Arch Comput Methods Eng (2009) 16: 3175DOI 10.1007/s11831-008-9028-8

O R I G I NA L PA P E R

Multiscale Methods for Composites: A Review

P. Kanout D.P. Boso J.L. Chaboche B.A. Schrefler

Received: 15 March 2008 / Accepted: 15 March 2008 / Published online: 22 January 2009 CIMNE, Barcelona, Spain 2009

Abstract Various multiscale methods are reviewed in thecontext of modelling mechanical and thermomechanical re-sponses of composites. They are developed both at the ma-terial level and at the structural analysis level, consideringsequential or integrated kinds of approaches. More specifi-cally, such schemes like periodic homogenization or meanfield approaches are compared and discussed, especially inthe context of non linear behaviour. Some recent develop-ments are considered, both in terms of numerical meth-ods (like FE2) and for more analytical approaches basedon Transformation Field Analysis, considering both thehomogenization and relocalisation steps in the multiscalemethodology. Several examples are shown.

1 Introduction

Composite materials are commonly applied in engineeringpractice. They allow to take advantage of the different prop-erties of the component materials, of the geometric structure

P. Kanout J.L. ChabocheThe French Aerospace Lab, 29 avenue de la Division Leclerc,92322 Chtillon, France

P. Kanoute-mail: pascale.kanoute@onera.fr

J.L. Chabochee-mail: Jean-Louis.Chaboche@onera.fr

D.P. Boso () B.A. SchreflerDepartment of Structural and Transportation Engineering,University of Padua, Via Marzolo 9, 35131 Padua, Italye-mail: boso@dic.unipd.it

B.A. Schreflere-mail: bas@dic.unipd.it

and of the interaction between the constituents to obtain atailored behaviour as a final result. Such composite materialsmay present both high stiffness and high damping, improvedstrength and toughness, improved thermal conductivity andelectrical permittivity, improved permeability, unusual phys-ical properties such as negative Poisson ratio, negative stiff-ness inclusions, etc.

Most composite materials are multiscale in nature, i.e.the scale of the constituents is of lower order than the scaleof the resulting material and structure. To fix the ideas, wespeak of the macroscopic scale as the particular scale inwhich we are interested in (e.g. at structural level), whilethe lower scales are referred to as microscopic scales (some-times an intermediate scale is called mesoscopic scale). Weexclude here scales at atomic level, which would require aseparate paper.

For most of the analyses of composite structures, effec-tive or homogenized material properties are used, insteadof taking into account the individual component proper-ties and geometrical arrangements. These effective proper-ties are usually difficult or expensive to measure and in thedesign stage the composition may vary substantially, mak-ing frequent measurements prohibitive. Hence a lot of effortwent into the development of mathematical and numericalmodels to derive homogenized material properties directlyfrom those of the constituents and from their microstruc-ture. Many engineering problems are solved at macroscopicscale with such homogenized properties. However, some-times such analyses are not accurate enough.

In principle it would be possible to refer directly to themicroscopic scale, but such microscopic models are oftenfar too complex to handle for the analysis of a large struc-ture. Further, the data obtained would be redundant andcomplicated procedures would be required to extract infor-mation of interest.

32 P. Kanout et al.

A way out is what is now commonly known as multi-scale modelling, where macroscopic and microscopic mod-els are coupled to take advantage of the efficiency of macro-scopic models and the accuracy of the microscopic models.The scope of such multiscale modelling is to design com-bined macroscopic-microscopic computational methods thatare more efficient than solving the full microscopic modeland at the same time give the information that we need tothe desired accuracy [56].

In the case of material multiscale modelling and in ho-mogenization in general, one usually proceeds from thelower scales upward, in order to obtain equivalent materialproperties. Alternatively, in the case of structural modellingit is important to be able to step down through the scales un-til the desired scale of the real, not homogenized, material isreached. This technique is often known as unsmearing, lo-calization or recovering method. Usually in a global analysisboth aspects need to be pursued, think for instance of a dam-age or fracture analysis. The procedure just described is thatof a serial coupling and represents some sort of data pass-ing up and down the scales. An alternative is the concurrentcoupling where both microscale and macroscale models arestrongly interwoven and have to be addressed continuouslyas the computation goes on. This is particularly the case innonlinear situations.

In this paper we will briefly review the most commonmethods applied to obtain equivalent properties and thenconsider full multiscale modelling. Both linear and non-linear aspects will be covered. Unsmearing is then dealt within some detail because it is often neglected while in manycases it is of fundamental importance.

In Sect. 2 usual bounds and other estimates are presented,starting from the Voigt and Reuss bounds. Section 3 il-lustrates asymptotic homogenisation and its numerical im-plementation. In Sect. 4 mean field approaches are dealtwith, while in Sect. 5 semi-analytical methods are described.In Sect. 6 sequential multiscale procedures are addressedand computational techniques are shown in some details inSect. 7. Recovery methods are presented in the final section.Examples of applications of the discussed methods can befound throughout the paper.

2 Bounds and Other Estimates

Over the last decades a large body of literature was de-veloped, which deals with the micromechanical modellingtechniques for heterogeneous materials. As far as the effec-tive properties are concerned, the various approaches maybe divided into two main categories, depending upon the mi-crostructure characteristics.

In case of composites with linear constitutive behaviour,if the microstructure is sufficiently regular to be considered

periodic, the effective properties may be determined in termsof unit cell problems with appropriate boundary conditions.This case is presented in Sect. 3. If the microstructure isnot regular the effective properties cannot be determined ex-actly. Thus the goal consists instead in the definition of therange of the possible effective behaviour in terms of bounds,which depend on some parameters characterizing the mi-crostructure, such as for instance the volume ratio of the in-clusions in a matrix. To this purpose many homogenisationmethods have been developed. We mention the pioneeringstudies by Voigt [175] and Reuss [151], who formulated rig-orous bounds for the effective moduli of composites withprescribed volume fraction. Some decades later Hashin andShtrikman [8890] presented an extension of the method,based on variational formulations. If the microstructure iscomposed of a matrix and spheric or spheroidal inclusions,the effective behaviour of composite can be obtained bymeans of the self-consistent method [24, 26, 91, 93, 102],for which an example will be shown in Sect. 7.6.

If the composite materials have non linear constitutivebehaviour, for periodic microstructure the effective prop-erties can still be obtained in terms of unit cell problemswith appropriate boundary conditions. For composites withrandom microstructures the first bounds are obtained byBishop and Hill [12, 13] for rigid perfectly plastic polycrys-tals. Some extensions of the self-consistent method are alsoavailable in literature, for example in Hill [94], Hutchinson[99] and Berveiller and Zaoui [11].

In the framework of non linear bounds we mention alsothe work by Willis and Talbot [166, 181], which providesextensions of the Hashin-Shtrikman variational principlesfor non linear composites. Their work is followed by theintroduction of several new variational principles makinguse of appropriately chosen linear comparison compos-ites, which allow the determination of Hashin-Shtrikmanand more general bounds and estimates, directly from cor-responding estimates for linear composites. These includethe variational principles of Ponte Castaeda [141, 142] andTalbot and Willis [167] for general classes of nonlinear com-posites, of Suquet [163] for power-law composites and Ol-son [138] for perfectly plastic composites.

2.1 The Voigt and Reuss Bounds

Consider an N -phase composite occupying the domain, with each phase occupying sub-domains (r) (r =1, . . . ,N), and let the energy-density function w(x,) andthe complementary energy function u(x, ) be expressedin terms of the homogeneous phase potentials w(r)() andu(r)( ) as

w(x,) =N

r=1(r)(x)w(r)(), (1)

Multiscale Methods for Composites: A Review 33

u(x, ) =N

r=1(r)(x)u(r)( ), (2)

where (r)(x) = 1 if x (r) or 0 otherwise, is the charac-teristic function of phase r . These functions define the mi-crostructure.

The characterization of the effective behaviour of a com-posite can be obtained from the principle of minimum po-tential energy. An effective potential W is defined for thecomposite from the average minimum potential energy de-fined as

W () = min(x)Kw(x,), (3)

where the McCauley brackets denote volume average on thedomain, = is the average of the actual strain field andK is the set of kinematically admissible strains. A similarformulation defines the stress potential U

U ( ) = min (x)Su(x, ), (4)

where = is the average of the actual stress field ,and S is the set of statically admissible stresses satisfying anaverage stress boundary condition.

For linear elastic composites, the effective strain andstress potentials may be written in terms of the effective elas-ticity tensor L and compliance tensor M in the followingway

W () = 12 : L : , (5)

U ( ) = 12 : M : . (6)

Usually Voigts analysis [175] is indicated as the first studyof the effective mechanical properties of composite solids,with a complementary contribution given later by Reuss[151]. Voigt assumes that the strain field within an aggre-gate sample of heterogeneous material is uniform, arrivingto the estimate

L =N

r=1c(r)L(r), (7)

where c(r) and L(r)are respectively the volume fraction andthe elasticity tensors of the r phase.

The dual assumption is made by Reuss, who approxi-mates the stress fields within the aggregate of a polycrys-talline material as uniform, obtaining

(L1)1 =[

N

r=1c(r)(L(r))1

]1. (8)

In the context of linear composites, Hill [92] and Paul [140]made the observation that the choice of a uniform strain fieldin the variational statement (3) for W leads to the rigorousupper bound

W () 0 are Y-periodic i.e. take thesame values on the opposite sides of the cell of periodicity.The term scaled with the n-th power of in (22)(26) iscalled term of order n.

