An augmented lagrangian method and a mortar approach to solve three-dimensional frictional contact problems

FEDERICO J. CAVALIERI; ALBERTO CARDONA

Sao Paulo

Congreso; 10th World congress on computational mechanics; 2012

Universidad de Sao Paulo

Contact mechanics is present in several mechanical engineering applications, and numerous works have been dedicated to the numerical solution of contact problems: for instance, the design of gears, metal forming processes, contact fatigue and others. New advances and techniques, including friction, large displacements, plasticity, and wear, are constantly being introduced. However, there is not yet a completely robust contact algorithm suitable for a wide range of applications in contact mechanics. The relative displacement of two contacting bodies is currently described in the framework of the Finite Element Method (FEM) using the so-called node-segment and segment-segment approaches. The main drawback of the node-segment approach is that it is not able to transmit a constant stress field from one body to the other when the meshes are non-conforming, i.e., it does not pass the contact patch tests, and therefore introduces errors in the solutions independently of the mesh discretisation of the contacting bodies. The contact approach, based on the mortar method, gives a smooth representation of the contact forces across the bodies interface, and can be used in arbitrarily curved 3D configurations. In this work, a finite element formulation using a mortar method to deal with the frictional contact problem between elastic bodies is presented. A mixed penalty-duality formulation from an augmented Lagrangian approach for treating the contact and the friction inequality constrains is used. The corresponding first and second variations of the saddle-point functional, which yields the internal force vector and the tangent matrix of the contact problem, are computed exactly. The algorithm has four main features: the nonlinear equations can be solved by means of a monolithic Newton scheme of simple implementation, problems with non-conforming meshes can be accurately solved, the results do not depend on the definition of any penalty parameters by the users and can be implemented very easily in a finite element program for nonlinear analysis without any change to the main structure of the code. Numerical examples for frictional contact problems in three-dimension illustrate the performance of the proposed method.