  This short chapter provides a fractional generalization of gradient
mechanics, an approach (originally advanced by the author in the mid 80s) that
has gained world-wide attention in the last decades due to its capability of
modeling pattern forming instabilities and size effects in materials, as well
as eliminating undesired elastic singularities. It is based on the
incorporation of higher-order gradients (in the form of Laplacians) in the
classical constitutive equations multiplied by appropriate internal lengths
accounting for the geometry/topology of underlying micro/nano structures. This
review will focus on the fractional generalization of the gradient elasticity
equations (GradEla) an extension of classical elasticity to incorporate the
Laplacian of Hookean stress by replacing the standard Laplacian by its
fractional counterpart. On introducing the resulting fractional constitutive
equation into the classical static equilibrium equation for the stress, a
fractional differential equation is obtained whose fundamental solutions are
derived by using the Greens function procedure. As an example, Kelvins problem
is analyzed within the aforementioned setting. Then, an extension to consider
constitutive equations for a restrictive class of nonlinear elastic
deformations and deformation theory of plasticity is pursued. Finally, the
methodology is applied for extending the authors higher-order diffusion theory
from the integer to the fractional case.
