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

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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 \(10^{-4}\,\text {s}^{-1}\) 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.

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References

  1. ABAQUS (2016) Abaqus v6.16 User’s Manual, version 6.16 Edition. ABAQUS Inc., Richmond

  2. Altynova M, Hu X, Daehn GS (1996) Increased ductility in high velocity electromagnetic ring expansion. Metall Trans A 27:1837–1844

    Article  Google Scholar 

  3. Barlat F (1987) Crystallographic texture, anisotropic yield surfaces and forming limits of sheet metals. Mater Sci Eng 91:55–72

    Article  Google Scholar 

  4. Bridgman PW (1952) Studies in large plastic flow and fracture, with special emphasis on the effects of hydrostatic pressure, vol 1. McGraw-Hill Book Company Inc, New York

    MATH  Google Scholar 

  5. Butuc MC, Gracio JJ, Barata da Rocha A (2003) A theoretical study on forming limit diagrams prediction. J Mater Process Technol 142(3):714–724

    Article  Google Scholar 

  6. Cazacu O (2019) New mathematical results and explicit expressions in terms of the stress components of Barlat, et al (1991) orthotropic yield criterion. Int J Solids Struct 176–177:86–95

    Article  Google Scholar 

  7. Cazacu O, Plunkett B, Barlat F (2006) Orthotropic yield criterion for hexagonal closed packed metals. Int J Plast 22(7):1171–1194

    Article  MATH  Google Scholar 

  8. Cazacu O, Revil-Baudard B, Chandola N (2019) Plasticity-damage couplings: from single crystal to polycrystalline materials. In: Solid mechanics and its applications. Springer International Publishing

  9. Chou PC, Carleone J (1977) The stability of shaped-charge jets. J Appl Phys 48(10):4187–4195

    Article  Google Scholar 

  10. Considère A (1885) Mémoire sur l’emploi du fer et de l’acier dans les constructions. Ann. Ponts et Chaussées 9:574–775

    Google Scholar 

  11. Drucker DC (1949) Relation of experiments to mathematical theories of plasticity. J Appl Mech 16:349–357

    Article  MathSciNet  MATH  Google Scholar 

  12. Dudzinski D, Molinari A (1991) Perturbation analysis of thermoviscoplastic instabilities in biaxial loading. Int J Solids Struct 27:601–628

    Article  MATH  Google Scholar 

  13. El Maï S, Mercier S, Petit J, Molinari A (2014) An extension of the linear stability analysis for the prediction of multiple necking during dynamic extension of round bar. Int J Solids Struct 51:3491–3507

    Article  Google Scholar 

  14. Fressengeas C, Molinari A (1985) Inertia and thermal effects on the localization of plastic flow. Acta Metall 33:387–396

    Article  Google Scholar 

  15. Fressengeas C, Molinari A (1994) Fragmentation of rapidly stretching sheets. Eur J Mech A Solids 13:251–268

    MATH  Google Scholar 

  16. Grady DE, Benson DA (1983) Fragmentation of metal rings by electromagnetic loading. Exp Mech 12:393–400

    Article  Google Scholar 

  17. Guduru PR, Freund LB (2002) The dynamics of multiple neck formation and fragmentation in high rate extension of ductile materials. Int J Solids Struct 39:5615–5632

    Article  MATH  Google Scholar 

  18. Gurson A (1977) Continuum theory of ductile rupture by void nucleation and growth. Part I: yield criteria and flow rules for porous ductile media. ASME J Eng Mater Technol 99:2–15

    Article  Google Scholar 

  19. Hershey AV (1954) The plasticity of an isotropic aggregate of anisotropic face centered cubic crystals. J Appl Mech 76:241–249

    Article  MATH  Google Scholar 

  20. Hill R (1948) A theory of the yielding and plastic flow of anisotropic metals. In: Proceedings of the royal society of London A: mathematical, physical and engineering sciences, vol 193. The Royal Society, pp 281–297

