En mécanique des matériaux et des structures, l’interface entre deux composants matériels ou deux éléments structuraux est traditionnellement et le plus souvent supposé parfaite. Au sens mécanique, une interface parfaite est une surface à travers laquelle le vecteur de déplacement et le vecteur de contrainte sont tous les deux continus. L’hypothèse des interfaces parfaites est inappropriée dans de nombreuses situations en mécanique. En effet, l’interface entre deux corps ou deux parties d’un corps est un endroit propice aux réactions physico-chimiques complexes et favorable à l’endommagement mécanique. L’intérêt pour les interfaces imparfaites devient depuis quelques années grandissant avec le développement des matériaux et structures nanométriques dans lesquels les interfaces et surfaces jouent un rôle prépondérant. A partir de la configuration de base où une interphase de faible épaisseur sépare deux phases, ce travail établit trois modèles d’interface imparfaite généraux qui permettent de remplacer l’interphase par une interface imparfaite dans les cas de la conduction thermique, de l’élasticité linéaire et de la piézoélectricité sans perturber les champs en questions à une erreur fixée près. La dérivation de ces modèles est basée sur le développement de Taylor et sur une approche originale de géométrie différentielle indépendante de tout système de coordonnées. Les trois modèles généraux permettent non seulement de mieux appréhender certains modèles phénoménologiques d’interface imparfaite mais aussi de décrire les effets d’interface que les modèles existants ne sont pas en mesure de prendre en compte. Les modèles d’interface imparfaite établis sont appliqués dans la détermination des propriétés effectives thermiques, élastiques et piézoélectriques d’un matériau composite constitué d’une matrice renforcée par des particules ou fibres enrobées d’une interphase. La méthode utilisée pour rendre compte des effets des interfaces imparfaites sur les propriétés effectives repose sur une condition d’équivalence énergétique qui ramène un matériau hétérogène avec interfaces imparfaites à un matériau hétérogène avec interfaces parfaites / In mechanics of materials and structures, the interface between two material components or two structural elements is traditionally and the most often assumed to be perfect. In mechanics, a perfect interface is a surface through which the displacement and stress vectors are continuous. The assumption of the perfect interfaces is inappropriate in many situations in mechanics. Indeed, the interface between two bodies or two parts of a body is a place propitious to complex physicochemical reactions and vulnerable to mechanical damage. The interest in imperfect interfaces has become for a few years growing with the development of nanometric materials and structures in which the interfaces and surfaces play a preponderant role. Starting from the basic configuration where an interphase of thin thickness separates two phases, this work establishes three general models of imperfect interface which make it possible to replace the interphase by an imperfect interface in the cases of thermal conduction, linear elasticity and piezoelectricity without disturbing the fields in questions to within a fixed error. The derivation of these models is based on the development of Taylor and an original coordinate-free approach of differential geometry. The three general models make it possible not only to get a better understanding of certain phenomenological models of imperfect interface but also to describe the effects of interface which the existing models are not able to take into account. The established models of imperfect interface are applied to determining the thermal, elastic and piezoelectric effective properties of composite materials consisting of a matrix reinforced by particles or fibers coated with an interphase. The method used to account for the effects of imperfect interfaces on the effective properties rests on an energy equivalency which brings back a heterogeneous material with imperfect interfaces to a heterogeneous material with perfect interfaces

In this work we study peridynamics, a non-local model in continuum me- chanics introduced by Silling (2000). The non-locality is reflected in the fact that points at finite distance exert a force upon each other. If, however, these points are more distant than a characteristic length called horizon, it is customary to assume that they do not interact. We compare peridynamics with elasticity, especially in the limit of small horizon. We restrict ourselves, concerning this vanishing non-locality, to variational formulation of time- independent processes. We compute a Γ-limit for homogeneous and isotropic solid in linear peridynamics. In some cases this Γ-limit coincides with linear elasticity and the Poisson ratio is equal to 1 4. We conclude by clarifying why in some situation the computed Γ-limit can differ from the linear elasticity. 1

