NUMERICAL STUDIES OF FRICTIONAL SLIDING BEHAVIOR AND INFLUENCES OF CONFINING PRESSURE ON ACOUSTIC ACTIVITIES IN COMPRESSION TESTS USING FEM/DEM

Transcription

1 NUMERICAL STUDIES OF FRICTIONAL SLIDING BEHAVIOR AND INFLUENCES OF CONFINING PRESSURE ON ACOUSTIC ACTIVITIES IN COMPRESSION TESTS USING FEM/DEM by Qi Zhao A thesis submitted in conformity with the requirements for the degree of Master of Applied Science Graduate Department of Civil Engineering University of Toronto Copyright 2013 by Qi Zhao

2 Numerical Studies of Frictional Sliding Behavior and Influences of Confining Pressure on Acoustic Activities in Compression Tests Using FEM/DEM Qi Zhao Master of Applied Science Graduate Department of Civil Engineering University of Toronto 2013 Abstract The combined finite-discrete element method (FEM/DEM) has been used to simulate processes of brittle fracturing and associated seismicity. With the newly extended FEM/DEM algorithm, two topics involving rock mechanics and geophysics are investigated. In the first topic, a velocityweakening law is implemented to investigate the initiation of frictional slip, and an innovative method that incorporates surface roughness with varying friction coefficients is introduced to examine the influences of surface roughness. Simulated results revealed detailed responses of stresses to the propagation of the slip front. In the second topic, acoustic activities induced in confined compression tests are simulated and quantitatively studied using the internal monitoring algorithm in FEM/DEM. It is shown that with increasing confinement, AE events are spatially more concentrated and temporally more separated, accompanied by a decreasing b-value. Moreover, interesting correlation between orientations of cracks and the mechanical behavior of the rock was observed. ii

3 Acknowledgments This thesis would not have accomplished without support and encouragement from many individuals, to whom I would like to express my sincerest gratitude. First, I would like to thank my supervisors, Prof. Giovanni Grasselli and Prof. Qinya Liu for their guidance, support, and keen enthusiasm in introducing me to the world of rock mechanics and geophysics. I would like to thank Prof. Maurice B. Dusseault from the University of Waterloo for leading me into the area of engineering geology and microseismicity three years ago and for his continuous advices ever since. Thanks also to Prof. Kaiwen Xia's enlightening advices on my thesis. I would also like to thank my fellow colleagues, Mr. Andrea Lisjak, Mr. Bryan Tatone, Mr. Omid K. Mahabadi, Miss Paola Costanza Miglietta, and Mr. Scott Briggs, for their friendship and support. Special thanks to Mr. Lisjak, Mr. Tatone, and Mr. Mahabadi for their remarkable previous contributions to FEM/DEM and patience in teaching me throughout the whole period of my study. Thanks to Mr. Briggs for setting up a fantastic desktop, it has smoothed my work significantly. Thanks also to Prof. Kaiwen Xia's students for their friendship. Finally, I would like to express my gratitude to my parents for their constant love, support, and encouragement. They provided great environment for my growth and educated me to look toward my life with a positive attitude, which carried me through many difficulties. iii

4 Table of Contents Acknowledgments... iii Table of Contents... iv List of Tables... vi List of Figures... vii Chapter 1 Thesis overview Thesis overview Problem descriptions and motivations Approaches Thesis structure... 2 Chapter 2 The combined finite-element and discrete-element method The combined finite-element and discrete-element method Numerical methods in rock mechanics Principles of FEM/DEM... 6 Chapter 3 Frictional slipping of rock interfaces Frictional slipping of rock interfaces General remarks on rock friction Constitutive laws of frictional sliding Implementation of velocity-weakening law in FEM/DEM Chapter 4 Numerical study of the onset of frictional slip Numerical study of the onset of frictional slip Brief review of recent studies on how things slip Model description and simulation processes Simulation results and discussions Linearly distributed normal loading iv

5 4.3.2 Uniformly distributed normal loading Conclusion Chapter 5 Numerical study of AE in compression tests of granite samples under confinement Numerical study of AE in compression tests of granite samples under confinement Role of AE in the study of rock fractures Model description and simulation processes Stress-strain behavior and AE Temporal evolution and spatial clustering of AE Dependence of b-value on confining pressure Influences of crack element inclination and mineral phase Conclusion and discussion Chapter 6 Concluding remarks and future work Concluding remarks and future work Overall conclusion Future work Bibliography Appendix A Appendix B v

6 List of Tables Table 4-1. Material properties of PMMA glass Table 4-2. Roughness values and their corresponding percentages of assignment Table 5-1. Material properties of Stanstead Granite sample used in the simulation Table 5-2. The variation of failure mode associated with increasing confining pressure vi

7 List of Figures Figure 2-1. Behavior of crack elements and the mesh structure after the implementation of crack elements... 7 Figure 2-2. Constitutive behavior of crack element... 8 Figure 3-1. Schematic diagram of a typical friction experiment and its corresponding forcedisplacement curve Figure 3-2. Slip-weakening law and velocity-weakening law Figure 3-3. An interacting couple consists of two elements sliding against each other in the FEM/DEM modelling Figure 3-4. Test model verifying the implementation of the dynamic friction law Figure 3-5. Comparisons between simulated result from FEM/DEM code with newly implemented friction law and the analytical prediction Figure 4-1. Onset of the frictional slip and the propagation of slip fronts Figure 4-2. Model set-up of a PMMA block slides on a rigid platform Figure 4-3. Velocity-weakening law implemented in the FEMDEM code Figure 4-4. Normal distribution of roughness values and their corresponding percentages used in the model Figure 4-5. Stress conditions (a) and apparent friction coefficients (b) along the sliding interface during the propagation of the slip fronts in homogeneous and microroughness models Figure 4-6. Comparisons between stress profiles from the homogeneous model and the microroughness model Figure 4-7. The evolution of stress profiles with time vii

8 Figure 4-8. Slip initiation time and the influence of microroughness of elements on the sliding interface Figure 4-9. Propagation velocity of the slip front initiated from the left edge Figure 5-1. Stress-strain curves of the FEM/DEM compression tests with increasing confining pressure from 0 to 30 MPa Figure 5-2. Stress-strain curves of the compression tests with increasing confining pressure from 0 to 10 MPa Figure 5-3. Location and magnitude of AE events Figure 5-4. The influence of confining pressure on peak axial stresses Figure 5-5. Logarithmic scale plot of the rates of acoustic events as function of the time before main ruptures and their best fitting lines Figure 5-6. Correlation integral versus source distance and best fitting lines for their linear portion Figure 5-7. Frequency-magnitude plots for each simulation Figure 5-8. Magnitudes and locations of five largest events in each simulation with different confinements Figure 5-9. b-value, p-value, and D-value plotted against confining pressure Figure Frequency count of the inclination of broken crack elements before the main rupture time Figure Frequency distribution of the inclination of broken crack elements at different phases and associated mechanical behavior in the simulation with σ 3 =10 MPa Figure Relative frequency of failed boundary types for the pre-peak portion of each simulation and their common events viii

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10 1 Chapter 1 Thesis overview 1 Thesis overview 1.1 Problem descriptions and motivations The dynamics of friction have been studied for hundreds of years, yet many aspects of the frictional sliding behavior are not understood (Rubinstein et al., 2004). An important aspect is the onset of frictional slip. Traditionally described as the transition from static to dynamic friction, recent studies show that the onset of frictional slip is controlled by three types of coherent crack-like fronts (Rubinstein et al., 2004; Ben-David et al., 2010; Zapperi, 2010; Kammer et al., 2012). The relationship between the local stress condition and the propagation of these slip fronts are examined (Ben-David et al., 2010; Kammer et al., 2012). A velocityweakening law is used in numerical simulations in this thesis to investigate the initiation of frictional slip. Moreover, an innovative model that represents surface roughness with varying friction coefficients is introduced, and influences of surface roughness to the onset of frictional slip are studied. Acoustic emission (AE) and microseismicity monitoring techniques have been applied in mining industry for decades, and recently, intensive interests are drawn to this area with the success of fracking techniques. Lisjak et al. (2013) introduced a numerical monitoring algorithm, and simulated a uniaxial compression test on granite. Simulated results are in good agreements with previous laboratory and numerical studies. To further understand the mechanical behavior of rocks under loading, a series of simulations of compression tests with confinement are carried out.

