Calculation of the forming limit diagram

Abstract

A mathematical model is presented to help understand sheet metal deformation during forming. The particular purpose of this model is to predict the forming limit diagram (FLD). The present model is an extension of a previous analysis by Jones and Gillis (JG)[1] in which the deformation is idealized into three phases: (I) homogeneous deformation up to maximum load; (II) deformation localization under constant load; (III) local necking with a precipitous drop in load. In phase III, the neck geometry is described by a Bridgman-type neck. The present model extends the JG theory, which was applied only to the right-hand side (RHS) of the FLD. The main difference in treating the two different sides of the FLD lies in the assumptions regarding the width direction deformations. For biaxial stretching (the RHS), the minor strain rate is assumed to be homogeneous throughout the process. However, for the left-hand side (LHS) of the FLD in the critical cross section, the minor strain rate is taken to be proportional to major strain rate. This is a critical difference from the JG approach and permits the LHS to be computed with good accuracy. Another important difference between this and the JG analysis is a more realistic neck geometry. At the inception of phase III, JG matched the phase II sheet thickness at the center of the neck, that is, at its minimum cross section. Here, the phase III neck matches the phase II sheet thickness at its ends, that is, at its maximum cross section. Although this may seem a minor point, it greatly improves the geometrical concept involved. Both the actual neck geometry and the criterion for determining the limit strain are modified from the earlier analysis in order to agree more closely with actual press shop practice. Results from this analysis are compared with the experimental ones for aluminum-killed (AK) steel and three aluminum alloys. These results are also compared to other theoretical calculations of the forming limit for AK steel. It is apparent that the present model is best. Unlike the other types of analyses, the present model predicts the limiting strain states for several materials very accurately without any adjustable parameters. This is certainly an unprecedented result. Using the mathematical model, the effects of varying material properties are studied. The properties considered are the strain-hardening exponent,n, the strain-rate sensitivity parameter,m, and the plastic anisotropy ratio,r, The important influence of these material properties upon the formability (level of the FLD) is affirmed.