hjb/101130

COMPOSITES ENGINEERING, Part II317.003

CONTINUUM MICROMECHANICS

OF MATERIALS

Helmut J. Bohm

Institut fur Leichtbau und Struktur-BiomechanikTU Wien

c©H.J. Bohm, ILSB/TUW, 1993, 2010

Contents

Notes iv

Remarks on Composites v

1 Introduction 1

1.1 Length Scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Homogenization and Localization: Basic Notions . . . . . . . . . . . . . . . 21.3 Material Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.4 Basic Modeling Strategies in Continuum Micromechanics of Materials . . . 5

2 Some Basic Analytical Approaches for Thermoelastic Composites 10

2.1 Rules of Mixture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2 The VFD Model of Dvorak and Bahei-el-Din . . . . . . . . . . . . . . . . . 112.3 The CSA and CCA Models . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3 Mean Field Methods 14

3.1 General Relations between Mean Fields in Thermoelastic Two-Phase Materials 143.2 Misfit Strains: Eshelby’s Solution . . . . . . . . . . . . . . . . . . . . . . . 173.3 Dilute Inhomogeneous Inclusions . . . . . . . . . . . . . . . . . . . . . . . 193.4 Mori–Tanaka Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.5 Self-Consistent Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.6 Other Mean Field and Related Estimates . . . . . . . . . . . . . . . . . . . 283.7 Mean Field Methods for Elastoplastic Composites . . . . . . . . . . . . . . 283.8 Mean Field Methods for Nonaligned Composites . . . . . . . . . . . . . . . 313.9 Mean Field Methods for Diffusion-Type Problems . . . . . . . . . . . . . . 32

4 Bounding Methods 33

4.1 Hill and Hashin–Shtrikman-Type Bounds . . . . . . . . . . . . . . . . . . . 334.2 Improved Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354.3 Bounds for the Nonlinear Behavior . . . . . . . . . . . . . . . . . . . . . . 364.4 Comparisons Between Some Mean Field and Bounding Predictions . . . . . 37

5 General Remarks on Modeling Approaches Based on Discrete Micro-

geometries 43

6 Periodic Microfield Models 47

6.1 Basic Concepts of Unit Cell Models . . . . . . . . . . . . . . . . . . . . . . 476.2 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

ii

6.3 Application of Loads and Evaluation of Microfields . . . . . . . . . . . . . 536.4 Unit Cell Models for Continuous Fiber Reinforced Composites . . . . . . . 576.5 Unit Cell Models for Short Fiber Reinforced Composites . . . . . . . . . . 616.6 Unit Cell Models for Particle Reinforced Composites . . . . . . . . . . . . 646.7 Unit Cell Models for Woven and Laminated Composites . . . . . . . . . . 686.8 Unit Cell Models for Porous and Cellular Materials . . . . . . . . . . . . . 70

7 Embedded Cell Models 73

8 Windowing Approaches 76

9 Multi-Scale Models 80

10 Closing Remarks 83

iii

Notes

In the following, Nye notation is used for mechanical variables, i.e., tensors of order 4 arewritten as 6 × 6-quasi-matrices, and stress-like as well as strain-like tensors of order 2 as6-quasi-vectors. These 6-vectors are connected to index notation by the relations

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 angles. Tensors of order 4 are denoted by bold upper caseletters, stress- and strain-like tensors of order 2 by bold lower case Greek letters, and 3-vectors by bold lower case letters. All other variables are taken to be scalars. Note thatthe use of the present notation requires that the 4th order tensors show orthotropic orhigher symmetry and that the coefficients of Eshelby and concentration tensors may differcompared to index notation. The contraction between a tensor of order 2 and a 3-vector isdenoted by the symbol “∗”, where [ζ ∗ n]i = ζijnj , and the tensorial product between twotensors of order 2 as well as the dyadic product between two vectors are denoted by thesymbol “⊗”, where [η ⊗ ζ]ijkl = ηijζkl, and [a⊗ n]ij = ainj, respectively. A superscript T

denotes the transpose of a tensor or vector.

Constituents (phases) are denoted by superscripts, with (p) standing for a general phase,(m) for a matrix, (i) for inhomogeneities, and (f) for fibers.

References to equations and section numbers beginning with a roman I are understoodto pertain to the first part of the lecture Composites Engineering, which is given byProf. F.G.Rammerstorfer.

For information on the research work in the field of continuum micromechanics ofmaterials carried out at the Institut fur Leichtbau und Struktur-Biomechanik see the webpages http://ilsb.tuwien.ac.at/ilfb/ilfb ra13.html.

iv

Remarks on Composites

Inhomogeneous materials consist of dissimilar constituents (or “phases”) that are distin-guishable at some length scale(s), called the characteristic length(s) of the inhomogeneities(e.g., the diameters of or distances between the reinforcements of composites). Each con-stituent shows different material properties (as, e.g., in composite and porous materials)and/or material orientations (as, e.g., in polycrystals) and may again be inhomogeneousat some smaller length scale.

There are numerous schemes for the classification and nomenclature of inhomogeneousand multiphase materials. The following list, which is neither comprehensive nor complete,gives a number of classification schemes which are relevant to the present course.

• “Production” route:

– “natural” composites (e.g. wood, bone, alloys containing precipitates or inclu-sions, in-situ composites, polycrystalline materials, many porous and cellularmaterials)

– “artificial” composites (composite materials in the strict sense1), foams, syntac-tic foams, functionally graded materials (FGMs)

• Matrix material (for “artificial” composites):

– polymer matrix composites (PMCs)thermoset or thermoplastic matrix; often referred to in terms of the reinforce-ment: GFRP, CFRP, hybrid composites . . .

– metal matrix composites (MMCs)often taken to include intermetallic matrix composites (IMCs), which are brittleat room temperature

– brittle matrix compositesceramic matrix composites (CMCs),glass matrix composites,carbon–carbon composites (CCCs)

1Such materials are designed to take advantage of the different properties of the constituents to achievesome overall behavior(s). Production-related issues of many composite materials and their usage inlightweight structures are covered in the ILSB course Leichtbau mit faserverstarkten Kunststoffen

given by I.C. Skrna–Jakl.

v

• Mesoscale topology (for “artificial” composites):

– laminates

– composites containing woven, knitted, braided reinforcements

• Microscale topology:

– matrix–inclusion topologies

– interwoven topologies (e.g. duplex steels; open cell foams)

– layered topologies

• Phase arrangement statistics

– statistically homogeneous phase arrangements

– statistically inhomogeneous (e.g. graded) phase arrangements

• Shape of the reinforcements (for matrix–inclusion topologies):

– continuously reinforced; typically unidirectional (UD) fibers

– long fiber reinforced; may signify either continuous fibers or fibers markedlyexceeding the critical length (compare eqn. (I–1.41))

– short fiber reinforced: aspect ratios are clearly larger than 1; length typicallycomparable to or exceeding the critical length

– particle reinforced: aspect ratios close to 1 (“equiaxed”)

– platelet reinforced: aspect ratios clearly smaller than 1

– hybrid reinforcements

• Orientation of the reinforcements (for matrix–inclusion topologies):

– aligned, unidirectional

– nonaligned (e.g. due to production processes); orientations may be described byorientation distribution functions (ODFs)

– random, planar random

• Relative size of the reinforcements (matrix–inclusion topologies):

– monodisperse sizes (e.g., equal fiber diameters)

– bidisperse sizes (e.g., two clusters of particle radii)

– polydisperse sizes

vi

Chapter 1

Introduction

1.1 Length Scales

Descriptions of the properties of composite materials have to account for at least two (butoften three or more) length scales2:

• Macroscale: length scale of the structure, component or sample

• Mesoscale: intermediate length scale (e.g., lamina level in layered composites)

• Microscale: length scale of the inhomogeneities, e.g., reinforcement diameters ordistances

In many cases, the constituents themselves are inhomogeneous at some smaller lengthscale, being, e.g., polycrystalline or porous. Figure 1.1 shows a schematic depiction of thevarious length scales in a hypothetical structure consisting of a stiffened shell made of ametal matrix composite.

The subject of the present lectures is continuum micromechanics, i.e., the study ofmechanical properties of inhomogeneous materials within the framework of continuum me-chanics while directly accounting for the phase arrangement at the microscale. Even thoughthe emphasis is put on the thermomechanical behavior of two-phase composites, there isa large body of literature applying analogous or related methods both to other physicalproperties, such as thermal conduction, and to a wide range of other inhomogeneous ma-terials.

In micromechanical approaches, the stress and strain fields in an inhomogeneous mate-rial are split into contributions corresponding to the different length scales, which may betermed “fast” and “slow” variables. It is assumed that these length scales are sufficientlydifferent so that

• the stress and strain fields at the smaller length scales (“fast variables”, microfields),ε(x) and σ(x), influence the macroscopic behavior at the larger length scales onlyvia their volume averages, but not via their fluctuations, and

2The nomenclature used here is far from universal, the naming of the length scales of inhomogeneousmaterials being notoriously inconsistent in the literature.

1

MACRO (structural)

MESO (laminate)

MICRO 2 (polycrystal)

MICRO 1 (composite)

SUBMICRO 2 (atomistic)

SUBMICRO 1 (nano)

Figure 1.1: Schematic depiction of length scales in a hypothetical laminated MMC.

• gradients of the stress and strain fields at larger length scales (“slow variables”,macrofields), 〈ε〉 and 〈σ〉, as well as compositional gradients are not significant atthe smaller length scales, where these fields appear constant and can be cast into theform of uniform “far field” or “applied” stresses and strains.

Viewed from the macroscale the behavior of a material that is inhomogeneous at somemicroscale can be described via an energetically equivalent homogenized continuum, pro-vided the separation between the two length scales is sufficiently large and the aboveconditions are met. If these relations between fast and slow variables are not fulfilled to asufficient degree (e.g., near free surfaces of anisotropic materials or at macroscopic inter-faces adjoined by at least one inhomogeneous material), continuum micromechanics mustbe used with extreme care and appropriate procedures must be employed.

1.2 Homogenization and Localization: Basic Notions

For regions of components or samples that do not exhibit significant macroscopic stressor strain gradients, the microscopic strain and stress fields, ε(x) and σ(x), and the cor-responding macroscopic responses, 〈ε〉 and 〈σ〉, can be formally linked by localizationrelations

ε(x) = A(x) 〈ε〉

σ(x) = B(x) 〈σ〉 (1.1)

and by homogenization relations of the type

〈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 . (1.2)

2

Here Ωs and Γs stand for the volume and the surface of a volume element, u(x) are thedeformation vectors, t(x) = σ(x) ∗ nΓ(x) the surface traction vectors, nΓ(x) the unit sur-face normal vectors, and ⊗ is the dyadic product of vectors. The tensors A(x) and B(x)in eqn. (1.1) are called mechanical strain and stress concentration tensors (Hill, 1963),respectively.

The surface integral formulation for 〈ε〉 given above holds only for perfect interfacesbetween the constituents, in which case the mean strains and stresses in a control volume,〈ε〉 and 〈σ〉, are fully determined by the surface displacements and tractions. Otherwisecorrection terms involving the displacement jumps across gaps or cracks must be accountedfor, see, e.g., Nemat-Nasser and Hori (1993). In the absence of body forces the microstressesσ(x) are self-equilibrated (but not necessarily zero). Equation (1.1) applies only to elasticbehavior, but can be easily modified to cover thermoelastic behavior, compare eqn. (3.5),and the extension to the nonlinear range (e.g., to elastoplastic materials described by se-cant or incremental plasticity) is formally rather straightforward, compare section 3.7.

The microgeometries of real inhomogeneous materials are at least to a certain extentrandom and, in nearly all practically relevant cases, are highly complex. Accordingly, exactexpressions for A(x), B(x), ε(x) and σ(x) cannot realistically be provided for nontrivialcomposites and approximations have to be introduced.

Typically, these approximations are based on the ergodic hypothesis, the heteroge-neous material being assumed to be statistically homogeneous. Accordingly, sufficientlylarge subvolumes selected randomly within a sample are taken to give rise to the sameeffective material properties which, in turn, correspond to the overall material properties3.Ideally, the homogenization volume should be chosen to be an appropriate representativevolume element (RVE), which is a volume element that is of sufficient size to contain allinformation necessary for describing the behavior of the composite. For discussions ofRVEs and the boundary conditions applied to them see, e.g., (Hashin, 1983; Markov, 2000;Bornert et al., 2001; Kanit et al., 2003). In practice, however, it may be very difficult toidentify RVEs and, accordingly, approximations to them are used in discrete microfieldanalysis. Such volume elements used for homogenization, denoted here as Ωs, should besufficiently large to allow a meaningful sampling of the microfields and sufficiently small forthe influence of macroscopic gradients to be negligible and for an analysis of the microfieldsto be possible.

1.3 Material Symmetries

The homogenized behavior of many multiphase materials can be idealized as being eitherstatistically isotropic (e.g., composites reinforced by spherical particles, randomly orientedparticles of general shape or randomly oriented fibers, many polycrystals, many porous and

3This requirement was symbolically denoted as MICRO≪MESO≪MACRO by Hashin (1983), whereMICRO and MACRO have their “usual” meanings and MESO stands for the size of the homogenizationvolume. Note that some inhomogeneous materials, e.g., graded materials, are not statistically homogeneous(and consequently non-ergodic) so that special procedures may be required to model them.

3

cellular materials, random mixtures of two phases) or statistically transversely isotropic(composites reinforced with aligned fibers or platelets, composites reinforced with non-aligned inhomogeneities showing a planar random or some other axisymmetric orientationdistribution function, etc.), compare Hashin (1983).

Statistically isotropic multiphase materials show the same macroscopic behavior in alldirections, and their effective elasticity and thermal expansion tensors take the forms (inNye notation)

E =

E11 E12 E12 0 0 0E12 E11 E12 0 0 0E12 E12 E11 0 0 00 0 0 E44 0 00 0 0 0 E44 00 0 0 0 0 E44 = 1

2(E11 − E12)

α =

ααα000

. (1.3)

Two independent parameters (chosen among moduli such as the effective Young’s modu-lus E, the effective Poisson number ν, the effective shear modulus G, the effective bulkmodulus K=E/3(1 − 2ν), or the effective Lame constants) are sufficient for describingtheir overall linear elastic behavior and one is required for the effective thermal expansionbehavior in the linear range, the coefficient of thermal expansion α = α11.

The effective elasticity and thermal expansion tensors for statistically transverselyisotropic materials have the structure

E =

E11 E12 E12 0 0 0E12 E22 E23 0 0 0E12 E23 E22 0 0 00 0 0 E44 0 00 0 0 0 E44 00 0 0 0 0 E66 = 1

2(E22 −E23)

α =

αl

αq

αq

000

, (1.4)

where 1 is the axial direction and 2–3 is the transverse plane of isotropy. Generally, thethermoelastic behavior of transversely isotropic materials is described by five independentelastic constants and two independent coefficients of thermal expansion. Appropriate elas-tic parameters for this purpose are, e.g., the axial and transverse effective Young’s moduli,El and Eq, the axial and transverse effective shear moduli, Glq and Gqt, the axial andtransverse effective Poisson numbers, νlq and νqt, as well as the effective transverse bulkmodulus KT=El/2[(1 − νqt)(El/Eq) − 2ν2

lq]. The transverse (“in-plane”) properties arerelated via Gqt=Eq/2(1 + νqt), but for general transversely isotropic materials there is nolinkage between the axial properties El, Glq and νlq. Both an axial and a transverse effec-tive coefficient of thermal expansion, αl = α11 and αq = α22, are required. For the specialcase of transversely isotropic aligned fibrous two-phase materials the relations

El = ξE(f)l + (1− ξ)E(m) +

4(ν(f)lq − ν

(m))2

(1/K(f)T − 1/K

(m)T )2

(

ξ

K(f)T

+1− ξ

K(m)T

−1

KT

)

νlq = ξν(f)lq + (1− ξ)ν(m) −

ν(f)lq − ν

(m)

1/K(f)T − 1/K

(m)T

(

ξ

K(f)T

+1− ξ

K(m)T

−1

KT

)

(1.5)

4

(Hill, 1964) allow the effective moduli El and νlq to be expressed by KT, some constituentproperties, and the fiber volume fraction ξ. Equations (1.5) can be used to reduce thenumber of independent effective elastic parameters required for describing unidirectionalcontinuously reinforced composites to three.

It is worth noting that the overall material symmetries of inhomogeneous materials andtheir effect on various physical properties can be treated in full analogy to the symmetriesof crystals as discussed, e.g., by Nye (1957). The effects of the overall symmetry of thephase arrangement on the overall mechanical behavior of inhomogeneous materials can bemarked4, especially on the post-yield and other nonlinear responses to mechanical loads,compare Nakamura and Suresh (1993) or Weissenbek (1994). Accordingly, even thoughthe overall behavior of actual multiphase materials may deviate to some minor degree fromthe ideal material symmetries discussed above, it is important to stay as close as possibleto the appropriate overall symmetry in any modeling effort.

Layer-type (“generalized plane stress”) material tensors for use with lamination andlayered shell theories (which are important for mesoscopic and macroscopic descriptions ofcomponents made of composites materials, compare part I of this lecture) can be obtainedfrom the fully three-dimensional material tensors given in eqns. (1.3) and (1.4) as

EL =

E11,L E12,L −− 0 0 0E12,L E22,L −− 0 0 0−− −− −− −− −− −−0 0 −− E44,L 0 00 0 −− 0 E55,L 00 0 −− 0 0 E66,L

αL =

αl

αq

−−000

, (1.6)

with Eij,L = Eij −Ej3

E33Ei3, compare Dorninger (1989).

1.4 Basic Modeling Strategies in Continuum Microme-

chanics of Materials

Homogenization methods aim at finding a volume element’s responses to prescribed me-chanical loads (typically far field stresses or far field strains) or temperature excursionsand to deduce from them the overall thermomechanical tensors or moduli. The moststraightforward application of such studies is materials characterization, i.e., simulatingthe material response under simple loading conditions such as uniaxial tensile tests. Anal-ysis of this kind can be performed by all approaches described here (with the possibleexception of bounding methods).

Many homogenization procedures can also be employed directly as micromechanicallybased constitutive material models at higher length scales, i.e., they can link general macro-scopic stress tensors to the corresponding strain tensors. This, of course, requires the ca-pability of describing the overall response of an inhomogeneous material under any loading

4Overall properties described by tensors of order 2, e.g., the thermal expansion response, are much lesssensitive to symmetry effects than the elastic behavior, compare Nye (1957).

5

condition and for any loading history. Compared to semi-empirical constitutive laws, asproposed, e.g., by Davis (1996), micromechanically based constitutive models have both aclear physical basis and an inherent capability for “zooming in” on the local phase stressesby using localization procedures. The latter allow the local response of the phases to befound when the macroscopic response or state of a sample or structure is known, e.g., forpredicting plastic yielding of the constituents or for assessing the initiation and progressof microscopic damage under given macroscopic loads.

In addition to materials characterization and constitutive modeling, there are a numberof other important applications of continuum micromechanics, among them studying localphenomena in inhomogeneous materials, e.g., the nucleation and growth of cracks, thestresses at intersections between macroscopic interfaces and free surfaces, the interactionsbetween phase transformations and microstresses, and the fields in the vicinity of cracktips. For the latter behaviors details of the microstructure tend to be of major importanceand can even determine the macroscopic response, an extreme case being the mechanicalstrength of brittle inhomogeneous materials.

Because for realistic phase distributions the analysis of the spatial variations of themicrofields in sufficiently large volume elements is typically beyond present capabilities,approximations have to be used. The majority of the resulting modeling approaches maybe classified as falling into two groups. The first of them comprises methods that de-scribe the microgeometries of inhomogeneous materials on the basis of (limited) statisticalinformation:

• Mean Field Approaches (MFAs) and related methods (see chapter 3): The microfieldswithin each constituent of an inhomogeneous material are approximated by theirphase averages 〈ε〉(p) and 〈σ〉(p), i.e., piecewise (phase-wise) uniform stress and strainfields are employed. Such models typically use information on the microscopic topol-ogy, the reinforcement shape and orientation, and, in some cases, on statistical de-scriptors of details of the phase geometry. The localization relations then take theform

〈ε〉(p) = A(p)〈ε〉

〈σ〉(p) = B(p)〈σ〉 (1.7)

and the homogenization relations can be written as

〈ε〉(p) = 1Ω(p)

∫

Ω(p)

ε(x) dΩ with 〈ε〉 =∑

pV (p)〈ε〉(p)

〈σ〉(p) = 1Ω(p)

∫

Ω(p)

σ(x) dΩ with 〈σ〉 =∑

pV (p)〈σ〉(p) (1.8)

where (p) stands for a given phase of the composite, Ω(p) is the corresponding phasevolume, and V (p) = Ω(p)/

∑

k Ω(k) is the volume fraction of the phase. Note that, incontrast to eqn. (1.1), for MFAs the phase concentration tensors A and B are notfunctions of the spatial coordinates within the volume element.Mean Field Approaches tend to be formulated in terms of the phase concentrationtensors, eqn. (1.7), and they pose relatively low computational requirements. They

6

have been highly successful in describing the thermoelastic response of compositesand other inhomogeneous materials; their use in modeling elastoplastic inhomoge-neous materials is the subject of ongoing research.

• Bounding Methods (see chapter 4): Variational principles are used to obtain upperand (in many cases) lower bounds on the overall elastic tensors, elastic moduli, secantmoduli, and other physical properties of inhomogeneous materials. Bounds — asidefrom their intrinsic interest — are important tools for assessing other models ofinhomogeneous materials. In addition, in most cases one of the bounds providesgood estimates for the physical property under consideration, even if the bounds arerather slack (Torquato, 1991). No variational bounds are available for the microfields.Many bounding methods are closely related to mean field estimates.

The second group of approximations are based on studying discrete microstructures andincludes the following groups of models, compare the sketches in fig. 1.2.

• Periodic Microfield Approaches (PMAs) or Periodic Homogenization (see chapter6): The inhomogeneous material is approximated by an infinitely extended periodicmodel material. The corresponding periodic microfields are typically obtained by an-alyzing unit cells (which may describe microgeometries of a wide range of complexity)via analytical or numerical methods. Periodic homogenization can handle constitu-tive modelling for both linear and nonlinear behaviors and, at present, is the mostflexible method of continuum micromechanics. The high resolution of the microfieldsprovided by PMAs can be highly useful for studying the initiation of damage at themicroscale. However, because they inherently give rise to periodic configurations ofall relevant fields, PMAs are not suited to investigating phenomena involving damagethat is localized rather than diffuse (e.g., the interaction of the microstructure witha macroscopic crack). Periodic microfield approaches can give detailed informationon the local stress and strain fields within a given unit cell.

• Embedded Cell or Embedding Approaches (ECAs; see chapter 7): The behavior ofthe inhomogeneous material is approximated by a model consisting of a “core”, i.e.,a resolved discrete phase arrangement, that is embedded within some outer regionto which far field loads are applied. The material properties of this outer region maybe described by some macroscopic constitutive law, they can be determined (quasi)self-consistently from the behavior of the core, or the embedding region may take theform of a coarse description and/or discretization of the phase arrangement. ECAsare highly useful for materials characterization, and they are usually the best choicefor modeling regions of special interest, such as crack tips and their surroundings,in inhomogeneous materials. Embedding approaches can be used when the lengthscales are not well resolved, e.g., when then far field loads or the composition of thematerial are not uniform. They can resolve local stress and strain fields in the coreregion at high detail but tend to be computationally expensive.

