Coupled estimates for the bulk and shear moduli of a two-dimensional isotropic elastic composite

J. Mrdr.

Ph_w. Solids Vol. 41, No. 5, pp. 937-980, Printed in GreatBritain.

~22-5~9~J93

1993.

$6.~~~~.~

Pergamon Press Ltd

COUPLED ESTIMATES FOR THE BULK AND SHEAR MODULI OF A TWO-DIMENSIONAL ISOTROPIC ELASTIC COMPOSITE A. V. CHERKAEV~ and Courant

Institute.

L. V. GIBIANSKY$

251 Mercer Street, New York, NY 10012, U.S.A.

(Receiuerd 27 January

1992)

ABSTRACT WII IMPROVE the classical Hashin-Shtrikman and Walpole estimates of the effective properties for an isotropic mixture assembled from two isotropic elastic materials. The planar elasticity problem is considered. Unlike the prior estimates which bound the bulk and shear moduli independently, our estimates are coupled and more restrictive. The set of the bulk modulus-shear modulus pairs turns out to be bounded in the plane of the values of these moduli by two straight lines (the Hashin-Shtrikman or Walpole bulk modulus estimates) and also by two fractional linear curves. To obtain the new estimates we use the translation method, which provides a general approach to both Hashin-Shtrikman (well-ordered materials) and WaIpoIe (badly-ordered materials) cases; the method also provides the estimates for anisotropic mixtures.

1.

INTRODUCTION

A PRIORI ESTIMATES of the effective properties for elastic composites are useful for many applications. Such bounds allow the prediction of the properties of composites with unknown structures, and show limits for the improvement of the mechanical properties of a mixture by changing the disposition of the components. In this paper we suggest new estimates for the effective properties of an isotropic composite made from two isotropic elastic materials with known properties. The materials are supposed to be mixed in an arbitrary way but with fixed volume fractions. The obtained estimates improve the classical ~ashin-~htrjkman and Walpole bounds ; they are the most restrictive among the known ones. The corresponding results for anisotropic composites are also obtained. To get the bounds we use the “translation method” suggested in the papers of LUKE and CUERKAEV (1984b, 1986a) (see also LURIE et u1., 1982; LUR~E and CHERKAEV, 1986b), MURAT and TARTAR (1985), TARTAR (1978,1985), and developed in a number of other papers (see for example, KOHN and STRANG, I986 ; FRANCFORT

? Permanent addresses : St Petersburg University of Ocean Technology, 3 Lotzmanskaja Street, St Petersburg, Russia. I/IA. F. loffe Physi~l-Technical Institute Academy of Sciences, Politechnicheskaja 26, St Petersburg 19402 1, Russia. 937

A. V. CHERKAEV and L. V.

938

GIBIANSKY

and MURAT, 1986; KOHN and MILTON, 1986; MILTON, 1990a,b, 1991; CHERKAEV and GIBIANSKY, 1991; GIBIANSKY and CHERKAEV,1986, 1987). We have discussed this approach in detail in our recent paper (CHERKAEV and GIBIANSKY, 1991), where the exact coupled bounds for the electromagnetic properties of an anisotropic composite have been found. Here we adopt the translation method for estimating the effective rigidity tensor for planar elasticity. Then we give an elementary proof of the known HashinShtrikman and Walpole estimates and apply the same method to prove the new coupled bounds for the shear and bulk moduli of an isotropic elastic composite ; we also discuss some estimates for the effective properties of anisotropic mixtures. The history of the problem and known results are outlined in the next section after introducing basic notations. We consider the plane problem of elasticity. The obtained results are also valid for the effective moduli of a transversely isotropic three-dimensional composite with arbitrary cylindrical inclusions.

2.

KNOWN BOLJNDSFOR THE MODULI OF THE ELASTIC COMPOSITES

In this section we describe the equations of plane elasticity, introduce give the statement of the problem and outline known results.

the notations,

2.1. Basic equations and notations We deal with the plane problem of elasticity. Let x = (x,, x2) be Cartesian coordinates, u = (u,, UJ be the displacement vector, E be the strain tensor and o be the stress tensor. The state of an isotropic body is characterized by the following system (see for example, LOVE, 1927) : the compatibility equations &II = Ul.X,l the equilibrium

and the equations

El2 = 621= 12(%.XZf~2.X,)~ 822= uz.x,;

(2.1)

equations

of Hooke’s

law

Here K = K(x), p = p(x) are the bulk and shear moduli of the plane elasticity strain problem) at the point x = (x,, x2). Remark 2.1 : The bulk modulus of the planar elasticity K is expressed as K = kfy/3,

in terms of the bulk modulus k of the three-dimensional theory. The planar modulus p does not differ from the one used in the three-dimensional theory.

(plane

(2.4) shear

Coupled

We will use the following

estimates

for moduli

of composites

939

tensor form of (2.1)-(2.3) E=:(VOU+(V@U)=), v*u=o,

O.=OT,

u = C(K, p)

(2.5)

“E,

where V is the two-dimensional Hamilton operator, C(K(x), p(x)) is the tensor of rigidity of an isotropic material-a fourth order symmetric positive definite tensor, @ is the sign of a tensor product and (*) and (es) are contractions with regard to one and two indices, respectively. We will also use the representation for the stress tensor r~ through the Airy stress function 4 (see LOVE, 1927)

(2.6) where (2.7) and R=

(2.8)

is the tensor of rotation by a right angle. It is convenient (see, for example, GIBIANSKY and CHERKAEV, 1986 ; LURIE et al., 1982; LURIE and CHERKAEV, 1984a, 1986b) to introduce the following orthonormal basis in the space of the second order tensors : a, = (i@i+j@j)/Jz,

a2 = (i@i-j@j)/$,

a3 = (i@j+jOi)/Jz,

a4 = (i@j-j@i)/Jz, a,

“aj

=

(2.9)

6,,

where i and j are the unit vectors of the Cartesian axis xi and x2, and 6, is the Kronecker symbol. The subspace of the symmetric second order tensors has the orthonormal basis a,, a2, a,. Symmetric strain and stress tensors have the following representation in this basis 3

3 E =

C

&jai,

c

i= 1

where the coefficients E, = &**a, = L(c:I, Jz c, = d**al = ‘(~7~~ Jz

E;, 0, are expressible +E~~),

+o,J,

=

a,a,,

1 i=

(2.10)

I

as

E2 = &**a2 = $(E,,

02 = Is “a2 = $(rs,,

-cz2),

-az2),

t3 = &**a3 = J- 2c,,, g3 = fs”a3 = J 2rr,,. (2.11)

940

A. V. CHERKAEV

In this basis Hooke’s

and L. V. GIBIANSKY

law has the form (see LOVE, 1927) 0, = 2K&,,

c2 =

[compare with (2.3)] and the isotropic diagonal matrix

fS3 =

2P%,

2j_lE,

(2.12)

tensor C(K, ,u) of rigidity is represented

by the

(2.13)

The elastic energy density

can be represented w,

or as a quadratic

either as a quadratic

form of strains (2.14)

=E-'C"E

form of stresses (2.15)

where

SEC-‘=

1 ~ 2K

0

0

0

1 -~ 2P

0

0

0

I ~-~ 2P

(2.16)

is the compliance tensor. It is convenient to introduce the nonsymmetric matrix [ = V 0 u of the gradient of displacement vector u. This matrix has the following representation in the basis (2.9) c=V@u,

i=

c[,ai,

i=1,2,3,4.

[,=[-*ai,

(2.17)

,= I Note that the first three components

of tensors

ji =

E,,

i=

c and E coincide 1,2.3,

(2.18)

and the fourth component of [ (cd # L, = 0) does not affect the equation of elasticity. Namely, the density of the elastic energy (2.14) as a function of the tensor i becomes W(i) = ( - * C’ - - [, where the (4 x 4) matrix

C’ has the diagonal r I -~ 2K C’ =

0 0 co

(2.19)

form 0

0

0

2:

0

0

0

2;

0

0

0

0

(2.20)

Coupledestimatesfor moduliof composites

941

in the introduced bases al, a2, a,, a,. A zero value of the fourth modulus of the matrix C’ corresponds to the zero value of energy associated with the rotationa degree of freedom of material points and provides the frame indifference of the energy. 2.2. Homogenization Let us consider a doubly-periodic structure in the (x,, ,x2) plane. The element of periodicity Q is supposed to be divided into two parts a, and 52, with the relative areas m I and m2 respectively n, un, (vol~,)/(vol~)

= 52, = ml,

ml+m2

(vol~~)/(~~ol~) = mZ, (2.21)

= 1,

where the symbol vol (9) means the area of the domain (*). Assume that these two parts are occupied by two isotropic elastic materials with the moduli (K,, ,u,) and (K,, ~1~)respectively, and investigate the homogenization problem, i.e. the problem of describing the medium’s effective properties. It is well known (CHRISTENSEN, 1979 ; SANCHEZ-PALENCIA, 1980) that the average behavior of a mixture is described by the homogenized equations of elasticity (e) = <:(VOu+(VOU)T)),

V*(a)

=o,

(G) = Co”(&).

