Bounds for the effective properties of heterogeneous plates

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European Journal of Mechanics A/Solids 28 (2009) 1051–1063

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European Journal of Mechanics A/Solids

journal homepage: www.elsevier .com/locate/e jmsol

Bounds for the effective properties of heterogeneous plates

Trung-Kien Nguyen a, Karam Sab a, Guy Bonnet b,*

a Universite Paris-Est, UR Navier, Ecole des Ponts ParisTech, 6-8 Avenue Blaise Pascal, Champs-sur-Marne, 77455 Marne-La-Vallee Cedex 2, Franceb Universite Paris-Est, Laboratoire de Modelisation et Simulation Multi Echelle (FRE3160 CNRS), 5 bd Descartes, 77454 Marne-La-Vallee Cedex 2, France

a r t i c l e i n f o

Article history:Received 14 November 2008Accepted 23 May 2009Available online 6 June 2009

Keywords:Hashin–Shtrikman variational principleG-operatorPlateFourier transformsSecond-order boundsEffective properties

* Corresponding author.E-mail address: [email protected] (G. B

0997-7538/$ – see front matter � 2009 Elsevier Masdoi:10.1016/j.euromechsol.2009.05.006

a b s t r a c t

This paper presents new bounds for heterogeneous plates which are similar to the well-known Hashin–Shtrikman bounds, but take into account plate boundary conditions. The Hashin–Shtrikman variationalprinciple is used with a self-adjoint Green-operator with traction-free boundary conditions proposed bythe authors. This variational formulation enables to derive lower and upper bounds for the effective in-plane and out-of-plane elastic properties of the plate. Two applications of the general theory areconsidered: first, in-plane invariant polarization fields are used to recover the ‘‘first-order’’ boundsproposed by Kolpakov [Kolpakov, A.G., 1999. Variational principles for stiffnesses of a non-homogeneousplate. J. Meth. Phys. Solids 47, 2075–2092] for general heterogeneous plates; next, ‘‘second-order bounds’’for n-phase plates whose constituents are statistically homogeneous in the in-plane directions areobtained. The results related to a two-phase material made of elastic isotropic materials are shown. The‘‘second-order’’ bounds for the plate elastic properties are compared with the plate properties ofhomogeneous plates made of materials having an elasticity tensor computed from ‘‘second-order’’Hashin–Shtrikman bounds in an infinite domain.

� 2009 Elsevier Masson SAS. All rights reserved.

1. Introduction

The prediction of the effective properties of heterogeneousmedia from the material properties and from the geometricalarrangement of their constituting phases requires the solution ofa boundary value problem on a representative volume. The char-acteristic size of this representative volume must be large enoughcompared to one of the microstructures to ensure that the materialcan accurately be treated as homogeneous with spatially constantaverages. If the geometrical arrangement of the heterogeneitieswithin the medium is perfectly described, solving the boundaryvalue problem is possible. Practically, however, only a few featuresof the geometry (typically the volume fraction of each phase andlow-order correlation functions) are known. In such a case, effectiveproperties cannot be obtained exactly, but lower and upper boundscan be derived from variational principles.

The application of variational methods to composite materialswas initiated by Hill (1952) who recovered the so-called Voigt andReuss bounds. The classical variational principles usually requireeither compatible strain fields or self-balanced stress fields. Hashinand Shtrikman (1962a, b) variational principles may be the mostwidely used since they lead to optimal bounds for the conductivity,bulk, and shear moduli of isotropic composites made of isotropic

onnet).

son SAS. All rights reserved.

constituents. These principles are expressed in terms of polariza-tion fields on which no constraint is imposed, contrarily to the caseof classical variational formulations. Hashin and Shtrikman (1967)subsequently extended those principles to inhomogeneous elasticbodies subject to polarization fields and mixed boundary condi-tions. Willis (1977, 1981) summarized and generalized the Hashin–Shtrikman variational principles by using a G-operator related to aninfinite medium.

The homogenization of elastic periodic plates has been studiedby many authors (Duvaut and Metellus, 1976; Caillerie, 1984; Kohnand Vogelius, 1984; Lewinski and Telega, 2000; Bourgeois et al.,1998; Cecchi and Sab, 2002a, b, 2004; Sab, 2003; Dallot and Sab,2008a, b), but few of previous works were performed to obtain thebounds of plates properties. In Kolpakov (1999); Kolpakov andSheremet (1999) and Kolpakov (1998) first-order (‘‘Voigt’’ and‘‘Reuss’’) bounds were obtained for plates and the question arisesnaturally to find if more sophisticated bounds can be found, andmore specifically if Hashin–Shtrikman bounds can be extended toplates. The answer to such a question is simple when the size of theheterogeneities is small compared to the thickness of the plate orwhen the thickness of the plate is small compared to the (in-plane)size of the heterogeneities. In the first case, bounds can be obtainedfor the plates containing isotropic composites made of isotropicconstituents by computing Hashin and Shtrikman bounds for thefilling material and by computing from it the stiffness properties ofthe plate. In the second case, the bounds can be computed from the

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Fig. 1. Example of a plate containing a random distribution of heterogeneities.

T.-K. Nguyen et al. / European Journal of Mechanics A/Solids 28 (2009) 1051–10631052

Hashin and Shtrikman bounds computed from the plane-stressproperties of the components.

Obtaining the bounds in the case of heterogeneities whose sizeis comparable to the plate thickness is the subject of the presentpaper. A main feature of the Hashin–Shtrikman principle is that ituses the G-operator of the homogeneous medium. If this principlemust be extended to plates, a necessary ingredient is the G-oper-ator taking into account the stress-free boundary conditions of theplate. A new method for the computation of the effective elasticproperties of a periodic plate was recently proposed by Nguyenet al. (2008). The theoretical derivations rely on a new G-operatorfor periodic media with traction-free boundary conditions, on aniterative algorithm and on the use of the Fast Fourier Transform(FFT), in the line of methods used for homogenization of periodicmedia (Moulinec and Suquet, 1994, 1998; Bonnet, 2007). This newmethod was proved numerically efficient to estimate the effectiveelastic properties of plates. Based on this previous work (Nguyenet al., 2008), the present paper aims at using a Hashin–Shtrikmanvariational principle in the case of heterogeneous plates forsupplying bounds of the effective in-plane and out-of-plane elasticstiffnesses by introducing into the variational principle theG-operator obtained by the authors.

Compared to the usual Hashin–Shtrikman bounds for infinitemedia, two main differences are taken into account here: thetraction-free conditions on the surfaces of the plate and the inho-mogeneous macroscopic strains and stresses for the case ofbending. As explained before, the case of interest is when thethickness of the plate and the size of heterogeneities are of thesame order. The thickness and the period of the plate are thereforeassumed to be of the same order and small compared to the in-plane typical (macroscopic) size L of the plate. The stiffnessconstants of the plate can be bounded by introducing suitablepolarization fields into the variational principle. Two applicationsof the theory are considered. In-plane invariant polarization fieldsare first used to recover the bounds proposed by Kolpakov (1999)for general heterogeneous plates. Secondly, n-phase plates whoseconstituents are distributed randomly but in a statistically uniformmanner in the in-plane directions are studied. The derived energyfunctional expressed in terms of the polarizations depends on thethickness-coordinate only and the computation of the effectiveelastic properties of the plate requires the use of a two-pointdistribution function. The example of a two-phase plate made ofisotropic materials is considered. The derived bounds for theeffective elastic properties of the plate are compared with theelastic properties of the plate computed from elasticity tensorsobtained from the classical Hashin–Shtrikman bounds (Hashin andShtrikman, 1962a, b, 1965) for an infinite two-phase medium. Itmust be emphasized that the computation of bounds presentedthereafter does not use the computation of effective properties ofa periodic medium, as effected in Nguyen et al. (2008). Similar tothe known HS bounds, the bounds are obtained directly from thevariational principle and from assumptions related to the two-point probability function of heterogeneities distribution, withoutensemble averaging or ‘‘cell computations’’. The Green tensor forthe plate is used with wave lengths large enough compared to thesize of heterogeneities.

The paper is organized as follows. In Section 2, the Hashin–Shtrikman variational principle (Hashin and Shtrikman, 1962a, b;Willis, 1977, 1981; Drugan and Willis, 1996) is recalled and it isshown that it can be used by introducing the G-operator for peri-odic media with traction-free boundary conditions (Nguyen et al.,2008), because this operator is self-adjoint. Thus completed, thisgeneral variational formulation enables to derive lower and upperbounds for the effective elastic properties of heterogeneous plates.Section 3 presents an application of the results derived in Section 2

by using polarization fields which are assumed to be invariantalong the in-plane directions. In Section 4, we apply the method torandom materials whose constituents are distributed randomly butin a statistically uniform manner in the in-plane directions. Thefunctional is expressed in terms of the in-plane invariant polari-zations. Then a randomly distributed two-phase plate made ofisotropic constituents is considered. An analysis of the effects of thesize of heterogeneity is finally performed.