11 2 1.2 Approaches This thesis aims at extending the abilities of the combined finite-discrete element method (FEM/DEM) in simulating mechanical behavior of rocks. Two aspects corresponding to the two problems elaborated above are studied independently. In order to study the frictional slip behavior, a velocity-weakening law is first implemented in the FEM/DEM code. Since only the sliding surface is of interest, no contact interactions are involved except for the sliding interface. Homogeneous model is first simulated to study the onset of frictional slip, and then a heterogeneous model is introduced to examine the influences of surface roughness on the onset of frictional slip. To investigate acoustic activities induced in a confined compression test, a series of confined compression tests are carried out based on previous studies by Mahabadi (2012); Mahabadi, Lisjak, et al. (2012); Lisjak et al. (2013). Statistic analysis including spatial and temporal distribution of acoustic activities, frequency-magnitude distribution, and crack orientations are systematically examined. 1.3 Thesis structure In this dissertation, topics are divided into six chapters. In Chapter 2, the FEM/DEM algorithm is briefly introduced and compared with other numerical methods. Two topics are then studied in the following chapters using the FEM/DEM methodology: frictional slip behavior and AE events in a conventional triaxial test. In Chapter 3, a literature review regarding frictional slip behavior is first provided, then the implementation and verification of a new friction law in FEM/DEM is described, and further simulations on frictional slip are based on this new version of the code. In Chapter 4, a homogeneous model is first tested to study the onset of frictional slip and compared with previous studies; then, a heterogeneous model which mimics surface roughness variation is simulated to investigate the influences of surface roughness on the onset of frictional slip. Different loading conditions are simulated as well.

12 3 Chapter 5 quantitatively studies the AE induced in confined compression tests. A series of static analyses are applied to the internally monitored AE events. The spatial distribution of AE events are assessed using the correlation integral, the temporal distribution is examined by AE rate, and the frequency-magnitude relationship is studied following the Gutenberg-Richter law. Moreover, variations of orientations of failed cracks corresponding to acoustic activities are investigated. Chapter 6 concludes studies carried out in this thesis and discusses improvements can be made in future work.

13 4 Chapter 2 The combined finite-element and discrete-element method 2 The combined finite-element and discrete-element method 2.1 Numerical methods in rock mechanics A numerical model that can successfully solve rock mechanics problems should have the ability of simulating rock friction, fracture, and the associated wave propagation (Mora and Place, 1994). Two classes of numerical modeling techniques are widely used in mechanical researches: continuum methods, for example, finite element methods (FEM) and finite difference methods (FDM); and discontinuum methods, for instance discrete element methods (DEM). Continuum approaches developed based on constitutive laws have been successfully applied to the assessment of global behavior of rock masses and the analysis of stress and deformation; however, due to the limitation of their continuum assumptions, explicit representation of fractures and fracturing processes is not straightforward in these methods (Mahabadi, Lisjak, et al., 2012). The limitations of continuum approaches motivated the development of DEM. DEM is based on interaction laws, and contact patterns of the DEM system can continuously change as the system deforms. Some numerical methods developed based on the concept of DEM are briefly reviewed below. Mora and Place (1994) used a lattice solid model to study the origin of the stick-slip instability which is responsible for earthquakes and is an common phenomena in laboratory shear tests (Brace and Byerlee, 1966; Byerlee, 1978; Scholz, 1998). Their model took into account rock friction, fracture and the associated seismic wave radiation, and simulation results agree with some of the laboratory test such as in Brune et al. (1993). Mora and Place (1998) later extended their two-dimensional lattice solid model to simulate faults with and without fault gouge, and

14 5 stick-slip behavior is observed in both cases. Their results provided the first comprehensive and quantitative possible explanation of the heat flow paradox and indicated that fault gouge has fundamental influence in the dynamics of earthquake faults (Mora and Place, 1994). Hazzard et al. (2002) used a bonded-particle model to simulate shear-type microseismic events induced by tunnel excavation in granite. In this model, individual particles are bonded together at points of contact, and a plane of weakness is included in the model which is subjected to increasing shear load while the normal load across the plane is held constant (Hazzard et al., 2002). Hazzard et al. (2002) showed how the AE event was preceeded by many small "foreshocks" and then followed by several discrete "aftershocks". Aharonov and Sparks (2004) used a 2D DEM to simulate the behavior of a simplified layer of grains. This method treats individual grains as inelastic disks which undergo linear and rotational accelerations. In this study, two phases of dynamical behavior are defined by the fluctuation of the shear-direction velocity of the loading platen: continuous motion and stick-slip motion (Aharonov and Sparks, 2004). Some recent numerical methods focused on interface sliding behavior are derived from the boundary element method (BEM), which are referred to as semi-analytical method (SAM) in Renouf et al. (2011). The advantage of BEM is that only a small region of the contacting bodies is to be meshed, leading to a dramatic decrease in computational complexity compared to FEM or DEM, in which discretization of the entire bulk is required (Spinu and Amarandei, 2012). FEM is used to study the detailed behavior of the sliding interface and stress distribution by Kammer et al. (2012), and their simulated results agree with laboratory tests in Ben-David et al. (2010). However, FEM have problem representing fractures and their propagation; meanwhile, DEM have known issues when dealing with deformability and fracture of discrete element: fracture initiation and growth is generally approximated by means of elasto-plastic material constitutive laws assigned to the rock blocks instead of being directly modeled (Mahabadi, Lisjak, et al., 2012). Moreover, all these numerical methods suffer three groups of limitations: the first group involves problems with proper choice of element sizes; the second group includes hardware problems; and the last one takes into account boundary difficulties (Łodygowski and Sumelka, 2006).

