Plasticity of Cellular Metals (Foams)

  • Thomas DaxnerEmail author
Part of the Engineering Materials book series (ENG.MAT.)


Cellular metals, e.g., made by solidification of molten metal foam, have interesting mechanical properties, among them high specific strength and stiffness coupled with inflammability and good damping properties. This makes them interesting for engineering applications which require the prediction of the onset of yielding under multi-axial stress states and the development of plastic strains over a strain range that may extend into the regime of full compaction of the foam micro-structure, as it is the case in applications for crash protection. This chapter investigates the micro-mechanical deformation mechanisms which govern the elasto-plastic behavior of cellular metals on the macro-mechanical level, where the cellular structure can be treated as a homogeneous material if the difference between the cell size and the component size is large enough. If this is the case suitable constitutive models can be applied for predicting the onset of macroscopic yielding, the evolution of plastic strains and the hardening behavior. Thus, a review of the most important material models proposed for simulating the effective elasto-plastic behavior of isotropic cellular metals is presented. This behavior is characterized by a distinct pressure sensitivity, which sets apart the behavior of cellular metals from the one of solid metals as described by classical (e.g., von Mises) theory of plasticity.


Cellular metal Metal foam Open-cell foam Closed-cell foam Yield surface Flow rule Hardening  Micromechanics Multi-axial loading Pressure dependent yielding 



The author would like to thank Lorna J. Gibson, Ronald E. Miller, Vikram S. Deshpande, and Wolfgang Ehlers for helpful discussions regarding their respective publications. The author also wants to thank Maria Steininger for her help with obtaining some of the publications cited in this article.


