We study a class of dynamic thermal sub-differential contact problems with friction, for time-depending long memory viscoelastic materials, with or without the clamped condition, which can be put into a general model of system defined by a second order evolution inequality, coupled with a first order evolution equation. We present and establish an existence and uniqueness result, by using general results on first order evolution inequality, with monotone operators and fixed point methods. Then a fully discrete scheme for numerical approximations and analysis of error order estimate are provided. Finally, various numerical computations in dimension two will be given.

time depending long memory thermo-viscoelasticity, sub-differential contact condition, non clamped condition, dynamic process, fixed point, evolution inequality, error estimate analysis, numerical simulations.