Géométrie des espaces de tenseurs Une approche effective appliquée à la mécanique des milieux continus

Abstract : Tensorial formulation of mechanical constitutive equations is a very important matter in continuum mechanics. For instance, the space of elastic tensors is a subspace of 4th order tensors with a natural SO(3) group action. More generaly, we have to study the geometry of a tensor space defined on R 3 , under O(3) group action. To describe such a geometry, we first have to exhibit its isotropy classes, also named symetry classes. Indeed, each tensor space possesses a finite number of isotropy classes. In this present work, we propose an original method to obtain isotropy classes of a given tensor space. As an illustration of this new method, we get for the first time the isotropy classes of a 8th order tensor space occuring in second strain-gradient elasticity theory. In the case of a real representation of a compact group, invariant algebra seperates the orbits. This observation motivates the purpose to find a finite generating set of poly- nomial invariants. For that purpose, we make use of the link between tensor spaces and spaces of binary forms, which belongs to the classical invariant theory. We thus have to deal with SL(2, C) group action. To obtain new results, we have reformulated and rein- terpreted effective approaches of Gordan’s algorithm, developped during XIXth century. We then obtain for the first time a minimal generating family of elasticity tensor space, and a generating family of piezoelectricity tensor space. Using linear algebra arguments, we were also able to get important relations of classical invariant theory, such as the Gordan’s series and the Abdesselam–Chipalkatti’s quadratic relations on transvectants.
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Marc Olive. Géométrie des espaces de tenseurs Une approche effective appliquée à la mécanique des milieux continus. Théorie des représentations [math.RT]. AMU, 2014. Français. ⟨tel-01165379⟩

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