Homogenization and field concentrations in heterogeneous media

Abstract

A multiscale characterization of the field concentrations inside composite and polycrystalline media is developed. We focus on gradient fields associated with the intensive quantities given by the temperature and the electric potential. In the linear regime these quantities are modeled by the solution of a second order elliptic partial differential equation with oscillatory coefficients. The characteristic length scale of the heterogeneity relative to the sample size is denoted by ε and the intensive quantity is denoted by u ε. Field concentrations are measured using the L p norm of the gradient field ||∇u ε||L p(D) for 2 ≤ p \u3c ∞. The analysis focuses on the case when 0 \u3c ε ≪ 1. Explicit lower bounds on lim inf ε→0 are developed. These bounds provide a way to rigorously assess field concentrations generated by the microgeometry without having to compute the actual field u ε. © 2006 Society for Industrial and Applied Mathematics