In der vorliegenden Arbeit wird ein gemischtes Element gegen Locking-Effekte untersucht. Dazu wird ein Fünf-Feld-Hu-Washizu-Funktional (CoFEM-Element) für lineare und quadratische Hexaeder-Elemente unter einer hyperelastischen, isotropen, polykonvexen sowie einer transversal-isotropen Materialformulierung implementiert. Die resultierenden nichtlinearen Gleichungen werden mithilfe eines Mehrebenen-NEWTON-RAPHSON-Verfahren unter Beachtung einer konsistenten Linearisierung gelöst. Als repräsentatives Beispiel der numerischen Untersuchungen dient der einseitig eingespannte Cook-Balken mit einer quadratischen Druckverteilung am Rand. Zur Beurteilung des CoFEM-Elements wird das räumliche Konvergenzverhalten für unterschiedliche Polynomgrade und für verschiedene Netze unter Beachtung der algorithmischen Effizienz untersucht. / This paper presents a mixed finite element formulation of Hu-Washizu type (CoFEM) designed to reduce locking effects with respect to a linear and quadratic approximation in space. We consider a hyperelastic, isotropic, polyconvex material formulation as well as transverse isotropy. The resulting nonlinear algebraic equations are solved with a multilevel NEWTON-RAPHSON method. As a numerical example serves a cook-like cantilever beam with a quadratic distribution of in-plane load on the Neumann boundary. We analyze the spatial convergence with respect to the polynomial degree of the underlying Lagrange polynomials and with respect to the level of mesh refinement in terms of algorithmic efficiency.

The variation of rock strength and stiffness, known as mechanical anisotropy, is expected at different scales: large (rock mass) - or small (intact rock) - scales. It is always mandatory for engineering applications built either on or in anisotropic rock masses to investigate the strength and deformation behavior of those masses. To achieve this goal, continuum-based constitutive models are presented to analyze the mechanical anisotropy. One of both implemented models is named ‘Transubi model’ which considers the transverse isotropic elasticity into bi-linear Mohr-Coulomb strain hardening/softening plastic framework. Experimental investigations and numerical simulations focused mainly on the influence of the mechanical anisotropy on the plastic zoning around excavated openings in laminated rocks. Later, the Transubi model was applied to a tunnel excavated in a shaly facies formation of bedded argillaceous Opalinus clay in an URL (FE-tunnel) to obtain the short-term stability insights. Overall, the research outcomes may have a prospective impact regarding the understanding of anisotropy of laminated, bedded and foliated rocks which improves the deformation behaviour predictability using continuum-based numerical modeling tools.

The present work describes investigations on the anisotropic strength behavior of rocks in the splitting tensile test (Brazilian test). Three transversely isotropic rocks (gneiss, slate and sandstone) were studied in the Lab. A total of more than 550 indirect tensile strength tests were conducted, with emphasis was placed on the investigation of the influence of the spatial position of anisotropic weakness plane to the direction of the load on the fracture strength and fracture or fracture mode. In parallel, analytical solutions were evaluated for stress distribution and developed 3D numerical models to study the stress distribution and the fracture mode at the transversely isotropic disc. There were new findings on the fracture mode of crack propagation, the influence of the disc thickness, the influence of the applying loading angle and angle of the loading-foliation for transversely isotropic material. / Inhalt der Arbeit sind Untersuchungen zum anisotropen Festigkeitsverhalten von Gesteinen beim Spaltzugversuch (Brazilian Test). Laborativ wurden drei transversalisotrope Gesteine (Granit, Schiefer und Sandstein) untersucht. Insgesamt wurden mehr als 550 Spaltzugversuche durchgeführt, wobei der Schwerpunkt auf die Untersuchung des Einflusses der räumlichen Lage der Anisotropieebene zur Richtung des Lasteintrages auf die Bruchfestigkeit und das Bruchbild bzw. den Bruchmodus gelegt wurde. Parallel dazu wurden analytische Lösungen zur Spannungsverteilung ausgewertet sowie numerische 3D-Modelle entwickelt, um die Spannungsverteilung sowie den Bruchmodus bei einer transversalisotropen Scheibe zu untersuchen. Es wurden neue Erkenntnisse zum Bruchmodus, der Rissausbreitung, des Einflusses der Scheibendicke, dem Einfluss des Lasteinleitungswinkel sowie des Winkels Lasteintrag - Anisotropieebene für transversalisotropes Material gewonnen.

