The solutions to certain elliptic boundary value problems have singularities with a typical structure near polyhedral corners. This structure can be exploited to devise an eigenvalue problem whose solution can be used to quantify the singularities of the given boundary value problem. It is necessary to parametrize a ball centered at the corner. There are different possibilities for a suitable parametrization; from the numerical point of view, spherical coordinates are not necessarily the best choice. This is why we do not specify a parametrization in this paper but present all results in a rather general form. We derive the eigenvalue problems that are associated with the Laplace and the linear elasticity problems and show interesting spectral properties. Finally, we discuss the necessity of widely accepted symmetry properties of the elasticity tensor. We show in an example that some of these properties are not only dispensable, but even invalid, although claimed in many standard books on linear elasticity.

The constant gamma in the strengthened Cauchy-Bunyakowskii-Schwarz inequality is a basic tool for constructing of two-level and multilevel preconditioning matrices. Therefore many authors consider estimates or computations of this quantity. In this paper the bilinear form arising from 3D linear elasticity problems is considered on a polyhedron. The cosine of the abstract angle between multilevel finite element subspaces is computed by a spectral analysis of a general eigenvalue problem. Octasection and bisection approaches are used for refining the triangulations. Tetrahedron, pentahedron and hexahedron meshes are considered. The dependence of the constant $\gamma$ on the Poisson ratio is presented graphically.

Dans cette thèse, nous nous sommes intéressés à la statique et la dynamique du mouillage de gouttes d’eau sur des substrats mous tels que des gels, encore connu sous le nom d’élastomouillage. Pour ce faire, nous avons d'abord développé une méthode quantitative de visualisation par strioscopie permettant de mesurer la déformation de la surface d'un film de gel transparent avec une précision élevée. Nous montrons que la déformation superficielle de films mous de silicone (PDMS) dépend de la taille des gouttelettes déposées ainsi que de l'épaisseur et de l’élasticité de ces films. Nous avons construit un modèle basé sur la théorie de l'élasticité linéaire tenant compte de la tension superficielle des gels qui prédit bien la forme et l’amplitude de la déformation de surface. Nous apportons aussi la preuve expérimentale et l'analyse théorique de l’importance de l'hystérèse de l’angle de contact dans la description de la déformation en démontrant que la force tangentielle due à la tension superficielle entre liquide et vapeur à la ligne de contact, souvent négligé, contrôle la déformation de la surface. La dynamique de mouillage est étudiée en dégonflant des gouttelettes sur des films de PDMS avec une épaisseur bien contrôlée. Il est démontré que la dissipation d'énergie dans le gel dépend fortement de l'épaisseur lorsque cette dernière est inférieure à 100 μm). L'effet de freinage viscoélastique et l'effet d'épaisseur sont bien rationalisés avec un modèle basé sur la viscoélasticité linéaire et une simple loi l'échelle qui tient compte de l'effet d'épaisseur capture très bien nos expériences. Enfin, nous démontrons que nous pouvons dériver et guider les gouttelettes en mouvement avec la conception de surfaces couvertes de couches de gels ayant des gradients d'épaisseur. / In this thesis, we aim at obtaining a better understanding of the statics and dynamics of the wetting of liquids on soft gels, otherwise known as elastowetting. First, we develop a quantitative Schlieren optics to measure the surface deformation of a transparent gel film with a high precision over large areas in real time. The long-range surface deformation of soft PDMS films is found to be dependent on the sessile droplet size, and the thickness and elasticity of the soft films. We build a model based on linear elasticity theory with the integration of the surface tension of soft materials that predicts the long-range surface deformation in excellent agreement with the data. We also bring the experimental proof and theoretical analysis of the importance of contact angle hysteresis in the description of the deformation of the surface of the gel. We demonstrate that the tangential component of the liquid-vapor surface tension at the contact line, whose contribution are often neglected, significantly affects the surface deformation. Wetting dynamics is investigated by deflating droplets on PDMS films with well-controlled thickness. It is shown that energy dissipation in the soft gel depends on the thickness when the latter is smaller than 100 μm. The viscoelastic braking effect and the thickness effect are both well rationalized with a model based on the theory of linear viscoelasticity and a simple scaling law accounting for the thickness effect captures very well our experiments. Finally, we demonstrate that we are able to guide moving droplets with coatings having a gradient of their thickness.

