[pt] Seja V um conjunto de n pontos no espaço euclidiano X de dimensão d. Pelo teorema de Johnson-Lindenstrauss, existe uma projeção entre X e Y, outro espaço de dimensão k bastante menor, com a propriedade que as distâncias entre imagens de pontos de V sejam mantidas dentro de um fator c arbitrariamente próximo de 1. O teorema apresenta uma relação entre d, k e c, indicando a possibilidade de dramáticas reduções de dimensão para representações fidedignas de V. A demonstração emprega as Grassmannianas, as variedades de subespaços de dimensão k em X. São construídas cartas e uma medida homogênea em relação à ação natural do grupo ortogonal na Grassmanniana. O resultado segue estimando através de gaussianas certas integrais de caráter fortemente geométrico. / [en] Let V be a set of n points in the Euclidean space X of dimension d. The Johnson-Lindenstrauss theorem states that there is a projection between X a and Y, another Euclidean space of a smaller dimension k, with the property that images of points of X under projection do not differ by more that a multiplicative factor c arbitrarily close to 1. The theorem presents a relation among d, k and c, indicating the possibility of dramatic dimensional reduction of very faithful representations of V. The proof makes use of Grassmanians, the manifolds consisting of subspaces of dimension k in X. In the text, charts are presented, together with a measure which is homogeneous with respect to the natural action of the orthogonal group on the Grassmanian. The result follows by taking estimates using gaussians of certain integrals with a strong geometric flavor.

Les travaux de la thèse visent à mettre en place une méthodologie innovante de caractérisation du comportement viscoplastique des matériaux métalliques sous chargement purement inertiel. Sous chargements mécaniques extrêmes (e.g., crash, impact ou explosions), leur comportement mécanique présente en effet pour nombre d’entre eux une sensibilité à la vitesse de déformation. Des approches dites statiquement déterminées sont majoritairement utilisées pour caractériser leur comportement, mais elles requièrent de nombreux essais dont les conditions expérimentales sont souvent contraintes comme par exemple l’homogénéité de la vitesse de déformation qui doit être maintenue constante en temps par exemple. En revanche, des approches dites statiquement indéterminées permettent l’exploitation d’essais mécaniques avec peu d’hypothèses (voire sans) sur les conditions d’essai. Une méthodologie fondée sur un essai d’impact purement intertiel est mise en oeuvre ici pour identifier le comportement viscoplastique de ces matériaux. Avec la Méthode des Champs Virtuels, la méthodologie permet l’identification des paramètres matériaux en exploitant uniquement la mesure des champs de déformation et d’accélération, potentiellement hétérogènes en temps et en espace. Ainsi, celui-ci peut être caractérisé sur une large gamme de déformations et de vitesses de déformation plastiques en procédant à un nombre limité d’expériences. La méthode repose sur le développement d’un simulateur d’images avancé permettant de définir au préalable l’ensemble du dispositif expérimental (géométrie de l’éprouvette et conditions expérimentales). Optimisées numériquement pour prescrire les paramètres d’essai critiques, les réalisations expérimentales menées sur un alliage de Titane utilisé dans l’industrie aéronautique ont permis d’identifier les paramètres d’un modèle de Johnson-Cook sur un spectre de déformations et de vitesses de déformation plastiques pré-déterminé. Les incertitudes de la mesure sont également intégrées et analysées dans ce travail. / This thesis aims at developing an innovative methodology for viscoplastic material behaviour characterization of metallic materials under purely inertial loads. Indeed, their mechanical behaviour under extreme conditions (e.g., crash, impact or explosions) is often rate-dependant. Statically determinate approaches are mainly used to characterize their behaviour. However, they require numerous tests for which testing conditions are strongly constrained, such as the strain rate which has to remain constant in time and space for instance. By contrast, statically undeterminate approaches enable test processing with a few (or without) hypotheses on experimental conditions. In this work, the Image-Based Inertial Impact test methodology has been extended to characterize the viscoplastic behaviour of metallic materials. Owing to the Virtual Fields Method, it enables the identification of constitutive material parameters with the sole knowledge of strain and acceleration fields (possibly heterogeneous in time and space). Therefore, constitutive models can be characterized over a wide range of plastic strain and strain rate, while the number of tests is limited. Tests design notably relies on the development of a synthetic images generator to determine the experimental setup (e.g., specimen geometry or testing conditions). Finally, experiments are carried out with optimized test configurations to identify Johnson-Cook parameters over a predicted range of plastic strain and strain rate for a titanium alloy widely used in aerospace industry. Identification uncertainties are also quantified and analysed in this work.

