JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS: Vol. 42, No. 2, FEBRUARY 1984

On the Existence of Solutions to Some Problems of Optimal Design for Bars and Plates 1 K. A. LURIE, 2 A. V. CHERKAEV, 3 AND A. V. FEDOROV 3

Abstract. The problem of the optimal control of the material characteristics of continuous media necessitates an extension of the initial class of materials to the set of composites assembled from elements belonging to the initial class. Such an extension guarantees the existence of an optimal control and is equivalent to the construction of the G-closure G U of the initial set U. In this paper, we consider some problems of constructing G-closures for the operators V. 29. V and V. V. 4 9 . . VV, where 29 and 49 denote self-adjoint tensors of 2nd and 4th rank, respectively, their components belonging to bounded sets of L~. These operators arise in the theory of the torsion of bars and in the theory of bending of thin plates. A procedure is suggested that provides estimates of some sets E containing GU. These estimates are expressed through weak limits of certain functions of the elements of the U-set) The estimates are based on the weak convergence of the elastic energy and, for operators of 4th order, also on the weak convergence of the second invariant of deformation, I2(e) =

w x x w r r - W Z x y ~ I 2 ( e O) - W xo ~ W yoy

o 2. --(Wxy)

For operators of 2nd order, we consider the problem of the control of the orientation of the principal axes of 29. The set E is constructed, and it is shown that its elements correspond, in a sense, to media with some well-determined microstructure (layered media). A number of problems for operators of 4th order are also considered. For these problems, the E-set is constructed for the control problem of the modulus of dilatation k, if the shear modulus/z is fixed and also for the control problem of the shear modulus/~ for fixed value of k. The problem of the orientation of the principal axes of elasticity is analyzed for a medium with cubic symmetry (semi-isotropic medium). The 1 The authors are indebted to Dr. U. E. Raitum for valuable discussions. 2 Senior Research Worker, A. F. Ioffe Physical-Technical Institute, Academy of Sciences of the USSR, Leningrad, USSR. 3 Research Worker, A. F. Ioffe Physical-Technical Institute, Academy of Sciences of the USSR, Leningrad, USSR. 247 0022-3239/84/0200-0247503.50/0 © 1984 Plenum Publishing Corporation

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description of the GU-set, with the aid of the weak limits of the U-elements, allows one to prove the existence of optimal solutions to a fairly wide class of problems with weakly continuous cost functionals. Some examples are considered in detail. Key Words. Sliding regimes, regularization, G-closure, composite materials, optimal design, existence of solutions, bars, plates.

1. Introduction As has been illustrated in Ref. 1, the problems of optimal design of elastic elements give birth to composite media that are known to remove contradictions arising otherwise within a system of necessary conditions of optimality. The absence of such contradictions does not, however, guarantee the existence of an optimal control. It may well turn out that necessary conditions of some different origin (not connected with the variation in a strip) would lead again to contradictions that invoke further complication of the internal structure of a composite. The rule of regularization (layered media), which is suggested in this paper, turns out to be exhaustive for a number of cases; for these cases, an optimal control actually exists within a corresponding class. The reasons for the absence of solutions to such problems, in their initial nonreguJarized form, are substantially the same as those inducing sliding regimes in optimal design problems for lumped-parameter systems. The main reason is that the initially accepted set of admissible controls is, in a sense, not closed. Regularization of such problems is therefore connected with some special closure of the admissible set of controls. Consider the set z(u) of solutions to the boundary-value problem defined by the operator L(u),

L(u)z=O; here, u denotes the control belonging to the admissible set U. Let

{z'}=[z(u~)} be some weakly convergent sequence of solutions. Its weak limit will be denoted by z °, Zi.-.-. Z °.

The element z ° may be associated with no element u of the initial set U. Following Ref. 2, we define G-closure of U as the set of controls ~ such

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that, for any z °, there exists some element

249

a e GU inducing

z o = z°(a).

The set

GU ={c,} is now G-closed in itself, and the problem of minimization of any weakly continuous functional I(z) possesses a solution within GU, the minimal value I(z °) being the same as the limit of the minimizing sequence {I(zi)}; that is,

I(z °) =lim I(zi), u~infU I(z(u))= umin I(z(u)). i~oo eGU For the above-mentioned problems, the absence of an optimal control is motivated by the fact that the commonly used initial classes of controls are not G-closed. The controls that give rise to minimizing sequences may characterize, for example, the elastic moduli of isotropic compounds. These moduli distributions become more and more inhomogeneous with each elementary volume, which leads finally to some limiting anisotropic material. The characteristics of this material are evidently dependent both on the characteristics of the compounds and the microstructure of the composite. Consider the problem

minI(z), IsV~?. uVzdX=Is ~?fdx, ze W~(S), rle lfV~(S),

(la)

f e CI(S),

(lb)

ue U: {ueL~(S), u_0}.

Here, the functional I(z) is lower semicontinuous; x = (xl . . . . . x,) denotes the set of independent variables; S is a region of R n. The G-closure of the U-set of isotropic controls,

U: {ueLo~(S), u_ ~0}, was described in Refs. 3, 4. The set GU is the set of matrices whose eigenvalues hi . . . . . )~n satisfy the inequalities hi_ < bl+U_/ ( U+"}- U - - - l~.n) < I~ 1 < ' ' "

~ t~, ~ U+.

(2)

If u denotes the characteristic of a continuous medium, then G U represents a set of anisotropic materials. In what follows (Section 2), some general techniques will be described which allow one to estimate the G-closures of different initial sets U of controls. In other words, estimates will be constructed for the invariants of the effective tensors ~°. The set of tensors satisfying those estimates is called E, GU C E.

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The approach mentioned above is applicable to a wide class of linear operators containing controls in their coefficients. We give a detailed description of the procedure for the operators 7.2@. V and V- V. aN.. VV of 2nd and 4th order, which are encountered in the problems of torsion of bars and bending of thin plates, respectively. Further, in Sections 3 and 4, the exactness of the estimates is investigated, that is, whether there exist composites corresponding to each point of the sets X. There are examples for which the correspondence between some composite and a point of X has not been established; that is, the coincidence between the E sets and the GU sets has not been proven. Nevertheless, the problem of the existence of an optimal control is solved for these examples, by virtue of certain limitations of the class of functionals to be minimized. The proof uses the observation that the strain and stress themselves do not uniquely determine all the components of the tensor N. In this sense, one is dealing with classes of equivalent materials; that is, the points of X can be represented by some smaller set of points, this set being realized by composites of well-defined structure. For a number of cases, a detailed description of the G-closed set GU will be given. For later calculations, we need a formula expressing the effective tensor ~° of elastic moduli of a layered composite made of two compounds, ~+ and ~_, whose concentrations are m and 1 - m, respectively (Refs. 1, 5). This relation is @° = D* -{m(1 - m)/nn.. ( ~ + + ~_ - @*).. nn]}A~. • nnnn.. AN. (3) Here, n denotes a unit vector normal to the layers; N* = r a N + + ( 1 - re)N_,

AN=~+-@_;

the symbol ( . . ) denotes a double convolution. An analogous formula holds for tensors of 2nd rank, @° = ~ * - { m ( 1 - m)/[n. (~+ + ~_ - ~*). n]}A~, nn. AN,

(4)

where the dot denotes a scalar product.

