy matrix. The elastic contrast of these constituents is c ≈ 21 and

the thermal expansion contrast takes a value of approximately 0.14, see table 2.

E ν α

[GPa] [ ] [1/K ×10−6]

matrix 3.5 0.35 36.0

reinforcements 74.0 0.2 4.9

TABLE 2: Constituent material parameters of epoxy matrix and E-glass reinforcementsused in generating figs. 6 to 12.

Figures 6 and 7 show predictions for the overall Young’s and shear moduli of a particle

reinforced glass/epoxy composite based on the above constituent material parameters.

The Hill bounds can be seen to be very slack. The Mori–Tanaka results (MTM) —

as mentioned before — coincide with the lower Hashin–Shtrikman bounds (H/S LB),

whereas the classical self-consistent scheme (CSCS) shows a typical behavior in that it

is close to one Hashin–Shtrikman bound at low volume fractions, approaches the other

at high volume fractions, and displays a transition behavior in the form of a sigmoid

curve inbetween. The three-point bounds (3PLB and 3PUB) given in figs. 6 and 7 apply

to monodisperse impenetrable spherical inhomogeneities, are based on formalisms de-

veloped by Beran/Molyneux (1966), Milton (1981) and Phan-Thien/Milton (1983), and

use expressions for the statistical parameters η and ζ reported in (Miller/Torquato,1990)

and (Torquato et al.,1987), respectively. Like Torquato’s second order estimates (3PE)

these expressions are available for reinforcement volume fractions up to ξ=0.650). In-

terestingly, for the case considered here, the results from the generalized self-consistent

scheme (GSCS), which as usual predicts a slightly stiffer behavior than the Mori–Tanaka

method (taking the “standard” case of stiff inhomogeneities in a compliant matrix), are

very close to the lower three-point bounds even though microgeometries corresponding

to the GSCS do not show monodisperse inhomogeneities.

Predictions for the effective coefficients of thermal expansion of statistically isotropic

glass-epoxy composites are given in fig. 8, the bounds being obtained from a relation

proposed by Levin (1967) and Schapery (1968) that can be evaluated with any bounds

or estimates for the effective bulk modulus. Here the Hashin–Shtrikman bounds and

the three-point bounds are used as basis for bounding the overall thermal expansion

behavior51). The CTEs corresponding to the self-consistent schemes were obtained by

50) This value approaches the maximum volume fraction achievable in random packings ofspheres, which is ξ = 0.64 (Weisstein,2000).

51) Upper bounds for the bulk modulus give rise to lower bounds for the CTEs and viceversa.

– 40 –

0.0

40.0

EF

FE

CT

IVE

YO

UN

Gs

MO

DU

LU

S [

GP

a]

0.0 0.2 0.4 0.6 0.8 1.0

PARTICLE VOLUME FRACTION []

Hill/Reuss LBHill/Voigt UBH/S LB; MTMH/S UB3PLB (mono/h)3PUB (mono/h)GSCS3PE (mono/h)CSCS

FIG.6: Bounds and estimates for the effective Young’s moduli of glass/epoxy particlereinforced composites as functions of the particle volume fraction.

0.0

20.0

EF

FE

CT

IVE

SH

EA

R M

OD

UL

US

[G

Pa]

0.0 0.2 0.4 0.6 0.8 1.0

PARTICLE VOLUME FRACTION []

Hill/Reuss LBHill/Voigt UBH/S LB; MTMH/S UB3PLB (mono/h)3PUB (mono/h)GSCS3PE (mono/h)CSCS

FIG.7: Bounds and estimates for the effective shear moduli of glass/epoxy particlereinforced composites as functions of the particle volume fraction.

– 41 –

0.0

20.0

40.0

EF

FE

CT

IVE

CT

E [

1/K

]

0.0 0.2 0.4 0.6 0.8 1.0

PARTICLE VOLUME FRACTION []

H/S UB; MTMH/S LB3PUB (mono/h)3PLB (mono/h)GSCS3PE (mono/h)CSCS

−6x

10

FIG.8: Bounds and estimates for the effective CTEs of glass/epoxy particle reinforcedcomposites as functions of the particle volume fraction.

using the mean field formalism discussed in section 2.1. Because they are based on the

same estimates for the effective bulk modulus, the results given for the GSCS and the

Mori–Tanaka-scheme agree with the upper Levin/Hashin–Shtrikman bounds.

In figs. 9 to 12 results are presented for the overall transverse Young’s moduli, overall

axial and transverse shear moduli, as well as overall axial and transverse coefficients of

thermal expansion of glass/epoxy composites reinforced by aligned continuous fibers52).

The three-point bounds shown correspond to the case of aligned identical impenetrable

circular cylinders, follow (Silnutzer,1972; Milton,1981), and use statistical parameters

evaluated by Torquato and Lado (1992). A qualitatively similar behavior to the particle

reinforced case is evident. It is noteworthy, however, that the overall transverse CTEs

in fig. 12 (marked as “tr”) at low fiber volume fractions are predicted by all models

to exceed the CTEs of both constituents. Such behavior is typical for continuously

reinforced composites and is caused by the axial constraint enforced by the stiff fibers.

Nearly the same axial CTEs (“ax”) are obtained for all models considered. No bounds

for the CTEs are given in fig. 12 because there is no equivalent to the Levin formula for

overall transversely isotropic materials.

52) In the case of the axial Young’s moduli the different estimates and bounds are essentiallyindistinguishable from each other (and from the rules of mixture result) for the material param-eters and scaling used in fig. 9 and are, accordingly, not shown. It is, however, worth notingthat the effective axial Young’s modulus of a composite reinforced by aligned continuous fibersis always equal to or larger than the rules of mixture result.

– 42 –

0.0

40.0

EF

FE

CT

IVE

YO

UN

Gs

MO

DU

LU

S [

GP

a]

0.0 0.2 0.4 0.6 0.8 1.0

FIBER VOLUME FRACTION []

Hill/Reuss LBHill/Voigt UBH/S LB; MTMH/S UB3PLB (mono/h)3PUB (mono/h)GSCS3PE (mono/h)CSCS

FIG.9: Bounds and estimates for the effective transverse Young’s moduli of glass/epoxycomposites with continuous aligned fiber reinforcement as functions of the fiber volumefraction.

0.0

10.0

20.0

30.0

EF

FE

CT

IVE

SH

EA

R M

OD

UL

US

[G

Pa]

0.0 0.2 0.4 0.6 0.8 1.0

FIBER VOLUME FRACTION []

Hill/Reuss LBHill/Voigt UBH/S LB; MTMH/S UB; MTM3PLB (mono/h)3PUB (mono/h)GSCS3PE (mono/h)CSCS

FIG.10: Bounds and estimates for the effective axial shear moduli of glass/epoxy com-posites with continuous aligned fiber reinforcement as functions of the fiber volume frac-tion.

– 43 –

0.0

10.0

20.0

30.0

EF

FE

CT

IVE

SH

EA

R M

OD

UL

US

[G

Pa]

0.0 0.2 0.4 0.6 0.8 1.0

FIBER VOLUME FRACTION []

Hill/Reuss LBHill/Voigt UBH/S LB; MTMH/S UB3PLB (mono/h)3PUB (mono/h)GSCS3PUB (mono/h)CSCS

FIG.11: Bounds and estimates for the effective transverse shear moduli of glass/epoxycomposites with continuous aligned fiber reinforcement as functions of the fiber volumefraction.

0.0

20.0

40.0

EF

FE

CT

IVE

CT

E [

1/K

]

0.0 0.2 0.4 0.6 0.8 1.0

FIBER VOLUME FRACTION [ ]

MTM ax.MTM tr.GSCS ax.GSCS tr.3PE (mono/h) ax.3PE (mono/h) tr.CSCS ax.CSCS tr.

−6x

10

FIG.12: Estimates for the effective axial and transverse CTEs of glass/epoxy compositeswith continuous aligned fiber reinforcement as functions of the fiber volume fraction.

– 44 –

In figs. 6 to 12, the classical self-consistent scheme is not in agreement with the three-

point bounds shown, which explicitly correspond to matrix–inclusion topologies. Con-

siderably better agreement with the CSCS is found by using three-point parameters of

the interpenetrating sphere or cylinder type (which can also describe cases where both

phases percolate, but are not as symmetrical with respect to the constituents as the

CSCS). From a practical point of view it is worth noting that despite their sophistication

improved bounds (and Torquato’s second order estimates) may give overly optimistic pre-

dictions for the overall moduli because they describe ideal two-phase materials, whereas

in actual materials it is practically impossible to avoid flaws such as porosity.

Before closing this chapter it should be mentioned that more complex responses may be

obtained when one or both of the constituents are transversely isotropic with a high stiff-

ness contrast between the axial and transverse directions (note that e.g. carbon fibers

typically exhibit transverse Young’s moduli that are much smaller than the Young’s

moduli of, say, metallic matrices, whereas the opposite holds true for the axial Young’s

moduli). Such effects may be especially marked in the nonlinear range. Also, the angu-

lar dependences of stiffnesses and CTEs in fiber reinforced materials can be quite rich,

compare e.g. (Pettermann et al.,1997).

– 45 –

4. GENERAL REMARKS ON APPROACHES BASED ON DISCRETE

MICROGEOMETRIES

In the present context, micromechanical approaches based on discrete microgeometries

encompass unit cell, embedding and windowing methods. Broadly speaking, these ap-

proaches trade off restrictions to the generality of the microstructures that can be studied

for the capabilities of using fine grained geometrical models and of resolving details of

the stress and strain fields at the length scale of the inhomogeneities. The main fields

of application of these methods are studying the nonlinear behavior of inhomogeneous

materials and evaluating the microscopic stress and strain fields of “model materials”

at high resolution. This information, on the one hand, may be required when the lo-

cal stress and strain fields fluctuate strongly and are, accordingly, difficult to describe

by averaged fields. On the other hand, it is important for understanding the damage

and failure behavior of inhomogeneous materials, which can depend on details of their

microgeometry.

4.1 Microgeometries

The generation of “simply periodic” microgeometries, such as periodic hexagonal ar-

rangements of fibers or cubic arrangements of spherical particles, compare chapter 5, is

rather straightforward. In order to obtain more realistic, and thus more complex, micro-

geometries for modeling inhomogeneous materials, one of two — often complementary

— modeling philosophies is typically followed, viz., using computer generated or “real

structure” phase arrangements.

Computer-Generated Microgeometries

Computer generated microgeometries may be classified into two groups, one of which is

based on generic random arrangements of a small to a fair number of reinforcements,

whereas the other aims at approximating some given phase distribution statistics.

The generation of generic random microgeometrical models typically involves random

addition methods53) and random perturbation or relaxation techniques. The reinforce-

ment volume fractions that can be reached by RSA methods tend to be moderate due to

jamming (geometrical frustration) and their representativeness for phase arrangements

generated by mixing processes is open to question (Stroeven et al.,2004). Major improve-

ments on both counts can be obtained by using RSA geometries as starting configurations

for random perturbation or hard core shaking models (HCSM, also referred to as dynamic

simulations), that apply small random displacements to each inhomogeneity to enforce

minimum spacings between individual reinforcements as the size of the volume element

53) In random addition (RSA) algorithms (also known as random sequential insertion or,in two-dimensional cases, as random sequential adhesion models, and sometimes referred toas static simulations), positions for new reinforcements are created by random processes, acandidate inhomogeneity being accepted if it does not “collide” with (approach too closely oroverlap with) any of the existing ones and rejected otherwise.

– 46 –

is reduced54). Methods of the above types can be used to generate statistically homoge-

neous distributions of reinforcements, see e.g. (Gusev,1997; Bohm/Han,2001), as well as

idealized clustered microgeometries (Segurado et al.,2003).

Another modeling strategy, based on simulated annealing and related procedures, aims

at generating “statistically reconstructed” microstructures55). This way accurate repre-

sentations of the microgeometry of a given composite can be provided on the base of

experimentally determined statistical descriptors.

Computer generated microgeometries have tended to employ idealized reinforcement

shapes, equiaxed particles embedded in a matrix, for example, being typically repre-

sented by spheres, and fibers by cylinders or prolate spheroids56) of appropriate aspect

ratio57). Microgeometries of composites reinforced by continuous fibers can be well de-

scribed this way, but for reinforcements with aspect ratio close to unity such simplified

phase arrangements may constitute a less satisfactory approximation, giving rise, for

example, to considerable weaker macroscopic hardening in models of ductile matrix com-

posites compared to polyhedral reinforcements of equivalent orientation and aspect ratio,

compare (Chawla/Chawla,2006).

By appropriately adjusting the algorithms, computer generated microgeometries may be

periodic or non-periodic, as required by the analysis method to be employed. Enforcing

periodicity (for use with periodic microfield methods) introduces long-range order into

phase arrangements and may make them less realistic.

Real Structure Microgeometries

Instead of generating them by computer algorithms, microgeometries may be chosen

to follow as closely as possible the phase arrangement in part of a given sample of

the material to be modeled, obtained from metallographic sections and serial sections,

tomographic data, etc., see e.g. (Fischmeister/Karlsson,1977; Terada/Kikuchi,1996; Li et

54) Alternatively, explicit models of production processes may be used for achieving realisticmicrostructures with elevated reinforcement volume fractions.

55) Statistically reconstructed microstructures are equivalent to real ones in a statisti-cal sense, i.e. they show equal or very similar phase distribution statistics. Algorithms forreconstructing matrix–inclusion and more general microgeometries that approximate prede-fined statistical descriptors are discussed e.g. in (Rintoul/Torquato,1997), (Torquato,1998),(Roberts/Garboczi,1999), (Zeman/Sejnoha,2001) and (Bochenek/Pyrz,2002). For in-depth dis-cussions of many of the underlying issues, such as statistical descriptions of inhomogeneousmaterials, see e.g. (Torquato,2002; Zeman,2003).

56) For uniform boundary conditions it can be shown that the overall elastic behavior ofmatrix–inclusion type composites can be bounded by approximating the actual shape of particlesby inner and outer envelopes of “smooth” shape (e.g., inscribed and circumscribed ellipsoids).This is known as the Hill modification theorem (or Hill’s comparison theorem, Hill’s auxiliarytheorem) see (Hill,1963; Huet et al.,1991). Approximations by ellipsoids typically work consid-erably better for convex than for non-convex particle shapes (Kachanov/Sevostionov,2005).

57) As an alternative to approximating the aspect ratios of real reinforcements, equal surfacevs. volume ratios S3/V 2 may be used in models, see e.g. (Kenesei et al.,2005).

– 47 –

al.,1999; Babout et al.,2004; Chawla/Chawla,2006). The resulting descriptions are often

referred to as “real structure” models.

Recently the generation of real structure models from pixel (digital images) or voxel

(tomographic) data on the geometries of inhomogeneous materials has been the focus of

considerable research efforts. It typically involves the steps of selecting an appropriate

volume element for analysis (“registration”) and of identifying the areas or volumes,

respectively, occupied by the material’s constituents by thresholding of the grey values

of the pixels or voxels (“segmentation”). At this stage pixel or voxel models (compare

section 4.2) can be generated directly or contouring procedures may be used for obtaining

“smooth” phase domains, the latter operation typically being manpower intensive and

difficult to automatize.

Sufficiently large real structure models can provide excellent discrete microstructures for

micromechanical modeling, but care58) is required to avoid bias in the choice of the regions

(“windows”) extracted for modeling. Typically some simplification of the experimental

data is necessary within the framework of discretizing engineering methods, e.g., with

respect to the shapes of the reinforcements and by not explicitly modeling reinforcements

the sizes of which fall below some threshold. Real structure microgeometries typically

are non-periodic, so they cannot be used directly in periodic microfield models.

Sizes of Volume Elements

For both computer generated phase arrangements and for real structure models the ques-

tion immediately arises what geometrical complexity (and thus what size of volume ele-

ment) is required for adequately representing the physical behavior of the inhomogeneous

material to be studied59).

There are two main approaches to assessing adequate sizes of volume elements. On the

one hand, the arrangement statistics of the microgeometry can be used, in correspondence

with the idea of “geometrical RVEs”, compare section 1.2. In this context adequate sizes

of model geometries may be estimated on the basis of experimentally obtained correlation

lengths (Bulsara et al.,1999) or covariances (Jeulin,2001) of the phase arrangement or by

posing the requirement of having at least two statistically independent inhomogeneities

in the cell (Zeman/Sejnoha,2001). On the other hand, statistical descriptors (e.g., the

integral range of a second order correlation function, compare (Jeulin,2001)), hierarchies

of bounds (resulting, e.g., from windowing analysis, compare section 7), the variations

in the predictions for some physical property (e.g., an effective modulus or the extremal

value of some microfield) or differences between predictions for some macroscopic be-

havior (e.g., a homogenized stress vs. strain curve) can be employed for checking the

adequacy of volume elements following the concept of “physical RVEs”. In addition,

58) Statistical assessments of the phase geometries of given volume elements are suitable forestablishing their suitability within the concept of geometrical RVEs, whereas at present thereappears to be no low-cost procedure for checking the compliance with the concept of physicalRVEs.

59) This requirement is sometimes phrased as finding a “minimum RVE” (Ren/Zheng,2004).

– 48 –

deviations from the required material symmetry of the overall response can be used for

checking the adequacy of the size of a given volume element. A comparison of different

geometry and microfield based criteria for choosing the size of model geometries was

given by Trias et al. (2007).

Volume elements fulfilling one or more of the above criteria with a given degree of ac-

curacy are in general not RVEs in the strict sense as discussed in section 1.2 and are

sometimes referred to as Statistical Representative Volume Elements (SRVEs). With

growing size of SRVEs the predicted micro and macro fields tend to become less sensitive

to the boundary conditions actually applied, compare (Shen/Brinson,2006).

Assessments based on geometrical parameters or on the overall elastic behavior typically

give rise to relatively small volume elements. For example, Zeman (2003) reported that

the transverse elastic behavior of continuously reinforced composites can be satisfactorily

described by unit cells containing reconstructed arrangements of 10 to 20 fibers.

The concept of “physical volume elements” implies that the size of model geometries de-

pends on the specific physical property to be studied. For the case of elastic statistically

isotropic composites with matrix–inclusion topology and sphere-like particles, Drugan

and Willis (1996) estimated that for approximating the overall moduli with errors of less

than 5% or less than 1%, respectively, (nonperiodic) volume elements with sizes of ap-

proximately two or five particle diameters are sufficient for any volume fraction; compare

also (Drugan,2000). When inelastic constituent behavior is present, however, a num-

ber of numerical studies (Zohdi,1999; Jiang et al.,2001; Bohm/Han,2001; Gitman,2007)

have indicated that larger volume elements may be required for satisfactorily approxi-

mating the overall symmetries and for obtaining good agreement between the responses

of (nominally) statistically equivalent phase arrangements, especially at elevated strains.

The reason for this behavior lies in the marked inhomogeneity of the strain fields typ-

ically present in such regimes, where contiguous zones of high strains tend to appear.

These often evolve to become considerably larger than individual reinforcements and

thus effectively introduce a new length scale into the problem. Analogous behavior was

reported for models of composites that explicitly account for microscopic damage, see

e.g. (Swaminathan/Ghosh 2006). Accordingly, the size of satisfactory multi-particle unit

cells tends to depend on the phase material behavior60).

In general, each model microgeometry may be viewed as being a member of a family

of phase arrangements, all of which are (approximate) realizations of some statistical

process. In the typical case, where individual model geometries obtained by a computer

model or from real structures are smaller than proper RVEs, ensemble averaging over

a number of results obtained from such volume elements can be used to estimate the

60) For elastoplastic matrices, weaker strain hardening tends to translate into geometricallymore complex volume elements, and very large model geometries are necessary when one ofthe phases shows softening, e.g. due to damage (Zohdi/Wriggers,2001). For the latter case, infact, eliminating the dependence of the homogenized response on the size of the SRVE may beimpossible (Gitman et al.,2007).In the case of elastic polycrystals, the anisotropy and shape of the grains were also reported toinfluence the required size of volume elements (Ren/Zheng,2004).

– 49 –

effective material properties, compare e.g. (Kanit et al.,2003; Stroeven et al., 2004).

The number of volume different elements required for a given accuracy of the ensemble

averages decreases as the sizes of the models increase (Khisaeva/Ostoja-Starzewski,2006).

4.2 Numerical Engineering Methods

The majority of published micromechanical studies of discrete microstructures have em-

ployed standard numerical engineering methods for resolving the microfields. Studies us-

ing Finite Difference (Adams/Doner,1967) and Finite Volume (Bansal/Pindera,2006) al-

gorithms, spring lattice models (see e.g. Ostoja-Starzewski,2002), the Boundary Element

Method (BEM; see e.g. Achenbach/Zhu,1989; Liu et al.,2005), the Finite Element Method

(FEM) and its developments such as Extended Finite Element Methods (XFEM; see e.g.

Sukumar et al.,2001), meshfree and particle methods (see e.g. Missoum-Benziane et al.,

2007), techniques using Fast Fourier Transforms (FFT; see e.g. Moulinec/Suquet,1994)

and Discrete Fourier Transforms (DFT; Muller,1996)61), as well as FE-based discrete

dislocation models (Cleveringa et al.,1997) have been reported. A number of more spe-

cialized approaches are discussed in connection with periodic microfield analysis, see

section 5.1. Generally speaking, spring lattice models tend to have advantages in han-

dling pure traction boundary conditions and in modeling the progress of microcracks due

to local (brittle) failure. Boundary elements tend to be at their best in studying geo-

metrically complex linear elastic problems. For all the above methods the characteristic

length of the discretization (“mesh size”) must, of course, be considerably smaller than

the microscale of a given problem in order to obtain spatially well resolved results.

At present, the FEM is the most commonly used numerical scheme for evaluating discrete

microgeometries, especially in the nonlinear range, where its flexibility, efficiency and

capability of supporting a wide range of constitutive models for the constituents and for

the interfaces between them are especially appreciated62). An additional asset of the

FEM in the context of continuum micromechanics is its ability to handle discontinuities

in the stress and strain components (which typically occur at interfaces between different

constituents) in a natural way via appropriately placed element boundaries.

Applications of the FEM to micromechanical problems tend to fall into four main groups,

compare fig. 13. In most published works the phase arrangements are discretized by an

often high number of “standard” continuum elements, the mesh being designed in such

a way that element boundaries (and, where required, special interface elements) are

positioned at all interfaces between constituents. Such an approach has the advantage

that in principle any microgeometry can be handled and that readily available commercial

61) FFT and DFT based methods are limited to periodic configurations, for which they tendto be highly efficient, compare e.g. (Michel et al.,1999). Due to their use of a fixed regular gridthey are well suited for real structure models and for studying the evolution of microstructures,see e.g. (Dreyer et al.,1999).

62) Constitutive models for constituents used in FEM based micromechanics have includeda wide range of elastoplastic, viscoelastoplastic and continuum damage mechanics type descrip-tions as well as crystal plasticity models (McHugh et al.,1993; Bruzzi et al.,2001).

– 50 –

a) b) d)c)

FIG.13: Sketch of FEM approaches used in micromechanics: a) discretization by stan-

dard elements, b) special hybrid elements, c) pixel/voxel discretization, d) “multiphase

elements”

FE packages may be used. On the one hand, the actual modeling of complex phase

configurations in many cases requires sophisticated and/or specialized preprocessors for

generating the mesh, a task that has been difficult to automatize for microgeometries.

The resulting stiffness matrices may show unfavorable conditioning due to suboptimal

element shapes and the requirement of satisfactory resolution of the microfields at local

“hot spots” (e.g. between closely neighboring reinforcements) can lead to very large

models indeed63). On the other hand, the possibility of providing for a refined mesh

where it is required is one of the strength of this discretization strategy.

Alternatively, a smaller number of special hybrid elements may be used, which are specif-

ically formulated to model the deformation, stress, and strain fields in an inhomogeneous

region consisting of a single inhomogeneity together with the surrounding matrix on the

basis of some appropriate analytical theory, see e.g. (Accorsi,1988). The most highly de-

veloped approach of this type at present is the Voronoi Finite Element Method (Ghosh

et al.,1996), in which the mesh for the hybrid elements is obtained by Voronoi tesse-

lations based on the positions of the reinforcements. Large planar multi-inhomogeneity

arrangements can be analyzed this way using a limited number of (albeit rather complex)

elements, and good accuracy as well as significant gains in efficiency have been claimed.

Numerous damage models have been implemented into the method and an extension to

three dimensions was reported recently (Ghosh/Moorthy,2004). Computational strate-

63) Note that even when quadratic shape functions are used, discretizing “thin” regionsof a phase (such as the matrix between two reinforcements) with two “layers” of elementsrestricts the variation of stresses and strains there to piecewise linear functions consisting oflinear sections. This is, at best, the barest minimum for capturing the local fields, which tendto show marked gradients exactly in such regions, especially in the elastoplastic regime. Themacroscopic and phase averaged fields tend to be much more forgiving with respect to thefineness of the discretization.

– 51 –

gies of this type are, of course, specifically tailored for inhomogeneous materials with

matrix–inclusion topologies.

A computational strategy that occupies an intermediate position between between the

above two approaches uses static condensation to remove the degrees of the interior nodes

of the inhomogeneities (or, in the case of coated reinforcements, of inhomogeneities and

interface) from the stiffness matrix of a heterogeneous volume element (Liu et al.,2005).

This method is, however, limited to linear analysis.

Especially when the phase arrangements to be studied are based on experimentally ob-

tained digital representations of actual microgeometries, a third approach for discretizing

microgeometries of interest. It consists of using a regular mesh that consists of rectangu-

lar or hexahedral elements of fixed size mesh and has the same resolution as the digital

data, each element being assigned to one of the constituents by operations such as thresh-

olding of the grey values of the corresponding pixel or voxel, respectively. Such models

have the advantage of being suitable for (semi-)automatic generation from appropriate

experimental data (metallographic sections, tomographic scans) and of avoiding ambigu-

ities in smoothing the digital data (which may be an issue if “standard” FE meshes are

employed to discretize data of this type). Obviously “voxel element” strategies lead to

ragged phase boundaries, which may give rise to some oscillatory behavior of the solu-

tions (Niebur et al.,1999), can cause very high local stress maxima (Terada et al.,1997),

may degrade accuracy, and restrict modeling of interfacial behavior. Such “digital im-

age based” (DIB) models are, however, claimed not to cause unacceptably large errors

even for relatively coarse discretizations, at least in the linear elastic range (Guldberg

et al.,1998). Good spatial resolution, i.e. a high number of pixels or voxels, of course, is

very beneficial in such models.

A fourth approach also uses regular FE meshes, but assigns phase properties at the inte-

gration point level of standard elements (“multiphase elements”, see e.g. Schmauder et

al.,1996; Quilici/Cailletaud,1999). Essentially, this amounts to trading off ragged bound-

aries at element edges for smeared-out (and typically degraded) microfields within those

elements that contain a phase boundary, because stress or strain discontinuities within

elements cannot be adequately handled by standard FE shape functions. With respect

to the element stiffnesses the latter concern can be much reduced by overintegrating ele-

ments containing phase boundaries, which leads to good approximations of integrals in-

volving non-smooth displacements by numerical quadrature, see (Zohdi/Wriggers,2001).

The resulting stress and strain distributions, however, remain smeared-out approxima-

tions in elements that contain phase boundaries, the rate of convergence of the microfields

being accordingly slow. The above drawbacks are avoided by algorithmic developments

such as Extended Finite Element methods, which use a background mesh that does not

have to conform to phase boundaries, which are accounted for, e.g., by appropriately

enriching the shape functions (Moes et al.,2003).

A fairly recent development for studying microgeometries involving a large number of

inhomogeneities by Finite Element methods involves programs specially geared towards

solving micromechanical problems. Such codes may be based on matrix-free iterative

– 52 –

solvers such as Conjugate Gradient (CG) methods, analytical solutions for the microfields

(e.g. constant strain approximations corresponding to the Hill upper bounds, eqn.(55))

being used as starting solutions to speed convergence64). For studies involving such

codes see e.g. (Gusev et al.,2000; Zohdi/Wriggers,2001). Fast solvers of this type open

the possibility of solving inverse problems, e.g. for finding optimal particle shapes for

given load cases and damage modes, compare (Zohdi,2003).

Special Finite Element formulations are typically required for asymptotic homogenization

(compare section 5.2) and for other schemes that concurrently solve the macroscopic and

microscopic problems, see e.g. (Urbanski,1999).

64) Essentially, in such a scheme the initial guess gives a good estimate of “long wavelength”contributions to the solution, and the CG iterations take care of short “wavelength” variations.

– 53 –

5. PERIODIC MICROFIELD APPROACHES

The following discussion concentrates on Periodic Microfield Approaches (PMAs) as used

in continuum micromechanics of materials, where they are used to describe the macro-

scopic and microscopic behavior of inhomogeneous materials by studying model materials

that have periodic microstructures. Closely related methods are used in other fields of

engineering for handling problems that involve two or more appropriate length scales, e.g.

for describing beam, plate and shell structures made up of repeating periodic units65).

The first two sections of this chapter cover some basic concepts of unit cell based PMAs.

Following this, applications to continuously reinforced composites, discontinuously rein-

forced composites, and some other inhomogeneous materials are discussed.

In the following discussion the main emphasis is on the small strain elastic and elasto-

plastic behavior; for homogenization methods geared specifically towards finite strain

problems see e.g. (Miehe et al.,1999) or (Terada et al.,2003).

5.1 Basic Concepts of Unit Cell Models

Periodic microfield approaches analyze the behavior of infinite (one, two- or three-

dimensional) periodic phase arrangements66) under the action of far field mechanical

loads or uniform temperature fields67). The most common approach for studying the

stress and strain fields in such periodic configurations is based on describing the micro-

geometry by a periodically repeating unit cell to which the investigations may be limited

without loss of information or generality, at least for static analysis68). The literature on

65) For unified treatments of continuum, shell-like and beam-like periodic structures see e.g.(Urbanski,1999; Pahr/Rammerstorfer,2006).

66) In the present work the designation “unit cell” is used for any phase arrangement thatgenerates a periodic microgeometry. Accordingly, a unit cell may comprise a part of a simpleperiodic base unit, a simple periodic base unit, a collective of simple periodic base units, or aphase arrangement of arbitrary geometrical complexity that shows translational periodicity.

67) Standard PMAs cannot handle macroscopic gradients in mechanical loads, temperatureor composition in any direction in which periodicity of the fields is prescribed. Gradients andfree boundaries can be handled, however, in directions where periodicity is not prescribed, atypical case being layer-type models that are nonperiodic in one direction and periodic in theother(s), see e.g. (Wittig/Allen,1994; Reiter et al.,1997; Weissenbek et al.,1997).

68) Periodic phase arrangements typically are not well suited to dynamic analysis, becausethey act as filters that exclude all waves with frequencies outside certain bands, see e.g.(Suzuki/Yu,1998). In addition, due to the boundary conditions required for obtaining peri-odicity, unit cell methods can only handle wavelengths that are smaller than or equal to therelevant cell dimension, and only standing waves can be dealt with.In analogy, in stability analysis unit cells can directly resolve only buckling modes of specificwavelengths, so that considerable care is required in using them for such purposes, compare(Vonach,2001; Pahr/Rammerstorfer,2006). However, bifurcation modes with wavelengths ex-ceeding the size of the unit cell can be resolved in periodic linear models by using the Blochtheorem from solid state physics (Gong et al.,2005).

– 54 –

such PMAs is fairly extensive, and well developed mathematical theories are available

on scale transitions in periodic structures, compare (Michel et al.,2001).

A wide variety of unit cells have been employed in published PMA studies, ranging from

geometries that describe simple periodic arrays of inhomogeneities to highly complex

phase arrangements, such as multi-inhomogeneity cells (“supercells”). For some simple

periodic phase arrangements it is also possible to find analytical solutions via series ex-

pansions that make explicit use of the periodicity (see e.g. Sangani/Lu,1987; Cohen,2004)

or via appropriate potential methods (Wang et al.,2000).

Even though most PMA studies in the literature have used standard numerical engineer-

ing methods as described in chapter 4, some more specialized approaches for evaluating

microscopic stress and strain fields are worth mentioning. One of them, known as the

Method of Cells (Aboudi,1989; Aboudi,1991), discretizes unit cells that correspond to

square arrangements of square fibers into four subcells, within each of which displace-

ments are approximated by low-order polynomials. Traction and displacement continuity

conditions at the faces of the subcells are imposed in an average sense and analytical

and/or semi-analytical approximations to the deformation fields are obtained in the elas-

tic and inelastic ranges. While using highly idealized microstructures and providing only

limited information on the microscopic stress and strain fields, the resulting models pose

relatively low computational requirements but have limited capabilities for handling ax-

ial shear. The Method of Cells has been used as a constitutive model for analyzing

structures made of continuously reinforced composites, see e.g. (Arenburg/Reddy,1991).

Developments of the algorithm led to the Generalized Method of Cells (Aboudi,1996),

which allows finer discretizations of unit cells for fiber and particle reinforced compos-

ites, reinforcement and matrix being essentially split into a number of “subregions” of

rectangular or hexahedral shape. For some comparisons with microfields obtained by

Finite Element based unit cells see e.g. (Iyer et al.,2000; Pahr/Arnold,2002). Recently,

further improvements of the Method of Cells have been reported, see (Aboudi et al.,2003;

Pindera et al.,2003).

