Functional methods and effective potentials for non-linear composites

Functional methods and effective potentials for non-linear composites

Journal of the Mechanics and Physics of Solids 48 (2000) 429±459 www.elsevier.com/locate/jmps Functional methods and e€ective potentials for non-line...
Download PDF
273KB Sizes 6 Downloads 22 Views
Report

Journal of the Mechanics and Physics of Solids 48 (2000) 429±459 www.elsevier.com/locate/jmps

Functional methods and e€ective potentials for non-linear composites Yves-Patrick Pellegrini a,*, Marc BartheÂleÂmy a, Gilles Perrin b Service de Physique de la MatieÁre CondenseÂe, Commissariat aÁ l'EÂnergie Atomique, BP12, 91680 BruyeÁres-le-ChaÃtel, France b Institut Franc° ais du PeÂtrole 1 et 4, avenue du Bois-PreÂau, 92852 Rueil-Malmaison Cedex, France a

Received 24 August 1998; received in revised form 26 May 1999

Abstract A formulation of variational principles in terms of functional integrals is proposed for any type of local plastic potentials. The minimization problem is reduced to the computation of a path integral. This integral can be used as a starting point for di€erent approximations. As a ®rst application, it is shown how to compute to second order the weak-disorder perturbative expansion of the e€ective potentials in random composite. The three-dimensional results of Suquet and Ponte-CastanÄeda (Suquet, P., Ponte-CastanÄeda, P., 1993. Small-contrast perturbation expansions for the e€ective properties of nonlinear composites. C. R. Acad. Sci. (Paris) Ser. II 317, 1515±1522) for the plastic dissipation potential with uniform applied tractions are retrieved and extended to any space dimension, taking correlations into account. In addition, the viscoplastic potential is also computed for uniform strain rates. # 2000 Elsevier Science Ltd. All rights reserved. Keywords: Variational principles; Inhomogeneous materials; Microstructure; Viscoplastic material; Constitutive behavior, principles

1. Introduction In the last decade, studies of disordered non-linear composites have forced new homogenization methods in Mechanics. Most such methods rely on rigorous

* Corresponding author. 0022-5096/00/$ - see front matter # 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 2 - 5 0 9 6 ( 9 9 ) 0 0 0 4 0 - X

430

Y.-P. Pellegrini et al. / J. Mech. Phys. Solids 48 (2000) 429±459

variational principles, from which exact optimal bounds can be deduced (PonteCastanÄeda and Suquet, 1998). These principles are either standard minimum energy principles, or more re®ned approaches using a linear reference material, be it homogeneous (Hashin and Shtrikman, 1962a, 1962b; Talbot and Willis, 1985), or even heterogeneous (Ponte-CastanÄeda, 1991; Suquet, 1993). In the study of non-linear dielectric media as well, few exact results were previously available, save for the case where the non-linearity in the response was treated as a perturbation of the linear behavior (Stroud and Hui, 1988). In a pioneering work, Blumenfeld and Bergman (1989, 1991) computed the e€ective dielectric permittivity of a strongly non-linear random medium to second order in a weak-disorder expansion; cf. also Bergman and Lee (1998). The perturbative expansion was carried out by use of the Green function associated to the linear problem. An analogous approach was subsequently undertaken by Suquet and Ponte-CastanÄeda (1993) on strongly non-linear elastic composites. These results were then further extended to composites with inclusions of complex shapes (see, e.g. Ponte-CastanÄeda and Suquet, 1998 for a review). Perturbative results are important as testing-benches for self-consistent e€ectivemedium theories. Linear self-consistent formulae (Budianski, 1965) are known to give qualitatively correct predictions even for a high contrast between the constituents of the random medium. They account for, e.g. the existence of a `rigidity threshold', i.e. a volumetric concentration of voids above which a porous linear elastic material loses its rigidity (Sahimi, 1998). In addition to the ful®lment of exact non-linear bounds (Gilormini, 1995), criteria of acceptability for a nonlinear self-consistent formula include the recovery of second-order perturbative results, and also constraints on the critical behavior near the rigidity threshold (Levy and Bergman, 1994) where ®eld ¯uctuations are enhanced. In Mechanics, a recent self-consistent e€ective-medium theory (Nebozhyn and Ponte-CastanÄeda, 1997) reproduces exact perturbative results to second order, but its critical ¯uctuations have not been studied yet. An equivalent theory has been proposed in electrostatics (Ponte-CastanÄeda and Kailasam, 1997). These result were derived from a variational principle. As a complement to classical means, functional methods originating from ®eld theory (for a book on functional methods see, e.g., Kleinert, 1995) and from the statistical physics of disordered systems (MeÂzard et al., 1987) were recently harnessed to contribute to e€ective-medium studies (BartheÂleÂmy and Orland, 1993; Parcollet et al., 1996). They have not yet spread in the mechanics community. Their major interest is that they not only provide a convenient workhorse for implementing variational principles under various constraints, but that they also lend themselves to approximations resulting in self-consistent e€ective-medium formulae (Budianski, 1965; Parcollet et al., 1996), some of them being of a new type (Pellegrini and BartheÂleÂmy, 1999). With such methods, the problem of the minimization of the potential becomes equivalent to the computation of a functional integral, as shown below. This single integral, which may be written down for all types of local potentials, can then be approximated using standard tools (saddle point methods, perturbative expansions, etc.).

Y.-P. Pellegrini et al. / J. Mech. Phys. Solids 48 (2000) 429±459

431

The purpose of this article is to introduce in detail the speci®c mathematical apparatus needed to apply functional methods to variational principles in mechanics. As a ®rst application, we re-derive the result of Suquet and PonteCastanÄeda (1993) for the weak-disorder expansion of a strongly non-linear viscoplastic composite assumed to be of the Norton type. We extend it to any space dimension, and show how to take correlations into account. We adopt an approach (BartheÂleÂmy and Orland, 1998) which utilizes the socalled `replica method' (Edwards and Anderson, 1975) developed in the framework of spin glass theory (MeÂzard et al., 1987). The replica method enables one to average over the disorder in a non-perturbative way, but is by no means compulsory in order to exploit the functional formulation. It is, however, a most natural route towards extensions to self-consistent formulae. Speci®c applications to self-consistent formulae for linear or non-linear materials (including the EMT) will be presented elsewhere. The paper is self-contained, and is organized as follows. Section 2 introduces the variational formulations of the problem and explains how to cast them under a functional form, either starting from the local viscoplastic potential (expressed in terms of the stress tensor) or from the plastic dissipation potential (expressed in terms of the strain rate tensor). We show that minimizing a potential amounts to computing the statistical average of the logarithm of some suitably de®ned partition function. The computation is carried out with the replica method, explained in the text. In Section 3, we specialize to the perturbative expansion, beginning with the plastic dissipation potential (Section 3.1.1). The results are then applied to the Norton law (Section 3.1.2). We retrieve the result of Suquet and Ponte-CastanÄeda (1993), generalized to any space dimension. The perturbative calculation of the viscoplastic potential, slightly more involved, is presented next, in Section 3.2. In both cases, the disorder is assumed to be site-disorder; the local potentials are statistically uncorrelated from point to point, for simpli®cation purposes. This restriction is overcome in Section 4, where spatial correlations (not necessarily isotropic) are introduced. Before concluding in Section 6, we comment on the range of applicability of the method (Section 5). Technical details are left to appendices. Appendix A is a brief reminder of the use of fourth-rank tensors in mechanics, and de®nes various fundamental fourthrank tensors employed hereafter. Various algebras and sub-algebras encountered in the paper are also de®ned in this appendix, which ought be read before the reader goes through our calculations. The determinant (det), trace (tr) and inversion (ÿ1) operators will always be indexed by a label indicating in which algebra or sub-algebra of operators they act. Useful formulae for restricted Gaussian integrals over vectors or second-rank tensors, which we could not ®nd in the literature, are derived in Appendix B. Inversion and determinant formulae for matrices in the replica space are given in Appendix C. Evaluations of functional integrals are relegated to Appendix D. The Legendre transform between the perturbative expansions of the viscoplastic and of the plastic dissipation potential is examined in Appendix E. Before going on, we state our notational conventions: unless otherwise