2. Estimates of the Set G U 2.1. Weak Convergence of the Strain Energy. Consider a thin plate whose material is characterized by the tensor @(x, y) of elastic moduli. This self-adjoint tensor (Refs. 1, 10) of 4th rank has positive eigenvalues. With this in mind, ~(x, y) may be chosen arbitrarily, but so that its components

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belong to some bounded sets of the Lo~(S)-space. The set of admissible values of ~ is denoted by U. To determine the G-closure of the initial U-set, we may introduce the sequence { ~ } of controls, bounded in L~(S)-space, and the corresponding sequences {e~}, {M ~} of deformations and moments, respectively, The latter may be treated (Ref. 6) as uniformaly bounded in some Lp(S), p > 2, and presumably some subsequences, weakly convergent in this space. Denote by e °, M ° the weak limits e'

Lv(S)

. e °,

Mi

Lp(S)

~ M °.

(5)

The functions M i, e i are connected by Hooke's taw,

Mi = ~ . . ei ; here and below, no summation is supposed over repeated upper indices, the elastic energy density being expressed as

N ~= M i.. e* =

e i.. ~ji.

e i.

We show that, for sufficiently smooth external loads, the following limiting relation holds: N ~=M~..e i

Lp/2(S)

~ M ° . . e o = N °"

Following the definition of a generalized solution, we have

VVwi,.@~,.VVrf dxdy=

f s q~Tidxdy,

° 2 Vrf ~ W2(S).

(6)

Assume that n~ _- ~ ( w ~- w°),

where ~(x, y) is sufficiently smooth and w ~, w° denote deflections corresponding to the deformations e ~, e °, respectively. With e = -VV w, we obtain

s~VVw~..

(VVw i - V V w °) dx dy

- - - I s ( w i - w°)VVwi'" ~ i .VV~dxdy -2

VVw~..N~..V~®V(w~-w °) dxdy+

~q(w~-w °) dxdy. (7)

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Let q e C2(S). Then (Ref. 6), for any strictly internal subregion S', S' C S, all wi(N ~) belong to some bounded set of Wp2 (S'), for some p > 2. The right-hand side of (7) obviously tends to vanish; and, in view of the arbitrariness of ~, M i'" ( e i - e °) ~

O,

tp/2(~") or N i -_ M i. . e i

~ L,/2(S')

M ° •. e o = N °.

The weak limits M °, e ° are obviously connected by some bounded linear operator ZP, since the operation of transition to a weak limit is linear. 4 It can be shown that the operator ~ is local; that is, M°(x)

= ~e°(x),

for any x E S.

Now, because of symmetry of the 2nd-rank tensors M °, e °, the operator is a self-adjoint tensor of the 4th rank (Refs. 1, 10), S f - ~ °,

M O = ~ O . . e °,

Using the weak convergence of the elastic energy,5 we estimate the tensor ~°. We define D* as the weak limit of the sequence {~i}, @i, . ~,. L~( S)

Note that the weak limit ~* and the effective tensor ~° of elastic moduli are essentially different, in general. The reader will observe in what follows that the tensor ~* serves as a tool providing necessary estimates of the elements of the tensor ~°. Then ~*-tensor is of advantage, since it is easily calculated for known compounds by simple averaging over the elements of the microstructure. For example, two materials ~+ and ~_ with concentrations rn and 1 - m produce D* = m~++ ( 1 - rn)~_. The sequence A i = 2(Mi..

ei

_Mi..

e o) = 2 ( e i - e O ) . . ~ i .

ei

is weakly convergent to zero, A i ..............

. O.

Lv:2(S) 4 One can see that S' may be replaced by S in the last limiting equation. 5 For scalar controls in the 2nd-order. problem, the estimates were obtained in Refs. 4, 7.

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Transforming A i, we obtain A i = (e i - e ° ) . . D i . . (e i -- e °) - eO.. 9 i . . e o +ei..~i..e

i

(8)

~0. Lp/2(S)

The first term on the left-hand side is nonnegative, since the same is true for the eigenvalues of ~i. Passing to the weak limit in (8), we get e o . . 9 o . . e o<~ e o . . ~ * . . e °, from which it follows that the eigenvalues of 9 * - 9 ° are nonnegative; that is, ,~j[@,_ ~o] >i 0, j = 1, 2, 3. (9) In quite the same way, we start with the limiting relationship B ' = 2(M'--

ei-M

." (9~) -1"- ( M ' - M

° . . e') = ( M i - M ° )

-M°"(gi)-l"'MO+MC'(~i)

-l"'Mi

°)

"0. Lp/2(id)

In view of the positive definiteness of the tensor ( ~ ) - 1 and the weak convergence of the expression M ~ . . ( 9 ~ ) - ~ . . M ~..M i = M ~ . . e i ........... , M °

eo=Mo..(~o)-I..M

o,

Lp/2(S)

we arrive at the inequality M o . . (9o)-1.. M o < M ° . . ( 9 - 1 ) * . . M °, where, by definition, (~)-I

. (9-1),. Lo~($ )

The eigenvalues of the tensor (9-1) * - (@o)-1 are obviously nonnegatire, that is, h i [ ( 9 - : ) , _ (~o)-,] I> 0,

j = 1, 2, 3.

(10)

The results obtained here can be applied, mutatis mutandis, to the 2nd-order operator V. z~. V; see Ref. 4. 2.2. Weak Convergence of the Second Invariant of Strain. we prove an important feature of the tensor

e ~= -VV wi; namely, its second invariant I2(e i) ~---WxxWyy i i -- ( W xiy) 2

Here,

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weakly converges in Lp/z(S), p > 2, to the second invariant

12(:) = wOXw°yy- ( w°Y) z of the tensor e ° = -VV w °. Observe that

212(e i) =e i . . ~ h . . e , i where ~i2 denotes the tensor of the 4th rank that is represented by 912 = a l a 1 - a 2 a 2 -

a3a3,

in the basis

at = (1/,/2)(ii +j]),

az = (1/,~)(ii-jj),

a3 = (I/,/2)(ij+]i).

It will be shown that e i " ~ I 2 "" ei

Lp/2(S)

- e ° " ~r2"' e°-

(11)

Consider the sequence e i " @I2 " " e i - e°" " ~ 2 " " e°,

(12)

and take some sufficiently smooth finite function = ~(x, y) to construct the expression

A = f s , e " ~x2" e dx dy = 2 f s ,[ WxxWyy- ( wxy)2] dx dy

= f ~[(a/ax)(wxwyy- wyWxy)-(a/ay)(WxWx,,- wywx~)]dx dy. d S

Double integration by parts provides the equivalent form

A = fs [(~rwx + ~x~wy)wr + (~xrWr - (rrwx) wx] dx dy.

( 13)

Multiplying (12) by ~ and transforming the result by double integration, we get

Is ~( ei" ~ z " e i - e° " ~2" " e°) dx dy = Is {2~xy( WixWy - W°xw~) i 2 - ~ x [ ( w yi ) 2 - ( w ,o) 2 l-~yr[(w~) -(w~)21} dxdy.

