Evaporation is a key process for the water exchange between soil and atmosphere, it is controlled by the internal water fluxes and surface vapor fluxes. The focus of this thesis is to visualize and quantify the multiphase flow processes during evaporation from porous media. The retained liquid films in surface roughness (thick-film flow) and angular corners (corner flow) have been found to facilitate and dominate evaporation. Using the representative 2D micromodels (artificial pore networks) with different surface roughness and pore structures, this thesis gives visualizations of the corner and thick-film flow during the evaporation process, presents the enhanced hydraulic continuity by corner and thick-film flow, and tests the validity of the SSC-model which assumes corner flow is dominant for the mass transport during evaporation. Surface roughness and wettability are proved both experimentally and theoretically to play a key role for the time and temperature behaviors of the evaporation process, besides, this thesis shows that for a consistent description of the time-dependent mass loss and the geometry of the corner/thick-film flow region, the fractality of the evaporation front must be taken into account.

Recently observed cumulation of unexpected collapses of slope-distant waste dumps in lignite mining areas of eastern germany re-initiated research of soil liquefaction. Especially it turned the question of internal initials that correspond to water rise. Parallel to laboritory tests and field experiments a micromechanical model should be developed, which can reproduce processes in the soil during saturation. In first approximation a partly saturated soil consists of two phases: the soil particles and the pore fluid. For micromechanical modeling a coupling of discontinuum particles) and continuum (fluid) is required. The soil particles can be simulated with the Discrete-Element-Method (DEM). For the pore fluid, which is assumed to be a mixture of liquid and gaseous fractions, Pore scale model with Finite Volumes (PFV) is used. At low water content liquid bridges (meniscii) arise between the particles that cause an apparent cohesion. The effect of the meniscii is considered by a correspondingly contact law in the DEM model. During the saturation of a soil both, cohesive effect and fluid bulk modulus, are reduced. In addition buoyancy acts on the particles during the process. The micromechanical modeling approach has the advantage, that just a few model parameters are needed. The numerical model shows pore fluid pressures during saturation process, that leads to a reduction of effective stress. It is investigated how much the reduction is regarding porosity, degree of saturation, stress conditions and grain shape. Furthermore the influence of model parameters as well as hydromechanics is investigated. The investigations are completed with another series of experiments under special conditions like integration of macropores, horizontal fixed model boundaries and abrupt saturation.

In der vorliegenden Arbeit werden verschiedene Randwertprobleme zur Beschreibung der Ausbreitung magnetotellurischer Felder mit Hilfe der Finite-Elemente-Methode numerisch gelöst. Die zwei- und dreidimensionalen Randwertprobleme zur Simulation des elektrischen oder des magnetischen Feldes, des magnetischen Vektorpotentials und des elektrischen Skalarpotentials, des magnetischen Vektorpotentials allein oder des anomalen magnetischen Vektorpotentials werden aus den Maxwell-Gleichungen hergeleitet. Auf Grundlage von Anwendung der Konvergenztheorie auf die Finite-Elemente-Lösung werden Konvergenzstudien für zweidimensionale Modelle des homogenen und des geschichteten Halbraums sowie für das dreidimensionale COMMEMI 3-D-2-Modell durchgeführt. Diese werden genutzt, um die Randwertprobleme hinsichtlich ihrer Effizienz zu bewerten. Außerdem liefern Konvergenzstudien eine Abschätzung des lokalen Fehlers der numerischen Lösung für ein realitätsnahes Modell des Vulkans Stromboli und seiner Umgebung, welches digitale Geländedaten enthält. / This thesis presents the numerical finite-element solution of different formulations of the magnetotelluric boundary value problem. Based on Maxwell\'s equations, the two-dimensional and three-dimensional boundary value problems are derived in terms of the electric or the magnetic field, the magnetic vector and the electric scalar potential, the magnetic vector potential only, or the anomalous magnetic vector potential. To evaluate their efficiency, convergence studies are performed for the two-dimensional models of the homogeneous and the layered halfspace as well as for the COMMEMI-3-D-2 model. Moreover, convergence studies yield estimates of the local error of the numerical solution for a close-to-reality model of Stromboli volcano incorporating digital terrain data.

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.

