Die engineers routinely use finite element analyses in the metal- forming industry to assess the formability of sheet-metal products long before the dies are built in order to save money in die build and tryout costs, as well as to address manufacturability issues early in the product design cycle. One of the most important ob- jectives in this assessment is to avoid necking and fracture of the sheet metal. Although the finite element method (FEM) does not directly predict whether or not she sheet metal will neck or tear during the forming of the product, it does predict the metal flow and the development of stress and strains throughout the forming process. The engineer determines the forming severity by compar- ing the predictions of the FEM to a forming limit criterion, which is a function of the sheet-metal properties and the forming history. Obviously, a critical factor in the success of FEM analysis is the reliability of this forming limit criterion.

The most commonly used method of gauging forming severity

with respect to necking is based on the forming limit diagram (FLD) developed by Keeler [1] and Goodwin [2] as illustrated in Fig. 1 for 2008 T4 Al. The diagram is composed of a curve in strain-space defined to characterize the forming limit of the mate- rial. As long as all strains on the part fall below this forming limit curve (FLC), that part will be free from necks. The forming limit is determined by forcing the material to follow linear strain paths as illustrated in the figure, and measure the strain on the material just before a neck appears.

Questions of the generality of the FLD began to arise in the late 1970s. Ghosh [3] reported that the forming limit of steel, sub- jected to biaxial prestrains, is different from the as-received ma- terial, depending on the magnitude of the prestrain. Similar obser- vations were made for other materials subjected to a range of prestrain conditions. Graf [4] reported the strain-path dependence of the FLC for a 2008 T4 aluminum alloy subjected to uniaxial, plane-strain, and biaxial prestrains, as shown in Fig. 2. This study consisted of 3 sets of equi-biaxial prestrain, 6 sets of uniaxial prestrain and 4 sets of near-plane-strain prestrain conditions.

Although these experiments raised questions about the general- ity of the FLC for the as-received material, the metal forming industry continued to use it through the mid-1990s with little regard 

to strain path. This lack of concern was justified for two reasons. First of all, the question of the ability of FEM to predict strains with sufficient accuracy was of far greater concern during that period than the question of the limitations of the conventional FLC. Second, most of the early applications were restricted to analyses of the first draw die, where it was believed the strain paths were sufficiently linear that the path dependent nature of the FLC would not play a role. As the application of the FEM was extended to analysis of hydro-forming, redraws and flanging op- erations, where the total strain path is significantly nonlinear, the limitations of the conventional FLD could no longer be ignored. Furthermore, nonlinear strain paths have been found to be much more common in the first draw die than first believed, resulting in costly errors in the assessment of forming severity.

A first impression of the strain-path dependence of the FLC is that it is very complex and somewhat intractable. As shown in Fig. 2, not only does the curve shift in strain space depending on the nature and magnitude of the prestrain, but it also changes shape. Many of the early experiments provided no quantified model to explain the effect. More recently, however, Barata da Rocha [5] and many others showed that the defect model devel- oped by Marciniak [6], originally proposed to explain the conven- tional strain-based FLC, also accurately predicts the observed strain-path effects.

Although the Marciniak model accounts for the path depen- dency of the FLC, the question of how to efficiently use it in the determination of formability is not simple. The problem is that a separate model is required for every element of the mesh used in the die design analysis. Each Marciniak analysis must take into account the distinct strain-path of each of the elements used in the die design in order to identify the instance of bifurcation between the strain rate in the defect and surrounding material. In practice, considerable time and cost could be saved by restricting the Mar- ciniak analyses to those areas where formability problems are an- ticipated, or where the strain path is very nonlinear. But even if only 5 percent of the elements in the die design were in a critical area, hundreds of Marciniak analyses would be required. And the danger exists that there may be unanticipated problem areas.

Although the Marciniak analysis presents a challenge, its costs could easily be justified by the benefit of solving the path depen- dent nature of the conventional forming limit criterion. However, another discovery suggests that these analyses are unnecessary. Kleemola [7] and independently, Arrieux [8] discovered that there exists a forming limit criterion based on the state of stress that isaccount the strain path, and argued that there were no systematic differences in the location of the stress-based curves with the magnetic or type of prestrain. A valid challenge to this argument is that the forming limit curves will naturally be squeezed together in stress space because of the saturation of the stress-strain rela- tion. Some opponents of the stress-based criterion suggest that this saturation is responsible for the apparent convergence of the form- ing limit curves in stress-space. This paper will review the data using error analysis to show the convergence in stress-space is not due to a saturation of the stress-strain relation, but is due to the fact that the final states of stress achieved in each of the tests are identical within the experimental uncertainty.