<p>In this study, we designed a framework of the mixed finite element method for solving a class of multi-layer viscoelastic contact systems with time-dependent Tresca friction conditions, which can be used for mechanical modeling of engineering problems such as asphalt pavement. Theoretically, we have proven the convergence properties of the numerical solution of the displacement field under the condition that the dual multipliers do not have uniqueness. Moreover, by combining the midpoint strategy, this numerical method still maintains stability in the numerical simulation of complex contact systems with elastic and viscoelastic layers, thus becoming a unified computational framework for such contact problems. Benefit from this, the numerical method can achieve mechanical simulations of asphalt pavements, thereby offering prospects for reference and application in actual industrial modeling.</p>