Influence of the sensitivity of plastic deformation to the third invariant on the stress state achievable during stretch forming of isotropic materials

For isotropic materials, the von Mises yield criterion is generally used to interpret bulge test data and assess formability. In this paper, we investigate the role played by the \({J}_{3}\) $J_3$ dependence of the plastic response on the behavior during stretch forming under pressure. To this end, we consider the isotropic yield criterion of Drucker, which involves a unique parameter c expressible solely in terms of the ratio between the yield stresses in shear and uniaxial tension, \({\tau }_{Y}/{\sigma }_{T}\) ${\tau }_Y/{\sigma }_T$ . In the case when \({\tau }_{Y}/{\sigma }_{T}=\sqrt{3}\) ${\tau }_Y/{\sigma }_T=\sqrt{3}$ , the parameter c = 0 and the von Mises yield criterion is recovered, otherwise Drucker’s criterion accounts for dependence on both \({J}_{2}\) $J_2$ and \({J}_{3}\) $J_3$ . First, an analytical estimate of the ratio of the principal stresses at the apex of the dome is deduced. It is demonstrated that the stress ratio depends on the parameter c, the deviation from an equibiaxial stress state induced by changing the die aspect ratio is more pronounced for materials with higher \({\tau }_{Y}/{\sigma }_{Y}\) ${\tau }_Y/{\sigma }_Y$ ratio. Finite element predictions using the yield criterion and isotropic hardening confirm the trends put into evidence theoretically. Moreover, the F.E. simulations show that there is a correlation between the value of the parameter c that describes the dependence on \({J}_{3}\) $J_3$ in the model and the strain paths that can be achieved in any given test, the level of plastic strains that develop in the dome, and the thickness reduction. Specifically, for a material characterized by c > 0 ( \({\tau }_{Y}/{\sigma }_{T}<1/\sqrt{3}\) ${\tau }_Y/{\sigma }_T<1/\sqrt{3}$ ) under elliptical bulging, at the apex the plastic strain ratio is greater than in the case of a von Mises material, while the stress ratio is less. On the other hand, for a material characterized by c < 0 ( \({\tau }_{Y}/{\sigma }_{T}>1/\sqrt{3}\) ${\tau }_Y/{\sigma }_T>1/\sqrt{3}$ ), the reverse holds true. The FE results also suggest that for certain isotropic materials neglecting the dependence of their plastic behavior on \({J}_{3}\) $J_3$ would lead to an underestimation of the thickness reduction.