Micropolar effects on the effective elastic properties and elastic fracture toughness of planar lattices

Effective elastic properties, and mode I elastic fracture toughness of three isotropic planar lattices - hexagonal, kagome, and triangular - are studied from a micropolar continuum perspective. The hexagonal lattice is bending dominated whereas the kagome and the triangular are stretching dominated for any prescribed macroscopic in-plane loading. Discrete asymptotic homogenization is applied to obtain analytical expressions for the topology governed effective micropolar elastic properties of each lattice and their scaling with relative density: the hexagonal lattice is found to possess the largest micropolar internal length parameter at any given relative density. The homogenized micropolar continuum is then discretized by a four-node finite element model to compute its Cauchy and couple-stress intensities for a central crack under mode I loading. Micropolar scaling exponents for mode I fracture toughness with relative density agree well with the estimates from the literature based on fully discrete beam network models. Hexagonal lattice exhibits an order of magnitude higher couple stress intensity factor when compared to the triangular and the kagome lattices of identical relative density, signifying the importance of micropolar effects in bending dominated architectures.