The Role of the Horizon in Modeling Failure due to Strain and Damage Localization with Peridynamics

This paper investigates the effect of the horizon size on failure due to strain and damage localization in the case where peridynamics is a nonlocal theory by its own, which corresponds to most bond-based peridynamics models. Two constitutive relationships are discussed: the microelastic brittle model and a progressive damage model. The usual practice with the microelastic brittle model is to fit the microelastic constant for a given horizon size so that elasticity is recovered. At the same time, the fracture energy provides the critical bond stretch. This methodology yields an indirect determination of the tensile strength of the material, that goes to infinity as the horizon size trends to zero. With the damage model, the stretch at the inception of damage can be obtained from the tensile strength. Then, a simple one-dimensional case of wave propagation and interactions in a bar is considered. For fixed values of the horizon, convergence with a refinement of the discretization is checked. The energy dissipated upon fracture is found to be a linear function of the horizon. It is also a function of the softening response. The horizon cannot be chosen arbitrarily, unless the softening parameter is adjusted to fit the fracture energy, like in the crack band model. Surprisingly, such a methodology is very seldom mentioned in the current literature dealing with fracture modeled by peridynamics.