2. Theoretical formulation

As it will be recalled thereafter from (Nguyen et al., 2008), theGreen tensor for the plate can be computed for a periodic plateunder the form of a series on the wave numbers associated with thedimensions of the rectangular chosen period. The result corre-sponds to a periodic repartition of singularities applied withina homogeneous plate. In order to obtain the bounds for repartitionsof heterogeneities which are not periodically distributed as in Fig. 1,the Green tensor must be obtained for an infinite domain. However,even if the Green tensor for the infinite homogeneous plate isobviously obtained when the sizes of the period tend to infinity,there is no closed-form solution of the Fourier integral thusobtained. From another point of view, the Green tensor will be usedin association via a convolution product with a correlation functionwhose support is practically finite. It is therefore possible to use theGreen tensor associated with the periodic plate to perform sucha convolution product with an error which can be as small asdesired. From another point of view, heterogeneities within theplate will be assumed random within the plate. One more time, dueto the finite support of the correlation function, the computation ofbounds is not affected by using a random distribution of elasticproperties within a period as soon as the correlation length is smallcompared to the period dimensions.

Finally, the problem which is considered here is the one related toa heterogeneous medium within a period, the heterogeneities beingreproduced periodically. Such a solution will be used in Section 4 forperiods large enough compared to the correlation length.

The unit cell Y that generates the plate by periodicity in the (x1,x2)-directions is the 3D domain:

Y ¼�

x˛R3; x ¼ ðx1; x2; x3Þ; xi˛�� li

2;

li2½; i ¼ 1; 2; 3

�(1)

The domain u ¼ � � l1=2; l1=2½� �l2=2; l2=2½� is the middlesurface of the cell. vu is the boundary of u andvYl ¼ vu�� ðt=2Þ; ðt=2Þ½ðl3 ¼ tÞ is the lateral boundary of Y. Thetop and bottom surfaces of the cell are vY� ¼ u� f�ðt=2Þg. Mixedconditions are defined along the boundaries of the cell as follows:periodic boundary conditions are assumed along vYl while traction-free boundary conditions are assumed along vY�. All formulationsare performed under the assumption of a linear elastic behaviorand small deformations of materials. The Greek indices belong to {1, 2}and the Latin indices to {1, 2, 3}.

T.-K. Nguyen et al. / European Journal of Mechanics A/Solids 28 (2009) 1051–1063 1053

The local elasticity problem defined on Y is as follows:8>>><>>>:sðxÞ,V ¼ 0; sðxÞ ¼ LðxÞ 3ðxÞ; 3ðxÞ ¼ Eþx3cþeðvperðxÞÞ;eðvperðxÞÞ ¼ vperðxÞ5sV;

sðxÞ,e3 ¼ 0 on vY�;

vperðxÞperiodic on vYl; sðxÞ,n anti-periodic on vYl; ð2Þ

where the nabla operator V is used to express the gradient anddivergence operators ðvper5V ¼ vper

i; j ei5ej; s,V ¼ sij; jeiÞ. ei is thevector of an orthogonal basis of the space, vper(x) is the (x1, x2)-periodic displacement field, e(vper(x)) is the corresponding strainfield (the symmetrical part of vperðxÞ5V), 3ðxÞ is the total strainfield, L(x) is the elasticity tensor and sðxÞ is the total stress field.The macroscopic membrane strains, E, and the macroscopiccurvatures, c, only have in-plane components (Ej3¼ 0, cj3¼ 0).

Once the solution of the boundary value problem (Eq. (2)) isobtained, the homogenized elastic properties of the plate arederived from the elastic strain energy expressed by:

W ¼ 12hsðxÞ 3ðxÞi ¼ 1

2ðNEþMcÞ; (3)

where the brackets C$D correspond to the following average oper-ator on the unit cell:

hf i ¼ 1juj

ZY

f ðxÞ dx: (4)

N is the membrane stress tensor and M is the flexure momenttensor:

Nab ¼D

sab

E; Mab ¼

Dx3sab

E: (5)

Eq. (3) can be interpreted as the macro-homogeneity conditionof Hill–Mandel which ensures that the macroscopic elastic strainenergy of a given representative volume element is equal to theintegral of the microscopic elastic strain energy within this volume.

Hence, the strain energy per unit area becomes

W ¼ 12hsðxÞ 3ðxÞi ¼ 1

2ðEAEþ 2EBcþ cDcÞ; (6)

where the homogenized elastic tensors (A, B, D) of the plate are asfollows:

N ¼ AEþ Bc and M ¼ BTEþ Dc; (7)

where the superscript T is the transposition operator.The elasticity problem (Eq. (2)) may be solved along the lines of

Suquet (1990) by introducing a homogeneous reference medium ofstiffness Lo and a polarization field s:8>>><>>>:

sðxÞ,V ¼ 0; sðxÞ ¼ Lo3ðxÞþ sðxÞ; 3ðxÞ ¼ Eþ x3cþeðvperðxÞÞ;eðvperðxÞÞ ¼ vperðxÞ5sV;

sðxÞ,e3 ¼ 0 on vY�;

vperðxÞperiodic on vYl; sðxÞ,n anti-periodic on vYl; ð8Þ

where the polarization field sðxÞ is given by:

sðxÞ ¼ dLðxÞ3ðxÞ with dLðxÞ ¼ LðxÞ � Lo: (9)

We note that the elasticity problem (Eq. (8)) with ðE; c; sÞ canbe split into two auxiliary problems:

� An auxiliary problem which is identical to the elasticityproblem (Eq. (8)) but in which s ¼ 0. It coincides with the setof equations (Eq. (2)) for L(x)¼ Lo and admits the fieldsðso; 3o; vper

o Þ as a solution.

� Another auxiliary problem identical to the elasticity problem(Eq. (8)) but in which E¼ 0 and c ¼ 0. This second auxiliaryproblem admits the fields ðsper; eðuperÞ; uperÞ as a solution.

As a result, the field’s solution of the elasticity problem (Eq. (8))with ðE;c; sÞ is the superposition of ðso; 3o; vper

o Þ withðsper; eðuperÞ; uperÞ.

2.1. Homogeneous solutions in the reference medium

This section is devoted to the first auxiliary problem introducedin the previous section. The fields ðsoðxÞ; 3oðxÞ; vper

o ðxÞÞ which aresolutions of the local elasticity problem (Eq. (2)) defined on thereference medium with stiffness Lo are derived from the followingsystem of equations:8>>><>>>:

soðxÞ,V¼0;soðxÞ¼Lo3oðxÞ; 3oðxÞ¼Eþx3cþeoðvpero ðxÞÞ;

eoðvpero ðxÞÞ¼vper

o ðxÞ5sV;

soðxÞ,e3¼0 on vY�;

vpero ðxÞperiodiconvYl; soðxÞ,n anti-periodiconvYl; ð10Þ

This problem has a trivial solution, ðsoðx3Þ; 3oðx3Þ;vpero ðx3ÞÞ, that

depends on x3 only. This trivial solution can be derived directly bysolving the ordinary differential equations in x3 for given boundaryconditions. To this aim, the stresses and strains are split into theirin-plane and out-of-plane components:8<:so

ðiÞ ¼�so

11; so22; so

12

�T; so

ðoÞ ¼�so

33; so23; so

13

�T;

3oðiÞ ¼ ð3

o11; 3o

22; 23o12Þ

T; 3oðoÞ ¼ ð3

o33; 23o

23; 23o13Þ

T;(11)

where the indices (i) and (o) indicate the in-plane and out-of-planecomponents, respectively. Similarly, the reference elasticity tensorLo is also split into four 3� 3 matrices: The in-plane componentsL(i)(i)

o , the coupling components L(i)(o)o and L(o)(i)

o , and the out-of-plane components L(o)(o)

o , L(i)(i)o and L(o)(o)

o are symmetrical whileL(i)(o)

o and L(o)(i)o verify (L(i)(o)

o )T¼ L(o)(i)o .

The constitutive equations thus become(soðiÞðx3Þ ¼ Lo

ðiÞðiÞ3oðiÞðx3Þ þ Lo

ðiÞðoÞ3oðoÞðx3Þ;

soðoÞðx3Þ ¼ Lo

ðoÞðiÞ3oðiÞðx3Þ þ Lo

ðoÞðoÞ3oðoÞðx3Þ:

(12)

One observes that the in-plane strains are equal to the macro-scopic strains, i.e., 3o

ðiÞðx3Þ ¼ EðiÞ þ x3cðiÞ, since the in-planecomponents of eo(vo

per) are zero. Moreover, from the balanceequations and the boundary conditions, one finds out that the out-of-plane stresses so

ðoÞðx3Þ are also zero. Hence, the out-of-planestrains can be expressed in terms of the in-plane strains:

3oðoÞðx3Þ ¼ �ðLo

ðoÞðoÞÞ�1Lo

ðoÞðiÞ3oðiÞðx3Þ: (13)

Substituting Eq. (13) into the first equation of Eq. (12), one linksthe in-plane stresses to the in-plane strains:

soðiÞðx3Þ ¼ Lo

ðsÞ3oðiÞðx3Þ; (14)

where

LoðsÞ ¼ Lo

ðiÞðiÞ � LoðiÞðoÞðL

oðoÞðoÞÞ

�1LoðoÞðiÞ ¼ ðL

o�1ðiÞðiÞÞ

�1 (15)

is the plane-stress elastic stiffness matrix of the reference medium.

