Exact estimates of conductivity of composites formed by two isotropically conducting media taken in prescribed proportion

Abstract

SynopsisThis paper is a sequel to [1-4]. We consider the problem of G-closure, i.e. the description of the set GU of effective tensors of conductivity for all possible mixtures assembled from a number of initially given components belonging to some fixed set U. Effective tensors are determined here in a sense of G-convergence relative to the operator ∇· D · ∇, of the elements DeU ∈ [5, 6].The G-closure problem for an arbitrary initial set U in the two-dimensional case has already been solved [3, 4]. It remained, however, unclear how to construct, in the most economic way, a composite with some prescribed effective conductivity, or, equivalently, how to describe the set GmU of composites which may be assembled from given components taken in some prescribed proportion. This problem is solved in what follows for a set U consisting of two isotropic materials possessing conductivities D+ = u+E and D− = u−E where 0<u−<u+<∞ and E ( = ii+jj) is a unit tensor.

The G-closure of two well-ordered, anisotropic conductors

“…The papers [4], [5], [9] completely solved the G-closure problem for two isotropic components using the method of compensated compactness (later renamed the translation method by Milton [6]). This article follows the approach of Kohn and Milton in [3] adapted to the case of anisotropic component materials.…”

On hierarchical structures and reiterated homogenization

“…Here, we only mention a few examples of optimal iterated structures. Concerning optimal iterated laminates for the conductivity problem see Lurie and Cherkaev [27], Milton [29], Schulgasser [42] and Tartar [45]. On optimal microstructures for the elasticity case in two-dimensions (transversely isotropic case) there exist periodic (i.e.…”

Optimum composite material design

“…The concept of composite media not only comes directly from the physical world but also provides a theoretically sound means for relaxation of variational problems -the problem of optimum topology design (see [5], [22], [12], [2] or [14]) in the first rank of importance. It is a classical result of the homogenization theory that composites can be replaced by a macroscopically homogeneous medium whose material constants -the so called effective constants or effective moduli -depend on the microgeometry in which the (*) Manuscrit received March 10, 94.…”

“…In the case of a scalar linear elliptic partial differential équation (the steady heat transfer équation), the G o -closure sets are known for mixtures of two phases ; one of the phases may be degenerate, i.e., a void ; see [14], [23], [16]. Ho wever, for the case of the System of PDE's of linear elasticity, only a partial information about the G o -closure sets is available so far ; namely we know how to minimize the complementary energy for a given single macroscopic stress field (see [1]).…”