Différentes applications de l'élasticité linéaire en géologie structurale sont présentées dans cette thèse à travers le développement de trois types de codes numériques. Le premier utilise la modélisation directe pour étudier les déplacements et champs de contraintes autour de zones faillées complexes. On montre que l'ajout de contraintes inégalitaires, telles que la friction de Coulomb, permet d'expliquer l'angle d'initiation des dominos dans les relais extensifs. L'ajout de matériaux hétérogènes et d'optimisations, telles la parallélisation sur processeurs multi-coeurs ainsi que la réduction de complexité des modèles, permettent l'étude de modèles beaucoup plus complexes. Le second type de code numérique utilise la modélisation inverse, aussi appelée estimation de paramètres. L'inversion linéaire de déplacements sur les failles ainsi que la détermination de paléo-contraintes utilisant une approche géomécanique sont développées. Le dernier type de code numérique concerne la restoration de structures complexes plissées et faillées. Il est notamment montré qu'une telle méthode permet de vérifier l'équilibre de coupes géologiques, ainsi que de retrouver la chronologie des failles. Finalement, nous montrons que ce même code permet de lisser des horizons 3D faillés, plissés et bruités en utilisant la géomécanique. / Different applications of linear elasticity in structural geology are presented in this thesis through the development of three types of numerical computer codes. The first one uses forward modeling to study displacement and perturbed stress fields around complexly faulted regions. We show that incorporating inequality constraints, such as static Coulomb friction, enables one to explain the angle of initiation of jogs in extensional relays. Adding heterogeneous material properties and optimizations, such as parallelization on multicore architectures and complexity reduction, admits more complex models. The second type deals with inverse modeling, also called parameter estimation. Linear slip inversion on faults with complex geometry, as well as paleo-stress inversion using a geomechanical approach, are developed. The last type of numerical computer code is dedicated to restoration of complexly folded and faulted structures. It is shown that this technique enables one to check balanced cross-sections, and also to retrieve fault chronology. Finally, we show that this code allows one to smooth noisy 3D interpreted faulted and folded horizons using geomechanics.

This is a documentation of the software package COCOS. The purpose of COCOS is the computation of corner singularities of elliptic equations in polyhedral corners and crack tips. COCOS provides a self-contained library for the generation of structured 2D finite element meshes, including various routines for mesh manipulation, as well as several algorithms for the solution of quadratic eigenvalue problems with Hamiltonian structure. These and further features will be described in this documentation.

This thesis involves the derivation of a constitutive model under large deformationtheory using Updated Lagrange method applied on an orthotropic material.Thefollowing aspects are included in this thesis work: introduction, theory, FEM implementation, derivation of constitutive model, calibration, result, discussion, conclusion and the future work. This thesis studies the deformation behavior of wood, which is widely used as aconstruction material, in an advanced and more detailed way by analyzing the mechanical properties of wood from both, the application in laboratory and theoreticalcalculation under large deformation theory. A non-linear elastic constitutive model is proposed, derived and calibrated using asimple inverse analysis procedure. The calibration process was performed to identify8 constitutive parameters A1 − A8 of the constitutive model by performing inverseanalysis against relevant experimental data acquired using the Aramis system. Theresults in the comparison were extracted from the specimen when it is both intangential orientation and radial orientation. The project work will be dedicated to the development of a Finite Element Method(FEM) code implemented in MATLAB scripts which was directly used to study themechanical properties of the orthotropic wood material when hyper-elastic behavioris assumed. The results will contain three parts: 1) study of the influence of pith location onthe load required to deform the specimen specimen, 2) reaction force comparisonof the model results against experimental results, and, 3) comparison of the GreenLagrangian strain pattern over the specimen between the experimental data and themodel’s results.

The mathematical characterization of the sound of a musical instrument still follows Schumann’s laws [1]. According to this theory, the resonances of the instrument body, “the formants”, filter the oscillations of the sound generator (e.g., strings) and produce the characteristic “timbre” of an instrument. This is a strong simplification of the actual situation. It applies to a point source and does not distinguish between a loudspeaker and a three-dimensional instrument. In this work we investigate Finite-Element-based numerical simulations of eigenfrequencies and eigenmodes of a tuning fork in order to capture the oscillation behavior of its eigenfrequencies. We model the tuning fork as an elastic solid body and solve an eigenvalue equation derived from a system of coupled equations from linear elasticity theory on an unstructured three-dimensional grid. The eigenvalue problem is solved using the preconditioned inverse iteration (PINVIT) method with an efficient geometric multigrid (GMG) preconditioner. The latter allows us to resolve the tuning fork with a high resolution grid, which is required to capture fine modes of the simulated eigenfrequencies. To verify our results, we compare them with measurement data obtained from an experimental modal analyses of a real reference tuning fork. It turns out that our model is sufficient to capture the first eight eigenmodes of a reference tuning fork, whose identification and reproduction by simulation is novel to the knowledge of the author.