432

Y.-P. Pellegrini et al. / J. Mech. Phys. Solids 48 (2000) 429±459

indicated, scalars are denoted by regular typefaces (e.g. a or A ); vectors are denoted by bold typefaces (a); tensors of rank two by sans-serif typefaces (a) and tensors of rank four by capital Blackboard typefaces (A). Tensor indices are denoted by roman letters (e.g. Aij ), whereas replica indices are Greek letters. The Einstein summation convention on repeated indices is used, except when the indices are underlined …Aii ˆ trA, whereas Aii denotes the ith diagonal element of A). Dyads (i.e. tensor products of vectors) are denoted omitting the tensor product operator: A ˆ ab is the dyad with components Aij ˆ ai bj : 2. The problem and its formulation with replicas 2.1. Variational principles and e€ective potentials We adopt here the presentation of Ponte-CastanÄeda and Suquet (1998). Let s …x† be the local stress (tensor) ®eld, and d(x) be the local strain rate (tensor) ®eld deriving from the local velocity (vector) ®eld v(x). The material domain, of volume V, is denoted by O: Then, the equilibrium equations on s read, for x 2 O (the upper index t denotes the transpose): r  s ˆ 0,

…1†

s ˆ ts:

…2†

In addition, dˆ

 1 rv ‡ t…rv† : 2

…3†

The local constitutive relations are expressed by means of either the viscoplastic potential cx or the plastic dissipation potential fx , as: sˆ

@fx …d †, @d

dˆ

@cx s †: …s @s s

…4†

The subscript x indicates that the potentials may vary from point to point in the material, according to the local material properties. Both local potentials are usually convex functions of the ®elds, and are linked by the (Legendre) duality relation   s† : fx …d † ˆ max s :d ÿ cx …s s

…5†

Classically, two types of boundary conditions are considered: 1. For boundary conditions of uniform traction (ut) on the boundary @O of the material, the heterogeneous material is shown to be macroscopically described by a pair of e€ective potentials C and F such that

Y.-P. Pellegrini et al. / J. Mech. Phys. Solids 48 (2000) 429±459

ÿ  Cut S^ ˆ

 s=

cx …s s †, min r:s s ˆ 0, s ˆ t s s  n ˆ S  n on @O

Fut …D † ˆ min fx …d †:

433

…6†

…7†

v=dˆD

(the overline denotes a volume average on O, and n is the outward normal to O). 2. For boundary conditions of uniform strain rate (us) D on the boundary, the e€ective potentials are: ÿ  Cus S^ ˆ

 s=

Fus …D † ˆ

cx …s s †, min r:s s ˆ 0, s ˆ t s s ˆ S^ fx …d †:

min

v vˆDx on @ O

…8†

…9†

For both types of boundary conditions, the e€ective potentials F and C are shown (Suquet, 1987; Willis, 1989) to be (Legendre) dual functions such that C…S † ‡ F…D † ˆ S:D, D ˆ

@ C…S † @F…D † , Sˆ : @S @D

…10†

In both cases S ˆ s and D ˆ d: Depending on the boundary conditions considered, the latter equalities are either de®nitions, or theorems (Hill, 1963). Therefore, only one equation in each pair (6) and (7) or (8) and (9) is sucient to characterize the e€ective homogenized material. The problem considered in this article consists in computing the e€ective potentials for disordered materials. Eqs. (6)±(9) refer to one sample (one particular realization of the disorder). A key assumption is that the e€ective potentials are self-averaging, i.e. that they do not depend on the sample at hand in the so-called `thermodynamic limit', where the size of the sample goes to in®nity. In other words, the material contains in the various regions of its bulk all the possible realizations of the disorder. Therefore, rather than carrying out the calculation of the e€ective potentials for one particular con®guration of the disorder (which is impossible), we arrive at the correct result by considering statistical averages. For instance, in the limit V 41 we can as well write, instead of Eq. (6): Cut …S † ˆ hCut …S †i ˆ

*



s=

+ cx …s s† : min r  s ˆ 0,s s ˆ ts s  n ˆ S  n on @O

…11†

434

Y.-P. Pellegrini et al. / J. Mech. Phys. Solids 48 (2000) 429±459

Note that the statistical averaging operator hi and the in®mum operator do not commute (inverting the operators would lead to the incorrect result that the e€ective potential is the average of the local potentials). 2.2. Functional representations The minimization problem is reformulated by extending to functionals the following straightforward result for scalar functions: …b 1 …12† miny2‰a,bŠ f…y † ˆ ÿ lim log dx eÿbf…x† : b41b a Note that this formula holds even if the minimum is met at several points in the interval. Each in®mum (6), (7), (8) or (9) is obtained for the state of the ®elds s or d which minimize the potentials, among all the possible states ful®lling the equilibrium and boundary constraints. In statistical mechanics, this problem is analogous to that of ®nding the lowest energy state E0 of some hamiltonian H[s ] functionally depending on some ®eld s…x†: In this case, E0 is given by 1 log Z, b41b

E0 ˆ ÿ lim

…13†

Z ˆ tr eÿbH ,

…14†

where the trace operator tr denotes the sum over all the allowed states of the system, i.e. the con®gurations of s. Indeed, the partition function Z is obviously dominated by exp…ÿbE0 † in the limit b 4 1, where E0  H‰s0 Š and s0 is the con®guration of s that minimizes H …b is the reciprocal of the temperature in statistical physics). The means of summing over all the possible states of a continuous ®eld subjected to constraints is to use functional integrals (also termed path integrals ) with suitably de®ned measures. With such a formalism, it is more convenient to use the variational formulations (7) and (8) where the boundary conditions are implemented through volume averages. We, therefore, write (with the additional statistical averages): 1. For uniform tractions (d is computed in terms of v by Eq. (3)), … 1 … † Fut D ˆ ÿ lim hlog Zut i, Zut ˆ Dvd…d ÿ D †eÿbfx …d † , b41b where d is the Dirac distribution and Y dv…x†, Dv ˆ x2O

…15†

…16†

Y.-P. Pellegrini et al. / J. Mech. Phys. Solids 48 (2000) 429±459

 Y  d… d ÿ D † ˆ d dij ÿ Dij :

435

…17†

iRj

For brevity, the constrained measure in Eq. (15) will be denoted by Dut v ˆ Dvd…d ÿ D †:

…18†

2. For uniform strain rates,

… 1 hlog Zus i, Zus ˆ Ds s d…r  s †d…sÅ ÿ S †eÿbcx …ss † : b41b

Cus …S † ˆ ÿ lim

…19†

with Ds s ˆ

Y

ds s …x†,

…20†

x2O

ds s …x† ˆ 2

d…dÿ1 †=4

"Y #" # ÿ  Y d sij …x† ÿ sji …x† dsij …x† , i
Y

d… r  s † ˆ

d…@ k ski …x††,

…21†

i,j

…22†

i,x2O

d…sÅ ÿ S † ˆ

Y ÿ  d s ij ÿ Sij :

…23†

iRj

The integration measure ds s …x† allows one to integrate over all symmetric stress tensors (cf. Appendix B). The normalization factor 2d…dÿ1†=4 in ds s …x†, where d is the space dimension, is of no importance but has been included for consistency with Appendix B. We set: Dus s ˆ Ds s d…r  s †d…sÅ ÿ S †:

…24†

2.3. The replica method Hereafter, we focus on Zut, but the following considerations apply mutatis mutandis to Zus. The statistical average of log Zut is a complicated quantity to evaluate directly. The replica method consists in writing  1ÿ r hZ ut i ÿ 1 , r40 r

hlog Zut i ˆ lim

…25†

and in computing hZ rut i with r an integer (in the replica literature, the number of replicas, r, is usually denoted by n. We do not use the latter in order to avoid confusions with the Norton exponent). At the end of the calculation, we take the

436

Y.-P. Pellegrini et al. / J. Mech. Phys. Solids 48 (2000) 429±459

limit r4 0, assuming the analytical continuation is legitimate, which we cannot prove in general. Therefore, introducing r replicas va of the ®eld v, the partition function

Z rut ˆ

r … Y

…

Dut va eÿbfx d

g†



aˆ1

r X ÿb fx …dg † …Y r Dut va e gˆ1 ˆ

…26†

aˆ1

is that of a system made of r replicas of the initial system. The greek labels (which exclude b, this symbol being devoted to the reciprocal of the `temperature') are replica labels running from 1 to r. The statistical average, therefore, is + * r X g† … ÿb fx d …Y r Dut va e gˆ1 : …27† hZ rut i ˆ aˆ1

For simpli®cation purposes, we shall limit ourselves to site-disordered materials where the local potentials are statistically uncorrelated from point to point (we explain in Section 4 how to overcome this restriction). By de®nition of site disorder, we have for any function F the property that  ÿ  ÿ ÿ ÿ   s…x†† F cy …s …y †† i ˆ hF cx …s s…x†† ihF cy …s …y †† i if x 6ˆ y: …28† hF cx …s For this type of disorder, the average in Eq. „ (27) isPreadily carried out. Replacing the volume integral by a Riemann sum dx4 v x , where v denotes the unit `in®nitesimal' volume element of the theory (here the size of v also is the statistical correlation radius within which the potential is constant), one obtains + * r X g† … ÿ…vb=V† f d …Y x r x,gˆ1 Dut va e hZ rut i ˆ aˆ1

r X + * …dg † ÿ…vb=V† f …Y x r Y gˆ1 Dut va e ˆ x

aˆ1

* ˆ

r X

…Y r Y ÿ…vb=V† gˆ1 Dut va e aˆ1

This ®nally yields:

x

+ fx …dg † :

