On G-Closure ~ corresponding case in Ref. 2. Based on these results, a more precise description of corresponding G-closures is presented.

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1 JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS: Vol. 53, No. 2, MAY 1987 ERRATA CORRIGE On G-Closure ~ K. A. LURIE 2 AND A. V. CHERKAEV 3 Abstract. We give a correct analysis of case (25) in Ref. 1 and of the corresponding case in Ref. 2. Based on these results, a more precise description of corresponding G-closures is presented. Key Words. G-closure, composite materials. 1. G-Closure of a Set of Two Anisotropically Conducting Materials In Ref. 1, we give a description of the G-closure of a set formed by two anisotropically conducting materials whose tensors of conductivity are given by 4 ~+=ot+a+a++/3+b+b+, a+<~/3+, (19) ~_= a_a_a_+/3_b_b_, a_<~/3_. (20) Assuming that ~ /3+ ~< ~_/3_, we must distinguish between two possible cases: a+ <~ a_, /3+ ~</3_ ; (24) a ~< a_, /3+/>/3_. (25) The corresponding G-closures are in both cases restricted by hyperbolic t The authors are indebted to F. Murat, Universit6 Pierre et Marie Curie, Paris, France for pointing out an error in the analysis of one of the specific cases encountered in Refs. 1 and 2. 2 Senior Research Fellow, A. F. Ioffe Physical-Technical Institute, Academy of Sciences of the USSR, Leningrad, USSR. 3 Research Fellow, A. F. Ioffe Physical-Technical Institute, Academy of Sciences of the USSR, Leningrad, USSR; Visiting Scholar, Department of Solid Mechanics, Technical University of Denmark, Lyngby, Denmark. 4 Here and below, unless otherwise stated, we use the notation and numeration of formulas and figures from Ref /87/ ,00/ Plenum Publishing Corporation

2 320 JOTA: VOL. 53, NO. 2, MAY 1987 segments (cf. Figs. 3 and 5) A~A2= 5+fl+, 5+<~Xl, A2~<fl+, (21) A1A2= a_fl_, c~_<~x~, A2~<fl_. (22) These segments are attained for polycrystals 5 constructed from materials (19), (20) taken separately; for each case, they form a part of the boundary of the GU-set. To obtain the other parts, the materials (19) and (20) should be intermingled to form a binary composite. The parts in question are given by different analytic expressions in cases (24) and (25). Consider the case (24), and assume that,)t 1 <~ A2. As for the case of two isotropic compounds (Section 1, Ref. 1), the corresponding part of the boundary may be determined from Ineqs. (26), (27): A2 <~ mfl+ + ( 1 - m)fl_, (26) All~ ma~.l + (1 -m)a -1. (27) Bearing (24) in mind, we eliminate the parameter m from these inequalities. The resulting inequalities, ~+5+(/3_-/3 ) 5+ ~ 5_fl_ -- 5+fl+ -- (5_ -- 5+)A2 ~ A1 ~ A2 ~ '#-' (El) generalize those of (4) and reduce to them for 5+ =/3+ = a, 5_ =/3_ =/3. The equality 5+5_(/3_-~+) A1 = (30) (5_ - 5+),~ 2 is satisfied for a layered composite of the first rank, with the normal to the layers oriented parallel to the principal axes a, a_ of the tensors ~+, ~_ corresponding to their smallest eigenvalues 5+, 5_. The GU-set is now s A simple proof of this statement is based on the relationships (Ref. 3) w~t(1)w~2(2) _ w~,2(1)w~(2 ) _a_ det(vws(1), VwS(2)) ~ det(v wo(1), VwO(2)), det(@ s. VwS(1), 9 s. VwS(2)) *-~ det(9 o. Vw (1), 90. Vw (2)), in which VwS(1),VwS(2) denote potential gradients associated with linearly independent loads fl, f2, respectively, and the symbol *--" denotes weak * convergence in L I(s) to the weak limits Vw (1), Vw (2). On the other hand, for 2 x 2 matrices, the following identity holds: det(9 Vw(1), 9-Vw(2)) = det 9. det(vw(1), Vw(2)). But, by the statement of the problem, det 9 s = a/3 = const(s), and the preceding limiting equations now show that det 9 = AIA 2 = det 9 s = aft.

JOTA: VOL. 53, NO. 2, MAY 1987 321 formed (Fig. 3) as a common part of the sets defined by Ineqs. (El) and (21), (22). Consider now the case (25). For this one, elimination of m from Ineqs.

