A general theory of nonlocal elasticity based on nonlocal gradients and connections with Eringen's model

TL;DR

This paper presents a comprehensive mathematical framework for nonlocal linear elasticity using nonlocal gradients, connecting it with Eringen's model and demonstrating convergence to classical elasticity under certain limits.

Contribution

It introduces a general nonlocal elasticity theory based on nonlocal gradients, establishes existence and uniqueness of solutions, and links it explicitly to Eringen's model.

Findings

The framework includes Eringen's model as a special case.

Existence and uniqueness of solutions are proven for various boundary conditions.

Solutions converge to classical elasticity as the interaction horizon vanishes or fractional parameter approaches one.

Abstract

We develop a general theory of nonlocal linear elasticity based on nonlocal gradients with general radial kernels. Starting from a nonlocal hyperelastic energy functional, we perform a formal linearization around the identity deformation to obtain a system of nonlocal linear elasticity equations. We establish the existence and uniqueness of weak solutions for both Dirichlet and Neumann boundary conditions, proving a general Korn-type inequality for nonlocal gradients. We show that this framework encompasses Eringen's nonlocal elasticity model as a particular case, establishing an explicit connection between the two formulations. Finally, we prove localization results demonstrating that solutions to the nonlocal problems converge to their classical local counterparts in two different regimes: as the interaction horizon vanishes and, in the fractional case, as the fractional parameter approaches one. These results provide a comprehensive and unified mathematical foundation for nonlocal elasticity theories.