Savoie University
Polytech Annecy-Chambéry
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.... 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.
This Note presents a comparison of some recently developed "second-order" homogenization estimates for two-dimensional, ideally plastic porous media subjected to plane strain conditions with corresponding yield analysis results using a... more
This Note presents a comparison of some recently developed "second-order" homogenization estimates for two-dimensional, ideally plastic porous media subjected to plane strain conditions with corresponding yield analysis results using a new linearization technique and systematically optimized finite elements meshes. Good qualitative agreement is found between the second-order theory and the yield analysis results for the shape of the yield surfaces, which exhibit a corner on the hydrostatic axis, as well as for the dependence of the effective flow stress in shear on the porosity, which is found to be non-analytic in the dilute limit. Both of these features are inconsistent with the predictions of the standard Gurson model. To cite this article: J. Pastor, P. Ponte Castañeda, C. R. Mecanique 330 (2002) 741-747. 2002 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS porous media / homogenization / limit analysis / optimization Critère de plasticité des matériaux poreux en déformation plane : Estimations du second ordre et résultats numériques Résumé Cette Note présente une comparaison entre d'une part, les estimations issues d'une récente théorie d'homogénéisation, dite de « deuxième ordre », pour les matériaux parfaitement plastiques poreux en déformation plane, et d'autre part, les résultats homologues obtenus par analyse limite grâce une nouvelle technique de linéarisation du problème et une optimisation systématique des maillages éléments finis utilisés. Qualitativement parlant on observe un bon accord entre les deux approches sur la forme de la surface limite, avec mise en évidence d'un point anguleux sur l'axe hydrostatique, et sur la dépendance de la contrainte équivalente en cisaillement avec la porosité, contrainte dont la limite pour les faibles porosités apparaît non analytique. Ces deux caractéristiques ne sont pas prévues par le modèle de Gurson standard. Pour citer cet article : J. Pastor, P. Ponte Castañeda, C. R. Mecanique 330 (2002) 741-747. 2002 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS matériaux poreux / homogénéisation / analyse limite / optimisation
In a companion paper (Part I) we investigated the plane stress case in order to compare it with specific tensile tests. In the present paper, the Gurson problem is investigated in other cases of symmetry, including the most general... more
In a companion paper (Part I) we investigated the plane stress case in order to compare it with specific tensile tests. In the present paper, the Gurson problem is investigated in other cases of symmetry, including the most general 3D-plane case. Once the problem is outlined, we will present its formulation in terms of limit analysis (LA), then a new method to obtain a linear programming (LP) problem very suitable for modern LP codes based on Interior Point algorithms. The results indicate that a Gurson-like formulation cannot correctly represent the general solution, as in the plane stress case of Part I. A realistic global formulation could be specified in terms of all loading parameters, without using the restriction induced by an equivalent stress form.
The well-known problem of the height limit of a Tresca or von Mises vertical slope of height h, subjected to the action of gravity stems naturally from Limit Analysis theory under the plane strain condition. Although the exact solution to... more
The well-known problem of the height limit of a Tresca or von Mises vertical slope of height h, subjected to the action of gravity stems naturally from Limit Analysis theory under the plane strain condition. Although the exact solution to this problem remains unknown, this paper aims to give new precise bounds using both the static and kinematic approaches and an Interior Point optimizer code. The constituent material is a homogeneous isotropic soil of weight per unit volume γ. It obeys the Tresca or von Mises criterion characterized by C cohesion. We show that the loading parameter to be optimized, γh/C, is found to be between 3.767 and 3.782, and finally, using a recent result of Lyamin and Sloan (Int. J. Numer. Meth. Engng. 2002; 55: 573), between 3.772 and 3.782. The proposed methods, combined with an Interior Point optimization code, prove that linearizing the problem remains efficient, and both rigorous and global: this point is the main objective of the present paper. Copyright © 2003 John Wiley & Sons, Ltd.
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.... 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.
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... 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).
- by Joseph PASTOR and +1
- •
- Mechanical Engineering, Limit Analysis
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... 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.
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... 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).
This paper proposes an original decomposition approach to the upper bound method of limit analysis. It is based on a mixed finite element approach and on a convex interior point solver using linear or quadratic discontinuous velocity... more
This paper proposes an original decomposition approach to the upper bound method of limit analysis. It is based on a mixed finite element approach and on a convex interior point solver using linear or quadratic discontinuous velocity fields. Presented in plane strain, this method appears to be rapidly convergent, as verified in the Tresca compressed bar problem in the linear velocity case. Then, using discontinuous quadratic velocity fields, the method is applied to the celebrated problem of the stability factor of a Tresca vertical slope: the upper bound is lowered to 3.7776-value to be compared with the best published lower bound 3.772-by succeeding in solving non-linear optimization problems with millions of variables and constraints.
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... 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.
This paper is devoted to the stability analysis of a vertical embankment in reinforced soil, assuming that a very large number of reinforcements are periodically distributed throughout the soil mass. The reinforced soil is modelled as a... more
This paper is devoted to the stability analysis of a vertical embankment in reinforced soil, assuming that a very large number of reinforcements are periodically distributed throughout the soil mass. The reinforced soil is modelled as a homogeneous medium that obeys a macroscopic yield condition.Two numerical formulations of the homogenized problem, derived from the lower and upper bound theorems of limit analysis, respectively, with a finite element discretization technique, are described. They both lead to a linear programming problem, which is carried out by means of XPRESS industrial LP code.The practical implementation of both the static and kinematic finite element programs on the case of a vertical reinforced earth wall results in close estimates of its failure height, which are in good agreement with available experimental data. This points to the ability of such programs to provide a rigorous evaluation of the limit loads of structures through the determination of lower bound and upper bound estimates sufficiently close to each other.
