Towards Fractional Gradient Elasticity

TL;DR

This paper extends gradient elasticity models by incorporating fractional derivatives to better describe non-local effects, providing explicit solutions and analyzing the impact on stress and displacement fields.

Contribution

It introduces two phenomenological models of fractional gradient elasticity using Caputo and Riesz derivatives, with explicit solutions and physical implications.

Findings

Stress equilibrium requires non-zero internal forces in Caputo models.

Displacement behavior near point loads depends on fractional order in Riesz models.

Explicit solutions demonstrate the effects of fractional derivatives on elasticity.

Abstract

An extension of gradient elasticity through the inclusion of spatial derivatives of fractional order to describe power-law type of non-locality is discussed. Two phenomenological possibilities are explored. The first is based on the Caputo fractional derivatives in one-dimension. The second involves the Riesz fractional derivative in three-dimensions. Explicit solutions of the corresponding fractional differential equations are obtained in both cases. In the first case it is shown that stress equilibrium in a Caputo elastic bar requires the existence of a non-zero internal body force to equilibrate it. In the second case, it is shown that in a Riesz type gradient elastic continuum under the action of a point load, the displacement may or may not be singular depending on the order of the fractional derivative assumed.