• Windowing Approaches (see chapter 8): Subregions of simple shape (“windows”) arechosen at random from a given phase arrangement and are subjected to macrohomo-geneous and/or mixed uniform boundary conditions. The former type of boundaryconditions give rise to lower and upper estimates and bounds for the moduli or ten-sors describing the overall behavior of a given inhomogeneous material, whereas the

7

PERIODIC APPROXIMATION,PHASE ARRANGEMENTUNIT CELL

EMBEDDED CONFIGURATION WINDOW

Figure 1.2: Schematic sketch of a random matrix–inclusion microstructure and of thevolume elements used by a periodic microfield method (which employs a slightly differ-ent periodic “model” microstructure), an embedding scheme and a windowing model forstudying this inhomogeneous material.

latter lead to estimates. At present the main fields of applications of windowingapproaches involve linear behaviors of inhomogeneous materials.

• For small inhomogeneous samples (i.e., samples the size of which exceeds the mi-croscale by not more than, say, an order of magnitude) the full microstructure canbe modeled and studied, see, e.g., Papka and Kyriakides (1994). In such samplesboundary condition effects can play a prominent role. Since no transitions betweenlength scales are involved in problems of this type, they are not micromechanicalmethods in the strict sense.

Further descriptions of inhomogeneous materials, such as rules of mixtures (isostrainand isostress models) and semi-empirical formulae like the Halpin–Tsai equations (Halpinand Kardos, 1976), which in most cases have a rather weak physical background and lim-ited predictive capabilities, are given a short discussion in chapter 2, where also a numberof physically based models, such as expressions for the self-similar composite sphere as-semblages (CSA) of Hashin (1962) and composite cylinder assemblages (CCA) of Hashinand Rosen (1964), are introduced.

For studying materials that are inhomogeneous at a number of (sufficiently different)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 exten-

8

sion of the above concepts. Such multi-scale models are the subjects of a short discussionin chapter 9.

All of the continuum micromechanical methods discussed in the present course notesare suitable for homogenization problems. Mean field and periodic microfield procedurescan be used for localization tasks without major restrictions, whereas bounding methodsare not targeted at localization. Embedding and windowing methods show boundary layersin the vicinity of the surface of the volume elements, where local fields are perturbed bythe embedding material or by the prescribed uniform boundary conditions.

The present course notes concentrate on “classical composites”, i.e., inhomogeneousmaterials showing matrix–inclusion microtopology, and they are limited to the special caseof two-phase composites. Most of the expressions and methods can, however, be extendedto multi-phase materials.

9

Chapter 2

Some Basic Analytical Approaches

for Thermoelastic Composites

The analytical approaches described in the present section are not mean field methods,but some of them can be used within a mean field framework. They are, in general,considerably simpler than mean field methods proper, but tend to be less accurate and/orless flexible.

2.1 Rules of Mixture

In the most general case, “rule of mixture” expressions for some scalar effective physicalproperty Ψ of a two-phase composite take the form

Ψ =[

ξ(Ψ(i))β + (1− ξ)(Ψ(m))β]

1β , (2.1)

where (i) and (m) denote inhomogeneities (fibers or particles) and matrix, respectively, andthe exponent β must be chosen to obtain a good fit to experimental data.

The rule of mixture expressions given in sections I–1.1.1 and I–1.3.2a are special casesof eqn.(2.1) with β = 1 (Voigt models) and β = −1 (Reuss models), respectively, which,in contrast to most other choices of β, have clear physical interpretations. Voigt-type ex-pressions correspond to full strain coupling of the phases (“springs in parallel”, isostrainmodels, arithmetic averages) and Reuss-type expressions to full stress coupling (“springsin series”, isostress models, harmonic averages), i.e., they describe the in-plane and out-of-plane behavior, respectively, of a layered material made up of two materials having thesame Poisson number. The usefulness of Voigt- and Reuss-type expressions for actual com-posites depends strongly on the given microtopology and material properties. They areclosely related to the Hill bounds, eqn. (4.1), and, with few exceptions, their predictionslie outside the Hashin–Shtrikman bounds discussed in section 4.1.

Even though it neglects Poisson effects, the Voigt-type expression

El = ξE(f)l + (1− ξ)E(m) (2.2)

is usually a very good approximation to the axial Young’s modulus of continuously rein-forced unidirectional composites. It can also be extended to nonlinear material behavior of

10

the phases, giving reasonable results, e.g., for the axial response of continuously reinforcedunidirectional MMCs. Furthermore, the assumption of strain coupling gives rise to thefollowing simple, but useful expression for the axial thermal expansion behavior of longfiber reinforced UD composites

αl =ξE

(f)l α

(f)l + (1− ξ)E(m)α

(m)l

ξE(f)l + (1− ξ)E(m)

. (2.3)

Reuss-type models for the overall behavior of particle reinforced materials or for the trans-verse and shear properties of continuously reinforced composites typically generate exces-sively soft macroscopic responses.

Rules of mixture can in principle be used to generate effective “elastic tensors” (andconsequently, using eqn.(3.13), concentration tensors) that may be employed in a meanfield framework. Because they do not intrinsically account for the relationships betweenthe engineering moduli, however, such procedures are either inconsistent or non-unique.

Another semi-empirical description of the overall properties of composites is given bythe Halpin–Tsai equations (Halpin and Kardos, 1976),

Ψ = Ψ(m) 1 + ξζΨηΨ

1− ξηΨwith ηΨ =

Ψ(i) −Ψ(m)

Ψ(i) + ζΨΨ(m), (2.4)

(compare section I–1.1.1), which can be viewed as a fit giving the correct behavior for thelimiting cases ξ = 0 and ξ = 1. The fitting parameter ζΨ depends on the microgeometry ofthe composite and on the physical property to be described. The Halpin–Tsai equationshave been used mostly in assessing experimental results and are rather limited in theirpredictive capabilities. For them, too, predictions for different engineering moduli are notnecessarily consistent in terms of the material’s overall symmetry.

2.2 The VFD Model of Dvorak and Bahei-el-Din

An approach that is closely related to the rules of mixture, but gives consistent and uniqueoverall material tensors for unidirectional continuously reinforced composites is known asthe Vanishing Fiber Diameter (VFD) model (Dvorak and Bahei-el Din, 1982). The physi-cal interpretation of the VFD model is a composite containing aligned and continuous, butinfinitely thin fibers (which strongly influence the axial effective behavior, but affect thetransverse behavior of the composite only via the macroscopic Poisson effect) in a matrix.

VFD expressions are obtained by employing Voigt-type formulae for the effective axialYoung’s modulus and the effective axial Poisson number, whereas Reuss-type expressionsare used for the axial and transverse shear moduli and for the transverse bulk modulus

El = ξE(f) + (1− ξ)E(m)

νlq = ξν(f) + (1− ξ)ν(m)

Glq = Gqt = [ξ/G(f) + (1− ξ)/G(m)]−1

KT = [ξ/K(f)T + (1− ξ)/K

(m)T ]−1 . (2.5)

11

These equations do not necessarily fulfill Hill’s relations, eqn (1.5).

The VFD model tends to underestimate the effective transverse stiffness of composites(typically some predictions lie outside the Hashin–Shtrikman bounds, see section 4.1) andthe mean hydrostatic stresses in the matrix. Due to its simplicity, however, it has beena popular description for continuously reinforced elastoplastic and viscoelastoplastic UDcomposites, giving good results for fiber dominated properties and reasonable predictionsfor the strain hardening behavior in matrix dominated deformation modes.

2.3 The CSA and CCA Models

The composite sphere assemblage (CSA) model of Hashin (1962) and the composite cylin-der assemblage (CCA) model of Hashin and Rosen (1964) are of considerable interestbecause they give exact expressions for some engineering moduli of special, but fairlyrealistic, phase arrangements that represent particle reinforced and aligned continuouslyreinforced composites, respectively. It may be noted that even though the derivations ofthe CCA and CSA methods do not involve phase averaged microstresses and strains, theresults can be directly interpreted in terms of mean fields.

Both CSA and CCA are based on analyzing control volumes tightly packed with ei-ther composite spheres or composite cylinders of different diameters. The cores of thesecomposite spheres/cylinders consist of the reinforcement, and the matrix is placed in aconcentric shell of the appropriate thickness to give the desired volume fractions, com-pare fig. 2.1. These representative volume elements are subjected to suitable homogeneousboundary conditions, from which the boundary conditions for a single cylinder or sphereare deduced. The appropriate differential equations governing elastic deformation of thecomposite spheres or cylinders are then solved and the overall moduli are obtained.

Figure 2.1: Microgeometry corresponding to the CCA and CSA methods (Hashin, 1983)

12

If this process is carried out for the radial deformation of composite spheres, the CSAexpression for the effective bulk modulus K of a particle reinforced composite results as

K = K(m) +ξ

1/(K(i) −K(m)) + 3(1− ξ)/(3K(m) + 4G(m)). (2.6)

For continuously reinforced unidirectional composites, the load cases of uniform axial ex-tension, radial deformation in the transverse plane, as well as uniform axial shearing dis-placements and tractions can be handled, giving rise to the CCA expressions

El = ξE(f)l + (1− ξ)E(m) +

4ξ(1− ξ)(ν(f)lq − ν

(m))2

(1− ξ)/K(f)T + ξ/K

(m)T + 1/G(m)

νlq = ξν(f)lq + (1− ξ)ν(m) +

ξ(1− ξ)(ν(f)lq − ν

(m))(1/K(m)T − 1/K

(f)T )

(1− ξ)/K(f)T + ξ/K

(m)T + 1/G(m)

KT = K(m)T +

ξ

1/(K(f)T −K

(m)T ) + (1− ξ)/(K

(m)T +G(m))

Glq = G(m) +ξ

1/(G(f)lq −G

(m)) + (1− ξ)/2G(m). (2.7)

Shear loading of composite spheres and transverse shear loading as well as transverse ex-tension of composite cylinders cannot be handled within this framework, so that the moduliG (in the case of particle reinforced composites) and Eq, Gqt as well as νqt (in the caseof continuously reinforced UD composites) must be evaluated by other means, the usualchoice being a three-phase self-consistent scheme, see section 3.5.

13

Chapter 3

Mean Field Methods

Mean field methods in continuum micromechanics aim at obtaining the overall propertiesof inhomogeneous materials, such as their overall elasticity and compliance tensors, E andC, respectively, and their overall tensor of coefficients of thermal expansion (CTEs), α, interms of the appropriate phase properties and of information on the phase topology andgeometry. The descriptions are based on phase averaged stress and strain fields in theconstituents, in terms of which localization is also carried out.

3.1 General Relations between Mean Fields in Ther-

moelastic Two-Phase Materials

In the present chapter, a number of general relations between the phase averaged fields,the overall material tensors and the phase concentration tensors are given and mean fieldmodels for dilute and non-dilute composites are introduced.

For thermoelastic materials, the overall stress–strain relations take the form

〈σ〉 = E〈ε〉+ ϑ∆T

〈ε〉 = C〈σ〉+ α∆T , (3.1)

where ϑ = −Eα is the overall specific thermal stress tensor (i.e., the overall stress responseof the fully constrained material to a purely thermal unit load) and ∆T is the (spatiallyhomogeneous) temperature difference with respect to some stress-free reference tempera-ture.

The phase averaged stresses and phase averaged strains in the material’s constituentscan be defined as

〈σ〉(m) =1

Ω(m)

∫

Ω(m)

σ(x)dΩ 〈σ〉(i) =1

Ω(i)

∫

Ω(i)

σ(x)dΩ

〈ε〉(m) =1

Ω(m)

∫

Ω(m)

ε(x)dΩ 〈ε〉(i) =1

Ω(i)

∫

Ω(i)

ε(x)dΩ , (3.2)

where Ω(m) and Ω(i) are the volumes taken up by matrix and reinforcement, respectively,with Ωs = Ω(m) + Ω(i). The phases themselves are taken to behave thermoelastically, i.e.,

14

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

〈ε〉(m) = C(m)〈σ〉(m) + α(m)∆T 〈ε〉(i) = C(i)〈σ〉(i) + α(i)∆T , (3.3)

where ϑ(m) = −E(m)α(m) and ϑ(i) = −E(i)α(i).

From the definition of volume averaging, eqn. (3.2), the relations between the phaseaveraged fields,

〈ε〉 = ξ〈ε〉(i) + (1− ξ)〈ε〉(m) = εa

〈σ〉 = ξ〈σ〉(i) + (1− ξ)〈σ〉(m) = σa , (3.4)

are immediately obtained, where ξ = V (i) = Ω(i)/Ωs stands for the volume fraction of thereinforcements and 1 − ξ = V (m) = Ω(m)/Ωs for the volume fraction of the matrix. εa

and σa are the far field (applied) homogeneous stress and strain fields, respectively, withεa = Cσa. Perfect interfaces between the phases are assumed in expressing the macro-scopic strain of the composite as the weighted sum of the phase averaged strains.

The phase averaged strains and stresses can be related to the overall strains and stressesby the phase strain and stress concentration tensors A, η, B, and β (Hill, 1963), respec-tively, which are defined for thermoelastic inhomogeneous materials by the expressions

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

〈σ〉(m) = B(m)〈σ〉+ β(m)

∆T 〈σ〉(i) = B(i)〈σ〉+ β(i)

∆T , (3.5)

compare eqn. (1.7) for the purely elastic case.

By using eqns. (3.4) and (3.5), the phase averaged phase strain and stress concentrationtensors (which will be called strain and stress concentration tensors, respectively, from nowon) can be shown to fulfill the relations

ξA(i) + (1− ξ)A(m) = I ξη(i) + (1− ξ)η(m) = o

ξB(i) + (1− ξ)B(m) = I ξβ(i)

+ (1− ξ)β(m)

= o , (3.6)

where I stands for the rank 4 (symmetric) unit tensor and o for the rank 2 null tensor.

The effective elasticity and compliance tensors, respectively, of the composite can beevaluated from the properties of the phases and from the concentration tensors using therelationships

E = ξE(i)A(i) + (1− ξ)E(m)A(m)

= E(m) + ξ[E(i) − E(m)]A(i) = E(i) + (1− ξ)[E(m) − E(i)]A(m) (3.7)

C = ξC(i)B(i) + (1− ξ)C(m)B(m)

= C(m) + ξ[C(i) −C(m)]B(i) = C(i) + (1− ξ)[C(m) −C(i)]B(m) (3.8)

15

and the effective thermal expansion coefficient tensor, α, and the specific thermal stresstensor, ϑ, can be obtained as

α = ξ[C(i)β(i)

+ α(i)] + (1− ξ)[C(m)β(m)

+ α(m)]

= ξα(i) + (1− ξ)α(m) + (1− ξ)[C(m) −C(i)]β(m)

= ξα(i) + (1− ξ)α(m) + ξ[C(i) −C(m)]β(i)

. (3.9)

ϑ = ξ[E(i)η(i) + ϑ(i)] + (1− ξ)[E(m)η(m) + ϑ(m)]

= ξϑ(i) + (1− ξ)ϑ(m) + (1− ξ)[E(m) − E(i)]η(m)

= ξϑ(i) + (1− ξ)ϑ(m) + ξ[E(i) − E(m)]η(i) . (3.10)

The above expressions can be derived by inserting eqns. (3.3) and (3.5) into eqns. (3.4)and comparing with eqns. (3.1). In addition, α can be obtained as

α = ξ(B(i))Tα(i) + (1− ξ)(B(m))Tα(m) , (3.11)

an expression known as the Levin (1967) formula.

By invoking the principle of virtual work relations were developed (Benveniste and Dvo-rak, 1990; Benveniste et al., 1991) which link the thermal strain concentration tensors, η(p),to the mechanical strain concentration tensors, A(p), and the thermal stress concentration

tensors, β(p)

, to the mechanical stress concentration tensors, B(p), respectively,

η(m) = [I− A(m)][E(i) − E(m)]−1[ϑ(m) − ϑ(i)]

η(i) = [I− A(i)][E(m) − E(i)]−1[ϑ(i) − ϑ(m)]

β(m)

= [I− B(m)][C(i) −C(m)]−1[α(m) −α(i)]

β(i)

= [I− B(i)][C(m) −C(i)]−1[α(i) −α(m)] . (3.12)

Furthermore, by using eqns. (3.7) and (3.8), the elastic strain and stress concentrationtensors can be obtained from the effective plus the phase elasticity and compliance tensors,respectively, as

A(m) =1

1− ξ[E(m) −E(i)]−1[E− E(i)] A(i) =

1

ξ[E(i) − E(m)]−1[E− E(m)]

B(m) =1

1− ξ[C(m) −C(i)]−1[C−C(i)] B(i) =

1

ξ[C(i) −C(m)]−1[C−C(m)] . (3.13)

Analogous expressions for the thermal strain and stress concentration tensors result as

η(m) =1

1− ξ[E(m) − E(i)]−1[ϑ− ξϑ(i) − (1− ξ)ϑ(m)]

η(i) =1

ξ[E(i) −E(m)]−1[ϑ− ξϑ(i) − (1− ξ)ϑ(m)]

β(m)

=1

1− ξ[C(m) −C(i)]−1[α− ξα(i) − (1− ξ)α(m)]

β(i)

=1

ξ[C(i) −C(m)]−1[α− ξα(i) − (1− ξ)α(m)] . (3.14)

16

In addition, the following set of equations can be derived, which provide relations betweenthe stress and strain concentration tensors of each phase

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

A(i) = C(i)B(i)E(m)[I + ξ(C(i) −C(m))B(i)E(m)]−1 = C(i)B(i)E (3.15)

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

B(i) = E(i)A(i)C(m)[I + ξ(E(i) − E(m))A(i)C(m)]−1 = E(i)A(i)C . (3.16)

From eqns. (3.12), (3.15) and (3.16) it is evident that the knowledge of one elastic phaseconcentration tensor is sufficient for describing the full thermoelastic behavior of a two-phase inhomogeneous material within the mean field framework5.

Finally, the energetic equivalence in elasticity between the behaviors on the micro- andmacroscales can be expressed by the relation

1

2〈σT ε〉 =

1

2Ω

∫

Ω

σT(x) ε(x) dΩ =1

2〈σ〉T〈ε〉 , (3.17)

where the σ(x) are general statically admissible stress fields and the ε(x) are generalkinematically admissible strain fields. Equation (3.17) is known as Hill’s macrohomogeneitycondition or the Mandel–Hill condition. It states that the volume average of the elasticenergy density evaluated on the basis of the microscopic fields must be equal to the elasticenergy density of the homogenized material.

3.2 Misfit Strains: Eshelby’s Solution

A large proportion of the mean field descriptions used in continuum micromechanics ofmaterials are based on the work of Eshelby (1957), who studied the stress and strain dis-tributions in homogeneous media that contain a subregion that spontaneously changes itsshape and/or size (undergoes a “transformation”) so that it no longer fits into its previ-ous space in the “parent medium”. Eshelby’s results show that if an elastic homogeneousellipsoidal inclusion (i.e., an inclusion consisting of the same material as the matrix) inan infinite matrix is subjected to a homogeneous strain εt (called the “stress-free strain”,“unconstrained strain”, “eigenstrain”, or “transformation strain”), the stress and strainstates in the constrained inclusion are uniform, i.e., σ(i) = 〈σ〉(i) and ε(i) = 〈ε〉(i) (Eshelby,1957). The uniform strain in the constrained inclusion (the “constrained strain”), εc, isrelated to the stress-free strain εt by the expression

εc = Sεt (3.18)

where S is referred to as the Eshelby tensor. For eqn. (3.18) to hold, εt may be any kindof eigenstrain which is uniform over the inclusion (e.g., a thermal strain or a strain dueto some phase transformation which involves no changes in the elastic constants of theinclusion).

5Similarly, n−1 elastic phase concentration tensors must be known for describing the overall thermoe-lastic behavior of an n-phase material.

17

For spheroidal inclusions (i.e., ellipsoids of rotation) in an isotropic matrix, S can beevaluated analytically and depends only on the Poisson’s ratio of the homogeneous material(or, in the case of inhomogeneous inclusions, on the Poisson’s ratio of the matrix) and onthe aspect ratio a of the inclusion. Under these conditions the nonzero components of theEshelby tensor can be expressed as

S(1, 1) =1

2(1− ν(m))

1− 2ν(m) +3a2 − 1

a2 − 1−

[

1− 2ν(m) +3a2

a2 − 1

]

g(a)

S(2, 2) = S(3, 3) =3

8(1− ν(m))

a2

a2 − 1+

1

4(1− ν(m))

[

1− 2ν(m) −9

4(a2 − 1)

]

g(a)

S(1, 2) = S(1, 3) = −1

2(1− ν(m))

1− 2ν(m) +1

a2 − 1+

[

1− 2ν(m) +3

2(a2 − 1)

]

g(a)

S(2, 1) = S(3, 1) = −1

2(1− ν(m))

a2

a2 − 1−

1

4(1− ν(m))

[

3a2

a2 − 1− (1− 2ν(m))

]

g(a)

S(2, 3) = S(3, 2) =1

4(1− ν(m))

a2

2(a2 − 1)−

[

1− 2ν(m) +3

4(a2 − 1)

]

g(a)

S(4, 4) = S(5, 5) =1

2(1− ν(m))

1− 2ν(m) −a2 + 1

a2 − 1−

1

2

[

1− 2ν(m) −3(a2 + 1)

a2 − 1

]

g(a)

S(6, 6) =1

2(1− ν(m))

a22(a2 − 1) +

[

1− 2ν(m) −3

4(a2 − 1)

]

g(a)

(3.19)

(Tandon and Weng, 1984). Here the 1-direction is the axis of rotation of the spheroid anda stands for the aspect ratio of the inclusions (i.e., a is given by the length of the axis ofrotation of the spheroids divided by their diameter, so that for continuous cylindrical fibersa → ∞, for spherical inclusions a = 1, and for infinitely thin circular disks or plateletsa→ 0). The function g(a) is given by the expressions

g =a

(a2 − 1)3/2[a(a2 − 1)1/2 − arcosh a]

for fiberlike (prolate) inclusions (a > 1) and

g =a

(1− a2)3/2[arccos a− a(1− a2)1/2]

for disklike (oblate) inclusions (a < 1).

For the special case of spherical inclusions (a = 1) in an isotropic matrix, the onlynonzero components of the Eshelby tensor are

S(1, 1) = S(2, 2) = S(3, 3) =7− 5ν(m)

15(1− ν)

S(1, 2) = S(2, 1) = S(1, 3) = S(3, 1) = S(2, 3) = S(3, 2) =5ν(m) − 1

15(1− ν(m))

S(4, 4) = S(5, 5) = S(6, 6) =2(4− 5ν(m))

15(1− ν(m))(3.20)

and for inclusions in the form of continuous fibers (a → ∞) of circular cross section theytake the form

18

S(2, 2) = S(3, 3) =5− 4ν(m)

8(1− ν(m))

S(2, 3) = S(3, 2) =4ν(m) − 1

8(1− ν(m))

S(2, 1) = S(3, 1) =ν(m)

2(1− ν(m))

S(4, 4) = S(5, 5) =1

2

S(6, 6) =3− 4ν(m)

4(1− ν(m)). (3.21)

Equations (3.19) to (3.21) follow the conventions of Nye notation as discussed in theintroductory notes, with the shear terms in the strain vector ε being the shear anglesγij = 2εij. A similar notation is employed by Pedersen (1983), but a number of standardworks dealing with misfitting inclusions, such as Mura (1987), use different conventionsand consequently give different expressions for S(4, 4), S(5, 5) and S(6, 6).

