A fractional nonlocal elasticity model is presented in this Note. This model can be understood as a possible generalization of Eringen's 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 onedimensional 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 Eringen's 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 Eringen's 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.

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We study the dynamics of a mass, sliding on a dry surface and impacting against a rigid wall through a viscoelastic body, that we model as a straight rod of negligible mass. The problem comprises a constitutive model of the viscoelastic body with fractional derivatives of stress and strain, restrictions on the coefficients that follow from Clausius–Duhem inequality, and discontinuous inequality constraint conditions imposed by the Coulomb friction model. We show that the dynamics of the problem is governed by a single integro-differential inclusion. By use of the slack variable algorithm the problem was solved numerically. The predictions of the model concerning the duration of the impact, maximal values of the impacting force and deformation, as well as the restitution coefficient are determined for several values of system parameters. Depending on the dry friction coefficient three different impact scripts are identified: rebound after the impact, capture in the approaching phase, and capture in the rebound phase.

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We study dynamics of a mass, moving on a straight line, and impacting against the rigid wall through a deformable body, that we model as a straight rod of negligible mass. The chosen constitutive model of the viscoelastic body comprises fractional derivatives of stress and strain and the restrictions on the coefficients that follow from Clausius Duhem inequality. We show that the dynamics of the problem is governed by a single differential equation of real order. The obtained equation was solved numerically. The comparison is made to the solution obtained by the Laplace transform and Post’s inversion formula. The predictions of the model concerning the duration of the impact, maximal values of the impacting force and deformation as well as the restitution coefficient are determined for several values of system parameters.