A Generalized Nonlocal Calculus with Application to the Peridynamics Model for Solid Mechanics

TL;DR

This paper develops a generalized nonlocal vector calculus framework, extending previous models, and applies it to reformulate the peridynamics equation of motion, demonstrating its broad applicability and potential for new nonlocal models.

Contribution

It introduces a general nonlocal calculus independent of specific models, unifies existing formulations, and demonstrates its application to peridynamics and potential for new nonlocal operators.

Findings

Unified nonlocal calculus operators as integral kernels

Reformulation of peridynamics using the new calculus

Introduction of new nonlocal operators for broader models

Abstract

A nonlocal vector calculus was introduced in [2] that has proved useful for the analysis of the peridynamics model of nonlocal mechanics and nonlocal diffusion models. A generalization is developed that provides a more general setting for the nonlocal vector calculus that is independent of particular nonlocal models. It is shown that general nonlocal calculus operators are integral operators with specific integral kernels. General nonlocal calculus properties are developed, including nonlocal integration by parts formula and Green's identities. The nonlocal vector calculus introduced in [2] is shown to be recoverable from the general formulation as a special example. This special nonlocal vector calculus is used to reformulate the peridynamics equation of motion in terms of the nonlocal gradient operator and its adjoint. A new example of nonlocal vector calculus operators is introduced, which shows the potential use of the general formulation for general nonlocal models.