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Investigation and comparative analysis of plastic instability criteria: application to forming limit diagrams


The prediction of forming limit diagrams (FLDs) is of significant interest to the sheet metal forming industry. Although a large variety of plastic instability criteria have been developed during the previous decades, there is still a lack of comparison of their respective theoretical bases. The aim of this paper is to present the theoretical formulations of a representative selection of diffuse necking and strain localization criteria based on the maximum force principle, the Marciniak–Kuczyński method and the bifurcation approach. The theoretical foundations and underlying assumptions for each criterion are specified prior to their application to several materials to determine the associated FLDs. The capability of the criteria to predict the formability of thin metal sheets is discussed and a classification of some of the criteria is attempted according to their order of occurrence in terms of the localization prediction.

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  1. 1.

    Keeler SP (1965) Determination of forming limits in automotive stampings. Sheet Met Ind 42(461):683–691

    Google Scholar 

  2. 2.

    Keeler SP, Backofen A (1963) Plastic instability and fracture in sheets stretched over rigid punches. Metall Trans A 56:25

    Google Scholar 

  3. 3.

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

    Google Scholar 

  4. 4.

    Swift HW (1952) Plastic instability under plane stress. J Mech Phys Solid 1(1):1–18

    Article  Google Scholar 

  5. 5.

    Hill R (1952) On discontinuous plastic states, with special reference to localized necking in thin sheets. J Mech Phys Solid 1(1):19–30

    Article  Google Scholar 

  6. 6.

    Hora P, Tong L, Reissner J (1996) A prediction method of ductile sheet metal failure in FE simulation. In: Numisheet 1996, Dearborn, Michigan, USA, pp 252–256

  7. 7.

    Mattiasson K, Sigvant M, Larson M (2006) Methods for forming limit prediction in ductile metal sheets. In: IDDRG 2006, Porto, Portugal, pp 1–9

  8. 8.

    Brunet M, Morestin F (2001) Experimental and analytical necking studies of anisotropic sheet metals. J Mater Process Technol 112:214–226

    Article  Google Scholar 

  9. 9.

    Ben Tahar M (2005) Contribution à l’étude et la simulation du procédé d’hydroformage. PhD Thesis, ENSMP, Nice

  10. 10.

    Drucker DC (1956) On uniqueness in the theory of plasticity. Q Appl Math 14:35–42

    MATH  MathSciNet  Google Scholar 

  11. 11.

    Hill R (1958) A general theory of uniqueness and stability in elastic–plastic solids. J Mech Phys Solid 6(3):236–249

    Article  MATH  Google Scholar 

  12. 12.

    Valanis KC (1989) Banding and stability in plastic materials. Acta Mech 79(1–2):113–141

    Article  MATH  MathSciNet  Google Scholar 

  13. 13.

    Rice JR (1976) The localization of plastic deformation. In: 14th International Congress on Theoretical and Applied Mechanics. North-Holland, Delft, Netherlands, pp 207–220

    Google Scholar 

  14. 14.

    Rudnicki JW, Rice JR (1975) Conditions for the localization of deformation in pressure-sensitive dilatant materials. J Mech Phys Solid 23(6):371–394

    Article  Google Scholar 

  15. 15.

    Stören S, Rice JR (1975) Localized necking in thin sheets. J Mech Phys Solid 23(6):421–441

    Article  MATH  Google Scholar 

  16. 16.

    Asaro RJ, Rice JR (1977) Strain localization in ductile single crystals. J Mech Phys Solid 25(5):309–338

    Article  MATH  Google Scholar 

  17. 17.

    Franz G, Abed-Meraim F, Ben Zineb T, Lemoine X, Berveiller M (2009) Role of intragranular microstructure development in the macroscopic behavior of multiphase steels in the context of changing strain paths. Mater Sci Eng A 517(1–2):300–311

    Article  Google Scholar 

  18. 18.

    Franz G, Abed-Meraim F, Ben Zineb T, Lemoine X, Berveiller M (2013) Strain localization analysis for single crystals and polycrystals: towards microstructure-ductility linkage. Int J Plast 48:1–33

    Article  Google Scholar 

  19. 19.

    Dudzinski D, Molinari A (1991) Perturbation analysis of thermoviscoplastic instabilities in biaxial loading. Int J Solids Struct 27(5):601–628

    Article  MATH  Google Scholar 

  20. 20.

    Lejeune A, Boudeau N, Gelin JC (2003) Influence of material and process parameters on bursting during hydroforming process. J Mater Process Technol 143–144(1):11–17

    Article  Google Scholar 

  21. 21.

    Benallal A (2000) Perturbation and stability of rate-dependent solids. Eur J Mech Solid 19(special issue):45–60

    Google Scholar 

  22. 22.

    Marciniak Z, Kuczyński K (1967) Limit strains in the processes of stretch-forming sheet metal. Int J Mech Sci 9(9):613–620

    Article  Google Scholar 

  23. 23.

    Ghosh AK (1974) Strain localization in the diffuse neck in sheet metal. Metall Mater Trans B 5(7):1607–1616

    Article  Google Scholar 

  24. 24.

    Reyes A, Hopperstad OS, Berstad T, Lademo OG (2008) Prediction of necking for two aluminium alloys under non-proportional loading by using a FE-based approach. Int J Mater Form 1(4):211–232

    Article  Google Scholar 

  25. 25.

