Stress-based Forming Limit Curves Research Articles

Orientation and Path Dependence of Formability in the Stress- and the Extended Stress-Based Forming Limit Curves

This article analyzes the formability data sets for aluminum killed steel (Laukonis, J. V., and Ghosh, A. K., 1978, “Effects of Strain Path Changes on the Formability of Sheet Metals,” Metall. Trans. A., 9, pp. 1849–1856), for Al 2008-T4 (Graf, A., and Hosford, W., 1993, “Effect of Changing Strain Paths on Forming Limit Diagrams of Al 2008-T4,” Metall. Trans. A, 24A, pp. 2503–2512) and for Al 6111-T4 (Graf, A., and Hosford, W., 1994, “The Influence of Strain-Path Changes on Forming Limit Diagrams of Al 6111 T4,” Int. J. Mech. Sci., 36, pp. 897–910). These articles present strain-based forming limit curves (ϵFLCs) for both as-received and prestrained sheets. Using phenomenological yield functions, and assuming isotropic hardening, the ϵFLCs are transformed into principal stress space to obtain stress-based forming limit curves (σFLCs) and the principal stresses are transformed into effective stress and mean stress space to obtain the extended stress-based forming limit curves (XSFLCs). A definition of path dependence for the σFLC and XSFLC is proposed and used to classify the obtained limit curves as path dependent or independent. The path dependence of forming limit stresses is observed for some of the prestrain paths. Based on the results, a novel criterion that, with a knowledge of the forming limit stresses of the as-received material, can be used to predict whether the limit stresses are path dependent or independent for a given prestrain path is proposed. The results also suggest that kinematic hardening and transient hardening effects may explain the path dependence observed in some of the prestrain paths.

Application of an Extended Stress-Based Forming Limit Curve to Predict Necking in Stretch Flange Forming

An extension of the stress-based forming limit curve (FLC) advanced by Stoughton (2000, “A General Forming Limit Criterion for Sheet Metal Forming,” Int. J. Mech. Sci., 42, pp. 1–27) is presented in this work. With the as-received strain-based FLCs and stress-strain curves for 1.6-mm-thick AA5754 and 1-mm-thick AA5182 aluminum alloy, stress-based FLCs are obtained. These curves are then transformed into extended stress-based forming limit curves (XSFLCs), which consist of the invariants, effective stress, and mean stress. By way of application, stretch flange forming of these aluminum alloy sheets is considered. The AA5754 stretch flange displays a circumferential crack during failure, whereas the AA5182 stretch flange fails through a radial crack at the edge of the cutout. It is shown that the necking predictions obtained using the strain- and stress-based FLCs in conjunction with shell element computations are inconsistent when compared with the experimental results. By comparing the results of the shell element computations with those in which the mesh comprises eight-noded solid elements, it is demonstrated that the plane stress approximation is not valid. The XSFLC is then used with results from the solid-element computations to predict the punch depths at the onset of necking. Furthermore, it is shown that the predictions of failure location and failure mode obtained using the XSFLC are in accord with the differences observed between the two alloys/gauges.

Prediction of necking of high strength steel tubes during hydroforming—multi-axial loading

This article studies tubular hydroforming of high strength low alloy (HSLA) and dual phase (DP600) straight tubes under the action of end feeding loads. Experiments demonstrate that higher end feed loads enhance the formability of the tubes and increase the internal fluid pressure for onset of necking and bursting. Because of the action of the internal pressure and the axial compressive load, the onset of localization (necking) is due to a complex three-dimensional state of stress. Using free expansion experiments, approximate upper and lower bound strain-based forming limit curves are determined for the tube materials. These limit curves, in turn, are used to derive upper and lower bound extended stress-based forming limit curves [Simha et al., Prediction of necking in tubular hydroforming using an extended stress-based FLC. Transactions of the ASME Journal of Engineering Materials and Technology 2007;129(1): 36–47]. In conjunction with finite element computations that use solid elements to model the tube, these stress-based limit curves are used to predict upper and lower bound necking pressures under the action of end feed loading. These predictions of necking pressures, when an appropriate coefficient of tube-die friction is used, are found to bracket the experimentally measured necking pressures. Computations using plane stress shell elements to model the tubes are shown to give erroneous results, since the plane stress approximation is not valid when tubes are hydroformed in a die.

