@Inbook{Jha2019wellposedbook,
author="Jha, Prashant K.
and Lipton, Robert",
editor="Voyiadjis, George Z.",
title="Well-Posed Nonlinear Nonlocal Fracture Models Associated with Double-Well Potentials",
bookTitle="Handbook of Nonlocal Continuum Mechanics for Materials and Structures",
year="2019",
publisher="Springer International Publishing",
address="Cham",
pages="1417--1456",
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 H{\"o}lder spaces and Sobolev spaces. The motivation for the H{\"o}lder 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.",
isbn="978-3-319-58729-5",
doi="10.1007/978-3-319-58729-5_40",
url="https://doi.org/10.1007/978-3-319-58729-5_40"
}