This book contains a systematical analysis of geometrical situations  leading to  contact pairs -- point-to-surface, surface-to-surface, point-to-curve, curve-to-curve and curve-to-surface.  Each contact pair  is inherited with a special coordinate system based on its geometrical properties such as a Gaussian surface coordinate system or a Serret-Frenet curve coordinate system.  The formulation in a covariant form allows in a straightforward fashion to consider various constitutive relations for a  certain pair such as anisotropy for both frictional and structural parts. Then standard methods well known in computational contact mechanics such as penalty, Lagrange multiplier methods, combination of both and others  are formulated in these coordinate systems. Such formulations require then the powerful apparatus of differential geometry of surfaces and curves as well as of convex analysis. The final goals of such transformations are  then ready-for-implementation numerical algorithms within the finite element method including any arbitrary discretization techniques such as high order and isogeometric finite elements, which are most convenient for the considered geometrical situation.

Differential Geometry of Surfaces and Curves.- Closest Point Projection Procedure and Corresponding Curvilinear Coordinate System.- Geometry and Kinematics of Contact.- Weak Formulation of Contact Conditions.- Contact Constraints and Constitutive Equations for Contact Tractions.- Linearization of the Weak Forms - Tangent Matrices in a Covariant Form.- Surface-To-Surface Contact - Various Aspects for Implementations.- Special Case of Implementation - Reduction into 2D Case.- Implementation of Contact Algorithms with High Order FE.- Anisotropic Adhesion-Friction Models - Implementation.- Experimental Validations of the Coupled Anistropi.- Various Aspects of Implementation of the Curve-To-Curve Contact Model.- 3D-Generalization of the Euler-Eytelwein Formula Considering Pitch.