CD> = (@>T,

(2.22)

Here the symbol ((.)) denotes averaging over the element of periodicity Q (2.23) and the tensor CO connecting the averaged stresses and strains is by definition the effective rigidity tensor of a composite. This tensor is in general an anisotropic positive definite fourth order symmetric tensor, It depends on the properties of the original materials, their volume fractions as well as on a geometrical structure of the composite and it is independent of loading. The energy density W stored in the composite is known to equal W; = (E>“CO”(&)

(2.24)

W, = (a)**S().*(o>,

(2.25)

or

where the effective compliance tensor So is determined as S, = C; ’ (see for example CHRISTENSEN,1979 ; SANCHEZ-PALENCIA, 1980). 2.3. The known estimates The problem of bounds for the elastic moduli has a long history. Several estimates were found for the three-dimensional problem and all of them had natural two-

A. V.

942

CHERKAEV

and

L. V. GIBIANSKY

dimensional analogues ; isotropy in planar elasticity corresponds to transverse isotropy in the three-dimensional problem for the structures with a cylindrical arrangement of pure materials. Here we summarize the known estimates for the two-dimensional problem and simplify it by using some new notations. 2.3.1. Hill’s estimates. The first bounds of the effective moduli I&, ,uOwere found by HILL (1952) who applied the REUSS (1929) and VOIGT (1928) estimates (see also WIENER, 1912) to elasticity. Hill’s bounds have the form (K-‘)-’

d K, B (K),

Note that the functions such type is averaged as

(j.~‘>~-’

< PLgd c/J>.

K(x), p(x) are piece-wise

(44)

constant;

any function

(2.26) d(x) of

(2.27)

= mIdI +m4,

where d, and dz are the values of this function in subdomains Q, and Q2, respectively. The estimates (2.26) can be represented in the following form JJ(&) 2 0, where the fractional

linear function

0”)

(2.28)

3 0,

y is determined

by the relation

y(d,,) = y(do, d,, d2) = - fL;!+;-’ _

d,d2(dm ‘).

(2.29)

0

Remark 2.2 : This fractional linear “Y-transformation” turns out to play an important role in the problem of the bounds for the effective properties ; see, for example, BERGMAN (1978), BERRYMAN(1982), MILTON (1985, 1991), MILTON and PHAN-THEN (1982), CHERKAEV and GIBIANSKY (1991). Here we use two representations of the “Y-transformation”. The first one is a function y(d,) of one argument do with implicit dependence on the parameters d,, d, and m, ; it maps do one-to-one into y(d,). The second one is an explicit function y(d,, d,, d2) of three arguments do, d,, d, and one implicit parameter m,. _ It is easy to verify that functions y(d,) and y(d,,, d,, d,) have the following properties : y(d,)

= y(d,,d,,dJ

= -d,,

y(dz)

= y(d,, d,, dz) = -d2,

y_(“do, Ed,, ad,) = ccy(do, d, >d2)> y(d,‘,d,‘,d,‘)

= yP’(do,d,,d,).

(2.30)

2.3.2. The Hashin-Shtrikman and Walpole estimates. HASHIN and SHTRIKMAN (1963) suggested the variational method which allowed them to take into account differential restrictions on stress and strain fields ; they found new bounds of the elastic moduli of an isotropic mixture made from isotropic materials.

Coupled estimates for moduli of composites

943

Originally, the Hashin-Shtrikman bounds were formulated for isotropic threedimensional mixtures ; however, in the paper by HASHIN (1965) the bounds were formulated for transversely isotropic composites with cylindrical inclusions as well. The above problem is just the case we will study here. The original materials were supposed to be “well-ordered”. This means that both bulk and shear moduli of the first material are bigger than those of the second material PI aru2,

The obtained

estimates

K,

>

(2.31)

K2.

have the form (see HASHIN, 1965)

where Kk,=K,+-m

I

Ml

(2.33)

, m2

K,--2+K2+~2

(2.34)

+K,+PI

Kz-K,

r&s= P2 +

ml --I--

m2(K*

=+----

+2p7)

+2(K2

1

(2.35)

+PUZ)

(2.36)

&,

By using the introduced “Y-transformation” ,& can be defined as the unique solutions

.Jals)

=

K2~2 ___ K2+%2’

the above values of the equations

k’,Pl Ybh)

= K,

KkS, KU& and

(2.37)

+ZP,

The Hashin-Shtrikman estimates are more restrictive than those of Hill. Moreover, they describe the minimal rectangle in the bulk modulus-shear modulus plane which contains the set of possible values of these moduli. It is proved that two opposite corner points A and C

of this rectangle (see Fig. 1) corresponding to maximal or minimal values of both moduli are attainable by special microstructures [i.e. there exist microstructures with such properties ; see FRANCFORT and MURAT (1986), LLJRIEand CHERKAEV (1984c), MILTON (1986), NORRIS (1985)]. WALPOLE (1966, 1969) considered the opposite case of “badly-ordered” original

A. V. CHERKAEV and

944

L. V. GIBIANSKY

V(b) FIG.

I. The

fractional

linear curves representing K,=9.

Here

AB’CD’

materials

the bounds of the Theorem

1’,=9,

is the Hashin-Shtrikman

K2=Ir

for the values of the parameters

/Lz=I

rectangle. the lines EAD and FBC’ correspond estimates.

to the new

when ~1 3

~2,

K,

<

(2.39)

K2.

He obtained the bounds for the effective moduli of an isotropic mixture by using a similar variational method. The two-dimensional Walpole estimates we are dealing with also have a simple representation in terms of the “Y-transformation” : K&dK,,
K&, Kl;y and ~6, PL;; satisfy the equations .YW&) =

(2.40)

I.&
~2,

_YVGJ)

:

= PI,

(2.41) Remark 2.3: Note that the cases (2.31) and (2.39) cover all possible relations between the elastic moduli of two original materials because one can order the materials so that PI >,

P2.

(2.42)

All mentioned estimates provide independent ranges for both bulk and shear moduli variation. They bound the set of these moduli by a rectangular domain in the (K,, pO) plane (see Figs 14). Remark 2.4: One can check that if K, = Kz = K then the Hashin-Shtrikman and Walpole rectangles degenerate into the interval on the straight line K,=K,

&=piv
(2.43)

Coupled

estimates

-61 -10 FIG.

for moduli

-5

p,=3,

Here A’BC’D is the Walpole rectangle,

945

i 5 Y(Ko)

0

2. The fractional linear curves representing the bounds K,=6,

of composites

of the Theorem

K,=9,

for the values of the parameters

/~~=l.

the lines BCH and ADG correspond to the new estimates.

If ,u, = p2 = p then the Hashin-Shtrikman the point

and Walpole

rectangles

Kks = K& = K. = K;;, = K;.

,uO = ,u,

degenerate

into (2.44)

The particular cases mentioned were studied by HILL (1963, 1964), LURIE et al. (1982), LURIE and CHERKAEV (1984a, 1986b), LIPTON (1988) and FRANCFORT and TARTAR (1991). This method was further developed and used in the papers by AVELLANEDA (1987), KOHN and LIPTON (1988) and KOHN and MILTON (1988).

01

0

2

4

FIG. 3. The bounds for the pair (Y(K,), Y&))

K,=9, Here AB’CD’ is the Hashin-Shtrikman Kublanov

estimates.

6

1

6

10 Y(Ko)

for the values of the parameters

~,=9,

K,=l,

/~*=l.

rectangle, the lines B-I-C and A-4-D correspond to the Miltonand the lines B-2-C and A-3-D show the estimates of the Theorem.

946

A. V. CHERKAEVand L. V.

GIBIANSKY

O.Ok--Y~K) 1

2

3

0

FIG. 4. The bounds for the pair fY(K,). U(g,)) for the values of the parameters K, = 6.

J’, = 3.

K> = 9,

/l> = 1.

Mere A’BC’D is the Walpole rectangle, the lines B-1-C and A-4-D correspond to the Milton-Kublanov estimates. and the lines B-2-C and A-3-D show the estimates of the Theorem.

estimates. The next step was taken by (1982) and BERRYMAN and MILTON (1988) who used a different variational method to obtain bounds for the effective properties of isotropic three-dimensional mixtures ; KUBLANOV and MILTON (1992) applied this method for the plane case in which we are interested here. The authors obtained an impressive result--they found coupled bulk-shear moduli estimates. In other words, the rectangular domain of the permitted values of the moduli was replaced by a smaller domain lying in this rectangle and bounded by some curves. In the well-ordered case (2.3 1) these estimates have the form : 2.3.3.

The Milton-P~aP1-Thien-Kublanov

MILTON and

PHAN-THIEN

Kks G &

d I%,

(2.45)

P:~K+V&i) d PO d I&U+ W&

where the values of ,u&+ (KC,) and ,&&+. (I&) are defined by the equations (2.46)

(2.47)

In the badly-ordered

case (2.39) the estimates

G,f d Ko d KY?,

Al&

can be presented

Wd d P”oG AIK- Wok

in the form : (2.48)

Coupled

where ,&-

(K,) and &k-

estimates

for moduli

of composites

(K,) are the solutions

Y(&- (Ko))= 2~z+K2-2---

of the equations

P2K2 KI--Kz P~YZ ~, _~2

941

K, y(KO) (y(K,)-~2)’

(2 49)

(2.50)

~h4h- (Kd) =

The Milton-Kublanov estimates are illustrated Fig. 3. It is remarkable that they pass through through Walpole points B = (K&, G),

by the curves B-l-C and A-4-D in Hashin-Shtrikman points A, C and

D = (Ktt, L&).

(2.51)

Therefore the Hashin-Shtrikman and Walpole points A, B, C, D turn out to be important for both well- and badly-ordered cases. Note that we have presented all the estimates in terms of the Y-transformation and in such a representation the estimates turn out to be independent of the volume fractions of the original materials.

NEW ESTIMATES

3.

In this paper we suggest estimates that are sharper than the known ones. The results are given by the following statement. Theorem: All the pairs (y(K,), y(,uJ) of values of the Y-transformation bulk KO and shear ,u~ moduli for an elastic isotropic composite are restricted inequalities (Hashin-Shtrikman-Walpole estimates of bulk modulus) : ~2

and by the inequalities

of the by the

PI

(3.1)

d APO) d

t;;M&))

(3.2)

G Y(PO) d

~4MKd)

(3.3)

~,MKJ-‘+%‘I ~,~,v&-‘+K~‘)-~,’

(3.4)

d

YWO)

d

: F, CWd)

in the well-ordered case (2.31), or by the inequalities ~x(.UG)) in the badly-ordered case (2.39) Here F, ( y(K,)) is defined as

F, MKJ)

=

-PI

+

948

A. V. CHBRKAEV

and

L. V. GIBMNSKY

where RI = (!-G’ -PI

1)(KIII~+K,~L+2~,1*)(Kz~L+KK2C1Z+2~,I_12),

(3.5)

BI = (K,~,+K,~L2+211,~Z)(K2+2~L2)(Kr’+~.2’) -(~2~~+$2~2+$-~,122)(K,+2~2)(Kr'+~1'),

(3.6)

’ fp, ‘).

