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Résumé
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Abstract
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Références bibliographiques
Bibliographic references
Cette thèse porte sur l'élasticité des corps minces bidimensionnels. Nous insistons sur les rapports entre les équations de l'élasticité et la géométrie.
Nous envisageons tout d'abord le cas des coques, qui sont définies comme les corps élastiques minces possédant une courbure au repos. On sait que le comportement élastique d'une coque est largement conditionné par la rigidité infinitésimale de sa surface moyenne : selon qu'il est possible ou non de déformer cette surface tout en conservant les longueurs de toutes les courbes inscrites, on dira que la coque est isométriquement déformable, ou inhibée. Nous interprétons la classification des surfaces de révolution due à Cohn-Vossen, et la généralisons aux surfaces quelconques. Nous mettons en évidence des courbes rigidifiantes.
Nous considérons ensuite la délamination des films minces comprimés~: sous certaines conditions mécaniques, ces films se décollent du substrat auxquels ils adhéraient. Nous étudions la fracture de l'interface film/substrat au moyen d'un modèle de fissure avec frottement de Coulomb entre les lèvres.
Des motifs de délamination en forme de fil de téléphone ont été largement observés expérimentalement. Nous les interprétons comme le résultat d'un flambage élastique secondaire dans les équations de Föppl-von Karman. Enfin, nous montrons que la structure des équations de Föppl-von Karman d'une part, et les propriétés de la fissure interfaciale d'autre part, permettent d'expliquer la stabilité des cloques de délamination.
Élasticité. Rigidité géométrique. Fracture interfaciale. Frottement. Ténacité dépendante de mode. Délamination. Motifs de délamination. Équations de Föppl-von Karman. Flambage élastique.
This thesis deals with the elasticity of thin (2D) elastic materials. The connection between geometry and the theory of elasticity is emphasised.
We first study shells, which are defined as thin elastic materials having some curvature at rest. It is known that the elastic behaviour of a shell depends to a large extent on the infinitesimal rigidity of its mean surface~: if this surface can be deformed without stretching, the shell is said to be bendable. If, contrarily, any deformation of the shell involves some extension of the mean surface, it is said to be infinitesimally rigid. The classification of shells of revolution by Cohn-Vossen is interpreted, and extended to arbitrary surfaces. Rigidifying curves are pointed out.
We also consider the delamination of compressed thin films. Under certain mechanical conditions, these films lift off the substrate to which they were bound. The fracture of the film/substrate interface is studied using a model of interfacial crack with Coulomb friction between the lips.
Delamination patterns looking like telephone cord have been widely observed in experiments. They are interpreted as the result of a secondary bifurcation in the FvK equations. Finally, it is shown that the stability of delamination blisters can be understood from the structure of the FvK equations and from the properties of the interfacial crack.
Elasticity. Geometric rigidity. Interfacial cracks. Friction. Mode-dependent toughness. Delamination. Delamination patterns. Föppl-von Karman equations. Elastic buckling.
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