Fractional Gradient Elasticity from Spatial Dispersion Law

TL;DR

This paper introduces a fractional gradient elasticity model derived from a lattice with power-law spatial dispersion, connecting microscopic lattice behavior to continuum equations with fractional Laplacians, expanding the understanding of weak non-locality in elasticity.

Contribution

It presents a novel derivation of fractional gradient elasticity from a lattice model with power-law dispersion, linking microscopic and continuum descriptions with fractional calculus.

Findings

Derived continuum equations with fractional Laplacians from lattice models.

Solved fractional elasticity equations for specific cases like sub-gradient and super-gradient elasticity.

Established the connection between lattice dispersion and weak non-locality in continuum mechanics.

Abstract

Non-local elasticity models in continuum mechanics can be treated with two different approaches: the gradient elasticity models (weak non-locality) and the integral non-local models (strong non-locality). This article focuses on the fractional generalization of gradient elasticity that allows us to describe a weak non-locality of power-law type. We suggest a lattice model with spatial dispersion of power-law type as a microscopic model of fractional gradient elastic continuum. We prove that the continuous limit maps the equations for lattice with this spatial dispersion into the continuum equations with fractional Laplacians in the Riesz form. A weak non-locality of power-law type in the non-local elasticity theory is derived from the fractional weak spatial dispersion in the lattice model. The suggested continuum equations, which are obtained from the lattice model, describe a fractional generalization of the gradient elasticity. These equations of fractional elasticity are solved for some special cases: sub-gradient elasticity and super-gradient elasticity.