…29†

Y.-P. Pellegrini et al. / J. Mech. Phys. Solids 48 (2000) 429±459

2

hZ rut i ˆ

*

r X ÿ…vb=V† fx …dg † 7 6… 5 4 dx gˆ1 log e Dut va exp , v aˆ1

…Y r

437

+3

…30†

where we reintroduce the integral notation in the exponent. This is the usual shorthand notation used in functional integral methods and has to be understood as the limit of a Riemann sum. This expression, completed by Eqs. (25) and (15), constitutes an exact functional representation of the variational problem (7) for the particular type of disorder under study, and is valid for any form of fx …d †: In other words, we recast a minimization problem under a computational form. Computing this integral exactly is equivalent to solving the minimization problem. We note that before averaging, the replicas are independent of one another, but are coupled after the average is carried out. Here appears an important feature of the replica method: the use of replicated ®elds allows one to average over disorder by transforming the disordered problem into a non-disordered one, but with couplings (if there were no couplings, averaging the logarithm would be a trivial matter). We are now left with a complicated ®eld-theory problem without disorder, consisting of r coupled ®elds, and we have to resort to approximations. In Section 3, we present a perturbative calculation of this integral. An important simpli®cation arises from the fact that hZ rut i always shows up in the form hZ rut i ˆ …Cbc †r f …r, b†, where f…0, x† ˆ 1 and is a non-homogeneous function of x, C is a b-independent constant, and c is an exponent linked to d, the space dimension. Therefore,   ÿ c 1  1 1ÿ r 1 hZ ut i ÿ 1 ˆ ÿ log Cb ‡ log f…r, b† ‡ O…r† , …31† ÿ br b r and the multiplicative factor …Cbc †r does not contribute to the limit b 4 1: Hereafter, such multiplicative factors will, therefore, systematically be dropped out of the calculations. This will be indicated by the proportionality symbol A appearing instead of the equal sign in equations.

3. Perturbative calculation We detail in this section the perturbative calculation for F ®rst, since it is simpler than that for C: The latter is examined next, once the reader has become acquainted with the method. In the following, we assume that the d-dimensional material is rigid-viscoplastic, described by the dual (in the sense of Eq. (5)) Norton potentials …1RnR1, m ˆ 1=n). ( fx …d † ˆ ym d m‡1 eq =…m ‡ 1 † , cx …s …32† s † ˆ on sn‡1 eq =…n ‡ 1 †, tr2 …d † ˆ 0

438

Y.-P. Pellegrini et al. / J. Mech. Phys. Solids 48 (2000) 429±459

with _ 0 =sn0 : ym ˆ s0 =_em 0 , on ˆ e The equivalent Mises norms seq and deq are:   1=2 1=2 d ÿ 1 … 0 2† d tr2 d tr2 …s 0 2 † , seq ˆ : deq ˆ d dÿ1

…33†

…34†

These are the d-dimensional counterparts of the usual three-dimensional de®nitions (for d ˆ 3, we recover the factors 2/3 and 3/2). The deviatoric part of the stress or strain rate tensor has been denoted by a prime: a 0 ˆ a ÿ tr2 …a†I=d, where a ˆ s or d. The additional incompressibility constraint tr2 …d † ˆ r  v ˆ 0, which characterize incompressible materials, can as well be accounted for by adding a term …K=2†tr2 …d †2 to fx …d†, and by letting the constant K 4 1 at the end of the calculation. Instead, we shall implement it directly by multiplying the measure Dut v by a factor d…r  v†: Both ways actually amount to the same. The constitutive parameters s0 and e_ 0 describe the material and are random functions of the position variable x, uncorrelated from point to point, according to our site-disorder hypothesis. The exponent n is taken to be constant in the material, for simpli®cation purposes (but this is not a limitation of the method).

3.1. The plastic dissipation potential 3.1.1. General framework Integral (30) is an exact expression for the e€ective potential. Its perturbation expansion is considered hereafter. The measure Dut va contains the term d…d ÿ D†: This means that the main contribution to this integral is coming from ®elds around D. Indeed, if we suppose that only the constant ®eld d ˆ D (i.e. v ˆ D  x such that tr2 …D† ˆ 0† contributes to the integral, we ®nd the trivial lowest-order result Fut …D † ˆ hfx …D †i:

…35†

We shall now seek the ®rst corrective term, and compute Eq. (30) by integrating over ®elds v such that d di€ers little from D. The perturbation expansion is obtained with Laplace's method for asymptotic expansions of integrals (Bender and Orszag, 1978). We therefore write dˆD‡e

…36†

and expand the logarithm in Eq. (30) in powers of e. The linear and quadratic terms are then kept inside the main exponential, and the rest is further expanded in e. We start from the expansion:

Y.-P. Pellegrini et al. / J. Mech. Phys. Solids 48 (2000) 429±459

1 00 fx …D ‡ e † ˆ fx …D † ‡ W ij0 eji ‡ W ij,kl eji elk ‡ O…e3 †, 2

439

…37†

with W ij0 …x†

@ fx ˆ @ dij …x†

00 W ij,kl …x†

, dˆD

@ 2 fx ˆ @dij …x†@dkl …x†

:

…38†

dˆD

With a change of integration variables such that v now denotes the correction to D  x, the expansion of hZ rut i takes the form (taking into account the incompressibility constraint): …Y   Dut va d…r  va † 1 ‡ O…e3 † hZ rut i ˆ heÿrb…v=V†fx …D † iV=v a

"  exp ÿ bhW 0 ie :

X a

ea ÿ

1bX 2 V ag

…

#

…39†

dx ea …x†:Mag :eg …x† :

The functional measure now is Dut v ˆ Dvd…e†, and e ˆ ‰rv ‡ t …rv†Š=2: Besides, we have introduced the matrix Mag ˆ M1 dag ÿ b…v=V †M2 , with M1 ˆ hW 00 ie

…40†

M2 ˆ hW 0 W 0 ie ÿ hW 0 ie hW 0 ie :

…41†

The notation hie stands for the weighted average hA…x†ie 

hA…x† eÿrb…v=V†fx …D † i ˆ hA…x†i ‡ O…r†: heÿrb…v=V†fx …D † i

…42†

Only the leading term in r is needed in Eq. (42) (a handwaving argument for this is that the sums over replicas in Eq. (39) are at least proportional to r; the contribution is null for r ˆ 0: They therefore already provide the linear terms in r required for evaluating Eq. (25)), and we shall therefore in practice identify hie with hi: Hereafter, square matrices in the replica space are denoted by a double overbar. The identity in the replica space is I ag ˆ dag , and we furthermore de®ne the matrix U ag ˆ 1 for all a, g, so that v M ˆ M1 I ÿ b M2 U: V

…43†

The second-order term in the weak-disorder expansion of hZ rut i is provided by the Gaussian integral (39) only, neglecting the corrections O…e3 † and higher. As in centered scalar Gaussian integrals, the term proportional to e3 vanishes upon integration, and the next non-zero correction in fact is an O…e4 †: The aim of the paper being to explain the functional method, we shall ignore such higher-order

440

Y.-P. Pellegrini et al. / J. Mech. Phys. Solids 48 (2000) 429±459

terms hereafter but they could as well be computed by pursuing Laplace's expansion scheme. The details of the integration over v with measure Dut v are left to Appendix D. Taking into account that e is expressed in terms of the gradient of v involves a change from v(x) to its Fourier components v(k), one ®nds hZ rut iAheÿrb…v=V†fx …D † iV=v

Y

(

k6ˆ0

h i )ÿ1=2 ÿ  det Qkà  kà  M  kà  Qkà , 2‰?kà Š, rep

à now calculate the determinant. De®ning where kà ˆ k=k and Qkà ˆ I ÿ kà k:We ÿ  M1 ˆ Qkà  kà  M1  kà  Qkà , M2 ˆ Qkà  …k  M2  k†  Qkà , we obtain, with the help of formulae (C2) and (C3) in Appendix C: h i ÿ  det Qkà  kà  M  kà  Qkà 2‰?kà Š, rep "  # ÿ12‰?kà Š v r ‡ O…r2 †: ˆ 1 ‡ log det …M1 † ÿ b tr2 M2  M1 V 2‰?kà Š

