Approximation of Solution for Unilateral Contact Problem with Locking Materials

This paper investigates a unilateral contact problem involving nonlocal Tresca friction between a locking material and a rigid foundation. Initially, we derive the mathematical model for the static process, formulate the problem as a variational inequality, and demonstrate the existence and uniqueness of the solution. In the second stage, we employ the penalty method to formulate the penalized problem. This involves two key modifications: first, the penalization of terms in the constitutive law, and second, the regularization of the contact condition, which is expressed as a variational equality instead of the Signorini contact law. Subsequently, we establish the convergence of the continuous penalty solution as the penalty parameter approaches infinity. Finally, we derive error estimates for the finite element discretization of the penalty method. By selecting the discretization parameter as , we obtain error bounds of the same order as those in the constrained problem. Our results confirm that the penalized formulation exhibits convergence properties consistent with the original constrained formulation.