(14)

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The right-hand side of the last equation tends to zero, since w ~ Wpz (S). This proves (11), because ~: is arbitrary. Remark 2.1. In view of the well-known statical-geometrical analogy (Ref. 8) for the corresponding problem of tension of a plate, one may establish the weak convergence of the second invariant of the stress tensor. Remark 2.2.

The weak convergence of the bilinear forms

M . . e = M,,,,e,,n +M,e,+2M,,,e,,t,

lz(e) = l,,,,e,t- e 2,

to the corresponding combinations of the weak limits M °-- e °, I2(e °) holds because these forms satisfy the following conditions: (i) they are invariant; (ii) across any boundary line dividing media with different rigidities, the jump of each form depends linearly on the jumps of the components of the tensors M, e. The latter feature follows immediately from the continuity conditions [M,,.]+=[e.,]+=[e.]+=O, which hold across a boundary line with normal n and tangent t. In view of the linearity of the operation of passing to the weak limit (that is, averaging over elements of the microstructure) and due to linearity of the jumps M . . e , I:z(e), relative to the jumps [Mr,]_+, [M,c]_+, [e,,]_+, the sequences of these forms are weakly convergent. 2.3. Estimates of the Effective Tensors ~o in Terms of Weakly Converging Tensors. The preceding sections provide the background for some important inequalities, derived below. We have

2 ( e i " ~ J i " e i - - e i " ~ i " e °)

-0,

(15)

Lp/2(:S)

2(e i. • ~,2-- e i - e i ..~x2..e °)

~0.

(16)

Lp/2(S)

Multiplying (16) by some fixed constant a and subtracting the result from (15), we obtain 2[ei,.(~i-af~z2)..e~-e~',(~i-a~z~).,e

°)

.0.

Lp/2(S)

Denote

and transform the last relationship in the following way:

(el-e°)"~i"(ei--e°)--e°"~i"e°+ei"£)i"e

i

Lp/2($)

-0.

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If the parameter a is chosen so that the eigenvalues of ~i are nonnegative for any number i, that is, I> 0,

(17)

then eO..~O..eO~
The latter inequality shows that the eigenvalues of ~ * - ~° are nonnegative, that is, A j [ ~ * - ~°] = X j [ ~ * - 9 °] >I 0.

(18)

By analogy, we arrive at the following estimate: X j [ ( ~ - l ) , _ (~o)-1] 1>O,

(19)

where =

= l imk(

,_

Inequality (18) does not depend on the parameter o~ and coincides with (9). Inequality (19) is actually dependent on a and allows one to obtain stronger estimates than (10); note that, for (~ =0, the result is the same. The parameter a should be chosen so as to provide the strongest possible estimates. However, the requirement (17) should be kept in mind. The restrictions imposed on the elements of the U-set define some relationship between the weak limits ~* and ( ~ - i ) . The technique of deriving such a relationship is illustrated below by examples. The final estimates characterize the class of composite media that can be assembled from the given materials (compounds). For a number of cases, these estimates are shown to be exact. Remark 2.3. For problems of the 2nd order, connected with the operator V. 2~. V, there remains only the statement of Section 1.1, dealing with convergence of the strain energy. The corresponding estimates can be obtained from (17)-(19) by setting a - - 0 . The set of tensors @° satisfying Ineqs. (17)-(19) is denoted by E. Clearly, G U C E, that is, any corresponding pair of weak limits M °, e ° are connected by the relationship (Hooke's law) M o = ~o.. e °,

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where ~ ° c £ . To demonstrate existence, it suffices to show that, for any point of Y., we are able to construct a corresponding composite, using only the initial set U of compounds. This is the same as establishing the coincidence of the sets £ and GU.

3. Vector Problems of the 2nd Order: Control of the Orientation of the Principal Axes of the Tensor of Material Constants

3.1. Construction of the Set ~. In this section we consider the problem of constructing the G-closure of the U-set for the problem of the control of the orientation of the principal axes of the tensor of material constants. The discussion follows in part the arguments applied in Ref. 4 to the problem (1). Let S denote a planar region. The solution w to the problem satisfies the integral identity

[email protected]=fs ~Tqdxdy,

V~?e Wza(S),

(20)

where ~ is the tensor of 2nd rank represented by

= d÷ala2+d_a2a>

(21)

Here, d÷, d_ denote fixed eigenvalues. The eigenvectors al, a2 are expressed through the control function ¢, according to the formulas at = ix cos ¢ + iy sin ~p,

a2 = -ix sin ~p+ iy cos ¢,

(22)

where ix, iy denote a fixed pair of unit vector. The material state equation is expressed by

M=~.Vw=~.e, where

e=Vw; or, in the basis ix, iy,

Mx = dllex + d12ey,

My --dlEex + dzzey.

(23)

Note that, for the problem to be discussed, the invariants dl 1+ d22 = d÷ + d_, of the ~-tensor are fixed.

dn d22- (d12) 2 =

d÷d_

(24)

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We consider some bounded sequence { i} of controls and the corresponding sequences {el}, { M i } , the latter being weakly convergent; namely, ei ~ e Lp(S)

°,

M i

Lp(S)

. M °,

p>2.

(25)

In Section 2.1, we introduced the symmetric tensor 9 ° , satisfying the condition M ° = 9 °. e °.

(26)

The problem that now arises requires description of some set £ containing the whole class G U of tensors ~o defined by the set U of admissible controls 9. This description can be obtained immediately with the aid of the estimates of Section 2. However, we follow here an alternative procedure, leading to the same results. The general approach of Section 2 will be applied in Section 4 to the investigation of the more complicated problem of 4th order. To begin, we estimate the value of the energy density, N i = M i. e i

~ N ° = M o . e °, Lp/2( S)

from below. For this, we rewrite N i in the form N i_ i i 2 i i i 2 i 2 -(1/dll){(M~) ' [ - [ d l l d 2 2 - ( d 1 2 ) ](ey) } - ( 1 / a l il ) [ ( M ~ )i 2 +

(27)

d+d_(e~)2].

Here, we used Eqs. (24). The resulting expression for N ~is a convex function i d ni > 0. The sequence {d~l} is bounded in L p ( S ) and may thereof Mix, e r, fore be considered as weakly convergent to some limit d*l. Note that d1"1 # d~l, in general. Making use of the weak convergence in (25) and the definition (26) of 9 °, M x = dl~ex ° . + d12ey, . .

