Microscopic Bifurcation and Macroscopic Localization in Periodic Cellular Solids: Elastoplastic Analysis Based on a Homogenization Theory

• Published in: Transactions of the Japan Society of Mechanical Engineers Series A Vol. 69; no. 686; pp. 1421 - 1428
• Main Authors: OKUMURA, Dai, OHNO, Nobutada, NOGUCHI, Hirohisa
• Format: Journal Article
• Language: Japanese
• Published: The Japan Society of Mechanical Engineers 2003
• Subjects: Bifurcation; Buckling; Finite Deformation Theory; Homogenization Theory; Periodic Material; Plasticity
• ISSN: 0387-5008; 1884-8338
• DOI: 10.1299/kikaia.69.1421

In this work, the microscopic bifurcation leading to macroscopic localization in elastoplastic, periodic cellular solids is studied. To this end, a general framework of microscopic bifurcation analysis is established on the basis of the up-dated Lagrangian type homogenization theory developed by the present authors. We thus derive the boundary value problem to find microscopic eigenmodes and the orthogonality indicating no influence of the eigenmodes on the macroscopic variation at the onset of microscopic bifurcation. Then, using a substructure-based finite element method, bifurcation analysis and post-bifurcation analysis are performed for cell aggregates of an elastoplastic honeycomb subject to in-plane uniaxial compression. It is shown that in macroscopically unstable states, microscopic bifurcation with multiplicity occurs due to the Bloch wave nature, if the periodic cell number is not small. It is also shown that the Bloch wave consists of longitudinal and transverse components: only the longitudinal one grows to the macroscopic localization of compression type in a cell row perpendicular to the loading axis.