Mathematical analysis of thermo-elastic-viscoplastic contact problems with time-fractional derivatives

Abstract

This paper presents a mathematical model for a quasi-static frictional contact problem between a thermo-elastic-viscoplastic body and a deformable foundation, incorporating memory effects via time-fractional derivatives in both the constitutive law and the heat conduction. The model is constructed inside a variational framework, resulting in a mathematical system comprising a fractional variational inequality that dictates the displacement's evolution, and an evolutionary equation for the thermal field. We use the Galerkin approximation approach, the Banach's fixed point theorem and certain analytical instruments associated with fractional variational inequalities to demonstrate the existence of a weak solution.