On the optimality of the variational linear comparison bounds for porous viscoplastic materials

Abstract

This paper is concerned with the optimality of the variational bounds of the Hashin-Shtrikman type (VHS) for nonlinear composites, first obtained by Ponte Castañeda (1991a) by means of the corresponding HS bounds for suitably optimized linear comparison composites (LCCs). For simplicity, this problem is addressed in the context of porous viscoplastic materials with incompressible, isotropic matrix phase and two-dimensional microstructures, subjected to plane-strain loading conditions. Although special, this case—exhibiting infinite heterogeneity contrast and compressible macroscopic response—is expected to be fully representative of more general three-dimensional porous materials, as well as more general two-phase, well-ordered composites. Thus, it is shown that the VHS bounds, which were originally derived for the class of nonlinear composites with statistically isotropic microstructures, are in fact attained over the larger class of microstructures with anisotropic nonlinear response but isotropic linear response. By appealing to an exact variational representation for the effective potential of finite-rank nonlinear laminates, it is shown that there exist certain values of the applied macroscopic stress for which the finite-rank laminate (closed-cell porous) microstructures attaining the linear HS bounds also attain the nonlinear VHS bound. Explicit results are obtained in the ideally plastic limit for the yield surface of the finite-rank laminates attaining the VHS bound. In particular, the results of the paper highlight the fact that bounds for nonlinear composites are much more sensitive to microstructural details than bounds for linear composites.