  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.
