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.