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

Authors:
Grégoire Allaire and Robert V. Kohn

Journal:
Quart. Appl. Math. **51** (1993), 675-699

MSC:
Primary 73B27; Secondary 35B27, 73K20, 73K40, 73V25

DOI:
https://doi.org/10.1090/qam/1247434

MathSciNet review:
MR1247434

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Abstract | References | Similar Articles | Additional Information

Abstract: 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 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.

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Additional Information

DOI:
https://doi.org/10.1090/qam/1247434

Article copyright:
© Copyright 1993
American Mathematical Society