Another micromechanical approach, the Transformation Field Analysis (Dvorak,1992),

allows the prediction of the nonlinear response of inhomogeneous materials based on

either mean field descriptions (compare the remarks in section 2.4) or on periodic mi-

crofield models for the elastic behavior. High computational efficiency is claimed for TFA

models involving piecewise uniform transformation fields (Dvorak et al.,1994), but very

fine discretizations of the phases are required to achieve high accuracy.

A further group of solution strategies for PMAs reported in the literature (Axelsen and

Pyrz,1995; Fond et al.,2001; Schjødt-Thomsen and Pyrz,2004) use numerically evaluated

equivalent inclusion approaches that account for interacting inhomogeneities as provided

e.g. by the work of Moschovidis and Mura (1975).

In typical periodic microfield approaches strains and stresses are split into constant

macroscopic strain and stress contributions (“slow variables”), 〈ε〉 and 〈σ〉, and peri-

odically varying microscopic fluctuations (“fast variables”), ε′(z) and σ′(z), in analogy

to eqn.(1). The position vectors, however, are denoted as z to indicate that the unit cell

– 55 –

s

u

uε,

s

<ε >

<ε >ε

∆us

A A AB B B A

s

s

s

s

Ul

FIG.14: Schematic depiction of the variation of the strains εs(s) and the displacements

us(s) along a section line (coordinate direction s) in some hypothetical inhomogeneous

material consisting of constituents A and B.

“lives” on the microscale. The volume integrals used to obtain averages, eqns.(3) and

(4), must, of course, be solved over the volume of the unit cell, ΩUC. Formal derivations

of the above relationships for periodically varying microstrains and microstresses show

that the work done by the fluctuating stress and strain contributions vanishes, compare

(Michel et al.,2001).

Evidently in periodic microfield approaches each unit of periodicity (unit cell) contributes

the same increment of the displacement vector ∆u and the macroscopic displacements

vary (multi)linearly. An idealized depiction of such a situation is presented in fig. 14,

which shows the variations of the strains εs(s) = 〈εs〉 + ε′s(s) and of the corresponding

displacements us(s) = 〈εs〉 s + u′

s(s) along some section line s in a hypothetical periodic

two-phase material consisting of constituents A and B. Obviously, the relation 〈εs〉 =

∆us/lU holds for linear displacement–strain relations, where lU stands for the length of

a unit cell in direction s and ∆us for the corresponding displacement increment. The

periodicity of the strains and of the displacements is immediately apparent.

5.2 Boundary Conditions, Application of Loads, Evaluation of Microfields

Boundary Conditions

Unit cells together with the boundary conditions (B.C.s) prescribed on them must gener-

ate valid tilings both for the undeformed geometry and for all deformed states pertinent

to the investigation. Accordingly, gaps and overlaps between neighboring unit cells as

– 56 –

well as unphysical constraints on the deformations must not be allowed. In order to

achieve this, the boundary conditions for the unit cells must be specified in such a way

that all deformation modes appropriate for the load cases to be studied can be attained.

The three major types of boundary conditions used in periodic microfield analysis are

periodicity, symmetry, and antisymmetry (or point symmetry) B.C.s69).

2

1 periodic boundarysymmetry boundarypoint symmetry boundary

symmetry center(pivot point)

AB

F

GC

H I

D

E

J

FIG.15: Periodic hexagonal array of circular inhomogeneities in a matrix and 11 unit

cells that can be used to describe at least some of the mechanical responses of this

arrangement under loads acting parallel to the coordinate axes.

Generally, for a given periodic phase arrangement unit cells are non-unique, the range

of possible shapes being especially wide when point or mirror symmetries are present in

the microgeometry. As an example, fig. 15 depicts a (planar) periodic hexagonal array

of circular inhomogeneities (fibers with out-of-plane orientation) and some of the unit

cells that can be used to study aspects of the mechanical behavior of this arrangement.

There are considerable differences in the sizes and capabilities of the unit cells shown.

In fig. 16 schematic sketches of periodicity, symmetry and antisymmetry B.C.s, are given

for two-dimensional unit cells. The sides and corner points of the cells are annotated; an

extension of this nomenclature to three dimensions is trivial.

69) For more formal treatments of boundary conditions for unit cells than given here seee.g. (Anthoine,1995; Michel et al.,1999). In the case of layer-type models, which can be gener-ated by specifying free surfaces at appropriate boundaries, macroscopic bending and twistingdeformations can be studied by appropriately modifying the above standard B.C.s. For modelswhere such macroscopic rotational degrees of freedom are present, “out-of-plane” shear loadsmust always be accompanied by appropriate bending moments to achieve stress equilibrium, seee.g. (Urbanski,1999).

– 57 –

NW

SE

NE

SW

NW

SW

NE

SE SW

NWNEUu’

Lu’

Pu’

N

W E

S

u’

SEu’

EW

N

S

u’NWu’NE

SEu’

W

S

Nu’NWNW

ENEu’

z

z

L

P

U

1

2

FIG.16: Sketch of periodicity (left), symmetry (center), and antisymmetry (right)boundary conditions as used with two-dimensional unit cells.

The most general boundary conditions for unit cells are periodicity (“toroidal”, “cyclic”)

B.C.s, which can handle any physically valid deformation state of the cell and, conse-

quently, of the inhomogeneous material to be modeled. Periodicity boundary conditions

make use of translatory symmetries of a geometry; in fig. 15 cells A to E belong to this

group. Because such unit cells tile the computational space by translation, neighboring

cells (and, consequently, the “opposite” faces of a given cell) must fit into each other like

the pieces of a jigsaw puzzle in both undeformed and deformed states. For the case of

quadrilateral two-dimensional unit cells this can be achieved by pairing opposite faces

and linking the corresponding degrees of freedom within each pair of faces. Using the

conventions for naming the vertices and faces shown in fig. 16 (left), the resulting equa-

tions for the periodically varying microscopic displacement vectors at the boundaries70)

u′, can be symbolically written as

u′

N(z1) = u′

S(z1) + u′

NW and u′

E(z2) = u′

W(z2) + u′

SE (59)

which implies

u′

NE = u′

NW + u′

SE .

Here z1 and z2 denote “corresponding positions” on the N and S faces and on the E and

W faces of the unit cell, respectively. Equation (59) slaves the microscopic displacements

of nodes on faces N and E to those of corresponding nodes on the “master faces” S and

W, with the nodes SE and NW acting as “master nodes”. The undeformed geometry

must fulfill the compatibility conditions

zE(z2)− zW(z2) = zSE − zSW = lEW and zN(z1)− zS(z1) = zNW − zSW = lNS (60)

70) In principle, all variables (i.e., for mechanical analysis, the displacements, strains andstresses) must be linked by appropriate periodicity conditions. When a displacement basedFE code is used, however, such conditions can be specified explicitly only for the displacementcomponents (including, where appropriate, rotation D.O.F.s), and the periodicity of the stressesand strains is fulfilled in an integral sense only. When stresses are periodic, unit cell boundarytractions are antiperiodic.For periodicity and antisymmetry boundary conditions additional errors may be introduced ifthe nodal stresses and strains are not averaged across cell boundaries, even though they oughtto be.

– 58 –

for master–slave pairs of nodes zE and zW as well as zN and zS, i.e. it must be pos-

sible to translate a master face into its corresponding slave face. In addition, for each

master–slave pair of faces the phase arrangements and the FE discretizations must be

compatible, compare (Swan,1994). With the exception of these conditions there is wide

latitude in choosing the shapes and the microgeometries of unit cells that use periodicity

boundary conditions71). Periodicity B.C.s generally are the least restrictive option for

multi-inhomogeneity unit cell models using phase arrangements obtained by statistics-

based algorithms (“supercells”) or on the basis of experimental techniques (real structure

based models).

In practice FE-based unit cell studies using periodicity boundary conditions can be rather

expensive in terms of computing time and memory requirements, because the multi-point

constraints required for implementing eqn.(59) or its equivalents tend to degrade the band

structure of the system matrix, especially in three-dimensional problems. In addition it

is worth noting that — especially for simple phase arrangements — considerable care

may be required to prevent over- and underconstraining due to inappropriate selection

of regions with periodicity boundary conditions, compare (Pettermann/Suresh,2000).

For rectangular and hexahedral unit cells in which the faces of the cell coincide with

symmetry planes of the phase arrangement and for which this property is retained for all

deformed states that are to be studied, periodicity B.C.s simplify to symmetry boundary

conditions. For the arrangement shown in fig. 16 (center) these B.C.s take the form

u′

E(z2) = u′

SE v′

N(z1) = v′

NW u′

W(z2) = 0 v′

S(z1) = 0 , (61)

where u′ and v′ are the components of the microscopic displacement vector u′. The

roles of slaves and masters are the same as in eqn.(59). Evidently, eqn.(61) enforces

the condition that pairs of master and slave faces must stay parallel throughout the

deformation history. Note that for symmetry B.C.s there are no requirements with

respect to compatibility of phase arrangements or meshes at different faces.

Symmetry boundary conditions are fairly easy to use and tend to give rise to small unit

cells for simply periodic phase arrangements, but the load cases that can be handled

are limited to uniform thermal loads, mechanical loads that act in directions normal to

one or more pairs of faces, and combinations of the above72). In fig. 15, unit cell F

71) Conditions analogous to eqn.(59) can be specified for regular area-filling two-dimensionalcells having an even number of sides, i.e. squares, rectangles and hexagons (compare cell E in fig.15), and for regular space-filling three-dimensional cells, i.e. right hexahedra, cubes (sc symme-try), rhombic dodekahedra (fcc symmetry) and regular tetrakaidekahedra (truncated octahedra,bcc symmetry). More complicated conditions have to be used for unit cells of less regular shape,compare e.g. (Cruz/Patera,1995; Estrin et al.,1999; Xia et al.,2003). If appropriate symmetriesare present within periodic unit cells, as may be the case with textile composites, these can beused to reduce the problem to arrays of smaller subcells, see e.g. (Tang/Whitcomb,2003).

72) Because these load cases include different normal loads acting on each pair of faces, load-ing by macroscopic extensional shear (obtained e.g. in the two-dimensional case by applyingnormal stresses σa to the vertical and −σa to the horizontal faces), hydrostatic loading, and

– 59 –

uses boundary conditions of this type73). Symmetry boundary conditions are typically

very useful for describing relatively simple microgeometries, but tend to be somewhat

restrictive for modeling phase arrangements generated by statistics-based algorithms or

obtained from micrographs.

Antisymmetry boundary conditions are even more limited in terms of the microgeometries

that they can handle, because they require the presence of centers of point symmetry

(“pivot points”). In contrast to symmetry boundary conditions, however, unit cells

employing them on all faces are subject to few restrictions to the load cases that can be

studied. Among the unit cells shown in fig. 15, cells G and H use point symmetry B.C.s

on all faces and can handle any in-plane deformation74). Alternatively, antisymmetry

B.C.s can be combined with symmetry B.C.s to obtain very small unit cells that are

restricted to loads acting normal to the symmetry faces, compare unit cells I and J in fig.

15. The right hand sketch in fig. 16 shows such a “reduced” unit cell, the antisymmetry

boundary conditions being applied on the E-side where a pivot point P is present. For

this configuration the boundary conditions

u′

U(s) + u′

L(−s) = 2u′

P v′

N(z1) = v′

NW = 2v′

P v′

S(z1) = 0 u′

W(z2) = 0 (62)

must be fulfilled, where u′

U(s) and u′

L(−s) are the deformation vectors of pairs of points

U and L that are positioned symmetrically on face E (where a local coordinate system

s centered on the pivot P is defined in analogy to fig. 24) and deform symmetrically

with respect to the pivot P. The undeformed geometry of such a face also must be

antisymmetric with respect to the pivot point P, i.e.

zL(s) + zU(−s) = 2zP , (63)

and the phase arrangements as well as the discretizations on both halves of the face

must be compatible. Three-dimensional unit cells employing combinations of symmetry

and point symmetry B.C.s can e.g. be used to advantage for studying cubic arrays of

particles, see (Weissenbek et al.,1994).

(hygro)thermal loading, symmetry B.C.s are typically sufficient for materials characterization.However, stress triaxialities obtained this way may correspond to different orientations with re-spect to the macroscopic material coordinate system (for example, extensional shear correspondsto simple shear in a coordinate system rotated by 45).

73) Whereas for a given periodic phase arrangement there are infinitely many equivalentunit cells with periodicity B.C.s, the number of “smallest” unit cells with symmetry B.C.s isalways limited. It is also worth noting that for an n-dimensional problem 2n unit cells withsymmetry B.C.s can be assembled (by mirroring operations) into a unit cell that is suitable forthe application of periodicity B.C.s.

74) Triangular unit cells similar to cell H in fig. 15 were used e.g. by Teply and Dvorak(1988) to study the transverse mechanical behavior of hexagonal arrays of fibers. Rectangularcells with point symmetries on each boundary were introduced in (Marketz/Fischer,1994) forperturbed square arrangements of inhomogeneities. The periodic cells D and E also show pointsymmetry on their faces.

– 60 –

Within the nomenclature shown in fig. 16 the displacements at the master nodes SE

and NW, u′

SE and u′

NW, control the overall deformation of the unit cell in the 1- and

2-directions and can be used to evaluate the macroscopic strain of the corresponding

periodic model material, compare eqn.(72). With respect to unit cell models that use

the FEM for evaluating the microfields, it is worth noting that all of the above types

of boundary conditions can be handled by any code that provides linear multipoint

constraints between degrees of freedom and thus allows the linking of three or more

D.O.F.s by linear equations.

Obviously, the description of real materials, which in general are not periodic, by peri-

odic model materials entails some geometrical approximations. These take the form of

periodicity constraints on computer generated generic phase arrangements or of appro-

priate modifications in the case of real structure microgeometries75), compare fig. 1. The

effects of such approximations in the vicinity of the cell surfaces, of course, diminish in

importance with growing size of the model.

Equations (59) to (63) refer to the usual case where local material descriptions are used

on the macroscale and the scale transition is handled via homogenized stress and strain

fields. Periodicity boundary conditions that are conceptually similar to eqn.(59) can be

devised for cases where gradient (nonlocal) theories are employed on the macroscale and

higher order stresses as well as strain gradients figure in coupling the length scales (Geers

et al.,2001).

Linking Macroscale and Microscale

The primary practical challenge in using periodic microfield approaches for modeling

inhomogeneous materials lies in choosing and generating suitable unit cells that — in

combination with appropriate boundary conditions — allow a realistic representation

of the actual microgeometries within available computational resources. The unit cells

must then be subjected to appropriate macroscopic stresses, strains and temperature

excursions. Whereas loads of the latter type generally do not pose major difficulties,

applying far field stresses or strains is not necessarily straightforward (note that the

variations of the stresses along the faces of a unit cell in general are not known a priori,

so that it is not possible to prescribe the boundary tractions via distributed loads).

The most versatile and elegant strategy for linking the macroscale and the microscale in

periodic microfield models is based on a mathematical framework known as asymptotic

homogenization, asymptotic expansion homogenization, or homogenization theory, see

e.g. (Suquet,1987). Instead of formulating the unit cell problem solely on the microscale,

75) The simplest way of “periodifying” a non-periodic rectangular or cuboidal volume el-ement consists in mirroring it with respect to 2 or 3 of its surfaces, respectively, and doingperiodic homogenization on the resulting larger unit cell. Such operations, however, may leadto undesired shapes of reinforcement and will change the arrangement statistics at least to someextent. The simpler expedient of specifying symmetry B.C.s for rectangular or cuboidal realstructure cells amounts to an even larger modification of the microgeometry and introducesmarked local constraints. For sufficiently large multi-inhomogeneity cells Terada et al. (2000)obtained useful results by applying periodicity boundary conditions to nonperiodic geometries.

– 61 –

macroscopic and microscopic coordinates, Z and z, respectively, are explicitly introduced.

They are related by the expression

zi = Zi/ε , (64)

where ε = `/L 1 is a scaling parameter, L and ` being characteristic lengths of the

macro- and microscales76). The displacement field in the unit cell can then be represented

by an asymptotic expansion of the type

ui(Z, z, ε) = u(0)i (Z) + ε u

(1)i (Z, z) + ε2 u

(2)i (Z, z) + H.O.T. , (65)

where the u(0)i are the effective or macroscopic displacements and u

(1)i stands for the 1st

order displacement perturbations due to the microstructure, which are assumed to vary

periodically77). Using the chain rule, i.e.

∂

∂xf(Z, z =

Z

ε) →

∂

∂Zf +

1

ε

∂

∂zf , (66)

the strains can be related to the displacements (in the case of small strains) as

εij(Z, z, ε) =1

2

(

∂

∂Zj

u(0)i +

∂

∂Zi

u(0)j

)

+

(

∂

∂zj

u(1)i +

∂

∂zi

u(1)j

)

+ε

2

(

∂

∂Zj

u(1)i +

∂

∂Zi

u(1)j

)

+

(

∂

∂zj

u(2)i +

∂

∂zi

u(2)j

)

+ H.O.T.

= ε(1)ij (Z, z) + ε ε

(2)ij (Z, z) + H.O.T. , (67)

where terms of the type ε(0)ij = 1

ε∂

∂zju

(0)i are deleted due to the underlying assumption

that the variations of slow variables are negligible at the microscale. The stresses are

expanded in analogy to the strains so that the equilibrium condition can be written as

( ∂

∂Zj

+1

ε

∂

∂zj

) [

σ(1)ij (Z, z) + ε σ

(2)ij (Z, z)

]

+ fi(Z) = 0 , (68)

the fi being macroscopic body forces. Working out eqn.(68) and sorting the contributions

by order of ε gives rise to a hierarchical system of partial differential equations.

∂

∂zj

σ(1)ij = 0 (order ε−1)

∂

∂Zj

σ(1)ij +

∂

∂zj

σ(2)ij + fi = 0 (order ε0) , (69)

76) Equation (64) may be viewed as “stretching” the microscale so it becomes comparableto the macroscale, f(x)→ f(Z,Z/ε) = f(Z, z).

77) The nomenclature used in eqns.(64) to (70) follows typical usage in asymptotic homoge-nization. It is more general than, but can be directly compared with, the one used in eqns.(1)to (63), where only microscopic coordinates are employed. Zero order terms can generally beidentified with macroscopic quantities and first order terms generally correspond to periodicallyvarying microscopic quantities.

– 62 –

the first of which gives rise to a boundary value problem at the unit cell level that is

referred to as the “micro equation”. By making a specific ansatz for the strains at unit

cell level and by volume averaging over the second equation in the system (69), the

“macro equation”, the microscopic and macroscopic fields can be linked such that the

homogenized elasticity tensor is obtained as

EHijkl =

1

ΩUC

∫

ΩUC

Eεijmn

(1

2

(

δmkδnl + δmlδnk

)

+∂

∂zl

χkmn

)

dΩ . (70)

Here ΩUC is the volume of the unit cell, Eεijkl(z) is the phase-level (microscopic) elasticity

tensor, and the “characteristic function” χkmn(z) describes the deformation modes of the

unit cell78) and, accordingly, relates the micro- and macrofields. Analogous expressions

can be derived for tangent modulus tensors in nonlinear analysis, compare e.g. (Ghosh

et al.,1996).

The above relations can be used as the basis of Finite Element algorithms that solve for

the characteristic function χkmn. For detailed discussions of asymptotic homogenization

methods within the framework of FEM-based micromechanics see e.g. (Jansson,1992;

Ghosh et al.,1996; Hassani/Hinton,1999; Chung et al.,2001).

Asymptotic homogenization allows to directly couple FE models on the macro- and

microscales, compare e.g. (Terada et al.,2003), an approach that has been used in a

number of multi-scale studies (compare chapter 8) and is sometimes referred to as FE2

method (Feyel/Chaboche,2000). A treatment of homogenization in the vicinity of macro-

scopic boundaries can be found in (Schrefler et al.,1997), an asymptotic homogenization

procedure for elastic composites that uses standard elements within a commercial FE

package was proposed by Banks-Sills et al. (1997). Asymptotic homogenization schemes

have also been involved for problems involving higher order stresses and strain gradients

(Kouznetsova et al.,2002)79).

When asymptotic homogenization is not used, it is good practice to apply far field stresses

and strains to a given unit cell via concentrated nodal forces or prescribed displacements

at the master nodes and/or pivots, an approach termed the method of macroscopic

degrees of freedom by Michel et al. (1999). Employing the divergence theorem and

using the nomenclature and configuration of fig. 16 (left), the forces to be applied to the

master nodes SE and NW, PSE and PNW, of a two-dimensional unit cell with periodicity

boundary conditions can be shown to be given by the surface integrals

PSE =

∫

ΓE

ta(z) dΓ PNW =

∫

ΓN

ta(z) dΓ , (71)

78) Even though eqns.(69) and (70) are derived from an explicit two-scale formulation neitherof them contains the scale parameter ε, see the discussion in (Chung et al.,2001).

79) Such methods are especially useful for problems in with the length scales are not wellspread; in them the “unit cells” do not necessarily remain periodic during the deformationprocess.

– 63 –

compare (Smit et al.,1998). Here ta(z) = σanΓ(z) stands for the homogeneous surface

traction vector corresponding to the applied (far field) stress field80) at some given point

z on the cell’s surface ΓUC, with nΓ(z) being the local surface normal. Equation (71)

can be generalized to require that each master node is loaded by a force corresponding

to the surface integral of the applied traction vectors over the face slaved to it via an

equivalent of eqn.(59). An analogous procedure holds for three-dimensional cases, and

symmetry as well as antisymmetry boundary conditions as described by eqns.(61) and

(62) can be handled similarly.

For applying far field strain boundary conditions within the method of macroscopic de-

grees of freedom, displacements are prescribed to the master nodes. These nodal displace-

ments must be obtained from the macroscopic strains via the appropriate displacement–

strain relations. For example, following the notation of eqns.(59) to (62), the displace-

ments to be prescribed to the master nodes 2 and 4 of the unit cells shown in fig. 16 can

be evaluated as

u′

SE = 〈ε〉11 lEW v′

SE = γ12 lEW u′

NW = γ21 lNS v′

NW = 〈ε〉22 lNS (72)

for the case of linear strain–displacement relations81), the reference lengths of the rectan-

gular unit cells being defined in eqn.(60). When FE methods are used, strain controlled

analysis is typically considerably easier to carry out by the method of macroscopic degrees

of freedom than stress controlled analysis82). In order to obtain the elastic tensors of

periodic phase arrangements, a sufficient number (depending on the overall symmetry of

the unit cell, compare section 1.3) of linearly independent load cases must be evaluated.

In general, the overall stress and strain tensors within a unit cell can be evaluated by

volume averaging83) (e.g. using some numerical integration scheme) or by using the equiv-

80) Note that the ta(z) are not identical with the actual local values of the tractions, t(z) atthe cell boundaries, but are equal to them over the cell face in an integral sense. For geometricallynonlinear analysis eqn.(71) must be applied to the current configuration. Because σa is constantit can be factored out of the surface integrals, so that the force applied to a master node PM

takes the form PM = σa

∫

ΓSnΓ(z) dΓ, which is the projection of the applied stress on the

current average surface normal of the slave face ΓS times the current area of the face.81) One of the two displacement components, v′

SE and u′

NW, is an integration constantdescribing solid body rotations of the unit cell and, accordingly, can be chosen at will. Thedisplacements at node NE are fully determined by those of the master nodes, compare eqn.(59).Analogous relations hold for three-dimensional cases. Equations (72) can be directly extendedto nonlinear strain–displacement relations and to handling deformation gradients.

82) Even though the loads acting on the master nodes obtained from eqn.(71) are in principleequilibrated, in stress controlled analysis solid body rotations may be induced through smallnumerical errors. When these solid body rotations are suppressed by deactivating degrees offreedom beyond those required for enforcing periodicity, the quality of the solutions may bedegraded or transformations of the averaged stresses and strains may be required.

83) For large strain analysis the consistent procedure is to first average the deformationgradients and then evaluate the strains from these averages.

– 64 –

alent surface integrals given in eqn.(3), i.e.

〈σ〉 =1

ΩUC

∫

ΩUC

σ(z) dΩ =1

ΩUC

∫

ΓUC

t(z)⊗ z dΓ

〈ε〉 =1

ΩUC

∫

ΩUC

ε(z) dΩ =1

2ΩUC

∫

ΓUC

(u(z)⊗ nΓ(z) + nΓ(z)⊗ u(z)) dΓ . (73)

In the case of rectangular or hexahedral unit cells that are aligned with the coordinate

axes, averaged engineering stress and strain components can, of course, be evaluated by

dividing the applied or reaction forces at the master nodes by the appropriate surface

areas and by dividing the displacements at the master nodes by the appropriate cell

lengths, respectively. Typical relations of this kind are

〈σ11〉 =P1,SE

ΓEand 〈ε〉11 =

u′

SE

lEW, (74)

where P1,SE is the 1-component of P1,SE.

For evaluating phase averaged quantities from unit cell models, it is usually possible to

use direct volume integration of eqn.(12) or similar expressions84). In many FE codes

this can be done by approximate numerical quadrature according to

〈f〉 =1

Ω

∫

Ω

f(z)dΩ ≈1

Ω

N∑

l=1

fl Ωl , (75)

where fl and Ωl are the function value and the integration weight (in terms of an inte-

gration point volume), respectively, associated with the l-th integration point within a

given integration volume Ω that contains N integration points85), When effective values

or phase averages are to be generated of variables that are nonlinear functions of the

stress or strain components (e.g. equivalent stresses, equivalent strains, stress triaxiali-

ties, stress principals), the variable must be evaluated first (e.g. at the integration points)

and volume averaging of the results performed subsequently, compare also the remarks in

section 2.4. Besides generating overall and phase averages, microscopic variables can also

be evaluated in terms of higher statistical moments and of distribution functions (“stress

spectra”, see e.g. Bornert et al.,1994; Bohm/Rammerstorfer,1995; Bohm/Han,2001).

84) It is of some practical interest that volume averaged and phase averaged microfieldsobtained from unit cells must fulfill all relations given in section 2.1 and can thus be convenientlyused to check the consistency of a given model by inserting them into eqns.(15). For finitedeformations, however, appropriate stress and strain measures must be used for this purpose(Nemat-Nasser,1999).

85) Such procedures may also be used for evaluating other volume integrals, e.g. in computingWeibull-type fracture probabilities for reinforcements (Antretter,1998; Eckschlager,2002).

– 65 –

5.3 Unit Cell Models for Continuously Reinforced Composites

Composites reinforced by continuous aligned fibers, which typically show a statistically

transversely isotropic overall behavior, can be studied relatively easily and efficiently

with unit cell methods. Materials characterization — with the exception of the overall

axial shear behavior — can be carried out with two-dimensional unit cell models employ-

ing generalized plane strain elements that use one global degree of freedom to describe

the axial deformation of the whole model86). Such elements are implemented in a num-

ber of commercial FE codes. For handling the overall axial shear response, however,

special elements (Adams/Crane,1984; Sørensen,1991) or three-dimensional models with

appropriate boundary conditions (Pettermann/Suresh,2000) are required87). In all of

the above cases generalized plane strain states of some type are described, i.e. the axial

strains are constant over the unit cell, while axial out-of-plane deformations are present.

Three-dimensional modeling is also relevant for the materials characterization of com-

posites reinforced by continuous aligned fibers when the effects of fiber misalignment,

of fiber waviness, or of broken fibers88) are to be studied, see e.g. (Mahishi,1986; Gar-

nich/Karami,2004). Unit cell models used for nonlinear constitutive modeling, of course,

must be fully three-dimensional and employ periodicity boundary conditions.

Figure 17 shows a number of periodic fiber arrangements for two-dimensional unit cell

modeling of aligned continuously reinforced composites, the periodic hexagonal (PH0)

and periodic square (PS0) arrays being “classical” simple periodic models. The mod-

els with hexagonal symmetry (PH0,CH1,RH2,CH3) show the appropriate transversely

isotropic thermoelastic overall behavior, whereas the other fiber arrangements shown in

fig. 17 are macroscopically tetragonal (PS0,CS7,CS8) or monoclinic (MS5). None of the

eight configurations gives a transversely isotropic response in the elastoplastic regime

under mechanical loads that are not axisymmetric with respect to the fiber direction, be-

cause the hexagonal symmetry of the phase arrangements is broken due to the resulting

distributions of the local material properties (the strain hardening and thus the instan-

taneous moduli at a given position depend on the load history the point has undergone),

compare also fig. 19.

Fiber arrangements of the type shown in fig. 17 can reproduce many features and ef-

fects of the fiber arrangements found in continuously reinforced composites. They are,

however, not statistically representative of the microstructures of most real composites,

86) Because the axial stiffness of composites reinforced by continuous aligned fibers canusually be satisfactorily described by Voigt type models, unit cell models of such materials havetended to concentrate on the transverse behavior. Plane strain models, however, cannot provideany information on the induced axial response of continuously reinforced composites and do notadequately describe the axial microstresses.

87) For linear elastic material behavior there is the additional option of making use of theformal analogy between out-of-plane (antiplane) shear and diffusion problems.

88) For investigating the axial failure behavior of continuously reinforced MMCs, statisticalmethods concentrating on fiber fragmentation or more elaborate spring lattice models, see e.g.(Zhou/Curtin,1995), have been used successfully.

– 66 –

PH0

CH1

RH2

CH3

PS0

MS5

CS7

CS8

FIG.17: Eight periodic fiber arrangements of fiber volume fraction ξ=0.475 for modelingcontinuously fiber reinforced composites (Bohm/Rammerstorfer,1995). Note that theminimum fiber distances are the same for all configurations except the simple periodicarrangements PH0 and PS0.

the exception being materials reinforced by undirectional monofilaments and produced

by fiber–foil techniques. Major improvements with respect to the realism of the micro-

geometries can be be obtained by multi-fiber unit cells (see e.g. Nakamura/Suresh,1993;

Moulinec/Suquet,1997; Gusev et al.,2000; Sejnoha/Zeman,2002) that use random in-

plane positions of the fibers, compare fig. 18.

In table 3 some thermoelastic properties of an aligned continuously reinforced ALTEX/Al

MMC as predicted by bounding methods, MFAs and unit cells using periodic hexagonal

(PH0), clustered hexagonal (CH1), and periodic square (PS0) arrangements as well as

the multi-fiber cell shown in fig. 18 (MFUC) are listed. For this material combination

all unit cell results fall within the Hashin–Shtrikman bounds for continuously reinforced

composites89), but the square arrangement gives predictions for the overall transverse

properties that, besides being clearly anisotropic, significantly violate the three-point

bounds for materials reinforced by identical aligned continuous fibers. Table 3 also shows

89) The constituents’ material properties underlying table 3 have a low elastic contrast ofc ≈ 3. In general the elastic stiffnesses obtained from square-type arrangements may to violatethe Hashin–Shtrikman bounds and usually lie outside the three-point bounds.

– 67 –

FIG.18: Multi-fiber unit cell for a continuously reinforced MMC (ξ=0.453) using 60quasi-randomly positioned fibers and symmetry B.C.s (Nakamura/Suresh,1993)

minor deviations of the results of the multi-fiber model from transversely isotropic elastic

behavior.

The configurations depicted in figs. 17 and 18 give nearly identical results for the over-

all elastoplastic behavior of continuously reinforced composites under axial mechanical

loading, and the predicted overall axial and transverse responses under thermal loading

are very similar. The overall elastoplastic behavior under transverse mechanical load-

ing, however, tends to be very sensitive to the fiber arrangement. As to simply periodic

models, the behavior of hexagonal arrangements is typically sandwiched between the

stiff (in 0–direction) and the compliant (in 45–direction) responses of periodic square

arrangements in both the elastic and elastoplastic ranges, see fig. 19. Multi-fiber unit

cells that approach statistical transverse isotropy such as fig. 18 tend to show notice-

ably stronger strain hardening than periodic hexagonal arrangements of the same fiber

volume fraction, compare also (Moulinec/Suquet,1997). Generally the macroscopic yield

surfaces of uniaxially reinforced MMCs are not of the Hill type, but rather follow bimodal

descriptions (Dvorak/Bahei-el-Din,1987).

The distributions of microstresses and microstrains in fibers and matrix tend to depend

markedly on the fiber arrangement, especially under thermal and transverse mechanical

loading90). In the plastic regime, the microscopic distributions of equivalent stresses

90) Irrespective of the types of load applied, for technologically relevant reinforcement volumefractions Eshelby’s (1957) result of uniform stresses and strains in dilute inhomogeneities holdsneither in the elastic nor the inelastic regimes for nondilute composites.