    Haddag B, Balan T, Abed-Meraim F (2007) Investigation of advanced strain-path dependent material models for sheet metal forming simulations. Int J Plast 23(6):951–979

    Article  MATH  Google Scholar 

  26. 26.

    Gurson AL (1977) Continuum theory of ductile rupture by void nucleation and growth: Part 1—yield criteria and flow rules for porous ductile media. J Eng Mater Technol Trans ASME 99(1):2–15

    Article  Google Scholar 

  27. 27.

    Needleman A, Rice JR (1978) Limits to ductility set by plastic flow localization. In: Koistinen DP, Wang NM (eds) Mechanics of sheet metal forming. Plenum, New York, pp 237–267

    Chapter  Google Scholar 

  28. 28.

    Tvergaard V, Needleman A (1984) Analysis of the cup–cone fracture in a round tensile bar. Acta Metall 32(1):157–169

    Article  Google Scholar 

  29. 29.

    Kachanov LM (1986) Introduction to continuum damage mechanics. Martinus Nijhoff, Dordrecht

  30. 30.

    Rabotnov YN (1969) Creeps problems in structural members. North-Holland, Amsterdam

    Google Scholar 

  31. 31.

    Lemaitre J (1985) Continuous damage mechanics model for ductile fracture. J Eng Mater Technol Trans ASME 107(1):83–89

    Article  MathSciNet  Google Scholar 

  32. 32.

    Lemaitre J, Chaboche JL (1990) Mechanics of solid materials. Cambridge University Press, Cambridge

  33. 33.

    Lemaitre J (1992) A course on damage mechanics. Springer, Berlin

    Book  MATH  Google Scholar 

  34. 34.

    Haddag B, Abed-Meraim F, Balan T (2009) Strain localization analysis using a large deformation anisotropic elastic–plastic model coupled with damage. Int J Plast 25(10):1970–1996

    Article  Google Scholar 

  35. 35.

    Habbad M (1994) Instabilités plastiques en élasto-plasticité anisotrope et grandes déformations. Ecole Centrale de Lyon, Lyon

    Google Scholar 

  36. 36.

    Hill R (2001) On the mechanics of localized necking in anisotropic sheet metals. J Mech Phys Solid 49(9):2055–2070

    Article  MATH  Google Scholar 

  37. 37.

    Hill R (1991) A theoretical perspective on in-plane forming of sheet metal. J Mech Phys Solid 39(2):295–307

    Article  MATH  Google Scholar 

  38. 38.

    Stout MG, Hecker SS (1983) Role of geometry in plastic instability and fracture of tubes and sheet. Mech Mater 2:23–31

    Article  Google Scholar 

  39. 39.

    Ramaekers JAH, Bongaerts PCP (1994) A note on the forming limit curve. In: Shemet’1994, Belfast, UK, pp 39–50

  40. 40.

    Aretz H (2004) Numerical restrictions of the modified maximum force criterion for prediction of forming limits in sheet metal forming. Model Simul Mater Sci Eng 12(4):677–692

    Article  Google Scholar 

  41. 41.

    Banabic D, Comsa DS, Jurco P, Wagner S, He S, Van Houtte P (2004) Prediction of forming limit curves from two anisotropic constitutive models. In: ESAFORM 2004 Conference. Springer, Trondheim, pp 455–459

    Google Scholar 

  42. 42.

    Hutchinson JW, Neale KW, Needleman A (1978) Sheet Necking-I. Validity of plane stress assumptions of the long-wavelength approximation. In: Koistinen DP, Wang NM (eds) Mechanics of sheet metal forming. Plenum, New York, pp 111–126

    Chapter  Google Scholar 

  43. 43.

    Aretz H (2007) Numerical analysis of diffuse and localized necking in orthotropic sheet metals. Int J Plast 23(5):798–840

    Article  MATH  Google Scholar 

  44. 44.

    Banabic D, Aretz H, Paraianu L (2005) Application of various FLD modelling approaches. Model Simul Mater Sci Eng 13:759–769

    Article  Google Scholar 

  45. 45.

    Abspoel M, Scholting ME, Droog JMM (2013) A new method for predicting forming limit curves from mechanical properties. J Mater Process Technol 213(5):759–769

    Article  Google Scholar 

  46. 46.

    Drucker DC (1950) Some implications of work hardening and ideal plasticity. Q Appl Math 7:411–418

    MATH  MathSciNet  Google Scholar 

  47. 47.

    Hadamard J (1903) Leçons sur la propagation des ondes et les équations de l’hydrodynamiques. Paris

  48. 48.

    Bigoni D, Hueckel T (1991) Uniqueness and localization—associative and non-associative elastoplasticity. Int J Solid Struct 28(2):197–213

    Article  MATH  MathSciNet  Google Scholar 

  49. 49.

    Neilsen MK, Schreyer HL (1993) Bifurcations in elastic–plastic materials. Int J Solids Struct 30(4):521–544

    Article  MATH  Google Scholar 

  50. 50.

    Tvergaard V (1980) Bifurcation and imperfection—sensitivity at necking instabilities. Z Angew Math Mech 60:T26–T34

    MATH  MathSciNet  Google Scholar 

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Correspondence to Farid Abed-Meraim.

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Abed-Meraim, F., Balan, T. & Altmeyer, G. Investigation and comparative analysis of plastic instability criteria: application to forming limit diagrams. Int J Adv Manuf Technol 71, 1247–1262 (2014).

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  • Necking
  • Strain localization
  • Anisotropic elasto-plasticity
  • Large strain
  • Formability
  • Forming limit diagram