Order relationships for boundary conditions effect in heterogeneous bodies smaller than the representative volume

J. Mrch.

Pergamon

P/y.s. Soliri.\, Vol. 42, No. 12. pp. 1995 201 1. lYY4 Copyright pi 1994 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0022 5096.94 $7.00+0.00

0022-5096(94)00053-O

ORDER RELATIONSHIPS FOR BOUNDARY CONDITIONS EFFECT IN HETEROGENEOUS BODIES SMALLER THAN THE REPRESENTATIVE VOLUME S. HAZANOV Ecole Polytechnique

and C. HUET

Fed&ale de Lausanne, Laboratoire Ecublens, CH-1015 Lausanne.

(Recrir~f

de Materiaux Switzerland

3 MUJ 199 3 ; in revised fiwn 4 Ju!,

de Construction,

MX-G

1994)

ABSTRACT Previous results by Huet [J. MC&. Plains. Solids 38,8 13-841 (lYYO)] for heterogeneous elastic bodies smaller than the representative volume and submitted to static or kinematic uniform boundary conditions are partly extended to the case of mixed boundary conditions. Apparent elastic modulus and compliance tensors defined through the energetic procedure on a single specimen are considered. Through a variational approach, it is shown that each of them is bounded on one side. For cases fulfilling the Hill condition, these apparent modulus and compliance tensors are reciprocals. This provides two-sided bounds on each one. In this case, for a body smaller than the representative volume, the apparent elasticity tensor for mixed boundary conditions falls between the tensors associated with the static and kinematic uniform boundary conditions. Illustrating examples are studied. Various possible fields of application and other extensions are considered.

NOMENCLATURE

a’ sym a 6, S’HF” C’R S,,‘c, S’Tcc sn”. cn,

domain occupied in space by a material body boundary of D position of a material point at time / stress tensor strain tensor displacement vector traction vector density on a surface spatial average of the variable a on a domain D tensor product (dyadic) twice contracted tensor product transpose of the tensor or matrix a symmetric part 4 (u + u’) of the tensor a potential energy and complementary energy functionals, respectively effective compliance and modulus tensors, respectively kinematic apparent compliance and modulus tensors, respectively static apparent compliance and modulus tensors, respectively mixed apparent compliance and modulus tensors, respectively I995

S. HAZANOV I.

and C. HUET

INTRODUCTION

The theory of evaluating the effective properties of random inhomogeneous materials through their constituent properties has experienced a great deal of development in the last 40 years. This is due to the pioneering works by Hill (1952, 1963) and Krijner (1953, 1958) and to the techniques of providing bounds on overall properties, elaborated in Hill (1952,1963), Paul (1960), Hashin and Shtrikman (1963), Kriiner (1972, 1977), Willis (1981), Willis and Talbot (1989) and Nemat-Nasser and Hori (1993). On the other hand, the practically important homogenization problem of heterogeneous elastic bodies smaller than the representative volume was first considered in a recent paper in this journal (Huet, 1990) and also in Huet (1991). The mentioned problem is especially in~por~nt in two cases frequently met in practice : (a) When the size of heterogeneities is so big that the representative volume of the material is too large in comparison with experimental possibilities (for instance dam concrete, where the representative volume reaches several cubic meters). (b) When the material is used in layers with thicknesses of the same order of magnitude as the maximal size of the heterogeneities (coatings, pavement, etc.) For such cases the concept of overall apparent properties was introduced in Huet (1990), and absolute bounds on properly defined apparent properties were established, helping to estimate the values of the effective properties in terms of the constituent properties. These apparent properties were based on two classic types of boundary conditions : uniform kinematic conditions and uniform static loading. In the past, these boundary conditions have been classically used for the theoretical study of the t$L’ctkepropertie.r based on specimens bigger than the representative volume. This is because they can be used very easily when dealing with theoretical or numerical studies and because effective properties are supposed to be independent of the boundary conditions. This is not the case on the experimental side. In laboratory practice, these two types of loading (especially the first one) are difficult to obtain, Usually, the mechanical testing of materials involves mixed boundary conditions. One of the typical examples is the classical tension and/or compression uniaxial test performed on a prism or a cylinder with free lateral faces. For instance, in most cases of uniaxial testing the plates of the testing machine are much stiffer than the material to be tested, meaning that the displacement vector is at least approxiinately imposed on the faces perpendicular to the loading axis, while the stress vector is imposed on the faces parallel to it which are free boundaries, or boundaries loaded by a fluid pressure. In this case. even the question of defining the apparent properties, and thus the meaning of the test results, yields conceptual difhculties. Thus, the problem of studying the possible definition of the apparent properties for the case of mixed boundary conditions and of establishing relationships between them and constituent properties or overall properties obtained in other loading conditions is interesting from both the practical and theoretical viewpoints. In this paper the procedure elaborated in Huet (1990) is extended to the mixed boundary problem and new bounds and order relationships are derived.