2.2. Self-adjoint property of the G-operator for the plate

This section is devoted to the second auxiliary problem intro-duced in Section 2 which is solved by the G-operator for the plate.


This second auxiliary problem corresponds to Eq. (8) in which E¼ 0and c ¼ 0:8>><>>:

sperðxÞ,V ¼ 0; sperðxÞ ¼ LoeðuperðxÞÞ þ sðxÞ;eðuperðxÞÞ ¼ uperðxÞ5sV;sperðxÞ,e3 ¼ 0 on vY�;uperðxÞ periodic on vYl; sperðxÞ,n anti-periodic on vYl;

(16)

where the polarization tensor is given by Eq. (9). Eq. (16) is solvedby using a G-operator which produces the total strain e(x) in termsof s:

e ¼ �Gs (17)

The expression of the G-operator was obtained by Nguyen et al.(2008). Following the procedure used in this paper, the field uper(x)solution of Eq. (16) is split into two terms: uper(x)¼ up(x)þuh(x).up(x) is obtained with the standard periodicity conditions in the x3-direction and the complementary term uh(x) enables to recover thestress-free boundary conditions. Likewise, the total strain and theG-operator are expressed as e¼ epþ eh and G ¼ Gp þ Gh where ep

and Gp correspond to the standard periodic problem and eh and Ghto the complementary problem.

The boundary value problem (Eq. (16)) can be solved byexpanding the polarization field sðxÞ˛L2ðYÞ into three-dimensionalFourier series (see Eq. (71)). The periodic strain field ep(x) solutionof Eq. (16) is derived from:�

spðxÞ,V ¼ 0; spðxÞ ¼ LoepðxÞ þ sðxÞ; epðxÞ ¼ upðxÞ5sV;upðxÞ periodic on vY ; spðxÞ,n anti-periodic on vY:

(18)

ep(x) can be explicitly derived in the Fourier space (see Suquet(1990); Moulinec and Suquet (1994) for details). It is then used todefine the corresponding self-adjoint periodic Gp-operator definedby (see Appendix A for details):

epðxÞ ¼ � 1jY j

ZY

Gpðx � x0Þsðx0Þ dx0: (19)

In contrast, the strain field eh(x) and the related operator GhðxÞare derived by solving the following equation:8><>:

shðxÞ,V ¼ 0; shðxÞ ¼ LoehðxÞ; ehðxÞ ¼ uhðxÞ5sV;sh

j3ðxÞ ¼ �spj3ðxÞ on vY�;

uhðxÞ periodic on vYl; shðxÞ,n anti-periodic on vYl:

(20)

Eq. (20) was solved by Nguyen et al. (2008) by using Fouriertransforms along (x1, x2)-directions. eh and the correspondingoperator Gh are linked by:

ehðxÞ ¼ � 1jY j

ZY

Gh

�~x � ~x0; x3; x03

�sðx0Þ dx0; (21)

where the Gh-operator is periodic in the (x1, x2)-directions andð~x � ~x0; x3; x03Þ holds for ðx1 � x01; x2 � x02; x3; x03Þ (see Appendix Afor details).

Since G ¼ Gp þ Gh and since G is a self-adjoint operator, Gh isalso a self-adjoint operator. The self-adjoint character of G can bedirectly inferred from Eq. (16). To this aim, let us consider a kine-matically admissible field, eðxÞ˛U, and a statically admissible stressfield, sðxÞ˛S, where:

U ¼ feðxÞ=eðxÞ ¼ uperðxÞ5sV; uperðxÞ periodicon vYlg; (22)

S ¼n

sðxÞ= s : V ¼ 0; s : e3

¼ 0 on vY�;s : n antiperiodic on vYl

o(23)

For all fields ðs1; e1Þ and ðs2; e2Þ solutions of Eq. (16) withpolarization fields s1 or s2, respectively, s1 and s2 are staticallyadmissible and e1 and e2 are kinematically admissible. FromGreen’s theorem, they verify:

hs1e2iY ¼ 0 and hs2e1iY ¼ 0: (24)

Substituting the stress field, s ¼ Loeþ s, into Eq. (24) yields thefollowing:

hs1e2iY ¼ �e1Loe2

Y ; hs2e1iY ¼ �

e2Loe1

Y : (25)

From the symmetry of the elasticity tensor Lo, we haveCs1e2DY ¼ Cs2e1DY . The G-operator links strains to polarizationfields:

e1 ¼ �Gs1 and e2 ¼ �Gs2: (26)

Therefore,

hs1Gs2iY ¼ hs2Gs1iY ¼e1Loe2

Y cs1; s2; (27)

which proves that G is a self-adjoint operator. Setting s1 ¼ s2 in Eq.(27) also shows that G is positive. Combining Eqs. (27) with (26)yields: G ¼ GLoG. Finally, since Gp is a self-adjoint operator, Gh alsois a self-adjoint operator and verifies the following:

hs1eh2iY ¼ hs2eh

1iY or hs1Ghs2iY ¼ hs2Ghs1iY : (28)

2.3. Hashin–Shtrikman functional and variational principle

As shown in the previous subsection, the Green-operator for theplate is self-adjoint. It is therefore possible to introduce this Green-operator into the Hashin–Shtrikman variational principle (Hashinand Shtrikman (1962a, 1962b, 1967; Willis, 1977, 1981).

The effective properties are obtained from the mean elasticenergy denoted by Weff:

Weff ¼12hs3i ¼ 1

2hs3oi: (29)

The Hashin–Shtrikman principle can be written by using thefunctional WðsÞ defined by:

WðsÞ ¼ 12

3oLo3oþ 1

2

D2s3o � sdL�1s� sGs

E: (30)

When s is the polarization field solution of the equilibriumequation, the value of the functional is equal to the mean energyWeff. This functional can be split into two partsWðsÞ ¼ Wo þW1ðsÞ, where:

Wo ¼12

3oLo3o ¼ 1

2

Zt=2

�t=2

3oðiÞðx3ÞLo

ðsÞ3oðiÞðx3Þ dx3; (31)

and:

W1ðsÞ ¼12

D2s3o � sdL�1s� sGs

E: (32)

Eq. (31) shows that Wo can be explicitly calculated and does notdepend on s.


The variational principle associated to the functional may beexpressed as follows:�

Weff ¼ MaxsðWo þW1ðsÞÞ dL > 0;Weff ¼ MinsðWo þW1ðsÞÞ dL < 0;

(33)

Hence Eq. (33) can supply lower and upper bounds for theeffective elastic properties of the plate. The functional W1ðsÞwill beconsidered in more detail in the following sections. Eq. (33) showsalso that the lower and upper bounds depend on the properties ofthe reference medium. An optimization on the reference mediumwill thus yield optimal bounds.

3. First-order bound estimates for (x1, x2)-invariantpolarization fields

Different bounds can be obtained for the properties of plates,depending on the assumptions of the distribution of the hetero-geneities. In this section we introduce (x1, x2)-invariant polarizationfields (i.e., sðxÞ ¼ sðx3Þ) in the variational formulation. As it will beshown thereafter, the solution will be closely related to the prop-erties of stratified plates. Therefore, a laminated plate with elasticproperties that are (x1, x2)-independent (i.e., L(x)¼ L(x3)) as inFig. 2 is first considered. In such a case, the fields solution ofEq. (2) is invariant in the (x1, x2)-directions: ðsðxÞ; 3ðxÞ; vðxÞÞ ¼ðsðx3Þ; 3ðx3Þ; vðx3ÞÞ. Hence, as is the case with homogeneous plates,the solution can be obtained analytically and the overall elasticproperties of the plate are given by:

ðA; B; DÞ ¼Zt=2

�t=2

�1; x3; x2

3

�LðsÞðx3Þ dx3; (34)

where it is recalled that the matrices A, B, and D were defined in Eq.(7) and where L(s)(x3) is the plane-stress elasticity matrix at x3

which verifies Eq. (15).Moreover, for any L

0(x3)� L(x3) we have�

A0; B0; D0�� ðA; B; DÞ; (35)

in the sense of the corresponding strain energy.Let us consider the more general case L(x)¼ L(x1, x2, x3). In such

a case the functional W1 in Eq. (32) is given by:

W1ðsÞ ¼12

Zt=2

�t=2

h2sðx3Þ3oðx3Þ � sðx3ÞdL�1ðx3Þsðx3Þ

� sðx3ÞGsðx3Þi

dx3; (36)

where fðx3Þ is the (x1, x2)-average of f(x1, x2, x3):

fðx3Þ ¼1juj

Zfðx1; x2; x3Þ dx1 dx2: (37)

Fig. 2. Example of a stratified distribution of polarization tensor (homogeneous ineach layer) which is used to compute the bounds.

