A mixed computational algorithm based on the updated Lagrangian description for plane elastic contact problems

Abstract

A mixed variational statement and corresponding finite element model are developed for an arbitrary plane body undergoing large deformations (i.e. large displacements, large rotations and small strains) under external loads using the updated Lagrangian formulation. The mixed finite element formulation allows the nodal displacements and stresses to be approximated independently. Inclusion of stresses as the nodal variables allows the satisfaction of stress equilibrium and contact boundary conditions more exactly than in the displacement finite element model.

Two algorithms are discussed for the analysis of a thin, uniformly loaded plate with a circular hole in contact with a pin. The different algorithms consider the separate cases of a rigid pin and a flexible pin, and use different methods to account for the computational difficulties that arise from the unknown contact area and the presence of friction between the pin and the plate. Both techniques use the nonlinear mixed finite element formulation. A number of different contact problems are solved using these two techniques.

A hybrid technique is presented that combines the numerical technique of the finite element method with the experimental technique of moire interferometry. The displacements at the pin-hole interface are measured from physical experiments and are then used as prescribed boundary conditions in the finite element analysis of the modeled problem, which in turn is used to compute the required stresses at the contact region. Results of this algorithm are compared with solutions obtained from strictly computational algorithms that are independent of the experimental data.