Numerical Analysis of Nonlocal Fracture Models

Abstract

In this work, 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 H&oumlder space as it allows fields which do not have well-defined derivatives. We show that O(Delta t + h^2 / epsilon^2) is the rate of convergence when the solution is in C^2 and O(Delta t + h^gamma / epsilon^2) when the solution is in H&oumlder space. h is the size of mesh, epsilon is the size of the horizon, and gamma in (0,1] is the H&oumlder 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. We also consider the model in one dimension. We show that the solution of peridynamic model converges to the solution of elastodynamic. The same is shown for the approximation of peridynamic model. We present some numerical results to verify the claims. This is a joint work with Dr. Robert Lipton.

Date
Jul 19, 2017 12:00 AM
Event
USNCCM14 2017
Location
Montreal, Canada
Prashant K. Jha
Prashant K. Jha
Research Associate

My research is driven by the application of mathematics and computational science to present-day relevant and challeng- ing problems. Specific areas of interest include mechanics of solids and granular media, computational oncology, and multiscale modeling.