The problem of brittle fracture of a 2D periodic porous material is considered. For the considered relative density range the cellular material approximation with beam elements becomes invalid and a novel analysis technique based on the discrete Fourier transform is suggested. Mode I fracture toughness is derived from the continuum analysis of the stress state in a perforated elastic plane with a macroscopic crack composed of a number of microcracks between the neighboring voids. The crack length is proved to be sufficiently large such that the K-field of the linear elastic fracture mechanics takes place. The solution is obtained in the form of superposition of weighted Green functions every one of which corresponds to a unit displacement jump at a single microcrack. The Green function solutions are found by a combination of the representative cell technique and the finite element method. As a result the initial problem for infinite plane is solved by multiple analysis of the repetitive periodicity cell. 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 the limiting case of low density the results are compared with known data for cellular materials.