  21. Hill R, Hutchinson JW (1975) Bifurcation phenomena in the plane tension test. J Mech Phys Solids 23:239–264

    Article  MathSciNet  MATH  Google Scholar 

  22. Hosford WF (1979) On yield loci of anisotropic cubic metals. In: Proceedings of the seventh North American metalworking research conference, pp 191–196

  23. Hutchinson JW, Neale K, Needleman A (1978) Mechanics of sheet metal forming. Plenum Press, New York/London, pp 269–285

    Book  Google Scholar 

  24. Jacques N (2020) An analytical model for necking strains in stretched plates under dynamic biaxial loading. Int J Solids Struct 200–201:198–212

    Article  Google Scholar 

  25. Jacques N, Rodríguez-Martínez JA (2020) Influence on strain-rate history effects on the development of necking instabilities under dynamic loading conditions. Submitted for Publication

  26. Jouve D (2013) Analytic study of plastic necking instabilities during plane tension tests. Eur J Mech A Solids 39:180–196

    Article  MathSciNet  MATH  Google Scholar 

  27. Jouve D (2015) Analytic study of the onset of plastic necking instabilities during biaxial tension tests on metallic plates. Eur J Mech A Solids 50:59–69

    Article  MATH  Google Scholar 

  28. Karpp RR, Simon J (1976) An estimate of the strength of a copper shaped charge jet and the effect of strength on the breakup of a stretching jet. Tech. rep., U.S. Army Ballistic Research Laboratories (BRL)

  29. Marciniak Z, Kuczyński K (1967) Limit strains in the processes of stretch-forming sheet metal. Int J Mech Sci 9(9):609–620

    Article  MATH  Google Scholar 

  30. Mercier S, Granier N, Molinari A, Llorca F, Buy F (2010) Multiple necking during the dynamic expansion of hemispherical metallic shells, from experiments to modelling. J Mech Phys Solids 58:955–982

    Article  MathSciNet  MATH  Google Scholar 

  31. Mercier S, Molinari A (2004) Analysis of multiple necking in rings under rapid radial expansion. Int J Impact Eng 30:403–419

    Article  Google Scholar 

  32. Mises RV (1928) Mechanik der plastischen formänderung von kristallen. ZAMM J Appl Math Mech/Zeitschrift für Angewandte Mathematik und Mechanik 8(3):161–185

    Article  MATH  Google Scholar 

  33. Niordson FL (1965) A unit for testing materials at high strain rates. Exp Mech 5:29–32

    Article  Google Scholar 

  34. N’souglo KE, Jacques N, Rodríguez-Martínez JA (2021) A three-pronged approach to predict the effect of plastic orthotropy on the formability of thin sheets subjected to dynamic biaxial stretching. J Mech Phys Solids 146:104189

    Article  MathSciNet  Google Scholar 

  35. N’souglo KE, Rodríguez-Martínez JA, Cazacu O (2020) The effect of tension-compression asymmetry on the formation of dynamic necking instabilities under plane strain stretching. Int J Plast 128:102656

  36. N’souglo KE, Srivastava A, Osovski S, Rodríguez-Martínez JA (2018) Random distributions of initial porosity trigger regular necking patterns at high strain rates. Proc R Soc A Math Phys Eng Sci 474:20170575

  37. Parmar A, Mellor PB (1978) Predictions of limit strains in sheet metal using a more general yield criterion. Int J Mech Sci 20(6):385–391

    Article  MATH  Google Scholar 

  38. Petit J, Jeanclaude V, Fressengeas C (2005) Breakup of copper shaped-charge jets: experiment, numerical simulations, and analytical modeling. J Appl Phys 98(12):123521

    Article  Google Scholar 

  39. Rodríguez-Martínez JA, Molinari A, Zaera R, Vadillo G, Fernández-Sáez J (2017) The critical neck spacing in ductile plates subjected to dynamic biaxial loading: on the interplay between loading path and inertia effects. Int J Solids Struct 108:74–84

