Microscopic Buckling Analysis of Cellular Solids Based on a Homogenization Theory of Finite Deformation

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The microscopic symmetric bifurcation buckling of cellular solids subjected to macroscopically uniform compression was studied. To begin with, describing the principle of virtual work for infinite periodic materials in the updated Lagrangian form, we built a homogenization theory of finite deformation, which satisfies the principle of material objectivity. Then, we stated a postulate that at the onset of microscopic symmetric bifurcation, microscopic velocity becomes spontaneous, yet changing the sign of such spontaneous velocity has no influence on the variation in macroscopic states. By applying the postulate to the homogenization theory, we derived the conditions to be satisfied at the onset of microscopic symmetric bifurcation. The homogenization theory and buckling conditions established were employed to analyze the in-plane biaxial buckling of an elastic hexagonal honeycomb. We thus found the following : Simple, double and triple bifurcations can take place, if the largest compressive load is transmitted in one, two and three directions of cell walls, respectively. At the double and triple bifurcation points, uniaxial buckling modes develop simultaneously in the two and three directions of cell walls and are linearly combined to generate a biaxial mode, Mode II, reported by Gibson and Ashby (1997) and a flower-like mode, Mode III, observed by Papka and Kyriakides (1999a). In other words, these biaxial modes result from the multiplicity of bifurcation, so that they have very complex cell-patterns in comparison with the uniaxial buckling mode, Mode I, occurring at the simple bifurcation points. Moreover, we showed that Modes II and III, as well as Mode I, are classified as microscopic symmetric bifurcation in spite of their very complex cell-patterns, since they are not macroscopically influenced by changing the sign of spontaneous perturbed velocity as described in the postulate.