3 JOTA: VOL. 53, NO. 2, MAY formed (Fig. 3) as a common part of the sets defined by Ineqs. (El) and (21), (22). Consider now the case (25). For this one, elimination of m from Ineqs. (26), (27) is no longer possible, and these inequalities alone fail to determine the required part of the boundary. In Ref. 1, this part was identified erroneously with (30), too. At the same time, as was pointed out by F. Murat, the curve /3+/3_(~_- ~+) A2 - a_/3_- a+/3+- (/3_-/3+)h1' (E2) drawn on a (hi, h2)-plane, is placed higher than that of (30); e.g., it provides greater values of h2 for the same values of hi. This curve corresponds also to a layered composite of the first rank, but this time the normal to the layers is oriented along the principal axes b+, b_ of the tensors 9+, 9_ associated with their largest eigenvalues/3+, ]3-. It can be shown that the curve (E2) actually represents the required part of the boundary. The proof is associated with the solution of the fairly complicated problem of building the G-closure of two anisotropically conducting materials taken in a precribed proportion; it will be given elsewhere. Here, we restrict ourselves to the statement that, in case (25), the GU-set is formed (Fig. 5) as a common part of the sets defined by h2<~ /3+/3_(~,_- ~+) ~-/3-- ~+/3+- (/3--/3+)hl (E3) and Ineqs. (21), (22). These results can be illustrated graphically in the plane generated either by the parameters (,~-1, h2) or by the parameters (hi, h;1), provided that we give up the earlier assumption hi ~< h:. Any anisotropic material whose 9-tensor possesses the eigenvalues dl, da is represented on each of these planes by two points passing each other under the interchange of symbols dl and d2: in the coordinates (h~ -1, ha), these points will be (dl 1, d2) and (dk ~, dl); and, in the coordinates (hi, h21), these points will be (dl, d~ 1) and (d2, d~-~). It is apparent that, if the point (a -1,/3) belongs to the boundary of the GU-set in the (hi 1, ha)-plane, then the same also holds for the point (/3-1, a). In the coordinates (X~ -1, h2), the material 9+ is now represented by the points N1 and N~ [Fig. El, case (24), and Fig. E2, case (25)]; the hyperbola (21) is represented by the straight line segment NIN~ passing through the origin; the material 9_ is represented by the points N2 and N~; and the hyperbola (22) is represented by the straight line segment NeN~. Connect now the points N1 and N2, as well as the points N~ and N~ by the straight line segments N~N2 and N~N~. Equation (30) shows that

4 322 JOTA: VOL. 53, NO. 2, MAY 1987 N, Fig. 1 % the first of these segments represents a layered first-rank composite with normal to the layers oriented along the principal axes a+, a_ of the compounds. The segment N~ N~ also represents a layered composite of the first rank, this time, however, with normal to the layers oriented along the principal axes b, b_. The straight line segment N1N2 represents a part of the boundary of the GU-set in case (24); and the segment N~N~ represents a part of the boundary in case (25). For each of these two cases, three sides of a quadrangle NIN2N~N~ represent a part of the GU-boundary; and the fourth side [N~N~ in case (24), N1N2 in case (25)] belongs to the inner part of GU. The reason is that the straight line segments mentioned above are not transformed into one another as the coordinates ;tl and A2 interchange. In order to complement the quadrangle to the complete GU-set, one should build the images of the straight line segments N~N2 and N~N'2 resulting from that interchange operation. The segment NIN2 is mapped onto a eurvilinear segment N~N'~, and the segment N~N~ is mapped onto a curvilinear segment N1N2 (Figs. El, E2). The set GU for both cases is

5 JOTA: VOL. 53, NO. 2, MAY Fig. 2 represented as a sum of quadrangles N~N2N~N~ and N~N2N~NI. This construction is easily reproduced also in the coordinates (A1, A ~l); for these, the segments N~N2 and N~N~ become curvilinear, and vice versa (see Figs. E3, E4). 2. G-Closure of an Arbitrary Set of Anisotropically Conducting Materials We first prove that the GU-set may be determined for this general case as a sum of the GU-sets borne by any pair of elements belonging to the initial set U. It suffices to consider the case when the set U consists of three compounds represented by the points N~, N2, N3 [Fig. E5 illustrates the case when each of the three pairs (N1, N2), (N2, N3), (N3, N1) satisfy Ir/eqs. (24)]. The mentioned sum, the curvilinear polygon NIN2N3QaQ2Q1, obviously belongs to GU. We will show that an inverse inclusion also holds. The polygon N1N2NaQ3Q2Q1 is a part of some larger polygon N1N2N4QaQ2QI, whose vertex N4 is determined as a point of intersection of the prolonged segment N~N2 and the ray Q3N3. This new polygon may

