StrucOpt manuscript No. èwill be inserted by the editorè Detecting stress æelds in an optimal structure I Two-dimensional case and analyzer A. Cherkae

Transcription

1 StrucOpt manuscript No. èwill be inserted by the editorè Detecting stress æelds in an optimal structure I Two-dimensional case and analyzer A. Cherkaev and _I. Kíucíuk Abstract In this paper, weinvestigate the stress æeld in optimal elastic structures by the necessary conditions of optimality, studying two-phase elastic composites in two dimensions. Necessary conditions show that an optimal design is characterized by three zones: Zone of pure weak Material ; Zone of pure strong Material ; Zone 3 Material and Material are mixed to form an optimal microstructure. To characterize these zones we introduce two rotationally invariant norms N and N of a stress tensor. The derived optimality conditions state that inequality N èè é constant holds in Zone ; inequality N èè é constant holds in Zone ; and two equalities hold simultaneously in Zone 3: N èè = constant ; in the Material ; N èè = constant in the Material. Using this result, we analyze suboptimal projects and ænd how close æelds there are to the regions of optimality. Introduction Structural optimization asks for the ëbest" layout of two elastic materials in a design domain. Problems of this kind are examples of nonconvex multivariable variational problems; they require minimization of an integral if a non-convex Lagrangian that depends on æelds, such as the stress æelds in elasticity. More speciæcally, the Lagrangian of the considered problem is a two-well function: A minimum of two convex functions èwellsè that correspond to the energies of each phase. The nonconvexity of the Lagrangian yields to nonexistence of a classical èvariationalè solution, to the necessity to build special highly oscillating minimizing sequences and develop special methods for relaxation of Received: August, 00 A. Cherkaev and _I. Kíucíuk Department of Mathematics, University of Utah, Salt Lake City, 0, USA cherk@math.utah.edu and kucuk@math.utah.edu the Lagrangian. The most popular method to handle the unstable problem is similar to the Ritz method: A class of special oscillating minimizing sequences called laminated of a rank, that is laminates made of laminates, is considered. The averaged properties of these structures can be analytically calculated. They depend on several geometric parameters which are adjusted to improve the solution. Another method ètranslation methodè is based on suæcient conditions of optimality: a lower bound of the nonconvex Lagrangian is build that leads to a smooth solution of the relaxed variational problem. This new solution is in fact the average stress æeld in an optimal structure. To build the bound, the diæerential properties of the solution èlike the divergence-free character of stressesè are replaced by a weaker requirements. Both methods were applied to several problems of structural optimization, including to the problem under study. They provide the complete picture of the solution when the minimum achieved on minimizing sequences matches the lower bound. However, both methods have serious limitations: The ëlaminates" method critically depends on the correctness of a priori guess about the class of proper structures, and the translation method can lead to violation of the diæerential properties of solution and to a slack bound. Therefore, a more universal method is needed to investigate the structural optimization problems that is free of strong a priori guesses. Looking for such method, we turn to a classical variational technique of strong local variations, developed in Lurie è975è and applied to the structural optimization in early papers: Lurie è975è and Lurie and Cherkaev è97è. Last years, this technique was further developed in the book Cherkaev è000è conducting designs were considered. The technique we used in this paper can be generalized to sum of the energies for multiple loading case Cherkaev et al. è99bè. In this paper, we develop the method of necessary conditions and apply it to elastic structures. The stress æeld in optimal elastic structures is investigated by means of special conditions of optimality. We consider the well investigated problem of the optimal layout of two-phase elastic composites in two dimensions. This allows us to compare the suggested method with the known ones. The discussed problem has already been studied from

2 various perspectives èsee recent review, Eschenauer and Olhoæ è00èè: The lower bound of the energy among all structures was calculated by means of the translation method, see Gibiansky and Cherkaev è997è; Allaire and Kohn è99è; Grabovsky è996aè. It was shown that the optimal structures are either second-rank laminates Gibiansky and Cherkaev è997è; Milton è9è or, in some special cases, Vigdergauz structures Vigdergauz è99è; Grabovsky and Kohn è995è or Hashin's coated spheres Hashin and Shtrikman è96è. It is possible that other microstructures exist that are optimal as well, which brings up the question: What do these structures have in common? Our analysis of optimal æelds answers this question. The necessary conditions approach supplement the mentioned results by analysis of the æelds in an optimal structure without a priori speciæcation of admissible geometric conægurations. We are not checking the properties of guessed geometries, but describing the æelds in all optimal structures. In this paper, we derive the optimality conditions and characterize the stress æelds in both materials mixed in the optimal design. We also analyze suboptimal projects to judge how close they are to the optimum. Finally, we compare the necessary conditions of optimality with suæcient conditions. The discussed technique can also be applied to more general problems: The minimization the energy of multi-material mixtures èsee Cherkaev è000èè or the minimization of other functionals. Formulation of the problem Consider a domain æ that is divided into two subdomains æ i ; i =;. The subdomains æ i are occupied by isotropic materials with bulk and shear moduli of i and i, respectively. Suppose that the volumes of æ and æ are æxed and a load is applied from the boundary. A linearly elastic structure at equilibrium and in the absence of body forces satisæes the following elasticity equations èsee, for example, Timoshenko è970è, Fung è965è, and Atkin and Fox è90èè ræèxè =0; æ = èru +èruèt è; æèxè =Sèxèèxè; èè u is a deæection vector, èxè, and æèxè are stress and strain æelds respectively and Sèxè is the compliance tensor that characterizes the material properties. The density of the energy is given by Gè; è = èxè :Sèxè :èxè; èè the second-rank stress tensors èxè belong to the set F s èxè of statically admissible stress æelds èsee èè.è F s èxè =fèxè jræèxè =0in æ; èxènèxè =tèxè on, T g: è3è Here, tèxè is surface traction, nèxè is the normal to the surface, and, T is the boundary component the traction is applied. In èè, Sèxè is a fourth-rank material compliance tensor of the heterogeneous material. The set of compliance tensors, S ad, consists of the tensors: Sèxè =èxès è ; è+è, èxèès è ; è; èè i and i are shear and bulk moduli of each material, and èxè isthecharacteristic function: 0 if x æ ; èxè = è5è if x æ : Using the principle of minimum total stress energy, we can reformulate the elasticity equation as the minimizer for the total energy Eèè Eèè = min èxèf s èxè Z Gè; èdæ: æ è6è the stress æeld èxè minimizes the total stress energy èor maximizes the total stiænessè. Consider now an optimization problem J = min Wèè; èxè Wèè =ëeèè+æ +æ è,èë; è7è æ and æ are the cost of Material and Material, respectively. The optimization problem è7è leaves the possibility of æne-scale oscillations of the control èxè èsee, for example, Cherkaev è000èè. Physically speaking, these oscillations require that the optimally designed structure is a composite. The composites properties represent limits of rapidly oscillating sequences of the original layout. To deal with the fast oscillating solutions to optimization problems, relaxation techniques are developed, see Cherkaev è000è. Relaxation essentially averages the solution of an optimization problem by replacing the heterogeneous medium with the composite with the optimal eæective properties. Matrix Representation for a Fourth Rank Plane Tensor Dealing with tensor calculations, it is convenient to transform tensors into matrices and vectors in a tensor space. Let us consider the following base: b èè = p 0 0, ; b èè = p 0 0 b è3è = p 0 0 : èè ;

