A Stress-Based Variational Model for Ductile Porous Materials and Its Extension Accounting for Lode Angle Effects

  • Long ChengEmail author
  • Vincent Monchiet
  • Géry de Saxcé
  • Djimédo Kondo


The aim of this work is to derive by homogenization techniques a macroscopic plastic model for porous materials with von Mises matrix. In contrast to the Gurson’s well known kinematical approach [19] applied to a hollow sphere, the proposed study proceeds by means of a statical limit analysis procedure, for which a suitable trial stress field is proposed. In the first part, the formulation of the stress variational model is developed, by considering the Hill’s variational principle, and introducing a Lagrange’s multiplier to solve the resulting saddle-point minimization problem. This methodology being opposite to the Gurson’s kinematical approach, complements the limit analysis methods for porous materials. The second part is devoted to an application of the proposed approach to the porous materials with von Mises matrix. To this end, an axisymmetric model is first studied by adopting a suitable trial stress field, which is composed by a heterogeneous part corresponding to the exact solution of hydrostatic loading and a homogeneous part for capturing the shear effects. We derive closed form formula which depends not only on the first and second invariant of the macroscopic stress tensor but also on the sign of the third invariant of the stress deviator. Moreover, an extension of the above axisymmetric model to the general case of non-axisymmetric loadings by introducing a more general trial stress field is studied. The established new yield locus explicitly depends on the effect of the third invariant (equivalently the Lode angle). The obtained results are fully discussed and compared to existing models, available numerical data and to Finite Elements results obtained from cell calculation carried out during the present study.


