<p> An investigation of general Materials of Differential Type [MDT], and Motions With Constant Stretch History [MCSH] is presented. Rivlin-Ericksen tensors An are shown to result from a Taylor series expansion of the relative strain tensor Ct(T). Internal constraint in MDT is discussed. General Solutions of Motions of Differential Type are worked out. Dynamically possible stresses are found for certain irrotational motions. Theorems regarding necessary and sufficient conditions for MCSH are proved. A class of MCSH is introduced, and an approximate MCSH is suggested. Necessary equations regarding gradients of a scalar-valued tensor function are derived. </p> / Thesis / Master of Engineering (MEngr)

The principal contribution of this dissertation is a theory of Strongly Orthotropic Continuum Mechanics that is derived entirely from an assertion of geometric strain indeterminacy. Implementable into the finite element method, it can resolve widespread kinematic misrepresentations and offer unique and purportedly exact strain-induced energies by removing the assumptions of strain tensor symmetry. This continuum theory births the proposal of a new class of physical tensors described as the Intrinsic Field Tensors capable of generalising the response of most classical mechanical metrics, a number of specialised formulations and the solutions shown to be kinematically intermediate. A series of numerical examples demonstrate Euclidean objectivity, material frame-indifference, patch test satisfaction, and agreement between the subsequent Material Principal Co-rotation and P??I??C decomposition methods that produce the intermediary stress/strain fields. The encompassing theory has wide applicability owing to its fundamental divergence from conventional mechanics, it offers non-trivial outcomes when applied to even very simple problems and its use of not the Eulerian, Lagrangian but the Intrinsic Frame generates previously unreported results in strongly orthotropic continua.

The principal contribution of this dissertation is a theory of Strongly Orthotropic Continuum Mechanics that is derived entirely from an assertion of geometric strain indeterminacy. Implementable into the finite element method, it can resolve widespread kinematic misrepresentations and offer unique and purportedly exact strain-induced energies by removing the assumptions of strain tensor symmetry. This continuum theory births the proposal of a new class of physical tensors described as the Intrinsic Field Tensors capable of generalising the response of most classical mechanical metrics, a number of specialised formulations and the solutions shown to be kinematically intermediate. A series of numerical examples demonstrate Euclidean objectivity, material frame-indifference, patch test satisfaction, and agreement between the subsequent Material Principal Co-rotation and P??I??C decomposition methods that produce the intermediary stress/strain fields. The encompassing theory has wide applicability owing to its fundamental divergence from conventional mechanics, it offers non-trivial outcomes when applied to even very simple problems and its use of not the Eulerian, Lagrangian but the Intrinsic Frame generates previously unreported results in strongly orthotropic continua.

Les outils de la géodésie spatiale sont aujourd'hui très fortement impliqués dans les études géophysiques. Le champ de déformations horizontales d'un site d'étude est fourni par les vecteurs déplacement ou par un champ tensoriel. Ce dernier possède l'avantage d'être indépendant de tout référentiel, contrairement à ce qui est nécessaire pour exprimer les vecteurs vitesse. Néanmoins, les méthodes de calcul de tenseurs dépendent souvent d'une décomposition arbitraire en figures élémentaires à partir des points de mesures géodésiques. De plus, la représentation de ces tenseurs selon leurs axes principaux est d'une lecture et d'une interprétation assez difficiles et nécessitent un certain entraînement. Cette thèse traite, dans un premier temps, le problème de fournir un champ continu de déformations sous la forme des tenseurs régulièrement répartis, de façon peu dépendante des points de mesure, et dans un deuxième temps, de fournir une représentation cartographique intuitive de ces tenseurs avec, pour la première fois, une représentation simultanée de leur degré de significativité. L'estimation des incertitudes de la déformation obtenue est analysée selon deux points de vue : d'une part, une méthode de Monte Carlo est appliquée pour la détermination des barres d'erreurs liées aux mesures, son résultat permet le calcul de degré de significativité des tenseurs par comparaison des valeurs de tenseurs par rapport à leurs incertitudes, et d'autre part, une estimation des contraintes imposées par la géométrie de distribution des points de mesures qui est ensuite combinée avec la première source d'erreur. La nouvelle approche de représentation a été analysée via une enquête auprès d'un groupe de géophysiciens, en leur fournissant plusieurs possibilités de représentations. En se basant sur les résultats de cette enquête, nous avons pu valider la nouvelle représentation qui permet de mettre en évidence certains aspects mal mis en évidence par la représentation classique, et donc le choix des éléments graphiques de la carte permettant de fournir la représentation la plus intuitive possible / Space geodesy tools are now strongly involved in geophysical studies. The horizontal deformation field for a region of interest is provided by two main methods : a velocity field and a strain tensor field. A strain tensors field solution has the advantage of being independent of the reference frame in which the velocities are expressed. Nevertheless, the current methods of calculation of a strain tensors field depend on the positioning of geodetic points. Furthermore, the current mapping method of tensors by their mains axis is not easy to read and to interpret, needing some training. This thesis is devoted to the problem of calculating a continuous field of regularly spaced strain tensors, and providing an intuitive mapping method of these tensors with a simultaneous representation of their significance level on the same map. The estimation of uncertainties related to the deformation field is made in two steps : firstly, a Monte Carlo method is applied for the calculation of uncertainties related to the measurements, its results allow to define the significance level of tensors by normalizing tensor's values with respect to their related uncertainties, then, the constraints coming from the distribution of the network of measurement points are calculated and combined with the first source of error. The new approach of mapping tensors was analyzed through an opinion survey by providing several possibilities of representation. The results of this opinion survey allowed us to validate this new mapping method by geophysicists for representing a deformation field, because it allows highlighting some aspects not well illustrated by the classical mapping method of tensors, and therefore choosing the graphical elements of the map which provide the best intuitive method of mapping a strain tensors field