This thesis covers different problems that arise due to crystal and pseudospin anisotropy present in 3d and 5d transition metal oxides. We demonstrate that the methods of computational quantum chemistry can be fruitfully used for quantitative studies of such problems. In Chapter 2, Chapter 3, and Chapter 7 we show that it is possible to reliably calculate local multiplet splittings fully ab initio, and therefore help to assign peaks in experimental spectra to corresponding electronic states. In a situation of large number of peaks due to low local symmetry such assignment using semi-empirical methods can be very tedious and non-unique. Moreover, in Chapter 4 we present a computational scheme for calculating intensities as observed in the resonant inelastic X-ray scattering and X-ray absorption experiments. In our scheme highly-excited core-hole states are calculated explicitly taking into account corresponding orbital relaxation and electron polarization. Computed Cu L-edge spectra for the Li2CuO2 compound reproduce all features present in experiment. Unbiased ab initio calculations allow us to unravel a delicate interplay between the distortion of the local ligand cage around the transition metal ions and the anisotropic electrostatic interactions due to second and farther coordination shells. As shown in Chapter 5 and Chapter 6 this interplay can lead to the counter intuitive multiplet structure, single-ion anisotropy, and magnetic g factors. The effect is quite general and may occur in compounds with large difference between charges of metal ions that form anisotropic environment around the transition metal, like Ir 4+ in plane versus Sr 2+ out of plane in the case of Sr2IrO4. An important aspect of the presented study is the mapping of the quantum chemistry results onto simpler physical models, namely extended Heisenberg model, providing an ab initio parametrization. In Chapter 5 we employ the effective Hamiltonian technique for extracting parameters of the anisotropic Heisenberg model with single-ion anisotropy in the case of quenched orbital moment and second-order spin-orbit coupling. Calculated strong easy-axis anisotropy of the same order of magnitude as the symmetric exchange is consistent with experimentally-observer all-in/all-out magnetic order. In Chapter 6 we introduce new flavour of the mapping procedure applicable to systems with first-order spin-orbit coupling, such as 5d 5 iridates based on analysis of the wavefunction and interaction with magnetic field. In Chapter 6 and Chapter 7 we use this new procedure to obtain parameters of the pseudospin anisotropic Heisenberg model. We find large antisymmetric exchange leading to the canted antiferromagnetic state in Sr2IrO4 and nearly ideal one-dimensional Heisenberg behaviour of the CaIrO3, both agree very well with experimental findings.

The acoustics of small cavities raises interest of the scientific community since it involves particular damping mechanisms. In fluid dynamics, when a small perturbation is propagating within a Newtonian and heat-conducting fluid bounded by a rigid and isothermal surface, viscous and thermal dissipative mechanisms are generated near the walls. Such effects can have significant impact on the acoustic behaviour of the system.<p>Several types of practical applications can be cited, among which: hearing aids, micro-electro-mechanical systems (transducers, microphones and loud-speakers), absorbing materials made of thin capillary nets or small pores, dissipative silencers, thermo-acoustic heat exchangers, or any kind of device bringing into play small resonant cavities filled with a dissipative fluid (micro-acoustics).<p><p>This study focuses on appropriated reductions of the physical equations, in order to enhance the efficiency of the numerical resolution without adversely affecting the accuracy. Moreover, the proposed strategies lead to numerically stable systems as they involve only one scalar partial order differential equation (or equivalent fluid equation). The emphasis is put on the physical aspect of those reductions, their range of applicability, benefits and drawbacks.<p>Two new reduced models are proposed to estimate the visco-thermal acoustic wave propagation. A first extension deals with waveguide geometries and relax the hypothesis of the fluid at rest. The second original formulation addresses visco-thermal acoustics in 3D arbitrary geometries. This model is based on different considerations coming from existing techniques as well as the estimation of a wall-distance field.<p><p>A second part aims at studying the acoustic behaviour of biphasic materials and more specifically poro-elastic materials. This type of acoustic component is widely used in industry because of their good absorbing properties in the medium- and high-frequency <p>ranges.<p>A preliminary bibliographic research deals with the derivation of the set of partial order differential equations that account for both fluid/structure interactions and the anisotropy of a given poro-elastic material. It has been shown that transversely orientated capillary materials (for instance catalyst substrates) can be simulated using the proposed reduction technique.<p>At last, the modelling of the acoustic transmission between two domains separated by perforated or micro-perforated plates or thin plates of poro-elastic materials is discussed. The analogy between the rigid perforated plate models with an equivalent fluid formulation has been presented. As a result, this model has been extended in order to account for flexural effects of the solid part.<p><p><p>Ce travail porte sur l'étude de certains phénomènes d'amortissements intervenant dans l'acoustique des petites cavités. En méchanique des fluides, lorsqu'une petite perturbation se propage au sein d'un fluide newtonien et caloporteur borné par un mur rigide et isotherme, ces mécanismes dissipatifs particuliers se localisent aux abords des parois et jouent un rôle significatif dans certaines situations.<p>Parmi les exemples d'applications pratiques, il est possible de citer les appareils d'aide auditive, les systèmes microélectromécaniques (transducteurs, microphones et haut-parleurs), les matériaux absorbants constitués de fins réseaux capillaires ou de pores aux dimensions réduites, les systèmes de silencieux, d'échangeurs de chaleur thermo-acoustiques ou tout autre appareil mettant en jeu des cavités résonantes aux dimensions réduites (micro-acoustique).