A Homogenization Method for Nonlinear Materials undergoing Large Deformation : 1st Report, Mathematical Formulations, Which Can Rigorously Satisfy the Assumption of Periodicity

• Published in: Transactions of the Japan Society of Mechanical Engineers Series A Vol. 64; no. 618; pp. 450 - 456
• Main Authors: MARUYAMA, Takuya, KUMAZAWA, Noriyoshi, OKADA, Hiroshi, FUKUI, Yasuyoshi
• Format: Journal Article
• Language: Japanese
• Published: The Japan Society of Mechanical Engineers 25.02.1998
• Subjects: Finite Element Method; Geometric Nonlinear Problem; Homogenization Method; Material Nonlincar Problem; Microstructure
• ISSN: 0387-5008; 1884-8338
• DOI: 10.1299/kikaia.64.450

The homogenization method has been known as a powerful tool to investigate the macroscopic material properties of composite materials, whose microstructures are periodic. This methodology can also be considered to be an advanced version of unit cell approach. However, the assumption of periodic structure becomes somewhat suspicious after the material has undergone a large deformation, because each unit cell also deforms finitely. In the context of homogenization method presented by Guedes and Kikuchi, the periodicity condition has to be satisfied rigorously. In this paper, a formulation which can rigorously satisfy the condition of periodicity even after the microstructure has undergone a finite deformation, is presented. It is accomplished by referring the initial undeformed configuration of microstructure, when we define the periodicity condition. Some simple numerical results of two-dimensional finite strain elasto-plastic problem are also presented and discussed. The outcome of the present research may find some application in analyzing Metal Matrix Composite Materials.