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Journal of the Mechanics and Physics of Solids 56 (2008) 742–771

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Entropy of microstructure

Victor L. Berdichevsky

Mechanical Engineering, Wayne State University, Detroit, MI 48202, USA

Received 15 February 2007; received in revised form 24 June 2007; accepted 5 July 2007

Abstract

Two points are made in this paper: first, energy of random structures is not determined uniquely by any finite set of the

characteristics of microstructure. The information lost is characterized by entropy of microstructure; it describes the

scattering of the values of energy. Therefore, entropy of microstructure is a key thermodynamic parameter in

phenomenological modeling of the behavior of random structures. Second, mathematical modeling of a random structure

is based on the construction of its probabilistic measure; a way to select the probabilistic measure from the experimental

data is outlined. The corresponding probabilistic measure is remarkably similar to that of classical statistical mechanics,

though the underlying physics is quite different. After the probabilistic measure is chosen, the entropy of microstructure

can be found from the analysis of the homogenization problem. Entropy of microstructure is computed in two example

problems. Applications to phenomenological modeling of work hardening are discussed.

r 2007 Elsevier Ltd. All rights reserved.

Keywords: Homogenization; Random structures; Scattering; Entropy; Work hardening

1. Entropy of microstructure

To model the macroscopic behavior of materials one has to specify the parameters describing the material,r1; . . . ;rk, and the dependence of energy, E, on these parameters and the thermodynamic entropy, S:

E ¼ EðS; r1; . . . ;rkÞ. (1)

For simplicity, we consider here macroscopically homogeneous states, otherwise Eq. (1) holds for thecorresponding densities.

As soon as the equation of state (1) is specified and some additional assumptions regarding the nature of theirreversible processes are made, the usual thermodynamic formalism yields a closed system of governingequations (see, e.g., De Groot and Mazur, 1962).

Modeling of the behavior of random structures faces the following difficulty. Consider, for example, thematerial shown in Fig. 1. This is a titanium alloy; it contains the precipitates of the average size of about10mm. If the microstructure of the alloy does not change in the course of deformation and the applied force issmall enough to cause pure elastic macroscopic deformations, then one can use linear elasticity to describe the

e front matter r 2007 Elsevier Ltd. All rights reserved.

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Fig. 1. Microstructure of a titanium alloy; the average size of the precipitates is about 10mm.

V.L. Berdichevsky / J. Mech. Phys. Solids 56 (2008) 742–771 743

response of the material, and

E ¼ 12

Cijkleijekl jV j þ E0ðSÞ, (2)

where eij and Cijkl are the components of the strain tensor and elastic moduli tensor, respectively, jV j thevolume of the region V occupied by the specimen, E0ðSÞ energy of the material at zero strains, small Latinindices run through values 1,2,3, and summation over repeated upper and low indices is always implied.

One can measure the elastic moduli experimentally. They depend on microstructure, however, since themicrostructure does not change in the course of deformation, these moduli are all that we need to know todescribe the material response.

The situation becomes essentially different if the microstructure changes. First of all, we have to say how tocharacterize the changes, i.e. we need to introduce the parameters describing the state of the microstructure.The number of such parameters is necessarily small not only because the model must be simple enough butalso due to the limited experimental data on the microstructure. In what follows, by r1; . . . ;rk we mean theseparameters. In case of microstructure of Fig. 1, we may use as such parameters the volume concentration ofthe precipitates, their average eccentricity, etc.

If all that one wishes to know is the elastic macroscopic response, then one needs to know only the values ofelastic moduli: they accumulate all necessary information on the microstructure. However, if one wishes tocharacterize also the changes in microstructure, i.e. the evolution of the parameters, r1; . . . ;rk, and follow theusual thermodynamic formalism, one has to specify how energy depends on r1; . . . ; rk. The problem thatarises is that energy of the specimen is not determined uniquely by any finite set of such parameters. This canbe seen from the mathematical setting of the homogenization problem considered in the next section. Since theparameters, r1; . . . ;rk, are all that we have at hand, we have to admit that energy, E, can take different valuesfor different samples even if these samples have the same values of parameters, r1; . . . ;rk. Thus, energybecomes a random number. We accept that, for given r1; . . . ;rk, energy has some probability densityfunction,1 f ðE j r1; . . . ;rkÞ.

1For simplicity, we consider adiabatic processes, and, since thermodynamic entropy is constant, it is dropped from the set of parameters.

ARTICLE IN PRESSV.L. Berdichevsky / J. Mech. Phys. Solids 56 (2008) 742–771744

We define entropy of microstructure, SmðE;r1; . . . ;rkÞ, by the Einstein-type formula2

f ðE j r1; . . . ;rkÞ ¼ c eSmðE;r1;...;rkÞ. (3)

Here c is the normalizing constant:Zc eSmðE;r1;...;rkÞ dE ¼ 1. (4)

Entropy of microstructure is defined up to an additive constant due to the presence of the constant, c, inEq. (3).

There are two qualitatively different situations in modeling of random structures: entropy ofmicrostructure, SmðE; r1; . . . ; rkÞ, can be a smooth function of E or it may have a sharp maximum. In thelatter case, as we will see from further examples, Sm contains a large factor, N,

SmðE; r1; . . . ; rkÞ ¼ NSmðE;r1; . . . ;rkÞ, (5)

while SmðE;r1; . . . ;rkÞ is a smooth function of E. In the further examples, the large parameter, N, has themeaning of a ‘‘number of inhomogeneities’’ in the random structure: N ¼ jV j=a3, a being the correlationradius of the microstructure.

In the case (5), the most probable value of energy, E, appears with overwhelming probability. At this valuefunction SmðE;r1; . . . ;rkÞ has maximum over E for fixed r1; . . . ;rk. The most probable value of energy, E, is afunction of r1; . . . ;rk,

E ¼ Eðr1; . . . ;rkÞ. (6)

Eq. (6) can be considered as the equation of state (1). Since energy, up to small fluctuations, is a function ofr1; . . . ;rk, the parameters, r1; . . . ;rk, may be viewed as the thermodynamic parameters of the system.

The case when entropy of microstructure, SmðE; r1; . . . ; rkÞ, does not have sharp maximum is different:energy becomes an independent parameter of state additional to the parameters, r1; . . . ; rk. To return to theusual framework of classical thermodynamics, we have to admit that there is an additional parameter of statewhich ‘‘absorbs’’ the arbitrariness of energy for given r1; . . . ;rk, entropy of microstructure,3 Sm, and

E ¼ EðSm;r1; . . . ;rkÞ. (7)

Function (7) can be viewed as the inversion of the function SmðE;r1; . . . ; rkÞ introduced by Eq. (3).4

To find entropy of microstructure experimentally, one has to consider many samples, to measure for eachsample the values of the parameters, E;r1; . . . ;rk, determine probability density function and computeentropy of microstructure from Eq. (3). In the case of crystal plasticity, when one is interested in modeling ofmotion and nucleation of crystal defects, energy can be found experimentally by comparing the amount ofheat needed to melt the sample and the corresponding defectless monocrystal (Taylor and Quinney, 1934).

If entropy has a sharp maximum and Eq. (5) holds, one may seek the joint probability density function ofthe parameters, E; r1; . . . ; rk, f ðE;r1; . . . ;rk). It is given by the formula

f ðE; r1; . . . ; rkÞ ¼ c1 eSmðE;r1;...;rkÞ, (8)

where function SmðE; r1; . . . ; rkÞ is the same as in Eq. (3) while the constant, c1, isZc1 e

SmðE;r1;...;rkÞ dE dr1 . . . drk ¼ 1.

Obviously, Eq. (3) follows from Eqs. (8) and (5) asymptotically as N !1.To find entropy of microstructure from theoretical reasoning, one has to study the homogenization problem

to which we proceed.

2Regarding Einstein’s formula see, e.g., Kubo (1965) or Berdichevsky (1997).3The necessity to include entropy of microstructure in the set of thermodynamic parameters appeared in modeling of plasticity of micro-

samples (Berdichevsky, 2005); it has become clear later (Berdichevsky, 2006a) that this situation is generic.4In fact, formula (3) is meaningful only in the case (5), and for smooth change of entropy it must be rectified; this is done further in

Section 4 (formula (23)).

ARTICLE IN PRESSV.L. Berdichevsky / J. Mech. Phys. Solids 56 (2008) 742–771 745

2. Homogenization problem

By homogenization one usually means a bridge between micro- and macroscopic properties of the materials.A breakthrough in understanding of homogenization occurred quite recently, in seventies, when it wasrecognized that homogenization of the media with periodic microstructure has an asymptotic nature andmay be treated as a two-scale asymptotic expansion. This was done independently by Sanchez-Palencia(1971, 1974), Bakhvalov (1974) and Babuska (1976). The first proof of the statement, which may be calledwithin the framework of the homogenization theory the homogenized equation existence theorem, was obtainedactually earlier by Freidlin (1964) who was motivated by pure mathematical issues. Note also the pioneeringworks on the existence of the homogenized equations by De Giorgi and Spagniola (1973), Bensoussan, Lionsand Papanicolaou (1975) and De Giorgi (1975). A complete asymptotic analysis of the problem was given byBakhvalov in a series of publications. Remarkably, the computation of the coefficients of the homogenizedequations and the local fields can be found from the solution of some cell problem (Bakhvalov, 1974, 1975). Inthe homogenization problems possessing the variational structure, the homogenized equations also possess avariational structure, while the homogenized Lagrangian is the minimum value of the cell variational problem inboth linear and nonlinear cases (Berdichevsky, 1975). The next important development was an extension of theasymptotic procedure to quasi-periodic and random structures given by Kozlov (1977, 1978). It turned out thathomogenization of random structures is also reduced to solution of some cell problem. Kozlov’s cell problemwas extended to nonlinear structures in Berdichevsky (1981). The reviews can be found in Bensoussan et al.(1978), Sanchez-Palencia (1980), Berdichevsky (1983), Bakhvalov and Panasenko (1984), Jikov et al. (1991),Milton (2002), Torquato (2002), Panasenko (2005). A relevant mathematical formalization is related to thenotion of G-convergence (Spagnolo, 1968) and G-convergence(De Giorgi, 1975, 1984) which were furtheradvanced in many studies (see Jikov et al., 1991). In contrast to periodic microstructures for which a completeunderstanding was achieved, the random structures, in spite of the progress made, are still poorly understood.