– 68 –

E∗

A E∗

T ν∗

A ν∗

T α∗

A α∗

T

[GPa] [GPa] [] [] [K−1×10−6] [K−1×10−6]

fibers 180.0 180.0 0.20 0.20 6.0 6.0

matrix 67.2 67.2 0.35 0.35 23.0 23.0

HS/lo 118.8 103.1 0.276 0.277 — —

HS/hi 119.3 107.1 0.279 0.394 — —

3PBm/lo 118.8 103.8 0.278 0.326 — —

3PBm/hi 118.9 104.5 0.279 0.347 — —

MTM 118.8 103.1 0.279 0.342 11.84 16.46

GSCS 118.8 103.9 0.279 0.337 11.84 16.46

2OEm 118.8 103.9 0.279 0.338 11.89 16.40

PH0 118.8 103.7 0.279 0.340 11.84 16.46

CH1 118.7 103.9 0.279 0.338 11.90 16.42

PS0/00 118.8 107.6 0.279 0.314 11.85 16.45

PS0/45 118.8 99.9 0.279 0.363 11.85 16.45

MFUC/00 118.8 104.8 0.278 0.334 11.90 16.31

MFUC/90 118.8 104.6 0.278 0.333 11.90 16.46

TABLE 3: Overall thermoelastic properties of a unidirectional continuously reinforcedALTEX/Al MMC (ξ=0.453) as predicted by the Hashin–Shtrikman (HS) and three-point(3PB) bounds, by the Mori–Tanaka method (MTM), by the generalized self-consistentscheme (GSCS), by Torquato’s second order estimates (2OE), as well as by unit cellanalysis using periodic arrangements shown in fig. 17 (PH0, CH1, PS0) and the multi-fiber cell displayed in fig. 18 (MFUC). For arrangement PS0 responses in the 0 and 45

directions, and for the multi-fiber cell responses in the 0 and 90 directions are given.

and equivalent plastic strains91), of hydrostatic stresses, and stress triaxialities tend

to be markedly inhomogeneous92), see e.g. fig. 20. This type of behavior typically is

most marked for applied transverse shear loads and for elastic–ideally plastic matrix

behavior. As a consequence of the inhomogeneity of the microfields, there tend to be

strong constrained plasticity effects in continuously reinforced MMCs and the onset of

damage in the matrix, of fracture of the fibers, and of interfacial decohesion at the fiber–

matrix interfaces show a strong dependence on the fiber arrangement. Marked local

irregularities in the microgeometry, such as clusters of closely approaching fibers and/or

“matrix islands”, as present e.g. in arrangement RH2 or in the multi-fiber unit cell shown

in fig. 18, are especially critical in this respect.

A further group of materials that are reinforced by continuous fibers and can be stud-

ied by unit cell methods are composites using woven, knitted, and braided reinforce-

91) Note that for regions of concentrated strains in inhomogeneous materials to be shearbands in the strict sense the constituent material behavior must display strain softening, e.g.due to damage.

92) The corresponding distribution functions, phase averages, and higher statistical momentsare also significantly influenced by the fiber arrangement, compare (Bohm/Rammerstorfer,1995).

– 69 –

0.0

0E+0

0 1

.00E

+01

2.0

0E+0

1 3

.00E

+01

4.0

0E+0

1 5

.00E

+01

6.0

0E+0

1

AP

PL

IED

ST

RE

SS

[M

Pa]

0.00E+00 5.00E-04 1.00E-03 1.50E-03 2.00E-03 2.50E-03 3.00E-03 3.50E-03 4.00E-03

STRAIN []

ALTEX FIBERAl99.9 MATRIXPH0/00PH0/90PS0/00PS0/45DN/00

Fiber

Matrix

PS/45

PH

DNPS/00

FIG.19: Transverse uniaxial response of a unidirectional continuously reinforced AL-TEX/Al MMC (ξ=0.453, elastoplastic matrix with linear isotropic hardening) as pre-dicted by unit cell models based on periodic hexagonal fiber arrangements (PH0) andperiodic square fiber arrangements (PS0), compare fig. 17, as well as by the multi-fiberunit cell shown in fig. 18 (MFUC).

EPS.EFF.PLASTIC

2.25E-02

1.75E-02

1.25E-02

7.50E-03

2.50E-03

FIG.20: Microscopic distributions of the accumulated equivalent plastic strains in thematrix of a transversely loaded unidirectional continuously reinforced ALTEX/Al MMC(ξ=0.453, elastoplastic matrix) under uniaxial loading in the horizontal direction as pre-dicted by the multi-fiber unit cell shown in fig. 18.

– 70 –

ments. Such materials are typically modeled by considering bundles of fibers (tows),

the properties of which are obtained by treating them as unidirectional composites of

high fiber volume fraction, that are embedded in pure matrix. Models of this type, see

e.g. (Chang/Kikuchi,1993; Dasgupta et al.,1996, Wang et al.,2007), can be considered

as multiscale approaches, compare chapter 8, in which the weave geometry proper is

considered at the mesoscale. The unit cells used for studying laminated and woven com-

posites are typically three-dimensional, usually have two free surfaces, and may have

macroscopic rotational degrees of freedom to describe warping and twisting. They tend

to be fairly complex geometrically and to be computationally expensive, especially for

nonlinear constituent behavior. One way of mitigating the computational costs consists

of making use of the symmetries of the unit cells of the weaves by introducing special

boundary conditions (Tang/Whitcomb,2003).

Finally, it is worth mentioning that unit cell approaches can be used for investigating

the local stress and strain fields in cross-ply laminates reinforced by monofilaments, see

e.g. (Lerch et al.,1991; Ismar/Schroter,2000).

5.4 Unit Cell Models for Discontinuously Reinforced Composites

The present section concentrates on unit cell models for many practically important types

of discontinuously reinforced composites, viz. short fiber and particle reinforced materials

that do not exhibit major compositional gradients or variations93). Due to the complex-

ity and irregularity of their three-dimensional phase arrangements discontinuously rein-

forced composites tend to be considerably more difficult to model by techniques based

on discrete microgeometries than are materials reinforced by aligned continuous fibers,

two-dimensional descriptions rarely being satisfactory.

Composites Reinforced by Aligned Short Fibers

Because of their relatively simple microgeometries short fiber composites with aligned

reinforcements have been the focus of considerable modeling efforts over the past decades.

The overall symmetry of such materials is transversely isotropic.

Most three-dimensional unit cell models of aligned short fiber reinforced composites re-

ported in the literature describe non-staggered or staggered cylindrical fibers in periodic

square or hexagonal arrangements94), compare fig. 21. Such approaches are relatively

simple conceptually and, when employing periodicity B.C.s, can in principle be used to

model the full thermomechanical response of aligned short fiber reinforced composites.

When limitations are accepted with respect to the load cases that can be studied, con-

siderably smaller unit cells that employ symmetry boundary conditions may be used

93) Platelet reinforced composites can generally be described in analogy to short fiber rein-forced ones.

94) Like in the case of models for continuously reinforced composites, hexagonal arrangementsare overall transversally isotropic in the elastic range, whereas square arrangements give rise totetragonal overall symmetry, i.e. the transverse overall properties are direction dependent.

– 71 –

FIG.21: Sketch of fiber positions in non-staggered hexagonal (left) and staggered square(right) periodic arrangements of aligned short fibers.

for staggered and non-staggered fiber arrangements, see e.g. (Levy/Papazian,1991) and

compare fig. 22. Analogous microgeometries based on periodic hexagonal arrangements

of non-staggered (Jarvstrat,1992) or staggered (Tucker/Liang,1999) fibers as well as

ellipsoidal unit cells aimed at describing composite ellipsoid assembly microstructures

(Jarvstrat,1993) were also proposed95).

FIG.22: Three-dimensional unit cells for short fibers employing symmetry boundaryconditions used for modeling non-staggered (top) and staggered (bottom) square ar-rangements of fibers (Weissenbek,1994).

Staggered and non-staggered unit cells of the above types, however, are rather restrictive

in terms of fiber arrangements and shapes, a difficulty that can be resolved by using much

larger multi-fiber unit cells that contain randomly positioned aligned fibers of different

lengths see e.g. (Ingber/Papathanasiou,1997; Gusev,2001; Liu et al.,2005).

For many materials characterization studies, a more economical alternative to the above

three-dimensional unit cells are axisymmetric models describing the axial behavior of

non-staggered or staggered arrays of aligned cylindrical short fibers in an approximate

way. The basic idea behind these models is to replace unit cells for square or hexagonal

arrangements by circular composite cylinders of equivalent cross sectional area (and

volume fraction) as sketched in fig. 23.

95) For a comparison between unit cell and analytical predictions for the overall elasticproperties for short fiber reinforced composites see e.g. (Tucker/Liang,1999).

– 72 –

FIG.23: Periodic arrays of aligned non-staggered (top) and staggered (bottom) shortfibers and corresponding axisymmetric cells (left: cross sections in transverse plane; right:sections parallel to fibers).

The resulting axisymmetric cells are not unit cells in the strict sense, because they overlap

and are not space filling96). In addition, they do not have the same transverse fiber

spacing as the corresponding three-dimensional arrangements and they are restricted to

handling load cases that lead to axisymmetric deformations (i.e. they cannot be used to

study the response to transverse normal or shear loading). They have the advantage,

however, of significantly reduced computational requirements.

NW

SE

NE

NW

NE

SE

W

S

N

W

N

E

S

E

UNDEFORMED DEFORMED

s

−s L

UU

P

L

r

z

r

z

P

~

~SW SW

FIG.24: Axisymmetric cell for staggered arrangement of short fibers: undeformed anddeformed shape.

Axisymmetric cell models employ symmetry boundary conditions for the top and bottom

faces of the cells, and the B.C.s at the cylinder surface are chosen to maintain the same

96) Analogous axisymmetric cells may also be used for studying aspects of the behavior ofcontinuously reinforced composites.

– 73 –

cross sectional area along the axial direction for an aggregate of cells. In the case of

non-staggered fibers this can be easily done by specifying symmetry-type boundary con-

ditions, whereas for staggered arrangements a pair of cells with “opposite” fiber positions

is considered, for which the total cross sectional area is required to be independent of

the axial coordinate97). Using the nomenclature of fig. 24 and the notation of eqns.(59)

to (62), nonlinear relations are obtained for the radial displacements, u′, together with

linear constraints for the axial displacements, v′,

(rU + u′

U)2 + (rL + u′

L)2 = 2(rP + u′

P)2 and v′

U + v′

L = 2v′

P . (76)

Here U and L are nodes on the E-side of the unit cell that are positioned symmetrically

with respect to the pivot point P. Nonlinear constraints of this type may not be available

or can be cumbersome to use in standard FE codes98). However, for most applications

the B.C.s for the radial displacements in eqn.(76) can be linearized without major loss

in accuracy, so that antisymmetry-type boundary conditions analogous to eqn.(62) are

obtained for the radial surface, i.e.

u′

U + u′

L = 2u′

P and v′

U + v′

L = 2v′

P . (77)

For the past 15 years axisymmetric cell models have been the workhorses of PMA studies

of short fiber reinforced composites, see e.g. (Povirk et al.,1992; Tvergaard,1994), and

they have been extended to cover composites containing aligned inhomogeneities of dif-

ferent sizes, shapes and aspect ratios (essentially by coupling two different cells instead of

two equal ones as shown in fig. 24), compare (Bohm et al.,1993; Weissenbek,1994; Tver-

gaard,2004). Typically, descriptions using staggered arrangements allow a wider range

of microgeometries to be covered and give somewhat more realistic descriptions of actual

composites.

Composites Reinforced by Nonaligned Short Fibers

Whereas a considerable number of unit cell studies of materials reinforced by aligned

discontinuous fibers have been reported, the literature contains few such models dealing

with nonaligned fibers, among them three-dimensional studies of composites reinforced

by alternatingly tilted misaligned fibers (Sørensen et al.,1995) and plane-stress models

describing planar random short fibers (Courage/Schreurs,1992). The situation with re-

spect to mean field models for such materials is not fully satisfactory for the elastic range,

either, compare section 2.7, while little work has been done on the inelastic thermome-

chanical behavior.

97) As originally proposed by Tvergaard (1990), the nonlinear displacement boundary con-ditions in eqn.(76) were combined with antisymmetry traction B.C.s for use with a hybrid FEformulation.

98) For an example of the use of the nonlinear BCs described in eqn.(76) with a commercialFE code see e.g. (Ishikawa et al.,2000), where a cell of truncated cone shape is employed todescribe bcc arrangements of particles.

– 74 –

FIG.25: Unit cell for a short fiber reinforced MMC (ξ=0.15 nominal) containing 15cylindrical short fibers of aspect ratio a = 5 in a quasi-random arrangement suitable forusing periodicity B.C.s (Bohm et al.,2002).

At present the most promising modeling strategy for nonaligned short fiber reinforced

composites consists of using three-dimensional multi-fiber unit cells in which the fibers

are randomly positioned and oriented such that the required ODF is approximately ful-

filled, see e.g. (Bohm et al.,2002; Lusti et al.,2002). Such multi-fiber unit cells can also

account for the distributions of fiber sizes and/or aspect ratios. This modeling approach,

however, tends to pose considerable challenges in generating appropriate fiber arrange-

ments at nondilute volume fractions due to geometrical frustration. The meshing of the

resulting phase arrangements for use with the FEM also is typically difficult, compare e.g.

(Shephard et al.,1995), and analyzing the mechanical response of the resulting cells re-

quires considerable computing power, especially for nonlinear constituent behavior. The

first studies of this type were accordingly restricted to the linear elastic range, where the

BEM has been found to answer well, see e.g. (Banerjee/Henry,1992).

As an example of a multi-fiber unit cell for a composite reinforced by nonaligned short

fibers, fig. 25 shows a cube-shaped cell that contains 15 randomly oriented cylindrical

fibers of aspect ratio a = 5, supports periodicity boundary conditions, was generated

by random sequential insertion, and is discretized by tetrahedral elements. The phase

arrangement is set up in such a way that spheroidal fibers of the same aspect ratio,

volume fractions, center points, and orientations can also be used in order to allow

a comparison of fiber shapes (Bohm et al.,2002). Unit cells corresponding to higher

fiber volume fractions can be generated (as well as meshed and solved in the linear

range) by the commercial micromechanics code PALMYRA (MatSim Ges.m.b.H., Zurich,

Switzerland).

– 75 –

E∗ ν∗

[GPa] []

SiC fibers 450.0 0.17

Al2618-T4 matrix 70.0 0.30

HS/lo 87.6 0.246

HS/hi 106.1 0.305

MTM 89.8 0.285

CSCS 91.2 0.284

KTM 90.3 0.285

MFUC/sph 89.4 0.285

MFUC/cyl 90.0 0.284

TABLE 4: Overall elastic properties of a SiC/Al MMC reinforced by randomly ori-ented fibers (a = 5, ξ=0.15) as predicted by the Hashin–Shtrikman (HS) bounds, byBenveniste’s (1987) Mori–Tanaka method (MTM), by Berryman’s (1980) classical self-consistent scheme (CSCS), by the Kuster–Toksoz (1974) model (KTM) and by multi-fiberunit cells of the type shown in fig. 25 (MFUC), which contain 15 spheroidal or cylindricalfibers (Bohm et al.,2002.

In table 4 predictions are given for the overall elastic response of a composite reinforced

by randomly positioned and oriented fibers obtained by analytical estimates and bounds

as well as results of the above type of unit cell, data of the latter type being provided for

both cylindrical and spheroidal inhomogeneities. Interestingly, considerable differences

were found between the predictions for reinforcement by spheroidal and spheroidal fibers

that have the same fiber volume fraction and aspect ratio, occupy the same positions and

show the same orientations, the spheroidal reinforcements leading to a softer response, es-

pecially in the inelastic regime. Generally very satisfactory results were obtained despite

the low number of fibers in the unit cell (chosen in view of the available computational re-

sources) and excellent agreement with analytical descriptions has been achieved in terms

of the orientation dependence of the stresses in individual fibers, compare fig. 5. It should

be noted, however, that from the point of view of the representativeness of the phase

arrangement and for use in the elastoplastic range unit cells containing a considerably

higher number of fibers are certainly desirable.

Composites Reinforced by Particles

Materials reinforced by statistically uniformly distributed particles show isotropic over-

all behavior, but there exist no simple periodic arrangements of particles that are both

inherently elastically isotropic and space filling99). In actual materials, which often show

rather irregular particle shapes, anisotropy in the microgeometry and in the overall re-

sponse may be introduced by processing effects, e.g. extrusion textures. Accordingly,

unit cell studies of particle reinforced composites that use generic phase arrangements

99) Even though pentagonal dodekahedra and icosahedra have the appropriate symmetryproperties (Christensen,1987), they are not space filling.

– 76 –

are subject to nontrivial tradeoffs between keeping computational requirements at mod-

erate levels (which favors simple particle shapes combined with two-dimensional or simple

three-dimensional microgeometries) and obtaining sufficient level of realism for a given

purpose (which often leads to requirements for unit cells containing a considerable num-

ber of particles of complex shape at random positions in three dimensions). In many

respects periodic microfield analysis of particle reinforced composites is subject to similar

constraints and use analogous approaches as work on short fiber reinforced composites.

2-particle cell2-particle cell

1-particle cell

2-particlecell

1-particlecell

1-part.cell

ReferenceVolume

FIG.26: Simple cubic, face centered cubic, and body centered cubic arrangements ofspheres with some simple unit cells (Weissenbek et al.,1994)

Most three-dimensional unit cell studies of generic microgeometries for particle reinforced

composites have been based on simple cubic (sc), face centered cubic (fcc) and body cen-

tered cubic (bcc) arrangements of spherical, cylindrical and cube-shaped reinforcements,

see e.g. (Hom/McMeeking,1991). By invoking the symmetries of these arrangements

and using symmetry as well as antisymmetry boundary conditions, relatively simple unit

cells can be obtained for materials characterization100), see e.g. (Weissenbek et al.,1994;

Sanders/Gibson,2003) and compare figs. 26 and 27. In addition, work employing hexago-

nal or tetrakaidekahedral particle configurations, see e.g. (Rodin,1993), and Voronoi cells

for cubic arrangements of particles (Li/Wongsto,2004), was reported.

FIG.27: Some unit cells for particle reinforced composites using cubic arrangements: s.c.arrangement of cubes, b.c.c. arrangement of cylinders and f.c.c. arrangement of spheres(Weissenbek et al.,1994)

100) Again, larger unit cells with periodicity rather than symmetry boundary conditionsare required for unrestricted modeling of the full thermomechanical response of cubic phasearrangements.

– 77 –

The overall elastic moduli obtained from cubic arrangements of particles101) do not nec-

essarily fulfill the Hashin–Shtrikman bounds and they typically lie outside the three-point

bounds, compare table 5. Of the unit cells shown in fig. 26, simple cubic models are the

easiest to handle, but they show a more marked anisotropy than bcc and fcc ones.

Recently, three-dimensional studies based on more complex phase arrangements have

appeared in the literature. Gusev (1997) used Finite Element methods in combination

with unit cells containing up to 64 randomly distributed particles to describe the over-

all behavior of elastic particle reinforced composites. Hexahedral unit cells containing

up to 10 particles in a perturbed cubic configuration (Watt et al.,1966) as well as cube

shaped cells incorporating 15 to 20 spherical particles at quasi-random positions (Bohm

et al.,1999, Bohm/Han,2001), compare fig. 28, were proposed for studying elastoplastic

particle reinforced MMCs and related materials. Three-dimensional simulations involv-

ing high numbers of particles for investigating composites with statistically uniform phase

arrangements have been reported for models of elastic composites (Michel et al.,1999)

and for studying brittle matrix composites that develop damage (Zohdi/Wriggers,2001).

In addition composites with statistically nonuniform arrangements of spherical reinforce-

ments were investigated via unit cells containing 7 clusters of 7 particles each (Segurado

et al.,2003).

FIG.28: Unit cell for a particle reinforced MMC (ξ=0.2 nominal) containing 20spherical particles in a quasi-random arrangement suitable for using periodicity B.C.s(Bohm/Han,2001)

101) Cubic symmetry is a special case of orthotropy with equal responses in all principaldirections and requires three independent moduli for describing the elastic behavior. For a dis-cussion of the elastic anisotropy of cubic materials see e.g. (Cazzani/Rovati,2003). The diffusionbehavior of materials with cubic symmetry is isotropic.

– 78 –

In analogy to short fiber reinforced materials, axisymmetric cell models with staggered

or non-staggered particles (compare fig. 21) can be used for materials characterization of

particle reinforced composites. Such approaches have been a mainstay of PMA modeling

of these materials, see e.g. (Bao et al.,1991; LLorca,1996). It is interesting to note

that by appropriate choices of the cells’ aspect ratios axisymmetric cell models can be

generated that in many ways correspond to simple, face centered, and body centered

cubic arrangements (Weissenbek,1994).

E∗ E∗[100] E∗[110] ν∗ α∗

[GPa] [GPa] [GPa] [] [K−1×10−6]

SiC particles 429.0 — — 0.17 4.30

Al99.9 matrix 67.2 — — 0.35 23.0

HS/lo 90.8 — — 0.286 16.8

HS/hi 114.6 — — 0.340 18.6

3PBm/lo 91.4 — — 0.323 18.5

3PBm/hi 93.9 — — 0.328 18.6

MTM 90.8 — — 0.329 18.6

GSCS 91.5 — — 0.327 18.6

2OEm 91.8 — — 0.327 18.6

sc — 96.4 90.5 — 18.7

fcc — 89.0 91.7 — 18.6

bcc — 90.0 90.6 — 18.6

axi/sc 95.4 — — 0.318 18.6/18.6

axi/fcc 88.1 — — 0.342 18.1/19.0

axi/bcc 87.9 — — 0.352 18.4/18.8

3/D MPUC 92.4 — — 0.326

2/D PST MPUC 85.5 — — 0.334

2/D PSE MPUC 98.7 — — 0.500

TABLE 5: Overall thermoelastic properties of a particle reinforced SiC/Al MMC(spherical particles, ξ=0.197) as predicted by the Hashin–Shtrikman (HS) and three-point(3PB) bounds, by the Mori–Tanaka method (MTM), by the generalized self-consistentscheme (GSCS), by Torquato’s second order estimates (2OE), as well as by unit cell anal-ysis using three-dimensional cubic arrangements, axisymmetric cells, three-dimensionalmulti-particle models, and two-dimensional multi-particle models using plane stress (2/DPST) and plane strain (2/D PSE) states.

Due to their relatively low computational requirements, planar unit cell models of parti-

cle reinforced materials can also be found in the literature. Generally, plane stress models

(which actually describe thin “reinforced sheets”) show a more compliant and plane strain

models (which correspond to reinforcement by aligned continuous out-of-plane fibers

rather than particles) show a stiffer overall response than three-dimensional descriptions,

compare table 5. With respect to modeling the elastoplastic overall behavior of parti-

cle reinforced materials, plane stress kinematics tend to give slightly better results than

– 79 –

plane strain descriptions, see e.g. (Weissenbek,1994). It must be stressed, however, that

no two-dimensional model can reliably provide satisfactory results in terms of the pre-

dicted microstress and microstrain distributions102), see table 5 and (Bohm/Han,2001).

Axisymmetric cell models typically provide considerably better results than planar ones.

Accordingly, extreme care on the part of the analyst is required in using planar unit cell

models for particle reinforced composites and quantitative agreement with experiments

cannot be expected. Evidently, three-dimensional (or axisymmetric) models should be

used for discontinuously reinforced materials wherever possible.

In table 5 bounds, MFA results, and PMA predictions for the overall thermoelastic mod-

uli of a particle reinforced SiC/Al MMC (elastic contrast c ≈ 6) are compared, the loading

directions for the cubic arrangements being identified by Miller indices. It is interesting

that — for the loading directions considered here, which do not span the full range of

possible responses of the configurations — none of the results from the cubic arrange-

ments falls within the three-point bounds for identical spherical inhomogeneities103). The

overall anisotropy of the simple cubic arrangement is marked, whereas the body centered

and face centered arrays deviate much less from overall isotropy. The predictions of the

axisymmetric models can be seen to be of comparable quality to those of the correspond-

ing cubic arrays. The results given for the three-dimensional multi-particle models are

ensemble averages over a number of unit cells and loading directions (compare the dis-

cussion in section 4.1), and they show very good agreement with the three-point bounds

and second order estimates, whereas there are marked differences to the plane stress and

plane strain models104). For the composite studied here, predicted overall moduli eval-

uated from the individual multi-particle cells differ by about 2%, and due to the use of

a periodic pseudo-random rather than a statistically homogeneous particle distribution,

the Drugan–Willis estimates for the errors in the overall moduli are exceeded slightly.

Table 6 lists predictions for the overall behavior of a particle reinforced glass-epoxy

composite (elastic contrast c ≈ 23) at much higher volume fraction, ξ=0.389, where the

three-point bounds are no longer particularly sharp. Both the three-point bounds and

Torquato’s second order estimates show some sensitivity to the size distribution of the

spherical reinforcements, monodisperse (m) and polydisperse (p) cases being compared.

102) Plane stress configurations tend to predict much higher levels of equivalent plastic strainsin the matrix and weaker overall hardening than do plane strain and generalized plane strainmodels using the same phase geometry. For a given particle volume fraction, results from three-dimensional arrangements typically lie between those of the above planar models. Evidently,the configurations of regions of concentrated strains that underlie this behavior depend stronglyon the geometrical constraints, compare also (Iung/Grange,1995; Ganser et al.,1998; Bohmet al.,1999, Shen/Lissenden,2002). For continuously reinforced composites under transverseloading the equivalent plastic strains tend to concentrate in bands oriented at 45 to the loadingdirections, whereas the patterns are qualitatively different in particle reinforced materials.

103) For cubic symmetries the extremal values of the Young’s modulus are found for loadingin the [100] and [111] directions (Nye,1957).

104) The comparisons given in tables 3 to 6 are strictly applicable only for the materialparameters used there, but do show typical trends.

– 80 –

E∗ ν∗ G∗ K∗ α∗

[GPa] [] [GPa] [GPa] [K−1×10−6]

glass particles 73.1 0.18 30.1 38.1 4.90

expoxy matrix 3.16 0.35 1.17 3.51 40.0

HS/lo 6.80 0.024 2.59 6.11 12.1

HS/hi 20.9 0.404 8.53 12.7 23.5

3PB(m)/lo 7.05 0.236 2.69 6.16 20.1

3PB(m)/hi 10.0 0.334 3.95 7.22 23.6

3PB(p)/lo 7.17 0.195 2.74 6.26 17.2

3PB(p)/hi 12.1 0.356 4.79 8.56 23.0

MTM 6.80 0.315 2.59 6.11 23.5

GSCS 7.23 0.303 2.77 6.11 23.5

2OE(m) 7.29 0.304 2.80 6.22 23.2

2OE(p) 7.52 0.304 2.88 6.39 22.6

3/D MPUC(m) 7.95 0.291 3.11 6.34 22.8

3/D MPUC(p) 8.06 0.288 3.11 6.36 22.7

TABLE 6: Overall elastic properties of a particle reinforced glass/epoxy composite(spherical particles, ξ=0.389) as predicted by the Hashin–Shtrikman (HS) and three-point(3PB) bounds, by the Mori–Tanaka method (MTM), by the generalized self-consistentscheme (GSCS), by Torquato’s second order estimates (2OE), as well as by multi-particleunit cells (MPUC) using monodisperse and polydisperse particle sizes.

Again there is very good agreement between the analytical results and the predictions

of multi-particle unit cells, which used 80 monodispersely or polydispersely sized quasi-

randomly positioned spherical particles. Both cells were generated and evaluated with

the FE-based program PALMYRA. On closer inspection the moduli listed for the unit

cells can be seen to deviate somewhat from the relationships pertaining to isotropic

materials, because ensemble averaged moduli from slightly anisotropic cells are given.

Like fiber reinforced MMCs, particle reinforced composites typically display highly in-

homogeneous distributions of the microstresses and microstrains in the nonlinear range,

compare fig. 29 (which shows the predicted equivalent accumulated plastic strains inside a

multi-particle model of an MMC). This behavior tends to give rise to microscopic “struc-

tures” that can be considerably larger than individual particles, leading to longer ranged

interactions between inhomogeneities105). The latter, in turn, underlie the need for larger

control volumes for studying composites with nonlinear matrix behavior mentioned in

chapter 4.

If the particles are highly irregular in shape or if their volume fraction markedly ex-

ceeds 0.5 (as is typically the case for cermets such as WC/Co), generic unit cell models

105) Due to the filtering effect of periodic phase arrangements mentioned in section 5.1, thelength of such features can exceed the size of the unit cell only in certain directions (parallel tothe faces and diagonals of the cells), which may induce a spurious direction dependence into thepredicted large-strain behavior of sub-RVE unit cells.

– 81 –

SECTION FRINGE PLOTEPS.EFF.PLASTIC INC.12

5.0000E-02

4.0000E-02

3.0000E-02

2.0000E-02

1.0000E-02

SCALAR MIN: 6.6202E-04SCALAR MAX: 8.1576E-02

FIG.29: Predictions for the equivalent plastic strains in a particle reinforced MMC(ξ=0.2 nominally) subjected to uniaxial tensile loading obtained by a unit cell with 20spherical particles in a quasi-random arrangement (Bohm/Han,2001)

employing relatively regular particle shapes may not result in very satisfactory micro-

geometries. For such materials a useful approach consists of basing PMA models on

modified real structures obtained from metallographic sections, compare e.g. (Fischmeis-

ter/Karlsson,1977). Due to the nature of the underlying experimental data, models of

this type have often taken the form of planar analysis, the limitations of which are dis-

cussed above. This difficulty can be overcome e.g. by voxel-based discretization schemes

using digitized serial sections, see (Terada/Kikuchi,1996) or microtomography data, see

e.g. Kenesei et al. (2006). Alternatively, more or less irregular particles identified by

such procedures may be approximated by equivalent ellipses or ellipsoids, compare (Li

et al.,1999).

5.5 PMA Models for Some Other Inhomogeneous Materials

In this section some additional applications of periodic microfield approaches are pre-

sented, which were selected mainly in view of discussing modeling concepts. The reader

should be aware that it touches only upon a very small fraction of the research activi-

ties in which unit cell methods have been applied to the study of the thermomechanical

properties of materials.

Generic PMA Models for Clustered, Graded, and Interwoven Materials

In some steels, the overall strength is limited by the presence of weak inhomogeneities

which must, accordingly, be avoided. In others, among them high speed tool steels (HSS),

carbidic particles are used to increase strength, hardness and wear resistance. Both of

– 82 –

the above types of steel may be modeled as matrix–inclusion-type composites, the inho-

mogeneities being dilutely dispersed in the former group. The carbide volume fraction in

HSS typically is non-dilute and the phase arrangements range from statistically homoge-

neous to highly clustered or layered mesogeometries. Limitations of available computing

power, however, have mostly restricted studies of clustered or layered configurations,

in which a considerable number of particles have to be modeled, to two-dimensional

descriptions.

A rather straightforward, but nevertheless flexible strategy for setting up generic planar

model geometries for such studies consists of tiling the computational plane with regular

hexagonal cells which are assigned to one of the constituents by statistical or deterministic

rules. If required, the shapes of the cells can be modified or randomly distorted and their

sizes can be changed to some extent to adjust phase volume fractions. Such a Hexagonal

Cell Tiling (HCT) concept can be used e.g. to generate unit cells for analyzing layer

structured HSSs, compare (Plankensteiner et al.,1997) and see fig. 30 (left), as well as

clustered or uniform arrangements of particles in a matrix106).

FIG.30: HCT unit cell models for a layer structured high-speed steel containingequiaxed and elongated carbides (left; Plankensteiner et al.,1997) and for a function-ally graded material (right).

Similar HCT concepts can be used to obtain generic planar geometries for modeling

functionally graded materials (FGMs), in which the phase volume fractions run through

the full physically possible range (Weissenbek et al.,1997; Reiter et al.,1997), so that

106) HCT and related models of composites with matrix–inclusion topology are, however,typically restricted to moderate reinforcement volume fractions and tend to be much moreregular than actual materials.The direct 3D equivalent to HCT consists of filling space with cube-shaped, dodekahedral ortetrakaidekahedral cells, which are assigned phase properties in analogy to HCT. A related,but geometrically more flexible approach based on tetrahedral regions was reported by Galli etal.(2008).

– 83 –

there are regions that show matrix–inclusion microtopologies and others that do not,

compare fig. 30 (right)107). A further application of such models has been the study of

the microstructure–property relationships in duplex steels, for which matrix–inclusion

and interwoven (bi-connected) phase arrangements were compared108), see (Siegmund et

al.,1995; Silberschmidt/Werner,2001). An FFT-based equivalent to the HCT models was

developed by Michel et al. (1999) explicitly for studying interwoven microgeometries.

PMA Models for Polycrystals

Most materials that are of technological interest are polycrystals and, accordingly, are

inhomogeneous at the length scale of the crystallites. In the specific case of metals

and metallic alloys phenomena associated with plastic flow and ductile failure can be

investigated at a number of length scales, compare e.g. (McDowell,2000), some of which

are amenable to study by continuum micromechanical methods, among them unit cell

analysis.

Micromechanical methods, especially analytical self-consistent approaches, have been

applied to investigating the mechanical behavior of polycrystals since the 1950s. The

more recent concept of studying polycrystals by periodic microfield approaches is quite

straightforward in principle — a planar or three-dimensional unit cell is partitioned into

subregions corresponding to grains the behavior of which is described by appropriate

anisotropic material models and to which suitable material as well as orientation pa-

rameters are assigned109). In practice the generation of realistic grain geometries may,

however, require sophisticated algorithms. In addition, the use of crystal plasticity for-

mulations for describing the material behavior of the individual grains tends to pose

considerable demands on computational resources, especially for three-dimensional mod-

els, see e.g. (Quilici/Cailletaud,1999).

Resolved microstructure models of polycrystals have, for example, successfully described

the evolution of texture in metals during forming processes. For a general discussion of

the issues involved in micromechanical models in crystal plasticity and related fields see

e.g the overview by Dawson (2000).

PMA Models for Two-Phase Single Crystal Superalloys

Nickel-base single crystal superalloys, which consist of a γ matrix phase containing

aligned cuboidal γ′ precipitates, are of major technical importance due to their creep

107) In contrast to the other inhomogeneous materials discussed in this report, FGMs andlayer-structured HSSs are not statistically homogeneous.

