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

J. C. Bellido, G. García-Sáez

TL;DR

This work develops a comprehensive, unified framework for nonlocal elasticity based on general radial nonlocal gradients $D_ ho$. By linearizing a nonlocal hyperelastic energy around the identity, it yields a nonlocal Navier–Lamé-type system and proves a Korn-type inequality to secure well-posedness under Dirichlet and Neumann conditions. A key result is that Eringen's model emerges as a special case when the nonlocal interaction kernel is chosen so that $ ilde{A}(|x|)=Q_ ho*Q_ ho$, linking gradient-based and stress-filtering formulations. The authors also establish localization via $oldsymbol{ extGamma}$-convergence in two regimes—the vanishing horizon and the fractional-to-local limit—thereby showing convergence to classical local elasticity or to fractional models, respectively, for both Dirichlet and Neumann problems. Collectively, the paper provides a rigorous mathematical foundation and a unifying perspective for nonlocal elasticity theories across general kernels, including connections to fractional and Eringen-type models with robust localization results that support model reduction to classical elasticity in appropriate limits.

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.

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

TL;DR

This work develops a comprehensive, unified framework for nonlocal elasticity based on general radial nonlocal gradients . By linearizing a nonlocal hyperelastic energy around the identity, it yields a nonlocal Navier–Lamé-type system and proves a Korn-type inequality to secure well-posedness under Dirichlet and Neumann conditions. A key result is that Eringen's model emerges as a special case when the nonlocal interaction kernel is chosen so that , linking gradient-based and stress-filtering formulations. The authors also establish localization via -convergence in two regimes—the vanishing horizon and the fractional-to-local limit—thereby showing convergence to classical local elasticity or to fractional models, respectively, for both Dirichlet and Neumann problems. Collectively, the paper provides a rigorous mathematical foundation and a unifying perspective for nonlocal elasticity theories across general kernels, including connections to fractional and Eringen-type models with robust localization results that support model reduction to classical elasticity in appropriate limits.

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.
Paper Structure (17 sections, 22 theorems, 163 equations)

This paper contains 17 sections, 22 theorems, 163 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Nonlocal gradients
  5. Definitions and nonlocal calculus
  6. Nonlocal Sobolev spaces
  7. Poincaré-Wirtinger inequality and Neumann boundary conditions
  8. A general model via linearization
  9. Existence and uniqueness of solutions
  10. Dirichlet boundary conditions for general nonlocal gradients
  11. Neumann boundary conditions for $\rho=\rho^s_\delta$
  12. Connection with Eringen's model
  13. Localization results
  14. Localization for the Dirichet problem
  15. Homogenization for general gradients
  16. ...and 2 more sections

Key Result

Lemma 2.3

Let $u\in C_c^\infty({\mathbb R}^n)$ and $v\in C_c^\infty({\mathbb R}^n;{\mathbb R}^n)$. Then,

Theorems & Definitions (26)

  • Definition 2.1: Nonlocal gradient
  • Definition 2.2
  • Lemma 2.3: Integration by parts
  • Theorem 2.4: General nonlocal fundamental theorem of calculus
  • Theorem 2.5
  • Theorem 2.6
  • Lemma 2.7: From nonlocal to local
  • Lemma 2.8: From local to nonlocal
  • Theorem 2.9: Poincaré inequality
  • Theorem 2.10
  • ...and 16 more