We formulate the major results of the homogenization theory of random structures for physical phenomenathat are modeled by the minimization problem:

IðuÞ ¼

ZV

1

2aijðxÞ

qu

qxi

qu

qxjd3x! min

uðxÞ:

ujqV¼uðbÞ

. (9)

The notation (9) means that one seeks the minimum of the functional IðuÞ with respect to all functions uðxÞ

which satisfy the boundary conditions uðxÞ ¼ uðbÞ at the boundary qV of the region V . The inhomogeneitiesare described by the dependence of the local characteristics, aij , on the space coordinates, x. For definitenesswe assume that V is a bounded region in three-dimensional space.

The asymptotic setting of the problem (9) is as follows. We prescribe some functions, aij0 ðyÞ, of the auxiliary

variables, y, defined in three-dimensional space Ry. These functions are random, i.e. they depend on the event,o, o 2 O, and there is a probabilistic measure on O. The coefficients, aij , in Eq. (9) are set to be equal to

aij ¼ aij0

x

e;o

� �,

and one considers the asymptotics of the minimizer in the variational problem (9) as e! 0. The random field,a

ij0 ðy;oÞ, is assumed to be ergodic. This feature holds if the correlations of a

ij0 at points y and y0 decay fast

enough as jy� y0j ! 1.It turns out that the minimizer in the variational problem (9) tends to some deterministic function, uðxÞ, as

e! 0. Function uðxÞ is the minimizer of the homogenized variational problem,

IðuÞ ¼

ZV

1

2a

ijeff

qu

qxi

qu

qxjd3x! min

uðxÞ:

ujqV¼uðbÞ

, (10)

where the coefficients, aijeff , are determined from Kozlov’s cell problem:

aijeffvivj ¼ min

c:hqc=qyii¼0a

ij0 ðy;oÞ vi þ

qcqyi

� �vj þ

qcqyj

� �� . (11)


Here h�i means average value over the space Ry: for any function, jðyÞ,

hjðyÞi ¼ liml!1

1

jBlj

ZBl

jðyÞd3y,

Bl being the ball of radius l, and vi are constant parameters. The cell problem determines not only theeffective characteristics but also the local fields.

Formally, the variational cell problem for random structures differs from that for periodic structures(Berdichevsky, 1975) by replacing the integration over the periodic cell by integration over the space Ry.

Remarkably, it does not matter which realization of the random field, aij0 ðy;oÞ, is used in Eq. (11): formula

(11) holds for (almost) any realization.Formula (11) determines the dependence of the solution on the probabilistic measure implicitly: the random

field is known if one knows its probabilistic measure; this measure enters (11) through a realization of therandom field, a

ij0 ðy;oÞ.

In numerical simulations, one has to replace the integration over the infinite space Ry by integration over afinite volume, V. Then, for sufficiently small e, as easy to see, Kozlov’s cell problem transforms to thevariational problem

1

2a

ijeffvivj ¼ min

uðxÞ:ujqV¼vix

i

1

jV j

ZV

1

2aijðxÞ

qu

qxi

qu

qxjd3x. (12)

Eq. (12) can be considered as the definition of the effective coefficients, aijeff , for finite volumes. For a finite

volume, the effective characteristics, aijeff , and, therefore, energy, depend on realization and may fluctuate.

The results mentioned show that the mathematical modeling of random structures is equivalent to choosingthe probabilistic measure in the space of physical characteristics. This choice, however, must be quite specificas discussed in the next section.

3. The choice of probabilistic measure

Consider again Fig. 1. To simplify the matter, let us assume that each precipitate is an isotropic ellipsoidwith the elastic properties different from the properties of the matrix. Then each precipitate is specified by thecoordinates of its center, the orientation and the semiaxes, i.e. by nine numbers. If there are N ellipsoids inregion V, then the entire structure has n ¼ 9N degrees of freedom, denote them by o ¼ fo1; . . . ;ong. The set,O, in n-dimensional space run by these parameters has quite complex geometry because we assumeadditionally that the ellipsoids do not overlap. The set O has a finite volume,

jOj ¼ZOdo1; . . . ;don �

ZOdo.

To model the microstructure we have to specify the probabilistic measure on O. This measure cruciallydepends on which parameters we are able to measure experimentally. Let, say, all that we know about themicrostructure is the volume concentration of the precipitates, r. The volume concentration is some functionof o1; . . . ;on, r ¼ Fðo1; . . . ;onÞ. Let the experimental values of r lie in the interval ½r;rþ Dr�. The admissiblevalues of o1; . . . ;on are in the region

rpFðo1; . . . ;onÞprþ Dr; o 2 O.

All points of this region should be assigned with the equal probability. For, if the probabilities were notequal for different points, we know about the microstructure more than we claimed. In essence, this isLaplace’s principle of insufficient reason. Tending Dr to zero, we arrive at the probability measure

f ðo1; . . . ;onÞ ¼ cdðr� Fðo1; . . . ;onÞÞ; o 2 O. (13)

Here dðrÞ is d-function, c the normalizing constant,

c

ZOdðr� Fðo1; . . . ;onÞÞdo ¼ 1. (14)


Such probability density can be interpreted as the conditional probability density (under conditionFðo1; . . . ;onÞ ¼ rÞ if the unconditional probability density on the set O is constant,

f ðo1; . . . ;onÞ ¼1

jOj. (15)

We see that the probability measure is quite similar to that in classical statistical mechanics. As in classicalstatistical mechanics the probability density is constant, while Eq. (13) is equivalent to microcanonicaldistribution with the function Fðo1; . . . ;onÞ playing the role of Hamilton’s function. The physical reasoningsbehind the constancy of the probability densities are quite different though: in statistical mechanics theprobability density is constant in the phase space due to the invariance of the probabilistic measure withrespect to Hamiltonian phase flow. Another difference, which yields various deviations from the relations ofclassical statistical mechanics, is that admissible o belong to a set with quite complex geometry.

Any additional information on microstructure changes the probabilistic measure. If we know the values,r1; . . . ;rk, of k characteristics, F1ðo1; . . . ;onÞ; . . . ;Fkðo1; . . . ;onÞ, then, repeating the previous reasoning weobtain

f ðoÞ ¼ cdðr1 � F1ðoÞÞ . . . dðrk � FkðoÞÞ; o 2 O, (16)

where

c

ZOdðr1 � F1ðoÞÞ � � � dðrk � FkðoÞÞdo ¼ 1. (17)

In addition, we may know from experiments the values of effective coefficients. In the case (9) this isequivalent to knowing the value of energy, denote it by HðoÞ. Then the measure is

f ðoÞ ¼ cdðE �HðoÞÞdðr1 � F1ðoÞÞ . . . dðrk � FkðoÞÞ; o 2 O. (18)

If the effective coefficients are all that we know, we obtain the probabilistic measure which is especially close tothe microcanonical distribution,

f ðoÞ ¼ cdðE �Hðo1; . . . ;onÞÞ; o 2 O. (19)

Another measure is obtained if one uses the maximum principle for the information entropy (Jaynes, 1957),

Sinf ¼ �Z

f ðoÞ log½f ðoÞ�do.

The major motivation for the maximum information entropy principle in statistical mechanics is that ityields the firmly established Gibbs’ distribution. In homogenization problems, Gibbs’ distribution can be usedas long as it approximates well the distribution (18). One can expect that this is the case in the limit n!1.For finite n the maximum information entropy principle brings the distribution which differs from Eq. (18).

Any macroscopic parameter is a function of o. Computation of probability distributions of macroscopicparameters is considered in the next section.

4. Entropy of microstructure and distribution of macroscopic parameters

So, let the probability density be given by formula (18). As in classical thermodynamics, we introduce the‘‘parameter volume’’,

GðE;r1; . . . ; rkÞ ¼

Zo2O;HðoÞpE;F1ðoÞpr1;...;FkðoÞprk

do,

and define the entropy of microstructure as

SmðE; r1; . . . ;rkÞ ¼ lnGðE;r1; . . . ; rkÞ

jOj. (20)

Entropy is dimensionless and, since GpjOj, always negative.


The constant, c, in Eq. (18) can be expressed in terms of derivatives of entropy or the parameter volume.Indeed, writing the parameter volume in terms of the step function, yðEÞ (yðEÞ ¼ 0 for Eo0, yðEÞ ¼ 1 forEX0)

GðE;r1; . . . ;rkÞ ¼

Zo2O

yðE �HðoÞÞyðr1 � F1ðoÞÞ . . . yðrk � FkðoÞÞdo,

and using that dyðEÞ=dE ¼ dðEÞ, we have

qkþ1GðE;r1; . . . ;rkÞ

qEqr1 . . . qrk

¼

Zo2O

dðE �HðoÞÞdðr1 � F1ðoÞÞ . . . dðrk � FkðoÞÞdo.

From this relation and the normalization condition for c,

c ¼qkþ1GðE; r1; . . . ; rkÞ

qEqr1 . . . qrk

!�1. (21)

In particular, if only the value of energy is known, and the probability density is Eq. (19), then

c ¼1

GE

; GE �dGðEÞdE

; GðEÞ �Zo2O;HðoÞpE

do. (22)

Let one choose samples with all possible values of o, and seek for the joint probability density function,f ðE; r1; . . . ;rkÞ, to observe the values E; r1; . . . ; rk of the characteristics, H ;F1; . . . ;Fk. According to Eq. (15),

f ðE; r1; . . . ; rkÞ ¼1

jOj

ZOdðE �HðoÞÞdðr1 � F1ðoÞÞ . . . dðrk � FkðoÞÞdo,

or,

f ðE; r1; . . . ; rkÞ ¼1

jOjqkþ1GðE; r1; . . . ; rkÞ

qEqr1 . . . qrk

.

