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Abstract.We consider the Laplace operator in a thin tube of R 3 with a Dirichlet condition on its boundary. We study asymptotically the spectrum of such an operator as the thickness of the tube's cross section goes to zero. In particular we analyse how the energy levels depend simultaneously on the curvature of the tube's central axis and on the rotation of the cross section with respect to the Frenet frame. The main argument is a Γ-convergence theorem for a suitable sequence of quadratic energies.Mathematics Subject Classification.

In this paper we apply the asymptotic expansion method to obtain a nonlinear adaptive elastic rod model. We first consider the model of Cowin and Hegedus with later modifications, and with a remodeling rate equation depending nonlinearly on the strain field and for a thin rod whose cross section is a function of a small parameter. Based on the asymptotic expansion method for the elastic case, we prove that, when the small parameter tends to zero the solution of the nonlinear adaptive elastic rod model converges to the leading term of its asymptotic expansion. Moreover, we show that this term is also the solution of a well-known simplified adaptive elastic model, with generalized Bernoulli-Navier equilibrium equations and a remodeling rate equation whose driving mechanism is the strain energy per unit volume, in good agreement with some of the models used in practice.

( 2 ) Communicated by P G CIARLET Abstract -In this work we obtain a gênerahzation of Timoshenko s beam theory by applying the asymptotic expansion method to a mixed vanational formulation of the three dimensional hneanzed elasticity model A classical subject of major discussion in this model is the proper définition of the so called Timoshenko s constants taking into account the f act that the shear stresses vary on each cross section Due to the technique employed we shall be able to define these constants in a clear way and show its dependence on the geometty of the cross section and on Poisson s ratio Finally we present several numencal examples showing the relationship between the classical and the new constants for different geometnes Resumé -En appliquant la methode des développements asymptotiques a un modèle variationnel mixte de l élasticité linéarisée on obtient une généralisation de la theorie de poutres de Timoshenko Associées a cette généralisation on obtient aussi une définition et une généralisation des constantes de Timoshenko tenant en compte la flexion additionnelle due a l effort tranchant La technique employee permet de demontier sa dependance par rapport a la geometrie et au coefficient de Poisson Finalement différents exemples numériques sont traites montrant la relation entre les nouvelles constantes et les constantes classiques pour différentes geometties NOTATIONSIn this work the summation convention on repeated indexes is used Latin indexes such as z, y, k, take values on the set {1,2,3} while Greek indexes such as a, (3, y, take values on the set {1,2}