Locking-Free Finite Elements for Unilateral Crack Problems in Elasticity

Abstract

Abstract. We consider mixed and hybrid variational formulations to the linearized elasticity system in domains with cracks. Inequality type conditions are prescribed at the crack faces which results in unilateral contact problems. The variational formulations are extended to the whole domain including the cracks which yields, for each problem, a smooth domain formulation. Mixed finite element methods such as PEERS or BDM methods are designed to avoid locking for nearly incompressible materials in plane elasticity. We study and implement discretizations based on such mixed finite element methods for the smooth domain formulations to the unilateral crack problems. We obtain convergence rates and optimal error estimates and we present some numerical experiments in agreement with the theoretical results.

Dual-Dual Formulation for a Contact Problem with Friction

“…Kunisch and Stadler [19] use Fenchel's duality theory to derive the dual problem with the friction force as additional Lagrange multiplier for a contact problem with Tresca friction. Furthermore, Belhachmi et al [6] present a dual formulation for some unilateral crack problems in elasticity. Here, the authors consider a contact problem where no friction occurs.…”