For the purpose of numerically simulating continuum mechanical structures, different types of material may be represented by the extreme values {<img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Cepsilon" />,1}, where 0&lt;<img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Cepsilon" /><img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Cll" />1, of a varying coefficient <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Calpha" /> in the governing equations. The paramter <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Cepsilon" /> is not allowed to vanish in order for the equations to be solvable, which means that the exact conditions are approximated. For example, for linear elasticity problems, presence of material is represented by the value <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Calpha" /> = 1, while <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Calpha" /> = <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Cepsilon" /> provides an approximation of void, meaning that material-free regions are approximated with a weak material. For acoustics applications, the value <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Calpha" /> = 1 corresponds to air and <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Calpha" /> = <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Cepsilon" /> to an approximation of sound-hard material using a dense fluid. Here we analyze the convergence properties of such material approximations as <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Cepsilon" />!0, and we employ this type of approximations to perform design optimization. In Paper I, we carry out boundary shape optimization of an acoustic horn. We suggest a shape parameterization based on a local, discrete curvature combined with a fixed mesh that does not conform to the generated shapes. The values of the coefficient <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Calpha" />, which enters in the governing equation, are obtained by projecting the generated shapes onto the underlying computational mesh. The optimized horns are smooth and exhibit good transmission properties. Due to the choice of parameterization, the smoothness of the designs is achieved without imposing severe restrictions on the design variables. In Paper II, we analyze the convergence properties of a linear elasticity problem in which void is approximated by a weak material. We show that the error introduced by the weak material approximation, after a finite element discretization, is bounded by terms that scale as <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Cepsilon" /> and <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Cepsilon" />1/2hs, where h is the mesh size and s depends on the order of the finite element basis functions. In addition, we show that the condition number of the system matrix scales inversely proportional to <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Cepsilon" />, and we also construct a left preconditioner that yields a system matrix with a condition number independent of <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Cepsilon" />. In Paper III, we observe that the standard sound-hard material approximation with <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Calpha" /> = <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Cepsilon" /> gives rise to ill-conditioned system matrices at certain wavenumbers due to resonances within the approximated sound-hard material. To cure this defect, we propose a stabilization scheme that makes the condition number of the system matrix independent of the wavenumber. In addition, we demonstrate that the stabilized formulation performs well in the context of design optimization of an acoustic waveguide transmission device. In Paper IV, we analyze the convergence properties of a wave propagation problem in which sound-hard material is approximated by a dense fluid. To avoid the occurrence of internal resonances, we generalize the stabilization scheme presented in Paper III. We show that the error between the solution obtained using the stabilized soundhard material approximation and the solution to the problem with exactly modeled sound-hard material is bounded proportionally to <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Cepsilon" />.