…44†

…45† …46†

…47†

Carrying out as in Eq. (25) the limit r4 0 in Eq. (44) it follows that       ÿ12‰?kà Š 1 1vX log…b† 1 … † tr2 M2  M1 ‡o ‡o : ÿ hlog Zut i ˆ hfx D i ÿ b 2 V k6ˆ0 b b …48† The sum over k 6ˆ 0 . . . (divided by V ) can now be replaced by a d-dimensional integral (divided by …2p†d † over all the Fourier modes k in the continuum limit, since the integrand is not singular at k ˆ 0: The ®nite size of the elementary cells of volume v is accounted for by an upper cut-o€ L in this integral. It is chosen such that … dk ˆ 1: …49† v d k
Y.-P. Pellegrini et al. / J. Mech. Phys. Solids 48 (2000) 429±459

441

This result is the second-order perturbation expansion valid for any potential fx …d † with the constraint tr2 d ˆ 0:

3.1.2. Application to the Norton law We evaluate the previous result (50) for the Norton potential (32). We write …51† ym ˆ hym i ‡ dym : p  ^  D† ^ ˆ tr4 …D ^ D† ^ ˆ 1, Eqs. (38), (40) and ^ ˆ D 0 = tr2 …D 0 2 †, so that tr2 …D De®ning D (41) entail: ^ D, ^ M1 ˆ m 1 J ‡ m 2 D

…52†

^ D, ^ M2 ˆ m 3 D

…53†

where m1 ˆ

dÿ1 hym iDmÿ1 eq , d

m2 ˆ …m ÿ 1 †m1 , m3 ˆ

d ÿ 1 2 2m hdym iDeq : d

…54† …55† …56†

Setting ÿ 2 ^  kj ^ 2  kà ÿ kà  D ^  kà , à 2 ˆ kà  D Dkà ˆ jQkà  D ^  k=jQ ^ à à and introducing the unit vector uà ˆ Qkà  D kà  D  kj, we obtain    m1 ÿ m1 à Qkà ÿ uà uà ‡ m2 Dkà ‡ uà u, M1 ˆ 2 2 à M2 ˆ m3 Dkà uà u:

…57†

…58† …59†

à are a pair of complementary orthogonal projectors in Since Qk ÿ uà uà and uà u, à (where Q à plays the role of the identity), the inverse of M1 , its product L…2‰?kŠ† k with M2 and the ®nal trace in Eq. (50) are readily obtained, resulting in: … dOkà m3 Dkà : …60† Fut …D † ˆ hfx …D †i ÿ Sd m1 ‡ 2m2 Dkà Now replacing m1, m2 and m3 by their values (54)±(56) ®nally yields

442

Y.-P. Pellegrini et al. / J. Mech. Phys. Solids 48 (2000) 429±459

Fut …D † ˆ

ym eff …D † m‡1 D , m ‡ 1 eq "

hdy2m i ym eff …D † ˆ hym i 1 ÿ hym i2

…61† …

dOkà …m ‡ 1 †Dkà hdy3m i ‡O Sd 1 ‡ 2…m ÿ 1 †Dkà hym i3

!# :

…62†

This result reduces to that of Suquet and Ponte-CastanÄeda (1993) for a space dimension d ˆ 3: 3.2. The viscoplastic potential We undertake a similar calculation for the viscoplastic potential. It can be checked (cf. Appendix E) that the resulting perturbative expressions (76) and (77) is linked by a Legendre transform to Eqs. (61) and (62), the one for the dissipation potential, though di€erent boundary conditions are considered. This is a consequence of our working in the thermodynamic limit where the volume of the system V 4 1 (in the calculations, this shows up essentially in that the Fourier sums are carried out with k > 0, rather than with k > …2p†=L, where L would be the typical system size). As noticed by Hill and Mandel in elasticity, the homogenized sti€ness tensor obtained with uniform strains on @ O, and the homogenized compliances tensor obtained with uniform stresses on @ O are inverse of one another up to terms of order O……x=L†3 † in three dimensions, where x is the typical size of the heterogeneities (Suquet, 1987). They, therefore, become exact inverses in the thermodynamic limit. The same e€ect is at play here. The viscoplastic potential can, thus, directly be obtained from the dissipation potential as done in Appendix E. However, we prefer to compute it from scratch because di€erences with the previous section show up, and it may be useful to see what these are for future purposes; indeed, as soon as one leaves exact perturbative expansions and deals with self-consistent estimates, one approximation scheme worked out separately for dual potentials quite often generates two estimates which are not duals of one another (a notable exception is the self-dual EMT theory). Treating in a symmetrical way the two potentials is then a more sensible thing to do than working with only one. 3.2.1. General framework We now have to compute

hZ rus i

2

*

r X

+3 g†

ÿb…v=V† cx …s …Y 6 … r 41 a gˆ1 dx log e ˆ Dut s exp v aˆ1

7 5

:

…63†

The perturbative expansion of the e€ective potential is carried out around the constant solution s ˆ S, writing s ˆ S ‡ s: The expansion of cx …S ‡ s† has the same form as Eq. (37). We now denote by W 0 and W the ®rst and second

Y.-P. Pellegrini et al. / J. Mech. Phys. Solids 48 (2000) 429±459

443

derivatives of cx …s s† evaluated at s ˆ S, respectively. Using s as integration variables, the expansion of hZ rus i is: …Y   D s s a 1 ‡ O … s3 † hZ rus iAheÿrb…v=V†cx …S † iV=v "

a

X v 1 1X sa ÿ b  exp ÿ b hW 0 ie : V 2 V ag a

…

# a

ag

…64†

g

dxs …x†:M :s …x† ,

where Ds s ˆ Ds sd…r  s†d…s† and Mag is de®ned as in Eqs. (40), (41) and (43). Once again, the details of the integration over s with measure Ds s are left to Appendix D. The implementation of the constraint r  s ˆ 0 involves a change from s…x† to its Fourier components s…k†: The result is: 8   9ÿ1=2   < = ÿ  Y ÿ1 0 4s , rep à à M det  k  M  k  Q : det Q hZ rus iAheÿrb…v=V†cx …S † iV=v kà kà ; : 4s 0 , rep 2‰?kà Š, rep k6ˆ0 …65† The determinants are once again straightforwardly obtained, and the limits r 40 and b 41 carried out as in the previous section. Setting now   ÿ1 …66† M1 ˆ Qkà  kà  M1 4s 0  kà  Qkà ,   ÿ1 ÿ1 M2 ˆ Qkà  kà  M1 4s 0 :M2 :M1 4s 0  kà  Qkà ,

…67†

the second-order perturbation expansion for the viscoplastic potential ®nally reads:   …    ÿ1 à 1 dOkà ÿ1 tr4 M2 :M1 4s 0 ÿ tr2 M2  M1 2‰?k Š : …68† Cus …S † ˆ hcx …S †i ÿ 2 Sd

3.2.2. Application to the Norton law p Writing on ˆ hon i ‡ don , and S^ ˆ S 0 = tr2 …S 0 2 †, we have ÿ  M1 ˆ m1 J ‡ m2 S^ S^ ˆ m1 J ÿ S^ S^ ‡ …m1 ‡ m2 †S^ S^ ^ M2 ˆ m3 S^ S,

…69† …70†

where m1 ˆ

d hon iSnÿ1 eq , dÿ1

…71†

444

Y.-P. Pellegrini et al. / J. Mech. Phys. Solids 48 (2000) 429±459

m2 ˆ …n ÿ 1 †m1 m3 ˆ

…72†

d hdo2n iS2n eq : dÿ1

…73†

Noting that J ÿ S^ S^ and S^ S^ are complementary orthogonal projectors in L…4s 0 †, ÿ1 ÿ1 ÿ1 ÿ1 we compute M1 4s 0 , M1 4s 0 :M2 and M1 4s 0 :M2 :M1 4s 0 : Now, introducing the unit à ^ à à uà and vector uà ˆ Qkà  S^  k=jQ kà  S  kj, we express M1 and M2 in terms of Qkà ÿ u uà uà and compute the rest. With h ÿ 2 i à 2 ˆ 1 ÿ 2 kà  S^ 2  kà ÿ kà  S^  kà , …74† S à ˆ 1 ÿ 2jQ à  S^  kj k

k

we arrive at Cus …S † ˆ hcx …S †i ÿ

1 2

…

dOkà m3 Skà : Sd m1 ‡ m2 SkÃ

…75†

Finally replacing m1, m2, m3 with their values (71)±(73) yields the desired expression for the e€ective Norton viscoplastic potential: Cus …S † ˆ

on eff …S † n‡1 S , n ‡ 1 eq "

1 hdo2n i on eff …S † ˆ hon i 1 ÿ 2 hon i2

…76† …

dOkà …n ‡ 1 †Skà hdo3n i ‡O Sd 1 ‡ …n ÿ 1 †Skà hon i3

!# :

