Variationally consistent quadratic finite element contact formulations for finite deformation contact problems on rough surfaces

• Extension of previous work of the authors on constitutive contact laws in combination with the dual mortar method for contact problems to quadratic finite elements.
• Suggestion of a Petrov Galerkin dual mortar approach for quadratic finite elements to avoid consistency errors in case of large curvatures or high gradients.
• Modification of the presented algorithms considering multi-point constraints like cyclic symmetry or directional blocking on the contact zone.
• Verification of the derived algorithms with industrial motivated examples including various contact zones, large deformations and plasticity.

Although different discrete formulations for contact problems have been widely studied during the last decade, the numerical simulation of complex industrial applications is still challenging. While suitable Lagrange multiplier based formulations are well-known for their consistency and stability in the case of classical model problems of Coulomb type, rough surface contact laws and additional multi-point constraints are much less understood. In this paper, we focus on a quadratic finite element approach for quasi-static calculations and extend ideas from our previous work on constitutive contact laws combined with suitable solutions for multi-point constraints like cyclic symmetry on the contact boundary. The popular dual mortar method is used to enforce the contact constraints in a variationally consistent way without increasing the algebraic system size. To avoid possible consistency errors of the dual mortar approach in case of large curvatures or gradients in the contact zone, an alternative quadratic Petrov–Galerkin mortar formulation is presented. Numerical examples demonstrate the robustness of the derived numerical algorithm. Special focus is set to industrial motivated applications involving large deformations and plastic effects as well as rough surfaces on the micro-scale.