15 6 2.2 Principles of FEM/DEM Since deformability can be well described by FEM and fractures can be explicitly represented by DEM, Munjiza et al. (1995); Munjiza et al. (1999); Munjiza (2004) introduced the combined finite-discrete element method (FEM/DEM). FEM/DEM discretizes models into discrete elements based on FEM meshes, and is able to model both continuum and discontinuum behavior, thus capturing the whole loading and failure path and the progressive damage process of fractured rocks (Mahabadi, Lisjak, et al., 2012). The FEM/DEM technique combines the advantage of discrete element methods (DEM) to capture the interaction and fracturing of different solids with finite element method (FEM) to describe their elastic deformation (Munjiza, 2004). A unique advantage of FEM/DEM is its ability of modeling the transition from continuum to discontinuum by means of fracture and fragmentation processes (Munjiza et al., 1999). As shown below, FEM/DEM is capable of simulating crack propagation and the corresponding particle motions. The fundamental governing equation can be expressed as: 2 x M R 2 t (1) where M is the mass matrix of the system, 2 x/ t 2 is the second order derivative of displacement, and R is a nodal force vector which includes external loads, interaction forces between bodies, deformation forces, viscous damping forces, and the crack bonding forces. In order to simulate the initiation and propagation of fractures, dedicated four-node cohesive elements, named as crack elements (Figure 2-1), are inserted between each pair of triangular elements. The failure processes are modeled in FEM/DEM following principles of non-linear elastic fracture mechanics (Dugdale, 1960; Barenblatt, 1962). Crack elements are enabled between all adjacent triangular pairs, thus arbitrary fractures can be captured within the constraints imposed by the original mesh topology. An important aspect illustrated in Figure 2-1b is that the model is meshed by the common Delaunay triangulation scheme, which has great advantage of avoiding formation of triangles with small included angles (Sloan, 1987). The resulted unstructured mesh has relatively high quality (Shewchuk, 1996), and most importantly, elements do not have a preferred orientation

16 7 which could result in anisotropic orientations of the generated crack elements. Moreover, when mesh is not duplicated, the FEM/DEM code can function as a time-explicit FEM which will be used in Chapter 3. Similar to other numerical techniques, FEM/DEM is sensitive to the applied temporal and spatial discretization. First of all, the numerical stability of the simulation is influenced by the element size and time step length; certain conditions have to be met for the result to be stable although not a quantitative criterion has been proposed yet. In order to reproduce a realistic fracture pattern, the characteristic mesh size needs to be smaller than grain size of the rock. Moreover, parameters such as contact penalty, tangential penalty, and damping coefficient will influence the stability of a simulation. These values used in simulations, which are usually empirical values, are optimized through tests. Figure 2-1. Behavior of crack elements and the mesh structure after the implementation of crack elements. (a) Behavior of crack elements under given nodal forces; (b) exaggerated mesh structure after the introduction of crack elements. The FEM/DEM mesh is developed based on a FEM mesh generated using a Delaunay triangulation scheme. Modified from Lisjak et al. (2013). The behaviors of three modes of cracking are simulated as follows: 1. Mode I (i.e. opening mode) fracturing is simulated through a cohesive crack model based on the Fictitious Crack Model originally proposed for concrete by Hillerborg et al. (1976). When the opening between two triangular elements reaches a critical value (o p ), which is related to intrinsic tensile strength (f t ), the normal bonding stress is gradually reduced until a residual opening value (o r ) is reached (Figure 2-2a).

17 8 2. Mode II (i.e. sliding mode) fracturing is simulated by a slip weakening model which resembles the model from Ida (1972). The critical slip (s p ) corresponds to the intrinsic shear strength (f s ) which is calculated according to the Mohr-Coulomb criterion: f s c tan (2) n where c is the internal cohesion, ϕ is the internal friction angle, and σ n is the normal stress acting across the fracture surface. While slip approaches s r, the tangential bonding stress is reduced to the residual value Lisjak, et al., 2012). f tan gradually as shown in Figure 2-2b (Mahabadi, r n 3. Mode I-II (i.e. mixed mode) fracturing is defined by a coupling criterion between crack opening and slip. As illustrated in Figure 2-2c, this mode describes a combination of shear and tensile deformation, and the failure criterion is defined by o o or p 2 s s sr p 2 1 (3) where o is opening distance and s is slipping distance. Figure 2-2. Constitutive behavior of crack element. (a) Mode I (opening mode), the crack element start to yield when normal stress reaches f t and breaks at residual opening O r ; (b) Mode II (sliding mode), the crack element yields under peak shear stress f s and breaks at critical shear distance S r where the residual shear stress f r is applied even after the breakage; (c) Graphic representation of the coupling relationship between crack opening and crack slip for Mode I-II, the mixed mode. Modified from Lisjak et al. (2013).

18 9 The shapes of the curves for mode I and mode II are based upon complete experimental stressstrain curves obtained for concrete in direct tension (Evans and Marathe, 1968; Munjiza et al., 1999). The specific fracture energy values G Ic and G IIc are related to the areas under these curves through: o r op G Ic ( o) do (4) s r s p G IIc [ ( s) f r ] ds (5) From an energetic point of view, strain energy is stored during the elastic deformation of the triangular elements, and once the material intrinsic strength is overcome, the release of the strain energy begins along with the initiation of a new fracture. Released energy is consumed by the fracturing process itself through G Ic and G IIc. If sliding occurs, part of the energy is dissipated by frictional work, and the excess of elastic strain energy is radiated as kinetic energy in the form of AE. Lisjak et al. (2013) introduced and validated the newly developed AE modeling methodology based on FEM/DEM, and here it is briefly reviewed for completion. Two approaches were considered to capture quantitative AE information from a FEM/DEM simulation. The first approach takes advantage of the discrete representation of the material and the explicit dynamic solver of the method, and the second approach is a standard seismic source inversion technique (Lisjak et al., 2013). The first approach will be described in the following context; meanwhile, the second approach is heavily influenced by the loading rate and the relatively diffuse damage pattern (Lisjak et al., 2013); therefore, due to its limited application, will not be discussed in this dissertation. The AE algorithm introduced by Lisjak et al. (2013) monitors the relative displacement of crack surfaces and record the kinetic energy of nodes in proximity of propagating fractures. For each acoustic events, four important parameters are numerically assessed: source location, fracture mode, initiation time, and seismic energy (Lisjak et al., 2013). The breakage of each crack element is assumed to be an AE event, and its location coincides with the centroid of the element itself. The fracture mode is derived from the relative displacement of the fracture edges according to the constitutive behavior illustrated in Figure 2-2.

19 10 Since the failure of the crack elements occur over a finite time interval, the initial time is assume to be the time at which the kinetic energy of the crack element reaches a maximum. The associated seismic energy, following algorithm, from Lisjak et al. (2013): E e, is calculated using the kinetic energy at the initiation time by the (i) When the peak strength of a crack element is reached, it starts to yield, and the kinetic energy of the four nodes of the crack element is stored as E k, y mivi, y, (6) 2 i 1 where m i and v i,y are the nodal mass and nodal velocity at the time of yielding, respectively. (ii) The kinetic energy evolving with time, E k (t), is monitored until the crack residual displacement is reached, where the element fails, and the change in kinetic energy is calculated at each time step using. (7) E k ( t) Ek ( t) Ek, y (iii) The seismic energy released from each crack breakage ( E ) is assume to be equal to the maximum value of E k (t) between the time of yielding ( T Ty ) and the time of failure ( T Tf ), thus corresponds to the initiation time, T i. e E e max E ( t). (8) [ T, T ] y f k (iv) Finally, the event magnitude, M e, is calculated from proposed by Gutenberg (1956) is used: E e. In this dissertation, the relationship 2 M e (log10 Ee 4.8). (9) 3 The AE monitoring technique described here only considers seismic energy radiated from the nucleation of new fracture within the intact rock material (Lisjak et al., 2013). Therefore, the seismic events derived from crack reactivation are not modeled which could be a huge disadvantage in modeling rock with pre-existing fractures.

