Improved bounds for the effective energy of nonlinear 3D conducting composites

Abstract

Recent variational inequalities of Talbot are used to improve the lower % and upper bounds for the effective energy of nonlinear 3-D two-phase conducting composites. The effective conductivity of the linear isotropic two-phase periodic conducting composite used as comparison material in the inequalities is computed through an asymptotic homogenization model by finite element analysis of the local problem on the three-dimensional cubic unit cell with one spherical inclusion. A brief mathematical description of the numerical method is included. Numerical calculations of the effective conducting linear property are compared with % Bruno's bounds. It shows that the numerical solution for the limit cases of superconducting % and empty inclusions improves the bounds when the inclusion volume fraction is greater than about $0.4$. It is natural to expect an improvement in the whole volume % fraction of Talbot's bounds for nonlinear % conducting composites when the numerical calculation is used instead of % bounds for the linear comparison problem, as is the case here.