    Article  Google Scholar 

  40. Rodríguez-Martínez JA, Vadillo G, Fernández-Sáez J, Molinari A (2013) Identification of the critical wavelength responsible for the fragmentation of ductile rings expanding at very high strain rates. J Mech Phys Solids 61:1357–1376

    Article  MathSciNet  Google Scholar 

  41. Rodríguez-Martínez JA, Vadillo G, Zaera R, Fernández-Sáez J, Rittel D (2015) An analysis of microstructural and thermal softening effects in dynamic necking. In: iUTAM Symposium on materials and interfaces under high strain rate and large deformation. Mechanics of materials, vol 80, pp 298–310

  42. Shenoy VB, Freund LB (1999) Necking bifurcations during high strain rate extension. J Mech Phys Solids 47:2209–2233

    Article  MATH  Google Scholar 

  43. Sowerby R, Duncan JL (1971) Failure in sheet metal in biaxial tension. Int J Mech Sci 13(3):217–229

    Article  Google Scholar 

  44. Stören S, Rice J (1975) Localized necking in thin sheets. J Mech Phys Solids 23:421–441

    Article  MATH  Google Scholar 

  45. Tvergaard V (1982) On localization in ductile materials containing spherical voids. Int J Fract 18(4):237–252

    Article  Google Scholar 

  46. Tvergaard V, Needleman A (1984) Analysis of the cup-cone fracture in a round tensile bar. Acta Metall 32:157–169

    Article  Google Scholar 

  47. Vaz-Romero A, Rodríguez-Martínez JA, Mercier S, Molinari A (2017) Multiple necking pattern in nonlinear elastic bars subjected to dynamic stretching: the role of defects and inertia. Int J Solids Struct 125:232–243

    Article  Google Scholar 

  48. Walsh JM (1984) Plastic instability and particulation in stretching metals jets. J Appl Phys 56:1997–2006

    Article  Google Scholar 

  49. Xavier M, Czarnota C, Jouve D, Mercier S, Dequiedt JL, Molinari A (2020) Extension of linear stability analysis for the dynamic stretching of plates: spatio-temporal evolution of the perturbation. Eur J Mech A Solids 79:103860

    Article  MathSciNet  MATH  Google Scholar 

  50. Xue Z, Vaziri A, Hutchinson JW (2008) Material aspects of dynamic neck retardation. J Mech Phys Solids 56:93–113

    Article  MATH  Google Scholar 

  51. Zaera R, Rodríguez-Martínez JA, Vadillo G, Fernández-Sáez J (2014) Dynamic necking in materials with strain induced martensitic transformation. J Mech Phys Solids 64:316–337

    Article  MathSciNet  Google Scholar 

  52. Zaera R, Rodríguez-Martínez JA, Vadillo G, Fernández-Sáez J, Molinari A (2015) Collective behaviour and spacing of necks in ductile plates subjected to dynamic biaxial loading. J Mech Phys Solids 85:245–269

    Article  MathSciNet  Google Scholar 

  53. Zhang H, Ravi-Chandar K (2010) On the dynamics of localization and fragmentation-IV. Expansion of Al 6061-O tubes. Int J Fract 163:41–65

    Article  MATH  Google Scholar 

  54. Zheng X, N’souglo KE, Rodríguez-Martínez JA, Srivastava A (2020) Dynamics of necking and fracture in ductile porous materials. J Appl Mech Trans ASME 87(4):41005

    Article  Google Scholar 

  55. Zhou F, Molinari JF, Ramesh KT (2006) An elasto-visco-plastic analysis of ductile expanding ring. Int J Impact Eng 33:880–891

    Article  Google Scholar 

Download references

Acknowledgements

OC acknowledges partial support provided by AFOSR grant FA9550-18-1-0517 and the support during her sabbatical at the UC3M through the Programa Cátedras de Excelencia UC3M-Santander.