3.2 Formalism of the Homogenisation Procedure

The necessary mathematical tools are the chain rule of dif-ferentiation with respect to the micro variable and averagingover a cell of periodicity.

We introduce the assumption (22)(26) into equations ofthe heterogeneous problem (15)(21) and make use of thechain rule and of the following notation (see also [153])

ddxi

f =(

xi+ 1

yi

)f = f,i(x) + 1

f,i(y). (27)

This equation explains also the notation used in the sequelfor differentiation with respect to local and global indepen-dent variables.

Because of (27) the balance equations split into termsof different orders (the terms of the same power of areequated to zero separately: e.g. (28) and (31) are of order1/).

For the equilibrium equation we have

0ij,j (y)(x,y) = 0, (28) 0ij,j (x)(x,y) + 1ij,j (y)(x,y) + fi(x) = 0, (29) 1ij,j (x)(x,y) + 2ij,j (y)(x,y) = 0, (30) We have similar expressions for the heat balance equation

q0i,i(y)(x,y) = 0, (31)q0i,i(x)(x,y) + q1i,i(y)(x,y) r(x) = 0, (32)q1i,i(x)(x,y) + q2i,i(y)(x,y) = 0, (33) .From (19) and (27) it follows that the main term of e inexpansions (25) depends not only on u0, but also on u1

e0ij (x, y) = u0(i,j)(x) + u1(i,j)(y) eij (x)(u0) + eij (y)(u1).(34)

36 P. Kanout et al.

The constitutive relationships (17) and (18) assume now theform

0ij (x,y) = aijkl(y)(ekl(x)(u0) + ekl(y)(u1)) ij (y)0,(35)

1ij (x,y) = aijkl(y)(ekl(x)(u1) + ekl(y)(u2)) ij (y)1,(36)

q0k (x,y) = Kkl(y)(0,l(x) + 1,l(y)), (37)q1k (x,y) = Kkl(y)(1,l(x) + 2,l(y)), (38) It can be seen that the terms of order n in the asymptoticexpansions for stresses (35), (36) and heat flux (37), (38)depend respectively on the displacement and temperatureterms of order n and n + 1. In this way the influence ofthe local perturbation on the global quantities is accountedfor. This is the reason why for instance we need u1(x,y)to define via the constitutive relationship the main term inexpansion (24) for stresses.

3.3 Global Solution

Referring separately to the terms of the same powers of leads to the following variational formulations for unknownsof successive order of the problem. Starting with the firstorder, it can be formally shown [73, 153] that u1(x,y) andsimilarly 1(x,y) may be represented in the following formwith separated variables

u1i (x,y) = epq(x)(u0(x))pqi (y) + Ci(x), (39)1(x,y) = 0,p(x)(x)p(y) + C(x). (40)

We will call the pqi (y) and p(y) the homogenisationfunctions for displacements and temperature respectively.

The zero order component of the equation of equilibrium(28) and of heat balance (31) in the light of (39) and (40)yields the following boundary value problems for functionsof homogenisation

find pqi VY such that: vi VY

Y

aijkl(y)(ipjq + pqi,j (y)(y))vk,l(y)(y)d = 0; (41)

find p VY such that: VY

Y

Kij (y)(ip + p,i(y)(y)),j (y)(y)d = 0, (42)

where VY is the subset of the space of kinematically admis-sible functions that contains the functions with equal valueson the opposite sides of the cell of periodicity Y . The six

vectors pq and the three scalar functions p depend onlyon the geometry of the cell of periodicity and on the valuesof the jumps of material coefficients across SJ . Functionsv(y) and (y) are usual test functions having the meaningof Y -periodic displacements and temperature fields respec-tively. They are used here to write explicitly the counterpartsof the expressions (28) and (31), in which the prescribed dif-ferentiations are understood in a weak sense.

The solutions pq and p of the local (i.e. definedfor a single cell of periodicity) boundary value problems(41) and (42) with periodic boundary condition can be in-terpreted as obtained for the cell subject to a unitary aver-age strain epq and, respectively, unitary average temperaturegradient ,p(y). The true value of perturbations are obtainedafter by scaling pq and p with true global strains (gra-dient of global temperature), as it is prescribed by (39) and(40).

In the asymptotic expansion for displacements (22) andfor temperature (23) the dependence on x alone occurs onlyin the first term. The independence on y of these functionscan be proved (see for example [153]). The functions de-pending only on x define the macro-behaviour of the struc-ture and we will call them global terms. To obtain the globalbehaviour of stresses and of heat flux the following meanvalues over the cell of periodicity are defined [153]

0ij (x) = |Y |1

Y

0ij (x,y)dY,(43)

q0(x) = |Y |1

Y

q0(x,y)dY.

Averaging of (35) and (37) results in the following, effectiveconstitutive relationships

0ij (x) = ahijklekl(u0) hij 0, q0i = khij 0,j . (44)In the above equations the effective material coefficients ap-pear. They are computed according to

ahijkl = |Y |1

Y

aijpq(y)(kplq + pqk,l(y)(y))dY, (45)

khij = |Y |1

Y

kip(y)(jp + j,p(y))dY, (46)

hij = |Y |1

Y

ij (y)dY. (47)

The macro-behaviour can be defined now by averaging firstorder terms in the equilibrium and flux balance equations(29), (32) and boundary conditions (20) and substitutingthen the averaged counterparts of stress and heat flux (43)(first order perturbations vanish in averaging of (29), (32)because of periodicity). Equations (17) and (18) should bereplaced by (44), while (21) has no more sense since we dealnow with homogeneous uncoupled thermo-elasticity.

Multiscale Methods for Composites: A Review 37

The heterogeneous structure can now be studied as a ho-mogeneous one with effective material coefficients givenby (45)(47) and global displacements, strains and averagestresses and heat fluxes can be computed. Then we go backto (35) for the local approximation of stresses. This last stepis the above mentioned unsmearing or localization.

3.4 Local Approximation of the Stress Vector

We note that the homogenisation approach results in two dif-ferent kinds of stress tensors. The first one is the averagestress field defined by (44). It represents the stress tensor forthe homogenized, equivalent but unreal body. Once the ef-fective material coefficients are known, the stress field andthe heat flux may be obtained from a standard F.E. structuraland heat transfer code.

The other stress field is associated with a family of uni-form states of strains epq(x)(u0) over each cell of periodicityY. This local stress is obtained by introducing (34) into (35)and results in

0ij (x,y) = aijkl(y)(kplq pqk,l(y))epq(x)(u0) ij (y)0.(48)

Because of (28) and (41) this tensor fulfils the equations ofequilibrium everywhere in Y. If needed, the stress descrip-tion can be completed with a higher order term in (24). Thisapproach will be dealt in Sect. 8.2.

Finally the local approximation of heat flux is as follows

q0j (y) = kij (y)(ip + p,i(y)(y))0,p(x). (49)

3.5 Numerical Implementation

It is now shown how the asymptotic homogenization can beimplemented in any finite element program [184]. The prob-lem to be solved refers to a unit cell shown for instance onthe right hand side of Fig. 1. For the purpose of a finite ele-ment solution it is convenient to use matrix notation for theabove introduced quantities. Accordingly the homogenisa-tion functions are ordered as defined by (50) and (51) re-spectively (the numbers in the superscripts in (50) and sub-scripts in (51) refer to the reference axes 1, 2, 3).

XT (y)[{11(y)}{22(y)}{33(y)}{12(y)}{23(y)}{13(y)}]36, (50)

T T (y) = {1(y)2(y)3(y)}13.This is in accordance with the ordering of strains and tem-peratures

e = {e11 e22 e33 e12 e23 e13 }T6 = {epq}T6 , (51) = {1, 2, 3}T3 = {p}T3 .

In the following the superscript e denotes the nodal valuein a finite element mesh of the unit cell. We have the usualrepresentations for each element

X(y) = N(y)Xe, T (y) = N(y)Te, (52)

where N contains the values of standard shape functions.It is easy to show that the variational formulation (41) can

be rewritten as follows

find X VY such that: v VY

Y

eT (v(y))D(y)(1 + LX(y))dY = 0. (53)

In the above L denotes the matrix of differential operators,D contains the material coefficients aijkl in the repetitivedomain. Matrix Xe which contains the values of homoge-nization functions at the nodes of the mesh is obtained as afinite element solution of (53). The equation to solve is thefollowing

KXe F = 0; (54)

where X is Y -periodic, with zero mean value over the cell,and

F =

Y

BT D(y), K =

Y

BT D(y)B, B = LN(y),(55)

D contains the material coefficients aijkl .It can be shown that X in (53) and thus (54) is a solution

of a boundary value problem, for which the loading consistsof unitary average strains over the cell. This is seen in theform of the first of (55), which forms a matrix. We solvethus six equations for six functions of homogenisation.

The variational formulation (42) can be represented in aform similar to (54), Te being Y -periodic, with given meanzero value over the cell

KTe + F = 0; (56)

where

F =

Y

BT K (y)dY ; KT =

Y

BT K (y)B dY ;(57)

B = LN(y).

K contains the conductivities kij of materials in the repet-itive domain. Differential operators in L are ordered suit-ably for the thermal problem.

The periodicity conditions can be taken into account us-ing Lagrange multipliers in the construction of a finite el-ement code. Also the requirements of the zero mean valuehas to be included in the program.

38 P. Kanout et al.

Having computed Xe, Te and by consequence u1 and 1one can derive the effective material coefficients, accordingto

Dh = |Y |1

Y

D(y)(1 + BXe)dY

Kh = |Y |1

Y

K (y)(1 + BT e)dY (58)

h = |Y |1

Y

(y)dY.

