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

TL;DR

This paper analyzes the convergence of finite difference methods applied to a nonlocal peridynamic fracture model, demonstrating stability and convergence rates through theoretical analysis and numerical simulations.

Contribution

It establishes the convergence rate of finite difference approximations for a state-based peridynamic fracture model and verifies it with numerical simulations.

Findings

Convergence rate of $C_t \, \Delta t + C_s h^\gamma/\epsilon^2$ for the finite difference scheme.

Semi-discrete approximations are stable over time.

Numerical simulations confirm the theoretical convergence rate.

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 perydynamic 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 H\"older continuous functions $C^{0,\gamma}$ with H\"older exponent $\gamma \in (0,1]$. Here the length scale of nonlocality is $\epsilon$, the size of time step is $\Delta t$ and the mesh size is $h$. The finite difference approximations are seen to converge to the H\"older solution at the rate $C_t \Delta t + C_s h^\gamma/\epsilon^2$ where the constants $C_t$ and $C_s$ 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 precracked samples subject to a bending load.