The limit diagram under hot sheet metal forming.

I. E. Keller ; D. S. Petukhov ; A. Kazantsev ; V. N. Trofimov

Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 22 (2018) no. 3, pp. 447-486.

Résumé

Methods of theoretical analysis and experimental verification of conditions of limit deformation are considered for a reasonable choice of the constitutive equations for mathematical modeling of processes of hot and warm treatment by pressure of sheet metal products with a deep drawing. Attention is focused on the forming limit curve of sheet metal on the plane of the principal strains (one of that corresponds to stretching, and the second can specify stretching or compression), the characteristic of the local state of the material corresponding to the critical growth of strain localization. Localization here is understood as a local thinning of the sheet and corresponds to diffuse form of localization. Other defects (shear bands, crack formation) develop from this limiting state or (formation of folds and wrinkles) are not local and require complete formulation of the problem. The forming limit curve (FLC) defines the conditions of realization of a technological process and can be theoretically predicted depending on the constitutive equations of plasticity, indicator of critical state and initial imperfections. The Marciniak–Kuczyński scheme is considered for getting FLC, where the sample has two zones of homogeneous strains and allows analytical reduction of the problem to the system of several ordinary differential equations solved numerically. The experimental methods assume testing by pressing a punch with a spherical or cylindrical tip into a specimen cut from a sheet. Depending on the depth of the lateral cutouts from the specimen, it can be provided tension or compression of the specimen in the transverse direction in these tests. Both approaches are analyzed as tools for selection and experimental verification of the constitutive model and the limit state indicator. They solve methodological problem of identification of mathematical models on a quite non-standard experiments involving strain localization. With the use of Marciniak–Kuczyński scheme the effect of a number of yield criteria for anisotropic sheet metal, hardening laws, damage accumulation models and criteria of viscous failure on qualitative and quantitative features of the FLC. To do this a proprietary algorithm has been developed. Experimental standard test methods of Hasek, Marciniak and Nakajima were implemented numerically in the software package LS-DYNA. The numerical FLD obtained were compared with theoretical and experimental ones. Possibilities of integration into Marciniak–Kuczyński scheme the dependence on temperature, strain rate and microstructure parameters for each basic rigid-plastic (scleronomous) model were discussed. It is noted this scheme is significantly limited by proportional changes of the main deformations in the sample outside and inside the strain localization zone. It is revealed this scheme is not adapted for determination of limit properties of the metals deformable in the conditions of deformation softening (aluminum, titanium alloys and some steels at temperatures of dynamic recrystallization). For a wider range of material deformation conditions, there is no alternative to the above-mentioned numerical method for predicting FLC. An open and relevant question is the description of the evolution of anisotropic plastic and fracture properties due to the anisotropic damage accumulation.

Détail

Keywords: sheet metal forming, forming limit diagram, viscous failure criteria, plastically anisotropic materials, material models, standard experiment.
Mots-clés : calculation

Bibliographie

[1] “Defects in thermo-mechanical processing”, Thermo-Mechanical Processing of Metallic Materials, Pergamon Materials Series, 11, eds. B. Verlinden, J. Driver, I. Samajdar, R. D. Doherty, Pergamon, 2007, 333–348 | DOI

[2] Marciniak Z., Duncan J. L., Hu S. J., Mechanics of Sheet Metal Forming, Butterworth Heinemann, London, 2002, xii+211 pp. | DOI

[3] Silva M. B., Isik K., Tekkaya A. E., Martins P. A. F., “Fracture Loci in Sheet Metal Forming: A Review”, Acta Metall. Sin. (Engl. Lett.), 28:12 (2015), 1415–1425 | DOI

[4] Ilyushin A. A., “The deformation of a visco-plastic solid”, Uchenye Zapiski Moskovskogo Gosudarstvennogo Universiteta. Mekhanika, 1940, no. 39, 3–81 (In Russian)

[5] Marciniak Z., Kuczyński K., “Limit strains in the processes of stretch-forming sheet metal”, Int. J. Mech. Sci., 9:9 (1967), 609–620 | DOI

[6] ISO 12004-2:2008. Metallic materials – Sheet and strip – Determination of forming-limit curves – Part $2:$ Determination of forming-limit curves in the laboratory, International Organization for Standardization, 2008, 27 pp. | DOI