3 3 b èiè = fb èiè ææg.inthis base, any symmetric forthrank tensor can be rewritten as a three by three matrix while any symmetric second-rank tensor can be represented by a three-dimensional vector. A second-rank stress tensor has the following representation, ææ = b èiè bèiè ææ = s b èè ææ + s b èè ææ + s 3b è3è ææ è9è ès s s 3 è T = s is a vector with the components, s =, p ; s = p and s 3 = + p ; è0è and an anisotropic fourth-rank compliance tensor S is replaced by a symmetric three by three matrix D =èd ij è: Components of D are determined by the following transformation, D ij = b èiè ææ S ææb èjè ; i; j =; : èè The components of D can be written as follows D = ès + S è,s ; D = S,S ; D 3 = ès + S è; èè D = ès + S è+s ; D = S + S ; D 33 = S : Particularly, a fourth-rank isotropic compliance tensor S is given by a diagonal matrix, D i = diag i ; i ; : è3è i Remark Although we used a new notation D for a transformed compliance tensor, the tensor notation S will also be used for transformed compliance tensors hereafter. 3 Variations 3. The Scheme of the Weierstrass Test We investigate the optimality of a design by means of necessary conditions of optimality, namely, the Weierstrasstype test. The Weierstrass-type structural variation is described next. An array of inænitesimal inclusions of an admissible material S n is implanted at a point x in the domain Fig. A cartoon for inclusions in a base material. The volume fractions of inclusions is inænitely small. æ occupied by a host material S h èsee Figure è. We compare the cost of the problem with and without the implant, and compute the increment of the cost: The diæerence between the functional corresponding to the design with and without implant. If the examined structure is optimal, the increment is nonnegative. Otherwise, the cost is reduced by the variation, and the design fails the Weierstrass-type test. The increment of energy depends on the shape of the region of variation and must stay positive no matter what this shape is. Accordingly, the shape must be adjusted to the æeld to minimize the increment that must remain nonnegative, see Lurie è975è. 3. Variation of Properties The usual way to perform the Weierstrass-type variation is to consider an elliptical inclusion and to adjust it, see Lurie è975è. However, we use a slightly diæerent procedure here. Namely, we consider the structural variation that is a dilute array èsecond-rank laminatesè of elongated inclusions following Cherkaev è000è. In this structure, the inclusions are made of the ènucleiè material S n, and the envelope is made of the host material S h. This array is implanted into the pure host medium S h. First, we describe properties of a second-rank laminate. This composite structure can be characterized by its eæective tensor S æ ; the eæective tensor that links the averaged stress and strain æelds in the ërepresentative region." This region is much larger than an individual inclusion, but much smaller than the region of variation. The eæective tensor S æ èm n è of the structure depends on geometrical parameters of the inclusion array; its derivation uses the continuity conditions on the boundaries of the inclusions, see Cherkaev è000è. In our notations, it

