Microscopic symmetric bifurcation analysis for cellular solids based on a homogenization theory of finite deformation (1st report, theory)

Abstract

In this paper, we analyze the microscopic symmetric bifurcation buckling of cellular solids subjected to macroscopically uniform compression. To this end, showing the principle of virtual work for periodic solids in the updated Lagrangian form, we build a homogenization theory of finite deformation, which satisfies the principle of material objectivity. Then, we state the following postulate: At a microscopic symmetric bifurcation point, microscopic displacement rate gets spontaneous, but changing the sign of the spontaneous displacement rate field has no influence on the variation of macroscopic states. By applying this postulate to the homogenization theory, we derive the conditions to be satisfied at the bifurcation point. The resulting conditions are discretized using a finite element method in order to employ the present theory in computational analysis. The finite element discretization also deals with a general case, which includes microscopic non-symmetric bifurcation as well as symmetric one.