[7] ASTM E2218-15. Standard Test Method for Determining Forming Limit Curves, ASTM International, West Conshohocken, PA, USA, 2015, 16 pp. | DOI

[8] Bogatov A. A., Mizhiritskii O. I., Smirnov S. V., Resurs plastichnosti metallov pri obrabotke davleniem [Plasticity resource at metal treatment under pressure], Metallurgiia, Moscow, 1984, 144 pp. (In Russian)

[9] Smirnov S. V., Shveikin V. P., Plastichnost' i deformiruemost' uglerodistykh stalei pri obrabotke davleniem [Plasticity and deformability of carbon steels under metal forming], UrO RAN, Ekaterinburg, 2009 (In Russian)

[10] Kalpin Yu. G., Perfilov V. I., Petrov P. A., Riabov V. A., Filippov Yu. K., Soprotivlenie deformatsii i plastichnost' pri obrabotke metallov davleniem [Deformation resistance and plasticity in the processing of metals by pressure], Mashinostroenie, Moscow, 2011, 244 pp. (In Russian)

[11] Kim J.B., Yang D.Y., “Prediction of wrinkling initiation in sheet metal forming processes”, Engineering Computations, 20:1 (2003), 6–39 | DOI | Zbl

[12] Freudenthal A. M., The Inelastic Behavior of Engineering Materials and Structures, John Wiley Sons, New York, 1950

[13] Clift S. E., Hartley P., Sturgess C. E. N., Rowe G. W., “Fracture prediction in plastic deformation processes”, Int. J. Mech. Sci., 32:1 (1990), 1–17 | DOI

[14] Cockcroft M. G., Latham D. J., “Ductility and the workability of metals”, J. Inst. Metals, 96 (1968), 33-39

[15] Oh S. I., Chen C. C., Kobayashi S., “Ductile fracture in axisymmetric extrusion and drawing. Part 2. Workability in extrusion and drawing”, J. Eng. Ind., 101:1 (1979), 36–44 | DOI

[16] McClintock F. A., “A Criterion for Ductile Fracture by the Growth of Holes”, J. Appl. Mech., 35:2 (1968), 363–371 | DOI

[17] Rice J. R., Tracey D. M., “On the ductile enlargement of voids in triaxial stress fields”, J. Mech. Phys. Solids, 17:3 (1969), 201–217 | DOI

[18] Johnson R., Cook W. H., “Fracture characteristics of three metals subjected to various strains, strain rates, temperatures and pressures”, Eng. Fract. Mech., 21:1 (1985), 31–48 | DOI

[19] Wierzbicki T., Bao Y., Lee Y.W., Bai Y., “Calibration and evaluation of seven fracture models”, Int. J. Mech. Sci., 47:4 (2005), 719–743 | DOI

[20] Bao Y., Wierzbicki T., “On fracture locus in the equivalent strain and stress triaxiality space”, Int. J. Mech. Sci., 46:1 (2004), 81–98 | DOI

[21] Wilkins M. L., Streit R. D., Reaugh J. E., Cumulative-strain-damage model of ductile fracture: simulation and prediction of engineering fracture tests, Technical Report UCRL-53058, Lawrence Livermore National Laboratory, October 3, 1980 Available at (July 11, 2018) https://www.osti.gov/servlets/purl/6628920-KUgBmG/

[22] Yang X., Lang L., Liu K., Guo C., “Modified MK model combined with ductile fracture criterion and its application in warm hydroforming”, Trans. Nonferrous Met. Soc. China, 25:10 (2015), 3389–3398 | DOI

[23] Takuda H., Mori K., Hatta N., “The application of some criteria for ductile fracture to the prediction of the forming limit of sheet metals”, J. Mat. Proc. Tech., 95:1–3 (1999), 116–121 | DOI

[24] Oyane M., Sato T., Okimoto K., Shima S., “Criteria for Ductile Fracture and Their Applications”, Journal of Mechanical Working Technology, 4:1 (1980), 65–81 | DOI

[25] Ko Y.K., Lee J.S., Huh H. et al., “Prediction of fracture in hubhole expanding process using a new ductile fracture criterion”, J. Mat. Proc. Tech., 187 (2007), 358–362 | DOI