…77†

4. Other types of disorder We brie¯y discuss here how to cope with non site-disordered models, with possibly anisotropic correlations. It suces to write the average of the exponential in Eq. (27) simply as 2 * + +3 * r r X X g g ÿb fx …d † ÿb f x …d † 5 4 e

gˆ1

ˆ exp log e

gˆ1

…78†

and to expand the logarithm as in Section 3.1.1, only keeping the quadratic terms in the outer exponential. Eq. (39) is unchanged, save for a slightly di€erent (but equivalent) form for the ®rst averaged prefactor, and for a modi®cation in the quadratic term in the main exponential. This term becomes … 1bX …79† dx dyea …x†:Mag …xjy†:ea …y †, ÿ 2 V a,g

Y.-P. Pellegrini et al. / J. Mech. Phys. Solids 48 (2000) 429±459

445

where Mag …xjy† ˆ hW 00 idag d…x ÿ y † ÿ

 b 0 hW …x†W 0 …y †i ÿ hW 0 …x†ihW 0 …y †i : V

…80†

This requires modi®cations in the calculations, but does not change their principle. One merely has to deal with summations over another pair of indices, namely the coordinates x and y. Now the determinants, e.g. in Eq. (44), become functional determinants and must simultaneously be evaluated as determinants of operators in the Fourier space, in addition to being operators on ®nite-dimensional vector spaces. However, this is not a serious problem in the perturbative calculation considered here since only traces are required in the ®nal second-order results. In the case of translation-invariant disorder, the kernel M ag …xjy† is diagonal in the Fourier representation and Eq. (44) still holds, provided that one writes M…k† (for its Fourier transform with respect to x ÿ y† instead of M. The two-point correlation function g…x† is de®ned by: hdym …x†dym …y †i ˆ hdy2m ig…x ÿ y †,

…81†

hdon …x†don …y †i ˆ hdo2n ig…x ÿ y †,

…82†

with g… x ˆ 0 † ˆ

…

dk …2p †d

g…k † ˆ 1:

…83†

The ®nal results are then only modi®ed as: " !# … hdy2m i dk …m ‡ 1 †g…k †Dkà hdy3m i ‡O , ym eff …D † ˆ hym i 1 ÿ hym i2 …2p †d 1 ‡ 2…m ÿ 1 †Dkà hym i3 "

1 hdo2n i on eff …S † ˆ hon i 1 ÿ 2 hon i2

…

dk …n ‡ 1 †g…k †Skà hdo3n i ‡O d 1 ‡ …n ÿ 1 †S hon i3 …2p † kÃ

…84†

!# :

…85†

The Legendre duality property once again applies, whatever the nature of the correlations, and the result is once again independent of the boundary conditions. The previous expressions for isotropic site-disorder are recovered using g…x† ˆ vd…x†: Ellipsoidal symmetry corresponds to a correlation function g…x† ˆ g…jZ  xj†, where Z is a symmetric constant tensor (Willis, 1977; Ponte-CastanÄeda and Suquet, 1995).

5. Discussion In this section, we address the range of applicability of the functional method.

446

Y.-P. Pellegrini et al. / J. Mech. Phys. Solids 48 (2000) 429±459

Firstly, a major concern of studies on heterogeneous media is about bounds. In principle, the above functional formulation should be apt to deliver bounds, at least as far as the initial expressions (15) and (19) are concerned. One might think, e.g. to use the convexity properties of the exponential in order to obtain inequalities on approximations to the potentials. However, to the knowledge of these authors, the replica method is an inadequate tool to use for subsequent calculations, if these are to preserve inequalities. Indeed, a convexity equality of the type ! ! r r X X m e aˆ1

Va

m

Va

re

aˆ1

…86†

where m is some normalized functional measure and V is some potential, valid as long as the potential is replicated r times, rr1, may not survive the limit r 4 0: In the zero-replica limit indeed, quantities always positive when rr1 may become negative, minima may transform into maxima (Parisi, 1984), and it is most often hard to conclude on inequalities. The replica method has been invented, and employed, to ®nd estimates (which can be of very good quality) to free energies when disorder is present, not bounds. Second, one must not confuse the initial functional formulation with the replica method, which we use as a tool in order to work out averages. These are distinct matters. Actually, there are two starting points for the calculations: either we transfer the statistical averages of the logarithm inside the functional integral by means of the replica method, and carry out approximations on the e€ective potential * + X ÿb fx …dg † 1 g , …87† ÿ log e b cf. Eq. (78). This is what we did in the paper. Either we carry out an approximation on the non-replicated initial functional integral (15), then take the logarithm, and average afterwards only. Save for perturbative expansions, the results will be di€erent. Third, all types of calculations feasible with classical methods (especially the perturbative ones) should translate into simple approximation schemes to the basic Eqs. (15) and (19). But we believe that the compact starting point of the functional formulation might enable one to obtain non-perturbative results more easily, and of di€erent nature, than by classical means. There resides its main interest. Finally, the trick consisting in writing the minimum of some functional as the limit of a particular functional integral could be applied elsewhere, e.g. to compute variational expressions such as those presented in Ref. (Ponte-CastanÄeda and Suquet, 1998) (computing a maximum can of course be done by identical means).

Y.-P. Pellegrini et al. / J. Mech. Phys. Solids 48 (2000) 429±459

447

6. Conclusion A ®eld-theoretic method in computing the e€ective properties of highly nonlinear composites has been introduced. The minimization problem was shown to be equivalent to the computation of a functional integral. We believe the main advantage of this formulation is that it is valid for all types of local potentials, and that it can be used as a convenient starting point for approximations. As a ®rst application, a weak-disorder second-order perturbative calculation of this functional integral was presented. The second-order results of Suquet and PonteCastanÄeda (1993) for the e€ective plastic dissipation potential (with uniform traction boundary conditions) were recovered. In addition, the second-order correction to the viscoplastic potential was also obtained, and these results were extended to correlated disorder. With this method, minimizing over a vector velocity ®eld, or a tensor stress ®eld, is found to be equally feasible. A natural extension of this work is to ®nd a well-behaved non-perturbative approximation. Self-consistent calculations are currently under study. Acknowledgements We thank P. Suquet and M. Garajeu for stimulating discussions, and for communicating to us very useful references, and J.-M. Diani for his interest in this work. Thanks are also due to Prof. J.R. Willis for useful comments.

Appendix A. Fourth-rank tensors in mechanics Before brie¯y reminding the reader about the main properties of fourth-rank tensors in mechanics, we need to set our notations about the various algebras and sub-algebras of linear operators that come into play. We designate by L(2) the algebra of linear operators (second-rank d  d square matrices) M:Rd 4Rd : Next, fourth-rank tensors appear in mechanics as representations of linear operators M:L…2† 4L…2†: They can, therefore, be considered as square matrices, the indices of which are pairs of indices, and their components are denoted by Mij,kl , such that i, j, k, l ˆ 1, . . . ,d: We denote the algebra they belong to by L(4). The product in L(4) is …M:N †ij,kl ˆ Mij,mn Nnm,kl :

…A1†

Hence, applying M to a second-rank tensor N, we have …M:N †ij ˆ Mij,kl Nlk :

…A2†

The identity I in L(4) is Iij,kl ˆ dil djk ,

…A3†

448

Y.-P. Pellegrini et al. / J. Mech. Phys. Solids 48 (2000) 429±459

and the trace is de®ned (and denoted) by tr4 M ˆ Mij,ji :

…A4†

Hence, tr4 I ˆ d 2 : The algebra L(2) can be split in the direct sum of that of symmetric traceless matrices (which we denote by L…2s 0 †), diagonal matrices proportional to the identity [L(2d )], and anti-symmetric matrices [L(2a )]. The algebra of symmetric matrices is the direct sum L…2s† ˆ L…2s 0 †  L…2d †: The algebra L(4), therefore, admits three remarkable sub-algebras, which we denote by L…4s 0 †, L(4d ), L(4a ), generated by the mutually orthogonal projectors J, K and Ia , respectively, de®ned by Jij,kl ˆ

 1 1ÿ dil djk ‡ dik djl ÿ dij dkl , 2 d

1 Kij,kl ˆ dij dkl , d I aij,kl ˆ

 1ÿ dil djk ÿ dik djl : 2

…A5†

…A6†

…A7†

They obey I ˆ Ia ‡ J ‡ K: Hence, the sub-algebras L…4s 0 †, L(4d ), L(4a ) are that of the endomorphisms on L…2s 0 †, L(2d ), L(2a ), respectively. The operators J, K, and Ia are the appropriate identity operators in each sub-algebra. We have tr4 J ˆ

d…d ‡ 1 † ÿ 1, 2

tr4 K ˆ 1, tr4 Ia ˆ

d…d ÿ 1 † : 2

…A8† …A9†

…A10†

These numbers count the number of eigenvalues of each operator. In L(4s ), the sub-algebra of the endomorphisms on L(2s ), the identity is Is ˆ J ‡ K: I sij,kl ˆ

 1ÿ dil djk ‡ dik djl , 2

…A11†

and tr4 Is ˆ d…d ‡ 1†=2: Note that L…4s 0 †  L…4d †  L…4s†, but the two sets are not equal. Finally note that the inverse (resp. determinant) in L(4) of an operator M 2 L…4s 0 †  L…4d †  L…4a† is the sum (resp. product) of the inverses (resp. determinants) of its constituents in their respective sub-algebra.