In the present case, the functional can be computed as fora material which is layered in the thickness of the plate because theintegrand in W1ðsÞ depends on x3 only. Let us therefore introducethe tensor L*(x3), related to a stratified plate, which is given by:

L*ðx3Þ ¼�

dL�1ðx3Þ��1þLo; (38)

The homogenized stiffnesses of the plate related to this tensorare given by (A*, B*, D*):

�A*; B*; D*

�¼

Zt=2

�t=2

�1; x3; x2

3

�L*ðsÞðx3Þ dx3: (39)

Eq. (33) yields8<: ðA; B; DÞ ��

A*; B*; D*�

cLo; dL > 0;

ðA; B; DÞ ��

A*; B*; D*�

cLo; dL < 0:(40)

The bounds for the homogenized elastic properties of the platevary with the reference medium and can therefore be optimized byan appropriate choice of the reference medium. To this aim, oneconsiders the following two limit problems: Lo tends to zero and Lo

tends to infinity.For the case Lo/N and dL< 0, with no loss of generality, one

can write Lo¼ bI where I is the unit matrix and b is a positivecoefficient which tends to infinity. One performs the followingtransformations:

dL�1ðx3Þ ¼�LðxÞ � Lo��1 ¼ �b�1

hI� b�1LðxÞ

i�1: (41)

By using the expansion:

ðI� XÞ�1¼ Iþ Xþ X2 þ/ cX; jjXjj < 1; (42)

Eq. (41) becomes

dL�1ðx3Þ ¼ �b�1hIþ b�1Lðx3Þ þ b�2L2ðx3Þ þ/

i: (43)

The high-order terms in Eq. (43) can be neglected when b/N.Hence, by making use of Eq. (42) again for ðdL�1ðx3ÞÞ�1, one obtains

L*ðx3Þ/Lðx3Þ when b/N: (44)

We prove next that the limit value obtained above, Lðx3Þ, is thebest bound. It means that this is the minimum value of L*(x3), i.e.,L*ðx3Þ � Lðx3Þ with dL< 0. This proof requires the followinginequality to be verified:�

dL�1ðx3Þ��1� Lðx3Þ � Lo ¼ dLðx3Þ: (45)

By substituting M¼�dL> 0 into Eq. (45), one recovers theinequality between the Voigt and Reuss bounds: M � ðM�1Þ�1.Therefore Eq. (45) is satisfied for any dL< 0 and the limit value Lðx3Þyields the optimal upper bound for the effective in-plane and out-of-plane elastic properties of the plate:

�A*; B*; D*

�þ¼

Zt=2

�t=2

�1; x3; x2

3

�LðsÞðx3Þ dx3 (46)

or

�A*; B*; D*

�þ¼

Zt=2

�t=2

�1; x3; x2

3

��L�1ðiÞðiÞ

��1ðx3Þ dx3: (47)


It means that this bound is computed as follows: in a first step,the mean value of the elasticity tensor for each plane parallel to themean plane of the plate is computed. This allows to define a strat-ified plate whose properties are these mean properties. The stiff-nesses of the plates are bounded by the stiffness of a homogeneousplate whose properties are obtained by the Voigt bound. It isworthwhile to notice that the value of L0 leading to the bound is notthe elasticity tensor used for the classical derivation of Hashin–Shtrikman bounds, which is finite (as in the next subsection).

The other bound can be obtained as follows: for the case Lo/0and dL> 0, from Eq. (38) follows L*ðx3Þ/ðL�1Þ�1. We prove nextthat ðL�1Þ�1 is the maximum value of L*(x3). To this aim, theinequality L*ðx3Þ � ðL�1Þ�1 with dL> 0 has to be verified.

Since for any G� 0, ðG�1Þ�1 is concave, the following relationholds for any G1�0 and G2� 0 and l˛½0; 1�:�ðlG1 þ ð1� lÞG2Þ

�1��1� lðG�1

1 Þ�1þð1� lÞðG�1

2 Þ�1: (48)

Substituting G1¼ dL, G2¼ Lo and l¼ 1/2 in the above equationyields��

dL þ Lo��1��1��

dL�1��1þ�

Lo�1��1

: (49)

Therefore, the limit value, ðL�1Þ�1, is the maximum value ofL*(x3) and the optimal lower bound for the effective elastic prop-erties of the plate is

�A*; B*; D*

��¼

Zt=2

�t=2

1; x3; x2

3

��L�1

ðiÞðiÞ��1ðx3Þ dx3: (50)

The optimal bounds obtained from Eqs. (50) and (46) with theuse of the Hashin–Shtrikman functional recover the bounds whichwere obtained by Kolpakov (1999) by using the Castigliano func-tional with a statically admissible in-plane stress field dependingon x3 and the Lagrange functional with a kinematically admissibleout-of-plane displacement field depending on x3.

These ‘‘first-order’’ bounds, like the Voigt and Reuss bounds foran infinite medium, may be far from each other, especially if theplate is made of materials which exhibit a significant mechanicalcontrast. By focusing, however, on the polarizations in the phasesthose bounds can be improved. We will do so in the next section byanalyzing random heterogeneous plates.

4. Second-order bounds for random plates

A random distribution of the heterogeneities within the plate isnow considered and information on second-order correlationfunction of the elastic properties will be used. When the charac-teristic size of the structural material elements is large comparedwith that of the microstructure, the material can accurately betreated as locally homogeneous with spatially constant averageproperties. Many studies have been performed on such materialsand variational principles used to derive lower and upper boundsfor their overall properties (see e.g. Hashin and Shtrikman (1962a,b); Willis (1977, 1981)).

Moreover, for some materials the assumption of statisticalhomogeneity does not apply. Functionally graded materials (FGM)are examples of such materials. The continuous spatial variation ofproperties in those materials prevents the use of the methodsdeveloped for statistically uniform heterogeneous materials. Forexample, the work of Luciano and Willis (2004) applied the theorypresented in Drugan and Willis (1996) to derive nonlocal consti-tutive equations for functionally graded materials. Several other

micromechanical models have been proposed for the analysis of theoverall thermomechanical properties of FGM (see e.g. Reiter andDvorak (1997, 1998); Suresh and Mortensen (1998)).

This section aims at estimating bounds for the effective elasticproperties of random plates. It is based on the Hashin–Shtrikmanfunctional formulated in Section 2. We assume that the constituentmaterials are randomly distributed and statistically homogeneousalong directions which are parallel to the plane (x1, x2) of the plateand that the in-plane dimensions of the plate are large compared toits thickness.

4.1. n-Phase random plates

Let us now consider a composite with a random microstructuremade of n phases whose geometrical distribution is characterizedby a. Let j be the set of microstructures and p(a) the probabilitydensity of a in j. Thus, any property, f, of the material is a functionof a and its ensemble average is defined as:

Ef ðaÞ ¼Zj

f ðaÞpðaÞ da: (51)

Then, the probability Pr(x) of finding the phase r at the location xis

PrðxÞ ¼ EIrðx; aÞ ¼Zj

Irðx; aÞpðaÞ da; (52)

where Ir(x, a) is the indicator function of the region containing thephase r, Ir(x, a)¼ 1 when x ˛ Yr and Ir(x, a)¼ 0 otherwise. Likewise,the two-point probability Prs(x, x0) of finding simultaneously thephase r at the location x and the phase s at the location x0 is

Prsðx; x0Þ ¼ EðIrðx; aÞIsðx0; aÞÞ ¼Zj

Irðx; aÞIsðx0; aÞpðaÞ da:

(53)

If phase r (where r ¼ 1; 2; .; n) is homogeneous and hasa modulus Lr, the elasticity tensor L(x) in sample a and its ensembleaverage are

Lðx; aÞ ¼X

rLrIrðx; aÞ; ELðx; aÞ ¼

Xr

LrPrðxÞ: (54)

Furthermore, we assume that the materials are statisticallyhomogeneous along the (x1, x2)-directions. The characteristicproperties are thus insensitive to translations along thesedirections:

PrðxÞ ¼ Prðx3Þ; Prsðx; x0Þ ¼ Prs

�~x � ~x0; x3; x03

�: (55)

Moreover, the polarization field in each phase is assumed to benot dependent of x1 and x2, i.e., srðxÞ ¼ srðx3Þ. In the following, thepolarization field is assumed to be a linear combination of theindicator functions and can therefore be expressed as:

sðx; aÞ ¼X

rsrðx3ÞIrðx; aÞ: (56)

As already mentioned, a common practice is to use theG-operator for a periodic medium in the case of statisticallyhomogeneous random media, as soon as the period is large enoughcompared to the correlation length (see e.g. Sab and Nedjar (2005)).This process will be used thereafter.


