Elastic and fracture properties of the two-dimensional triangular and square lattices

Lattice models are finding increasing use in modeling the elastic and fracture behaviour of inhomogeneous or multi-phase systems. The elastic and failure properties of a rotationally invariant formulation of the bond-bending model on the two-dimensional triangular (with first-neighbour couplings) and on a novel version of the square lattice (involving first- and second-neighbour couplings) are examined. Expressions for the elastic constants of the bond-bending model on the above mentioned lattices are given in terms of the two- and three-body force constants. The tensile failure surface of the bond-bending model on the triangular and square lattices is calculated, and displays some degree of anisotropy in both cases. The central-force model (with a zero bond-bending constant) on the triangular lattice shows the highest degree of anisotropy, namely 50%. The presence of the bond-bending coupling constant improves the degree of isotropy of the tensile failure surface for both the triangular and square lattices. It is therefore concluded that the role of the bond-bending coupling constant is to provide for a more uniform energy distribution among the bonds, so that preferential bond cleavage does not occur upon the application of stress.