4 has the form: S æ èm n è=s h + m n, èsn, S h è, +è, m n èæèæè æ, ; èè m n is the volume fraction of the nuclei and matrix æ depends on the structure of the array. In the base èè, the coordinate axes coincide with the directions of lamination, matrix æèæè is æèæè = h h h + h 0 0 æ, C A ; 0 æ : æ, 0 è5è Here, the inner parameter æ is the relative elongation of inclusions in the array; this parameter is responsible for anisotropy of eæective tensor S æ : Remark For the considered problem, the eæective tensor S æ èm n èwas calculated in the papers Gibiansky and Cherkaev è997è; Francfort and Murat è96è; Milton è96è; Bendse et al. è993è. In order to compute it, a two-step procedure is used: First, the eæective property of a laminate made from initial materials is computed; then the eæective property of a laminate made from the host material and the obtained laminate is computed by the same procedure. After proper simpliæcations, one arrives at the formula èè. Suppose now that the volume fraction m n is inænitesimal, m n = æm. The eæective property S æ èm + æmè can be calculated via Taylor's expansion: eæective properties æs is introduced. Instead, we consider the standard variation of the inænitesimal volume fraction æmèxè: Oèæè; if kx, x0 k éæ; æmèxè = æm C 0; if kx, èæè èè x 0 kæ: and the corresponding stationary optimality conditions. Remark 3 Tensor æs depends on two parameters of variation: The angle between the layers and the relative elongation æ. In the next calculations, we assume that the axes of the tensor æs in base èè èsee è3èè are codirected with the principal axes of matrix. Later, in Section 6 it will be clear that the obtained necessary conditions are the strongest ones, thus justifying this assumption. Namely, we will observe that ænal results coincide with the extension obtained from suæcient conditions èboundsè. 3.3 Increment The cost consists of the increment of energy due to the variation of æs and the direct cost of the variation èi.e., the change in the total cost due to change of quantities of the materials usedè. When replacing the host material S h èwith the speciæc cost æ h è is replaced with the material of S n èwith the speciæc cost æ n è, the direct cost is proportional to the diæerence of the costs of these materials: èæ n, æ h è æm: è9è S æ èm + æmè =S æ èmè+æm d dm S æèmè+oèæmè è6è The increment of the energy æe can be computed as: æeèæè = s T Vèæèsæm + G0èæmè: è0è Substituting the value of S æ èmè from èè into è6è, we obtain æs = Væm+ oèæmè; S æ èæmè = S h + æs; V =, ès n, S h è, + æ æ, ; è7è and æ is deæned in è5è. Next, compute variation of energy caused by the variation è7è of properties æs. Since we have replaced the array of inclusions by its eæective tensor S æ èmè that continuously depends on æm, we can compute the variation of the cost using the classical variational of stationary conditions: There is no need to consider the continuity conditions on the boundary of the inclusions because they are already taken into account when the tensor of V is given in è7è. Remark Notice that we consider a weak èstationaryè variation of the eæective properties æs in contrast with the strong variations of the point-wise properties on the boundary of the inclusions in the array. The weak variation of the properties is easy to compute: the stress æelds remain constant since the main term of the variation of order of æm is independent of their variations. This remark justiæes the use of eæective tensors in the variations. Increment æe depends on parameter æ. We obtain the following expression for æeèæè using è7è: æeèæè = V s + V s + V 33 s 3 +V s s +V 3 s s 3 +V 3 s s 3 æm; èè

5 5 and the components of V depend on æ as follows is According to è3è, the optimal increment in energy V = è h, n èè n è h + h è+ h h èè h + h è h Kèæè V 3 = èæ, èè h, n èè n, h èè h + h è Kèæè V = h, n h n V 33 = è h, n èè n è h + h è+ h h èè h + h è ; h Kèæè V = 0; V 3 =0 Kèæè = C æèæ, è + n n h è h + h è+ h h è h + h èè n + n è+ n n h ; in which the constant C is equal to C = h h è h, n èè n, h è: We remind, that the coordinate axes are directed along the principal stresses. Thus, s = è x 0 y è T is used in the calculations of èè. Finally, the cost of the variation, æj æm, is expressed as an explicit function of æ: æj æm =èæ n, æ h + æeèæèè æm: èè Since all variations èincluding the most sensitive oneè lead to the nonnegative increment æj, the optimality condition of the ith material becomes: æj ès i ; S j ; j è=æ n, æ h + min æ æeèæè 0; è3è æj ès i ; S j ; i èisavariation caused by adding material S j into the material S i ; and i is the æeld in the host material S i The Most Sensitive Variations The calculation of the optimal variation is slightly different inthe case of well- and badly ordered materials. Let us number the materials so that é. The wellordered case corresponds to the moduli of materials so that é ; otherwise, it is called badly ordered case. We demonstrate calculations of the well-ordered case hereafter unless otherwise stated. f 0 =min æ æeèæè: èè Here the inner parameter æ is used to adjust the structure to the acting stress æeld. To compute an optimal variation, we take the derivative of æeèæè with respect to æ in èè, and obtain the following values of æ's as zeros : æ =, xè h + n è h + y è h + h è n ; è5è h è x, y èè n, h è æ = xè h + n è h, y è h + h è n : è6è h è x + y èè h, n è The optimal values of æ must belong to the interval of ë0; ë. If none of the è5è and è6è belongs to è0; è, the optimal values are at the end points, i.e., æ =0oræ =: There are two cases to consider depending on the sign of h, n. Increment f 0 is deæned as f a 0 and f b 0 for the corresponding cases: Case A: If h, n é 0, then f 0 is f a 0 = é é: Eè0è if y x é K; Eèæ è if K y x K; Eèè if y x é K ; when h, n é 0. Here K = è n + h è h è h + h è n é : è7è èè Case B: If h, n é 0, then optimal value æ 0 always belongs to the boundary of the interval ë0; ë. The minimal æe is reached at the end points, i.e., æ 0 is either 0 or. f b 0 = é : Eèè if x y ; Eè0è if x y ; when h, n é 0 for which K é. è9è Note that these cases correspond to physically diæerent situations under the given conditions: Case A corresponds to inclusions of a strong material added to a weak material and Case B corresponds to the inverse situation. The next section will illustrate this point using a speciæc example.

