Effect of the third invariant on the formation of necking instabilities in ductile plates subjected to plane strain tension

In this paper, we have investigated the effect of the third invariant of the stress deviator on the formation of necking instabilities in isotropic metallic plates subjected to plane strain tension. For that purpose, we have performed finite element calculations and linear stability analysis for initial equivalent strain rates ranging from \(10^{-4}\,\text {s}^{-1}\) to \(8 \cdot 10^{4}\,\text {s}^{-1}\). The plastic behavior of the material has been described with the isotropic Drucker (J Appl Mech 16:349–357, 1949) yield criterion, which depends on both the second and third invariant of the stress deviator, and a parameter c which determines the ratio between the yield stresses in uniaxial tension and in pure shear \(\sigma _T / \tau _Y\). For \(c=0\), Drucker (J Appl Mech 16:349–357, 1949) yield criterion reduces to the von Mises (ZAMM J Appl Math Mech/Zeitschrift für Angewandte Mathematik und Mechanik 8(3):161–185, 1928) yield criterion while for \(c=81/66\), the Hershey–Hosford (J Appl Mech 76:241–249, 1954; Proceedings of the seventh North American metalworking research conference, 1979) \(\left( m=6\right)\) yield criterion is recovered. The results obtained with both finite element calculations and linear stability analysis show the same overall trends and there is also quantitative agreement for most of the loading rates considered. In the quasi-static regime, while the specimen elongation when necking occurs is virtually insensitive to the value of the parameter c, both finite element results and analytical calculations using Considère (Ann Ponts et Chaussées 9:574–775, 1885) criterion show that the necking strain increases as the parameter c decreases, bringing out the effect of the third invariant of the stress deviator on the formation of quasi-static necks. In contrast, at high initial equivalent strain rates, when the influence of inertia on the necking process becomes important, both finite element simulations and linear stability analysis show that the effect of the third invariant is reversed, notably for long necking wavelengths, with the specimen elongation when necking occurs increasing as the parameter c increases, and the necking strain decreasing as the parameter c decreases.