Papers by Malorie Trillat
Je tiens avant tout à remercier Monsieur le Professeur J. Pastor pour son dynamisme et son implic... more Je tiens avant tout à remercier Monsieur le Professeur J. Pastor pour son dynamisme et son implication dans son rôle de directeur de thèse. Sans l'aide de ses analyses scientifiques rigoureuses, cette thèse ne serait pas ce qu'elle est. J'exprime toute ma gratitude à Monsieur P. Francescato qui, malgré son manque de disponibilité inhérent à ses responsabilités vis-à-vis de la filière Matériaux Composites de l'Ecole, a trouvé du temps à m'accorder en tant que co-directeur de thèse. Ses suggestions pertinentes m'ont aidée. Que Monsieur J. Salençon, Professeur à l'Ecole Polytechnique et membre de l'Académie des Sciences, qui m'a fait l'honneur de s'intéresser à mon travail et de présider mon jury de thèse trouve ici l'expression de ma reconnaissance. Ses remarques perspicaces et son sens du contact peu commun m'ont été précieux. Je remercie Monsieur S. Turgeman, Professeur à l'Université Joseph Fourier, pour l'intérêt qu'il a porté à ce travail en acceptant d'être rapporteur. Ses remarques et ses commentaires pertinents ont réellement contribué à améliorer cette thèse. Je suis également particulièrement heureuse que Monsieur G. Maier, Professeur au Politecnico di Milano ait porté attention à mon travail en acceptant d'être le second rapporteur. J'adresse mes profonds remerciements à Monsieur J.-B. Leblond, Professeur à l'Université Pierre et Marie Curie et correspondant de l'Académie des Sciences. Ses avancées dans le domaine de la rupture des matériaux poreux font de lui un spécialiste des plus reconnus. Ses remarques ciblées et ses conseils ont démontré son intérêt pour ce travail. La dernière partie de ce travail a été réalisée en collaboration avec le CORE
This thesis concerns the study of the yield criterion of the porous materials using the homogenei... more This thesis concerns the study of the yield criterion of the porous materials using the homogeneization, the limit analysis and the interior point optimization. The yield criterion of a porous material using Gurson's model, the most widely accepted for such materials in elasto-platic codes, is investigated. The Gurson model idealizes the porous material as a single cavity in a homothetic cell composed of a rigid plastic Mises material, called the Representative Volume Element (RVE) in the following. In this model, the cavities don't have any interactions or coalescence. Then we use the two limit analysis approaches, via a discretization of the model in finite elements. They lead to non-linear optimization problems, solved either by two commercial (???j'ai change le “cormmecial”)codes, XA or MOSEK, both optimization codes based on so-called interior point methods. For porous materials with cylindrical cavities, the Gurson criterion appears to be insufficient. In the generalized plane strain case, an analytical expression of this true criterion must take the form of a function of the loading parameters, at least in a three-dimensional representation. Conversely, for a porous material with spherical cavities, a full 3D model is worked out. The Gurson approach is slightly improved and, for the first time, it is validated by our rigorous static and kinematic approaches. The study of a RVE with 35 cylindrical random cavities confirms a bimodal criterion in generalized plane strain. In plane stress loading, the RVE is not representative. A nonlinear interior point method for solving stress based upper bound problems is proposed. To solve the problem, we used a convex optimizer, written on Matlab, developed at CORE (Centre of Operation Research and Econometrics) of Louvain la Neuve in Belgium. Assuming a linear, continuous or discontinuous virtual velocity field, the method appears to be efficient and general. This method is straightforward, needing only the yield criterion as information on the material
This thesis concerns the study of the yield criterion of the porous materials using the homogenei... more This thesis concerns the study of the yield criterion of the porous materials using the homogeneization, the limit analysis and the interior point optimization. The yield criterion of a porous material using Gurson's model, the most widely accepted for such materials in elasto-platic codes, is investigated. The Gurson model idealizes the porous material as a single cavity in a homothetic cell composed of a rigid plastic Mises material, called the Representative Volume Element (RVE) in the following. In this model, the cavities don't have any interactions or coalescence. Then we use the two limit analysis approaches, via a discretization of the model in finite elements. They lead to non-linear optimization problems, solved either by two commercial (???j'ai change le “cormmecial”)codes, XA or MOSEK, both optimization codes based on so-called interior point methods. For porous materials with cylindrical cavities, the Gurson criterion appears to be insufficient. In the gener...