The peridynamic theory is a generalization of classical continuum mechanics and takes into account the interaction between material points separated by a finite distance within a peridynamic horizon &#948;. The parameter &#948; corresponds to a length scale and is treated as a material property related to the microstructure of the body. Since the balance of linear momentum is written in terms of an integral equation that remains valid in the presence of discontinuities, the peridynamic theory is suitable for studying the material behavior in regions with singularities. The first part of this work concerns the evaluation of the properties of a linear elastic peridynamic material in the context of a three-dimensional state-based peridynamic theory, which uses the difference displacement quotient field in the neighborhood of a material point and considers both length and relative angle changes. This material model is based upon a free energy function that contains four material constants, being, therefore, different from other peridynamic models found in the literature, which contain only two material constants. Using convergence results of the peridynamic theory to the classical linear elasticity theory in the limit of small horizons and a correspondence argument between the free energy function and the strain energy density function from the classical theory, expressions were obtained previously relating three peridynamic constants to the classical elastic constants of an isotropic linear elastic material. To calculate the fourth peridynamic material constant, which couples both bond length and relative angle changes, the correspondence argument is used once again together with the strain field of a linearly elastic beam subjected to pure bending. The expression for the fourth constant is obtained in terms of the Poisson\'s ratio and the shear elastic modulus of the classical theory. The validity of this expression is confirmed through the consideration of other experiments in mechanics, such as bending of a beam by terminal loads and anti-plane shear of a circular cylinder. In particular, numerical results indicate that the expressions for the constants are independent of the experiment chosen. The second part of this work concerns an investigation of the behavior of a one-dimensional linearly elastic bar of length L in the context of the peridynamic theory; especially, near the ends of the bar, where it is expected that the behavior of the peridynamic bar may be very different from the behavior of a classical linear elastic bar. The bar is in equilibrium without body force, is fixed at one end, and is subjected to an imposed displacement at the other end. The bar has micromodulus C, which is related to the Young\'s modulus E in the classical theory through different expressions found in the literature. Depending on the expression for C, the displacement field may be singular near the ends, which is in contrast to the linear behavior of the displacement field observed in classical linear elasticity. In spite of the above, it is also shown that the peridynamic displacement field converges to its classical counterpart as the peridynamic horizon tends to zero. / A teoria peridinâmica é uma generalização da teoria clássica da mecânica do contínuo e considera a interação de pontos materiais devido a forças que agem a uma distância finita entre si, além da qual considera-se nula a força de interação. Por ter o balanço de momento linear formulado como uma equação integral que permanece válida na presença de descontinuidades, a teoria peridinâmica é adequada para o estudo do comportamento de materiais em regiões com singularidades. A primeira parte deste trabalho consiste no cálculo das propriedades de um material peridinâmico elástico linear no contexto de uma teoria peridinâmica de estado, linearmente elástica e tridimensional, que utiliza o campo quociente de deslocamento relativo na vizinhança de um ponto material e leva em conta mudanças relativas angulares e de comprimento. Esse modelo utiliza uma função energia livre que apresenta quatro constantes materiais, sendo, portanto, diferente de outros modelos peridinâmicos investigados na literatura, os quais contêm somente duas constantes materiais. Utilizando resultados de convergência da teoria peridinâmica para a teoria de elasticidade linear clássica no limite de pequenos horizontes e um argumento de correspondência entre as funções energia livre proposta e densidade de energia de deformação da teoria clássica, expressões para três constantes peridinâmicas foram obtidas em função das constantes de um material elástico e isotrópico da teoria clássica. O argumento de correspondêmcia, em conjunto com o campo de deformações de uma viga submetida à flexão pura, é utilizado para calcular a quarta constante peridinâmica do material, que relaciona mudanças angulares relativas e de comprimentos das ligações entre as partículas. Obtem-se uma expressão para a quarta constante em termos do coeficiente de Poisson e do módulo de elasticidade ao cisalhamento da teoria clássica. A validade dessa expressão é confirmada por meio da consideração de outros experimentos da mecânica, tais como flexão de um viga por cargas terminais e cisalhamento anti-plano de um eixo cilíndrico. Em particular, os resultados numéricos indicam que as expressões para as constantes são independentes do experimento escolhido. A segunda parte deste trabalho consiste em uma investigação do comportamento de uma barra unidimensional linearmente elástica de comprimento L no contexto da teoria peridinâmica; especialmente, próximo às extremidades da barra, onde espera-se que o comportamento da barra peridinâmica possa ser muito diferente do comportamento de uma barra elástica linear clássica. A barra está em equilíbrio e sem força de corpo, fixa em uma extremidade, e sujeita a deslocamento imposto na outra extremidade. A barra possui micromódulo C, o qual está relacionado ao módulo de Young E da teoria clássica por meio de diferentes expressões encontradas na literatura. Dependendo da expressão para C, o campo de deslocamento pode ser singular próximo às extremidades, o que contrasta com o comportamento linear do campo de deslocamento observado na elasticidade linear clássica. Apesar disso, é mostrado também que o campo de deslocamento peridinâmico converge para o campo de deslocamento da teoria clássica quando o horizonte peridinâmico tende a zero.

