Updated Lagrangian formulation of contact problems using variational inequalities

This article is devoted to the formulation and solution of general frictional contact problems in elasto-plastic solids undergoing large deformations using variational inequalities. An updated Lagrangian formulation is adopted to develop the incremental variational inequality representing this class of problems over the loading history. The Jaumann objective stress rate is incorporated in the formulation of the elasto-plastic constitutive equations to account for large rotations, while Coulombʹs law is used to model the friction forces. The resulting variational inequality is treated using mathematical programming in association with a newly developed successive approximation scheme. This scheme, which is based upon the regularization of the frictional work, is used to impose the active contact constraints identi ed to calculate the incremental changes in the displacement eld. The newly developed approach o ers the advantages of reducing the active number of variables which is highly desirable in non-linear elasto-plastic problems. The merits of the formulations are demonstrated by application to an illustrative example and to the analysis of the deep drawing process.