Title : A study of time-fractional variational inequalities with application to frictional contact problems
Abstract:
This paper deals with the mathematical analysis of a quasistatic frictional contact problem arising from the interaction between a thermo-viscoelastic body and a thermally conductive foundation. The mechanical behavior of the material is described by a fractional Kelvin–Voigt constitutive law, which incorporates memory effects through a time-fractional derivative. In parallel, the heat transfer process is governed by a fractional evolution equation associated with the temperature field. The contact conditions on the potential contact boundary are modeled by the Signorini condition combined with a Coulomb-type dry friction law, leading to a nonlinear and nonsmooth coupling between the mechanical and thermal fields. A rigorous variational formulation of the problem is first established within an appropriate functional framework. Then, the existence of a weak solution is proved by combining several analytical tools, including monotone operator theory, properties of the Caputo fractional derivative, the Galerkin approximation method, and the Banach fixed point theorem. These techniques allow us to handle both the nonlinearities and the fractional nature of the model. Finally, numerical simulations are carried out to demonstrate the effectiveness and accuracy of the proposed approach. The results highlight the influence of fractional parameters on the mechanical response and temperature distribution, confirming the relevance of the model for applications involving memory-dependent materials.
