Elastic plastic shakedown of 3D periodic heterogeneous media: a direct numerical approach

A numerical direct method is presented for analyzing shakedown of three-dimensional heterogeneously periodic and elastic-perfectly plastic media. The main objective is to predict the long-term behavior of such media submitted to variable loads by coupling the periodic homogenization and shakedown theories. The macroscopic domains of external loads—the macroscopic strains or stresses—are obtained by solving the shakedown problem on a unit cell, representative of the heterogeneities. This coupled problem leads to a nonlinear constrained optimization problem. The developed method is tested on a layered material, on a perforated sheet, and on a material composed of spherical elastic inclusions. The results are compared with those of analytical and numerical analyses.