Funding

JAR-M and KEN acknowledge the financial support provided by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme. Project PURPOSE, grant agreement 758056.

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Appendices

Appendix A: Stability analysis results and finite element calculations with prescribed stretch rate

In this section, we show finite element results and stability analysis predictions obtained imposing the initial axial strain rate \({\dot{\varepsilon }}^0_{xx}\) as the loading condition, instead of the initial equivalent strain rate \(\dot{{\bar{\varepsilon }}}^0\), as in all other calculations performed in this work (see Sect. 3.1). The results correspond to Material 2 (\(c=-1.5\)), Material 3 (\(c=81/66\)) and Material 4 (\(c=2.25\)). The stability analysis predictions are obtained with the critical cumulative instability index \(I_c=3.75\). We also include finite element results for Material 2 imposing the initial equivalent strain rate.

Figure 17a, b show the evolution of the necking strain \({\bar{\varepsilon }}^{\textit{neck}}\) and the necking stretch \({F}^{\textit{neck}}_{xx}\) with \(L^0/h^0\) for \({\dot{\varepsilon }}^0_{xx}=4000\,\text {s}^{-1}\). The results obtained for the three materials are very similar to those reported Fig. 10a, b for an imposed initial equivalent strain rate of \(\dot{{\bar{\varepsilon }}}^0=4000\,\text {s}^{-1}\). Moreover, Fig. 18a, b display the \({\bar{\varepsilon }}^{\textit{neck}} - L^0/h^0\) and \({F}^{\textit{neck}}_{xx}-L^0/h^0\) curves for \({\dot{\varepsilon }}^0_{xx}=80{,}000\,\text {s}^{-1}\). The results are also quantitatively similar to those reported in Fig. 13a, b for \(\dot{{\bar{\varepsilon }}}^0=80{,}000\,\text {s}^{-1}\), suggesting that imposing either the initial equivalent strain rate or the initial stretch rate in the calculations yields similar results. However, there are qualitative differences on the effect of the parameter c on the necking strain and the necking stretch. Specifically, both finite element calculations and stability analysis predictions presented in Fig. 18a show that for \({\dot{\varepsilon }}^0_{xx}=80{,}000\,\text {s}^{-1}\) the necking strain is greater as the parameter c decreases (i.e. as the rate of accumulation of plastic deformation increases), while this is not the case when the initial equivalent strain rate is imposed, since the smaller necking strain for long wavelengths corresponds to \(c=-1.5\) (see Fig. 13a). In addition, Fig. 18b shows that, while the relative order of the \({F}^{\textit{neck}}_{xx}-L^0/h^0\) curves for the three materials is the same obtained for \(\dot{{\bar{\varepsilon }}}^0=80{,}000\,\text {s}^{-1}\), the difference in the necking stretch for the whole range of wavelengths is less (see Fig. 13b).

Fig. 17
figure 17

Comparison between finite element results (FEM) and linear stability analysis predictions (LSA) obtained with Drucker yield criterion [11] for Material 2 (\(c=-1.5\) and \(\sigma _T / \tau _Y=1.674\)), Material 3 (Hershey–Hosford for a BCC material, \(c=81/66\) and \(\sigma _T / \tau _Y=1.790\)) and Material 4 (\(c=2.25\) and \(\sigma _T / \tau _Y=1.852\)), respectively. The initial stretch rate along the axial direction is \({\dot{\varepsilon }}^0_{xx}=4000\,\text {s}^{-1}\). a Necking strain \({\bar{\varepsilon }}^{\textit{neck}}\) versus \(L^0/h^0\). b Necking stretch \({F}^{\textit{neck}}_{xx}\) versus \(L^0/h^0\). In the finite element simulations the amplitude of the imperfection is \(\Delta =0.2{\%}\). In the linear stability analysis the critical cumulative instability index is \(I_c=3.75\). We also include finite element results for Material 2 (\(c=-1.5\)) imposing the initial equivalent strain rate \(\dot{{\bar{\varepsilon }}}^0=4000\,\text {s}^{-1}\)