. M.y =. d12ex . +. d22e r,

we obtain [in view of the convexity of the right-hand side of (27)] the following estimate: N ° >>-( 1 / a * ~ ) [ ( M ~ )2+ d+a_(ey)2] = ( 1 / d , 1 ) [ ( d ~ l e ~ x +d12ey) o o 2 + d + d _ ( e yo) 2 ] -__ e°'A1 • e °,

where --[tIA*

"~r (d~1)2

Al--t~/tq1~Ld~ld~ 2

d~ld~2

t

(d~2)2+d+d_.J"

(28)

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By the same argument, starting from the representation N i= (1/d~2)[(M~)2+ d+d_(e~)21,

instead of (27), we arrive at the estimate NO>_.(1/d,2)[(d~2eO +d22ey)+d+d_(e:,) o o 2] = e o -A2"e °, o

(29)

where

=,.,..,[(a,2)o

2+

a+a_ d ,2a 2l

On the other hand, according to the definition N ° = e o . @o. e o, Ineqs. (28), (29) express the nonnegativeness of the eigenvalues of the matrices @ ° - A1, @° - A:. The corresponding conditions are easily obtained, since the matrices @°, At, A2 have the same principal axes. Denoting with )h, ~,2 the eigenvalues of the @°-matrix, we obtain

d+0<,_]

LO The nonnegativeness of the eigenvalues is expressed by ~1 <~d*l,

(30)

A21 <~d*l/d+d_.

(31)

By a similar argument, we get the conditions ;tz <- d*zz,

(32)

)iT ~ <<,d*2/ d+d_.

(33)

To obtain the required relationships, it remains to connect d*x and d*2 with one another. Note that d~l + d~2 = d+d_;

see (24); this relationship is presented also in the weak limit, d*l + d~2 -" d+d_.

The system (30)-(33) may be rewritten in the form d+d_/(d+ + d _ - d*l ) <~;tl <~d ' l , d+ d_/ d*l <~)t 2 <~d+ + d_ - d*l.

(34)

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The parameter d*~ of the system (34) varies within the limits d - ~< d1"1 ~ d+.

(35)

Each admissible value of dl*~ gives rise to some rectangular region of possible values of )q, )tz (Fig. 1). The points A for which ;tz=d++d_-d*l

A1 = d*l, define the curve

hi+h2= d++d-,

d - ~ A1,

12<~ d+.

(36)

The points C for which a~ = &d_/(d+ + d_-d*~),

a2 = d+d_/d**

define the curve (37)

1/x~ + 1/a2 = l / d + + 1/d_. Lastly, the points B: hi = d'n,

A2 = d+d_/d*~

and 9: A1 = d+d_/(d+ + d _ - d*l ),

A2 = d+ + d _ - d*l

belong to the same curve

(38)

A1A2 = d+d_.

~2

C

Fig. 1

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261

a

d.

Fig. 2 The planar domain E between the curves (36) and (37), see Fig. 2, contains the required GU-set, E = { ~ : ~ = Aaalal + Aza2a2, AI+A2~ d+ + d_, 1/hl + l / h 2 1/d++l/d_}. 3.2. Description ot the Set of Materials Equivalent to the GU-Set. In this section, we show that each point of Y~may be set into correspondence with some well-determined weak limiting process; that is, physically speaking, it may be set into correspondence with some quite definite microstructure of the medium. The proof will be performed in two stages. First, we show that the intermediate curve (38) is provided by the layered composite for which the principal axes a~, a2 of the compounds are parallel and perpendicular to the direction of the layers, respectively (Fig. 3). The layers of the first compound have their largest eigenvalue d+ corresponding to the al-direction, and the layers of the second compound, to the a2-direction (Fig. 3). In fact, substitution of the tensors (see Fig. 3) @~= d+ala~ + d-a2a2, ~z = d+aaa3 + d-a4a4

into Eq. (4) shows that the tensor ~° equals ~° = [rod+ + (1 - rn)d_]tt +{d+d_/[md+ + (1 - m)d_]}nn,

which is in accordance with (38).

(39)

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la,(d~) ]r~ l -3

0-q

Fig. 3 Definite representations of the microstructure corresponding to the curves (36) and (37) and to the space between them have not been given. This explicit representation would mean the same as the coincidence of the sets E and GU. For this reason, the second stage of the proof, which follows the technique used in Re/. 4, will set a medium of the type (38) into correspondence with any admissible pair of weak limits of M i, ek Actually, we observe that the vector M ° would remain unchanged if we replace ~ ° by ~ ° + ~±, where ~Z

" e ° ~ 0.

This arbitrariness allows one to set a vast class of media into correspondence with any admissible pair M °, e °. It is to be proven that, within this class, the material (38) may also be found. For any weak limits M °, e °, the following relationship holds: (M°) 2 - (hi + A 2 ) M °" e ° + A i A 2 ( e ° ) 2 = 0, (MO)2 = MO. M°,

(cO)2 __ e ° . e °.

(40)

This relationship follows from the state equation (23) by elimination of the angle prescribing the orientation of the principal axes of ~°. For given values of the parameters

2/2 = ( M ° ) 2 / ( e ° ) 2,

2/cos ~ = M °" e°/(e°) 2,

(41)

the curve (40) in the hi, X2-plane (Fig. 4) represents the class of media that are equivalent to one another, in the following sense: for any two points 1, 2 belonging to this curve, there exist tensors ~1, ~2 such that, by suitable orientation of their principal axes, one is free to satisfy the state equation (23) for given M °, e °.

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263

O.

,t.

j-d_~(00) Fig. 4

The parameters y, cos ~Osatisfy the following conditions: d_ ~< y cos 0 ~< d+ cos 2 i]/~ d+.

(42)

We shall prove that, under these conditions, the curve (40) and any curve of the set (36)-(38) have a common point, whose abscissa ;tl belongs to the interval [d_, d+]. The form of the curve (40) (see Fig. 4) shows that it suffices to prove the existence of an intersection between the curves (40), (38). Let /3 be the abscissa (Fig. 4) of this point. The condition of intersection is given by fl2,y COS ~b-- ( y 2 +

d+d_)/3 + d+d_y cos ~0= 0.

This quadratic equation has a nonnegative discriminant. We show that both real roots belong to the interval [d_, d+]. It suffices to prove (see Fig. 4) that, for d_, the curve (40) goes below the point a and that, for )tl = d+, it goes above the point b. Both requirements are expressed by the single inequality A1=

7 2 - (d++ d_)y cos qJ+d+d_<~O, which may be deduced on the basis of the relationship (d++ d_)M', e' = (d++ d_)[ d+( e'. a] )2..[_a_( e'. a~2)z]

= d2+(e'. a~ )2+ a2(e ,. a~)2+ d+d_[(e', a~ )2+ (e'. = (M') 2 + d+d_(e') 2,

a~) 2]

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where (Mi)Z=MCM

i,

( e i ) 2 = e i . e i.

This equality holds for any compound forming the microstructure. Passing to the weak limit and using weak convergence M i. e i

~

Lp/2(S)

M ° . eo

and convexity of the right-hand side, we obtain (d+ + d _ ) M °. e ° >I ( M ° ) 2 + d+d_(e°) 2.

