Numerical convergence of finite difference approximations for state based peridynamic fracture models

Abstract

In this work, we study the finite difference approximation for a class of nonlocal fracture models. The nonlocal model is initially elastic but beyond a critical strain the material softens with increasing strain. This model is formulated as a state-based peridynamic model using two potentials: one associated with hydrostatic strain and the other associated with tensile strain. We show that the dynamic evolution is well-posed in the space of Hölder continuous functions C0,γ with Hölder exponent γ∈(0,1]. Here the length scale of nonlocality is ϵ, the size of time step is Δt and the mesh size is h. The finite difference approximations are seen to converge to the Hölder solution at the rate CtΔt+Cshγ∕ϵ2 where the constants Ct and Cs are independent of the discretization. The semi-discrete approximations are found to be stable with time. We present numerical simulations for crack propagation that computationally verify the theoretically predicted convergence rate. We also present numerical simulations for crack propagation in pre-cracked samples subject to a bending load.