The previous definitions are introduced to enable the analysis ofthe ensemble average of the functional (Eq. (30)). Since Wo does notdepend on s, the total energy WðsÞ and W1ðsÞ have the samestationary point. Averaging the functional (Eq. (32)) now yields:

EW1ðsÞ ¼12<2X

rsrðx3Þ3oðx3ÞPrðx3Þ

�X

rsrðx3ÞdL�1

r Prðx3Þsrðx3Þ

�X

r

Xs

srðx3Þ��Yj�1

ZY

Gp

�~x�~x0; x3�x03

�Prs

��

~x�~x0; x3; x03

�ss�x03�

dx0�X

r

Xs

srðx3Þ��Y j�1

�ZY

Gh

�~x�~x0; x3; x

03

�Prs

�~x�~x0; x3; x

03

�ss�x03�

dx0>:

(57)

where 3oðx3Þ is the total strain field estimated in the referencemedium. It is a deterministic function of x3.

−5 −4 −3 −2 −1 0 1 2 3 4 50.45

0.5

0.55

0.6

0.65

0.7

x1/2R

P2

2(r)

Fig. 3. The two-point probability function, P22(r), c1¼0.3.

4.2. Choice of the polarization tensor

We remind that the search for lower and upper bounds for theeffective elastic properties of the plate can be performed by esti-mating the stationary value of the functional (Eq. (57)). To this aim,one discretizes the plate along the x3 and x3

0 directions with N3

points each. The polarization field in the phase is approximated bya piecewise function:

srðx3Þ ¼X

msm

r Imðx3Þ; (58)

where Im is the indicator function of the finite intervals whoseunion yields the part of the x3-axis which is inside the plate.Therefore, the functional (Eq. (57)) can be rewritten as follows:

EW1xt

2N3

Xm

"Xr

2smr 3o

mPmr �

Xr

smr dL�1

r Pmr sm

r �Xm0

Xr

�X

ssm

r1

N3juj

Zu

�Gmm0

p

�~x0�þ Gmm0

h

�~x0��

Pmm0rs

�~x0�sm0

s d~x0

35;(59)

where Gmm0p ð~x0Þ ¼ Gpð~x0; x3ðmÞ � x03ðm0ÞÞ; Gmm0

h ð~x0Þ ¼ Ghð~x0; x3ðmÞ; x03ðm0ÞÞ, Pmm0

rs ð~x0Þ ¼ Prsð~x0; x3ðmÞ; x03ðm0ÞÞ, 3om ¼ 3oðx3ðmÞÞ,

Prm¼ Pr(x3(m)).

We notice that the integral expressions in Eq. (57) can becomputed in the Fourier space with the wave vectors (k1, k2) usingParseval’s theorem. For m; m0 ¼ 1; .; N3 points, Eq. (59) yields:

EW1 ¼t

2N3

�2TTF� TTMlT � TTMpT � TTMhT

�; (60)

where F, Ml, Mp and Mh are linear and quadratic operators of thefunctional (Eq. (59)) and where the vector T is made of the dis-cretized values of the polarization fields whose values are uniformin each discretized interval (see Eq. (58)):

T¼ðs11;.;s1

n;.;sN31 ;.;sN3

n Þwith smr ¼srðx3ðmÞÞ; (61)

where the polarization field in each phase is written under the formof a six-component vector (3D-problem): (s11, s22, s33, s23, s13, s12)T.

The vector F is computed from the one-point probability func-tion and from the strain field produced in the reference medium3oðx3Þ. 3oðx3Þ also is represented by a six-component vector (311

o , 322o ,

333o , 2323

o , 2313o , 2312

o )T. The symmetrical squared matrix Ml is deter-mined from the one-point probability function, the mechanicalproperties of the constituent materials and the mechanicalproperties of the reference medium. The squared matrices Mp andMh are computed from the operators Gp and Gh and from thetwo-point probability function Prs. Moreover, we notice that Mp

and Mh are symmetrical matrices, since Gp and Gh are self-adjoint operators which verify: Gijkl

p (x�x0)¼Gklijp (x0 � x) and

Ghijklð~x � ~x0; x3; x03Þ ¼ Gh

klijð~x0 � ~x; x03; x3Þ.The functional (Eq. (60)) is stationary when the following

stationary condition is verified:

Ts ¼�Ml þMp þMh

��1F: (62)

Hence, the stationary value of the average of the energy func-tional is obtained as:

EWs1 ¼

t2N3

TTs F: (63)

The above value, used with Eq. (31), allows the computation ofbounds for the effective elastic properties of the random plate.Those bounds vary with the choice of the reference medium and anappropriate choice of the elasticity tensor Lo enables the optimi-zation of those bounds. The present theory was derived in thegeneral case of random plates whose constituent materials arestatistically homogeneous in the (x1, x2)-directions. To validate thesteps described above, the simple case of an isotropic material isconsidered next. The estimated bounds for the homogenized elasticstiffnesses of a Love–Kirchhoff plate will enable to study the effectsof the size of the heterogeneities on the values obtained for thebounds.

5. Application

5.1. Description of the microstructure

For performing an application of the results obtained previously,a Love–Kirchhoff plate made of two isotropic material phases isnow considered and the different probability functions must be

−5 −4 −3 −2 −1 0 1 2 3 4 50.1

0.15

0.2

0.25

0.3

0.35

0.4

x1/2R

P2

2 (r)

Fig. 4. The two-point probability function, P22(r), c1¼0.64.

0 0.1 0.2 0.3 0.4 0.5 0.61

2

3

4

5

6

7

8

9

10

11

c1

No

rm

alized

b

en

din

g stiffn

ess

UHSUpper boundLHSLower boundα=0.7

Fig. 6. Comparison of the normalized bounds D1111/D1111 (E1) for the bending stiffnessin the case of small inclusions as a function of the concentration of material 1, E2/E1¼10, t/2R¼ 20, l1/2R¼ 20.


defined. The one-point probability function Pr(x) is the volumefraction cr and the two-point probability function is invariant bytranslation and rotation. Expressions derived analytically as well asnumerically to estimate the two-point probability function exist fornumerous random material distributions (see e.g. Torquato (2001)).In the following, we consider a microstructure with a matrix con-taining a random distribution of non-overlapping spherical parti-cles, for which the two-point probability function is given asfollows (see e.g. Drugan and Willis (1996)):

Prsðjx � x0jÞ ¼ crcs þ crðdrs � csÞ hðjx � x0jÞ; (64)

where cr, cs are the volume fractions of the phases r and s, respec-tively. This probability function is expressed in terms of the relativedistance between two sampled points, namely r¼ jx� x0j. Thefunction h(r) can be expressed under an exponential form as(Drugan, 2003):

hðrÞ ¼ e�

ra; (65)

0 0.1 0.2 0.3 0.4 0.5 0.61

2

3

4

5

6

7

8

9

10

11

c1

No

rm

alized

m

em

bran

e stiffn

ess

UHSUpper boundLHSLower boundα=0.7

Fig. 5. Comparison of the normalized bounds A1111/A1111 (A1) for the membrane stiff-ness in the case of small inclusions as a function of the concentration of material 1, E2/E1¼10, t/2R¼ 20, l1/2R¼ 20.

where the coefficient a is a function of the radius R and of thevolume fraction c1 of the spherical inclusions (phase 1):

a2 ¼R25F1c1H1þ2G1

�H1ð2þc1Þ�c1

�~c1=c1

�2=3

9�H1þ~c21

��10G1ð1�c1ÞH1

;

(66)

where

~c1 ¼ c1 �1

16c2

1; F1 ¼32

~c21

1� 0:7117~c1 � 0:114~c2

1

��

1� ~c1

�4 : (67)

G1 ¼ 12F1

�1� ~c1

�2

~c1

�2þ ~c1

�; H1 ¼ 1þ 2~c1: (68)

0 0.1 0.2 0.3 0.4 0.5 0.60.98

1

1.02

1.04

1.06

1.08

1.1

1.12

1.14

No

rm

alized

m

em

bran

e stiffn

ess, A

/A

UH

S

c1

l1 /2R=2.5

l1/2R=5.0

l1/2R=10

Fig. 7. Normalized upper bound of the membrane stiffness (normalization withrespect to the computation at l*¼ 20), E2/E1¼10, t/2R¼ 5.

