Homogenization of an elasticity problem in domains with a net of slender bars near surface

We consider an elasticity problem in a domain Ω⁽⁾=ΩF⁽⁾, where Ω is an open bounded domain in R³, F⁽⁾ is a connected nonperiodic set in Ω like a net of slender bars, and is a parameter characterizing the microstructure of the domain. We consider the case of a surface distribution of the set F⁽⁾, i.e., for sufficiently small , the set F⁽⁾ is concentrated in arbitrary small neighbourhood of a surface Γ. Under a hypothesis on the asymptotic behaviour of the energy functional, we obtain the macroscopic (homogenized) model. To cite this article: M. Goncharenko, L. Pankratov, C. R. Mecanique 331 (2003).     

Développement en polynômes de chaos d'un modèle lagrangien d'écoulement autour d'un profilPolynomial chaos expansion of a Lagrangian model for the flow around an airfoil

A polynomial chaos (PC) expansion a the Lagrangian model for the stochastic incompressible inviscid flow around an airfoil is presented. The flow field is modeled using a distribution of lumped vortices on the airfoil surface while the wake is modeled with Lagrangian point vortices. An original technique is proposed for the computation of the PC coefficients of the velocities induced by the vortices. Two computational examples for random airfoil motions are provided to illustrate the capability of the method to deal with complex situations. To cite this article: O. Le Maître, C. R. Mecanique 334 (2006).     

On nano-scale hydrodynamic lubrication models

Current magnetic head sliders and other micromechanisms involve gas lubrication flows with gap thicknesses in the nanometer range and stepped shapes fabricated by lithographic methods. In mechanical simulations, rarefaction effects are accounted for by models that propose Poiseuille flow factors which exhibit singularities as the pressure tends to zero or +∞. In this Note we show that these models are indeed mathematically well-posed, even in the case of discontinuous gap thickness functions. Our results cover popular models that were not previously analyzed in the literature, such as the Fukui–Kaneko model and the second-order model, among others. To cite this article: G. Buscaglia et al., C. R. Mecanique 333 (2005).     

Diagramme de Laguerre

This Note presents a generalization of a known fast and robust algorithm of incremental construction of the Delaunay triangulation to the case of the regular triangulation of points in ${\mathbb{R}}^d$. In particular, the transport formula of simplex circumball centers are naturally extended to the case of the regular triangulation. The associated Laguerre diagram can then be obtained by duality from the regular triangulation. Some numerical examples of Laguerre diagrams in three dimensions are given. To cite this article: H. Borouchaki et al., C. R. Mecanique 333 (2005).     

Couplage dans les alliages à mémoire de formeCoupling in shape memory alloys.

This Note concerns the study of the micromechanic behavior of shape memory alloys. The advantage of this model permits the coupling between the martensitic transformation and microstructural evolution observed after cycling. The model makes it possible to obtain consecutives equations, which explain at the same time, mechanical properties and the changing structures during the transformation. It provides original physical results on the global behavior of shape memory alloys. To cite this article: A. Alhamany et al., C. R. Mecanique 332 (2004).     

Méthode particulaire anisotrope pour des écoulements de fluide visqueuxAnisotropic particle method for viscous flows

The anisotropic particle method has been extended to the case of viscous flows. The moment transport equation is modified to account for viscous effects. The diffusion term has been evaluated by using the PSE method and the particle moments. The modified transport equation includes geometrical moments for which a specific transport equation has been introduced. The study of the evolution of two corotating vortices allowed the comparison of the anisotropic particle method with the usual particle method. To cite this article: A. Beaudoin et al., C. R. Mecanique 332 (2004).     

Domaine numérique du produit et de la bimultiplication

In this paper, we present an extension of Bouldin's result (1970) concerning the numerical range of the product of two operators and that are commuting and for which one of the set or consists of positive numbers. We also prove that if or is a subnormal operator on a separable Hilbert space, then

where is the operator bimultiplication and is the convex hull.

RÉSUMÉ. Dans ce travail, nous améliorons un résultat de Bouldin (1970) concernant la localisation de le domaine numérique du produit de deux opérateurs et sur un espace de Hilbert lorsque et commutent et est constitué de réels strictement positifs. Dans le cas où ou est un opérateur sous normal sur un espace de Hilbert séparable, nous montrons que

où est l'opérateur produit ou bimultiplication et est l'enveloppe convexe.

     

Computing Green's function of elasticity in a half-plane with impedance boundary condition

This Note presents an effective and accurate method for numerical calculation of the Green's function G associated with the time harmonic elasticity system in a half-plane, where an impedance boundary condition is considered. The need to compute this function arises when studying wave propagation in underground mining and seismological engineering. To theoretically obtain this Green's function, we have drawn our inspiration from the paper by Durán et al. (2005), where the Green's function for the Helmholtz equation has been computed. The method consists in applying a partial Fourier transform, which allows an explicit calculation of the so-called spectral Green's function. In order to compute its inverse Fourier transform, we separate as a sum of two terms. The first is associated with the whole plane, whereas the second takes into account the half-plane and the boundary conditions. The first term corresponds to the Green's function of the well known time-harmonic elasticity system in (cf. J. Dompierre, Thesis). The second term is separated as a sum of three terms, where two of them contain singularities in the spectral variable (pseudo-poles and poles) and the other is regular and decreasing at infinity. The inverse Fourier transform of the singular terms are analytically computed, whereas the regular one is numerically obtained via an FFT algorithm. We present a numerical result. Moreover, we show that, under some conditions, a fourth additional slowness appears and which could produce a new surface wave. To cite this article: M. Durán et al., C. R. Mecanique 334 (2006).