Y.-P. Pellegrini et al. / J. Mech. Phys. Solids 48 (2000) 429±459

449

Appendix B. Gaussian integrals B.1. Gaussian integrals over vector ®elds We start from the well-known d-dimensional Gaussian integral over a vector ®eld l (Kleinert, 1995). If M is a d  d positive-de®nite symmetric matrix, and b any vector, then …

#1=2   "  …2p †d 1 1 ÿ12 exp b  M  b , dll exp ÿ l  M  l ‡ b  l ˆ 2 2 det2 …M † 

…B1†

Q where the integration measure is dll ˆ i dli : This formula is meaningless if M is not invertible in L(2). However, let kà be a à where M? is invertible unit vector, b now be a real vector, and M ˆ M? ‡ E2 kà k, à only in L…2‰?kŠ†, the set of the endomorphisms which operate in the subspace à det2 …M† ˆ E2 det à …M? †, à Then, Mÿ12 ˆ Mÿ12‰?kŠ ‡ Eÿ2 kà k, of vectors orthogonal to k: ? 2‰?kŠ and   … 1 dll exp ÿ l  M  l ‡ ib  l 2 #1=2  "  …2p †d 1 1 ÿ à 2 ÿ12‰?kà Š exp ÿ b  M bÿ 2 bk : …B2† ˆ 2 E det2‰?kà Š …M? † 2 2E Letting E4 0 and using the representation of the Dirac distribution 2 1 d…x† ˆ lim p eÿ1=2…x=E† , E 4 0 2pE2

we obtain   … 1 dll exp ÿ l  M?  l ‡ ib  l 2 #1=2  "  ÿ  …2p †d‡1 1 exp ÿ b  Mÿ12‰?kà Š  b d b  kà : ˆ 2 det2‰?kà Š …M? † We de®ne the integration measure over the vectors orthogonal to kà to be ÿ  dll ˆ d l  kà dll: ‰?kà Š

…B3†

…B4†

…B5†

Multiplying both sides of Eq. (B4) by exp…ib  a†, where a is a real vector, and integrating over b yields

450

Y.-P. Pellegrini et al. / J. Mech. Phys. Solids 48 (2000) 429±459

…



1 db exp ÿ b  Mÿ12‰?kà Š  b ‡ ib  a 2 ‰?kà Š



31=2 …   …2p †ÿ…d‡1 † 1  5 dll db exp ÿ l  M?  l ‡ ib  …l ‡ a † ˆ4 ÿ1 à 2 det2‰?kà Š M? 2‰?k Š 2

31=2 …   …2p †ÿ…d‡1 † 1  5 dll exp ÿ l  M?  l d…a ‡ l † ˆ …2p † 4 ÿ1 à 2 det2‰?kà Š M? 2‰?k Š 2

d

31=2   …2p †dÿ1 1   5 exp ÿ a  M?  a , ˆ4 ÿ1 à 2 det2‰?kà Š M? 2‰?k Š 2

…B6†

Q ÿ1 à where d…x† ˆ i d…x i †: Note that we can exchange M? 2‰?kŠ and M? in Eq. (B6). Moreover, M? can be written M? ˆ Qkà  M  Qkà , where Qkà is the projector Qkà ˆ I ÿ à for some M 2 L…2†: Besides, the b are now orthogonal to kà (because of the kà k, integration measure) so that b  Qkà  M  Qkà  b ˆ b  M  b: Finally, the result can be analytically continued to complex values of a. Changing the names of the variables, we therefore arrive at the formula   … 1 dll‰?kà Š exp ÿ l  M  l ‡ b  l 2 #1=2  "  …2p †dÿ1 ÿ1 ‰ à Š 1 ÿ 2 ? k ÿ  exp b  Qkà  M  Qkà b : …B7† ˆ 2 det2‰?kà Š Qkà  M  QkÃ

B.2. Gaussian integrals over tensors of rank 2 The generic Gaussian integral over all matrices s 2 L…2† is #1=2   "   2 … …2p †d 1 1 exp b:Mÿ14 :b , …B8† ds exp ÿ s:M:s ‡ b:s ˆ det4 …M † 2 2 Q where ds ˆ ij dsij is the appropriate measure, M is a symmetric matrix of L(4), i.e. is such that Mij,kl ˆ Mkl,ij : This formula is a direct consequence of Eq. (B1) via 2 a mapping of s onto a column vector in Rd : Once again, the integral is meaningless in general if M is not invertible in L(4). We apply the same method as in Section B.1.

Y.-P. Pellegrini et al. / J. Mech. Phys. Solids 48 (2000) 429±459

451

B.2.1. Gaussian integrals over symmetric matrices Let b 2 L…2† be real. We consider M ˆ Ms ‡ …E2 =2†Ia , where Ms 2 L…4s† and is 4s ‡ …2=E2 †Ia and det4 …M† ˆ det4s …M†  invertible in L(4s ). Then Mÿ14 ˆ Mÿ1 s d…dÿ1†=2 2 …E =2† : Hence, from Eq. (B8),   … 1 ds exp ÿ s:M:s ‡ ib:s 2 #1=2 "   2 …2p †d 1 1 1 a ÿ14s …B9† ˆ d…dÿ1 †=4 exp 2 b:Ms :b ÿ E2 b:I :b , ÿ det4s …Ms † E2 =2 P But b:Ia :b ˆ …1=2† i
…B11†

i,j
Multiplying both sides of Eq. (B10) by exp…ib:a†, and integrating with respect to b with db yields   … 1 ÿ14s ds b exp ÿ b:Ms :b ‡ ia:b 2 "

2

…

†

…2p †ÿd ÿd dÿ1 =2 ÿ  ˆ 4s det4s Mÿ1 s " ˆ …2p †

d2

2

#1=2 …

…

†

…2p †ÿd ÿd dÿ1 =2 ÿ  4s det4s Mÿ1 s



1 db ds exp ÿ s:Ms :s ‡ i…s ‡ a†:b 2 #1=2 …



  1 ds exp ÿ s:Ms :s d…s ‡ a† 2

#1=2   …2p †d…d‡1†=2 1 ÿ  exp ÿ a:M s :a , …B12† ˆ 4s 2 det4s Mÿ1 s Q where d…x† ˆ ij d…x ij †: Whence the generic result for Gaussian integrals over "

452

Y.-P. Pellegrini et al. / J. Mech. Phys. Solids 48 (2000) 429±459

symmetric matrices, with M 2 L…4†: …

#1=2   "  …2p †d…d‡1 †=2 1 1 s s †ÿ14s … b: I exp :M:I :b : ds s exp ÿ s:M:s ‡ b:s ˆ 2 2 det4s …Is :M:Is † 

…B13† B.2.2. Gaussian integrals over traceless symmetric matrices If Is :M:Is is not invertible in L(4s ), but only in L…4s 0 † for instance, the procedure can be repeated: let us assume that M ˆ Ms 0 ‡ E2 K, where Ms 0 2 ÿ1 L…4s 0 †: Starting from Eq. (B13) with b4 ib, using …Is :M:Is †ÿ14s ˆ Ms 0 4s 0 ‡ K=E2 , s s 2 det4s …I :M:I † ˆ det4s 0 …Ms 0 †E and letting E4 0, one ®nds: #1=2    "  … d…d‡1 †=2‡1 1 d…2p † 1 ÿ14s 0 exp ÿ b:Ms 0 :b d…tr2 b†: ds s exp ÿ s:Ms 0 :s ‡ ib:s ˆ det4s 0 …Ms 0 † 2 2 …B14† We de®ne the integration measure over symmetric traceless tensors to be ds 0 s ˆ d 1=2 d…tr2 s† ds s:

…B15†

As one easily checks, we have, with Eq. (B11) #" " # … Y Y d…d‡1 †=2 d…dÿ1 †=4 … † … † 2p d…sii † d…sij ‡ sji † : ds x exp is:x ˆ 2 i

…B16†

i
Multiplying Eq. (B14) by exp…ia:b†, and integrating over b with measure ds b, we deduce, using Ms 0 ˆ J:M:J, the result for Gaussian integrals over symmetric traceless tensors: #1=2   "   … …2p †d…d‡1†=2ÿ1 1 1 ÿ1 0 exp b:…J:M:J † 4s :b : ds 0 s exp ÿ s:M:s ‡ b:s ˆ det4s 0 …J:M:J † 2 2 …B17† B.2.3. Other Gaussian integrals For the sake of completeness, we ®nally give without demonstration the results for Gaussian integrals over antisymmetric tensors, with measure #" " # Y Y d…dÿ1 †=4 …B18† d…sii † d…sij ‡ sji † ds, da s ˆ 2 i

i
and over diagonal tensors proportional to the identity, of the type sI=d, with measure

Y.-P. Pellegrini et al. / J. Mech. Phys. Solids 48 (2000) 429±459

dd s ˆ d ÿ1=2 ds

453

…

dsd…s ÿ sI=d†:

…B19†

One ®nds …

#1=2   "  … † …2p †d dÿ1 =2 1 1 a a †ÿ14a … b: I exp :M:I :b da s exp ÿ s:M:s ‡ b:s ˆ 2 2 det4a …I a :M:Ia † 

…B20† …

   1=2  1 2p 1 ÿ14d exp b:…K:M:K † :b : dd s exp ÿ s:M:s ‡ b:s ˆ 2 det4d …K:M:K † 2 …B21† 

The last identity is trivial (scalar Gaussian integral).