As already pointed out, any thermoelastic finite element pro-gram can be used to obtain global displacements and tem-peratures. In a linear case the unit cell problem has to besolved only once. The structural or thermo-mechanical prob-lem can then be performed separately, using these effec-tive properties: it is hence a two step procedure on differentscales, a homogenization step on the unit cell and a standardthermo-mechanical problem on the structural level. How-ever the method permits one to obtain also local strains andstresses as shown above and dealt with more extensively inSects. 7 and 8. For this unsmearing procedure the gradientsof temperatures and strains are needed in the regions of in-terest, see (48) and (49). Strains and temperature gradientsare directly obtained from the finite element interpolation.To present the graphs of stresses and heat fluxes over thesingle cell nodal projection can be used. To assure continu-ity of tangential stresses, this projection should be extendedto patches of cells, see Sect. 8.1.

3.6 Boundary Effects and Time Dependent Problems

The periodicity condition used to find the local perturbationis strictly applicable only inside the body. We have hencethe solution of a (thermo-)elasticity problem based on anassumed stress or displacement field which is valid nearlyeverywhere in the region occupied by the body under inves-tigation, except on the boundary. The use of material coef-ficients based on the assumption of periodicity in the globalsolution (where the real boundary conditions are imposed)may implicitly impose some unrealistic constraints close tothe boundary. This problem can be solved by some correc-tions which change the solution in the domain close to theboundary while away from the boundary the correction fieldshould asymptotically decrease. Such a correction can beobtained by replacing the expansion (22) by

u(x) u0(x) + (u1(x,y) + b1(x, z)) + , (59)where z represents the coordinates of an additional systemwith the origin on the external surface, the axis z3 directednormal to the boundary surface with the other two axes ori-ented tangential to the surface and b is the boundary layercorrection. Vector b should vanish exponentially when z3

goes to infinity. The procedure is described in detail in [110]and has been implemented by making use of one cell of pe-riodicity and infinite elements with homogenized materialproperties around it. The use of infinite elements assures thedesired exponential decay. Although the procedure is theo-retically necessary near free edges, it is seldom applied inpractical engineering problems.

An alternative is the window method, where the mi-crostructure with effective properties is embedded in a ma-terial with constant properties, subjected at the outer sideto the boundary conditions wanted. In case of an appliedconstant strain field mixed b.c. at the real boundary are ob-tained [85].

Asymptotic theory of homogenization may involve alsothe time scale. This is the case in thermo-mechanical prob-lems with non stationary heat conduction or in presence ofimpulsive heat loads or in problems with impact loads. Inthe first case correctors for the initial conditions are usefulwhile in the other two cases spatial and temporal multiscalemethods should be applied. The interested reader is referredto [21, 42, 111, 183] and references therein.

4 Mean Field Approaches and the Non Linear Case

The development of homogenisation procedures allowingto compute the non-linear effective behaviour of heteroge-neous materials depending on the properties of their con-stituents or microstructure is a field of research in constantevolution. The focus here is on mean field methods whichare based on the Eshelby inclusion solution. In these ap-proaches, well adapted for heterogeneous materials with arandom microstructure distribution, average fields are con-sidered for each phase in the material.

The first incursions into the non-linear homogenisationdealt with crystal plasticity. The estimate of the elastic limitof polycrystals from one of a single crystal was started bySachs [152] in 1928 and developed a short time after byTaylor [169]. Later, Lin [113] extended the model of Tay-lor by imposing the rate of total strain to be uniform in thepolycristal. These first approaches are thus specific to crystalplasticity and formulated in the context of the physical met-allurgy. In the beginning of the sixties an important changetook place when Eshelby [58] gave the solution to the prob-lem of the inclusion through the concept of eigenstrains.

4.1 The Model of Krner

Budiansky et al. [27] used in 1960 the solution of the prob-lem of inclusion provided by Eshelby to model the initialstages of elasto-plasticity of polycrystals. The authors as-sumed that only the most favourably oriented grains devel-oped plasticity whereas the matrix remained elastic. This

Multiscale Methods for Composites: A Review 39

situation is then assimilated to the inclusion problem. More-over, by assuming that only a small quantity of grains tendto yield, the authors used the approximation of the dilutemedium. Later, Krner [102], initiator of the self-consistentmodel in elasticity, proposed to apply it to elasto-plasticity.Contrary to its predecessors who designed the matrix of theEshelby problem as the whole of non plasticized grains, heassimilated it to the effective equivalent medium itself, thatis, to the overall polycristal. The localisation rule obtainedis written in the following form:

= + 2(1 )(Ep p). (60)In this expression and are respectively the local andoverall stress tensors, Ep and p the overall and the localplastic strain tensors, the overall shear modulus, and is a parameter depending on the spatial distribution of thegrains and their elastic properties. When all inclusions areassumed to be unit spheres (i.e. arising from an isotropicspatial distribution) with isotropic elasticity, we have:

= 215

7 5v1 v . (61)

This localisation rule can easily be extended to more gen-eral situations. The model of Krner suffers from the sameinsufficiencies as the models of Taylor or Lin providing anestimate that is generally too stiff.

4.2 Hills Incremental Model

The alternative formulation suggested by Hill [94, 95] a fewyears later is based on a linearization of the local constitutivelaw. His idea is to bring back the non-linear problem of ho-mogenisation in the scale transition from the single crystal tothe polycristal to what he knows well, the linear homogeni-sation. For that, he linearizes the local constitutive law bybringing it back in an incremental form:

(x) = L(x) : (x), (62)where L is the tangent modulus.

In this linearized expression, the instantaneous modulusis of elasto-plastic nature and not only elastic. For the samereasons as Krner, he made the choice of the self-consistentestimate to obtain the macroscopic behaviour of the poly-cristal. The effective moduli L assigned to the reference ho-mogeneous medium is then given by the following expres-sion:

L = L : A =

r

crLr : Ar ,

where Ar is the strain localisation tensor, and cr the volumefraction of the phase r of the composite material. Therefore,the localisation rule is written as:

s = Bs(Ls) : , s = As(Ls) : E. (63)

The strain and stress localisation tensors are given from thesolution of a dilute problem for a single inclusion of phase s,embedded in a large volume of a homogeneous medium L:

As = [I + P : (Ls L)]1, Bs : L = Ls : As , (64)where the polarisation tensor P is defined by:

P = 14 | |

x=1H(x)1.x3dSx (65)

with H(x) = [x 1 x]sym, = x.L.x and character-izing the ellipsoidal geometry (on the unit sphere x = 1).Let us note the anisotropic character of the polarisation ten-sor P , due to its determination in terms of the referencestiffness (the overall tangent stiffness in the self-consistentscheme), which is anisotropic, even for an isotropic elasto-plastic material. Therefore it has to be determined numeri-cally at each step of an incremental simulation.

In this approach, the overall tangent behaviour can bewritten as:

= L(E) : E. (66)The elasto-plastic nature of the instantaneous moduli leadsto more realistic estimates (less stiff) than those given by theKrner or Taylor model. The model of Hill was extended byHutchinson [99] in 1976 to visco-plasticity by using tangentcreep compliances instead of elasto-plastic ones. He noticedthat, for power law creep, Hills formulation may be inte-grated into a total one making use of the secant creep com-pliances.

4.3 The Secant, Tangent and Affine Formulations

Following the work of Hill, several authors quickly realisedthat there are several possible ways to linearize the constitu-tive laws and that the incremental formulation of Hill is notunique. Thus, a few years later, several papers (Berveillerand Zaoui [11], Tandon and Weng [168]) were published,in the field of the non-linear homogenisation based on asecant linearization of the constitutive law. In the modelof Berveiller and Zaoui, the secant linearization consists inreplacing the local constitutive law (x) = L(x) : ((x) p(x)) by (x) = Lsec() : (x) where Lsec() is the secantmodule. In the particular case of a spherical inclusion withan isotropic behaviour, these authors have shown that theobtained localisation rule is formally identical to the modelof Krner, but with a reduction of the elastic moduli of thereference homogeneous medium by a factor characterisingthe evolution of the secant moduli of the effective medium.Moreover, this factor decreases very quickly as the plasticdeformation grows.

Molinari et al. [129] proposed in 1987 a new method forestimating the visco-plastic behaviour of polycrystals which

40 P. Kanout et al.

is based on a tangent linearization of the local constitutivelaws:

(x) = L(x) : (x) + (x), (67)

where L is the tangent modulus of the phase medium sub-jected to a pre-stress . For the homogenisation, like theirpredecessors, these authors used the self-consistent scheme.Initially, in order to lighten the calculations, Molinari et al.[129] assumed that the polarisation tensor is isotropic.

Later, a development was proposed by Lebensohn andTom [108] without using this assumption. In these two for-mulations, the definition of the effective characteristics Land which define the overall response of the material isnevertheless given only in the case of a power-law constitu-tive equation (with the same exponent for all the phases andall the slip systems). These authors thus thought to guaran-tee the property L = d/dE for the overall relation, itself apower law with the same exponent. It has been shown re-cently by Masson [119] and Masson et al. [120] that theeffective moduli and compliances using the tangent localquantities are in general not any more tangent. Moreover,they showed that it is not necessary to use a power law only.This observation has motivated the affine formulation pro-posed by Masson et al. [120]. As Molinari et al. [129], theyused a tangent linearization of the local constitutive laws:

(x) = L(x) : ((x) (x));(68)

(x) = S(x) : (x) + (x),

where is the pre-strain. However the affine formulationis based on a linear thermoelastic reference medium. Thetangent effective moduli and the pre-stress are then definedusing the following expressions:

L = L : A =

r

crLr : Ar ,(69)

= L : , = tB : =

r

cr : B tr : r .

