The plane problem of crack resistance in elastic materials with doubly periodic system of voids is investigated. The relatively high fractional density (medium and low porosity) of above 0.6 which is typical for the partially-sintered materials is addressed. The stress field in the voided plane with embedded Mode I crack of arbitrary length bridging the voids is evaluated. In the framework of stress criterion of fracture this field allows the determination of the fracture toughness of the considered material with periodic microstructure in terms of the solid (parent) material tensile strength.
The mathematical modeling of this problem is complicated due to the fact that the ''ligaments'' between the voids are so thick that they cannot be treated as beams. As a result, theoretical methods used to evaluate the fracture toughness of low-density cellular materials are not applicable. Accordingly, the present problem is solved by means of a novel analysis which is based on the combined use of the representative cell method, based on the discrete Fourier transform, in conjunction with a higher-order theory and a micromechanical model. These combined three approaches allow the performance of an accurate continuum mechanics analysis of the highly non-uniform stress field in the voided material with a crack.
The analysis of this field for different crack lengths has shown which length must be employed in the fracture toughness evaluation. The dependence of the fracture toughness upon the relative density is examined and found to vary linearly in the majority of the considered density range. This result agrees with the known experimental observations and other theoretical predictions.