3.3 Dilute Inhomogeneous Inclusions

Mean field methods for dilute inhomogeneous matrix–inclusion composites typically aim atmaking use of Eshelby’s expressions for the fields in a homogeneous inclusion subjected toan eigenstrain by using the concept of an equivalent homogeneous inclusion. This strategyinvolves replacing the actual inhomogeneous inclusion (often referred to as an inhomogene-ity), which has different material properties than the matrix and which is subjected to agiven unconstrained eigenstrain εt, with a (fictitious) “equivalent” homogeneous inclusionon which a (fictitious) “equivalent” eigenstrain ετ is made to act. This equivalent eigen-strain must be chosen in such a way that the same stress and strain fields are obtained inthe actual inhomogeneous inclusion and in the fictitious homogeneous inclusion.

Following the strategy outlined above, the case of inhomogeneous inclusions is handledby investigating an equivalent homogeneous inclusion subjected to an appropriate equiva-lent eigenstrain ετ . The latter is chosen in such a way that the inhomogeneous inclusionand the equivalent homogeneous inclusion attain the same stress state σ(i) and the sameconstrained strain εc (Eshelby, 1957; Withers et al., 1989). When σ(i) is expressed in termsof the elastic strain in the inhomogeneity or inclusion, this condition translates into theequality

σ(i) = E(i)(εc − εt) = E(m)(εc − ετ ) . (3.22)

Here εc − εt and εc − ετ are the elastic strains in the inhomogeneity and the equivalenthomogeneous inclusion, respectively. Obviously, in the general case the stress-free strainswill be different for the equivalent inclusion and the real inhomogeneity, εt 6= ετ .

A typical route (but not the only one) for obtaining mean field descriptions of inhomo-geneous materials subjected to an eigenstrain consists of first expressing the appropriate

19

equivalent eigenstrain of an equivalent homogeneous inclusion in terms of the constituents’material properties, the shape of the inclusion, and the stress-free strain of the inhomo-geneous inclusion. This equivalent eigenstrain is then used to generate the expressionsfor the average phase stresses and strains. Because eqn. (3.18) was derived for a generalhomogeneous inclusion problem, it will also hold for the equivalent homogeneous inclusion,for which it takes the form

εc = Sετ . (3.23)

By inserting this expression into eqn. (3.22) the relation

σ(i) = E(i)(Sετ − εt) = E(m)(S− I)ετ (3.24)

is obtained, which can be rearranged to express the equivalent eigenstrain ετ acting on thehomogeneous inclusion in terms of the known stress-free eigenstrain εt of the real inclusionas

ετ = [(E(i) −E(m))S + E(m)]−1E(i)εt . (3.25)

By substituting eqn. (3.25) into the right hand side of eqn. (3.24), the stress in the inho-mogeneity, σ(i), is finally obtained as

σ(i) = E(m)(S− I)[(E(i) −E(m))S + E(m)]−1E(i)εt

= E(m)RdilE(i)εt , (3.26)

where the tensor Rdil is defined as

Rdil = (S− I)[(E(i) − E(m))S + E(m)]−1 . (3.27)

As intended, eqn. (3.26) expresses the average stress in an inhomogeneous ellipsoidal in-clusion in terms of the Eshelby tensor S (and thus of the inhomogeneity aspect ratio a),of the elastic tensors of matrix and inhomogeneity, and of the stress-free strain εt, all ofwhich are known.

If a uniform external stress σa is applied to an inhomogeneous elastic matrix–inclusionsystem, the stress in the inhomogeneity, σ(i), will be a superposition of this applied stressand of some additional stress caused by the constraining effect of the surrounding matrixon the inhomogeneity. Such problems can be treated by extending of the strategy followedin eqns. (3.23) to (3.27) and introducing an equivalent homogeneous inclusion subjected toboth the applied stress σa and to a suitable equivalent eigenstrain ετ . Again, this equiv-alent eigenstrain is chosen in such a way that the stress, σ(i), and the constrained strain,εc, in the inhomogeneity are the same in the inhomogeneous and the equivalent homo-geneous cases. Alternatively, a uniform far field strain εa may be used as far field load.The equivalent eigenstrain depends on both the applied stress σa (or applied strain εa)and, if present, on the eigenstrain of the inclusion, εt, i.e., ετ = ετ (σ

a, εt) or ετ = ετ (εa, εt).

By writing the stress in the inhomogeneity as E(i)(εa+εc−εt) and that in the equivalenthomogeneous inclusion as E(m)(εa + εc − ετ ), i.e., in terms of the elastic tensors and theelastic strains, the relation

σ(i) = E(i)(εa + εc − εt) = E(m)(εa + εc − ετ ) (3.28)

20

is obtained. Inserting Eshelby’s result in the form of eqns. (3.23) to (3.27) and solving forthe equivalent eigenstrain ετ yields

ετ = [(E(i) − E(m))S + E(m)]−1[(E(m) −E(i))εa + E(i)εt] , (3.29)

which, as required, contains a contribution due to the applied strain εa as well as a term dueto the unconstrained eigenstrain εt. Substituting eqn. (3.29) into the r.h.s. of eqn. (3.28)allows to express the stress in the inhomogeneity as

σ(i) = E(m)I + (S− I)[(E(i) − E(m))S + E(m)]−1(E(m) − E(i))εa +

E(m)(S− I)[(E(i) − E(m))S + E(m)]−1E(i)εt . (3.30)

Because for a single inhomogeneity in an infinite matrix the relationship εa = Cσa =C(m)σa holds, eqn. (3.30) can be rewritten as

σ(i) = E(m)I + (S− I)[(E(i) −E(m))S + E(m)]−1(E(m) −E(i))C(m)σa +

E(m)(S− I)[(E(i) − E(m))S + E(m)]−1E(i)εt

= E(m)[I + Rdil(E(m) −E(i))]C(m)σa + E(m)RdilE

(i)εt (3.31)

where Rdil is defined by eqn. (3.27).

By comparing the definition of the inhomogeneity stress concentration tensor B(i),eqns. (1.7) and (3.5), with eqn. (3.31), by equating the effective stress in the compos-ite, 〈σ〉, with the applied stress, σa, and by setting the transformation strain to zero,εt = 0, an expression for the stress concentration tensor of a dilute composite can bedirectly obtained as

B(i)dil = E(m)[I + Rdil(E

(m) − E(i))]C(m) . (3.32)

The thermal stress concentration tensor for the inhomogeneities, β(i)

, can be found bysetting σa = 〈σ〉 = 0 and inserting the thermal mismatch strain for the unconstrainedeigenstrain in eqn. (3.31), i.e., εt = (α(i) − α(m))∆T , and then comparing with eqn. (3.2),which gives

β(i)dil = E(m)RdilE

(i)(α(i) −α(m)) . (3.33)

The matrix stress concentration tensors corresponding to eqns. (3.32) and (3.33) can beobtained easily by using eqn. (3.6).

Alternative expressions for the stress and strain concentration tensors of dilute compos-ites can be obtained by studying an equivalent inclusion in the absence of transformationstrains, i.e., εt = 0 (Hill, 1965; Benveniste, 1987). Under this condition, the strain in theinhomogeneity becomes

ε(i) = 〈ε〉(i) = εa + εc = εa + Sετ (3.34)

and the equivalent to eqn. (3.28) takes the form

σ(i) = E(i)(εa + εc) = E(m)(εa + εc − ετ ) , (3.35)

which, together with eqn. (3.34), provides the expression

E(i)〈ε〉(i) = E(m)(〈ε〉(i) − ετ ) . (3.36)

21

Solving eqn. (3.36) for ετ and substituting into eqn. (3.34) leads to the expression

〈ε〉(i) = [I + SC(m)(E(i) −E(m))]−1εa , (3.37)

from which the dilute inhomogeneity strain concentration tensor follows directly as

A(i)dil = [I + SC(m)(E(i) −E(m))]−1 . (3.38)

By setting 〈ε〉(i) = C(i)〈σ〉(i) and using εa = C(m)σa, the dilute stress concentration tensorfor the inhomogeneities is found from eqn. (3.37) as

B(i)dil = E(i)[I + SC(m)(E(i) −E(m))]−1C(m) . (3.39)

The expressions for stress and strain concentration tensors given in this section werederived for a single inclusion in an infinite matrix and hold for inhomogeneities that are di-lutely dispersed in the matrix and thus do not “feel” any effects of their neighbors (i.e., theyare loaded by the unperturbed applied stress σa). Accordingly, the dilute concentration

tensors for the inhomogeneities, A(i)dil, B

(i)dil and β

(i)dil, are independent of the inhomogeneity

volume fraction ξ. The inhomogeneity volume fraction does, however, enter the corre-sponding expressions for the matrix concentration tensors obtained via eqn. (3.6).

The overall elastic and thermal expansion tensors of dilute matrix–inclusion systemscan be easily obtained by plugging the concentration tensors derived above, eqns. (3.32),(3.33), (3.38) and/or (3.39), into the relations (3.7) to (3.9). It must be kept in mind,however, that the resulting expressions only hold for dilute composites with inhomogeneityvolume fractions in the range ξ ≪ 0.1.

It is worth noting that, within the framework of the equivalent inclusion approach,eqns. (3.19) to (3.21) can be used for any inhomogeneous inclusion in an isotropic ma-trix regardless of the inhomogeneity’s material symmetry. Analytical expressions for theEshelby tensor have also been given for spheroidal inclusions when the matrix shows trans-versely isotropic (Withers, 1989) or cubic material symmetry (Mura, 1987), provided thematerial axes of both constituents are aligned with the orientations of the inhomogeneities.In cases where no analytical solutions are available, the Eshelby tensor can evaluated nu-merically, compare Gavazzi and Lagoudas (1990).

In general, the stress and strain fields outside an inclusion in an infinite matrix arenot uniform on the microscale (Eshelby, 1959) and can be described via the exterior pointEshelby tensor, see, e.g., Ju and Sun (1999). In mean field theories, which aim to linkthe average fields in matrix and inclusions with the overall response of inhomogeneousmaterials, however, it is only the average matrix stresses and strains that are of interest.

3.4 Mori–Tanaka Estimates

Theoretical descriptions of the overall thermoelastic behavior of composites with reinforce-ment volume fractions of more than a few percent must account for the interaction betweeninhomogeneities, i.e., for the effects of the surrounding inhomogeneities on the stress and

22

strain fields experienced by a given fiber or particle. One way for achieving this consists ofapproximating the stresses acting on an inhomogeneity, which may be viewed as perturba-tion stresses caused by the presence of other inhomogeneities superimposed on the appliedfar field stress, by an appropriate average matrix stress. The idea of combining such a con-cept of an average matrix stress with Eshelby-type equivalent inclusion approaches goesback to Brown and Stobbs (1971) as well as Mori and Tanaka (1973). Effective field theo-ries of this type are generically referred to as Mori–Tanaka methods. By construction theydo not invoke explicit (e.g., pair-wise) interactions between “individual” inhomogeneities,but rather operate on a level of collective interactions.

It was pointed out by Benveniste (1987) that in the isothermal case the central assump-tion involved in Mori–Tanaka approaches can be denoted as

〈ε〉(i) = A(i)dil〈ε〉

(m)

〈σ〉(i) = B(i)dil〈σ〉

(m) . (3.40)

Thus, the methodology developed for dilute inhomogeneities is retained and the interac-tions with the surrounding inhomogeneities are accounted for by suitably modifying thestresses or strains acting on each inhomogeneity. Equation (3.40) may thus be viewed as amodification of eqn. (1.7) in which the macroscopic strain or stress, 〈ε〉 or 〈σ〉, is replacedby the phase averaged matrix strain or stress, 〈ε〉(m) and 〈σ〉(m), respectively. In a furtherstep suitable expressions for 〈ε〉(m) and/or 〈σ〉(m) must be introduced into the scheme.

Following Benveniste (1987), inserting eqns. (3.40) into eqns. (3.4) leads directly to rela-tions between the macroscopic strains and stresses on the one hand and the inhomogeneitystrains and stresses on the other hand

〈ε〉 = ξ〈ε〉(i) + (1− ξ)〈ε〉(m) = [ξA(i)dil + (1− ξ)I]〈ε〉(m)

= [ξA(i)dil + (1− ξ)I][A

(i)dil]

−1〈ε〉(i)

〈σ〉 = ξ〈σ〉(i) + (1− ξ)〈σ〉(m) = [ξB(i)dil + (1− ξ)I]〈σ〉(m)

= [ξB(i)dil + (1− ξ)I][B

(i)dil]

−1〈σ〉(i) . (3.41)

Equations (3.41) can be rearranged into the form

〈ε〉(i) = A(i)dil[(1− ξ)I + ξA

(i)dil]

−1〈ε〉

〈σ〉(i) = B(i)dil[(1− ξ)I + ξB

(i)dil]

−1〈σ〉 , (3.42)

from which Mori–Tanaka-expressions for the inhomogeneity strain and stress concentrationtensors for non-dilute composites follow directly 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 . (3.43)

Finally, the corresponding expressions for the matrix strain and stress concentration tensorscan be found by inserting eqns. (3.43) into eqns. (3.41) to give

A(m)M = [(1− ξ)I + ξA

(i)dil]

−1

B(m)M = [(1− ξ)I + ξB

(i)dil]

−1 . (3.44)

23

Equations (3.43) and (3.44) automatically fulfill conditions (3.6) and may be evaluatedwith any strain and stress concentration tensors pertaining to dilute inhomogeneities in amatrix. If the equivalent inclusion expressions, eqns. (3.38) and (3.39), are employed, thestrain and stress concentration tensors for the non-dilute composite 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) . (3.45)

The expressions for the effective elasticity and compliance tensors of the composite quotedin (Benveniste, 1987; Benveniste and Dvorak, 1990) can be recovered as

EM = E(m) + ξ[E(i) − E(m)]A(i)dil[(1− ξ)I + ξA

(i)dil]

−1

CM = C(m) + ξ[C(i) −C(m)]B(i)dil[(1− ξ)I + ξB

(i)dil]

−1 (3.46)

by substituting eqns. (3.45) into eqns. (3.7) and (3.8). Finally, the specific thermal stressand thermal expansion coefficient tensors follow from eqns. (3.10) and (3.9) as

ϑM = ε(m) + ξ[E(i) − E(m)]A(i)dil[(1− ξ)I + ξA

(i)dil]

−1[E(i) −E(m)]−1[ϑ(i) − ϑ(m)]

αM = α(m) + ξ[C(i) −C(m)]B(i)dil[(1− ξ)I + ξB

(i)dil]

−1[C(i) −C(m)]−1[α(i) −α(m)] .

(3.47)

Benveniste’s interpretation of the Mori–Tanaka approach, eqns. (3.43) to (3.47), isequivalent to a method developed by Tandon and Weng (1984), which directly gives ex-pressions for the overall elasticity tensor of two-phase inhomogeneous materials as

ET = E(m)

I− ξ[(E(i) − E(m))(

S− ξ(S− I))

+ E(m)]−1[E(i) − E(m)]

−1. (3.48)

Because eqn. (3.48) does not explicitly use the compliance tensor of the inhomogeneities,C(i), it can be modified in a straightforward way to describe the macroscopic stiffness ofporous materials by setting E(i) → 0, giving rise to the relationship

Epor = E(m)[

I +ξ

1− ξ(I− S)−1

]

−1. (3.49)

which, however, should not be used for void volume fractions that are in excess of, say,ξ = 0.256.

For realistic stiffness ratios between inhomogeneities and matrix (“elastic contrasts”),a number of further Mori–Tanaka-type theories (which use different derivations and obtaindifferent formulas for the concentration tensors), evaluate “to the same numbers” for theconcentration and overall elastic tensors as Benveniste’s approach, among them Pedersen’sMean Field Theory (Pedersen, 1983) and Wakashima’s method (Wakashima et al., 1988).

6Mori–Tanaka theories are based on the assumption that the shape of the inhomogeneities can bedescribed by ellipsoids of a given aspect ratio throughout the deformation history. In porous materialswith high void volume fractions deformation at the microscale takes place mainly by bending and bucklingof cell walls or struts (Gibson and Ashby, 1988). Such effects cannot be described by Mori–Tanaka models,which, consequently, tend to overestimate the effective stiffness of cellular materials by far.

24

As is evident from their derivation, Mori–Tanaka-type theories at all volume fractionsdescribe composites consisting of aligned ellipsoidal inhomogeneities embedded in a ma-trix, i.e., inhomogeneous materials with a distinct matrix–inclusion microtopology. Moreprecisely, it was shown by Ponte Castaneda and Willis (1995) that Mori–Tanaka methodsare a special case of so-called Hashin–Shtrikman variational estimates in which the spatialarrangement of the inhomogeneities follows an “ellipsoidal distribution” with the same as-pect ratio and orientation as the inhomogeneities themselves, compare fig. 3.1.

Figure 3.1: Sketch of ellipsoidal inhomogeneities in an aligned ellipsoidally distributedspatial arrangement as used implicitly in Mori–Tanaka-type approaches (a=2.0).

Mori–Tanaka predictions for the overall Young’s and shear moduli of composites rein-forced by aligned or spherical reinforcements tend to be on the low side (see the comparisonsin section 4.1), but generally are highly useful estimates. For discussions of the range ofvalidity of Mori–Tanaka theories for elastic inhomogeneous two-phase materials see, e.g.,Christensen et al. (1992). In addition to “standard” Mori–Tanaka methods describingmicrogeometries with aligned inhomogeneities having the same material properties, proce-dures have been developed for extending the Mori–Tanaka method to nonaligned or hybridcomposites (i.e., materials with more than one inclusion phase), compare section 3.8.

Mori–Tanaka-type theories can be implemented into computer programs in a straight-forward way. Because they are explicit algorithms, all that is required are matrix additions,multiplications, and inversions plus expressions for the Eshelby tensor. This makes themimportant tools for evaluating the stiffness and thermal expansion properties of inhomo-geneous materials that show a matrix–inclusion topology with aligned inhomogeneities orvoids. Mori–Tanaka-type approaches for thermoelastoplastic materials are discussed insection 3.7.

3.5 Self-Consistent Estimates

An alternative way of extending the expressions for the elastic properties of dilute two-phasematerials derived in section 3.3 to non-dilute volume fractions starts by rewriting eqn. (3.38)for an inhomogeneity that is surrounded by the “effective medium” (i.e., the composite

25

itself) instead of the matrix (i.e., E(m) → E and C(m) → C, so that A(i)dil → A

(i)dil(E,C) and

B(i)dil → B

(i)dil(E,C)), compare fig. 3.2. The results of the above formal procedure may be

inserted into eqn. (3.7), giving rise to the relationship

ESC = E(m) + ξ(E(i) − E(m))A(i)dil

= E(m) + ξ(E(i) − E(m))[I + SC(E(i) − E)]−1 . (3.50)

Equation (3.50) can be interpreted as an implicit nonlinear system of equations for theunknown elastic tensors E=ESC and C=CSC, which describe the effective medium. Thissystem can be solved by self-consistent iterative schemes of the type

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

−1

Cn+1 =[

En+1

]

−1. (3.51)

The Eshelby tensor Sn in eqn. (3.51) describes the response of an inhomogeneity embeddedin the n-th iteration of the effective medium; it must be recomputed for each iteration7.

effeffmmm

i i i i

σ σ σσa tot a a(m)

GSCSCSCSMTMdilute

Figure 3.2: Schematic comparison of the Eshelby method for dilute composites, Mori–Tanaka approaches and classical as well as generalized self-consistent schemes.

This approach is known as the two-phase or classical self-consistent scheme (CSCS),see, e.g., Hill (1965), and its predictions differ noticeably from those of Mori–Tanaka meth-ods in being close to one of the Hashin–Shtrikman bounds (compare section 4.1) for lowreinforcement volume fractions, but close to the other bound for reinforcement volumefractions approaching unity (compare figs. 4.1 to 4.7). Generally, two-phase self-consistentschemes are best suited to describing the overall properties of two-phase materials thatdo not show a matrix–inclusion microtopology at some or all of the volume fractions ofinterest. Essentially, the microstructures described by the CSCS are characterized by in-terpenetrating or granular phases around ξ = 0.5, with one of the materials acting asthe matrix for ξ → 0 and the other for ξ → 1. Consequently, the CSCS is typically notthe best choice for describing composites showing a matrix–inclusion topology, but is wellsuited to studying Functionally Graded Materials (FGMs) in which the volume fractionof a constituent can vary from 0 to 1 through the thickness of a sample. Multi-phaseversions of the CSCS are important methods for describing polycrystals. Being implicitand requiring the iterative solution of nonlinear equations, the CSCS is computationallymore demanding than Mori–Tanaka approaches.

7For aligned non-spherical inhomogeneities the effective medium shows transversely isotropic behavior,so that an appropriate procedure must be used for evaluating the Eshelby tensor.

26

A more elaborate self-consistent approach, the three-phase or generalized self-consistentscheme (GSCS) of Christensen and Lo (1979, 1986), describes inhomogeneities surroundedby a matrix layer (of a thickness appropriate for obtaining the required reinforcementvolume fraction) embedded in an effective medium, compare fig. 3.2. By considering thedifferential equations describing the elastic response of such three-phase regions underappropriate boundary conditions, equations for the overall elastic moduli can be obtained.For composites with spherical reinforcements the predictions for the effective bulk modulusK obtained with such a procedure correspond to the CSA results, and for compositesreinforced by unidirectional continuous fibers the CCA results for the effective transversebulk modulus KT, the effective axial shear modulus Glq and the effective axial Young’smodulus El are recovered. Beyond these results, in the case of a matrix reinforced withspherical inhomogeneities (which shows statistically isotropic overall behavior and is a veryuseful model for composites reinforced by approximately equiaxed particles) the followingquadratic equation is obtained for the effective shear modulus

(

GGSC

G(m)

)2

A+

(

GGSC

G(m)

)

B +D = 0 (3.52)

with

A = 8[G(i)/G(m) − 1](4− 5ν(m))η1ξ103 − 2[63(G(i)/G(m) − 1)η2 + 2η1η3]ξ

73

+252[G(i)/G(m) − 1]η2ξ53 − 25[G(i)/G(m) − 1](7− 12ν(m) + 8ν(m)2)η2ξ

+4(7− 10ν(m))η2η3

B = −4[G(i)/G(m) − 1](1− 5ν(m))η1ξ103 + 4[63(G(i)/G(m) − 1)η2 + 2η1η3]ξ

73

−504[G(i)/G(m) − 1]η2ξ53 + 150[G(i)/G(m) − 1](3− ν(m))ν(m)η2ξ

+3(15ν(m) − 7)η2η3

D = 4[G(i)/G(m) − 1](5ν(m) − 7)η1ξ103 − 2[63(G(i)/G(m) − 1)η2 + 2η1η3]ξ

73

+252[G(i)/G(m) − 1]η2ξ53 + 25[G(i)/G(m) − 1](ν(m)2 − 7)η2ξ − (7 + 5ν(m))η2η3

η1 = [G(i)/G(m) − 1](7− 10ν(m))(7 + 5ν(i)) + 105(ν(i) − ν(m))

η2 = [G(i)/G(m) − 1](7 + 5ν(i)) + 35(1− ν(i))

η3 = [G(i)/G(m) − 1](8− 10ν(m)) + 15(1− ν(m)) .

