Multiscale analysis of non-periodic stress state in composites with periodic microstructure

The size of a fine scale domain to be employed for accurate evaluation of a strongly non-periodic stress distribution in composites with periodic microstructure can be relatively large. Consequently, direct numerical simulation approach may become impractical. A method suggested to overcome this difficulty hinges on the Discrete Fourier Transform. It allows to determine a non-periodic stress field in the fine scale domain that includes many periodic cells by analysis of a single representative (repetitive) cell with Bloch type boundary conditions. The analysis is carried out with respect to the complex-valued displacements and stress transforms, and actual values of the stress strain field components at any point of the fine scale domain are given by the inverse transformation. The method is illustrated by the analysis of crack problems for the two-dimensional periodically voided material.