Singular and non-singular eigenvectors for the Gaudin model

We present a method to construct a basis of singular and non-singular common eigenvectors for Gaudin Hamiltonians in a tensor product module of the Lie algebra SL(2). The subset of singular vectors is completely described by analogy with covariant differential operators. The relation between singular eigenvectors and the Bethe Ansatz is discussed. In each weight subspace the set of singular eigenvectors is completed to a basis, by a family of non-singular eigenvectors. We discuss also the generalization of this method to the case of an arbitrary Lie algebra.Comment: 19 page

Scaling Navier-Stokes Equation in Nanotubes

On one hand, classical Monte Carlo and molecular dynamics (MD) simulations have been very useful in the study of liquids in nanotubes, enabling a wide variety of properties to be calculated in intuitive agreement with experiments. On the other hand, recent studies indicate that the theory of continuum breaks down only at the nanometer level; consequently flows through nanotubes still can be investigated with Navier-Stokes equations if we take suitable boundary conditions into account. The aim of this paper is to study the statics and dynamics of liquids in nanotubes by using methods of non-linear continuum mechanics. We assume that the nanotube is filled with only a liquid phase; by using a second gradient theory the static profile of the liquid density in the tube is analytically obtained and compared with the profile issued from molecular dynamics simulation. Inside the tube there are two domains: a thin layer near the solid wall where the liquid density is non-uniform and a central core where the liquid density is uniform. In the dynamic case a closed form analytic solution seems to be no more possible, but by a scaling argument it is shown that, in the tube, two distinct domains connected at their frontiers still exist. The thin inhomogeneous layer near the solid wall can be interpreted in relation with the Navier length when the liquid slips on the boundary as it is expected by experiments and molecular dynamics calculations.Comment: 27 page

On the influence of local fluctuations in volume fraction of constituents on the effective properties of nonlinear composites. Application to porous materials

International audienceComposite materials often exhibit local fluctuations in the volume fraction of their individual constituents. This paper studies the influence of such small fluctuations on the effective properties of composites. A general asymptotic expansion of these properties in terms of powers of the amplitude of the fluctuations is given first. Then, this general result is applied to porous materials. As is well-known, the effective yield surface of ductile voided materials is accurately described by Gurson's criterion. Suitable extensions for viscoplastic solids have also been proposed. The question addressed in the present study pertains to nonuniform distributions of voids in a typical volume element or in other words to the presence of matrix-rich and pore-rich zones in the material. It is shown numerically and analytically that such deviations from a uniform distribution result in a weakening of the macroscopic carrying capacity of the material

A semi-analytical model for the behavior of saturated viscoplastic materials containing two populations of voids of different sizes

International audienceThis paper presents a micromechanical model for a porous viscoplastic material containing two populations of pressurized voids of different sizes. Three scales are distinguished: the microscopic scale (corresponding to the size of the small voids), the mesoscopic scale (corresponding to the size of the large voids) and the macroscopic scale. It is assumed that the first homogenization step is performed at the microscopic scale, and, at the mesoscopic scale, the matrix is taken to be homogeneous and compressible. At the mesoscopic scale, the second homogenization step, on which the present study focuses, is based on a simplified representative volume element: a hollow sphere containing a pressurized void surrounded by a nonlinear viscoplastic compressible matrix. The nonlinear behavior of the matrix, which is expressed using the results obtained in the first homogenization step, is approached using a modified secant linearization procedure involving the discretization of the hollow sphere into concentric layers. Each layer has uniform secant moduli. The predictions of the model are compared with the more accurate numerical results obtained using the finite element method. Good agreement is found to exist with all the macroscopic stress triaxialities and all the porosity and nonlinearity values studied

Porosity dependence of thermal conductivity in UO2 nuclear fuels

International audienc

{W\cal W}-Gauge Structures and their Anomalies:An Algebraic Approach

Starting from flat two-dimensional gauge potentials we propose the notion of ${\cal W}$-gauge structure in terms of a nilpotent BRS differential algebra. The decomposition of the underlying Lie algebra with respect to an $SL(2)$ subalgebra is crucial for the discussion conformal covariance, in particular the appearance of a projective connection. Different $SL(2)$ embeddings lead to various ${\cal W}$-gauge structures. We present a general soldering procedure which allows to express zero curvature conditions for the ${\cal W}$-currents in terms of conformally covariant differential operators acting on the ${\cal W}$ gauge fields and to obtain, at the same time, the complete nilpotent BRS differential algebra generated by ${\cal W}$-currents, gauge fields and the ghost fields corresponding to ${\cal W}$-diffeomorphisms. As illustrations we treat the cases of $SL(2)$ itself and to the two different $SL(2)$ embeddings in $SL(3)$, {\it viz.} the ${\cal W}_3^{(1)}$- and ${\cal W}_3^{(2)}$-gauge structures, in some detail. In these cases we determine algebraically ${\cal W}$-anomalies as solutions of the consistency conditions and discuss their Chern-Simons origin.Comment: 46 pages,LaTe