Appendix C. Inverse and determinant in replica space In replica space, A and B being replica-independent operators pertaining to some algebra A, the inverse and determinant of a matrix of the type M ˆ AI ‡ BU,

…C1†

read M

ÿ1A,

rep

ˆ Aÿ1A I ÿ …A ‡ rB †ÿ1A BAÿ1A U,

…C2†

ÿ  h i rÿ1 det M ˆ det…A † det…A ‡ rB † ˆ 1 ‡ r log det…A † ‡ tr …BAÿ1A † ‡ O…r2 †

A, rep

A

A

A

A

…C3† The expansion derives from the operator identity (Kleinert, 1995) log det ˆ tr log:

Appendix D. Gaussian integrals over the velocity and stress ®elds D.1. Integration over the velocity ®eld We detail here the steps leading to Eq. (44). The Dirac distribution implementing the constraint r  v ˆ 0 in Eq. (39) is exponentiated ®rst; we introduce a scalar ®eld l and write:  …  … … † d r  v A Dl exp i dxl…x†r  v…x† : …D1† Then we Fourier transform the integrand. By de®nition, for any function f (which

454

Y.-P. Pellegrini et al. / J. Mech. Phys. Solids 48 (2000) 429±459

will be l or s), the Fourier transform is: … … dk ÿikx f…x† ˆ e f…k †: f…k † ˆ dx eikx f…x†, …2p †d

…D2†

Since f …x† is real, we write for k 6ˆ 0: 1  f…k † ˆ p f 2

…1 †

1  f… ÿ k † ˆ p f 2

…k † ‡ if

…1 †

…2 †

…k † ÿ if

 …k † ,

…2 †

…D3†

 …k † ,

…D4†

where f (1) and f (2) are real functions of the wavevector; also, we set f…k ˆ 0† ˆ f …1† …0†: This decomposition is carried out for all la …k† and va …k†: Introducing the set K‡ ˆ fk=k1 > 0g [ fk=k1 ˆ 0, k2 > 0g [ . . . [ fk=k1 ˆ 0, . . ., kdÿ1 ˆ 0, kd > 0g, a functional measure Df over f…x† thus becomes in the Fourier representation Df ˆ

Y df…x†Adf x

…1 †

…0 †

Y

df

…1 †

…k †df

…2 †

 …k † :

…D5†

k2K‡

Note that f …1† and f …2† are de®ned only for k 2 K‡ : By Parseval's identity, we have … …  dk  …2 † … † …D6† dxl…x†r  v…x† ˆ l …k †v…1 † …k †  k ÿ l 1 …k †v…2 † …k †  k : d K‡ …2p † As usual in functional methods, the latter integral is to be understood as a Riemann discrete sum by applying the correspondence: … dk 1X 4 : …D7† d V k …2p † In particular, this allows one to separate the contribution of the mode k ˆ 0: The argument of the exponential in Eq. (39) is likewise transformed into a sum of Fourier modes via … dxea …x†:Mag :eg …x†

ˆ

 1 a…1 † 1 X  a …1 † e …0 †:Mag :eg…1 † …0 † ‡ e …k †:Mag :eg…1 † …k † ‡ ea…2 † …k †:Mag :eg…2 † …k † V V ‡ k2K

ˆ

 1 X  a …1 † v …k †  …k  Mag  k †:vg…1 † …k † ‡ va…2 † …k †  …k  Mag  k †  vg…2 † …k † : V ‡ k2K

…D8†

Y.-P. Pellegrini et al. / J. Mech. Phys. Solids 48 (2000) 429±459

455

In the last equality, we have used the symmetry of Mag with respect to its tensor indices. Note that e…1† ˆ ÿ‰kv…2† ‡ v…2† kŠ=2 and e…2† ˆ ‰kv…1† ‡ v…1† kŠ=2: Because of the constraints d…ea †, the linear terms ea disappear from the exponential. There is no dependence of the integrand with respect to l…1† …0† nor to v…1† …0†, so that integration over these variables only yields harmless b-independent in®nite multiplicative factors which we drop out, since they do not contribute to the ®nal result in the limit b 4 1: The reason is the same as that invoked in conjunction with Eq. (31). Such physically irrelevant (because multiplicative) in®nities are often encountered when dealing with functional integrals. They are related to (here) unimportant normalization questions. We therefore „ see that for each k 6ˆ 0, mutually„ complex conjugate partial integrals A…k† ˆ dla…2† …k† dva…1† …k† [. . .] and A …k† ˆ dla…1† …k† dva…2† …k† [. . . ] show up in pairs, and can be evaluated independently from one another. Renaming, in A…k†, the integration variables into k-independent dla and dva , we arrive at hZ rut iAheÿrb…v=V†fx …D † iV=v

Y k2K

where A…k † ˆ

… Y

! a

dl dv

a

a

jA…k †j2 ,

…D9†

‡

"

1 b X a i X a a v  …k  Mag  k †  vg ‡ l …v  k † exp ÿ 2 2 V ag V a

#

…D10† A…k †A

…" Y a

A…k †A

det

# " # 1 b X a a ag g … † … † dv d v  k exp ÿ v  kM k v 2 V 2 ag

2‰?k Š, rep

a

h

iÿ1=2 ÿ  , Qkà  kà  M  kà  QkÃ

…D11†

…D12†

and Qkà ˆ I ÿ kà kà with kà ˆ k=k: In the step from Eq. (D11) to Eq. (D12) we used formula (B7), trivially generalized to the replica space, and discarded, among other irrelevant factors, a power of the modulus k. Eq. (44) ®nally follows from the fact that A…k† ˆ A…ÿk†:

D.2. Integration over the stress ®eld We now detail the steps leading to Eq. (65), which di€er little from above. Because of our considering the Norton viscoplastic potential (which does not depend on sm † for explicit applications, the matrix M1 given in Eqs. (52) and (53) is not invertible in L(4s ), but only in L…4s 0 †: As a consequence (cf. Eq. (C2)), M is not invertible in L…4s, rep†, but only in L…4s 0 , rep†: This leads to peculiarities that have to be taken into account but does not change the principle of the

456

Y.-P. Pellegrini et al. / J. Mech. Phys. Solids 48 (2000) 429±459

calculation. The equivalent of formula (65) for M invertible in L…4s, rep† is provided hereafter for the sake of completeness. The Dirac distribution implementing the constraint r  s ˆ 0 in the measure Ds s is exponentiated by the introduction of a vector ®eld l as:  …  …   d…r  s †A Dll exp i dxll…x†  r  s…x† !  …†   … †  1 X  …1 † …2 † 2 1 : s …k †: kll …k † ÿ s …k †: kll …k † ˆ Dll exp i V k2K‡

…D13†

…

A reasoning paralleling that in Section D.1 yields: Y jA…k †j2 , hZ rus iAheÿrb…v=V†cx …S † iV=v

…D14†

k2K‡

where now A…k † ˆ

… Y a

! a

dll ds s

a

"

1 b X a ag g i X a s :M :s ‡ s :…klla † exp ÿ 2 2 V ag V a

# …D15†

) ! ( a ÿ ÿ1=2 … Y   1 k2 X ag a a g ÿ1 0 4s , rep à à † k  l dll exp ÿ l  k  …M A…k †Adet4s 0 , rep M 2 b ag a ‰?kà Š …D16† ÿ ÿ1=2 A…k †Adet4s 0 , rep M

 det

2‰?kà Š, rep



Qkà  kà  M

ÿ14s 0 ,



rep

 kà  QkÃ

ÿ1=2 ,

…D17†

à and kà ˆ k=k: In the ®rst step from Eq. (D15) to Eq. (D16), we and Qkà ˆ I ÿ kà k, used the formula (B14) generalized to the replica space. The resulting factor d…tr2 kll†Ad…kà  l † was absorbed in the measure dlla‰?kŠÃ de®ned in Eq. (B5). Next, we appealed to Eq. (B7), also extended to replica space. Eq. (65) follows. Had M been invertible in L(4s, rep), one would have found ( h i )ÿ1=2 ÿ  Y ÿ14s, rep r ÿrb…v=V†cx …S † V=v i  kà …D18† det M det kà  M hZ iAhe k6ˆ0