The analogous expression for the overall transverse shear modulus of composites reinforcedby aligned continuous fibers (which show statistically transversely isotropic overall behav-ior) takes the form

(

GGSC,qt

G(m)

)2

A +

(

GGSC,qt

G(m)

)

B +D = 0 (3.53)

with

A = 3ξ(1− ξ)2[8[G(f)/G(m) − 1][G(f)/G(m) + ηf ]

+[(G(f)/G(m))ηm + ηfηm − ((G(f)/G(m))ηm − ηf)ξ3]

×[ξηm(G(f)/G(m) − 1)− ((G(f)/G(m))ηm + 1)]

B = −6ξ(1− ξ)2[G(f)/G(m) − 1][G(f)/G(m) + ηf ]

+[(G(f)/G(m))ηm + (G(f)/G(m) − 1)ξ + 1]

×[(ηm − 1)(G(f)/G(m) + ηf)− 2ξ3((G(f)/G(m))ηm − ηf)]

+ξ(ηm + 1)[G(f)/G(m) − 1][G(f)/G(m) + ηf + ((G(f)/G(m))ηm − ηf)ξ3]

27

D = 3ξ(1− ξ)2[G(f)/G(m) − 1][G(f)/G(m) + ηf ]

+[(G(f)/G(m))ηm + (G(f)/G(m) − 1)ξ + 1]

×[G(f)/G(m) + ηf + ((G(f)/G(m))ηm − ηf)ξ3]

ηm = 3− 4ν(m)

ηf = 3− 4ν(f) .

Generalized self-consistent schemes for general aligned spheroidal inhomogeneities can bederived from energy considerations (Huang and Hu, 1995), the resulting expressions beingagain implicit and somewhat unhandy. For “standard” composites, i.e., materials consist-ing of stiff inhomogeneities embedded in a soft matrix, some of the effective elastic moduli(K, KT) obtained from the above scheme coincide with the corresponding Mori–Tanakaresults, whereas others (e.g., E and G) are higher, compare figs. 4.1 to 4.7.

3.6 Other Mean Field and Related Estimates

The most important mean field approaches besides effective field and self-consistent meth-ods are differential schemes (McLaughlin, 1977; Norris, 1985). These effective mediumtheories may be envisaged as involving repeated cycles of adding small concentrations ofinhomogeneities to a material and then homogenizing. Following Hashin (1988) the overallelastic tensors can then be described by the differential equations

dED

dξ=

1

1− ξ(E(i) − ED)Adil

dCD

dξ=

1

1− ξ(C(i) −CD)Bdil (3.54)

with the initial conditions ED=E(m) and CD=C(m), respectively, at ξ=0. In analogy toeqn. (3.50) Adil and Bdil depend on ED. In general, eqns. (3.54) must be integrated nu-merically, e.g., by Runge–Kutta schemes. Differential schemes describe matrix–inclusionmicrogeometries with polydisperse distributions of the sizes of the inhomogeneities (corre-sponding to the repeated homogenization steps.

Torquato (1991) developed second order estimates for the thermoelastic properties oftwo-phase materials, which correspond to the same phase arrangements and use the samethree-point microstructural parameters η and ζ as discussed for improved bounds in section4.2. These estimates lie between the corresponding bounds and give the best analyticalpredictions currently available for the overall thermoelastic response of inhomogeneous ma-terials that show the appropriate microstructures and are free of flaws.

3.7 Mean Field Methods for Elastoplastic Composites

The Mori–Tanaka theories discussed in section 3.4 can be extended in a relatively straight-forward way to describing inhomogeneous materials that contain at least one elastoplasticor thermoelastoplastic constituent.

28

In the literature two main lines of development of mean field approaches to modelingelastoplastic inhomogeneous materials can be found. On the one hand, secant plastic-ity (deformation theory) concepts have been used, see, e.g., Tandon and Weng (1988) orDunn and Ledbetter (1997), and, on the other hand, incremental plasticity models havebeen developed, compare, e.g., Lagoudas et al. (1991), Karayaka and Sehitoglu (1993) orPettermann et al. (1999). Whereas methods based on secant plasticity are limited to radial(or approximately radial) loading paths of the constituents in stress space8 and, accord-ingly, are not suitable for use as micromechanically based constitutive models, incrementalplasticity approaches are not subject to limitations in this respect. However, especiallyfor matrix dominated deformation modes, the secant formulations are typically superior tomean field algorithms based on incremental plasticity in the post-yield regime, where thelatter algorithms show a tendency towards overestimating overall strain hardening. Thisproblem has been an active field of research, compare Suquet (1997), and, for statisticallyisotropic composites, recent algorithmic modifications have led to marked improvementsin this respect (Doghri and Ouaar, 2003).

Incremental mean field methods can be formulated on the basis of strain and stress ratetensors for elastoplastic phases (p), d〈ε〉(p) and d〈σ〉(p), which can be expressed in analogyto eqn. (3.5) as

d〈ε〉(p) = A(p)t d〈ε〉+ η

(p)t dT

d〈σ〉(p) = B(p)t d〈σ〉+ β

(p)t dT , (3.55)

where d〈ε〉 stands for the macroscopic strain rate tensor, d〈σ〉 for the macroscopic stress

rate tensor, and dT for a homogeneous temperature rate. A(p)t , η

(p)t , B

(p)t , and β

(p)t are

instantaneous strain and stress concentration tensors, respectively. Assuming the inhomo-geneities to show elastic and the matrix to show elastoplastic material behavior9, the overallinstantaneous (tangent) stiffness tensor can be written in terms of the phase properties andthe instantaneous concentration tensors as

Et = 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 , (3.56)

compare eqns. (3.7) and (3.8). Using the Mori–Tanaka formalism of Benveniste (1987), theinstantaneous 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 , (3.57)

compare eqn. (3.45). Expressions for the instantaneous thermal concentration tensors andinstantaneous coefficients of thermal expansion can be derived in analogy to the corre-

8In deformation plasticity elastoplastic behavior is approximated by nonlinear elastic behavior. As aconsequence such descriptions are restricted to monotonous loading and radial loading paths in stress spaceat the constituent level, a condition that is typically violated to some extent in the phases of elastoplasticinhomogeneous materials, even if the overall loading paths are perfectly radial, see, e.g., Pettermann (1997).This effect is due to changes in the accommodation of the phase stresses and strains in inhomogeneousmaterials upon yielding of a constituent.

9Analogous expressions can be derived for elastoplastic inhomogeneities in an elastic matrix or, ingeneral, for composites containing any required number of elastoplastic phases.

29

sponding thermoelastic relations. It should be noted that eqns. (3.57) use the instanta-neous Eshelby tensor St, which depends on the current state of the matrix material andin general has to be evaluated numerically.

Formulations of eqns. (3.55) to (3.57) that are directly suitable for implementation asmicromechanically based constitutive models at the integration point level within FiniteElement codes can be obtained by replacing rates such as d〈σ〉(p) with the correspondingfinite increments, e.g., ∆〈σ〉(p). It is worth noting that in the resulting incremental Mori–Tanaka methods no assumptions on the overall yield loci and the overall flow potential aremade, the effective material behavior being entirely determined by the incremental meanfield equations and the constitutive behavior of the phases. As a consequence, mappingof the stresses onto the yield surface cannot be handled at the level of the homogenizedmaterial and radial return mapping must be applied to the matrix at the microscale in-stead. Accordingly, the constitutive equations describing the overall behavior cannot beintegrated directly (as is the case for homogeneous elastoplastic materials), and iterativealgorithms are required. Extended versions of such algorithms can also handle thermalexpansion effects10 and temperature dependent material parameters.

Mean field based descriptions of elastoplastic inhomogeneous materials tend to overes-timate the macroscopic flow stress of composites because, in general, the phase average ofthe mean equivalent stress in an elastoplastic phase is greater than the equivalent stressgenerated from the corresponding phase averages of the stress components, i.e., the relation

〈σ2eff〉

(p) =3

2〈sT s〉(p) ≥

3

2〈sT 〉(p)〈s〉(p) , (3.58)

holds, where s stands for the deviatoric part of the stress tensor11. For isotropic overallbehavior and isotropic constituent properties this difficulty can be mitigated by evaluatingthe mean von Mises equivalent stresses from energy considerations (Qiu and Weng, 1992)or by using statistically based theories (Buryachenko,1996). Furthermore, because theyassume elastoplastic phases to yield as a whole once the corresponding equivalent stressσ

(p)eff exceeds the yield stress, mean field approaches predict sharp transitions from elastic

to plastic states instead of the actual, gradual progress of yielded regions at the microscale.

Incremental Mori–Tanaka methods, especially when they incorporate the improvementsintroduced, e.g., by Doghri and Ouaar (2003), offer a combination of useful accuracy, flexi-bility in terms of inhomogeneity geometries, and relatively low computational requirements.

Self-consistent schemes can also be combined with secant and incremental approachesto describing elastoplastic inhomogeneous materials, see, e.g., Hill (1965), Hutchinson(1970) or Berveiller and Zaoui (1981). These procedures (as well as their weaknessesand strengths) are closely related to those discussed above for the Mori–Tanaka method.

10Elastoplastic inhomogeneous materials such as metal matrix composites show a hysteretic thermalexpansion response, i.e., there are no thermal expansion coefficients in the strict sense.

11Evaluating the von Mises equivalent stresses from the phase averaged stress components also leads topredictions by standard mean field methods that materials with spherical reinforcements will not yield un-der overall hydrostatic loads and homogeneous temperature loads. This is in contradiction to experimentaland other theoretical results.

30

In addition, mean field approaches have been employed to obtain estimates for the nonlin-ear response of inhomogeneous materials due to microscopic damage or to combinations ofdamage and plasticity, see, e.g., Tohgo and Chou (1996) or Guo et al. (1997).

As an alternative to directly extending mean field theories into secant or incrementalplasticity, they can also be combined with Dvorak’s transformation field analysis (Dvorak,1992) in order to obtain descriptions of the overall behavior of inhomogeneous materialsin the plastic range, see, e.g., Plankensteiner (2000). Such approaches are well suited forbeing used as material models in FE codes, are very attractive in terms of computationalrequirements, but tend to strongly overestimate the overall strain hardening of elastoplasticcomposites because they use elastic accommodation of microstresses and strains through-out the loading history.

3.8 Mean Field Methods for Nonaligned Composites

The statistics of the microgeometries of nonaligned and hybrid matrix–inclusion compos-ites can be typically described on the basis of orientation distribution functions (ODFs)and aspect ratio or length distribution functions (LDFs) of the reinforcements.

Mori–Tanaka methods have been extended to describing the elastic behavior of suchmaterials, the typical starting point being dilute concentration tensors B

(i)∠dil (ϕ, ψ, θ) for

reinforcements having orientations with respect to the global coordinate system that aredescribed by the Euler angles ϕ, ψ and θ. A phase averaged dilute fiber stress concentrationtensor can be obtained by orientational averaging as

B(i)

dil =

∫ 2π

0

∫ 2π

0

∫ π

0

B(i)∠dil (ϕ, ψ, θ) ρ(ϕ, ψ, θ) dϕ dψ dθ , (3.59)

where the orientation distribution function ρ(ϕ, ψ, θ) is normalized to give ρ = 1. The corestatement of the Mori–Tanaka approach, eqn. (3.40), can then be written in the form

〈σ〉(i) = B(i)

dil〈σ〉(m) = Bdil B

(m)

M 〈σ〉 = B(i)

M〈σ〉 , (3.60)

which allows the phase concentration tensors B(m)

M and B(i)

M to be evaluated in analogy toeqns. (3.43) to (3.45). The stress 〈σ〉(i) obtained from eqn. (3.60) is an average over allinhomogeneities, irrespective of their orientation, and, accordingly, contains only limitedinformation. The average stresses in inhomogeneities of a given orientation (ϕ, ψ, θ), which

is of considerably more interest, can be evaluated as 〈σ〉(i)∠ = B(i)∠dil B

(m)

M 〈σ〉.

“Extended” Mori–Tanaka methods for modeling the elastic behavior of microstructuresthat contain nonaligned inhomogeneities were developed by a number of authors, see, e.g.,Benveniste (1990), Dunn and Ledbetter (1997), Pettermann et al. (1997) or Mlekusch(1999). These models differ mainly in the algorithms employed for orientational or config-urational averaging. Methods of this type tend to be very useful for many practical appli-cations. They are, however, of a somewhat ad-hoc nature and can lead to non-symmetric

31

effective stiffness tensors12 in certain situations (Benveniste et al., 1991; Ferrari, 1991).The latter unphysical behavior also appears in multi-phase Mori–Tanaka models in whichaligned inhomogeneities show both different material behaviors and different aspect ratios.

For the special case of randomly oriented fibers or platelets of a given aspect ratio,“symmetrized” dilute strain concentration tensors may be constructed, giving rise to theso-called Wu tensors (Wu, 1966). They can be combined with Mori–Tanaka methods (Ben-veniste, 1987) or classical self-consistent schemes (Berryman, 1980) to describe compositeswith randomly oriented phases of the matrix–inclusion and interwoven or granular types,respectively. Due to the overall isotropic behavior of composites reinforced by randomlyoriented fibers or platelets, their elastic behavior must comply with the “standard” Hashin–Shtrikman bounds (Hashin and Shtrikman, 1963), compare section 4.1.

A simpler approach to describing the macroscopic behavior of composites with non-aligned reinforcements is the use of laminate analogies. In such models the composite isapproximated as a laminate consisting of unidirectionally reinforced layers that are ap-propriately oriented for handling one fiber orientation each (Fu and Lauke, 1998). Thefiber orientation distributions are, accordingly, approximated by orientational averagingat the level of the elastic tensors. Within such a “laminate analogy approach” mean fieldexpressions can be used for the individual layers to handle fiber aspect ratios. The result-ing schemes can be useful especially for describing composites with planar random fiberorientations (Huang, 2001).

3.9 Mean Field Methods for Diffusion-Type Problems

There are strong conceptual and algorithmic analogies between inhomogeneity problemsdescribing elastostatic behavior and diffusive transport processes, see, e.g., Hashin (1983).Specifically, the effects of dilute inhomogeneous inclusions in diffusion-type problems (e.g.,heat conduction, electrical conduction, diffusion of moisture) as well as in some coupledproblems (e.g., inhomogeneous materials with at least one piezoelectric constituent) canbe described in analogy to eqns. (3.18) to (3.39) via a depolarization or “diffusion Eshelbytensor”13.

For non-dilute cases, analoga of the Mori–Tanaka, self-consistent and differential schemescan be derived, see, e.g., Hatta and Taya (1986), Miloh and Benveniste (1988), Dunn andTaya (1993), Chen (1997) as well as Torquato (2002).

12The reason for these difficulties lies in the aligned ellipsoidal symmetry of the phase arrangement thatis inherent to Mori–Tanaka methods.

13In contrast to the “mechanical Eshelby tensor”, which is of order 4, the depolarization tensor is oforder 2, and for the case of spheroidal inhomogeneities embedded in an isotropic matrix depends only onthe formers’ aspect ratio. Explicit expressions for the elements of depolarization tensors are given, e.g., byHatta and Taya (1986).

32

Chapter 4

Bounding Methods

Whereas mean field methods, unit cell approaches, embedding strategies and windowingmodels can typically be used for both homogenization and localization tasks, boundingmethods are restricted to homogenization. Discussions in this section again are restrictedto materials consisting of two perfectly bonded constituents. Accordingly, they do notaccount for any flaws (e.g., porosity) that may degrade the macroscopic behavior of actualcomposites

Rigorous bounds for the overall elastic properties of inhomogeneous materials can beobtained from appropriate variational (minimum energy) principles. In the following, onlyoutlines of bounding methods are given; formal treatments were given, e.g., by Nemat-Nasser and Hori (1993), Ponte Castaneda and Suquet (1998), Markov (2000), Bornertet al. (2001), Milton (2002) as well as Torquato (2002).

4.1 Hill and Hashin–Shtrikman-Type Bounds

Classical expressions for the minimum potential energy and the minimum complementaryenergy in combination with uniform stress and strain trial functions lead to the simplestvariational bounding expressions, the upper bounds of Voigt (1889) and the lower boundsof Reuss (1929). In their tensorial form they are known as the Hill bounds and can bewritten as

[

∑

(p)V (p)C(p)

]

−1

≤ E ≤∑

(p)V (p)E(p) . (4.1)

These bounds, while universal and very simple, do not contain any information on themicrogeometry of an inhomogeneous material beyond the phase volume fractions, and aretypically too slack to be of much practical use14, but in contrast to the Hashin–Shtrikmanand higher order bounds they also hold for (non-pathological) control volumes that are toosmall to be proper RVEs.

14The bounds on the Young’s moduli obtained from eqn. (4.1) are equivalent to Voigt and Reuss expres-sions in terms of the corresponding phase moduli (compare section 2.1) only if the constituents have equalPoisson’s ratios. Due to the homogeneous stress and strain assumptions used for obtaining the Hill bounds,

the corresponding phase strain and stress concentration tensors are A(p)V =I and B

(p)R =I, respectively.

33

Much tighter bounds can be obtained from variational expressions due to Hashin andShtrikman (1962), which are based on the principle of minimum potential energy. They areformulated in terms of a homogeneous reference material and of polarization fields τ(x)that describe the difference between the microscopic stress fields in the inhomogeneousmaterial (elasticity tensor E(x)) and the reference material (elasticity tensor E0),

〈σ〉 = E〈ε〉 = E0〈ε〉+ 〈τ(x)〉 with τ(x) = (E(x)− E0) ε(x) . (4.2)

Exact polarization fields are highly heterogeneous, but simplified approximations may beemployed for constructing trial fields for use in variational theorems. By choosing phase-wise constant polarizations and optimizing them, Hashin–Shtrikman-type bounds are ob-tained. The Hashin–Shtrikman bounds proper (Hashin and Shtrikman, 1963) apply toinhomogeneous materials with statistically isotropic overall symmetry and are given interms of bounds on the effective bulk and shear moduli. For two-phase materials thatfulfill the condition (K(m) −K(i))(G(m) − G(i)) > 0 the bounds on the effective moduli takethe form

KL.B.HS = K(m) +

ξ

1/(K(i) −K(m)) + 3(1− ξ)/(3K(m) + 4G(m))

KU.B.HS = K(i) +

1− ξ

1/(K(m) −K(i)) + 3ξ/(3K(i) + 4G(i))

GL.B.HS = G(m) +

ξ

1/(G(i) −G(m)) + 6(1− ξ)(K(m) + 2G(m))/[5G(m)(3K(m) + 4G(m))]

GU.B.HS = G(i) +

1− ξ

1/(G(m) −G(i)) + 6ξ(K(i) + 2G(i))/[5G(i)(3K(i) + 4G(i))]. (4.3)

For obtaining bounds on the effective Young’s moduli from these data see Hashin (1983),and for bounding expressions on the Poisson numbers see Zimmerman (1992).

Another set of Hashin–Shtrikman bounds were developed for statistically transverselyisotropic composites reinforced by aligned continuous fibers15. For the overall moduli KT,El, νlq, and Glq they are equivalent to the corresponding CCA expressions16, see eqn. (2.7).For the transverse shear modulus Gqt the expressions

GL.B.qt HS

= G(m) +ξ

1/(G(f)qt −G

(m)) + (1− ξ)(K(m)T + 2G(m))/2G(m)(K

(m)T +G(m))

GU.B.qt HS

= G(m)

(

1 +ξ(1 + β(m))

ρ− ξ[1 + 3(1− ξ)2β(m)2/(λξ3 + 1)]

)

for K(f)T > K

(m)T , G

(f)qt > G(m)

15These Hashin–Shtrikman bounds apply to transversely isotropic two-phase materials in which thephases are continuous in the “axial” direction.

16For KT, El and Glq the CCA expressions are lower bounds and for νlq they may be either bound.In order to obtain the corresponding other bounds the roles of inhomogeneities and matrix have to beexchanged in eqn. (2.7).

34

GL.B.qt HS

= G(m)

(

1 +ξ(1 + β(m))

ρ− ξ[1 + 3(1− ξ)2β(m)2/(λξ3 − β(m))]

)

GU.B.qt HS

= G(m) +ξ

1/(G(f)qt −G

(m)) + (1− ξ)(K(m)T + 2G(m))/2G(m)(K

(m)T +G(m))

for K(f)T < K

(m)T , G

(f)qt < G(m) (4.4)

hold, where

λ = (β(m) − γβ(f))/(1 + γβ(f))

ρ = (γ + β(m))/(γ − 1)

β(f) = K(f)T /(K

(f)T + 2G

(f)qt)

β(m) = 1/(3− 4ν(m))

γ = G(f)qt/G

(m)

Further developments led to the Willis (1977) bounds, which pertain to aligned ellipsoidalmicrogeometries, compare fig. 3.1 (and thus to transversely isotropic overall behavior).They include the Hashin–Shtrikman bounds for isotropic and continuously reinforced com-posites as special cases. From the practical point of view it is of interest that for two-phasematerials Mori–Tanaka estimates correspond to one of the Willis bounds, compare Weng(1990), and the other bound can be obtained after a “color inversion” (i.e., after exchangingthe roles of inhomogeneities and matrix)17. Accordingly, the bounds can be evaluated forfairly general phase geometries by matrix algebra such as eqns. (3.46) and (3.48). Hashin–Shtrikman-type bounds can also be derived for simple periodic phase arrangements, see,e.g., Nemat-Nasser and Hori (1993).

4.2 Improved Bounds

When more complex trial functions are used in variational bounding, their optimizationrequires statistical information on the phase arrangement in the form of n-point correlationfunctions18. This way improved bounds can be generated that are significantly tighter thanHashin–Shtrikman-type expressions (which in this context can be classified as two-pointbounds).

Three-point bounds for statistically isotropic two-phase materials can be formulatedin such a way that the information on the phase arrangement statistics is contained intwo three-point microstructural parameters, η(ξ) and ζ(ξ). In this framework, the Beranand Molyneux (1966) bounds on the effective bulk modulus of macroscopically isotropic

17Put more precisely, the lower Willis bounds are obtained from Mori–Tanaka methods when the morecompliant material is used as matrix phase, and the upper bounds when it is used as inclusion phase.

18For discussions on statistical descriptions of phase arrangements see, e.g., Torquato (1991), Pyrz andBochenek (1998) as well as Torquato (2002).

35

materials can be written as

KL.B.3P =

[

〈1/K〉 −4ξ(1− ξ)(1/K(i) − 1/K(m))2

4〈1/K〉+ 3〈1/G〉ζ

]

−1

KU.B.3P =

[

〈K〉 −3ξ(1− ξ)(K(i) −K(m))2

3〈K〉+ 4〈G〉ζ

]

(4.5)

and the Phan-Thien and Milton (1983) bounds for the overall shear modulus take the form

GL.B.3P =

[

〈G〉 −6ξ(1− ξ)(1/G(i) − 1/G(m))2

6〈G〉+ Ξ−1

]

GU.B.3P =

[

〈G〉 −6ξ(1− ξ)(1/G(i) − 1/G(m))2

6〈G〉+ Θ

]

(4.6)

with

Ξ = 5⟨ 1

G

⟩

ζ

⟨ 6

K−

1

G

⟩

ζ+

⟨ 1

G

⟩

η

⟨ 2

K+

21

G

⟩

ζ

⟨128

K+

99

G

⟩

ζ+45

⟨ 1

G

⟩

η

Θ =3〈G〉η〈6K + 7G〉ζ − 5〈G〉2ζ

2〈K −G〉ζ + 5〈G〉η

〈f〉 = (1− ξ)f (m) + ξf (i)

˜〈f〉 = ξf (m) + (1− ξ)f (m)

〈f〉η = (1− η)f (m) + ηf (i)

〈f〉ζ = (1− ζ)f (m) + ζf (i) .

