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Follower forces on a surface are defined as the forces which are keeping their constant directions with regard to the surface. A typical example is pressure, which is acting always in the direction of the normal. The linearization is necessary in order to solve a problem using an implicit scheme in the case of large deformations. Consider, for example, pressure p acting on a surface s, defined by the vector r(ξ 1 , ξ 2 ). In this case, a weak form is written as:Formally, as shown in [1], [2] the following linearization can be formulated aswhich is recovering an apparent unsymmetrical structure with regard to ∆u and δu and, therefore, corresponding unsymmetrical tangent matrix. Though, the system is initially conservative an initially looking unsymmetrical structure is derived due to the linearization of the normal in the Cartesian coordinate system. A special complicated transformation for integrals with closed boundaries was necessary to show the symmetry in this case, see in [3], [4]. Nevertheless, as mentioned by Simo in [5], linearization in a covariant form is always leading to the symmetric structure for conservative systems. Application of the covariant derivation has become a standard tool within the geometrically exact theory of contact interaction, see in [6], [7]. The corresponding symmetry of tangent matrices for all conservative cases is then automatically fulfilled, moreover, they are obtained for all contact pairs such as surface-to-surface, curve-to-curve, curve-to-surface in a close covariant form. Using these transformations, one can treat contact tractions not as unknown variables to be computed, but as given external forces which are keeping their direction in the local coordinate system. This case we can call an inverse contact algorithm. The main results of the geometrically exact theory for contact interactions for surface-to-surface contact pairs are outlined first in order to obtain all relationships for the inverse contact algorithm. Two bodies are coming into contact if a slave point r penetrates at least at the closest distance into the master surface ρ. This point is computed via the Closest Point Projection (CPP) procedure, see [6], [7]. Afterward, a special local coordinate system related to the master surface at the penetration point C is defined.This coordinate system is defined such that the convective coordinates ξ 1 , ξ 2 , ξ 3 are defining the measures of contact interactions. A value of the penetration ξ 3 is exactly the third coordinate in the surface coordinate system. Increments of the surface convective coordinates ∆ξ 1 and ∆ξ 2 are tangential measures. The relative motion of the slave point with regards to the master surface is considered in order to formulate the rule of computation for ∆ξ 1 and ∆ξ 2 . This leads to the difference of the slave point velocity r s and the master point velocity ρ projected onto the tangent plane with basis vectors ρ i . Thus, increments ∆ξ 1 and ∆ξ 2 are integral measures of the tangential ratesξ 1 ,ξ 2 , see more details in monographs [6...

Numerical approaches are in case of contact problems mainly dealing with additional terms enforcing constraints. Within the Nitsche approach the inclusion of constraints for the non-penetration and equilibrium of stresses of the contacting bodies is carried out in a fully variational sense. Taking into account a specific choice and the physical meaning of the encountered Lagrange multipliers two different schemes for the Nitsche formulation are obtained. Both types of the Nitsche approach are implemented in a nonlinear element and verification with numerical examples is done. Nitsche formulationContact problems within numerical approaches like the finite element method are mainly dealing with well known contact formulations like the Penalty or Lagrange multipliers method. For some particular problems these methods are not sufficient and thus other approaches like the so-called Nitsche approach have to be investigated. In the current contribution two different Nitsche schemes applicable to contact problems derived in the sense of optimization theory are represented. Starting in analogy to optimization and enforcing constraints in a fully variational sense, thus, dealing with two different sets of Lagrange multipliers for the considered constraints of non-penetration and of equilibrium of stresses of the contacting bodies within the Nitsche approach the following functional is gainedHere W M and W S denote the work of the structural part of the problem for the master and slave body, respectively, W D is the non-penetration constraint (ξ 3 = 0) either in Penalty or Lagrange formalities and the last two terms are representing the constraint of the equilibrium of stresses between the contacting bodies (σ M · n M + σ S · n S = 0). The fully variational inclusion of the equilibrium constraint is achieved by considering a set of Lagrange multipliers on the master (µ M ) and on the slave surface (µ S ). In order to obtain the afore-mentioned two different Nitsche formulations out of equation (1) the choice of the Lagrange multipliers µ and the physical meaning of the Lagrange multipliers λ used for non-penetration constraints have to be cleared. Choice of Lagrange multipliers set µConsidering equation (1) in variational form without frictional parts of the normal stress vector (σ · n → σ n n) and then excluding all terms including the strong form of equilibrium of the bodies and the non-penetration constraint the following expression remains and has to be fulfilledThis requires the definition for µ as. ρ M and r S define the position vectors on the master and slave surface. Lagrange multiplier λ for non-penetration constraintTaking the standard Lagrangian contact formulation in variational form into accountand excluding in the same manner as before terms including strong forms the requirement of σ n = λ is obtained. Furtheron, using the same procedure for the standard Penalty contact formulation σ n = ǫ N ξ 3 is received with ǫ N representing the penalty factor. As a result the Lagrange multiplier λ for the non-penetrat...

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