(3.7)

‘Jl = 2(K, -Kz>pS(K, The function

F2( y(K,))

’ +pz ‘)(K,

is equal to (3.8)

where az = ~P,(P~-P.I)(K, Bz = (K,

(3.10)

+~LI)(K~+~~*I)c~~-~I(K~+~I)(KI+~~~), y2 =

F,( y(K,))

(3.9)

+P,)(Kz+II,)~

(3.11)

pz,uCI:(K, - Kz).

is equal to (3.12)

where ~3 = B1=

(3.13)

2~2(~,--2)(K2+-~)(K,+~2),

(3.14)

(~2+~2)(K,+2~~)~,--112(K,+1*2)(~2+2~2),

(3.15)

~3 = P,/J;(K,-K,). F4(

y(K,))

is equal to (3.16)

where

^, y4

=

2(K> - K,)d(Kz

’ +pr

‘)(K2

’ +,u;

‘1.

(3.19)

i = 1, 2, 3, 4 in the inequalities Note that the limiting curves J(P”) = E;(y(K,)), (3.2), (3.3) are fractional linear functions. As for any fractional linear curve, each of them can be uniquely determined by the coordinates of any three points it passes through. There are some remarkable points of the curves. Namely the curve

Coupled y(&

= ~~(~(~~))

estimates

A = CY@GS)~

.Y(&iS))?

D = (Y@-&),

rG&)), Y(P,))

F=

Y(k,))

-11,);

(3.20)

the points

= (0, --111);

(fine ADG, Fig. 2) passes through A = (.~(Kds),

Y(,&))>

D = (Y(G),

Y(&v)),

‘)-’ I),

G = (Y((K-

Y&))

B = (YP&)?

YGGXJ))~

C = (.YUGis),

l’(/&)).

“Y(A)) = c-f&

(3.21)

the points

= (0, -PZ) ;

(line BCH, Fig. 2) passes through

wG)t

the points

.Y(&s)),

(y(@-‘)-I),

ff =

949

Y(P%)),

C = (.Y(K”Hs),

the curve F4(y(&))

= c--K,,

(line FBC, Fig. 1) passes through B = (Y@v),

the curve Fxj”(&))

of composites

(line EAD, Fig. 1) passes through

E = (~(k’,), the curve F2( y(K,))

for moduli

(3.22)

the points

-Pd.

(3.23)

The Y-transfo~ation (2.29) itsetf is also the fractional Iinear function of the effective moduli. Therefore, the corresponding bounds in the plane of the effective moduh (&, ,u,,) are given atso by fractional linear functions. Let us illustrate these results by some numerical calculations. Figure 3 shows all mentioned bounds in terms of the Y-transformation y(K,), y(,nO) when the moduli have the values K, =9,

p, =9,

K, = 1,

/lI = 1

(3.24)

(well-ordered materials). On this figure AB’CD’ is the Hashin-Shtrikman rectangle [bounds (2.32), (2.37)], lines B-I-C and A-4-D correspond to the Milton-KubIanov bounds (2.45)~(2.47), and the lines B-2-C and A-3-D show the estimates (3.2), (3.4)(3.11) of the Theorem. The same pictures in the (&, P,-> plane for the volume fractions tn, = 11~~= 0.5 are presented in Fig. 5. The case of a badly-ordered material for the parameters K, = 6,

p, = 3,

K, = 9,

pz = I

(3.25)

is illustrated by Fig. 4 [in the _Y(&), JJ(,u~) plane] and by Fig. 6 [in the (K,, plo) plane for the same volume fractions ml = m2 = 0.51. On these figures A 'BC'D is the Walpole rectangle [bounds (2.40)-(2.41)], lines A-4-D and B-l-C correspond to the MiltonKublanov bounds (2.48)-(2.50), and the lines A-3-D and B-2-Crepresent the estimates (3.3), (3.12)-(3.19) of the Theorem. Note, that the lines A-3-D and A-4-D practically coincide for our choice of parameters.

A. V. CHEKKAEVand L. V. GILSIANSKY

950

3.0-

6’

C.

lizzil. 1

2

3

2.02.5-

D

I3 A

4

D’

FIG. 5. The same as Fig. 3, but in the (K,, Qplane.

One can check that the obtained

estimates

permit the negative

K0-PO

Poisson

ratio v,) (3.26)

v”= FG

for a mixture, prepared from materials with positive ones. This seemed unnatural until the paper of MILTON (1992) appeared where such mixtures (in particular, laminated composites) were constructed. Moreover, these mixtures possess the Poisson ratio close to the minimal value permitted by our estimates as was shown by MILTON (1992). In Fig. 7 line 1 shows the function v,(h) for the Poisson ratio attainable by Milton’s structures, the lines 2, 3 and 4 represent the lower bounds of v0 following from the estimates (3.2), (3.4))(3.1 I), from the Milton-Kublanov bounds

PO

1.75

C’

B 2

1.651.70-

lb01 222

A d

4 -1111:

z24

1 C

3 D

7.26

7.26

i 7.30 K,

FIG. 6. The same as Fig. 4, but in the (K,, p,,)-plane.

Coupled estimates for moduli of composites

a02

0.00

a04

951

,

CL06

a10

h

Frc. 7. The line 1 shows the function v,(h) for the Poisson ratio attainable by the Milton structures ; the lines 2, 3 and 4 represent the lower bounds of Y,,following from the estimates of the Theorem, the MiltonKublanov estimates, and the Hashin-Shtrikman estimates, respectively. It is assumed that

(2.45))(2.47), and from the Hashin-Shtrikman estimates (2.32))(2.37) respectively. It is assumed here that the Poisson ratios vr and v2 of the original materials have the same value and h is the ratio of the rigidities of the first and second materials

K-l.4

1 i=

h=I(I=E!

1,2,

K2 4.

(3.27)

A’

THE TRANSLATION METHOD

To prove the Theorem we use the version detail in our previous paper (CHERKAEV and section, Note that although we concentrate mainly the inequalities of the type (4.35) that we anisotropic effective tensor as well. The method (CHERKAEV and GIBIANSKY, based on the lower estimate of the functional

of the translation

method

described in it in this

GIBIANSKY, 1991) ; we outline

on the bounds for isotropic mixtures, are going to obtain are valid for the 1991;

LURIE and CHERKAEV, 1986b)

is

Z,

N

I=

1 w,

i= I

(4.1)

This functional is equal to the sum of the values of elastic energy IV, stored in the element of periodicity of a composite which is exposed to N linearly independent external stress or strain fields with fixed mean values. The energy functional is used because its value is equal to the energy stored by an equivalent homogeneous medium

A. V. CHERKAEV and L. V. GIBIANSKY

952

in the uniform field. The equivalent medium is characterized by the tensor of the effective properties, and the uniform external field coincides with the mean value of the field in the composite. Clearly, the lower estimate of the functional (4.1) also provides the estimates of the effective tensor we are interested in. Below we get the estimates of the functional (4.1) independent of the microstructure of a mixture and extract the geometrically independent estimates for the effective moduli from them.

4.1. Functionals Now we will specify the functional of the type (4.1) which attains minimal values at the boundary of the set of pairs (K,, pO). We discuss here functionals providing estimates for various components of the boundary. To obtain the lower estimate for the bulk modulus one can expose the composite to an external hydrostatic strain &h= &,a, [see (2. IO)] because the energy of an isotropic composite under the action of this field is proportional to the effective bulk modulus KO [see (2.13), (2.14)]. It is convenient for the following procedure to express the energy as a function of the gradient [ = V 0 u of the displacement vector u [see (2.17) and (2.18)]. The corresponding functional has the form [see (2.19) and (2.20)] : F” = ([(x)--C’(x)*-c(x)> if (i(x)>**ar

= 61,

= 2K,s:,

c(x) E (2.18), (2.5), .*az

= 0,

(i(x))**a3

= 0,

(4.2)

where E, is a given constant. It is clear that the lower estimate of the functional (4.2) gives the lower estimate of the effective bulk modulus KO,because the amplitude E , of the hydrostatic strain field is assumed to be fixed. To get an upper estimate of this modulus, we need to expose the composite to a hydrostatic stress (Th= a,a, ; this makes the stored energy proportional to I/K”. The lower estimate of the corresponding functional gives us the upper estimate of K,,. So, this time we minimize the functional I” = (o(_x)~~S(x)~~o(x)) +)~(2.5),

= (2Ko))‘o;,

(G(X)) = alal,

(4.3)

where 0, is a given constant. We will see that the exact estimates of these functionals provide the HashinShtrikman estimates for the bulk modulus. Similarly, to obtain the lower estimate for the shear modulus of a mixture one can examine the energy stored in a composite exposed to the shear-type trial strain. This way we obtain an estimate on one of the two shear moduli of the mixture which is anisotropic in general. However, the other shear modulus can have an arbitrary value and the energy functionals of the types (4.2) and (4.3) are not sensitive to its value. To provide the isotropy of the mixture we should also care about the reaction of the composite on the orthogonal shear field. So, to estimate the shear modulus of an

Coupled estimates for moduli of composites

isotropic of energy E = E2a2 gradients

953

composite we should minimize the functional equal to the sum of two values stored by the medium under the action of two trial orthogonal shear fields and E’ = c3a3. Again we express the energy in terms of the corresponding c = V @ u and r = V @ u’ of the trial displacement vectors u and u’ * - C’(x) * * i(x) + i’(x) * - C’(x) - * l’(x))

Zii = (i(x)

= 2/.&&5 + s:>, if

i(x),

i’(x) E (2.18)

(2.Q

(i(x)>

**a,

= 0,

(i(x))*-a2

= Ed,

({(x))-*a3

= 0,



*-a,

= 0,

(i’(x))*-a,

= 0,

(i’(x))**a,

= Ed,

(4.4)

where s2 and s3 are fixed constants. To find the upper estimate of the shear modulus we use the functional equal to the sum of two energies stored in a composite exposed to two orthogonal shear stresses cr = 02a2 and 0’ = 03a3 I”” = (n(x)

- - S(x) - * a(x) + a’(x) * * S(x) * - a’(x)) = &i+(i:),

if

a,a’~(2.5),

(a)

= 02a2,

(a’)

= 03a3.

