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