Fracture toughness of 2D low density periodic material

Straightlinear Mode I and Mode II cracks in elastic plane weakened by a double periodic system of voids are considered. A macrocrack is generated by a number of microcracks crossing the necks between neighboring voids. The fracture toughness is expressed in terms of the rupture modulus of bulk material. The stress state for the infinite plane with a finite-length macrocrack is determined by a new method based on the combination of the discrete Fourier transform and the finite element method. The initial problem for the infinite plane is reduced to multiple analysis of the repetitive periodicity cell with a single void. An optimization of the voids shape for fixed relative density is performed in order to design the material with improved fracture toughness. Several types of voids arrangement are considered. In particular, it was found, that the voids of optimal shape in the case of 4-fold symmetry (square voids arrangement) provide the largest Mode I fracture toughness while for the Mode II fracture toughness the case of 6-fold symmetry (hexagonal voids arrangement) is to be prefered.