Table 1Relative error on the normalized stiffnesses of a random plate for different values ofthe period length l1, E1/E2¼1/10, t/2R¼ 5, c1¼0.64.

l1/2R r (A1111) (%) r (D1111) (%)

Lower bound Upper bound Lower bound Upper bound

5 1.82% 4.17% 2.51% 5.26%7.5 0.54% 1.32% 0.98% 1.82%10 0.12% 0.34% 0.38% 0.35%15 �0.06% �0.15% �0.01% �0.05%20 0 0 0 0


This expression is consistent with the numerical resultsobtained for example by Torquato and Stell (1985) who used thePercus and Yevick (1958) model of the statistical problem withrandom hard sphere distributions corrected by Verlet and Weis(1972) (see Drugan (2003) for details).

Figs. 3 and 4 display the profiles of the two-point probabilityfunction along the in-plane direction in which the in-plane coor-dinate is normalized with respect to the diameter of the sphericalinclusions. Moreover, since the inclusions cannot overlap, theirconcentration has an upper bound. This value is considered to be0.64 (Torquato, 2001). One observes that the probability functionsdecrease and are then stationary. Fig. 4 shows that the minimumin-plane correlation length is 10R, above which a plateau appears.

5.2. Numerical results

To apply numerically the theory described in the previoussection, the computations are performed at the limit of a planedeformation problem in planes containing x2-direction. A dis-cretization of the space in the (x1, x3)-directions is necessary. Thecoordinates of the discretized points are defined as follows:

xj ¼ jljNj; with j ¼ �

Nj

2; .; 0; .;

Nj

2�1 ðj ¼ 1; 3Þ: (69)

This discretization also is appropriate to use the FastFourier Transform (FFT), hence Nj¼ 2p, p ˛ Nþ. The computa-tion of the FFT corresponds to the wave-number order

2 3 4 50.955

0.96

0.965

0.97

0.975

0.98

0.985

0.99

0.995

1

No

rm

alized

m

em

bran

e stiffn

ess

Fig. 8. Normalized upper and lower bounds of the membrane stiffness as a function of thebounds for infinite media), E2/E1¼10, l1/2R¼ 10.

nj ¼ ½0; .; ðNj=2Þ�1;�ðNj=2Þ; .; �1�. The components of thewave vectors are then defined as follows: kj¼ 2pnj/lj.

The homogenized elastic stiffnesses of the plate are obtainedby an appropriate choice of the macroscopic strains ðE; cÞ and bythe computation of the extreme values of the energy functionalWo and EW1

s. This approximation will be compared with thestiffness properties of homogeneous plates whose elasticitytensors are computed from the classical Hashin–Shtrikmanbounds in infinite medium (UHS for the upper bound and LHS forthe lower bound).

Figs. 5 and 6 display comparisons of the bounds for thehomogenized membrane and bending stiffnesses obtained fromthe theory developed in this work with the stiffnesses derived fromthe classic solution of Hashin–Shtrikman in a case where theheterogeneities are small compared to the thickness of the plate.The values are expressed in terms of the concentration of phase 1and normalized with the stiffnesses produced in a homogeneousmedium with properties of phase 1. A discretization withN1¼N3¼ 27 which ensures the convergence of the solution wasused for the computations. For comparison, values between thebounds are introduced, which are given by: Lo¼ aL1þ (1 – a)L2

with a¼ 0.7.It must be noticed that our optimized reference media for lower

and upper bounds are identical to those computed for Hashin–Shtrikman bounds (Hashin and Shtrikman, 1962a, b), i.e., Lo¼ L1

and Lo¼ L2 (L1< L2), respectively. Figs. 5 and 6 allow to check thatthe stiffnesses computed from the classical Hashin–Shtrikmanbounds for infinite media are recovered by our computation, whichis the waited result in the case where the plate thickness is largecompared to the size of heterogeneities (here t/2R¼ 20).

After this first verification, a second study is performed toanalyze the effect of the size of the period used in the computationof the Green tensor, when it is compared to the radius of hetero-geneities (related to the correlation length). Indeed, the Greentensor for the plate which is known only for a periodic mediummust be used for a length l1 of the period which is large enoughcompared to the thickness of the plate and to the size of hetero-geneities. The convergence of the stiffnesses of the plate for anincreasing length of the period is now considered.

In a first step, the upper bound for the membrane stiffness isobtained for different values of l1. Different period lengths were

6 7 8 9 10t/2R

normalized stiffness at higher bound

normalized stiffness at lower bound

normalized thickness (normalization with respect to the stiffness obtained from H–S

2 3 4 5 6 7 8 9 100.92

0.93

0.94

0.95

0.96

0.97

0.98

0.99

1

t/2R

No

rm

alized

b

en

din

g stiffn

ess

normalized stiffness at higher boundnormalized stiffness at lower bound

Fig. 9. Normalized upper and lower bounds of the curvature stiffness as a function of the normalized stiffness (normalization with respect to the stiffness obtained from H–Sbounds for infinite media), E2/E1¼10, l1/2R¼ 10.


considered and it was found that it is not necessary to increase l1 atvalues higher than l1¼20R. The stiffnesses were obtained fordifferent values of the normalized in-plane period defined byl*¼ l1/2R. They are reported in Fig. 7 where the upper bound for themembrane stiffness normalized with respect to its value for thecomputation at l*¼ 20, A¼ A1111(l*)/A1111(l*¼ 20) is given asa function of the concentration of material 1 for different values ofl*. It shows that the convergence is slower at higher concentrationof inclusions.

To precise the level of convergence, a ‘‘relative error’’ r is definedfrom the normalized stiffnesses M(l*) (M being A111 or D1111) by therelationship:

Relative error r ð%Þ ¼M�

l*��M

�l* ¼ 20

�M�

l* ¼ 20� � 100%: (70)

The results obtained for this relative error r are presented inTable 1 for the case c1¼0.64 (for which the convergence is theslowest) showing that a relative error inferior to 0.5% is producedfor l*¼ l1/2R¼ 10, which will be adopted in the following to studythe effect of the size of the heterogeneities with respect to the ratiot/2R. These results confirm that the Green tensor for the periodicplate can be used to study random plates as soon as the size of theperiod is large enough compared to the size of the heterogeneities.

Now, it is possible to reach the main objective of this work,which is to compute the Hashin–Shtrikman bounds for the plateand to compare these bounds with those obtained by using theelastic properties of plates computed from the Hashin–Shtrikmanbounds of the elastic moduli related to an infinite medium. Thebounds for the plate are compared for different values of thevariable t/2R, which represents the ratio between the thickness ofthe plate and the size of heterogeneities. As shown previously, thestiffnesses of the plate computed from the Hashin–Shtrikmanbounds in infinite media are recovered from the boundscomputed for the plate when the thickness is large enoughcompared to the inclusion size. A comparison is therefore madeby computing the normalized bound B*

p ¼ Bp=BHS where Bp is thebound for A1111 or D1111 computed from previous considerationsand BHS is the bound obtained by introducing into the plate

properties of the effective moduli produced by Hashin–Shtrikmanbounds for infinite media.

These normalized bounds are computed in terms of t/2R andreported in Figs. 8 and 9.

It can be seen that for large values of the thickness, thenormalized stiffness becomes constant, and that the relativedifference between Bp and BHS becomes small (typically less than1%) when the ratio t=2R is large enough. The residual differenceis of the order of the acceptation threshold chosen for the rela-tive error induced by a limited period length. All results showthat an estimation of the bounds for the plate from the HSbound for infinite media overestimates plate bounds, for bothupper and lower bounds. This result confirms again that thebounds for the effective properties of the plate can be computedfrom the usual Hashin–Shtrikman bounds as soon as the thick-ness is large enough, compared to the size of the heterogeneities.The difference between both bounds increases when the ratiot=2R decreases below a threshold around 6. The ratio betweenbounds reaches smaller values for the upper bound and for thebending stiffness, but the difference between plate bounds andplate properties computed from Hashin–Shtrikman bounds forinfinite media still remains inferior to 10% in all cases. This resultis compatible with the results obtained by other authors inrelation to the minimum size of the ‘‘Representative VolumeElement’’ which is often found around five to six times theheterogeneity size.

6. Conclusions

This paper has presented the results obtained from theHashin–Shtrikman variational principle for heterogeneous plateswhen using within the variational principle a G-operator withstress-free boundary conditions. Two applications were consid-ered. First, a polarization which does not depend on the positionwithin a plane parallel to the plate is introduced within thevariational principle. The bounds which are thus obtained coin-cide with ‘‘first-order’’ refined estimates of the Voigt and Reussbounds proposed by Kolpakov (1999) for heterogeneous plates.The optimized polarization fields which are used for reaching the


bounds are related to null and infinite elastic moduli of thereference medium introduced within the variational principle.

In the second application, the Hashin–Shtrikman variationalformulation is used for random heterogeneous plates. Theassumption of a statistically uniform distribution of the materialsalong the in-plane directions of the plate yielded a simplifiedexpression of the functional in terms of the in-plane invariantpolarizations.