Submitted by Maria Cristina (library@lncc.br) on 2017-12-12T10:49:35Z No. of bitstreams: 1 Thesis - Thiago Quinelato.pdf: 2369263 bytes, checksum: 6a1ac9e2d37bb0377981785cfa50683c (MD5) / Approved for entry into archive by Maria Cristina (library@lncc.br) on 2017-12-12T10:50:02Z (GMT) No. of bitstreams: 1 Thesis - Thiago Quinelato.pdf: 2369263 bytes, checksum: 6a1ac9e2d37bb0377981785cfa50683c (MD5) / Made available in DSpace on 2017-12-12T10:50:14Z (GMT). No. of bitstreams: 1 Thesis - Thiago Quinelato.pdf: 2369263 bytes, checksum: 6a1ac9e2d37bb0377981785cfa50683c (MD5) Previous issue date: 2017-03-01 / Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) / Esta tese é focada no desenvolvimento e na análise de aproximações em dimensão finita das equações que descrevem problemas de elasticidade linear e poroelasticidade. A estratégia de aproximação é baseada em formulações de elementos finitos mistas hibridas desses problemas e a construção dos espaços de dimensão finita é guiada por várias propriedades desejadas: continuidade das trações (conservação do momento linear), simetria do tensor de tensão (conservação do momento angular), número reduzido de graus de liberdade globais e robustez sob distorção de malha. A principal dificuldade está relacionada com o atendimento simultâneo da condição inf-sup e da simetria do tensor de tensão. O ultimo requisito é relaxado, sendo satisfeito de maneira fraca pela introdução de um multiplicador de Lagrange. A maior contribuição é o desenvolvimento e a análise de espaços de dimensão finita estáveis para aproximação mista dos problemas de elasticidade linear e poroelasticidade em malhas quadrilaterais arbitrárias. Esses espaços são capazes de fornecer convergência com taxa ótima do campo de tensão na norma H(div) em malhas de quadriláteros arbitrários, o que é provado pela análise numérica e confirmado por experimentação. / This thesis is focused on the development and analysis of finite dimensional approximations of the equations describing linear elasticity and poroelasticity problems. The approximation strategy is based on mixed hybrid finite element formulations of those problems and the construction of the finite dimensional spaces is guided by several desired properties: continuity of the tractions (conservation of linear momentum), symmetry of the stress tensor (conservation of angular momentum), reduced number of global degrees of freedom, and robustness under mesh distortion. The main difficulty is related with the simultaneous fulfillment of the inf-sup condition and the symmetry of the stress tensor. The last requirement is relaxed, being enforced in the weak sense through the introduction of a Lagrange multiplier. The main contribution is the development and analysis of stable finite dimensional spaces for mixed approximation of linear elasticity and poroelasticity problems on arbitrary quadrilateral meshes. These spaces are capable of providing optimal order convergence of the stress field in the H(div)-norm on meshes of arbitrary quadrilaterals, which is proved by numerical analysis and confirmed by experimentation.

Aujourd’hui, l’un des principaux défis dans l’industrie pétrolière, et particulièrement dans le domaine de l’exploration, est l’exploitation des nouvelles ressources dans des zones structuralement complexes.Nous savons que la géométrie et le glissement le long des failles actives modifie la distribution locale des contraintes. La connaissance du champ de contrainte perturbé actuel est importante pour l’étude des tremblements de Terre, pour la planification de forages ainsi que pour la prédiction de la fracturation induite par l’hydro-fracturation et la prédiction de la réactivation des fractures. Les contraintes perturbées passées, quant à elles sont responsables du développement des fractures naturelles (préexistantes). La détection et la modélisation de celles-ci sont essentielles tant dans le domaine pétrolier (migration et piégeage des fluides) pour une exploitation efficace et à moindre coût des réserves naturelles.Comprendre et quantifier le développement spatial et temporel de ces contraintes a un impact économique non négligeable. L'analyse des paléo-contraintes a été introduite dans un premier temps par Anderson (1905 & 1942) de manière intuitive, puis dans le milieu du siècle dernier, Wallace (1951) et Bott ( 1959) proposèrent les simples postulats que le champ de contrainte est homogène et que la direction du rejet est parallèle à la traction projetée sur le plan de faille. Beaucoup de méthodes d’inversion de contraintes reposent aujourd’hui sur ce principe.Afin d’étudier la validité de l’hypothèse Wallace et Bott, une comparaison avec les vecteurs de glissement générés à partir de modèles numériques (BEM) est effectuée. En testant l’influence de multiples paramètres (géométrie, contraintes au limites du modèle, friction, coefficient de poisson, demi-espace, pression de fluide dans la faille), il est montré que les failles à géométries complexes soumises à certaines contraintes aux limites peuvent engendrer des vecteurs glissements présentant des écarts important avec les la plus grande contraintes cisaillantes résolue sur le plan de faille. A l’inverse, la présence d’une forte friction de glissement permet, dans certaines conditions, de valider l’hypothèse de Wallace et Bott. On s’attache ensuite à