<p><p>L'étude proposée ici se focalise sur des stratégies de réduction appropriées des équations physiques, ceci afin d'améliorer l'efficacité du modèle tout en conservant une précision acceptable. Les techniques présentées aboutissent à des systèmes numériquement stables mettant en jeu une seule équation scalaire (ou équation fluide équivalent). Ainsi, l'accent est porté sur l'aspect physique des réductions, leurs domaines d'application, avantages et inconvénients.<p>Deux modèles originaux sont proposés afin de prédire la propagation acoustique visco-thermique. Une première extension permet d'évaluer la pression acoustique au sein de géométries particulières de type guides d'onde en présence d'un écoulement hydrodynamique. La seconde formulation présentée s'intéresse à l'acoustique dans des domaines 3D arbitraires. Cette méthode se base sur des considérations conjointes de modèles réduits existants ainsi que sur l'estimation d'un champ de distance à la plus proche paroi.<p><p>Dans une seconde partie, nous nous proposons d'étudier le comportement acoustique de matériaux biphasique et plus précisément les matériaux poro-élastiques (très utilisés dans l'industrie en raison de leurs caractéristiques absorbantes dans les domaines des moyennes et hautes fréquences).<p>Une étude bibliographique préliminaire nous a permis d'exprimer l'ensemble des équations aux dérivées <p>partielles modélisant à la fois les interactions fluide/structure et l'anisotropie générale des matériaux. <p>Cette réflexion nous a permis d'aboutir à un modèle de matériau isotrope transverse intéressant, combinant le modèle fluide proposé et la formulation acousto-élastique équivalente. Ainsi la modélisation de structures capillaires orientées (comme les matériaux utilisés dans les catalyseurs automobiles) s'en trouve grandement simplifiée.<p>Enfin, la transmission acoustique intervenant entre deux domaines fluides séparés par une plaque perforée ou micro-perforée ou bien une couche de matériau poreux a été étudiée. L'analogie entre les modèlisations existantes et un modèle générique de fluide équivalent a été mise en évidence. Pour finir, cette formulation a été étendue afin de prendre en compte les effets de flexion de la partie solide. / Doctorat en Sciences de l'ingénieur / info:eu-repo/semantics/nonPublished

This thesis covers different problems that arise due to crystal and pseudospin anisotropy present in 3d and 5d transition metal oxides. We demonstrate that the methods of computational quantum chemistry can be fruitfully used for quantitative studies of such problems. In Chapter 2, Chapter 3, and Chapter 7 we show that it is possible to reliably calculate local multiplet splittings fully ab initio, and therefore help to assign peaks in experimental spectra to corresponding electronic states. In a situation of large number of peaks due to low local symmetry such assignment using semi-empirical methods can be very tedious and non-unique. Moreover, in Chapter 4 we present a computational scheme for calculating intensities as observed in the resonant inelastic X-ray scattering and X-ray absorption experiments. In our scheme highly-excited core-hole states are calculated explicitly taking into account corresponding orbital relaxation and electron polarization. Computed Cu L-edge spectra for the Li2CuO2 compound reproduce all features present in experiment. Unbiased ab initio calculations allow us to unravel a delicate interplay between the distortion of the local ligand cage around the transition metal ions and the anisotropic electrostatic interactions due to second and farther coordination shells. As shown in Chapter 5 and Chapter 6 this interplay can lead to the counter intuitive multiplet structure, single-ion anisotropy, and magnetic g factors. The effect is quite general and may occur in compounds with large difference between charges of metal ions that form anisotropic environment around the transition metal, like Ir 4+ in plane versus Sr 2+ out of plane in the case of Sr2IrO4. An important aspect of the presented study is the mapping of the quantum chemistry results onto simpler physical models, namely extended Heisenberg model, providing an ab initio parametrization. In Chapter 5 we employ the effective Hamiltonian technique for extracting parameters of the anisotropic Heisenberg model with single-ion anisotropy in the case of quenched orbital moment and second-order spin-orbit coupling. Calculated strong easy-axis anisotropy of the same order of magnitude as the symmetric exchange is consistent with experimentally-observer all-in/all-out magnetic order. In Chapter 6 we introduce new flavour of the mapping procedure applicable to systems with first-order spin-orbit coupling, such as 5d 5 iridates based on analysis of the wavefunction and interaction with magnetic field. In Chapter 6 and Chapter 7 we use this new procedure to obtain parameters of the pseudospin anisotropic Heisenberg model. We find large antisymmetric exchange leading to the canted antiferromagnetic state in Sr2IrO4 and nearly ideal one-dimensional Heisenberg behaviour of the CaIrO3, both agree very well with experimental findings.

In der vorliegenden Arbeit wird ein gemischtes Element gegen Locking-Effekte untersucht. Dazu wird ein Fünf-Feld-Hu-Washizu-Funktional (CoFEM-Element) für lineare und quadratische Hexaeder-Elemente unter einer hyperelastischen, isotropen, polykonvexen sowie einer transversal-isotropen Materialformulierung implementiert. Die resultierenden nichtlinearen Gleichungen werden mithilfe eines Mehrebenen-NEWTON-RAPHSON-Verfahren unter Beachtung einer konsistenten Linearisierung gelöst. Als repräsentatives Beispiel der numerischen Untersuchungen dient der einseitig eingespannte Cook-Balken mit einer quadratischen Druckverteilung am Rand. Zur Beurteilung des CoFEM-Elements wird das räumliche Konvergenzverhalten für unterschiedliche Polynomgrade und für verschiedene Netze unter Beachtung der algorithmischen Effizienz untersucht. / This paper presents a mixed finite element formulation of Hu-Washizu type (CoFEM) designed to reduce locking effects with respect to a linear and quadratic approximation in space. We consider a hyperelastic, isotropic, polyconvex material formulation as well as transverse isotropy. The resulting nonlinear algebraic equations are solved with a multilevel NEWTON-RAPHSON method. As a numerical example serves a cook-like cantilever beam with a quadratic distribution of in-plane load on the Neumann boundary. We analyze the spatial convergence with respect to the polynomial degree of the underlying Lagrange polynomials and with respect to the level of mesh refinement in terms of algorithmic efficiency.

