Analysis of three-dimensional sheet metal forming processes using continuous surface and contact treatment

Abstract

In large deformation problems as in sheet metal forming, the contact between nodal points and tools surfaces must be carefully checked in order to obtain stable and reliable solutions. Especially, in the FE analysis of sheet forming , the normal vector, tangential vector and the gap between a penetrated node and the tool surface needed for the iterative procedure should be readily provided from the mathematical information of the tool surface. In the present study, a general method for the mathematical description of arbitrarily-shaped tool surfaces is proposed by introducing the parametric and non-parametric patch approaches. The cubic Ferguson patch is employed to describe the tool surface in the parametric patch approach and an efficient contact search algorithm is suggested to make this parametric patch approach more realistic for the FE analysis. A new concept of the continuous contact treatment has been proposed in which the FEM mesh at each step is globally smoothed to form a smooth surface patches of net form. New normal scheme so-called continuous sheet surface normal scheme has been suggested in connection with the continuous contact treatment in the case of the parametric patch approach. The non-parametric patch approach introduces a cubic blending function as in the case of the parametric patch approach to allow $C^1$ continuity. In the approach, an array of equi-spaced points is required to construct the whole tool surface. An efficient normal scheme so-called z-projection combined with mixed normal is newly suggested to remove the drawbacks of the traditional z-projection scheme in contact treatment. In spite of various advantages of the rigid-plastic formulation based on the membrane theory, the conventional membrane formulation is not appropriate for problems which involve the considerable effect of bending and shape change. As an improved version of conventional membrane element, a node-based BEAM (abbreviated from Bending Energy Augmented...