[26] Lou Y., Huh H., Lim S., Pack K., “New ductile fracture criterion for prediction of fracture forming limit diagrams of sheet metals”, Int. J. Sol. Struct., 49:25 (2012), 3605–3615 | DOI

[27] Ghosh A. K., “A criterion for ductile fracture in sheets under biaxial loading”, Metall. Trans. A, 7:4 (1976), 523–533 | DOI

[28] Norris (Jr.) D. M., Reaugh J. E., Moran B., Quinones D. F., “A Plastic-Strain, Mean-Stress Criterion for Ductile Fracture”, J. Eng. Mater. Technol., 100:3 (1978), 279–286 | DOI

[29] Atkins A. G., “Possible explanation for unexpected departures in hydrostatic tension-fracture strain relations”, Metal Science, 15:2 (1981), 81–83 | DOI

[30] Weck A., Wilkinson D. S., “Experimental investigation of void coalescence in metallic sheets containing laser drilled holes”, Acta Materialia, 56:8 (2008), 1774–1784 | DOI

[31] Li H., Fu M. W., Lu J., Yang H., “Ductile fracture: Experiments and computations”, Int. J. Plast., 27:2 (2011), 147–180 | DOI

[32] Vlasov A.V., “O primenenii kriteriia Kokrofta–Letema dlia prognozirovaniia razrusheniia pri kholodnoi ob"emnoi shtampovke”, Izv. Tulsk. Gos. Un-ta. Tekhn. Nauki, 2017, no. 11–1, 46–58 (In Russian)

[33] Atkins A. G., “Fracture mechanics and metal forming. Damage mechanics and the local approach of yesterday and today”, Fracture research in retrospective, eds. H. P. Rossmanith, Balkema, Rotterdam, 1997, 327–350

[34] Botkin A. V., Valiev R. Z., Stepin P. S., Baymukhametov A. Kh., “Evaluation of Metals Damage at Cold Plastic Deformation at Kokroft–Latam Fracture Model”, Materials Defamation and Damage, 2011, no. 7, 17–22 (In Russian)

[35] Isik K., Silva M. B., Tekkaya A. E., Martins P. A. F., “Formability limits by fracture in sheet metal forming”, J. Mat. Proc. Tech., 214:8 (2014), 1557–1565 | DOI

[36] Silva M. B., Isik K., Tekkaya A. E. et al., “Fracture Toughness and Failure Limits in Sheet Metal Forming”, J. Mat. Proc. Tech., 234:8 (2016), 1557–1565 | DOI

[37] Atkins A. G., “Fracture in forming”, J. Mat. Proc. Tech., 56:1–4 (1996), 609–618 | DOI

[38] Tvergaard V., Needleman A., “Analysis of the cup-cone fracture in a round tensile bar”, Acta Metallurgica, 32:1 (1984), 157–169 | DOI

[39] Vlasov A.V., Gerasimov D.A., “Realization of the Gurson-Tvergaard-Needleman model for calculation the processes of cold bulk forging of incompressible materials”, Izv. Vyssh. Uchebn. Zaved., Mashinostr., 2017, no. 8 (689), 8–17 (In Russian) | DOI

[40] Gurson A. L., “Continuum theory of ductile rupture by void nucleation and growth. I. Yield criteria and flow rules for porous ductile media”, J. Eng. Mater. Technol., 99:1 (1977), 2–15 | DOI

[41] Needleman A., Triantafyllidis N., “Void growth and local necking in biaxially stretched sheets”, J. Eng. Mater. Technol., 100:2 (1978), 164–169 | DOI

[42] Nahshon K., Hutchinson J. W., “Modification of the Gurson Model for shear failure”, Eur. J. Mech., A, Solids, 27:1 (2008), 1–17 | DOI | Zbl

[43] Andrade F. X. C., Feucht M., Haufe A., Neukamm F., “An incremental stress state dependent damage model for ductile failure prediction”, Int. J. Fract., 200:1–2 (2016), 127–150 | DOI

[44] Andrade F. X. C., Feucht M., Haufe A., On the Prediction of Material Failure in LS-DYNA ®: A Comparison Between GISSMO and DIEM, The 13th LS-DYNA International Conference (June 8–10, 2014, Dearborn, MI), 2014 Available at (July 11, 2018) https://goo.gl/AQUhP9

