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The exact coupled bounds for effective tensors of electrical and magnetic properties of two-component two-dimensional composites
Andrej V. Cherkaev (Courant Institute of Mathematical Sciences), Leonid V. Gibiansky (Courant Institute of Mathematical Sciences)
Abstract: Synopsis This paper describes the effective properties of a two-dimensional two-component dielectric composite. Each component is supposed to be isotropic and is characterised by a pair of constants of electrical and magnetic permeabilities. Effective properties of an arbitrary anisotropic composite depend on its microstructure and are characterised by a pair of symmetric tensors of electrical and magnetic permeabilities. Here we construct the so-called Gm-closure which is the set of all possible values of these pairs, corresponding to all possible microstructures with prescribed volume fractions of the given components. The exact bounds of the Gm-closure are obtained and the microstructure corresponding to each point of the Gm-closure set is determined.