20 11 After the breakage of crack elements, discrete interacting elements are generated and contact detection and interaction algorithms are used to define the behavior of interacting distinct elements. In FEM/DEM, the Munjiza-NBS (No Binary Search) contact detection algorithm (Munjiza and Andrews, 1998; Munjiza, 2004) is implemented. The NBS algorithm is one of the most efficient contact detection algorithms in which the total CPU time needed to detect all contacting couples varies linearly with the total number of discrete elements (Mahabadi, Lisjak, et al., 2012). As soon as contacting couples are detected, contact interaction is performed to calculate forces between discrete bodies. Based on the assumption that contacting couples tend to penetrate into each other, a potential function method is used for the interaction algorithm (Mahabadi, Lisjak, et al., 2012). Distributed contact forces are generate according to the size and shape of the overlap between elements, and are quantitatively defined by the contact penalty (compressive stiffness), p C, as described in Mahabadi, Lisjak, et al. (2012). Mahabadi, Lisjak, et al. (2012) introduced a Coulomb-type friction law in the interaction algorithm to address quasi-static rock engineering problems. To distinguish static and dynamic friction, the tangential stress σ tan is calculated using sliding distance and tangential penalty (p T ): tan, (10) s p T where t is the relative sliding distance of elements. V rel is the relative sliding s V rel velocity and Δt is the time step size. If σ tan is larger than the frictional force µσ N, where µ is the friction coefficient and σ N is the normal stress, the tangential stress and sliding distance are updated and the friction mechanism involves tan N tan, (11) tan and s tan. (12) p T

21 12 The two-dimensional FEM/DEM code, namely Y-Geo (Mahabadi, Lisjak, et al., 2012), is used in this thesis. The FEM/DEM Graphic User Interface (Y-GUI) developed by Mahabadi et al. (2010) is used to assign element properties and boundary conditions to models. Parameters assigned for all simulations are listed in Appendix A. Numerical parameters such as penalties, damping, time step size, and element size are all related to numerical stability and calculation efficiency. Similar to FEM, small time step and small element size can achieve better numerical stability but are more computationally expensive. On the other hand, large penalties and damping coefficient will lead to an instable simulation, and these parameters are calibrated based on previous experience.

22 13 Chapter 3 Frictional slipping of rock interfaces 3 Frictional slipping of rock interfaces 3.1 General remarks on rock friction Figure 3-1a is a schematic diagram of a typical friction experiment system from Byerlee (1978). Spring AB, attached to the rider, is pulled to the right at point B with a velocity V. A schematic curve of the frictional force as a function of displacement of point B is shown in Figure 3-1b. Forces at point C, D, and G are known as the initial, maximum and residual friction respectively (Byerlee, 1978). Once the maximum point D is reached, the rider may suddenly slip and the force in the spring will suddenly decrease to point E. The force will then increase again until sudden slip takes place once more at point F (Byerlee, 1978). However, there are a number of ways in which the force-displacement curves may differ from Figure 3-1b. The motion of the rider may initially occur by microslip (Byerlee, 1978), and recent studies regarding the onset of frictional slip indicate the importance of microslips. There may be cycles of stick-slip before the maximum friction is reached, and the maximum and residual friction may be identical at high pressure (Byerlee, 1978). Moreover, the presence of a gauge layer may alter the slip behavior and make the frictional values ambiguous to be determined. Byerlee (1978) concluded that at low stresses where most civil engineering problems are involved, friction depends mostly on roughness, meanwhile, at intermediate and high stresses conditions, roughness has little or no effect on friction; instead, it was found that at high pressure, the shear and normal stresses during sliding could be closely approximated by linear law. Byerlee (1978) summarized previous experimental studies and give a set of equations for frictions at different stress conditions.

23 14 At intermediate to high normal stress conditions, for normal stress up to 2 kbar, the shear stress required to cause sliding is given approximately by the equation n ; (13) and for normal stresses above 2 kbar the friction is given approximately by n. (14) The equations are reported to be valid for surfaces with different topography, and rock type have little to no effect on friction (Byerlee, 1978). Figure 3-1. Schematic diagram of a typical friction experiment and its corresponding force-displacement curve. a) Schematic diagram of a typical friction experiment system which consists a spring attached to a rider, and the rider with mass m is free to slide on a rigid flat plane; b) schematic diagram of the frictional force as a function of displacement of the rider; from Byerlee (1978). 3.2 Constitutive laws of frictional sliding Non-constant motions are often observed in sliding systems such as squeaking machinery, oscillating violin strings, and the unstable fault slip on the boundaries of the Earth's tectonic plates (Rice and Ruina, 1983). This type of unstable motion, termed "stick-slip", is a common engineering phenomenon and was first described by Bowden and Leben (1938, 1939) from their

24 15 studies based on a number of different solid surfaces, although it has been observed in earlier experiments such as in Bridgman (1936). The unstable stick-slip motion observed in the frictional sliding of geological materials was first proposed by Brace and Byerlee (1966). They suggested the stick-slip behavior as one possible mechanism for shallow focus earthquakes. When two rock surfaces slide over one another in laboratory experiment, the motion is usually jerky, and the jerky motion is termed "stick-slip" (Brace and Byerlee, 1966). Stick-slip occurred under a wide range of conditions including water saturated and air-dried conditions (Brace and Byerlee, 1966). A quantitative understanding of the sliding instability started after the critical value of the hardening modulus (critical stiffness) was precisely formulated by Rudnicki (1977). Byerlee (1978) reviewed the results of previous studies on friction of rocks and proposed the formulations (known as the Byerlee's Law) for stresses that are required to initiate sliding. Classically, kinetic friction coefficient µ k is considered as a constant according to the Coulomb's law. It has been shown experimentally that this is a great simplification under most scenarios; however, some dynamic features, such as the hysteresis of friction, are not considered (Liang and Feeny, 1995). After Brace and Byerlee (1966) pointed out the stick-slip instability as an mechanism for earthquakes, a great deal of researches was focused on the physics of rock friction, and it was found that most of the statements in traditional models had to be revised (Scholz, 1998). Three interrelated phenomenon were observed and studied: 1) Dieterich (1972) first noted that static friction µ s depends on the history of sliding and increases logarithmically with the time two surfaces are held in stationary contact. It is also suggested that the time-dependent manner is only observed for rough surfaces that were separated by a thin layer of displacement-produced wear particles (Dieterich, 1972). Dieterich (1978) later combined the concepts of time-dependent friction and velocity-dependent friction. 2) Scholz et al. (1972) stated that under steady-state sliding, the dynamic friction µ d depends logarithmically on the slip rate V, with a coefficient that can be either positive (velocity

25 16 strengthening) or negative (velocity weakening), depending on the rock type and certain other parameters such as temperature and pressure (Stesky et al., 1974). However, only velocity weakening can produce instability (Rice and Ruina, 1983). 3) When a slipping interface is subjected to a sudden change in the loading velocity, the frictional properties change to new values over a characteristic slipping distance D c, which is interpreted as the slip necessary to rebuild the microscopic contacts between the two rough surfaces (Dieterich, 1978). It is generally agreed that, static friction is higher than dynamic friction because contacts are older and contact size decreases as velocity increases. Based on these interpretations, Dieterich (1979, 1981) integrated experimental and theoretical results into an unified constitutive theory in which the slip rate V appears explicitly in the friction equation and the frictional strength evolves with a characteristic time set by the mean lifetime D c / V of the surface contacts. Dieterich and Conrad (1984) further developed the constitutive theory by taking into account the effect of humidity. Scholz (1990) stated that the time dependence of µ s and the velocity dependence of µ k are related behavior (as cited in Scholz, 1998). Several forms of rate- and state-variable constitutive laws (R&S laws) are developed to model laboratory observations of rock friction. The Dieterich-Ruina law proposed by Dieterich (1979) and developed/simplified by Rice and Ruina (1983); Ruina (1983) is recognized to have the best agreement with experimental data (Scholz, 1998). V V 0 a ln b ln, (15) V0 d c 0 where µ 0 and V 0 are base friction and velocity, which are nearly independent of rock type and temperature (Byerlee, 1978). a and b are related to material properties, ( a b) 0 indicates a intrinsically stable condition, and ( a b) 0 represent a velocity-weakening situation which allow seismic events to nucleate (Scholz, 1998). d c is critical slip distance, and θ is the state variable. In this case, the state variable evolves according to Ruina (1983): d V V ln, (16) dt dc d c

