Tenseur métrique

En géométrie, et plus particulièrement en géométrie différentielle, le tenseur métrique est un tenseur d'ordre 2 permettant de définir le produit scalaire de deux vecteurs en chaque point d'un espace, et qui est utilisé pour la mesure des longueurs et des angles. Il généralise le théorème de Pythagore. Dans un système de coordonnées donné, le tenseur métrique peut se représenter comme une matrice symétrique, généralement notée G, pour ne pas confondre la matrice (en majuscule) et le tenseur métrique g. Dans ce qui suit, la convention de sommation d'Einstein est utilisée. Définition Le tenseur métrique d'un espace vectoriel E de dimension finie n est un tenseur covariant de rang 2 (c'est-à-dire une forme bilinéaire) défini sur E : \begin{align} g , : , &E\times E &\to &,,, \R \ &(\mathbf{u},\mathbf{v}) &\mapsto &,,, g(\mathbf{u},\mathbf{v}) \end{align} g est :

symétrique : \forall \

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Harmonic Maps on Smooth Metric Measure Spaces and on Riemannian Polyhedra

Zahra Sinaei

This thesis is a study of harmonic maps in two different settings. The first part is concerned with harmonic maps from smooth metric measure spaces to Riemannian manifolds. The second part is study of harmonic maps from Riemannian polyhedra to non-positively curved (locally) geodesic spaces in the sense of Alexandrov. The first part is organized as follows. We begin by defining a notion of harmonicity, and justify- ing the definition by checking it against pre-existing definitions and results in special cases. There are two main theorems in this section. The first is Theorem 0.1.1, which is the general- ization of the Shoen-Yau theorem [SY76] in our setting. The second is on the convergence of harmonic maps between Riemannian manifolds. Specifically we will show that if fi : Mi → N are a sequence of harmonic maps between Riemannian manifolds, and if the manifolds Mi converge to a smooth metric measure space M in the measured Gromov-Hausdorff topology, then the fi converge to a harmonic map f : M → N. This is the content of Theorem 0.1.2 In the second part, we prove Liouville-type theorems for harmonic maps under two different assumptions on the source space. First we prove the analogue of the Schoen-Yau theorem on a complete (smooth) pseudomanifolds with non-negative Ricci curvature. To this end we gen- eralize some Liouville- type theorems for subharmonic functions from [Yau76]. Then we study 2-parabolic admissible Riemannian polyhedra and prove vanishing results for subharmonic functions and harmonic maps on 2-parabolic pseudomanifolds.

EPFL2013

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Cascading gravity: Extending the Dvali-Gabadadze-Porrati model to higher dimension

Michele Redi

We present a generalization of the Dvali-Gabadadze-Porrati scenario to higher codimensions which, unlike previous attempts, is free of ghost instabilities. The 4D propagator is made regular by embedding our visible 3-brane within a 4-brane, each with their own induced gravity terms, in a flat 6D bulk. The model is ghost-free if the tension on the 3-brane is larger than a certain critical value, while the induced metric remains flat. The gravitational force law "cascades" from a 6D behavior at the largest distances followed by a 5D and finally a 4D regime at the shortest scales.

2008

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Damage-driven strain localisation in networks of fibres: A computational homogenisation approach

Pablo Antolin Sanchez, Felipe Figueredo Rocha

In many applications, such as textiles, fibreglass, paper and several kinds of biological fibrous tissues, the main load-bearing constituents at the micro-scale are arranged as a fibre network. In these materials, rupture is usually driven by micro-mechanical failure mechanisms, and strain localisation due to progressive damage evolution in the fibres is the main cause of macro-scale instability. We propose a strain-driven computational homogenisation formulationbased on Representative Volume Element (RVE), within a framework in which micro-scale fibre damage can lead to macro-scale localisation phenomena. The mechanical stiffness considered here for the fibrous structure system is due to: i) an intra-fibre mechanism in which each fibre is axially stretched, and as a result, it can suffer damage; ii) an inter-fibre mechanism in which the stiffness results from the variation of the relative angle between pairs of fibres. The homogenised tangent tensor, which comes from the contribution of these two mechanisms, is required to detect the so-called bifurcation point at the macro-scale, through the spectral analysis of the acoustic tensor. This analysis can precisely determine the instant at which the macro-scale problem becomes ill-posed. At such a point, the spectral analysis provides information about the macro-scale failure pattern (unit normal and crack-opening vectors). Special attention is devoted to present the theoretical fundamentals rigorously in the light of variational formulations for multi-scale models. Also, the impact of a recent derived more general boundary condition for fibre networks is assessed in the context of materials undergoing softening. Numerical examples showing the suitability of the present methodology are also shown and discussed.

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