In terms of entropy (20) the joint probability density is

f ðE; r1; . . . ; rkÞ ¼1

jOjqkþ1

qEqr1 . . . qrk

eSmðE;r1;...;rkÞ. (23)

If entropy contains a large parameter, as in Eq. (5), then formula (23) is asymptotically equivalent toEq. (8).5 Otherwise, Eq. (23) is an exact relation which replaces Eq. (8) if entropy is a smooth function ofenergy.6 It should be used to find entropy from the experimental data in that case.

Since the parameter space has a finite volume, jOj, it is worthy to consider the complementary parametervolume, G�, defined as

G� ¼ jOj � G,

and the complementary entropy,

S�m ¼ logG�

jOj.

The parameter volume, G, increases as energy grows, and so does entropy, Sm. The complementaryparameter volume, G�, and the complementary entropy, S�m, decay as energy increases.

In summary, to model a random structure means to construct its probabilistic measure; we split all that weknow about the microstructure into two categories: all that we know and all that we do not know; theparameters describing what we do not know, o, are endowed with the constant probability.

5For positive temperatures, see next section.6Note the similarity with the theory of large thermodynamic fluctuations in Hamiltonian ergodic systems with a small number of degrees

of freedom (Berdichevsky, 1997).


As soon as a probabilistic measure on O is known, one may ask the probabilistic questions: What is theprobability distribution of the effective coefficients? What is the probability distribution of the local fields? etc.

The situation in modeling of random structures is quite similar to the mathematical modeling in otherphysical circumstances where we design the model, find the solution, and check the outcomes against theexperimental results. In the case of composite materials, we construct the probabilistic measure. If thecoincidence of the effective characteristics, their fluctuations, local fields, etc., with the experimental data issatisfactory, the model captures the physical reality indeed.

5. Temperature of microstructure and equipartition law

In this section we consider the case when only energy is known. Then,

SmðEÞ ¼ lnGðEÞjOj

; S�mðEÞ ¼ lnG�ðEÞjOj

. (24)

We define the temperature of microstructure, Tm, by the ‘‘thermodynamic relation’’,

1

TmðEÞ¼

qSmðEÞqE

. (25)

Accordingly, the complementary temperature is

1

T�mðEÞ¼

qS�mðEÞqE

.

Temperature of microstructure, TmðEÞ, is always positive because GðEÞ (and, thus, SmðEÞÞ increases asenergy grows. The complementary temperature, T�mðEÞ, is always negative because G�ðEÞ and S�mðEÞ aredecaying functions of energy.

In general, entropies introduced by formulas (3) and (20) are different: entropy (20) is always increasingwhile entropy in Eq. (3) may have a maximum. Therefore, further we use for entropy in Eq. (3) the notation~SmðEÞ. One can define the corresponding temperature of microstructure, ~TmðEÞ, as

1

~TmðEÞ¼

q ~SmðEÞqE

.

If entropy ~SmðEÞ has a maximum at a point, E, its derivative, and, accordingly, ~TmðEÞ, changes the sign:~TmðEÞ is positive for EoE and negative for E4E.We will show for a bar with random structure (see Section 6 and Appendix A) that

~SmðEÞ ¼ SmðEÞ for EoE; ~SmðEÞ ¼ S�mðEÞ for E4E.

Accordingly, ~TmðEÞ coincide with TmðEÞ for EoE and with T�mðEÞ for E4E. Negative temperature statescorrespond to larger energies and are ‘‘hotter’’ than positive temperature states as in other branches of physics(Onsager, 1949).

The physical interpretation of temperature in classical thermodynamics is based on the equipartition law.The analogous relation for microstructures is

o1qH

qo1

� �¼ � � � ¼ oN

qH

qoN

� �¼ Tm, (26)

where h�i means averaging over the surface HðoÞ ¼ E. The equipartition of energy would hold if thesurface HðoÞ ¼ E was the boundary of the region HðoÞpE;o 2 O. However, usually, this is not the case:the set, O, may have its own boundaries due to the geometrical constraints on the parameters, o. Forexample, if the parameters, o, are the position vectors of the inclusions in a composite, the set, O, looks like acheese with many holes. Therefore, some corrections may appear in Eq. (26). These corrections and thephysical meaning of the energy equipartition depend on the meaning of the parameters, o. Consider someexamples.

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Fig. 2. An elastic plate weakened by a series of notches.

V.L. Berdichevsky / J. Mech. Phys. Solids 56 (2008) 742–771750

Let an elastic rectangular plate be weakened by a series of notches (Fig. 2). One side of the plate is clamped,at the opposite side some constant displacement is given. We know the experimental value of the effectivecoefficient/energy. The lengths of the notches, l1; l2; . . . ; ln, play the role of the parameters, o. Each lengthchanges within the limits, 0plspa, s ¼ 1; . . . ; n, a being the width of the plate. If the length of any of thenotches is equal to a, then the energy of the plate is zero: the plate is split into two disconnected pieces.Derivative of energy with respect to ls has the meaning of the energy release rate of the sth notch, Es. We aregoing to show that the average values, hlsEsi, are the same for all notches, and, moreover, they are equal to thecomplementary temperature of the microstructure:

hl1E1i ¼ hl2E2i ¼ � � � ¼ hlnEni ¼ T�m. (27)

Energy release rate of each notch is negative (since the elastic energy of the plate decays as a notchelongates), therefore, not surprisingly, the complementary temperature, which is negative, enters Eq. (27).

To prove Eq. (27) consider the average,

hl1E1i ¼1

GE

Z a

0

. . .

Z a

0

l1qH

ql1dðE �Hðl1; . . . ; lnÞÞdl1 . . . dln.

It can be rewritten as

hl1E1i ¼ �1

GE

Z a

0

. . .

Z a

0

l1qql1

yðE �Hðl1; . . . ; lnÞÞdl1 . . . dln.

Integrating by parts over l1 we have

hl1E1i ¼GGE

�1

GE

Z a

0

. . .

Z a

0

ayðE �Hða; l2; . . . ; lnÞÞdl2 . . . dln.


Since Hða; l2; . . . ; lnÞ ¼ 0,

hl1E1i ¼G� jOjGE

¼ �G�

GE

¼G�

G�E¼ T�m.

The same result we obtain for hl2E2i; . . . ; hlnEni. Thus Eq. (27) holds true.A weak version of Eq. (26),

o1qH

qo1

� �¼ � � � ¼ oN

qH

qoN

� �, (28)

may valid for some structures just due to symmetry reasons. Let, for example, a periodic cell consists ofN boxes, B1; . . . ;BN . Each box has its own heat conductivity, ak, which may take any values within the limits,a� and aþ. Heat conductivities play the role of parameters, o, and the set, O, is the product of N segments,½a�; aþ�. Energy

7 is determined by the variational problem

Hða1; . . . ; aNÞ ¼ minperiodic c

XN

k¼1

1

2ak

ZBk

vi þqcqyi

� �vi þ

qcqyi

� �dV . (29)

Since

qH

qak

¼1

2

ZBk

vi þq �cqyi

!vi þ

q �cqyi

!dV ,

�c being the minimizer in the variational problem (29), the number, hakqH=qaki (no summation over k), hasthe meaning of the averaged energy of the kth box.

If all boxes are congruent and can be obtained one from another by translation, then equipartition of energy(28) holds due to symmetry. If the boxes are not congruent, then some ‘‘generalized equipartition’’ takes place.To obtain it, we note, that in accordance with Eqs. (19), (22),

a1qH

qa1

� �¼

1

GE

ZO

a1qH

qa1dðE �Hða1; . . . ; aNÞÞda1 . . . daN

¼ �1

GE

ZO

a1qyðE �Hða1; . . . ; aN ÞÞ

qa1da1 da2 . . . daN .

Integrating by parts, we find

a1qH

qa1

� �¼ �

1

GE

Z½aþyðE �Hðaþ; a

0ÞÞ � a�yðE �Hða�; a0ÞÞ�da0 þ

GGE

. (30)

Here a0 ¼ fa2; . . . ; aNg, da0 ¼ da2 . . . daN .Denote by GsðE; aÞ the parameter volume of the sth box,

GsðE; aÞ ¼

ZOyðE �Hða1; . . . ; as�1; a; asþ1; . . . ; aNÞÞda1 . . . das�1 dasþ1 . . . daN . (31)

It relates to probability density function, f sðE; aÞ, of the parameter, as, for a given value of energy

f sðE; aÞ ¼ hdða� akÞi ¼1

GE

ZOdða� asÞdðE �Hða1; . . . ; aN ÞÞda1 . . . daN

as

f sðE; aÞ ¼1

GE

qqE

GsðE; aÞ. (32)

7In case of heat conductivity, it would be more appropriate to call it dissipation, but this is of no importance for our consideration.


Then, from Eqs. (30) and (31),

henergy of the first boxi þaþG1ðE; aþÞ � a�G1ðE; a�Þ

GE

¼ Tm. (33)

Similar relations hold for all other boxes. Formula (33) determines the meaning of equipartition of energy inthis particular problem: not the averaged energies are the same and equal to the temperature ofmicrostructure, but the energies corrected by a term related to the parameter volumes of the extreme valuesof heat conductivity. In the case of elastic bar considered further this correction is a function of temperature.Therefore, the average energy of each box, though not equal to the microstructure temperature, is a universalfunction of the microstructure temperature.

6. Entropy of an elastic bar

In this section the notions introduced above are illustrated by an example where all calculations can be doneanalytically. Consider a bar made up of a sequence of homogeneous sections. The number of sections is N,their Young’s moduli are a1; a2; . . . ; aN , the length of each section is D. The Young’s moduli, a1; a2; . . . ; aN ,play the role of parameters, o. The left end of the bar is clamped, the right end is shifted for uL. The truedisplacement is the minimizing function in the variational problem:

IðuÞ ¼1

2

Z L

0

aðxÞdu

dx

� �2

dx! minuðxÞ:

uð0Þ¼0; uðLÞ¼uL

. (34)

The function aðxÞ is piece-wise constant on the segment ½0;L� ðL ¼ NDÞ and takes the value as whenðs� 1ÞDpxpsD.