6 m 324 JOTA: VOL. 53, NO. 2, MAY 1987 Fig. 3 be thought of as the set GO, borne by two materials represented by the points N1 and N4. But it is apparent that GUC GU, since the material corresponding to the point N2 can be obtained as a layered first-rank composite assembled of N1 material, and N4 material, and the material N3 may be obtained as a polycrystal borne by an anisotropic compound N4. This all implies that a curvilinear open polygon QaQ2QININ2 forms a part of the GU-boundary; at the same time, it represents a part of the boundary of the sum of the GU-sets, borne by the pairs (N~, N2), (N2, N3), (N3, Nt) of compounds. The same conclusion follows in a similar way also for an open curvilinear polygon Q~Q2Q3N3N2. This completes the proof. The G-closure of an arbitrary initial set of anisotropic compounds is now constructed as the least set containing U and convex in both the coordinates (h~-1, h2) and the coordinates (a~, hj1). All the points of GU may be realized by some layered media assembled only of those representatives of the U-set which belong to the GU-boundary. Since the GU-set

7 JOTA" VOL. 53, NO. 2, MAY ~ 4 Fig. 4 includes media of arbitrary microstructure, it means, in particular, that any such material is equivalent (in the sense of its effective properties) to some layered composite. The boundary points of GU are modelled by layered composites of the first rank, and the inner points are modelled by composites of the second rank. 3. G-Closure of a Set of Materials with Cubic Symmetry Having the Same Dilatation Moduli in the Problem of Bending of Thin Elastic Plates Consider the problem of the G-closure for a set U including two materials (~-tensors) are given by the relationships 6 ~+ = kalat + tz2+a~+a~+ +/z3+a~+a~+, (35a) ~ = (/z2+ + k)a2 a~+ + (/z3+ + k)a~+a~+ A -- S $ -- ~ s S =/a,2+a2+a2+ -t-/z3+a3+a3+, (35b) 6 From this point on, we use the notation of Ref. 2.

8 326 JOTA: VOL. 53, NO. 2, MAY 1987 / Q,\/ //'z/ -o Fig. 5 ~- = kalal + 1~2_a~_a~_ + ~3_a~_a~_, (36a) ~- = (~2-4- k )a~_a~_ 4- (~3-4- k )a~_a~_ =/~2-a2-a2- + ~3-a3-a3-. We may always assume that g2+ <~ g3+, g2- <~ 53- (36b) and that $ s The notation a2+,..., a3- reflects the assumption that the orientation of the principal axes of the eigentensors a~+,..., a~_ of both materials may be different within different pieces of those materials which constitute the assembled composite (a polycrystal).

9 JOTA: VOL. 53, NO. 2, MAY This problem is quite similar to that discussed in Sections 1 and 2. In a similar fashion, we must distinguish between two cases: ~2+ ~ /'~2--, ~'~3+ ~</~3--, (E4) the last two inequalities being analogs of Ineqs. (24) and (25) of Ref. 1, respectively. The GU-set is now built as the intersection of the set 2o 3o bounded in the (/~20,/23o)-plane by two hyperbolas, and of the sets /.~2+ ~ ~ ]~20 ~ ~30 ~" ~3 _, (E6) ) 3o if (E4) holds, and ~2+ ~ ~20 ~ ~30 ~ ~--~/~3+, (E7) if (E5) holds. The last two inequalities are analogous to Ineqs. (El) and (E3), respectively. The equality sign in the second of Ineqs. (E6) corresponds to a laminated composite assembled of the materials (35a) and (36a), oriented within the layers in such a way that the main axes of the tensors a~+, a~_ would coincide with the normal and the tangent to the layers, respectively. The equality sign in the next to the last of Ineqs. (E7) corresponds to a laminated composite, with the normal and the tangent to the layers oriented along the principal axes of tensors a~+, a~_ of the compounds. The results obtained in Sections 1 and 2 can also be extended to the present case: in particular, it is apparent that any composite belonging to the GU-set formed by two initially given materials (35a) and (36a) is equivalent to some laminated composite of the first rank or the second rank. (E5) References 1. LURIE, K. A., and CHERKAEV, A. V., G-Closure of a Set of Anisotropically Conducting Media in the Two-Dimensional Case, Journal of Optimization Theory and Applications, Vol. 42, pp , LURIE, K. A., and CHERKAEV, A. V., G-Closure of Some Particular Sets of Admissible Material Characteristics for the Problem of Bending of Thin Elastic Plates, Journal of Optimization Theory and Applications, Vol. 42, pp , RESHETNYAK, YU. G., Stability Theorems for Mappings with Bounded Excursions, Siberian Mathematical Journal, Vol. 9, pp , 1968 (in Russian).

(x 1, y 1 ) = (x 2, y 2 ) if and only if x 1 = x 2 and y 1 = y 2.

(x 1, y 1 ) = (x 2, y 2 ) if and only if x 1 = x 2 and y 1 = y 2. 1. Complex numbers A complex number z is defined as an ordered pair z = (x, y), where x and y are a pair of real numbers. In usual notation, we write z = x + iy, where i is a symbol. The operations of