26 17 and it is called the slip law because no evolution occurs at zero sliding velocity, i.e. d / dt 0 when V 0 (Beeler et al., 1994). A similar rate- and state-variable constitutive law was introduced by Dieterich (1986) which proposed the relationship know as the slowness (ageing) evolution equation: V0 V0 ln 1 ln 1 0 a b, (17) V d c where the state variable obeys the slowness law, which indicates that at steady state, the state variable is proportional to slowness i.e. d / dt 0 when d c / V (Scholz, 1998). d V 1, (18) dt d c In these R&S friction laws, first logarithmic term represents the direct effect of friction, while the second accounts for the evolution of the state variable (Tinti et al., 2004). They differ only for the evolution relation which defines the temporal behavior of the state variable, and in the slowness law formulation, the state variable has the physical meaning of an average contact time between the sliding surfaces (Dieterich, 1986; Roy and Marone, 1996; Bizzarri and Cocco, 2003). The average contact time is defined as V by Scholz (1998). ss d c / Note that the friction coefficient here is defined as the ratio between shear stress and effective normal stress ( / ). Moreover, the critical slipping distance d c varies from few micros to tens of micros depends on surface conditions (Dieterich, 1986). The R&S laws can be used to simulated repeated seismic cycles because they imply fault restrengthening after the dynamic failure (Rice, 1993; Cocco and Bizzarri, 2002). Moreover, Baumberger et al. (1994); Heslot et al. (1994) showed that the R&S laws could extend to low velocities where a sliding object comes to a halt. Another category of friction laws are slip-weakening (SW) laws. Ida (1972) proposed a slipweakening model with a finite continuous solution based on the assumption that cohesive force depends only on the relative displacement between the two sides. Andrews (1985) describe this

27 18 model with the following SW law (Figure 3-2a), which is widely used such as in Bizzarri and Cocco (2003); Tinti et al. (2004); Bizzarri (2010). Du u ( u f ), Du d0 d0, (19) f, Du d0 where friction strength has the value τ u at zero slip, decreases linearly to τ f as slip increases to d 0, and is constant at τ f for larger values of slip. Physically, τ u is the upper yield stress, τ f is the final (kinetic) friction, and d 0 is the characteristic slip-weakening distance (Bizzarri and Cocco, 2003). Many approaches are proposed to estimate the critical slipping distance (d c or d 0 ), and most of them are based on kinematic rupture models (Tinti et al., 2004). Fukuyama et al. (2003); Mikumo et al. (2003) estimated the critical slip weakening distance from slip velocity functions, which although was agreed by Tinti et al. (2004), is affected by an error of about 50% (Mikumo et al., 2003; Tinti et al., 2004). Velocity weakening laws are also (Figure 3-2b) used to simulate the transition from static to dynamic friction, such as in Kammer et al. (2012) which will be discussed in the following section. An inherent ambiguity of these friction laws is that reference values (e.g. V 0 and µ 0 ) are usually arbitrarily chosen. Although simulation results give close approximations to the reality, their relationship with material properties are not well understood, and more quantitative studies are required. In a velocity-weakening situation, the friction decreases with increasing velocity which indicates a fastening slip, however, the slip eventually will stop and leads to instability. The process of slipping involves the interactions between the driving force and friction force resisting it. A detailed review of the stick-slip behavior based on the system shown in Figure 3-1 is provided by Bo and Pavelescu (1982), and will not be further discussed here. Moreover, instabilities associated with velocity-weakening were further investigated by Rice and Ruina (1983), and several new aspects of rock friction including memory effects and dilatancy were studied as well (Rummel and Fairhurst, 1970). High-precision devices were developed by T. E. Tullis and Weeks (1986) which allowed more detailed measurements for a wide range of materials under variable sliding conditions.

28 Traction (τ) Friction coefficient (µ) 19 a) b) τ u µ s τ(d 0 ) = τ f µ k Slip distance Slide Velocity, v Figure 3-2. Slip-weakening law and velocity-weakening law. a) Schematic diagram of the SW law which expresses the evolution of traction with increasing slip distance; b) Schematic diagram of the velocityweakening law which provides a smooth transition from static friction to dynamic friction. 3.3 Implementation of velocity-weakening law in FEM/DEM Velocity weakening is a necessity for slip instability (Rice and Ruina, 1983). Kammer et al. (2012) implemented the following velocity weakening friction law to insure a smooth transition from the static friction coefficient µ s to the kinetic friction coefficient µ k, ( v) ( )exp( v ( ) / ), (20) k s k s k where µ s is the static friction, µ k is the kinetic friction, and ω is the transition parameter (Kammer et al., 2012). The friction depends on µ s, µ k, ω, and v, among which only v is a variable. Numerical implementation of this friction law is fairly straightforward using finite-element method and the result shown by Kammer et al. (2012) agreed well with laboratory tests by Ben- David et al. (2010). Because only velocity is a variable, static friction coefficient (µ s ), kinetic friction coefficient (µ k ), and transition parameter (ω) are specified as constants. Naturally, update the friction coefficient according to current sliding velocity will enable the velocity-weakening function in FEM/DEM.

29 20 Figure 3-3. An interacting couple consists of two elements sliding against each other in the FEM/DEM modelling. Element ΔABC is moving with velocity v relative to element ΔDEF. v x and v y are horizontal and vertical velocities respectively, meanwhile, v t and v n are tangential and normal velocities respectively. AB and ED are sliding surfaces where AB has a slope of tan(α 1 ) and ED has a slope of tan(α 2 ). Consider an interacting element couple consists of two triangle elements (Figure 3-3), the upper element (ΔABC) is moving with velocity v relative to the lower element (ΔDEF). v and its x- and y-components are calculated in the original FEM/DEM code. A method used to determine the relative sliding velocity v t is proposed as follows, which will be used to calculate the dynamic friction coefficient at each time step. An important feature of the discrete-element method is that elements can penetrate into each other, and when forces are unevenly distributed upon the upper element, points A and B will penetrate into different depths which leads to a difference between the tilt angles of sliding edges (α 1 for edge AB and α 2 for edge ED). It is feasible to take the average value of the tilt angles of the two sliding interfaces as the sliding angle: ( 1 2)/ 2, (21) where α 1 and α 2 can be calculated according to coordinates of corresponding nodes,

30 21 tan( 1 ) y x B A, and (22) tan( 2 ) y x B E y x A y x E D. (23) D Thus, tangential and normal velocities v t can be expressed using calculated values as v t v cos( ), and (24) v n v sin( ). (25) Although coordinates are recorded in FEM/DEM code, the method proposed above is not yet applicable. Another key requirement here is to determine which nodes are included in the sliding interfaces. In a single interacting couple, only two nodes are involved, but in a real simulation that involves many elements, it is necessary to identify to which sliding interfaces these nodes belong. Nevertheless, this method seems to be an approachable way to implement the new friction law in FEM/DEM to handle problems involving frictional slipping. On the other hand, for a specific type of question in which only horizontal sliding occurs, the tangential velocity can be approximated by the horizontal velocity ( v v ), and to verify this assumption, a simple example is used (Figure 3-4). t x a) b) Figure 3-4. Test model verifying the implementation of the dynamic friction law. a) The initial state of the model, the upper block has an initial velocity of 0.2 mm/ms, and the lower block is a fixed platform. The static friction coefficient is 0.6, and the kinetic friction coefficient is 0.4. A down-pointing acceleration of 0.01 m/s 2 is applied to the mode domain to ensure a constant normal stress. b) The final states of the model, blue block represents the result using dynamic friction law (sliding velocity v x ), and red block is the result from original FEM/DEM code using friction coefficient 0.6.

