Les phénomènes liés au comportement à la rupture de la glace sous impact sont fréquents dans le génie civil, pour les structures offshore, et les processus de dégivrage. Pour réduire les dommages causés par l'impact de la glace et optimiser la conception des structures ou des machines, l'étude sur le comportement à la rupture dynamique de la glace sous impact est nécessaire. Ces travaux de thèse portent donc sur la propagation dynamique des fissures dans la glace sous impact. Une série d'expériences d'impact est réalisée avec un dispositif de barres de Hopkinson. La température est contrôlée par une chambre de refroidissement. Le processus dynamique de la rupture de la glace est enregistré avec une caméra à grande vitesse et ensuite analysé par des méthodes d'analyse d'images. La méthode des éléments finis étendus complète cette analyse pour évaluer la ténacité dynamique. Au premier abord, le comportement dynamique de la glace sous impact est étudié avec des échantillons cylindriques afin d'établir la relation contrainte-déformation dynamique qui sera utilisée dans les simulations numériques plus tard. Nous avons observé de multi-fissuration dans les expériences sur les échantillons cylindriques mais son étude est trop difficile à mener. Pour mieux comprendre la propagation des fissures dans la glace, des échantillons rectangulaires avec une pré-fissure sont employés. En ajustant la vitesse d'impact on aboutit à la rupture des spécimens avec une fissure principale à partir de la pré-fissure. L'histoire de la propagation de fissure et de sa vitesse sont évaluées par analyse d'images basée sur les niveaux de gris et par corrélation d'images. La vitesse de propagation de la fissure principale est identifiée dans la plage de 450 à 610 m/s ce qui confirme les résultats précédents. Elle varie légèrement au cours de la propagation, dans un premier temps elle augmente et se maintient constante ensuite et diminue à la fin. Les paramètres obtenus expérimentalement, tels que la vitesse d'impact et la vitesse de propagation de fissure, sont utilisés pour la simulation avec la méthode des éléments finis étendus. La ténacité d'initiation dynamique et la ténacité dynamique en propagation de fissure sont déterminées lorsque la simulation correspond aux expériences. Les résultats indiquent que la ténacité dynamique en propagation de fissure est linéaire vis à vis de la vitesse de propagation et semble indépendante de la température dans l'intervalle -15 à -1 degrés.

The phenomena relating to the fracture behaviour of ice under impact loading are common in civil engineering, for offshore structures, and de-ice processes. To reduce the damage caused by ice impact and to optimize the design of structures or machines, the investigation on the dynamic fracture behaviour of ice under impact loading is needed. This work focuses on the dynamic crack propagation in ice under impact loading. A series of impact experiments is conducted with the Split Hopkinson Pressure Bar. The temperature is controlled by a cooling chamber. The dynamic…

Advisors/Committee Members: Nélias, Daniel (thesis director).

In this dissertation we study non-negative self-adjoint Laplace type operators acting on smooth sections of a vector bundle. First, we assume base manifolds are compact, boundaryless, and Riemannian. We start from the Fourier integral operator representation of half-wave operators, continue with spectral zeta functions, heat and resolvent trace asymptotic expansions, and end with the quantitative Wodzicki residue method. In particular, all of the asymptotic coefficients of the microlocalized spectral counting function can be explicitly given and clearly interpreted. With the auxiliary pseudo-differential operators ranging all smooth endomorphisms of the given bundle, we obtain certain asymptotic estimates about the integral kernel of heat operators. As applications, we study spectral asymptotics of Dirac type operators such as characterizing those for which the second coefficient vanishes. Next, we assume vector bundles are trivial and base manifolds are Euclidean domains, and study non-negative self-adjoint extensions of the Laplace operator which acts component-wise on compactly supported smooth functions. Using finite propagation speed estimates for wave equations and explicit Fourier Tauberian theorems obtained by Yuri Safarov, we establish the principle of not feeling the boundary estimates for the heat kernel of these operators. In particular, the implied constants are independent of self-adjoint extensions. As a by-product, we affirmatively answer a question about upper estimate for the Neumann heat kernel. Finally, we study some specific values of the spectral zeta function of two-dimensional Dirichlet Laplacians such as spectral determinant and Casimir energy. For numerical purposes we substantially improve the short-time Dirichlet heat trace asymptotics for polygons. This could be used to measure the spectral determinant and Casimir energy of polygons whenever the first several hundred or one thousand Dirichlet eigenvalues are known with high precision by other means.

Os escoamentos bifásicos incompressíveis água-óleo são modelados por um sistema de equações diferenciais parciais nas incógnitas velocidade, pressão e saturação. Neste trabalho é apresentado um novo método numérico em duas dimensões espaciais para a resolução numérica da equação de transporte de massa. Suas principais características são: eficiência computacional, boa precisão numérica e ausência de oscilações espúrias na presença de dados inicias descontínuos. O modelo matemático usado no desenvolvimento do método numérico se baseia na combinação e extensão das idéias dos esquemas centrais de Lax-Friedrichs, Rusanov, Nessyahu, Tadmor e Kurganov para duas dimensões espaciais, isto é, usa diferenças centradas, velocidade local de propagação e o algoritmo REA. O novo esquema central obtido tem precisão de segunda ordem e admite uma formulação semi-discreta, tornando a sua pequena difusão numérica independente do tamanho do passo de tempo usado para a evolução da equação diferencial. Isto garante a principal propriedade apresentada pelo novo método numérico: nenhuma difusão numérica extra é inserida se o passo de tempo for reduzido. Este comportamento é muito importante quando aplicado a problemas de escoamentos multifásicos em reservatórios de petróleo altamente heterogêneos, pois a alta heterogeneidade do meio poroso introduz uma grande variabilidade no campo de velocidades que exige, por sua vez, uma redução no passo de tempo. Sua formulação final, bastante simples, é um sistema de equações diferenciais ordinárias, na incógnita saturação, para cada célula do domínio computacional definido. Estas equações são então resolvidas pelo método de Runge-Kutta de segunda ordem na sua forma explícita. Os resultados numéricos obtidos são bastante precisos quando comparados com o esquema central totalmente discreto Nessyahu-Tadmor de segunda ordem. Estes resultados indicam que, ao usar o esquema de Nessyahu-Tadmor, torna-se necessário multiplicar por 16 o número de células da malha computacional para se obter uma solução comparável à solução apresentada pelo novo método numérico.

Incompressible two-phase flows aremodeled by a non-linear system of partial differential equations involving velocity, pressure and saturation. In this work we present a new numerical method in two spatial dimensions to solve the fluid transport equation. Its main features are computational eficiency, accuracy of solutions and absense of spurious oscillations with descontinuous initial conditions. The mathematical model used to develop this numerical method combines and extends ideias from Lax-Friedrichs, Rusanov, Nessyahu, Tadmor and Kurganovs central schemes to two spatial dimensions, i.e., it uses central differences, local speed of propagation and the REA algorithm. This new central scheme has second order accuracy and allows for a semi-discrete formulation, which makes its small numerical diffusion independent of the time step size used to integrate the differential equation. This provides the main feature of this new central scheme: no extra…

Advisors/Committee Members: Helio Pedro Amaral Souto, Frederico Furtado, Steven Dufour, César Guilherme de Almeida, Luis Felipe Feres Pereira.