Je tiens avant tout à remercier Monsieur le Professeur J. Pastor pour son dynamisme et son implic... more Je tiens avant tout à remercier Monsieur le Professeur J. Pastor pour son dynamisme et son implication dans son rôle de directeur de thèse. Sans l'aide de ses analyses scientifiques rigoureuses, cette thèse ne serait pas ce qu'elle est. J'exprime toute ma gratitude à Monsieur P. Francescato qui, malgré son manque de disponibilité inhérent à ses responsabilités vis-à-vis de la filière Matériaux Composites de l'Ecole, a trouvé du temps à m'accorder en tant que co-directeur de thèse. Ses suggestions pertinentes m'ont aidée. Que Monsieur J. Salençon, Professeur à l'Ecole Polytechnique et membre de l'Académie des Sciences, qui m'a fait l'honneur de s'intéresser à mon travail et de présider mon jury de thèse trouve ici l'expression de ma reconnaissance. Ses remarques perspicaces et son sens du contact peu commun m'ont été précieux. Je remercie Monsieur S. Turgeman, Professeur à l'Université Joseph Fourier, pour l'intérêt qu'il a porté à ce travail en acceptant d'être rapporteur. Ses remarques et ses commentaires pertinents ont réellement contribué à améliorer cette thèse. Je suis également particulièrement heureuse que Monsieur G. Maier, Professeur au Politecnico di Milano ait porté attention à mon travail en acceptant d'être le second rapporteur. J'adresse mes profonds remerciements à Monsieur J.-B. Leblond, Professeur à l'Université Pierre et Marie Curie et correspondant de l'Académie des Sciences. Ses avancées dans le domaine de la rupture des matériaux poreux font de lui un spécialiste des plus reconnus. Ses remarques ciblées et ses conseils ont démontré son intérêt pour ce travail. La dernière partie de ce travail a été réalisée en collaboration avec le CORE
European Journal of …, 2007
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… Ultimate Limit State, 2008
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9th European …, 2008
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European Journal of Mechanics - A/Solids, 2009
A fully kinematical, mixed finite element approach based on a recent interior point method for co... more A fully kinematical, mixed finite element approach based on a recent interior point method for convex optimization is proposed to solve the limit analysis problem involving homogeneous Gurson materials. It uses continuous or discontinuous quadratic velocity fields as virtual variables, with no hypothesis on a stress field. Its modus operandi is deduced from the Karush-Kuhn-Tucker optimality conditions of the mathematical problem, providing an example of cross-fertilization between mechanics and mathematical programming. This method is used to solve two classical problems for the von Mises plasticity criterion as a test case, and for the Gurson criterion for which analytical solutions do not exist. Using only the original plasticity criterion as material data, the method proposed appears robust and efficient, providing very tight bounds on the limit loadings investigated.
Engineering Fracture Mechanics, 2008
First, we summarize our convex optimization method to solve the static approach of limit analysis... more First, we summarize our convex optimization method to solve the static approach of limit analysis. Then, we present the main features of a quadratic extension of a recently proposed mixed finite element method of the kinematic approach. Both methods are applied to obtain precise solutions to a forming problem with Gurson and Drucker-Prager materials. Finally, in order to analyze the criterion of ''Porous Drucker-Prager'' materials, the Gurson micro-macro model involving a Drucker-Prager matrix containing cylindrical cavities is investigated. Comparing previous results shows, among other things, a similarity in the compression case not always observed for the ''Porous von Mises'' material between cylindrical and spherical cases.
European Journal of Mechanics - A/Solids, 2004
The ductile failure of porous metallic materials is studied here using the lower and upper bound ... more The ductile failure of porous metallic materials is studied here using the lower and upper bound methods of Limit Analysis (LA), a problem treated by Gurson with his famous kinematical approach in 1977. The present work is devoted to determining the strength of porous materials with long circular cylindrical cavities in the case of plane stress. The numerical methods developed here use the Hill-Mandel method based on the homogenization theory of heterogeneous media within the Limit Analysis framework. The use of kinematical and static approaches gave an excellent estimation of the yield criterion for all the cases studied. The existence of a corner of the yield surface on the hydrostatic axis is pointed out, a corner not predicted by the preceding authors, except Rousselier and Ponte Castañeda. The numerical results then are compared with yield domain expressions proposed by different authors and with experimental results of uniaxial tensile strength of aluminum ramdomly perforated sheets. The results show that the Richmond and Smelser model was the most accurate in terms of our predictions. In a companion paper (part II) other cases of symmetry are investigated.