4s, rep

2, rep

instead. Appendix E. The Legendre transform Though the plastic dissipation potential (Eqs. (61) and (62)) and the viscoplastic

Y.-P. Pellegrini et al. / J. Mech. Phys. Solids 48 (2000) 429±459

457

potential (Eqs. (76) and (77)) have been obtained for di€erent boundary conditions, we show here that they are linked by the Legendre transform (10). Let us deduce Eqs. (61) and (62) from Eqs. (76) and (77), for instance, assuming that Eq. (10) holds between both. We write F…D† ˆ F0 …D† ‡ dF…D†, C…S† ˆ C0 …S† ‡ dC…S†, where F0 …D† ˆ hfx …D†i, C0 …S† ˆ hcx …S†i are the leading terms in the perturbative expansions, and dF…D† and dC…S† are the corrective terms. It is straightforward to check that F0 …D† and C0 …S† are Legendre duals: F0 …D† ‡ C0 …S† ˆ S:D: Hence, F…D † ˆ S:D ÿ C0 …S † ÿ dC…S † ˆ F0 …D † ÿ dC…S †,

…E1†

so that dF…D† ˆ ÿdC…S†: Moreover, since S ˆ @F…D†=@ D ˆ @F0 …D†=@ D ‡ @ dF…D†=@ D, we obtain second order   @F0 …D † : …E2† F…D † ' F0 …D † ÿ dC @D The Legendre transform of C0 …S† ˆ hon iSn‡1 eq =…n ‡ 1† reads, with m ˆ 1=n: F 0 …D † ˆ

hon iÿm Dm‡1 eq , …m ‡ 1 †

tr2 …D † ˆ 0:

…E3†

In addition, S0 '

dÿ1 hon iÿm Dmÿ1 eq D, d

Seq ' hon iÿm Dm eq :

…E4†

These expressions for S 0 and Seq are correct to ®rst order only. However, ^ symmetry considerations show that S 0 is always proportional to D. Thus, S^ ˆ D, whence from the de®nitions (57) and (74), Skà ˆ 1 ÿ 2Dkà :

…E5†

The weak-disorder expansion of F…D† has to be expressed in terms of ym ˆ oÿm n ˆ hym i ‡ dym : To second order, we have " # 1 m ‡ 1 hdy2m i hdo2n i 1 hdy2m i ÿ1=m ' 2 : …E6† 1‡ , hon i ' hym i 2 2 2 2 m hym i m hym i2 hon i Gathering these results into the perturbative expansion (E2) of F…D† ®nally yields Eqs. (61) and (62) back, as announced.

References BartheÂleÂmy, M., Orland, H., 1993. Replica ®eld theory for composite media. J. Phys. I (France) 3, 2171±2177.

458

Y.-P. Pellegrini et al. / J. Mech. Phys. Solids 48 (2000) 429±459

BartheÂleÂmy, M., Orland, H., 1998. A path-integral approach to e€ective non-linear medium. Eur. Phys. J B 6, 537±541. Bender, C.M., Orszag, S.A., 1978. Advanced Mathematical Methods for Scientists and Engineers. McGraw-Hill, Singapore. Bergman, D.J., Lee, H.-C., 1998. Comment on strongly nonlinear composite dielectrics: a perturbation method for ®nding the potential ®eld and bulk e€ective properties, in press. Blumenfeld, R., Bergman, D.J., 1989. Exact calculation to second order of the e€ective dielectric constant of a strongly nonlinear inhomogeneous composite. Phys. Rev B 40, 1987±1989. Blumenfeld, R., Bergman, D.J., 1991. Strongly nonlinear composite dielectrics: a perturbation method for ®nding the potential ®eld and bulk e€ective properties. Phys. Rev B 44, 7378±7386. Budianski, B., 1965. On the elastic moduli of some heterogeneous materials. J. Mech. Phys. Solids 13, 223±227. Edwards, S.F., Anderson, P.W., 1975. Theory of Spin Glasses. J. Phys F 5, 965±974. Gilormini, P., 1995. Insusance de l'extension classique du modeÁle autocoheÂrent au comportement non-lineÂaire. C.R. Acad. Sci. Paris, SeÂrie IIb 320, 115±122. Hashin, Z., Shtrikman, S., 1962a. On some variational principles in anisotropic and nonhomogeneous elasticity. J. Mech. Phys. Sol 10, 335±342. Hashin, Z., Shtrikman, S., 1962b. A variational approach to the theory of the elastic behavior of polycrystals. J. Mech. Phys. Sol 10, 343±352. Hill, R., 1963. Elastic properties of reinforced solids: some theoretical principles. J. Mech. Phys. Sol 11, 357±372. Kleinert, H., 1995. Path Integrals. 2nd ed., World Scienti®c, Singapore. Levy, O., Bergman, D.J., 1994. Critical behavior of the weakly nonlinear conductivity and ¯icker noise of two-component composites. Phys. Rev. B 50, 3652±3660. MeÂzard, M., Parisi, G. and Virasoro, M., 1987. Spin Glass Theory and Beyond. World Scienti®c, Singapore. Parcollet, O., BartheÂleÂmy, M., ZeÂrah, G., 1996. Replica treatment of the e€ective elastic behavior of a composite. Phys. Rev. B 53, 3161±3170. Parisi, G., 1984. An introduction to the statistical mechanics of amorphous systems. In: Zuber, J.B., Stora, R. (Eds.), Les Houches, Session XXXIX, 1982 Ð DeÂveloppements ReÂcents en TheÂorie des Champs et MeÂcanique Statistique/Recent Advances in Field Theory and Statistical Mechanics. North-Holland, Amsterdam, pp. 473±523. Pellegrini, Y.-P., BartheÂleÂmy, M., 1999. Self-consistent e€ective-medium approximations with path integrals. Submitted. Ponte-CastanÄeda, P., 1991. The e€ective mechanical properties of nonlinear isotropic composites. J. Mech. Phys. Sol 39, 45±71. Ponte-CastanÄeda, P., Suquet, P., 1995. On the e€ective mechanical behavior of weakly inhomogeneous nonlinear materials. Eur. J. Mech., A/Solids 14, 205±236. Ponte-CasatnÄeda, P., Kailasam, M., 1997. Nonlinear electrical conductivity in heterogeneous media. Proc. R. Soc. London 453, 793±816. Nebozhyn, M.V., Ponte-CastanÄeda, P., 1998. Exact second-order estimates of the self-consistent type for nonlinear composite materials. Mechanics of Materials 28, 9±22. Ponte-CastanÄeda, P., Suquet, P., 1998. Nonlinear composites. Adv. Appl. Mech. 34, 171±302, and references therein. Sahimi, M., 1998. Non-linear and non-local transport processes in heterogeneous media: from longrange correlated percolation to fracture and materials breakdown. Phys. Rep. 306, 213±395, and references therein. Stroud, D., Hui, P.M., 1988. Nonlinear susceptibilities of granular matter. Phys. Rev. B 37, 8719±8724. Suquet, P., 1987. Elements of homogenization for inelastic solid mechanics. In: Sanchez-Palencia, E., Zaoui, A. (Eds.), Homogenization techniques for composite media, Lecture Notes in Physics 272. Springer±Verlag, Berlin, pp. 193±278. Suquet, P., Ponte-CastanÄeda, P., 1993. Small-contrast perturbation expansions for the e€ective properties of nonlinear composites. C. R. Acad. Sci. (Paris) Ser. II 317, 1515±1522.

Y.-P. Pellegrini et al. / J. Mech. Phys. Solids 48 (2000) 429±459

459

Suquet, P., 1993. Overall potentials and extremal surfaces of power-law or ideally plastic materials. J. Mech. Phys. Sol 41, 981±1002. Talbot, D.R.S., Willis, J.R., 1985. Variational principles for inhomogeneous nonlinear media. IMA J. Appl. Math 35, 39±54. Willis, J.R., 1977. Bounds and self-consistent estimates for the overall properties of anisotropic composites. J. Mech. Phys. Solids 25, 185±202. Willis, J.R., 1989. The structure of overall constitutive relations for a class of nonlinear composites. IMA J. Appl. Math 43, 231±242.