6 6 3. Necessary Conditions Let us determine the regions of permitted æelds for an optimal design using the Weierstrass variations or the conditions è7è - è9è.. Suppose that an inclusion of second material S èstrong materialè is placed into domain æ èweak materialè. The corresponding increment æj ès ; S ; è is deæned by è3è: æj ès ; S ; è=æ, æ + f b 0èS ; S ; è 0; è30è is the æeld at a point of domain æ of the ærst material. Since S and S are given, the inequality in è30è depends only on the stress æeld. The set of stresses that satisæes the condition è30è is called F : If F, material S is optimal.. When the inclusion of ærst material æ èweak materialè is placed into domain æ èstrong materialè, the increment is deæned by è3è æj ès ; S ; è=æ, æ + f a 0èS ; S ; è 0; è3è is the æeld at apoint of domain æ. The set of stresses that satisæes the condition è3è is called F : If F, then the material S is optimal èsee footnote èèè. Remark 5 When the increment due to the inclusion of the strong material into the weak material is calculated, the formula in è7è is modiæed as: æs =,Væm+ oèæmè; V =, ès, S è, + æ æ, while subscript h in è5è becomes: h =. The union F = F ë F of two sets of optimality of the ærst and second materials does not coincide with the whole range of. The remaining part, F f, is called the forbidden region. Here, none of the materials is optimal. Fields in an optimal design should never belong to this region to matter what the external loading is. If the external loading belongs to F f, the pointwise æelds are still in their regions F and F, but the mean æeld matches the external load. This phenomenon is the reason of formation a structure or a composite material in the optimal designs. Remark 6 According to the theory of quasiconvexity, the solution èstress æeldè of the variational problem è6è is oscillatory if the Lagrangian èè is not quasiconvex, see for example Cherkaev è000è. The forbidden region F f necessarily belongs to the region the Lagrangian is not quasiconvex. Optimality is understood in the sense of satisfaction of necessary conditions. Remark 7 Dealing with special necessary conditions, one cannot guarantee that there are no other tests that are more restrictive. If this were the case, the forbidden region would be larger than F f.for the considered problem, however, the results are ænal as it follows from the results of Section 6. Details of the calculations are shown explicitly next when x y é 0. Assume that the axes are oriented so that xy = 0. In this case, x and y become the eigenvalues of. The set F of stresses optimal in the material S and set F of stresses optimal in S are determined by è30è and è3è, respectively: F = f j æj ès ; S ; è=æ, æ + Eèè 0g : è3è F = f j æj ès ; S ; è=æ, æ + é é: Eè0è if y é K; x Eèæ è if y K; x 9 9 é= é= é; 0 é; Eèæ è =, è, èè + è è + + è è x + y è : è33è è3è When x y é 0, parameter æ lies in the interval of è0,è. Therefore, Eèæ èisreplacedby Eèæ è in è33è which yields to the expression: Eèæ è =, è, èè + è è + + è è x, y è : è35è The obtained results can be interpreted for the condition of x y é 0asfollows: æ Formula è3è corresponds to the variation when the inclusion of the strong material is implanted into a weak material. The most sensitive variation corresponds to a ællet èlaminatesè that is placed along the direction of maximal æeld. Thus, the nuclei ëexposes itself" to the maximal eæect of the variation. æ Formula è33è corresponds to the variation when the inclusion of the weak material is implanted into the strong material. The most sensitive variation corresponds either to the family of nuclei in the second rank lamination èif the ratio of x and y is smaller that a limit valueè or a ællet èif the ratio of x and y is larger than this valueè. Thus, the weak material ëhides itself" to maximize the eæects of the variation.

7 7 s y 6 x s y x Fig. The stress æelds when the materials have zero Poisson ratio. Results. Range of Admissible Fields Based on the results of Section 3.3., we observe that the boundaries of F and F consist of the two following components: Two symmetric curves Eèæè = A x + B y +C x y = B x + A y +c x y = è36è the constant coeæcients A, B and C are determined by the material constants; this component corresponds to optimal values æ = 0 or æ = of Eèæè. The other component is represented by two families of straight lines Eèæ i è= D è x + y è = D è x, y è = è37è the constant coeæcients D and D are determined by the material constants; this component corresponds to stationary values of æ in è5è and è6è. Well-ordered case: In this case, the curves in è36è are ellipses since A é 0, B é 0, and AB é C. The optimality regions are presented in Figure and Figure 3. Region F of optimality of the weak and cheap material lies inside of intersection of two smaller ellipses; this region is shown in Figure 3 by the curve with crosses on it. Region F of optimality of the strong and expensive material lies outside of the linear envelope stretched on two larger ellipses; this region is shown by the curve with circles on it. The boundary contains elliptical and straight components. Forbidden region F f lies between F and F. In Figure 3èaè, we ærst plot the curves determined by stationary values 0,, æ,andæ of æ, using èaè Region F 6 lies inside of the region given by crosses, while region F lies outside of the region given by circles. The forbidden region F f lies between the regions. 6 y x èbè The asymptotic case when the ratio between the material properties is large. Fig. 3 Optimal æelds for well-ordered case: é and é.