comparer les résultats des inversions de contraintes basées sur l’hypothèse de Wallace et Bott (appelé méthode d’inversion classique de contraintes) avec une méthode géomécanique. Pour cela, une faille à géométrie complexe est utilisée dans une étude de sensibilité (contraintes aux limites, friction, échantillonnage) permettant d’analyser l’incertitude des résultats des deux méthodes d’inversion. Cette analyse est ensuite confrontée à l’étude d’un cas de terrain, montrant les avantages et inconvénients des méthodes d’inversions classiques de contraintes.Un des principaux défis de l’industrie pétrolière est l’exploitation des ressources des zones structuralement complexes telles que les réservoirs naturellement fracturés. Connaitre l’état de contraintes hétérogène passé permet d’optimiser la modélisation de ces fractures naturelles. Etant donné que les glissements sur les failles sont difficiles à observer dans les réservoirs pétroliers, les données de d’orientation de fractures (joints, failles, stylolites) sont naturellement prises en compte lors de l’inversion des contraintes. On montre, en utilisant divers exemples de terrain et d’industrie, que dans de tels cas, l’utilisation d’inversions basée sur la mécanique est beaucoup plus appropriée. Cependant, il est parfois difficile de déterminer le type cinématique de fracture observée le long des puits, et très souvent, les zones étudiées ont subi une tectonique polyphasée. La dernière partie vise donc à résoudre le problème des données de types cinématiques non identifiables (joints, failles, stylolites…) et étend parallèlement l’inversion mécanique des contraintes à la séparation de phases tectoniques. / Today, one of the main challenges in the oil industry, especially during the exploration phase, is the exploitation of new resources in structurally complex areas such as naturally fractured reservoirs, salt diapirs, mountain ranges, and unconventional reservoirs.We know that the geometry and sliding along active faults modifies the local stress distribution. Knowing the present day perturbed stress field is important for the study of earthquakes, for the planning of the borehole drilling and stability as well as for the prediction of fractures induced by hydro-fracturing and reactivation of natural fractures. In the other side, perturbed paleostress are responsible for the development of (pre-existing) natural fractures. The detection and modeling of the latter, are essential both in the oil industry (migration and trapping of fluids) for a cost efficient recovery of natural reserves.Understanding and quantifying the spatial and temporal development of the stress distribution has a significant economic and environmental impact. The analysis of paleo-constraints was intuitively introduced first by Anderson (1905 & 1942), then in the middle of the last century, Wallace (1951) and Bott (1959) proposed the simple hypothesis that (i) The stress field is homogeneous in space and constant in time, and that (ii) the slip direction is parallel to the traction projected on the fault plane which gives the direction of the shear stress. Many stress inversion methods are based on this hypothesis while recent studies raise doubts as to their compatibility with rock mechanics.In order to investigate the validity of the Wallace and Bott hypothesis, a comparison with vectors of slip generated with numerical models (BEM) is performed. By testing the influence of multiple parameters (geometry, boundary conditions, friction, Poisson’s coefficient , half-space, fault fluid pressure), it is shown that the complex geometry faults subject to specific boundary conditions can yield slip vectors with significant discrepancies with the maximum shear stress resolved on the fault plane. Conversely, the presence of a high sliding friction, allows under certain conditions, to validate the hypothesis of Wallace and Bott.We then focus on the task to compare the results of stress inversions based on the assumption of Wallace and Bott (called classical stress inversion methods) to a geomechanical method. For this, a complex fault geometry is used in a sensitivity analysis (boundary conditions, friction, sampling) to evaluate the uncertainty of the results of the two inversion methods. This analysis is then compared to a case study, Chimney Rock (Utah, USA), showing the advantages and disadvantages of the classical stress inversion methods.One of the main challenges of the oil industry is the exploitation of resource in structurally complex oil fields such as naturally fractured reservoirs. Knowing the heterogeneous paleostress allows to optimize the modeling of these natural fractures. Since slip on faults is hardly observed in petroleum reservoirs, fracture orientation data (joints, faults, stylolites) are naturally taken into account during the inversion of stresses. It is shown, using various field and industry examples, that in such cases the use of mechanical stress inversions is much more appropriate.However, it is sometimes difficult to determine the fracture kinematics observed along wellbores, and very often the studied regions underwent multiple tectonic phases. The final section aims to address the problem of data with unknown kinematic (joints, faults, stylolites ...) and expends the mechanical stress inversion to the separation of tectonic phases.

