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

Abstract

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 s −1 to 8 • 10 4 s −1 . The plastic behavior of the material has been described with the isotropic Drucker yield criterion [11], 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 σ T /τ Y . For c = 0, Drucker yield criterion [11] reduces to the von Mises yield criterion [32] while for c = 81/66, the Hershey-Hosford (m = 6) yield criterion [19,22] 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 criterion [10] 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.

Theoretical predictions of dynamic necking formability of ductile metallic sheets with evolving plastic anisotropy and tension-compression asymmetry

“…This technique is based on the superposition of a small perturbation on the homogeneous solution of the problem, so that if the perturbation grows faster than the background solution, the plastic flow is unstable and a non-homogeneous, neck-like deformation field can develop. We have used the same 2D approach employed in [43,51,63] so that the stress multiaxiality effects that develop inside the necked section have been approximated using Bridgman [11] correction. The fundamental solution of the problem is composed of 16 equations which are nondimensionalized, perturbed using the frozen coefficients method and linearized (note that in the case of N'souglo et al [42] the number of equations was 17 because the energy balance was taken into account).…”

Theoretical predictions of dynamic necking formability of ductile metallic sheets with evolving plastic anisotropy and tension-compression asymmetry

“…Linear stability analysis: this technique is based on the superposition of a small perturbation on the homogeneous solution of the problem, so that if the perturbation grows faster than the background solution, the plastic flow is unstable and a non-homogeneous, neck-like deformation field can develop. We have used the same 2D approach employed in [51,33,39] so that the stress multiaxiality effects that develop inside the necked section have been approximated using Bridgman [8] correction. The fundamental solution of the problem is composed of 16 equations which are nondimensionalized, perturbed using the frozen coefficients method and linearized (note that in the case of N'souglo et al [32] the number of equations was 17 because the energy balance was taken into account).…”

A new analytical model to predict the formation of necking instabilities in porous plates subjected to dynamic biaxial loading

“…Very recently, Rodríguez-Martínez et al (2021) developed a linear stability analysis for plates subjected to plane strain tension modelled with the isotropic Drucker (1949) yield criterion, which depends on both the second and third invariant of the stress deviator. The analysis was performed for effective strain rates ranging from 10 −4 s −1 to 8 • 10 4 s −1 , and the theoretical results were compared with uni-cell finite element calculations, as in N'souglo et al (2020, 2021).…”

“…'souglo et al (2020, 2021)). Note that the normalized perturbation wavelength is usually referred to as necking wavelength(N'souglo et al, 2020;Rodríguez-Martínez et al, 2021). The root of the polynomial with physical meaning is the one with the greatest positive real part(Dudzinski and Molinari, 1991), hereinafter denoted by η+ .…”

“…The root of the polynomial with physical meaning is the one with the greatest positive real part(Dudzinski and Molinari, 1991), hereinafter denoted by η+ . According to the notation used by ElMaï et al (2014) andRodríguez-Martínez et al (2021), among others, η+ will be referred to as instantaneous instability index, and it will be used to compute the so-called cumulative instability index I = t 0 η+ ε1 dτ , which tracks the history of the instantaneous grow rate of all the growing modes.The cumulative instability index generally provides better predictions than the instantaneous instability index for the critical strain leading to necking formation at high strain rates (see the recent paper of Rodríguez-Martínez et al (2021)). Assuming that a perturbation mode turns into a necking mode when the cumulative instability index reaches a critical value which depends on the loading path and the strain rate, see Section 3.1, the linear stability analysis is used in Section 3 to construct forming limit diagrams which are compared with unit-cell finite element simulations performed in ABAQUS/Explicit (2019) to demonstrate the ability of the analytical model to capture the effect of porosity on necking localization (this is same methodology used byN'souglo et al (2021) to assess the ability of the linear stability analysis to capture the effect of anisotropy on necking localization in materials described withHill (1948) yield criterion).…”