Espaces de Thom et contre-exemples de J.L. Verdier et M. Goresky

Résumé SoitS une variété algébrique complexe singulière, de dimension réelle 2s. M.H. Schwartz et R. Mac-Pherson ont défini des classes caractéristiques, généralisation des classes de Chern, dans l'homologie deS (de telles classes n'existent pas en cohomologie). D'autre part l'homomorphisme de PoincaréH ^2s−⋆ (S)→H _⋆ (S) n'est en géneral, ni injectif, ni surjectif. Cet homomorphisme se factorise par l'homologie d'intersectionIH _⋆ (S). Il est naturel de se demander quel est le “comportement” des classes deS (classes de M.H. Schwartz-R. Mac-Pherson) vis-à-vis du morphisme canonique α:IH _⋆(S)→H_⋆(S). J. L. Verdier a construit un exemple dans lequel, le morphisme canonique α n'étant pas injectif, les classes deS peuvent ètre réalisées de plusieurs manières comme images de classes de Chern de variétés lisses, désingularisations deS, et dont l'homologie est isomorphe àIH _⋆ (S). M. Goresky a construit une variation de cet exemple dans laquelle les classes de Chern ne sont pas dans l'image de α. Nous montrons que ces deux exemples sont cas particuliers d'une même situation:S est un espace de Thom associé à un plongement d'une variétéB dans un espaceIP ^k . L'essentiel de cet article a été écrit lors d'un séjour des auteurs à l'Université du Rio Grande do Sul (Porto-Alegre-Brésil), sur invitation de M. Sebastiani. Nous le remercions ici, ainsi que l'Université de Porto-Alegre, de leur accueil et de leur hospitalité      

Une modélisation thermomécanique unidimensionnelle de la propagation d'un front de changement de phase dans un monocristal d'AMFA one-dimensional thermomechanical modeling of phase change front propagation in a SMA monocrystal

We propose a behavioral modeling that takes into account the thermomechanical couplings accompanying the phase transition in single-crystal CuZnAl samples. The goal of this model is to put forward the significant role played by the heat diffusion in the propagation mode of the phase change fronts. Numerical simulations showed the existence of such a phase change front and predicted the calorimetric and kinematic effects accompanying its propagation. In particular, an inversion of the propagation way during a creep test and caused by an increase of the room temperature was correctly simulated by the model. To cite this article: A. Chrysochoos et al., C. R. Mecanique 331 (2003).     

Response of periodic structures due to moving loadsRéponse de structures périodiques à des charges mobiles

A method is proposed to calculate the response of periodic structures subjected to moving loads. It is based on the Floquet decomposition which allows the restriction of the analysis for the overall system to a generic cell. The main contribution of the approach presented hereafter is that the response is directly deduced from transfer functions in the space-wavenumber domain calculated in an unbounded generic cell. Moreover, the equivalence of this new solution with the response of invariant structures obtained using Fourier transforms is established. To cite this article: H. Chebli et al., C. R. Mecanique 334 (2006).     

Convergence of Perry and Shanno's Memoryless Quasi-Newton Method for Nonconvex Optimization Problems

1.IntroductionWeconsidertheunconstrainedoptimizationproblem:minj(x).(1.1)acRewheref:R"-- Riscontinuouslydifferelltiable.PerryandShanno'smemorylesssquasiNewtonmethodisoftenusedtosolvetheproblem(1.1)whennislarge.ItwasoriginatedfromtheworksofPerry(1977[l])an…     

Model equations for the Eiffel Tower profile: historical perspective and new results

Model equations for the shape of the Eiffel Tower are investigated. One model purported to be based on Eiffel's writing does not give a tower with the correct curvature. A second popular model not connected with Eiffel's writings provides a fair approximation to the tower's skyline profile of 29 contiguous panels. Reported here is a third model derived from Eiffel's concern about wind loads on the tower, as documented in his communication to the French Civil Engineering Society on 30 March 1885. The result is a nonlinear, integro-differential equation which is solved to yield an exponential tower profile. It is further verified that, as Eiffel wrote, “in reality the curve exterior of the tower reproduces, at a determined scale, the same curve of the moments produced by the wind”. An analysis of the actual tower profile shows that it is composed of two piecewise continuous exponentials with different growth rates. This is explained by specific safety factors for wind loading that Eiffel & Company incorporated in the design of the free-standing tower. To cite this article: P. Weidman, I. Pinelis, C. R. Mecanique 332 (2004).     

Asymptotics of eigenfrequencies of an elastic body with a heavy and hard peak-shaped inclusion

A heavy and hard peak-shaped inclusion in an elastic body provokes to concentration of eigenvalues in the low-frequency range of the spectrum and localization of the corresponding eigenmodes near the peak tip. To cite this article: S.A. Nazarov, C. R. Mecanique 335 (2007).     

Analysis of viscoelastic contact with normal compliance,friction and wear diffusionAnalyse d'un problème viscoélastique avec compliance normale,frottement et diffusion d'usure

We consider a quasistatic problem of frictional contact between a viscoelastic body and a moving foundation. The contact is with wear and is modeled by normal compliance and a law of dry friction. The novelty in the model is that it allows for the diffusion of the wear debris over the potential contact surface. Such kind of phenomena arise in orthopaedic biomechanics and influence the properties of joint prosthesis. We derive a weak formulation of the problem and state that, under a smallness assumption on the problem data, there exists a unique weak solution for the model. To cite this article: M. Shillor et al., C. R. Mecanique 331 (2003).     

A two scale method for modulated vibration modes of large,nearly repetitive,structuresUne méthode asymptotique à deux échelles pour les modes de vibrations modulés des longues structures presque répétitives

Nearly repetitive structures can present at least two kinds of vibration modes: localized modes and modulated ones. In this Note, the multiple scale method is applied to characterize a packet of modulated modes. In this respect, only small size problems are to be solved: periodic problems posed on a few basic cells and amplitude equations, which define a sort of homogenized model for this type of modes. It is established that the influence of the non-repetitive part of the structure is accounted by a boundary condition. To cite this article: E.M. Daya et al., C. R. Mecanique 331 (2003).     

Continuum modeling for the modulated vibration modes of large repetitive structuresModèle continu pour les modes de vibrations modulés des longues structures répétitives

By homogenization theory, one can predict the vibrations of long repetitive structures in the low frequency range. Beyond this range, many modes have a modulated shape. Based on a multiple scale analysis, a continuum model is presented, that is able to account for this class of modes. This model involves a real coefficient that can be computed from the finite element resolution of problems defined on a few basic cells. An application in 2D elasticity is presented. To cite this article: E.M. Daya et al., C. R. Mecanique 330 (2002) 333–338.