A teoria de elasticidade linear clássica é utilizada no modelamento de problemas da física médica relacionados com a determinação de parâmetros elásticos de tecidos biológicos a partir da medição in vivo, ou, in vitro dos deslocamentos, ou, das deformações. Baseados em observações experimentais, as quais revelam que os tecidos biológicos anômalos têm comportamento mecânico diferente dos tecidos biológicos sadios, os pesquisadores têm modelado estes tecidos como sólidos elástico-lineares, isotrópicos, heterogêneos e incompressíveis. Neste trabalho, analisa-se uma classe de problemas planos relacionados à determinação do módulo de elasticidade ao cisalhamento µ de tecidos biológicos e propõe-se um procedimento numérico não-iterativo para obter soluções aproximadas para estes problemas a partir de campos de deslocamentos conhecidos de ensaios possíveis de serem realizados em laboratório. Os ensaios são estáticos e são simulados numericamente via método dos elementos finitos. Apresentam-se resultados obtidos das distribuições de µ em cilindros retos, longos e de seção retangular contendo inclusões cilíndricas circulares centradas, ou, excêntricas. Consideram-se inclusões mais, ou, menos rígidas do que o meio elástico circundante. Adicionalmente, os resultados obtidos no presente trabalho são comparados com resultados de outros pesquisadores que utilizam ensaios dinâmicos. Neste sentido, dois casos de inclusão circular centrada são resolvidos com as condições de contorno adaptadas do caso dinâmico para o caso estático. Resolve-se finalmente o caso de uma inclusão de forma geométrica complexa e seis vezes mais rígida do que o entorno. O cilindro contendo esta inclusão está submetido às condições de contorno propostas neste trabalho e também às condições de contorno adaptadas do caso dinâmico. Em todos os casos analisados os resultados são satisfatórios, apesar do emprego de uma quantidade reduzida de elementos finitos na reconstrução de µ. Deve-se ressaltar que nenhum método de regularização foi utilizado para tratar os deslocamentos obtidos dos ensaios simulados. Este trabalho é de grande interesse na detecção de tumores cancerígenos, tais como tumores nos seios e na próstata, e no diagnóstico diferenciado de tecidos biológicos. / The theory of classical linear elasticity is used to model of problems in medical physics that are related to the determination of elastic parameters of biological tissues from the measurement in vivo, or, in vitro of either displacements or strains. Based on experimental observations, which indicate that the abnormal biological tissues have different mechanical behavior from normal biological tissues, researchers have modeled these tissues as an incompressible, heterogeneous, and isotropic linear elastic solid. In this work a class of plane problems related to the determination of the shear elastic modulus µ of biological tissues is examined. A non-iterative numerical procedure to obtain an approximate solution to these problems from known displacement fields is proposed. The displacement fields are obtained from experiments that are possible to reproduce in laboratory. The experiments are quasi-static and are simulated numerically using the finite element method. Results for the distribution of µ in long, straight cylinders of rectangular cross-sections, containing either centered or eccentric circular inclusions that are more, or, less stiff than the surrounding elastic medium, are presented. Additionally, the results obtained in this study are compared with results of other researchers who use dynamical experiments. In this sense, two cases of centered circular inclusions are solved by using an adaptation of the dynamical case to the static case. Finally, the case of an inclusion with a complex geometry that is six times more rigid than the surrounding medium is solved. In all cases analyzed, the results are satisfactory, despite the fact that they were obtained with a reduced number of finite elements. It should be noted that no method of regularization has been used to treat the displacement data obtained from the simulated experiments. This work is of great interest in the detection of cancerous tumours, such as those in the breasts and in the prostate, and in the differential diagnosis of biological tissues.

