Application of the asymptotic homogenization method for composites with irregular material configurations

The asymptotic homogenization method is applied to composites whose microstructural configurations reveal irregularity. Although the method usually requires periodic boundary conditions for microstructures, i.e., volume elements (RVE), actual composite materials do not have geometrical periodicity when arbitrary regions are taken as RVEs. After reviewing the theory of the mathematical homogenization method, we show that the homogenization for periodic media be appled to arbitrary statistically homogeneous media. Using several sizes of unit cells in numerical analyses, the convergence trends of macroscopic and microscopic variables are considered, and the theoretical results are justified. Also, we try to show how to determine the RVE as a unit cell for a particular type of composites. For this purpose, the systematic modeling technique utilizing digitized images is extensively utilized to properly model the irregular material configurations for the homogenization analyses and to take into account the actual microstructural geometry in evaluating the micro and macroscopic variables. Furthermore, we briefly discuss the image-based evaluation of the results in the linear and nonlinear homogenization.