Predicting breakage on a die radius with a straight bend axis during sheet forming

As the use of high strength sheet steels for automotive applications increases, there have been increasing occurrences of breakage on die radii at strains that are less than predicted by a conventional forming limit curve. To develop a better understanding of the problem, data from an angular stretch bend test were analyzed to determine the applied stress on a bend that caused breakage for a range of die radius, sheet thickness, and steel grade. The forming limit stress was calculated using a modified version of Stoughton's stress-based forming limit curve. The ratio of the applied failure stress to the forming limit stress is defined as the failure stress ratio. Regression analysis was used to determine the effect of the sheet thickness to die radius ratio ( T/ R) on the failure stress ratio for the different steel grades ranging from DDQ+ to DP980. The results were grouped due to commonality among various steel grades. Three predictive equations for (1) DP800, (2) DP980, and (3) all the other steel grades describe the behavior for all the steels in the study. It is found that breakage on a die radius for high strength steels such as DP600, DP800, and DP980 is typically a result of low values of the forming limit stress.

Computational analysis of stress-based forming limit curves

This article, through computational analyses, examines the validity of using the stress-based and extended stress-based forming limit curves to predict the onset of necking during proportional loading of sheet metal. To this end, a model material consisting of a homogeneous zone and a zone that has voids (material inhomogeneity) is proposed and used to simulate necking under plane strain and uni-axial stress load paths. Results of the in-plane loading computations are used to construct a strain-based formability limit curve for the model material. This limit curve is transformed into principal stress space using the procedure due to Stoughton [Stoughton, T.B., 2000. A general forming limit criterion for sheet metal forming. International Journal of Mechanical Sciences 42, 1–27]. The stress-based limit curve is then transformed into equivalent stress and mean stress space to obtain an Extended Stress-Based Limit Curve (XSFLC). When subjected to three-dimensional loading, the model material is observed to display a variety of responses. From these responses, a criterion for the applicability of the XSFLC to predict the onset of necking in the model material when it is subjected to three-dimensional loading is obtained. In the context of straight tube hydroforming, to provide support for the use of the XSFLC, it is demonstrated that the criterion is satisfied.

Prediction of Necking in Tubular Hydroforming Using an Extended Stress-Based Forming Limit Curve

This paper presents an extended stress-based forming limit curve (XSFLC) that can be used to predict the onset of necking in sheet metal loaded under non-proportional load paths, as well as under three-dimensional stress states. The conventional strain-based ϵFLC is transformed into the stress-based FLC advanced by Stoughton (1999, Int. J. Mech. Sci., 42, pp. 1–27). This, in turn, is converted into the XSFLC, which is characterized by the two invariants, mean stress and equivalent stress. Assuming that the stress states at the onset of necking under plane stress loading are equivalent to those under three-dimensional loading, the XSFLC is used in conjunction with finite element computations to predict the onset of necking during tubular hydroforming. Hydroforming of straight and pre-bent tubes of EN-AW 5018 aluminum alloy and DP 600 steel are considered. Experiments carried out with these geometries and alloys are described and modeled using finite element computations. These computations, in conjunction with the XSFLC, allow quantitative predictions of necking pressures; and these predictions are found to agree to within 10% of the experimentally obtained necking pressures. The computations also provide a prediction of final failure location with remarkable accuracy. In some cases, the predictions using the XSFLC show some discrepancies when compared with the experimental results, and this paper addresses potential causes for these discrepancies. Potential improvements to the framework of the XSFLC are also discussed.