(4.7)

ζ(ξ) and η(ξ) can in principle be evaluated for any given microstructure. Analyticalexpressions or tabulated data in terms of the reinforcement volume fraction ξ are availablefor a number of generic microgeometries of practical importance, among them statisticallyhomogeneous isotropic materials containing identical, bidisperse and polydisperse impene-trable (“hard”) spheres (describing matrix–inclusion composites) as well as monodisperseinterpenetrating spheres (“Boolean models” that can describe many interpenetrating phasearrangements), and statistically homogeneous transversely isotropic materials reinforced byimpenetrable or interpenetrating aligned cylinders.

References to some three-point bounds and to a number of expressions for η(ξ) and ζ(ξ)applicable to matrix–inclusion composites can be found in section 4.4, where results frommean field and bounding approaches are compared. For a review of higher order boundsfor elastic (as well as other) properties of inhomogeneous materials see Torquato (1991).

4.3 Bounds for the Nonlinear Behavior

Analoga to the Hill bounds for nonlinear inhomogeneous materials were introduced byBishop and Hill (1951). For polycrystals the nonlinear equivalents to Voigt and Reuss

36

expressions are usually referred to as Taylor and Sachs bounds, respectively.

In analogy to mean field estimates for elastoplastic material behavior, nonlinear boundsare 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., up-per or lower bounds, depending on the material combination) on the nonlinear mechanicalbehavior of inhomogeneous materials. An important development was the derivation byPonte Castaneda (1992) of a variational principle that allows upper bounds on specificstress-strain responses of elastoplastic inhomogeneous materials to be generated on thebasis of upper bounds on the elastic tensors19 by using a series of inhomogeneous referencematerials, the properties of which have to be obtained by optimization procedures for eachstrain level. Essentially, the variational principle guarantees the best choice for the com-parison material at a given load. The Ponte Castaneda bounds are closely related to meanfield approaches using improved secant plasticity methods (compare section 3.7).

The study of bounds — like the development of improved estimates — for the overallnonlinear mechanical behavior of inhomogeneous materials has been an active field ofresearch during the past decades, see the recent reviews by Suquet (1997), Ponte Castanedaand Suquet (1998) as well as Willis (2000).

4.4 Comparisons Between Some Mean Field and Boun-

ding Predictions

In order to give some idea of the type of predictions that can be obtained by differentmean field (and related) approaches and by bounding methods for the thermomechanicalresponses of inhomogeneous thermoelastic materials, results for the overall elastic moduliand coefficients of thermal expansion are presented as functions of the reinforcement vol-ume fraction ξ in this section. All comparisons are based on E-glass particles or fibersembedded in an epoxy matrix. The elastic contrast of these constituents is c ≈ 21 and thethermal expansion contrast takes a value of approximately 0.14, see table 4.1.

Table 4.1: Constituent material parameters of epoxy matrix and E-glass reinforcementsused in generating figs. 4.1 to 4.7.

E[GPa] ν[ ] α[1/K]matrix 3.5 0.35 36.0reinforcements 74.0 0.2 4.9

Figures 4.1 and 4.2 show predictions for the overall Young’s and shear moduli of aparticle reinforced composite using the above material parameters. The Mori–Tanaka re-sults (MTM) — as mentioned before — coincide with the lower Hashin–Shtrikman (H/S)

19The Ponte Castaneda bounds are rigorous for nonlinear elastic inhomogeneous materials and, on thebasis of deformation theory, are very good approximations for materials with at least one elastoplastic con-stituent. Applying the Ponte Castaneda variational procedure to elastic lower bounds does not necessarilylead to a lower bound for the inelastic behavior.

37

0.0

40.0

80.0

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 boundsH/S LB; MTMH/S UB3PLB (mono/h)3PUB (mono/h)GSCS3PE (mono/h)CSCSDS

Figure 4.1: Bounds and estimates for the effective Young’s moduli of glass/epoxy particlereinforced composites as functions of the particle volume fraction.

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

PARTICLE VOLUME FRACTION [ ]

Hill boundsH/S LB; MTMH/S UB3PLB (mono/h)3PUB (mono/h)GSCS3PE (mono/h)CSCSDS

Figure 4.2: Bounds and estimates for the effective shear moduli of glass/epoxy particlereinforced composites as functions of the particle volume fraction.

bounds, whereas the classical self-consistent scheme (CSCS) shows a typical behavior inthat it is close to one Hashin–Shtrikman bound at low volume fractions, approaches theother at high volume fractions, and displays a transition behavior in the form of a sigmoidcurve in-between. The three-point bounds (3PLB and 3PUB) shown in figs. 4.1 and 4.2 are

38

based on formalisms developed by Beran and Molyneux (1966), Milton (1981) as well asPhan-Thien and Milton (1983). Like the three-point estimates (3PE) they were evaluatedfor impenetrable spherical particles of equal size and use expressions for the statisticalparameters η(ξ) and ζ(ξ) that were reported by Miller and Torquato (1990) and Torquatoet al. (1987), respectively. These expressions are available for reinforcement volume frac-tions up to ξ=0.6. The results from the generalized self-consistent scheme (GSCS) liebetween the Mori–Tanaka predictions and the three-point estimates, whereas the differen-tial scheme (DS) provides the stiffest predictions except the upper bounds.

Predictions for the coefficients of thermal expansion of statistically isotropic inhomo-geneous materials are presented in fig. 4.3, the bounds being obtained by evaluating aversion of Levin’s formula, eqn.(3.11), with bounds on the effective bulk modulus. Herethe Hashin–Shtrikman bounds and the three-point bounds are used as the basis of thisprocedure. The CTEs corresponding to the self-consistent schemes were obtained withthe mean field formalism discussed in section 3.1. Because they are based on the sameestimates for the effective bulk modulus, the results given for the GSCS and the Mori–Tanaka-scheme agree with the upper Levin/Hashin–Shtrikman bounds.

0.0

20.0

40.0

EF

FE

CT

IVE

CT

E [

1/K

x 1

0 ]-6

0.0 0.2 0.4 0.6 0.8 1.0

PARTICLE VOLUME FRACTION [ ]

H/S LBH/S UB; MTM3PLB (mono/h)3PUB (mono/h)GSCS3PE (mono/h)CSCSDS

Figure 4.3: Bounds and estimates for the effective CTEs of glass/epoxy particle reinforcedcomposites as functions of the particle volume fraction.

In figs. 4.3 to 4.7 results are presented for the overall transverse Young’s moduli, over-all axial and transverse shear moduli, as well as overall axial and transverse coefficients ofthermal expansion of composites reinforced by aligned continuous fibers20. The results forthe three-point bounds shown follow the formalism of Silnutzer (1972) and Milton (1981),correspond to the case of aligned impenetrable circular cylinders of equal diameter and

20In the case of the axial Young’s moduli the estimates and bounds are nearly indistinguishable fromeach other and from the rule of mixture result, eqn.(2.2), for the material parameters and scaling used infig. 4.4 and are, accordingly, not given here.

39

0.0

40.0

80.0

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 boundsH/S LB; MTMH/S UB3PB LB (mono/h)3PB LB (mono/h)GSCS3PE (mono/h)CSCSDS

Figure 4.4: Bounds and estimates for the effective transverse Young’s moduli of glass/epoxyfiber reinforced composites as functions of the fiber volume fraction.

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 boundsH/S LB; MTMH/S UB3PLB (mono/h)3PUB (mono/h)GSCS3PE (mono/h)CSCSDS

Figure 4.5: Bounds and estimates for the effective axial shear moduli of glass/epoxy fiberreinforced composites as functions of the fiber volume fraction.

use statistical parameters evaluated by Torquato and Lado (1992) for fiber volume frac-tions ξ / 0.7. Generally, a qualitatively similar behavior to the particle reinforced caseis evident. It is noteworthy, however, that the overall transverse CTEs in fig. 4.7 at lowfiber volume fractions exceeds the CTEs of both constituents. Such behavior is typical for

40

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 boundsH/S LB; MTMH/S UB3PLB (mono/h)3PUB (mono/h)GSCS3PE (mono/h)CSCSDS

Figure 4.6: Bounds and estimates for the effective transverse shear moduli of glass/epoxyfiber reinforced composites as functions of the fiber volume fraction.

0.0

20.0

40.0

EF

FE

CT

IVE

CT

E [

1/K

x 1

0 ]-6

0.0 0.2 0.4 0.6 0.8 1.0

FIBER VOLUME FRACTION [ ]

MTM, GSCS ax.MTM, GSCS tr.3PE tr. (mono/h)CSCS ax.CSCS tr.DS ax.DS tr.

Figure 4.7: Estimates for the effective axial and transverse CTEs of glass/epoxy fiberreinforced composites as functions of the fiber volume fraction.

continuously reinforced composites and is caused by the marked axial constraint enforcedby the fibers.

In figs. 4.1 to 4.7 the classical self-consistent scheme — as expected — is not in goodagreement with the three-point bounds shown, which explicitly correspond to matrix–

41

inclusion topologies. Considerably better agreement with the CSCS is found by usingthree-point parameters of the interpenetrating sphere or cylinder type (which can also de-scribe cases where both phases percolate, but are not as symmetrical with respect to theconstituents as the CSCS). From a practical point of view it is worth noting that despitetheir sophistication improved bounds (and higher order estimates) may give overly opti-mistic predictions for the overall moduli because they describe ideal composites, whereasin actual materials it is practically impossible to avoid flaws such as porosity.

42

Chapter 5

General Remarks on Modeling

Approaches Based on Discrete

Microgeometries

In the present context, micromechanical approaches based on discrete microgeometries en-compass unit cell, embedding and windowing methods. Broadly speaking, these approachestrade off restrictions to the generality of the microstructures that can be studied for thecapabilities of using fine grained geometrical models and of resolving details of the stressand strain fields at the length scale of the inhomogeneities. The main fields of applica-tion of these methods are studying the nonlinear behavior of inhomogeneous materials andevaluating the microfields at high resolution. Even though in many cases the latter typeof information is not necessary for describing the macroscopic behavior of inhomogeneousmaterials, their damage and failure behavior can depend on details of the microgeometry.

There are two, often complementary, philosophies for modeling inhomogeneous ma-terials via discrete microgeometries. One of them is based on studying generic phasearrangements, which may range from simple periodic arrays of fibers or particles to highlycomplex microgeometries involving a considerable number of reinforcements, the positionsof which are chosen to approximate some statistical phase distribution. Complex phasearrangements of the latter type can be generated by appropriate statistically based algo-rithms such as random insertion methods or numerical annealing procedures, which giverise to quasi-random and “statistically reconstructed” microstructures21, respectively. Suchmicrogeometries tend to employ idealized inclusion shapes, equiaxed particles embeddedin a matrix, for example, being represented by spheres22.

Alternatively, microgeometries may be chosen to follow as closely as possible the phasearrangement of a given sample of the material to be modeled, obtained, e.g., from met-

21As understood here, statistically reconstructed microstructures are equivalent to real ones in a statis-tical sense, i.e., they show equal or very similar phase distribution statistics. For algorithms for recon-structing matrix–inclusion and more general microgeometries see, e.g., Rintoul and Torquato (1997) andTorquato (1998).

22For uniform boundary conditions it can be shown that the overall elastic behavior of matrix–inclusion-type composites can be bounded by approximating the actual shape of particles by inner and outer en-velopes of “smooth” (e.g., ellipsoidal) shape. This is known as the Hill modification theorem (Hill, 1963;Huet et al., 1990).

43

allographic sections, serial sections or tomographic data (Terada and Kikuchi, 1996; Liet al., 1999; Babout et al., 2004; Chawla and Chawla, 2006). The resulting descriptionsare termed “real structure” models.

Unless simple periodic phase arrangements are considered, for both of the above ap-proaches to obtaining microgeometries the question immediately arises as to how complex(and thus “large”) the model geometry must be in order to adequately capture the phys-ical behavior of the material to be studied. For the case of elastic statistically isotropiccomposites 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, volume elements with sizes of roughly two or five inclusiondiameters are sufficient for any volume fraction23. Alternatively, adequate sizes of modelgeometries for elastic studies may be estimated on the basis of experimentally obtainedcorrelation lengths (Bulsara et al., 1999; Jeulin, 2001), by comparing statistical distributionfunctions of actual and model microgeometries (Zeman and Sejnoha, 2001), or by usingwindowing approaches (compare section 8) to bounding the overall response from aboveand below. For nonlinear matrix behavior a number of numerical studies (Zohdi, 1999;Jiang et al., 2001; Bohm and Han, 2001) have indicated that substantially larger volumeelements may be necessary for satisfactorily approximating the required overall symmetryand for obtaining good agreement between the responses of (nominally) statistically equiv-alent phase arrangements, especially at elevated overall inelastic strains. This indicatesthat the size of satisfactory multi-inclusion unit cells depends markedly on the phase ma-terial behavior24.

The majority of published micromechanical studies of discrete microstructures haveemployed standard numerical engineering methods for resolving the microfields, studiesusing Finite Difference (FD) algorithms, compare Adams and Doner (1967), spring lat-tice models, compare Ostoja-Starzewski (1996), the Boundary Element Method (BEM),compare Achenbach and Zhu (1989), the Finite Element Method (FEM), techniques usingFast Fourier Transforms (FFT), compare Moulinec and Suquet (1994), and Discrete FourierTransforms (DFT), compare Muller (1996), as well as FE-based discrete dislocation models(Cleveringa et al., 1997) having been reported. A number of more specialized approachesare discussed in connection with periodic microfield analysis, see section 6. Generallyspeaking, spring lattice models tend to have advantages in handling pure traction bound-ary conditions and in modeling the progress of microcracks due to local (brittle) failure.Boundary elements tend to be at their best in studying geometrically complex linear elasticproblems. For all the above methods the characteristic length of the discretization (“meshsize”) must be considerably smaller than the microscale of a given problem in order toobtain spatially well resolved results.

At present, the FEM is the most popular numerical scheme for evaluating discretemicrogeometries, especially in the nonlinear range, where its flexibility and capability of

23Note that a volume element fulfilling a condition of this type is not necessarily an RVE as discussedin section 1.2.

24For elastoplastic matrices, weaker strain hardening translates into unit cells requiring a higher numberof inhomogeneities, and very large volume elements may be necessary when the matrix shows strainsoftening and/or localization.

44

supporting a wide range of constitutive models for the constituents and for the interfacesbetween them are especially appreciated. An additional asset of the FEM in the context ofcontinuum micromechanics is its ability to handle discontinuities in the stress and straincomponents (which typically occur at interfaces between different constituents) in a natu-ral way via appropriately placed element boundaries.

Applications of the FEM to micromechanical studies tend to fall into four main groups,compare fig. 5.1. In most published works the phase arrangements are discretized by anoften high number of “standard” continuum elements, the mesh being designed in sucha way that element boundaries (and, where appropriate, special interface elements) arepositioned at all interfaces between constituents. Such an approach has the advantagethat in principle any microgeometry can be handled and that readily available commercialFE packages may be used. However, the actual modeling of complex phase configurationsin many cases requires sophisticated and/or specialized preprocessors for generating themesh, a task that has been difficult to automate, and the resulting stiffness matrices mayshow unfavorable conditioning due to suboptimal element shapes.

Alternatively, a smaller number of special hybrid elements may be used, which arespecifically formulated to model the deformation, stress, and strain fields in an inhomoge-neous region consisting of a single inhomogeneity or void together with the surroundingmatrix on the basis of some appropriate analytical theory. The most highly developed ap-proach 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 tessellation based on thepositions of the reinforcements. Large planar multi-inclusion arrangements can be analyzedthis way using a limited number of (albeit rather complex) elements, and good accuracy aswell as significant gains in efficiency have been claimed. Computational strategies of thistype are specifically tailored for inhomogeneous materials with matrix–inclusion topologies.

a) b) d)c)

Figure 5.1: Sketch of FEM approaches used in micromechanics: a) discretization by stan-dard elements, b) special hybrid elements, c) pixel/voxel discretization, d) “multiphaseelements”

45

Especially when the phase arrangements to be studied are based on digital images ofactual microgeometries, a third approach to discretizing microgeometries of interest. Itconsists of using a regular square or hexahedral mesh that has the same resolution as thedigital image, each element being assigned to one of the constituents by operations suchas thresholding of the grey values of the corresponding pixel or voxel, respectively. Suchmeshes have the advantage of allowing a straightforward automatic model generation fromappropriate experimental data (metallographic sections, tomographic scans) and of avoid-ing ambiguities in smoothing the digital data (which are generally present if a “standard”FE mesh is employed to discretize experimental data of this type). Obviously “voxel ele-ment” strategies lead to ragged phase boundaries, which may give rise to some oscillatorybehavior of the solutions (Niebur et al., 1999) and can lead to very high local stress maxima(Terada et al., 1997). Such “digital image based” (DIB) models are, however, claimed notto cause unacceptably large errors in the predicted effective behavior, even for relativelycoarse discretizations, at least in the linear elastic range (Guldberg et al., 1998). Goodspatial resolution, i.e., a high number of pixels or voxels, of course, is very beneficial tosuch models.

A fourth approach also uses regular FE meshes, but assigns phase properties at theintegration point level of standard elements (“multiphase elements”), see, e.g., Schmauderet al. (1996). Essentially, this amounts to trading off ragged boundaries at element edgesfor smeared-out (and typically degraded) microfields within those elements that containa phase boundary, because stress or strain discontinuities within elements cannot be ade-quately handled by standard FE shape functions. With respect to the element stiffnessesthe latter concern can be much reduced by overintegrating elements containing phaseboundaries, which leads to good approximations of integrals involving non-smooth dis-placements by numerical quadrature, see Zohdi and Wriggers (2001). The resulting stressand strain distributions, however, remain smeared-out approximations in elements thatcontain phase boundaries.

A fairly recent development for studying microgeometries involving a large number ofinhomogeneities (tens to thousands) by Finite Element methods involves programs spe-cially geared towards solving micromechanical problems. Such codes may be based onmatrix-free iterative solvers such as Conjugate Gradient (CG) methods, analytical solu-tions for the microfields (e.g. constant strain approximations corresponding to the upperHill bounds, eqn. (4.1)), being used as starting solutions to speed convergence25. For stud-ies involving such programs see, e.g., Gusev (1997) or Zohdi and Wriggers (2001).

25Essentially, in such a scheme the initial guess gives a good estimate of “long wavelength” contributionsto the solution, and the CG iterations take care of “short wavelength” variations.

46

Chapter 6

Periodic Microfield Models

Periodic Microfield Approaches (PMAs) aim to describe the macroscopic and microscopicbehavior of inhomogeneous materials by studying model materials that have periodic mi-crostructures. The first two parts of this chapter cover some basic concepts of unit cellbased PMAs. Following this, applications to some types of composites and other inhomo-geneous materials are discussed.

6.1 Basic Concepts of Unit Cell Models

Periodic microfield approaches analyze the behavior of infinite (one, two- or three-dimen-sional) periodic arrangements of constituents making up a given inhomogeneous materialunder the action of far field mechanical loads or uniform temperature fields26. The mostcommon approach to studying the stress and strain fields in such periodic configurations isbased on partitioning the microgeometry into periodically repeating unit cells to which theinvestigations may be limited without loss of information or generality, at least for staticanalysis27. A wide variety of unit cells have been employed in published PMA studies, rang-ing from simple periodic arrays of inhomogeneities to highly complex phase arrangements,such as multi-inclusion cells (“supercells”). For some simple periodic phase arrangementsit is also possible to find analytical solutions via series expansions that make explicit use ofthe periodicity (Sangani and Lu, 1987) or via appropriate potential methods (Wang et al.,2000).

Even though most PMA studies in the literature have used standard numerical engi-neering methods as described in chapter 5, some more specialized approaches to evaluating

26Standard PMAs cannot handle free boundaries or macroscopic gradients in mechanical loads, temper-atures or composition in any direction in which the fields are periodic. If such gradients or free boundariesare to be studied, however, in many cases “layer models” can be generated that are non-periodic in onedirection and periodic in the other(s), see, e.g., Reiter and Dvorak (1998) and Weissenbek et al. (1997).

27Periodic phase arrangements, especially simply periodic microstructures, typically have to be used withextreme care in dynamic analysis, because they act as filters that exclude all waves with frequencies that donot fall within certain bands, see, e.g., Suzuki and Yu (1998). In addition, due to the boundary conditionsrequired for obtaining periodicity, unit cell analysis can only handle wavelengths that are smaller than orequal to the appropriate cell dimension, and mechanical waves are “locked in” within the cell instead ofbeing allowed to pass through and continue into the far field. For the same reason, unit cell based methodstypically can resolve only subsets of the buckling modes and buckling loads in stability analysis (dependingon the geometry of the cell), so that considerable care is required in using them for such purposes.

47

the mechanical behavior of periodic geometries are worth mentioning. One of them, knownas the Method of Cells (Aboudi, 1989), discretizes unit cells that correspond to square ar-rangements of square fibers into four subcells, within each of which displacements areapproximated by low-order polynomials. Traction and displacement continuity conditionsat 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 elastic and inelasticranges. Even though they use highly idealized microstructures, are restricted in handlingaxial shear and provide only limited information on the microscopic stress and strain fields,the resulting models pose relatively low computational requirements. The Method of Cellshas been used as a constitutive model for analyzing structures made of continuously rein-forced composites, see, e.g., Arenburg and Reddy (1991). Developments of the algorithmled to the Generalized Method of Cells (Aboudi, 2004), which allows finer discretizationsof unit cells for fiber and particle reinforced composites, reinforcement and matrix beingessentially split into a number of “subregions” of rectangular or hexahedral shape. Forsome comparisons with microfields obtained by Finite Element based unit cells see Pahrand Arnold (2002).

Another micromechanical approach, the Transformation Field Analysis of Dvorak (1992),allows the prediction of the nonlinear response of inhomogeneous materials based on ei-ther mean field descriptions (compare the remarks in section 3.7) or on periodic microfieldmodels for the elastic behavior. Provided sufficiently fine discretizations of the phases areemployed the use of essentially elastic accommodation has been found to be acceptable,and high computational efficiency is claimed for the method (Dvorak et al., 1994). A fur-ther solution strategy for PMAs uses numerically evaluated equivalent inclusion approachesthat explicitly account for interacting inhomogeneities, see, e.g., Fond et al. (2001).

In typical periodic microfield approaches strains and stresses are approximated as thesum of uniform macroscopic contributions (slow variables), 〈ε〉 and 〈σ〉, and periodicallyvarying microscopic fluctuations (fast variables), ε′ and σ′, respectively, i.e.,

ε(z) = 〈ε〉+ ε′(z)

σ(z) = 〈σ〉+ σ′(z) . (6.1)

Here z is a “microscopic coordinate” that has sufficient resolution for describing the vari-ations on the microscale. For suitably defined unit cells, ΩUC, eqn. (1.2) implies that themicroscopic fluctuations must fulfill the relations

1

ΩUC

∫

ΩUC

ε′(z) dΩ = 0 and1

ΩUC

∫

ΩUC

σ′(z) dΩ = 0 (6.2)

As a consequence of eqn. (6.1), in periodic microfield approaches each unit of periodicity(unit cell) contributes the same increment of the displacement vector ∆u and the macro-scopic displacements vary (multi)linearly.