Ordering of small heterogeneous bodies 2.

ENERGETIC

APPARENT

1997

PROPERTIES

2.1. Families of elasticity problems Let us consider a hypothetical set of mechanical boundary problems k, k = 1 to n. Let us give the superscript k to all the data and variables involved in the Problem k. In many cases, Problem k may be defined by a given set of vector data {Fdk, Td”,Pd’} applied to a given body D. It may also be defined by a more general procedure. For instance, a relationship between some components of the displacement and traction vectors may be imposed on the boundary. In the first case, Fdkdenotes for Problem k the volume force vector imposed at each point XE D, rdk denotes the displacement vector imposed on the part aD, of the boundary 8D of that same body D, and Pdk the traction vector imposed on the part (?D,k of this boundary, with, for each Problem k: 3Drk u dDOA.= aD.

(1)

Let us now consider three problems relating to the same heterogeneous smaller, in general, than the representative volume. Problem

Problem

Problem

body D

1 : D is subjected to kinematic uniform boundary conditions [E”-KUBC in the sense of Huet (1990)], with the data {Fd’ = 0 in D, cd’ = E’* x on 8D} and solution fields {{‘, E’, c’} . 2: D is subjected to static uniform boundary conditions [a”-SUBC in the sense of Huet (1990)], with the data {Fd2 = 0 in D, Pd2 = 4 - n on 8D) and solution fields {c’, E’, c’} 3 : D is subjected to mixed boundary conditions of any kind with zero accelerations and volume forces and with solution fields {c3, c3, 0’1. This problem may be of the first kind defined above or of a more general type.

In the following we consider isothermal equilibrium elasticity problems only. Thus, the accelerations are zero. In addition, we consider only problems for which the displacements are continuous and the volume forces Fd are zero everywhere in D. Without any restriction, the elastic energy @)kfor every problem k is defined by

where S and C are the local compliance and modulus tensors, respectively. They are fourth rank tensors with the usual symmetries and with components being functions of the coordinates X. For Problems 1 and 2, that are in E”-KUBC and a”-SUBC respectively, we define the apparent compliance and modulus tensors Sflpp,S:pp, CPP, CZPPin the same fashion as in Huet (1990). To simplify the reading of the equations, in the following we skip the superscript ypp), keeping at the same time the subscripts g and E, related to the kind of boundary conditions under consideration. Hence the definitions are : (a’)

= C,:(k?‘)

= C,:E”;

S, = C,’ ;

for&-KUBC,

(3)

1998 (c2)

= s,

For heterogeneous fourth order tensors

: (2)

S. HAZANOV

and C. HUET

= s, : (f’ ;

C, = S;’ :

(4)

solids C,, S,., C, and S, are well defined and positive with the usual symmetries. Since the Hill condition (g:i:)

= (o):(c:>

is satisfied in E”-KUBC or a”-SUBC in terms of (g> and (a) only : @, =

for&--SUBC.

;c,: :

= (fzi):,? respectively,

(El) : (c:‘} =

;sr:

(o’}

or

(Y”:(E)

definite

(5)

the elastic energy can be expressed

: (G’) =

;cf:

E0: co,

(6) (7)

From here it follows that for Problems 1 and 2 the mechanical and the energetic definitions of the apparent properties-correspondingly (3) and (4) and (6) and (7)) coincide. This does not hold in general for mixed boundary conditions (Problem 3), but it is still possible to use at will the mechanical or the energetic definitions. In the next section the apparent properties for Problem 3 will be defined through an energetic procedure. similar to the one used above.