The effective Young modulus is defined by the equation

H ¼ minu

IðuÞ ¼1

2aeff e2L; e �

uL

L.

We consider three situations. In the first one each Young’s modulus, ak, can take any value in the segment

a�pakpaþ (35)

and this is the only information which we have about the microstructure. The set, O, is a product of N

segments, ½a�; aþ�, and

jOj ¼ ½a�N ; ½a� � aþ � a�.

Since we do not provide more information about the microstructure, each admissible sequence, fa1; . . . ; aNg, isendowed with the same probability: if the probabilities were different, then we know more than we claimed.So, the probabilistic measure on O is

f ða1; . . . ; anÞ ¼1

½a�N¼ const,

and we can determine the frequency (the probability) of measuring a certain value of energy or the effectiveYoung’s modulus (energy is in one-to-one correspondence with the effective Young’s modulus for a givenaverage strain, eÞ.

The second situation is that the value of energy/effective Young’s modulus is known. Thus, the moduli,a1; . . . ; an, have the distribution on O,

f ða1; . . . ; anÞ ¼1

GE

dðE �Hða1; . . . ; anÞÞ.

In the third case, instead of the effective modulus, we have another information on the microstructure: weknow the arithmetic average of the local moduli:

1

Nða1 þ � � � þ aNÞ ¼ r. (36)


Thus, the probability distribution is

f ða1; . . . ; anÞ ¼ const � d r�1

Nða1 þ � � � þ aNÞ

� �.

We begin with the first case.The ‘‘micro-problem’’ (34) has, obviously, the minimizer

uðxÞ ¼ P

Z x

0

dx

aðxÞ,

where P is a constant determined by the boundary condition

uðLÞ ¼ P

Z L

0

dx

aðxÞ.

Therefore, the effective Young’s modulus is given by the formula

aeff ¼1

1

N

1

a1þ

1

a2þ � � � þ

1

aN

� �. (37)

The effective Young’s modulus can take any value in the segment ½a�; aþ�,

a�paeffpaþ. (38)

The value aeff ¼ a can be obtained, for example, when all a1; . . . ; aN are equal to a. Accordingly, energychanges within the limits

E�pEpEþ; E� ¼12a�e2L; Eþ ¼

12aþe2L.

Denote the probability density function of the effective Young’s modulus by f ðtÞ. The correspondingprobability distribution of energy is f ðE= 1

2e2LÞ= 1

2e2L. Function f ðtÞ is finite when t 2 ½a�; aþ� and is equal to

zero outside ½a�; aþ�. The asymptotics of f ðtÞ for large N, as is explained in Appendix A, has the form

f ðtÞ ¼1

t2

ffiffiffiffiffiffiffiffiffiffiffiN

2pS00

reNSðtÞ; a�ptpaþ. (39)

Here SðtÞ is the solution of the variational problem,

SðtÞ ¼ minx

Sðt;xÞ; Sðt;xÞ ¼ lnM ex=a �x

t, (40)

M is mathematical expectation: for any function, jðaÞ,

Mj �1

½a�

Z aþ

a�

jðaÞda.

The point �x where Sðt;xÞ reaches its minimum value is determined from the equationR aþa�ð1=aÞe �x=a daR aþa�

e �x=a da¼

1

t, (41)

it is a function of t : �x ¼ �xðtÞ. The second derivative, q2Sðt; xÞ=qx2, at the point x ¼ �xðtÞ, is denoted by S00; it isalso a function of t.

The dependence of the dimensionless solution of Eq. (41), �x, on t is shown in Fig. 3 for a�=aþ ¼ 0:1 (in allfigures the parameters which have the dimension of a, in particular, �x and t, are referenced to aþÞ.

When t changes from a� to aþ, the solution, �x, runs over the real axis from +1 to �1.The corresponding function SðtÞ is presented in Fig. 4.

ARTICLE IN PRESS

-0.5

-1

-1.5

-2

-2.5

-3

0.2 0.4 0.6 0.8 1

Fig. 4. Dependence of entropy on the value of the effective coefficient for a�=aþ ¼ 0:1.

X

10

7.5

5

2.5

-2.5

-5

-7.5

-10

0.2 0.4 0.6 0.8 1

t

Fig. 3. Dependence of the solution of Eq. (41) on t for a�=aþ ¼ 0:1.

V.L. Berdichevsky / J. Mech. Phys. Solids 56 (2008) 742–771754

To find the most probable value of the effective coefficient, i.e. the value of t for which SðtÞ reaches itsmaximum, we note that

dSðtÞ

dt¼

dSðt; �xðtÞÞ

dt¼�x

t2. (42)

Hence, at the point where dSðtÞ=dt ¼ 0, we have �x ¼ 0. From Eq. (41) we see that the maximum value of thefunction SðtÞ is reached at the point

am:p: ¼ M1

a

� ��1. (43)

The point where the microstructure entropy reaches its maximum corresponds to the most probable value ofthe effective Young’s modulus.

The maximum of the probability distribution (39) is very sharp for large N; the bigger N, the sharperprobability distribution; a comparison of the probability distributions of various N is shown in Fig. 5.

Total entropy of the microstructure, ~Sm, according to Eq. (39), is

~SmðEÞ ¼ NSE

12e2L

!, (44)

where SðtÞ is the function (40). It is shown in Appendix A that, for positive �x, the dependence ofmicrostructure temperature on energy is determined from the relation

1

Tm¼

dSmðEÞdE

¼d ~SmðEÞdE

. (45)

ARTICLE IN PRESS

1

0.8

0.6

0.4

0.2

10.80.60.40.2

Fig. 5. Probability density functions of the effective coefficient for various numbers of inhomogeneities, N, N ¼ 10 (upper curve),

N ¼ 100 (middle curve), N ¼ 1000 (bottom curve). Probability density functions are normalized to have the unit maximum value.


Therefore, from Eqs. (45), (44), and (42),

1

Tm¼

dðNSðtÞÞ

dt

dt

dE¼

N12e2L

�xðtÞ

t2

t¼E=ð1=2Þe2L

¼1

12e2D

�xðtÞ

t2

t¼E=ð1=2Þe2L

. (46)

Hence, if N;L and E tend to infinity in such a way that energy per unit length, E=L, and the segment size,D ¼ L=N, are finite, temperature of microstructure is finite. For �xo0, the microstructure temperature goes toinfinity in this limit (see Appendix A).

The complementary temperature is finite and given by Eq. (46) for �xo0,

1

T�m¼

dS�mðEÞdE

¼d ~SmðEÞdE

¼1

12e2D

�xðtÞ

t2

t¼E=ð1=2Þe2L

, (47)

T�m tends to �1 as N !1 for �x40.Consider now the second case when the value of energy is known. An interesting question here is:

What is the distribution of local Young’s modulus, say, a1, if one knows the value of the effective Young’smodulus, aeff .

It is shown in Appendix B that the probability density function of each Young modulus becomesexponential,

f ðajaeff Þ ¼ c eb=a, (48)

where c is a normalizing coefficient while b is a function of aeff determined from the equation

Mð1=aÞ eb=a

M eb=a¼

1

aeff. (49)

This probability density function is homogeneous only when b ¼ 0, i.e. when the effective coefficient has thevalue, a,

1

a¼M

1

a�

1

½a�

Z aþ

a�

da

a.

In the third case we seek for the probability density of the effective Young’s modulus under condition thatthe arithmetic average of the local moduli is known,

f ðt j rÞ ¼ cM dðt� aeff Þd r�a1 þ � � � þ aN

N

� �h i. (50)


Here the constant c is determined by the normalizing conditionZf ðt j rÞdt ¼ 1.

Its asymptotics as N !1 is given by the formula (Appendix C)

f ðt j rÞ ¼ const eNSðt;rÞ, (51)

where Sðt;rÞ is the minimum value in the variational problem:

Sðt;rÞ ¼ minx;x

Sðt;r;x; xÞ, (52)

Sðt;r;x; xÞ ¼ �x

tþ xrþ ln

Z aþ

a�

ex=a�xa da

½a�. (53)

The minimizing point, f �x; �xg, of the variational problem (52) is determined by the system of two nonlinearequations:R aþ

a�ð1=aÞ e �x=a��xa daR aþa�

e �x=a��xa da¼

1

t;

R aþa�

a e �x=a��xa daR aþa�

e �x=a��xa da¼ r. (54)

The most probable value corresponds to the point where

d

dtSðt;r; �xðtÞ; �xðtÞÞ ¼ 0.

Since

d

dtSðt;r; �xðtÞ; �xðtÞÞ ¼

�x

t2

at this point, we have �x ¼ 0. So, for �x ¼ 0, t is the value of the most probable effective coefficient, am:p:. FromEqs. (54), for a given r and �x ¼ 0, �x must be found from the equationR aþ

a�ae�

�xa daR aþa�

e��xa da¼ r. (55)

If Eq. (55) is solved with respect to �x, and this value of �x, �xðrÞ, is inserted in the first Eq. (54), one obtains thedependence of the most probable value of the effective coefficient, am:p:, on r:

am:p: ¼

R aþa�

e��xðrÞa daR aþ

a�1ae��xðrÞa da

. (56)

The relation (56) can be interpreted, similarly to Eq. (43), as an inverse mathematical expectation of a�1,

am:p: ¼ ~M1

a

� ��1, (57)

with an ‘‘effective’’ probabilistic density

~f ðaÞ ¼e�

�xðrÞaR aþa�

e��xðrÞa da. (58)

If r coincides with an ‘‘average value’’ of the ‘‘unconstrained’’ case,

r ¼1

½a�

Z aþ

a�

a da ¼a� þ aþ

2,

then, from Eq. (55), �x ¼ 0, and, from Eq. (58), ~f ðaÞ ¼ const ¼ ½a��1; hence M ¼ ~M, and the most probableeffective coefficient (57) coincides with Eq. (43). For all other values of r, the most probable effective

ARTICLE IN PRESS

1

0.8

0.6

0.4

0.2

0.2 0.4 0.6 0.8 1

Fig. 6. Dependence of the most probable effective coefficient on the value of the parameter r (the coefficients are referenced to aþ, and the

ratio a�=aþ is 0:1).


coefficient differs from Eq. (43). The dependence of the most probable effective coefficient on r is shown inFig. 6.

