Generalized Newton methods for crack problems with nonpenetration condition

A class of semismooth Newton methods for unilaterally constrained variational problems modeling cracks under a nonpenetration condition is introduced and investigated. On the continuous level, a penalization technique is applied that allows to argue generalized differentiability of the nonlinear mapping associated to its first‐order optimality characterization. It is shown that the corresponding semismooth Newton method converges locally superlinearly. For the discrete version of the problem, fast local as well as global and monotonous convergence of a discrete semismooth Newton method are proved. A comprehensive report on numerical tests for the two‐dimensional Lamé problem with three collinear cracks under the nonpenetration condition ends the article.