Numerical Analysis of Nonlocal Fracture Models

Abstract

In this talk, we present the numerical analysis of a bond-based peridynamic model. Model is nonlinear and is characterized by two wells. One well, near zero strain, correspond to the linear elastic interaction whereas second well corresponds to the softening of material. We present finite element analysis when the solution is in C^1 and C^2. We also analyze the finite difference approximation when the solution is in Hölder space. We consider Hölder space as it allows fields which do not have well-defined derivatives. We show O(h^2/epsilon^2) rate of convergence when the solution is in C^2 and O(h^gamma/epsilon^2) when the solution is in Hölder space. h is the size of mesh, epsilon is the size of the horizon, and gamma in (0,1] is the Hölder exponent. We show that the instability in the peridynamic model is due to softening of bonds and occurs when sufficiently large number of bonds have strain above critical strain.