A stabilized Lagrange multiplier method for the enriched finite-element approximation of Tresca contact problems of cracked elastic bodies

•The LPS technique is adapted to the nonlinear problem of small strain elastostatics with X-FEM Tresca frictional contact.•Existence and uniqueness results are obtained for the approximated Tresca frictional contact problem in elasticity.•A priori error estimates of three hybrid discrete formulations are given.•Numerical tests are performed on a cracked plate with a crack in contact over a portion of the lips.•Convergence curve with non-structured meshes are shown and the numerical choice of stabilization parameter is explained.

In this paper we propose a local projection stabilized Lagrange multiplier method in order to approximate the two-dimensional linear elastostatics unilateral contact problem with Tresca friction in the framework of the eXtended Finite Element Method X-FEM. This last method allows to perform finite-element computations on cracked domains by using meshes of the non-cracked domain. The advantage of the used stabilization technique is to affect only the equation on multipliers and thus to be equation independent. We study the existence, uniqueness and a priori error estimate of three hybrid discrete formulations.