31 22 In this example, the model consists of two blocks in which the upper block is assign with an initial velocity ( v mm/ms), and the lower block is a fixed rigid platen. When the simulation starts, no horizontal forces are applied and vertical acceleration is constant that the sliding block will stop due to friction. The simulation has been run twice, first with static friction law and second with dynamic friction law. According to the velocity-weakening law, the friction will increase (from µ ) with the decreasing velocity which slows the velocity even more, and at the point the block stops, friction coefficient reaches. Since in original code the static friction coefficient is used during the whole simulation, the sliding block will stop earlier. The simulated result agree with this general prediction (Figure 3-4), quantitative comparisons are applied to further verify this method as follows. Figure 3-5. Comparisons between simulated result from FEM/DEM code with newly implemented friction law and the analytical prediction. Blue dashed curves are analytical predictions, black curves are FEM/DEM simulation results with velocity-weakening law using v x, and purple curves are FEM/DEM simulation results with velocity-weakening law using v.

32 23 The exact same model is calculated analytically for comparison purpose, and the analytical solution is included in Appendix B. Figure 3-5 demonstrated results from FEM/DEM code using velocity-weakening law with v (purple) and v x (black), and theoretical predictions (blue). Results using v in the velocity-weakening law deviates from the prediction significantly especially at low velocities, and the oscillation due to the vertical velocity remains at the end of the simulation; the resulted displacement is not accurate either. Meanwhile, when using horizontal portion of relative velocity (v x ) in the velocity-weakening law, the simulation results agree with the predictions very well. Based on the discussions above, it is plausible to extend the approximation to situations where the relative slipping directions of the interfaces are known (i.e. α is known). Discussions above indicate a simple way to implement the velocity-weakening law in FEM/DEM. More interestingly, assign different friction coefficients to elements at sliding surfaces can mimic the variation of roughness along the interface. The influence of microroughness on nucleation of slip front can be investigated.

33 24 Chapter 4 Numerical study of the onset of frictional slip 4 Numerical study of the onset of frictional slip 4.1 Brief review of recent studies on how things slip Frictional slip is initiated by the rapid rupture of discrete contacts on sliding surfaces, and understanding of the mechanisms of how interface rupture takes place is limited by our knowledge of material properties, strength, and stability of the rough surface (Ben-David et al., 2010). Constitutive laws discussed in Section 3.2 provide close descriptions to the frictional slip behavior; however, unexpected variations of μ s have been reported, and recent studies indicate that the key to understand this phenomenon lies in the dynamics of slip initiation (Ben-David and Fineberg, 2011). Ohnaka and Kuwahara (1990) generalized the dynamic instabilities into three phases: (I) the static formation of a slip failure nucleus, (II) subsequent quasi-static steady crack growth, and (III) quasi-static but accelerating crack growth up to a critical crack length at which the instability occurs (Ohnaka and Kuwahara, 1990). When an existing crack is involved in the slip nucleation, only phases II and III will occur. The crack growth rate in phase II is proportional to the applied strain rate, which has been confirmed by experiments (Ohnaka and Kuwahara, 1990). However in practice, it is difficult to experimentally determine the onset and the propagation of unstable rupture. Hirose and Achenbach (1991) assume that the point at which the crack-tip velocity begins to accelerate rapidly to a value close to sonic velocities is the beginning of unstable rupture. In recent researches on how planar interfaces slide, much attention was paid to the nucleation and propagation of slipping fronts (Rubinstein et al., 2004; Ben-David et al., 2010; Zapperi, 2010;

34 25 Ben-David and Fineberg, 2011; Kammer et al., 2012). The developed measuring techniques and the usage of optical materials advanced the knowledge in this area rapidly. Onset of frictional slip is traditionally described as the transition from static to dynamic friction however, recent studies show that it is controlled by coherent crack-like fronts (Rubinstein et al., 2004; Ben-David et al., 2010; Zapperi, 2010; Kammer et al., 2012). The relationship between the local stress condition and the propagation of these slip fronts are examined with innovative methodologies (Rubinstein et al., 2004; Ben-David et al., 2010). To have a better understanding of how things slip, Ben-David et al. (2010); Zapperi (2010) developed experiments to examine the microscopic behavior of frictional fronts. These new laboratory measurement techniques are overturning some long-held assumptions related to the processes controlling friction (Zapperi, 2010). Figure 4-1 illustrates the conditions on the sliding interface when the frictional slip occurs. It is shown that during shear sliding, the shear stress distribution at interface in generally nonuniform, and the shear strength is reached only at a narrow zone which might cause interface rupture (Kammer et al., 2012). These results indicate that the nonuniformity plays an key role in the onset of frictional slip. Kammer et al. (2012) conducted a finite-element analysis on the propagation of slip fronts with a velocity-weakening law, and emphasized the relationship between the ununiformaly distributed stress condition and propagation of slip fronts. These innovative analysis on slip fronts and the onset of frictional slip bridge the gap between the microscopic interactions that define local frictional resistance and the resulting macroscopic motion in the slip of large bodies (Ben-David et al., 2010). It is convenient to simplify the slip problem by considering uniform stress profile along the interface, however, any geometrical discontinuity, material mismatch, or existence of edges will lead to substantial nonuniformity (Ben-David et al., 2010). Laboratory experiment and numerical simulation results mentioned here indicated the key role of stress nonuniformity to frictional stability and dynamics (Ben-David et al., 2010; Kammer et al., 2012). It is found that speeds of frictional fronts increases with the local ratio of shear and normal stress, ( x) / ( x), and three distinguish regimes of rupture dynamics are classified based on stress condition (Rubinstein et al., 2004; Ben-David et al., 2010):

35 26 (a) slow ruptures, ( x)/ ( x) 0. 5; (b) sub-rayleigh ruptures, 0.5 ( x)/ ( x) 0. 8 ; and (c) supershear ruptures, ( x)/ ( x) Slow ruptures propagate far below material wave speed, sub-rayleigh ruptures propagate up to the Rayleigh wave speed, and supershear ruptures surpass the shear wave speed. Overall motion of the sliding blocks occur until either slow rupture or sub-rayleigh rupture traverses the entire interface (Rubinstein et al., 2004). Transitions between these modes are observed in field and numerical simulations, yet the exact mechanisms for transitions of spontaneously nucleated ruptures from sub-rayleigh to supershear rupture speed are not clear (Xia et al., 2004). It is also pointed out that, nucleation locations are often regions where ( x) / ( x) is maximal hence, either low normal stress or high shear stress can influence the location and initiation of rupture fronts (Ben-David et al., 2010). Stress ratio mentioned above is from the measurements made before slip occurs, Kammer et al. (2012) termed this ratio the static stress ratio, and proposed the concept of dynamic stress ratio which is measured at the slip tip as the slip front propagates. The correlation between acceleration and deceleration slip fronts with dynamic stress ratio is observed which was not shown by the static stress ratio (Kammer et al., 2012). Figure 4-1. Onset of the frictional slip and the propagation of slip fronts. Slip occurs by the nucleation of a detachment region (yellow area) and the propagation of detachment fronts (green line), from Zapperi (2010).

