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Conditions de compatibilité en mécanique des solides

Abstract : The compatibility conditions, associated with partial differential equation of deformable bodies, are used as a guideline of the present work. The basic idea, first presented by Darboux in the context of the general theory of surfaces, consists in replacing the Christoffel symbols by vectors called the Darboux vectors. These vectors are related to rotations in a similar way as the instantaneous rotation vectors in rigid body dynamics. The compatibility conditions are revisited here in the framework of large strain. Two systems of decoupled partial differential equation allow to obtain the displacement of the deformed body by successive integrations. Our results show the validity of the developed tools. An original investigation of three-dimensional Riemann manifolds, with the same curvature as a sphere, is carried out. The theory of surfaces is also studied by introducing the Darboux vectors. A surface is rebuilt from his two fundamental forms in accordance with the Bonnet theorem. The particular study of a minimal surface leads to an efficient building process from the knowledge of the boundary. A new concept, called sister minimal surface, is introduced and its application is developed in the case of two examples. Finally the equivalence between the cancellation of the Riemann-Christoffel curvature tensor in a shell and the Gauss-Codazzi-Mainardi conditions on its mean surface is established. Further developments of the present work would be concerned with the rigid body, treated as a six-dimensional Riemann manifold.
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Submitted on : Thursday, April 16, 2009 - 12:43:28 PM
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Danielle Léonard Fortuné. Conditions de compatibilité en mécanique des solides. Mécanique [physics.med-ph]. Université de Poitiers, 2008. Français. ⟨tel-00375897⟩

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