(4.5)

Here g’2 and o3 are given constants. We show below that the lower estimates of these functionals lead to the Hashin-Shtrikman and Walpole bounds for the shear modulus. In order to get coupled shear-bulk estimates one can expose a composite to three different fields : a hydrostatic field and two orthogonal shear fields. We have a choice between stress and strain trial fields (two shear fields are supposed to be of the same nature to provide isotropy of the mixture). Therefore the following functions should be considered : p

= zc $ pi >

(4.6)

I”“” = Z”+Z”“,

(4.7)

Z”K = Z” + ZK>

(4.8)

Zb@J= ZC+ Z”“.

(4.9)

The lower estimate of each of these functionals gives some component of the boundary. We should distinguish the cases of well- and badly-ordered materials. The lower estimates of the functionals Zrii and I”“” provide the lower and upper estimates of the convex combination of the effective bulk and shear moduli because these functionals depend linearly on these moduli (the functional Zrri) or on their inverse values (the functional ZuU”). In the well-ordered case (2.31), however, the points of maximal and minimal values of both moduli (the Hashin-Shtrikman points A and C in Fig. 1) are attainable by special microstructures (see FRANCFORT and MURAT, 1986 ; LURIE and CHERKAEV, 1984~ ; MILTON, 1986 ; NORRIS, 1985) ; and it is clear that the estimates of these functionals cannot improve the classical inequalities (2.32) and (2.37).

A. V. CHERKAEV

954

and L. V. GIBEANSKY

On the other hand, minimization of the fLln~tiona~s IaT; or Ii”” demands minimization of one of the moduli and maximization of the other one. We show below that for well-ordered materials this leads to coupled estimates of the moduh which are more restrictive than the Hashin-Shtrikman ones. In the badly-ordered case (2.39) we face the opposite situation: the estimates of the functionals FCCand I”“” Improve the Walpole bounds, and the estimates of the functionals I”“’ and Ciao leads to known ones.

III this section we outline the ideas of the estimating procedure (for details see and ~~~~AN~~Y,~ 991; LURE and ~~ERKA~V,~ 986b; MILTON, I99&,1991) and give the final formulae. First. note that each of the functionals described above is a quadratic form of the fields. It can be represented in the form CWERKAEV

I = (e*D(x)

(4.10)

*e>,

(4.1 I)

eEEK,

where e is a vector composed from the coefficients of tensors of gradients [ or stresses CTin the basis (2.9) ; EK is a set of the doubly periodic vectors possessing prescribed mean values and satisfying some differential restrictions. For components of stress these restrictions are given by the equilibrium equations V* CT= 0, and for gradients [ they express the CompatibiIity conditions < = V 0 u. The matrix D is a piece-wise constant block diagonal matrix composed from the coefficients of the material tensors in the basis of (2.9>. For example, the matrix D and the vector e for the functional ICic have the forms D = D”‘” L=

e = Ii,i’,C”l

C’

0

0

0

C’

0

I 0

0

C’ 1

= [(ilriL,5J,541,

EK = {e : < = Vu,

i’ = Vu’,

(4.12)

,

ti;,i;,i%,i>),

K’~,ii,i’~~i’;~l,

{e) = e,, e -!&periodic.

iy” = Vu”.

Here the vectors u, u’, u” correspond to the three different displacement tensors 5, <‘, i” will be resolved in the basis (2.9). Similarly, the matrix vector e for the functional frrn have the form C’

0

0

0

s

0

L0

0

s I

D = Din0 -‘

e = EK=

j . (4.13) fields ; the D and the

(4.14)

,

K,o,a’l = [Kl,C2,i3,i4), (~I,~?,~~), ~~‘13~i,d3)l~ je:[=Vu,

Vo = 0, Vo' = 0, {e) = eo,e-Q-periodic.),

and so on. Our goal is to find the infimum and over all mi~rostru~tures with prescribed

of the functional volume fractions

(4.15)

I over all fields e E EK of the components.

Coupledestimates formoduliof composites

955

To obtain the simplest estimate of this infimum it is sufficient to note (see, for example, LURIEand CHERKAEV,1986b) that the integrand of the functional (4. IO)is the convex function of the vector e and the inverse matrix D- ‘. It is assumed that the matrix D is non-negative. Due to the convexity of the integrand in the variables e and DP ’ the estimate of the functional (4. IO)has the form Ia

(4.16)

(e)*(D-‘>-‘*
At the same time the value of the functional of the mixture :

lis determined

by the effective properties

I = (e) - (Do) *(e).

(4.17)

Comparing (4.16) and (4.17) which are valid for any vector estimates of the moduli, of the type derived by HILL (1952),

(e}, we end up with

Do 3 (D-l)-‘.

(4.18)

Note that this estimating procedure does not take into account the differential restrictions (4.11) and the estimates are valid but, in general, not exact. The translation method is based on the theory of quasiconvexity (see, for example, DACOROGNA,1982) and the theory of compensated compactness (TARTAR, 1979). Its main idea is to use the integral corollaries of the linear differential restrictions (4.1 I). Namely, it is possible to find quasiaj%ze (DACOROGNA,1982 ; TARTAR, 1985) quadratic functions of the fields e 4(e) = e*T.e possessing

the property

(4.19)

of affine functions

(4(e)>

= 4((e)>

or

(e*T.e)

= (c)*T*(e)

(4.20)

for every field e E EK. Here T is a so called translation matrix which is a constant matrix depending on the differential restriction (4. I 1). Such matrices can depend on several free parameters. The construction of quasiaffine bilinear and quadratic functions for the problem under study (i.e. for stress and strain fields) is discussed in the next section. Let us describe the procedure of obtaining the “translation estimates” (see CHERKAEV and GIBIANSKY, 1991 ; LURIE and CHERKAEV,1986b; MILTON, 1991). First, assuming that quasiaffine functions are already found we add a function 4(e) to the functional I and then subtract it : I=

(e*(D-T(x)*e)+(e*T(x)*e).

(4.21)

Then we estimate the first term by using the above described Hill’s procedure second term by using the equality (4.20). This leads to the inequality (e>*DO’(e)

3 (e).[((D-T)-‘)-‘+T]*(e).

Similar to the simplest case, we may refer to the arbitrariness and pass to the matrix inequalities : D0 9 ((D-T)P’)m’+T.

and the

(4.22) of the acting

fields e (4.23)

A. V.

956

CHERKAEV and

L. V.

GIMANSKY

As before, it is assumed here that the piece-wise constant matrix negative at any point. This leads to the restrictions on the coefficients matrix T : D, -T

D(x) -T is nonof the translation

>, 0,

(4.24)

D?-T>O.

(4.25)

To transform the estimates we use the matrix form of the above described [see (2.29)] Y-transformation introduced by MILTON (1991) and by CHERKAEV and GIBIANSKY (1991) Y(D,)

= ((D)-D,-D,)-m,m,(D,-D,)*(D,-(D))-’.(D,-D,).

If the matrix (D, -DJ resented in a surprisingly 1991; and Appendix)

(4.26)

does not degenerate, then the estimates (4.22) can be repsimple form (see MILTON, 1991 ; CHERKAEVand GIBIANSKY,

(4.27)

Y(D,) +T 2 0. However, in the problem under study we eigendirections and eigenvalues of the matrices C’ for any material has the fourth eigenvalue C’ is in its turn the diagonal block of matrices (4.6), (4.8) and (4.9). Degeneration of the matrices satisfy the relationships P,.D, det[P*(D,

face the situation, when some of the D , and D2 coincide because the matrix equal to zero [see (2.20)]. The matrix D used in the functionals (4.2), (4.4), differences DI-D, means that these

*P; = P,.D2.Pf, -Dz)*PT]

(4.28)

# 0.

Here PI is a projector on the degenerated subspace of the matrix (D, - DJ and P is a projector on the nondegenerated subspace of this matrix which is the complement to the projector P, : POP,

(4.29)

=E,

where E is a unit matrix. We assume that these subspaces the matrices D, and D2 : PJ.D,.PT From

are the eigensubspaces

for

=PI*DZ.PT

=O.

(4.30)

=PI*D2.P:

=O.

(4.31)

(2.20) we have also PI.D,.P;

Remark 4.1 : We also meet the degenerated matrix D, -D2 in homogenization problems dealing with composites made from materials with equal bulk or shear moduli. Such problems were studied in the already mentioned papers of FRANCFORT and TARTAR (1991), HILL (1964), LIPTON (1988), LURIE et al. (1982) and LURIE and CHERKAEV (1984aj. Note that these cases do not imply the equation (4.3 1). The homogenized problem for a mixture made from materials satisfying (4.28) was studied in the paper by LURIE and CHERKAEV (1984a). It was found that the effective

Coupled

estimates

for muduli

957

of composites

properties tensor Do possesses the properties PI*D,-P:=P,*DI.P::=P*.D2.PT, PI~D,.PT=PI-D1,PT=“PPioD?.PT=O

(4.32)

independently of the microstructure of the mixture. In other words, if the property tensors of components have the same eigenvalues and eigendirections, then the effective tensor of a composite also possesses the same eigenvalues and eigendirections. This implies that the tensor transformation Y is undefhred in the whole space because it includes a ratio of two zeros [see (4,26), (4.32)f. However, it can be defined in the nondegenerated subspace of the difference (D t - Dz). Instead of the inequalities (4.23) we will use here (see the Appendix) the system of estimates : the equality (432) and the inequaliiy P*DO’P’

>, P((D-T)~-“)~V’PT+P-T*PT.

