International Journal of Material Forming

, Volume 12, Issue 2, pp 241–255 | Cite as

Ductility prediction of substrate-supported metal layers based on rate-independent crystal plasticity theory

  • Mohamed Ben BettaiebEmail author
  • Farid Abed-Meraim
Thematic Issue: Advances in Material Forming Simulation


In several modern technological applications, the formability of functional metal components is often limited by the occurrence of localized necking. To retard the onset of such undesirable plastic instabilities and, hence, to improve formability, elastomer substrates are sometimes adhered to these metal components. The current paper aims to numerically investigate the impact of such elastomer substrates on the formability enhancement of the resulting bilayer. To this end, both the bifurcation theory and the initial imperfection approach are used to predict the inception of localized necking in substrate-supported metal layers. The full-constraint Taylor scale-transition scheme is used to derive the mechanical behavior of a representative volume element of the metal layer from the behavior of its microscopic constituents (the single crystals). The mechanical behavior of the elastomer substrate follows the neo-Hookean hyperelastic model. The adherence between the two layers is assumed to be perfect. Through numerical simulations, it is shown that bonding an elastomer layer to a metal layer allows significant enhancement in formability, especially in the negative range of strain paths. These results highlight the benefits of adding elastomer substrates to thin metal components in several technological applications. Also, it is shown that the limit strains predicted by the initial imperfection approach tend towards the bifurcation predictions as the size of the geometric imperfection in the metal layer reduces.


Substrate-supported metal layers Forming limit diagrams Localized necking Neo-Hookean model Rate-independent crystal plasticity Bifurcation and imperfection analyses 


Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


  1. 1.
    Tekoğlu C, Hutchinson JW, Pardoen T (2015) On localization and void coalescence as a precursor to ductile fracture. Phil Trans R Soc A 373:2038Google Scholar
  2. 2.
    K. Saanouni (2012) Damage mechanics in metal forming. Advanced modeling and numerical simulation. ISTE/Wiley, LondonGoogle Scholar
  3. 3.
    Keeler SP, Backofen WA (1963) Plastic instability at fracture in sheets stretched over rigid punches. Trans ASM 56:25Google Scholar
  4. 4.
    Goodwin GM (1968) Application of the strain analysis to sheet metal forming in the press shop. Metallurgia Italiana 60:767Google Scholar
  5. 5.
    Kotkunde N, Krishna G, Shenoy SK, Gupta AK, Singh SK (2015) Experimental and theoretical investigation of forming limit diagram for Ti-6Al-4 V alloy at warm condition. Int J Mater Form 10(2):255–266. CrossRefGoogle Scholar
  6. 6.
    Strano M, Colosimo BM (2006) Logistic regression analysis for experimental determination of forming limit diagrams. Int J Mach Tool Manu 46(6):673–682. CrossRefGoogle Scholar
  7. 7.
    Rice JR (1976) The localization of plastic deformation. The localization of plastic deformation. In: Koiter W T (ed) Theoretical and Applied Mechanics (Proceedings of the 14th International Congress on Theoretical and Applied Mechanics, Delft, vol 1. North-Holland Publishing Co., Amsterdam, pp 207–220Google Scholar
  8. 8.
    Marciniak Z, Kuczynski K (1967) Limit strains in the processes of stretch-forming sheet metal. Int J Mech Sci 9(9):609–620. CrossRefzbMATHGoogle Scholar
  9. 9.
    Chiu SL, Leu J, Ho PS (1994) Fracture of metal‐polymer line structures. I. Semiflexible polyimide. J Appl Phys 76(9):5136–5142. CrossRefGoogle Scholar
  10. 10.
    Hommel M, Kraft O (2001) Deformation behavior of thin copper films on deformable substrates. Acta Mater 49(19):3935–3947. CrossRefGoogle Scholar
  11. 11.
    Alaca BE, Saif MTA, Sehitoglu H (2002) On the interface debond at the edge of a thin film on a thick substrate. Acta Mater 50(5):1197–1209. CrossRefGoogle Scholar
  12. 12.
    Xue ZY, Hutchinson JW (2007) Neck retardation and enhanced energy absorption in metal–elastomer bilayers. Mech Mater 39(5):473–487. CrossRefGoogle Scholar
  13. 13.
    Xue ZY, Hutchinson JW (2008) Neck development in metal/elastomer bilayers under dynamic stretchings. Int J Solids Struct 45(13):3769–3778. CrossRefzbMATHGoogle Scholar
  14. 14.
    Ben Bettaieb M, Abed-Meraim F (2015) Investigation of localized necking in substrate-supported metal layers: Comparison of bifurcation and imperfection analyses. Int J Plast 65:168–190. CrossRefGoogle Scholar
  15. 15.
    Rudnicki JW, Rice JR (1975) Conditions for the localization of 937 deformation in pressure-sensitive dilatant materials. J. Mech. 938 Phys. Solids 23(6):371–394.
  16. 16.
    J.W. Hutchinson, K.W. Neale (1978) Sheet Necking- II. Time-Independent Behavior. In: Koistinen DP, Wang NM (eds) Mechanics of sheet metal forming. Plenum Publishing Corporation, New York, pp 127–153Google Scholar
  17. 17.
    Borja RI, Wren JR (1993) Discrete micromechanics of elastoplastic crystals. Int J Numer Methods Eng 36(22):3815–3840.
  18. 18.
    Akpama HK, Ben Bettaieb M, Abed-Meraim F (2016) Numerical integration of rate-independent BCC single crystal plasticity models: comparative study of two classes of numerical algorithms. Int J Numer Methods Eng 108(5):363–422. MathSciNetCrossRefGoogle Scholar
  19. 19.
    Anand L, Kothari M (1996) A computational procedure for rate-independent crystal plasticity. J Mech Phys Solids 44(4):525–558. MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Hunter SC (1979) Some exact solutions in the theory of finite elasticity for incompressible neo-Hookean materials. Int J Mech Sci 21(4):203–211. CrossRefzbMATHGoogle Scholar
  21. 21.
    Hutchinson JW, Neale KW, Needleman A (1978) Sheet Necking- I. Time-independent behavior. In: Koistinen DP, Wang N-M (eds) Mechanics of sheet metal forming, Plenum Publishing Corporation, New york, pp 111–126Google Scholar
  22. 22.
    Changchun Z, Hongpo Z, Jianqiang H, Wanzhao C (2003) MEMS Devices Technol. 10 (30)(2003)Google Scholar
  23. 23.
    McShane GJ, Stewart C, Aronson MT, Wadley HNG, Fleck NA, Deshpande VS (2008) Dynamic rupture of polymer–metal bilayer plates. Int J Solids Struct 45(16):4407–4426. CrossRefzbMATHGoogle Scholar
  24. 24.
    Men Y, Wang S, Jia H, Wu Z, Li L, Zhang C (2013) Experimental study on tensile bifurcation of nanoscale Cu film bonded to polyethylene terephthalate substrate. Thin Solid Films 548:371–376. CrossRefGoogle Scholar
  25. 25.
    Szyndler J, Madej L (2014) Effect of number of grains and boundary conditions on digital material representation deformation under plane strain. Arch Civ Mech Eng 14(3):360–369. CrossRefGoogle Scholar
  26. 26.
    Houdaigui FE, Forest S, Gourgues A-F, Jeulin D (2007) On the size of the representative volume element for isotropic elastic polycrystalline copper. In: Bai Y (ed) IUTAM Symposium on mechanical behavior and micro-mechanicsof nanostructured materials. Springer, Dordrecht pp 171–180Google Scholar
  27. 27.
    Yoshida K, Kuroda M (2012) Comparison of bifurcation and imperfection analyses of localized necking in rate-independent polycrystalline sheets. Int J Solids Struct 49(15-16):2073–2084. CrossRefGoogle Scholar
  28. 28.
    Amirkhizi AV, Isaacs J, McGee J, Nemat-Nasser S (2006) An experimentally-based viscoelastic constitutive model for polyurea, including pressure and temperature effects. Philos Mag 86(36):5847–5866. CrossRefGoogle Scholar
  29. 29.
    Hutchinson JW (1970) Elastic-Plastic Behaviour of Polycrystalline Metals and Composites. Proc R Soc Lond A 319(1537):247–272. CrossRefGoogle Scholar
  30. 30.
    Yoshida K, Brenner R, Bacroix B, Bouvier S (2009) Effect of regularization of Schmid law on self-consistent estimates for rate-independent plasticity of polycrystals. Eur J Mech – A/Solids 28(5):905–915. CrossRefzbMATHGoogle Scholar
  31. 31.
    Zhou Y, Neale KW (1995) Predictions of forming limit diagrams using a rate-sensitive crystal plasticity model. Int J Mech Sci 37(1):1–20. CrossRefzbMATHGoogle Scholar
  32. 32.
    Signorelli JW, Bertinetti MA, Turner PA (2009) Predictions of forming limit diagrams using a rate-dependent polycrystal self-consistent plasticity model. Int J Plast 25(1):1–25. CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag France SAS, part of Springer Nature 2018

Authors and Affiliations

  1. 1.LEM3, UMR CNRS 7239Arts et Métiers ParisTechMetz Cedex 3France
  2. 2.DAMAS, Laboratory of Excellence on Design of Alloy Metals for low-mAss StructuresUniversité de LorraineLorraineFrance

Personalised recommendations