Dynamic analysis of a soft-contact problem using viscoelastic and fractional-elastic rheological models

The paper presents mathematical formulation of a dynamic soft-contact problem consisting of a material point impacting rheological structures. Two different models were analyzed. The first model concerns a viscoelastic structure while the second one applies to the so-called fractional-elastic scheme. In both cases the formulation of the problem needs the notion of variational inequalities describing unilateral constraints. Moreover, the fractional-elastic model contains an element defined via fractional derivatives (spring-pot) that complicates the numerical solution. Thanks to the methods used in this paper, both problems were described mathematically applying appropriate systems of algebraic-differential equations. The solution of the fractional-elastic problem involving a fractional-differential equation (FDE) was obtained with the use of time-discretization schemes proposed in the literature. Selected numerical results of the initial-value problem solution, modelling a material point falling freely and impacting the rheological structures were presented.