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Propagation des ondes dans les plaques multicouches : le modèle du Bending-Gradient et la méthode des développements asymptotiques

Abstract : This thesis is dedicated to the modelling of plane wave propagation in infinite multilayered plates, in the context of linear elasticity. The aim of this work is to find an analytical or semi-analytical approximation of the wave dispersion relations when the ratio of the thickness to the wavelength is small. The dispersion relations, linking the angular frequency and the wave number, provide key information about the propagation characteristics of the wave modes. Two methods are proposed in this thesis: the Bending-Gradient model and the asymptotic expansion method. The relevance of these methods is tested by comparing their predictions to those of well-known plate theories, and to reference results computed using the finite element method. Preliminarily, the first part of the thesis is devoted to the mathematical justification of the Bending-Gradient theory in the static framework using variational methods. The first step is to identify the mathematical spaces in which the variational problems of the Bending-Gradient are well posed. A series of existence and uniqueness theorems of the corresponding solutions are then formulated and proved. The second part is dedicated to the formulation of the equations of motion of the Bending-Gradient theory. Numerical simulations are realized for different types of layer stacks to assess the ability of this model to correctly predict the propagation of flexural waves. The third part is concerned with the asymptotic analysis of the three-dimensional equations of motion, carried out using the asymptotic expansion method, the small parameter being the ratio of the thickness to the wavelength. Assuming that the three-dimensional fields can be written as expansions in power of the small parameter, a series of problems which can be solved recursively is obtained. The validity of this method is evaluated by comparison with the finite element method.
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Nadine Bejjani. Propagation des ondes dans les plaques multicouches : le modèle du Bending-Gradient et la méthode des développements asymptotiques. Autre. Université Paris-Est; Université Saint-Joseph (Beyrouth). Ecole supérieure d'ingénieurs de Beyrouth, 2019. Français. ⟨NNT : 2019PESC1025⟩. ⟨tel-02915304⟩

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