Using the self-consistent scheme in the case of similar andaligned ellipsoidal inclusions, the localisation rule can nowbe written as:

r = Ar : [E + P : (Lr : L : )]. (70)

We have also the overall constitutive equation as:

= L : (E ). (71)

As mentioned in Masson and Zaoui [120], the operator Ldoes not correspond exactly with the current tangent overallstiffness of the overall stress-strain response of the compos-ite. In this model, as in the incremental tangent formulation,

the stiffness tensors (anisotropic), the localisation and polar-isation tensors have to be evaluated at each step at a givenload. Moreover, the eigenstrains and the tangent stiffness aretaken as uniform inside each phase, but the stress and strainfields are not necessarily uniform.

The development of variational approaches [165, 181]for a behaviour deriving from a single potential (non-linearelasticity or viscosity) has made possible to compare someof these non-linear estimates to rigorous bounds of Hashin-Shtrikman type. It has been shown by Gilormini [79] thatboth the incremental and the classical secant formulationslead to estimates which are too stiff and can even violatethese bounds in some cases. More recently, Chaboche et al.[37, 40] have used some of these procedures in the context ofcomposite materials, and compared their estimates to unit-cell finite element calculations using periodic boundary con-ditions. The obtained main results are illustrated in Sect. 5.4.

4.4 The Beta Rule

In the same category of the modified Krner approach bya secant method, we have a slightly different one, usefulto describe the transition in local accommodation betweenphases, from the small inelastic strains to very large ones.

Introduced by Cailletaud [29], this rule replaces p andEp in (60) by an accommodation variable called s , andits average B = s css . The evolution rule for s is givenby the classical Armstrong-Frederick rule of plasticity:

s = ps Dsps . (72)It allows to recover Krner elastic accommodation for smallplastic strains, and Sachs for larger ones, when plasticstrains are important in most of the phases. Compared to themethods described just above, we may notice the followingadvantages:

the correct applicability under any cyclic or multi-axialloading;

the much lower computational cost than in incremental oraffine methods.

Using comparisons with 3D finite element analyses, the rule has been shown by Cailletaud [30] and Cailletaud andPilvin [31] as having self-consistent overall properties.

4.5 Higher Order Theories

A modified secant theory has been proposed by Suquet[164]. It is based on second-order moments in each individ-ual phase of the linear comparison solid. The use of second-order moments has also been considered by Hu [97] andBuryachenko [28]. It has been shown by Suquet [164] andPonte Castaeda and Suquet [147] that this secant theorybased on second-order moments coincides with the Ponte

Multiscale Methods for Composites: A Review 41

Castaedas [142] variational procedure; this property en-sures that the resulting estimates do not lead to the viola-tion of any bound. More recently, the second-order momentshave been used by Brenner et al. [22] in the affine formula-tion to take into account the intra-phase heterogeneity.

On the other hand, new linearizations have been searchedfor, with the objective of generating softer estimates evenwhen using the classical reference quantities. This is the sec-ond order procedure proposed by Ponte Castaeda [143, 144]which makes use of a second-order Taylor development ofthe strain or stress potentials. The comparison of the second-order method to numerical simulations (Moulinec and Su-quet [132]) has shown that quite accurate estimates are ob-tained even at high values of heterogeneity contrast, includ-ing the cases of rigid reinforcements or porous materials.

However, it has been demonstrated recently by Leroy andPonte Castaeda [112] that the second-order estimates canviolate the Hashin and Shtrikman bounds [89] near the per-colation limit. Moreover, earlier, Nebozhyn and Ponte Cas-taeda [135] have pointed out the non convex character ofthe predictions for the effective yield surface in the caseof porous materials with an incompressible matrix phase.Later, Ponte Castaeda [145] has then proposed an im-proved second order method incorporating field fluctuations.The method delivers then nonlinear estimates that are ex-act to second order in heterogeneity. The new homogenisa-tion technique is motivated by the fact established by Suquet[164] that the original second order method uses the first mo-ments, or averages of the fields over the phases while secondmoments of the fields are known to be important when thefield fluctuations are significant, as it is the case near perco-lation.

Several applications of this new second order methodwhich incorporates both first and second-moment infor-mation are then presented later for porous and rigidly re-inforced composites (Ponte Castaeda [146]). The resultsshow that the new theory satisfies rigorous bonds, even nearthe percolation limit, where field fluctuations become im-portant, case for which the original second order estimatesfailed. The case of porous materials with an incompressiblematrix is also now well estimated with the new method, andin general the new estimates appear in better agreement withnumerical simulations taken from the literature.

5 Semi-analytical Methods

Semi-analytical methods can be defined as direct micro-macro procedures for which the local constitutive equationsand criteria are evaluated at the local scale using explicit re-lations to establish the link between the macroscopic andthe microscopic fields. Such method correlates the overallmacroscopic behaviour with microstructural responses. The

analytical relations are usually based on mean field proce-dures. One advantage is that the local inelastic constitutivelaws are totally free. Contrarily, in most variational methodsor higher order theories like discussed in Sect. 4.5, there isthe assumption of a unique potential, therefore limiting theapplication to non linear elasticity or deformation plasticity.The following sections give an overview of some relevantworks from the literature applying this type of procedures.

5.1 The Transformation Fields Analysis

Regarding approximate schemes, the Transformation FieldAnalysis proposed by Dvorak and co-workers can be seen asan elegant way of reducing the number of macroscopic inter-nal variables by assuming the microscopic fields of internalvariables to be piecewise uniform. The method has been de-veloped initially for elasto-plastic composites (Dvorak andRao [52], Dvorak and Bahei-El-Din [50], Dvorak et al. [54],Teply and Dvorak [170]). Its formalisation by Dvorak andBenveniste [51] and Dvorak [49] has provided a theoreticalbasis for further works.

The Transformation Field analysis is based on the idea ofa purely elastic redistribution of the macroscopic stress andstrain, and of the local eigenstresses or eigenstrains. TFAis a generalised way of writing explicit scale transitions,based on purely elastic interactions between sub-volumesof the RVE, and it can be shown that many classical ap-proaches (self-consistent, Sachs, Taylor, Krner, etc.) arespecial cases. In this method, the plastic strain and the ther-mal expansion are considered as given eigenstrains. The lo-cal stress field is then determined from the fields of eigen-strain by solving linear problems with eigenstrains and theplastic strain field which is updated by the flow rule.

Considering a representative volume V of an heteroge-neous material where the size of the inhomogeneities issmall compared to that of V , the volume V is subdividedinto several local sub-domains Vr, r = 1,2, . . . ,N , such thateach contains one phase material. The local constitutive re-lations in each sub-volume are written in the following form:

r (x) = Lr : r (x) + r (x), (73)r (x) = er (x) + r (x), r (x) = Lr : r (x),r (x) denotes a prescribed distribution of local eigenstrainsand r (x) is the corresponding eigenstress field. The eigen-strain and eigenstress fields may consist of contributions ofdistinct physical origin like thermal strains, plastic strainsand transformation strains. The relation between the localand overall fields is given by the following localisation rule:

r = Ar : E +

s

Drs : s ,(74)

r = Br :

s

Frs : s

42 P. Kanout et al.

Dsr and Fsr are the transformation influence tensors(fourth rank tensors). All the tensors Ar ,Br ,Drs,Frs de-pend on the local and overall elastic moduli, and on theshape and the volume fraction of the phases, and can bederived once, independently of the inelastic process.

Let us note that expressions (74) are simple discretiza-tions of exact expressions involving the non-local elas-tic Green operator (x,x) of the homogeneous elasticmedium. For instance:

(x) = A(x) : + 1V

V

D(x,x) : (x)dx, (75)

where D(x,x) = (x,x) : L(x) gives the strain at point xcreated by a transformation strain (x) at point x.

In cases where two phases are considered, with one sub-volume only for each phase, these quantities Ar and Drs (re-spectively Br and Frs) can be expressed in closed form bymeans of Eshelby tensor. Contrarily, when using a decompo-sition into several sub-domains for each phase, these tensorscan be determined by solving a set of linear problems (6for the concentration tensors and 6N for the influence ten-sors), by a finite element method (Dvorak, [55], Carrre etal., [36]).

This formalism has been used in different works withmore complex behaviour (thermo-visco-plasticity, damage,interface debonding) by Dvorak et al. ([53, 55]) and by othergroups namely Kattan and Voyiadjis [100], Chaboche et al.[39], Fish et al. [70], Fish and Yu [69].

In the paper of Chaboche et al. [39] the transformationfield analysis has been extended to take into account the non-linearities due to changing local behaviour induced by tem-perature or damage. A generalised eigenstrain which takesinto account thermal and damage effects on the elastic be-haviour is defined such as at the local scale the local consti-tutive equation

r = Lr : (r r pr ) (76)is replaced by:

r = Lr : (r GEr ), (77)where Lr is the average damaged stiffness in the sub-volumeand the thermal strain. The generalised eigenstrain is consid-ered as an eigenstrain of the same nature as the plastic andthermal ones:

GEr = (thr + pr + dr ), (78)where the damage strain dr = (IL1s : Ls)(s ps s ).

Then the localisation rule is written as follows:

r = Ar : +

s

Drs : Ls : GEs . (79)

It is also demonstrated in [35] that by considering a largenumber sub-volumes the method captures the local stressand inelastic strain fields with good accuracy in comparisonto finite-element simulations.