108) In HSSs and FGMs particulate inhomogeneities are present at least in part of the volumefraction range, so that planar models are less than ideal for them. Duplex steels, in contrast,typically show elongated aligned grains, for which generalized plane strain kinematics are sat-isfactory. Analogous considerations hold for polycrystalline aggregates consisting of alignedcolumnar grains, see e.g. (Chimani/Morwald,1999).

109) Symmetry boundaries passing through grains must be avoided in such models becausethey give rise to “twin-like” pairs of grains that are unphysical in most situations.

– 84 –

resistance at high temperatures. The most important field of application of superalloys

is in the hot sections of gas turbines. Because these materials show relatively regu-

lar microstructures that change under load, a process known as rafting, there has been

considerable interest in micromechanical modeling for elucidating their thermomechan-

ical behavior and for better understanding their microstructural evolution. The elastic

behavior of two-phase superalloys can be handled relatively easily by hexahedral unit

cells with appropriately oriented anisotropic phases. In the inelastic range, however, the

highly constrained plastic flow in the γ channels is difficult to describe even by crystal

plasticity models, see e.g. (Nouailhas/Cailletaud,1996).

PMA Models for Porous and Cellular Materials, Cancellous Bone, Wood

Elastoplastic porous materials have been the subject of a considerable number of PMA

studies due to their relevance to the ductile damage and failure behavior of metals110).

Generally, modeling concepts are closely related to those employed for particle rein-

forced composites, compare section 2.4, the main difference being that the shapes of the

voids may evolve significantly through the loading history111). Accordingly, axisymmetric

cells, see e.g. (Koplik/Needleman,1988; Pardoen/Hutchinson,2000; Garajeu et al.,2000)

and three-dimensional unit cells based on cubic arrangements of voids, compare e.g.

(McMeeking/Hom,1990; Segurado et al.,2002), have been used in the majority of pub-

lished unit cell studies of porous materials. Recently, studies of void growth in ductile

materials based on multi-void cells and using geometry data from serial sectioning were

reported (Shan/Gokhale,2001).

In cellular materials, such as foams, wood and cancellous bone, the volume fraction

of the solid phase is low (often amounting to a few percent or less). The voids may

be topologically connected (open cell foams), unconnected (closed cell foams, syntactic

foams), or both connected and unconnected voids may be present (as in hollow sphere

foams).

The macroscopic tensile and compressive stress–strain responses of cellular materials

usually are very different, with the latter typically showing three regimes. At small

strains the materials display a finite stiffness, the linear elastic range often being very

small. At higher stains gross shape changes of the cells take place, with bending, elastic

110) Many models describing ductile damage in metals, see e.g. (Rice/Tracey,1969; Gur-son,1977; Tvergaard/Needleman,1984; Gologanu et al.,1997; Kailasam et al.,2000; Benz-erga/Besson,2001), are based on micromechanical studies of porous ductile materials and onthe growth of (pre-existing) voids in elastoplastic matrices.

111) Additional complexity is introduced by void size effects (Tvergaard,1996) and void coa-lescence (Faleskog/Shih,1997). The evolution of the shapes of initially spheroidal voids undernon-hydrostatic loads has also been the subject of intensive studies by mean field type meth-ods, see e.g. (Kailasam/Ponte Castaneda,1998; Kailasam et al.,2000). Models of the latter typeare typically based on the assumption that initially spherical pores will stay ellipsoids through-out the deformation history, which studies using axisymmetric cells (Garajeu et al.,2000) haveshown to be an excellent approximation for axisymmetric tensile load cases. For compressiveloading, however, initially spherical pores may evolve into markedly different shapes (Seguradoet al.,2002).

– 85 –

buckling, plastic buckling, and brittle failure of cell walls or struts playing major roles on

the microscale. On the macroscale this regime is characterized by a stress plateau, which

gives rise to the typically excellent energy absorption properties of cellular materials. At

even higher strains the effective stiffness rises sharply as the cellular structure is crushed

so that cell walls or struts are in contact. No plateau region is present under tensile or

shear loading.

Periodic microfield methods are well suited to studying the thermomechanical behavior

of cellular materials. The widely used results of Gibson and Ashby (1988), in which

thermomechanical moduli and other overall physical properties of cellular materials are

expressed as power laws in terms of the relative density, e.g.,

E∗

E(m)∝

( ρ∗

ρ(m)

)n

, (78)

were derived by analytically studying specific arrangements of beams (for open cell foams)

and plates (for closed cell foams)112). When the finite strain behavior of cellular materials

is to be studied by periodic microfield methods, large deformations of cells and contact

between cell walls/struts must be typically provided for113). In addition, models must

be sufficiently large so that appropriate nontrivial deformation patterns can develop114).

The geometrically most simple cellular materials are regular honeycombs, which can be

modeled by planar hexagonal cell models. Despite their geometrical simplicity regular

honeycombs can show complex deformation patterns, the resolution of which requires

the provision of sufficiently large unit cells, compare e.g. (Daxner et al.,2002). Further-

more, they tend to be subject to marked localization effects under compressive “in-plane”

loading, see e.g. (Papka/Kyriakides,1994). Somewhat less ordered two-dimensional ar-

rangements have been used for studying the crushing behavior of soft woods, see e.g.

(Holmberg et al.,1999), and highly irregular planar arrangements, compare fig. 31, can

be used to study aspects of the geometry dependence of the mechanical response of

cellular materials such as metallic foams.

For three-dimensional studies of closed cell foams various generic microstructures

based on cubic cells, see e.g. (Hollister et al.,1991), truncated cubes plus small cubes

(Santosa/Wierzbicki,1998), rhombic dodekahedra, regular tetrakaidekahedra (Grenest-

edt,1998; Simone/Gibson,1998; Ableidinger,2000), compare fig. 32, as well as Kelvin,

112) The majority of these geometries can be extended to give proper unit cells that describeperiodic phase arrangements. The behavior of the Gibson–Ashby models, however, is dominatedby local bending. As a consequence, n=2 in eqn.(78) is obtained for open cell foams and thepredicted overall moduli tend to be lower than those of many actual microgeometries.

113) Note that this may imply providing for periodic contact.114) For perfectly regular structures such as hexagonal honeycombs the minimum size of a

unit cell for capturing bifurcation effects can be obtained from extended homogenization theory(Saiki et al.,2002), and Bloch wave theory (Gong et al.,2005) may be used to study long wavebuckling modes.Symmetry B.C.s must not be specified on cell walls that can be expected to bend or buckle.

– 86 –

X

Y

Z X

Y

Z

FIG.31: Planar periodic unit cell for studying irregular cellular materials (Daxner etal.,2002).

Williams and Wearie–Phelan115) (Kraynik/Reinelt,1996; Bitsche,2005) geometries have

been used. In models of this type the cell walls are typically described via thin shells.

Analogous regular microgeometries have formed the basis for analytical and numerical

models for open cell foams, see e.g. (Zhu et al.,1997; Shulmeister et al.,1998; Vajjhala

et al.,2000), the struts connecting the nodes of the cellular structures being typically

modeled as beams116). In addition, tetrahedral arrangements (Sihn/Roy,2004) and cubic

arrangements with additional “reinforcements” as well as perturbed strut configurations

(Luxner et al.,2005) have been used for studying open cell microstructures.

A recent development in the modeling of metallic and polymer foams have been unit

cell models that employ voxel-type discretizations for open-cell and closed-cell foams117),

which are based on three-dimensional microstructures of the actual materials obtained by

tomographic methods, see e.g. (Maire et al.,2000; Roberts/Garboczi,2001). Alternatively,

image processing methods my be used to generate a surface model of the solid phase in

the volume element to allow meshing with “standard FE techniques, compare (Youssef

et al.,2005). Because the microstructures extracted from the experiments are, in general,

115) In Kelvin, Williams and Wearie–Phelan foams some of the cell walls are spatially distortedin order to minimize the total surface, whereas in polyhedral foams all cell walls are planar.These small distortions lead to noticeable differences in the overall elastic response at low solidvolume fractions.

116) For unit cell models employing structural finite elements such as shells and beams peri-odicity, symmetry and antisymmetry boundary conditions, eqns.(59) to (63), must be extendedto also cover the rotational degrees of freedom of the boundary nodes, see e.g. (Bitsche,2005).

117) Roberts and Garbozci (2001) estimated the “systematic discretization error” to be ofthe order of 10% in terms of the overall moduli for voxel-based models of elastic open-cell andclosed-cell foams.

– 87 –

FIG.32: Microstructure of a closed cell foam modeled by regular tetrakaidekahedra(truncated octahedra).

not periodic, windowing approaches using homogeneous strain boundary conditions are

often preferred to unit cells in such models.

A further group of cellular materials amenable to PMA modeling are syntactic foams

(i.e. hollow spheres embedded in a solid matrix) and hollow sphere foams (in which the

spaces between the spheres are “empty”), for which axisymmetric or three-dimensional

cell models describing cubic arrangements of spheres can be used, see e.g. (Rammerstor-

fer/Bohm,2000; Sanders/Gibson,2003).

The effects of details of the microgeometries of cellular materials (e.g. thickness distribu-

tions and geometrical imperfections or flaws of cell walls or struts, the Plateau borders

formed at the intersections of cell walls), which can considerably influence the overall be-

havior (a typical example being the very small elastic ranges of metallic foams), have been

an active field of research, see e.g. (Grenestedt,1998; Daxner,2002). Even though unit cell

analysis of cellular materials with realistic microgeometries tends to be rather complex

and numerically demanding due to these materials’ tendency to deform by local mecha-

nisms and instabilities118), analytical solutions have been reported for the linear elastic

behavior of some simply periodic phase arrangements, see e.g. (Warren/Kraynik,1991).

A further type of cellular material, cancellous (or spongy) bone, has attracted consid-

erable research interest for the past twenty years119). Cancellous bone shows a wide

range of microstructures, which can be idealized as beam or beam–plate configurations

(Gibson,1985), and the solid phase again is an inhomogeneous material at a lower length

118) Interestingly, instability phenomena in the cell walls of closed cell foams may even occurunder uniaxial tensile macroscopic loading, see (Ableidinger,2000).

119) Like many materials of biological origin, bone is an inhomogeneous material at a numberof length scales. The solid phase of spongy bone is inhomogeneous and can also be studied withmicromechanical methods.

– 88 –

scale. In studying the mechanical behavior of cancellous bone, large three-dimensional

unit cell models based on tomographic scans of actual samples and using voxel-based

discretization schemes have become fairly widely used, see e.g. (Hollister et al.,1994; van

Rietbergen et al.,1999).

5.6 PMA Models for Diffusion-Type Problems

Periodic microfield methods analogous to those discussed in sections 5.1 to 5.5 can be

used to study linear diffusion-type problems of the types mentioned in section 2.9, Laplace

solvers (typically for use with heat conduction problems) being available with many FE

packages. Specialized solvers are, however, required in some cases, e.g. when studying

the frequency dependent dielectric properties of composites via complex potentials, see

e.g. (Krakovsky/Myroshynchenko,2002).

There are, however, intrinsic difficulties in employing periodic microfields for transport

problems where nonlinear conductivity/resistivity behavior of the constituents is present.

Whereas in solid mechanics, material nonlinearities are typically formulated terms of

the “generalized intensity” or “generalized flux” fields, i.e. the microscopic strains and

stresses, nonlinear resistivities or diffusivities in transport problems typically depend on

the direct variable, e.g. the temperature in heat conduction, compare table 1. As is ev-

ident from fig. 14, in PMAs the generalized intensities and fluxes are periodic, whereas

the direct variables consist of fluctuating plus linear contributions, so that they accu-

mulate from cell to cell. As a consequence, in solid mechanics problems the position

dependence of material properties is periodic also in the presence of nonlinearities, but

this is not the case in transport problems. Accordingly, even though seemingly workable

solutions, e.g. in terms of temperature fields, can be obtained from unit cells in which

temperature dependent material properties are specified, the results do not correspond

to proper periodic solutions. Periodic homogenization approaches should not be used in

such cases, embedded cell models being more suitable for studying such problems.

– 89 –

6. EMBEDDING APPROACHES AND RELATED METHODS

Like periodic microfield methods, Embedded Cell Approaches (ECAs) aim at predicting

the microfields in inhomogeneous materials at high spatial resolution. For this purpose

they use models consisting of a geometrically fully resolved core (“local heterogeneous

region”), which can range from relatively simple configurations to complex phase ar-

rangements, that is embedded in an outer region for which a smeared out behavior is

used and in which the phases are not (or at least not fully) resolved, compare fig. 33.

This embedding region (“effective region”, “effective layer”) serves mainly for transmit-

ting the applied loads into the core. The material description used for the outer region

must be chosen such that it is consistent with the homogenized behavior and symmetry

of the material described by the core, so that errors in the macrosocpic accommodation

of stresses and strains are kept to a minimum.

REGION

CORE(discrete phase arrangement)

EMBEDDING

INTERFACE betweencore and embedding region

FIG.33: Schematic depiction of the arrangement of core, embedding region, and inter-face in an embedded cell approach.

The resulting modeling strategy avoids some of the drawbacks of PMAs, especially the

requirement that the geometry and all microfields must be strictly periodic120).

An intrinsic feature of embedding models are “boundary layers” that occur at the “inter-

faces” between the core and the embedding region121). and perturb the local stress and

120) Embedding models can be used without intrinsic restrictions at or in the vicinity of freesurfaces and interfaces, they can handle gradients in composition and loads, they can be usedto study the interaction of macrocracks with microstructures, and they support temperaturedependent conductivities in thermal analysis. The cores of embedding models do not necessarilyhave to be RVEs.

121) The interfaces between core and embedding region obviously are a consequence of themodeling approach and do not have any physical significance. The resulting boundary layerstypically have a thickness of, say, a particle diameter for elastic materials, but they may belonger ranged for nonlinear material behavior.

– 90 –

strain fields. Provision must be made to keep regions of specific interest, such as crack

tips and process zones, away from the core’s boundaries122).

Although some analytical methods such as classical and generalized self-consistent

schemes, see section 2.3, may be viewed as embedding schemes, most ECAs with more

complex core microgeometries have used numerical engineering methods in order to re-

solve discrete microstructures in the core region.

Three basic types of embedding approaches can be found in the literature. One of them

uses discrete phase arrangements in both the core region and in the surrounding material,

the latter, however, being discretized much more coarsely, see e.g. (Sautter et al.,1993).

Mesh superposition techniques, which use a coarse mesh over the macroscopic model of

some structure or sample together with a geometrically independent, much finer mesh in

regions of interest (Takano et al.,2001) are conceptually similar to the above modeling

strategy. If appropriately graded meshes are used models of the above types are not

subject to marked boundary layers between core and embedding regions.

In the second group of embedding methods the behavior of the outer regions is described

via appropriate smeared-out material models that suitably approximate the macroscopic

behavior of the inhomogeneous material. These typically take the form of semiempirical

or micromechanically based constitutive laws that are prescribed a priori for the embed-

ding zone. Such constitutive laws and the material parameters used with them must be

be chosen to correspond in a sufficiently close way to the homogenized behavior of the

core123). This way, conceptually simple models are obtained that are very well suited for

studying local phenomena such as the stress and strain distributions in the vicinity of

crack tips or at macroscopic interfaces in composites (Chimani et al.,1997), the growth of

cracks in inhomogeneous materials (Wulf et al.,1996; Ableidinger,2000), or damage due

to the processing of composites (Monaghan/Brazil,1998).

The third type of embedding scheme uses the homogenized thermomechanical response

of the core to determine the effective behavior of the surrounding medium, giving rise

to models of the self-consistent type, which are mainly employed for materials char-

acterization. The use of such approaches is predicated on the availability of suitable

parameterizable constitutive laws for the embedding material that show the appropriate

122) When crack tips are to be modeled they obviously must not be allowed to progress intothe boundary layers. Similarly, the boundary layers and embedding regions may interfere withextended regions of concentrated strains, with shear bands, and with the progress of damage.In transient analysis the boundary layers will in general lead to reflection and/or refraction ofwaves. Special transition layers can, however, be introduced to mitigate boundary layer effectsin the latter case.

123) Care is required in modeling the behavior of elastoplastic inhomogeneous materials withcontinuous reinforcements via two-dimensional models, because the homogenized response ofthe material (and thus, of the core) is anisotropic and, consequently, cannot be well describedby von Mises plasticity.It is worth noting that by restricting damage effects to the core of a model and prescribingthe behavior of the damage-free composite to the embedding material, the volume fractions ofdamaged regions can in principle be controlled in ECA models, compare e.g. (Bao,1992).

– 91 –

material symmetry and that can follow the core’s instantaneous homogenized behavior

with sufficient accuracy for all load cases and for any loading history. This require-

ment typically can be fulfilled easily in the linear range (see e.g. Chen et al.,1994), but

leads to considerable difficulties when at least one of the constituents shows elastoplas-

tic or viscoplastic material behavior124). Accordingly, approximations have to be used

(the consequences of which may be difficult to assess in view of the nonlinearity and

path dependence of elastoplastic material behavior)125), and models of this type are best

termed “quasi-self-consistent schemes”. Such approaches are discussed e.g. in (Bornert

et al.,1994; Dong/Schmauder,1996). When a multi-particle unit cell is used as the core

in a quasi-self-consistent embedding scheme, the relaxation of the periodicity constraints

tends to make the overall responses of the embedded configuration softer than that of

the same periodic arrangement in a PMA (Bruzzi et al.,2001)126).

1

2

3 1

2

3

FIG.34: Embedded cell model of a CT specimen for studying crack progress in an opencell metallic foam (Ableidinger,2000).

124) Typically, the effective yielding behavior of the (inhomogeneous) core shows some de-pendence on the first stress invariant, and, for low plastic strains, the homogenized responseof the core tends to be strongly influenced by the fraction(s) of the elastoplastic constituent(s)that have actually yielded. In addition, in many cases anisotropy of the yielding and hardeningresponses is introduced by the phase arrangement of the core (e.g. aligned fibers). At present,there appears to be no constitutive law that, on the one hand, can fully account for these phe-nomena (compare e.g. Dvorak,1991) and, on the other hand, has the capability of being adaptedto the instantaneous response of the core by adjusting appropriate free parameters.

125) In case a uniaxial stress–strain diagram is adapted to the core’s behavior and multiaxialstresses are handled via standard metal plasticity relations (e.g. using von Mises or Hill-typeplasticity), such approximations are implicitly invoked.

126) In the study mentioned above, the difference between PMA and ECA results is relativelysmall. Essentially identical responses can be expected from the two types of model if the phasearrangement is a proper RVE and the embedding material can fully describe the homogenizedbehavior of the core.

– 92 –

For materials characterization embedded cells can be loaded by homogeneous stresses

or strains applied to the outer boundaries. If problems involving macrocracks (or re-

lated questions) are to be studied, however, it is often preferable to impose displacement

boundary conditions corresponding to the far field behavior of suitable analytical solu-

tions (e.g. for the displacement field around a crack tip). Embedded cell models also

allow complete samples to be considered in “simulated experiments”, see e.g. fig. 34,

which shows an embedding model consisting of one half of a CT specimen made of an

open cell foam (Ableidinger,2000). In it, the embedding region is described as a homog-

enized elastic continuum, in the core region a highly idealized beam model is used for

the open cell foam, and the crack is assumed to run along the symmetry plane.

The core and embedding regions may be planar, axisymmetric or fully three-dimensional,

and symmetries present in the geometries can, as usual, be employed to reduce the size

of the model, compare (Peters,1997). Effective and phase averaged stresses and strains

from embedded cell models are best evaluated from eqn.(73) or its equivalents, and in

the case of geometrically complex phase arrangements it is good practice to use only the

central regions of the core for this purpose in order to avoid boundary layers.

A modeling approach that is closely related to embedding models uses coupling condi-

tions between a macromodel that typically employs smeared-out material data with a

geometrically fully resolved micromodel. Such submodeling techniques may, on the one

hand, be implemented “off-line” such that the macromodel is solved for first and the

micromodel is studied in a separate analysis, the macromodel being only used to pro-

vide suitable boundary conditions, see e.g. (Heness et al.,1999; Varadi et al.,1999). On

the other hand, geometrically fully resolved regions may be activated automatically in

regions where the conditions for periodic homogenization fail to be met (e.g. near crack

tips), see e.g. Ghosh et al. (2001), where loads are introduced into the core regions via

transition elements.

Like unit cell models, embedding and related approaches approaches are also well suited

for handling diffusion-type problems.

– 93 –

7. WINDOWING APPROACHES

A further group of methods for studying microstructures that were obtained experimen-

tally or were computer generated to follow some given phase distribution statistics is

subsumed here as “windowing approaches”. These methods are based on extracting sim-

ply shaped mesoscopic regions (“test windows”) at random positions (and with random

orientations) from an inhomogeneous material, compare fig. 35, and subjecting them to

nonperiodic boundary conditions that are guaranteed to fulfill the macrohomogeneity

condition, eqn.(23).

FIG.35: Schematic representation of a windowing approach — four square windowsof different size obtained from a material consisting of a matrix reinforced by alignedcontinuous fibers are shown.

By using the Gauss theorem and assuming small strain conditions as well as perfect

interfaces between the constituents the macrohomogeneity condition can be formulated

in terms of a surface integral as

∫

Ωs

[

t(x)− 〈σ〉nΓ(x)] [

u(x)− 〈ε〉x]

dΓ = 0 (79)

(Hazanov,1998). This condition trivially holds when homogeneous stress (“uniform

static”, “uniform traction”, “uniform natural”) or homogeneous strain (“uniform kine-

matic”, “uniform displacement”, “uniform essential”) boundary conditions are used, both

of which set one of the vectors in the integrand of eqn.(79) identically to zero. In ad-

dition, eqn.(79) can be fulfilled by mixed uniform boundary conditions that make the

two vectors orthogonal127) and by periodic B.C.s, in which the contributions of coupled

127) Mixed uniform boundary conditions can be strictly realized only in materials that haveorthotropic or higher symmetry (Hazanov,1998) and must be applied in the appropriate materialcoordinate system.

– 94 –

pairs of faces cancel out. In general, homogeneous strain and mixed uniform boundary

conditions are more straightforward to apply than periodic B.C.s because there are no

dependences between phase geometries or discretizations on “opposite” faces, and the

simpler constraint equations can make for reduced computational requirements. Pre-

scribing uniform homogeneous tangential tractions, however, is not supported by most

preprocessors.

Windowing approaches using homogeneous strain and traction boundary conditions allow

to obtain estimates for and rigorous bounds on the overall responses of inhomogeneous

materials on the basis of randomly extracted volume elements that may be too small to

be proper RVEs128). In fact, windowing methods typically aim at describing the apparent

responses of inhomogeneous materials when it is known that the available volume ele-

ments are too small to be proper RVEs. When “non-pathological” SRVEs are subjected

to homogeneous stress and homogeneous strain boundary conditions, respectively, non-

trivial lower and upper estimates are obtained, compare e.g. (Nemat-Nasser/Hori,1993),

and ensemble averages of such estimates provide lower and upper bounds on the overall

elastic (and diffusion-type129)) behavior of the material.

It was shown by Hazanov and Huet (1994) that for elastic material behavior estimates

obtained with mixed uniform boundary conditions are bounded by the lower and up-

per estimates generated with macrouniform boundary conditions for the same volume

element, and Ostoja-Starzewski (2006) gave order relations for apparent elastic moduli

obtained with the above types of B.C.s. Suquet (1987) proved that elastic tensors ob-

tained with periodicity boundary conditions always lie between results generated with

uniform traction and uniform strain boundary conditions. Pahr and Zysset (2008) pro-

posed a set of mixed uniform boundary conditions that leads to equal predictions as do

periodicity B.C.s for periodic volume elements with phase arrangements of orthotropic or

higher symmetry; these B.C.s are referred to as “periodicity compatible mixed uniform

boundary conditions”.

By definition, for proper RVEs the lower and upper estimates and bounds as well as esti-

mates obtained with mixed uniform and periodic boundary conditions must coincide130),

(Hill,1963), compare also section 1.2. The closeness of the upper and lower bounds ob-

128) Such “sub-RVE” volume elements are also referred to as Test Volume Elements (TVEs)or Statistical Reference Volume Elements (SRVEs). Bounds obtained from such volume elementsare known as mesoscale bounds.

129) For diffusion-type problems homogeneous normal flux and homogeneous gradient bound-ary conditions must be applied, see e.g. (Jiang et al.,2002; Kanit et al.,2006). These give riseto lower and upper estimates and bounds, respectively, of the effective conduction or diffusiontensors.

130) Note that, when a given volume element is to be checked as to how closely it approachesbeing an RVE, either the complete apparent/effective tensors must be evaluated or differentloading conditions (e.g. load cases with different macroscopic stress triaxialities or different loadorientations) must be sampled.Equal lower and upper mesoscale bounds are, however, difficult to obtain in practice, the con-vergence of the two bounds being especially slow for softer inclusions in a stiffer matrix.

– 95 –

tained for the macroscopic properties can be used to check the suitability of volume

elements of a given size for describing the effective (as opposed to apparent) mechanical

response of an inhomogeneous material, see e.g. (Ostoja-Starzewski,1998). Hierarchies

of bounds for a given microstructure can be generated by using series of windows of

increasing size, which allows to assess the dependence of the predicted overall moduli

on the size of the windows. This concept of hierarchical bounds obtained from samples

that are smaller than RVEs was verified by planar (Amieur,1994) and three-dimensional

(Guidoum,1994) modeling schemes and by experimental studies (Huet,1999). Volume

elements of increasing size subjected to periodicity or symmetry boundary conditions

have been reported to give rise to hierarchies of estimates, see e.g. (Sautter et al.,1993;

Terada et al.,2000).

Recently windowing approaches giving rise to hierarchical bounds have been extended

into the nonlinear range (Jiang et al.,2001, Jiang et al.,2002), where the bounding prop-

erties of the two types of homogeneous boundary conditions can be proven within the

context of nonlinear elasticity (and thus also hold for the deformation theory of plas-

ticity). No rigorous proofs, however, appear to be available for incremental plastic-

ity. A discussion of windowing bounds in a finite strain elasticity setting is given in

Khisaeva/Ostoja-Starzewski (2006). Windowing approaches employing mixed uniform

boundary conditions can be used for generating estimates of the macroscopic behavior of

elastoplastic inhomogeneous materials. The restrictred load paths that can be followed

by such an approach, however, allow little more than materials characterization to be

carried out.

For relatively small windows (having a ratio between inhomogeneity diameter and window

size of, say, 5) different boundary conditions, especially uniform kinematic and uniform

static B.C.s, can give rise to marked differences in the distributions of the microfields,

especially the plastic strains. When larger windows are used, however, increasingly sim-

ilar statistical distributions of the microfields are obtained (Shen/Brinson,2006). Again,

improved predictions of the overall properties can be obtained by ensemble averaging

over a number predictions obtained from small volume elements with the same B.C.s.

A conceptually different class of windowing models were reported by Soppa et al.(2003),

who subjected rectangular test windows to experimentally determined boundary dis-

placements. The main interest in such approaches lies in correlating microstrain fields

obtained by experiments and by modelling.

Windowing concepts have also been used to determine statistical material property fields,

compare e.g. (Ostoja-Starzewski,1993), and to studying the statistics of the apparent

material properties of inhomogeneous materials obtained from moving-window mesoscale

TVEs, see e.g. (Graham/Baxter,2001).

– 96 –

8. MULTI-SCALE MODELS

The micromechanical methods discussed in chapters 2 to 7 are each designed to handle

a single scale transition between a lower and a higher length scale (“microscale” and

“macroscale”), overall responses being obtained by homogenization and local fields by

localization. Many inhomogeneous materials, however, show more than two clearly dis-

tinct characteristic length scales, typical examples being laminated composites, woven

composites reinforced by tows containing many thin fibers, materials in which there are

well defined clusters of particles, as well as many biomaterials. In such cases an obvious

modeling strategy is a hierarchical (or multi-scale) approach that uses a sequence of scale

transitions, i.e. the material response at any given length scale is described on the basis of

the homogenized behavior of the next lower one131). Figure 36 schematically shows such

a hierarchical model for a particle reinforced composite with a clustered mesostructure.

transitionscale

#1transitionscale

#2

MACROSCALE MESOSCALE MICROSCALE(sample) (particle clusters) (particles in matrix)

FIG.36: Schematic representation of a multi-scale approach for studying a materialconsisting of clustered particles in a matrix. Two scale transitions, macro←→meso andmeso←→micro, are used.

From a technical point of view, homogenization and/or localization of the thermome-

chanical responses of an inhomogeneous material to link a given pair of length scales

is equivalent to finding an energetically equivalent homogeneous material by microme-

chanical approaches as discussed in chapters 2 and 5 to 7. A multi-scale model can be

viewed as a sequence of such scale transitions and, accordingly, all of the above methods,

i.e. mean field, unit cell, embedding, and some windowing approaches, may be used as

131) Describing the material behavior at lower length scales by a homogenized model impliesthat characteristic lengths differ by, say, an order of magnitude or more, so that valid volumescan be defined for homogenization. It is not technically correct to employ hierarchical approacheswithin “bands” of more or less continuous distributions of length scales, as can be found e.g. insome metallic foams with highly disperse cell sizes.

– 97 –

“building blocks” at any level within hierarchical schemes132). Such multi-scale model-

ing strategies have the additional advantage of allowing the behavior of the constituents

at all lower length scales to be assessed via the corresponding localization relations133).

Because local stresses and strains at the lower length scales can differ markedly from

the applied macroscopic conditions in both magnitude and orientation, it is advisable

that the homogenized descriptions of the thermomechanical responses at all length scales

(with the possible exception of the top one) take the form of proper constitutive models

that are capable of handling any loading conditions and any loading history. In most

cases, this requirement can be readily fulfilled by micromechanically based constitutive

models that employ mean field methods or periodic microfield approaches.

Among the continuum mechanical multi-scale descriptions of the thermomechanical be-

havior of inhomogeneous materials reported in the literature, some combine mean field

methods at the higher length scale with mean field (Hu et al.,1998; Tszeng,1998) or

periodic microfield approaches (Gonzalez/LLorca,2000) at the lower length scale. The

most common strategy for multi-scale modeling, however, uses Finite Element based unit

cell or embedding methods at the topmost length scale, which implies that the homoge-

nized material models describing the lower level(s) of the hierarchy must take the form

of micromechanically based constitutive laws that can be evaluated at each integration

point.

This requirement typically does not give rise to problematic computational workloads in

the elastic range, where the superposition principle can be used to obtain the full ho-

mogenized elasticity and thermal expansion tensors from a limited number of load cases

applied to an appropriate unit cell134). For simulating the thermomechanical response of

nonlinear inhomogeneous materials, however, essentially a full micromechanical submodel

has to be maintained and solved at each integration point in order to account for the his-

tory dependence of the local responses135). Even though within such a framework the use

of sophisticated unit cell based models at the lower length scales usually is a very expen-

132) Micromechanical methods that by construction describe inhomogeneous materials witha wide range of length scales, e.g. differential schemes, are, however, of limited suitability formulti-scale analysis.

133) It should be noted, however, that generating predictions for all relevant fields at two ormore length scales can give rise to very large amounts of data, especially if models based onnumerical discretizing methods are used at one or more of the length scales.

134) In the elastic range domain decomposition techniques can be used to formulate the prob-lems at the lower length scales in such a way that they are well suited for parallel process-ing, allowing the development of computationally highly efficient multi-scale procedures, see(Oden/Zohdi,1997).

135) Some approaches have been reported that aim at partially “decoupling” models at thehigher and lower length scales by parameterizing results from unit cell analyses and using thestored data together with some interpolation scheme as an “approximate constitutive model”,see e.g. (Terada/Kikuchi,1996; Ganser,1998; Schrefler et al.,1999; Ghosh et al.,2001). The maindifficulty in such modeling strategies lies in handling load history and load path dependencesto account for elastoplastic constituents or microscopic damage to constituents. Appropriaterepresentation of general multiaxial stress and/or strain states also tends to be a challenge.

– 98 –

Eps_eff,p_(m)

3.5000E-03

3.0000E-03

2.5000E-03

2.0000E-03

1.5000E-03

SCALAR MIN: -6.8514E-04SCALAR MAX: 1.3667E-02

FIG.37: Phase averaged microscopic equivalent plastic strains in the matrix within theparticle-poor regions of a cluster-structured high speed steel under mechanical loading aspredicted by a mesoscopic unit cell model combined with an incremental Mori–Tanakamodel at the microscale (Plankensteiner,2000).

sive proposition in terms of computational requirements, work of this type was reported

e.g. by (Wu et al.,1989), and asymptotic homogenization has been used successfully in

this context for two-dimensional descriptions (Ghosh et al.,1996). Lower (but by no

means negligible) computational costs can be achieved by using mean field models in lieu

of constitutive models at the integration point level. Applications for elastoplastic com-

posites136) have included incremental Mori–Tanaka methods, see e.g. (Pettermann,1997),

mean field based versions of the Transformation Field Analysis, see e.g. (Fish/Shek,1998;

Plankensteiner,2000), and the nonuniform TFA (Michel/Suquet,2004). It is also possible

to handle the “lowest” scale transition by a combination of a mean field or TFA-type

homogenization procedure and unit-cell based localization algorithms, the latter being

used only in regions of special interest (Fish et al.,1997).

