Abstract
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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Benzerga, A.A., Besson, J.: Plastic potentials for anisotropic porous solids. Eur. J. Mech. A, Solids 20, 397–434 (2001)
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)
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)
Garajeu, M., Suquet, P.: Effective properties of porous ideally plastic or viscoplastic materials containing rigid particles. J. Mech. Phys. Solids 45, 873–902 (1997)
Gologanu, M.: Etude quelques problèmes de rupture ductile des métaux. Thèse de doctorat, Université Paris-6 (1997)
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)
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)
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)
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)
Leblond, J.B., Perrin, G., Suquet, P.: Exact results and approximate models for porous viscoplastic solids. Int. J. Plast. 10, 213–235 (1994)
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)
MOSEK ApS: C/O Symbion Science Park, Fruebjergvej 3, Box 16, 2100 Copenhagen ϕ, Denmark (2002)
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)
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)
Pastor, F., Kondo, D., Pastor, J.: Numerical limit analysis bounds for ductile porous media with oblate voids. Mech. Res. Commun. 38, 250–254 (2011)
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)
Pastor, J.: Analyse limite: détermination numérique de solutions statiques complètes. Application au talus vertical. J. Méc. Appl. 2, 167–196 (1978)
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)
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)
Thai-The, H., Francescato, P., Pastor, J.: Limit analysis of unidirectional porous media. Mech. Res. Commun. 25, 535–542 (1998)
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)
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)
Trillat, M., Pastor, J.: Limit analysis and Gurson’s model. Eur. J. Mech. A, Solids 24, 800–819 (2005)
Tvergaard, V.: Influence of voids on shear band instabilities under plane strain conditions. Int. J. Fract. Mech. 17, 389–407 (1981)
Tvergaard, V., Needleman, A.: Analysis of the cup-cone fracture in a round tensile bar. Acta Metall. 32, 157–169 (1984)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer Science+Business Media Dordrecht
About this paper
Cite this paper
Pastor, F., Thoré, P., Kondo, D., Pastor, J. (2013). Limit Analysis and Conic Programming for Gurson-Type Spheroid Problems. In: de Saxcé, G., Oueslati, A., Charkaluk, E., Tritsch, JB. (eds) Limit State of Materials and Structures. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-5425-6_12
Download citation
DOI: https://doi.org/10.1007/978-94-007-5425-6_12
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-007-5424-9
Online ISBN: 978-94-007-5425-6
eBook Packages: EngineeringEngineering (R0)