The present work describes investigations on the anisotropic strength behavior of rocks in the splitting tensile test (Brazilian test). Three transversely isotropic rocks (gneiss, slate and sandstone) were studied in the Lab. A total of more than 550 indirect tensile strength tests were conducted, with emphasis was placed on the investigation of the influence of the spatial position of anisotropic weakness plane to the direction of the load on the fracture strength and fracture or fracture mode. In parallel, analytical solutions were evaluated for stress distribution and developed 3D numerical models to study the stress distribution and the fracture mode at the transversely isotropic disc. There were new findings on the fracture mode of crack propagation, the influence of the disc thickness, the influence of the applying loading angle and angle of the loading-foliation for transversely isotropic material.:ACKNOWLEDGMENTS 5 ABSTRACT 7 TABLE OF CONTENTS 9 LIST OF FIGURES 13 LIST OF TABLES 19 I. INTRODUCTION 21 Objective of this work 22 Scope of work 23 Research procedure 23 Significance of the work 24 Layout 24 1 STATE OF THE ART 27 1.1 Review of the Brazilian tensile strength test 27 1.1.1 General overview 27 1.1.2 Development of the Brazilian tensile strength test 29 1.1.3 The Brazilian tensile strength test on anisotropic rocks 31 1.1.4 Summary 32 1.2 Analytical aspects 33 1.2.1 Hypotheses for the conventional Brazilian test 34 1.2.2 Failure criteria 36 1.2.3 Crack initiation and propagation 39 1.2.4 Summary 41 1.3 Numerical considerations 41 1.3.1 Numerical methods 42 1.3.2 Summary 42 1.4 Conclusion 43 2 DIAMETRAL COMPRESSION IN A SOLID DISC – COMPILATION OF ANALYTICAL AND SEMI-ANALYTICAL SOLUTIONS 45 2.1 Introduction 45 2.2 Diametral compressive stress distribution in an isotropic elastic disc 45 2.2.1 Elastic theory of line load 46 2.2.2 2D analytical solutions 47 2.2.3 3D disc under line and diametral compressive distributed loads 55 2.2.4 3D solution under diametral compressive distributed load 56 2.3 Stress and strain in an isotropic solid disc 59 2.4 Stress and strain in anisotropic rocks 61 2.5 Conclusion 65 3 LABORATORY TESTS 69 3.1 Introduction 69 3.2 Laboratory test program 70 3.3 Sample preparation 71 3.4 Ultrasonic measurements 72 3.5 Uniaxial and triaxial compression tests 73 3.5.1 Uniaxial compression test 73 3.5.2 Triaxial compression tests 74 3.6 Brazilian tensile strength tests 76 3.6.1 Test apparatus 76 3.6.2 Laboratory test results 77 3.6.3 Interpretation of the test results 89 3.7 Conclusion 96 4 NUMERICAL SIMULATION OF ISOTROPIC MATERIALS - COMPARISON WITH ANALYTICAL SOLUTIONS 97 4.1 Introduction 97 4.2 Numerical simulation of isotropic materials 97 4.2.1 FLAC3D simulation program 97 4.2.2 Simulation procedure 98 4.2.3 Numerical model setup 98 4.2.4 Influence of mesh type 99 4.2.5 Influence of specimen thickness 100 4.2.6 Influence of Poisson’s ratio 102 4.2.7 Influence of loading angle (2) 106 4.2.8 Comparison of 3D analytical and numerical results 110 4.2.9 Influence of stress concentration at the loading jaws 112 4.3 Comparison with experimental results of Postaer Sandstone (FG.Ss) 112 4.4 Conclusion 114 5 NUMERICAL SIMULATION OF ANISOTROPIC MATERIALS - COMPARISON WITH LABORATORY TESTS 117 5.1 Introduction 117 5.2 General procedure for simulating the Brazilian test using FLAC3D 117 5.2.1 Conceptual model 119 5.2.2 Boundary Conditions 119 5.2.3 Numerical model set-up 120 5.3 Constitutive model 121 5.3.1 Choice of constitutive model 121 5.3.2 Bilinear Strain-Hardening/Softening Ubiquitous-Joint Model [98] 121 5.4 Parameter calibration 124 5.4.1 Material parameters used 124 5.4.2 Contact between disc and loading jaws 126 5.4.3 Post-failure deformation properties 128 5.4.4 Tension cut-off 129 5.5 Numerical simulation results 131 5.5.1 Introduction 131 5.5.2 Stress distribution and failure state 133 5.5.3 Stress state in an isotropic elastic medium with arbitrary orientation planes 136 5.5.4 Plasticity states 139 5.5.5 Damage and fracture process 141 5.5.6 Fracture patterns – Comparison of lab results and numerical simulations 148 5.6 Tensile strength – Comparison of lab results and numerical simulations 149 5.6.1 Tensile strength of Le.Gs Gneiss 150 5.6.2 Tensile strength of My.Sc Slate 