Figure 37 shows a result obtained by applying a relatively straightforward multi-scale

model to the materials characterization of a cluster-structured high speed steel. At the

mesoscale, a unit cell approach is used in which particle-rich clusters and particle-poor

regions are described as separate phases. Each of the latter is treated as a particle

reinforced MMC with appropriate reinforcement volume fraction at the microscale, an

incremental Mori–Tanaka model being employed for homogenization. Such an approach

136) Multi-scale approaches using micromechanical models at the integration point level maylead to spurious stress and strain fields close to interior boundaries at the higher length scale,e.g. at interfaces between reinforced and monolithic regions in selectively reinforced components,because in such regions the conditions for standard homogenization methods typically are notfulfilled locally, compare e.g. (Chimani et al.,1997).

– 99 –

not only predicts macroscopic stress–strain relationships, but also allows the distribu-

tions of phase averaged microscopic variables to be evaluated, compare (Plankensteiner

et al.,1998)137). Alternatively, multiscale approaches may be rather “loosely coupled”,

essentially linking together quite different models to obtain an overall result, see e.g.

(Onck/van der Giessen,1997).

For a number of years there has been a surge of research interest in developing and

applying algorithms for the multi-scale and hierarchical continuum modeling of in-

homogeneous materials, see e.g. (Lee/Ghosh,1996; Zohdi et al.,1996; Fish/Shek,2000;

Feyel/Chaboche,2000; Ibrahimbegovic/Markovic,2003; Moes et al.,2003). A recent de-

velopment are multi-scale methods in which the computational domain is adaptively

split into regions resolved at appropriate length scale. In an FE-based framework de-

veloped Ghosh et al. (2001) “non-critical” regions are described by continuum elements

with homogenized material properties, and at zones of high macroscopic stress and strain

gradients PMA models are automatically activated at the integration points to monitor

the phase behavior at the microscale. If violations of the homogenization conditions (e.g.

onset of damage, mesoscopic interfaces, free edges) are detected, embedded models of the

fully resolved microstructure are activated at the appropriate positions. This strategy

allows detailed studies of critical regions in inhomogeneous materials, e.g. near free edges.

In an alternative approach, homogenization models employing generalized continua have

been proposed for handling regions close to free surfaces and zones with high local field

gradients (Feyel,2003).

Finally, it should be noted that multi-scale approaches are not limited to using the

“standard” methods of continuum micromechanics as implicitly assumed above, see e.g.

the discussions on metal plasticity in (Tadmor et al.,2000). Especially the capability of

the Finite Element method of handling highly complex constitutive descriptions has been

used to build multi-scale descriptions that employ, for example, material models based

on atomistics (Shenoy et al.,1998). Evidently, the distinction between micromechanical

studies using sophisticated constitutive models for the phases on the one hand and multi-

scale modeling on the other hand as used here is to some extent a matter of definition

(and personal preferences).

137) When assessing results from multi-scale analysis using mean field methods at the smallestlength scale, such as fig. 37, it must be kept in mind that the maximum stresses and strainsshown are the maxima of the local phase averaged values and not maximum microscopic stressesas obtained by “standard” micromechanical unit cell studies.

– 100 –

9. CLOSING REMARKS

The research field of continuum micromechanics of materials has enjoyed considerable

success in the past four decades in furthering the understanding of the thermomechanical

behavior of inhomogeneous materials and in providing predictive tools for engineers and

materials scientists. However, its methods are subject to some practical limitations that

should be kept in mind when employing them.

All the methods discussed in sections 2 to 7 implicitly use the assumption that the

constituents of the inhomogeneous material to be studied can be treated as homogeneous

at the lower length scale, which, of course, does not necessarily hold. When the length

scale of the inhomogeneities in a constituent is much smaller than the length scale of the

phase arrangement to be studied, the above assumption is valid and multi-scale models

(“successive homogenization”) as discussed in chapter 8 can be used, see e.g. Carrere et al.

(2004). However, no rigorous theory appears to be available at present for handling scale

transitions in materials that do not fulfill the above requirement (e.g. particle reinforced

composites in which the grain size of the matrix is comparable to or even larger than

the size of the particles138)). Typically the best that can be done in such cases is either

to use sufficiently large models with resolved microgeometry (a computationally very

expensive proposition) or to employ homogenized phase properties and be aware of the

approximation that is introduced139).

A major practical difficulty in the use of continuum micromechanics approaches (which

in many cases is closely related to the questions mentioned above) has been identifying

appropriate constitutive models and obtaining material parameters for the constituents.

Typically, available data pertain to the behavior of the bulk materials (as measured from

homogeneous macroscopic samples), whereas the actual requirement is for parameters

and, in some cases, constitutive theories that describe the in-situ response of the con-

stituents at the microscale. For example, with respect to MMCs, on the one hand there

is considerable evidence that classical continuum plasticity theories (in which there is no

absolute length scale) cannot adequately describe the behavior of metallic materials in

the presence of strong strain gradients, where geometrically necessary dislocations can

markedly influence the material behavior, see e.g. (Hutchinson,2000). On the other hand,

it is well known that the presence of reinforcements can lead to accelerated aging and

refinement of the grain size in the matrix (“secondary strengthening”). Furthermore, in

many cases reasonably accurate material parameters are essentially not available (e.g.

strength parameters for most interfaces in inhomogeneous materials). In fact, the dearth

of dependable material parameters is one of the reasons why predictions of the strength

of inhomogeneous materials by micromechanical methods tend to be a considerable chal-

lenge.

138) For this special case unit cell or embedding models employing anisotropic or crystalplasticity models for the phases may be used which, however, can become very large when thestatistics of grain orientation are accounted for.

139) It is worth noting that bounds evaluated using such an ad-hoc procedure are no longerrigorous — such results should be treated as lower and upper estimates.

– 101 –

Another point that should be kept in mind is that continuum micromechanical descrip-

tions in most cases do not have absolute length scales unless a length scale is introduced,

typically via the constitutive model(s) of one (or more) of the constituents140). In this

vein, absolute length scales, on the one hand, may be provided explicitly via discrete dis-

location models (Cleveringa et al.,1997), via gradient or nonlocal constitutive laws, see

e.g. (Tomita et al.,2000; Niordson/Tvergaard,2002), or via (nonlocal) damage models.

On the other hand, they may be introduced in an ad-hoc fashion, e.g. by adjusting the

phase material parameters to account for grain sizes via the Hall–Petch effect. Analysts

should also be aware that absolute length scales may be introduced inadvertently into

a model by mesh dependence effects of discretizing numerical methods, a “classical” ex-

ample being strain localization due to softening141) material behavior of a constituent,

compare also the discussions in the Appendix. When multi-scale models are used spe-

cial care may be necessary to avoid introducing inconsistent length scales at different

modeling levels.

In addition it is worth noting that usually the macroscopic response of inhomogeneous

materials is much less sensitive to the phase arrangement (and to modeling approxima-

tions) than are the distributions of the microfields. Consequently, whereas good agree-

ment in the overall behavior of a given model with “benchmark” theoretical results or

experimental data typically indicates that the phase averages of stresses and strains are

described satisfactorily, this does not necessarily imply that the higher statistical mo-

ments of the stress and strain distributions (and thus the heterogeneity of these fields)

are captured correctly.

It is important to be aware that work in the field of micromechanics of materials invariably

involves finding viable compromises in terms of the complexity of the models, which, on

the one hand, have to be able to account (at least approximatively) for the physical

phenomena relevant to the given problem, and, on the other hand, must be sufficiently

simple to allow solutions to be obtained within the relevant constraints of time, cost, and

computational resources. Obviously, actual problems cannot be solved without recourse

to various approximations and tradeoffs — the important point is to be aware of them and

to account for them in interpreting and assessing results. It is worth keeping in mind

that there is no such thing as a “best micromechanical approach” for all applications

and that models, while indispensable in understanding the behavior of inhomogeneous

materials, are “just” models and do not reflect the full complexities of real materials.

140) One important exception are models for studying macroscopic cracks (e.g. via embeddedcells), where the crack length does introduce an absolute length scale.Note the behavior of nanocomposites the behavior typically is “interface dominated”. Appro-priately accounting for this behavior may require the introduction of an absolute length scale.Within the framework of mean field methods this may be achieved by introducing surface termsinto the Eshelby, see e.g. Duan et al. (2005).

141) Note that here the expression “localization” is used to describe a physical phenomenonin materials and not a procedure in micromechanical modeling strategy as in section 1.2.

– 102 –

APPENDIX: SOME ASPECTS OF MODELING DAMAGE AT THE

MICROLEVEL

Modeling efforts aimed at studying the damage and failure behavior of inhomogeneous

materials may take the form of microscopic or macroscopic approaches. Methods be-

longing to the latter group often are based to some extent on micromechanical con-

cepts and encompass, among others, the “classical” failure surfaces for continuously

reinforced laminae such as the Tsai–Wu criterion (Tsai/Wu,1971) and macroscopic con-

tinuum damage models of various degrees of complexity, see e.g. (Chaboche et al.,1998;

Voyiadjis/Kattan,1999). Also on the macroscopic level the Considere criterion may be

used in combination with micromechanical models to check for (macroscopic) plastic in-

stability when studying the ductility of inhomogeneous elastoplastic materials, see e.g.

(Gonzalez/LLorca,1996). In what follows, however, only the modeling of damage on the

microscale in the context of (mainly Finite Element based) micromechanical methods of

the types discussed in chapters 4 to 8 are considered.

General Remarks

When damage and failure of inhomogeneous materials are to be studied at the microscale,

it is physically intuitive to represent them as acting at the level of the constituents and of

the interfaces between them, where appropriate damage and fracture models have to be

provided. All of the resulting microscopic failure mechanisms contribute to the energy ab-

sorption during crack growth. Taking MMCs as an example, this concept translates into

viewing damage and failure as involving a mutual interaction and competition between

ductile damage of the matrix, brittle cleavage of the reinforcements, and decohesion of

the interfaces, whereas in polycrystals intergranular or intragranular failure are typical

damage mechanisms. The dominant microscopic damage process is determined by the

individual constituents’ strength and stiffness parameters, by the microgeometry, and by

the loading conditions, compare Needleman et al. (1993). In fact, it may be markedly in-

fluenced by details of the microgeometry, such as particle shape and size, see e.g. Miserez

et al. (2004), a behavior which strongly supports the concept of using micromechanics

for damage modeling.

Generally, macrocracks as well as microcracks that have lengths comparable to the mi-

croscale of a given phase arrangement must be resolved explicitly in micromechanical

studies142). Smaller cracks and voids, however, can be treated in a homogenized man-

ner as damage. For describing cracks in micromechanical models “standard” FE models

of cracks may be used; for an overview covering some of the relevant methods see e.g.

(Liebowitz et al.,1995). “Standard” criteria for the initiation and/or growth of cracks,

such as J -integral, CTOD or CTOA criteria, etc., can also be used.

If only information on the initiation of damage is required it may be sufficient to evalu-

ate appropriate “damage relevant fields”, such as the maximum principal stress in brittle

142) Crack-like features that develop due to the failure of contiguous sets of elements in adamage-type model fulfill this requirement.

– 103 –

materials, a ductile damage indicator, or the traction vectors at interfaces, without ac-

counting for the loss of stiffness that accompanies the evolution of damage. For the

use of a ductile damage indicator in this context see e.g. (Gunawardena et al.,1993;

Bohm/Rammerstorfer,1996). Alternatively, it may be of interest to study microstruc-

tures containing flaws (e.g. fiber breaks, matrix flaws) in order to assess the resulting

microfields, see e.g. (Tsotsis/Weitsman,1990), where J -integrals were evaluated for spe-

cific matrix crack configurations.

If, however, the progress of damage and the interaction of damage modes at the con-

stituent level as well as final failure are to be studied, two major modeling strategies are

available at present. They are, on the one hand, the “volume oriented” damage-based

approaches used in continuum damage mechanics and related models and, on the other

hand, “surface oriented” methods that aim at following the progress of individual cracks.

The remainder of the present section is devoted to a short discussion of some aspects of

these two groups of methods.

When constituent damage at the microscale is modeled by continuum damage mechanics

(CDM) and related approaches (lumped together as CDM-type models in what follows),

the implicit assumption is made that the characteristic length scale of inhomogeneities

underlying the damage (pores, microcracks, etc.) is much smaller than the characteristic

length scale of phase arrangement to be studied. Essentially such models amount to

multi-scale methods as discussed in chapter 8. This group of approaches includes ductile

rupture models for describing ductile damage, see e.g. (Gurson,1977; Rousselier,1987),

and damage indicator triggered (DDIT) techniques, in which the stiffness of an integra-

tion point is drastically reduced or the corresponding element is removed when some

critical value of an appropriate damage indicator is reached. CDM-type methods can

account for the stiffness degradation due to the evolution of damage, which tends to

give rise to localization and may lead to the formation of proper shear bands within the

microscopic phase arrangement.

A numerical difficulty typically encountered with CDM-type models lies in their tendency

to provide solutions that show a marked mesh dependence143). The characteristic size

of the localized or failed region is typically found to be comparable to the dimensions

of one element, regardless of the actual degree of mesh refinement, and the predicted

energies for crack formation accordingly tend to decrease with decreasing mesh size,

compare fig. 38. This behavior can be avoided or at least mitigated by using theories

that are suitably regularized by enrichment with higher order gradients, see e.g. (Geers et

al.,2001), by nonlocal averaging of the rate of an appropriate damage variable, compare

(Tvergaard/Needleman,1995; Jirasek/Rolshoven,2002), or by the use of time dependent

formulations involving rate effects (e.g. viscoelasticity, viscoplasticity), compare (Needle-

man,1987). All of the above procedures explicitly introduce an absolute length scale

143) Within classical continuum theory strain softening material behavior leads to ill-posedboundary value problems, i.e. the governing equations lose their ellipticity or hyperbolicity,giving rise to the mesh sensitivity of the solutions, compare e.g. (Baaser/Gross,2002). In orderto provide an objective description of localization, the problem has to be regularized.

– 104 –

into the problem144) A rather ad-hoc alternative to the above regularization methods

consists in selecting a mesh size that leads to physically reasonable sizes of localization

features145).

increasing numberof elements

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1

forc

e-ra

tioR

F2/R

F2,m

ax

displacement-ratio U2/U2,max

≈ 2.500 el.≈ 6.000 el.≈ 21.000 el.

1

FIG.38: Normalized overall force–displacement responses obtained for a column with ahole under transverse loading with a local Gurson–Tvergaard-Needleman ductile rupturemodel (Drabek,2006). A marked mesh dependence of the results is evident.

Surface-oriented numerical strategies for simulating the progress of cracks emphasize

the concept that the nucleation and growth of cracks correspond to the creation of

new free surface. In most such models crack growth involves breaking the connectiv-

ity between neighboring elements, e.g. by node release techniques (Bakuckas et al.,1993;

Narasimhan,1994) or by cohesive surface elements, compare e.g. (Elices et al.,2002). Due

to the restriction of the growth of cracks to element boundaries, in models of this type a

mesh dependence of the predicted crack paths tends to arise unless the latter are known

a priori. This difficulty can be counteracted by using very fine and irregular discretiza-

tions or by introducing remeshing schemes that locally adapt the element geometry for

each crack growth increment, compare e.g. (Fish et al.,1999; Bouchard et al.,2003) and

thus provide realistic crack paths. Both of these remedies tend to lead to complex and/or

large models, so that in micromechanics of materials crack opening approaches have been

reported mainly for interfacial cracks and for phase arrangements where the crack paths

stay can be assumed to follow some plane of symmetry.

The mesh dependence of the crack paths predicted by surface oriented approaches can be

removed by using specially formulated solid elements, known as embedded discontinuity

144) A precondition for the proper regularization by higher order gradients or by nonlocalaveraging is that the discretization must be finer than than this absolute length scale.

145) Local damage procedures may be interpreted as introducing a field of characteristiclengths via the local element sizes. Material parameters identified on the basis of such pro-cedures pertain to a given element size only.

– 105 –

elements, which employ appropriately enriched shape functions. Such elements, on the

one hand, must allow discontinuities to develop at any position and in any direction within

each element and, on the other hand, must provide for continuous crack paths across

element boundaries, see e.g. (Jirasek,2000; Alfaiate et al.,2002). The actual opening

of cracks is typically handled by formulations of the cohesive surface type. Embedded

discontinuity elements at present are in a developmental stage, are typically not available

in standard FE packages, and up till have been little used in the context of continuum

micromechanics.

Modeling of Reinforcement Damage and Failure

Particulate and fibrous reinforcements in composites typically fail by brittle cleavage.

Within the framework of micromechanical models using the FEM, the failure of such

phases can be modeled by following the extension of a crack through the the reinforce-

ment, which implies that complex crack paths may have to be resolved, or by employ-

ing descriptions in which the crack opens instantaneously along some predefined “crack

plane”.

There are two groups of approaches for controlling the activation of cracks that are mod-

eled as opening instantaneously. When deterministic (Rankine) criteria are used, a parti-

cle or fiber is induced to fail once a critical level of some stress or strain measure is reached

or some fracture probability is exceeded. Studies using normal and maximum principal

stress criteria, respectively, within such a framework were published e.g. by Berns et al.

(1998) and Ghosh/Moorthy (1998). In “fully probabilistic” models a random process

(Monte-Carlo-type procedure) is provided which “decides” if a given reinforcement will

fail at a given load level on the basis of appropriate fracture probabilities, compare e.g.

(Pandorf,2001; Eckschlager,2002). Within micromechanical descriptions such fracture

probabilities can be evaluated from the predicted distributions of the microstresses by

Weibull-type weakest-link models146) that were originally developed for describing the

failure statistics of bulk brittle materials. A simple Weibull expression for evaluating the

failure probability Pn of a given particle n may take the form (Eckschlager,2002)

Pn = 1− exp

−1

V0

∫

Ωn:σ1>0

(σ1(x)

σ0

)m

dΩ

, (80)

where σ1(x) is the microscopic field of maximum principal stresses in the inhomogeneity

and Ω:σ1>0 is that part of the particle volume in which the maximum principal stress

146) Whereas there is agreement that the weakest-link behavior of long fibers is largely dueto surface defects, the physical interpretation is not as clear for Weibull models of particlesembedded in a ductile matrix. Weibull models based on eqn.(80) or similar expressions, however,have a number of advantages for the latter class of applications, compare (Wallin et al.,1987),not the least of which is their capability for describing particle size effects. It should be noted,however, that simple Weibull-type models were reported to give predictions for the dependenceof the number of fracture events per volume on particle size and volume fraction that do notagree with acoustic emission data from ductile matrix composites (Mummery et al.,1993).

– 106 –

is tensile. Equation (80) involves three material parameters, viz. σ0, which stands for

the particle’s average strength, m, which is known as the Weibull modulus, and V0, a

control volume. Expressions such as eqn.(80) assign a failure probability to a particle or

short fiber as a whole147), which fits well with the concept of “instantaneous cleavage”

of particles. Typically procedures of this type predict the onset of damage at rather

low applied loads, with some of the reinforcements failing at low fracture probabilities.

Because each simulation run provides a different realization of the underlying statistics,

the results from fully probabilistic models for a given microgeometry are best evaluated

in terms of ensemble averages over a number of anlyses.

In models of discontinuously reinforced composites that use axisymmetric cells, a natural

choice for the position of a predefined crack is a symmetry plane that is oriented normal to

the axis of rotation (which coincides with the loading direction) and runs through the cen-

ter of the inhomogeneity, see e.g. (Bao,1992; Davis et al.,1998; Gonzalez/LLorca,2000).

For more general arrangements of particles or short fibers the most realistic approach

would be to consider the stress state of each particle in selecting the crack plane148), so

that perturbations due to (possibly failed) neighboring particles can be accounted for.

Due to their high computational requirements up till now such algorithms have only been

reported for planar arrangements of ellipsoidal reinforcements, the crack being taken to

pass through the position of maximum principal stress in the particle and to be oriented

normally to the local maximum principal stress at this point (Ghosh et al.,2001).

For three-dimensional particle arrangements a simplified concept for approximating the

orientations and positions of cracks within spherical particles can be based on the as-

sumption that they pass through the particle center and are oriented normal to the

average maximum principal stress in the inhomogeneity or to the macroscopic maximum

principal stress (Eckschlager et al.,2001). Tomography data obtained on a model sys-

tem consisting of aluminum reinforced by spheres of zirconia/silica at ξ=0.04 (Babout et

al.,2004), however, show cracks that are oriented normally to the macroscopic maximum

principcal stress but do not tend to run through the centers of particles.

The alternatives to the above approach of instantaneous particle cracking consist in using

either cohesive surface models for opening predefined cracks in reinforcements, see e.g.

(Finot et al.,1994; Tvergaard,1994), or in applying continuum damage models to describ-

ing failure of the reinforcements, see e.g. (Zohdi/Wriggers,2001; Mishnaevsky,2004). Such

models are capable of resolving the growth sequence of a crack rather than “popping it

open” instantaneously. Whereas predefined crack paths in the reinforcements have typ-

147) In modeling the fracture of continuous fibers data on the positions of transverse cracksare obviously required. Such information can be obtained by evaluating Weibull fracture prob-abilities for cylindrical subvolumes of the fibers.

148) For small monocrystalline particulates the crystallographic orientations can also be ex-pected to play a role in determining the crack plane, thus introducing a statistical element intothe problem. In particles of irregular shape local stress concentrations must be expected to playa role in determining the positions of cracks and for elongated particles or fibers the appearanceof multiple cracks must be provided for. For experimental results on the orientation of cracksin particles see e.g. (Mawsouf,2000).

– 107 –

ically been used with cohesive surfaces, no such provision is required in the CDM-type

models, which can describe geometrically complex cracks149). Both approaches, however,

do not introduce effects of the absolute size of the reinforcements into micromechanical

models.

Modeling of Matrix Failure

The type of matrix failure found in composites obviously depends on the matrix ma-

terial and thus on the composite system considered. In the case of metallic matrices

ductile damage and failure must be considered, which either takes place by void nucle-

ation, growth, and coalescence (void impingement) or by the formation of void sheets,

compare e.g. (Horstemeyer et al.,2000). The former mechanism dominates at high stress

triaxialities σm/σeq, whereas the latter one leads to shear failure in the low triaxiality

regime.

Over the past two decades the dominant approach for modeling the failure of ductile

matrices in metal matrix composites have been descriptions based on the ductile rup-

ture model of Gurson (1977) and its improved versions (Tvergaard/Needleman,1984;

Garajeu,1995). Models of this type are implemented in a number of commercial Finite

Element codes150). For micromechanical studies that employ Gurson-type models for

describing ductile failure of the matrix of MMCs see e.g. Needleman et al. (1993), Geni

and Kikuchi (1999), and Hu et al. (2007). A version of the ductile damage model of

Rousselier (1987) was used in micromechanical models by Drabek and Bohm (2004).

In the context of micromechanics element elimination models triggered by a ductile dam-

age indicator (“DDIT models”) have been used as an alternative to “ductile rupture”

models of the above type151). The damage indicator most commonly used for such pur-

poses was proposed by Gunawardena et al. (1991) on the basis of earlier studies on void

growth (Rice/Tracey,1969; Hancock/Mackenzie,1976) and takes the form

Did =

εeq,p∫

0

exp(

32σm/σeq

)

1.65 εfdεeq,p , (81)

149) From the physical point of view, continuum damage models are most appropriate insituations where a process zone, e.g. due to microcracks, is present. Because particle cleavagetends to produce “clean” cracks, it should preferably be handed by surface based models.

150) Gurson models are are based on the behavior of spherical voids and are best suited forhandling ductile damage in situations of high stress triaxaiality. A Gurson-type model thatallows for evolution of the void shapes was developed by Gologanu et al.(1997), and Nahshonand Hutchinson (2008) extended a Gurson-type model to the low-triaxiality situation of shearfailure.

151) In contrast to the physically interpretable damage variables used in ductile rupture andstandard CDM models, which allow the effect of damage on all material properties to be handledconsistently, only two values of the damage parameter Did defined in eqn.(81) have a directphysical interpretation: 0 indicates no damage and 1 ductile failure. Accordingly, ductile damageindicators per se cannot follow the degradation of stiffness as damage develops.

– 108 –

where εeq,p stands for the accumulated equivalent plastic strain, σm for the mean (hy-

drostatic) stress, σeq for the equivalent (von Mises) stress, and εf for the uniaxial fail-

ure strain of the material152). Like most ductile rupture models the damage indicator,

eqn.(18), was derived for the high stress triaxiality regime153), which is the more relevant

one for micromechanical analysis of composites on account of the pronounced tendency

towards highly constrained plasticity. Recently ductile damage indicators have been

developed that can also account for local shear failure, see e.g. (Wilkins,1999).

DDIT-based element elimination schemes in the context of continuum micromechan-

ics of inhomogeneous materials were reported e.g. in (Schmauder et al.,1996; Berns et

al.,1998). Other criteria used for controlling element elimination in ductile matrices have

included strain energy criteria (Mahishi/Adams,1982), the so called s-criterion (Spiegler

et al.,1990), plastic strain criteria (Adams,1974), and stress triaxiality criteria (Steinkopff

et al.,1995). Wulf et al. (1996) reported comparisons between predictions obtained by

planar embedded cell models using element elimination models controlled by stress triax-

iality, plastic strain, and Rice–Tracey criteria, the latter giving the best agreement with

experimental results for that special case. The Finite Element code ABAQUS provides

a set of hybrid schemes for modeling ductile and shear damage in metals, which combine

a damage indicator-type concept for detecting “damage initiation”154) with phenomeno-

logical CDM-like algorithms for describing the evolution of damage, leading finally to

element elimination.

The use of nonlocal versions of ductile rupture models and DDIT schemes, which allow

the mesh dependence of the solutions to be reduced or eliminated, was first proposed

for micromechanical applications by Tvergaard and Needleman (1995). These methods

have seen further development recently155) see e.g. (Hu et al.,2004; Drabek/Bohm,2004;

Drabek,2006).

152) The main advantage of this ductile damage indicator lies in requiring only one mate-rial parameter, εf , which can be evaluated experimentally in a relatively straightforward way,compare the discussion in Fischer et al. (1995). Alternative ductile damage indicators weredeveloped e.g. by McClintock (1978) and Oyane (1972).

153) For a discussion of the validity of eqn.(81) in the low to negative triaxiality regime, wherethe damage mechanism is shear failure rather than void growth and coalescence, see (Ganser etal., 2001).

154) In this context the expression “damage initation” is not used in the sense of metalsphysics, but rather signifies a state where damage effects on the behavior of some materialvolume become noticeable.

155) As mentioned before, all elements subject to damage must be smaller than the charac-teristic length in nonlocal averaging. The average mesh size, however, should not be too muchsmaller than the characteristic length, either, because otherwise the high number of neighborscontributing to the nonlocal averages will make the algorithms too slow. These restrictions canconsiderably constrain the applicability of nonlocal averaging methods.Note that at present there is no agreement whether the characteristic length in ductile damage isa physical material property or a numerical modeling parameter. If, however, the characteristiclength is assumed to be a physical parameter determined by void nucleation from small defects itmust be inversely proportional to the third root of the defects’ volume density (Rousselier,1987).

– 109 –

Alternatively, cohesive surface models may be used for modeling ductile failure, especially

in cases where the crack path is known a priori, a typical example being ductile cracks

propagating along a plane of symmetry, see e.g. Finot et al. (1994). A possible drawback

of cohesive surface models for studying ductile failure in a continuum micromechanics

framework, however, has been revealed by recent studies, which indicate that the deco-

hesion parameters tend to depend on the stress triaxialities, see e.g. (Chen eta al.,2001).

Approaches for modeling matrix failure that do not fall into the above groups were

reported, among others, by Biner (1996), who studied creep damage to the matrix of

an MMC using Tvergaard’s (1984) grain boundary cavitation model, and by Mahishi

(1986), who combined a node release technique with an energy release criterion to study

cracks in polymer matrices156). Composites with brittle elastic matrix behavior can

be investigated via fracture mechanics based methods see e.g. (Ismar/Reinert,1997), or

Weibull-type criteria may be employed (Brockmuller et al.,1995).

Finally, it may be mentioned that in the case of cyclic loading shakedown theorems

may be used on the microscale to assess if elastic shakedown may take place in an

inhomogeneous material or if low cycle fatigue will act on the microscale, see (Bohm,1993;

Schwabe,2000).

Modeling of Interface Failure

Probably the most commonly used description for the debonding of a reinforcement from

a matrix in numerical micromechanics has been a cohesive surface model developed by

Needleman (1987), It is based on an interface potential that specifies the dependence of

the normal and tangential interfacial tractions on the normal and tangential displace-

ment differences across the interface and was extended by Tvergaard (1990) to handle

the modeling of tangential separation. Interfacial behavior of this type is typically de-

scribed in FE codes via special interface elements, which in many cases are also capable

of handling contact and friction between reinforcement and matrix once interfacial de-

cohesion has occurred. Some examples of micromechanical studies employing algorithms

of the Needleman–Tvergaard type are (Nutt/Needleman,1987; Needleman et al.,1993;

Haddadi/Teodosiu,1999). A modified version of the above interfacial decohesion model

was developed by Michel and Suquet (1994), and another related technique is reported

in (Zhong/Knauss,2000).

An alternative approach suitable for describing very weakly bonded reinforcements con-

sists in using standard contact elements at the interface, the composite being held to-

gether in a “shrink fit” by the thermal stresses caused by cooling down from the processing

temperature, see e.g. (Gunawardena et al.,1993; Sherwood/Quimby,1995). As an exten-

sion of this type of description, Coulomb-type models, i.e. a linkage between normal and

shear stresses analogous to that employed for dry friction, may be used for the interface

(Benabou et al.,2002).

156) Because crack paths in inhomogeneous materials may be rather complicated, appropriatecriteria for determining the crack direction have to be employed and remeshing typically isnecessary when node release techniques are used, compare (Vejen/Pyrz,2002).

– 110 –

Other interfacial decohesion models reported in the literature include strain softening in-

terfacial springs (Vosbeek et al.,1993; Li/Wisnom,1994), a node release technique based

on fracture mechanics solutions for bimaterial criterion (Moshev/Kozhevnikova,2002).

Interphases, i.e. interfacial layers of finite thickness, were studied by various ficti-

tious crack techniques, see e.g. (Ismar/Schmitt,1991; McHugh/Connolly,1994; Walter

et al.,1997) and thermomechanically based continuum damage models (Benabou et

al.,2002).

A rather different approach has been developed specifically for accounting for the stiffness

loss due to debonding within mean field models. It is based on describing debonded

reinforcements by perfectly bonded (transversally isotropic) “fictitious inhomogeneities”

of appropriately reduced stiffness. Within such a framework, the fraction of debonded

reinforcements may be controlled via Weibull-type models, see e.g. (Zhao/Weng,1995;

Sun et al.,2003).

Models Involving Multiple Failure Modes

Whereas an appreciable number of the works mentioned above involve two interacting

micromechanical failure modes, only few attempts to concurrently model matrix, rein-

forcement and interfacial failure have been published. For two-dimensional studies of

this type describing ductile matrix composites see e.g. (Berns et al.,1998; Pandorf,2000),

where generic particle arrangements in tool steels are compared with respect to their frac-

ture properties, and (Hu et al.,2007), where particle reinforced metal matrix composites

are investigated.

– 111 –

REFERENCES

Ableidinger A.: Some Aspects of the Fracture Behavior of Metal Foams. Diploma Thesis,Vienna University of Technology, Vienna, Austria, 2000.

Aboudi J.: Micromechanical Analysis of Composites by the Method of Cells;Appl.Mech.Rev. 42, 193–221, 1989.

Aboudi J.: Mechanics of Composite Materials. Elsevier, Amsterdam, The Netherlands,1991.

Aboudi J.: Micromechanical Analysis of Composites by the Method of Cells — Update;Appl.Mech.Rev. 49, S83–S91, 1996.

Aboudi J., Pindera M.J., Arnold S.M.: Higher-Order Theory for Periodic MultiphaseMaterials with Inelastic Phases; Int.J.Plast. 19, 805–847, 2003.

Accorsi M.L.: A Method for Modeling Microstructural Material Discontinuities in a Fi-nite Element Analysis; Int.J.Num.Meth.Engng. 26, 2187–2197, 1988.

Achenbach J.D., Zhu H.: Effect of Interfacial Zone on Mechanical Behavior and Failureof Fiber-Reinforced Composites; J.Mech.Phys.Sol. 37, 381–393, 1989.

Adams D.F.: A Micromechanical Analysis of Crack Propagation in an Elastoplastic Com-posite Material; Fibre Sci.Technol. 7, 237–256, 1974.

Adams D.F., Crane D.A.: Finite Element Micromechanical Analysis of a UnidirectionalComposite Including Longitudinal Shear Loading; Comput.Struct. 18, 1153–1165, 1984.

Adams D.F., Doner D.R.: Transverse Normal Loading of a Uni-Directional Composite;J.Compos.Mater. 1, 152–164, 1967.

Advani S.G., Tucker C.L.: The Use of Tensors to Describe and Predict Fiber Orientationin Short Fiber Composites; J.Rheol. 31, 751–784, 1987.

Alfaiate J., Wells G.N., Sluys L.J.: On the Use of Embedded Discontinuity Elements withCrack Path Continuity for Mode-I and Mixed Mode Fracture; Engng.Fract.Mech. 69,661–686, 2002.

