Limit Analysis and Conic Programming for Gurson-Type Spheroid Problems

  • F. Pastor
  • P. Thoré
  • D. Kondo
  • J. Pastor
Conference paper


In his famous 1977-paper, Gurson used the kinematic approach of Limit Analysis (LA) about the hollow sphere model with a von Mises solid matrix. The computation led to a macroscopic yield function of the “Porous von Mises”-type materials. Several extensions have been further proposed in the literature, such as those accounting for void shape effects by Gologanu et al. (J. Eng. Mater. Technol. 116:290–297, 1994; Continuum Micromechanics, Springer, Berlin, 1997), among others. To obtain pertinent lower and upper bounds to the exact solutions in terms of LA, we have revisited our existing kinematic and static 3D-FEM codes for spherical cavities to take into account the model with confocal spheroid cavity and boundary. In both cases, the optimized formulations have allowed to obtain an excellent efficiency of the resulting codes. A first comparison with the Gurson criterion does not only show an improvement of the previous results but points out that the real solution to the hollow sphere model problem depends on the third invariant of the stress tensor. A second series of tests is presented for oblate cavities, in order to analyze the above-mentioned works in terms of bound and efficiency.


Spherical Cavity Tetrahedral Element Macroscopic Stress Conic Constraint Gurson Model 
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  1. 1.
    Benzerga, A.A., Besson, J.: Plastic potentials for anisotropic porous solids. Eur. J. Mech. A, Solids 20, 397–434 (2001) zbMATHCrossRefGoogle Scholar
  2. 2.
    Danas, K., Idiart, M.I., Castañeda, P.P.: A homogenization-based constitutive model for isotropic viscoplastic porous media. Int. J. Solids Struct. 45, 3392–3409 (2008) zbMATHCrossRefGoogle Scholar
  3. 3.
    Francescato, P., Pastor, J., Riveill-Reydet, B.: Ductile failure of cylindrically porous materials. Part i: plane stress problem and experimental results. Eur. J. Mech. A, Solids 23, 181–190 (2004) zbMATHCrossRefGoogle Scholar
  4. 4.
    Garajeu, M., Suquet, P.: Effective properties of porous ideally plastic or viscoplastic materials containing rigid particles. J. Mech. Phys. Solids 45, 873–902 (1997) MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Gologanu, M.: Etude quelques problèmes de rupture ductile des métaux. Thèse de doctorat, Université Paris-6 (1997) Google Scholar
  6. 6.
    Gologanu, M., Leblond, J., Perrin, G., Devaux, J.: Approximate models for ductile metals containing non-spherical voids—case of axisymmetric oblate ellipsoidal cavities. J. Eng. Mater. Technol. 116, 290–297 (1994) CrossRefGoogle Scholar
  7. 7.
    Gologanu, M., Leblond, J., Perrin, G., Devaux, J.: Recent extensions of gurson’s model for porous ductile metals. In: Suquet, P. (ed.) Continuum Micromechanics. Springer, Berlin (1997) Google Scholar
  8. 8.
    Gurson, A.L.: 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, 2–15 (1977) CrossRefGoogle Scholar
  9. 9.
    Kammoun, Z., Pastor, F., Smaoui, H., Pastor, J.: Large static problem in numerical limit analysis: a decomposition approach. Int. J. Numer. Anal. Methods Geomech. 34, 1960–1980 (2010) CrossRefGoogle Scholar
  10. 10.
    Leblond, J.B., Perrin, G., Suquet, P.: Exact results and approximate models for porous viscoplastic solids. Int. J. Plast. 10, 213–235 (1994) zbMATHCrossRefGoogle Scholar
  11. 11.
    Lee, B., Mear, M.: Axisymmetric deformation of power-law solids containing a dilute concentration of aligned spheroidal voids. J. Mech. Phys. Solids 40, 1805–1836 (1992) CrossRefGoogle Scholar
  12. 12.
    MOSEK ApS: C/O Symbion Science Park, Fruebjergvej 3, Box 16, 2100 Copenhagen ϕ, Denmark (2002) Google Scholar
  13. 13.
    Monchiet, V., Cazacu, O., Charkaluk, E., Kondo, D.: Macroscopic yield criteria for plastic anisotropic materials containing spheroidal voids. Int. J. Plast. 24, 1158–1189 (2008) zbMATHCrossRefGoogle Scholar
  14. 14.
    Monchiet, V., Charkaluk, E., Kondo, D.: An improvement of Gurson-type models of porous materials by using Eshelby-like trial velocity fields. C. R., Méc. 335, 32–41 (2007) zbMATHCrossRefGoogle Scholar
  15. 15.
    Pastor, F., Kondo, D., Pastor, J.: Numerical limit analysis bounds for ductile porous media with oblate voids. Mech. Res. Commun. 38, 250–254 (2011) CrossRefGoogle Scholar
  16. 16.
    Pastor, F., Loute, E., Pastor, J.: Limit analysis and convex programming: a decomposition approach of the kinematical mixed method. Int. J. Numer. Methods Eng. 78, 254–274 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Pastor, J.: Analyse limite: détermination numérique de solutions statiques complètes. Application au talus vertical. J. Méc. Appl. 2, 167–196 (1978) Google Scholar
  18. 18.
    Pastor, J., Castaneda, P.P.: Yield criteria for porous media in plane strain: second-order estimates versus numerical results. C. R., Méc. 330, 741–747 (2002) zbMATHCrossRefGoogle Scholar
  19. 19.
    Pastor, J., Francescato, P., Trillat, M., Loute, E., Rousselier, G.: Ductile failure of cylindrically porous materials. part II: other cases of symmetry. Eur. J. Mech. A, Solids 23, 191–201 (2004) zbMATHCrossRefGoogle Scholar
  20. 20.
    Thai-The, H., Francescato, P., Pastor, J.: Limit analysis of unidirectional porous media. Mech. Res. Commun. 25, 535–542 (1998) zbMATHCrossRefGoogle Scholar
  21. 21.
    Thoré, P., Pastor, F., Kondo, D., Pastor, J.: Hollow sphere models, conic programming and third stress invariant. Eur. J. Mech., A Solids (2010, in press) Google Scholar
  22. 22.
    Thoré, P., Pastor, F., Pastor, J., Kondo, D.: Closed form solutions for the hollow sphere model with Coulomb and Drucker-Prager materials under isotropic loadings. C. R. Méc., Acad. Sc. Paris 337, 260–267 (2009) zbMATHGoogle Scholar
  23. 23.
    Trillat, M., Pastor, J.: Limit analysis and Gurson’s model. Eur. J. Mech. A, Solids 24, 800–819 (2005) zbMATHCrossRefGoogle Scholar
  24. 24.
    Tvergaard, V.: Influence of voids on shear band instabilities under plane strain conditions. Int. J. Fract. Mech. 17, 389–407 (1981) CrossRefGoogle Scholar
  25. 25.
    Tvergaard, V., Needleman, A.: Analysis of the cup-cone fracture in a round tensile bar. Acta Metall. 32, 157–169 (1984) CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.Laboratoire de mécanique de Lille (LML)UMR 8107 CNRSVilleneuve d’AscqFrance
  2. 2.Laboratoire LOCIEPolytech’ Annecy-Chambéry, Université de SavoieLe Bourget du LacFrance
  3. 3.Institut D’Alembert, Université Pierre et Marie CurieUMR 7190 CNRSParis Cedex 05France

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