  In this paper we consider extensions of the gradient elasticity models
proposed earlier by the second author to describe materials with fractional
non-locality and fractality using the techniques developed recently by the
first author. We derive a generalization of three-dimensional continuum
gradient elasticity theory, starting from integral relations and assuming a
weak non-locality of power-law type that gives constitutive relations with
fractional Laplacian terms, by utilizing the fractional Taylor series in
wave-vector space. In the sequel we consider non-linear field equations with
fractional derivatives of non-integer order to describe nonlinear elastic
effects for gradient materials with power-law long-range interactions in the
framework of weak non-locality approximation. The special constitutive
relationship that we elaborate on, can form the basis for developing a
fractional extension of deformation theory of gradient plasticity. Using the
perturbation method, we obtain corrections to the constitutive relations of
linear fractional gradient elasticity, when the perturbations are caused by
weak deviations from linear elasticity or by fractional gradient non-locality.
Finally we discuss fractal materials described by continuum models with
non-integer dimensional spaces. Using the recently suggested vector calculus
for non-integer dimensional spaces, we consider problems of fractal gradient
elasticity.