  1. 1.
    Abrate, S.: Criteria for yielding or failure of cellular materials. J. Sandw. Struct. Mat. 10(1), 5–51 (2008)CrossRefGoogle Scholar
  2. 2.
    Bitsche, R.: Space-filling polyhedra as mechanical models for solidified dry foams. Vienna University of Technology, Vienna, Diploma thesis (2005)Google Scholar
  3. 3.
    Chen, C.: Manual for a UMAT user subroutine. Technical Report CUED/C-MICROMECH/TR.4, Department of Engineering, Cambridge University, Cambridge (1998)Google Scholar
  4. 4.
    Chen, C., Lu, T.J.: A phenomenological framework of constitutive modelling for incompressible and compressible elasto-plastic solids. Int. J. Sol. Struct. 37, 7769–7786 (2000)CrossRefzbMATHGoogle Scholar
  5. 5.
    Combaz, E., Bacciarini, C., Charvet, R., Dufour, W., Dauphin, F., Mortensen, A.: Yield surface of polyurethane and aluminium replicated foam. Acta Mater 58, 5168–5183 (2010)CrossRefGoogle Scholar
  6. 6.
    Combaz, E., Bacciarini, C., Charvet, R., Dufour, W., Mortensen, A.: Multiaxial yield behaviour of Al replicated foam. J. Mech. Phys. Solids 59, 1777–1793 (2011)CrossRefGoogle Scholar
  7. 7.
    Dassault Systèmes (2012) Abaqus 6.12 Theory Manual. Dassault Systèmes Simulia Corp., Providence, Rhode Island.Google Scholar
  8. 8.
    Daxner, T., Denzer, R., Böhm, H.J., Rammerstorfer, F.G., Maier, M.: Simulation des elasto-plastischen Verhaltens von Metallschaum mit Hilfe von 2D und 3D Einheitszellen-Modellen. Mater-wiss u Werkst-techn 31, 447–450 (2000)CrossRefGoogle Scholar
  9. 9.
    Daxner, T., Böhm, H.J., Rammerstorfer, F.G.: Numerical investigation of local yielding in metallic foams. In: Banhart, J., Fleck, N.A., Mortensen, A. (eds.) Cellular Metals: Manufacture, pp. 413–418. Properties, Applications, Verlag MIT, Berlin (2003)Google Scholar
  10. 10.
    Daxner, T., Bitsche, R., Böhm, H.: Space-filling polyhedra as mechanical models for solidified dry foams. Mater. Trans. 47(9), 2213–2218 (2006)CrossRefGoogle Scholar
  11. 11.
    Deshpande, V.S., Fleck, N.A.: Isotropic constitutive models for metallic foams. J. Mech. Phys. Solids 48, 1253–1283 (2000)CrossRefzbMATHGoogle Scholar
  12. 12.
    Droste, A.: Beschreibung und Anwendung eines elastisch-plastischen Materialmodells mit Schädigung für hochporöse Metallschäume. Ph.D. thesis, University of Stuttgart, Stuttgart (2002)Google Scholar
  13. 13.
    Ehlers, W., Avci, O.: Stress-dependent hardening and failure surfaces of dry sand. Int. J. Numer. Anal. Meth. Geomech. 37, 787–809 (2013)CrossRefGoogle Scholar
  14. 14.
    Ehlers, W., Droste, A.: A continuum model for highly porous aluminium foam. Techn. Mech. 19(4):341–350 (1999a)Google Scholar
  15. 15.
    Ehlers, W., Droste, A.: FE simulation of metal foams based on the macroscopic approach of the theory of porous media. In: Banhart, J., Ashby, M.F., Fleck, N.A. (eds.) Metal Foams and Porous Metal Structures, pp. 299–302. Verlag MIT, Bremen (1999b)Google Scholar
  16. 16.
    Ehlers, W., Müllerschön, H., Klar, O.: On the behaviour of aluminium foams under uniaxial and multiaxial loading. In: Banhart, J., Ashby, M.F., Fleck, N.A. (eds.) Metal Foams and Porous Metal Structures, pp. 255–262. Verlag MIT, Bremen (1999)Google Scholar
  17. 17.
    Gibson, L.J., Ashby, M.F.: Cellular Solids: Structure and Properties, 2nd edn. Cambridge University Press, Cambridge, New York (1997)Google Scholar
  18. 18.
    Gibson, L.J., Ashby, M.F., Zhang, J., Triantafillou, T.C.: Failure surfaces for cellular materials under multiaxial loads—I. Modelling. Int J Mech Sci 31(9), 635–663 (1989)Google Scholar
  19. 19.
    Gong, L., Kyriakides, S.: Compressive response of open cell foams. Part II: Initiation and evolution of crushing. Int. J. Sol. Struct. 42(5–6):1381–1399 (2005)Google Scholar
  20. 20.
    Gong, L., Kyriakides, S., Triantafyllidis, N.: On the stability of Kelvin cell foams under compressive loads. J. Mech. Phys. Solids 53, 771–794 (2005)CrossRefzbMATHGoogle Scholar
  21. 21.
    Gradinger, R.C.: Das mechanische Verhalten von Aluminiumschaum bei Druck—und Crushbelastung – Experimente und numerische Simulation. Vienna University of Technology, Vienna, Diploma thesis (1997)Google Scholar
  22. 22.
    Hallquist, J.O.: LS DYNA Theoretical Manual. Livermore Software Technology Corporation, Livermore, CA (1998)Google Scholar
  23. 23.
    Hanssen, A.G.: Structural crashworthiness of aluminium foam-based components. Ph.D. thesis, Norges Tekniske Høgskole, Trondheim, Norway (2000)Google Scholar
  24. 24.
    Hanssen, A.G., Hopperstad, O.S., Langseth, M., Ilstad, H.: Validation of constitutive models applicable to aluminium foams. Int. J. Mech. Sci. 44(2), 359–406 (2002)CrossRefGoogle Scholar
  25. 25.
    Laroussi, M., Sab, K., Alaoui, A.: Foam mechanics: nonlinear response of an elastic 3D-periodic microstructure. Int. J. Sol. Struct. 39(13–14), 3599–3623 (2002)CrossRefzbMATHGoogle Scholar
  26. 26.
    Lubliner, J.: Plasticity Theory. Macmillan Publishing Company, New York (1990)Google Scholar
  27. 27.
    Miller, R.E.: A continuum plasticity model for the constitutive and indentation behaviour of foamed metals. Int. J. Mech. Sci. 42, 729–754 (2000)CrossRefzbMATHGoogle Scholar
  28. 28.
    Schreyer, H.L., Zuo, Q.H., Maji, A.K.: An anisotropic plasticity model for foams and honeycomb. J. Eng. Mech. ASCE 120(9), 1913–1930 (1994)CrossRefGoogle Scholar
  29. 29.
    Seitzberger, M., Rammerstorfer, F.G., Degischer, H.P., Gradinger, R.: Crushing of axially compressed steel tubes filled with aluminium foam. Acta Mechanica 125, 93–105 (1997)CrossRefzbMATHGoogle Scholar
  30. 30.
    Shahbeyk, S.: Yield/failure criteria, constitutive models, and crashworthiness applications of metal foams. In: Dukhan, N. (ed.) Metal Foams: Fundamentals and Applications, pp. 131–214. DEStech Publications, Lancaster, Pennsylvania (2013)Google Scholar
  31. 31.
    Shim, V.P.W., Tay, B.Y., Stronge, W.J.: Dynamic crushing of strain-softening cellular structures—a one-dimensional analysis. J. Eng. Mat. Tech. ASME 112, 398–405 (1990)CrossRefGoogle Scholar
  32. 32.
    Todt, M.: Long wave instabilities in periodic structures. Vienna University of Technology, Vienna, Diploma thesis (2008)Google Scholar
  33. 33.
    Zhang, J., Kikuchi, N., Li, V., Yee, A., Nusholtz, G.: Constitutive modeling of polymeric foam material subjected to dynamic crash loading. Int. J. Impact. Eng. 21(5), 369–386 (1998)Google Scholar

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© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.CAE Simulation & SolutionsViennaAustria

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