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Fractional Gradient Elasticity from Spatial Dispersion Law

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.