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

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.

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.

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

Dans cette thèse, on s'intéresse au comportement effectif non linéaire issu de l'homogénéisation d'un milieu désordonné de fibres dans le plan. A partir d'une hypothèse de déformation homogène, on obtient une loi de comportement non linéaire isotrope par intégration sur les orientations de fibres. La particularité du modèle réside dans la représentation du flambement des fibres par une loi de comportement de type ressort avec raideurs asymétriques en traction/compression. Dans un premier temps, la limite d'une raideur nulle en compression est étudiée numériquement par éléments finis sur des géométries planes fissurées. On montre que le champ de contrainte solution est constitué d'un ensemble de graphe de force de traction en équilibre avec le chargement. En particulier, le caractère unilatéral révèle une interaction forte des fissures : la raideur en traction d'une éprouvette fissuré peut-être nulle quand bien même la percolation des fissures n'est pas atteinte. Par ailleurs, on montre que le champ de contrainte est partitionné en régions auto-équilibrées et libres de contrainte Pour finir, on propose une application au calcul des bornes énergétiques de la solution élastique d'un problème aux limites. En perspective, on met en évidence le caractère bi-module d'un voile de fibres de verre avec des mesures de champs de déplacement par corrélations d'images. Un dispositif d'Arcan est mis en place pour exercer un chargement anisotrope sur le voile. Les champs de déplacement mesurés sont utilisés pour identifier une loi de comportement du matériau. Un résultats préliminaire montre que le caractère unilatéral du voile est d'autant plus prononcé que sa densité est faible. / An homogenized non linear mechanical behavior of a 2D disordered fiber mats is considered. Under the homogeneous strain assumptions, a continuum description of an isotropic non linear media is derived by integration over all fibers orientations. The fiber non linearity due to buckling is featured by a lower spring constant in compression than in traction. First, the response of a sheet containing multiple cracks with different geometries is explored numerically in the case of a tension-field material, corresponding to the singular limit where the compressive spring constants goes to zero. It is reported that the stress solution is composed of a set of pattern built from tensile branches in equilibrium with the boundary conditions. In particular, the unilateral feature reveal a strong interaction between multiple cracks : the tensile stiffness of a cracked sheet can be zero even though the sheet is made up of a single piece. Besides the pattern of tensile branches, it is reported a partitioning of the stress into self-equilibrated and stress free regions. Finally, an application is proposed to find good estimates of the energetic bounds of the boundary value problem. As an experimental perspective to this work, it is proposed to estimates the bi-modulus feature of a manufactured fiber glass veil using digital image correlations. An Arcan device is set up to enforce an anisotropic loading on the veil. Measured displacement fields are used to identify a material constitutive law. As a preliminary results, it is reported that the unilateral feature is noticeable in light weight fiber glass veil.