8 0 5 y x 5 0 è5è è6è. Next, we follow the conditions in Section 3. to select among these stationary values. In Figure 3èbè, the asymptotic case is shown. One of the materials has inænite compliance or becomes void and the optimization problem becomes the problem of topology optimization Bendse è995è. The region of optimality F of the void ëmaterial" shrinks to zero as expected. Elliptic components of the boundary of F disappear. Finally, we observe that ellipses in Figure 3 are rotated by angles depending on Poisson ratio èe.g., when Poisson ratio is zero, C = 0 in è36è, see Figure the ellipses are mutually orthogonalè. 5 èaè Region F 0 lies inside of the region given by crosses, while region F lies outside of the region given by circles, and in between lies the forbidden region F f : y 0 s s 0 x Badly ordered case: In the badly ordered case, A and B in è36è have diæerent signs, and the elliptical components of the boundaries in well-ordered case become hyperbolic. The situation is more tedious than in the wellordered case. Each material plays both a role of strong or weak one, depending on the type of æeld : If the æeld is close to hydrostatic, the ordering of the bulk moduli deænes what material is called strong or weak and if the æeld is close to shear, the ordering of the shear moduli deænes the ordering of materials. The cheaper material is always optimal in the proximity of the origin èthat corresponds to small magnitude of the stressè. The results are presented at Figure èaè. The boundary of the region of optimality for the more expensive material corresponds to the intersection of two sharp hyperbolas in è36è; this region is shown by a curve with circles on it. The boundary of the region of optimality for the cheaper material corresponds to the convex envelope stretched on two other hyperbolas; this region is shown by a curve with crosses. The straight part of the envelope corresponds to optimality of implant that is formed as a second-rank dilute composite. When the cost of both materials is the equal, the regions degenerates. One can verify that all boundaries become straight lines passing through the origin. 0 èbè The asymptotic case when the ratio between the material properties is large. Fig. Optimal æelds for badly ordered case: é and é. Optimal æelds and optimal structures We already mentioned that the optimal design problem we are dealing with was investigated in a number of papers. Particularly,itwas found that the variety of optimal structures consists of second-rank laminates. Also, the Vigdergauz structures are equally optimal if x y é 0 and one of the phases is void. Our analysis allows us to explain the optimality of these diæerent geometries. Suppose that a periodic optimal structure is submerged into a constant external stress æeld. If the external æeld belongs to the forbidden region F f, the local æelds belong to the boundaries of the permitted regions, F or F, and the averaged stress is equal to the applied one.

9 9 y Rank-one F Rank-two F Rank-one x the corner point of its permitted region, being isotropic: x = y. The æelds in the envelope vary from point to point, but they remain on the straight component of the boundary of F! Here, we do not demonstrate the bulky calculations that conærm these results; the reader can check them using the formulas in the previous section. Remark The optimality of the mentioned structures was independently veriæed by the suæcient conditions. This optimality now proves that our variations are the most sensitive ones. Indeed, if a more sensitive variation existed, it would correspond to a larger forbidden region; the æelds in mentioned structures would be in that larger region and therefore would be nonoptimal which contradicts suæcient conditions. Fig. 5 The stress æelds when the materials have zero Poisson ratio: Case when det 0. The second-rank laminates are optimal, if the external æeld belongs to the triangle ërank-two" in Figure 5, that allows the representation = m + m + m 3 è3è m i 0; m + m + m 3 =. Here, stress corresponds to the ærst èweakè material located in the inner layer of the second-rank structure, and it belongs to the corner point of the boundary of its permitted region: = æ, see Figure 5. In è3è, stress corresponds to the second èstrongè material located in the other inner layer of the second-rank structure. The æelds and are compatible detè, è = 0. This æeld corresponds to that point of the boundary of permitted region F the elliptical component meets the straightcomponent èsee Figure 5è. Finally, stress corresponds to the second èstrongè material located in the outer layer of the second-rank structure. This æeld is not compatible with neither nor but it is compatible with their weighted sum: detè s, è = 0 s = m =èm + m è + m =èm + m è. This æeld corresponds to a point the straight component of the boundary of region F èsee Figure 5è. Simple laminates are optimal when the external æeld is anisotropic, and belongs to curved trapezoids ërankone" in Figure 5: It combines as a sum of two æelds; each of them belong to the elliptical component of the boundary of the correspondent permitted region and they are compatible. The æelds in Vigdergauz structures or in Hashin-Shtrikman ëcoated spheres" are diæerent, however they also belong to the boundaries of the permitted regions. The æeld in the nucleus again belongs to Structures with inænitesimal elements The optimality requirements force the æelds to obey an additional condition: they must belong to the boundaries of the permitted regions if the external æeld is in the forbidden region. This requirement cannot be enforced in a ëbulky" domain ælled with one material since we can control only the curve that divide the domains of diæerent material. Therefore, we conclude that the optimal structure should, in general, not contain ëbulky domains" that a dividing curve should have inænitely often wiggles, Cherkaev è000è, unless the structure is strictly periodic and the load is strictly constant. The optimal structure becomes a composite in which æelds belong to the boundaries èf = constant èorèf = constant è at each point èsee Table èèè.. Norms For centuries, architects and engineers knew that stress æelds should be distributed evenly over an optimal structure; thus the overstressed regions require more reinforcements, while the understressed regions can be lightened. This common sense requirement of optimality of equally stressed designs can be stated mathematically in terms of a norm of a stress æeld. We consider the well-ordered case. Our results can be rewritten in the following form: The boundaries of F correspond to constancy of two norms: Table Table of the optimal design. Average æeld 0 Optimal Material 0 F Material 0 F Material 0 F f Composite in every

