Geometric properties of 2D and 3D unilateral large slip contact operators

In order to treat the unilateral (frictionless) large slip contact between a deformable body and a smooth rigid obstacle, geometrical properties associated to smooth curves (2D) and surfaces (3D) are investigated. Starting from basic differential geometry properties of smooth 2D curves, the kinematics of node-to-node or node-to-facet contact is generalized: the projection of a potential contactor point (striker) on any parametrizable curve and its first variation are derived. Expressions for the contact gap and its first and second variations are then calculated to obtain force equilibrium conditions and associated consistent linearization. These results are extended to 3D geometries: intrinsic properties of smooth surfaces, related to their first and second fundamental forms are introduced and serve to characterize the projection of a striker on the surface. The signed contact gap and its first two variations are then expressed explicitly. These results are specialized to the particular case where cubic Hermite curves and surfaces are considered. The resulting contact element features are illustrated with a Hertz contact benchmark problem and with a 3D model of the human patella (knee-cap) sliding over the femur during knee flexion.