Nonlocal strong forms of thin plate, gradient elasticity, magneto-electro-elasticity and phase field fracture by nonlocal operator method

TL;DR

This paper introduces a unified nonlocal operator method to derive nonlocal strong forms for various physical models, simplifying the process and enabling efficient conversion from local to nonlocal formulations, with applications in elasticity, thin plates, and fracture modeling.

Contribution

It presents a simple, general variational approach using nonlocal operators to derive nonlocal forms for multiple physical problems, including elasticity and fracture.

Findings

Nonlocal elasticity form matches dual-horizon non-ordinary state-based peridynamics.

The method efficiently converts local models into nonlocal forms.

Numerical examples validate the nonlocal elasticity and thin plate models.

Abstract

The derivation of nonlocal strong forms for many physical problems remains cumbersome in traditional methods. In this paper, we apply the variational principle/weighted residual method based on nonlocal operator method for the derivation of nonlocal forms for elasticity, thin plate, gradient elasticity, electro-magneto-elasticity and phase field fracture method. The nonlocal governing equations are expressed as integral form on support and dual-support. The first example shows that the nonlocal elasticity has the same form as dual-horizon non-ordinary state-based peridynamics. The derivation is simple and general and it can convert efficiently many local physical models into their corresponding nonlocal forms. In addition, a criterion based on the instability of the nonlocal gradient is proposed for the fracture modelling in linear elasticity. Several numerical examples are presented to validate nonlocal elasticity and the nonlocal thin plate .