To simplify the formulae, we now skip the superscript 3 when considering the mixed boundary problem. Let us assume that we can find a family of such mixed boundary problems for which we can formally define a mixed apparent compliancr termr S,,, by an energetic procedure involving the actual mean stress (cr), i.e. through :

with S,,, a fourth rank tensor remaining the same for all the problems of the family. Since the volume forces are supposed to be zero, this means that, through the classical equations relating the volume averages to the boundary conditions, we would have : 1 CD= 2+,

:

n

P.
sym (P x X) dt; : ii ,‘/> i ii‘ <~I)

(9)

where the field variables on the boundary are, at least in principle, controllable or measurable in the experiment. This condition has to be checked for each particular family under consideration. In the case of mixed boundary conditions of the first kind involving aD,, and 07D:, this can also be written as : Pd*{dX

110)

Ordering

of small heterogeneous

bodies

1999

From its definitions in (8) and (IO), one can conclude that @‘, an integral of a positive definite function, is positive definite. Thus, S,,, is also a positive definite tensor. On the other hand, let us suppose that we can find another family of Problems 3 for which we can-for each mixed boundary problem of the family-define in the same fashion a mixed apparent modulus tensor C,, by an energetic procedure using the actual mean strain (a) :

It must be realized that, for general cases of mixed boundary conditions, the definitions (8))( 11) may have a formal sense only. This is because, in a real experiment, we can obtain only a limited set of components of the apparent modulus and compliance tensors. In order to obtain other components, additional loading cases should be considered at the same time, that may not belong to the same family of mixed boundary problems. Nevertheless, in particular examples, it is possible to derive useful information from these formal definitions. We will come back to this point in Section 4. As is well known and already discussed in Huet (1990), the energetic compliances and moduli are not, in the general case, equal to the corresponding mechanical ones, defined by the corresponding apparent Hooke laws when the latter can be defined for the family of mixed boundary problems under consideration. This equality occurs if and only if the Hill condition (5) holds. For mixed boundary conditions, this Hill condition is not fulfilled in the general case. However, it can be fulfilled for particular problems involving adequately defined geometries and loading conditions, among which some classical testing conditions and corresponding families of problems may be found. Also, in the general case, S,,,, and C,, as energetically defined above are not mutual reciprocals. For the sake of clarity, let us first illustrate the mixed problem and the introduced definitions on an example. 2.3. An example

qfmiked

louding

Let us consider the following mixed boundary value problem. A uniform displacement component t;’ is applied without friction in the direction of the vertical axis OX, at the upper and lower ends i?o’, and do; of a prismatic specimen (Fig. 1). In this section, the subscripts 1,2,3 on 5” and Pd,e"and 6” relate to their components along x,, s2, xj. Denoting as I, the length of the specimen along OX,, one applies (Fig. 1) : gy = E;, I, ; P: =o:,n, =O;

Pi =ay,n, =0

(12)

at the upper face aD’, and <;’ = 0;

P$ = CT&n, = 0;

P: = CT!,n, = 0

(13)

2000

S. HAZANOV

and C. HUET

r-------I

Fig. 1

at the lower face dDy. In (12), E’,, is a given scalar constant, that can be taken at will within the considered family of problems. The values CJ~, and cri, are taken as zeros. On the lateral face with the normal vector n2 one applies : -d I Cl = &II,YI ;

P; = dz;

and at the lateral face with the normal

Pj = ~7’:~n, = 0

vector n3 :

P: = o-&n3 = 0;

g =e;,.x,;

(14)

P: = ~7:~n3 = 0.

(15)

Here & is another given scalar constant, that can be taken at will within the considered family of problems. This is a particular family of problems belonging to a more general family in which uniform conditions are chosen for any three of the six displacement or traction vector components. In the sequel, we name these conditions “orthogonal uniform mixed boundary conditions”. Simple analysis shows that, independently from the material, we have (a, I ) = 4 I ; Taking have

into account

(022) = a&.

(16)

the equilibrium

and the boundary

conditions

(012)

= (0.23) = (013)

= 0.