(4.33)

Note, however, that the restrictions (4.24) and (4.25) on the translation matrices T should be valid in the whole space, not only on nonzero subspace of the differences (DL-Dz). To simplify the inequality (4.33) we also can use the Y-transformation; the result has the form (see the Appendix) Y(P*D,.PT)l’P*T’PT+(P.T.Pr).fPIofX)I-T).PTI-’

*(P,*T.PT)

2 0, (4.34)

where Y(P*D,*P’)

= P*(
-mlwlZP.(D,-D*).PT-(P-(D~-{D))~PT)-’.P.(D,-Dz),PT. Remember now that P, *D, *PT = PL *D,*PT, leads to the further simphfication of the estimates Y(P~DO~P’r)-i-P*T.PT-(P-T.P;).[Pi

(4.35)

see (4.28). The equality (4.31)

*T*P&‘.(P,

eT.PT) 2 0.

(4.36)

This relation means that all eigenvalues of the matrix in the left-hand side are nonnegative. The scalar consequence of this inequality which is used for establishing the bounds has the form det~Y(P~T)o.PT)+P-T-PT-{P~T*P~)*~F~_~T*P~3-~*(P~

-T*PT)] 2 0,

for at1 translation matrices T : TE (4.19) (4.20) (4.24), (4.25).

(4.37)

The parameters of the symmetric matrix T should be chosen in order to make the estimates (4.37) most restrictive. To do this we have to determine these parameters from the requirement that the maximum number of eigenvalues of the matrices D, -T and DZ- T vanish (see CHERKAEV and GIRIANSKY, 1991) rank (D, -T) -!-rank (Dz -T)

A. V.

958

is to be minimized

CHERKAEV

over all translation T~(4.19),

5.

and

L. V.

matrices

GIBIANSKY

T :

(4.20), (4.24), (4.25).

(4.38)

CONSTRUCTION OF QUASIAFFINE FUNCTIONS

To use the translation method for the problem under study we need to find the set of bilinear quasiaffine functions of stresses and strains. In this section we determine such functions depending on two strain fields, two stress fields, and the bilinear function depending on stress and strain fields.

5.1. The strain-sfrain

quasiclfJine

,functions

We are looking for the quasiaffine functions in the space of strains. To use the already known results we will find them in an enlarged class of gradients of displacements. Remember that the tensors of strains and gradients of displacements are connected by the relationships (2.18). So, we expand the space of strains by adding the fourth orthogonal coordinate equal to the antisymmetric part of the gradient of displacement [see (2.17)-(2.20)]. We will use the well-known result about weak convergence of the Jacobian J of two potentials N’, and IL’? J = det (Viz’,, VN?) = V\r , * R * V\zl>, where R is a rotation

through

(5.1)

a right angle R=i@j-jai.

(5.2)

According to RESHE~NYAK (1967) (see also DACOROGNA, 1982; KOHN and STRANG, 1986 ; TARTAR, 1985) the Jacobian (5.1) is a quasiaffine function of its arguments : (J)

= (V\~,*R.VM~~)

To find the bilinear quasiaffine consider the bilinear form

function

(5.3)

= (VN~,)*R*(V~~). of the gradients

of displacements

let us

(5.4) of the gradients

[ = V Q u, [’ = V 0 u’ for any two displacement u = (uli+uzj>,

u’ = (u’,i+u;j).

vectors (5.5)

This function is an arbitrary linear combination of Jacobians of pairs of components ’ u;. It is clear from the previous note that 4([, [‘) is a quasiaffine function 1.4I, U?, UI, for any values of the parameters t:, i = 1, 2, 3,4 of this linear combination. The expression (5.4) can be written in the form

Coupledestimates formoduliof composites

where the matrix @Ccis represented

959

in the basis a ,, . . . , a4 [see (2.9)] as follows t1

c

-t2 @cc(tl,t2,t3rt4)=

f2

f3

-t,

_t

1 -t4

1

_i’ _i’

_t 3

f4

4

I

--f3

(5.7)

2

t,

t, J

and parameters t,, i = 1, 2, 3, 4 are linearly independent linear combinations of parameters t:, i = 1, 2, 3,4. Remark 5.1 : Particularly, to estimate the energy of a composite (the functional Zi) it is sufficient to use a quadratic form of gradients [ instead of a bilinear one. This quadratic form is associated with the symmetric part Tr = a([, Qs of the matrix @Cc described above. It has the form Ti(t,)

= @“r(t,,O,O,O).

(5.8)

This function ALLAIRE and

written in different terms has been used for obtaining estimates by KOHN (1992), GIBIANSKY and CHERKAEV(1986, 1987), KOHN and STRANC (1986), LURIE et al. (1982), LURIE and CHERKAEV(1984a, 1986b) and FRANCFORTand MURAT (1986). 5.2. The stress-stress

quasiaflne

functions

Let us note first that any stress tensor has the following 1927) ” RT-~*R=V@V~ through

the Airy stress function

x ; therefore

(5.9)

v=Vx.

(5.10)

form V@v*-@(‘a-V@v’

(5.11)

is quasiaffine for any vector fields v, v’, as shown in the previous the bilinear form of the stresses 0, C+ (RT.O.R)..@,ii.

(see LOVE,

we have

Rr-0.R=V@v, On the other hand the bilinear

representation

.(R. a-RT)

section.

=~..(RTORT)..~ii..(ROR)..a’

Therefore,

(5.12)

is also quasiaffine. Remember now that due to the equilibrium conditions CJis a symmetric matrix, therefore 0 -*a4 = 0 identically. It means that the quasiaffine function (5.12) can be represented as a bilinear function 0 * w”(t, of the first three coefficients

) t2, tx) - d

(5.13)

ci, 0;. i = 1, 2, 3 [see (2.10)], with the matrix

WU(t,,t2,t3)=

rI

tl -t2

-t,

t2

-t3

-t4

t3 t4 1 -t,

)

J

(5.14)

A. V.

960

which is the upper left (3 x 3) minor basis (a,, a,, as, a,). 5.3. The strain-stress Let us consider

quasiafine

L. V. GIBIANSKY

CHERKAEV and

of the matrix

(RT @ RT) *. mn **(R @ R) in the

functions

again the bilinear

quasiaffine

function

[[email protected]

(5.15)

of the arguments [ and Vv = RT *c *R which are gradients of the vector potentials. We are to rewrite this form as a function of the stress tensor and the tensor of the gradient of displacement. We also should take into account the condition of symmetry we get the following quasiof the stress tensor r~- * a4 = 0. After obvious calculations affine bilinear form : (p”’ = o. (pi . [,

(5.16)

where the (3 x 4) matrix
a,“[ = W(t,,

t2, t3, t4) =

[ ;;

;

-);

-;;I

(5.17)

in the mentioned basis (2.9). Here the three-dimensional vector 0 and the fourdimensional vector i are the coefficients of tensors 0 and [ in the bases (a,, a2, a,) and (a,, al, a3, a,) respectively.

6.

PROOF OF THE HASHIN-SHTRIKMAN AND WALPOLE ESTIMATES

In this section we get the Hashin-Shtrikman and Walpole bounds by using a regular procedure of the translation method [see also MILTON (1990b, 1991) where similar results were obtained]. We prove it here for the completeness of the discussion and also for the demonstration of a regular procedure on simple examples.

6.1. Bounds for the bulk modulus 6.1.1. The lower estimate. To estimate the functional I’ we need the symmetric translation strain-strain matrix Tb [see (5.8)]. To get the result we use the inequality (4.37) where we should substitute the following matrices Di=Di=C:,

i=1,2,

T = T’(t), 0

0

0

1

0

0

0

0

1

0

1

P’=

0

L 1

Pi = (0

0

0

1)

)

(6.1)

Coupled

estimates

for moduli

961

of composites

[see (4.2) and (58)f. The conditions (4.24) and (4.25) of the positivity of the matrices C’, - Tr(r) and C; - Tr(t) have the form

1

[2Ki-t

0

0 0

0

2Pift

0

0 1

2pj-kt 0

-t0

’ ,.

i

’ i= 1,2

(6.2)

and lead to the scalar inequalities t Q 0,

t 2 -min

(fit, ~~1 = --Fz.

(4.3)

The estimate (4.36) for the isotropic matrix D,i - C0 associated with the functional I[ has the form YG’Ko)f t 0 0

I

0

0

Y(2Puo)- t 0

0 YPPO) - t

and leads to the inequality for the bulk modulus K0 y(2Ko) >, -4

1 20

tE (6.3).

(6.4)

(6.5)

The most restrictive bound corresponds to the critical value [see (6.3) and (4.3&)] t = t* = --I4

16.6)

of parameter t. This bound coincides with the Hash~n-Sht~kman bound for the bulk modulus [see (2.32)-(2.37), (2.40)-(2.41)].

and Walpole lower

6.1.2. The upper estimate. The upper estimate for the bulk modulus can be obtained analogously using the functional I” instead of II and the quasiaffine function associated with the translation matrix TO@,) = W’(ti,

0,

0,

This matrix is a symmetric part of the matrix W’ft,, the matrices Dj, i = 0, 1, 2 are equal Di = D: = Sj,

0).

