Limit analysis of hollow spheres or spheroids with Hill orthotropic matrix

Franck Pastor; Joseph Pastor; Djimedo Kondo

doi:10.1016/j.crme.2011.11.011

Limit analysis of hollow spheres or spheroids with Hill orthotropic matrix

Franck Pastor ¹; Joseph Pastor ²; Djimedo Kondo ³

¹ Laboratoire de mécanique de Lille, UMR 8107 CNRS, université de Lille 1, cité scientifique, 59655 Villeneuve dʼAscq, France
² Laboratoire LOCIE, UMR 5271 CNRS, université de Savoie, 73376 Le Bourget du Lac, France
³ Université Pierre-et-Marie-Curie, institut DʼAlembert, UMR 7190 CNRS, 4, place Jussieu, 75252 Paris cedex 05, France

Comptes Rendus. Mécanique, Volume 340 (2012) no. 3, pp. 120-129.

Abstract
Résumé

Recent theoretical studies of the literature are concerned by the hollow sphere or spheroid (confocal) problems with orthotropic Hill type matrix. They have been developed in the framework of the limit analysis kinematical approach by using very simple trial velocity fields. The present Note provides, through numerical upper and lower bounds, a rigorous assessment of the approximate criteria derived in these theoretical works. To this end, existing static 3D codes for a von Mises matrix have been easily extended to the orthotropic case. Conversely, instead of the non-obvious extension of the existing kinematic codes, a new original mixed approach has been elaborated on the basis of the plane strain structure formulation earlier developed by F. Pastor (2007). Indeed, such a formulation does not need the expressions of the unit dissipated powers. Interestingly, it delivers a numerical code better conditioned and notably more rapid than the previous one, while preserving the rigorous upper bound character of the corresponding numerical results. The efficiency of the whole approach is first demonstrated through comparisons of the results to the analytical upper bounds of Benzerga and Besson (2001) or Monchiet et al. (2008) in the case of spherical voids in the Hill matrix. Moreover, we provide upper and lower bounds results for the hollow spheroid with the Hill matrix which are compared to those of Monchiet et al. (2008).

De récentes études dans la littérature ont porté sur lʼétablissement de critères macroscopiques de milieux plastiques orthotropes de type Hill contenant des vides sphériques ou sphéroïdaux. Pour évaluer ces critères, nous avons étendu les codes statiques et cinématiques 3D existants du cas isotrope à lʼanisotrope. Les codes statiques ont pu lʼêtre assez aisément, lʼapproche cinématique étant plus problématique du fait des indispensables discontinuités de vitesses. Pour cette raison une nouvelle et originale approche mixte a donc été élaborée sur la base de la formulation proposée par F. Pastor (2007) pour les structures en déformation plane, cette formulation ayant lʼavantage de ne pas nécessiter lʼexpression des puissances dissipées unitaires. Cette approche mixte a débouché sur un code mieux conditionné et significativement plus rapide que le code précédent, sans perdre pour autant le caractère de borne supérieure rigoureuse du résultat. La pertinence de la nouvelle approche a été dʼabord démontrée sur le cas dʼune matrice de von Mises, obtenu comme cas particulier de la matrice de Hill. Puis, des bornes numériques sont fournies et comparées au critère établi par Benzerga et Besson (2001) et Monchiet et al. (2008) dans le cas de la matrice orthotrope avec une cavité sphérique. Enfin, les prédictions de Monchiet et al. (2008) dans le cas dʼune cavité sphéroïdale plongée dans la matrice orthotrope sont évaluées et discutées.

Received: 2011-10-15
Accepted: 2011-11-28
Published online: 2012-02-06

DOI: 10.1016/j.crme.2011.11.011

Keywords: Porous media, Anisotropy, Hill matrix, Gurson approach, Hollow spheroid model, Limit analysis, 3D-FEM, Conic optimization
Mots-clés : Mileux poreux, Anisotropie, Matrice de Hill, Approche Gurson, Modèle du sphéroïde creux, Analyse limite, 3D-MEF, Optimization conique

Author's affiliations:

Franck Pastor ¹; Joseph Pastor ²; Djimedo Kondo ³

¹ Laboratoire de mécanique de Lille, UMR 8107 CNRS, université de Lille 1, cité scientifique, 59655 Villeneuve dʼAscq, France
² Laboratoire LOCIE, UMR 5271 CNRS, université de Savoie, 73376 Le Bourget du Lac, France
³ Université Pierre-et-Marie-Curie, institut DʼAlembert, UMR 7190 CNRS, 4, place Jussieu, 75252 Paris cedex 05, France

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     author = {Franck Pastor and Joseph Pastor and Djimedo Kondo},
     title = {Limit analysis of hollow spheres or spheroids with {Hill} orthotropic matrix},
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Franck Pastor; Joseph Pastor; Djimedo Kondo. Limit analysis of hollow spheres or spheroids with Hill orthotropic matrix. Comptes Rendus. Mécanique, Volume 340 (2012) no. 3, pp. 120-129. doi : 10.1016/j.crme.2011.11.011. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2011.11.011/

References
Cited by

[1] 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. Mat. Technol., Volume 99 (1977), pp. 2-15

[2] M. Gologanu; J.B. 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

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

[4] M. Gologanu; J.B. Leblond; G. Perrin; J. Devaux Recent extensions of Gursonʼs model for porous ductile metals (P. Suquet, ed.), Continuum Micromechanics, Springer-Verlag, 1997

[5] 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

[6] 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

[7] A.A. Benzerga; J. Besson Plastic potentials for anisotropic porous solids, Eur. J. Mech. A Solids, Volume 20 (2001), pp. 397-434

[8] 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

[9] S.M. Keralavarma; A.A. Benzerga A constitutive model for plastically anisotropic solids with non-spherical voids, J. Mech. Phys. Solids, Volume 58 (2010), pp. 874-901

[10] A. Benzerga; J.B. Leblond Ductile fracture by void growth to coalescence, Adv. Appl. Mech., Volume 44 (2010), pp. 169-305

[11] 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

[12] 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

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

[14] 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

[15] F. Pastor; D. Kondo; J. Pastor Numerical limit analysis bounds for ductile porous media with oblate voids, Mech. Res. Comm., Volume 38 (2011), pp. 350-354

[16] F. Pastor, D. Kondo, Assessment of models accounting for voids shape effects on ductile behavior: A conic programming limit analysis approach, submitted for publication.

[17] MOSEK ApS C/O Symbion Science Park, Fruebjergvej 3, Box 16, 2100 Copenhagen ϕ, Denmark, email: info@mosek.com, 2011.

[18] R. Hill A theory of yielding and plastic flow of anisotropic solids, Proc. R. Soc. Lond. Ser. A, Volume 193 (1948), pp. 281-297

[19] F. Pastor, Résolution par des méthodes de point intérieur de problèmes de programmation convexe posés par lʼanalyse limite, Thèse de doctorat, Facultés universitaires Notre-Dame de la Paix, Namur, 2007.

[20] 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

[21] J. Salençon Théorie de la plasticité pour les applications à la mécanique des sols, Eyrolles, Paris, 1974

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

[23] F. Pastor; E. Loute Solving limit analysis problems: an interior-point method, Comm. Numer. Methods Engrg., Volume 21 (2005) no. 11, pp. 631-642

[24] D. Radenkovic; Q.S. Nguyen La dualité des théorèmes limites pour une structure en matériau rigide-plastique standard, Arch. Mech., Volume 24 (1972) no. 5–6, pp. 991-998

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