Liquefaction is the phenomena when there is loss of strength in saturated and cohesion-less soils because of increased pore water pressures and hence reduced effective stresses due to dynamic loading. It is a phenomenon in which the strength and stiffness of a soil is reduced by earthquake shaking or other rapid loading. In this study, after the short review of liquefaction definition, the models of prediction and estimation of liquefaction were considered. Application of numerical modelling with two major software (FLAC & PLAXIS) for the Wildlife site liquefaction, under superstition earthquake in 1987 were compared and analysed. Third step was started with introduction of Fuzzy logic and neural network as two common intelligent mathematical methods. These two patterns for prediction of soil liquefaction were combined. The “Neural network- Fuzzy logic-Liquefaction- Prediction” (NFLP) was applied for liquefaction prediction in Wildlife site. The results show the powerful prediction of liquefaction happening with high degree of accuracy in this case.

The thesis describes the application of anchors in mining and tunneling and gives an up-to-date overview about anchor types, design principles and the interaction mechanisms between anchors and rockmass. A constitutive model was developed, implemented and tested for the 2- and 3-dimensional numerical codes FLAC and FLAC3D to simulate non-linear anchor behaviour including unloading and reloading. The interaction between rockbolts and rockmass was studied in detail via numerical simulations for 5 Vietnamese coal mines. An extended version of the so-called c-Φ reduction method and a new introduced reinforcement factor were applied to quantify the effect of bolting. Mine specific and generalised relations were deduced to quantify the influence of anchor length and distance between anchors on the effect of bolting.

