Microscopic symmetric bifurcation condition of cellular solids based on a homogenization theory of finite deformation
The presented analysis is motivated by recent observations on complex deformation modes of cellular materials such as foams or honeycombs. While buckling conditions describing macroscopic instabilities are well established, the authors claim to be the first to give a condition applicable to the microscopic buckling of periodic solids with geometrical symmetries. Starting from the principle of virtual work in an updated Lagrangian form, a homogenization theory (finite deformations and usual volume averaging) is considered which satisfies the principle of material objectivity and filters out the macroscopic spin. Central to the subsequent analysis is a postulate that ``at the onset of microscopic symmetric bifurcation, perturbed velocities become spontaneous, yet changing the sign of such perturbed velocities has no influence on the variation in macroscopic state.'' With this postulate, conditions to be satisfied at the onset of bifurcation are derived and are shown to have the similar or even the same forms as known from classical bifurcation analysis. The finite element discretization and subsequent numerical analysis are briefly described and illustrated on in-plane biaxial buckling of an elastic hexagonal honeycomb subject to macroscopic uniform compression. It is shown that three kinds of experimentally observed buckling modes of honeycombs (uniaxial/mode I, biaxial/mode II and flower-like/mode III) can be obtained by the presented analytical and numerical analysis. 14 figures illustrate the findings, and four short appendices elucidate details omitted for brevity in the main text.
scientific article; zbMATH DE number 1788084
scientific article