Implementation of the Nitsche approach for various contact kinematics

Abstract

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...