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Pastor F.,Athenee royal Victor Horta | Pastor J.,CNRS Laboratory of Design Optimisation and Environmental Engineering | Kondo D.,CNRS Jean Le Rond dAlembert Institute
Comptes Rendus - Mecanique | Year: 2015

The paper is devoted to a numerical Limit Analysis of a hollow cylindrical model with a Coulomb solid matrix (of confocal boundaries) considered in the case of a generalized plane strain. To this end, the static approach of Pastor et al. (2008) [18] for Drucker-Prager materials is first extended to Coulomb problems. A new mixed-but rigorously kinematic-code is elaborated for Coulomb problems in the present case of symmetry, resulting also in a conic programming approach. Owing to the good conditioning of the resulting optimization problems, both methods give very close bounds by allowing highly refined meshes, as verified by comparing to existing exact solutions. In a second part, using the identity of Tresca (as special case of Coulomb) and von Mises materials in plane strain, the codes are used to assess the corresponding results of Mariani and Corigliano (2001) [13] and of Madou and Leblond (2012) [11] for circular and elliptic cylindrical voids in a von Mises matrix. Finally, the Coulomb problem is investigated, also in terms of projections on the coordinate planes of the principal macroscopic stresses. © 2015. Source


Shen W.Q.,Lille Laboratory of Mechanics | Pastor F.,Athenee royal Victor Horta | Kondo D.,University Pierre and Marie Curie
Theoretical and Applied Fracture Mechanics | Year: 2013

In the framework of limit analysis theory, a macroscopic yield function has been recently established by [14] for ductile porous materials having a Green type matrix. The present study aims at improving this macroscopic criterion by considering Eshelby-like velocity fields for the limit analysis of a hollow sphere subjected to uniform strain rate boundary conditions. The newly derived criteria are assessed by comparing their predictions with data obtained from numerical limit analysis computations (lower and upper bounds). © 2014 Elsevier Ltd. Source


Pastor F.,Athenee royal Victor Horta | Anoukou K.,University Pierre and Marie Curie | Pastor J.,CNRS Laboratory of Design Optimisation and Environmental Engineering | Kondo D.,University Pierre and Marie Curie
Journal of the Mechanics and Physics of Solids | Year: 2016

This second part of the two-part study is devoted to the numerical Limit Analysis of a hollow sphere model with a Mohr-Coulomb matrix and its use for the assessment of theoretical results. Brief background and fundamental of the static and kinematic approaches in the context of numerical limit analysis are first recalled. We then present the hollow sphere model, together with its axisymmetric FEM discretization and its mechanical position. A conic programming adaptation of a previous iterative static approach, based on a piecewise linearization (PWL) of the plasticity criterion, was first realized. Unfortunately, the resulting code, no more than the PWL one, did not allow sufficiently refined meshes for loss of convergence of the conic optimizer. This problem was solved by using the projection algorithm of Ben Tal and Nemriovski (BTN) and the (interior point) linear programming code XA. For the kinematic approach, a first conic adaptation appeared also inefficient. Then, an original mixed (but fully kinematic) approach dedicated to the general Mohr-Coulomb axisymmetric problem was elaborated. The final conic mixed code appears much more robust than the classic one when using the conic code MOSEK, allowing us to take into account refined numerical meshes. After a fine validation in the case of spherical cavities and isotropic loadings (for which the exact solution is known) and comparison to previous (partial) results, numerical lower and upper bounds (a posteriori verified) of the macroscopic strength are provided. These bounds are used to assess and validate the theoretical results of the companion (part I) paper. Effects of the friction angle as well as that of the porosity are illustrated. © 2016 Elsevier Ltd. All rights reserved. Source


Anoukou K.,University Pierre and Marie Curie | Pastor F.,Athenee royal Victor Horta | Dufrenoy P.,Lille Laboratory of Mechanics | Kondo D.,University Pierre and Marie Curie
Journal of the Mechanics and Physics of Solids | Year: 2016

The present two-part study aims at investigating the specific effects of Mohr-Coulomb matrix on the strength of ductile porous materials by using a kinematic limit analysis approach. While in the Part II, static and kinematic bounds are numerically derived and used for validation purpose, the present Part I focuses on the theoretical formulation of a macroscopic strength criterion for porous Mohr-Coulomb materials. To this end, we consider a hollow sphere model with a rigid perfectly plastic Mohr-Coulomb matrix, subjected to axisymmetric uniform strain rate boundary conditions. Taking advantage of an appropriate family of three-parameter trial velocity fields accounting for the specific plastic deformation mechanisms of the Mohr-Coulomb matrix, we then provide a solution of the constrained minimization problem required for the determination of the macroscopic dissipation function. The macroscopic strength criterion is then obtained by means of the Lagrangian method combined with Karush-Kuhn-Tucker conditions. After a careful analysis and discussion of the plastic admissibility condition associated to the Mohr-Coulomb criterion, the above procedure leads to a parametric closed-form expression of the macroscopic strength criterion. The latter explicitly shows a dependence on the three stress invariants. In the special case of a friction angle equal to zero, the established criterion reduced to recently available results for porous Tresca materials. Finally, both effects of matrix friction angle and porosity are briefly illustrated and, for completeness, the macroscopic plastic flow rule and the voids evolution law are fully furnished. © 2016 Elsevier Ltd. All rights reserved. Source


Pastor F.,Athenee royal Victor Horta | Kondo D.,University Pierre and Marie Curie | Pastor J.,CNRS Laboratory of Design Optimisation and Environmental Engineering
International Journal for Numerical Methods in Engineering | Year: 2013

The first purpose of this paper is the numerical formulation of the three general limit analysis methods for problems involving pressure-sensitive materials, that is, the static, classic, and mixed kinematic methods applied to problems with Drucker-Prager, Mises-Schleicher, or Green materials. In each case, quadratic or rotated quadratic cone programming is considered to solve the final optimization problems, leading to original and efficient numerical formulations. As a second purpose, the resulting codes are applied to non-classic 3D problems, that is, the Gurson-like hollow sphere problem with these materials as matrices. To this end are first presented the 3D finite element implementations of the static and kinematic classic methods of limit analysis together with a mixed method formulated to give also a purely kinematic result. Discontinuous stress and velocity fields are included in the analysis. The static and the two kinematic approaches are compared afterwards in the hydrostatic loading case whose exact solution is known for the three cases of matrix. Then, the static and the mixed approaches are used to assess the available approximate criteria for porous Drucker-Prager, Mises-Schleicher, and Green materials. © 2013 John Wiley & Sons, Ltd. Source

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