The paper investigates the efficiency, stability, and accuracy of finite element solutions based on the mortar approach in solving dynamic contact problems that involve unsaturated soils. We employ a fully coupled finite element solution and a recently developed bounding surface model for describing the hydro-mechanical hysteresis of unsaturated soils. The paper discusses a key source of instability in numerical simulations of this class of problems which is stress overshooting. We evaluate the efficiency of a solution for this problem and discuss the impact of this solution on improving the accuracy and stability of numerical simulations. Numerical simulations are presented for verification and analysis of efficiency, stability, and accuracy.