Large-scale Multiscale Analysis for Microscopic Buckling and Macroscopic Instability of Periodic Cellular Solids Based on a Homogenization Theory

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In this study, first, a general framework was developed to analyze microscopic bifurcation and post-bifurcation behavior of periodic cellular solids. The framework was built on the basis of a two-scale theory, called a homogenization theory, of the updated Lagrangian type. The eigenmode problem of microscopic bifurcation and the orthogonality to be satisfied by the eigenmodes were thus derived. It was shown that 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.Second, by use of the framework mentioned above, bifurcation and post-bifurcation analysis were performed for cell aggregates of an elastoplastic hexagonal honeycomb subject to in-plane compression. Thus, demonstrating a basic, long-wave eigenmode of microscopic bifurcation under uniaxial compression, it was shown that the eigenmode has the longitudinal component dominant … More to the transverse component and consequently causes microscopic buckling to localize in a cell row perpendicular to the loading axis. It was 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.Third, long-wave and short-wave buckling of elastic square honeycombs subject to in-plane biaxial compression were analyzed using the two-scale theory. By taking cell aggregates to be periodic units, the bifurcation and post-bifurcation behavior were analyzed to discuss the dependence of buckling stress on periodic length. It was shown that buckling stress decreases as periodic length increases, and that very-long-wave buckling occurs just after the onset of macroscopic instability if the periodic length is sufficiently long. Then, a simple formula to evaluate the very-long-wave buckling stress under in-plane biaxial compression was derived by exploring the macroscopic instability condition in the light of the two-scale analysis. The resulting formula was verified using an energy method. Less