155 5.7 Summary and Review 159 5.7.1 Potential failure state deduced from pure elastic considerations 159 5.7.2 Tensile strength distribution 160 5.7.3 Tensile strength – determining the anisotropy factor 161 5.7.4 Tensile strength – different procedures - different results 163 6 CONCLUSION AND RECOMMENDATIONS 165 APPENDICES 171 Appendix 3.1 - Fracture patterns in FG.Ss samples 171 Appendix 3.2 - Fracture patterns in FG.Gs samples 177 Appendix 3.3 - Fracture patterns in Le.Gs samples 183 Appendix 3.4 - Fracture patterns in My.Sc samples 190 Appendix 4.1 - Influence of loading angle 197 Appendix 4.2 - Influence of material properties 203 Appendix 5.1 - Failure zone state in Le.Gs Gneiss 209 Appendix 5.2: Failure zone state in My.Sc Slate 216 REFERENCES 223 / Inhalt der Arbeit sind Untersuchungen zum anisotropen Festigkeitsverhalten von Gesteinen beim Spaltzugversuch (Brazilian Test). Laborativ wurden drei transversalisotrope Gesteine (Granit, Schiefer und Sandstein) untersucht. Insgesamt wurden mehr als 550 Spaltzugversuche durchgeführt, wobei der Schwerpunkt auf die Untersuchung des Einflusses der räumlichen Lage der Anisotropieebene zur Richtung des Lasteintrages auf die Bruchfestigkeit und das Bruchbild bzw. den Bruchmodus gelegt wurde. Parallel dazu wurden analytische Lösungen zur Spannungsverteilung ausgewertet sowie numerische 3D-Modelle entwickelt, um die Spannungsverteilung sowie den Bruchmodus bei einer transversalisotropen Scheibe zu untersuchen. Es wurden neue Erkenntnisse zum Bruchmodus, der Rissausbreitung, des Einflusses der Scheibendicke, dem Einfluss des Lasteinleitungswinkel sowie des Winkels Lasteintrag - Anisotropieebene für transversalisotropes Material gewonnen.:ACKNOWLEDGMENTS 5 ABSTRACT 7 TABLE OF CONTENTS 9 LIST OF FIGURES 13 LIST OF TABLES 19 I. INTRODUCTION 21 Objective of this work 22 Scope of work 23 Research procedure 23 Significance of the work 24 Layout 24 1 STATE OF THE ART 27 1.1 Review of the Brazilian tensile strength test 27 1.1.1 General overview 27 1.1.2 Development of the Brazilian tensile strength test 29 1.1.3 The Brazilian tensile strength test on anisotropic rocks 31 1.1.4 Summary 32 1.2 Analytical aspects 33 1.2.1 Hypotheses for the conventional Brazilian test 34 1.2.2 Failure criteria 36 1.2.3 Crack initiation and propagation 39 1.2.4 Summary 41 1.3 Numerical considerations 41 1.3.1 Numerical methods 42 1.3.2 Summary 42 1.4 Conclusion 43 2 DIAMETRAL COMPRESSION IN A SOLID DISC – COMPILATION OF ANALYTICAL AND SEMI-ANALYTICAL SOLUTIONS 45 2.1 Introduction 45 2.2 Diametral compressive stress distribution in an isotropic elastic disc 45 2.2.1 Elastic theory of line load 46 2.2.2 2D analytical solutions 47 2.2.3 3D disc under line and diametral compressive distributed loads 55 2.2.4 3D solution under diametral compressive distributed load 56 2.3 Stress and strain in an isotropic solid disc 59 2.4 Stress and strain in anisotropic rocks 61 2.5 Conclusion 65 3 LABORATORY TESTS 69 3.1 Introduction 69 3.2 Laboratory test program 70 3.3 Sample preparation 71 3.4 Ultrasonic measurements 72 3.5 Uniaxial and triaxial compression tests 73 3.5.1 Uniaxial compression test 73 3.5.2 Triaxial compression tests 74 3.6 Brazilian tensile strength tests 76 3.6.1 Test apparatus 76 3.6.2 Laboratory test results 77 3.6.3 Interpretation of the test results 89 3.7 Conclusion 96 4 NUMERICAL SIMULATION OF ISOTROPIC MATERIALS - COMPARISON WITH ANALYTICAL SOLUTIONS 97 4.1 Introduction 97 4.2 Numerical simulation of isotropic materials 97 4.2.1 FLAC3D simulation program 97 4.2.2 Simulation procedure 98 4.2.3 Numerical model setup 98 4.2.4 Influence of mesh type 99 4.2.5 Influence of specimen thickness 100 4.2.6 Influence of Poisson’s ratio 102 4.2.7 Influence of loading angle (2) 106 4.2.8 Comparison of 3D analytical and numerical results 110 4.2.9 Influence of stress concentration at the loading jaws 112 4.3 Comparison with experimental results of Postaer Sandstone (FG.Ss) 112 4.4 Conclusion 114 5 NUMERICAL SIMULATION OF ANISOTROPIC MATERIALS - COMPARISON WITH LABORATORY TESTS 117 5.1 Introduction 117 5.2 General procedure for simulating the Brazilian test using FLAC3D 117 5.2.1 Conceptual model 119 5.2.2 Boundary Conditions 119 5.2.3 Numerical model set-up 120 5.3 Constitutive model 121 5.3.1 