Fracture analysis of cellular materials: A strain gradient model

Abstract

A generalized continuum model is developed for cellular materials based on the equivalence of strain energy at the macro- and microscale. It is rather similar to the strain gradient theory, but has a well-defined characteristic length, namely, the cell size. The continuum model enables one to use powerful analytical methods to investigate fracture of cellular materials. The near-tip asymptotic fields and full-field solutions are obtained for cellular materials with hexagonal, triangular, or square lattice. Using the same strain-energy equivalence at the macro- and microscale, displacements and rotation of discrete cell walls are estimated from the continuum near-tip asymptotic fields. By postulating a maximum-tensile-stress failure criterion of cell walls, the fracture toughness of cellular materials is estimated to be proportional to the thickness h of cell walls and inversely proportional to √L, where L is the cell size. Moreover, the mixed-mode fracture toughness can be simply obtained from the fracture toughness in pure mode 1 and mode II, once the mode mixity is known. It is established that, with the same mass density, the hexagonal or triangular lattice in a cellular material can provide much higher fracture toughness than the square lattice.