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Explicit optimal bounds on the elastic energy of a two-phase composite in two space dimensions

Grégoire Allaire, Robert V. Kohn

1993
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Quarterly of Applied Mathematics
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This paper is concerned with two-dimensional, linearly elastic, composite materials made by mixing two isotropic components. For given volume fractions and average strain, we establish explicit optimal upper and lower bounds on the effective energy quadratic form. There are two different approaches to this problem, one based on the "Hashin-Shtrikman variational principle" and the other on the "translation method". We implement both. The Hashin-Shtrikman principle applies only when the component
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... when the component materials are "well-ordered", i.e., when the smaller shear and bulk moduli belong to the same material. The translation method, however, requires no such hypothesis. As a consequence, our optimal bounds are valid even when the component materials are not well-ordered. Analogous results have previously been obtained by Gibianski and Cherkaev in the context of the plate equation. 0. Introduction. The macroscopic properties of a linearly elastic composite material are described by its tensor of effective moduli (Hooke's law) a*. This fourthorder tensor depends on the microgeometry of the mixture as well as on the elastic properties of the components. There is a large body of literature concerning the estimation of a* in terms of statistical information on the microstructure; see, e.g., [9, 39, 41]. Recently a related but somewhat different question has received much attention: given a fixed collection of component materials, can one describe all composites o* achievable by mixing these components in prescribed volume fraction? Known as the "(7-closure problem", this question arises naturally from problems of structural optimization; see, e.g., [24, 26, 33] . A complete answer is available only in a few special cases; see, e.g., [13, 25, 27] . Much more is known about the analogous question 675 676 GREGOIRE ALLAIRE and ROBERT V. KOHN for scalar phenomena (heat conduction, electrical resistance, etc.); see, e.g., [16, 28, 38], While the full G-closure problem remains beyond reach in the context of elasticity, there has been remarkable progress concerning optimal bounds on the elastic energy quadratic form [1-6, 14, 15, 17, 21, 23, 30]. A specific consequence of that work is the following: consider two isotropic materials with bulk modulus xr, , k2 and shear modulus /ij, n2, which are to be mixed with volume fractions 6{ and 02 respectively. Assume moreover that the components are "well-ordered", i.e., that either K{ < k2, < /u2 or k2 < k{ , n2 < /i, . Then one can identify the largest and smallest possible effective energy of a* as functions of the macroscopic strain. In other words, one can identify functions f±(n{, /i2, k1 , k2, dx, d2, £,) such that (o.i) and such that both inequalities can be achieved (for any £) by suitable microstructures (which depend on £). We call f± "optimal bounds on the elastic energy", since /_ is evidently the smallest and / the largest function for which (0.1) can hold. Clearly (0.1) improves upon the well-known harmonic and arithmetic mean bounds, known as Paul's or the Voigt-Reuss bounds: We emphasize that the symmetry of a* is not restricted in (0.1). As might be expected, the extremal composites are isotropic only when £ is isotropic; in that case (0.1) reduces to the well-known Hashin-Shtrikman bound on the effective bulk modulus of an isotropic composite [19] , Energy bounds of the form (0.1) offer partial information about the G-closure problem, since they specify the extreme values of the linear functions o* -► , £) for every second-order tensor £. In addition, such bounds are of use in their own right. The best developed application is to structural optimization, where (0.1) or its analogue for complementary energy permits the solution of problems involving compliance as a design criterion [3, 7] , Other potential applications include coherent phase transitions [22] and modelling the accumulation of damage [11] , We remark that (0.1) represents only a special case of [4]: that paper actually identifies optimal upper and lower bounds for any sum of energies {o*E,x , £,} H + (c*£" , £"} • The work just summarized has, alas, two significant shortcomings. First, the optimal upper and lower bounds f± are not given explicitly, rather, they are given as the extremal values of certain finite-dimensional optimizations. Second, the restriction that the component materials be well-ordered is unnatural: it is forced by the method of analysis (which is based on the Hashin-Shtrikman variational principle), not by anything intrinsic to the problem. The goal of the present work is to redress these difficulties in the special case of two space dimensions. We implement this as follows. First, we review briefly the optimal energy bound from the Hashin-Shtrikman variational principle, specialized to the case of two wellordered isotropic components in two space dimensions. Here we follow [2, 21, 23] rather than [4, 5] , so f± is given as the extremal value of a convex but nonsmooth

doi:10.1090/qam/1247434
fatcat:voeg65esr5cidkl7twuwxpak4u