Comptes Rendus
Limit analysis and lower/upper bounds to the macroscopic criterion of Drucker–Prager materials with spheroidal voids
Comptes Rendus. Mécanique, Volume 342 (2014) no. 2, pp. 96-105.

The paper is devoted to a numerical Limit Analysis of a hollow spheroidal model with a Drucker–Prager solid matrix, for several values of the corresponding friction angle ϕ. In the first part of this study, the static and the mixed kinematic 3D-codes recently evaluated in [1] are modified to use the geometry defined in [2] for spheroidal cavities in the context of a von Mises matrix. The results in terms of macroscopic criteria are satisfactory for low and medium values of ϕ, but not enough for ϕ=30° in the highly compressive part of the criterion. To improve these results, an original mixed approach, dedicated to the axisymmetric case, was elaborated with a specific discontinuous quadratic velocity field associated with the triangular finite element. Despite the less good conditioning inherent to the axisymmetric modelization, the resulting conic programming problem appears quite efficient, allowing one take into account numerical discretization refinements unreachable with the corresponding 3D mixed code. After a first validation in the case of spherical cavities whose exact solution is known, the final results for spheroidal voids are given for three usual values of the friction angle and two values of the cavity aspect factor.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crme.2013.12.002
Keywords: Gurson-type models, Spheroidal voids, Micromechanics, Limit analysis, Upper and lower bounds, Conic programming

Franck Pastor 1; Djimedo Kondo 2

1 Athénée royal Victor-Horta, rue de la Rhétorique, 16, Bruxelles, Belgium
2 Institut Jean-Le-Rond-d'Alembert, UMR 7190 CNRS, UPMC, 75252 Paris cedex 05, France
@article{CRMECA_2014__342_2_96_0,
     author = {Franck Pastor and Djimedo Kondo},
     title = {Limit analysis and lower/upper bounds to the macroscopic criterion of {Drucker{\textendash}Prager} materials with spheroidal voids},
     journal = {Comptes Rendus. M\'ecanique},
     pages = {96--105},
     publisher = {Elsevier},
     volume = {342},
     number = {2},
     year = {2014},
     doi = {10.1016/j.crme.2013.12.002},
     language = {en},
}
TY  - JOUR
AU  - Franck Pastor
AU  - Djimedo Kondo
TI  - Limit analysis and lower/upper bounds to the macroscopic criterion of Drucker–Prager materials with spheroidal voids
JO  - Comptes Rendus. Mécanique
PY  - 2014
SP  - 96
EP  - 105
VL  - 342
IS  - 2
PB  - Elsevier
DO  - 10.1016/j.crme.2013.12.002
LA  - en
ID  - CRMECA_2014__342_2_96_0
ER  - 
%0 Journal Article
%A Franck Pastor
%A Djimedo Kondo
%T Limit analysis and lower/upper bounds to the macroscopic criterion of Drucker–Prager materials with spheroidal voids
%J Comptes Rendus. Mécanique
%D 2014
%P 96-105
%V 342
%N 2
%I Elsevier
%R 10.1016/j.crme.2013.12.002
%G en
%F CRMECA_2014__342_2_96_0
Franck Pastor; Djimedo Kondo. Limit analysis and lower/upper bounds to the macroscopic criterion of Drucker–Prager materials with spheroidal voids. Comptes Rendus. Mécanique, Volume 342 (2014) no. 2, pp. 96-105. doi : 10.1016/j.crme.2013.12.002. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2013.12.002/

[1] F. Pastor; D. Kondo; J. Pastor 3D-FEM formulations of limit analysis methods for porous pressure-sensitive materials, Int. J. Numer. Methods Eng., Volume 95 (2013), pp. 847-870

[2] F. Pastor; D. Kondo Assessment of hollow spheroid models for ductile failure prediction by limit analysis and conic programming, Eur. J. Mech. A, Solids, Volume 38 (2013), pp. 100-114

[3] A.L. Gurson 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., Volume 99 (1977), pp. 2-15

[4] J. Lee; J. Oung Yield functions and flow rules for porous pressure-dependent strain-hardening polymeric materials, J. Appl. Mech., Volume 67 (2000), pp. 288-297