Allen D.H., Lee J.W.: The Effective Thermoelastic Properties of Whisker-ReinforcedComposites as Functions of Material Forming Parameters; in “Micromechanics and In-homogeneity” (Eds. G.J.Weng, M.Taya, H.Abe), pp. 17–40; Springer–Verlag, New York,NY, 1990.

Amieur M.: Etude numerique et experimentale des effets d’echelle et de conditions auxlimites sur des eprouvettes de beton n’ayant pas le volume representatif. Doctoral Thesis,Ecole Polytecnique Federale de Lausanne, Lausanne, Switzerland, 1994.

Anthoine A.: Derivation of the In-Plane Elastic Characteristics of Masonry ThroughHomogenization Theory; Int.J.Sol.Struct. 32, 137–163, 1995.

Antretter T.: Micromechanical Modeling of High Speed Steel. VDI–Verlag (Reihe 18, Nr.232), Dusseldorf, Germany, 1998.

Arenburg R.T., Reddy J.N.: Analysis of Metal-Matrix Composite Structures — I. Mi-cromechanics Constitutive Theory; Comput.Struct. 40, 1357–1368, 1991.

Avellaneda M., Milton G.W.: Optimal Bounds on the Effective Bulk Modulus of Poly-crystals; J.Appl.Math. 49, 824–837, 1989.

Axelsen M.S., Pyrz R.: Correlation Between Fracture Toughness and the Microstruc-ture Morphology in Transversely Loaded Unidirectional Composites; in “Microstructure–Property Interactions in Composite Materials” (Ed. R.Pyrz), pp. 15–26; Kluwer Aca-demic Publishers, Dordrecht, The Netherlands, 1995.

– 112 –

Baaser H., Gross D.: On the Limitations of CDM Approaches to Ductile Fracture Prob-lems; in “Proc. Fifth World Congress on Computational Mechanics (WCCM V)” (Eds.:H.A.Mang, F.G.Rammerstorfer, J.Eberhardsteiner), paper # 81462; Vienna Universityof Technology, Vienna, 2002.

Babout L., Brechet Y., Maire E., Fougeres R.: On the Competition between ParticleFracture and Particle Decohesion in Metal Matrix Composites; Acta mater. 52, 4517–4525, 2004.

Bakuckas J.G., Tan T.M., Lau A.C.W., Awerbuch J.: A Numerical Model for Pre-dicting Crack Path and Modes of Damage in Unidirectional Metal Matrix Composites;J.Reinf.Plast.Compos. 12, 341–358, 1993.

Banerjee P.K., Henry D.P.: Elastic Analysis of Three-Dimensional Solids with FiberInclusions by BEM; Int.J.Sol.Struct. 29, 2423–2440, 1992.

Banks-Sills L., Leiderman V., Fang D.: On the Effect of Particle Shape and Orientationon Elastic Properties of Metal Matrix Composites; Composites 28B, 456–481, 1997.

Bansal Y., Pindera M.J.: Finite-Volume Direct Averaging Micromechanics of Heteroge-neous Materials with Elastic-Plastic Phases; Int.J.Plast. 22, 775–825, 2005.

Bao G.: Damage Due to Fracture of Brittle Reinforcements in a Ductile Matrix;Acta.metall.mater. 40, 2547–2555, 1992.

Bao G., Hutchinson J.W., McMeeking R.M.: Particle Reinforcement of Ductile MatricesAgainst Plastic Flow and Creep; Acta metall.mater. 39, 1871–1882, 1991.

Benabou L., Benseddiq N., Naıt-Abdelaziz M.: Comparative Analysis of Damage atInterfaces of Composites; Composites 33B, 215–224, 2002.

Benveniste Y.: A New Approach to the Application of Mori–Tanaka’s Theory in Com-posite Materials; Mech.Mater. 6, 147–157, 1987.

Benveniste Y.: Some Remarks on Three Micromechanical Models in Composite Media;J.Appl.Mech. 57, 474–476, 1990.

Benveniste Y., Dvorak G.J.: On a Correspondence Between Mechanical and Ther-mal Effects in Two-Phase Composites; in “Micromechanics and Inhomogeneity” (Eds.G.J.Weng, M.Taya, H.Abe), pp. 65–82; Springer–Verlag, New York, NY, 1990.

Benveniste Y., Dvorak G.J., Chen T.: On Diagonal and Elastic Symmetry of the Approx-imate Effective Stiffness Tensor of Heterogeneous Media; J.Mech.Phys.Sol. 39, 927–946,1991.

Benzerga A.A., Besson J.: Plastic Potentials for Anisotropic Porous Solids; Eur.J.Mech.A/Solids 20, 397–434, 2001.

Beran M.J., Molyneux J.: Use of Classical Variational Principles to Determine Boundsfor the Effective Bulk Modulus in Heterogeneous Media; Quart.Appl.Math. 24, 107–118,1966.

Berns H., Melander A., Weichert D., Asnafi N., Broeckmann C., Gross-Weege A.: A NewMaterial for Cold Forging Tools; Comput.Mater.Sci. 11, 166–188, 1998.

Berryman J.G.: Long-Wavelength Propagation in Composite Elastic Media, II. Ellip-soidal Inclusions; J.Acoust.Soc.Amer. 68, 1820–1831, 1980.

Berryman J.G., Berge P.A.: Critique of Two Explicit Schemes for Estimating ElasticProperties of Multiphase Composites; Mech.Mater. 22, 149–164, 1996.

Berveiller M., Zaoui A.: An Extension of the Self-Consistent Scheme to Plastically Flow-ing Polycrystals; J.Mech.Phys.Sol. 26, 325–344, 1979.

Bhattacharyya A., Weng G.J.: The Elastoplastic Behavior of a Class of Two-PhaseComposites Containing Rigid Inclusions; Appl.Mech.Rev. 47, S45–S65, 1994.

– 113 –

Biner S.B.: A Finite Element Method Analysis of the Role of Interface Behavior in theCreep Rupture Characteristics of a Discontinuously Reinforced Composite with SlidingGrain Boundaries; Mater.Sci.Engng. A208, 239–248, 1996.

Bisegna P., Luciano R.: Bounds on the Off-Diagonal Coefficients of the HomogenizedConstitutive Tensor of a Composite Material; Mech.Res.Comm. 23, 239–246, 1996.

Bishop J.F.W., Hill R.: A Theory of the Plastic Distortion of a Polycrystalline AggregateUnder Combined Stress; Phil.Mag. 42, 414–427, 1951.

Bitsche R.: Space Filling Polyhedra as Mechanical Models for Solidified Dry Foams.Diploma Thesis, Vienna University of Technology, Vienna, Austria, 2005.

Bochenek B., Pyrz R.: Reconstruction Methodology for Planar and Spatial Random Mi-crostructures; in “New Challenges in Mesomechanics” (Eds. R.Pyrz, J.Schjødt-Thomsen,J.C.Rauhe, T.Thomsen, L.R.Jensen), pp. 565–572; Aalborg University, Aalborg, Den-mark, 2002.

Bohm H.J.: Numerical Investigation of Microplasticity Effects in Unidirectional Long-fiber Reinforced Metal Matrix Composites; Modell.Simul.Mater.Sci.Engng. 1, 649–671,1993.

Bohm H.J.: Notes on Some Mean Field Approaches for Composites. CDL–FMD Report1–1998, Vienna University of Technology, Vienna, Austria, 1998.

Bohm H.J.: A Short Introduction to Continuum Micromechanics; in “Mechanics of Mi-crostructured Materials” (Ed. H.J.Bohm), pp. 1–40; CISM Courses and Lectures Vol.464, Springer–Verlag, Vienna, 2004.

Bohm H.J., Han W.: Comparisons Between Three-Dimensional and Two-DimensionalMulti-Particle Unit Cell Models for Particle Reinforced MMCs; Modell.Simul.Mater.Sci.Engng. 9, 47–65, 2001.

Bohm H.J., Rammerstorfer F.G.: Fiber Arrangement Effects on the Microscale Stressesof Continuously Reinforced MMCs; in “Microstructure–Property Interactions in Com-posite Materials” (Ed. R.Pyrz), pp. 51–62; Kluwer Academic Publishers, Dordrecht, TheNetherlands, 1995.

Bohm H.J., Rammerstorfer F.G.: Influence of the Micro-Arrangement on Matrix andFiber Damage in Continuously Reinforced MMCs; in “Micromechanics of Plasticity andDamage of Multiphase Materials” (Eds. A.Pineau, A.Zaoui), pp. 19–26; Kluwer, Dor-drecht, The Netherlands, 1996.

Bohm H.J., Eckschlager A., Han W.: Modeling of Phase Arrangement Effects in HighSpeed Tool Steels; in “Tool Steels in the Next Century” (Eds. F.Jeglitsch, R.Ebner,H.Leitner), pp. 147–156; Montanuniversitat Leoben, Leoben, Austria, 1999.

Bohm H.J., Eckschlager A., Han W.: Multi-Inclusion Unit Cell Models for Metal MatrixComposites with Randomly Oriented Discontinuous Reinforcements; Comput.Mater.Sci.25, 42–53, 2002.

Bohm H.J., Rammerstorfer F.G., Weissenbek E.: Some Simple Models for Microme-chanical Investigations of Fiber Arrangement Effects in MMCs; Comput.Mater.Sci. 1,177–194, 1993.

Bornert M.: Morphologie microstructurale et comportement mecanique; characterisationsexperimentales, approches par bornes et estimations autocoherentes generalisees. Doc-toral Thesis, Ecole Nationale des Ponts et Chaussees, Paris, France, 1996.

Bornert M.: Homogeneisation des milieux aleatoires: bornes et estimations; in “Ho-mogeneisation en mecanique des materiaux 1. Materiaux aleatoires elastiques et milieuxperiodiques” (Eds. M.Bornert, T.Bretheau, P.Gilormini), pp. 133–221; Editions Hermes,Paris, 2001.

– 114 –

Bornert M., Suquet P.: Proprietes non lineaires des composites: Approches par les poten-tiels; in “Homogeneisation en mecanique des materiaux 2. Comportements non lineaireset problemes ouverts” (Eds. M.Bornert, T.Bretheau, P.Gilormini), pp. 45–90; EditionsHermes, Paris, 2001.

Bornert M., Stolz C., Zaoui A.: Morphologically Representative Pattern-Based Boundingin Elasticity; J.Mech.Phys.Sol. 44, 307–331, 1996.

Bornert M., Bretheau T., Gilormini P. (Eds.): Homogeneisation en mecanique desmateriaux 1. Materiaux aleatoires elastiques et milieux periodiques. Editions Hermes,Paris, 2001.

Bornert M., Bretheau T., Gilormini P. (Eds.): Homogeneisation en mecanique desmateriaux 2. Comportements non lineaires et problemes ouverts. Editions Hermes, Paris,2001.

Bornert M., Herve E., Stolz C., Zaoui A.: Self Consistent Approaches and Strain Het-erogeneities in Two-Phase Elastoplastic Materials; Appl.Mech.Rev. 47, S66–S76, 1994.

Brockmuller K.M., Bernhardi O., Maier M.: Determination of Fracture Stress andStrain of Highly Oriented Oriented Short Fibre-Reinforced Composites Using a FractureMechanics-Based Iterative Finite-Element method; J.Mater.Sci. 30, 481–487, 1995.

Brown L.M., Stobbs W.M.: The Work-Hardening of Copper–Silica. I. A Model Basedon Internal Stresses, with no Plastic Relaxation; Phil.Mag. 23, 1185–1199, 1971.

Bruzzi M.S., McHugh P.E., O’Rourke F., Linder T.: Micromechanical Modelling of theStatic and Cyclic Loading of an Al21244-SiC MMC; Int.J.Plast. 17, 565–599, 2001.

Bouchard P.O., Bay F., Chastel Y.: Numerical Modelling of Crack Propagation: Au-tomatic Remeshing and Comparison of Different Criteria; Comput.Meth.Appl.Mech.Engng. 192, 3887–3908, 2003.

Bulsara V.N., Talreja R., Qu J.: Damage Initiation under Transverse Loading of Unidi-rectional Composites with Arbitrarily Distributed Fibers; Compos.Sci.Technol. 59, 673–682, 1999.

Buryachenko V.A.: The Overall Elastoplastic Behavior of Multiphase Materials withIsotropic Components; Acta Mech. 119, 93–117, 1996.

Camacho C.W., Tucker C.L., Yalvac S., McGee R.L.: Stiffness and Thermal ExpansionPredictions for Hybrid Short Fiber Composites; Polym.Compos. 11, 229–239, 1990.

Carrere N., Valle R., Bretheau T., Chaboche J.L.: Multiscale Analysis of the TransverseProperties of Ti-Based Matrix Composites Reinforced by SiC Fibers: From the GrainScale to the Macroscopic Scale; Int.J.Plast. 20, 783–810, 2004.

Cazzani A., Rovati M.: Extrema of Young’s Modulus for Cubic and Transversely IsotropicSolids; Int.J.Sol.Struct. 40, 1713–1744, 2003.

Chaboche J.L., Kanoute P.: Sur les approximations “isotrope et “anisotrope del’operateur tangent pour les methodes tangentes incrementale et affine; C.R.Mecanique331, 857–864, 2003.

Chaboche Y.L., Kruch S., Pottier T.: Micromechanics versus Macromechanics: a Com-bined Approach for Metal Matrix Composite Constitutive Modeling; Eur.J.Mech. A/Solids17, 885–908, 1998.

Chaboche J.L., Kruch S., Maire J.F., Pottier T.: Towards a Micromechanics BasedInelastic and Damage Modelling of Composites; Int.J.Plast. 17, 411–439, 2001.

Chang W., Kikuchi N.: Analysis of Residual Stresses Developed in the Curing ofLiquid Molding; in “Advanced Computational Methods for Material Modeling” (Eds.D.J.Benson, R.A.Asaro), pp. 99–113; AMD–Vol.180/PVP–Vol.268, ASME, New York,NY, 1993.

– 115 –

Chawla N., Chawla K.K.: Microstructure Based Modeling of the Deformation Behaviorof Particle Reinforced Metal Matrix Composites; J.Mater.Sci. 41, 913–925, 2006.

Chen C.R., Scheider I., Siegmund T., Tatschl A., Kolednik O., Fischer F.D.: FractureInitiation and Crack Growth — Cohesive Zone Modeling and Stereoscopic Measurements;in “Proceedings of the ICF 10” (Eds. K.Ravi-Chandar et al.), paper # 100409P; 2001.

Chen H.R., Yang Q.S., Williams F.W.: A Self-Consistent Finite Element Approach tothe Inclusion Problem; Comput.Mater.Sci. 2, 301–307, 1994.

Chen T.: Exact Moduli and Bounds of Two-Phase Composites with Coupled MultifieldLinear Responses; J.Mech.Phys.Sol. 45, 385–398, 1997.

Chimani C.M., Morwald K.: Micromechanical Investigation of the Hot Ductility Behaviorof Steel; ISIJ Int. 39, 1194–1197, 1999.

Chimani C.M., Bohm H.J., Rammerstorfer F.G.: On Stress Singularities at Free Edgesof Bimaterial Junctions — A Micromechanical Study; Scr.mater. 36, 943–947, 1997.

Christensen R.M.: Sufficient Symmetry Conditions for Isotropy of the Elastic ModuliTensor; J.Appl.Mech. 54, 772–777, 1987.

Christensen R.M., Lo K.H.: Solutions for Effective Shear Properties Three Phase Sphereand Cylinder Models; J.Mech.Phys.Sol. 27, 315–330, 1979.

Christensen R.M., Lo K.H.: Erratum to Christensen and Lo, 1979; J.Mech.Phys.Sol. 34,639, 1986.

Christensen R.M., Schantz H., Shapiro J.: On the Range of Validity of the Mori–TanakaMethod; J.Mech.Phys.Sol. 40, 69–73, 1992.

Chung P.W., Tamma K.K., Namburu R.R.: Asymptotic Expansion Homogenization forHeterogeneous Media: Computational Issues and Applications; Composites 32A, 1291–1301, 2001.

Cleveringa H.H.M., van der Giessen E., Needleman A.: Comparison of Discrete Disloca-tion and Continuum Plasticity Predictions for a Composite Material; Acta mater. 45,3163–3179, 1997.

Clyne T.W., Withers P.J.: An Introduction to Metal Matrix Composites. CambridgeUniversity Press, Cambridge, UK, 1993.

Courage W.M.G., Schreurs P.J.G: Effective Material Parameters for Composites withRandomly Oriented Short Fibers; Comput.Struct. 44, 1179–1185, 1992.

Cohen I.: Simple Algebraic Approximations for the Effective Elastic Moduli of CubicArrays of Spheres; J.Mech.Phys.Sol. 52, 2167–2183, 2004.

Corvasce F., Lipinski P., Berveiller M.: The Effects of Thermal, Plastic and ElasticStress Concentrations on the Overall Behavior of Metal Matrix Composites; in “InelasticDeformation of Composite Materials” (Ed. G.J.Dvorak), pp. 389–408; Springer–Verlag,New York, NY, 1991.

Cox H.L.: The Elasticity and Strength of Paper and Other Fibrous Materials;Brit.J.Appl.Phys. 3, 72–79, 1952.

Cruz M.E., Patera A.T.: A Parallel Monte-Carlo Finite Element Procedure for the Anal-ysis of Multicomponent Random Media; Int.J.Num.Meth.Engng. 38, 1087–1121, 1995.

Dasgupta A., Agarwal R., Bhandarkar S.: Three-Dimensional Modeling of Woven-Fabric Composites for Effective Thermo-Mechanical and Thermal Properties; Com-pos.Sci.Technol. 56, 209–223, 1996.

Davis L.C.: Flow Rule for the Plastic Deformation of Particulate Metal Matrix Compos-ites; Comput.Mater.Sci. 6, 310–318, 1996.

– 116 –

Davis L.C., Andres C., Allison J.E.: Microstructure and Strengthening of Metal MatrixComposites; Mater.Sci.Engng. A249, 40–45, 1998.

Dawson P.R.: Computational Crystal Plasticity; Int.J.Sol.Struct. 37, 115–130, 2000.

Daxner T.: Multi-Scale Modeling and Simulation of Metallic Foams. VDI–Verlag (Reihe18, Nr. 285), Dusseldorf, Germany, 2003.

Daxner T., Bohm H.J., Seitzberger M., Rammerstorfer F.G.: Modeling of Cellular Ma-terials; in “Handbook of Cellular Metals” (Eds.: H.P.Degischer, B.Kriszt), pp. 245–280;Wiley–VCH, Weinheim, Germany, 2002.

Delannay L., Doghri I., Pierard O.: Prediction of Tension–Compression Cycles in Mul-tiphase Steel Using a Modified Mean-Field Model; Int.J.Sol.Struct. 44, 7291–7306, 2007.

Doghri I., Friebel C.: Effective Elasto-Plastic Properties of Inclusion-Reinforced Com-posites. Study of Shape, Orientation and Cyclic Response; Mech.Mater. 37, 45–68,2005.

Doghri I., Ouaar A.: Homogenization of Two-Phase Elasto-Plastic Composite Materialsand Structures: Study of Tangent Operators, Cyclic Plasticity and Numerical Algorithms;Int.J.Sol.Struct. 40, 1681–1712, 2003.

Doghri I., Tinel L.: Micromechanical Modeling and Computation of Elasto-Plastic Ma-terials Reinforced with Distributed-Orientation Fibers; Int.J.Plast. 21, 1919–1940, 2005.

Dong M., Schmauder S.: Modeling of Metal Matrix Composites by a Self-ConsistentEmbedded Cell Model; Acta mater. 44, 2465–2478, 1996.

Drabek T.: Modeling of Matrix Damage in Particle Reinforced Ductile Matrix Compos-ites. VDI–Verlag (Reihe 18, Nr. 302), Dusseldorf, Germany, 2006.

Drabek T., Bohm H.J.: Implementation of Nonlocal Damage Models in Commercial FE-Codes; in “Computational Mechanics. Proceedings of the Sixth World Congress on Com-putational Mechanics” (Eds.: Z.H.Yao, M.W.Yuan, W.X.Zhong), paper # 262; TsinghuaUniversity Press and Springer–Verlag, Beijing, 2004.

Dreyer W., Muller W.H., Olschewski J.: An Approximate Analytical 2D-Solution for theStresses and Strains in Eigenstrained Cubic Materials; Acta Mech. 136, 171–192, 1999.

Drugan W.J.: Micromechanics-Based Variational Estimates for a Higher-Order NonlocalConstitutive Equation and Optimal Choice of Effective Moduli for Elastic Composites;J.Mech.Phys.Sol. 48, 1359–1387, 2000.

Drugan W.J., Willis J.R.: A Micromechanics-Based Nonlocal Constitutive Equa-tion and Estimates of Representative Volume Element Size for Elastic Composites;J.Mech.Phys.Sol. 44, 497–524, 1996.

Duan H.L., Wang J., Huang Z.P., Karihaloo B.L.: Size-Dependent Effective Elas-tic Constants of Solids Containing Nano-Inhomogeneities with Interface Stress;J.Mech.Phys.Sol. 53, 1574–1596, 2005.

Duan H.L., Karihaloo B.L., Wang J., Yi X.: Effective Conductivities of HeterogeneousMedia Containing Multiple Inclusions with Various Spatial Distributions; Phys.Rev.B73, 174203, 2006.

Dunn M.L., Ledbetter H.: Elastic–Plastic Behavior of Textured Short-Fiber Composites;Acta mater. 45, 3327–3340, 1997.

Dunn M.L., Taya M.: Micromechanics Predictions of the Effective Electroelastic Moduliof Piezoelectric Composites; Int.J.Sol.Struct. 30, 161–175, 1993.

Duschlbauer D.: Computational Simulation of the Thermal Conductivity of MMCs un-der Consideration of the Inclusion–Matrix Interface. VDI–Verlag (Reihe 5, Nr. 561),Dusseldorf, Germany, 2004.

– 117 –

Duschlbauer D., Pettermann H.E., Bohm H.J.: Mori–Tanaka Based Evaluation of Inclu-sion Stresses in Composites with Nonaligned Reinforcements; Scr.mater. 48, 223–228,2003.

Duschlbauer D., Pettermann H.E., Bohm H.J.: Heat Conduction of a Spheroidal Inho-mogeneity with Imperfectly Bonded Interface; J.Appl.Phys. 94, 1539–1549, 2003.

Dvorak G.J.: Plasticity Theories for Fibrous Composite Materials; in “Metal MatrixComposites: Mechanisms and Properties” (Eds. R.K.Everett, R.J.Arsenault), pp. 1–77;Academic Press, Boston, MA, 1991.

Dvorak G.J.: Transformation Field Analysis of Inelastic Composite Materials;Proc.Roy.Soc.London A437, 311–327, 1992.

Dvorak G.J., Bahei-el-Din Y.A.: A Bimodal Plasticity Theory of Fibrous CompositeMaterials; Acta Mech. 69, 219–241, 1987.

Dvorak G.J., Benveniste Y.: On Transformation Strains and Uniform Fields in Multi-phase Elastic Media; Proc.Roy.Soc.London A437, 291–310, 1992.

Dvorak G.J., Bahei-el-Din Y.A., Wafa A.M.: Implementation of the Transformation FieldAnalysis for Inelastic Composite Materials; Comput.Mech. 14, 201–228, 1994.

Eckschlager A.: Simulation of Particle Failure in Particle Reinforced Ductile MatrixComposites. Doctoral Thesis, Vienna University of Technology, Vienna, Austria, 2002.

Eckschlager A., Bohm H.J., Han W.: Modeling of Brittle Particle Failure in ParticleReinforced Ductile Matrix Composites by 3D Unit Cells; in “Proceedings of the 13thInternational Conference on Composite Materials (ICCM–13)” (Ed. Y.Zhang), PaperNo. 1342; Scientific and Technical Documents Publishing House, Beijing, 2001.

Eduljee R.F., McCullough R.L.: Elastic Properties of Composites; in “Materials Sci-ence and Technology Vol.13: Structure and Properties of Composites” (Eds. R.W.Cahn,P.Haasen, E.J.Kramer), pp. 381–474; VCH, Weinheim, Germany, 1993.

Elices M., Guinea G.V., Gomez J., Planas J.: The Cohesive Zone Model: Advantages,Limitations and Challenges; Engng.Fract.Mech. 69, 137–163, 2002.

Eshelby J.D.: The Determination of the Elastic Field of an Ellipsoidal Inclusion andRelated Problems; Proc.Roy.Soc.London A241, 376–396, 1957.

Eshelby J.D.: The Elastic Field Outside an Ellipsoidal Inclusion; Proc.Roy.Soc.LondonA252, 561–569, 1959.

Estevez R., Maire E., Franciosi P., Wilkinson D.S.: Effect of Particle Clustering on theStrengthening versus Damage Rivalry in Particulate Reinforced Elastic Plastic Materials:A 3-D Analysis from a Self-Consistent Modelling; Eur.J.Mech. A/Solids 18, 785–804,1999

Estrin Y., Arndt S., Heilmaier M., Brechet Y.: Deformation Behavior of Particle-Strengthened Alloys: A Voronoi Mesh Approach; Acta mater. 47, 595–606, 1999.

Faleskog J., Shih C.F.: Micromechanics of Coalescence — I.: Synergistic Effects ofElasticity, Plastic Yielding and Multi-Size-Scale Voids; J.Mech.Phys.Sol. 45, 21–50,1997.

Ferrari M.: Asymmetry and the High Concentration Limit of the Mori–Tanaka EffectiveMedium Theory; Mech.Mater. 11, 251–256, 1991.

Feyel F.: A Multilevel Finite Element Method (FE2) to Describe the Response of HighlyNon-Linear Structures Using Generalized Continua; Comput.Meth.Appl.Mech.Engng.192, 3233–3244, 2003.

Feyel F., Chaboche J.L.: FE2 Multiscale Approach for Modelling the ElastoviscoplasticBehaviour of Long Fiber SiC/Ti Composite Materials; Comput.Meth.Appl.Mech.Engng.183, 309–330, 2000.

– 118 –

Finot M.A., Shen Y.L., Needleman A., Suresh S.: Micromechanical Modeling of Re-inforcement Fracture in Particle-Reinforced Metal-Matrix Composites; Metall.Trans.A25A, 2403–2420, 1994.

Fischer F.D., Kolednik O., Shan G.X., Rammerstorfer F.G.: A Note on Calibration ofDuctile Failure Damage Indicators; Int.J.Fract. 73, 345–357, 1995.

Fischmeister H., Karlsson B.: Plastizitatseigenschaften grob-zweiphasiger Werkstoffe;Z.Metallk. 68, 311–327, 1977.

Fish J., Shek K.: Computational Plasticity and Viscoplasticity for Composite Materialsand Structures; Composites 29B, 613–619, 1998.

Fish J., Shek K.: Multiscale Analysis of Composite Materials and Structures; Com-pos.Sci.Technol. 60, 2547–2556, 2000.

Fish J., Shephard M.S., Beall M.W.: Automated Multiscale Fracture Analysis; in “Dis-cretization Methods in Structural Mechanics” (Eds. H.A.Mang, F.G.Rammerstorfer), pp.249–256; Kluwer, Dordrecht, The Netherlands, 1999.

Fish J., Shek K., Pandheeradi M., Shephard M.S.: Computational Plasticity for Com-posite Structures Based on Mathematical Homogenization: Theory and Practice; Com-put.Meth.Appl.Mech.Engng. 148, 53–73, 1997.

Fond C., Riccardi A., Schirrer R., Montheillet F.: Mechanical Interaction between Spher-ical Inhomogeneities: An Assessment of a Method Based on the Equivalent Inclusion;Eur.J.Mech. A/Solids 20, 59–75, 2001.

Fu S.Y., Lauke B.: The Elastic Modulus of Misaligned Short-Fiber-Reinforced Polymers;Compos.Sci.Technol. 58, 389–400, 1998.

Fukuda H., Chou T.W.: A Probabilistic Theory of the Strength of Short-Fibre Compositeswith Variable Fibre Length and Orientation; J.Mater.Sci. 17, 1003–1007, 1982.

Fukuda H., Kawata K.: On Young’s Modulus of Short Fiber Composites; FibreSci.Technol. 7, 207–222, 1974.

Galli M., Botsis J., Janczak–Rusch J.: An Elastoplastic Three-Dimensional Homogeniza-tion Model for Particle Reinforced Composites; Comput.Mater.Sci. 41, 312–321, 2008.

Ganser H.P.: Large Strain Behavior of Two-Phase Materials. VDI–Verlag (Reihe 5, Nr.528), Dusseldorf, Germany, 1998.

Ganser H.P., Fischer F.D., Werner E.A.: Large Strain Behaviour of Two-Phase Materialswith Random Inclusions; Comput.Mater.Sci. 11, 221–226, 1998.

Ganser H.P., Atkins A.G., Kolednik O., Fischer F.D., Richard O.: Upsetting of Cylinders:A Comparison of Two Different Damage Indicators; J.Engng.Mater.Technol. 123, 94–99, 2001.

Garnich M.R., Karami G.: Finite Element Micromechanics for Stiffness and Strength ofWavy Fiber Composites, J.Compos.Mater. 38, 273–292, 2004.

Garajeu M.: Contribution a l’etude du comportement non lineaire de milieux poreux avecou sens renfort. Doctoral Thesis, Universite de la Mediteranee, Marseille, France, 1995.

Garajeu M., Michel J.C., Suquet P.: A Micromechanical Approach of Damage inViscoplastic Materials by Evolution in Size, Shape and Distribution of Voids; Com-put.Meth.Appl.Mech.Engng. 183, 223–246, 2000.

Gavazzi A.C., Lagoudas D.C.: On the Numerical Evaluation of Eshelby’s Tensor and itsApplication to Elastoplastic Fibrous Composites; Comput.Mech. 7, 12–19, 1990.

– 119 –

Geers M.G.D., Kouznetsova V., Brekelmans W.A.M.: Micro–Macro Scale Transitionsfor Engineering Materials in the Presence of Size Effects; in “Trends in ComputationalStructural Mechanics” (Eds. W.A.Wall, K.U.Bletzinger, K.Schweizerhof), pp. 118–127;CIMNE, Barcelona, Spain, 2001.

Geers M.G.D., Engelen R.A.B., Ubachs R.J.M.: On the Numerical Modelling of DuctileDamage with an Implicit Gradient-Enhanced Formulation; Rev.Eur.Elem.Fin. 10, 173–191, 2001.

Geni M., Kikuchi M.: Void Configuration under Constrained Deformation in DuctileMatrix Materials; Comput.Mater.Sci. 16, 391–403, 1999.

Ghosh S., Moorthy S.: Particle Fracture Simulation in Non-Uniform Microstructures ofMetal-Matrix Composites; Acta mater. 46, 965–982, 1998.

Ghosh S., Moorthy S.: Three Dimensional Voronoi Cell Finite Element Model for Mi-crostructures with Ellipsoidal Heterogeneities; Comput.Mech. 34, 510–531, 2004.

Ghosh S., Lee K.H., Moorthy S.: Two Scale Analysis of Heterogeneous Elastic-PlasticMaterials with Asymptotic Homogenization and Voronoi Cell Finite Element Model;Comput.Meth.Appl.Mech.Engng. 132, 63–116, 1996.

Ghosh S., Lee K.H., Raghavan P.: A Multi-Level Computational Model for Multi-ScaleAnalysis in Composite and Porous Materials; Int.J.Sol.Struct. 38, 2335–2385, 2001.

Ghosh S., Nowak Z., Lee K.H.: Quantitative Characterization and Modeling of CompositeMicrostructures by Voronoi Cells; Acta mater. 45, 2215–2234, 1997.

Gibson L.J.: The Mechanical Behavior of Cancellous Bone; J.Biomech. 18, 317–328,1985.

Gibson L.J., Ashby M.F.: Cellular Solids: Structure and Properties. Pergamon Press,Oxford, UK, 1988.

Gilormini P.: Insuffisance de l’extension classique du modele auto-coherent au comporte-ment non-lineaire; C.R.Acad.Sci.Paris, serie IIb 320, 115–122, 1995.

Gitman I.M., Askes H., Sluys L.J.: Representative Volume: Existence and Size Determi-nation; Engng.Fract.Mech. 74, 2518–2534, 2007.

Gologanu M., Leblond J.P., Perrin G., Devaux J.: Recent Extensions of Gurson’s Modelfor Porous Ductile Materials; in “Continuum Micromechanics” (Ed. P.Suquet), pp. 61–130; Springer–Verlag, Vienna, Austria, 1997.

Gommers B., Verpoest I., van Houtte P.: The Mori–Tanaka Method Applied to TextileComposite Materials; Acta mater. 46, 2223–2235, 1998.

Gong L., Kyriakides S., Triantafyllidis N.: On the Stability of Kelvin Cell Foams underCompressive Loads; J.Mech.Phys.Sol. 53, 771–794, 2005.

Gonzalez C., LLorca J.: Prediction of the Tensile Stress–Strain Curve and Ductility inSiC/Al Composites; Scr.mater. 35, 91–97, 1996.

Gonzalez C., LLorca J.: A Self-Consistent Approach to the Elasto-Plastic Behavior ofTwo-Phase Materials Including Damage; J.Mech.Phys.Sol. 48, 675–692, 2000.

Graham L.L., Baxter S.C.: Simulation of Local Material Properties Based on Moving-Window GMC; Prob.Engng.Mech. 16, 295–305, 2001.