An idealized depiction of such a situation is presented in fig. 6.1, which shows the pe-riodic variations of the strains εs(s) = 〈εs〉+ ε′s(s) and of the corresponding displacementsus(s) = 〈εs〉 s + u′s(s) along the length s of a generic one-dimensional periodic two-phasematerial consisting of constituents A and B. The periodicity of the strains and of the

48

displacements is immediately apparent, the unit of periodicity and the corresponding dis-placement increment being marked as cU and ∆us, respectively.

s

uε,

<ε >s

<ε >ε

u

∆us

A A AB B B A

s

ss

s

cu

Figure 6.1: Schematic depiction of the variation of the strains εs(s) and the displacementsus(s) along a generic one-dimensional composite (coordinate s) consisting of constituentsA and B. Symmetry points of εs(s) and us(s) are indicated by small circles.

6.2 Boundary Conditions

The proper use of unit cell based methods requires that the cells together with the bound-ary conditions (BC) prescribed on them generate valid tilings both for the undeformedgeometry and for all deformed states pertinent to the problem (i.e., gaps and overlapsbetween neighboring unit cells as well as unphysical constraints on the deformations mustnot be possible). In order to achieve this, the boundary conditions for the unit cells mustbe specified in such a way that all deformation modes appropriate for the load cases tobe studied can be attained. The three major types of boundary conditions used in peri-odic microfield analysis are periodicity, symmetry, and antisymmetry (or point symmetry)BC28.

Generally, for a given periodic phase arrangement unit cells are non-unique, the rangeof possibilities being especially wide when point or mirror symmetries are present in themicrogeometry. As an example, fig. 6.2 depicts a hexagonal array of planar circular inho-mogeneities and some of the unit cells that can be used to study the mechanical behaviorof this arrangement. In fig. 6.3 the three basic types of boundary conditions are sketchedfor two-dimensional unit cells. The sides and corner points of the cells are annotated andan extension of this nomenclature to three-dimensional cases is shown in fig. 6.5.

28In addition, free surface boundary conditions may be used for layer-type models. For a more formaltreatment of boundary conditions for unit cells see, e.g., Michel et al. (1999).

49

2

1 periodic boundarysymmetry boundarypoint symmetry boundary

symmetry center(pivot point)

AB

F

GC

H I

D

E

J

Figure 6.2: Periodic hexagonal array of circular inhomogeneities in a matrix and 10 unitcells that can be used to describe the mechanical responses of this arrangement under loadsacting parallel to the coordinate axes.

The most general boundary conditions for unit cells are periodicity BC, which canhandle any possible deformed state of the cell and, consequently, of the inhomogeneousmaterial to be modeled. In fig. 6.2 cells A to E belong to this group. Because suchunit cells tile the computational space by translation, neighboring cells (and, consequently,opposite faces of a given cell) must fit into each other like the pieces of a jigsaw puzzlein both undeformed and deformed states. Periodic phase arrangements can be describedby sets of periodicity vectors pn, where the number of periodicity vectors, N , equals thespatial dimension of the problem. The minimum volume of unit cells pertaining to sucha set of periodicity vectors is well defined. There is, however, a wide variety shapes ofminimum volume unit cells for a given periodic phase arrangement, as is indicated byfig. 6.4. The surface of any unit cell for use with periodicity boundary conditions mustconsist of at least N pairs of faces (or parts of faces) Γk. The surfaces making up such apair must be identical but are shifted relative to each other by “shift vectors” ck whichare linear combinations of the periodicity vectors. In fig. 6.4 pairs of faces are marked bybeing drawn with the same line style.

NW

SE

NE

SW

NW

SW

NE

SE SW

NW NEUu

Lu

Pu

NEu

NEu

SEu SEu

N

W E

S

u

EW

N

S

uNW

W

S

NuNWNW

E

z

z

L

P

U

1

2

Figure 6.3: Sketch of periodicity, symmetry, and antisymmetry boundary conditions asused with two-dimensional unit cells.

50

p

p + p

p1

2

1 2

Figure 6.4: Six different minimum unit cells for a two-dimensional periodic matrix–inclusion medium with two (slightly) non-orthogonal translation vectors p1 and p2. Pairedfaces (or parts of faces) are marked by identical line styles and inhomogeneities inside theunit cells are set off by darker shading.

In the small strain regime the most general formulation of periodicity boundary condi-tions for the pair of surfaces Γk can be given as

u(sk + ck)− u(sk) = 〈ε〉 ∗ ck , (6.3)

where sk are position vectors on the faces making up the pair. For the special case ofquadrilateral two-dimensional unit cells, such as the one shown in fig. 6.3 (left) this leadsto the expressions

uN(s1) = uS(s1) + uNW uE(s2) = uW(s2) + uSE , (6.4)

where sk denotes the position in a local coordinate system on the pair of faces making upΓk and vertex SW is assumed to be fixed. Equations (6.4) directly imply that

uNE = uNW + uSE .

For numerical analysis the two faces making up a pair Γk must be discretized in a com-patible way, i.e., the nodal points on them must be positioned at equal values of the “facecoordinates” sk. Equations (6.3) then become sets of linear constraints each of which linksthree nodal displacement DOFs29. Comparing eqns.(6.3) and (6.4) shows that the displace-ments of the “master nodes”, SE and NW, contain the information on the macroscopicstrain tensor 〈ε〉. In addition, the displacements of the master nodes and of faces S andW fully control the displacements of the “slave faces” N and E.

29In principle, all variables (i.e., for mechanical analysis the displacements, strains and stresses) must belinked by appropriate periodicity conditions. When a displacement based FE code is used, however, suchconditions can be specified explicitly only for the displacement components (including, where appropriate,rotational DOFs). The periodicity of the stresses and strains is fulfilled approximately, mainly because intypical implementations nodal stresses and strains are not averaged across cell boundaries, even thoughthey ought to be.The periodicity of stresses leads to antiperiodic tractions at unit cell boundaries.

51

Conditions analogous to eqn. (6.4) can be specified for any appropriately chosen regularspace-filling two-dimensional cell having an even number of sides (squares, rectangles, andhexagons) and for three-dimensional cells with an even number of faces (cubes, hexahe-dra, rhombic dodekahedra, and regular tetrakaidekahedra), compare fig. 6.5. The schemecan also be extended to unit cells of less regular shape, compare Cruz and Patera (1995).Periodicity BC generally are the least restrictive option for unit cell models using phasearrangements obtained by statistically based algorithms (“supercells”). In practice FE-based unit cell analysis with periodicity boundary conditions can be rather expensive interms of computing time and memory requirements, because the linear equations betweenthree boundary DOFs corresponding tend to markedly degrade the band structure of thesystem matrix, especially in three-dimensional problems. Compared to other discrete mi-crostructure approaches, periodic homogenization typically shows the fastest convergencein terms of cell sizes, see, e.g., El Houdaigui et al. (2007).

Figure 6.5: Sketch of three-dimensional unit cell with nomenclature of vertices, edges andfaces.

For two-dimensional rectangular and hexahedral unit cells in which all faces of the cellcoincide with symmetry planes of the phase arrangement and for load cases that do notinterfere with these mirror symmetries, periodicity BC simplify to symmetry boundaryconditions. For the cell shown in fig. 6.3 (center) these take the form

uE(s2) = uSE vN(s1) = uNW uW(s2) = 0 vS(s1) = 0 , (6.5)

where u and v stand for the displacement components in 1- and 2-directions, respectively.Again the vertices SE and NW play the role of master nodes, but the constraint equationsare much simpler than eqn.(6.4), involving only 2 DOFs. Volume elements using symmetryboundary conditions must be rectangles or regular hexahedra, compare cell G in fig. 6.2;the extension of eqn.(6.5) to three-dimensional problems is straightforward. The load casesthat can be handled with symmetry BC are restricted to uniform thermal loads, normalmechanical loads acting on one or more pairs of faces, and combinations of the above30.A modified version of symmetry boundary conditions can be used to model the bend-ing of layer models, see, e.g., Weissenbek et al. (1997). Symmetry boundary conditions

30Because the load cases that can be handled include uniaxial loading normal to the faces, loading byextensional shear (obtained, e.g., in the two-dimensional case by applying normal stresses σa to the verticaland −σa to the horizontal faces), biaxial loading, hydrostatic loading and thermal loading, symmetry BCin many cases are sufficient for materials characterization.

52

are very useful for describing relatively simple microgeometries, but tend to be restrictivein modeling phase arrangements generated by statistically based algorithms because inho-mogeneities crossed by the cell boundaries must be symmetric with respect to the cell faces.

Antisymmetry boundary conditions are even more limited in terms of the microgeome-tries 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 em-ploying them on all faces are subject to few restrictions to the load cases that can behandled. Among the unit cells shown in fig. 6.2, cells G and H use point symmetry BC onall faces and can handle any in-plane deformation31. Alternatively, antisymmetry BC canbe combined with symmetry BC to obtain very small unit cells that are restricted to loadsacting normal to the symmetry faces, compare unit cells I and J in fig. 6.2. The sketchin fig. 6.3 (right) also shows such a small unit cell, the antisymmetry boundary conditionsbeing applied to the E-side only, where a pivot point P is present. For this configuration,the boundary conditions can be denoted as

uU(sP) + uL(−sP) = 2uP

vN(s1) = vNW = 2vP vS(s1) = 0 uW(s2) = 0 , (6.6)

where uU(sP) and uL(−sP) are the deformation vectors of pairs of points U and L that arepositioned symmetrically on face E (where a local coordinate sP centered on the pivot P isdefined in analogy to fig. 6.11) and the displacements of which are point symmetric withrespect to P. The phase arrangement, the undeformed geometry and the discretization ofsuch a face must also be antisymmetric with respect to the pivot point P. Three-dimensionalunit cells employing combinations of symmetry and point symmetry BC can be used toadvantage for studying cubic arrays of particles, see, e.g., Weissenbek et al. (1994).

6.3 Application of Loads and Evaluation of Microfields

The primary practical challenge in using periodic microfield approaches for modeling in-homogeneous materials lies in choosing and generating suitable unit cells that — in com-bination with appropriate boundary conditions — allow an accurate representation of theactual microgeometries within available computational resources. These unit cells mustthen be subjected to appropriate macroscopic stresses, strains and temperature excur-sions. Whereas loads of the latter type generally do not pose a problem, applying far fieldstresses or strains is not necessary trivial (note that the variations of the stresses along thefaces of a unit cell in general are not known a priori, so that it is not possible to prescribeappropriate boundary tractions via distributed loads).

31Unit cells used with antisymmetry boundary conditions may have odd numbers of faces. Triangularunit cells similar to cell H in fig. 6.2 were, e.g., used by Teply and Dvorak (1988) to study the transverse me-chanical behavior of hexagonal arrays of fibers. Rectangular cells with point symmetries on each boundarywere introduced by Marketz and Fischer (1994) for perturbed square arrangements of inhomogeneities.

53

Asymptotic Homogenization

The most versatile and elegant strategy for linking the macroscale and microscale in unitcell analysis is based on a mathematical framework known as homogenization theory orasymptotic homogenization, see, e.g., Suquet (1987). Macroscopic and microscopic coor-dinates, Z and z, respectively, are explicitly introduced. The macroscopic coordinates, Z,correspond to the ”standard” coordinates x, whereas the microscopic coordinates z arescaled up as

zi = xi/ǫ (6.7)

in order to zoom in on the local behavior. Here the ratio between the characteristic lengthsof the macroscale, L, and the microscale, ℓ, is used as scaling parameter ǫ = ℓ/L≪ 1.

The displacement field in the unit cell can then be represented by an asymptotic ex-pansion of the type

ui(Z, z, ǫ) = u(0)i (Z) + ǫ u

(1)i (Z, z) + ǫ2 u

(2)i (Z, z) + H.O.T. , (6.8)

where the u(0)i are the effective or macroscopic displacements and u

(1)i stands for the peri-

odically varying displacement perturbations due to the microstructure32. Using the chainrule, i.e.,

∂

∂xf(Z(x), z(x), ǫ) →

∂

∂Zf +

1

ǫ

∂

∂zf , (6.9)

the strains can be related to the displacements (in the small strain regime) as

εij(Z, z, ǫ) =1

2

(

∂

∂Zju

(0)i +

∂

∂Ziu

(0)j

)

+

(

∂

∂zju

(1)i +

∂

∂ziu

(1)j

)

+ǫ

2

(

∂

∂Zju

(1)i +

∂

∂Ziu

(1)j

)

+

(

∂

∂zju

(2)i +

∂

∂ziu

(2)j

)

+ H.O.T.

= ε(1)ij (Z, z) + ǫ ε

(2)ij (Z, z) + H.O.T. , (6.10)

where terms of the type ε(0)ij = 1

ǫ∂

∂zju

(0)i are deleted due to the underlying assumption that

the variations of slow variables are negligible at the microscale. The strains of order O(ǫ0)can be split into slow and fast contributions,

ε(1)ij =

1

2

(

∂

∂Zju

(0)i +

∂

∂Ziu

(0)j

)

and ε(1)ij =

1

2

(

∂

∂zju

(1)i +

∂

∂ziu

(1)j

)

, (6.11)

respectively. The stresses can be expanded into the expression

σij(Z, z, ǫ) = σ(1)ij (Z, z) + ǫ σ

(2)ij (Z, z) + H.O.T. . (6.12)

Using the two-scale assumption and, as a consequence, eqn. (6.9), the equilibriumequations take the form

( ∂

∂Zj

+1

ǫ

∂

∂zj

)

σij(Z, z, ǫ) + fi(Z) = 0 , (6.13)

32The nomenclature used in eqns. (6.7) to (6.17) follows typical usage in asymptotic homogenization.It is more general than but can be directly compared to the one used in eqns. (6.1) to (6.6), where nomacroscopic coordinate Z is employed. Zero order terms can generally be identified with macroscopicquantities and first order terms generally correspond to periodically varying microscopic quantities.

54

the fi being macroscopic body forces. By inserting eqn. (6.12) into this expression and sort-ing the resulting terms by order of ǫ a hierarchical system of partial differential equationsis obtained

∂

∂zj

σ(1)ij = 0 (order ǫ−1)

∂

∂Zj

σ(1)ij +

∂

∂zj

σ(2)ij + fi = 0 (order ǫ0) , (6.14)

the first of which gives rise to a boundary value problem at the unit cell level that isreferred to as the “micro equation”. Using eqn. (6.11) and elastic stress–strain relations,

an ansatz for the strains ε(1)ij can be made in the form

ε(1)ij = ε

(1)ij + ε

(1)ij =

(

Iijmn +∂

∂zjχimn

)

ε(1)mn (6.15)

so that the microstresses appearing in the micro equation can be expressed as

σ(1)ij = Eijkl(z)

(

Iklmn +∂

∂zlχkmn

)

ε(1)mn , (6.16)

where Eijkl(z) is the phase-level elasticity tensor, Iijkl is the 4th-order unit tensor, andthe “characteristic function” χijk(z) describes the deformation modes of the unit cell. Thesecond equation of the system (6.14), the ”macro equation”, after volume averaging overthe unit cell, allows to link the macroscopic and microscopic fields. The homogenizedelasticity tensor is then obtained as

Eijkl =1

ΩUC

∫

ΩUC

Eijkl(z)(

Iklmn +∂

∂zlχkmn(z)

)

dΩ . (6.17)

Analogous expressions can be derived for the tangent modulus tensors used in nonlinearanalysis, compare Ghosh et al. (1996).

These relations can be used as the basis of Finite Element algorithms that solve forthe characteristic function χijk, a task that typically has required special analysis codes.For detailed discussions of asymptotic homogenization methods within the framework ofFEM-based micromechanics see, e.g., Hollister et al. (1991), Ghosh et al. (1996) or Hassaniand Hinton (1999). An asymptotic homogenization procedure for elastic composites thatuses standard elements within a commercial FE package is described by Banks-Sills andLeiderman (1997). On the basis of homogenization theory macroscopic problems can besolved without explicitly specifying homogenized constitutive laws, so that simultaneoustwo-scale analysis can be carried out, see, e.g., Ghosh et al. (1996), Feyel (2003) or Teradaet al. (2003).

Method of Macroscopic Degrees of Freedom

When asymptotic homogenization is not used, it is good practice to apply far field stressesand strains to a given unit cell via concentrated nodal forces or prescribed displacements,

55

respectively, at the master nodes and/or pivots, an approach termed the “method of macro-scopic degrees of freedom” by Michel et al. (1999). Employing the divergence theorem andusing the nomenclature and configuration of fig. 6.3 (left), for load controlled analysis theforces to be applied to the master nodes SE and NW, PSE and PNW, of a two-dimensionalunit cell with periodicity boundary conditions can be shown to be given by the surfaceintegrals

PSE =

∫

ΓE

ta(z) dΓ PNW =

∫

ΓN

ta(z) dΓ . (6.18)

Here ta(s) = σa∗nΓ(s) stands for the homogeneous surface traction vector corresponding tothe applied (far field) stress field33 at some given point s on the cell’s surface ΓUC, and nΓ(s)is the local unit normal vector to the appropriate face. Equation (6.18) can be generalizedto require that each master node is loaded by a force corresponding to the surface integralof the surface traction vectors over the face slaved to it via an equivalent of eqns. (6.18),compare Smit et al. (1998). An analogous procedure holds for three-dimensional cases. Forgeometrically nonlinear analysis eqns. (6.18) must be applied to the current configuration.

For applying far field strains to periodic volume elements, the displacements to be pre-scribed to the master nodes must be obtained from the macroscopic strains via appropriatedisplacement–strain relations. For example, using the notation of eqns. (6.4), the displace-ments to be prescribed to the master nodes NE and SW of the unit cell shown in fig. 6.3(left) can be evaluated from eqn. (6.3) as

uSE = εa11 c1 vSE = γa

12 c1

uNW = γa21 c2 vNW = εa

22 c2 (6.19)

for an applied strain εa and linear displacement–strain relations34. Typically strain con-trolled unit cell models are easier to handle than stress controlled ones.

In general, the overall stress and strain tensors within a unit cell can be evaluated byvolume averaging (e.g., using some numerical integration scheme) or by using the equivalentsurface integrals given in eqn. (1.2), i.e.,

〈σ〉 =1

ΩUC

∫

ΩUC

σ(z) dΩ =1

ΩUC

∫

ΓUC

t⊗ z dΓ

〈ε〉 =1

ΩUC

∫

ΩUC

ε(z) dΩ =1

2ΩUC

∫

ΓUC

(u⊗ nΓ + nΓ ⊗ u) dΓ . (6.20)

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 dividingthe applied or reaction forces at the master nodes by the appropriate surface areas and bydividing the displacements at the master nodes by the appropriate cell lengths, respectively.

33Note that the ta(s) are not identical with the actual local values of the tractions, t(s), at the cell

boundaries, but are equal to them in an integral sense over the cell face.34The displacements at node NE are fully determined by those of the master nodes, compare eqns. (6.4),

so that one of the variables vSE and uNW is not independent. Analogous relations hold for three-dimensionalcases.

56

In order to obtain three-dimensional homogenized elastic tensors with the method ofmacroscopic degrees of freedom six suitable, linearly independent load cases must be solvedfor.

For evaluating phase averaged quantities from unit cell analysis, it is good practice touse direct volume integration according to eqn. (1.8) in analogy to the left hand terms ineqn. (6.20)35. In many FE codes this can be done by approximate numerical quadratureaccording to

〈f〉 =1

Ω

∫

Ω

f(z)dΩ ≈1

Ω

N∑

l=1

fl Ωl (6.21)

Here fl and Ωl are the function value and the integration weight (in terms of the volumeof the integration point), respectively, associated with the l-th integration point within agiven integration volume Ω that contains N integration points. When macroscopic valuesor phase averages are to be generated of variables that are nonlinear functions of the stressor strain components (e.g., equivalent stresses, equivalent strains or stress triaxialities),only direct volume averaging should be used, because evaluation of nonlinear variables onthe basis of averaged components may lead to unacceptable inaccuracies, compare alsoeqn. (3.58).

6.4 Unit Cell Models for Continuous Fiber Reinforced

Composites

Composites reinforced by continuous aligned fibers typically show a statistically trans-versely isotropic overall behavior and can be studied well with periodic homogenization.Materials characterization with the exception of the overall axial shear behavior can be car-ried out with two-dimensional unit cell models employing generalized plane strain elementsthat use a global degree of freedom to describe the axial deformation of the whole model36.Such elements are implemented in a number of commercial FE codes. For handling theoverall axial shear response, however, special elements (Adams and Crane, 1984) or three-dimensional analysis with appropriate periodicity boundary conditions (Pettermann andSuresh, 2000) are required. It is worth noting that in all the above cases generalized planestrain states of some type are described, i.e., the axial strains are constant over the unit cellalthough out-of-plane deformations are present. Three-dimensional analysis is necessaryfor studying composites reinforced by continuous aligned fibers when the effects of fibermisalignment or of broken fibers are to be studied, see, e.g., Mahishi (1986).

35It is of some practical interest that volume averaged and phase averaged microfields obtained from unitcells must fulfill all relations given in section 3.1 and can thus be conveniently used to check the consistencyof a given model by inserting them into eqns. (3.4). Note, however, that for finite deformations appropriatestress and strain measures must be used for this purpose (Nemat-Nasser, 1999).

36Because the axial stiffness of composites reinforced by continuous aligned fibers can usually be sat-isfactorily described by Voigt-type models, compare eqn. (2.2), unit cell models of such materials havetended to concentrate on the transverse behavior. Plain strain models, however, can describe neither theaxial constraints nor the axial components of microstresses and microstrains.

57

PH0

CH1

RH2

CH3

PS0

MS5

CS7

CS8

Figure 6.6: Eight periodic fiber arrangements of fiber volume fraction ξ=0.475 for modelingcontinuously fiber reinforced composites (Bohm and Rammerstorfer, 1995).

Basic generalized plane strain models of continuously reinforced composites make useof simple periodic fiber arrangements as shown in fig. 6.6, all of which can be described by

EPS.EFF.PLASTIC INC.800

2.25E-02

1.75E-02

1.25E-02

7.50E-03

2.50E-03

Figure 6.7: Microscopic distributions of the accumulated equivalent plastic strain in thematrix of a transversely loaded unidirectional continuously reinforced ALTEX/Al MMC(ξ=0.453) as predicted by a multi-fiber unit cell (arrangement DN).

58

rather small unit cells using symmetry and/or point symmetry BC. The simplest amongthese arrangements are the periodic hexagonal (PH0) and periodic square (PS0) arrays.Models with hexagonal symmetry (PH0,CH1,RH2,CH3) give rise to transversely isotropicthermoelastic overall behavior, whereas the other fiber arrangements shown in fig. 6.6 havetetragonal (PS0,CS7,CS8) or monoclinic (MS5) overall symmetry. Elastoplastic behaviorof the matrix causes the macroscopic symmetries of the fiber arrangements to degradeunder most loads due to the low symmetry of the local material properties, the materialstate depending on the load history a given point has undergone, compare also fig. 6.8. Inmany cases simple periodic microgeometries do not provide fully satisfactory descriptionsof fiber reinforced materials, most of which show at least some randomness in the fiberpositions. Much improved models can be obtained by periodic multi-fiber unit cells thatemploy quasi-random fiber positions. Such models can either use symmetry BC, compareNakamura and Suresh (1993) and fig. 6.7, or periodicity BC (Zeman and Sejnoha, 2007).