[45] Chaboche J. L., “Continuum damage mechanics: present state and future trends”, Nucl. Eng. Design, 105:1 (1987), 19–33 | DOI

[46] Lee H., Peng K. E., Wang J., “An anisotropic damage criterion for deformation instability and its application to forming limit analysis of metal plates”, Eng. Fract. Mech., 21:5 (1985), 1031–1054 | DOI

[47] Volkov I. A., Igumnov L. A., Vvedenie v kontinual'nuiu mekhaniku povrezhdennoi sredy [Introduction into continuum mechanics of damaged media], Fizmatlit, Moscow, 2017, 304 pp. (In Russian)

[48] Murakami S., Continuum Damage Mechanics. A Continuum Mechanics Approach to the Analysis of Damage and Fracture, Solid Mechanics and Its Applications, 185, Springer, Netherlands, 2012, xxix+402 pp. | DOI

[49] Zhu Y. Y., Cescotto S., “A fully coupled elasto-visco-plastic damage theory for anisotropic materials”, Int. J. Sol. Struct., 32:11 (1995), 1607–1641 | DOI | Zbl

[50] Badreddine H., Saanouni K., Nguyen T. D., “Damage anisotropy and its effect on the plastic anisotropy evolution under finite strains”, Int. J. Sol. Struct., 63 (2015), 11–31 | DOI

[51] Badreddine H., Labergère C., Saanouni K., “Ductile damage prediction in sheet and bulk metal forming”, Comptes Rendus Mécanique, 344:4–5 (2016), 296–318 | DOI

[52] Simo J. C., Ju J. W., “On continuum damage-elastoplasticity at finite strains. A computational framework”, Computational Mechanics, 5:5 (1989), 375–400 | DOI | Zbl

[53] Voyiadjis G. Z., Abu Al-Rub R. K., Palazotto A. N., “Thermodynamic framework for coupling of non-local viscoplasticity and non-local anisotropic viscodamage for dynamic localization problems using gradient theory”, Int. J. Plast., 20:6 (2004), 981–1038 | DOI | Zbl

[54] Lemaitre J., “A continuous damage mechanics model for ductile fracture”, J. Eng. Mater. Technol., 107:1 (1985), 83–89 | DOI

[55] Heibel S., Nester W., Clausmeyer T., Tekkaya A.E., “Influence of Different Yield Loci on Failure Prediction with Damage Models”, J. Physics: IOP Conf. Series, 896 (2017), 012081 | DOI

[56] Hill R., “A theory of the yielding and plastic flow of anisotropic metals”, Proc. R. Soc. Lond., Ser. A, 193:1033 (1948), 281–297 | DOI | MR | Zbl

[57] Hill R., “Theoretical plasticity of textured aggregates”, Math. Proc. Camb. Philos. Soc., 85:1 (1979), 179–191 | DOI | MR | Zbl

[58] Hill R., “Constitutive dual potentials in classical plasticity”, J. Mech. Phys. Solids, 35:1 (1987), 23–33 | DOI | MR | Zbl

[59] Hill R., “Constitutive modelling of orthotropic plasticity in sheet metals”, J. Mech. Phys. Solids, 38:3 (1990), 405–417 | DOI | MR | Zbl

[60] Hill R., “A user-friendly theory of orthotropic plasticity in sheet metals”, Int. J. Mech. Sci., 35:1 (1993), 19–25 | DOI | Zbl

[61] Barlat F., Lian J., “Plastic behavior and stretchability of sheet metals. Part II. A yield function for orthotropic sheets under plane stress conditions”, Int. J. Plast., 5:2 (1989), 51–66 | DOI

[62] Barlat F., Lege D. J., Brem J. C., “A 6-component yield function for anisotropic materials”, Int. J. Plast., 7:7 (1991), 693–712 | DOI

[63] Barlat F., Maeda Y., Chung K. et al., “Yield function development for aluminum alloy sheets”, J. Mech. Phys. Solids, 45:11 (1997), 1727–1763 | DOI

[64] Barlat F., Brem J. C., Yoon J. W. et al., “Plane stress yield function for aluminum alloy sheets. Part 1: Theory”, Int. J. Plast., 19:9 (2003), 1297–1319 | DOI | Zbl