10 0 P N è è= p Eèè; N è è= é é: and peè0è if y é K; x peèæ è if y x K; 9 é= é; è39è = x 0 is the stress æeld in 0 y æ, and = x 0 is the stress æeld in 0 y æ ; and K is given in èè. In è39è, Eè0è and Eèè are given in è36è, and Eèæ è is given in è37è. The reader can check that N and N are indeed rotationally invariant norms of two-by-two symmetric tensors: The axioms of norms are satisæed. In the onedimensional case, both N and N degenerate to the absolute value of the stress. In the regions composites are used these norms are constant in each material in an optimal structure N constant ; or N constant : è0è The mean æeld varies due to the variation of concentrations. The mixtures should be inænitesimal æne, otherwise one cannot preserve the constancy of the norm inside the solid region of a material. Remark 9 Notice that the equation è0è is fulælled every in the design domain. These conditions are satisæed as equalities in the whole region of composites. The conditions are nonlocal: the variation of the loading in a small subdomain leads to an evenly distributed variation of optimal æelds and corresponding optimal microstructures in all design domain. This feature, natural for optimal design, is a mark of quasiconvex envelope, see Cherkaev è000è since the Euler-Lagrange equations of the corresponding variational problem lose ellipticity that is the reason for localization. Topology optimization Finally, we observe that in the asymptotic case when one of the materials is void èthe case of topology optimization, see Figure 3èbè, the small ellipses in the center become a point at the origin: N = 0. The non-void material must be never understressed in an optimal design: Its norm becomes N = j x j + j y j and the inequality holds s N èæ, æ è ; x æ : èè + Fig. 6 Plate design domain, load and boundary conditions. 5 Analyzer The obtained necessary conditions can be used to quantitative examine suboptimal projects. A design may have stresses whose norm is quite close to a constant every. This design would approximately solve the optimality conditions and therefore would be suboptimal. The geometry of the design structure can be arbitrary; only the æelds in it need to be checked. The necessary conditions provide a tool that we call an analyzer to examine the given structure and to judge how close to an optimal one is it. Using the analyzer, we can see how far from the constancy is the corresponding norm of the stress æeld. The use of it is similar to the use of color maps of a failure criterion that show how close the stresses are to the limit. 5. An example We illustrate the use of the analyzer by the following example. Consider a plane domain, a 5cmæ5cm plate with Poisson ratio 0.3 and Young's modulus e; the plate is loaded as shown in Figure 6. The caverns of unknown shape will be introduced and the results will be compared using the analyzer. We deal with an asymptotic case one of the materials is void, or with the problem of topology optimization. Another application of this problem can be seen in Kíucíuk è00è a plate for a circle, square and two rectangles is examined with diæerent material properties. In this case, it is expected that N =0;N = j x j + j y j; the norm N must be constant every in an optimal projection if x y é 0, that trace of tensor is constant, and if x y é 0, then Von Mises stress is constant èsee Section.è. This corresponds to the earlier results of Bendse è995è; Grabovsky è996bè. We consider examples of suboptimal design for planar structures, and compare the shapes of a circle, square and P

11 Von Mises stress distribution for a deformed structure when a circle is included. Fig ture Von Mises stress distribution for a suboptimal struc- We obtain the following normalized energies W for each structure in Figure 7-Figure 9; W = 0:3967 in Figure 7, W = 0:350 in Figure, and W = 0:53 in Figure 9. This calculation con rms that that the structure in Figure 9 is the best from the compared set. Checking the norms given in (39) allows us to describe quantitatively suboptimal structures and to point to the places the norm derivated from the constant. One can use this as a tool to decide if a structure is close to optimal Supplement: Comparison with Su cient Conditions Fig Von Mises stress distribution for a deformed structure when a combination of di erent geometries is included. Fig. four ellipses with the same areas, following the ideas of Cherkaev et al. (99a). We nd that the structure in Figure 9 better approximates the necessary conditions than the others. Comparing this norm in Figure -Figure 9 for various structures, we observe that the deviation from the constancy of the norm N( ) is smaller in Figure 9 than the others. It is not claimed to be the optimal but it represents a reasonable suboptimal design. In the true optimal solution the norm N( ) is expected to be constant every. 6. Translation method The analysis of admissible regions is based on special necessary conditions; in principle, these regions could be decreased by more sophisticated conditions. Here we show that in fact the results cannot be improved, by comparing necessary conditions and su cient conditions. The su cient conditions obtained by the Translation method (see the discussion of the method and its development in the books Cherkaev and Kohn (997), Cherkaev (000), and Milton (00)) are used here to derive the information of the admissible regions. Contrary to necessary conditions, the su cient conditions correspond to admissible regions that could, in principle, be abridged by more sensitive conditions. Both methods provide the two-side bounds for admissible regions. In setting of two-dimensional elasticity, the translation method exploits the nonuniqueness of the de nition