Zoning map for Johnson City, Tennessee created February 18, 2000 by Johnson City GIS. The guide to zoning districts can be found in a box on the lower left corner. The color coded key and additional information is included along the bottom. Arterial and collector streets are also denoted using empty versus solid circles. Physical copy resides with Johnson City, Geographic Information Systems Division. Scale - 1" = 2000' / https://dc.etsu.edu/rare-maps/1058/thumbnail.jpg

Zoning map for Johnson City, Tennessee created November 6, 2013 by Johnson City GIS. The guide to zoning districts can be found in a box on the lower left corner. The color coded key and additional information is included along the bottom. Arterial and collector streets are also denoted using empty versus solid circles. Physical copy resides with Johnson City, Geographic Information Systems Division. Scale - 1" = 2000' / https://dc.etsu.edu/rare-maps/1059/thumbnail.jpg

Zoning map for Johnson City, Tennessee created March 9, 2007 by Johnson City GIS. The guide to zoning districts can be found in a box on the lower left corner. The color coded key and additional information is included along the bottom. Arterial and collector streets are also denoted using empty versus solid circles. Physical copy resides with Johnson City, Geographic Information Systems Division. Scale - 1" = 2500' / https://dc.etsu.edu/rare-maps/1060/thumbnail.jpg

Zoning map for Johnson City, Tennessee created July 2, 2021 by Johnson City GIS. The guide to zoning districts can be found in a box on the side of map. The color coded key and additional information is included. An inset of downtown is featured in the bottom left corner. Physical copy resides with Johnson City, Geographic Information Systems Division. Scale 1" = 2000' / https://dc.etsu.edu/rare-maps/1061/thumbnail.jpg

Street map of Johnson City, Tennessee created on June 18, 1993 by Johnson City GIS. This map denotes the highways and roads of Johnson City as they were in 1993. No scale is included. Physical copy resides with Johnson City, Geographic Information Systems Division. / https://dc.etsu.edu/rare-maps/1062/thumbnail.jpg

Produced by the Johnson City GIS Division on July 25, 1997, this map denotes the streets of Johnson City. The legend includes fire stations, neighborhoods, and schools. A city street index is also included. The map was designed by Gregory Plumb, GIS Coordinator and Ann Howland, GIS Database Manager. Physical copy resides with Johnson City, Geographic Information Systems Division. Scale - 1:24,000 1" = .4 mile / https://dc.etsu.edu/rare-maps/1063/thumbnail.jpg

Produced by the Johnson City GIS Division on April 23, 1998, this map denotes the streets of Johnson City. The legend includes fire stations, neighborhoods, and schools. A city street index is also included. The map was designed by Gregory Plumb, GIS Coordinator and Ann Howland, GIS Database Specialist. Physical copy resides with Johnson City, Geographic Information Systems Division. Scale - 1:24,000 - 1" = .4 mile / https://dc.etsu.edu/rare-maps/1064/thumbnail.jpg

Produced by the Johnson City GIS Division on December 17, 2014, this map denotes the streets of Johnson City. The legend includes fire stations, hospitals, city parks, and schools. A city street index is also included along the left side edge. Physical copy resides with Johnson City, Geographic Information Systems Division. Scale - 1" = 2000' / https://dc.etsu.edu/rare-maps/1065/thumbnail.jpg