Choice of constitutive model 121 5.3.2 Bilinear Strain-Hardening/Softening Ubiquitous-Joint Model [98] 121 5.4 Parameter calibration 124 5.4.1 Material parameters used 124 5.4.2 Contact between disc and loading jaws 126 5.4.3 Post-failure deformation properties 128 5.4.4 Tension cut-off 129 5.5 Numerical simulation results 131 5.5.1 Introduction 131 5.5.2 Stress distribution and failure state 133 5.5.3 Stress state in an isotropic elastic medium with arbitrary orientation planes 136 5.5.4 Plasticity states 139 5.5.5 Damage and fracture process 141 5.5.6 Fracture patterns – Comparison of lab results and numerical simulations 148 5.6 Tensile strength – Comparison of lab results and numerical simulations 149 5.6.1 Tensile strength of Le.Gs Gneiss 150 5.6.2 Tensile strength of My.Sc Slate 155 5.7 Summary and Review 159 5.7.1 Potential failure state deduced from pure elastic considerations 159 5.7.2 Tensile strength distribution 160 5.7.3 Tensile strength – determining the anisotropy factor 161 5.7.4 Tensile strength – different procedures - different results 163 6 CONCLUSION AND RECOMMENDATIONS 165 APPENDICES 171 Appendix 3.1 - Fracture patterns in FG.Ss samples 171 Appendix 3.2 - Fracture patterns in FG.Gs samples 177 Appendix 3.3 - Fracture patterns in Le.Gs samples 183 Appendix 3.4 - Fracture patterns in My.Sc samples 190 Appendix 4.1 - Influence of loading angle 197 Appendix 4.2 - Influence of material properties 203 Appendix 5.1 - Failure zone state in Le.Gs Gneiss 209 Appendix 5.2: Failure zone state in My.Sc Slate 216 REFERENCES 223

This thesis provides a feasibility study on the determination of formation anisotropy parameters from logging-while-drilling (LWD) borehole acoustic measurements. For this reason, the wave propagation in fluid-filled boreholes surrounded by transverse isotropic (TI) formations is investigated in great detail using the finite-difference method. While the focus is put on quadrupole waves, the sensitivities of monopole and flexural waves are evaluated as well. All three wave types are considered with/without the presence of an LWD tool. Moreover, anisotropy-induced mode contaminants are discussed for various TI configurations. In addition, the well-known plane wave Alford rotation has been generalized to cylindrical borehole waves of any order, except for the monopole. This formulation has been extended to allow for non-orthogonal multipole firings, and associated inversion methods have been developed to compute formation shear principal velocities and accompanying polarization directions, utilizing various LWD (cross-) quadrupole measurements.:1 Introduction 1.1 Borehole acoustic configurations 1.2 Wave propagation in a fluid-filled borehole in the absence of a logging tool 1.3 Wave propagation in a fluid-filled borehole in the presence of a logging tool 1.4 Anisotropy 2 Theory 2.1 Stiffness and compliance tensor 2.1.1 Triclinic symmetry 2.1.2 Monoclinic symmetry 2.1.3 Orthotropic symmetry 2.1.4 Transverse isotropic (TI) symmetry 2.1.5 Isotropy 2.2 Reference frames 2.3 Seismic wave equations for a linear elastic, anisotropic medium 2.3.1 Basic equations 2.3.2 Integral transforms 2.3.3 Christoffel equation 2.3.4 Phase slowness surfaces 2.3.5 Group velocity 2.4 Solution in cylindrical coordinates for the borehole geometry 2.4.1 Special case: vertical transverse isotropy (VTI) 2.4.2 General case: triclinic symmetry 3 Finite-difference modeling of wave propagation in anisotropic media 3.1 Finite-difference method 3.2 Spatial finite-difference grids 3.2.1 Standard staggered grid 3.2.2 Lebedev grid 3.3 Heterogeneous media 3.4 Finite-difference properties and grid dispersion 3.5 Initial conditions 3.6 Boundary conditions 3.7 Parallelization 3.8 Finite-difference parameters 4 Wave propagation in fluid-filled boreholes surrounded by TI media 4.1 Vertical transverse isotropy (VTI) 4.1.1 Monopole excitation 4.1.2 Dipole excitation 4.1.3 Quadrupole excitation 4.1.4 Summary 4.2 Horizontal transverse isotropy (HTI) 4.2.1 Monopole excitation 4.2.2 Theory of cross-multipole shear wave splitting 4.2.3 Dipole excitation 4.2.4 Quadrupole excitation 4.2.5 Hexapole waves 4.2.6 Summary 4.3 Tilted transverse isotropy (TTI) 4.3.1 Monopole excitation 4.3.2 Dipole excitation 4.3.3 Quadrupole excitation 