Well-posed nonlinear nonlocal fracture models associated with double-well potentials

Abstract

In this chapter, we consider a generic class of bond-based nonlocal nonlinear potentials and formulate the evolution over suitable function spaces. The peridynamic potential considered in this work is a differentiable version of the original bond-based model introduced in Silling (J Mech Phys Solids 48(1):175- 209, 2000). The potential associated with the model has two wells where one well corresponds to linear elastic behavior and the other corresponds to brittle fracture (see Lipton (J Elast 117(1):21-50, 2014; 124(2):143-191, 2016)). The parameters in the potential can be directly related to the elastic tensor and fracture toughness. In this chapter we show that well-posed formulations of the model can be developed over different function spaces. Here we will consider formulations posed over Hölder spaces and Sobolev spaces. The motivation for the Hölder space formulation is to show a priori convergence for the discrete finite difference method. The motivation for the Sobolev formulation is to show a priori convergence for the finite element method. In the following chapter we will show that the discrete approximations converge to well-posed evolutions. The associated convergence rates are given explicitly in terms of time step and the size of the spatial mesh.