Nitsche method for contact with Coulomb friction: Existence results for the static and dynamic finite element formulations

Mixed and Nitsche’s discretizations of Coulomb frictional contact-mechanics for mixed dimensional poromechanical models

“…It results that the stabilized mixed method is also well-posed for the Tresca model. For the Coulomb frictional model, an existence result can also be derived for the mean-Nitsche's formulation (and stabilized mixed method) following [23].…”

“…Furthermore, exploiting their facewise constant approximation, the Lagrange multipliers can be eliminated. It leads to a new Nistche type method (called hereafter the mean-Nitsche's method) based on face average tractions and displacement jumps, bridging the gap between the Nitsche method introduced in [24,20,23] and the mixed formulation (see also the related works [32,34,35]). The links between the mixed and mean-Nitsche's formulations of the contact mechanics are carefully investigated in this work.…”

“…Conversely to bilateral (persistent) contact, Signorini's problem involves non-linear boundary conditions associated to unilateral contact, with an unknown actual contact region. The extension to the Tresca friction model is analysed in [20] and to the Coulomb friction model in [23]. Following the pioneering paper of R. Stenberg [52], various recent works have been dedicated to improve our understanding of the link between Nitsche's and stabilized mixed methods: see [33] for domain decomposition, see [32,34] for frictionless contact and [35] for Tresca friction.…”

“…The mixed formulation is first recalled in Section 3.1, then we introduce the stabilized mixed discretization and carefully investigate its mean-Nitsche's equivalent formulation in Subsection 3.2 as well as its links with the mixed formulation. The Nitsche's discretization introduced in [24,20,23] is recalled in Subsection 3.3. Section 4 compares the performance of the three discretizations on stand alone contact mechanical test cases with various fracture networks.…”

Rotational self-friction problem of elastic rods

“…The unilateral contact problem with non-local Coulomb friction has been studied by Renard [23]. For static and dynamic unilateral contact problems with Coulomb friction, where the existence and/or uniqueness of solutions for discretized problems have been proved, we can refer to [6].…”