Grenestedt J.L.: Influence of Wavy Imperfections in Cell Walls on Elastic Stiffness ofCellular Solids; J.Mech.Phys.Sol. 46, 29–50, 1998.

Gross D., Seelig T.: Bruchmechanik mit einer Einfuhrung in die Mikromechanik.Springer–Verlag, Berlin, 2001.

– 120 –

Guidoum A.: Simulation numerique 3D des comportements des betons en tant que com-posites granulaires. Doctoral Thesis, Ecole Polytecnique Federale de Lausanne, Lau-sanne, Switzerland, 1994.

Guldberg R.E., Hollister S.J., Charras G.T.: The Accuracy of Digital Image-Based FiniteElement Models; J.Biomech.Engng. 120, 289–295, 1998.

Gunawardena S.R., Jansson S., Leckie F.A.: Transverse Ductility of Metal MatrixComposites; in “Failure Mechanisms in High Temperature Composite Materials” (Eds.K.Haritos, G.Newaz, S.Mall), pp. 23–30; AD–Vol.22/AMD–Vol.122, ASME, New York,NY, 1991.

Gunawardena S.R., Jansson S., Leckie F.A.: Modeling of Anisotropic Behavior of WeaklyBonded Fiber Reinforced MMC’s; Acta metall.mater. 41, 3147–3156, 1993.

Guo G., Fitoussi J., Baptiste D., Sicot N., Wolff C.: Modelling of Damage Behavior ofa Short-Fiber Reinforced Composite Structure by the Finite Element Analysis Using aMicro–Macro Law; Int.J.Dam.Mech. 6, 278–299, 1997.

Gurson A.L.: Continuum Theory of Ductile Rupture by Void Nucleation andGrowth: Part I — Yield Criteria and Flow Rules for Porous Ductile Media;J.Engng.Mater.Technol. 99, 2–15, 1977.

Gusev A.A.: Representative Volume Element Size for Elastic Composites: A NumericalStudy; J.Mech.Phys.Sol. 45, 1449–1459, 1997.

Gusev A.A.: Numerical Identification of the Potential of Whisker- and Platelet-FilledPolymers; Macromolecules 34, 3081–3093, 2001.

Gusev A.A., Hine P.J., Ward I.M.: Fiber Packing and Elastic Properties of a Trans-versely Random Unidirectional Glass/Epoxy Composite; Compos.Sci.Technol. 60, 535–541, 2000.

Haddadi H., Teodosiu C.: 3D-Analysis of the Effect of Interfacial Debonding on thePlastic Behaviour of Two-Phase Composites; Comput.Mater.Sci. 16, 315–322, 1999.

Halpin J.C., Kardos J.L.: The Halpin–Tsai Equations: A Review; Polym.Engng.Sci. 16,344–351, 1976.

Hancock J.W., Mackenzie A.C.: On the Mechanisms of Ductile Failure in High-StrengthSteels Subjected to Multi-Axial Stress-States; J.Mech.Phys.Sol. 24, 147–169, 1976.

Hashin Z.: The Elastic Moduli of Heterogeneous Materials; J.Appl.Mech. 29, 143–150,1962.

Hashin Z.: Analysis of Composite Materials — A Survey; J.Appl.Mech. 50, 481–505,1983.

Hashin Z.: The Differential Scheme and its Application to Cracked Materials;J.Mech.Phys.Sol. 36, 719–733, 1988.

Hashin Z., Rosen B.W.: The Elastic Moduli of Fiber-Reinforced Materials; J.Appl.Mech.31, 223–232, 1964.

Hashin Z., Shtrikman S.: On Some Variational Principles in Anisotropic and Nonhomo-geneous Elasticity; J.Mech.Phys.Sol. 10, 335–342, 1962.

Hashin Z., Shtrikman S.: A Variational Approach to the Theory of the Elastic Behaviorof Multiphase Materials; J.Mech.Phys.Sol. 11, 127–140, 1963.

Hassani B., Hinton E.: Homogenization and Structural Topology Optimization. Springer–Verlag, London, UK, 1999.

Hatta H., Taya M.: Equivalent Inclusion Method for Steady State Heat Conduction inComposites; Int.J.Engng.Sci. 24, 1159–1172, 1986.

– 121 –

Hazanov S.: Hill Condition and Overall Properties of Composites; Arch.Appl.Mech. 68,385–394, 1998.

Hazanov S., Huet C.: Order Relationships for Boundary Condition Effects in Heteroge-neous Bodies Smaller than the Representative Volume; J.Mech.Phys.Sol. 41, 1995–2011,1994.

Heness G.L., Ben-Nissan B., Gan L.H., Mai Y.W.: Development of a Finite ElementMicromodel for Metal Matrix Composites; Comput.Mater.Sci. 13, 259–269, 1999.

Herve E., Zaoui A.: n-Layered Inclusion-Based Micromechanical Modelling; Int.J.Engng.Sci. 31, 1–10, 1993.

Herve E., Stolz C., Zaoui A.: A propos de’l assemblage des spheres composites de Hashin;C.R.Acad.Sci.Paris, serie II 313, 857–862, 1991.

Hill R.: The Elastic Behaviour of a Crystalline Aggregate. Proc.Phys.Soc. A65, 349–354,1952.

Hill R.: Elastic Properties of Reinforced Solids: Some Theoretical Principles;J.Mech.Phys.Sol. 11, 357–372, 1963.

Hill R.: Theory of Mechanical Properties of Fibre-Strengthened Materials: I. ElasticBehaviour; J.Mech.Phys.Sol. 12, 199–212, 1964.

Hill R.: A Self Consistent Mechanics of Composite Materials; J.Mech.Phys.Sol. 13,213–222, 1965.

Hill R.: The Essential Structure of Constitutive Laws for Metal Composites and Poly-crystals; J.Mech.Phys.Sol. 15, 79–95, 1967.

Hollister S.J., Brennan J.M., Kikuchi N.: A Homogenization Sampling Procedure forCalculating Trabecular Bone Effective Stiffness and Tissue Level Stress; J.Biomech. 27,433–444, 1994.

Hollister S.J., Fyhrie D.P., Jepsen K.J., Goldstein S.A.: Application of HomogenizationTheory to the Study of Trabecular Bone Mechanics; J.Biomech. 24, 825–839, 1991.

Holmberg S., Persson K., Petersson H.: Nonlinear Mechanical Behaviour and Analysisof Wood and Fibre Materials; Comput.Struct. 72, 459–480, 1999.

Hom C.L., McMeeking R.M.: Plastic Flow in Ductile Materials Containing a CubicArray of Rigid Spheres; Int.J.Plast. 7, 255–274, 1991.

Horstemeyer M.F., Matalanis M.M., Sieber A.M., Botos M.L.: Micromechanical FiniteElement Calculations of of Temperature and Void Configuration Effects on Void Growthand Coalescence; Int.J.Plast. 16, 979–1013, 2000.

Hu C., Moorthy S., Ghosh S.: A Voronoi Cell Finite Element for Ductile Damage inMMCs; in “Proceedings of NUMIFORM 2004” (Eds. S.Ghosh, J.M.Castro, J.K.Lee), pp.1893–1898; American Institute of Physics, Melville, NY, 2004.

Hu C., Bai J., Ghosh S.: Micromechanical and Macroscopic Models of Ductile Fracture inParticle Reinforced Metallic Materials; Modell.Simul.Mater.Sci.Engng. 15, S377–S392,2007.

Hu G.K.: A Method of Plasticity for General Aligned Spheroidal Void or Fiber-ReinforcedComposites; Int.J.Plast. 12, 439–449, 1996.

Hu G.K., Weng G.J.: The Connections between the Double-Inclusion Model and thePonte Castaneda–Willis, Mori–Tanaka, and Kuster–Toksoz Models; Mech.Mater. 32,495–503, 2000.

Hu G.K., Weng G.J.: Some Reflections on the Mori–Tanaka and Ponte Castaneda–WillisMethods with Randomly Oriented Ellipsoidal Inclusions; Acta Mech. 140, 31–40, 2000.

– 122 –

Hu G.K., Guo G., Baptiste D.: A Micromechanical Model of Influence of Particle Frac-ture and Particle Cluster on Mechanical Properties of Metal Matrix Composites; Com-put.Mater.Sci. 9, 420–430, 1998.

Huang J.H.: Some Closed-Form Solutions for Effective Moduli of Composites ContainingRandomly Oriented Short Fibers; Mater.Sci.Engng. A315, 11–20, 2001.

Huang J.H., Yu J.S.: Electroelastic Eshelby Tensors for an Ellipsoidal Piezoelectric In-clusion; Compos.Engng. 4, 1169–1182, 1994.

Huang Y., Hu K.X.: A Generalized Self-Consistent Mechanics Method for Solids Con-taining Elliptical Inclusions; J.Appl.Mech. 62, 566–572, 1995.

Huet C.: Application of Variational Concepts to Sizes Effects in Elastic HeterogeneousBodies; J.Mech.Phys.Sol. 38, 813–841, 1990.

Huet C.: Coupled Size and Boundary-Condition Effects in Viscoelastic Heterogeneousand Composite Bodies; Mech.Mater. 31, 787–829, 1999.

Huet C., Navi P., Roelfstra P.E.: A Homogenization Technique Based on Hill’s Modifi-cation Theorem; in “Continuum Models and Discrete Systems” (Ed. G.A.Maugin), pp.135–143; Longman, Harlow, UK, 1991.

Hutchinson J.W.: Elastic-Plastic Behavior of Polycrystalline Metals and Composites;Proc.Roy.Soc.London A319, 247–272, 1970.

Hutchinson J.W.: Plasticity on the Micron Scale; Int.J.Sol.Struct. 37, 225–238, 2000.

Ibrahimbegovic A., Markovic D.: Strong Coupling Methods in Multi-Phase andMulti-Scale Modeling of Inelastic Behavior of Heterogeneous Structures; Com-put.Meth.Appl.Mech.Engng. 192, 3089–3107, 2003.

Ingber M.S., Papathanasiou T.D.: A Parallel-Supercomputing Investigation of the Stiff-ness of Aligned Short-Fiber-Reinforced Composites Using the Boundary Element Method;Int.J.Num.Meth..Engng. 40, 3477–3491, 1997.

Ishikawa N., Parks D., Socrate S., Kurihara M.: Micromechanical Modeling of Ferrite–Pearlite Steels Using Finite Element Unit Cell Models; ISIJ Int. 40, 1170–1179, 2000.

Ismar H., Reinert U.: Modelling and Simulation of the Macromechanical Nonlinear Be-havior of Fibre-Reinforced Ceramics on the Basis of a Micromechanical–Statistical Ma-terial Description; Acta Mech. 120, 47–60, 1997.

Ismar H., Schmitt U.: Simulation von Einspiel- und Rißbildungsvorgangen in Faserver-bundwerkstoffen mit metallischer Matrix; Arch.Appl.Mech. 61, 18–29, 1991.

Ismar H., Schroter F.: Three-Dimensional Finite Element Analysis of the MechanicalBehavior of Cross Ply-Reinforced Aluminum; Mech.Mater. 32, 329–338, 2000.

Iung T., Grange M.: Mechanical Behavior of Two-Phase Materials Investigated by theFinite Element Method: Necessity of Three-Dimensional Modeling; Mater.Sci.Engng.A201, L8–L11, 1995.

Iyer S.K., Lissenden C.J., Arnold S.M.: Local and Overall Flow in Composites Predictedby Micromechanics; Composites 31B, 327–343, 2000.

Jansson S.: Homogenized Nonlinear Constitutive Properties and Local Stress Concentra-tions for Composites with Periodic Internal Structure; Int.J.Sol.Struct. 29, 2181–2200,1992.

Jarvstrat N.: Homogenization and the Mechanical Behavior of Metal Composites; in“Residual Stresses – III” (Eds. H.Fujiwara, T.Abe, K.Tanaka), Vol.1, pp. 70–75; Elsevier,London, UK, 1992.

Jarvstrat N.: An Ellipsoidal Unit Cell for the Calculation of Micro-Stresses in ShortFibre Composites; Comput.Mater.Sci. 1, 203–212;

– 123 –

Jayaraman K., Kortschot M.T.: Correction to the Fukuda–Kawata Young’s Modulus The-ory and the Fukuda–Chou Strength Theory for Short Fibre-Reinforced Composite Mate-rials; J.Mater.Sci. 31, 2059–2064, 1996.

Jeulin D.: Random Structure Models for Homogenization and Fracture Statistics; in“Mechanics of Random and Multiscale Microstructures” (Eds. D.Jeulin, M.Ostoja–Starzewski), pp. 33–91; Springer–Verlag, Vienna, 2001.

Jiang M., Ostoja-Starzewski M., Jasiuk I.: Scale-Dependent Bounds on Effective Elasto-plastic Response of Random Composites; J.Mech.Phys.Sol. 49, 655–673, 2001.

Jiang M., Ostoja-Starzewski M., Jasiuk I.: Apparent Elastic and Elastoplastic Behaviorof Periodic Composites; Int.J.Sol.Struct. 39, 199–212, 2002.

Jirasek M.: Comparative Study on Finite Elements with Embedded Discontinuities; Com-put.Meth.Appl.Mech.Engng. 188, 307–330, 2000.

Jirasek M., Rolshoven S.: Comparison of Integral-Type Nonlocal Plasticity Models forStrain-Softening Materials; Int.J.Engng.Sci. 41, 1553–1602, 2003.

Johannesson B., Pedersen O.B.: Analytical Determination of the Average Eshelby Tensorfor Transversely Isotropic Fiber Orientation Distributions; Acta mater. 46, 3165–3173,1998.

Ju J.W., Sun L.Z.: A Novel Formulation for the Exterior Point Eshelby’s Tensor of anEllipsoidal Inclusion; J.Appl.Mech. 66, 570–574, 1999.

Ju J.W., Sun L.Z.: Effective Elastoplastic Behavior of Metal Matrix Composites Con-taining Randomly Located Aligned Spheroidal Inhomogeneities. Part I: Micromechanics-Based Formulation; Int.J.Sol.Struct. 38, 183–201, 2001.

Kachanov M., Sevostianov I.: On Quantitative Characterization of Microstructures andEffective Properties; Int.J.Sol.Struct. 42, 309–336, 2005.

Kachanov M., Tsukrov I., Shafiro B.: Effective Moduli of Solids with Cavities of VariousShapes; Appl.Mech.Rev. 47, S151–S174, 1994.

Kailasam M., Ponte Castaneda P.: A General Constitutive Theory for a Linear andNonlinear Particulate Media with Microstructure Evolution; J.Mech.Phys.Sol. 46, 427–465, 1998.

Kailasam M., Aravas N., Ponte Castaneda P.: Porous Metals with Developing Anisotropy:Constitutive Models, Computational Issues and Applications to Deformation Processing;Comput.Model.Engng.Sci. 1, 105–118, 2000.

Kanit T., Forest S., Gallier I., Mounoury V., Jeulin D.: Determination of the Size ofthe Representative Volume Element for Random Composites: Statistical and NumericalApproach; Int.J.Sol.Struct. 40, 3647–3679, 2003.

Kanit T., N’Guyen F., Forest S., Jeulin D., Reed M., Singleton S.: Apparent and Effec-tive Physical Properties of Heterogeneous Materials: Representativity of Samples of TwoMaterials from Food Industry; Comput.Meth.Appl.Mech.Engng. 195, 3960–3982, 2006.

Karayaka M., Sehitoglu H.: Thermomechanical Deformation Modeling of Al2xxx-T4/SiCp Composites; Acta metall.mater. 41, 175–189, 1993.

Kenesei P., Biermann H., Borbely A.: Structure–Property Relationship in Particle Re-inforced Metal-Matrix Composites Based on Holotomography; Scr.mater. 53, 787–791,2005.

Kenesei P., Biermann H., Borbely A.: Estimation of Elastic Properties of Particle Re-inforced Metal-Matrix Composites Based on Tomographic Images; Adv.Engng.Mater. 8,500–506, 2006.

– 124 –

Khisaeva Z.F., Ostoja-Starzewski M.: Mesoscale Bounds in Finite Elasticity and Ther-moelasticity of Random Composites; Proc.Roy.Soc.London A462, 1167–1180, 2006.

Khisaeva Z.F., Ostoja-Starzewski M.: On the Size of RVE in Finite Elasticity of RandomComposites; J.Elast. 85, 153–173, 2006.

Kouznetsova V.G., Geers M.G.D., Brekelmans W.A.M.: Multi-Scale Constitutive Mod-eling of Heterogeneous Materials with a Gradient-Enhanced Computational Homogeniza-tion Scheme; Int.J.Num.Meth.Engng. 54, 1235–1260, 2002.

Krakovsky I., Myroshnychenko V.: Modelling Dielectric Properties of Composites byFinite Element Method; J.Appl.Phys. 92, 6743–6748, 2002.

Kraynik A.M., Reinelt D.A.: Linear Elastic Behavior of Dry Soap Foams; J.ColloidInterf.Sci. 181, 511–520, 1996.

Kreher W.S.: Residual Stresses and Stored Elastic Energy of Composites and Polycrys-tals; J.Mech.Phys.Sol. 38, 115–128, 1990.

Kroner E.: Self-Consistent Scheme and Graded Disorder in Polycrystal Elasticity;J.Phys.F 8, 2261–2267, 1978.

Kuster G.T., Toksoz M.N.: Velocity and Attenuation of Seismic Waves in Two-PhaseMedia: I. Theoretical Formulation; Geophysics 39, 587–606, 1974.

Lagoudas D.C., Gavazzi A.C., Nigam H.: Elastoplastic Behavior of Metal Matrix Com-posites Based on Incremental Plasticity and the Mori–Tanaka Averaging Scheme; Com-put.Mech. 8, 193–203, 1991.

Lee H.K., Simunovic S.: Modeling of Progressive Damage in Aligned and Randomly Ori-ented Discontinuous Fiber Polymer Matrix Composites; Composites 31B, 77–86, 2000.

Lee K.H., Ghosh S.: Small Deformation Multi-Scale Analysis of Heterogeneous Mate-rials with the Voronoi Cell Finite Element Model and Homogenization Theory; Com-put.Mater.Sci. 7, 131–146, 1996.

Lerch B.A., Melis M.E., Tong M.: Experimental and Analytical Analysis of the Stress–Strain Behavior in a [90/0]2S SiC/Ti-15-3 Laminate. NASA Technical MemorandumNASA–TM–104470, 1991.

Levin V.M.: On the Coefficients of Thermal Expansion of Heterogeneous Materials;Mech.Sol. 2, 58–61, 1967.

Levy A., Papazian J.M.: Elastoplastic Finite Element Analysis of Short-Fiber-ReinforcedSiC/Al Composites: Effects of Thermal Treatment; Acta metall.mater. 39, 2255–2266,1991.

Li D.S., Wisnom M.R.: Unidirectional Tensile Stress–Strain Response of BP-SiC FiberReinforced Ti-6Al-4V; J.Compos.Technol.Res. 16, 225–233, 1994.

Li M., Ghosh S., Richmond O., Weiland H., Rouns T.N.: Three Dimensional Character-ization and Modeling of Particle Reinforced Metal Matrix Composites, Part II: DamageCharacterization; Mater.Sci.Engng. A266, 221–240, 1999.

Li S.G., Wongsto A.: Unit Cells for Micromechanical Analyses of Particle-ReinforcedComposites; Mech.Mater. 36, 543–572, 2004.

Li Y.Y., Cui J.Z.: The Multi-Scale Computational Method for the Mechanics Parametersof the Materials with Random Distribution of Multi-Scale Grains; Compos.Sci.Technol.65, 1447–1458, 2005.

Liebowitz H., Sandhu J.S., Lee J.D., Menandro F.C.M.: Computational Fracture Me-chanics: Research and Application; Engng.Fract.Mech. 50, 653–670, 1995.

– 125 –

Lielens G., Pirotte P., Couniot A., Dupret F., Keunings R.: Prediction of Thermo-Mechanical Properties for Compression-Moulded Composites; Composites 29A, 63–70,1997.Liu D.S., Chen C.Y., Chiou D.Y.: 3-D Modeling of a Composite Material Rein-forced with Multiple Thickly Coated Particles Using the Infinite Element Method; Com-put.Model.Engngn.Sci. 9, 179–192, 2005.

Liu L., James R.D., Leo P.H.: Periodic Inclusion–Matrix Microstructures with ConstantField Inclusions; Metall.Mater.Trans. 38A, 781–787, 2007.

Liu Y.J., Nishimura N., Otani Y., Takahashi T., Chen X.L., Munakata H.: A FastBoundary Element Method for the Analysis of Fiber-Reinforced Composites Based onRigid-Inclusion Model; J.Appl.Mech. 72, 115–128, 2005.

LLorca J.: A Numerical Analysis of the Damage Mechanisms in Metal-Matrix Compositesunder Cyclic Deformation; Comput.Mater.Sci. 7, 118–122, 1996.

Lubarda V.A., Markenscoff X.: On the Absence of Eshelby Property for Non-EllipsoidalInclusions; Int.J.Sol.Struct. 35, 3405–3411, 1998.

Lusti H.R., Hine P.J., Gusev A.A.: Direct Numerical Predictions for the Elastic andThermoelastic Properties of Short Fibre Composites; Compos.Sci.Technol. 62, 1927–1934, 2002.

Luxner M.H., Stampfl J., Pettermann H.E.: Finite Element Modeling Concepts andLinear Analyses of 3D Regular Open Cell Structures; J.Mater.Sci.40, 5859–5866, 2005.

Mahishi J.M.: An Integrated Micromechanical and Macromechanical Approach to Frac-ture Behavior of Fiber-Reinforced Composites; Engng.Fract.Mech. 25, 197–228, 1986.

Mahishi J.M., Adams D.F.: Micromechanical Predictions of Crack Initiation, Propaga-tion and Crack Growth Resistance in Boron/Aluminum Composites; J.Compos.Mater.16, 457–469, 1982.

Maire E., Wattebled F., Buffiere J.Y., Peix G.: Deformation of a Metallic Foam Studiedby X-Ray Computed Tomography and Finite Element Calculations; in “Metal MatrixComposites and Metallic Foams” (Eds. T.W.Clyne, F.Simancik), pp. 68–73; Wiley–VCH,Weinheim, Germany, 2000.

Mandel J.: Une generalisation de la theorie de la plasticite de W.T. Koiter;Int.J.Sol.Struct. 1, 273–295, 1965.

Marketz F., Fischer F.D.: Micromechanical Modelling of Stress-Assisted MartensiticTransformation; Modell.Simul.Mater.Sci.Engng. 2, 1017–1046, 1994.

Markov K.: Elementary Micromechanics of Heterogeneous Media; in “Heterogeneous Me-dia: Micromechanics Modeling Methods and Simulations” (Eds. K.Markov, L.Preziosi),pp. 1–162; Birkhauser, Boston, MA, 2000.

Markov K., Preziosi L. (Eds.): Heterogeneous Media: Micromechanics Modeling Methodsand Simulations. Birkhauser, Boston, MA, 2000.

Masson R., Bornert M., Suquet P., Zaoui A.: An Affine Formulation for the Predictionof the Effective Properties of Nonlinear Composites and Polycrystals; J.Mech.Phys.Sol.48, 1203–1227, 2000.

Mawsouf N.M.: A Micromechanical Mechanism of Fracture Initiation in DiscontinuouslyReinforced Metal Matrix Composite; Mater.Charact. 44, 321–327, 2000.

McClintock F.A.: A Criterion for Ductile Fracture by the Growth of Holes; J.Appl.Mech.35, 363–371, 1968.

McDowell D.L.: Modeling and Experiments in Plasticity; Int.J.Sol.Struct. 37, 293–310,2000.

– 126 –

McHugh P.E., Connolly P.: Modelling the Thermo-Mechanical Behavior of an Al Alloy-SiCp Composite. Effects of Particle Shape and Microscale Failure; Comput.Mater.Sci.3, 199–206, 1994.

McHugh P.E., Asaro R.J., Shih C.F.: Computational Modeling of Metal-Matrix Compos-ite Materials, Parts I to IV; Acta metall.mater. 41, 1461–1510, 1993.

McLaughlin R.: A Study of the Differential Scheme for Composite Materials;Int.J.Engng.Sci. 15, 237–244, 1977.

McMeeking R., Hom C.L.: Finite Element Analysis of Void Growth in Elastic-PlasticMaterials; Int.J.Fract. 42, 1–19, 1990.

Michel J.C.: Theorie des modules effectifs. Approximations de Voigt et de Reuss; in“Homogeneisation en mecanique des materiaux 1. Materiaux aleatoires elastiques etmilieux periodiques” (Eds. M.Bornert, T.Bretheau, P.Gilormini), pp. 41–56; EditionsHermes, Paris, 2001.

Michel J.C., Suquet P.: An Analytical and Numerical Study of the Overall Behaviour ofMetal-Matrix Composites; Modell.Simul.Mater.Sci.Engng. 2, 637–658, 1994.

Michel J.C., Suquet P.: Computational Analysis of Nonlinear Composite Structures Usingthe Nonuniform Transformation Field Analysis; Comput.Meth.Appl.Mech.Engng. 193,5477–5502, 2004.

Michel J.C., Moulinec H., Suquet P.: Effective Properties of Composite Materials withPeriodic Microstructure: A Computational Approach; Comput.Meth.Appl.Mech.Engng.172, 109–143, 1999.

Michel J.C., Moulinec H., Suquet P.: Composites a microstructure periodique; in “Ho-mogeneisation en mecanique des materiaux 1. Materiaux aleatoires elastiques et milieuxperiodiques” (Eds. M.Bornert, T.Bretheau, P.Gilormini), pp. 57–94; Editions Hermes,Paris, 2001.

Miehe C., Schroder J., Schotte J.: Computational Homogenization Analysis in Fi-nite Plasticity. Simulation of Texture Development in Polycrystalline Materials; Com-put.Meth.Appl.Mech.Engng. 171, 387–418, 1999.

Miller C.A., Torquato S.: Effective Conductivity of Hard Sphere Suspensions;J.Appl.Phys. 68, 5486–5493, 1990.

Miloh T., Benveniste Y.: A Generalized Self-Consistent Method for the Effective Conduc-tivity of Composites with Ellipsoidal Inclusions and Cracked Bodies; J.Appl.Phys. 63,789–7796, 1988.

Milton G.W.: Bounds on the Electromagnetic, Elastic, and Other Properties of Two-Component Composites; Phys.Rev.Lett. 46, 542–545, 1981.

Milton G.W.: The Theory of Composites. Cambridge University Press, Cambridge, UK,2002.

Miserez A., Rossoll A., Mortensen A.: Fracture of Aluminium Reinforced with DenselyPacked Ceramic Particles: Link between the Local and the Total Work of Fracture; Actamater. 52, 1337–1351, 2004.

Mishnaevsky L.L.: Three-Dimensional Numerical Testing of Microstructures of ParticleReinforced Composites; Acta mater. 52, 4177–4188, 2004.

Missoum-Benziane D., Ryckelynck D., Chinesta F.: A New Fully Coupled Two-ScalesModelling for Mechanical Problems Involving Microstructure: The 95/5 Technique; Com-put.Meth.Appl.Mech.Engng. 196, 2325–2337, 2007.

Mlekusch B.: Thermoelastic Properties of Short-Fibre-Reinforced Thermoplastics; Com-pos.Sci.Technol. 59, 911–923, 1999.

– 127 –

Moes N., Cloirec M., Cartraud P., Remacle J.F.: A Computational Approach to HandleComplex Microstructure Geometries; Comput.Meth.Appl.Mech.Engng. 192, 3163–3177,2003.

Molinari A., Canova G.R., Ahzi S.: A Self-Consistent Approach for Large DeformationViscoplasticity; Acta metall. 35, 2983–2984, 1987.

Monaghan J., Brazil D.: Modeling the Sub-Subsurface Damage Associated with the Ma-chining of a Particle Reinforced MMC; Comput.Mater.Sci. 9, 99–107, 1997.

Mori T., Tanaka K.: Average Stress in the Matrix and Average Elastic Energy of Mate-rials with Misfitting Inclusions; Acta metall. 21, 571–574, 1973.

Moschovidis Z.A., Mura T.: Two Ellipsoidal Inhomogeneities by the Equivalent InclusionMethod; J.Appl.Mech. 42, 847–852, 1975.

Moshev V.V., Kozhevnikova L.L.: Structural Cell of Particulate Elastomeric Compositesunder Extension and Compression; Int.J.Sol.Struct. 39, 449–465, 2002.

Moulinec H., Suquet P.: A Fast Numerical Method for Computing the Linear and Non-linear Mechanical Properties of Composites; C.R.Acad.Sci.Paris, serie II 318, 1417–1423,1994.

Moulinec H., Suquet P.: A Numerical Method for Computing the Overall Response ofNonlinear Composites with Complex Microstructure. Comput.Meth.Appl.Mech.Engng.157, 69–94, 1997.

Muller W.H.: Mathematical versus Experimental Stress Analysis of Inhomogeneities inSolids; J.Phys.IV 6, C1-139–C1-148, 1996.

Mummery P., Derby B., Scruby C.B.: Acoustic Emission from Particulate-ReinforcedMetal Matrix Composites; Act metall.mater. 41, 1431–1445, 1993.

Mura T.: Micromechanics of Defects in Solids. Martinus Nijhoff, Dordrecht, The Nether-lands, 1987.

Nadeau J.C., Ferrari M.: On Optimal Zeroth-Order Bounds with Application to Hashin–Shtrikman Bounds and Anisotropy Parameters; Int.J.Sol.Struct. 38, 7945–7965, 2001.

Nakamura T., Suresh S.: Effects of Thermal Residual Stresses and Fiber Packing onDeformation of Metal–Matrix Composites; Acta metall.mater. 41, 1665–1681, 1993.

Narasimhan R.: A Numerical Study of Fracture Initiation in a Ductile Population of 2ndPhase Particles. I. Static Loading; Engng.Fract.Mech. 47, 919–934, 1994.

Nahshon K., Hutchinson J.W.: Modification of the Gurson Model for Shear Failure;Eur.J.Mech. A/Solids 27, 1–17, 2008.

Needleman A.: A Continuum Model for Void Nucleation by Inclusion Debonding;J.Appl.Mech. 54, 525–531, 1987.

Needleman A.: Material Rate Dependence and Mesh Sensitivity in Localization Problems;Comput.Meth.Appl.Mech.Engng. 67, 68–85, 1987.

Needleman A., Nutt S.R., Suresh S., Tvergaard V.: Matrix, Reinforcement, andInterfacial Failure: in “Fundamentals of Metal Matrix Composites” (Eds. S.Suresh,A.Mortensen, A.Needleman), pp. 233–250; Butterworth–Heinemann, Boston, MA, 1993.

Nemat-Nasser S.: Averaging Theorems in Finite Deformation Plasticity;, Mech.Mater.31, 493–523, 1999.

Nemat-Nasser S., Hori M.: Micromechanics: Overall Properties of Heterogeneous Solids.North–Holland, Amsterdam, The Netherlands, 1993.

Niebur G.L., Yuen J.C., Hsia A.C., Keaveny T.M. Convergence Behavior of High-Resolution Finite Element Models of Trabecular Bone; J.Biomech.Engng. 121, 629–635,1999.

– 128 –

Niordson C.F., Tvergaard V.: Nonlocal Plasticity Effects on Fibre Debonding in aWhisker-Reinforced Metal; Eur.J.Mech. A/Solids 21, 239–248, 2002.

Norris A.N.: A Differential Scheme for the Effective Moduli of Composites; Mech.Mater.4, 1–16, 1985.

Nouailhas D., Cailletaud G.: Finite Element Analysis of the Mechanical Behavior ofTwo-Phase Single-Crystal Superalloys; Scr.mater. 34, 565–571, 1996.

Nutt S.R., Needleman A.: Void Nucleation at Fiber Ends in Al–SiC Composites;Scr.metall. 21, 705–710, 1987.

Nye J.F.: Physical Properties of Crystals, Their Representation by Tensors and Matrices.Clarendon, Oxford, UK, 1957.

Oden T.J., Zohdi T.I.: Analysis and Adaptive Modeling of Highly Heterogeneous ElasticStructures; Comput.Meth.Appl.Mech.Engng. 148, 367–391, 1997.

Onck P., van der Giessen E.: Microstructurally-Based Modelling of Intergranular CreepFracture Using Grain Elements; Mech.Mater. 26, 109–126, 1997.

Ostoja-Starzewski M.: Micromechanics as a Basis for Random Elastic Continuum Ap-proximations; Probab.Engng.Mech. 8, 107–114, 1993.

Ostoja-Starzewski M.: Random Field Models of Heterogeneous Materials; Int.J.Sol.Struct.35, 2429–2455, 1998.

Ostoja-Starzewski M.: Mechanics of Random Materials. Stochastics, Scale Effects andComputation; in “Mechanics of Random and Multiscale Microstructures” (Eds. D.Jeulin,M.Ostoja–Starzewski), pp. 93–161; Springer–Verlag, Vienna, 2001.

Ostoja-Starzewski M.: Lattice Models in Micromechanics; Appl.Mech.Rev. 55, 35–60,2002.

Ostoja-Starzewski M.: Material Spatial Randomness: From Statistical to RepresentativeVolume Element; Probab.Engng.Mech. 21, 112–131, 2006.

Oyane M.: Criteria of Ductile Fracture Strain; Bull.JSME 15, 1507–1513, 1972.

