In this thesis we use various techniques to investigate the occurrence of trapped modes in the presence of thin obstacles. Physically trapped modes are oscillations of finite energy in a fluid which is unbounded in at least one direction. These oscillations mainly occur locally to some structure and decay to zero at large distances away from it. Trapped modes are important as they have been found to exist in a wide range of physical situations. We consider a number of problems in two and three dimensions including waveguides containing bodies and arrays of identical structures. A modified residue calculus technique, a variational technique and a method based on the truncation of matched eigenfunction expansions are used to solve the problems, with numerous results presented.

The problem of two-dimensional acoustic scattering of an incident plane wave by a semi-infinite lattice is solved. The problem is first considered for sound-soft cylinders whose size is small compared to the wavelength of the incident field. In this case the formulation leads to a scalar Wiener--Hopf equation, and this in turn is solved via the discrete Wiener--Hopf technique. We then deal with a more complex case which arises either by imposing Neumann boundary condition on the cylinders' surface or by increasing their radii. This gives rise to a matrix Wiener--Hopf equation, and we present a method of solution that does not require the explicit factorisation of the kernel. In both situations, a complete description of the far field is given and a conservation of energy condition is obtained. For certain sets of parameters (`pass bands'), a portion of the incident energy propagates through the lattice in the form of a Bloch wave. For other parameters (`stop bands' or `band gaps'), no such transmission is possible, and all of the incident field energy is reflected away from the lattice.

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La problématique récente de la conception de métamatériaux a renouvelé l'intérêt dans les théories de l'homogénéisation en régime dynamique. En particulier, la théorie de l'homogénéisation élastodynamique initiée par J.R. Willis a reçu une attention particulière suite à des travaux sur l'invisibilité élastique. La présente thèse reformule la théorie de Willis dans le cas des milieux périodiques, examine ses implications et évalue sa pertinence physique au sens de quelques ``conditions d'homogénéisabilité'' qui sont suggérées. En se basant sur les résultats de cette première partie, des développements asymptotiques approximatifs de la théorie de Willis sont explorés en relation avec les théories à gradient. Une condition nécessaire de convergence montre alors que toutes les branches optiques de la courbe de dispersion sont omises quand des développements asymptotiques de Taylor de basse fréquence et de longue longueur d'onde sont déployés. Enfin, une nouvelle théorie de l'homogénéisation est proposée. On montre qu'elle généralise la théorie de Willis et qu'elle l'améliore en moyenne fréquence de sorte qu'on retrouve certaines branches optiques omises auparavant. On montre également que le milieu homogène effectif défini par la nouvelle théorie est un milieu généralisé dont les champs satisfont une version élastodynamique généralisée du lemme de Hill-Mandel / The recent issue of metamaterials design has renewed the interest in homogenization theories under dynamic loadings. In particular, the elastodynamic homogenization theory initiated by J.R. Willis has gained special attention while studying elastic cloaking. The present thesis reformulates Willis theory for periodic media, investigates its outcome and assesses its physical suitability in the sense of a few suggested ``homogenizability conditions''. Based on the results of this first part, approximate asymptotic expansions of Willis theory are explored in connection with strain-gradient media. A necessary convergence condition then shows that all optical dispersion branches are lost when long-wavelength low-frequency Taylor asymptotic expansions are carried out. Finally, a new homogenization theory is proposed to generalize Willis theory and improve it at finite frequencies in such a way that selected optical branches, formerly lost, are recovered. It is also proven that the outcome of the new theory is an effective homogeneous generalized continuum satisfying a generalized elastodynamic version of Hill-Mandel lemma

The static and dynamic response of lattice materials is investigated to disclose and control the connection between microstructure and effective behavior. The analytical methods developed in the thesis aim at providing a new understanding of material instabilities and strain localizations as well as effective tools for controlling wave propagation in lattice structures. The time-harmonic dynamics of arbitrary beam lattices, deforming flexurally and axially in a plane, is formulated analytically to analyze the influence of the mechanical parameters on the dispersion properties of the spectrum of Floquet-Bloch waves. Several forms of dynamic localizations are shown to occur for in-plane wave propagation of grid-like elastic lattices. It is demonstrated that lattices of rods, despite being `simple' structures, can exhibit a completely different channeled response depending on the characteristics of the forcing source (i.e. frequency and direction) as well as on the slenderness of the elastic links. It is also shown how the lattice parameters can be tuned to attain specific dispersion properties, such as flat bands and sharp Dirac cones. In the research field of material instabilities, a key result proposed in this thesis is the development of both static and dynamic homogenization methods capable of accounting for second-order effects in the macroscopic response of prestressed lattices. These methods, the former based on an incremental strain-energy equivalence and the latter based on the asymptotic analysis of lattice waves, allow the identification of the incremental constitutive operator capturing the macroscopic incremental response of arbitrary lattice configurations. The homogenization framework has allowed the systematic analysis of prestress-induced phenomena on the incremental response of both the lattice structure and its `effective' elastic solid, which in turn has enabled the identification of the complex interplay between microstructure, prestress, loss of ellipticity (shear band formation) and short-wavelength bifurcations. Potential new applications for the control of wave propagation are also shown to be possible by leveraging the inclusion of second-order terms in the incremental dynamics. In particular, the tunability of the prestress state in a square lattice structure has been exploited to obtain dynamic interfaces with designable transmission properties. The interface can be introduced in a material domain by selectively prestressing the desired set of ligaments and the prestress level can be tuned to achieve total reflection, negative refraction, and wave channeling. The obtained results open new possibilities for the realization of engineered materials endowed with a desired constitutive response, as well as to enable the identification of novel dynamic material instabilities.

Dans cette thèse, nous présentons des résultats d’homogénéisation périodique d’un problème spectral et de l’équation d’ondes de Bloch. Il permet de modéliser les ondes à basse et haute fréquences. La partie modèle à basse fréquence est bien connu et n’est pas donc abordée dans ce travail. A contrario ; la partie à haute fréquence du modèle, qui représente des oscillations aux échelles microscopiques et macroscopiques, est un problème laissé ouvert. En particulier, les conditions aux limites de l’équation macroscopique à hautes fréquences établies dans [36] n’étaient pas connues avant le début de la thèse. Ce dernier travail apporte trois contributions principales. Les deux premières contributions, portent sur le comportement asymptotique du problème d’homogénéisation périodique du problème spectral et de l’équation des ondes en une dimension. La troisième contribution consiste en une extension du modèle du problème spectral posé dans une bande bi dimensionnelle et bornée. Le résultat d’homogénéisation comprend des effets de couche limite qui se produisent dans les conditions aux limites de l’équation macroscopique à haute fréquence. / In this dissertation, we present the periodic homogenization of a spectral problem and the waveequation with periodic rapidly varying coefficients in a bounded domain. The asymptotic behavioris addressed based on a method of Bloch wave homogenization. It allows modeling both the lowand high frequency waves. The low frequency part is well-known and it is not a new point here.In the opposite, the high frequency part of the model, which represents oscillations occurringat the microscopic and macroscopic scales, was not well understood. Especially, the boundaryconditions of the high-frequency macroscopic equation established in [36] were not known prior to thecommencement of thesis. The latter brings three main contributions. The first two contributions, areabout the asymptotic behavior of the periodic homogenization of the spectral problem and waveequation in one-dimension. The third contribution consists in an extension of the model for thespectral problem to a thin two-dimensional bounded strip Ω = (0; _) _ (0; ") _ R2. The homogenizationresult includes boundary layer effects occurring in the boundary conditions of the high-frequencymacroscopic equation.