Negligible microstructure in a class of porous nonlinear materials via variational bounds

Abstract

Communicated by I. SevostianovThis work is devoted to the estimation of the effective energy density of porous nonlinear materials by means of variational bounds. Because of the unavailability of an improved upper bound, attention is turned to the improved bound obtained by considering constant nonzero polarization fields in the formal lower bound, which follows from the generalization to nonlinear behavior of the Hashin-Shtrikman variational principles. A particular class of nonlinearity is considered which is relevant in different physical situations. Also, 11 different microstructures are considered. Several computational experiments performed show that the elementary upper bound and the improved lower bound are indistinguishable, suggesting that nonlinearity dominates over the microstructural effects. In other words, at least for the nonlinearity considered here, the influence of microstructure on the effective behavior is negligible. So, in this case, there is no need to search for an improved upper bound, as the elementary one provides a simple but accurate estimate of the effective energy density.