Employing of the discrete Fourier transform for evaluation of crack-tip field in periodic materials

An approach for numerical evaluation of a self-similar crack-tip field for a long (semi-infinite) crack embedded in a material with periodic microstructure is suggested. The conditions at the boundaries of a rectangular domain around the tip are formulated by the use of K-field for the homogeneous material possessing effective elastic properties and then the finite discrete Fourier transform is applied. This allows to replace standard analysis of a large periodic domain with many cells by the analysis of a single repetitive cell in the transform space which can be carried out by any numerical method. Consequently, the volume of calculations in comparison with the standard approach is reduced and the problem of a macrocrack embedded in a material with fine microstructure can be addressed without simplifying assumptions. The accuracy of the proposed approach is verified by a comparison with the analytical solution for a crack embedded in a homogeneous plane. Application of the suggested method is given for a crack in a two-dimensional periodically voided material with triangular isotropic layout. The cell problem is resolved by the finite element method. The fracture toughness of the material in the framework of stress criterion for crack propagation is determined and its dependence upon the material relative density is investigated. A comparison of the fracture toughnesses of the solid and voided materials has shown for which parameter combinations voided ones will provide better crack resistance.