Multivalued fractional differential equations as a model for an impact of two bodies

Abstract

Impact models of two colliding bodies are usually generated by fundamental physical and geometrical principles. Initially, the number of unknown variables is not the same as the number of equations expressing the principles, so the models should be complemented with an appropriate set of constitutive equations which contain enough information on the physical properties of the system and therefore allow accurate predictions of its behavior. Along these lines we here discuss a problem when a block, moving along a line on a dry surface, impinges against another block being at rest, through a deformable straight rod of negligible mass. Among the variety of all possible choices that can be used, we suggest the constitutive model of the viscoelastic body with fractional derivatives of stress and strain, restrictions on the coefficients that follow from Clausius–Duhem inequality, and the Coulomb friction law given in the set-valued form. Owing to the presence of dry friction and the proposed fractional model, known as the fractional Zener model, the problem belongs to the class of set-valued fractional differential equations (or multivalued differential equations of arbitrary real order) leading to the equivalent Cauchy problem given in terms of two coupled integro-differential inclusions, for which the existence result ensuring the contractible solution set exists. By use of the combinatorial analysis of the problem, we identify 11 imaginable scripts and separate 10 feasible ones that were confirmed numerically by a procedure that combines nonlocal and nonsmooth modules.