TY - JOUR

T1 - Fractional differential calculus for 3D mechanically based non-local elasticity

AU - Di Paola, Mario

AU - Zingales, Massimiliano

AU - Di Paola, Mario

AU - Zingales, Massimiliano

PY - 2011

Y1 - 2011

N2 - 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

AB - 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

KW - Central marchaud fractional derivatives

KW - Fractional differential calculus

KW - Fractional finite differences

KW - Long-range interactions

KW - Non-local elasticity

KW - Central marchaud fractional derivatives

KW - Fractional differential calculus

KW - Fractional finite differences

KW - Long-range interactions

KW - Non-local elasticity

UR - http://hdl.handle.net/10447/63777

M3 - Article

VL - 9

SP - 579

EP - 597

JO - International Journal for Multiscale Computational Engineering

JF - International Journal for Multiscale Computational Engineering

SN - 1543-1649

ER -