7. Small Gaussian disturbances of homogeneous media

Another example where the computations can be done analytically is the case of small Gaussiandisturbances of homogeneous media. Consider the variational problem

IðuÞ ¼

ZV

1

2aijðx;oÞ

qu

qxi

qu

qxjd3x�

ZqV

qiniudA! min

uðxÞ, (59)

where qi are some constants. We take the Neumann type problem (59) instead of the Dirichlet type problem(9) to simplify some further technicalities.8

The minimum value in the variational problem (59) is negative and equal to

�I ¼ �12að�1Þijeff qiqjjV j.

This formula can be considered as a definition of the effective coefficients, aijeff , a

ð�1Þijeff being the inverse tensor

to aijeff . Energy is introduced as

E ¼ � �I=jV j ¼ 12að�1Þijeff qiqj.

Let the local fields aijðx;oÞ be small disturbances of the homogeneous field:

aijðx;oÞ ¼ aijþ a0ijðx;oÞ,

where aij are some constants while a0ijðx;oÞ are small and have some order e. Without loss of generality we canput Ma0ijðx;oÞ ¼ 0. In addition, for simplification of further formulas, we accept that the space average of thefields a0ijðx;oÞ vanishes for each o

ha0ijðx;oÞi ¼ 0. (60)

In this section h�i denotes the space average over the region V : for any function j,

hji �1

jV j

ZV

jd3x.

The minimizing function in the variational problem (59) can be presented as

u ¼ uþ u0, (61)

8Note that the entropy of microstructure is a non-local characteristic and may depend on the type of the boundary conditions.


where u is the solution of the homogeneous problem:

aij uj ¼ qi; uj �qu

qxj; u ¼ a

ð�1Þij qixj þ const, (62)

while u0 is small. The minimizing function in the variational problem (59) is determined up to an arbitraryconstant, therefore the constant in Eq. (62) can be put equal to zero. By the same reason, function u0 can beassumed satisfying the constraint

hu0i ¼ 0. (63)

In what follows, ui are considered as some constants found from the homogeneous problem (62).Plugging Eq. (61) in Eq. (59) we have

Iðuþ u0Þ ¼

ZV

1

2ðaijþ a0ijðx;oÞÞ ui þ

qu0

qxi

� �uj þ

qu0

qxj

� �d3x�

ZqV

qiniðuþ u0ÞdA

¼1

2aij uiujjV j þ

ZV

1

2a0ijðx;oÞuiuj d

3xþ

ZV

aij ui

qu0

qxjd3x

þ

ZV

a0ijðx;oÞui

qu0

qxjd3xþ

ZV

1

2aij qu0

qxi

qu0

qxjd3x

þ

ZV

1

2a0ijðx;oÞ

qu0

qxi

qu0

qxjd3x�

ZqV

qiniudA�

ZqV

qiniu0 dA.

The sum of the first and the seventh terms is equal to � 12

aij uiujjV j due to Eq. (62); we denote it by�EjV j; E � 1

2aij uiuj. The second term vanishes due to Eq. (60). The sum of the third and the eighth terms

vanishes due to Eq. (62). The sixth term is small compared with the fifth one and can be dropped. Finally, thevariational problem (59) is reduced to the problem

Iðuþ u0Þ ¼ �EjV j þ ~Iðu0Þ; ~Iðu0Þ ! minu0

,

~Iðu0Þ ¼

ZV

1

2aij qu0

qxi

qu0

qxjd3xþ

ZV

a0ijðx;oÞui

qu0

qxjd3x.

Obviously, min ~Iðu0Þ�e2, and the only term dropped is on the higher order, e3.We are going to show that, under some assumptions on the random field a0ijðx;oÞ, the probability density

function of energy is given by the formulas

f ðEÞ ¼ c eNSðEÞ; N ¼ jV j=a3, (64)

where a is the correlation radius of the local characteristics,

SðEÞ ¼ minz

SðE; zÞ (65)

and

SðE; zÞ ¼ ðE � E0Þz�

Z þ1�1

ln 1þ zupuqBpjql

ðZsÞZjZl

apqZpZq

!d3Z. (66)

Functions BpjqlðZsÞ are connected to the correlation tensor of the random field a0ijðx;oÞ and defined further

by Eq. (82).The derivation of Eq. (64) is based on the formula for probability density function of the minimum values of

stochastic quadratic functional, 12ðAu; uÞ � ðl; uÞ (Berdichevsky, 1999):

f ðEÞ ¼1

2pi

Z i1

�i1

eEz

ZMðe�1=2ðAu;uÞþi

ffiffizpðl;uÞÞDAu

�dz, (67)

where DAu is the ‘‘volume element’’ in the space of functions uðxÞ. For reader’s convenience the necessaryexplanations regarding this formula are summarized in Appendices D and E and placed in ElectronicAnnex (see also the upcoming new edition of the monograph Berdichevsky, 1983); its applications to


other problems can be found in (Le and Berdichevsky, 2001; Berdichevsky and Le, 2002; Berdichevsky,2002, 2007).

According to Eq. (67),

f ðEÞ ¼1

2pi

Z i1

�i1

eðE�EÞjV jz

ZMðe

�R

V12aijðqu0=qxiÞðqu0=qxj Þd3x�i

ffiffizp R

Va0ij ðx;oÞuiðqu0=qxj Þ d3x

ÞDAu0 �

dz. (68)

Let the field a0ijðx;oÞ be Gaussian. By definition, that means that for any deterministic functions, jpq,

M eiR

Va0pqðx;oÞjpqðxÞ d

3x¼ e�ð1=2Þ

RV

RV

Bpqkl ðx; ~xÞjpqðxÞjkl ð ~xÞ dx d ~x, (69)

where Bpqklðx; ~xÞ is the correlation tensor,

Bpqklðx; ~xÞ ¼Ma0pqðx;oÞa0klð ~x;oÞ

and ZV

ZV

Bpqklðx; ~xÞjpqðxÞjklð ~xÞdxd ~x ¼M

ZV

a0pqðx;oÞjpqðxÞd3x

� �2

.

For Gaussian fields formula (68) takes the form:

f ðEÞ ¼1

2pi

Z i1

�i1

eðE�EÞjV jz

�

Ze�R

V12apqðqu0=qxpÞðqu0=qxqÞ d3x�z1

2upuq

RV

RV

Bpjql ðx; ~xÞðqu0=qxj ÞðxÞðqu0=qxl Þð ~xÞ dx d ~xDAu0 �

dz.

ð70Þ

To compute the functional integral appeared in Eq. (70) we have to further specify the random field a0ijðx;oÞ.Let V be a box of the size L : jxijpL=2, and yi be the dimensionless coordinates, yi ¼ 2pxi=L; jyijpp.Consider the Fourier series expansion of the functions a0ijðy;oÞ :

a0mjðy;oÞ ¼Xk2Z0

amjðkÞðoÞ e

iky, (71)

where k is the vector of the integer grid, Z0, with the excluded origin9: k ¼ fk1; k2; k3g, ki¼ 0;1;2; . . .,

kikia0 and ky � kiyi. The coefficients, amjðkÞðoÞ, are some random numbers with zero mean defined at each

point of the grid, Z0. Functions, a0mjðy;oÞ, are real-valued, therefore their Fourier coefficients, amjðkÞ, obey the

conditions

amjðkÞ ¼ a

mjð�kÞ, (72)

bar denotes complex conjugation.Let us present the variable of integration in the functional integral, u0, in terms of the Fourier series,

u0 ¼X

Z0

uðkÞ eiky. (73)

The term corresponding to the origin of the grid, k ¼ 0, is absent in Eq. (73) due to the condition (63). Forreal-valued u0,

uðkÞ ¼ uð�kÞ. (74)

The Fourier coefficients, uðkÞ, may be viewed as the coordinates of the functions u0ðyÞ in the basis eiky.Integration in the functional space may be conducted over the coordinates, uðkÞ. To this end we have to selectthe independent coordinates since not all uðkÞ are independent due to the constraint (74). This can be done inthe following way: we split the grid, Z0, into two subgrids, Z0þ and Z0�, which do not overlap and form

9The origin is excluded due to Eq. (60).


together the entire grid, Z0 ¼ Z0þ þ Z0�, and possess the property: if k 2 Z0þ, then �k 2 Z0�. Such a partitioncan be done in various ways, and its particular choice is not essential for what follows. The complexcoordinates, uðkÞ, for k 2 Z0þ are independent. One can use as the real-valued coordinates in the functionalspace the real and imaginary parts of uðkÞ for k 2 Z0þ which we denote by u0ðkÞ and u00ðkÞ.

The quadratic functional,ZV

apq qu0

qxp

qu0

qxqd3x,

in terms of the coordinates, uðkÞ, takes the form:ZV

apq qu0

qxp

qu0

qxqd3x ¼

L

2p

Zjyi jpp

apq qu0

qyp

qu0

qyqd3y ¼

L

2p

Zjyi jpp

apq qu0

qyp

qu0

qyqd3y

¼L

2p

Zjyi jpp

Xk;m2Z0

apqkpmquðkÞ eiðk�mÞyuðmÞ d

3y

¼ Lð2pÞ2Xk2Z0

apqkpkquðkÞuðkÞ ¼ 2Lð2pÞ2X

k2Z0þ

apqkpkquðkÞuðkÞ

¼ 2Lð2pÞ2X

k2Z0þ

apqkpkqððu0ðkÞÞ

2þ ðu00ðkÞÞ

2Þ. ð75Þ

Similarly,

upuq

ZV

ZV

Bpjqlðx; ~xÞqu0

qxjðxÞ

qu0

qxlð ~xÞdxd ~x

¼L

2p

� �4

upuq

Zjyi jpp

Zj ~yi jpp

Bpjqlðy; ~yÞqu0

qyjðxÞ

qu0

q ~ylð ~yÞdyd ~y

¼L

2p

� �4

upuq

Zjyi jpp

Zj ~yi jpp

Bpjqlðy; ~yÞX

k;m2Z0

kjuðkÞ eikymluðmÞ e

�im ~y dyd ~y.

