Spatially fractional-order viscoelasticity, non-locality and a new kind of anisotropy

TL;DR

This paper explores how space-fractional viscoelastic equations introduce a new form of anisotropy due to non-local interactions, with explicit solutions for specific cases.

Contribution

It introduces a novel anisotropy arising from space-fractional derivatives and derives constitutive equations and explicit solutions for these non-local viscoelastic models.

Findings

Space-fractional equations of order less than 2 enable new anisotropic behaviors.

Explicit fundamental solutions are constructed for certain isotropic and anisotropic cases.

The study links non-locality with angular dependence in stress-strain interactions.

Abstract

Spatial non-locality of space-fractional viscoelastic equations of motion is studied. Relaxation effects are accounted for by replacing second-order time derivatives by lower-order fractional derivatives and their generalizations. It is shown that space-fractional equations of motion of an order strictly less than 2 allow for a new kind anisotropy, associated with angular dependence of non-local interactions between stress and strain at different material points. Constitutive equations of such viscoelastic media are determined. Explicit fundamental solutions of the Cauchy problem are constructed for some cases isotropic and anisotropic non-locality.