= (013)

for P,, P2, P3 we

(17)

The value of (cr, ,) is given by (18) On the other hand, calculating (o:e>

(C : c) one arrives in a similar fashion

= (U,,E;,+&E??)

= c’,,((T,,)+(EZZ)&

at the expression

= (a>:(c).

(19)

This means that the Hill condition holds for the studied family of problems, and the same for all orthogonal uniform mixed boundary conditions. Due to the Hill condition and to the linearity of the problem, and since there are only two loading parameters E’,, and CT;? to vary in the studied family of problems, one may write additionally due to (17) :

Ordering of small heterogeneous bodies

are constant In W), Smykl

coefficients with the dimension of compliance. The values can be found by measuring the transverse disof (&,2), <&31>, <&23) placements t2 and t3 at the sample’s surface. One can see that only 11 apparent compliances can be evaluated from several tests by varying E’,, and ai2 within the studied family of problems. In terms of (g), the energy expression (8) can be rewritten due to (17) as: ;@ =

&,,,I

(~,,)‘+sll2222

(~22)2fx11122

(21)

(~1lX~22).

One of the principal aims of the present research is to establish some order relationships between the coefficients Sm,,k, and C,, S,, C,:, S, in both cases of complete and incomplete quadratic forms for the macroscopic free energy. As we will see, this can still be done by means of the classical variational principles of elasticity.

3.

ORDER RELATIONSHIPS BETWEEN THE APPARENT PROPERTIES IN UNIFORM AND MIXED BOUNDARY CONDITIONS

3.1. EIustic energy expressed

as a complete quadratic jbrm

qf the mean stress tensor

In the above example some components of (0) were not present in the energy expression and the quadratic form (21) was incomplete. However, let us consider first a family of mixed problems for which the elastic energy is a complete quadratic form of the components of the mean stress tensor. Since throughout this paper volume forces and accelerations are supposed to be zero for each mixed boundary problem, the field (o(x)} which is divergence free everywhere in D is an admissible stress field for Problem 1. This is because the latter involves no stress requirement on the boundary (30, equal here to dD,, Hence the minimum theorem for the complementary energy Fi (5) of linear elasticity for Problem 1 gives, because of the value (E” * x) of [” in Problem 1, 1

-F;(a)---

S:a:adI’+

2,s G---F;@‘)

=-;

(a *n) (co x) dC s ,JD

S:o’:a’dV+ sD

(a’ - n) s iD

- (E”

* x)

dC.

(22)

2002

S. HAZANOV

and C. HUE7

However. it is known that, for the actual solution. the complement~~ry energy with the negative sign equals the potential energy F,! (E’) for the same Problem I, see for instance Huet (1990) : (23)

Fl(n’)=F;1(e’)~~:C,:r”:r:o.

where C is, for a body smaller than the representative volume, the well defined apparent elastic moduli tensor under E”-KUBC introduced in Huet [1990. (3) and

(611. On the other hand, from (8) we get. for a mixed problem belonging to the family of problems corresponding to the definition of the apparent overall compliance S,,,, -F:(a)

= -Q-t

zz-

= -

’ (~*n).(&“..~)dC ! i /)

‘I’s,,,, :(0) : ((T) i-E” : ps”,m: (c-7): (CT)+

s

(aen) x sd2:

i0

Vd’: (5),

(24)

where 4Dand S,, are defined by (8). Due to (22) this yieids

Hence A, a scalar function of the tensor (a), has an upper bound when maintained fixed and (0) is varied within the considered family. One has ?A --=iT(a) which is equal to zero for (0)

(26)

S”,, : (0) fe”.

equal to CF*such that (27)

s,,,, : (i* = 2. As a positive write

definite

E” is

symmetric

fourth

rank tensor,

5* = (S,,)

S,,, is invertible

and one can

’ : E0

(28)

and further.

(29) Thus, for any fixed 8, A experiences (28). This yields

a maximum

for the value

5*

of

{rr)

given by

Ordering

of small heterogeneous

A mzLx= A (a*) = ;(s,,)‘-’ Inequality (30) can be rewritten rank tensors :

:&o:.z* < ;C,::E*:E”

as an order relationship

c,

e%,,)-’ < Introducing

a mixed apparent

2003

bodies

modulus

V&o.

between

(30) the two fourth

‘de,.

