Fracture toughness of self-similar hierarchical material

Abstract

Two-dimensional materials with self-similar second-order hierarchy are considered. The microstructure with fourfold cubic symmetry is generated by a periodic system of voids in brittle parent material, in one case, macrovoids are filled by a microvoided phase, and in the second one, this phase surrounds hollow macrovoids. A crack is embedded in a rectangular periodic domain subjected to the K-field boundary conditions, and the fracture toughness is determined by the analysis of stresses in the crack-tip vicinity. A novel approach based on the discrete Fourier transform reduces the computational cost of the problem. The analysis of the domain, consisting of several hundreds repeating patterns, is reduced to the multiple analysis of a single representative cell in the Fourier transform space.A parametric study is carried out and, in particular, the influence of parent material redistribution between the hierarchical levels is examined. It is found that for the material with filled macrovoids increasing the relative stiffness at the second hierarchical level diminishes the fracture toughness, while for the second layout the effect is opposite. A comparison with non-hierarchical voided materials of the same relative density emphasizes the role of the length scale parameter of a layout.