Analysis and Numerical Simulation for a Time-Fractional Contact Problem with History-Dependent Operator in Thermo-Viscoelasticity

The objective of this work is to study a frictionless contact scenario involving a thermo-viscoelastic body and a thermally conductive foundation. The history-dependent constitutive law involves time-fractional derivatives of the Caputo type to describe displacement behavior. The modeling of contact involves the utilization of the normal response condition. We formulate the problem variationally and prove the existence of its weak solution by employing a combination of techniques, such as the theory of monotone operators, Caputo derivatives, the Galerkin method, and the Banach fixed-point theorem. Further, we introduce a fully discrete scheme to solve the problem numerically. Under certain solution regularity assumptions, we derive an optimal order error estimate of the discretization. Finally, we provide numerical simulations which show the efficiency of the method and illustrate the theoretical error estimate.