36 27 The heterogeneity of stress over the interface reveals the very detailed factors of frictional slip processes; however, new approaches are needed to better understand this phenomenon. Some promising approaches are suggested to understand this heterogeneous stress profiles across the contact interface, such as the mesoscopic friction models, in which the elastic body is considered to be a set of discrete blocks connected by springs. 4.2 Model description and simulation processes The two-dimensional model consists of a rectangular isotropic elastic body (W=200 mm, H=50 mm) and a fixed rigid plane underneath (Figure 4-2). Corners of the elastic body are chopped to avoid stress singularities at edges. Since sliding of two interfaces are of interests, nodes in the model are not duplicated, and the FEM/DEM functions as a FEM for the elastic body, similar to the method used in Kammer et al. (2012); however, interacting algorithms are involved between the elastic body and the rigid platform. The isotropic body is modeled with properties of acrylic poly (methyl methacrylate) (PMMA) glass, whose properties are listed in Table 4-1, according to Ben-David et al. (2010); Kammer et al. (2012). Friction coefficients and transition parameter listed in Table 4-1 correspond to the velocity-weakening law in Figure 4-3. The simulation is divided into three stages to accomplish a similar set-up as in Kammer et al. (2012) (Figure 4-2). In stage one, a down-pointing vertical velocity is imposed at the upper boundary and the resulted displacement is the equivalent of a linearly distributed load. The upper boundary is then fixed in vertical direction to hold the normal load. During stage two, no other disturbances are applied, and velocity field introduced in stage one is damped out so the model reaches an equilibrium state. In the last stage, a uniform horizontal velocity pointing to the right, v x, is applied to the upper boundary through the whole stage period. Relatively low horizontal velocity is applied to ensure a quasi-static condition (Kammer et al., 2012; Lisjak et al., 2013). Simulations are run with ten times the critical damping value in first two stages to dissipate the oscillation introduced by the vertically moving upper boundary. Numerical damping attenuates the dynamic motion of elements (i.e., velocity). In first two stages, stress distribution is not

37 28 influenced by the damping factor. However, at the last stage, critical damping is applied to preserve the dynamic activities induced by the onset of slip. The critical damping value (D c ) is calculated using the following equation proposed by Munjiza (2004), D c 2h E (26) where h is the characteristic element size, h 2 mm. Figure 4-2. Model set-up of a PMMA block slides on a rigid platform. A rectangular isotropic elastic body in contact with a rigid plane is first imposed by a linearly distributed velocity, and then a uniformly distributed shearing velocity is applied to the upper boundary. Elements in the model possess different roughness values and are color-coded according to Table 4-2. In order to produce randomly distributed roughness along the interface, eleven element property sets are used among which only the friction coefficient is different, and increasing friction is color-coded with increasing color saturation (Table 4-2). Scholz (1968b) illustrated a model with inhomogeneity by using a randomly distributed stress according to a normal distribution. Similar concept is used here to mimic interface heterogeneity. Roughness values ranging from 0.7 to 1.0 are used with an interval of 0.05 and the abundance of a specific friction coefficient follows a modified normal distribution. f 2 [20( s 0.85)] 1 2 ( s ) e. (27) 2

38 29 Abundances of elements with different roughness values assigned to the model are integrated values (i.e. areas covered by corresponding portions of the curve). This form of normal distribution is chosen so that the variance is 1, and the property set with average roughness value μ=0.85 being the most abundant one. A homogeneous model using property set 4 in (μ=0.85) is simulated for comparison. Moreover, different loading conditions are tested to examine the influences of surface roughness. Table 4-1. Material properties of PMMA glass, from Kammer et al. (2012). Parameter Value & unit Material Young's Modulus, E 2.6 GPa Poisson's ratio, υ 0.37 Density, ρ 1200 kg/m 3 P-wave speed, v p 1584 m/s S-wave speed, v s 890 m/s Critical damping coefficient, D c 7065 (kg/m s) Friction Law Static friction coefficient, µ s 0.85; [0.7,1]* Kinetic friction coefficient, µ k 0.7 Transition parameter, ω 0.1 m 2 /s 2 * Static friction coefficient is constant in homogeneous model and varies in a range in the microroughness model. Table 4-2. Roughness values and their corresponding percentages of assignment Property Set ID µ s color

39 Normal distribution Percentage Friction Coefficient Slip velocity v (m/s) µs=1 µs=0.95 µs=0.9 µs=0.85 µs=0.8 µs=0.75 µs=0.7 Figure 4-3. Velocity-weakening law implemented in the FEMDEM code. Static friction coefficients are associated with property sets, and the friction law provides a smooth transition from static coefficient (µ s ) to the dynamic friction coefficient (µ k =0.7). Roughness Figure 4-4. Normal distribution of roughness values and their corresponding percentages used in the model. These percentages are then assigned randomly to elements in the model. 4.3 Simulation results and discussions Linearly distributed normal loading First set of simulations are carried out based on Kammer et al. (2012) to test the capability of FEM/DEM in simulating frictional slip nucleation. A linearly distributed velocity ( v y mm/ms, 2 v y mm/ms) is applied to the upper boundary for 1 ms to establish a nonuniformly distribution of stress on the sliding interface. This normal loading condition allows

40 31 a spontaneous nucleation of slip away from edges (Kammer et al., 2012). Moreover, the shearing velocity is used as 10-5 of the P-wave speed ( v mm/ms). Even though corners are chopped to avoid stress concentration, small amount of stress concentration still exists at lower corners. The anomaly not only influence local stress conditions, but also hinders the thorough propagation of the slip front. Components of the 2-D stress tensor of the elements along the sliding interface are demonstrated in Figure 4-5 ( yx xy, xx, and yy ), where τ and σ yy are used to calculate the apparent friction coefficient (µ a ). σ yy varies very little during the propagation of slip fronts, the normal stress to the vertical direction (σ xx ) and the traction (τ), however, reveal compelling details of the slip behavior. µ a is calculated as the local ratio of shear and normal stress, ( x)/ ( x), and when the apparent friction coefficient reaches the static friction coefficient (µ s ) assigned to the element, a slip event initiates. Although experiment suggested that local µ a may well over µ s (Ben-David et al., 2010; Ben-David and Fineberg, 2011), in an idealized numerical simulation, this situation is not considered. The initiating point of the slip is detected at location x x i yy 81 mm, and although slip fronts propagate in both directions, only the left front is further investigated. Similar to Kammer et al. (2012), the initiation location is away from the edge where the apparent friction coefficient reaches maximum. In Figure 4-5, stress conditions and apparent friction coefficients along the interface with different locations of left slip front (x L = 15 mm and x L = 19 mm) are shown to demonstrate the influences of slip propagation on stress conditions. Because of the high normal loading, it takes relatively long to simulate a slip nucleation in these models, and the velocity and stresses of elements on the interface are monitored every 0.02 ms which yields a 100 m/s uncertainty when slip front propagation speed is examined. Nevertheless, it is shown that the variation of surface roughness does not affect the nucleation time or the propagation speed of slip fronts in this set of simulations therefore, in Figure 4-5, figures in same column represent the same time point. To further investigate the influences of surface roughness, models with uniform normal loading and gentle shear loading is simulated and monitored in detail in the next section.

