Elastoplastic microscopic bifurcation and post-bifurcation behavior of periodic cellular solids

In this study, a general framework is developed to analyze microscopic bifurcation and post-bifurcation behavior of elastoplastic, periodic cellular solids. The framework is built on the basis of a two-scale theory, called a homogenization theory, of the updated Lagrangian type. We thus derive the eigenmode problem of microscopic bifurcation and the orthogonality to be satisfied by the eigenmodes. The orthogonality allows the macroscopic increments to be independent of the eigenmodes, resulting in a simple procedure of the elastoplastic post-bifurcation analysis based on the notion of comparison solids. By use of this framework, then, bifurcation and post-bifurcation analysis are performed for cell aggregates of an elastoplastic honeycomb subject to in-plane compression. Thus, demonstrating a basic, long-wave eigenmode of microscopic bifurcation under uniaxial compression, it is shown that the eigenmode has the longitudinal component dominant to the transverse component and consequently causes microscopic buckling to localize in a cell row perpendicular to the loading axis. It is further shown that under equi-biaxial compression, the flower-like buckling mode having occurred in a macroscopically stable state changes into an asymmetric, long-wave mode due to the sextuple bifurcation in a macroscopically unstable state, leading to the localization of microscopic buckling in deltaic areas.