Interface decohesion effects in composites were intro-duced later by Dvorak and Zhang [53]. The authors repre-sent stress changes due to local debonding under increasingoverall loads by damage equivalent eigenstrains which ad-just local stresses in the affected volumes to values impliedby the selected decohesion models. Interaction between thestill bonded and partially debonded phases at any damagestate is described by transformation influence functions us-ing TFA.

The TFA has been used by Fish et al. [70] to analyse acomposite structure by the finite element method. These au-thors found a good agreement between the two-point aver-aging scheme (plane TFA with uniform plane strain in thematrix) and a more refined computation (multi-point incre-mental homogenisation).

However, it has long been recognised by Dvorak himself(Teply and Dvorak [170] and confirmed by others (Suquet[162]; Chaboche et al. [39]; Michel et al. [123]) that the ap-plication of the TFA to two-phase systems may require, fornon-linear heterogeneous materials, a sub-division of eachindividual phase into several sub-domains to obtain a sat-isfactory description of the effective behaviour. As a con-sequence, the number of internals variables needed in theeffective constitutive relations, although finite, can be pro-hibitively high. The need for a finer sub-division of the phaseresults from the intrinsic non-uniformity of the plastic strainfield which can be highly heterogeneous even within a singlematerial phase.

5.2 Non Uniform TFA

In order to better take into account the non-linear redistrib-utions induced by plasticity, the TFA method uses a subdi-vision of each individual phase into several sub-domains, inwhich are assumed uniform transformation strains (or eigen-strains, or plastic strains). With the aim of reducing the num-ber of such sub-volumes, the NTFA (Non-Uniform Trans-formation Field Analysis) was proposed by Michel and Su-quet ([123, 124]). The plastic strain is decomposed on a fi-nite set of plastic modes which can present large deviationsfrom uniformity:

an(x) =

k

ank k(x). (80)

The ank are now coefficients of the decomposition (that maydepend on time) and the k(x) are the non-uniform transfor-mation fields, defined independently on each phase. Thesemodes k are non uniform (not even piecewise uniform)and are meant to capture the salient features of the plastic

Multiscale Methods for Composites: A Review 43

flow modes. They are determined once (most often by solv-ing specific numerical problems on given unit cells) and arenormalised with keq = 1 in order for ank to be homoge-neous to a plastic strain.

The constitutive relations are expressed in terms of scalarprojections on the modes (there is a direct analogy with crys-tal plasticity):

k = : k, ek = : k, eank = an : k.(81)

The NTFA localisation rule becomes then:

(x) = A(x) : +

l

D k(x)anl (82)

that can be rewritten, after multiplication by k and averag-ing over V :

ek = ak : +

l

Dklanl , (83)

where the second order tensor ak and the influence factorDkl are defined as:

ak = AT : k, Dkl = k : (D l ). (84)

Having assumed that all phases are elastically isotropic,characterised by a bulk modulus k and a shear modulusGk , the resolved shear stress is given by:

k = 2Gk(ek eank ). (85)

As we will see in Sect. 6.2 one interest of the TFA proce-dure is to contain a general writing for macroscopic con-stitutive equations based on the micromechanical evolutionlaws. This property is partly retained with NTFA method. In[123] there are two versions proposed: the uncoupled model: the reduced macroscopic state vari-

ables of the model are the overall strain , the set of allthe ank and a set of tensorial variables k associated witheach mode (and representing the field of internal variablesat the local scale).

the coupled model: in order to reduce the number of in-ternal variables, this version attaches an internal variabler to each phase, and not to each mode. It also couples

the different modes supported by the same phase, by aquadratic average like:

Aanr =(

M(r)

k=1|Aank |2

)1/2, (86)

where the Aank are thermodynamic forces associated withthe r , and M(r) is the number of modes in phase r .

The explicit forms of the two corresponding macroscopicmodels are not indicated here. They assume a restrictionof inelastic local constitutive equations of the phases to aGeneralised Standard Material format (Halphen and Nguyen[86]), and to a linear kinematic hardening. The interestedreader is invited to report to Michel and Suquet ([123, 124])for additional information.

An example is given here, taken from (Michel and Suquet[124] ), for a plane strain unit cell (square periodicity), witha cylindrical elastic fibre (Ef = 400 GPa, f = 0.2) andan elasto-plastic matrix with non-linear isotropic hardening(Em = 75 GPa, m = 0.3, 0 = 75 MPa, h = 416.5 MPa,m = 0.3895), such that eq R(p) = 0 +hpm. There are 3modes introduced, corresponding to 3 independent monoto-nous inelastic analyses with the same constitutive behaviour(made by FFT or by FE), uniaxial tension in directions 1 and2, as well as pure shear.

Figure 2 shows the plastic mode 1, with its 4 compo-nents 111,

122,

112,

133. Figure 3a indicates the results for

uniaxial (1) and shear (2) loadings for the reference cal-culations (very fine mesh), for TFA (with 1 sub-volumeby phase), for NTFA with the coupled version (with the 3modes). Very clearly TFA delivers a much too stiff macro-scopic response, though NTFA correlates very well with thereference solution. In Fig. 3b there are comparisons for 3other loading conditions.

The advantage of NTFA over classical TFA is evident,from the above example and others. However, there are stillsome limitations, like:

the need for a number of initial numerical elasto-plasticsolutions and the corresponding selection of the relevantmodes for a given situation. Moreover, only the coupledversion seems to work well for combined loadings, notcorresponding to the reference modes (see examples in[123]);

Fig. 2 Plastic mode 1, with its4 components (with permissionof P. Suquet)

44 P. Kanout et al.

Fig. 3 Comparisons betweenTFA, NTFA and the referencecomputation for 5 loadingconditions (with permission ofP. Suquet)

the situations where cyclic conditions or non proportionalloading conditions are taking place, or non-isothermalones, which was not the case in the shown example;

the GSM format that is needed, which limits the possibleflexibility of the local constitutive behaviour (non-linearkinematic hardening, visco-plasticity. . .).Another advantage of the NTFA method is its ability to

easily recover the local fields, in a natural way, after the solu-tion of the overall structural boundary value problem. Sucha capability is illustrated by examples given in Michel andSuquet [124]. See also Sect. 8 that is devoted to the locali-sation methods.

5.3 The Methods of Cells or Generalised Methods of Cells

Another quite popular method in the literature is the Meth-ods of Cells of Aboudi ([1, 2]). As in the TransformationField Analysis all the inelastic strains (thermal expansion,plasticity, visco-plasticity) are considered as eigenstrains,assumed to be known before applying the micro-macro tran-sition rule. The method consists of a discretization of the(periodic) unit cell as rectangular (in 2D) or parallelepiped(in 3D) sub-domains.

In his first works, the representative cell consists of foursub-cells; one is occupied by the fibrous material whereasthe other three are occupied by the matrix. The generalisedcell method (Paley and Aboudi [139]) is the generalisationto any number of sub-cells. In these methods local equilib-rium/compatibility equations are solved in an average sensefor uniform eigenstrains in each sub-cell. Continuity of trac-tion and displacement rate on an average basis are imposedat the interfaces between the constituents. The local equilib-rium is guaranteed by the assumption that the velocity factoris linearly expanded in terms of the local coordinates of thesub-cell. This forms the relation between the microscopicand the macroscopic strains, through the relevant concentra-tion factors. The overall behaviour is finally expressed as a

constitutive relation between the average stress, strain andplastic strain in the conjunction of the elastic stiffness tensorof the composite.

The method looks as an approximation of an hybrid for-mulation of the finite element method. In the context of uni-directional composites, the generalised cell method allowsaccurate and efficient analysis of the impact of fibre shapeand arrangement on the inelastic macroscopic response ofthe composite as demonstrated by Arnold et al. [7]. The pre-dictive capability of the method in various applications hasbeen summarised by Aboudi [4].

Damage effects also have been incorporated in the modelby Aboudi [2] to treat interface debonding or by Voyiad-jis and Deliktas [177], using the local incremental damagemodel of Voyiadjis and Park [178].

However, despite the accuracy of the method in mod-elling the inelastic macroscopic response of periodic com-posites, the accuracy with which local stress and strain fieldsare captured is not as good, not allowing the possibilityof following precisely the evolution of damage at the localscale. Aboudi et al. [5] have recently proposed a new methodfor periodic multiphase materials to correct these effects. Aspreviously, the approximate solution for the displacementfield within each sub-cell is constructed based on volumet-ric averaging of the equilibrium equations together with theimposition of periodic boundary conditions, on both the dis-placement and traction continuity conditions in an averagesense between the cells and sub-cells used to characterizethe microstructure of the material.

The particularity of this new approach is to approximatethe displacement field within each sub-cell, by quadraticfunctions expressed in local coordinates. The capability ofthe method to capture both the macroscopic response andthe local stress and inelastic strain fields is shown on a uni-directional gr/al composite by comparisons to finite elementanalyses and analytical models.

Multiscale Methods for Composites: A Review 45

5.4 Corrected TFA

As it has been recalled in the Sect. 5.1, the original TFAmethod does not lead to satisfactory estimates of the globaland local behaviour of the composite when it is applied totwo-phase medium using plastic strains uniform over eachof the two phases. One way to reduce the excessive stiffnessof the TFA localization rule, is to take into account the in-trinsic non-uniformity of the plastic strain as it is done bySuquet and Michel (see Sect. 5.2). Another approach, basedon the use of an asymptotic stiffness tensor, is proposed in1998 by Pottier [148] and Chaboche et al. [39]. The pro-posed corrected method consists of writing the total elasticlocalisation rule with corrected values for the eigenstrains.

