Non-local elasticity theory as a continuous limit of 3D networks of pointwise interacting masses

TL;DR

This paper develops a nonlocal elasticity model as a continuum limit of a large network of point masses with interactions, providing a bridge between discrete particle systems and continuous elastic media.

Contribution

It introduces a nonlocal elasticity theory derived as the asymptotic limit of 3D particle networks with pointwise interactions, extending classical elasticity models.

Findings

Derivation of a nonlocal homogenized system of equations

Asymptotic analysis of particle systems as the number of particles increases

Establishment of a continuum limit for nonlocal elastic interactions

Abstract

Small oscillations of an elastic system of point masses (particles) with a nonlocal interaction are considered. We study the asymptotic behavior of the system, when number of particles tends to infinity, and the distances between them and the forces of interaction tends to zero. The first term of the asymptotic is described by the homogenized system of equations, which is a nonlocal model of oscillations of elastic medium.