Estimates for the asymptotic convergence of a non- isothermal linear elasticity with friction

Abstract

This work is devoted to the asymptotic analysis of a dynamic thermoelastic prob- lem in a thin domain with Tresca friction conditions imposed on part of its bound- ary. The mathematical model couples the equations of linear elasticity with a heat equation in order to describe the interaction between mechanical deformations and thermal effects. First, a weak variational formulation of the problem is established in suitable functional spaces. The existence and uniqueness of weak solutions are recalled using tools from convex analysis and variational inequalities. To investigate the behavior of the system as the thickness parameter ε tends to zero, a rescaling transformation is introduced to convert the original problem posed on an ε-dependent domain into an equivalent problem defined on a fixed domain. This approach allows the derivation of uniform a priori estimates independent of ε. Using compactness arguments and weak convergence techniques, convergence results are established and the corresponding limit problem is derived. The obtained model provides a rigorous and simplified description of the asymptotic behavior of thermoelastic systems with Tresca friction in thin domains.