It must be mentioned that the necessity of using considerations of equivalence decreases the number of problems subject to regularization. This procedure fails, in particular, when applied to a structure simultaneously working under a variety of conditions (e.g., in static equilibrium, free vibration or buckling, etc.). For all of these problems, one has to deal with a number of e°-vectors, which makes the equivalence procedure inapplicable. The same conclusion holds for problems containing restrictions on the controls. To sum up, we arrive at the following conclusions. (i) Whatever the sequence {~0t} of admissible controls and the corresponding sequences {M~}, {el}, the weak limits M °, e ° are connected by the relationship M o = 9 ° . e °. Here, the set G U of tensors 9 ° belongs to the E-set, G U C Y.; see (36), (37). (ii) Each element ~ c E may be set into correspondence with some equivalent class of materials. In this class, there exists a material described as a layered composite (39). These conclusions guarantee the existence of an optimal control within a vast class G U of arbitrary composites that are constructed of elements belonging to the initially given U-set. The optimal elements are represented by layered composites. (iii) Existence of an optimal control can be guaranteed for any weakly continuous functional of solutions to the initial optimization problem.

4. Tensor Problems of 4th Order: Bending of Thin Plates

4.1. Tensor ~o Partly Known. In the following sections, we regularize some problems of the theory of bending of thin plates. Estimates obtained

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above for the GU-set are realized. However, for the problems treated below, the tensor 9 ° is partly known, which simplifies considerably the analysis. Consider a plate whose material is characterized by the tensor 9 of the elastic moduli (Refs. 1, 10) d = d l a l a l + d2a2a2+ d3a3a3,

where dl, a~ denote its eigenvalues and eigentensors, respectively. Suppose that the elements of the admissible set U of materials have the same value dl and the same eigentensor a~. Then, the same will hold for the effective tensor; that is, 9 ° will allow the following representation: (43)

9 ° = d l a l a l + d~a~a~ + d~a°3a~.

Consider the weakly convergent sequence of deformations {e~}, e~--.e o, and the corresponding sequence of moments, M i = 9 i . . ei---.M ° =

9 °.. e °,

(44)

where M ~= d i e ' a 1 + d2e2a2 i ~ ~ + d3e3ae, i i i

eji _- e i " • aj.i

(45)

Taking the convolution of the last tensorial equation with the constant tensor a~, we obtain M i..al

=

i o dle~--.d~ex

= M

o,

.a~,

where e~ = e ° • • a l . Expanding the tensor 9 ° over the basis al, a2, a3, where a2, a3 are orthogonal to al, a 1.-a 2= a l''a 3=0,

we arrive at the relationship 6 M ° = d~a,a; . e ° ,

d~j= d~,

i , j = 1,2,3.

Taking the convolution with a~, and using (43), (45), we get dle~ = dlje~, 6 Here and below, summation is supposed over repeated lower indices.

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which shows (in view of the arbitrariness of e °) that d71 = dl,

d72 = d73 = 0;

Eq. (43) is thereby proved. In particular, if the set U = {~}, = k a l a l +/~ (a2az + a3a3), al = (1/,f'2)(ii + j j ) ,

a2 = ( 1 / x / 2 ) ( i i - / j ) ,

(46)

a3 = (1/x/-2)( i] + ]i),

of isotropic materials is characterized by the same dilatation modulus k and different values of the shear modulus/z, then the effective tensor ~° can be represented as ~ ° = k a l a l + l . ~ a ~ a ~ + ~z;a;a;,

(47)

where a~, a; denote certain deviators. The latter tensor corresponds to some semi-isotropic medium. If the set U = {~} includes semi-isotropic media with the same value of the dilatation modulus, then ~° allows the following general representation: ~0

o

o

o

o

o

o

--- k a l a l + l~2aza2 + l~3aaa3.

Note that the effective medium remains semi-isotropic. If the elements of the initial class of isotropic media differ only by the values k of their dilatation moduli, then the effective tensor ~° allows the representation ~5° = k ° a l a l + I~ ( a2az + aaa3),

(48)

which is typical for an isotropic medium (see Ref. 11). Here, we have the case in which the fixed eigenvalues of the initial compounds are double eigenvalues, which does not influence the main limiting result. In what follows, we give a number of applications of these conclusions and the estimates of Section 2 to some optimal control problems of bending of thin plates. 4.2. Control of the Dilatation Modulus ot an Isotropic Medium. Consider a plate made of two isotropic materials with the same fixed values of the shear moduli/~ and some different (and fixed) values k+ and k_ of the dilatation moduli. The tensors of elastic moduli of the compounds are given by ~4 = k+aaat + t~( a2az + aaa3), ~ - = k-a1 at + ~ ( a2a2 + a3aa),

(49)

where al denotes a spherical tensor and az, a3 are the orthogonal deviators.

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The set G U of ~°-tensors (see Section 4.1), G U = { ~ ° : 9 °= k°alal+~(a2az+a3a3), k_<~ k°<~ k+}, represents a class of isotropic media with some intermediate value of the dilatation modulus k °. The results of Section 2 provide some estimates of k °, expressed through the value k* of the weak limit of the sequence {U}, that is, through the average v~tlue of k over the elements of the microstructure. To obtain these estimates, we make use of Ineqs. (18), (19). First, we calculate the tensors ~o, ~ , , (~-1).. We have ~° = (k ° - o0alal + (tz + a)(aza2+ a3a3),

~* = ( k * - a )al al(I.* + a )( a2a2 + a3a3), ( ~ - 1 ) , = [ 1 / ( k - a)]*alal + [1/(/~ + oO](azaz+ a3a3), where

[1/(k-

~)]* =

The tensors ~ * - ~ °

~k[1/(k~-

~)].

and ( ~ - t ) , _ ( ~ 0 ) - 1 are equal to

~*-~°=(k*-k°)alal, ( ~ - 1 ) , _ (~o)-1 = {[1/(k - a ) ] * - 1/(k ° - ce)}alal, and the estimates (18), (19) take the form

k*>-k o,

[1/(k-o,)]*~ l/(k°-~)

or

{ [ 1 / ( k - a)],}-1 ~
(50)

According to (17), the parameter a belongs to the interval

-tx<~a<~k_.

(51)

It remains now to establish the relationship between the weak limits k* and [ 1 / ( k - a ) ] * . We observe that the terms k i of the weakly convergent sequence {k i} accept either the value k+ or k_. Therefore, ( k + - k i ) ( k ~ - k _ ) = [( k+ - ~ ) - ( k ' -

,,)][(k

~- a) - (k_ - ~)] = 0.

Dividing this equation by k ~ - a and passing to the weak limit, we obtain ( k * - a ) - (k+ + k _ - 2 a ) + (k+- ot)(k_- a)[1/(k - a)]* = 0.

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This relationship allows one to eliminate the value {[1/(k-o0]*} -1 from Ineq. (50). We arrive at the following estimate: (k+-a)(k_-a)/[(k++k_-2a)-(k*-a)]+a<~k°<~k*.

(52)

For each a satisfying (51), Ineqs. (52) outline a region on the ( k °, k *) -plane. Considering the a-family of such regions, one can verify easily that the strongest estimate is provided by a =--/~,

and the corresponding region is determined as the intersection of those for which o, ~ [ - ~ , k_].

For

we obtain (k+ + tz)(k- + tz)/(k+ + k- + lz - k * ) - i z <~k °<~ k*,

k_<~ k * ~< k+.

