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The precise determination of the forming limit strains of a sheet metal is necessary in order to accurately predict the onset of failure in sheet metal forming processes. The method used to detect the onset of necking (i.e. the necking criterion) is a most important factor in formability analysis as it significantly affects the forming limit strains. Many different necking criteria have been developed, which take advantage of the flexibility and high resolution of digital image correlation (DIC) strain measurements. Several time-dependent and time/geometry-dependent necking criteria were carefully investigated in order to evaluate their ability to reliably and consistently detect the onset of necking of TRIP780 sheet specimens that were stretch-formed over a hemispherical punch. It was found that the geometry-dependent, surface slope criterion was the most robust and consistent criterion of those evaluated.

The conventional forming limit curve (FLC) is significantly strain path-dependent and therefore is not valid for formability evaluation of sheet metal parts that undergo nonlinear loading paths during the forming process. The stress-based forming limit curve (SFLC) is path-independent for all but very large prestrains and is a promising tool for formability evaluation. The SFLC is an ideal failure criterion for virtual forming simulations but it cannot be easily used on the shop floor as there is no straightforward experimental method to measure stresses in stamped parts. This paper presents a theoretical basis for predicting the effective limit strain curve (ELSC) using the Marciniak and Kuczynski (MK) analysis (Int J Mech Sci 9:609–620, 1967, Int J Mech Sci 15:789–805, 1973). Since the in-plane strain components are sufficient to calculate the effective strain, the ELSC can easily be determined from strains measured in the stamping plant, and therefore it is a better alternative to the SFLC for formability evaluation. This model was validated using experimental data for AISI-1012 steel (Molaei 1999) and AA-2008-T4 aluminum alloys Graf and Hosford (Metall Trans 24A:2503–2512, 1993). Predicted results showed that, similar to SFLC, the ELSC remains practically unchanged for a significant range of prestrain values under various bilinear loading paths, but some strain-path dependence can be observed for significant magnitudes of the effective prestrain (ee ≥ 0.37 for AISI-1012 steel and ee ≥ 0.25 for AA-2008-T4 aluminum).

Pre-bending is a critical process required prior to hydroforming. The bending has an effect on the tube thickness and strain which will use up a portion of the formability of the as-received tube. To compensate for this loss of formability, a multi-objective optimization method was applied to improve the hydroforming process after pre-bending. A multi-objective genetic algorithm (MOGA) and Kriging surrogate model were used to optimize the loading path. The Kriging model was used to replace the finite element simulation in constraint handling. The optimal loading parameters in the hydroforming process were obtained for a tube that was previously bent 90°, and showed an improvement in reducing the corner radii of the part at the extrados and intrados of the bend (8.73 mm and 11.24 mm for the extrados and intrados of the bend, respectively). The corresponding corner fill expansion (CFE) was improved by 16.7% (or 1.79 mm) compared to the maximum expansion of 10.73 mm obtained experimentally.

The prediction of the forming limits of sheet metals typically assumes plane stress conditions that are really only valid for open die stamping or processes with negligible out-of-plane stresses. In fact, many industrial sheet metal forming processes lead to significant compressive stresses at the sheet surface, and therefore the effects of the through-thickness stress on the formability of sheet metals cannot be ignored. Moreover, predictions of forming limit curves (FLC) that assume plane stress conditions may not be valid when the forming process involves non-negligible out-of-plane stresses. For this reason a new model was developed to predict FLC for general, three-dimensional stress states. Marciniak and Kuczynski (Int J Mech Sci 9:609-620, 1967) first proposed an analytical method to predict the FLC in 1967, known as the MK method, and this approach has been used for decades to accurately predict FLC for plane stress sheet forming applications. In this work, the conventional MK analysis was extended to include the through-thickness principal stress component (σ3), and its effect on the formability of different grades of sheet metal was investigated in terms of the ratio of the third to the first principal stress components (\( \beta = {{{{\sigma_3}}} \left/ {{{\sigma_1}}} \right.} \)). The FLC was predicted for plane stress conditions (β = 0) as well as cases with different compressive through-thickness stress values (β ≠ 0) in order to study the influence of β on the FLC in three-dimensional stress conditions. An analysis was also carried out to determine how the sensitivity of the FLC prediction to the through-thickness stress component changes with variations in the strain hardening coefficient, in the strain rate sensitivity, in plastic anisotropy, in grain size and in sheet thickness. It was found that the out-of-plane stress always has an effect on the position of the FLC in principal strain space. However, the analysis also showed that among the factors considered in this paper, the strain hardening coefficient has the most significant effect on the dependency of FLC to the through-thickness stress, while the strain rate sensitivity coefficient has the least influence on this sensitivity.

Path-dependent forming limits have been computed for sheet metals undergoing various combinations of plane stress loading conditions. This paper presents a theoretical model for prediction of stress-based forming limit curves (SFLC) based on the Marciniak and Kuczynski (MK) model. Acceptable agreement was observed between calculated forming limit curves (FLC) and experimental data for AISI-1012 steel (Molaei 1999) and AA-2008-T4 alloys (Graf and Hosford Metallurgical Trans 24A:2503–2512, 1993). In this paper, the path dependency of SFLCs predicted for different non-proportional loading histories has been investigated. For a range of prestrain values in different bilinear loading paths, the SFLC remains practically unchanged. However, some strain path dependency is observed for large values of prestrain (\( \bar{\varepsilon } \geqslant 0.35 \) for AISI-1012 steel) and for abrupt changes in strain path. Nevertheless, the SFLC remains a good failure criterion for virtual forming simulations because the path dependency of SFLCs is much less significant than that of strain-based FLCs.