Fractional differential calculus for 3D mechanically based non-local elasticity

This paper aims to formulate the three-dimensional (3D) problem of non-local elasticity in terms of fractional differential operators. The non-local continuum is framed in the context of the mechanically based non-local elasticity established by the authors in a previous study; Non-local interactions are expressed in terms of central body forces depending on the relative displacement between non-adjacent volume elements as well as on the product of interacting volumes. The non-local, long-range interactions are assumed to be proportional to a power-law decaying function of the interaction distance. It is shown that, as far as an unbounded domain is considered, the elastic equilibrium problem is ruled by a vector fractional differential operator that corresponds to a new generalized expression of a fractional operator referred to as the central Marchaud fractional derivative (CMFD). It is also shown that for bounded solids the corresponding integral operators contain only the integral term of the CMFD and no divergent terms on the boundary appear for a one-dimensional solid case. This aspect is crucial since the mechanical boundary conditions may be easily enforced as in classical local elasticity theory