(31)

tensor of the stress kind C,,, through (32)

and using S;: defined in (3) as the reciprocal equivalent sets of order relationships :

G, d

of C,,, one obtains

c, ; s,:< s,,.

the two following

(33) qf the mean strain tensor

3.2, Elastic energJ$ e.upressed as a complete quadratkform

Let us pass now to the case when the elastic energy can be written as a complete quadratic form of the mean strain tensor. We will proceed in the same fashion as in Section 3.1, but using the minimum theorem for the potential energy F6f (.Z) of linear elasticity for Problem 2. The strain field E for the mixed boundary problem is an admissible strain field for Problem 2 since it derives from a displacement field that is continuous over D, and since Problem 2 involves no displacement requirement on the boundary. Thus we have :

= -

;cmr; :

(8) : (E)

+a*:

nx
(34) Thus the function

B, defined as

B= -~c,,:(E):(&)+60:(E), experiences

a maximum

when (8) is varied at constant

6’.

(35)

2004

S. HAZANOV

and C. HUET

Then the procedure similar to that for the function foilowing order relationships :

A in Section

3.1 leads to the

(Cm,:) -’ d %

(36)

cod c,,:; s,,,:< s,,

(37)

or equivalently,

where S,, is defined as (C,,> _I.

When the Hill condition, expressed by (5) and yielding the equality between the macroscopic elastic energy and the microscopic one, is valid, the mechanical apparent compliance and modulus tensors S,, and C, defined for the given mixed boundary problem as (E) =

s, : (0)

are each equal to the corresponding and C, are reciprocal, one has

;

(CT) =

energetic

c, : (E)

one, respectively.

s,, = s,, = c, ’ ; c,,: = c, = s, Then recalling

(38)

(33) and (37) we come to the following

’.

two-sided

ccd c,, G c, ; s, < s, < s,.

Moreover,

since S,

(39) bounds

: (40)

Inequalities (40) show that for the chosen class of mixed problems, the overall mixed properties always fall between the static and the kinematic ones. It must be noted that the Hill condition holds for classical uniaxial testing of prismatic or cylindrical specimens. More generally, it holds for orthogonal uniform mixed boundary conditions, like the one studied above in Section 2.3.

3.4. E.qnlicit scalar meaning qf‘the tensor order refationships The tensor relationships (33) and (37) in fact mean that the tensor quadratic forms Q, = (S,, - S,> : E’ : 8’ and Q2 = (C,,, - C,) : CT’: o” from Sections 3.1 and 3.2 should be positive semi definite. However, as 6’ and E” are not vectors but second order tensors and as their coefficients have four indices, it will be useful to pass to appropriately defined matrix operations in order to obtain the corresponding relationships between the scalar components of the modulus or compliance tensors. Let us introduce the Voigt-Mande~ matrix r~presen~tion of the Hooke law :

Ordering

or, in the symbolic

matrix

of small heterogeneous

2005

bodies

form, ii=c:o,

and reciprocally,

with, in the matrix

sense,

Then the free energy expressions

(8) and (11) will be written

respectively

as (45)

and

Hence, for this particular choice of the matrix representation for the Hooke law, the tensor quadratic form for the energy, (11) [correspondingly (S)] is equal to the matrix quadratic form, written through the vectors of strains (stresses), introduced above, its kernel matrix being that of (41) (or correspondingly its reciprocal). The same holds for every tensor quadratic form K = (A : E) : E [correspondingly for L = (B : a) : a], that will be equal to the matrix quadratic form : K=(A:E):E=~~A& or correspondingly

to

(47)

2006

S.HAZANOVandC.HUET

L = (B : a) : r~ = sTB d.

