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.