Dans la nature les plantes sont sans cesse soumises à des sollicitations mécaniques qui affectent et modifient leur croissance. Un aspect remarquable de cette réponse est qu’elle n’est pas seulement locale mais non-locale : la flexion d’une tige ou d’une branche inhibe rapidement la croissance loin de la zone sollicitée. Cette observation suggère l'existence d'un signal pouvant se propager à travers toute la plante. Parmi les différentes hypothèses, il a été suggéré que ce signal pouvait être purement mécanique, et provenir d’un couplage hydro/mécanique entre la déformation du tissu et la pression de l’eau contenue dans le système vasculaire de la plante. L’objectif de cette thèse est de comprendre l’origine physique de ce couplage par une approche biomimétique. Pour cela, nous avons développé des branches artificielles micro-fluidiques possédant des caractéristiques mécaniques et hydrauliques similaires à celles d'une branche d'arbre. Nous avons montré que la flexion de ces branches génère une surpression globale non-nulle dans le système, qui varie comme le carré de la déformation longitudinale. Un modèle simple basé sur un mécanisme analogue à l’ovalisation des tubes permet de prédire cette réponse poroélastique non-linéaire et d’identifier le paramètre physique clé pilotant cette réponse en pression : le module de compressibilité de la branche. A la lumière de ces résultats, des expériences sur des branches d'arbre ont ensuite été conduites et des signaux similaires sont obtenus et comparés au modèle théorique. La similitude suggère le caractère générique du mécanisme physique identifié pour la génération de signaux hydraulique dans les plantes. / Plants are constantly subjected to external mechanical loads such as wind or touch and respond to these stimuli by modifying their growth and development. A fascinating feature of this mechanical-induced-growth response is that it is not only local, but also non-local: bending locally a stem or a branch can induce a very rapid modification of the growth far away from the stimulated area, suggesting the existence of a signal that propagates across the whole plant. The nature and origin of this signal is still not understood, but it has been suggested recently that it could be purely mechanical and originate from the coupling between the local deformation of the tissues and the water pressure in the vascular system. The objective of this work is to understand the origin of this hydro/mechanical coupling using a biomimetic approach. Artificial microfluidic branches have been developed, that incorporate the mechanical and hydraulic key features of natural ones. We show that the bending of these branches generates a steady overpressure in the whole system, which varies quadratically with the bending deformation. A simple model based on a mechanism analogue to tube ovalization enables us to predict this non-linear poroelastic response, and identify the key physical parameter at play, namely the elastic bulk modulus of the branch. Further experiments conducted on natural tree branches reveal the same phenomenology. Once rescaled by the model prediction, both the biomimetic and natural branches falls on the same master curve, showing the universality of the identified mechanism for the generation of hydraulic signals in plants.