[65] Barlat F., Aretz H., Yoon J.W. et al., “Linear transfomation-based anisotropic yield functions”, Int. J. Plast., 21:5 (2005), 1009–1039 | DOI | Zbl

[66] Karafillis A. P., Boyce M. C., “A general anisotropic yield criterton using bounds and a transformation weighting tensor”, J. Mech. Phys. Solids, 41:12 (1993), 1859–1886 | DOI | Zbl

[67] Vial-Edwards C., “Yield loci of FCC and BCC sheet metals”, Int. J. Plast., 13:5 (1997), 521–531 | DOI | Zbl

[68] Banabic D., Sheet metal forming processes. Constitutive Modelling and Numerical Simulation, Springer, 2010, xv+301 pp. | DOI

[69] Multiscale Modelling in Sheet Metal Forming, eds. Banabic D., Springer, 2016, xiii+405 pp. | DOI

[70] ASTM E8/E8M–16a. Standard Test Methods for Tension Testing of Metallic Materials, ASTM International, West Conshohocken, PA, USA, 2016, 30 pp. | DOI

[71] ASTM E517-00. Standard Test Method for Plastic Strain Ratio $r$ for Sheet Metal, ASTM International, West Conshohocken, PA, USA, 2010, 8 pp. | DOI

[72] Voce E., “The relationship between stress and strain for homogeneous deformation”, J. Inst. Met., 74 (1948), 537–562

[73] Butuc M. C., Teodosiu C., Barlat F., Gracio J. J., “Analysis of sheet metal formability through isotropic and kinematic hardening models”, Eur. J. Mech., A, Solids, 30:4 (2011), 532–546 | DOI | Zbl

[74] Sung J. H., Kim J. H., Wagoner R. H., “A plastic constitutive equation incorporating strain, strain-rate, and temperature”, Int. J. Plast., 26:12 (2010), 1746–1771 | DOI | Zbl

[75] Swift H. W., “Plastic Instability under Plane Stress”, J. Mech. Phys. Solids, 1:1 (1952), 1–18 | DOI

[76] Hu P., Ma N., Liu L., Zhu Y., Theories, Methods and Numerical Technology of Sheet Metal Cold and Hot Forming. Analysis, Simulation and Engineering Applications, Springer, London, 2013, xiv+210 pp. | DOI

[77] Schotten K., Mathematische Beschreibung der Fließkurve bei der Warmumformung verschiedener Stähle, Dissertation Rheinisch-Westfälische Technische Hochschule Aachen, Shaker Verlag, 2000 Available at (July 11, 2018) (In German) http://publications.rwth-aachen.de/record/58390

[78] Tong L., Stahel S., Hora P., “Modeling for the FE-Simulation of Warm Metal Forming Processes”, AIP Conference Proceedings, 778:1 (2005), 625–629 | DOI

[79] Khan A.S., Baig M., “Anisotropic responses, constitutive modeling and the effects of strain-rate and temperature on the formability of an aluminum alloy”, Int. J. Plast., 27:4 (2011), 522–538 | DOI | Zbl

[80] Khan A. S., Liang R., “Behaviors of three BCC metal over a wide range of strain rates and temperatures: experiments and modeling”, Int. J. Plast., 15:10 (1999), 1089–1109 | DOI | Zbl

[81] Khan A. S., Liang R., “Behaviors of three BCC metals during non-proportional multi-axial loadings: experiments and modeling”, Int. J. Plast., 16:12 (2000), 1443–1458 | DOI | Zbl

[82] Khan A. S., Suh Y. S., Kazmi R., “Quasi-static and dynamic loading responses and constitutive modeling of titanium alloys”, Int. J. Plast., 20:12 (2004), 2233–2248 | DOI | Zbl

[83] Zener C., Hollomon J. H., “Effect of Strain Rate Upon Plastic Flow of Steel”, J. Appl. Phys., 15:1 (1944), 22–32 | DOI

[84] Puchi-Cabrera E. S. , Staia M. H., Guerin J. D. et al., “An experimental analysis and modeling of the work-softening transient due to dynamic recrystallization”, Int. J. Plast., 54 (2014), 113–131 | DOI