Pahr D.H., Arnold S.M.: The Applicability of the Generalized Method of Cells for Ana-lyzing Discontinuously Reinforced Composites. Composites 33B, 153–170, 2002.

Pahr D.H., Rammerstorfer F.G.: Buckling of Honeycomb Sandwiches: Periodic FiniteElement Considerations; Comput.Model.Engng.Sci. 12, 229–241, 2006.

Pahr D.H., Zysset P.K.: Influence of Boundary Conditions on Computed Apparent Elas-tic Properties of Cancellous Bone; Biomech.Model.Mechanobiol., in print, 2008; doi:10.1007/s10237-007-0109-7.

Pandorf R.: Ein Beitrag zur FE-Simulation des Kriechens partikelverstarkter metallis-cher Werkstoffe. VDI–Verlag (Reihe 5, Nr. 585), Dusseldorf, Germany, 2000.

Papka S.D., Kyriakides S.: In-Plane Compressive Response and Crushing of Honeycomb;J.Mech.Phys.Sol. 42, 1499–1532, 1994.

Pardoen T., Hutchinson J.W.: An Extended Model for Void Growth Coalescence;J.Mech.Phys.Sol. 48, 2467–2512, 2000.

Pedersen O.B.: Thermoelasticity and Plasticity of Composites — I. Mean Field Theory;Acta metall. 31, 1795–1808, 1983.

Pedersen O.B., Withers P.J.: Iterative Estimates of Internal Stresses in Short-Fibre MetalMatrix Composites; Phil.Mag. A65, 1217–1233, 1992.

Peters P.W.M.: Modelling of Matrix Cracking in Fiber Reinforced Ceramics; Key En-gng.Mater. 127–131, 1093–1100, 1997.

– 129 –

Pettermann H.E.: Derivation and Finite Element Implementation of Constitutive Mate-rial Laws for Multiphase Composites Based on Mori–Tanaka Approaches. VDI–Verlag(Reihe 18, Nr. 217), Dusseldorf, Germany, 1997.

Pettermann H.E., Suresh S.: A Comprehensive Unit Cell Model — A Study of CoupledEffects in Piezoelectric 1–3 Composites; Int.J.Sol.Struct. 37, 5447–5464, 2000.

Pettermann H.E., Bohm H.J., Rammerstorfer F.G.: Some Direction Dependent Proper-ties of Matrix–Inclusion Type Composites with Given Reinforcement Orientation Distri-butions; Composites 28B, 253–265, 1997.

Phan-Thien N., Milton G.W.: New Third-Order Bounds on the Effective Moduli of N -Phase Composites; Quart.Appl.Math. 41, 59–74, 1983.

Pierard O., Friebel C., Doghri I.: Mean-Field Homogenization of Multi-Phase Thermo-Elastic Composites: A General Framework and its Validation; Compos.Sci.Technol. 64,1587–1603, 2004.

Pindera M.J., Aboudi J., Arnold S.M.: Analysis of Locally Irregular Composites UsingHigh-Fidelity Generalized Method of Cells; AIAA J. 41, 2331–2340, 2003.

Plankensteiner A.F.: Multiscale Treatment of Heterogeneous Nonlinear Solids and Struc-tures. VDI–Verlag (Reihe 18, Nr. 248), Dusseldorf, Germany, 2000.

Plankensteiner A.F., Bohm H.J., Rammerstorfer F.G., Pettermann H.E.: MultiscaleModeling of Highly Heterogeneous Particulate MMCs; J.Physique IV 8, Pr-301–Pr-308,1998.

Plankensteiner A.F., Bohm H.J., Rammerstorfer F.G., Buryachenko V.A., Hackl G.:Modeling of Layer-Structured High Speed Steel; Acta mater. 45, 1875–1887, 1997.

Ponte Castaneda P.: New Variational Principles in Plasticity and their Application toComposite Materials; J.Mech.Phys.Sol. 40, 1757–1788, 1992.

Ponte Castaneda P.: A Second-Order Theory for Non-Linear Composite Materials; C.R.Acad.Sci.Paris, serie IIb 322, 3–10, 1996.

Ponte Castaneda P., Suquet P.: Nonlinear Composites; in “Advances in Applied Mechan-ics 34” (Eds. E.van der Giessen, T.Y.Wu), pp. 171–302; Academic Press, New York, NY,1998.

Ponte Castaneda P., Willis J.R.: The Effect of Spatial Distribution on the EffectiveBehavior of Composite Materials and Cracked Media; J.Mech.Phys.Sol. 43, 1919–1951,1995.

Povirk G.L., Nutt S.R., Needleman A.: Analysis of Creep in Thermally Cycled Al/SiCComposites; Scr.metall.mater. 26, 461–66, 1992.

Pyrz R.: Correlation of Microstructure Variability and Local Stress Filed in Two-PhaseMaterials; Mater.Sci.Engng. A177, 253–259, 1994.

Pyrz R., Bochenek B.: Topological Disorder of Microstructure and its Relation to theStress Field; Int.J.Sol.Struct. 35, 2413–2427, 1998.

Qiu Y.P., Weng G.J.: A Theory of Plasticity for Porous Materials and Particle-Reinforced Composites; J.Appl.Mech. 59, 261–268, 1992.

Qu J., Cherkaoui M.: Fundamentals of Micromechanics of Solids. John Wiley, NewYork, NY, 2006.

Quilici S., Cailletaud G.: FE Simulation of Macro-, Meso- and Microscales in Polycrys-talline Plasticity; Comput.Mater.Sci. 16, 383–390, 1999.

Raabe D.: Computational Materials Science. Wiley–VCH, Weinheim, Germany, 1998.

– 130 –

Rammerstorfer F.G., Bohm H.J.: Finite Element Methods in Micromechanics of Com-posites and Foam Materials; in “Computational Mechanics for the Twenty First Century”(Ed. B.H.V.Topping), pp. 145–164; Saxe–Coburg Publications, Edinburgh, UK, 2000.

Reiter T.J., Dvorak G.J., Tvergaard V.: Micromechanical Models for Graded CompositeMaterials; J.Mech.Phys.Sol. 45, 1281–1302, 1997.

Ren Z.Y., Zheng Q.S.: Effects of Grain Sizes, Shapes, and Distribution on MinimumSizes of Representative Volume Elements of Cubic Polycrystals; Mech.Mater. 36, 1217–1229, 2004.

Reuss A.: Berechnung der Fließgrenze von Mischkristallen auf Grund der Plastizitatsbe-dingung fur Einkristalle; ZAMM 9, 49–58, 1929.

Rice J.R., Tracey D.M.: On the Ductile Enlargement of Voids in Triaxial Stress Fields;J.Mech.Phys.Sol. 17, 201–217, 1969.

Rintoul M., Torquato S.: Reconstruction of the Structure of Dispersions; J.Colloid In-terf.Sci. 186, 467–476, 1997.

Roberts A.P., Garboczi E.J.: Elastic Properties of a Tungsten–Silver Composite by Re-construction and Computation; J.Mech.Phys.Sol. 47, 2029–2055, 1999.

Roberts A.P., Garboczi E.J.: Elastic Moduli of Model Random Three-DimensionalClosed-Cell Cellular Solids; Acta mater. 49, 189–197, 2001.

Rodin G.J.: The Overall Elastic Response of Materials Containing Spherical Inhomo-geneities; Int.J.Sol.Struct. 30, 1849–1863, 1993.

Rousselier G.: Ductile Fracture Models and their Potential in Local Approach of Fracture;Nucl.Engng.Design 105, 97–111, 1987.

Sab K.: On the Homogenization and the Simulation of Random Materials; Eur.J.Mech.A/Solids 11, 585–607, 1992.

Saiki I., Terada K., Ikeda K., Hori M.: Appropriate Number of Unit Cells in a Repre-sentative Volume Element for Micro-Structural Bifurcation Encountered in a Multi-ScaleModeling; Comput.Meth.Appl.Mech.Engng. 191, 2561–2585, 2002.

Sanders W., Gibson L.J.: Mechanics of Hollow Sphere Foams; Mater.Sci.Engng. A347,70–85, 2003.

Sangani A., Lu W.: Elastic Coefficients of Composites Containing Spherical Inclusionsin a Periodic Array; J.Mech.Phys.Sol. 35, 1–21, 1987.

Santosa S.P., Wierzbicki T.: On the Modeling of Crush Behavior of a Closed-Cell Alu-minum Foam Structure; J.Mech.Phys.Sol. 46, 645–669, 1998.

Sautter M., Dietrich C., Poech M.H., Schmauder S., Fischmeister H.F.: Finite ElementModelling of a Transverse-Loaded Fibre Composite: Effects of Section Size and Net Den-sity; Comput.Mater.Sci. 1, 225–233, 1993.

Schapery R.A.: Thermal Expansion Coefficients of Composite Materials Based on EnergyPrinciples; J.Compos.Mater. 2, 380–404, 1968.

Schjødt-Thomsen J., Pyrz R.: Influence of Statistical Cell Description on the Local Strainand Overall Properties of Cellular Materials; in “Proceedings of NUMIFORM 2004”(Eds. S.Ghosh, J.M.Castro, J.K.Lee), pp. 1630–1635; American Institute of Physics,Melville, NY, 2004.

Schmauder S., Wulf J., Steinkopff T., Fischmeister H.: Micromechanics of Plasticityand Damage in an Al/SiC Metal Matrix Composite; in “Micromechanics of Plasticityand Damage of Multiphase Materials” (Eds. A.Pineau, A.Zaoui), pp. 255–262; Kluwer,Dordrecht, The Netherlands, 1996.

– 131 –

Schrefler B.A., Lefik M., Galvanetto U.: Correctors in a Beam Model for UnidirectionalComposites; Compos.Mater.Struct. 4, 159–190, 1997.

Schrefler B.A., Galvanetto U., Pellegrino C., Ohmenhauser F.: Global Non-Linear Be-havior of Periodic Composite Materials; in “Discretization Methods in Structural Me-chanics” (Eds. H.A.Mang, F.G.Rammerstorfer), pp. 265–272; Kluwer, Dordrecht, TheNetherlands, 1999.

Schwabe F.: Einspieluntersuchungen von Verbundwerkstoffen mit periodischer Mikrostruk-tur. Doctoral Thesis, Rheinisch-Westfalische Technische Hochschule, Aachen, Germany,2000.

Segurado J., LLorca J., Gonzalez C.: On the Accuracy of Mean-Field Approaches toSimulate the Plastic Deformation of Composites; Scr.Mater. 46, 525–529, 2002.

Segurado J., Gonzalez C., LLorca J.: A Numerical Investigation of the Effect of ParticleClustering on the Mechanical Properties of Composites; Acta mater. 51, 2355–2369,2003.Segurado J., Parteder E., Plankensteiner A.F., Bohm H.J.: Micromechanical Studies ofthe Densification of Porous Molybdenum; Mater.Sci.Engng. A333, 270–278, 2002.

Sejnoha M., Zeman J.: Overall Viscoelastic Response of Random Fibrous Com-posites with Statistically Quasi Uniform Distribution of Reinforcements; Com-put.Meth.Appl.Mech.Engng. 191, 5027–5044, 2002.

Shan Z., Gokhale A.M.: Micromechanics of Complex Three-Dimensional Microstruc-tures; Acta mater. 49, 2001–2015, 2001.

Shen H., Lissenden C.J.: 3D Finite Element Analysis of Particle-Reinforced Aluminum;Mater.Sci.Engng. A338, 271–281, 1997.

Shen H., Brinson L.C.: A Numerical Investigation of the Effect of Boundary Conditionsand Representative Volume Element Size for Porous Titanium; J.Mech.Mater.Struct. 1,1179–1204, 2006.

Shenoy V.B., Miller R., Tadmor E.B., Phillips R., Ortiz M.: Quasicontinuum Models ofInterfacial Structures and Deformation; Phys.Rev.Lett. 80, 742–745, 1998.

Shephard M.S., Beall M.W., Garimella R., Wentorf R.: Automatic Construction of 3/DModels in Multiple Scale Analysis; Comput.Mech. 17, 196–207, 1995.

Sherwood J.A., Quimby H.M.: Micromechanical Modeling of Damage Growth in Tita-nium Based Metal-Matrix Composites; Comput.Struct. 56, 505–54, 1995.

Shulmeister V., van der Burg M.W.D., van der Giessen E., Marissen R.: A NumericalStudy of Large Deformations of Low-Density Elastomeric Open-Cell Foams; Mech.Mater.30, 125–140, 1998.

Siegmund T., Werner E., Fischer F.D.: On the Thermomechanical Deformation Behaviorof Duplex-Type Materials; J.Mech.Phys.Sol. 43, 495–552, 1995.

Siegmund T., Cipra R., Liakus J., Wang B., LaForest M., Fatz A.: Processing–Microstructure–Property Relationships in Short Fiber Reinforced Carbon–Carbon Com-posite System; in “Mechanics of Microstructured Materials” (Ed. H.J.Bohm), pp. 235–258; CISM Courses and Lectures Vol. 464, Springer–Verlag, Vienna, 2004.

Sihn S., Roy A.K.: Modeling and Prediction of Bulk Properties of Open-Cell CarbonFoam; J.Mech.Phys.Sol. 52, 167–191, 2005.

Silberschmidt V.V., Werner E.A.: Analyses of Thermal Stresses’ Evolution in Ferritic–Austenitic Duplex Steels; in “Thermal Stresses 2001” (Eds. Y.Tanigawa, R.B.Hetnarski,N.Noda), pp. 327–330; Osaka Prefecture University, Osaka, Japan, 2001.

– 132 –

Silnutzer N.: Effective Constants of Statistically Homogeneous Materials. Doctoral The-sis, University of Pennsylvania, Philadelphia, PA, 1972.

Simone A.E., Gibson L.J.: Deformation Characteristics of Metal Foams; Acta mater.46, 2139–2150, 1998.

Smit R.J.M., Brekelmans W.A.M., Meijer H.E.H.: Prediction of the Mechanical Be-havior of Non-Linear Heterogeneous Systems by Multi-Level Finite Element Modeling;Comput.Meth.Appl.Mech.Engng. 155, 181–192, 1998.

Soppa E., Schmauder S., Fischer G., Brollo J., Weber U.: Deformation and Damage inAl/Al−2O3; Comput.Mater.Sci. 28, 574–586, 2003.

Sørensen N.J.: A Planar Type Analysis for the Elastic-Plastic Behaviour of ContinuousFibre-Reinforced Metal-Matrix Composites Under Longitudinal Shearing and CombinedLoading. DCAMM Report No.418, Technical University of Denmark, Lyngby, 1991.

Sørensen N.J., Suresh S., Tvergaard V., Needleman A.: Effects of Reinforcement Ori-entation on the Tensile Response of Metal Matrix Composites; Mater.Sci.Engng. A197,1–10, 1995.

Spiegler R., Schmauder S., Fischmeister H.F.: Simulation of Discrete Void Formation ina WC–Co Microstructure; in “Finite Elements in Engineering Applications 1990”, pp.21–42;INTES GmbH, Stuttgart, Germany, 1990.

Steinkopff T., Sautter M., Wulf J.: Mehrphasige Finite Elemente in der Verformungs-und Versagensanalyse grob mehrphasiger Werkstoffe; Arch.Appl.Mech. 65, 496–506,1995.

Stroeven M., Askes H., Sluys L.J.: Numerical Determination of Representative Volumesfor Granular Materials; Comput.Meth.Appl.Mech.Engng. 193, 3221–3238, 2004.

Sukumar N., Chopp D.L., Moes N., Belytschko T.: Modeling Holes and Inclusions byLevel Sets in the Extended Finite Element Method; Comput.Meth.Appl.Mech.Engng.190, 6183–6900, 2001.

Sun L.Z., Ju J.W., Liu H.T.: Elastoplastic Modeling of Metal Matrix Composites withEvolutionary Particle Debonding; Mech.Mater. 35, 559–569, 2003.

Suquet P.M.: Elements of Homogenization for Inelastic Solid Mechanics; in “Homoge-nization Techniques in Composite Media” (Eds. E.Sanchez-Palencia, A.Zaoui), pp. 194–278; Springer–Verlag, Berlin, Germany, 1987.

Suquet P.M.: Effective Properties of Nonlinear Composites; in “Continuum Microme-chanics” (Ed. P.Suquet), pp. 197–264; Springer–Verlag, Vienna, Austria, 1997.

Suzuki T., Yu P.K.L.: Complex Elastic Wave Band Structures in Three-DimensionalPeriodic Elastic Media; J.Mech.Phys.Sol. 46, 115–138, 1998.

Swaminathan S., Ghosh S.: Statistically Equivalent Representative Volume Elementsfor Unidirectional Composite Microstructures: Part II — With Interfacial Debonding;J.Compos.Mater. 40, 605–621, 2006.

Swaminathan S., Ghosh S., Pagano N.J.: Statistically Equivalent Representative VolumeElements for Unidirectional Composite Microstructures: Part I — Without Damage;J.Compos.Mater. 40, 583–604, 2006.

Swan C.C.: Techniques for Stress- and Strain-Controlled Homogenization of InelasticPeriodic Composites; Comput.Meth.Appl.Mech.Engng. 117, 249–267, 1994.

Tadmor E.B., Phillips R., Ortiz M.: Hierarchical Modeling in the Mechanics of Materials;Int.J.Sol.Struct. 37, 379–390, 2000.

Takano N., Zako M., Okazaki T.: Efficient Modeling of Microscopic Heterogeneity and Lo-cal Crack in Composite Materials by Finite Element Mesh Superposition Method; JSMEInt.J. Srs.A 44, 602–609, 2001.

– 133 –

Talbot D.R.S., Willis J.R.: Variational Principles for Inhomogeneous Non-Linear Media;J.Appl.Math. 35, 39–54, 1985.

Talbot D.R.S., Willis J.R.: Bounds of Third Order for the Overall Response of NonlinearComposites; J.Mech.Phys.Sol. 45, 87–111, 1997.

Tandon G.P., Weng G.J.: The Effect of Aspect Ratio of Inclusions on the Elastic Prop-erties of Unidirectionally Aligned Composites; Polym.Compos. 5, 327–333, 1984.

Tandon G.P., Weng G.J.: A Theory of Particle-Reinforced Plasticity; J.Appl.Mech. 55,126–135, 1988.

Tang X., Whitcomb J.D.: General Techniques for Exploiting Periodicity and Symmetriesin Micromechanics Analysis of Textile Composites; J.Compos.Mater. 37, 1167–1189,2003.

Taya M., Mori T.: Dislocations Punched Out Around a Short Fibre in MMC Subjectedto Uniform Temperature Change; Acta metall. 35, 155–162, 1987.

Taya M., Armstrong W.D., Dunn M.L., Mori T.: Analytical Study on DimensionalChanges in Thermally Cycled Metal Matrix Composites; Mater.Sci.Engng. A143, 143–154, 1991.

Teply J.L., Dvorak G.J.: Bounds on Overall Instantaneous Properties of Elastic-PlasticComposites; J.Mech.Phys.Sol. 36, 29–58, 1988.

Terada K., Kikuchi N.: Microstructural Design of Composites Using the HomogenizationMethod and Digital Images; Mater.Sci.Res.Int. 2, 65–72, 1996.

Terada K., Kikuchi N.: Nonlinear Homogenization Method for Practical Applications; in“Computational Methods in Micromechanics” (Eds. S.Ghosh, M.Ostoja-Starzewski), pp.1–16; AMD–Vol.212/MD–Vol.62, ASME, New York, NY, 1996.

Terada K., Miura T., Kikuchi N.: Digital Image-Based Modeling Applied to the Homog-enization Analysis of Composite Materials; Comput.Mech. 20, 331–346, 1997.

Terada K., Hori M., Kyoya T., Kikuchi N.: Simulation of the Multi-Scale Convergencein Computational Homogenization Approach; Int.J.Sol.Struct. 37, 2285–2311, 2000.

Terada K., Saiki I., Matsui K., Yamakawa Y.: Two-Scale Kinematics and Linearizationfor Simultaneous Two-Scale Analysis of Periodic Heterogeneous Solids at Finite Strain;Comput.Meth.Appl.Mech.Engng. 192, 3531–3563, 2003.

Tohgo K., Chou T.W.: Incremental Theory of Particulate-Reinforced Composites Includ-ing Debonding Damage; JSME Int.J. Srs.A 39, 389–397, 1996.

Toll S.: Interpolative Aggregate Model for Short-Fibre Composite; J.Compos.Mater. 26,1767–1782, 1992.

Tomita Y., Higa Y., Fujimoto T.: Modeling and Estimation of Deformation Behavior ofParticle Reinforced Metal-Matrix Composite; Int.J.Mech.Sci. 42, 2249–2260, 2000.

Torquato S.: Random Heterogeneous Media: Microstructure and Improved Bounds onEffective Properties; Appl.Mech.Rev. 44, 37–75, 1991.

Torquato S.: Morphology and Effective Properties of Disordered Heterogeneous Media;Int.J.Sol.Struct. 35, 2385–2406, 1998.

Torquato S.: Effective Stiffness Tensor of Composite Media: II. Applications to IsotropicDispersions; J.Mech.Phys.Sol. 46, 1411–1440, 1998.

Torquato S.: Random Heterogeneous Media. Springer–Verlag, New York, NY, 2002.

Torquato S., Hyun S.: Effective-Medium Approximation for Composite Media: RealizableSingle-Scale Dispersions; J.Appl.Phys. 89, 1725–1729, 2001.

– 134 –

Torquato S., Lado F.: Improved Bounds on the Effective Moduli of Random Arrays ofCylinders; J.Appl.Mech. 59, 1–6, 1992.

Torquato S., Lado F., Smith P.A.: Bulk Properties of Two-Phase Disordered Media. IV.Mechanical Properties of Suspensions of Penetrable Spheres at Nondilute Concentrations;J.Chem.Phys. 86, 6388–6392, 1987.

Trias D., Costa J., Turon A., Hurtado J.F.: Determination of the Critical Size of a Sta-tistical Representative Volume Element (SRVE) for Carbon Reinforced Polymers; Actamater. 54, 3471–3484, 2006.

Tsai S.W., Wu E.M.: A General Theory of Strength for Anisotropic Materials;J.Compos.Mater. 5, 38–80, 1971.

Tsotsis T.K., Weitsman Y.: Energy Release Rates for Cracks Caused by Moisture Ab-sorption in Graphite/Epoxy Composites; J.Compos.Mater. 24, 483–496, 1990.

Tszeng T.C.: The Effects of Particle Clustering on the Mechanical Behavior of ParticleReinforced Composites; Composites 29B, 299–308, 1998.

Tucker C.L., Liang E.: Stiffness Predictions for Unidirectional Short-Fiber Composites:Review and Evaluation; Compos.Sci.Technol. 59, 655–671, 1999.

Tvergaard V.: Constitutive Relations for Creep in Polycrystals with Boundary Cavita-tion; Acta metall. 32, 1977–1990, 1984.

Tvergaard V.: Analysis of Tensile Properties for a Whisker-Reinforced Metal-MatrixComposite; Acta metall.mater. 38, 185–194, 1990.

Tvergaard V.: Fibre Debonding and Breakage in a Whisker-Reinforced Metal.Mater.Sci.Engng. A190, 215–222, 1994.

Tvergaard V.: Effect of Void Size Difference on Growth and Cavitation Instabilities;J.Mech.Phys.Sol. 44, 1237–1253, 1996.

Tvergaard V.: Breakage and Debonding of Short Brittle Fibres Among Particulates in aMetal Matrix; Mater.Sci.Engng. A369, 192–200, 2004.

Tvergaard V., Needleman A.: Analysis of the Cup-Cone Fracture in a Round TensileBar; Acta metall. 32, 157–169, 1984.

Tvergaard V., Needleman A.: Effects of Nonlocal Damage in Porous Plastic Solids;Int.J.Sol.Struct. 32, 1063–1077, 1995.

Urbanski A.J.: Unified, Finite Element Based Approach to the Problem of Homogeni-sation of Structural Members with Periodic Microstructure; in ”Solids, Structures andCoupled Problems in Engineering” (Ed. W.Wunderlich); TU Munchen, Munich 1999.

Vajjhala S., Kraynik A.M., Gibson L.J.: A Cellular Solid Model for Modulus Reductiondue to Resorption of Trabeculae in Bone; J.Biomech.Engng. 122, 511–515, 2000.

van Rietbergen B., Muller R., Ulrich D., Ruegsegger P., Huiskes R.: Tissue Stressesand Strain in Trabeculae of a Canine Proximal Femur Can be Quantified from ComputerReconstructions; J.Biomech. 32, 165–173, 1999.

Varadi K., Neder Z., Friedrich K., Flock J.: Finite-Element Analysis of a Polymer Com-posite Subjected to Ball Indentation; Compos.Sci.Technol. 59, 271–281, 1999.

Vejen N., Pyrz R.: Transverse Crack Growth in Glass/Epoxy Composites with ExactlyPositioned Long Fibers. Part II: Numerical; Composites 33B, 279–290, 2002.

Voigt W.: Uber die Beziehung zwischen den beiden Elasticitats-Constanten isotroperKorper; Ann.Phys. 38, 573–587, 1889.

Vonach W.K.: A General Solution to the Wrinkling Problem of Sandwiches. VDI–Verlag(Reihe 18, Nr. 268), Dusseldorf, Germany, 2001.

– 135 –

Vosbeek P.H.J., Meurs P.F.M., Schreurs P.J.G.: Micro-Mechanical Constitutive Modelsfor Composite Materials with Interfaces; in “Proc.1st Forum of Young European Re-searchers”, pp. 223–228; Universite de Liege, Liege, Belgium, 1993.

Voyiadjis G.Z., Kattan P.I.: Advances in Damage Mechanics: Metals and Metal MatrixComposites. Elsevier, Amsterdam, 1999.

Wakashima K., Tsukamoto H., Choi B.H.: Elastic and Thermoelastic Properties of MetalMatrix Composites with Discontinuous Fibers or Particles: Theoretical Guidelines to-wards Materials Tailoring; in “The Korea–Japan Metals Symposium on Composite Ma-terials”, pp. 102–115; The Korean Institute of Metals, Seoul, Korea, 1988.

Wallin K., Saario T., Torronen K.: Fracture of Brittle Particles in a Ductile Matrix;Int.J.Fract. 32, 201–209, 1987.

Walpole L.J.: On Bounds for the Overall Elastic Moduli of Inhomogeneous Systems —II; J.Mech.Phys.Sol. 14, 289–301, 1966.

Walter M.E., Ravichandran G., Ortiz M.: Computational Modeling of Damage Evolutionin Unidirectional Fiber Reinforced Ceramic Matrix Composites; Comput.Mech. 20, 192–198, 1997.

Wang J., Andreasen J.H., Karihaloo B.L.: The Solution of an Inhomogeneity in a FinitePlane Region and its Application to Composite Materials; Compos.Sci.Technol. 60, 75–82, 2000.

Wang X.F., Wang X.W., Zhou G.M., Zhou C.W.: Multi-Scale Analysis of 3D WovenComposite Based on Periodicity Boundary Conditions; J.Compos.Mater. 41, 1773–1788,2007.

Warren W.E., Kraynik A.M.: The Nonlinear Elastic Properties of Open-Cell Foams;J.Appl.Mech. 58, 376–381, 1991.

Watt D.F., Xu X.Q., Lloyd D.J.: Effects of Particle Morphology and Spacing on theStrain Fields in a Plastically Deforming Matrix; Acta mater. 44, 789–799, 1996.

Weissenbek E.: Finite Element Modelling of Discontinuously Reinforced Metal MatrixComposites. VDI–Verlag (Reihe 18, Nr. 164), Dusseldorf, Germany, 1994.

Weissenbek E., Bohm H.J., Rammerstorfer F.G.: Micromechanical Investigations of Ar-rangement Effects in Particle Reinforced Metal Matrix Composites; Comput.Mater.Sci.3, 263–278, 1994.

Weissenbek E., Pettermann H.E., Suresh S.: Numerical Simulation of Plastic Deforma-tion in Compositionally Graded Metal–Ceramic Structures; Acta mater. 45, 3401–3417,1997.

Weisstein E.W.: Sphere Packing; http://mathworld.wolfram.com/SpherePacking.html;2000.

Weng G.J.: The Theoretical Connection Between Mori–Tanaka Theory and the Hashin–Shtrikman–Walpole Bounds; Int.J.Engng.Sci. 28, 1111–1120, 1990.

Wilkins M.L.: Computer Simulation of Dynamic Phenomena. Springer–Verlag, Berlin,1999.

Willis J.R.: Bounds and Self-Consistent Estimates for the Overall Properties ofAnisotropic Composites; J.Mech.Phys.Sol. 25, 185–202, 1977.

Willis J.R.: The Overall Response of Nonlinear Composite Media; Eur.J.Mech. A/Solids19, S165–S184, 2000.

Withers P.J.: The Determination of the Elastic Field of an Ellipsoidal Inclusion in aTransversely Isotropic Medium, and its Relevance to Composite Materials; Phil.Mag.A59, 759–781, 1989.

– 136 –

Withers P.J., Stobbs W.M., Pedersen O.B.: The Application of the Eshelby Method ofInternal Stress Determination to Short Fibre Metal Matrix Composites; Acta metall. 37,3061–3084, 1989.

Wittig L.A., Allen D.H.: Modeling the Effect of Oxidation on Damage in SiC/Ti-15-3Metal Matrix Composites; J.Engng.Mater.Technol. 116, 421–427, 1994.

Wu J.F., Shephard M.S., Dvorak G.J., Bahei-el-Din Y.A.: A Material Model for theFinite Element Analysis of Metal Matrix Composites; Compos.Sci.Technol. 35, 347–366, 1989.

Wu T.T: The Effect of Inclusion Shape on the Elastic Moduli of a Two-Phase Material;Int.J.Sol.Struct. 2, 1–8, 1966.

Wulf J., Steinkopff T., Fischmeister H.: FE-Simulation of Crack Paths in the Real Mi-crostructure of an Al(6061)/SiC Composite; Acta mater. 44, 1765–1779, 1996.

Xia Z., Zhang Y., Ellyin F.: A Unified Periodical Boundary Conditions for RepresentativeVolume Elements of Composites and Applications; Int.J.Sol.Struct. 40, 1907–1921, 2003.

Youssef S., Maire E., Gaertner R.: Finite Element Modelling of the Actual Structure ofCellular Materials Determined by X-Ray Tomography; Acta mater. 53, 719–730, 2005.

Zaoui A.: Changement d’echelle: motivation et methodologie; in “Homogeneisation enmecanique des materiaux 1. Materiaux aleatoires elastiques et milieux periodiques” (Eds.M.Bornert, T.Bretheau, P.Gilormini), pp. 19–40; Editions Hermes, Paris, 2001.

Zaoui A.: Plasticite: Approches en champ moyen; in “Homogeneisation en mecaniquedes materiaux 2. Comportements non lineaires et problemes ouverts” (Eds. M.Bornert,T.Bretheau, P.Gilormini), pp. 17–44; Editions Hermes, Paris, 2001.

Zaoui A.: Continuum Micromechanics: Survey; J.Engng.Mech. 128, 808–816, 2002.

Zeman J.: Analysis of Composite Materials with Random Microstructure. Doctoral The-sis, Czech Technical University, Prague, Czech Republic, 2003.

Zeman J., Sejnoha M.: Numerical Evaluation of Effective Elastic Properties of GraphiteFiber Tow Impregnated by Polymer Matrix; J.Mech.Phys.Sol. 49, 69–90, 2001.

Zhao Y.H., Weng G.J.: A Theory of Inclusion Debonding and Its Influence on the Stress–Strain Relations of a Ductile Matrix Composite; Int.J.Dam.Mech. 4, 196–211, 1995.

Zhong X.A., Knauss W.G.: Effects of Particle Interaction and Size Variation on DamageEvolution in Filled Elastomers; Mech.Compos.Mater.Struct. 7, 35–53, 2000.

Zhou S.J., Curtin W.A.: Failure of Fiber Composites: A Lattice Green Function Ap-proach; Acta metall.mater. 43, 3093–3104, 1995.

Zhu H.X., Mills N.J., Knott J.F.: Analysis of the High Strain Compression of Open-CellFoams; J.Mech.Phys.Sol. 45, 1875–1904, 1997.

Zimmerman R.W.: Hashin–Shtrikman Bounds on the Poisson Ratio of a CompositeMaterial; Mech.Res.Comm. 19, 563–569, 1992.

Zohdi T.I.: A Model for Simulating the Deterioration of Structural-Scale Material Re-sponses of Microheterogeneous Solids; in “Euromech Colloqium 402 — Micromechanicsof Fracture Processes” (Eds. D.Gross, F.D.Fischer, E.van der Giessen), pp. 91–92; TUDarmstadt, Darmstadt, Germany, 1999

Zohdi T.I.: Constrained Inverse Formulations in Random Material Design; Com-put.Meth.Appl.Mech.Engng. 192, 3179–3194, 2003.

Zohdi T.I., Wriggers P.: A Model for Simulating the Deterioration of Structural-Scale Ma-terial Responses of Microheterogeneous Solids; Comput.Meth.Appl.Mech.Engng. 190,2803–2823, 2001.

– 137 –

Zohdi T.I., Oden J.T., Rodin G.J.: Hierarchical Modeling of Heterogeneous Bodies; Com-put.Meth.Appl.Mech.Engng. 138, 273–289, 1996.

– 138 –