4.3.4 Summary 4.4 Anisotropy-induced mode contaminants 4.4.1 Vertical transverse isotropy (VTI) 4.4.2 Horizontal transverse isotropy (HTI) 4.4.3 Tilted transverse isotropy (TTI) 4.4.4 Summary 5 Inversion methods 5.1 Vertical transverse isotropy (VTI) 5.2 Horizontal transverse isotropy (HTI) 5.2.1 Inverse generalized Alford rotation 5.2.2 Inversion method based on dipole excitations 5.2.3 Inversion method based on quadrupole excitations 5.3 Tilted transverse isotropy (TTI) 5.4 Challenges in real measurements 5.4.1 Signal-to-noise ratio (SNR) 5.4.2 Tool eccentricity 6 Conclusions References List of Abbreviations and Symbols List of Figures List of Tables A Integral transforms A.1 Laplace transform A.2 Spatial Fourier transform A.3 Azimuthal Fourier transform A.4 Meijer transform B Stiffness and compliance tensor B.1 Rotation between reference frames B.2 Cylindrical coordinates C Christoffel equation C.1 Cartesian coordinates C.2 Cylindrical coordinates D Processing of borehole acoustic waveform array data D.1 Time-domain methods D.2 Frequency-domain methods D.2.1 Weighted spectral semblance method D.2.2 Modified matrix pencil method

Die Beeinflussung der Magnetisierungsrichtung magnetischer Dünnschichten mittels einer elektrischen Spannung, anstatt eines elektrischen Stromes, ist vielversprechend für die energieeffizientere Nutzung magnetischer Bauelemente in der Datenspeicherung und in anderen Technologien. Ein neuartiger Ansatz für solche magnetoelektrischen Materialien ist die Kontrolle des Magnetismus über elektrochemische Reaktionen. Die elektrische Spannung wird dabei über einen flüssigen oder festen Elektrolyten an die magnetische Schicht angelegt, und elektrochemische Grenzflächenreaktionen führen zur reversiblen Kontrolle magnetischer Eigenschaften. Bisher wurden dazu hauptsächlich Schichten mit senkrechter magnetischer Anisotropie untersucht und zur Charakterisierung nur die Magnetisierungskurven bei angelegter elektrischer Spannung aufgenommen. Für ein tieferes Verständnis der zugrunde liegenden Mechanismen ist eine zusätzliche Untersuchung der magnetischen Mikrostruktur notwendig. Im Rahmen der Arbeit wurde eine elektrochemische Zelle für flüssige Elektrolyten entwickelt, die mit magneto-optischer Kerr-Mikroskopie kompatibel ist. Mit dieser Messzelle wurden in situ Untersuchungen des Einflusses elektrochemischer Reaktionen auf die magnetischen Eigenschaften von FeOx/Fe-Dünnschichtsystemen durchgeführt. Die mittels Sputtern hergestellten FeOx/Fe Schichten zeigen eine uniaxiale magnetische Anisotropie mit der Magnetisierungsrichtung in der Ebene. Winkelaufgelöste Kerr-Mikroskopiemesssungen zeigten einen magnetisch blockierten Zustand entlang der harten Achse, der eine erhöhte Koerzivität und Remanenz aufweist. Dieser konnte auf die Wechselwirkung zwischen geladenen Néel-Domänenwand-Ausläufern zurückgeführt werden. Anhysteretische Messungen entlang der magnetisch harten Achse ermöglichten die Quantifizierung der uniaxialen Anisotropiekonstanten KU. Bei Anlegen einer elektrischen Spannung an die FeOx/Fe Schichten in einem 1 mol/l LiOH Elektrolyten wurde eine reversible elektrochemische Umwandlung von FeOx zu metallischen Fe mittels in-situ Ramanspektroskopie nachgewiesen. Diese Umwandlung führt gleichzeitig zu einer Aufhebung des blockierten magnetischen Zustands. Dadurch konnte ein reversibles An- und Ausschalten der Koerzivität und Remanenz erreicht werden. Über in situ Kerrmikroskopiemessungen konnte nachgewiesen werden, dass gleichzeitig mit der Abnahme der Koerzivität bei der elektrochemischen Umwandlung auch eine Erhöhung von KU und eine Vergrößerung der magnetischen Domänen auftritt. Mit diesen Analysen konnte die Verringerung der Wechselwirkungen zwischen den Néel-Domänenwand-Ausläufern als Ursache für die elektrische Kontrolle der Koerzivität aufgedeckt werden. Weiterhin spielt eine verringerte Pinningkraft der magnetischen Pinningzentren durch die FeOx – Fe-Umwandlung eine Rolle. Die elektrochemische Kontrolle der Koerzivität erlaubte es, bei einem geringen magnetischen Feld ein 180°-Schalten der Magnetisierung über eine elektrische Spannung zu erreichen. Die dazu benötigte Schaltenergie wurde zu 121 mJ pro 38.5 mm2 in 60 s abgeschätzt, was sehr vielversprechend für eine Steigerung der