Finite element analysis of the deep drawing process using variational inequalities

Abstract

This paper is devoted to the development and implementation of a reliable and consistent method to treat frictional contact in elastoplastic solids undergoing large deformation using variational inequalities. In this context, an updated Lagrangian formulation is employed to provide the incremental variational inequality corresponding to this class of problems over the loading history. The large rotations encountered during deformation are treated using the Jaumann objective stress rate tensor and friction was assumed to follow Coulomb's law. 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 identified to calculate the incremental changes in the displacement field. The newly developed approach offers the advantage of being able to reduce the active number of variables, leading to improved and enhanced computational economy. The merits of the formulations are demonstrated by application to the deep drawing process. Specifically, the work examined the effect of interfacial friction upon punch load, the von Mises stress trajectory, the successive deformed geometry, the springback and the unloading residual stresses in the sheet metal.