On the fractional generalization of Eringenʼs nonlocal elasticity for wave propagation

A fractional nonlocal elasticity model is presented in this Note. This model can be understood as a possible generalization of Eringenʼs nonlocal elastic model, with a free non-integer derivative in the stress–strain fractional order differential equation. This model only contains a single length scale and the fractional derivative order as parameters. The kernel of this integral-based nonlocal model is explicitly given for various fractional derivative orders. The dynamical properties of this new model are investigated for a one-dimensional problem. It is possible to obtain an analytical dispersive equation for the axial wave problem, which is parameterized by the fractional derivative order. The fractional derivative order of this generalized fractional Eringenʼs law is then calibrated with the dispersive wave properties of the Born–Kármán model of lattice dynamics and appears to be greater than the one of the usual Eringenʼs model. An excellent matching of the dispersive curve of the Born–Kármán model of lattice dynamics is obtained with such generalized integral-based nonlocal model.