Sur l'identification des modules elastiques

Abstract : The aim of this thesis is the identification of elastic moduli in an inhomogenous structure the framework of small deformations and linear elasticity. This is an inverse problem, where one seeks to determine the interior distribution of elastic moduli from simultanous force and displacement boundary measurements. The first chapter introduces the mathematical problem in analogy with the identification problem of the electrical conductivity. The application which gives consistency to this problem is the Dirichlet-to-Neumann data map. This application is equivalent to the application of "strain energy" and its expression depends on the Green function of the domain. The stability and and uniqueness questions are posed. The problem ill-posed and therefore weak stability is to be expected. The uniqueness question gives some problems in case of anisotropy. The second chapter presents a numerical method of reconstruction based on the notion of error on the constitutive law. The method uses extensively a decomposition of the intervening fields using eigenelastic moduli and eigentensors. These are the "eigenvalues" and "eigenvectors" of the elasticity tensor. The third chapter illustrates the minimization process of the error on the constitutive law in order to obtain distributions of elastic moduli. The examples are in bidimensional elasticity with isotropic or cubic symmetry. An inverse problem related to the measurement of residual stresses in presented in an appendix. It concerns the problem of reconstructing residual stresses from boundary X ray measurements after matter removal.
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  • HAL Id : tel-00346591, version 1

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Andrei Constantinescu. Sur l'identification des modules elastiques. Mathématiques [math]. Ecole Polytechnique X, 1994. Français. ⟨tel-00346591⟩

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