(53) The set E of admissible points (k °, k*) is shown in Fig. 5, where the curve A corresponds to the first of Eqs. (53) and the curve B corresponds to the second of Eqs. (53). The A-curve tends to the B-curve as the parameter /.~ increases to infinity.

k

~

k

Fig. 5

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It is easy to show that the A-curve corresponds to the layered medium assembled from the initially given materials. Substituting the tensors (49) into Eq. (3), and denoting by k * = mk+ + ( 1 - m ) k _ the weak limit of k, we obtain ~ ° = [ ( k+ + p. )( k_ + tz ) / ( k+ + k_ + p~ - k*) - i~ ] a l a l + lx ( a2a2 + a3a3).

(54) Note that the orientation of the layers is of no importance here, since the term in Eq. (3) containing n, [m(1 - m)/{nn.. [ m ~ _ + (1 - m) ~+]. • nn}]A@. • nnnn" A@ = [m(1-- m)(k+ + k_)2(al .. nn)2alaa]/{[mk_ + ( 1 - m)k+] x (ai'" nn)Z+tx[(a2 "" nn)2+(a3 "" nn)2]} = {Ira(1 - m)(k+ + k_)2]/[mk_ + (1 - m) k+ + g]}alal, is seen to be independent of this vector, which is obviously connected with the exceptional feature of the spherical tensor al. As to the points of the set £ [see (53)] not belonging to the A-curve, the problem of their representation remains open. v However, being able to approximate the A-curve, we are in a position to solve some important problems of optimal design. For example, consider the problem of maximizing the strain energy of a plate made of the materials described by Eqs. (49), under the additional condition that the average value of k*, the parameter Is k* dx dy, is given. The latter assumption is the same as prescribing the amount of the k+material,

f k* dx dy = meas{S[k_+ too(k+- k_)]}.

(55)

s

Now, the functions k ° and k*, subject to Ineqs. (53), are treated as controls. For k* prescribed, the value k°(k*) obviously accepts its maximal value allowed by (53), that is, the value belonging to the A-curve (Fig. 5). The optimal medium is approximated by layered 'composites according to (54). Note that the effective constants of this medium do not depend on the orientation of the layers. They are determined only by the concentration. 7If the set of materials with moduli ke[k_, k+] is at our disposal, then we are able to approximate all points of £ with the aid of layered components. To do so, it is sufficient to take layers with the moduli k_ and kl, where kl is some intermediate value of k. For this case, £ = GU.

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The necessary condition of optimality assumes the form 6k*[ e 2 (dk ° / d k *) - ~] I> 0, where x denotes the Lagrange multiplier associated with the restriction (55). Note that there are domains occupied by k+ and k_ materials, as well as domains in which [(k+ +/~) (k_ + ~)/(k÷ + k_ +/z - k*)]I 2 (e) = 2x. Also, note that el = e. . a , = ( 1 / 4 2 ) I 1 ( e).

Having the set of materials with k e [k_, k+] availailable, we are in a position to solve the problem of minimizing the strain energy under the restriction (55). It is easy to see that the materials for which k ° = k* (B-curve in Fig. 5) are now candidates for optimality. These media may well be considered as belonging to the initial class. The necessary condition of optimality of the intermediate control, e2 - ~ = 0 , expresses the constancy of the first invariant of deformation, I~ (e) = 2~.

4.3. Control of the Orientation of the Principal Axes of Elasticity of a Semi-lsotropic Medium. Consider the set U of semi-isotropic materials having the same prescribed values tz- and tz+ of the shear moduli and the value k of the dilatation modulus. The tensor @i is determined as [see (46), (47)] +/z-a2a2 +/z+a3a 3. The angle ~ of inclination of the principal axis of the deviator, a2, = ( 1 / 4 2 ) ( i ~ i ' - j ' f ) ,

to the x-direction is considered as control. According to Section 4.1, the effective tensor @° = k a ~ a l + tz~a~a~ + t z ~ a ~ a ;

characterizes some semi-isotropic medium. To evaluate ~°, we represent ~i in some fixed basis al, a2, a3 [Eq. (46)], ~i

i i i = kal al +/z22a2a2 q-/z 23 (aEa3 q- aaa2) + ~ 33aaa3.

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One can easily see that the expressions i i /*22 +/*33 i

(56)

=/*++/*--,

i

/* z2/.33 - (/*/~3)2=/*+/*-

(57)

represent the invariants of Ni and do not depend on i. The set E of tensors N° can be estimated with the aid of Ineqs. (17)-(19). We calculate the tensors ~*, ( ~ - 1 ) , in the basis al, a~, a;, ~ * = ( k - a )alal + (/*2*2+ a)a~a~ +/**23( a~a; + a~a~ )

ot)a3a3, o

°r" ( / * 3 3

at-

o

[1/(k - a)]al al + [(/z 33 + a) a2az -/*23 (a2a3 + a 3a2) + (/*~2

"1-

o o i i ~)aea3 ]/[/*22/*33

i 2 i +/*~33) + a 2] -(/*23) +a(/*z2

+[(/*~3 + a)a~a~ +/*~3 (a~a; + aga~)

=[1/(k-a)]alal

+ (/*~2 + a)a;a; ]/[/*+/*- + a (/*+ +/*-) + a2]. Passing to the weak limit, we obtain ( ~ - l ) * = [ 1 / ( k - a ) ] a l a l + [ ( / * * 3 + a ) a 2 a zo +(/*'2

..1_

o + / * 2 3* (a2a3

o o +a~a~)

o o a )a3a3 ]/[/*+/*- + a(tz+ + /*-) + a 2].

According to (17), the parameter a should belong to the interval [-/*, k]. As in Section 3.1, we arrive at the most exact estimates of £ ( a ) if we set a=k in (18), (19). The latter inequalities now yield ~j[(/.22

:g

o ~ --/d,2)a2a2

o

*

ai{[/*+/*_ + k(/z+ +/*_) o

o

o

° ° + (/*33 , -/*3)a3a3 o o o ] >I O, + /*23( a2a3 + a3az)

+ k2]-l[(/z3*3

o

+(/**z+k)a3a3]-(/*~ + k )

--1

(58)

° ° +/*23 , (a2a3 . . .+ .a3a2) + k)aga2 o

o

o

o

~_

a 2 a z - ( / * ~ +k)a3a3}~--O,

which imply that /*~ +/*; --<-/*~'2+/*~'3 =/*+ +/*_, (/*~ + k) -~ + (~; + k) -~ <_ (/*+ + k) -1 + (**++ k) -1, and (/*'2 -/*~) (/*'3 +/*; ) ~ 0**)2 >--O, [(/* ~'3 + k ) - ( / * + + k)(/*_ + k)/(/*~ + k)] x [(/**2 + k) - (/*+ + k)(/*_ + k)/(/*; + k)] i> 0.

(59)

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In view of Eqs. (56), (57), we obtain the following final estimates: (/z+ + k)(~_ + k)/[(/~+ + ~_ + 2k) - (/z*2 + k)] <~/~ + k ~
+ k) =

+

k).

The normal to the layers coincides with the principal axes of the deviators a2, a3 for the first and the second compounds, respectively.

4.4. Control of the Shear Modulus of an Isotropic Medium Construction o| the Set ~. Consider a plate made of isotropic materials having a fixed value k of the dilatation modulus and different values/z > 0 of the shear moduli, belonging to the interval

0< _< 11 11L s)

Zoo(S).