[5] T.F. Guo; J. Faleskog; C.F. Shih Continuum modeling of a porous solid with pressure sensitive dilatant matrix, J. Mech. Phys. Solids, Volume 56 (2008), pp. 2188-2212

[6] P. Thoré; F. Pastor; J. Pastor Hollow sphere models, conic programming and third stress invariant, Eur. J. Mech. A, Solids, Volume 30 (2011), pp. 63-71

[7] M. Gologanu; J. Leblond Approximate models for ductile metals containing non-spherical voids – case of axisymmetric prolate ellipsoidal cavities, J. Mech. Phys. Solids, Volume 41 (1993) no. 11, pp. 1723-1754

[8] M. Gologanu; J. Leblond; G. Perrin; J. Devaux Approximate models for ductile metals containing non-spherical voids – case of axisymmetric oblate ellipsoidal cavities, J. Eng. Mater. Technol., Volume 116 (1994), pp. 290-297

[9] M. Garajeu; P. Suquet Effective properties of porous ideally plastic or viscoplastic materials containing rigid particles, J. Mech. Phys. Solids, Volume 45 (1997), pp. 873-902

[10] V. Monchiet; E. Charkaluk; D. Kondo An improvement of Gurson-type models of porous materials by using Eshelby-like trial velocity fields, C. R. Mecanique, Volume 335 (2007), pp. 32-41

[11] V. Monchiet; O. Cazacu; E. Charkaluk; D. Kondo Macroscopic yield criteria for plastic anisotropic materials containing spheroidal voids, Int. J. Plast., Volume 24 (2008), pp. 1158-1189

[12] V. Monchiet, E. Charkaluk, D. Kondo, Macroscopic yield criteria for ductile materials containing spheroidal voids: an Eshelby-like velocity fields approach, Mech. Mater., . | DOI

[13] H. Thai-The; P. Francescato; J. Pastor Limit analysis of unidirectional porous media, Mech. Res. Commun., Volume 25 (1998), pp. 535-542

[14] P. Francescato; J. Pastor; B. Riveill-Reydet Ductile failure of cylindrically porous materials. Part I: Plane stress problem and experimental results, Eur. J. Mech. A, Solids, Volume 23 (2004), pp. 181-190

[15] J. Pastor; P. Francescato; M. Trillat; E. Loute; G. Rousselier Ductile failure of cylindrically porous materials. Part II: Other cases of symmetry, Eur. J. Mech. A, Solids, Volume 23 (2004), pp. 191-201

[16] M. Trillat; J. Pastor Limit analysis and Gurson's model, Eur. J. Mech. A, Solids, Volume 24 (2005), pp. 800-819

[17] J. Salençon Calcul à la rupture et analyse limite, Presses des Ponts et Chaussées, Paris, 1983

[18] E. Anderheggen; H. Knopfel Finite element limit analysis using linear programming, Int. J. Solids Struct., Volume 8 (1972), pp. 1413-1431

[19] K. Krabbenhoft; A. Lyamin; M. Hijaj; S. Sloan A new discontinuous upper bound limit analysis formulation, Int. J. Numer. Methods Eng., Volume 63 (2005), pp. 1069-1088

[20] F. Pastor Résolution par des méthodes de point intérieur de problèmes de programmation convexe posés par l'analyse limite, facultés universitaires Notre-Dame-de-la-Paix, Namur, Belgium, 2007 (thèse de doctorat)

[21] F. Pastor; E. Loute; J. Pastor; M. Trillat Mixed method and convex optimization for limit analysis of homogeneous Gurson materials: a kinematical approach, Eur. J. Mech. A, Solids, Volume 28 (2009), pp. 25-35

[22] A. Makrodimopoulos; C.M. Martin Upper bound limit analysis using discontinuous quadratic displacement fields, Commun. Numer. Methods Eng., Volume 24 (2008), pp. 911-927

[23] MOSEK ApS, 2010 http://www.mosek.com (C/O Symbion Science Park, Fruebjergvej 3, Box 16, 2010 Copenhagen ϕ, Denmark)

[24] P. Thoré; F. Pastor; J. Pastor; D. Kondo Closed form solutions for the hollow sphere model with Coulomb and Drucker–Prager materials under isotropic loadings, C. R. Mecanique, Volume 337 (2009), pp. 260-267

Cited by Sources:

Comments - Policy