Solvability of nonlocal systems related to peridynamics

TL;DR

This paper investigates the solvability and regularity of a nonlocal elasticity system derived from peridynamics, establishing well-posedness and optimal Sobolev regularity using Hilbert space methods.

Contribution

It proves well-posedness and regularity results for a nonlocal coupled system from peridynamics, extending understanding of such systems with non-symmetric and symmetric kernels.

Findings

Proved well-posedness of the nonlocal system.

Demonstrated optimal local Sobolev regularity of solutions.

Identified energy spaces coinciding with fractional Sobolev spaces for specific kernels.

Abstract

In this work, we study the Dirichlet problem associated with a strongly coupled system of nonlocal equations. The system of equations comes from a linearization of a model of peridynamics, a nonlocal model of elasticity. It is a nonlocal analogue of the Navier-Lam\'e system of classical elasticity. The leading operator is an integro-differential operator characterized by a distinctive matrix kernel which is used to couple differences of components of a vector field. The paper's main contributions are proving well-posedness of the system of equations and demonstrating optimal local Sobolev regularity of solutions. We apply Hilbert space techniques for well-posedness. The result holds for systems associated with kernels that give rise to non-symmetric bilinear forms. The regularity result holds for systems with symmetric kernels that may be supported only on a cone. For some specific kernels associated energy spaces are shown to coincide with standard fractional Sobolev spaces.