Plusieurs lois de comportement mécaniques possèdent une formulation tensorielle, comme c'est le cas pour l'élasticité où intervient un espace de tenseurs d'ordre 4, noté Ela. La classification des matériaux élastiques passent par la nécessité de décrire l'espace des orbites ELA/SO(3). Plus généralement, on étudie la géométrie d'un espace de tenseurs sur $mathbb{R}^{3}$, via l'action du groupe O(3). Cette géométrie est caractérisée par ses classes d'isotropies, ou encore classes de symétries. Tout espace de tenseurs possède en effet un nombre fini de classes d'isotropies. Nous proposons alors une méthode originale et générale pour obtenir ces classes d'istropie. Nous avons ainsi pu obtenir pour la première fois les classes d'isotropie d'un espace de tenseurs d'ordre 8 intervenant en théorie de l'élasticité linéaire du second-gradient de la déformation.Pour une représentation réelle d'un groupe compact, l'algèbre des polynômes invariants sépare les orbites, d'où la recherche d'une famille génératrice minimale de cette algèbre. Pour cela, on exploitant le lien entre les espaces de tenseurs et les espaces de formes binaires. Nous avons ainsi repris et ré-interprété les approches effectives de cette théorie, développées par Gordan au 19ième siècle. Cette ré-interprétation nous a permis d'obtenir de nombreux résultats, dont une famille génératrice minimale d'invariants pour l'élasticité mais aussi pour la piézoélectricté. Nous avons pu retrouver d'une façon simple les séries de Gordan, ainsi que des relations plus récentes d'Abdesselam--Chipalkatti sur les transvectants de formes binaires. / Tensorial formulation of mechanical constitutive equations is a very important matter in continuum mechanics. For instance, the space of elastic tensors is a subspace of 4th order tensors with a natural SO(3) group action. More generaly, we have to study the geometry of a tensor space defined on $mathbb{R}^{3}$, under O(3) group action.To describe such a geometry, we first have to exhibit its isotropy classes, also named symetry classes. Indeed, each tensor space possesses a finite number of isotropy classes. In this present work, we propose an original method to obtain isotropy classes of a given tensor space. As an illustration of this new method, we get for the first time the isotropy classes of a 8th order tensor space occuring in second strain-gradient elasticity theory. In the case of a real representation of a compact group, invariant algebra seperates the orbits. This observation motivates the purpose to find a finite generating set of polynomial invariants. For that purpose, we make use of the link between tensor spaces and spaces of binary forms, which belongs to the classical invariant theory. We thus have to deal with SL(2,$mathbb{C}$) group action. To obtain new results, we have reformulated and reinterpreted effective approaches of Gordan's algorithm, developped during XIXth century. We then obtain for the first time a minimal generating family of elasticity tensor space, and a generating family of piezoelectricity tensor space. Using linear algebra arguments, we were also able to get important relations of classical invariant theory, such as the Gordan's series and the Abdesselam--Chipalkatti's quadratic relations on transvectants.

D’un point de vue mécanique, la peau est une structure multicouche complexeayant des propriétés viscoélastique, non-linéaire, quasi-incompressible, anisotrope eten état de précontrainte. Le travail présenté dans cette thèse associe expérimentation,modélisation et identification numérique et se distingue en particulier parl’utilisation d’un dispositif d’extensométrie développé au laboratoire et adapté à desmesures in vivo non invasives. Des tests ex vivo ont cependant été réalisés égalementà titre de comparaison et de validation. Une attention particulière a été portée à latension cutanée initiale (ou naturelle). Les essais in vivo ont permis d’obtenir desréponses force – déplacement sous différentes configurations angulaires, d’intensitéet pour diverses localisations corporelles. Les essais ex vivo ont quant à eux permisd’estimer l’état de contrainte initiale par la mesure des forces nécessaires à la remiseen tension d’explants. Ces différents essais expérimentaux ont été modélisés en utilisantdeux lois de comportement : la loi d’Ogden du premier ordre permettant dedécrire un comportement hyperélastique isotrope et la loi d’Holzapfel-Gasser-Ogden(HGO) décrivant un comportement hyper élastique anisotrope. Cette dernière a étéimplémentée sous l’interface utilisateur du logiciel ANSYS. Les paramètres caractéristiquesdes zones cutanées testées ont été identifiés par méthode inverse. L’influencede la compressibilité de la peau sur son comportement mécanique est mise en évidence.Au final, les travaux de cette thèse ont été appliqués au lambeau d’avancementde type V-Y qui est une technique de suture pratiquée pour combler les pertes desubstance.229 / From a mechanical point of view, the human skin is a complex multilayerstructure with viscoelastic, non-linear and anisotropic properties and a pre-stressstate. The work presented in this thesis combines experimentation, modeling andnumerical identification and distinguishes especially by the use of an extensometerdevice developed in the laboratory and suitable for non-invasive in vivo measurements.Ex vivo tests were however also performed for comparison and validation.Particular attention was paid to the initial skin tension. in vivo tests allowed theobtaining of load – displacement responses for different angular configurations, intensitiesand body locations. ex vivo tests in turn allowed the estimation of the stateof initial stress by measuring the forces necessary for the re-tension of the explants.These different experimental tests were modeled using two constitutive laws : thefirst order Ogden law allowing the description of an isotropic hyperelastic behavior,and the Holzapfel-Gasser-Ogden’ law (HGO) allowing the description of an anisotropichyperelastic behavior. The latter was implemented in the user interface ofANSYS software. The characteristics parameters of the skin areas tested were identifiedby the reverse method. The influence of the compressibility of the skin on itsmechanical behavior is highlighted. Finally, the work of this thesis were applied toan advancement flap of V-Y type.