In table 6.1 thermoelastic moduli of an aligned continuously reinforced ALTEX/AlMMC as predicted by bounding methods, MFAs and unit cells methods using arrange-ments PH0, CH1 and PS0 (compare fig. 6.6) as well as DN (see fig. 6.7) are listed. For thismaterial combination all unit cell results (even PS0, which is not transversally isotropic)

Table 6.1: Overall thermoelastic moduli of a unidirectional continuously reinforced AL-TEX/Al MMC (ξ=0.453) as predicted by the Hashin–Shtrikman (HS) and three-point(3PB) bounds, by the Mori–Tanaka method (MTM) and the generalized self-consistentscheme (GSCS), by the differential scheme (DS), by Torquato’s three-point estimates(3PE), as well as by unit cell analysis using periodic arrangements shown in fig. 6.6 (PH0,CH1, PS0) and the multi-fiber cell displayed in fig. 6.7 (DN). For arrangement PS0 re-sponses in the 0 and 45, and for arrangement DN responses in the 0 and 90 directionsare listed.

E∗

l E∗

q ν∗lq ν∗qt α∗

l α∗

q

[GPa] [GPa] [] [] [K−1×10−6] [K−1×10−6]fibers 180.0 180.0 0.20 0.20 6.0 6.0matrix 67.2 67.2 0.35 0.35 23.0 23.0HS/lo 118.8 103.1 0.276 0.277 — —HS/hi 119.3 107.1 0.279 0.394 — —3PB/lo 118.8 103.8 0.278 0.326 — —3PB/hi 118.9 104.5 0.279 0.347 — —MTM 118.8 103.1 0.279 0.342 11.84 16.46GSCS 118.8 103.9 0.279 0.337 11.84 16.46DS 118.8 103.9 0.278 0.339 11.94 16.353PE 118.8 103.9 0.279 0.338 11.89 16.40PH0 118.8 103.7 0.279 0.340 11.84 16.46CH1 118.7 103.9 0.279 0.338 11.90 16.42PS0/00 118.8 107.6 0.279 0.314 11.85 16.45PS0/45 118.8 99.9 0.279 0.363 11.85 16.45DN/00 118.8 104.8 0.278 0.334 11.90 16.31DN/90 118.8 104.6 0.278 0.333 11.90 16.46

59

fall within the Hashin–Shtrikman bounds, but the predictions for the square arrangementshow clear in-plane anisotropy and do not follow the three-point bounds. The resultsfor the multi-fiber arrangement indicate some minor deviation from transversely isotropicmacroscopic behavior.

The fiber arrangements shown in figs. 6.6 and 6.7 give nearly identical results for theoverall thermoelastoplastic behavior of continuously reinforced composites under axial me-chanical loading, and the predicted overall axial and transverse responses under thermalloading are also very similar. The overall behavior under transverse mechanical loading,however, depends markedly on the phase arrangement, see fig. 6.8. For fiber arrangementsof tetragonal or lower symmetry (e.g., PS0, MS5, CS7 and CS8) the predicted transversestiffnesses depend strongly on the loading direction, and the behavior of the hexagonalarrangements is sandwiched between the stiff (0) and the compliant (45) responses ofperiodic square arrangements in both the elastic and elastoplastic ranges. Multi-fiber unitcells such as fig. 6.7 that approach statistical transverse isotropy tend to show noticeablystronger strain hardening compared to periodic hexagonal arrangements.

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

Figure 6.8: Transverse elastoplastic response of a unidirectional continuously reinforcedALTEX/Al MMC (ξ=0.453, elastoplastic matrix with linear hardening) to transverse uni-axial loading as predicted by unit cell models PH0, PS0 and DN.

The distributions of microstresses and microstrains in fibers and matrix typically de-pend markedly on the fiber arrangement, especially under thermal and transverse mechan-ical loading. In the plastic regime, the microscopic distributions of equivalent stressesand equivalent plastic strains, of hydrostatic stresses, and of stress triaxialities tend to bestrongly inhomogeneous, compare fig. 6.7. As a consequence, the onset of ductile damagein the matrix (or of brittle damage in brittle matrix composites), of brittle failure of thefibers and of interfacial decohesion at the fiber–matrix interfaces show a strong dependenceon the fiber arrangement.

60

6.5 Unit Cell Models for Short Fiber Reinforced Com-

posites

The overall symmetry of short fiber reinforced composites in many cases is isotropic (forrandom fiber orientations) or transversely isotropic (for aligned fibers, planar random fibersand other fiber arrangements with axisymmetric orientation distributions). However, pro-cessing conditions can give rise to a wide range of fiber orientation distributions and,consequently, lower overall symmetries (Allen and Lee, 1990). The thermoelastic and ther-moelastoplastic behavior of aligned short fiber reinforced composites has been successfullyestimated by Mori–Tanaka methods, which can also be extended to nonaligned fibers andreinforcements showing an aspect ratio distribution, compare section 3.8. Such mean-fieldapproaches are, however, limited in resolving details of fiber arrangements, especially forinelastic material behavior. At present the most powerful tools for studying the influenceof fiber shapes and orientations, of clustering effects, of the interaction of fibers of differentsizes, and of local stress and strain fields between neighboring fibers are periodic microfieldmethods.

In contrast to continuously reinforced composites, the phase arrangements of dis-continuously reinforced materials are inherently three-dimensional. The simplest three-dimensional unit cell models of aligned short fiber reinforced composites have used peri-odic square arrangements of non-staggered or staggered aligned fibers37, see, e.g., Levyand Papazian (1991) and compare fig. 6.9. Somewhat more complex three-dimensionalmodels can be based on periodic arrays of alternatingly tilted misaligned fibers (Sørensenet al., 1995). Such approaches are relatively simple but rather restrictive in terms of fiber

Figure 6.9: Three-dimensional unit cells for modeling non-staggered (left) and staggered(right) square arrangements of aligned short fibers. Shaded cells require symmetry BC,cells outlined in bold can use periodicity BC.

37Such square arrangements give rise to tetragonal overall symmetry, and, consequently, the transverseoverall properties are direction dependent.

61

arrangements that can be handled. When larger unit cells supporting periodicity boundaryconditions are used the full thermomechanical behavior of the composites can be studied.

For many materials characterization studies, a more economical alternative to the abovethree-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. Thebasic idea behind these models is to replace unit cells for square or hexagonal arrangementsby circular composite cylinders of equivalent cross sectional area (and volume fraction) assketched in 6.10. The resulting axisymmetric cells are not unit cells in the strict sense,because they overlap and are not space filling. In addition, they do not have the sametransverse fiber spacing as the corresponding three-dimensional arrangements and cannotbe used to study the response to most transverse mechanical loading conditions. However,they have the advantage of significantly reduced computational requirements. Symmetryboundary conditions are used for the top and bottom faces of these cells, and the BCs atthe “outer” (circumferential) surface are chosen to maintain the same cross sectional areaalong the axial direction for an aggregate of cells.

Figure 6.10: 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).

In the case of non-staggered fibers this can be easily done by specifying symmetry-typeboundary conditions for the outer surfaces, whereas for staggered arrangements a pair ofcells with different fiber positions is considered, for which the total cross sectional area isrequired to be independent of the axial coordinate. Using the nomenclature of fig. 6.11and the notation of eqns. (6.4) to (6.6), this leads to nonlinear relations for the radialdisplacements u and linear constraints for the axial displacements v at the outer surface,

(rU + uU)2 + (rL + uL)2 = 2(rP + uP)2 and vU + vL = 2vP , (6.22)

respectively, where r is the radius of the undeformed cell. By choosing the two cells makingup the pair to be antisymmetric with respect to a pivot point P the arrangement can be

62

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

zP

~

~SW SW

Figure 6.11: Axisymmetric cell for staggered arrangement of short fibers: undeformed anddeformed shapes.

described by a single cell with an antisymmetric outer (E-) face, U and L being nodeson this face that are positioned symmetrically with respect to the pivot, P. Nonlinearconstraints such as eqn. (6.22), however, are not widely available and may be cumbersometo use in a given FE code. For most applications the BC for the radial displacementsin eqn. (6.22) can be linearized without major loss in accuracy, so that antisymmetryboundary conditions analogous to eqn. (6.6) are obtained for the outer surface,

uU + uL = 2uP and vU + vL = 2vP , (6.23)

For many years axisymmetric unit cell models of the types shown in fig. 6.10 have been theworkhorses of PMA studies of short fiber reinforced composites, see, e.g., (Povirk et al.,1992) or Tvergaard (1994). Typically, descriptions using staggered arrangements allowa wider range of microgeometries to be covered, compare Bohm et al. (1993) or Tver-gaard (2003), and give more realistic descriptions of actual composites. Both staggeredand non-staggered axisymmetric models can be extended to study a considerable range ofarrangements incorporating fibers of different size and/or aspect ratio by foregoing the useof point symmetric geometries and coupling two or more different cells via the conditionof keeping the cross sectional area of the aggregate independent of the axial coordinate(Bohm et al., 1993).

Three-dimensional unit cells and axisymmetric cells have been used successfully forstudying the nonlinear thermomechanical behavior of aligned short fiber reinforced MMCs,e.g., with respect to the their stress–strain responses, to the pseudo-Bauschinger effect, andto thermal residual stresses. They have provided valuable insight into causes and effectsof matrix, interface and fiber damage.

Over the past 15 years unit cells containing fibers that follow some prescribed orien-tation distribution function (and where appropriate, fiber size and/or aspect ratio dis-tribution functions), have become feasible for non-dilute fiber volume fractions. Suchapproaches, however, tend to pose considerable challenges in generating appropriate fiberarrangements at non-dilute volume fractions due to geometrical frustration. The first stud-ies of this type were restricted to the linear elastic range, where the BEM has been foundto answer well, see, e.g., Banerjee and Henry (1992). Meshing such configurations for usewith the FEM can be difficult, compare Shephard et al. (1995), and analyzing the me-

63

Figure 6.12: Unit cell for a composite reinforced by randomly oriented short fibers (Bohmet al., 2002). The nominal fiber volume fraction is ξ = 0.15 and the 15 cylindrical fibersin the cell have a nominal aspect ratio of a = 5.

chanical response of the resulting cells requires considerable computing power, especiallyfor nonlinear material behavior. As an example, fig. 6.12 shows a unit cell that contains15 randomly oriented cylindrical fibers of aspect ratio 5, supports periodicity boundaryconditions, and is discretized by tetrahedral elements.

6.6 Unit Cell Models for Particle Reinforced Com-

posites

Particle reinforced composites are typically assumed to show a statistically isotropic over-all response, and mean field models are available for describing the overall thermoelasticbehavior for the case of spherical inhomogeneities. Extensions of the mean field solutionsinto the nonlinear range are available, compare section 3.7, but they tend to be subject tolimitations in predicting the overall thermomechanical response in the post-yield regime.In addition, mean field methods for particle reinforced composites cannot account for manyparticle shape, clustering, and size distribution effects and cannot resolve local fluctuationsof the stress and strain fields.

A difficulty in using unit cell approaches to modeling particle reinforced materials withoverall isotropic behavior is due to the fact that even for spherical inhomogeneities nosimple periodic phase arrangement exists that is inherently elastically isotropic. Togetherwith the wide variation in microgeometries and particle shapes in actual materials, thiscauses studies using generic phase arrangements to be subject to potentially tricky trade-offs between keeping computational requirements at manageable levels (favoring simpleparticle shapes combined with two-dimensional models or simple three-dimensional micro-geometries) and obtaining sufficiently realistic models for a given purpose (often leadingto requirements for unit cells containing a high number of particles of complex shape at

64

randomly selected positions). In many respects, however, periodic microfield models of par-ticle reinforced composites are subject to similar constraints and use analogous approachesas work on short fiber reinforced composites.

Most three-dimensional unit cell studies of generic microgeometries for particle rein-forced composites have been based on simple cubic (sc), face centered cubic (fcc) and bodycentered cubic (bcc) arrangements of spherical, cylindrical and cube-shaped particles, com-pare Hom and McMeeking (1991). By invoking the symmetries of these arrangements andusing symmetry as well as antisymmetry boundary conditions, relatively simple unit cellsfor materials characterization can be obtained38, compare Weissenbek et al. (1994) and seefigs. 6.13 and 6.14. In addition, work employing hexagonal or tetrakaidekahedral particlearrangements has been reported, see, e.g., Rodin (1993). Of the above arrangements, sim-ple cubic models are probably the easiest to handle, but they show a marked anisotropy,much more so than fcc and bcc configurations.

2-particle cell2-particle cell

1-particle cell

2-particlecell

1-particlecell

1-part.cell

ReferenceVolume

Figure 6.13: Simple cubic, face centered cubic, and body centered cubic arrangementsWeissenbek et al. (1994).

Figure 6.14: Some unit cells for particle reinforced composites using cubic arrangements ofinhomogeneities: s.c. arrangement of cubes, b.c.c. arrangement of cylinders and f.c.c. ar-rangement of spheres Weissenbek et al. (1994).

During the past 15 years three-dimensional studies based on more complex phase ar-rangements have begun to appear in the literature. Gusev (1997) used Finite Element

38Again, considerably larger unit cells with periodicity rather than symmetry boundary conditions arerequired for unrestricted modeling of the full thermomechanical response of cubic phase arrangements.

65

methods in combination with unit cells containing up to 64 statistically positioned parti-cles to describe the overall behavior of elastic particle reinforced composites. Hexahedralunit cells containing up to 10 particles in a perturbed cubic configuration (Watt et al.,1996) as well as cube shaped cells incorporating at least 15 spherical particles in quasi-random arrangements (Bohm et al., 1999; Bohm and Han, 2001), compare fig. 6.15, orclusters of particles (Segurado et al., 2003) were proposed for studying elastoplastic parti-cle reinforced MMCs and related materials. Three-dimensional simulations involving highnumbers of particles have been reported for investigating elastic composites (Michel et al.,1999), for studying brittle matrix composites that develop damage (Zohdi and Wriggers,2001), and for rubber reinforced polymers (Fond et al., 2001).

Figure 6.15: Unit cell for a particle reinforced MMC (ξ=0.2) containing 20 spherical par-ticles in a quasi-random arrangement suitable for using periodicity BC (Bohm and Han,2001).

In analogy to short fiber reinforced materials, compare fig. 6.10, axisymmetric cell mod-els with staggered or non-staggered particles can be used for materials characterization ofparticle reinforced composites. By appropriate choice of the dimensions of the axisymmet-ric cells, sc, fcc and bcc arrangements of particles can be approximated. Axisymmetriccell models have been a mainstay of PMA modeling of materials containing particulateinhomogeneities, see, e.g., Bao et al. (1991).

Due to their relatively low computational requirements, planar unit cell models of parti-cle reinforced materials have also been used to a considerable extent. Typically, plane stressmodels (which actually describe “reinforced sheets” or the stress states at the surface ofinhomogeneous bodies) show a more compliant and plane strain models (which correspondto reinforcement by aligned continuous fibers rather than particles) show a stiffer over-all response than three-dimensional descriptions, compare table 6.2. With respect to theoverall behavior, plane stress analysis may be preferable to plane strain analysis, compareWeissenbek (1994), but no two-dimensional model gives satisfactory results in terms of thepredicted microstress and microstrain distributions (Bohm and Han, 2001). Axisymmetriccell models typically provide considerably better results than planar ones.

66

Table 6.2: Overall thermoelastic properties of a particle reinforced SiC/Al MMC (spher-ical particles, ξ=0.2) as predicted by the Hashin–Shtrikman (HS) and third order (3PB)bounds, by the Mori–Tanaka method (MTM), by the generalized self-consistent scheme(GSCS), by the differential scheme (DS), by Torquato’s three-point estimates (3PE), aswell as by unit cell analysis using three-dimensional cubic arrangements, axisymmetriccells approximating sc, fcc and bcc arrangements, two-dimensional multi-particle modelsbased on plane stress (2/D PST) and plane strain (2/D PSE) kinematics, as well as thethree-dimensional multi-particle model shown in fig. 6.15.

E∗ E∗[100] E∗[110] ν∗ α∗

l

[GPa] [GPa] [GPa] [] [K−1×10−6]particles 429.0 — — 0.17 4.3matrix 67.2 — — 0.35 23.0HS/lo 90.8 — — 0.286 16.8HS/hi 114.6 — — 0.340 18.63PB/lo 91.4 — — 0.323 18.53PB/hi 93.9 — — 0.328 18.6MTM 90.8 — — 0.329 18.6GSCS 91.5 — — 0.327 18.6DS 92.7 — — 0.326 18.43PE 91.8 — — 0.327 18.6sc — 96.4 90.5 18.7fcc — 89.0 91.7 18.6bcc — 90.0 90.6 18.6axi/sc 95.4 — — 18.6/18.6axi/fcc 88.1 — — 18.1/19.0axi/bcc 87.9 — — 18.4/18.82/D PST 85.5 — — 0.3342/D PSE 98.7 — — 0.5003/D 92.4 — — 0.326

In table 6.2 bounds, MFA results, and a number of PMA predictions for the overallthermoelastic moduli of a particle reinforced SiC/Al MMC are compared, the loading di-rections for the cubic arrangements being identified by Miller indices. It is noteworthy that— for the loading directions considered here, which do not span the full range of possibleresponses of the cells — none of the cubic arrangements gives results that fall within thethird order bounds for identical spherical particles. The overall anisotropy of the simplecubic arrangement is marked, whereas the body centered and face centered arrays deviatemuch less from overall isotropy. The predictions of the axisymmetric analysis can be seento be of comparable quality in terms of the elastic moduli to those of the correspondingcubic arrays, but typically give an anisotropic thermal expansion behavior. The resultslisted for the three-dimensional multi-particle models are ensemble averages over a numberof unit cells and loading directions, and they show very good agreement with the three-point bounds and estimates, whereas there are marked differences to the plane stress andplane strain models.

67

Similarly to fiber reinforced MMCs, particle reinforced composites typically displayhighly inhomogeneous microscopic stress and strain distributions, especially for the matrixin the nonlinear range39. For example, fig. 6.16, shows the predicted equivalent plasticstrains of the elastoplastic matrix inside a multi-particle unit cell model of an MMC. Theconcentrations of plastic strains can lead to shear bands when the matrix material displayssoftening behavior.

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

Figure 6.16: Predictions for the equivalent plastic strains in a particle reinforced MMC(ξ=0.2) subjected to uniaxial tensile loading obtained by a unit cell with 20 sphericalparticles in a quasi-random arrangement (Bohm and Han, 2001).

If the particles are irregular in shape or if their volume fraction markedly exceeds a valueof 0.5 (as is the case, e.g., for cermets such as WC/Co), generic unit cell models employingrelatively regular particle shapes may not result in very satisfactory microgeometries. Forsuch materials a typical approach consists of basing PMA models on a “real structure”obtained from a metallographic section, compare Fischmeister and Karlsson (1977). Dueto the nature of the underlying experimental data, models of this type often take the formof planar analysis, the limitations of which have been discussed above.

6.7 Unit Cell Models for Woven and Laminated Com-

posites

Periodic microfield methods play an important role in studying composites with woven,braided or knitted reinforcements, where the reinforcing phase takes the form of textile-likestructures consisting of bundles of fibers (tows). Unit cell models of woven composites are

39Plane stress models tend to predict much higher levels of equivalent plastic strains in the matrix (andthus weaker hardening) than do plane strain and generalized plane strain models of the same reinforcementvolume fraction, and the results from three-dimensional arrangements typically lying between those of theabove groups of planar models.

68

TowsCompositeMatrix

Figure 6.17: A unit cell for modeling a plain weave lamina via symmetry boundary condi-tions. The tow region (left), the matrix region (center) and the “assembled” unit cell areshown.

typically based on modeling fiber bundles as a “mesophase” with smeared out materialproperties, which, in turn, are obtained from analyzing unidirectionally continuously rein-forced composites40. Such matrix–tow level unit cells in many cases describe one woven ply,so that free surfaces are used at the bottom and top faces. Symmetry boundary conditionscan be used for materials characterization, but for “full” homogenization modified peri-odicity boundary conditions are required, which introduce macroscopic rotational degreesof freedom that allow to describe warping and twisting of the woven lamina or laminae.Unit cells for woven composites tend to be fairly complex geometrically, even when onlysymmetry BC are used, compare fig. 6.17. They tend to be computationally expensive,especially for nonlinear constituent behavior.

Another group of composite materials that can be studied to advantage by unit cellmethods are laminates consisting of plies the thickness of which is not much greater thanthe fiber diameter and which, accordingly, contain only few fibers in thickness direction.Figure 6.18 shows unit cells for modeling the mechanical behavior of monofilament rein-

o90 plies

β

o0 ply

Figure 6.18: Two unit cells for modeling cross ply laminates and one for modeling angleply laminates.

40This modeling strategy obviously is a type of multiscale modeling.

69

forced cross ply composites with two (left) and one (center) fiber layers per ply, respectively,and for modeling angle ply laminates (right).

6.8 Unit Cell Models for Porous and Cellular Mate-

rials

Elastoplastic porous materials have been the subject of a considerable number of PMAstudies due to their relevance to the ductile damage and failure of metallic materials41.Generally, modeling concepts for porous materials are closely related to those employed forparticle reinforced composites, the main difference being that the shapes of the voids mayevolve significantly through the loading history42. Three-dimensional unit cells based oncubic arrangements of voids and axisymmetric cells as discussed in section 6.6 have beenused in the majority of the pertinent PMA studies.

In cellular materials, such as foams and cancellous bone, the volume fraction of the solidphase is low (often amounting to no more than a few percent) and the void phase may betopologically connected (open cell foams), unconnected (closed cell foams) or both of theabove (e.g., hollow sphere foams). Such materials often display a small linear range, and athigher strains gross shape changes of the cells typically take place. This behavior is espe-cially pronounced for compressive loading, where elastic buckling (e.g., in polymer foams),plastic buckling (e.g., in metal foams) and brittle failure (e.g., in cancellous bone) of strutsor cell walls can play major roles. Accordingly, unit cells for such materials often requirespecial provision for handling large deformations of and contact between cell walls43. Inaddition, care must be taken that boundaries on which symmetry BCs are specified do notcoincide with cell walls that can be expected to show instabilities. Finally, models mustbe sufficiently large so that nontrivial deformation patterns can develop, compare Daxner(2003).

The geometrically most simple cellular materials are regular honeycombs, which canbe modeled by planar hexagonal cell models. Somewhat less ordered two-dimensional ar-rangements have been used for studying the crushing behavior of soft woods (Holmberget al., 1999), and highly irregular planar arrangements, compare fig. 6.19, can be used tostudy many aspects of the geometry dependence of the mechanical response of cellularmaterials, such as metallic foams.

41Most constitutive models describing ductile damage and failure of metals are based on micromechanicalconsiderations, among them Rice and Tracey (1969), Gurson (1977), Tvergaard and Needleman (1984) aswell as Gologanu et al. (1997).

42The evolution of the shapes of initially spheroidal voids under non-hydrostatic loads has been thesubject of intensive studies by mean field type methods, compare Kailasam et al. (2000). Such modelsare based on the assumption that initially spherical pores will stay ellipsoids throughout the deformationhistory, which axisymmetric cell analysis (Garajeu et al., 2000) has shown to be an excellent approximationfor axisymmetric tensile load cases. For compressive loading, however, initially spherical pores may evolveinto markedly different shapes (Segurado et al., 2002).