Fig. 18
figure 18

Comparison between finite element results (FEM) and linear stability analysis predictions (LSA) obtained with Drucker yield criterion [11] for Material 2 (\(c=-1.5\) and \(\sigma _T / \tau _Y=1.674\)), Material 3 (Hershey–Hosford for a BCC material, \(c=81/66\) and \(\sigma _T / \tau _Y=1.790\)) and Material 4 (\(c=2.25\) and \(\sigma _T / \tau _Y=1.852\)), respectively. The initial stretch rate along the axial direction is \({\dot{\varepsilon }}^0_{xx}=80{,}000\,\text {s}^{-1}\). a Necking strain \({\bar{\varepsilon }}^{\textit{neck}}\) versus \(L^0/h^0\). b Necking stretch \({F}^{\textit{neck}}_{xx}\) versus \(L^0/h^0\). In the finite element simulations the amplitude of the imperfection is \(\Delta =0.2{\%}\). In the linear stability analysis the critical cumulative instability index is \(I_c=3.75\). We also include finite element results for Material 2 (\(c=-1.5\)) imposing the initial equivalent strain rate \(\dot{{\bar{\varepsilon }}}^0=80{,}000\,\text {s}^{-1}\)

Appendix B: Additional comparison between critical instantaneous instability index and critical cumulative instability index

In this section, we present additional comparisons between the predictions obtained with the critical instantaneous instability index \({\hat{\eta }}^+_c=19\) and the critical cumulative instability index \(I_c=3.75\) for Material 1 (von Mises material, \(c=0\)). Namely, Fig. 19 shows the evolution of the necking strain \({\bar{\varepsilon }}^{\textit{neck}}\) with \(L^0/h^0\) for five different initial equivalent strain rates \(\dot{{\bar{\varepsilon }}}^0=400\,\text {s}^{-1}\), \(4000\,\text {s}^{-1}\), \(10{,}000\,\text {s}^{-1}\), \(20{,}000\,\text {s}^{-1}\) and \(80{,}000\,\text {s}^{-1}\). The stability analysis results are compared with finite element calculations.

For the lower initial equivalent strain rates, \(400\,\text {s}^{-1}\), \(4000\,\text {s}^{-1}\) and \(10{,}000\,\text {s}^{-1}\), the predictions of the stability analysis with both criteria are similar (especially for \(400\,\text {s}^{-1}\)), and find qualitative and quantitative agreement with the finite element calculations, being the predictions obtained with \({\hat{\eta }}^+_c=19\) closer to the numerical simulations. On the other hand, for greater initial equivalent strain rates, there are important differences between the results obtained with \({\hat{\eta }}^+_c=19\) and \(I_c=3.75\) so that, contrary to what occurs at lower strain rates, for long wavelengths the instantaneous instability index predicts values of the necking strain greater than the cumulative instability index (see also Fig. 6). The different results obtained with both criteria are particularly noticeable for \(80{,}000\,\text {s}^{-1}\), for this strain rate the predictions with \(I_c=3.75\) being much closer to the finite element calculations than the results obtained with \({\hat{\eta }}^+_c=19\), showing that the analytical predictions based on the instantaneous instability index overestimate the role of inertia on the necking strain.