The effective tensor 9 ° of the elastic moduli is given by [see (47)] ~° = kalal + l~a~_a~ + l~a~a~, where a~, a~ represent a pair of orthogonal deviators. Following the procedure described earlier, we estimate the parameters of ~° in terms of those describing the weakly limiting tensor 9*. One calculates the tensors ~°, ~*, (~-1), as follows: ~o = ( k - ot)a,al + (tx~ + a)a~a~ + (tz~ + a)a~a~, ~* = ( k - a)alal + (l~ + a)*( a~a2 + a~a~ ), (~--1)* =

alal +

(a~a~ +a~a~),

where [1/(/z + a)]* = lim[1/(/x' + a)]. The parameter a belongs to the interval -tz_<~a<~k.

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From (18), we obtain the estimate h / [ ~ * - ~°] = Aj[(/** -/z~)a~a~ + (1** - Iz~ )a3a~3] >I O. Taking/x2O ~~/x3,o for definiteness, we obtain ~ <~t*~ ~
(60)

From Ineq. (19), we have Aj{{[1/(/z + oe)]*- 1/(t*~ - a)}a~2a~2 +{[1/(/* + a)]* - 1/(t*; + a)}a~3a; } >- O, and we deduce that 1/(/z; + a) ~< 1/(/z~ + oe) <~[1/(/, + oe)]*.

(61)

It remains to connect the weak limits ~* and [1/(/, + a)]*. We begin with the inequality [/*++ a - (/*~+ a)][/*_ + a - (/zi + a)] ~<0.

(62)

Upon division by/z~+ a, we get /z* + o~- (/z+ +/z_ + 2c~) + (/,+ + a)(/,_ + a ) [ l / ( / , + ~)]* <~0. Now, the weak limit [1/(/, + a)]* may be eliminated from (61). This result and Ineq. (60) yield 0 , + + ~)(t*- + ~)/(~*+ + ~*- + ~ - ~**)- ~ -< , 4 -< ~*; <- ~**.

These inequalities, combined with the obvious relationship /~_ ~
determine the £(~)-set of possible values of the parameters describing the ~°-tensor. The strongest estimate is provided by the value ~= k, o

z~

o

t * - <. (t*+ + k ) ( ~ , _ + k ) / ( ~ + + ~ _ + k - t* *) - k <- ~*2 ~ t* 3 <- ~ * <- t*+.

or finally tz_<.(tz++k)(tz_+k)/(tz++tx_+k-l~;)-k<~lz~<~l.~;~tz+.

(63)

The points of the A-curve (Fig. 6), p.~ + k = (tz+ + k ) ( t z - + k)/(l~+ + lz- + k - l z ~ ) ,

(64)

can be shown to represent a layered composite made of isotropic layers with the moduli (k,/~+) and (k,/z_).

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?,

Fig. 6 For such a composite, the effective moduli are calculated according to (3), 9 ° = k a l a l +((~+ + k)(/~_ + k)/[rn(i.~_ + k ) + (1 - m)(/x+ + k)] - k } a 2 a 2 + [m/~+ + (1 - m ) ~ - ] a a a 3 .

Elimination of m provides the relationship (64) between the eigenvalues /z~,/z~. Note that tz~ =/~* = l i m / i . weak

The mechanical interpretation of other points of ~ may be obtained with the aid of layers with the moduli (k,/z_) and (k,/.q), where /z_
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Here,

~i=(tzi+k)(a2a2+a3a3),

"_Mi=(t~i+k) devei,

A~r° = (/z°+ k) dev e°; therefore, ~i

e i = 2~i.. dev e' = (/z i + k)(dev el) 2,

(dev ei)2 = dev e ~- .dev e i. We use the inequality ~ + k t> [(/z ~+ k) z + (/~+ + k) (/.~_ + k)]/(~+ + tz- + 2k) = [(fii)2+ afl]/(a + ~),

(66)

following from (62) under the notation

fii=tzi+k ,

o~= / z _ + k,

/3 = ~ + + k .

From (66) and (65), one obtains

(a + fl ) ~ i . . dev e i i> ( ~ i ) 2 + afl(dev #)2 = []~li +,](afl)B.. dev ei] 2, (67) where the operator

B

= a2a3-

a3a2

is determined in the space a2, a3 of deviators. Its action is the same as the rotation of the principal axes of the deviator by the angle ~'/4. Using (65) and taking into account the convexity of the fight-hand side of (67), one arrives at the inequality (~ + / 3 ) ~ r°" .dev e°>~[l('l°+4(o43)B ".dev e°] 2.

(68)

On the other hand, Hooke's law, ~o=~O..eO=

~O..dev e°

= (l~ + k)e2a2+ (I~ +_k)e3e3 = ae2a2+ be3a3, where

a=tz~ +k,B=lz~ +k, provides the relationship

(~o)2_ (a + b)~l °.. dev e ° + ab(dev e°) 2 = 0.

(69)

Equation (69) represents an analog of Eq. (40) for the plate problem. For this problem, the conclusions concerning equivalence, made in Section 3.2,

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Eq. (40), still hold. By analogy with Section 3.2, we introduce the notation ,~2 =

(~o)2/(de v cO)2,

y cos $ = 3~r°'" dev e°/(dev e°) 2.

The inequalities a < 7 cos ¢ <~/3 cos z ~O
(70)

are easily checked by the same analogy. Now, Eq. (69) can be rewritten as b = 3,[(y-a

cos ~O)/(y cos $ - a ) ] ,

(71)

and the A-curve is represented by (72)

b = a +/3 - a / 3 / a .

The curves (71) and (72) are illustrated by Fig. 7. The left branch of (71) intersects the A-curve at two points. To prove this, it suffices to observe that the condition of intersection, a2(y cos ~ - a -/3) + a [ - y 2 + (a + f l ) y cos ~O+ a/3]- 2/3y cos ~ = 0, represents the quadratic equation whose discriminant [-3, 2`- ~/3 + (a +/3)3' cos ~]2-4a/3~2 sin 2 q~ is nonnegative, in view of Ineq. (68). For zero value of the discriminant, the curve (71) contacts the A-curve.

tq

/

f

,~÷ Fig. 7

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277

On the other hand, along the curve (71), the following relationship holds: (b'+l)ycos~+b'a-b=O, b' =db/da, (73) and can be obtained from (71) by differentiation. If the parameters y 2 and 3' cos 0 are connected by the condition [,y2+ 0/18 + (Or +18)~ COS ~]2 + 4a18(')'2-- ,y2 COS2 I#) = 0,

which is just (68) taken as an equality, then these parameters may be eliminated from (71), (73). We arrive at the relationship

(b')ZF'(a)-2b'[F(a)F(b)-2oq3(a-b)2]+ F2(b) =0, where

F(x) = ( x - a ) ( x - t8). The function (72) is readily seen to satisfy this condition. The above proof appears to be analogous to that given in Section 3.2 for the problem of the orientation of the principal axes. At the same time, there is considerable difference between the two approaches. In fact, now the curve of equivalence (68) may contact the A-curve, which is the boundary of the X-set. On the other hand, for the problem considered in Section 3.2, the curve of equivalence always intersects the boundary of the X-set, as well as curves similar to those of (36)-(38); see Fig. 2. For the former case, the A-curve, corresponding to a layered composite, is a limiting curve due to the condition of contact. Conversely, for the case of Section 2.2, the set X includes a vast class of materials represented by the admissible curves (36)-(38) (see Fig. 2), which all intersect the curve of equivalence. These materials have the same physical meanings. They also do not possess any limiting feature. The case in which the curve of equivalence has only one common point with the X-set, for the problem of Section 3.2, means that this point is either a or b in Fig. 2, which are the corner points of GU, corresponding to the initial anisotropic material.