(6.7)

t2, t3, t4). For this functional,

i = 0, I, 2.

(6.8)

The difference Dz -D2 is nondegenerate, therefore we can use estimates in the form (4.27). Now the restrictions (4.24), (4.25) have the matrix form p 1

-2K, 0

L

0

t’

’ -‘-+r, 2!4

0

0 0 J-3-q 2&

1

>O

J

(6.9)

A. V. CHERKAEV

962

and L. V. GIBIANSKY

or scalar form

The estimate

for an isotropic

‘.

(6.10)

>, 0.

(6.11)

effective tensor S,, becomes , 0

0

0

L The corresponding

0

0

1 Y LjE 0’

0

scalar inequality

for a bulk modulus y

(2+1 0

+tl

-tl K0

3 0

(6.12)

becomes the most restrictive when the parameter t, is chosen as t , = t: = - (1/2~ ,) [see (4.38) and (6.10)]. By using the properties of the Y-transformation (2.30) it can be represented in the form (6.13)

YWO) d PI. This bound coincides with the HashinShtrikman bulk modulus [see (2.32)-(2.37), (2.40)-(2.41)].

and Walpole

upper bounds

for the

6.2. Bounds for the shear modulus 6.2.1. The lower estimate. We begin with the functional for the shear modulus. We use the formulae (4.22)-(4.38), as an eight-dimensional vector

ICCto get the lower bound where e should be treated (6.14)

e = 111,12,r1,14,1’,,1;,1~,~~~, where ii, ii, i = 1, 2, 3, 4 are the components of gradients matrices Di, i = 0, 1, 2 have the block-diagonal form

i, <’ in the basis (2.9). The

(6. IS) The projector P [see (4.28)-(4.32)] on nonzero subspace the complementary projector P, have block forms

( 1 Pi

p = p;; =

0

0 Pi’

where Pi and Pi are defined by (6.1).

of the matrix

0 PI = py E ( Pi0

Pi’ 1

(D, - DJ and

(6.16)

Coupled

estimates

for moduli

963

of composites

Let us construct a quasiaffine quadratic form of the vector e. Consider form of this vector associated with the block matrix

Tr;

=

@cc(t;, t;, t;, t;,

w(t:,

t;, t:, t;;)

w(t:,

w(t:,

t;, tf, t:> > ’

t:, t:, ti)

the quadratic

(6.17)

where the matrices @cc are defined by the formula (5.7). It is clear that this function is quasiaffine as a sum of quasiaffine functions. The matrix Tii depends on 16 parameters t{, i, j = 1,2,3,4. However, the matrix of a quadratic form has to be symmetric ; this reduces the number of parameters to six. We will also assume that t i = t: because of the isotropy of the mixture. It also turns out to be sufficient for the shear modulus estimates, to make three of the five remaining parameters t/ zero, and to deal with the matrix -t,

Tii(t,,

tJ =

I

0

0

0

0

0

0

f2

0

t,

0

0

0

0

t2

0

0

0

t,

0

0

-t2

0

0

0

0

0

-t,

-t2

0

0

0

0

0

0

-t2

-t,

0

0

0

0

0

-t2

0

0

t,

0

0

0

12

0

0

0

0

t1

0

t2

0

0

0

0

0

0

-tl

2

(6.18)

which depends only on two parameters t,, t2. The restrictions (4.24), (4.25) have the matrix form Di-Tii(t,,

t2) 2K,+t, 0

0

0

0

0

2~i-tl

0

0

0

0

2/.Lu,-t,

0

0

0 0

0 0

tl t2

t2

0

t2

0

-

f2

0

2K,+t, 0

0

0

0

0

0

0

i= I,2

(6.19)

and the scalar form (2K,+t,)t,

-t:

(2pi-tt1)2-tt:

3 0 3 0

i= 1,2,

(6.20)

A. V. CHERKAEVand L. V. GIBIANSKY

964

or

(2K,i”+t,)t,-t:

2 0,

(2&in-t1)*-t:

(6.21)

2 03

where

min{K,,K2),

Kmin

=

Ain

= min

{PI,

P21 =

(6.22)

P2.

t ,, t2 correspond

The limiting values t:, t; of parameters [see (4.38)] ; they are equal to

to the equalities

in (6.21)

$22

(6.23)

‘l” = K,,,+~/L~’ 2cLl(Kmin +PLZ) Kmin+2p2

t’ = Substituting y(Pn

.

these values into the estimates

-

Do. p(c) +pii

. Tri.

t:’

(4.36) which now take the form

Tri. p?‘)

. [py.

0

0

0

2Y(kJ+ t:

0

0

0

2Y(h> + t:

0

0

0

0

0

- t:

0

tz*

2y(K,) + $

=

pl< _ (pii.

(6.24)

Tii

.

PI”]

1 . (p?

.

Tn. pii’)

I

i

0

2y(Ko) + ‘:i

- tT2

0

0

0

t2*

- t;

0

0

0

> 0,

(6.25)

I

we get the scalar inequality 2Y(Po)+tf-t?

= 2Y(Po)-

2K,,,c12 > o Km,“+2p2 ,’ .

(6.26)

Coupled

estimates

for moduli

965

of composites

It coincides with the Hashin-Shtrikman bound on the shear modulus in the well-ordered case (2.31) and with the Walpole bound (2.40)-(2.41) ordered case (2.39).

(2.32)-(2.37) in the badly-

6.2.2. The upper estimate. To obtain the upper bound for the shear modulus of the mixture we use the same procedure for estimating the functional I”“. In this case (6.27)

e = {~1,~2,~3,~;,&~;} is a six-dimensional vector consisting a’; the (6 x 6) matrices D.=D”“= I

of the components

of two stress tensors

(6.28)

i=O,1,2

I

0 and

are block-diagonal. The difference (D1 -D2) is nondegenerate, therefore we can use the estimate in the form (4.27). We construct the matrix T”” of a quasiaffine quadratic function of the vector e in the same way as Tri using the bilinear quasiaffine form au0 instead of the form #

Tuu(tl, tz) =

I-t* I

The restrictions

0 0

t1 0

t, 0

o

o

o

0 _t

-t2 0 0

t,0

o

0

I

0 0

-t,0

t2 0

0

I,0

II

(4.24), (4.25) now have the matrix form

&+t,

0

1 1

(6.29)

’

0

0

0

0

0

0

0

-f2

0

t2

0

0

0

I

0

Dy-TuO(t,,

1 -2P,

ti

0

0

0

0

I 2/Li

t1

t2) =

3 0,

&,+t,

0

I

0

0

0

t2

1

---I

’

2Pi

0

0

-I2

i=

0

1,2,

0

0

1 -2/Ai

tl (6.30)

or the scalar form 1 ‘I ’

-2K,,,,,’

(6.3 1)

966

A. V. CHERKAEV

and L. V.

(Z;a&t,)lr.

GIBIANSKY

3 0,

(6.32)

where K,,,,, = max {K,, Kz}, k,,

(6.33)

= max (PI, ,k} = pl,

The critical values t:, tT of the parameters

t,, t2 equal (6.34)

The estimate

(4.27) becomes

Y

(1 I

(!

1 y jE

0 Y(DT)

0

0

-t;”

TK,

0

+tt:

+T""(t;", t;) = 0 0

0

0

0

0

- t;

0

tz*

0

0

0

0

0

0

t;

0

- t?*

1

0

0

0 > 0.

y

0 0

1

&Lo

0

+t:

(6.35)

I

( 1 (2 1+t: y

- 0

-@Lo

It leads to the scalar inequality

(6.36)

Coupled

estimates

for moduli

of composites

961

or [see (2.30)] (6.37)

this one coinciding with the Hashin-Shtrikman bound [see (2.32))(2.37)] in the wellordered case (2.31) and with the Walpole bound [see (2.40))(2.41)] in the badlyordered case (2.39).

7.

COUPLED BULK-SHEAR MODULI BOUNDS

Here we prove new estimates translation method.

formulated

in the Theorem

by using the described

7.1. The well-ordered case 7.1.1. The lower estimate. Let us first estimate the functional Ioii which allows us to obtain the coupled bound, as mentioned above (Section 4). For this functional the vector e is the following 11-dimensional vector

The matrices

Di, i = 0, 1, 2 have the block-diagonal

D. I

and the projectors

z

form

DPi’ = I

(7.2)

P and PI are also block-diagonal

matrices

(7.3)

where E is the unit (3 x 3) matrix and Pi, Pi are defined translation matrix Turi(tl, t2, t,, t4) with the block form,

Tuii(t,,

t2, tj, t4) =

by (6.1). We use the

WU( - t,, 0, 0,O)

wyo, -t,,

@(O,

-tx,O,O)

w~(-t2,0,0,0)

@qO,O,o,-t,)

aqo,

0, -t,,

wyo,

aq

0)

0,O)

o,o, - t4)

W(0, 0, -t,, 0) ,

- t*, o,o, 0) 1 (7.4)

where@’

= (CD”oT.

968

A. V.

The restrictions DPii-Tuii(t,,

CHERKAEV

and L. V.

GIBIANSKY

(4.24) and (4.25) have the form

t2, t3, t4) =

&+t,

0

0

0

t3

0

0

t3

0

0

1

1

0

\

2,ui -tl

0

0

1 -2PL, t’

0

0

0

0

t3

0

2K,+ t2

0

0

t3

0

0

0

2/G- f2

0

0

0

0

0

0

0

0

f3

0

0

0

0

0

t3

0

0

0

0

0

t3 0

0

0 0 0

-t,

0

0

0

0

t3

f3

0 0

0 0 0

t2

-t4

0

-

t4

2Ki+

0

0

0

0

0

0

t>

2Pi-

t2

0

0

0

t4

0

t4

0

0

-

0 0 0 0 0

t3

0

0 0

-t3 0

0

t4

f4

0

-t4 0 0

0

0

0

0

0

0

h- t2 0 0 i= 1,2.