[85] Nemat–Nasser S., “Experimentally-based micromechanical modeling of metal plasticity with homogenization form micro-to-macro-scale properties”, IUTAM Symposium on Micro- and Macrostructural Aspects of Thermoplasticity, 1999, 101–113

[86] Panicker S. S., Panda S. K., “Formability Analysis of AA5754 Alloy at Warm Condition: Appraisal of Strain Rate Sensitive Index”, Materials Today: Proceedings, 2:4–5 (2015), 1996–2004 | DOI

[87] Lang L., Du P., Liu B., Cai G., Liu K., “Pressure rate controlled unified constitutive equations based on microstructure evolution for warm hydroforming”, J. All. Compounds, 574 (2013), 41-48 | DOI

[88] Chen H., Cao Ch., Guo L., Lin H., “Hot deformation mechanism and microstructure evolution of TC11 titanium alloy in $\beta$ field”, Trans. Nonferrous Met. Soc. China, 18:5 (2008), 1021-1027 | DOI

[89] Quan G., Liu K., Zhou J., Chen B., “Dynamic softening behaviors of 7075 aluminum alloy”, Trans. Nonferrous Met. Soc. China, 199 (2009), 537–541 | DOI

[90] Shesterikov S. A., Yumasheva M. A., “Concretization of the equation of state in the theory of creep”, Izv. Akad. Nauk SSSR, Seriya Mekhanika Tverdogo Tela, 1984, no. 1, 86–91 (In Russian)

[91] Alexandrov S., Mishuris G., “Viscoplasticity with a saturation stress: distinguishing features of the model”, Arch. Appl. Mech., 77:1 (2007), 35–47 | DOI | MR | Zbl

[92] Hill R., “On Discontinuous Plastic States, with Special Reference to Localized Necking in Thin Sheets”, J. Mech. Phys. Solids, 1:1 (1952), 19-30 | DOI | MR

[93] Hutchinson J. W., Neale K. W., “Sheet necking: II. Time-independent behavior”, Mechanics of sheet metal forming, eds. D. P. Koistinen, N. M. Wang, Springer, Boston, MA, 1978, 111–126 | DOI

[94] Kotkunde N., Srinivasan S., Krishna G., Gupta A.K., Singh S.K., “Influence of material models on theoretical forming limit diagram prediction for Ti-6Al-4V alloy under warm condition”, Trans. Nonferrous Met. Soc. China, 26 (2016), 736–746 | DOI

[95] Chu C.-C., “An analysis of localized necking in punch stretching”, Int. J. Sol. Struct., 16:10 (1980), 913–931 | DOI | Zbl

[96] Davies R. W., Grant G. J., Khaleel M. A. et all., “Forming-limit diagrams of aluminum tailor-welded blank weld material”, Metall. Mat. Trans. A, 32:2 (2001), 275–283 | DOI

[97] Graf A., Hosford W. F., “Calculations of Forming Limit Diagrams”, Metall. Trans. A, 21:1 (1990), 87–94 | DOI

[98] Lian J., Baudelet B., “Forming Limit Diagram of Sheet Metal in the Negative Minor Strain Region”, Mat. Sci. Eng., 86 (1987), 137–144

[99] Lian J., Barlat F., Baudelet B., “Plastic behaviour and stretchability of sheet metals. Part II: Effect of yield surface shape on sheet forming limit”, Int. J. Plast., 5:2 (1989), 131–147 | DOI

[100] Bruschi S., Altan T., Banabic D., Bariani P.F. et al., “Testing and modelling of material behaviour and formability in sheet metal forming”, CIRP Annals — Manufacturing Technology, 63:2 (2014), 727–749 | DOI

[101] Altmeyer G., “Theoretical and Numerical Comparison of Limit Point Bifurcation and Maximum Force Criteria. Application to the Prediction of Diffuse Necking”, Modeling and Numerical Simulation of Material Science, 3:1 (2013), 39–47 | DOI

[102] Khan A. S., Liu H., “Strain rate and temperature dependent fracture criteria for isotropic and anisotropic metals”, Int. J. Plast., 37 (2012), 1–15 | DOI