43The “standard” analytical models for the thermomechanical behavior of cellular materials (Gibsonand Ashby, 1988) use analytical unit cells.

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X

Y

Z X

Y

Z

Figure 6.19: Planar periodic unit cell for studying irregular cellular materials (Daxner,2003).

Three-dimensional studies of closed cell foams have used generic microgeometries basedon cubic arrangements of spherical voids (Hollister et al., 1991), of truncated cubes plussmall cubes (Santosa and Wierzbicki, 1998), of rhombic dodekahedra, and of regulartetrakaidekahedra (Simone and Gibson, 1998), compare fig. 6.20, as well as on randomarrangements of multiple spherical pores (Smit et al., 1999). Analogous model geometrieshave been reported for modeling open cell foams.

The effects of details of the microgeometries of cellular materials (e.g., thickness distri-butions and geometrical imperfections or flaws of cell walls or struts), which can consider-ably influence the overall behavior (a typical example being the very small elastic rangesof metallic foams), have been an active field of research over the past decade, see, e.g.,Grenestedt (1998) and Daxner (2003). Unit cell models of cellular materials with real-istic microgeometries often are rather complex and numerically demanding due to thesematerials’ tendency to deform by local mechanisms and instabilities. However, analytical

Figure 6.20: Foam microstructure modeled by regular tetrakaidekahedra (truncated octa-hedra).

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solutions have been reported for some simple periodic phase arrangements, compare, e.g.,Warren and Kraynik (1991).

The modeling of a special cellular material, cancellous (or spongy) bone, has attractedconsiderable research interest for the past 25 years. Cancellous bone shows a wide rangeof microstructures, which can be idealized as beam or beam–plate configurations (Gibson,1985). The solid phase of cancellous bone is an inhomogeneous material at a lower lengthscale. In studying the mechanical behavior of cancellous bone, large three-dimensional unitcell models based on tomographic scans of actual samples and using voxel-based discretiza-tion schemes have become fairly widely used, compare Hollister et al. (1991).

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Chapter 7

Embedded Cell Models

Like periodic microfield methods, Embedded Cell Approaches (ECAs) aim at predictingthe microfields in inhomogeneous materials at high spatial resolution. For this purpose theyuse models consisting of a core (or “local heterogeneous region”), the configuration of whichcan range from rather simple to highly detailed phase arrangements, that is embedded inan outer region serving mainly for transmitting the applied loads, compare fig. 7.1. Thismodeling strategy avoids some of the drawbacks of PMAs, especially the requirement thatthe geometry and all microfields must be strictly periodic44. In modeling the core, similarconsiderations hold as for PMAs, especially with respect to the overall symmetry and totwo-dimensional vs. three-dimensional analysis. Of course, a material description must bechosen for the outer region that is compatible to the smeared-out behavior of the core, sothat errors in the accommodation of stresses and strains are avoided. Some care is alsorequired with respect to spurious boundary layers which may occur at the “interfaces”between the core and the surrounding material45. Although some analytical methods suchas classical and generalized self-consistent schemes, see section 3.5, may be viewed asembedding schemes, most ECAs with more complex cores have been based on numericalengineering methods.

Three basic types of embedding approaches can be found in the literature. One of themuses discrete phase arrangements in both the core region and in the surrounding material,the latter, however, being discretized by a much coarser FE mesh, see, e.g., Sautter et al.(1993). Such models, which in some ways approach descriptions of a full sample with arefined mesh in an interior region, to a large extent avoid boundary layers between coreand outer regions, but tend to be relatively expensive computationally.

In the second group of embedding methods the behavior of the outer regions is describedvia appropriate smeared-out constitutive models. In the simplest case these take the formof semi-empirical or micromechanically based constitutive laws that are prescribed a priorifor the embedding zone and which must be chosen to correspond closely to the overall

44Embedding analysis can be used without intrinsic restrictions at or in the vicinity of free surfacesand interfaces, it can handle gradients in composition and loads, and it can be employed for studying theinteraction of macrocracks with microstructures. Also, the requirement of sufficiently separated lengthscales, compare section 1.1, does not necessarily apply to embedded models.

45These interfaces are a consequence of the modeling approach only and do not have any physicalbackground. Such boundary layers typically have a thickness of, say, an inhomogeneity diameter forelastic materials, but they may be longer ranged for nonlinear material behavior.

73

Figure 7.1: Schematic depiction of the arrangement of core and embedding region in anembedded cell model of a tensile test specimen.

behavior of the core. This way, conceptually simple models are obtained that are verywell suited to studying local phenomena such as the stress and strain distributions in thevicinity of crack tips or at macroscopic interfaces in composites (Chimani et al., 1997) orthe growth of cracks in inhomogeneous materials (van der Giessen and Tvergaard, 1994;Wulf et al., 1996; Motz et al., 2001; Gonzalez and LLorca, 2007). A closely related ap-proach uses appropriate coupling conditions (which may be implemented via the boundaryconditions of the micromodel) to link a macromodel employing smeared-out material datawith a micromodel describing discrete constituents (Varadi et al., 1999).

The third type of embedding schemes employ the homogenized thermomechanical re-sponse of the core to determine the effective behavior of the surrounding medium, givingrise to models of the self-consistent type, which are mainly employed for materials char-acterization. The use of such approaches is, of course, predicated on the availability ofsuitable parameterizable constitutive laws for the embedding material that can follow thecore’s instantaneous homogenized behavior with high accuracy for all load cases and for anyloading history. This requirement can typically be fulfilled easily in the linear range, seee.g., Chen (1997), but leads to considerable difficulties when at least one of the constituentsshows elastoplastic or viscoplastic material behavior46. Accordingly, approximations haveto be used (the consequences of which may be difficult to assess in view of the nonlinearityand path dependence of elastoplastic material behavior) and models of this type are besttermed “quasi-self-consistent schemes”. Such approaches were discussed, e.g., by Bornertet al. (1994) and by Dong and Schmauder (1996). They are the only embedding methodsthat do micromechanics in the strict sense in that they handle scale transitions.

46Typically, the effective yielding behavior of elastoplastic composites shows some dependence on thefirst stress invariant, and, for low plastic strains, the homogenized response of the core tends to be stronglyinfluenced by the fractions of the elastoplastic constituent(s) that have actually yielded. In addition, inmany cases anisotropies of the yielding and hardening behavior are introduced by the phase geometry (e.g.,aligned fibers) and by the phase arrangement of the core. Finding constitutive laws that, on the one hand,can satisfactorily account for such phenomena and, on the other hand, have the capability of being easilyadapted to the instantaneous responses of the core by adjusting free parameters, poses a major obstaclefor quasi-self-consistent schemes in nonlinear regimes.

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For materials characterization embedded cells can be loaded by homogeneous stressesor strains applied to the outer boundaries. If cracks or similar problems are to be studied,however, it is preferable to impose displacement boundary conditions corresponding to thefar field behavior of suitable analytical solutions (e.g., to the displacement field of a cracktip). Alternatively, complete samples can be considered in “simulated experiments”, assketched in fig. 7.1.

Effective and phase averaged stresses and strains from embedded cell analysis are bestevaluated with eqn. (6.20) or its equivalents, and in the case of geometrically complexphase arrangements it is good practice to use only the central regions of the core for thispurpose in order to avoid possible boundary layers.

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Chapter 8

Windowing Approaches

The aim of windowing methods is to obtain estimates or bounds for the macroscopic prop-erties of inhomogeneous materials on the basis of non-periodic volume elements that arereferred to as “mesoscopic test windows” or, shorter, as “windows”. These volume elementshave simple shapes, are extracted at random positions and with random orientations froman inhomogeneous medium, and are smaller than RVEs, compare fig. 8.1. Because theresults of windowing pertain to individual samples rather than to a material, they arereferred to as apparent (rather than effective) properties.

Figure 8.1: Schematic depiction of a composite and four rectangular windows of equal size.

Windowing methods are based on a surface integral version of the Hill condition,eqn. (3.17), that takes the form

∫

Γ

[

t(x)− 〈σ〉 ∗ nΓ(x)]T[

u(x)− 〈ε〉 ∗ x]

dΓ = 0 , (8.1)

see, e.g., Hazanov (1998). In general there are four ways of fulfilling this equation, threeof them being based on uniform boundary conditions (Hazanov and Amieur, 1995; Ostoja-Starzewski, 2006).

First, the traction term in eqn. (8.1) can be set to zero by specifying appropriateNeumann boundary conditions for the tractions t(x). This can be achieved by prescribing

76

a given macroscopically homogeneous stress tensor σa to all faces of the volume element,

t(x) = σa ∗ nΓ(x) ∀x ∈ Γ , (8.2)

leading to statically uniform boundary conditions (SUBC). Second, the right hand termin eqn. (8.1) can be set to zero by imposing a given macroscopically homogeneous straintensor εa on all boundaries,

u(x) = εa ∗ x ∀x ∈ Γ , (8.3)

resulting in kinematically uniform boundary conditions (KUBC). Third, mixed uniformboundary conditions (MUBC) may be specified, which enforce the scalar product underthe integral to vanish separately for each face Γk making up the surface of the volumeelement,

[t(x)− 〈σ〉 ∗ nΓ(x)]T[

u(x)− 〈ε〉 ∗ x]

dΓ = 0 ∀x ∈ Γk . (8.4)

This involves appropriate combinations of traction and strain components that are uniformover a given face of the volume element, but not macroscopically homogeneous. Mixeduniform boundary conditions that fulfill eqns. (8.1) and (8.4) must be orthogonal in thefluctuating contributions (Hazanov and Amieur, 1995).

Finally, the fluctuations of non-uniform boundary fields can be made to cancel out bypairing parallel faces of the volume element such that they show identical fluctuations butsurface normals of opposite orientations. This strategy, which does not involve homoge-neous fields, is used by the periodicity boundary conditions discussed in chapter 6.

Macrohomogeneous boundary conditions following eqns. (8.2) and (8.3) can be shownto give rise to lower and upper estimates, respectively, for the overall elastic stiffness of agiven mesoscopic volume element (Nemat-Nasser and Hori, 1993). Ensemble averages ofsuch estimates obtained from windows of comparable size provide lower and upper boundson the overall effective tensors of these volume elements. These bounds are sometimesreferred to as mesoscale bounds. By definition, for RVEs the lower and upper estimatesand bounds on the overall elastic properties must coincide (Hill, 1963). Accordingly, hier-archies of bounds that are generated from sets of windows of different sizes (Huet, 1999)can bring out the effects of the size of the volume elements and indicate the size of properrepresentative volume elements.

Equations (8.4) can be fulfilled by a range of different MUBC, resulting in differentestimates for the apparent macroscopic tensors. A specific set of mixed uniform boundaryconditions that avoids prescribing nonzero boundary tractions was proposed by Pahr andZysset (2008) for obtaining the apparent elastic tensors of cellular materials. Table 8.1 liststhese six load cases for volume elements that have the shape of right hexahedra aligned withthe coordinate system. Their edge lengths in the 1-, 2- and 3-directions are c1, c2 and c3,respectively. The components of the prescribed strain tensor are denoted as εa

ij and thoseof the prescribed traction vector as τ a

i . When applied to periodic volume elements withorthotropic phase arrangements, these MUBC were found to give the same predictions forthe macroscopic elasticity tensor as periodic homogenization. Accordingly, they are called

77

Table 8.1: The six linearly independent uniform strain load cases making up the periodicitycompatible mixed boundary conditions (PMUBC) proposed by Pahr and Zysset (2008) andthe pertinent boundary conditions for thermal loading; East, West, North, South, Top, andBottom denote the faces of the hexahedral volume element, compare fig. 6.5, and the ciare the lengths of its edges.

East West North South Top Bottom

Tensile 1 u1 = εa11c1/2 u1 = −εa

11c1/2 u2 = 0 u2 = 0 u3 = 0 u3 = 0

τa2 = τa

3 = 0 τa2 = τa

3 = 0 τa1 = τa

3 = 0 τa1 = τa

3 = 0 τa1 = τa

2 = 0 τa1 = τa

2 = 0

Tensile 2 u1 = 0 u1 = 0 u2 = εa22c2/2 u2 = −εa

22c2/2 u3 = 0 u3 = 0

τa2 = τa

3 = 0 τa2 = τa

3 = 0 τa1 = τa

3 = 0 τa1 = τa

3 = 0 τa1 = τa

2 = 0 τa1 = τa

2 = 0

Tensile 3 u1 = 0 u1 = 0 u2 = 0 u2 = 0 u3 = εa33c3/2 u3 = −εa

33c3/2

τa2 = τa

3 = 0 τa2 = τa

3 = 0 τa1 = τa

3 = 0 τa1 = τa

3 = 0 τa1 = τa

2 = 0 τa1 = τa

2 = 0

Shear 12 u2 = εa21c1/2 u2 = −εa

21c1/2 u1 = εa12c2/2 u1 = −εa

12c2/2 u3 = 0 u3 = 0

u3 = 0, τa1 = 0 u3 = 0, τa

1 = 0 u3 = 0, τa2 = 0 u3 = 0, τa

2 = 0 τa1 = τa

2 = 0 τa1 = τa

2 = 0

Shear 13 u3 = εa31c1/2 u3 = −εa

31c1/2 u2 = 0 u2 = 0 u1 = εa13c3/2 u1 = −εa

13c3/2

u2 = 0, τa1 = 0 u2 = 0, τa

1 = 0 τa1 = τa

3 = 0 τa1 = τa

3 = 0 u2 = 0, τa3 = 0 u2 = 0, τa

3 = 0

Shear 23 u1 = 0 u1 = 0 u3 = εa32c2/2 u3 = −εa

32c2/2 u2 = εa23c3/2 u2 = −εa

23c3/2

τa2 = τa

3 = 0 τa2 = τa

3 = 0 u1 = 0, τa2 = 0 u1 = 0, τa

2 = 0 u1 = 0, τa3 = 0 u1 = 0, τa

3 = 0

Thermal u1 = 0 u1 = 0 u2 = 0 u2 = 0 u3 = 0 u3 = 0

Loading τa2 = τa

3 = 0 τa2 = τa

3 = 0 τa1 = τa

3 = 0 τa1 = τa

3 = 0 τa1 = τa

2 = 0 τa1 = τa

2 = 0

periodicity compatible mixed uniform boundary conditions (PMUBC).

The concept of periodicity compatible mixed uniform boundary conditions can be ex-tended to thermoelasticity by adding a load case that constrains all displacements normalto the faces of the volume element, sets all in-plane tractions to zero, and applies a uniformtemperature increment ∆T , see table 8.1. This allows to evaluate the volume averaged spe-cific thermal stress tensor 〈ϑ〉, from which the apparent thermal expansion tensor can beobtained as α = −C〈ϑ〉.

It has been shown that the PMUBC listed in table 8.1 give valid results for sub-orthotropic periodic unit cells (Pahr and Bohm, 2008), which indicates that they are alsoapplicable to non-periodic volume elements provided the sub-orthotropic contributions totheir overall symmetry is relatively small. The PMUBC accordingly offer an attractive op-tion for evaluating estimates of the macroscopic elasticity and thermal expansion tensorsof periodic and non-periodic volume elements.

The generation of lower and upper estimates by windowing using SUBC and KUBCcan be shown to be valid in the context of nonlinear elasticity and deformation plasticity(Jiang et al., 2001). PMUBC, however, rely on the superposition principle for obtainingeffective tensors and, accordingly, cannot be used for generating tangent tensors. Theycan, however, be employed for some materials characterization tasks for elastoplastic inho-mogeneous materials (Pahr and Bohm, 2008). Like embedding methods windowing givesrise to perturbed boundary layers near the surfaces of the volume element, which mayinfluence phase averages of microfields. The principal strength of windowing methods liesin providing an approach to studying the linear behavior of non-periodic volume elements.

78

79

Chapter 9

Multi-Scale Models

The micromechanical methods discussed in chapters 3, 6, and 8 are designed to han-dle a single scale transition between a lower and a higher length scale (“microscale” and“macroscale”), overall responses being obtained by homogenization and/or local fields bylocalization. Many inhomogeneous materials, however, show more than two clearly dis-tinct characteristic length scales, typical examples being laminated and woven composites,materials in which there are well defined clusters of particles, as well as most biomaterials.In such cases an obvious modeling strategy is a hierarchical (or multi-scale) approach thatuses a sequence of scale transitions, i.e., the material response at any given length scaleis described on the basis of the homogenized behavior of the next lower one47 as depictedschematically in fig. 9.1.

transitionscale

#1transitionscale

#2

MACROSCALE MESOSCALE MICROSCALE(sample) (particle clusters) (particles in matrix)

Figure 9.1: Schematic representation of a multi-scale approach to studying a materialconsisting of clustered inhomogeneities in a matrix. Two scale transitions, macro←→mesoand meso←→micro, are used.

A multi-scale model can be viewed as a sequence of scale transitions and suitable mi-cromechanical models, i.e., mean field, unit cell, windowing and, to some extent, embeddingapproaches, may be used as “building blocks” at any level within hierarchical schemes. Such

47Describing the material behavior at lower length scales by a homogenized model implies that charac-teristic lengths differ by, say, two orders of magnitude or more, so that valid volumes can be defined forhomogenization. It is not technically correct to employ hierarchical approaches within “bands” of more orless less continuous distributions of length scales, as can be found, e.g., in some metallic foams with highlydisperse cell sizes.

80

multi-scale modeling strategies have the additional advantage of allowing the behavior ofthe constituents at all lower length scales to be assessed via the corresponding localizationrelations.

Among the continuum mechanical multi-scale descriptions of the thermomechanicalbehavior of inhomogeneous materials reported in the literature, some combine mean fieldmethods at the higher length scale with mean field (Hu et al., 1998; Tszeng, 1998) or pe-riodic microfield approaches (Gonzalez and LLorca, 2000) at the lower length scale. Themost common strategy for multi-scale modeling, however, uses Finite Element based unitcell 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 ofmicromechanically based constitutive laws that can be evaluated at each integration point.This requirement typically does not give rise to extreme computational workloads in theelastic range, where the superposition principle can be used to obtain the full homoge-nized elasticity and thermal expansion tensors with a limited number of modeling runs.For simulating the thermomechanical response of nonlinear inhomogeneous materials, how-ever, essentially a full micromechanical submodel has to be maintained and solved at eachintegration point in order to account for the history dependence of the local responses48.Although within such a framework the use of sophisticated unit cell based models at thelower length scales tends to be an expensive proposition in terms of computational require-ments, considerable work of this type has been reported (Belsky et al., 1995; Ghosh et al.,1996; Smit et al., 1998; Feyel, 2003). Lower (but by no means negligible) computationalcosts can be achieved by using constitutive models based on mean field approaches, e.g., in-cremental Mori–Tanaka methods. In recent advanced models the type of micromechanicalmodel used for the lowest scale transition is automatically adapted to the local geometryand/or gradients of the fields, embedding methods being used at the most critical locations(Raghavan and Ghosh, 2004).

Figure 9.2 shows a result obtained by applying a multiscale approach that uses anincremental Mori–Tanaka method, compare section 3.7, at the integration point level of ameso-scale unit cell model for describing the elastoplastic behavior of a cluster-structuredhigh speed steel. The particle-rich and particle poor regions used for the description at themesoscale are treated as particle reinforced MMCs with appropriate reinforcement volumefractions. Such an approach not only predicts macroscopic stress–strain curves, but alsoallows the mesoscopic distributions of phase averaged microscopic variables to be evaluated,compare Plankensteiner (2000).

Finally, it is worth noting that multi-scale approaches are not limited to using the“standard” methods of continuum micromechanics as discussed above. Especially the ca-pability of the Finite Element method of handling highly complex constitutive descriptionshas been used to build multi-scale descriptions that employ, among others, material modelsbased on crystal plasticity (McHugh et al., 1993), discrete dislocation plasticity (Cleveringaet al., 1997) and atomistics (Tadmor et al., 2000).

48Some approaches, however, have aimed a relaxing this requirement by using suitable precomputedresults for the lower homogenization step, compare Ganser (1998).

81

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

Figure 9.2: Phase averaged microscopic equivalent plastic strains in the matrix withinthe inhomogeneity-poor regions of a cluster-structured high speed steel under mechanicalloading as predicted by a mesoscopic unit cell model combined with an incremental Mori–Tanaka model at the microscale (Plankensteiner, 2000).

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Chapter 10

Closing Remarks

Methods of continuum micromechanics of materials have enjoyed considerable success inthe past five decades in furthering the understanding of the thermomechanical behaviorof inhomogeneous materials and in providing predictive tools for engineers and materialsscientists. However, they are subject to some practical limitations that should be kept inmind when employing them.

All the methods discussed in chapters 3 to 8 implicitly use the assumption that theconstituents of the inhomogeneous material to be studied can be treated as homogeneous,which, of course, is not necessarily true. When the length scale of the inhomogeneities in aconstituent is much smaller than the length scale of the phase arrangement to be studied,multi-scale models as discussed in section 9 can be used. No rigorous analytical theoryappears to be available at present for handling scale transitions in materials that do notfulfill this requirement (e.g., particle reinforced composites in which the grain size of thematrix is comparable to or even larger than the size of the particles). Typically the bestthat can be done in such cases is either to use sufficiently large models with resolved mi-crogeometries (a computationally expensive proposition) or to employ homogenized phaseproperties and be aware of the approximation that is introduced.

A major practical difficulty in the use of continuum micromechanics approaches (whichin many cases is closely related to the questions mentioned above) has been obtainingappropriate constitutive models and material parameters for the constituents. Typically,available data pertain to the behavior of the bulk materials (as measured from homoge-neous macroscopic samples), whereas what is actually required are parameters (and, insome cases, constitutive theories) describing the in-situ response of the phases at the mi-croscale. In fact, the dearth of dependable material parameters is one of the reasons whypredictions of the strength of inhomogeneous materials by micromechanical methods tendto be a considerable challenge.

Another point that should be kept in mind is that continuum micromechanical descrip-tions in most cases do not have absolute length scales unless these are introduced via theconstitutive models of the constituents. Absolute length scales can be provided “explic-itly” via discrete dislocation models, via gradient or nonlocal constitutive laws and damagemodels, or “implicitly”, e.g., by adjusting the phase material parameters to account forgrain sizes via the Hall–Petch effect. Analysts should also be aware that absolute length

83

scales may be introduced inadvertently into a model by mesh dependence effects of dis-cretizing numerical methods, an important issue being strain localization due to softeningmaterial behavior of a constituent. When multi-scale models are used special care may benecessary to avoid introducing inconsistent length scales at different modeling levels.

In addition it is worth noting that usually the macroscopic response of inhomogeneousmaterials is much less sensitive to the phase arrangement (and to modeling approxima-tions) than are the distributions of the microfields. Consequently, whereas good agreementin the overall behavior of a given model with “benchmark” theoretical results or experi-mental data typically indicates that the phase averages of stresses and strains are describedsatisfactorily, this does not necessarily imply that the stress and strain distributions havebeen captured correctly.

It is important to be aware that work in the field of micromechanics of materials invari-ably 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 physicalphenomena relevant to the given problem, and, on the other hand, must be sufficientlysimple to allow solutions to be obtained within the relevant constraints of time, cost, andcomputational resources. Obviously, actual problems cannot be solved without recourse tovarious approximations and tradeoffs — the important point is to be aware of them. It isworth keeping in mind that there is no such thing as a “best micromechanical approach”to all applications and that models, while indispensable in understanding the behavior ofinhomogeneous materials, are “just” models and do not reflect the full complexities of realmaterials.

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