Denote by Bpjqlðk;mÞ the quantities,

Bpjqlðk;mÞ ¼

1

ð2pÞ6

Zj ~yi jpp

Bpjqlðy; ~yÞ eiðky�m ~yÞ dyd ~y.

Obviously,

Bpjqlðk;mÞ ¼

1

ð2pÞ6

Zj ~yi jpp

Ma0pjðy;oÞa0qlð ~y;oÞ eiðky�m ~yÞ dyd ~y

¼1

ð2pÞ6

Zj ~yi jpp

MX

r;s2Z0

apjðrÞðoÞ e

�iryaqlðsÞðoÞ e

isy eiðky�m ~yÞ dyd ~y ¼MapjðkÞa

qlðmÞ. ð76Þ

Let aqlðkÞ be statistically independent for various k 2 Z0þ. Then B

pjqlðk;mÞ ¼ 0 for mak. Denote B

pjqlðk;mÞ for m ¼ k

by BpjqlðkÞ , B

pjqlðkÞ ¼Ma

pjðkÞa

qlðkÞ. Obviously, B

pjqlðkÞ ¼ B

qlpj

ðkÞ ¼ Bqlpjð�kÞ, and B

pjqlðkÞ are real for ðp; jÞ ¼ ðq; lÞ. We assume that

they are real for all values of indices. Finally,

upuq

ZV

ZV

Bpjqlðx; ~xÞqu0

qxjðxÞ

qu0

qxlð ~xÞdxd ~x

¼ L4ð2pÞ2upuq

Xk2Z0

BpjqlðkÞ kjkluðkÞuðkÞ

¼ 2L4ð2pÞ2upuq

Xk2Z0

þ

BpjqlðkÞ kjklððu

0ðkÞÞ

2þ ðu00ðkÞÞ

2Þ.


Let us introduce the coordinates, vðkÞ, putting vðkÞ ¼ u0ðkÞ for k 2 Z0þ and vðkÞ ¼ u00ðkÞ for k 2 Z0�. Then, the lastrelation can be written as

upuq

ZV

ZV

Bpjqlðx; ~xÞqu0

qxjðxÞ

qu0

qxlð ~xÞdxd ~x ¼ L4ð2pÞ2

Xk2Z0

upuqBpjqlðkÞ kjklv

2ðkÞ. (77)

In the same way, from Eq. (75) we haveZV

apq qu0

qxp

qu0

qxqd3x ¼ Lð2pÞ2

Xk2Z0

apqkpkqv2ðkÞ.

Therefore, the functional integral to be computed can be written asZe�ð1=2Þ

RV

apqðqu0=qxpÞðqu0=qxqÞd3x�zð1=2Þupuq

RV

RV

Bpjql ðx; ~xÞðqu0=qxj ÞðxÞðqu0=qxl Þð ~xÞ dx d ~xDAu0

¼

Ze�ð1=2ÞLð2pÞ2

Pk2Z0

apqkpkqv2ðkÞ�zð1=2ÞL4ð2pÞ2

Pk2Z0

upuqBpjql

ðkÞkjklv

2ðkÞ

�Yk2Z0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiLð2pÞ2apqkpkq

qdvðkÞffiffiffiffiffiffi2pp . ð78Þ

The infinite-dimensional integral (78) should be understood as a limit of the corresponding finite-dimensional integrals. Denote the function of z, Eq. (78), by exp½FðzÞ�. Since the infinite-dimensional integral(78) splits into the product of one-dimensional integrals

eFðzÞ ¼Yk2Z0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiLð2pÞ2apqkpkq

Lð2pÞ2apqkpkq þ zL4ð2pÞ2upuqBpjqlðkÞ kjkl

;

vuut (79)

or

FðzÞ ¼ �Xk2Z0

ln 1þ zL3upuqB

pjqlðkÞ kjkl

apqkpkq

!. (80)

In a typical situation, BpjqlðkÞ decay exponentially for large k2

¼ ksks,

BpjqlðkÞ �d

3 e�d2k2 (81)

d being the correlation radius. Indeed, since amjðkÞ are statistically independent for various k, Bpjqlðy; ~yÞ are

functions only of the difference, y� ~y :

Bpjqlðy; ~yÞ ¼Ma0pjðy;oÞa0qlð ~y;oÞ ¼Ma0pjðy;oÞa0qlð ~y;oÞ

¼MX

k;m2Z0

apjðkÞðoÞ e

ikyaqlðmÞðoÞ e

�im ~y

¼MXk2Z0

apjðkÞðoÞa

qlðkÞðoÞ e

ikðy� ~yÞ ¼Xk2Z0

Bpjql

ðkÞ eikðy� ~yÞ.

The correlation tensor, Bpjqlðy� ~yÞ, is a periodic function of x ¼ y� ~y with the Fourier coefficients, Bpjql

ðkÞ . If

Bpjqlðy� ~yÞ are practically vanishing for jy� ~yj4d, for example, Bpjqlðy� ~yÞ behaves as exp½�x2=2d2�, then,


for small d, the estimate (81) holds true:

Bpjql

ðkÞ ¼1

ð2pÞ3

Zjyi jpp

BpjqlðxÞ eikx d3x�Z

e�ð1=2d2Þxsxsþiksxs

d3x

�Ys¼3s¼1

Z þ1�1

e�ð1=2d2Þx2þiksx dx ¼ ð

ffiffiffiffiffiffi2pp

dÞ3 e�ð1=2Þd2k2 .

Therefore, we assume that BpjqlðkÞ can be presented in the form

BpjqlðkÞ ¼ d3Bpjql

ðdksÞ, (82)

where BpjqlðZsÞ is a smooth function of Zs decaying at infinity fast enough to provide the convergence of the

integral which further appears in Eq. (83).Denote by z1 the variable z1 ¼ zL3d3. Then, for d! 0 and the finite z1, the sum (80) converges to the

integral

F0ðz1Þ ¼ �1

d3

Z þ1�1

ln 1þ z1upuqBpjql

ðZsÞZjZl

apqZpZq

!d3Z. (83)

Let us change the variable of integration in Eq. (70) z! z1: Then the integral (70) takes the form

f ðEÞ ¼1

2piL3d3

Z i1

�i1

eð1=d3ÞSðE;zÞ dz,

where SðE; zÞ is given by Eq. (66). The asymptotics of this integral as d! 0 in a generic case is Eq. (64) whereN ¼ d�3 and SðEÞ is determined from Eq. (65).

8. On phenomenology of work hardening

Plastic deformation is a result of the microstructure evolution. Therefore, the preceding discussion suggestssome rectification of work hardening phenomenology which involves microstructure entropy andmicrostructure temperature (the entropy and the temperature of the dislocation networks). First we recallthe necessary facts from plasticity theory.

In classical plasticity theory, solids are characterized by displacement field, thermodynamic entropy density,S, and plastic deformations, eðpÞij . To obtain a closed system of equations from thermodynamic reasoning, onehas to specify the internal energy density, U , and dissipation density, D (see, e.g., Sedov, 1971; Kamenjarzh,1996; Berdichevsky, 2006b). For example, von Mises theory corresponds to the assumptions,

U ¼ U0ðeðeÞij ;SÞ; D ¼ Kð_eðpÞij _e

ðpÞij Þ

1=2, (84)

where _eðpÞij � deðpÞij =dt, and K is the yield stress. In the case of work hardening, one often assumes (see review inHill, 1950) that K is a function of plastic work, Wp, determined by the loading history from the equation

dWp

dt¼ sij

deðpÞij

dt, (85)

sij being the stress tensor. This assumption closes the system of equations.To make a link to microstructure evolution, we note (see Berdichevsky, 2006a) that the internal energy

density of a crystal or polycrystal is a sum of energy density, U0ðeðeÞij ;SÞ, and energy of microstructure, Um.

Then the first law of thermodynamics yields the equation (we assume for simplicity that the solid is non-heat-conducting)

TdS

dtþ

dUm

dt¼ sij

deðpÞij

dt. (86)


Here T is thermodynamic temperature. The plastic work rate, the right-hand side of Eq. (86), goes to heatingthe body, the first term in the left-hand side and to latent heat (the second term in the left-hand side).According to Taylor and Quinney (1934),

dUm

dt¼ Ksij

deðpÞij

dt, (87)

where K is a coefficient of the order 0.1. According to Eqs. (85) and (87), one can set

Wp ¼ KUm, (88)

and, if K is given as a function of Wp or Um, one has a closed model of work hardening.The situation changes if one wishes to describe an interplay between the evolution of microstructure and

work hardening. Let the microstructure be characterized by only one parameter, the total length of dislocationlines per unit volume, r. Moreover, let r be a thermodynamic parameter, i.e. Um ¼ UmðrÞ. It was shown inBerdichevsky (2006a)10 that the Voce law of work hardening corresponds to the function

UmðrÞ ¼ km ln1

1�

ffiffiffiffiffirrs

r �

ffiffiffiffiffirrs

r0BB@

1CCA, (89)

where k and rs are the phenomenological constants, m shear modulus.It is hard to expect, however, that r is the only thermodynamic parameter, i.e. Um is a function of r only. If

we are able to measure in the experiments only r and Um, then, most probably, energy, Um, is not determineduniquely by r, and we need to include an additional parameter, entropy of microstructure, which compensatesfor such a shortened description of the microstructure. It is too early to speculate on the possible constitutiveequations due to insufficient experimental data for Um and r, but the following general framework seemsplausible.