Yield Surface Stress Deviator Hollow Sphere Stress Triaxiality Lode Angle 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    Barthélémy JF, Dormieux L (2003) Détermination du critère de rupture macroscopique d’un milieux poreux par homogénéisation non linéaire. Comptes Rendus Mécanique 331:271–276CrossRefzbMATHGoogle Scholar
  2. 2.
    Benzerga AA, Besson J (2001) Plastic potentials for anisotropic porous solids. Eur J Mech A/Solids 20:397–434CrossRefzbMATHGoogle Scholar
  3. 3.
    Cazacu O, Revil-Baudard B, Lebensohn R, Garajeu M (2014) On the combined effect of pressure and third invariant on yielding of porous solids with von Mises matrix. J Appl Mech 80(6):064501CrossRefGoogle Scholar
  4. 4.
    Cheng L (2013) Homogenization of porous media with plastic matrix and non-associated flow rule by variational methods, Ph.D. thesis, University Lille 1Google Scholar
  5. 5.
    Cheng L, Guo TF (2007) Void interaction and coalescence in polymeric materials. Int J Solids Struct 44:1787–1808CrossRefzbMATHGoogle Scholar
  6. 6.
    Cheng L, de Saxcé G, Kondo D (2014) A stress variational model for ductile porous materials. Int J Plast 55:133–151CrossRefGoogle Scholar
  7. 7.
    Cheng L, Jia y, Oueslati A, de Saxcé G, Kondo D (2014) A bipotential-based limit analysis and homogenization of ductile porous materials with non-associated Drucker-Prager matrix, submitted to J Mech Phys SolidsGoogle Scholar
  8. 8.
    de Saxcé G (1992) Une généralisation de l’inégalité de Fenchel et ses applications aux lois constitutives. C R Acad Sci Paris Sér II 314:125–129zbMATHGoogle Scholar
  9. 9.
    de Saxcé G, Feng ZQ (1991) New inequality and functional for contact friction: the implicit standard material approach. Mech Struct Mach 19:301–325CrossRefGoogle Scholar
  10. 10.
    de Saxcé G, Bousshine L (1993) On the extension of limit analysis  theorems to the non-associated flow rules in soils and to the contact with Coulomb’s friction. In: XI Polish conference on computer methods in mechanics. Kielce, pp 815–822Google Scholar
  11. 11.
    de Saxcé G, Bousshine L (1998) Limit analysis theorems for the implicit standard materials: application to the unilateral contact with dry friction and the non associated flow rules in soils and rocks. Int J Mech Sci 40(4):387–398CrossRefzbMATHGoogle Scholar
  12. 12.
    Ekeland I, Temam R (1975) Convex analysis and variational problems. North Holland Publisher, AmsterdamGoogle Scholar
  13. 13.
    Fenchel W (1949) On conjugate convex functions. Can J Math 1:73–77CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Gao X, Zhang T, Hayden M, Roe C (2009) Effects of the stress state on plasticity and ductile failure of an aluminum 5083 alloy. Int J Plast 25:2366–2382CrossRefGoogle Scholar
  15. 15.
    Garajeu M, Suquet P (1997) Effective properties of porous ideally plastic or viscoplastic materials containing rigid particles. J Mech Phys Solids 45:873–902CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Gologanu M, Leblond JB, Perrin G, Devaux J (1997) Recent extensions of Gurson’s model for porous ductile metals. In: Suquet P (ed) Continuum micromechanics. Springer, New YorkGoogle Scholar
  17. 17.
    Green RJ (1972) A plasticity theory for porous solids. Int J Mech Phys Solids 14:215–224CrossRefzbMATHGoogle Scholar
  18. 18.
    Guo TF, Faleskog J, Shih CF (2008) Continuum modeling of a porous solid with pressure-sensitive dilatant matrix. J Mech Phys Solids 56:2188–2212CrossRefzbMATHGoogle Scholar
  19. 19.
    Gurson AL (1977) Continuum theory of ductile rupture by void nucleation and growth—part I: yield criteria and flow rules for porous ductile media. J Eng Mater Technol 99:2–15CrossRefGoogle Scholar
  20. 20.
    Hill R (1950) Mathematical theory of plasticity. Oxford University Press, LondonzbMATHGoogle Scholar
  21. 21.
    Jeong HY (2002) A new yield function and a hydrostatic stress-controlled model for porous solids with pressure-sensitive matrices. Int J Mech Phys Solids 32:3669–3691Google Scholar
  22. 22.
    Jeong HY, Pan J (1995) A macroscopic constitutive law for porous solids with pressure-sensitive matrices and its applications to plastic flow localization. Int J Mech Phys Solids 39:1385–1403Google Scholar
  23. 23.
    Li Z, Fu MW, Lua J, Yang H (2011) Ductile fracture: experiments and computations. Int J Plast 27:147–180CrossRefGoogle Scholar
  24. 24.
    Maghous S, Dormieux L, Barthélémy JF (2009) Micromechanical approach to the strength properties of frictional geomaterials. Eur J Mech A/Solids 28:179–188CrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    Monchiet V, Charkaluk E, Kondo D (2007) An improvement of Gurson-type models of porous materials by Eshelby-like trial velocity fields. Comptes Rendus Mécanique 335:32–41CrossRefzbMATHGoogle Scholar
  26. 26.
    Monchiet V, Cazacu O, Kondo D (2008) Macroscopic yield criteria for plastic anisotropic materials containing spheroidal voids. Int J Plast 24:1158–1189CrossRefzbMATHGoogle Scholar
  27. 27.
    Moreau JJ (2003) Fonctionnelles convexes. Istituto Poligrafico e Zecca dello Stato, RomeGoogle Scholar
  28. 28.
    Nahshon K, Hutchinson JW (2008) Modification of the Gurson model for shear failure. Eur J Mech A/Solids 27:1–27CrossRefzbMATHGoogle Scholar
  29. 29.
    Rockafellar RT (1970) Convex analysis. Princeton University Press, PrincetonzbMATHGoogle Scholar
  30. 30.
    Save MA, Massonnet CE, de Saxcé G (1997) Plastic limit analysis of plates, shells and disks. Elsevier, New YorkzbMATHGoogle Scholar
  31. 31.
    Sun Y, Wang D (1989) A lower bound approach to the yield loci of porous materials. Acta Mechanica Sinica 5:237–243CrossRefGoogle Scholar
  32. 32.
    Trillat M, Pastor J (2005) Limit analysis and Gurson’s model. Eur J Mech A/Solids 24:800–819CrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Long Cheng
    • 1
    Email author
  • Vincent Monchiet
    • 2
  • Géry de Saxcé
    • 1
  • Djimédo Kondo
    • 3
  1. 1.Laboratoire de Mécanique de Lille (UMR CNRS 8107)Université de Lille 1, Cité ScientifiqueVilleneuve d’AscqFrance
  2. 2.Laboratoire Modlisation et Simulation Multi Echelles (UMR CNRS 8208)Université Paris-EstMarne la ValléeFrance
  3. 3.Institut Jean Le Rond d’Alembert (UMR CNRS 7190)Université Pierre et Marie CurieParisFrance

Personalised recommendations