On Implementing Boundary Conditions for a Rate-Form Quasi-Static Contact Problem with Friction: A Node-to-Facet Finite Element Approach

A quasi-static geometrically non-linear initial-boundary value problem is considered in a rate form for investigating deformation of a solid. Within the framework of the finite element method, an approach is proposed to computationally implement unilateral contact conditions for this problem with friction defined by the Coulomb-Siebel's law. Such formulations are often encountered in simulating technological processing operations based on severe inelastic deformations. The approach consider nodes and facets as contacting discrete fragments, where the nodes correspond to a body representing a deformable billet, and the facets refer to a rigid tool performing a processing program. Modeling contact in this way reduces to imposing special kinematic and quasi-static constraints in the nodes at each time slice. Procedurally, these constraints can be realized by the well-known standard modifications of a resolving system of linear algebraic equations which establish relationships between nodal velocities and force rates. In this paper, we provide a detailed description of main techniques required for this task and formulate a variant of an algorithm executing them in an appropriate sequence.