12 of the elastic moduli, see Cherkaev et al. è99è. This nonuniqueness allows us to consider the ëtranslated" tensor of elastic moduli instead of the original tensor and bound this new tensor by the harmonic mean bound. Then the parameter of the translation is adjusted to make the bound stronger. The elasticity tensors S are translated by using a multiple of a special tensor T called translator, S trans = S, tt ; èè t is a constant. Formally, the translator is the isotropic tensor of two-dimensional elasticity with opposite bulk and shear moduli: T =and T =,. The mentioned nonuniqueness leads to the following property: the stress æeld of any periodic structure loaded by homogeneous external forces at inænity stays the same if the elasticity tensors S i are replaced by Si trans no matter what the magnitude t of the translator is, providing that the translated tensors Si trans are positive deænite; S i, tt 0 i: è3è 6. Basic formulas In order to compare necessary conditions with the translation method, we need to compute the æelds in the structures predicted by this method. Earlier a similar problem was considered in Grabovsky è996aè and it is discussed in Cherkaev è000è; Milton è00è. Relaxation of the problem in è7è is based on the following essentially algebraic procedure: Consider the Lagrangian in è6è and neglect all diæerential constraints on the stress æeld. Then the stress tensor become constant in each material, since the energy of each well is convex. The minimization problem è6è becomes an algebraic problem CèFè = Fèè = min F s 0 èxè ëfèè+æ m + æ m ë; X i m i T i S ièxè i and F s 0èxè =f i j G = X i m i i, 0 =0g; èè è5è è6è i is the æeld in the ith material subject only to integral restrictions è6è, and m i is the æxed fraction of the ith material in the design domain. This lower bound of the energy in èè is the convex envelope of the Lagrangian G. Let us compute the æelds that solve the problem èè. To minimize F under the constraint è6è, i èf + Gè =m i ès i i + è =0; è7è is the Lagrange multiplier by è6è. From è6è and è7è, we ænd: =,S 0 ; and èè i = S, i S 0 : è9è S = è X i m i S, i!, : Substituting è9è into èè, we obtain the lower bound that is the convex envelope of the multi-well Lagrangian. CèFè = T 0 S 0 Fèè : è50è The translation method improves the bound by using the nonuniqueness of S è generally, by using the quasiconvex but not convex functions, see the discussion in Cherkaev è000èè. The formal application of the translation bound is easy: we simply replace the elasticity tensors in è50è and è9è by the translated tensors as in èè, subject to the constraint è3è. In order to make this bound eæective, we maximize the right-hand side with respect to t; and obtain Fèè S h = S, T ; = max t:è3èholds T 0 ès h + T è 0 ; è5è è X i m i ès i, T è,!, : è5è Since all involved elasticity tensors correspond to the isotropic elasticity, we replace the elastic moduli: and è3è leads to is replaced by, t; is replaced by + t T = ft j,minf ; gt minf ; g: è53è

13 3 6.3 Comparing the æelds The obtained formulas assume that the volume fractions in the mixture are æxed. In contrary, the necessary conditions do not deal with volume fractions, only with the cost of materials. To compare the æelds in structures, we need to exclude volume fraction m from the formulas for translation bounds by computing the minimum over it. The dependence of the translation parameter t must be excluded as well. This leads to the following problem LèFè =min m max tt T 0 ès h + T è 0 + m æ + m æ ; è5è S h is given in è5è. After the optimal parameters m and t are computed as function of the load 0, these values are used to compute the æelds using the translation analog of è9è. The solution of the inner maximization problem in è5è is =0; 0 for all t T: è55è First, assume that optimal values of t belong to the set è53è. The conditions in è55è are observed by the following t's as a function of the æeld t =, xb, y a x d + y c ; è56è A is a function of the material properties. Notice that in è60è the signs of the second derivatives depend on æeld 0 and the material properties. In the well-ordered case when it is assumed that é and é ;wehave t opt = é : t if x é y ; t if x é y : è6è èfor the badly-ordered case, the roots t and t switch placesè. To calculate the optimal values of m, we impose conditions similar to the ones given in è55è. Hence, we have m t = è + èk, p l ak k =aèæ, æ è, è, èx and l =è + èè + èèa y, b x è k; è6è and the optimal volume fraction corresponding to t is m t = è + è~ k, a ~ k p ~l ~k =aèæ, æ è, è, è y and ~ l =è + èè + èèa x, b y è ~ k: è63è t =, yb + x a x c + y d ; è57è 6. Asymptotic: Topology optimization a =,, + ; b =, +, ; c =, ; d =è, +, + èm, + + : è5è è59è The correct sense of these extremum points is monitored by the second æ ææææ æ ææææ æ ææææ t=t æ ææææ æ ææææ æ ææææ t=t = ma è, mè èy, xèè, èè, è ; ma è, mè è y, xèè, èè, è ; è60è First, we examine the case of topology optimization: = =0.Thecase can be examined analytically. In this case, the optimal values of t take only limiting values, see Cherkaev è000è: t =, and t =. Case: t =, The optimal values m of m are m = j x + y j s è + è èæ, æ è è6è The condition m ë0; ë yields to the inequality for the stresses: s èæ, æ è 0 j x + y j : è65è + It shows that the region of optimal æelds is bounded by an inclined strip in the plane of eigenvalues x and y.

14 0 y 6 y x x 6 0 Fig. 0 The bound found from the necessary conditions is approached from above by using the translation bound for, max été max. Fig. The bound found from the necessary conditions is approached from below by using the translation bound for, max été max. Case: t = The optimal values m of m are m = j x, y j s è + è èæ, æ è : è66è Since m ë0; ë, the corresponding inequality for x and y is satisæed s 0 j x, y j èæ, æ è : + è67è It shows that the region of optimal æelds is bounded by an orthogonal inclined strip, too. Two intersections of the two inequalities è65è and è67è deæne the diamondlike forbidden region. Remark 0 The inequalities è65è and è67è match the necessary condition èè that deænes the same the diamondlike domain in the plane of eigenvalues. This proves the extreme character of these conditions and, consequently, the extreme character of the used variations. Notice the opposite signs of inequalities in the necessary condition èè and suæcient conditions; è65è and è67è. 6.5 General case In Figure 0 and Figure, we show the æelds in the materials according to the Translation bounds. The family of the curves correspond to diæerent values of t. For each æxed value of t T the bound of the energy is minimized over m; optimal m is given by è6è and è63è and is subjected to the condition of being in the interval of è0,è. The calculation of the bound was conducted using MAPLE. One can observe from these ægures, how the forbidden region obtained from suæcient conditions matches the region calculated from necessary conditions. For any æxed t the suæcient condition corresponds to the boundary of the forbidden interval given by an ellipse; the ellipse elongates following the variation of t; limiting values of t correspond to an extremely elongated ellipse which degenerates into either a pair of parallel lines, Figure 0 or into a ænite interval, Figure. In Figure 0 and Figure, the family of lines tends to the bound and the lines always belong to the permitted region. The forbidden region corresponds to the intersection of all ellipses that are the violation of a suæcient condition. The envelope of the ellipses matches the forbidden region found by the necessary conditions, approximating it from outside. On the contrary, the necessary conditions approximate the same region from inside as a union of all forbidden regions corresponding to a given type of the variation. Acknowledgements The authors gratefully acknowledge the anonymous referee whose suggestion helped to improve the exposition. This work was supported by NSF Grant No: DMS and by ARO Grant No: 363-MA.

