Lattice Model of Fractional Gradient and Integral Elasticity: Long-Range Interaction of Grunwald-Letnikov-Riesz Type

TL;DR

This paper introduces a lattice model with long-range power-law interactions linked to fractional derivatives, enabling a unified approach to fractional gradient and integral elasticity with analytical solutions and numerical simulation advantages.

Contribution

It proposes a novel lattice model with long-range interactions that map to fractional continuum equations, unifying fractional gradient and integral elasticity descriptions.

Findings

The model allows analytical solutions of fractional elasticity equations.

It facilitates numerical simulations due to finite difference definitions.

The continuum limit maps to equations with Riesz derivatives.

Abstract

Lattice model with long-range interaction of power-law type that is connected with difference of non-integer order is suggested. The continuous limit maps the equations of motion of lattice particles into continuum equations with fractional Grunwald-Letnikov-Riesz derivatives. The suggested continuum equations describe fractional generalizations of the gradient and integral elasticity. The proposed type of long-range interaction allows us to have united approach to describe of lattice models for the fractional gradient and fractional integral elasticity. Additional important advantages of this approach are the following: (1) It is possible to use this model of long-range interaction in numerical simulations since this type of interactions and the Grunwald-Letnikov derivatives are defined by generalized finite difference; (2) The suggested model of long-range interaction leads to an equation containing the sum of the Grunwald-Letnikov derivatives, which is equal the Riesz's derivative. This fact allows us to get particular analytical solutions of fractional elasticity equations.