Singular Behavior of Electric Field of High Contrast Concentrated Composites

A heterogeneous medium of constituents with vastly different mechanical properties, whose inhomogeneities are in close proximity to each other, is considered. The gradient of the solution to the corresponding problem exhibits singular behavior (blow up) with respect to the distance between inhomogeneities. This paper introduces a concise procedure for capturing the leading term of gradient's asymptotics precisely. This procedure is based on a thorough study of the system's energy. The developed methodology allows for straightforward generalization to heterogeneous media with a nonlinear constitutive description

Explicit corrector in homogenization of monotone operators and its application to nonlinear dielectric elastomer composites

This paper concerns the rigorous periodic homogenization for a weakly coupled electroelastic system of a nonlinear electrostatic equation with an elastic equation enriched with electrostriction. Such coupling is employed to describe dielectric elastomers or deformable (elastic) dielectrics. It is shown that the effective response of the system consists of a homogeneous dielectric elastomer described by a nonlinear weakly coupled system of PDEs whose coefficients depend on the coefficients of the original heterogeneous material, the geometry of the composite and the periodicity of the original microstructure. The approach developed here for this nonlinear problem allows obtaining an explicit corrector result for the homogenization of monotone operators with minimal regularity assumptions. Two $L^p-$gradient estimates for elastic systems with discontinuous coefficients are also obtained.Comment: We provide a new proof to extend the explicit first-order corrector result in the first version of this paper. The new explicit corrector result (cf. Theorem 1) holds globally and unifies the previous classical correct results in homogenization of the divergence equation (both linear and nonlinear). New references are added. Comments are welcome

Asymptotic analysis of an array of closely spaced absolutely conductive inclusions

We consider the conductivity problem in an array structure with square closely spaced absolutely conductive inclusions of the high concentra- tion, i.e. the concentration of inclusions is assumed to be close to 1. The problem depends on two small parameters: ", the ratio of the period of the micro-structure to the characteristic macroscopic size, and  , the ratio of the thickness of the strips of the array structure and the period of the micro- structure. The complete asymptotic expansion of the solution to problem is constructed and justified