15 5 References Allaire, G. and Kohn, R. V. è99è. Optimal lower bounds on the elastic energy of a composite made from two non wellordered isotropic materials. Quarterly of Applied Mathematics, LIIèè:3í333. Atkin, R. J. and Fox, N. è90è. An Introduction to the Theory of Elasticity. Longman Mathematical Texts, New York. Bendse, M. P., Daz, A. R., and Kikuchi, N. è993è. Topology and generalized layout optimization of elastic structures. In Topology design of structures èsesimbra, 99è, pages 59í 05. Kluwer Academic Publishers Group, Dordrecht. Bendse, M. P. è995è. Optimization of Structural Topology, Shape, and Material. Springer-Verlag, New York; Berlin; Heidelberg. Cherkaev, A. V. è000è. Variational Methods for Structural Optimization, volume 0. Springer-Verlag, New York; Berlin; Heidelberg. Cherkaev, A. V., Grabovsky, Y., Movchan, A. B., and Serkov, S. K. è99aè. The cavity of the optimal shape under shear stresses. Int. J. Solids Struct., 35è33è:39í0. Cherkaev, A. V. and Kohn, R. V., editors è997è. Topics in the mathematical modelling of composite materials. Birkhíauser Boston Inc., Cambridge, MA. Cherkaev, A. V., Krog, L. A., and Kucuk, I. è99bè. Staple optimal design of two-dimensional elastic structures. Control and Cybernetics, 7èè:65í. Cherkaev, A. V., Lurie, K. A., and Milton, G. W. è99è. Invariant properties of the stress in plane elasticity and equivalence classes of composites. Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, 3è90è:59í59. Eschenauer, H. A. and Olhoæ, N. è00è. Topology optimization of continuum structures: A review. Appl Mech Rev, 5èè:33í39. Francfort, G. and Murat, F. è96è. Homogenization and optimal bounds in linear elasticity. Arch. Rat. Mech. Anal., 9:307í33. Fung, Y. C. è965è. Foundation of Solid Mechanics. Prentice-Hall, Upper Saddle River, NJ 075. Gibiansky, L. V. and Cherkaev, A. V. è9è. Design of composite plates of extremal rigidity. Report 9, Ioæe Physicotechnical Institute, Acad of Sc, USSR, Leningrad, USSR. English translation in Gibiansky and Cherkaev è997è. Gibiansky, L. V. and Cherkaev, A. V. è997è. Design of composite plates of extremal rigidity. In Cherkaev, A. V. and Kohn, R. V., editors, Topics in the Mathematical Modelling of Composite Materials, volume 3 of Progress in nonlinear diæerential equations and their applications, pages 95í37. Birkhíauser Boston, Boston, MA. Translation. The original publication in Gibiansky and Cherkaev è9è. Grabovsky, Y. è996aè. Bounds and extremal microstructures for two-component composites: A uniæed treatment based on the translation method. Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, 5è97è:95í95. Grabovsky, Y. è996bè. Explicit solution of an optimal design problem with non-aæne displacement boundary conditions. Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, 5:909í9. Grabovsky, Y. and Kohn, R. V. è995è. Anisotropy of the Vigdergauz microstructure. Journal of Applied Mechanics, 6èè:063í065. Hashin, Z. and Shtrikman, S. è96è. A variational approach to the theory of the elastic behavior of polycrystals. J. Mech. Phys. Solids, 0:33í35. Kíucíuk, I. è00è. Variational approach to optimization of elastic structures. PhD thesis, University of Utah. Lurie, K. A. è975è. Optimal Control in Problems of Mathematical Physics. Nauka, Moscow, Russia. in Russian. Lurie, K. A. and Cherkaev, A. V. è97è. Nonhomogeneous bar of extremal torsional rigidity. In Nonlinear Problems in Structural Mechanics. Structural Optimization, pages 6í6. Naukova Dumka, Kiev, Ukraine. Milton, G. W. è9è. Bounds on the complex permittivity of a two-component composite material. Journal of Applied Physics, 5:56í593. Milton, G. W. è96è. Modelling the properties of composites by laminates. In Ericksen, J. et al., editors, Homogenization and Eæective Moduli of Materials and Media èminneapolis, Minn., 9è95è, pages 50í75, New York; Berlin; Heidelberg. Springer-Verlag. Milton, G. W. è00è. The Theory of Composites. 6. Cambridge Monographs on Applied and Computational Mathematics, Cambridge, UK. In Print. Timoshenko, S. P. è970è. Theory of Elasticity. McGraw-Hill, New York. Vigdergauz, S. B. è99è. Regular structures with extremal elastic properties. MTT, è3è:57í63.