Le contexte de cette thèse est la théorie de l'élasticité non linéaire, appelée également "élasticité finie". On y présente des résultats concernant la propagation d'ondes d'amplitude finie dans des matériaux élastiques non linéaires soumis à une grande déformation statique homogène. Bien que les matériaux considérés soient isotropes, lors de la propagation d'ondes un comportement anisotrope dû à la déformation statique se manifeste. Après un rappel des équations de base de l'élasticité non linéaire (Chapitre 1), on considère tout d'abord la classe générale des matériaux incompressibles. Pour ces matériaux, on montre que la propagation d'ondes transversales polarisées linéairement est possible pour des choix appropriés des directions de polarisation et de propagation. De plus, on propose des généralisations des modèles classiques de "Mooney-Rivlin" et "néo-Hookéen" qui conduisent à de nouvelles solutions. Bien que le contexte soit tri-dimensionnel, il s'avère que toutes ces ondes sont régies par des équations d'ondes scalaires non linéaires uni-dimensionelles. Dans le cas de solutions du type ondes simples, on met en évidence une propriété remarquable du flux et de la densité d'énergie. Dans les Chapitres 3 et 4, on se limite à un modèle particulier de matériaux compressibles appelé "modèle restreint de Blatz-Ko", qui est une version compressible du modèle néo-Hookéen. En milieu infini (Chapitre 3), on montre que des ondes transversales polarisées linéairement, faisant intervenir deux variables spatiales, peuvent se propager. Bien que la théorie soit non linéaire, le champ de déplacement de ces ondes est régi par une version anisotrope de l'équation d'onde bi-dimensionnelle classique. En particulier, on présente des solutions à symétrie "cylindrique elliptique" analogues aux ondes cylindriques. Comme cas particulier, on obtient aussi des ondes planes inhomogènes atténuées à la fois dans l'espace et dans le temps. De plus, on montre que diverses superpositions appropriées de solutions sont possibles. Dans chaque cas, on étudie les propriétés du flux et de la densité d'énergie. En particulier, dans le cas de superpositions il s'avère que des termes d'interactions interviennent dans les expressions de la densité et du flux d'énergie. Finalement (Chapitre 4), on présente une solution exacte qui constitue une généralisation non linéaire de l'onde de Love classique. On considère ici un espace semi-infini, appelé "substrat" recouvert par une couche. Le substrat et la couche sont constitués de deux matériaux restreints de Blatz-Ko pré-déformés. L'onde non linéaire de Love est constituée d'un mouvement non atténué dans la couche et d'une onde plane inhomogène dans le substrat, choisies de manière à satisfaire aux conditions aux limites. La relation de dispersion qui en résulte est analysée en détail. On présente de plus des propriétés générales du flux et de la densité d'énergie dans le substrat et dans la couche. The context of this thesis is the non linear elasticity theory, also called "finite elasticity". Results are obtained for finite-amplitude waves in non linear elastic materials which are first subjected to a large homogeneous static deformation. Although the materials are assumed to be isotropic, anisotropic behaviour for wave propagation is induced by the static deformation. After recalling the basic equations of the non linear elasticity theory (Chapter 1), we first consider general incompressible materials. For such materials, linearly polarized transverse plane waves solutions are obtained for adequate choices of the polarization and propagation directions (Chapter 2). Also, extensions of the classical Mooney-Rivlin and neo-Hookean models are introduced, for which more solutions are obtained. Although we use the full three dimensional elasticity theory, it turns out that all these waves are governed by scalar one-dimensional non linear wave equations. In the case of simple wave solutions of these equations, a remarkable property of the energy flux and energy density is exhibited. In Chapter 3 and 4, a special model of compressible material is considered: the special Blatz-Ko model, which is a compressible counterpart of the incompressible neo-Hookean model. In unbounded media (Chapter 3), linearly polarized two-dimensional transverse waves are obtained. Although the theory is non linear, the displacement field of these waves is governed by a linear equation which may be seen as an anisotropic version of the classical two-dimensional wave equation. In particular, solutions analogous to cylindrical waves, but with an "elliptic cylindrical symmetry" are presented. Special solutions representing "damped inhomogeneous plane waves" are also derived: such waves are attenuated both in space and time. Moreover, various appropriate superpositions of solutions are shown to be possible. In each case, the properties of the energy density and the energy flux are investigated. In particular, in the case of superpositions, it is seen that interaction terms enter the expressions for the energy density and the energy flux. Finally (Chapter 4), an exact finite-amplitude Love wave solution is presented. Here, an half-space, called "substrate", is assumed to be covered by a layer, both made of different prestrained special Blatz-Ko materials. The Love surface wave solution consists of an unattenuated wave motion in the layer and an inhomogeneous plane wave in the substrate, which are combined to satisfy the exact boundary conditions. A dispersion relation is obtained and analysed. General properties of the energy flux and the energy density in the substrate and the layer are exhibited.