(4X)

This associated matrix quadratic form is constructed through the vector of strains (stresses) introduced in (4). The kernel matrix A, called here the VoigttMandel matrix associated to the tensor quadratic form (A : E) : E. is composed from the components of the fourth order tensor A, in the fashion shown in (41) for C. The same for the kernel matrix fi, associated to the quadratic form (B : CT): c. Then the semi-positivity of the tensor quadratic form (A: E) :E [respectively (B : 0) : a] is governed by the semi-positivity of the principal minors of its associated matrix A (resp. B). For the quadratic forms Q, and Q2 mentioned at the beginning of this section. the associated VoigttMandel matrices will have the form of that from (41) with every element C,,,,, replaced by the element (C’,,,,- C,),,,, [or respectively by (&,, - S, )J. The introduced procedure makes it possible to write down the tensor inequalities as a set of explicit scalar inequalities. For instance, (33), (38) and (40) will then provide the bounds for the diagonal elements : C ln(illilG C, l,il ;

s, //// d s,,,,,,,,,

(49)

C hi,,, < C’,,,,,,,,;

&,,,,,, G s,,,,,.

(50)

Stilli d s,,,,,, d S,T,,,,

(51)

C,,,,, G Cn,,,,, d C; ilii; Other scalar relationships cipal minors of the matrices

will be obtained

A = C,,,,-C,

and

by computing

successively

I?%= S,,,n-S,.

all the prin-

(52)

We will obtain thus a set of scalar order relationships between the components of mixed apparent compliance tensor S,,, defined for the mixed boundary problem and those of the kinematic apparent compliance tensor S, defined for the E(‘-KUBC problem. In the same fashion we will obtain a set of scalar relationships between the components of mixed apparent compliance tensor C,, defined for the mixed boundary problem and those of the static apparent compliance tensor C, defined for the G”SUBC problem. Another advantage of the proposed relationship (41) is that the relationships based on it are independent of the frame of reference provided that the stress and strain components change according to the rules of second order tensors and the components of the associated VoigtGMandel matrix according to those for the fourth order tensors. This invariance follows from the invariance of the free energy of the body. 3.5. Or&r In the expressed in practice In such determined

relutionshipsJor

the cues

Lcith incomplete

energy- espressions

above sections, we studied the simplest case where the elastic energy is as a complete quadratic form of the mean stress or strain tensors. However, it is not always the case, as we have seen in Section 2.3 (21). mixed problems several components of the tensors S,,, or C,,, cannot be in experiments. Nevertheless, it can be easily verified that for the con-

Ordering

of small heterogeneous

bodies

2007

sidered class of problems, the same optilnization procedure as in Sections 3.1 and 3.2 is possible and it leads to the conclusion of semi-positivity ofcorresponding incomplete quadratic forms. This, according to the results of Section 3.4, means that the principal minors of the associated Voigt-Mandel matrices must be non-negative. It can be verified that an associated matrix of an incomplete quadratic form is always a sub-matrix of the general (complete) associated matrix in the sense described in Sections 3.1 and 3.2 above. This sub-matrix is composed of those components of the general matrix that correspond to non-trivial
4. We will illustrate

EXAMPLES

the results of Section

4.1. Uniaxinl t-qevimcnt

3.4 by three simple examples.

in 30

Let us study a particular mixed problem where only component (ax3) of the mean strain tensor contributes to the energy expression (11). In this case the Hill condition hoIds as it was mentioned above. Then the corresponding Voigt-M~ndel sub-matrix of the matrix associated to the quadratic form (C,,- C,) : 2: co will be of order I and contain only one element, that is (Cn,,,,,,,-C ,,&. Thus from Section 3.4 and Inequality (40) follows a simple two-sided bound on the strain defined apparent modulus Cm33i3:

4.2. Orthgonal

unifot-m mixed boundary conditions

in 30

Let us continue the analysis of our example from Section 2.3. In this case the Hill condition holds and we will use the results of Sections 3.3 and 3.4. In this case the energy expression (21) call be written in terms of as

2008

S. HAZANOV

and C. HUE7

+2snll22

(~“,,123Cm22,2+Cm2223CI

,13)).