A simple example with isotropic materials was considered andallowed to predict the effect of the size of the heterogeneities on‘‘second-order’’ bounds of the effective properties of plates. Thestatistical distribution of the heterogeneities is entirely defined byits two-point probability function as introduced by Torquato (2001)and does not require Monte Carlo computations or ensembleaveragings. The ‘‘second-order’’ bounds for the homogenizedelastic properties of the plate were compared with those obtainedby computing plate properties from classical Hashin–Shtrikman(HS) bounds for elastic effective properties (Hashin and Shtrikman,1962a, b, 1965). A natural estimation of bounds for elastic(membrane and stiffness) plate stiffnesses is indeed to compute theHS bounds for elastic properties in an infinite medium and tointroduce these bounding elastic properties into plate stiffnesses.Our results show that such a procedure leads to correct results fora ratio ‘‘thickness’’/‘‘heterogeneity size’’ which is greater thana threshold of around 6. For such a ratio being lower, the estimationof bounds by such a procedure overestimates the bounds for plateproperties.

Appendix A. Solution of the auxiliary problem andcalculation of the related operators

The Fourier method is used to solve the problem (Eq. (16)) forwhich any periodic function g(x) can be expanded into the Fourierseries,

gðxÞ ¼X

k

bgðkÞ eikx; bgðkÞ ¼ 1jYj

ZY

gðxÞ e�ikx dx; (71)

where k¼ (k1, k2, k3) or ~k ¼ ðk1; k2Þ used in the sequel denotes thediscrete wave vectors arranged along a discrete lattice havinga period 2p/li (l3¼ t) in the direction xi.

To solve the boundary value problem (Eq. (16)), the solutionfields are split into two components: a standard periodic compo-nent obtained from Eq. (18) and a complementary componentderived from Eq. (20) to recover boundary conditions. For theperiodic solution field, the resolution of Eq. (18) can be performedusing the Fourier transforms (Suquet, 1990; Moulinec and Suquet,1994). In fact, after Fourier transform of the equilibrium equation,of the constitutive equation and of the compatibility relation, fol-lowed by elimination of bsp

ijðkÞ between the equations, the periodicstrains of Eq. (18) can then be obtained in the Fourier space bymeans of the periodic Gp-operator associated to the homogeneousreference medium with stiffness Lo,

bepðkÞ ¼ �bGpðkÞ : bsðkÞ cks0; bepð0Þ ¼ 0: (72)

In real space, the periodic strains ep(x) are given by Eq. (19)under a convolution product formulated according to definition(71). The Fourier components of the Gp-operator are explicitly givenin Mura (1991) for different types of anisotropy for the homoge-neous medium. For an isotropic material with Lame coefficients (l,m), it takes the form:

bGpijkl ¼

1

4m��kj2

�dikkjkl þ djkkikl þ dilkjkk þ djlkikk

�

� lþ m

mðlþ 2mÞkikjkkkl��kj4 : (73)

It should be noted that Gp is a self-adjoint operator and due toparity of the tensor bGpðkÞ, the following property is verified:Gpðx � x0Þ ¼ Gpðx0 � xÞ.

Furthermore, the resolution of the complementary problem (Eq.(20)) is performed using the Fourier transformations along theperiodic directions (x1, x2) such that the solution field can bedefined as follows:�

uhðxÞ;ehðxÞ;shðxÞ�¼X

~k

~uh�

~k;x3

�;~eh�

~k;x3

�;~sh�

~k;x3

��eikaxa :

(74)

Using the Fourier transforms along (x1, x2), the strain field isrelated to the displacement field by: ~ehð~k;x3Þ¼ ~uhð~k;x3Þ5sV0

where the operator V0 is defined by V0 ¼½ik1 ik2 v3�T. The operator v

indicates the partial differentiation with respect to the coordinatesubscript that follows. Supposing that the material is isotropic, theequilibrium equation shðxÞ,V¼0 becomes ~shð~k;x3Þ,V0 ¼0 whichleads to

½ðlþ mÞV05sV0 þ mðV0,V0Þ1�~uh�

~k; x3

�¼ 0; (75)

where,

V05sV0 ¼

264 �k21 �k1k2 ik1v3

�k1k2 �k22 ik2v3

ik1v3 ik2v3 v233

375; V0,V0

¼ v233 �

�k2

1 þ k22

�: (76)

The solution of Eq. (75) is

~uh�

~k; x3

�¼�

aþ þ x3bþ�

esx3 þ�

a� þ x3b��

e�sx3 ; (77)

where:

a� ¼

8><>:a�1a�2

�1is

�a�1 k1 þ a�2 k2 � ilþ3m

lþmb�3�9>=>;; b� ¼

8><>:�ik1

s b�3�ik2

s b�3b�3

9>=>;;s ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2

1 þ k22

q: (78)

The six complex coefficients (a1�, a2

�, b3�) are obtained from the

six boundary conditions defined in Eq. (20) which is rewritten asfollows:

~shj3

~k; �t

2

�¼ �

Xk3

bspj3ðkÞe

�ik3t2: (79)

More precisely, the 3 boundary conditions at the top face(x3¼ t/2) and the 3 boundary conditions at the bottom face(x3¼�t/2) of the unit cell enable to construct a system of 6equations for determining 6 coefficients of the vectorx ¼ ðaþ1 ; a�1 ; aþ2 ; a�2 ; bþ3 ; b�3 Þ,

Kx ¼ q; (80)

where

K ¼

26666666666664

ms

�s2 þ k2

1

�est

2 �ms

�s2 þ k2

1

�e�st

2msk1k2est

2 �msk1k2e�st

2 2msik1g�est

2 2msik1gþe�st

2

ms

�s2 þ k2

1

�e�st

2 �ms

�s2 þ k2

1

�est

2msk1k2e�st

2 �msk1k2est

2 �2msik1gþe�st

2 �2msik1g�est

2

msk1k2est

2 �msk1k2e�st

2ms

�s2 þ k2

2

�est

2 �ms

�s2 þ k2

2

�e�st

2 2msik2g�est

2 2msik2gþe�st

2

msk1k2e�st

2 �msk1k2est

2ms

�s2 þ k2

2

�e�st

2 �ms

�s2 þ k2

2

�est

2 �2msik2gþe�st

2 �2msik2g�est

2

�2mik1est2 �2mik1e�st

2 �2mik2est2 �2mik2e�st

2 2mh�est2 �2mhþe�st

2

�2mik1e�st2 �2mik1est

2 �2mik2e�st2 �2mik2est

2 �2mhþe�st2 2mh�est

2

37777777777775; (81)


with g� ¼ sðt=2Þ � m=ðlþ mÞ, h� ¼ sðt=2Þ � ðlþ 2mÞ=ðlþ mÞ, andthe components of the vector q¼ (q1

þ, q1–, q2

þ, q2–, q3

þ, q3–) being

defined in Eq. (84). The closed-form solution of Eq. (80) can befound in Nguyen et al. (2008).

Moreover, the strain ~ehð~k; x3Þ for j~kjs0 (s s 0) is obtained fromthe compatibility equation. It can be expressed by using a matrixPð~k; x3Þ and the vector xð~kÞ as of form:

~eh�

~k; x3

�¼ P

�~k; x3

�x�

~k�¼ P

�~k; x3

�K�1

�~k�

q�

~k�; (82)

P ¼

2666666666664

ik1esx3 ik1e�sx3 0 0 �k21s x3esx3

k21s x3e�sx3

0 0 ik2esx3 ik2e�sx3 �k22s x3esx3

k22s x3e�sx3

�ik1esx3 �ik1e�sx3 �ik2esx3 �ik2e�sx3

�x3s� 2m

lþm

�esx3 �

�x3sþ 2m

lþm

�e�sx3

k1k22s esx3 �k1k2

2s e�sx3s2þk2

22s esx3 �s2þk2

22s e�sx3 ik2

s

�x3s� m

lþm

�esx3 ik2

s

�x3sþ m

lþm

�e�sx3

s2þk21

2s esx3 �s2þk21

2s e�sx3 k1k22s esx3 �k1k2

2s e�sx3 ik1s

�x3s� m

lþm

�esx3 ik1

s

�x3sþ m

lþm

�e�sx3

ik22 esx3 ik2

2 e�sx3 ik12 esx3 ik1

2 e�sx3 �k1k2s x3esx3 k1k2

s x3e�sx3

3777777777775; (83)

whereand the components of the vector qð~kÞ ¼ ðqþ1 ; q�1 ; qþ2 ; q�2 ; qþ3 ; q�3 Þare derived from the boundary conditions (Eq. (79)). They areexpressed in terms of the periodic stresses, bspðkÞ, in the Fourierspace as:

q�j ¼ �Xk3

e�ik3t2 bsp

j3ðkÞ ¼ �Xk3

e�ik3t2 bdj3mnðkÞbsmnðkÞ; (84)

where bdðkÞ ¼ I� LobGpðkÞ.It may be noticed that bdðkÞ is calculated with k s 0 and it is

reduced to the unity matrix I for k¼ 0. Therefore, the relation (84)allows to construct a matrix S(k) constituted from the producte�ik3ðt=2Þbdj3mnðkÞ according to the corresponding vector qð~kÞleading to:

q�

~k�¼ �

Xk3

SðkÞbsðkÞ: (85)

Substituting the above expression into Eq. (82) and taking intoaccount the Fourier transformation in x03 of the polarization tensorlead to

~eh�

~k; x3

�¼ �1

t

Zt=2

�t=2

~Gh

�~k; x3; x03

�~s�

~k; x03�

dx03; (86)

where ~Ghð~k; x3; x03Þ at the given points (x3, x03) is defined by

~Gh

�~k; x3; x03

�¼ P

�~k; x3

�K�1

�~k�X

k3

SðkÞe�ik3x03 : (87)

It should be noted that this expression is calculated with ~ks0,its value at ~k ¼ 0 is also considered from the corresponding strains.Moreover, taking into account the definition (71), the comple-mentary strains can be written in the real space as in Eq. (21).

