巨視的材料不安定性のマルチスケールモデリングにおける微視的構造周期のブロック対角化法による同定

Abstract

By employing the block-diagonalization method in the group-theoretic bifurcation theory, we develop a general procedure to estimate the number of periodic microstructures (cells) that should be contained in a representative volume element, in which the microstructural analysis in the nonlinear multi-scale homogenization analysis is performed. Since the problem of interest essentially involves the non-convexity of the total potential energy, we first provide some remarks on the multi-scale modeling strategy within the framework of the nonlinear homogenization theory. Next, knowing that the number of cells restricts possible bifurcation modes, we perform the bifurcation analysis for periodic microstructures by means of the block-diagonalization method. To demonstrate the proposed procedure, several numerical analyses are conducted on a cellular solid, which is one of the typical examples whose microstructural instability causes the macroscopic material instability. The proposed method can readily be applied to the characterization of the mechanical behaviors caused by micro-macro-coupled instability.