An existence and uniqueness theorem for time-dependent flexural shells
Résumé
In this paper we define a priori what is a natural two-dimensional model for a time-dependent flexural shell. As expected, this model takes the form of a set of hyperbolic variational equations posed over the space of admissible linearized inextensional displacements, and a set of initial conditions. The paramount difficulty arising when studying the existence and uniqueness of solutions of this model is that the the main function space does not satisfy the assumptions required to implement Galerkin method. We thus provide an alternate formulation of the studied problem using a suitable penalty scheme. The function space where the new problem is posed meets all the requirements to implement Galerkin method and we are thus in a position to prove the existence and uniqueness of strong solutions of the considered "penalized" problem. As a last step, we show that the family of "penalized" solutions converge to the unique weak solution of the original problem. For sake of completeness, in the final part of the paper, we also provide an existence and uniqueness theorem in the case where the linearly elastic shell under consideration is an elliptic membrane shell.
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