Background Controlled mechanical ventilation with randomly variable tidal volume patterns has been shown to improve gas exchange and respiratory system mechanics compared to conventional ventilation in numerous experimental models of acute respiratory distress syndrome (ARDS). Multiple mechanisms have been proposed to explain this phenomenon called stochastic resonance. The recruitment of collapsed lung regions has been proposed as the dominant mechanism, but the role of respiratory system recruitment and de-recruitment dynamics during variable ventilation and the influence of periodic instead of random variation has not been elucidated. Objectives The primary objective of this thesis was to investigate the effects of periodic tidal volume patterns during variable ventilation on functional parameters with a special focus on gas exchange, respiratory system mechanics and cardiovascular interactions. Further aims were to elucidate the relationship between recruitment and de-recruitment dynamics and recruitment effects of random variable ventilation as well as the impact of an excessive increase in pattern period during variable ventilation on respiratory system mechanics. Finally, the relationship between recruitment effects during variable ventilation and the recruitment and de-recruitment dynamics as well as the ability of random variable ventilation to prevent de-recruitment are to be clarified. Methods Recruitment and de-recruitment dynamics were investigated based on the analysis of the time course of dynamic respiratory system elastance in a double-hit model of ARDS in pigs, a model of lung inflammation in rats, and in silico. The effects of periodic variable ventilation were studied for a wide range of pattern periods using a non-linear computational model of respiratory system mechanics, and in two experimental studies: Partial pressure of oxygen in arterial blood (PaO2) was the primary outcome of the longitudinal study during six hours of therapy in a double-hit model of ARDS in pigs. A cross-over study in a hydrochloric acid-induced model of ARDS in rats was performed to investigate the effects of periodic variable ventilation on baroreflex and respiratory sinus arrhythmia in context of the improvement of the primary end-point PaO2. In both studies, tidal volume patterns were chosen to have main periods overlapping with the dynamics of cardiovascular and respiratory sub-systems. Results and Discussion Periodic variable ventilation, but not random variable ventilation, improved PaO2 compared to conventional ventilation in the double hit model of ARDS. In both experimental studies, variable ventilation independent of pattern period improved respiratory system elastance. The study in silico indicated that periodic patterns have no additional positive effect on respiratory system mechanics compared to random patterns, but will attenuate recruitment for an excessive increase in pattern period. Baroreflex and respiratory sinus arrhythmia were affected by periodic tidal volume patterns in the acid-induced ARDS model; however, pattern period was associated with a decrease in PaO2. Recruitment and de-recruitment dynamics in the experimental model were similar to values derived by analysis of dynamic computed tomography according to literature. In the computational study, re-cruitment during random variable ventilation was maximised for specific values of recruitment and de-recruitment dynamics. Recruitment dynamics were lower during random variable ventilation compared to conventional recruitment manoeuvres, however in the range of de-recruitment dynamics of the respective model. Consequently, random variable ventilation with a coefficient of variation of 30 % was sufficient to prevent an increase of respiratory system elastance during ventilation in the study on acute lung inflammation in rats. Conclusion The asymmetry between recruitment and de-recruitment dynamics, which could be quantified by the analysis of the time course of dynamic elastance, was associated with recruitment during random variable ventilation in numerical simulations. Periodic variable ventilation improved arterial oxygenation to a clinically relevant extent without concomitant improvement of lung recruitment compared to random variable ventilation in a double-hit model of ARDS. Cardiovascular-respiratory interactions and asymmetry of recruitment and de-recruitment dynamics were not associated with this improvement. / Hintergrund In zahlreichen experimentellen Modellen des Akuten Atemnotsyndroms (ARDS) konnte gezeigt werden, dass die kontrollierte maschinelle Beatmung mit zufällig variablen Tidalvolumen pro Atemzug den Gasaustausch und die Atemmechanik im Vergleich zur konventionellen maschinellen Beatmung deutlich verbessert. Es wurden mehrere Mechanismen zur Erklärung dieses Phänomens, der Stochastischen Resonanz, vorgeschlagen. Die Wiedereröffnung kollabierter Lungenareale (Rekrutierung) ist dabei als dominanter Mechanismus der variablen Beatmung identifiziert wurden. Die Rolle der Dynamik von Rekrutierung und Derekrutierung sowie der Einfluss von Periodizität an Stelle von Zufälligkeit in der Sequenz der Tidalvolumina während Zufälliger Variabler Maschineller Beatmung (ZVB) wurde bisher lediglich in numerischen Simulationen evaluiert. Fragestellung Hauptziel dieser Arbeit war es, die Auswirkungen der Periodischen Variablen Maschinellen Beatmung (PVB) auf Gasaustausch, Mechanik des Respiratorischen Systems sowie Kardiovaskulärer Wechselwirkungen zu untersuchen. Ferner sollten mögliche Mechanismen der PVB identifiziert werden. Der Zusammenhang zwischen der Rekrutierungsdynamik und den Rekrutierungseffekten der ZVB sowie den Auswirkungen einer übermäßigen Erhöhung der Periodendauer während der PVB auf die Mechanik des Respiratorischen System war ebenfalls zu untersuchen. Ferner war der Zusammenhang zwischen den Rekrutierungseffekten bei der ZVB und der Dynamik der Rekrutierung / Derekrutierung des Respiratorischen Systems zu untersuchen. Material und Methoden In einem nichtlinearen numerischen Modell der Atemmechanik wurden die Auswirkungen der PVB für einen breiten Bereich von Periodendauern untersucht. Die Dynamik der Rekrutierung und Derekrutierung der Lunge wurde basierend auf der Analyse des Zeitverlaufs der dynamischen Elastance des Respiratorischen Systems in einem Doppelhit-Modell des ARDS im Schwein, einem Modell der Lungenentzündung in der Ratte sowie in silico untersucht. Die Effekte der PVB auf Gasaustausch und Atemmechanik wurden in zwei experimentelle Studien in verschiedenen Modellen des experimentellen ARDS untersucht: Der Partialdruck von Sauerstoff im arteriellen Blut (PaO2 ) war die primäre Zielgröße in der Längsschnittuntersuchung während der sechsstündigen Therapie des experimentellen ARDS am Hausschwein, welches induziert wurde durch wiederholte Auswaschung von Surfaktant mit anschließender beatmungsinduzierter Lungenschädigung. In einer Cross-over-Studie an einem salzsäureinduzierten Modell des ARDS in Ratten wurden die Auswirkungen der PVB auf Baroreflex- und respiratorische Sinusarrhythmie im Zusammenhang mit dem primären Endpunkt PaO2 untersucht. Ergebnisse und Diskussion PVB jedoch nicht die ZVB, verbesserte den PaO2 im Vergleich zur konventionellen maschinellen Beatmung im Doppelhit-Modell des ARDS während sechstündiger Therapie. In beiden Studien verbesserte die PVB unabhängig von der Periodendauer die Elastance des Respiratorischen Systems. Die Simulationen am Computermodell bestätigten, dass periodische Muster keinen zusätzlichen positiven Effekt auf die Mechanik des Atmungssystems im Vergleich zu zufälligen Mustern haben, aber die Rekrutierung während Variabler Maschineller Beatmung für eine übermäßige Erhöhung der Periodendauer abschwächen können. Baroreflex und Respiratorische Sinusarrhythmie wurden durch periodische Sequenz aufeinander folgender Tidalvolumina im säure-induzierten ARDS-Modell beeinflusst, jedoch war die Musterperiode mit einem Rückgang des PaO2 assoziiert. Die im experimentellen Modell bestimmte Dynamik der Rekrutierung und Derekrutierung bestätigte aus der Literatur bekannte Werte, die durch die Analyse der dynamischen Computertomographie gewonnen wurden. In der numerischen Modell-Studie zeigte sich, dass die Rekrutierung während der ZVB für bestimmte Verhältnisse zwischen Rekrutierungs- und Derekrutierungsdynamik (Asymmetrie) maximiert werden. Die Dynamik der Rekrutierung war bei der ZVB im Vergleich zu herkömmlichen Rekrutierungsmanövern geringer, jedoch innerhalb des Wertebereichs der Dynamik der Rekrutierung des jeweiligen Modells. Folglich konnte durch ZVB mit einem Variationskoeffizienten von 30 % die Derekru- tierung der Lunge in einem Modell der akuten Lungenentzündung verhindert werden. Schlussfolgerung Die Asymmetrie zwischen der Dynamik der Rekrutierung und Derekrutierung der Lunge, die durch die Analyse des Zeitverlaufs der dynamischen Elastance quantifiziert werden konnte, war mit der Rekrutierung während der Zufälligen Variablen Beatmung in numerischen Simulationen assoziiert. Die Periodisch Variable Beatmung verbesserte die arterielle Oxygenierung in einem klinisch relevanten Umfang ohne gleichzeitige Verbesserung der Lungenrekrutierung im Vergleich zur Zufälligen Variablen Beatmung in einem Doppelhit-Modell des ARDS am Schwein. Weder Kardiovaskulär-respiratorische Wechselwirkungen noch die Asymmetrien der Rekrutierungs- und Derekruitierungsdynamik standen mit dieser Verbesserung im Zusammenhang.