For j~kj ¼ 0, the membrane strains ~ehab are null due to the peri-

odicity on vYl of the displacement field, i.e. ~ehabð~k ¼ 0; x3Þ ¼ 0. The

calculation of ~ehj3ð~k ¼ 0; x3Þ is also performed by using the

boundary condition (Eq. (79)) and the differential equation system

(Eq. (75)). This gives the expressions of the out-of-plane strain fieldas follows,

~eh33

�~k ¼ 0; x3

�¼ � 1

lþ 2m

Xk3

bsp33

�~k ¼ 0; k3

�eik3

t2; (88)

~eha3

�~k ¼ 0; x3

�¼ � 1

2m

Xk3

bspa3

�~k ¼ 0; k3

�eik3

t2: (89)

For estimating the values of bspj3 at ~k ¼ 0, it is noted that the out-

of-plane stresses, bspj3, are null for the case of ~k ¼ 0 and k3 s 0. As

a consequence, the calculation of the out-of-plane complementarystrains in Eqs. (88) and (89) is reduced to the case of k¼ 0 wherebsp

j3ðk ¼ 0Þ ¼ bsj3ðk ¼ 0Þ:

~eh33

�~k ¼ 0; x3

�¼ �

bs33ðk ¼ 0Þlþ 2m

;

~eha3

�~k ¼ 0; x3

�¼ �

bsa3ðk ¼ 0Þ2m

: (90)

It allows to determine the non-null components of the ~Gh at~k ¼ 0 as follows: ~Gh

3333ð~k ¼ 0; x3Þ ¼ 1=ðlþ 2mÞ and~Gh

a3a3ð~k ¼ 0; x3Þ ¼ 1=2m.

References

Bonnet, G., 2007. Effective properties of elastic periodic composite media withfibers. J. Mech. Phys. Solids 55 (5), 881–899.


Bourgeois, S., Debordes, O., Patou, P., 1998. Homogeneisation et plasticite des pla-ques minces. Rev. Eur. Elem. Finis. 7, 39–54.

Caillerie, D.,1984. Thin elastic and periodic plates. Math. Methods Appl. Sci. 6,159–191.Cecchi, A., Sab, K., 2002a. A multi-parameter homogenization study for modelling

elastic masonry. Eur. J. Mech. A. Solids 21, 249–268.Cecchi, A., Sab, K., 2002b. Out of plane model for heterogenous periodic materials:

the case of masonry. Eur. J. Mech. A. Solids 21, 715–746.Cecchi, A., Sab, K., 2004. A comparison between a 3D discrete model and two

homogenized plate models for periodic elastic brickword. Int. J. Solids Struct. 41,2259–2276.

Dallot, J., Sab, K., 2008a. Limit analysis of multi-layered plates. Part I: the homog-enized Love–Kirchhoff model. J. Mech. Phys. Solids 56, 561–580.

Dallot, J., Sab, K., 2008b. Limit analysis of multi-layered plates. Part II: shear effects.J. Mech. Phys. Solids 56, 581–612.

Drugan, W.J., 2003. Two exact micromechanics-based nonlocal constitutive equationsfor random linear elastic composite materials. J. Mech. Phys. Solids 51, 1745–1772.

Drugan, W.J., Willis, J.R., 1996. A micromechanics-based nonlocal constitutiveequation and estimates of representative volume element size for elasticcomposites. J. Mech. Phys. Solids 44, 497–524.

Duvaut, G., Metellus, A.M., 1976. Homogeneisation d’une plaque mince en flexiondes structures periodique et symetrique. C.R. Acad. Sci. Ser. A. 283, 947–950.

Hashin, Z., Shtrikman, S., 1962a. On some variational principles in anisotropic andnonhomogeneous elasticity. J. Mech. Phys. Solids 10, 335–342.

Hashin, Z., Shtrikman, S., 1962b. A variational approach to the theory of elasticbehavior of polycrystals. J. Mech. Phys. Solids 10, 343–352.

Hashin, Z., Shtrikman, S., 1965. On elastic behavior of fibre reinforced materials ofarbitrary transverse phase geometry. J. Mech. Phys. Solids 13, 119–134.

Hashin, Z., Shtrikman, S., 1967. Variational principles of elasticity in terms of thepolarization. Int. J. Eng. Sci. 5, 213–223.

Hill, R., 1952. The elastic behavior of a crystalline aggregate. Proc. Phys. Soc. 65,349–354.

Kohn, R.V., Vogelius, M., 1984. A new model for thin plates with rapidly varyingthickness. Int. J. Solids Struct. 20, 333–350.

Kolpakov, A.G., 1998. Variational principles for stiffnesses of a non-homogeneousbeams. J. Mech. Phys. Solids 46, 1039–1053.

Kolpakov, A.G., 1999. Variational principles for stiffnesses of a non-homogeneousplate. J. Mech. Phys. Solids 47, 2075–2092.

Kolpakov, A.G., Sheremet, I.G., 1999. The stiffnesses of non-homogeneous plates. J.Appl. Math. Mech. 63, 633–640.

Lewinski, T., Telega, J., 2000. Plates, Laminates and Shells: Asymptotic Analysis andHomogenization. World Scientific Publishing.

Luciano, R., Willis, J.R., 2004. Non-local constitutive equations for functionallygraded materials. Mech. Mater. 36, 1195–1206.

Moulinec, H., Suquet, P., 1994. A fast numerical method for computing the linearand nonlinear properties of composites. C.R. Acad. Sci. 318, 1417–1423.

Moulinec, H., Suquet, P., 1998. A numerical method for computing the overallresponse of nonlinear composites with complex microstructure. Comput.Methods Appl. Mech. Eng. 157, 69–94.

Mura, T., 1991. Micromechanics of Deflects in Solids. Kluwer Academic Publishers.Nguyen, T.K., Sab, K., Bonnet, G., 2008. Green’s operator for a periodic medium with

traction-free boundary conditions and computation of the effective propertiesof thin plates. Int. J. Solids Struct. 45 (25–26), 6518–6534.

Percus, J.K., Yevick, G.J., 1958. Analysis of classical statistical mechanics by means ofcollective coordinates. Phys. Rev. 110, 1–13.

Reiter, T., Dvorak, G.J., 1997. Micromechanical models for graded composite mate-rials. J. Mech. Phys. Solids 45, 1281–1302.

Reiter, T., Dvorak, G.J., 1998. Micromechanical models for graded composite mate-rials: II. Thermomechanical loading. J. Mech. Phys. Solids 46, 1655–1673.

Sab, K., 2003. Yield design of thin periodic plates by a homogenization techniqueand an application to masonry wall. Comptes Rendus Mecanique 331,641–646.

Sab, K., Nedjar, B., 2005. Periodization of random media and representative volumeelement size for linear composites. Comptes Rendus Mecanique 333, 187–195.

Suquet, P., 1990. Une methode simplifiee pour le calcul des proprietes elastiques demateriaux heterogenes a structure periodique. C.R. Acad. Sci. 311, 769–774.

Suresh, S., Mortensen, A., 1998. Fundamentals of Functionally Graded Materials.Maney Material, Science Publ.

Torquato, S., 2001. Random Heterogeneous Materials– Microstructure and Macro-scopic Properties. Springer, New York.

Torquato, S., Stell, G., 1985. Microstructure of two-phase random media. V. The n-point matrix probability functions for impenetrable spheres. J. Chem. Phys. 82,980–987.

Verlet, L., Weis, J.J., 1972. Equilibrium theory of simple liquids. Phys. Rev. A 5,939–952.

Willis, J.R., 1977. Bounds and self-consistent estimates for the overall properties ofanisotropic composites. J. Mech. Phys. Solids 25, 185–202.

Willis, J.R., 1981. Variational and related methods for the overall properties ofcomposites. Adv. Appl. Mech. 21, 1–78.

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Bounds for the effective properties of heterogeneous plates