European Journal of Mechanics-A/ …, Jan 1, 2009
A fully kinematical, mixed finite element approach based on a recent interior point method for co... more A fully kinematical, mixed finite element approach based on a recent interior point method for convex optimization is proposed to solve the limit analysis problem involving homogeneous Gurson materials. It uses continuous or discontinuous quadratic velocity fields as virtual variables, with no hypothesis on a stress field. Its modus operandi is deduced from the Karush-Kuhn-Tucker optimality conditions of the mathematical problem, providing an example of cross-fertilization between mechanics and mathematical programming. This method is used to solve two classical problems for the von Mises plasticity criterion as a test case, and for the Gurson criterion for which analytical solutions do not exist. Using only the original plasticity criterion as material data, the method proposed appears robust and efficient, providing very tight bounds on the limit loadings investigated.
Comptes Rendus Mecanique, Jan 1, 2006
A nonlinear interior point method associated with the kinematic theorem of limit analysis is prop... more A nonlinear interior point method associated with the kinematic theorem of limit analysis is proposed. Associating these two tools enables one to determine an upper bound of the limit loading of a Gurson material structure from the knowledge of the sole yield criterion. We present the main features of the interior point algorithm and an original method providing a rigorous kinematic bound from a stress formulation of the problem. This method is tested by solving in plane strain the problem of a Gurson infinite bar compressed between rough rigid plates. To cite this article: F. Pastor et al., C. R. Mecanique 334 (2006). 2006 Académie des sciences. Published by Elsevier SAS. All rights reserved. Résumé Méthode cinématique par les contraintes et optimisation convexe : cas du matériau de Gurson. Nous proposons une méthode d'optimisation de type point intérieur associée au théorème cinématique de l'analyse limite. L'association de ces deux outils permet de déterminer la borne cinématique du chargement limite d'une structure en matériau de Gurson à partir de la connaissance du seul critère de plasticité. On expose d'abord brièvement la méthode « point intérieur » de résolution d'un problème comportant des conditions linéaires et des conditions non linéaires, puis une formulation originale et rigoureuse, en contraintes, de l'approche cinématique. Cette méthode est validée en résolvant en déformation plane le problème d'une barre en matériau de Gurson comprimée entre deux plateaux rigides rugueux. Pour citer cet article : F. Pastor et al., C. R. Mecanique 334 (2006).
Comptes Rendus Mecanique, Jan 1, 2006
Extending a previous work on the Gurson model for a 'porous von Mises' material, the present stud... more Extending a previous work on the Gurson model for a 'porous von Mises' material, the present study first focuses on the yield criterion of a 'porous Drucker-Prager' material with spherical cavities. On the basis of the Gurson micro-macro model and a second order conic programming (SOCP) formulation, calculated inner and outer approaches to the criterion are very close, providing a reliable estimate of the yield criterion. Comparison with an analytical criterion recently proposed by Barthélémy and Dormieuxfrom a nonlinear homogenization method-shows both excellent agreement when considering tensile average boundary conditions and substantial improvement under compressive conditions. Then the results of an analogous study in the case of cylindrical cavities in plane strain are presented. It is worth noting that obtaining these results was made possible by using MOSEK, a recent commercial SOCP code, whose impressive efficiency was already seen in our previous works. To cite this article: M. Trillat et al., C. R. Mecanique 334 (2006). Résumé Analyse limite et optimisation conique : étude d'un matériau de Drucker-Prager poreux. Via l'extension d'un travail précédent sur le modèle micro-macro de Gurson portant sur le cas d'un matériau de von Mises poreux, la présente étude concerne le critère de « Drucker-Prager poreux ». En utilisant le modèle de Gurson et une formulation en optimisation conique du second ordre (SOCP), les approches intérieure et extérieure obtenues sont très proches, donnant ainsi une estimation fiable du critère recherché. La comparaison avec un critère analytique récemment proposé par Barthélémy et Dormieux-via une méthode d'homogénéisation non linéaire-montre à la fois une excellente concordance sous déformation moyenne de traction et une substantielle amélioration dans le cas compressif. Sont donnés ensuite les résultats de la même étude, en déformation plane, pour un matériau à cavités cylindriques. Il faut noter enfin que l'obtention de ces résultats a été rendue possible par l'utilisation de MOSEK, code de SOCP récent et d'une efficacité impressionnante, déjà constatée dans nos travaux précédents. Pour citer cet article : M. Trillat et al., C. R. Mecanique 334 (2006).
9th European …, Jan 1, 2008
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European Journal of Mechanics A-solids, Jan 1, 2005
The yield criterion of a porous material using Gurson's model [Gurson, A.L., 1977. Continuum theo... more 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. Technol. 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 cavities [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., Rousselier, 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.
Mechanics Research Communications, Jan 1, 2006
The yield criterion of a porous material satisfying Gurson criterion conditions is studied here. ... more The yield criterion of a porous material satisfying Gurson criterion conditions is studied here. We use the twofold limit analysis approach applied to a representative volume element of the porous media and recent optimization codes. Both upper and lower bounds are very close, and they give quasi-exact solutions. As a result, the Gurson approach is slightly improved and, for the first time, validated by a rigorous, full 3D static approach.
