Lattice Model with Power-Law Spatial Dispersion for Fractional Elasticity

TL;DR

This paper introduces a lattice model with power-law spatial dispersion that, in the continuum limit, leads to fractional elasticity equations, providing a microscopic basis for non-local elastic behavior with fractional derivatives.

Contribution

It develops a lattice-based microscopic model that maps to fractional continuum elasticity equations, bridging discrete and continuous descriptions of non-local elastic materials.

Findings

Derivation of fractional elasticity equations from lattice models.

Solutions provided for static cases of fractional integral and gradient elasticity.

Establishment of a link between microscopic lattice behavior and macroscopic fractional models.

Abstract

A lattice model with a spatial dispersion corresponding to a power-law type is suggested. This model serves as a microscopic model for elastic continuum with power-law non-locality. We prove that the continuous limit maps of the equations for the lattice with the power-law spatial dispersion into the continuum equations with fractional generalizations of the Laplacian operators. The suggested continuum equations, which are obtained from the lattice model, are fractional generalizations of the integral and gradient elasticity models. These equations of fractional elasticity are solved for two special static cases: fractional integral elasticity and fractional gradient elasticity.