W-algebras from symplectomorphisms

It is shown how $W$-algebras emerge from very peculiar canonical transformations with respect to the canonical symplectic structure on a compact Riemann surface. The action of smooth diffeomorphisms of the cotangent bundle on suitable generating functions is written in the BRS framework while a $W$-symmetry is exhibited. Subsequently, the complex structure of the symmetry spaces is studied and the related BRS properties are discussed. The specific example of the so-called $W_3$-algebra is treated in relation to some other different approaches.Comment: LaTex, 25 pages, no figures, to appear in Journ. Math. Phy

The role of complex structures in w-symmetry

In a symplectic framework, the infinitesimal action of symplectomorphisms together with suitable reparametrizations of the two dimensional complex base space generate some type of W-algebras. It turns out that complex structures parametrized by Beltrami differentials play an important role in this context. The construction parallels very closely two dimensional Lagrangian conformal models where Beltrami differentials are fundamental.Comment: LaTex, 34 pages, no figures, to be published in Nucl. Phys.

Modèle viscoplastique pour un monocristal poreux cubique sous chargement purement hydrostatique

Ce travail concerne la modélisation du comportement viscoplastique d'un monocristal poreux, constitué d'une matrice continue, dans laquelle sont distribuées de façon uniforme des cavités dont la taille caractéristique est petite devant celle du cristal environnant. Ce type de microstructure peut se rencontrer dans certains aciers inoxydables austénitiques irradiés, où des cavités peuvent apparaître à l'intérieur des grains de ces polycristaux (voir par exemple [Garner, F. A., 2012. Radiation Damage in Austenitic Steels. Comprehensive Nuclear Materials, 33-95]). La matrice cristalline est prise à symétrie cubique et trois types de familles sont considérées successivement : les cubiques faces centrées, les cubiques centrées et les ioniques. Un chargement en contrainte effective hydrostatique est considéré. La méthode des stratifiés de rang infini de [Idiart, M.I., 2008. Modeling the macroscopic behavior of two-phase nonlinear composites by infinite-rank laminates. J. Mech. Phys. Solids 56, 2599-2617] est mise en ?uvre. Des développements analytiques permettent d'écrire le potentiel effectif en contrainte en fonction d'une contrainte hydrostatique d'écoulement qui dépend de la microstructure, des paramètres matériaux locaux et du chargement. Des simulations numériques à base de transformées de Fourier rapides (méthode FFT de [Moulinec, H., Suquet, P., 1994. A fast numerical method for computing the linear and nonlinear properties of composites. C. R. Acad. Sci. Paris II 318, 1417?1423]) sont réalisées sur des microstructures tridimensionnelles poreuses périodiques sous chargement en contrainte effective hydrostatique. Les trois types de familles énumérées ci-dessus sont considérés successivement. Différentes porosités et différentes valeurs de l'exposant de fluage sont traitées. Un bon accord est obtenu entre les résultats FFT et ceux du modèle basé sur la méthode des stratifiés. Cette comparaison permet de proposer une forme analytique pour la contrainte hydrostatique d'écoulement. Les résultats sont en accord avec des résultats de la littérature ([Han, X., Besson, J., Forest, S., Tanguy, B., Bugat, S., 2013. A yield function for single crystals containing voids. Int. J. Solids Struct. 50, 2115?2131], [Mbiakop, A., Constantinescu, A., Danas, K., 2015. An analytical model for porous single crystals with ellipsoidal voids. J. Mech. Phys. Solids 84, 436?467], [Paux, J., Morin, L., Brenner, R., Kondo, D., 2015. European Journal of Mechanics A/Solids 51, 1-10])

Modélisation du comportement des combustibles à particules : caractérisation d'un volume élémentaire représentatif pour un milieu hétérogène aléatoire

Les combustibles à particules sont constitués de particules sphériques d'oxyde d'uranium enrobé de plusieurs couches de confinement, noyées dans une matrice graphite. Pour prendre en compte l'influence de la répartition aléatoire des particules sur les chargements thermomécaniques locaux, une modélisation multi-échelles est nécessaire. Le choix s'est porté vers la méthode des éléments-finis au carré, où interviennent deux échelles distinctes de discrétisation : une structure « macroscopique» homogène dont les propriétés en chaque point d'intégration sont calculées sur une seconde structure «microscopique» hétérogène (Volume Elémentaire Représentatif). La première partie de l'étude vise à caractériser la microstructure aléatoire par un indicateur morphologique basé sur la distribution des distances minimales entre les centres des particules. Le comportement élastique des VER, obtenu par calcul éléments finis, a ensuite été comparé à un modèle analytique. Enfin, nous avons défini des indicateurs de représentativité thermique et mécanique basés sur les modes de rupture des particules sous irradiation