A proper description of quasi-brittle failure within the frame of continuum Mechanics can only be achieved by models based on so-called generalised continua. This thesis focuses on a strain gradient generalised continuum and provides a specific methodology to derive corresponding models which account for the essential features of quasi-brittle failure. This methodology is discussed by means of four peer-reviewed journal articles. Furthermore, an extensive overview of the state of the art in the field of generalised continua is given at the beginning of the thesis. This overview discusses phenomenological extensions of standard Continuum Mechanics towards generalised continua together with corresponding homogenisation strategies for materials with periodic or random microstructure. / Eine geeignete, kontinuumsmechanische Beschreibung quasi-spröden Versagens ist nur unter Verwendung verallgemeinerter Kontinuumstheorien möglich. In dieser Habilitationsschrift stehen sogenannte Gradientenkontinua im Vordergrund. Für diese wird eine Methodik vorgeschlagen, welche die Herleitung von Modellen erlaubt, die in der Lage sind, quasi-sprödes Versagen adäquat abzubilden. Diese Methodik wird anhand von vier Publikationen dargestellt und diskutiert. Ein umfangreicher Überblick über den Stand der Forschung auf dem Gebiet der veralgemeinerten Kontinuumstheorien wird am Anfang der Habilitationschrift gegeben. Dabei werden neben phänomenologischen Ansätzen zur Ableitung verallgemeinerter Kontinuumstheorien auch die entsprechenden Homogenisierungskonzepte dargestellt. Letztere werden für Materialien mit periodischer Mikrostruktur und für Materialien mit zufälliger Mikrostruktur diskutiert.

A proper description of quasi-brittle failure within the frame of continuum Mechanics can only be achieved by models based on so-called generalised continua. This thesis focuses on a strain gradient generalised continuum and provides a specific methodology to derive corresponding models which account for the essential features of quasi-brittle failure. This methodology is discussed by means of four peer-reviewed journal articles. Furthermore, an extensive overview of the state of the art in the field of generalised continua is given at the beginning of the thesis. This overview discusses phenomenological extensions of standard Continuum Mechanics towards generalised continua together with corresponding homogenisation strategies for materials with periodic or random microstructure.:1 Introduction 7 2 Generalised Continua - a journey 9 2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2 Principal classes of generalised continua . . . . . . . . . . . . . . . . . 10 2.2.1 Polar field theories and their relatives . . . . . . . . . . . . . . 10 2.2.2 Non-local continua . . . . . . . . . . . . . . . . . . . . . . . . 14 2.3 Generalised continua by explicit homogenisation . . . . . . . . . . . . 15 2.3.1 Random micro-structures . . . . . . . . . . . . . . . . . . . . . 15 2.3.2 Periodic micro-structures . . . . . . . . . . . . . . . . . . . . . 18 2.3.3 Generalised homogenisation based on polynomials . . . . . . 20 3 Modelling of quasi-brittle failure 25 3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.3 Discussion of main results . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.4 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 Bibliography 29 4 Collection of articles reflecting the author’s contribution 35 / Eine geeignete, kontinuumsmechanische Beschreibung quasi-spröden Versagens ist nur unter Verwendung verallgemeinerter Kontinuumstheorien möglich. In dieser Habilitationsschrift stehen sogenannte Gradientenkontinua im Vordergrund. Für diese wird eine Methodik vorgeschlagen, welche die Herleitung von Modellen erlaubt, die in der Lage sind, quasi-sprödes Versagen adäquat abzubilden. Diese Methodik wird anhand von vier Publikationen dargestellt und diskutiert. Ein umfangreicher Überblick über den Stand der Forschung auf dem Gebiet der veralgemeinerten Kontinuumstheorien wird am Anfang der Habilitationschrift gegeben. Dabei werden neben phänomenologischen Ansätzen zur Ableitung verallgemeinerter Kontinuumstheorien auch die entsprechenden Homogenisierungskonzepte dargestellt. Letztere werden für Materialien mit periodischer Mikrostruktur und für Materialien mit zufälliger Mikrostruktur diskutiert.:1 Introduction 7 2 Generalised Continua - a journey 9 2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2 Principal classes of generalised continua . . . . . . . . . . . . . . . . . 10 2.2.1 Polar field theories and their relatives . . . . . . . . . . . . . . 10 2.2.2 Non-local continua . . . . . . . . . . . . . . . . . . . . . . . . 14 2.3 Generalised continua by explicit homogenisation . . . . . . . . . . . . 15 2.3.1 Random micro-structures . . . . . . . . . . . . . . . . . . . . . 15 2.3.2 Periodic micro-structures . . . . . . . . . . . . . . . . . . . . . 18 2.3.3 Generalised homogenisation based on polynomials . . . . . . 20 3 Modelling of quasi-brittle failure 25 3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.3 Discussion of main results . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.4 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 Bibliography 29 4 Collection of articles reflecting the author’s contribution 35