Predicting the effective elastic properties of a composite material based on the elastic properties of the constituent materials is extremely difficult, even when the microstructure is known. However, there are cases where certain properties in constituents always carry over to a composite, regardless of the microstructure of the composite. We call such instances exact relations. The general theory of exact relations allows us to find all of these instances in a wide variety of contexts including elasticity, conductivity, and piezoelectricity. We combine this theory with ideas from representation theory to find all exact relations for fiber-reinforced polycrystalline composites. We further extend these ideas to the concept of links. When two composites have the same microstructure but different constituent materials, their effective tensors may be related. We use the theory of exact relations to find such relations, which we call links. In this work we describe a special set of links between elasticity tensors of fiber-reinforced polycrystalline composites. These links allow us to generalize certain results from specific examples to generate new information about this widely-used class of composites. In particular, we apply the link to obtain information about composites made from two transversely isotropic materials and polycrystals made from one orthotropic material. / Mathematics

Composite structures, such as laminated beams, plates and shells, are widely used in the automotive, aerospace and marine industries due to their superior specific strength and tailor-able mechanical properties. Because of their use in a wide range of applications, and their commonplace in the engineering design community, the need to accurately predict their behavior to external stimuli is crucial. We consider in this thesis the application of the least-squares finite element method (LSFEM) to problems of static deformations of laminated and sandwich plates and transient plane stress deformations of sandwich beams. Models are derived to express the governing equations of linear elasticity in terms of layer-wise continuous variables for composite plates and beams, which allow inter-laminar continuity conditions at layer interfaces to be satisfied. When Legendre-Gauss-Lobatto (LGL) basis functions with the LGL nodes taken as integration points are used to approximate the unknown field variables, the methodology yields a system of discrete equations with a symmetric positive definite coefficient matrix. The main goal of this research is to determine the efficacy of the LSFEM in accurately predicting stresses in laminated composites when subjected to both quasi-static and transient surface tractions. Convergence of the numerical algorithms with respect to the LGL basis functions in space and time (when applicable) is also considered and explored. In the transient analysis of sandwich beams, we study the sensitivity of the first failure load to the beam's aspect ratio (AR), facesheet-core thickness ratio (FCTR) and facesheet-core stiffness ratio (FCSR). We then explore how failure of sandwich beams is affected by considering facesheet and core materials with different in-plane and transverse stiffness ratios. Computed results are compared to available analytical solutions, published results and those found by using the commercial FE software ABAQUS where appropriate / Master of Science / Composite materials are formed by combining two or more materials on a macroscopic scale such that they have better engineering properties than either material individually. They are usually in the form of a laminate comprised of numerous plies with each ply having unidirectional fibers. Laminates are used in all sorts of engineering applications, ranging from boat hulls, racing car bodies and storage tanks. Unlike their homogeneous material counterparts, such as metals, laminated composites present structural designers and analysts a number of computational challenges. Chief among these challenges is the satisfaction of the so-called continuity conditions, which require certain quantities to be continuous at the interfaces of the composite's layers. In this thesis, we use a mathematical model, called a state-space model, that allows us to simultaneously solve for these quantities in the composite structure's domain and satisfy the continuity conditions at layer interfaces. To solve the governing equations that are derived from this model, we use a numerical technique called the least-squares method which seeks to minimize the squares of the governing equations and the associated side condition residuals over the computational domain. With this mathematical model and numerical method, we investigate static and dynamic deformations of laminated composites structures. The goal of this thesis is to determine the efficacy of the proposed methodology in predicting stresses in laminated composite structures when subjected to static and transient mechanical loading.

The three-dimensional linear theory of piezo-elasticity is used to analyse steady state vibrations of a simply supported rectangular laminated composite plate with piezoelectric (PZT) actuator and sensor patches either embedded in it or bonded to the its surfaces. It is assumed that different layers are perfectly bonded to each other. The method of Fourier series is used to find an analytical solution of the problem. The analytical solution is then applied to study the shape control of a steadily vibrating composite plate by exciting different regions of a PZT actuator. Numerical results for a thin and a thick plate containing one embedded actuator layer and one embedded sensor layer are presented. For the former case, the optimum location of the centroid of the excited rectangular region that will require minimum voltage to control the out-of-plane displacements is determined. Keeping the location of the centroid and the shape of the excited region fixed, we ascertain the voltage required as a function of the length of its diagonal to nullify the deflections of the plate. The maximum shear stress at the interface between the sensor and the lamina is found to be lower than that between the actuator and the lamina. The point of maximum output voltage from the sensor coincides with that of its peak out-of-plane displacement. The variations of displacement and stress components through the thickness for the thin and thick plates are similar. The transient finite deformations of a neo-Hookean beam or plate with PZT patches bonded to its upper and lower surfaces are simulated by the finite element method. The constitutive relation for the piezoelectric material is taken to be linear in the Green-Lagrange strain tensor but quadratic in the driving voltage. A code using 8-noded brick elements has been developed and validated by comparing computed results with either analytical solutions or experimental observations. The code is then used to study flexural waves generated by PZT actuators and propagating through a cantilever beam both with and without a defect in it. The computed results are compared with test observations and with the published results for the linear elastic beam. The effects of both geometrical and material nonlinearities are discussed. A simple feedback control algorithm is shown to annul the motion of a neo-Hookean plate subjected to an impulsive load. / Ph. D.