Let Um be a function of r and Sm, or Sm be a function of Um and r. It is natural to assume that SmðUm;rÞincreases with Um for a fixed r. Therefore,

Tm ¼qUmðr;SmÞ

qSm40.

Besides, energy of microstructure, Um, increases if the total length of dislocations, r, grows, so qUm=qr isalso positive.

Suppose first that the stresses are zero. For any given values of Sm and r, if they do not correspond to theminimum of Um, some relaxation process should occur in the material. In this process Um goes to itsminimum value. In the relaxation process the total length of dislocation lines decreases. Besides, themicroscopic chaos measured by Sm should also decrease. According to Eq. (86),

TdS

dt¼ �Tm

dSm

dt�

qUm

qrdrdt

. (90)

The relaxation process causes heating, and the right-hand side of Eq. (90) must be positive. The simplestrelaxation law is

dSm

dt¼ �l1Tm � l12

qUm

qr;

drdt¼ �l21Tm � l2

qUm

qr. (91)

The coefficients in Eq. (91) must be such that the right-hand side of Eq. (90) is positive.

10Note a misprint in formula (18) of this paper: the inequality sigh must be inverted.


The plastic work acts as a ‘‘driving force’’ which increases Sm and r. Therefore, for non-zero stresses, onecan put

dSm

dt¼ �l1Tm � l12

qUm

qrþ m1s

ijdeðpÞij

dt,

drdt¼ �l21Tm � l2

qUm

qrþ m2s

ijdeðpÞij

dt

with some positive coefficients, m1;m2. The yield stress, as was suggested by Taylor, is a linear function offfiffiffirp

,while other relations of classical plasticity theory remain unchanged. That brings a closed system of equations.In particular, if l1 ¼ l12 ¼ l21 ¼ l2 ¼ m1 ¼ 0 and m2 ¼ KðqUm=qrÞ

�1, while Um is given by Eq. (89), we arriveat the Voce law.

This closure attempt suggests that the experiments are now in order to shed light on the real behavior of theparameters involved.

Appendix A. Derivation of Eq. (39)

In derivation of the asymptotic formulas like Eq. (39) it is convenient to deal with a smoother probabilitycharacteristics of the probability distribution like

gðtÞ ¼ Probabilityfaeffptg. (92)

Obviously,

f ðtÞ ¼dgðtÞ

dt. (93)

The definition (92) can be written as

gðtÞ ¼My t�N

a�11 þ � � � þ a�1N

� �. (94)

The step function can be presented as the following integral in the complex plane z ¼ xþ iy:

yðtÞ ¼1

2pi

Z aþi1

a�i1etz dz

z. (95)

The integral is taken along the line ½a� i1; aþ i1� in the right half-plane, a40. The integrand is ananalytical function whose absolute values tend to zero along any line x ¼ Re z ¼ const. Therefore the line canbe moved arbitrarily in the right half-plane.

Plugging Eq. (95) into Eq. (92) we have

gðtÞ ¼1

2piM

Z aþi1

a�i1

1

zetz�Nz=ða�1

1þ��þa�1

NÞ dz. (96)

Let us make a change of variables z!a�11þ��þa�1

Nt

z. Then

gðtÞ ¼1

2piM

Z aþi1

a�i1

1

zeða�11þ��þa�1

NÞz�Nz=t dz. (97)

Changing the order of computation the mathematical expectation and the integral over z we have

gðtÞ ¼1

2pi

Z aþi1

a�i1

1

zeNSðt;zÞ dz; Sðt; zÞ ¼ lnM ez=a �

z

t. (98)

It is more convenient to deal with another integral which converges absolutely,

GðtÞ ¼1

2pi

Z aþi1

a�i1

1

z2eNSðt;zÞ dz. (99)


Function gðtÞ is expressed in terms of GðtÞ as

gðtÞ ¼dG

dt

t2

N. (100)

To find the asymptotics of the integral (99) as N !1 we use the method of steepest descent. To this end, wenote that on the real axis the function, Sðt; zÞ, is real and strictly convex. Therefore, for each value of t it hasthe only minimum. The minimum can be achieved though at �1 or þ1. If the point of minimum, �x, is finite,it is the solution of the equation,

Mð1=aÞ e �x=a

M e �x=a¼

1

t. (101)

The point of minimum is a function of t : �x ¼ �xðtÞ. Function Sðt; �xðtÞÞ is denoted also as SðtÞ.Function SðtÞ is negative. Indeed, for any convex function, jðxÞ,

jðxÞ � jð0Þ ¼Z x

0

djðxÞdx

dxpdjðxÞdx

x,

because its second derivative is positive and, thus, djðxÞ=dx is non-decreasing function of x. Function jðxÞ ¼logM ex=a is a convex function of x (since its second derivativeR aþ

a�ð1=a2Þ ex=a da

R aþa�

ex=a da� ðR aþ

a�ð1=aÞ ex=a daÞ2

ðR aþ

a�ex=a daÞ2

is positive11), therefore, for any x,

logM ex=apMð1=aÞ ex=a

M ex=ax.

Plugging here instead of x the solution, �xðtÞ, of Eq. (101), we obtain SðtÞp0 as claimed.Function of x,R aþ

a�ð1=aÞ ex=a daR aþa�

ex=a da, (102)

is an increasing function of x because its derivative is positive (see the footnote). The ratio (102) tends to 1=a�when x!�1 and to 1=aþ as x!þ1. Accordingly, Eq. (101) has a solution when a�otoaþ. Let t be fromthis interval, and let �x be positive. We choose a in Eq. (99) equal to �x. In the vicinity of the point �x,

Sðt; zÞ ’ Sðt; �xÞ þ1

2

d2S

dz2

�x

ðz� �xÞ2. (103)

The second derivative, d2S=dz2, at the point �x is a real number,

d2S

dz2¼

Mð1=a2Þ e �x=aM e �x=a � ðMð1=aÞ e �x=aÞ2

ðM e �x=aÞ2

,

which is a positive (see the footnote) function of t. We denote it by S00. Emphasize that S00ad2SðtÞ=dt2. On theline ½ �x� i1; �xþ i1� function ReSðt; zÞ has, according to Eq. (103), a maximum at the point y ¼ 0 :

ReSðt; �xþ iyÞ ¼ ln jM e �x=aþiy=aj ��x

t’ SðtÞ �

1

2S00y2; SðtÞ � lnM e �xðtÞ=a �

�xðtÞ

t.

11Positiveness follows from the inequality (we give it in a more general setting assuming that the random number, a, has some

probability density function, f ðaÞ):

M 1ae �x=a

� 2¼R

1ae �x=af ðaÞda

� 2¼

Re �x=2a

a

ffiffiffiffiffiffiffiffiffif ðaÞ

pe �x=2a


pda

� �2pR

e �x=a

a2f ðaÞda

Re �x=af ðaÞda ¼M e �x=a

a2M e �x=a:


The values of ReSðt; �xþ iyÞ at any point y is less than ReSðt; �xÞ ¼ SðtÞ :

ReSðt; �xþ iyÞ ¼ ln jM e �x=aþiy=aj ��x

t

p lnMj e �x=aþiy=aj ��x

t¼ lnM e �x=a �

�x

t¼ SðtÞ. ð104Þ

More instructive estimate is to use the Cauchy inequality, ðR

fgÞ2pR

f 2R

g2,

jM e �x=aþiy=aj2 ¼

Ze �x=a cos

y

af ðaÞda

� �2

þ

Ze �x=a sin

y

af ðaÞda

� �2

¼

Ze �x=2a


pe �x=2a


pcos

y

ada

� �2

þ

Ze �x=2a


pe �x=2a


psin

y

ada

� �2

pZ

e �x=af ðaÞda

Zrf ðaÞcos2

y

adaþ

Ze �x=af ðaÞda

Ze �x=af ðaÞsin2

y

ada

¼

Ze �x=af ðaÞda

� �2

¼ ðM e �x=aÞ2. ð105Þ

In the Cauchy inequality, the equality sign holds only if the functions f and g are proportional. In theestimate (105), the functions involved are not proportional, thus we can conclude that

ReSðt; �xþ iyÞoSðt; �xÞ for ya0. (106)

Moreover, jM e �x=aþiy=aj decays as jyj�1, and ReSðt; �xþ iyÞ becomes negative for sufficiently large jyj :

jM e �x=aþiy=aj ¼

Z aþ

a�

e �x=a eiy=af ðaÞda

¼

Z aþ

a�

e �x=af ðaÞa2 1

ydðeiy=aÞ

¼1

y½e �x=af ðaÞa2 eiy=a�aþa� �

1

y

Z aþ

a�

eiy=a d

daðe �x=af ðaÞa2Þ

pconst

jyj¼

c

jyj. ð107Þ

Split the integral (99) into two parts, for jyjpe and jyjXe

GðtÞ ¼1

2p

Z e

�e

1

ð �xþ iyÞ2eNðSðt; �xÞ�ð1=2ÞS00y2Þ dyþ

1

2p

ZjyjXe

1

ð �xþ iyÞ2eNSðt; �xþiyÞ dy (108)

According to Eq. (107), for any constant d, we can choose a constant, l, such that ReSðt; �xþ iyÞod forjyjXl. On the finite segment jyjpl, function of y, ReSðt; �xþ iyÞ, is continuous, does not exceed Sðt; �xÞ andtakes the value Sðt; �xÞ only at one point, y ¼ 0. Therefore, there is a small e, such that ReSðt; �xþiyÞoReSðt; �xþ ieÞ for eojyjol. We see that ReSðt; �xþ iyÞoReSðt; �xþ ieÞ for all jyjXe. Therefore the secondintegral in (108) can be estimated as

1

2p

ZjyjXe

1

ð �xþ iyÞ2eNSðt; �xþiyÞ dy

p 1

2peNRe Sðt; �xþieÞ2

Z 10

dy

�x2 þ y2. (109)

Note that

1

2p

Z e

�e

1

ð �xþ iyÞ2e�ð1=2ÞNS00y2 dy ¼

1

2p

Z 1�1

e�ð1=2ÞNS00y2

ð �xþ iyÞ2dy�

1

2p

ZjyjXe

e�ð1=2ÞNS00y2

ð �xþ iyÞ2dy. (110)


In the first integral,

1

2p

Z 1�1

e�ð1=2ÞNS00y2ð �x2 � y2 � 2i �xyÞ

ð �x2 þ y2Þ2

dy ¼1

2p

Z 1�1

�x2 � y2

ð �x2 þ y2Þ2e�ð1=2ÞNS00y2 dy,

we make the change of variables, y! y=ffiffiffiffiffiffiffiffiffiNS00p

. We obtain

1

2pffiffiffiffiffiffiffiffiffiNS00p

Z 1�1

�x2 �y2

NS00

ð �x2 þy2

NS00Þ2e�ð1=2Þy

2dy.

