Analytic calculation of the stresses in an ensiled granular mediumCalcul analytique des contraintes dans un matériau granulaire ensilé

We present in this Note a new analytic approach, of continuous medium type, which improves the Janssen theory and enables us to calculate the stresses in an ensiled granular medium. This approach is based on the two dimensional equilibrium equations, coupled with the Mohr–Coulomb criterion and a slip condition at the walls of the silo. An analytic resolution is developed to compute the stresses for cohesive and non cohesive materials in the whole silo. To cite this article: O. Millet et al., C. R. Mecanique 334 (2006).     

Limit analysis and conic programming: ‘porous Drucker–Prager’ material and Gurson's modelAnalyse limite et optimisation conique : étude d'un matériau de Drucker–Prager poreux

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 Dormieux—from 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).     

Plasticity criterion for porous medium with cylindrical voidUn critère de plasticité pour un matériau poreux à vide cylindrique

A simple Gurson-based yield criterion for porous materials with cylindrical voids in plane stress is proposed. With no adjustable parameters, it compares quite satisfactorily with recent numerical data by Francescato et al. for different porosities. It is non-analytic with respect to the porosity, and displays an angulous point. To cite this article: Y.-P. Pellegrini, C. R. Mecanique 330 (2002) 763–768.     

Approche globale,minima relatifs et Critère d'Amorçage en Mécanique de la RuptureGlobal approach,relative minima and yield criterion in Fracture Mechanics

We show that it is possible to predict the onset of cracking by searching relative minima of the total energy of a body, the yield criterion depending on the choice of the energies. In particular in the case where the bulk energy is elastic and the surface energy depends on the jump of the displacement, we prove that the yield criterion is formulated in terms of the stress tensor and of intrinsic curve type. To cite this article: J. Laverne, J.-J. Marigo, C. R. Mecanique 332 (2004).     

Limit analysis and homogenization of porous materials with Mohr–Coulomb matrix. Part I: Theoretical formulation

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.     

Un algorithme efficace d'intégration plastique pour un matériau obéissant au critère anisotrope de HillAn efficient algorithm for plastic integration of material satisfying the Hill's anisotropic yield criterion.

This Note deals with an efficient algorithm to carry out the plastic integration and compute the stresses due to large strains for materials satisfying the Hill's anisotropic yield criterion. The classical algorithm of plastic integration such as ‘Return Mapping Method’ is largely used for nonlinear analyses of structures and numerical simulations of forming processes, but it requires an iterative schema and may have convergence problems. A new direct algorithm based on a scalar method is developed which allows us to directly obtain the plastic multiplier without an iteration procedure; thus the computation time is largely reduced and the numerical problems are avoided. To cite this article: I. Titeux et al., C. R. Mecanique 332 (2004).     

Electrophoresis of a 2-particle cluster near a plane boundaryElectrophorèse de deux particules en présence d'une paroi plane

Particle–boundary and particle–particle interactions in Electrophoresis are examined by considering a 2-particle cluster near a plane boundary. The advocated treatment holds for two insulating particles of arbitrary shapes and zeta potential functions and resorts to 13 boundary-integral equations. Preliminary results reveal that, depending upon the addressed velocity nature (translational or angular), wall–particle may be stronger or weaker than particle–particle interactions. To cite this article: A. Sellier, C. R. Mecanique 331 (2003).     

Anisotropic behaviour law for sheets used in stamping:: A comparative study of steel and aluminium

For a car manufacturer, reducing the weight of vehicles is an obvious aim. Replacing steel by aluminium moves towards that goal. Unfortunately, aluminium's stamping numerical simulation results are not yet as reliable as those of steel. Punch-strength and spring-back phenomena are not correctly described. This study on aluminium validates the behaviour law Hill 48 quadratic yield criterion with both isotropic and kinematic hardening. It is based on the yield surface and on associated experimental tests (uniaxial test, plane tensile test, plane compression and tensile shearing). To cite this article: J.-J. Sinou, B. Macquaire, C. R. Mecanique 331 (2003).     

Flow past an axisymmetric body embedded in a saturated porous mediumÉcoulement derrière un corps à symétrie axiale plongé dans un milieu poreux saturé

The problem of viscous fluid past an axisymmetric body embedded in a fluid saturated porous medium is studied using the Brinkman's extension. A general formula for the drag on the body is derived in the form of a limit of an expression involving the stream function characterizing the flow. The flow past an axisymmetric approximate sphere is also considered. The stream function in this case is obtained in terms of Bessel functions and Gegenbauer's functions. The drag acting on the body is evaluated by using the formula derived. Its variation is studied with respect to geometric and permeability parameters. The special cases of flow past a sphere and a spheroid are obtained from the present analysis. To cite this article: D. Srinivasa Charya, J.V. Ramana Murthy, C. R. Mecanique 330 (2002) 417–423.     

Limit analysis and homogenization of porous materials with Mohr–Coulomb matrix. Part II: Numerical bounds and assessment of the theoretical model

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.     