Consider a composite made of elasto-plastic phases withps as the plastic deformation in the phase s. The correction

consists of writing the total elastic localisation rule with cor-rected values for the eigenstrains as follows:

r = Ar : E +

s

Drs : Ks : ps . (87)

Moreover, we assume the asymptotic tangent stiffness ofthe local constitutive equation to be known, such as pr =(L

pr )

1 : r . In practice, it is obtained from the knowledgeof the hardening rules in each phase. For instance, in thecase of rate dependent plasticity, its modulus is the kine-matic hardening parameter (that plays a role in the largestrain domain).

The corrected tensors Ks are then determined by the rateforms of the localisation rule:

r = Ar : E +

s

Drs : Ks : ps (88)

with the corresponding elastic tangent one:

r = Ar : E, (89)where the asymptotic tangent concentration tensor Ar is de-fined by the following expression:

Ar = [I + P : (Lr L)]1 (90)and the elasto-plastic tangent stiffness by L1r = (Lpr )1 +L1r .

It is then shown that the following expression can be ob-tained:

s

Drs : Ks : (L1s L1s ) : Ls : As = Ar Ar . (91)

In this equation, the operators As ,Drs are constant and eval-uated once from the elastic properties of the constituents.The tensors Ks ,Ls ,As , are also constant and delivered fromthe knowledge of the asymptotic tangent stiffness.

Moreover, the system in Ks in (91) is indeterminate. Butchoosing Ks = I for the system with the smallest plasticstrain, its solution can be easily obtained.

We have to notice that the asymptotic tangent stiffnessis known only in its isotropic part, because the direction ofplastic flow is not known in advance. Therefore, the tensorKs is determined once, as an isotropic tangent corrector. De-spite the standard TFA formalism, the proposed modifica-tion does not follow exactly the Levin-Mandel equation.

5.5 Methods Comparisons

In order to compare some of the different methods listedhere, we have performed a numerical study on compositematerials in the context of asymptotic homogenisation. As-suming a periodic structure for the composite made up ofmany repetitive unit cells, reference solutions for the over-all and the local constitutive behaviour of the composite areobtained. The exact responses can then be used to evaluatethe accuracy of different homogenisation methods.

The results presented here are focussed on a two-phaseparticle composite.

The inclusion volume fraction is taken as 0.3 and thespherical inclusion is elastic isotropic with Ef = 400 GPa,f = 0.2 and the matrix isotropic elasticity is given byEf = 75 GPa, f = 0.3 exactly the same conditions consid-ered by Michel and Suquet [122]. The constitutive behaviourof the matrix is described by a Von Mises elasto-plasticitywith a power law for isotropic hardening:

eq Hp 0 0 (92)with H = 416 MPa, = 0.3895 and 0 = 75 MPa. In (92) pis the accumulated plastic strain defined by p =

23 p : p .

This case corresponds to a configuration considered by someothers authors (Michel and Suquet [122]; Gonzales andLlorca, [80]; Doghri and Ouaar [47]). The particle reparti-tion is assumed regularly distributed. For all the simulations,the interface between the particle and the matrix is consid-ered as perfect.

The composite has a square/hexagonal arrangement ofparticles leading to an isotropic spatial repartition. The cor-responding 3D unit cell can be nevertheless correctly ap-proximated by an axisymmetric unit cell (Chaboche et al.[40]).

The composite behaviour with a random distribution isobtained through the finite element analysis of a periodiccubic unit cell containing a random dispersion of 38 non-overlapping identical spheres. A compromise between theaccuracy of the solution and the computer time to solve theproblem (in the context of a sequential computation) haslead to this choice. The final particle arrangement is stati-cally isotropic (all the directions in the unit cell are macro-scopically equivalent). Figure 4 shows the local accumulated

46 P. Kanout et al.

Fig. 4 Iso-values of the plastic strain in the RVE of the composite with a regular microstructure

plastic strains obtained in the composite with a regular mi-crostructure.

The global stress/strain responses obtained with vari-ous mean-field methods (the analytical TFA, the Hills in-cremental, the Affine and finally the corrected TFA ap-proaches) have been then compared to the reference overallstress/strain responses delivered by the finite element simu-lation on the unit cells. In each application of the mean fieldmethods that will be presented here, the homogenisation ruleof Mori-Tanaka (Mori and Tanaka [131]) has been chosen.

We have to remind that in the incremental and affinemethods, the tangent polarisation tensor P is anisotropic be-cause of the anisotropic character of L. It has then to beevaluated numerically, at each time step of an incrementalor affine simulation. However, the anisotropic stiffness canbe replaced by an isotropic approximation (Chaboche andKanout [37]):

Ls = 3ksJ + 2sK (93)with J = 13 (1 1),K = I J. Assuming an isotropic VonMises plastic flow, we have 2s = 2shps /h where h =3s + hps , with the current plastic modulus hps defined ashps = eq/p. The incremental and the affine method have

been evaluated using the complete anisotropic polarisationtensor and this isotropic approximation.

Other isotropic approximations can be also considerednamely the following isotropic form based on the generalprojection method (Bornert, [17]; Doghri and Ouaar, [47]):Lisos = 3ksJ + 2tsK with ts/s = (4 + hps /hs)/5.The results presented in Fig. 5 show the comparisons be-tween the reference FE solution assuming a regular mi-

crostructure of the composite and several mean field meth-ods.

Regarding the Fig. 5, the following remarks can be done:

The affine, the incremental and the TFA procedures givean extremely stiff response. The affine delivers a slightlysofter response than the TFA and the incremental meth-ods.

The isotropic approximations of the affine and incremen-tal procedure, deliver much softer responses, much morein accordance with the finite element reference solution,but slightly softer. The same results has been obtainedfor the incremental method (Gonzales and Llorca, [80];Doghri and Ouaar, [47] for randomly distributed particle).

The best approximations of the global behaviour of thecomposite are given by the corrected TFA approach andthe incremental method based on an isotropic approxima-tion of the polarisation tensor.

6 Sequential Multiscale Procedures

When dealing with the inelastic analysis of a structural com-ponent, when the overall behaviour explicitly takes into ac-count information and processes present at the lower scale,there are currently two main approaches. In what follows wewill call them:

the sequential multiscale procedures, in which the mul-tiscale analysis, or the micro-to-macro homogenisationprocess, is made separately from the structural analysis.Prior treatments then define a set of overall constitutiveequations which format could be guided by the scalechange method. Material parameters in these equations

Multiscale Methods for Composites: A Review 47

Fig. 5 Comparison between thereference FE solution andvarious mean field methods.Case of a power law matrixcomposite

are subsequently identified with microscopic or macro-scopic results, either true experiments or virtual experi-ments performed through systematic applications of themultiscale method. Such an approach is currently the oneused when we define the effective properties of a compos-ite or multiphase material from the knowledge of the cor-responding local properties and the phase arrangement;

the integrated multiscale procedures, in which all thecomplexity of the local microstructure (at least its finestrepresentation) is present during all the analysis of thestructural component, without summarising it in someoverall constitutive framework. In such cases, at eachstep of the non-linear boundary value problem, the actualoverall response of each material point asks for the mi-croscale response, through the localisation/homogenisa-tion process. To some extent, through an averagingscheme, the internal variables in the overall boundaryvalue problem are those of the microscale unit cells usedin this integrated multiscale numerical procedure.

In what follows, we summarise a narrow selection ofworks done along the lines of the so-called sequentialprocedure. We consider also as a sequential approach theones in which the macroscopic constitutive model are ex-pressed more or less analytically through the local constitu-tive equations, provided the scale change method (localisa-tion/homogenisation rule) is taken in its simplest form. Ap-

plications of TFA methodologies in Sect. 5.4 appear of thiskind, as there are some approximations, eventually someoverall correction factors identified from numerical exper-iments. More sophisticated local representations, like TFAwith many sub-domains, the NTFA method, the HFGMC,the use of FFT techniques or multilevel finite elements, areconsidered as integrated multiscale procedures. They are il-lustrated in Sect. 6.2.

6.1 Unit Cell Methods

The class of so-called unit cell methods can be classified inthe category of sequential multiscale procedures. In thesemethods, numerical computations involving a detailed rep-resentative volume element (RVE) are used to induce macro-scopic model.

These approaches have been used by many authors. Wecan quote the works of Christman et al. [44] who have stud-ied the deformation characteristics of ceramic whisker andparticulate reinforced metal matrix composites using unitcell formulations to deliver the overall constitutive responseof the composite. Brockenbrough et al. [23] presented in1991 numerical results on the effects of fibre distribution andfibre cross-section geometry on the deformation of metal-matrix composite reinforced with continuous fibres. Naka-mura and Suresh [134] studied the combined effects of ther-mal residual stresses and fibre packing on deformation of

48 P. Kanout et al.

metal matrix composites using unit cell finite element calcu-lations. Tvergaard [173] uses the same technique to analysethe tensile properties for whiskers-reinforced metal matrixcomposites. More recently, the works of Van der Sluis [174]lie within this framework.

In the same period a lot of works have been carried out inmetal matrix composites to determine the thermally-inducedresidual stresses at the interface fibre/matrix and to evaluatetheir influence on the behaviour of this interface debondingcapability namely, Bhm [14], Mital et al. [128], Gunawar-dena [84], Allen et al. [6], Arnold and Wilt [8]. Sorensenand Talreja [160] analyse the influence of the fibre distribu-tion on the initiation of matrix cracks.

More recently, different works have been carried out withmore complex unit cells. We can quote for instance Bhm etal. [15] who use a multi-inclusion unit cell approach to studythe elastic and elasto-plastic behaviour of metal matrix com-posites reinforced by randomly oriented short fibres. Com-parisons between three-dimensional and two-dimensionalmulti-particle unit cell models were also performed (Bhmand Han [16]) for particle reinforced metal matrix compos-ites. The recent works of Gonzales et al. [83] lie within thiscontext.