Explicit Description of the Structure of Materials Belonging to the GU-Set. The demonstration of existence of optimal controls for the problem discussed here may also be given without any reference to equivalence considerations. We will show that, to each point of X [see (63)], one can associate a material with a definite microstructure, or alternatively that the equality

Z=GU holds.

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For this purpose, we consider a layered composite assembled from the initially given isotropic compounds, so that its tensor ~ belongs to the boundary of the GU-set; that is [see (46)-(47)], (74)

@ = k a l al + 1~2a2az + l~3a3a3,

where ~2 + k = (~+ + k)(~_ + k ) l ( ~ + + ~_ +

k - ~3);

see Fig. 8, curve A. Now, consider the layered structure of 2nd rank made of semi-isotropic materials, taken as compounds that differ from one another only by the orientation of the principal axes of elasticity. In particular, take the composite constructed by the rule of Section 3.2 (Fig. 9). For this, the eigenvalues /z~,/~ satisfy the relationship (see Section 3.2) ( ~ + k)0z; + k) = (~2 + k)(~3 + k),

illustrated by the N-curve in Fig. 8. Here,/z2,/% represent the eigenvalues of the composite of first rank. To each point n of the A-curve (boundary of the GU-set), there corresponds the N-curve that belongs to the GU-set. The union of N-curves, initiated by all the points n of the A-curve, obviously coincides with the GU-set. Remark 4.1. A similar procedure may be used to maintain the coincidence of the ~ and G U sets for the problem (1) considered in Refs. 3, 4.

j ~ .....

i\//

/~-

/~, Fig, 8

/<,

/<+ /V

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x

X

>~ X

279

~

X

X

X

.~

~

.,><,, \

,e<..>~

Fig. 9

Remark 4.2. In Ref. 1, the solution was given of the regularized problem of control of the shear modulus /z. The results obtained here suggest the conjecture that the optimal control really exists. We are now free to solve various optimization problems, including those with isoperimetric restrictions concerning the amount of rigid (soft) material. The corresponding necessary conditions resemble those obtained in Ref. 1 and have the form

[(/z~)2/(tz+ + k)(/z_ + k)]e~ + e3z = t = const, (tz~ -/x~)e~e~ =0. Here, i denotes the Lagrange multiplier associated with the isoperimetric condition. It should be added that the constructive description of the GU-set given in Section 4.4 allows one to solve a vast class of optimal design problems, for which it guarantees the existence of an optimal control.

5. Conclusions

The proof of existence of optimal controls, for all problems considered in the paper, includes two stages. First, we prove in Section 2 the inclusion GUCE, where E is a set prescribed by the estimates of Section 2. For any sequence {9i}, 9 i ~ U, the weak limits M °, e ° of the corresponding sequences {M~}, {e i} are connected by the tensor 9 °, 9 ° ~ E. The set GU of tensors 9 ° is therefore a subset of ~. In Sections 3 and 4, the possibility of inverse inclusion GUDE is investigated. For the problems considered in Section 4.4 and partly in

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Section 4.2, any tensor ~ ° ~ E invokes a special sequence ~ i c U (layered medium), such that the weak limits M °, e ° of the corresponding sequences {M~}, {e i} are connected by the tensor ~°. For the problem discussed in Sections 3 and 4.3, the inclusion G U D is guaranteed by considerations of equivalence (see Ref. 4). Whenever both inclusions exist, they provide the coincidence of the two sets, namely, G U =~.

It now follows that any weak limit w ° of solutions w(~ ~) to the initially posed problem, ~ ~ U, is evoked by some control ~ ° ~ G U = E,

where the set E allows a simple representation. To guarantee the existence of solutions to the problem of minimization of an arbitrary weakly continuous functional l(w), it is sufficient to establish the existence of the element w ° itself. This element is clearly induced by the control 9 ° belonging to the set E. Any point of E is approximated (either explicitly or by equivalence) by the sequences {~}, ~ U. To solve the corresponding optimization problems, one should reformulate them by extension of the admissible set U of controls, up to the set E. Except for important qualitative information concerning the properties of the optimal media, this extension is supposed to provide effective computational schemes. At the same time, the procedure of Section 2 may be useful for the estimation of the effective constants of arbitrary composites.

References 1. LURIE, K. A., CHERKAEV, A. V., and FEDOROV, A. V., Regularization of Optimal Design Problems for Bars and Plates, Parts 1 and 2, Journal of

Optimization Theory and Applications, Vol. 37, No. 4, 1982. 2. MARINO, A., and SPAGNOLO, S., Un Tipo di Approssimazione dell'Operatore ~'~ ~s O/Oxi(aij(x)O/Oxj) con Operatori Y.i o/oxi(a(x)O/axi), Annali della Scuola

Normale Superiore di Pisa, Vol. 23, pp. 657-673, 1969. 3. TARTAR, L , Problemes de Controle des Coefficients dans des Equations aux DerivdesPartielles, Springer-Verlag, New York, New York, pp. 420-426, 1975. 4. RAITUM, U. E., The Extension of Extremal Problems Connected with a Linear Elliptic Equation (in Russian), Doklady Akademii Nauk SSSR, Vol. 243, No.

2, 1978.

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5. MCCONNELL, W. H., On the Approximation of Elliptic Operators with Discontinuous Coefficients, Annali della Scuola Normale Superiore di Pisa, Serie N, Vol. 3, pp. 121-137, 1976. 6. LADYZHENSKAYA, O. A., and URALTSEVA, N. N., Linear and Quasilinear Equations of Elliptic Type (in Russian), Nauka, Moscow, USSR, 1973. 7. CEA, J., and MALANOWSKI, K., An Example of Min-Max Problem in Partial Differential Equations, SIAM Journal of Control, Vol. 8, 1970. 8 LURIE, A. I., Theory of Elasticity (in Russian), Nauka, Moscow, USSR, 1970. 9. LURIE, K. A., Optimal Control in Problems of Mathematical Physics (in Russian), Nauka, Moscow, USSR, 1975. 10. LURIE, K. A., Some Problems of Optimal Bending and Tension for Elastic Plates (in Russian), Izvestija Akademii Nauk SSSR, Mekhanika Tverdogo Tela, No. 6, 1979. 11. HILL, R., Elastic Properties of Reinforced Solids: Some Theoretical Principles, Journal of Mechanics and Physics of Solids, Vol. 11, No. 5, 1963.