Fractional Gradient Elasticity from Spatial Dispersion Law

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.

24 pages, LaTeX, typos removed

References in corpus (3)

• Fractional dynamics of systems with long-range interaction
• Review of Some Promising Fractional Physical Models
• Fractional Power-Law Spatial Dispersion in Electrodynamics

Cited by in corpus (5)

• Non-Standard Extensions of Gradient Elasticity: Fractional Non-Locality, Memory and Fractality
• Lattice with Long-Range Interaction of Power-Law Type for Fractional Non-Local Elasticity
• Fractional Liouville Equation on Lattice Phase-Space
• Large Lattice Fractional Fokker-Planck Equation
• Fractional Diffusion Equations for Lattice and Continuum: Grunwald-Letnikov Differences and Derivatives Approach