6.2 Macroscopic Constitutive Equations Obtainedfrom Analytical Homogenisation Techniques

Many homogenisation techniques are nowadays available,generating a microscopically based stress-strain relation.Most of this techniques consider only small elastic or elasto-plastic deformations on both micro-macro levels. They pro-vide adequate predictions of the overall behaviour of com-posites or polycrystalline polymers. They are mainly basedon mean-field approaches (see Sect. 4) or on semi-analyticalmethods (see Sect. 5). The focus here is on purely macro-scopic constitutive equations which are inferred from otheranalytical homogenisation techniques. Such models con-tain some microscopic features although they are written aspurely phenomenological equations at the global scale.

Chaboche et al. [38] proposed a thermo-elastic-visco-plastic constitutive equation for metal matrix compositeswith short fibres built up from a simplified micromechanicalanalysis. The paper considers the undamaged state of com-posite with small strains. Using the Transformation FieldAnalysis (TFA) developed a few years ago by Dvorak andBenveniste (see Sect. 5.1) a completely macroscopic con-stitutive model is derived. TFA is a very general manner ofwriting explicit scale transitions, based on purely elastic in-teractions between sub volumes of the RVE. Considering anelastic elasto-visco-plastic matrix and an elastic fibre, theyield surface is defined by the Von Mises criterion and thevisco-plasticity by the normality rule and a power law:

fr = r Xr y, a =(

32

a : Id : a)1/2

, (94)

pr =

fr

K

nfr

r. (95)

Moreover, a multi-kinematic non linear hardening rule isconsidered:

Xr =

q

Xqr , Xqr = Cq : pr q : Xqr pr , (96)

y,n,K are material coefficients, possibly depending ontemperature. Cq and q, q = 1,2, etc., are hardening anddynamic recovery tensors. The macroscopic constitutiveequation set is derived by substituting the local stress of thematrix in its local plasticity criteria by its expression givenby the TFA localisation rule. The formulation obtained leadsto a well established model in the thermodynamic frame-work. The model has been used to simulate the visco-plasticresponse of a short fibre SIC/Al composite. The fairly goodcorrelation with numerical results, obtained with the peri-odic homogenisation method, has led to apply this formula-tion in the analysis of structural components.

Later on, Pottier [148] has used the same methodologyfor long fibre composites and has introduced an anisotropicmacroscopic damage. To have a damage directly at the het-erogeneities scale, Carrre et al. ([34, 36]) proposed an ex-tended formulation. To that end the extension of TFA pro-posed by Chaboche et al. [34] is used to take into accountthe non-linearities due to changing elastic local behaviour(temperature or damage induced). Let us consider the fol-lowing localisation rule:

r = Ar : +

s

Drs : Ls : GEs (97)

with GEr = (thr + pr + dr ) (see Sect. 5.1).This leads to a system linking and . By regrouping the

terms in , we obtain a simple matrix system:

r = Br : Ar (T T0)

k

Hkr : pk(98)

with Br = {K1B}r , Ar = {K1A}r , Hkr = {K1Hk}r .K is a square matrix (2 2), B, A, and Hr are (1 2) ma-trices. Each term of these matrices is a fourth order tensordepending on the spatial distribution of the inclusions andon the elastic properties of the fibres and the matrix.

Finally, the macroscopic constitutive equations are writ-ten as follows:

= L(T ,D) : (E Ep Eth),

Ep =

r

{crBTr : pr crBTr : Sr :

k

Hkr : pk},

=

r

{crBTr : r}

r

{crBTr : Sr : Ar}(99)

Multiscale Methods for Composites: A Review 49

with L(T ,D) = (S0 +S)1 where S0 is the initial macro-scopic compliance tensor, and S the modification of com-pliance by temperature T and damage D. The r and arerespectively the thermal expansion tensor of the phase andthe macroscopic one.

Among the existing models, we can also quote the worksof Voyiadjis and Kattan [176] and Kattan and Voyiadjis[100] who use micromechanical considerations to correlateoverall and local damage effect of fibre reinforced compos-ite materials. An overall fourth-rank damage effect tensoris introduced using the assumption of elastic energy equiv-alence to account for the overall damage of the compositesystem. In addition, two local fourth-rank damage effect ten-sors are also considered to convey the damage effects in thecomposite phases. Both scales are correlated together usinganalytical scale transitions. The effective stress concept isthen used to describe the evolution of the internal parame-ters.

In the context of granular materials, this type of ap-proaches has usually been used. Some examples of recentstudies are given in references ([32, 41, 104, 121]).

7 Multiscale Computational Techniques

Finite element analysis of simple heterogeneous periodic orquasi-periodic materials is nowadays a routine exercise thatcan be easily handled at the level of the representative vol-ume element (RVE). In the last few years, in the context ofstructural inelastic analysis, a variety of direct micro-macromethods has been developed. These approaches estimate therelevant stress-strain relationship at a macroscopic point byperforming separate calculations on the RVE, assigned tothat macroscopic point. The analysis on the RVE is per-formed using the finite element method, in Smit [158], Smitet al., [159], Feyel [6165], Miehe et al. [126], Terada andKikuchi [171], the Vorono cell method in Ghosh et al. [78]or the fast Fourier transforms in Moulinec and Suquet [132].

Although these methods are computationally expensive,they offer the possibility of computing the macro-structuralresponse of heterogeneous materials with an arbitrary mi-croscopic geometry and constitutive behaviour. The follow-ing sections give an overview of the most relevant worksfrom the literature applying this type of procedures.

7.1 Multiscale FE Models

In most of the direct micro-macro methods in literature, theanalysis on the RVE, assigned to the macroscopic point, ismade by finite elements. The idea of using directly a fi-nite element discretization of the microstructure, linked tothe macroscopic scale, using homogenisation rules, was firstproposed by Renard et al. [150] in 1987. Its complete gen-eralisation and implementation in a general purpose finite

element code was done in 1998 by Feyel, in the contextof cyclic visco-plastic and damage analysis of componentsmade in a metal matrix composite. Later on numerous au-thors will use it.

Terada and Kikuchi [171] generalized the two-scale mod-elling scheme for the analysis of heterogeneous media withfine periodic microstructures by using variational state-ments. By means of the generalized variational principleof Hu-Washizu [98] applied in the framework of non-linearelasticity, they have shown that the global-local type com-putational schemes can be unified in association with thehomogenisation procedure for general non-linear problems.

The two-scale derived variational problem is then solvedusing the Newton-Raphson iterative scheme. In spite of thesuccessful computations for a heterogeneous elasto-plasticbody, the numerical analysis leads to underlying difficultiesin performing general class of non-linear multiscale analy-ses. One of these is that, since the microscopic problems aresolved at each Gauss point of the FE mesh of the overallstructure, the deformation histories at time tn must be storeduntil the equilibrium state at current time tn+1 is obtained.The non-linearities require large computational resources inpractical applications.

Miehe et al. [127] present a theoretical and computa-tional framework for the treatment of a homogenised macro-continuum with locally attached microstructure. The pro-posed concept is applied to the simulation of texture evo-lution in polycrystalline metals, where the micro-structureconsists of a representative assembly of single crystal grains.The deformation of this micro-structure is then coupledwith the local deformation at a typical material point of themacro-continuum. In a process driven by the deformationof the macro-continuum, this coupling can be defined bythree alternative constraints of the micro-structure deforma-tion: zero fluctuation in the domain (Taylor-type assump-tion); zero fluctuation on the boundary; or periodic fluctu-ations on the boundary. These boundary conditions induce,in combination with a given constitutive model for the ma-terials constituents, a static equilibrium state of the micro-structure at a certain stage of the deformation process.

The macroscopic extensive variables such as stresses anddissipation are then defined as volume averages of their mi-croscopic counterparts defined by the equilibrium state ofthe micro-structure. In the proposed procedure, these aver-ages are evaluated in a straightforward manner. This coversthe set-up and the solution of two locally coupled bound-ary value problems for the finite deformations of the micro-continuum and the pointwise associated micro-structure, re-spectively.

In the same period, a general method called FE2 was in-troduced by Feyel [6165], which consists in describing thebehaviour of heterogeneous structures by using a multiscalefinite element model. To each integration point at the macro-scopic scale a representative volume element is assigned and

50 P. Kanout et al.

a separate finite element computation is performed simul-taneously. This procedure does not require to specify themacroscopic constitutive behaviour which is deduced fromthe non-linearities in the behaviour of the associated mi-crostructure. Once the relevant mechanical scales are cho-sen, the model is built up using three main ingredients:

1) A modelling of the mechanical behaviour at the lowerscale (the RVE)

2) A localisation rule which determines the local solutionsinside the unit cell, for any given overall strain

3) A homogenisation rule giving the macroscopic stresstensor, knowing the micromechanical stress state.

In principle any localization/homogenisation rule can beused for points 13, but the most common method is the pe-riodic homogenisation of Sect. 3, which is considered here.Two F.E. meshes are needed: one for the cell of periodicityand the other one for the macroscopic structure. The com-putation is carried out simultaneously on both scales.

In the cell of periodicity located in each Gauss point ofthe macroscopic structure the method allows to compute thestress tensor at time t , knowing the strain and strain rate atthat time and the mechanical history since the beginning on.In classical phenomenological models at macroscopic scalethe mechanical history is taken into account through a setof internal variables. Here the internal variable set is con-structed by assembling all microscopic data required by theFE computation at lower level. This includes of course mi-croscopic internal variables needed to describe dissipativephenomena.