[103] Khan A. S., Suh Y. S., Kazmi R., “Quasi-static and dynamic loading responses and constitutive modeling of titanium alloys”, Int. J. Plast., 20:12 (2004), 2233–2248 | DOI | Zbl

[104] Safari M., Hoseinipour S. J., Azodic H.D., Yousefzadeha Sh., “Experimental and Theoretical Investigation of Forming Limit Diagram (FLD) and Forming Limit Stress Diagram (FLSD) For Aluminum Alloy 3105”, International Conference On Advances In Materials And Processing Technologies (AMPT2010) (Paris, France, 24–27 October 2010), AIP Conference Proceedings, 1315, ed. F. Chinesta, Y. Chastel, M. El Mansori, 2011, 45–50 | DOI

[105] Park N., Huh H., Lim S. J., Lou Y. et al., “Fracture-based Forming Limit Criteria for Anisotropic Materials in Sheet Metal Forming”, Int. J. Plast., 96 (2017), 1–35 | DOI

[106] Pourboghrat F., Venkatesan S., Carsley J. E., “LDR and hydroforming limit for deep drawing of AA5754 aluminum sheet”, J. Manuf. Process., 15:4 (2013), 600–615 | DOI

[107] Beaudoin A. J., Dawson P. R., Mathur K. K. et al., “Application of Polycrystal Plasticity to Sheet Forming”, Computer Methods in Applied Mechanics and Engineering, 117:1–2 (1994), 49–70 | DOI | Zbl

[108] Brunet M., Morestin F., “Experimental and analytical necking studies of anisotropic sheet metals”, J. Mat. Proc. Tech., 112:2–3 (2001), 214–226 | DOI

[109] Izosimova S. V., “The study of the influence of the shape of the workpiece on the accuracy of the limit strain diagram construction”, Student Science Spring: Engineering Technologies, Bauman MSTU, Moscow Available at (July 11, 2018) (In Russian) http://studvesna.qform3d.ru/?go=articles&id=810

[110] Hecker S. S., “Simple technique for determining forming limit curves”, Sheet Metal Industries, 5:9 (1975), 671–676

[111] Djavanroodi F., Derogar A., “Experimental and numerical evaluation of forming limit diagram for Ti6Al4V titanium and Al6061-T6 aluminum alloys sheets”, Materials and Design, 31:10 (2010), 4866–4875 | DOI

[112] Tadros A. K., Mellor P. B., “An experimental study of the in-plane stretching of sheet metal”, Int. J. Mech. Sci., 20:2 (1978), 121–134 | DOI

[113] Gronostajski J., Dolny A., “Determination of forming limit curves by means of Marciniak punch”, Memories Sci. Rev. Metal., 4 (1980), 570–578

[114] Raghavan K. S., “A simple technique to generate in-plane forming limit curves and selected applications”, Metall. Trans. A, 26:8 (1995), 2075–2084 | DOI

[115] Kuroda M., Tvergaard V., “Forming limit diagrams for anisotropic metal sheets with different yield criteria”, Int. J. Sol. Struct., 37:37 (2000), 5037–5059 | DOI | Zbl

[116] Avila A. F., Vieira E. L. S., “Proposing a better forming limit diagram prediction: a comparative study”, J. Mat. Proc. Tech., 141:1 (2003), 101–108 | DOI

[117] Wang L., Lee T. C., “The effect of yield criteria on the forming limit curve prediction and the deep drawing process simulation”, International Journal of Machine Tools and Manufacture, 46:9 (2006), 988–995 | DOI

[118] Kumar S. D., Jeyasingh J. J. V., Amjith T. R., “Development of Nakazima Test Simulation Tool for Forming Limit Diagram Generation of Aluminium Alloys”, International Journal of Engineering Studies and Technical Approach, 1:10 (2015), 37–45 Available at (July 11, 2018) http://ijesta.com/upcomingissue/05.10.2015.pdf

[119] Xu F., Zhao S. D., Han X. L., “Use of a modified Gurson model for the failure behavior of the clinched joint on Al6061 sheet”, Fatigue Fract. Engng. Mater. Struct., 37:3 (2014), 335–348 | DOI

[120] Safdarian R., “Stress based forming limit diagram for formability characterization of 6061 aluminum”, Trans. Nonferrous Met. Soc. China, 26:9 (2016), 2433–2441 | DOI