The time-independent and -dependent damage characteristics of brittle rocks inside the pre-failure range have been investigated using numerical simulations and lab testing. Grain-based discrete element models have been developed to simulate both, time-independent and -dependent damage evolution leading to ultimate failure of sandstone and granite, respectively. The models take into account elastic grain and elasto-plastic contact deformation, inter- and intra-granular fracturing and lifetime prediction on the basis of subcritical crack growth. The time-independent mechanical behavior of Coconino sandstone and Lac du Bonnet granite during uniaxial compression tests, Brazilian splitting tests and fracture toughness tests was simulated. Triaxial compression tests and fracture toughness tests for Kirchberg II granite and fracture patterns tests for Eibenstock II granite were carried out in laboratory to perform time-independent damage and failure criterion analysis. The corresponding simulations showed reasonable damage phenomena compared with experimental results. Damage indices were deduced and were applied for different time-independent simulations. Based on calibrations of the time-independent damage simulations of selected brittle rocks, Charles equation and Hillig-Charles equation, which are generally used to describe subcritical crack growth, were implemented into the numerical code to simulate time-dependent damage. One-edged crack growth in Coconino sandstone specimen due to stress corrosion has been analyzed theoretically and numerically. Uniaxial compressive creep tests for Lac du Bonnet granite were simulated and time-dependent behavior in terms of the damage process during primary, secondary and tertiary creep until final failure characterized by macroscopic fracturing was discussed in detail. Subsequent to this, the time-dependent Mode-I crack growth tests and uniaxial compressive creep tests for Kirchberg II granite were carried out and the corresponding simulations were performed. Simulation results are in good agreement with experimental observations. In addition, damage indices and time-dependent fracture development were monitored and illustrated. The developed approach was applied to two potential practical applications: the damage analysis of a sandstone landscape arch and a tunnel. Finally, the results are summarized and recommendations for future work are proposed.

‘Coupled process’ implies that one process affects the initiation and progress of the others and vice versa. The deformation and damage behaviors of rock under loading process change the fluid flow field within it, and lead to altering in permeable characteristics; on the other side inner fluid flow leads to altering in pore pressure and effective stress of rock matrix and flow by influencing stress strain behavior of rock. Therefore, responses of rock to natural or man-made perturbations cannot be predicted with confidence by considering each process independently. As far as hydro-mechanical behavior of rock is concerned, the researchers have always been making efforts to develop the model which can represent the permeable characteristics as well as stress-strain behaviors during the entire damage process. A brittle low porous granite was chosen as the study object in this thesis, the aim is to establish a corresponding constitutive law including the relation between permeability evolution and mechanical deformation as well as the rock failure behavior under hydro-mechanical coupled conditions based on own hydro-mechanical coupled lab tests. The main research works of this thesis are as follows: 1. The fluid flow and mechanical theoretical models have been reviewed and the theoretical methods to solve hydro-mechanical coupled problems of porous medium such as flow equations, elasto-plastic constitutive law, and Biot coupled control equations have been summarized. 2. A series of laboratory tests have been conducted on the granite from Erzgebirge–Vogtland region within the Saxothuringian segment of Central Europe, including: permeability measurements, ultrasonic wave speed measurements, Brazilian tests, uniaxial and triaxial compression tests. A hydro-mechanical coupled testing system has been designed and used to conduct drained, undrained triaxial compression tests and permeability evolution measurements during complete loading process. A set of physical and mechanical parameters were obtained. 3. Based on analyzing the complete stress-strain curves obtained from triaxial compression tests and Hoek-Brown failure criterion, a modified elemental elasto-plastic constitutive law was developed which can represent strength degradation and volume dilation considering the influence of confining pressure. 4. The mechanism of HM-coupled behavior according to the Biot theory of elastic porous medium is summarized. A trilinear evolution rule for Biot’s coefficient based on the laboratory observations was deduced to eliminate the error in predicting rock strength caused by constant Biot’s coefficient. 5. The permeability evolution of low porous rock during the failure process was described based on literature data and own measurements, a general rule for the permeability evolution was developed for the laboratory scale, a strong linear relation between permeability and volumetrical strain was observed and a linear function was extracted to predict permeability evolution during loading process based on own measurements. 6. By combining modified constitutive law, the trilinear Biot’s coefficient evolution model and the linear relationship between permeability and volumetrical strain, a fully hydro-mechanical coupled numerical simulation scheme was developed and implemented in FLAC3D. A series of numerical simulations of triaxial compression test considering the hydro-mechanical coupling were performed with FLAC3D. And a good agreement was found between the numerical simulation results and the laboratory measurements under 20 MPa confining pressure and 10 MPa fluid pressure, the feasibility of this fully hydro-mechanical coupled model was proven.