0

0

2l-et2

0

0

t2

t4

’

>/ 0,

(7.5)

1

Let us find the critical values of parameters t,, . . , t4. We mention first, that each of the matrices [Dpri - Toii], i = 1,2 is divided into the direct sum of four blocks

The matrix AFbc here denotes a diagonal minor of the matrix of elements standing on the intersection of lines and columns

[Dpii -Toil] ; it consists with numbers a, 6,c.

Coupled

The matrices

estimates

on the right-hand

for moduli

of composites

969

side of (7.6) have the following

form

:

t3

h-t*

,

t3

t4

,- 1 -2pi A?,4.,, I

t1

-

r

A!.738 I

t3

-t3

II t3

t2 -t4

t3

t2

J

(7.8)

t‘j ’ t2,

t3

_ -

(7.7)

’

-t3

t4

1 -2Pi

.

2Pi-

2iYi+t,

t3 i

t4

t3

-t4

(7.9)

)

2Ki+t2, (7.10)

The critical

values tj+ of the parameters

ti are determined

det Af.‘,”

from the following

system

= 0,

(7.11)

detA~,s~‘o = 0,

(7.12)

detA,,

24.11

_

- 0,

(7.13)

det At9 = 0.

(7.14)

They are equal to 1 t:=

-2K,+

(K,

-K2)d(K,

+C1,)Gc

[K,1L,(K,~,+K,~2+2~,~2)(K2+2~2)(Kr+CL2)

fP*)

(7.15) ’

(7.16)

(7.17) t4* = &-ttZ*. One can also check that such a choice of parameters minimal sum of ranks of the matrices [Dyii-Tuii(t:, (tf, t?, t?, t$)J, and the non-negativity of these matrices.

(7.18) ti, i = 1, . . ,4 provides the tT, tf, tt)] and [Dzii-Tua

970

A. V.

The estimate det [y(p”ii

. D;;;i . (pG)T)+pG

where for isotropic . D;;‘”

L. V.

GIBIANSKY

(4.37) becomes .T"i; . (pG>T

_(p~;;.T"i'.(p~)T)

y(p”:’

and

CHEKKAEV

.[p~~i.T”“.(p;ii)T]

tensors

1 .(pyii .T"i'.(p"i;)T)]

2

0,

(7.19)

Do

.(pdi)T)+p"i'

.TU;i

. (p”ii)T

_ (p";: .Tcii . (p?‘)‘)

. [pF

.T"'i . (p;X>T]

0

. (p;ii

.Tdi

. (~4')')

=

0

0

t3*(t2*+ tt)

0

tz* 0

ti* (tz* + t4*) t:

0

0

0

0

0

0

0

0

fP(fZ*+ t‘?)

tz*

0

0

0

0 0

0

- t;

0

0

0

0

0 0

tT(t2*+ t$)

0

0

0

0

0

2110 + t:

0

0

0

- t4*

t‘f

0

0

0

tz*

2Pco+t2*

0

t4*‘__. t2* -tT2

0

0

0

t4*

0

0

0

- t;

-g2 f:-27-.

0

0

0

+

0

0

- t:

2K,

2&

+

%o+tz* 0

1 2 0.

(7.20)

0

%-4l+9

The matrix on the left-hand side of (7.20) can also be presented four blocks of the type (7.7)-(7.10)

as a direct sum of

Coupled y(p”i:

. D;;‘;

.

estimates

for moduli

of composites

971

(pXi)T)+pdC .T"rr .(pdi)'

_ (pd:.Td6* . (pF)T) . [p$i .Tbii

.

(py>T]-

1. (p;ri

.Tdi

. (p”ii)T)

= A 0‘vx9 @ A;v4 @ A;”

@ A$*,

(7.21)

where

A;539

=

(7.22) 1

Kao)+

Ax4 _ ~3,’ _ o-o-

*2

@t:+t,

ffyf2*+ t4*)

t2*

-

*z*

i

1 1

A&s 0

=

e

YCGO)+ t2*

t4*

One can check that the most restrictive come from the inequality

(7.24)

YGGO) + t2* ’

estimates

(7.23)

on the set of the pairs (y(K,),

det (AA-5%9)3 0.

y(po))

(7.25)

The last inequality gives us the lower estimate (3.2) and (3.4) of the Theorem. The boundary of the set of possible pairs (y(K,), y(po)) which satisfy this estimate is defined by the fraction linear curve y(,uo) = F, (y(K,)). The direct calculation shows that it passes through the points (3.20).

7.1.2. The upper estimate. To study this case we deal with the functional liua. The proof of the estimate is the same as in the previous case. Here the vector e equals e= The matrices

(7.26)

{il,i2,i3,i4,al,a2,a3,~~,~~,~~}.

Di,i= 1,2 have the block diagonal

form

D I =D?‘= I

projectors

(7.27)

P and PI have the block forms p = piuu=

, PI =Pi""=(P$

0 O),

(7.28)

972

A.

V.

CHEKKAEV

and

t.

V.

GIBIANSKY

the matrix TSrm(t,, tZt f,, fp) is chosen in the following block form

TCbd(t,,t2,t3, t4)=

@y-t,,O,o,o)

W(O,-t,,O,O)

W(O,O, -t,,O)

mui(O,-bO,O)

W"(-t,,o,O,o)

W"(O,o,O,t,)

W”(O,O, 0, t4)

WU( - tz, o,o, 0) 1

r @(O, 0, - t.?,0)

. (7.29)

As in the previous case each of the (10 x 10) matrices Dp - Tso”, D$, -Tioa(tl,

’

t2, tg, t,) =

ZKi+

t,

0

0

2Pivtt

0

0

0

0

t3

0

0

2!-4-tr

0

0

0

0

0

t1

0

0

t3

0

0

0

&+t2 I

0 0 0

0

0

0

13

0

t3

0

0

0

0

0

-t3

0

t3

0

0

0

0

f3

0

0

0

t3

0

0

0

0

0

0

0

t3

0

0

0

13

0

0

0

0

-t3 0

0

0

0

0

14

1 -2j.ti 0

t2 --1 2pj

0

t, 0

0

0

- 14

& +tz

3

(7.30)

0

0

I

--t4

0

0

0

1

0

2,Ll,-tz 0

---1 2pi - t2 1

can be represented as a direct sum of four blocks D;ii -Tuii(t,,

tZ, t3, t4) =I _s&‘,~,‘~ @ A>3 @ &.a @ Ap.7.Y,

(7.3 1)

Coupled

estimates

for moduli

973

of composites

where

(7.32)

(7.33)

(7.34)

The critical values t: of four parameters Ii of the translation matrix are determined from the system of two equations detA:a’ = 0,

(7.35)

detA$’ = 0,

(7.36)

and two scalar equations following from the equality (7.37)

rankAt7.9 = 1. They equal

(7.38) (7.39)

(7.40)

(7.41) Such a choice of parameters provides the non-negativity Tair(t?, t:, t?, tf) and DTri -Tuii(t:, tf, t:, tf). We obtain the upper bound from the inequality det [y(p”ii . &ii

of the matrices Dyii-

. (pdI)T) + pdi . T”ii . (PM)’

_ (poti . Tdi . (pyCl)T) . [p?K . ~“ii . (p:K)T] - 1. (pr;ii . ~di . (pdi)T)]

2 0.

(7.42)

914

A.

V.

and

CHEKKAEV

L. V. GIHANSKY

The matrix in the square brackets is divided into four uncoupled blocks for isotropic tensors Do; the most restrictive estimate comes from one of them det Ail.‘.’ > 0.

(7.43)

This inequality can be rewritten in the form (3.2), (3.8) ; calculation boundary fraction linear curve passes through the points (3.21). 7.2. The badly-ordered

shows that the

case

The procedure does not change in this case, the only difference for parameters in the translation method. 7.2.1. The lower estimate. vector e equals

We estimate

is in the expressions

ICC;. The

the functional

12-dimensional

(7.44) The matrices

Di, i = 1, 2 have the form D =D’;;= / L

The projectors

i=

1,2.

(7.45)

P and PI are the block matrices PC

0

0

0

pi

0

: 0

0

Pi I

““I p = p‘ai =

“11 P, = Py. =

)

I Pi 0

Pi 0

Pi 0

and the matrix T has the block form

““I

T4i'(t,,f2rfs,f4)

=

I7

W’(-t,,O,O,O)

W(O,-t,,O,O)

wyo,o,-t,,O)

@(O,

@q-t,,O,o,o)

~~~(O,O,O,-t,)

~~~(O,O,O, -t4)

~‘i~(-tr,O,O,O)

!

-t3,0,0)

wqo,o,

Each of the (12 x 12) matrices minors

-t3,0)

(7.46)

(7.47 1

DF -Tici, i = I, 2 is a direct sum of the diagonal

where 2K,+t, A,1.“,l

I =

(7.49)

t3

t3

2f-e t, --f3 t3

-1s 2K,+t, t4

ts 14 t2

,

(7.50)

Coupled

estimates

for moduli

r 2/l-t,

A,$7,‘o =

-tj

-t3

-t4

t,

A,

-t, -t4

-t4

The critical values t,? of the parameters mined by the conditions

(7.51)

9

2K,+tz

2/l-t2

L -1,

975

1

-t4

t3

tj

-

-tj t2

-t3

1.6. I I _

of composites

(7.52)

.

2P, - t2 I

ti of the translation

matrix

Tici are deter-

det Af5,12 = 0,

(7.53)

detAz,5,”

(7.54)

= 0,

rank A$‘,”

= 1,

(7.55)

and equal tl*

~P,cL:W~-K,)

=_-_-----____ PIIK~+PZ)(KI

+W--P~WI &-d(~,W2+~2)-~22K,

t”~~__-------------_--~~, PII(K~+P~)(KI

____)

(7.56)

+1*2M2+21*2) +~2))

+244-~2W,

(7.57)

+~dW2+21*2)

ti* = Jt:(2&-t2*),

(7.58)

t$ =2/L,-@.

