Limit and shakedown analysis of periodic heterogeneous media

The paper deals with heterogeneous media with periodic arrangement of the inclusions (fiber-reinforced composites, perforated plates) and ductile components, subjected to either monotonic or variable repeated loading. The fundamentals of the theoretical static and kinematic approaches to the determination of the macroscopic yield strength of the equivalent homogenized material are reviewed. Outlined are also numerical methods that give lower and upper bounds for the collapse multiplier of the representative volume (RV) based on linear or nonlinear programming. Then such methods are extended so as to cover the case of the shakedown analysis of RVs subjected to macroscopic stresses varying within a given domain. Numerical examples prove the effectiveness of the proposed algorithms through comparisons with both analytical and experimental available results.