The time-independent and -dependent damage characteristics of brittle rocks inside the pre-failure range have been investigated using numerical simulations and lab testing. Grain-based discrete element models have been developed to simulate both, time-independent and -dependent damage evolution leading to ultimate failure of sandstone and granite, respectively. The models take into account elastic grain and elasto-plastic contact deformation, inter- and intra-granular fracturing and lifetime prediction on the basis of subcritical crack growth. The time-independent mechanical behavior of Coconino sandstone and Lac du Bonnet granite during uniaxial compression tests, Brazilian splitting tests and fracture toughness tests was simulated. Triaxial compression tests and fracture toughness tests for Kirchberg II granite and fracture patterns tests for Eibenstock II granite were carried out in laboratory to perform time-independent damage and failure criterion analysis. The corresponding simulations showed reasonable damage phenomena compared with experimental results. Damage indices were deduced and were applied for different time-independent simulations. Based on calibrations of the time-independent damage simulations of selected brittle rocks, Charles equation and Hillig-Charles equation, which are generally used to describe subcritical crack growth, were implemented into the numerical code to simulate time-dependent damage. One-edged crack growth in Coconino sandstone specimen due to stress corrosion has been analyzed theoretically and numerically. Uniaxial compressive creep tests for Lac du Bonnet granite were simulated and time-dependent behavior in terms of the damage process during primary, secondary and tertiary creep until final failure characterized by macroscopic fracturing was discussed in detail. Subsequent to this, the time-dependent Mode-I crack growth tests and uniaxial compressive creep tests for Kirchberg II granite were carried out and the corresponding simulations were performed. Simulation results are in good agreement with experimental observations. In addition, damage indices and time-dependent fracture development were monitored and illustrated. The developed approach was applied to two potential practical applications: the damage analysis of a sandstone landscape arch and a tunnel. Finally, the results are summarized and recommendations for future work are proposed.:1 Introduction 2 State of the art 3 Time-independent damage analysis 4 Time-dependent damage analysis 5 Applications of numerical models . 6 Conclusions and outlook References

The presented work documents the influence of Voronoi block size and shape as well as internal mesh size on the calibrated fracture toughness KIC. It is documented that Voronoi based procedures have an inevitable error of up to ± 30%. On the other hand, this approach is able to reproduce complex fracture pattern in a realistic manner with reasonable computational power. The work propose a KIC calibration procedure and documents based on the comparison with lab tests, that crack propagation, fracture pattern as well as stress-strain behavior of brittle solids can be duplicated by calibrated Voronoi based DEM simulations. The thesis also documents a swelling law for the DEM code UDEC including parameter determination and validation on lab tests with swelling cement. Finally, calibrated concrete models with one or two holes under different boundary conditions are used to predict swelling induced cracking. Numerical predictions were compared with corresponding lab tests and showed satisfying results.

‘Coupled process’ implies that one process affects the initiation and progress of the others and vice versa. The deformation and damage behaviors of rock under loading process change the fluid flow field within it, and lead to altering in permeable characteristics; on the other side inner fluid flow leads to altering in pore pressure and effective stress of rock matrix and flow by influencing stress strain behavior of rock. Therefore, responses of rock to natural or man-made perturbations cannot be predicted with confidence by considering each process independently. As far as hydro-mechanical behavior of rock is concerned, the researchers have always been making efforts to develop the model which can represent the permeable characteristics as well as stress-strain behaviors during the entire damage process. A brittle low porous granite was chosen as the study object in this thesis, the aim is to establish a corresponding constitutive law including the relation between permeability evolution and mechanical deformation as well as the rock failure behavior under hydro-mechanical coupled conditions based on own hydro-mechanical coupled lab tests. The main research works of this thesis are as follows: 1. The fluid flow and mechanical theoretical models have been reviewed and the theoretical methods to solve hydro-mechanical coupled problems of porous medium such as flow equations, elasto-plastic constitutive law, and Biot coupled control equations have been summarized. 2. A series of laboratory tests have been conducted on the granite from Erzgebirge–Vogtland region within the Saxothuringian segment of Central Europe, including: permeability measurements, ultrasonic wave speed measurements, Brazilian tests, uniaxial and triaxial compression tests. A hydro-mechanical coupled testing system has been designed and used to conduct drained, undrained triaxial compression tests and permeability evolution measurements during complete loading process. A set of physical and mechanical parameters were obtained. 3. Based on analyzing the complete stress-strain curves obtained from triaxial compression tests and Hoek-Brown failure criterion, a modified elemental elasto-plastic constitutive law was developed which can represent strength degradation and volume dilation considering the influence of confining pressure. 4. The mechanism of HM-coupled behavior according to the Biot theory of elastic porous medium is summarized. A trilinear evolution rule for Biot’s coefficient based on the laboratory observations was deduced to eliminate the error in predicting rock strength caused by constant Biot’s coefficient. 5. The permeability evolution of low porous rock during the failure process was described based on literature data and own measurements, a general rule for the permeability evolution was developed for the laboratory scale, a strong linear relation between permeability and volumetrical strain was observed and a linear function was extracted to predict permeability evolution during loading process based on own measurements. 6. By combining modified constitutive law, the trilinear Biot’s coefficient evolution model and the linear relationship between permeability and volumetrical strain, a fully hydro-mechanical coupled numerical simulation scheme was developed and implemented in FLAC3D. A series of numerical simulations of triaxial compression test considering the hydro-mechanical coupling were performed with FLAC3D. And a good agreement was found between the numerical simulation results and the laboratory measurements under 20 MPa confining pressure and 10 MPa fluid pressure, the feasibility of this fully hydro-mechanical coupled model was proven.