Geometrically exact covariant approach for contact between curves

For curve-to-curve contact situation a geometrically exact description in a covariant form is developed. The contact kinematics, the variational formulation and the constitutive relations for contact tractions are described in a specially defined curvilinear coordinate system based on the closest point projection (CPP) procedure. The fundamental problems about the solvability of the CPP procedure for the curve-to-curve case are investigated in detail as well. The introduced coordinate system is independent of the choice of a contacting curve. This allows to describe all possible relative motions of both curves including not only normal and tangential interactions, but also rotational interaction between curves representing e.g. circular cross sections of beams. All necessary derivations for the iterative solution method such as linearization of weak forms and the return-mapping schemes are fulfilled in a covariant form for the arbitrary distance between the curves. This allows to apply the developed theory to both contact cases: when bodies are contacting by their edges and to contact between beams. Another advantage is the complete independence concerning the order of approximation involved in finite element construction. The numerical examples are chosen to verify the proposed theory and to compare different cases for both, the beam-to-beam and the edge-to-edge contact cases as well as to illustrate the possibilities of the theory to describe relative contact kinematics.