(7.59)

As in the previous case they provide the minimal sum of the ranks of the matrices Df:‘_T”‘(@, @-, tf, tf) and D:2;r_Tiri($, t;, t_T, tz) and non-negativity of these matrices. The matrix [y(p;<< s D$’ a (p;
1 . (PPi . Tii; . (P’c’)‘)J for the isotropic

tensor DO also has a block diagonal structure. One could check that the most restrictive estimates correspond to vanishing of the determinant of the block Ah.4x9 det z4A.4,y3 0. This condition leads to the inequality curve passes through the points (3.22).

(3.3), (3.12) of the Theorem;

7.2.2. The U~J.W estimate. For this estimate is a quadratic form of the vector e e= The matrices

(7.60)

we consider

lo,,a2,a~,~‘,,~i,~;,o’;,crZ,

the functional

ai;>.

the boundary

I”““, which

(7.61)

Di, i = 1, 2 are defined by the expression

D 1 = D”“” i =

(7.62)

976

A. V. CHERKAEV and L. V. GIBIANSKY

the matrix T is

T”““(t, >t2, t?, td =

W”(-t,,O,O,O)

W”(0,

W”(0,

0,O)

WO( - t2,0, 0, 0)

W”(O,O, - t‘$,0)

: W”(O,O, 0, - t3)

W”(O,O, - t4, 0)

W6( - tz, o,o,0)

-t;,

-t,,O,0,0)

W”(O,O,O,-t,)

(7.63) The difference (D, - D2) is nondegenerate, form (4.27). Each of the (9 x 9) matrices sented as a direct sum of four blocks D[ii _Tii’ 1

therefore we can use the estimates in the [D,Y’-TbOb(t,, t2, t,, t,)] i = 1,2 is repre-

= A I!.5,9 @ Ai?4 cij A;*’ @ A;,‘,

(7.64)

where

(7.65)

A;,4

zz

(7.66)

(7.67)

(7.68)

The critical values t,+ of parameters from the equations

t, of the translation

matrix

T”“” are obtained

detA>‘x” = 0

(7.69)

det Af4 = 0,

(7.70)

det Azx4 = 0,

(7.71)

detA?*

(7.72)

= 0,

.

Coupled

estimates

for moduli

977

of composites

and equal

(7.75) (7.76)

t4* = 1/(2p,)-t:. The estimates

come from the inequality det [Y(Dr)

+T”““(t,*,

The most restrictive ones correspond AA.539 of the matrix [Y(D,“““) + T”““(t:,

t$, t?, t$)]

to vanishing t;, t:, t$)]

B 0.

(7.77)

of the determinant

of the block (7.78)

detAA,5*9 B 0. This condition leads to the inequality (3.3), (3.16) of the Theorem. that the boundary curve passes through the points (3.23). This statement concludes the proof of the Theorem.

One can check

ACKNOWLEDGEMENTS We are grateful to G. W. Milton for useful discussions and for additional references and to K. A. Lurie for helpful comments that improved the text. We are grateful for support from the Packard foundation through a fellowship awarded to G. W. Milton.

REFERENCES

Design of Structures (edited by M. BENDSGE and C. A. MOTA SOARES), p. 207. Kluwer, Utrecht.

ALLAIRE, G. and KOHN, R. V.

1992

Optimal

AVELLANEDA, M. BERGMAN, D. J. BERRYMAN, J. G.

1987 1978 1982

SIAM J. appt. Math. 47, 1216-1228. Phys. Rep. C 43, 377407. Elastic Wave Scattering and Propagation by V. K. VARADAN and p. 111. Ann Arbor, MI.

(edited V. V. VARADAN),

A. V. CHERKAEVand L. V.

978 BERRYMAN, J. G. and MILTON G. W. CHERKAEV. A. V. and

GIBIANSKY

1988

J. Phys. D: Appl. Phys. 21, 87-94.

1992

Proc. R. Sot. Edinburgh 122A, 93-125.

CHRISTENSEN.R. M.

1979

DACORO~NA, B.

1982

FRANCFORT, G. and MURAT, F. FRANCFORT. G. and TARTAR, L. GIBIANSKY, L. V. and CHERKAEV, A. V.

1986 1991

GIBIANSKY, L. V. and CHERKAEV, A. V.

1987

HASHIN, Z. HASHIN, Z. and SHTRIKMAN, S. HILL, R. HILL, R. HILL. R. KOHN, R. V. and LIPTON, R. KOHN, R. V. and MILTON, G. W.

1965 1963

Mechanics of Composite Mutevials. Wiley, New York. Weak Cnntinuity and Weak Lower Semicontinuity of Non-linear Functionals (Lecture Notes in Math. 922). Springer, New York. Arch. Rat. Mech. Anal. 94, 307-334. Muthematical Prohhms in Mechanics, t. 3 12. Series I, pp. 301-307. C.R. Acad. Sci., Paris. Design of’ composite plates of e.utremal rigidity, Preprint 914. Ioffe Physicotechnical Institute, Leningrad. Microstructures of’ composites of’ e.utremal rigidity und exact estimates af.provided enerqy~ density, Preprint I 115. Ioffe Physicotechuical Institute, Leningrad. J. Me&. Phys. Solids 13, 1 19-l 34. J. Me& Phys. Solids 11, 127-140.

KOHN, R. V. and MILTON, G. W. KOHN, R. V. and STRANC;, G

1988

KU~LANOV, L. and MILTON, G. W

1992

LIPTON, R. Love. A. E. H.

1988 1927

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l984a

LURIE, K. A. and CHEKKAEV. A. V

1984~

LURIE, K. A. and CHERKAEV. A. V.

l986a

LURIE, K. A. and CHERKAEV, A. V. LUREI, K. A., CHERKAEV, A. V. and FEDOKOV, A. V. MILTON, G. W.

1986b

Proc. R. Sot. Edinburgh 99A, 71-87 (first version Preprint 783. Ioffe Physicotechnical Institute, Leningrad. 1982). The problem of ,fbrmation of’ optimal isotropic, multiphuse composite, Preprint 895. Ioffe Physicotechnical Institute, Leningrad. Proc. R. Sot. Edinburgh 104A, 21-38 (first version Preprint 894, Ioffe Physicotechnical Institute, Leningrad, 1984). Adv. Mech. (Poland) 9, 3381.

1982

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1981

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APPENDIX Here we show the transformation of the inequality (4.23). Consider first the case of the nondegenerate matrix (D, -D,). We literally follow here the paper by MILTON (1991) and begin with the inequality (4.23). Observe that for any matrix X and a constant m,, mz, m,+mz = I we have

980

A. V. CHERKAEVand L. V. GIBIANSKY

m,X+m,E-m,m,(X-E)*(m,E+m,X))‘.(X-E)

= [(m,X+m,E)

*(m,E+m,X)-m,m,(X-E)*(X-E)]*(m,E+m,X)-’

= (m,+m,)*X

.(m,E+m,X))’ Setting X = (D, -T) . (D: -T)

’ and multiplying

= (m,X -‘+m,E)-

‘.

(AI)

on the right by (Dz - T) produces

(m,(D,-T)“+m,(D,-T)-’ =m,D,+m,D,-T-m,m,(D,-D,).(m,D,+m,D,-T)-’.(D,-D,). Combining

(A.2)

this with (4.23) we obtain

D, > m,D, +mD-nz,m,(D, The tensor transformation

-D,)*(m,D,

+m,D>-T))’

Y (4.26) can be rewritten

*(D, -Dz).

(A.3)

in the form

D,=m,D,+m,D,-m,m,(D,-D,).(m,D,+m,Dz+Y)~’.(D,-D,).

(4.4)

Comparing (A.3) and (A.4) and assuming (D, -DJ is nondegenerate gives the bound (4.27). Lets assume now that the matrices of the original materials have equal eigendirections and eigenvalues. To get the estimate for this case let us project the inequality (A.3) by means of P. We have P*D,.PT

3 P.(m,D,

+m,D,)*PT -m,m,P*(D,-Dz)*(m,D,+m,D,-T)-‘*(D,-D,)*PT.

Now we represent

the matrix (m,D, + m ,D, -T)

(m,D,+m,D,-T)

(A.5)

in the block form

( 1

= ;

;

,

c4.6)

where K = P*(w~~D,+~,D,-T)*P~,

(A.7)

L = P*(m,D,

(A.8)

+m,D,-T)*P;,

M = P,*(m,D,+m,D,-T).P;,

(A.9)

and suppose that the matrix M is nondegenerate. One can check that written in the similar block form has the upper left block K’ K’zz (K-L.M-

the inverse matrix being

‘L-r)‘,

(A.lO)

Substituting this into the (A.5) and keeping in mind that P * D, * P.1 = P * D, * PI = 0 and I’, . (D I -DJ * PI = 0 [see (4.28)] we come to the inequality P.Do*PT

> P.(m,D,+m,D,).PT--m,mZ[P.(D,

-Dz).P7]

.[P.(m,Li,+m,Dz-T).P~-(P.T.P:) .(P,.(m,D,+m,D,-T).P:).(P,.T.PT)]~’.[P,.(D,-D?).PT]. Comparing P.Do.PT

(A.1 1) and the following

form of the relation

= P.(m,D,+m,D,).P’-m,m,[P.(D,

(A.ll) (4.35)

-Dz)*PT]

~[P~(~,D,+~,D,)~P’~+Y(P~D,~P~~)]~~-[P~(D,-D,).P~] and assuming the nondegeneracy (4.36).

expression

of the difference

[P * (D, - DJ. P’] we end up

(A.I~) with

the