In this research we consider developing a reservoir simulator capable of simulating complex coupled poromechanical processes on massively parallel computers. A variety of problems arising from petroleum and environmental engineering inherently necessitate the understanding of interactions between fluid flow and solid mechanics. Examples in petroleum engineering include reservoir compaction, wellbore collapse, sand production, and hydraulic fracturing. In environmental engineering, surface subsidence, carbon sequestration, and waste disposal are also coupled poromechanical processes. These economically and environmentally important problems motivate the active pursuit of robust, efficient, and accurate simulation tools for coupled poromechanical problems. Three coupling approaches are currently employed in the reservoir simulation community to solve the poromechanics system, namely, the fully implicit coupling (FIM), the explicit coupling, and the iterative coupling. The choice of the coupling scheme significantly affects the efficiency of the simulator and the accuracy of the solution. We adopt the fixed-stress iterative coupling scheme to solve the coupled system due to its advantages over the other two. Unlike the explicit coupling, the fixed-stress split has been theoretically proven to converge to the FIM for linear poroelasticity model. In addition, it is more efficient and easier to implement than the FIM. Our computational results indicate that this approach is also valid for multiphase flow. We discretize the quasi-static linear elasticity model for geomechanics in space using the continuous Galerkin (CG) finite element method (FEM) on general hexahedral grids. Fluid flow models are discretized by locally mass conservative schemes, specifically, the mixed finite element method (MFE) for the equation of state compositional flow on Cartesian grids and the multipoint flux mixed finite element method (MFMFE) for the single phase and two-phase flows on general hexahedral grids. While both the MFE and the MFMFE generate cell-centered stencils for pressure, the MFMFE has advantages in handling full tensor permeabilities and general geometry and boundary conditions. The MFMFE also obtains accurate fluxes at cell interfaces. These characteristics enable the simulation of more practical problems. For many reservoir simulation applications, for instance, the carbon sequestration simulation, we need to account for thermal effects on the compositional flow phase behavior and the solid structure stress evolution. We explicitly couple the poromechanics equations to a simplified energy conservation equation. A time-split scheme is used to solve heat convection and conduction successively. For the convection equation, a higher order Godunov method is employed to capture the sharp temperature front; for the conduction equation, the MFE is utilized. Simulations of coupled poromechanical or thermoporomechanical processes in field scales with high resolution usually require parallel computing capabilities. The flow models, the geomechanics model, and the thermodynamics model are modularized in the Integrated Parallel Accurate Reservoir Simulator (IPARS) which has been developed at the Center for Subsurface Modeling at the University of Texas at Austin. The IPARS framework handles structured (logically rectangular) grids and was originally designed for element-based data communication, such as the pressure data in the flow models. To parallelize the node-based geomechanics model, we enhance the capabilities of the IPARS framework for node-based data communication. Because the geomechanics linear system is more costly to solve than those of flow and thermodynamics models, the performance of linear solvers for the geomechanics model largely dictates the speed and scalability of the coupled simulator. We use the generalized minimal residual (GMRES) solver with the BoomerAMG preconditioner from the hypre library and the geometric multigrid (GMG) solver from the UG4 software toolbox to solve the geomechanics linear system. Additionally, the multilevel k-way mesh partitioning algorithm from METIS is used to generate high quality mesh partitioning to improve solver performance. Numerical examples of coupled poromechanics and thermoporomechanics simulations are presented to show the capabilities of the coupled simulator in solving practical problems accurately and efficiently. These examples include a real carbon sequestration field case with stress-dependent permeability, a synthetic thermoporoelastic reservoir simulation, poroelasticity simulations on highly distorted hexahedral grids, and parallel scalability tests on a massively parallel computer. / text