Though the solution to the limit analysis problem of the hollow sphere model—with a von Mises matrix and under spherical symmetry—is well known, it is not available, to our knowledge, for both isotropic loadings (tension and compression)... more
Though the solution to the limit analysis problem of the hollow sphere model—with a von Mises matrix and under spherical symmetry—is well known, it is not available, to our knowledge, for both isotropic loadings (tension and compression) in the case of a Coulomb matrix and partially for a Drucker–Prager matrix. In the present Note, we establish in a unified framework, for this class of materials, closed-form solutions for stress and strain fields in a hollow sphere under external isotropic tension and compression. These analytical results not only give useful reference solutions, but can also be considered as a part of a trial velocity field in the hollow sphere submitted to an arbitrary loading. Comparisons with 3D finite element-based limit analysis approaches and with recent results in the literature are provided. In addition to the established analytical results, we present a rigorous evaluation of a recent Gurson-type macroscopic criterion corresponding to the Drucker–Prager hollow sphere under an arbitrary loading, by means of the previous 3D limit analysis codes. To cite this article: Ph. Thoré et al., C. R. Mecanique 337 (2009).Alors que la solution du problème d'analyse limite du modèle de la sphère creuse—à matrice de von Mises et en symétrie sphérique—est bien connue, elle n'est pas, à notre connaissance, disponible dans les deux cas de chargement isotrope (traction, compression) pour une matrice solide de Coulomb et partiellement pour une matrice de Drucker–Prager. Dans la présente Note, nous établissons dans un cadre unifié et pour cette classe de matériaux, les solutions exactes (champs de contraintes, de déformation) pour une sphère creuse soumise à une traction ou compression isotrope externe. Ces résultats analytiques sont non seulement utiles comme solutions de référence, mais elles peuvent aussi être considérées comme partie de champ d'essai en vitesse pour le modèle de la sphère creuse soumise à un chargement arbitraire. On fournit une comparaison avec des approches 3D par éléments finis du problème d'analyse limite, ainsi qu'avec de récents résultats dans la littérature. Outre les solutions analytiques établies, nous présentons une évaluation d'un récent critère de plasticité macroscopique correspondant à la sphère creuse de Drucker–Prager sous chargement quelconque, ceci à l'aide des précédents codes d'analyse limite 3D. Pour citer cet article : Ph. Thoré et al., C. R. Mecanique 337 (2009).
Le problème de la rupture ductile des matériaux métalliques poreux est analysé à l'aide ici des deux méthodes de l'Analyse Limite (A.L.), problème dont Gurson a donné en 1977 une approche cinématique célèbre. Le présent travail concerne... more
Le problème de la rupture ductile des matériaux métalliques poreux est analysé à l'aide ici des deux méthodes de l'Analyse Limite (A.L.), problème dont Gurson a donné en 1977 une approche cinématique célèbre. Le présent travail concerne l'étude des matériaux à cavités cylindriques dans le cas de la contrainte plane. Il est fondé sur la méthode d'homogénéisation des milieux hétérogènes en Analyse Limite dite de Hill–Mandel et sur la discrétisation en éléments finis du modèle originel de Gurson. L'utilisation systématique des approches statique et cinématique nous a permis de préciser très finement le critère de plasticité dans les cas de symétrie traités. Ainsi avons-nous pu situer les critères analytiques ou semi-analytiques proposés par plusieurs auteurs vis-à-vis de l'A.L. Il ressort, entre autres, de cette analyse que le critère de Richmond semble «le plus exact » au sens de l'Analyse Limite.The ductile failure of porous metallic materials is studied here using both Limit Analysis (LA) methods, a problem treated by Gurson with his famous kinematic 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 LA framework. The use of kinematic and static approaches gave an excellent estimation of the yield criterion for all the cases studied. The numerical results based on LA methods have been compared with analytical and semi-analytical yield domain expressions proposed by different authors. The results show that the Richmond model was the most accurate in terms of our predictions.
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... 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).
- by Malorie Trillat and +1
- •
- Mechanical Engineering, Limit Analysis
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... 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).
The effect of asymmetry with respect to sliding direction has been recently proved to be important in the analysis of static contact interactions of some automotive mechanical structures. In the present paper, we consider a nonclassical... more
The effect of asymmetry with respect to sliding direction has been recently proved to be important in the analysis of static contact interactions of some automotive mechanical structures. In the present paper, we consider a nonclassical friction law, based on a dissymmetry of friction in a given sliding direction. A simple analytical micro-model has been adopted to derive the macroscopic model via averaging. Not only does it allow the law macroscopic characteristics to be determined but it also theoretically proves the existence of such dissymmetry of friction. The thermodynamic admissibility of the law is then shown from the derivation of a generalized standard model including friction. To complete the analysis, the existence of asymmetric friction conditions is confirmed from an experimental point of view. We then compare theoretical computations resulting from the analytical model with experimental results.
- by Joseph PASTOR and +2
- •
- Civil Engineering, Applied Mathematics
The paper is devoted to a numerical limit analysis of a hollow spheroidal model with a von Mises solid matrix. To this purpose, existing kinematic and static 3D-FEM codes for the case of spherical cavities have been modified and improved... more
The paper is devoted to a numerical limit analysis of a hollow spheroidal model with a von Mises solid matrix. To this purpose, existing kinematic and static 3D-FEM codes for the case of spherical cavities have been modified and improved to account for the model of a spheroidal cavity confocal with the external spheroidal boundary. The optimized conic programming formulations and the resulting codes appear to be very efficient. This framework is then applied to the derivation of numerical upper and lower anisotropic bounds in the case of an oblate void. The numerical results obtained from a series of tests are presented and allow to assess the accuracy of closed-form expressions of the macroscopic criteria proposed by for porous media with oblate voids.