Bifurcation Analysis Versus Maximum Force Criteria in Formability Limit Assessment of Stretched Metal Sheets

Abstract

The present contribution deals with the prediction of diffuse necking in the context of forming and stretching of metal sheets. For this purpose, two approaches are investigated, namely bifurcation and the maximum force principle, with a systematic comparison of their respective ability to predict necking. While the bifurcation approach is of quite general applicability, some restrictions are shown for the application of maximum force conditions. Although the predictions of the two approaches are identical for particular loading paths and constitutive models, they are generally different, which is even the case for elasticity, confirming the distinct nature of the two concepts. Closed-form expressions of the critical stress and strain states are derived for both criteria in elasto-plasticity and rigid-plasticity for a variety of hardening models. The resulting useful formulas in rigidplasticity are shown to also accurately represent the elasto-plastic critical states for small ratios of the hardening modulus with respect to Young's modulus. Finally, the well-known expression of Swift's diffuse necking criterion, whose foundations are attributed in the literature to the maximum force principle, is shown here to originate from the bifurcation approach instead, providing a sound justification for it.

Prediction of necking in thin sheet metals using an elastic–plastic model coupled with ductile damage and bifurcation criteria

“…Furthermore, the critical necking strains given by the closed-form solutions of the Swift’52 criterion are equivalent to those obtained with the GB and LPB criteria. These results are fully consistent with those reported in Abed-Meraim et al. (2014b), where theoretical links between the Swift’52 diffuse necking criterion and the GB criterion were established.…”

“…This criterion can be expressed as F·1=0 and F·2=0, where the directions 1 and 2 correspond to the major and minor directions, respectively (see Figure 11). Within the framework of rigid flow theory of plasticity (without coupling with damage) and isotropic hardening, the above condition leads to the following general form of the Swift’52 criterion (Abed-Meraim et al., 2014b) where σ1 and σ2 are the principal Cauchy stress components associated with the in-plane forces F 1 and F 2 , respectively.…”

Numerical investigation of necking in perforated sheets using the periodic homogenization approach

“…Prediction of necking, for three particular strain-path ratios, using the different necking criteria: 5 provides the FLDs predicted by using the three different necking criteria. As can be seen, the forming limit curve given by MFC reveals to be a horizontal line, which also coincides with the predictions given by GBC for the three particular strain-path ratios:    ;  , and   (see, e.g., Abed-Meraim et al, 2014). The RBC is able to determine limit strains at localized necking only in the range of negative strain-path ratios.…”

“…Condition (33) suggests that diffuse necking occurs when components 1 R and 2 R reach their maximum values simultaneously. However, the satisfaction of this simultaneous condition is only possible for two particular strain paths (uniaxial and equibiaxial strain paths), as it has been experimentally and theoretically demonstrated in Habbad (1994) and Abed-Meraim et al (2014). Accordingly, to be able to predict the onset of diffuse necking for the whole range of strain paths, we only consider the first condition in Eq.…”