The local analysis yields the macroscopic (average)stresses and strains, defined as

ij = ij = 1V

V

ij dV,

(100)Eij = eij = 1

V

V

eij dV

also known as Bishop-Hill relations [12].The integrated approach can be easily implemented

in classical finite element codes, based upon a Newton-Raphson algorithm to handle all non-linearities. It consistsin fact of a sequence of Newton algorithms at global andlocal level.

For its optimum performances, the tangent stiffness ma-trix at macroscopic level has to be computed. For this calcu-lation we need the macroscopic (algorithmic) tangent matrixin the Gauss points which can be computed from the varia-tion of the average strains and stresses of (100) as

Dmacro = E

(101)

where denotes the increment of the quantity between timet and time t + t . This computation depends of course on

the homogenisation theory used, and on its finite elementimplementation.

In the case of the periodic homogenisation theory it isconvenient to add at local level (mesh of the cell of period-icity) some degrees of freedom to the elements correspond-ing to average or macroscopic strain E. It is recalled that atlocal level the unknown displacements are the periodic partu1 of the total displacement u on the cell. u1 is obtainedthrough (54) and (39). These, together with the first of (100)are contained in the condensation procedure outlined below,proposed in [62].

The deformation tensor is computed by (34) and may bewritten as

e0(x, y) = e(x)(u

0) + e(y)(u1) = E + e(y)(u

1). (102)

The B matrix needed in this case, called B, is similar to theusual one, except that a new part comes from degrees offreedom associated with E (these degrees of freedom areput at the end of the whole degrees of freedom list)B = (B 1), (103)where B denotes as usual the standard symmetric gradient ofthe shape functions. According to our choice, E has associ-ated degrees of freedom and hence the associated reactionsgive the mean stress , to be multiplied by the volume ofthe cell.

The assembled tangent stiffness matrix at the cell scalecan be written as

K =

cellBT DBd. (104)

Taking into account (103) leads to

K =

e

[BT DB BT D

DB D

]d =

[k GTG H

], (105)

where D represents the tangent matrix given by all micro-scopic phenomenological constitutive equations.

Recalling the meaning of the additional d.o.f. and the as-sociated reactions, the macroscopic tangent matrix Dmacro isthen nothing but a condensation of the previous matrix ontothe degrees of freedom associated to E (at most 6 degreesof freedom shared across the whole microscopic mesh) in-serted at the end of the list. That is

Dmacro = 1volume

(H Gk1GT ), (106)

k =

cellBTDBd, G =

cellDBd,

(107)H =

cellDd.

This condensed matrix is then very easy to compute. Then,depending on the complexity of the macroscopic structure,

Multiscale Methods for Composites: A Review 51

this model may lead to large computations. Parallel com-puting is proposed to overcome this difficulty and has beenapplied to the analysis of a reinforced bling (bladed ring),by Feyel and Chaboche [62].

The method has been used in generalised continua [64] totreat the cases for which the classical periodic media theorycannot be applied, for instance where the size of the RVEis not very small compared to the size of the structure itselfand also compared to the size of the mechanical gradients.However, the present FE2 framework does not treat specif-ically edge effects and may not deliver good solutions nearedges.

A similar multi-level finite element modelling has beendeveloped by Smit [158], Smit et al. [159] and applied morerecently in the large strain context. The method has alsobeen extended to higher-order continua, by Kouznetsova etal. [101].

Another multi scale computational strategy which makesuse of the homogenisation theory is proposed by Lade-vze and co-workers [105107]. In that case the structure isconsidered as an assembly of substructures and interfaces.The junction between the macro and micro scales takesplace only at the interfaces. Namely, in the recent works ofMarkovic and Ibrahimbegovic [118], a multi-level finite ele-ment procedure is applied for modelling the inelastic behav-iour of heterogeneous materials. In the paper strongly cou-pled scales are considered, where the finite element methodis used at both scales. Yet, the micro-scale is not infinitivelysmaller than the macro scale. The coupling of the scales isobtained trough the framework of localised Lagrangian mul-tipliers. The macro mesh plays the role of a frame which isconnected to the micro mesh through the Lagrangian multi-pliers.

The works of Fish [66] may also be classified in this cat-egory of models. The multiscale computational procedureproposed by Fish is a superposition based method. A hier-archical decomposition of the solution space u is made intoglobal uG and local uL effects, u = uL + uG. Enforcementof solution compatibility is obtained by prescribing homo-geneous boundary conditions on uL at the global interfaceGL. A multilevel solution scheme, termed as the s-versionof the finite element method has then been developed whereeach level is discretized using a finite element mesh of arbi-trary element size and polynomial order. Selection of the in-terface GL is one of the critical issues in this superpositionbased method. A mathematical analysis aimed at quantify-ing pollution effects on localized phenomena on the globalstructural behaviour and identifying the optimal location ofthe interface has also been carried out by Babuska [9] andFish [67].

7.2 The Vorono Cell Method

One of the first simultaneous global-local computationalprocedures has been carried out by Ghosh and co-workers[78] with the Vorono cell finite element method (VCFEM).In this method, the finite element mesh evolves naturally byDirichlet Tessellation of a representative structure. This is aprocess of subdivision of space, determined by set of points,in such a way that each point has associated with it a regionthat is closer to it than to any other. The subdivided regionsare called Voronoi cells. The cells may be identified withbasic structural elements in a heterogeneous microstructure.They represent regions of immediate influence for each het-erogeneity and also define neighbour regions by the cellfacets. The germinating points are then replaced by hetero-geneities with shape, size and orientation. The discretizationshould account for these features and avoid intersection ofthe Vorono cell edges with the heterogeneities.

A mesh generator accounting for arbitrariness in shape,size and spatial distribution of inclusions has also been in-troduced by Ghosh and Mulkopadhyay [77]. Tessellation ofa microstructural representative material element discretizesthe domain into a network of multi-sided convex Voronoipolygons or cells. Each Vorono cell contains at most onesecond phase inclusion.

Formulations have been developed to directly treat mul-tiple phase polygons as elements in a finite element modelby Ghosh and Mulkhopadhyay [77] for linear elasticity andby Ghosh and Liu [75] for micro-polar thermo-elasticity andfor elastic-plastic problems in Moorthy at al. [130] and Goshand Moorthy [76]. Several applications of the method toaccount for non-linear composite materials have been pre-sented in recent papers. The main drawback of this methodis its time and memory consuming.

7.3 Models Based on Fast Fourier Transform

Walker et al. [179] used in 1994 the periodic microstruc-ture assumption together with Fourier series approximationto analyse the non-linear visco-plastic behaviour of fibrouscomposites. The unit cell is discretized into triangular sub-volumes. In each step of the loading history the total strain atany point is governed by an integral equation. The strain andstress fields within the repeating unit cell are obtained us-ing Fourier series approximations. The non-linearity arisingfrom the visco-plastic behaviour of the material constituentsis treated as a fictitious body force in the governing inte-gral. The authors have also shown that the problem can bewritten using Greens function approaches, a more generalmethod because it doesnt need the periodic assumption, andfor which the numerical resolution converges more rapidly.

Fotiu and Nemat-Nasser [72] have also used Fourierseries approximation to estimate the overall properties of

52 P. Kanout et al.

elasto-visco-plastic periodic composites in the context of theperiodic distribution of inclusions assumption. The formu-lation is presented in three dimensions as well as for plane-stress and plane-strain problems. The two methods have theparticularity to employ quite accurate field representationwithin the repeating cell but can be computational time con-suming

To avoid the difficulty due to meshing in case of complexmicrostructure, when solving the local problem by FEMmethods, and to make direct use of microstructure imagesof heterogeneous materials, Moulinec and Suquet [132] pro-posed in 1998 an alternative method based on Fast FourierTransforms. FFT algorithms require data sampled in a gridof regular spacing, allowing the direct use of digital imagesof the microstructure. A specificity of the formulation is theuse of an iterative method not requiring the formation of astiffness matrix. Introducing an homogeneous reference ma-terial with elastic stiffness c0, they have shown that the lo-cal problem to be solved on a typical volume element canbe re-written as an implicit integral equation (the so-calledLippman-Schwinger equation). It takes the form:(u(x)) = 0 (c(x) c0) : (u(x)) + E, (108)where and E are respectively the local and the overallstrain tensors. This expression becomes in the Fourier space:

(u)() = 0() : (c(x) c0) : (u)(), = 0,(109)

(0) = E,where denotes the convolution product and where 0 is theGreen operator associated with c0 which is explicitly knownin Fourier space. An iterative scheme is then derived fromthe previous equation:

(ui+1) = 0 ((c c0) : (ui )) + E. (110)

Noting that a convolution product in real space becomes amere product in Fourier space:

i+1() = i () 0() : (c : i )(), = 0,(111)

i+1(0) = E.This method requires a number of iterations roughly pro-portional to the contrast between the mechanical propertiesof the phases and thus is inadequate for highly contrastedor infinitely contrasted composites. The advantages of themethod are the following:

Images of microstructures can be directly used in theanalysis, which avoids meshing the microstructure.

The iterative procedure does not require the formation orinversion of a stiffness matrix.

The convergence is fast.The method present the following limitations:

The convergence is not ensured for materials containingvoids or rigid inclusions.

The number of degrees of freedom is high in comparisonwith the FEM. The method can be applied only on com-puters with high memory capabilities.

To allow the application of this method for contrasted mate-rials, Eyre and Milton [59] improved the procedure by intro-ducing an accel