On elementary theories of linear elastic beams,plates and shells (review paper)

Summary This paper presents a review of the elementary theories on the bending of straight and curved beams, on plates and shells, using asymptotic approximations of the basic linearized equations of elasticity in three dimensions. The maximun norm has been chosen to specify the orders of magnitude of the quantities involved. The expansions are given as usual in terms of the small geometrical parameter characterizing the thinness of the structure. Most of the ideas and results are well known. Nevertheless, in the cases where more than one small parameter may be involved, such as small curvatures (shallow structures) or the small loading parameter used to linearize the equations of elasticity, the discussion on the limits of validity of the different theories lead to some interesting newer aspects. Moreover, the main ideas presented in this paper concerning multiple parameter expansions may be applied to discuss the behaviour of the structures and to obtain valuable analytical results in more complicated situations such as moderate and strong anisotropy, dynamic problems, stability etc.

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Zusammenfassung Die vorliegende Arbeit enthält einen Ueberblick über die elementaren Theorien der Biegung von geraden und krummen Stäben, von Platten und Schalen unter Verwendung von asymptotischen Approximationen der dreidimensionalen linearisierten Grundgleichungen der Elastizität, Zur Festlegung der Grössenordnung der massgebenden Funktionen wurde die Maximum-Norm verwendet. Die asymptotischen Entwicklungen sind wie üblich in Abhängigkeit des kleinen geometrischen Parameters gegeben, der die dünne Struktur charakterisiert. Die meisten Ideen und Resultate sind gut bekannt. Dennoch in den mehrparametrigen Fällen, so etwa bei kleinen Krümmungen (flache Strukturen) oder bei der expliziten Berücksichtigung des kleinen Belastungparameters, der zur Linearisierung der elastizitätsgleichungen verwendet wurde, führt die Diskussion über die Gültigkeitsgrenzen der verschiedenen Theorien zu einigen neueren und interessanten Aspekten. Ausserdem können die in der vorliegenden Arbeit dargelegten Hauptideen über mehrparametrige Entwicklungen zur Diskussion und zur Herleitung wertvoller Resultate über das Verhalten dünner Tragwerke in komplizierten Fällen wie mässige und starke Anisotropie, dynamische Probleme, Stabilität usw. übertragen werden.

     

The Rayleigh–Stokes problem for an edge in an Oldroyd-B fluid

The velocity fields corresponding to an incompressible fluid of Oldroyd-B type subject to a linear flow within an infinite edge are determined for all values of the relaxation and retardation times. The well known solution for a Navier–Stokes fluid, as well as those corresponding to a Maxwell fluid and a second grade one, appears as a limiting case of our solutions. To cite this article: C. Fetecau, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 979–984.     

Partitions lineaires arguesiennes d'un espace vectoriel

In this work, we study linear and desarguesian partitions of a finite dimensional vector space over a skew-field K. When K is finite, we describe the set of all these partitions as a homogeneous space of the general linear group and we give an enumeration formula.     

Regularity of Coupled Two-Dimensional Nonlinear Fokker-Planck and Navier-Stokes Systems

We consider systems of particles coupled with fluids. The particles are described by the evolution of their density, and the fluid is described by the Navier-Stokes equations. The particles add stress to the fluid and the fluid carries and deforms the particles. Because the particles perform rapid random motion, we assume that the density of particles is carried by a time average of the fluid velocity. The resulting coupled system is shown to have smooth solutions at all values of parameters, in two spatial dimensions.     

Theoretical determination of the ablation rate of metals in multiple-nanosecond laser pulses irradiation regime

A detailed understanding of the physical determinants of the ablation rate in multiple nanosecond laser pulses regime is of key importance for technological applications such as patterning and pulsed-laser deposition. Here, theoretical modeling is employed to investigate the ablation of thick metallic plates by intense, multiple nanosecond laser pulses. A new photo-thermal model is proposed, in which the complex phenomena associated to the ablation process are accounted for as supplementary terms of the classical heat equation. The pulsed laser ablation in the nanosecond regime is considered as a competition between thermal vapourization and melt ejection under the action of the plasma recoil pressure. Computer simulations using the photo-thermal model presented here and the comparison of the theoretical results with experiment indicate two different mechanisms that contribute to the decrease of the ablation efficiency. First, during the ablation process the vapour/plasma plume expanding above the irradiated target attenuates the laser beam that reaches the sample, leading to a marked decrease of the ablation efficiency. Additional attenuation of the laser beam incident on the sample is produced due to the heating of the plasma by the absorption of the laser beam into the plasma plume. The second mechanism by which the ablation efficiency decreases consists of the reduction of the incident laser intensity with the lateral area, and of the melt ejection velocity with the depth of the hole.     

Subspace Interpolation with Applications to Elliptic Regularity

In this paper, we prove new embedding results by means of subspace interpolation theory and apply them to establishing regularity estimates for the biharmonic Dirichlet problem and for the Stokes and the Navier–Stokes systems on polygonal domains. The main result of the paper gives a stability estimate for the biharmonic problem at the threshold index of smoothness. The classic regularity estimates for the biharmonic problem are deduced as a simple corollary of the main result. The subspace interpolation tools and techniques presented in this paper can be applied to establishing sharp regularity estimates for other elliptic boundary value problems on polygonal domains.