Stability of nonlocal Dirichlet integrals and implications for peridynamic correspondence material modeling

TL;DR

This paper investigates the stability of nonlocal Dirichlet integrals, revealing that their coercivity depends on the choice of interaction kernels, which impacts the modeling of nonlocal materials like peridynamics.

Contribution

It extends analysis of nonlocal gradient operators by identifying conditions under which the associated energy functional is stable, challenging previous assumptions.

Findings

Stability depends on the choice of nonlocal kernels.

Some kernels ensure coercivity, others do not.

Implications for nonlocal material modeling, especially peridynamics.

Abstract

Nonlocal gradient operators are basic elements of nonlocal vector calculus that play important roles in nonlocal modeling and analysis. In this work, we extend earlier analysis on nonlocal gradient operators. In particular, we study a nonlocal Dirichlet integral that is given by a quadratic energy functional based on nonlocal gradients. Our main finding, which differs from claims made in previous studies, is that the coercivity and stability of this nonlocal continuum energy functional may hold for some properly chosen nonlocal interaction kernels but may fail for some other ones. This can be significant for possible applications of nonlocal gradient operators in various nonlocal models. In particular, we discuss some important implications for the peridynamic correspondence material models.