The thesis describes the application of anchors in mining and tunneling and gives an up-to-date overview about anchor types, design principles and the interaction mechanisms between anchors and rockmass. A constitutive model was developed, implemented and tested for the 2- and 3-dimensional numerical codes FLAC and FLAC3D to simulate non-linear anchor behaviour including unloading and reloading. The interaction between rockbolts and rockmass was studied in detail via numerical simulations for 5 Vietnamese coal mines. An extended version of the so-called c-Φ reduction method and a new introduced reinforcement factor were applied to quantify the effect of bolting. Mine specific and generalised relations were deduced to quantify the influence of anchor length and distance between anchors on the effect of bolting.

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

In der vorliegenden Arbeit werden verschiedene Randwertprobleme zur Beschreibung der Ausbreitung magnetotellurischer Felder mit Hilfe der Finite-Elemente-Methode numerisch gelöst. Die zwei- und dreidimensionalen Randwertprobleme zur Simulation des elektrischen oder des magnetischen Feldes, des magnetischen Vektorpotentials und des elektrischen Skalarpotentials, des magnetischen Vektorpotentials allein oder des anomalen magnetischen Vektorpotentials werden aus den Maxwell-Gleichungen hergeleitet. Auf Grundlage von Anwendung der Konvergenztheorie auf die Finite-Elemente-Lösung werden Konvergenzstudien für zweidimensionale Modelle des homogenen und des geschichteten Halbraums sowie für das dreidimensionale COMMEMI 3-D-2-Modell durchgeführt. Diese werden genutzt, um die Randwertprobleme hinsichtlich ihrer Effizienz zu bewerten. Außerdem liefern Konvergenzstudien eine Abschätzung des lokalen Fehlers der numerischen Lösung für ein realitätsnahes Modell des Vulkans Stromboli und seiner Umgebung, welches digitale Geländedaten enthält. / This thesis presents the numerical finite-element solution of different formulations of the magnetotelluric boundary value problem. Based on Maxwell\'s equations, the two-dimensional and three-dimensional boundary value problems are derived in terms of the electric or the magnetic field, the magnetic vector and the electric scalar potential, the magnetic vector potential only, or the anomalous magnetic vector potential. To evaluate their efficiency, convergence studies are performed for the two-dimensional models of the homogeneous and the layered halfspace as well as for the COMMEMI-3-D-2 model. Moreover, convergence studies yield estimates of the local error of the numerical solution for a close-to-reality model of Stromboli volcano incorporating digital terrain data.