Mechanics Research Communications, 2006
The yield criterion of a porous material satisfying Gurson criterion conditions is studied here. ... more The yield criterion of a porous material satisfying Gurson criterion conditions is studied here. We use the twofold limit analysis approach applied to a representative volume element of the porous media and recent optimization codes. Both upper and lower bounds are very close, and they give quasi-exact solutions. As a result, the Gurson approach is slightly improved and, for the first time, validated by a rigorous, full 3D static approach.
European Journal of Mechanics - A/Solids, 2005
The yield criterion of a porous material using Gurson's model . Continuum theory of ductile ruptu... more The yield criterion of a porous material using Gurson's model . 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. Technol. 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 cavities [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., Rousselier, G., 2004. Ductile failure of cylindrically porous materials. Part 2: Other cases of symmetry. Eur. J. Mech. A Solids 23, [191][192][193][194][195][196][197][198][199][200][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.
Comptes Rendus Mécanique, 2006
A nonlinear interior point method associated with the kinematic theorem of limit analysis is prop... more A nonlinear interior point method associated with the kinematic theorem of limit analysis is proposed. Associating these two tools enables one to determine an upper bound of the limit loading of a Gurson material structure from the knowledge of the sole yield criterion. We present the main features of the interior point algorithm and an original method providing a rigorous kinematic bound from a stress formulation of the problem. This method is tested by solving in plane strain the problem of a Gurson infinite bar compressed between rough rigid plates. To cite this article: F. Pastor et al., C. R. Mecanique 334 (2006). 2006 Académie des sciences. Published by Elsevier SAS. All rights reserved. Résumé Méthode cinématique par les contraintes et optimisation convexe : cas du matériau de Gurson. Nous proposons une méthode d'optimisation de type point intérieur associée au théorème cinématique de l'analyse limite. L'association de ces deux outils permet de déterminer la borne cinématique du chargement limite d'une structure en matériau de Gurson à partir de la connaissance du seul critère de plasticité. On expose d'abord brièvement la méthode « point intérieur » de résolution d'un problème comportant des conditions linéaires et des conditions non linéaires, puis une formulation originale et rigoureuse, en contraintes, de l'approche cinématique. Cette méthode est validée en résolvant en déformation plane le problème d'une barre en matériau de Gurson comprimée entre deux plateaux rigides rugueux. Pour citer cet article : F. Pastor et al., C. R. Mecanique 334 (2006).
Comptes Rendus Mécanique, 2006
Extending a previous work on the Gurson model for a 'porous von Mises' material, the present stud... more Extending a previous work on the Gurson model for a 'porous von Mises' material, the present study first focuses on the yield criterion of a 'porous Drucker-Prager' material with spherical cavities. On the basis of the Gurson micro-macro model and a second order conic programming (SOCP) formulation, calculated inner and outer approaches to the criterion are very close, providing a reliable estimate of the yield criterion. Comparison with an analytical criterion recently proposed by Barthélémy and Dormieuxfrom a nonlinear homogenization method-shows both excellent agreement when considering tensile average boundary conditions and substantial improvement under compressive conditions. Then the results of an analogous study in the case of cylindrical cavities in plane strain are presented. It is worth noting that obtaining these results was made possible by using MOSEK, a recent commercial SOCP code, whose impressive efficiency was already seen in our previous works. To cite this article: M. Trillat et al., C. R. Mecanique 334 (2006). Résumé Analyse limite et optimisation conique : étude d'un matériau de Drucker-Prager poreux. Via l'extension d'un travail précédent sur le modèle micro-macro de Gurson portant sur le cas d'un matériau de von Mises poreux, la présente étude concerne le critère de « Drucker-Prager poreux ». En utilisant le modèle de Gurson et une formulation en optimisation conique du second ordre (SOCP), les approches intérieure et extérieure obtenues sont très proches, donnant ainsi une estimation fiable du critère recherché. La comparaison avec un critère analytique récemment proposé par Barthélémy et Dormieux-via une méthode d'homogénéisation non linéaire-montre à la fois une excellente concordance sous déformation moyenne de traction et une substantielle amélioration dans le cas compressif. Sont donnés ensuite les résultats de la même étude, en déformation plane, pour un matériau à cavités cylindriques. Il faut noter enfin que l'obtention de ces résultats a été rendue possible par l'utilisation de MOSEK, code de SOCP récent et d'une efficacité impressionnante, déjà constatée dans nos travaux précédents. Pour citer cet article : M. Trillat et al., C. R. Mecanique 334 (2006).
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Papers by Malorie Trillat