Plane Strain Bending of Cylindrical Sectors of Admissible Compressible Hyperelastic Materials

In the theory of nonlinear elasticity of rubber-like materials, if a homogeneous isotropic compressible material is described by a strain–energy function that is a homogeneous function of the principal stretches, then the equations of equilibrium for axisymmetric deformations reduce to a separable first-order ordinary differential equation. For a particular class of such strain–energy functions, this property is used to obtain a general parametric solution to the equilibrium equation for plane strain bending of cylindrical sectors. Specification of the arbitrary function that appears in such strain–energy functions yields some parametric solutions. In some cases, the parameter can be eliminated to yield closed-form solutions in implicit or explicit form. Other possible forms for the arbitrary constitutive function that are likely to yield such solutions are also indicated.     

Crack paths in three-dimensional elastic solids. i: two-term expansion of the stress intensity factors—application to crack path stability in hydraulic fracturing

The aim of this work is to lay theoretical foundations for the prediction of crack paths within the theory of quasistatic LEFM under the most general hypotheses: arbitrary three-dimensional geometry, arbitrary loading. This objective requires to derive the expression of the stress intensity factors along the crack front after an arbitrary infinitesimal propagation. Only the first two terms of their expansion in powers of the crack extension length δ, proportional to δ⁰ = 1 and δfn1fn2, are considered in this paper. Fully general formulae for these terms are obtained by combining arguments of dimensional analysis (scale changes) and regularity properties (continuity, differentiability) of the stresses at a fixed, given point with respect to δ for δ = 0 derived from the Bueckner–Rice weight function theory. This notably allows to extend the Cotterell–Rice criterion for stable rectilinear propagation of a mode I crack under plane strain conditions to the three-dimensional case. As an application, a penny-shaped crack induced by hydraulic fracturing is considered. Conclusions concerning the influence of the orientation and depth of such a crack upon the stability of its coplanar propagation seem to be compatible with experimental evidence.     

Self-switching of displacement waves in elastic nonlinearly deformed materialsSelf-commutation d'ondes de déplacement dans les milieux élastiques non-linéaires

The problem of self-switching plane waves in elastic nonlinearly deformed materials is formulated. Reduced and evolution equations, which describe the interaction of two waves the power pumping wave and the faint signal wave are obtained. For the case of wave numbers matching the pumping and signal waves, a procedure of finding the exact solution of evolution equations is described. The solution is expressed by elliptic Jacobi functions. The existence of the power wave self-switching is shown and commented. To cite this article: J. Rushchitsky, C. R. Mecanique 330 (2002) 175–180.     

Réduction a priori de modèles thermomécaniquesAn a priori model reduction method for thermomechanical problems

A model reduction method is proposed for finite element models. A previous computation of the state of the structure is not necessary. Residuals defined over the entire time interval and the Karhunen–Loève method provide basis functions. A non-incremental algorithm, from the LATIN method, is used to compute this basis functions. Because of the non-incremental feature, the reduced order model is representative for a large evolution of the state of the structure. To cite this article: D. Ryckelynck, C. R. Mecanique 330 (2002) 499–505.     

Saturated and unsaturated behaviour modelling of Meuse–Haute/Marne argillite

Poromechanical behaviour modelling of the Callovo–Oxfordian argillite under saturated and partially saturated conditions is proposed using the equivalent stress concept. In comparison with the previous works on this rock, the particular form of the yield criterion and the plastic flow potential proposed here help to better describe the rock behaviour in tension–stress paths. The evolution of the poroelastic parameters due to the induced cracks is also considered in a simple way. Due to its physical nature, different from classical soils, the partially saturated behaviour of this rock could not be correctly described by any of the pre-existing theories usually used for the partially saturated porous media (soils). Based upon experimental results on this rock and developing some ideas proposed by other authors, an extension of the saturated elastic–plastic model in unsaturated field is proposed. The key hypothesis of this extension is the evolution of the Biot’s coefficient as a function of the suction, justified by laboratory results and micromechanical analyses. The predictions of the model in saturated and partially saturated conditions are compared with laboratory results and a good general agreement is found.     

Étude par la PIV des structures à la périphérie d'une cavité rotor–stator aspiréeStudy of structures on the periphery of a rotor–stator cavity with suction by means of PIV

We experimentally study the three-dimensional structure of the spiral instability observed on the periphery of a opened rotor–stator cavity with suction. We use the stereoscopic Particles Images Velocimetry technique which gives sequences of maps of the three velocity components of fluid particles located in a meridian plane. This sequence is acquired with a low frequency during the running of a great number of structures. The originality of this study consists in an optimized temporal reclassifying of the maps on a single period that allows a three-dimensional reconstitution of the instability. To cite this article: D. Rémy et al., C. R. Mecanique 332 (2004).     

Stress-based upper-bound method and convex optimization: case of the Gurson materialMéthode cinématique par les contraintes et optimisation convexe : cas du matériau de Gurson

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).     

Yield design of thin periodic plates by a homogenization technique and an application to masonry wallsCalcul à la rupture des plaques minces périodiques par une technique d'homogénéisation et une application aux murs en maçonnerie

A homogenization method for determining overall yield strength properties of thin periodic plates from their local strength properties is proposed within the framework of the yield design theory. The proposed method is applied to the determination of the in-plane and out of plane strength criterion for masonry described as a regular assemblage of infinitely resistant bricks separated by Coulomb interfaces. To cite this article: K. Sab, C. R. Mecanique 331 (2003).