(54)

The coefficients in (54) are combinations of the moduli and the compliances, corresponding to the 11 compliance components that, as was established above, can be evaluated experimentally in this case. Hence, the quadratic form of the elastic energy in this problem is complete neither for the mean stresses nor for the mean strains. The VoigttMandel sub-matrices, associated to the incomplete quadratic forms will be

Hence, inequalities (40), together with the results of Section 3.5 will provide sided bounds on the apparent mixed compliances :

the two-

Corresponding bounds on the apparent mixed moduli can be easily received from the second relationship of inequalities (40), by expressing there the apparent mixed compliances through the mixed apparent moduli. 4.3. A 20 example Let us take now the 2D case, supposing only that the component Then the corresponding VoigttMandel sub-matrices will be :

(o12) is zero.

and inequalities (33), now taken in their scalar form, lead to the lower bounds apparent mixed compliances :

on the

Ordering

of small heterogeneous

bodies

2009

The bounds on the apparent mixed moduli can be obtained in the same manner as was indicated in the previous example. The proposed examples show that the general procedure and results of Section 3 are easily applied in particular problems and lead to simple bounds like those expressed by inequalities (53) (.56)-(58) and (60)-(62).

5.

DISCUSSION

AND CONCLUSION

In this paper some of the results of Huet (1990) concerning the static and kinematic apparent properties for a given particular specimen smaller than the representative volume have been extended to mixed problems, and so to cases much nearer to laboratory practice and engineering problems. The apparent elasticity tensor for mixed boundary conditions is shown to fall between the apparent tensors associated with the static and kinematic uniform conditions [see the bounds given by (33), (37) and (40)]. The importance of the two classical kinds-kinematic and static-of uniform boundary conditions is confirmed, since they provide two-sided bounds to all the other cases. Reciprocally, changing the viewpoint, the mixed boundary conditions considered here provide one-sided bounds to the apparent properties defined in Huet (1990) for the uniform boundary conditions. Another important although already known fact is that experimental mixed conditions satisfying the Hill conditions can be looked for and found independently from the properties and particular microstructure of the material, provided the continuity of the displacement field can be assumed. This can be achieved through the use of the general relationships between volume averages and external surface conditions recalled in Section 4 of Huet (1990). Numerical simulations performed in our laboratory for particular models of a material with randomly distributed inclusions representing concrete specimens confirm the analytical results in both the 2D and 3D cases. In addition, they make them more precise, allowing the estimation of the distances between bounds, and between one bound and the solution to a particular problem. These numerical results, extending the ones already presented in Amieur et al. (1993), are not given here because of the lack of space and will be presented in a forthcoming paper. Since the numerical simulations involve huge calculations subjected to several kinds of errors, the analytical bounds obtained here can be applied directly in order to validate the numerical results and their mutual consistency. It has been observed in the numerical simulations that, for specimens large enough to be considered as almost having the representative volume, the apparent properties in static and kinematic uniform properties are almost equal, and thus almost equal to the effective properties of the material. Since, from the analytical results obtained here, all other experimental loading conditions give apparent properties lying between these two uniform ones, this confirms the generally accepted idea that the effective modulus is independent of the boundary conditions, at least in the cases for which the Hill condition is valid. Since the effective properties always lie between the statistical apparent ones (ensem-

‘010

S. HAZANOV and C. H CJET

ble averages) obtained for the two kinds of uniform boundary conditions (see Huet, 1990), it can be conjectured, from the results obtained in the present paper, that statistical considerations applied to the above results will lead to further extensions and possibilities of applications. In particular, for a given kind of material. it seems possible to define tnixed experimental conditions providing statistical results very near the effective properties, even when specimens much smaller than the representative volume are used. This, which is now under study, would be very useful for the testing machine design. and also for several difficult problems like the testing of concretes, such as dam concretes, involving coarse aggregates by comparison with the size 01 the specimens that can be submitted to testing. Another important field of ~pplic~tion opened by the presented results seems to be the problem of ch~inillg the homogeni~~~tion procedures performed at several microscopic levels on heterogeneous materials like concrete involving wide granulometric distributions. This is because, in these materials, the matrix between coarse grains is itself a heterogeneous mortar which also has to be homogenized in a previous step. However, for compact materials with high density of grains. the spaces between coarse grains are smaller than the representative volume and are submitted to mixed boundary conditions (see Huet. 1993). This type of application is also presently under study. both from the analytical and numerical viewpoints.

WC are grateful to the National Swiss Foundation for Scientific Research subsidies nos 2117962.89 and 20-32206.91 and to the Swiss Foundation for Cement Research for financial support to this work. The authors would also like to acknowledge the helpful comments received frotn Prof. J. Willis and an anonymous referee which significantly contributed in the improvement of the final form of the present paper.

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