As N !1, this integral is equal to

1

2pffiffiffiffiffiffiffiffiffiNS00p

1

�x2

Z 1�1

e�ð1=2Þy2dy ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

2pNS00

r1

�x2. (111)

The second integral in Eq. (110) does not exceed the number,

2

Z 10

dy

ð �x2 þ y2Þ2e�ð1=2ÞNS00e2 ,

which is asymptotically smaller than Eq. (111). Similarly the right-hand side of Eq. (109), which is in the orderof exp½NReSðt; �xþ ieÞ�, is asymptotically smaller than Eq. (111) (recall that ReSðt; �xþ ieÞoSðt; �xÞ). Finallywe obtain in the leading approximation

GðtÞ ¼1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2pNS00ðtÞp

�x2eNSðtÞ. (112)

Differentiating Eq. (112) and taking into account that the derivative of the prefactor gives a negligiblecontribution, from Eqs. (100) and (42) we have

gðtÞ ¼eNSðtÞffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2pNS00p

�xðtÞfor �x40. (113)

If �xo0, then the line of integration ½aþ i1; aþ i1� in Eq. (99) must be moved in the left half plane. Thesingularity of the integrand at z ¼ 0 causes the addition of the residual at z ¼ 0 :

GðtÞ ¼1

2pi

Z aþi1

aþi1

1

z2eNSðt;zÞ dzþ

d eNSðt;zÞ

dz

z¼0

¼1

2pi

Z aþi1

aþi1

1

z2eNSðt;zÞ dzþN M

1

a�

1

t

� �; ao0. ð114Þ

From Eqs. (100) and (114),

gðtÞ ¼1

2pi

Z aþi1

aþi1

1

zeNSðt;zÞ dzþ 1; ao0. (115)

Applying the steepest descent method to the integral in Eq. (115) we find

gðtÞ ¼eNSðtÞffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2pNS00p

�xðtÞþ 1; �xo0. (116)

Probability density function, f ðtÞ ¼ dgðtÞ=dt, is obtained by differentiation of Eqs. (113) and (116) with theuse of Eq. (42). We arrive at the universal relation holding for both positive and negative �x :

f ðtÞ ¼N eNSðtÞffiffiffiffiffiffiffiffiffiffiffiffiffiffi2pNS00p

t2. (117)

If toa� or t4aþ, then f ðtÞ ¼ 0. This follows directly from the definition of gðtÞ and yðtÞ or can be derivedfrom Eqs. (98) or (99). For example, if toa�, then Sðt; xÞ ! �1 as x!1; on the other hand, ReSðt;xþ iyÞpSðt; xÞ; and moving the line of integration by increasing a to infinity, we see that GðtÞ ¼ 0 for toa�.


It is seen from the derivation that we consider the limit N !1 for a fixed nonzero number, �x, thereforeeNSðtÞ=

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi2pNS00p

�x in Eqs. (113) and (116) is exponentially small due to negativeness of entropy. In the formulafor the probability density (117) the singularity, 1= �x, disappears, and Eq. (117) holds for all a�otoaþ.

As follows from Eq. (117),

~SmðEÞ ¼ NSðtÞjt¼E=ð1=2Þe2L

On the other hand, for the parameter volumes we have

GðEÞ ¼ jOjgE

12e2L

!; G�ðEÞ ¼ jOj 1� g

E12e2L

! !. (118)

Therefore,

SmðEÞ ¼ logGðEÞjOj¼ log g

E12e2L

!¼

NSðtÞjt¼E=ð1=2Þe2L for �x40;

eNSðtÞffiffiffiffiffiffiffiffiffiffiffiffiffiffi2pNS00p

�xðtÞ

t¼E=ð1=2Þe2L

for �xo0:

8>><>>:

Here we neglected the logarithm of the prefactor in Eq. (113) as small compared to the leading term, NS, andused that the first term in Eq. (116) is much smaller than unity due to negativeness of SðtÞ: Accordingly,

~SmðEÞ ¼ SmðEÞ for EoE.

Similarly,

S�mðEÞ ¼ logG�ðEÞjOj¼ log 1� g

E12e2L

! !¼�

eNSðtÞffiffiffiffiffiffiffiffiffiffiffiffiffiffi2pNS00p

�xðtÞ

t¼E=ð1=2Þe2L

for �x40;

NSðtÞjt¼E=ð1=2Þe2L for �xo0:

8>><>>:

Hence, as was claimed in Section 5,

~SmðEÞ ¼ S�mðEÞ for E4E.

Accordingly, ~TmðEÞ � ðd ~SmðEÞ=dEÞ�1, coincides with TmðEÞ for EoE and with T�mðEÞ for E4E.

Appendix B. Derivation of Eq. (48)

To derive Eq. (48) we note that the probability density function, f ðajaeff Þ, of Young modulus, a1, is

f ðajaeff Þ ¼ c

Zdða� a1Þd aeff �

N

1

a1þ � � � þ

1

aN

0BB@

1CCAda1 . . . daN

¼ c

Zd aeff �

N

1

aþ

1

a2þ � � � þ

1

aN

0BB@

1CCAda2 . . . daN ,

where c is the normalizing constant. We will compute the function F ðajaeff Þ such that

f ðajaeff Þ ¼qF ðajaeff Þ

qaeff.

Obviously,

F ðajaeff Þ ¼ c

Zy aeff �

N

1

aþ

1

a2þ � � � þ

1

aN

0BB@

1CCAda2 . . . daN .


Proceeding as in the previous appendix, we have

F ðajaeff Þ ¼ c

Zdz

2pize

zaeff�ðNz=ð1aþ1

a2þ��þ 1

aNÞÞda2 . . . daN

or, changing z! zð1aþ � � � þ 1

aNÞ=aeff ,

F ðajaeff Þ ¼ c

Zdz

2pizeð1aþ��þ

1aNÞz�N 1

aeffzda2 . . . daN

¼ c

Zdz

2pizez=a e

ðN�1Þ lnð 1½a�

R aþ

a�ez=a daÞ�Nz

aeff da2 . . . daN .

Applying the steepest descent method, we obtain

F ðajaeff Þ ¼ const eb=a,

where b is the point of minimum of the function

ln1

½a�

Z aþ

a�

ex=a da�x

aeff;

this point is determined by Eq. (49).

Appendix C. Derivation of Eqs. (51)–(53)

In the case when we know the value of the arithmetic average of the local moduli, we have for theprobability distribution of the effective coefficient, aeff ,

gðt; rÞ ¼ Probabilityfaeffptg

¼ c

Z aþ

a�

� � �

Z aþ

a�

y t�N

a�11 þ � � � þ a�1N

� �d r�

a1 þ � � � þ aN

N

� �da1 . . . daN ,

where

c

Z aþ

a�

� � �

Z aþ

a�

d r�a1 þ � � � þ aN

N

� �da1 . . . daN ¼ 1 (119)

and

f ðt;rÞ ¼qgðt;rÞ

qt.

Using the presentation for the step function (95) and the corresponding presentation for d-function,

dðEÞ ¼1

2pi

Z i1

�i1

eEz dz, (120)

we obtain

gðt;rÞ ¼ c

Z aþ

a�

� � �

Z aþ

a�

Z i1

�i1

Z i1

�i1

dz

2pizdz2pi

eztþzr�ðNz=a�11þ��þa�1

NÞ�zða1þ��þaN=NÞda1 . . . daN

¼ c

Z aþ

a�

� � �

Z aþ

a�

Z i1

�i1

Z i1

�i1

dz

2pizN dz2pi

eða�11þ��þa�1

NÞzþNzr�Nz�zða1þ��þaN Þda1 . . . daN

¼ cN

Z i1

�i1

Z i1

�i1

dz

2pizdz2pi

eNSðt;r;z;zÞ, ð121Þ

where Sðt;r; z; zÞ is the function (53). The integral (121) contains a large parameter, N, and its asymptotics iscomputed by the steepest descent method as in Appendix A.


The constant c can be found from Eq. (119) by means of the presentation of the d- function (120):

c�1 ¼

Z aþ

a�

� � �

Z aþ

a�

d r�a1 þ � � � þ aN

N

� �da1 . . . daN

¼

Z aþ

a�

� � �

Z aþ

a�

Z i1

�i1

1

2piezr e�zða1þ��þaN=NÞdzda1 . . . daN

¼N

2pi

Z i1

�i1

eNzrZ aþ

a�

e�za da

� �N

dz.

This integral can be computed exactly, but we need only in its asymptotic value as N !1. As before, we canuse the steepest descent method to obtain

c�1 ¼N

2pi

Z i1

�i1

eNðzrþln

R aþ

a�e�za daÞ

dz

¼1ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2pNS00p eNS1ðx

�;rÞ, ð122Þ

where x� is the minimizing point of the function

S1ðx;rÞ ¼ xrþ ln

Z aþ

a�

e�xa da.

Appendices D and E. Supplementary data

Supplementary data associated with this article can be found in the online version at doi:10.1016/j.jmps.2007.07.004.

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