Micro-macro modeling of nonlinear mechanical behaviors based on the homogenization theory

A multiscale modeling method based on the mathematical theory of homogenization is introduced for studying the micro-macro coupled mechanical behaviors. After describing a general nature of two-scale boundary value problem for nonlinear elasticity in terms of the generalized variational principle, the multiscale mathematical modeling for several microscopically defined mechanical behaviors are provided by using the generalized convergence theorems in the homogenization theory. With some numerical examples for each modeling, the mathematical structure of this modeling is interpreted in mechanics' points of view. That is, the macrostructure refers its material response to the corresponding microscopic mechanical behavior which defines microscopic boundary value problem. Due to the generality of this modeling method and the numerical algorithm, the micro -macro coupling effects of heterogeneous media can be analyzed to clarify the highly complex mechanisms in nonlinear or inelastic responses.