Comptes Rendus
Limit analysis of hollow spheres or spheroids with Hill orthotropic matrix
Comptes Rendus. Mécanique, Volume 340 (2012) no. 3, pp. 120-129.

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.

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Published online:
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
Mot clés : Mileux poreux, Anisotropie, Matrice de Hill, Approche Gurson, Modèle du sphéroïde creux, Analyse limite, 3D-MEF, Optimization conique

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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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/

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