Energieeffizienz magnetischer Bauelemente ist. Die elektrochemische Umwandlung von FeOx zu Fe in 1 mol/l LiOH wurde auf das Schichtsystem FeOx/Fe/IrMn mit unidirektionaler Anisotropie in der Schichtebene angewandt. In diesem System kommt es durch die Kopplung von Ferromagnet/Antiferromagnet zu einer unidirektionalen Verschiebung der Hysterese (Exchange Bias). Hier konnte erstmals eine nichtflüchtige, elektrische Kontrolle des Exchange Bias erreicht werden. Mit XPS und dem Vergleich mit einem Model für den Exchange Bias wurde die elecktrochemisch-induzierte Schichtdickenvariation des Ferro-magneten als Ursache aufgedeckt. Die elektrochemische Kontrolle des Exchange Bias ermöglichte eine laterale Strukturierung der damit assoziierten magnetischen Domänen. Damit wurde hier eine neue Struk-turierungsmethode für unidirektionale Systeme vorgestellt. Gegenüber konventionellen Strukturierungsmethoden (beispielweise über Ionenbombardierung) kann so eine elektrochemische Strukturierung vorteilhaft sein, da sie bei Umgebungsbedingungen und ohne Vakuumtechnik funktioniert. Eine unidirektionale Anisotropie mit der Magnetisierungsrichtung senkrecht zur Schichtebene wird im System GdOx/Pd/Co/Pd/NiO erzielt. In diesem System wird ausgenutzt, dass das Anlegen einer elektrischen Spannung über elektrochemische-Reaktionen zur H-Anlagerung in den Co- und Pd-Schichten führt, was eine Änderung der senkrechten magnetischen Anisotrope zur Folge hat. Im Schichtsystem mit Co als ferromagnetischer und NiO als antiferromagnetischer Schicht konnte erstmals mittels einer elektrischen Spannung eine senkrecht zur Schichtebene ausgeprägte Exchange-Bias-Hysterese reversibel an- und ausgeschaltet werden. Für mehrere Zyklen werden Effekte beobachtet, die Trainings-Effekten an konventionellen Exchange-Bias-Systemen ähneln. Das Anlegen einer elektrischen Spannung an GdOx/Pd/GdCo/Pd/NiO mit GdCo als ferrimagnetischer Lage führt zu einer Umkehrung der Exchange-Bias-Hysterese und deren vorzeichenbehafteter Verschiebung, welches auf die Umkehrung der magnetischen Ausrichtung der Untergitter zurückgeführt wird.:1 Introduction 1 2 Fundamentals of magnetic thin films 5 3 State of the Art 11 3.1 Voltage control of magnetism 11 3.2 Electrochemical control of magnetism 14 4 Methods 21 4.1 Film fabrication 21 4.2 Ex-situ and in-situ analytical characterization 22 4.3 Electrochemical characterization 24 4.4 Magneto-optical Kerr Magnetometry and Microscopy 25 5 Combining Kerr microscopy and electrochemistry – the in situ cell 29 6 In-plane uniaxial anisotropy and blocked domain state in FeOx/Fe thin films 33 6.1 Microstructure and composition 33 6.2 Magnetically blocked state in pristine FeOx/Fe thin film 34 7 Voltage control of FeOx/Fe thin films with in-plane uniaxial anisotropy 41 7.1 Voltage control of hysteresis by electrochemical reduction of FeOx 41 7.2 Inverse scaling of coercivity and anisotropy revealed by anhysteresis 44 7.3 Voltage control of magnetic domains 46 7.4 Magnetic de-blocking due to change of Néel wall interactions 48 7.5 Switching of magnetization by a low voltage and energy efficiency 51 7.6 Energy efficiency and application potential for data storage and actuation 53 7.7 Interim conclusion 54 8 Voltage control of FeOx/Fe/IrMn thin films with in-plane unidirectional anisotropy 57 8.1 Characterization of the pristine state exchange biased thin films 57 8.2 Electrochemical modification of EB – voltage dependency 58 8.3 Electrochemical modification of EB – time dependency 61 8.4 Model for voltage control of EB by electrochemistry 64 8.5 Non-volatile and reversible voltage control of exchange bias 65 8.6 Nonvolatile change of oxidation state and layer thickness 67 8.7 Electrochemical patterning of EB and magnetic domain state 68 8.8 Interim conclusion 70 9 Voltage control of magnetic thin films with perpendicular unidirectional anisotropy 73 9.1 Co thin films with perpendicular unidirectional anisotropy 73 9.2 Voltage control of EB in Co/Pd/NiO thin films 73 9.3 Interim conclusion 81 9.4 Voltage control of ferrimagnetic GdCo/Pd/NiO thin film 81 10 Evaluation with regard to perspective applications 83 11 Summary 87 12 Appendix 93 References 107 List of Figures 121 Publication List 123 Acknowledgments 125 Symbols 127 Statement of Authorship 129