(A) rigid-plastic finite element formulation of continuum elements and its application to sheet metal workin

Abstract

The present study is concerned with a rigid-plastic finite element analysis using continuum elements based on incremental formulation for three- dimensional sheet metal forming processes. In sheet metal deformation, the displacement for each step is considerably large even though the effective strain increment is very small. For such large displacement problems, geometric nonlinearity must be considered. In the elastic-plastic finite element method using continuum elements, general incremental formulations to include the geometric nonlinearity are available. However, in the conventional rigid-plastic finite element analysis using continuum elements, the geometric nonlinearity has not been considered properly during an incremental time step. In this work, in order to incorporate geometric nonlinearity to rigid-plastic continuum elements during a step, the convected coordinate system is introduced. Since the material is incompressible, the penalty method is used for fulfillment of the incompressibility requirements. In order to avoid ill-conditions that worsen as penalty values are increased and to diminish the accumulated error with decreasing penalty values, the total type approach for the volume constraint is augmented to the updated Lagrangian formulation. The formulation is then extended to cover the orthotropic anisotropy in sheet metals using Hill``s quadratic yield function. In order to consider the effects of shape change and rotation of the planar anisotropic axis in the derivation of finite element equations using continuum elements, a curvilinear local convected coordinate system is employed. The anisotropic axes are updated using an algorithm based on polar decomposition. For the purpose of applying more realistic blankholding force conditions, a moving blankholder solution procedure is introduced that permits exact control of the blankholding force as in the experiments. In the analysis, many nodes are abruptly brought into contact or released co...