Finite element approximation of nonlocal fracture models

Abstract

In this talk, we present our new results on the numerical analysis of nonlocal fracture models. We begin by giving a brief introduction to the Peridynamic theory and the nonlocal potentials considered in our work. We consider a force interaction characterized by a double well potential. Here, one well, near zero strain, corresponds to the linear response of a material, and the other well, for large strain, corresponds to the softening of a material. We show the existence of a regularized model with evolving displacement field in either Hölder space or Sobolev space. Assuming exact solutions in Hölder space, we obtain apriori error estimates due to finite difference approximation. We show that the error converges to zero, uniformly in time, in the mean square norm. The rate depends on the nonlocal length scale and is proportional to O(Delta t + h^gamma / epsilon^2). Here h is the size of mesh, epsilon is the nonlocal length scale, Delta t is the size of time step, and gamma in (0,1] is the Hölder exponent. C is the constant independent of mesh size and size of time step and may depend on nonlocal length scale through the norm of the exact solution. We also study the finite element approximation and show that the error uniformly converges to zero at the rate O(Delta t + h^2/epsilon^2). We consider piecewise linear continuous elements. Theoretical claims are supported by numerical results. This is a joint work with Dr. Robert Lipton and is funded by the US Army Research Office under grant/award number W911NF1610456.