Fig. 19
figure 19

Necking strain \({\bar{\varepsilon }}^{\textit{neck}}\) versus \(L^0/h^0\). Comparison between finite element results (FEM) and linear stability analysis predictions (LSA) obtained with Material 1 (von Mises material, \(c=0\) and \(\sigma _T / \tau _Y=\sqrt{3}\)). The initial equivalent strain rate is: a \(\dot{{\bar{\varepsilon }}}^0=400\,\text {s}^{-1}\), b \(\dot{{\bar{\varepsilon }}}^0=4000\,\text {s}^{-1}\), c \(\dot{{\bar{\varepsilon }}}^0=10{,}000\,\text {s}^{-1}\), d \(\dot{{\bar{\varepsilon }}}^0=20{,}000\,\text {s}^{-1}\) and e \(\dot{{\bar{\varepsilon }}}^0=80{,}000\,\text {s}^{-1}\). In the finite element simulations the amplitude of the imperfection is \(\Delta =0.2{\%}\). Linear stability analysis results are shown for the critical cumulative instability index \(I_c=3.75\), and the critical instantaneous instability index \({\hat{\eta }}^+_c=19\)

Appendix C: Stability analysis results with strain rate dependent critical cumulative instability index

In this section, we compare finite element results with stability analysis predictions obtained using strain rate dependent critical cumulative index. The calculations are performed with Material 1 (von Mises material, \(c=0\) and \(\sigma _T / \tau _Y=\sqrt{3}\)), and the results correspond to four different initial equivalent strain rates: \(400\,\text {s}^{-1}\), \(10{,}000\,\text {s}^{-1}\), \(20{,}000\,\text {s}^{-1}\) and \(80{,}000\,\text {s}^{-1}\). The critical cumulative instability index is taken to be a second order polynomial \(I_c=F+ G \, \dot{{\bar{\varepsilon }}}^0 + H \left( \dot{{\bar{\varepsilon }}}^0\right) ^2\), whose coefficients \(F=0.96454\), \(G=8.883 \cdot 10^{-5}\) and \(H=-4.7983 \cdot 10^{-10}\) have been determined obtaining the value of \(I_c\) for three initial equivalent strain rates, \(400\,\text {s}^{-1}\), \(40{,}000\,\text {s}^{-1}\) and \(80{,}000\,\text {s}^{-1}\), following the methodology discussed in Sect. 6.1. Similar procedure has been recently used by Jacques and Rodríguez-Martínez [25] to determine the evolution of the critical cumulative index with the strain rate, to predict multiple necking formation in viscoplastic metallic bars subjected to dynamic stretching.

Figure 20 shows that the stability analysis predictions obtained with strain rate dependent critical cumulative index are in quantitative agreement with the unit-cell calculations, within the whole range of wavelengths investigated, and for all the strain rates considered (including strain rates other than those used to calibrate \(I_c\)). It becomes apparent that accounting for the strain rate dependence of \(I_c\) improves the predictions of the stability analysis, yet at the expense of needing additional calibration data. A throughout discussion of the pros and cons of considering the functional dependence of \(I_c\) on the strain rate is left for a future work.

Fig. 20
figure 20

Necking strain \({\bar{\varepsilon }}^{\textit{neck}}\) versus \(L^0/h^0\). Comparison between finite element results (FEM) and linear stability analysis predictions (LSA) obtained for Material 1 (von Mises material, \(c=0\) and \(\sigma _T / \tau _Y=\sqrt{3}\)). The results correspond to four different initial equivalent strain rates \(\dot{{\bar{\varepsilon }}}^0=400\,\text {s}^{-1}\), \(10{,}000\,\text {s}^{-1}\), \(20{,}000\,\text {s}^{-1}\) and \(80{,}000\,\text {s}^{-1}\). In the finite element simulations the amplitude of the imperfection is \(\Delta =0.2{\%}\). In the linear stability analysis the critical cumulative instability index is taken strain rate dependent \(I_c=0.96454+8.883 \cdot 10^{-5} \, \dot{{\bar{\varepsilon }}}^0-4.7983 \cdot 10^{-10} \cdot \left( \dot{{\bar{\varepsilon }}}^0\right) ^2\)

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Rodríguez-Martínez, J.A., Cazacu, O., Chandola, N. et al. Effect of the third invariant on the formation of necking instabilities in ductile plates subjected to plane strain tension. Meccanica 56, 1789–1818 (2021). https://doi.org/10.1007/s11012-021-01330-6

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