European Journal of Mechanics A/Solids 24 (2005) 800–819 Limit analysis and Gurson’s model Malorie Trillat, Joseph Pastor ∗ Laboratoire Optimisation de la Conception et Ingénierie de l’Environnement (LOCIE), École Supérieure d’Ingénieurs de Chambéry (ESIGEC), Université de Savoie, 73376 Le Bourget du Lac, France Received 28 January 2005; accepted 15 June 2005 Available online 2 August 2005 Abstract The yield criterion of a porous material using Gurson’s model [Gurson, A.L., 1977. Continuum theory of ductile rupture by void nucleation and growth – Part I: Yield criteria and flow rules for porous ductile media. ASME J. Engrg. Mater. Tech- nol. 99, 2–15] is investigated herein. Both methods of Limit Analysis are applied using linear and conic programming codes for solving resulting non-linear optimization problems. First, the results obtained for a porous media with cylindrical cavi- ties [Francescato, P., Pastor, J., Riveill-Reydet, B., 2004. Ductile failure of cylindrically porous materials. Part 1: Plane stress problem and experimental results. Eur. J. Mech. A Solids 23, 181–190; Pastor, J., Francescato, P., Trillat, M.,Loute, E., Rous- selier, G., 2004. Ductile failure of cylindrically porous materials. Part 2: Other cases of symmetry. Eur. J. Mech. A Solids 23, 191–201] are summarized, showing that the Gurson expression is too restrictive in this case. Then the hollow sphere problem is investigated, in the axisymmetrical and in the three-dimensional (3D) cases. A plane mesh of discontinuous triangular elements is used to model the hollow sphere as RVE in the axisymmetrical example. This first model does not provide a very precise yield criterion. Then a full 3D model is applied (using discontinuous tetrahedral elements), thus solving nearly exactly the general three-dimensional problem. Several examples of loadings are investigated in order to test the final criterion in a variety of situations. As a result, the Gurson approach is slightly improved and, for the first time, it is validated by our rigorous static and kinematic approaches.  2005 Elsevier SAS. All rights reserved. Keywords: Porous media; Gurson’s model; Limit analysis; Lower and upper approach; Linear and conic programming 1. Introduction Concerning the ductile failure of porous materials, Gurson’s criterion (Gurson, 1977) is the most widely accepted because it is based on a homogenization method and on the kinematic approach of limit analysis. The plastic domain is approached from the outside by a semi-analytical approach, proved to be an upper bound approach by Leblond (2003). Gurson’s model treats a hollow sphere with macroscopic strain imposed on the boundary. The criterion that he proposed for a rigid plastic isotropic matrix around a spherical cavity is expressed as follows: Σ 2 eqv 3k 2 + 2f cosh  √ 3 Σ m 2k  = 1 + f 2 (1) * Corresponding author. E-mail address: Joseph.Pastor@univ-savoie.fr (J. Pastor). 0997-7538/$ – see front matter  2005 Elsevier SAS. All rights reserved. doi:10.1016/j.euromechsol.2005.06.003