41 32 The traction value (τ) maximizes at the location of the slip front, and the upper limit of the traction increases while travelling with slip front. As a result, traction values form a parabolic shaped curve which maximizes at the slip front and minimizes at the location of nucleation. At the initiating stage (Figure 4-5 i & iv), the traction at left edge is negative which is caused by the leftward lateral expanding induced by high normal loading, and as the shear loading applies continuously, the traction becomes positive. Here we preserve the sign of τ in µ s, and a negative µ s indicates a potential of lateral sliding to the left. i) x i = 81 mm ii) x L = 15 mm iii) x L = 19 mm iv) x i = 81 mm v) x L = 15 mm vi) x L = 19 mm Figure 4-5. Stress conditions (a) and apparent friction coefficients (b) along the sliding interface during the propagation of the slip fronts in homogeneous and microroughness models. i), ii), and iii) are results from the homogeneous model, and iv), v), and vi) are results from microroughness model. Black dashed lines indicate the static friction coefficient in the homogeneous model, and solid magenta curves represent varying static friction coefficient in the microroughness model. Blue dashed line marks the division between compressive (negative) and tensile (positive) stresses, and blue solid line represents the initial and current locations of the slip front. i) & iv) are the initiate point where the slip nucleates at x i = 81 mm; in ii) & v) the left slip front propagates to x L = 15 mm; and in iii) & vi) the left slip front propagates to x L = 19 mm. Values closer than 2 mm to edges are omitted.

42 33 σ yy vary very little especially for elements away from the edge, and the general trend is controlled by the linearly distributed normal loading. However, σ yy of lower corners are highly influenced by the shearing loading. The lateral shearing velocity to the right applies a rotation effect that increases the σ yy at the right side of the model and decreases the σ yy at the left side, and this effect, establishes the lowest point of σ yy at x = 81 mm, which accounts for the nucleation away from edges. σ xx is highly influenced by the slip propagation as well. After the slip front propagations through a location, local σ xx cumulated before the slip releases first, and as the slip front propagates further, compressive stress becomes tensile stress. σ xx values also form a parabolic shaped curve along the interface which maximizes at the middle of left and right slip fronts. The variation in σ xx reflects the elastic deformation introduced by the propagation of the slip front, and similar partially relaxation of elastic stress is observed in experimental model by Baumberger et al. (1994). Moreover, the strong response of σ xx to the propagation of the slip front could be used as a monitoring criterion for the propagation of the slip front. The influence of microroughness is visible only in the σ xx profile (Figure 4-5 vi). At location 24 mm where the highest static friction coefficient is simulated, σ xx deviates from the parabolic curve and drops to zero from tensile stress, and σ xx values of vicinity elements are lowed compared with the results from the homogeneous model. This observation indicates that the region with high roughness recovers from the influences of the propagation of the slip front quicker, and the roughness-induced stress perturbation may influence the healing of a natural fault. The initiation location is controlled by the shear and normal loading conditions, and the high value and linearly distributed normal stress limited the influences of surface roughness. Byerlee (1978) suggested that at high normal stress, roughness has little or no effect on friction. Therefore, taking stress profiles with the left slip front at x 19 mm as examples, τ, σ xx, and σ yy are examined separately against roughness profile (Figure 4-6) to study the influences of surface roughness. L x

43 34 a) b) c) Figure 4-6. Comparisons between stress profiles from the homogeneous model and the microroughness model. The homogeneous model is used as the reference, and values plotted in figures are calculated by subtracting values generated by the homogeneous model from values simulated by the microroughness model. x L marks the location of the slip front (x L = 19 mm), and color specifications are same as in Figure 4-5. The variation of the surface roughness is not influential to the distribution of σ xx and σ yy at approximately 10 mm ahead of the slip front, and two largest variations are seen at the slip tip and at the location with highest roughness. There is less normal compressive stress (σ yy ) at the slip tip, accompanied by a stronger release of σ xx. Meanwhile, elements that have been disturbed by the propagation of the slip front will involve in stick-slip sequences, and surface roughness becomes more influential at this stage. The element with highest static friction coefficient appears to recover from the disturbance faster than other locations where normal stresses begin to accumulate again. Moreover, the large variation at the end of each profile may be caused by the stress concentration at the right corner. The slip tip influences the stress conditions in its vicinity, and the variation of stresses ahead of the slip front may determine the propagation speed. This fact agrees with the importance of stress conditions ahead of the slip tip emphasized by Kammer et al. (2012) Uniformly distributed normal loading In this set of simulations, 2 v mm/ms and v mm/ms. The shearing velocity is 1 y y used as 10-6 v p. Velocity and stresses of elements on the sliding interface are monitored at each time step so that a accurate slip onset detection can be applied, and stress conditions can be

44 35 examined in detail. This loading condition is similar to Ben-David et al. (2010), and the evolution of stress conditions over time are displayed in Figure 4-7. The global shear loading applied on the upper boundary reduces the normal loading on the left side of the sliding interface and increases that on the right side. Since either increasing τ or decreasing σ yy can induce slip nucleation, the relationship between these two quantities is essential to determine the slip dynamics (Ben-David et al., 2010). The onsets at the left corner ( x 95 mm), which agrees with experimental results from i Rubinstein et al. (2004) and Ben-David et al. (2010). However, a closer look at the velocities of elements on the surface indicates that under this loading condition, multiple simultaneous nucleation points occur in addition to the propagation of the slip front initiated from the left corner. The interface is more readily to slide because the normal loading is relatively small, and with slight increase in traction, the static friction can be easily overcome. Due to this fact, a global-wise slip front propagation speed is not applicable, and the propagation speed of the slip front is examined before it is interrupted by other nucleation points. a) b) c) Figure 4-7. The evolution of stress profiles with time. Four time points are chosen for both models, and solid lines represent simulation results from the homogeneous model and dashed lines are simulated by the microroughness model. (1) t = 0 ms, when the shear loading is first applied and not yet influencing stresses on the sliding interface; (2) t = 1 ms, when the slip nucleates at the left corner; (3) t = 1.2 ms, the slip front propagates to the right; (4) t = 1.4 ms, which is the time right before the slip front merges with other slip nucleation points.

45 36 a) b) Figure 4-8. Slip initiation time and the influence of microroughness of elements on the sliding interface. (a) Slip nucleation time of each element on the sliding interface for the homogeneous model (blue) and microroughness model (black); (b) Differences of slip onset time (Δt i ) plotted against the variation of surface roughness, the homogeneous model is used as the reference, and the Δt i is calculated by subtracting slip onset time simulated by the homogeneous mode from that of the microroughness model. Slip onset time are recorded and shown in Figure 4-8a. The slip first nucleates at the left corner, and before this slip front encounters another nucleation at approximately x 75 mm, its propagation speed is calculated (Figure 4-9). The slip front propagation speed started as slow front and accelerates to sub-rayleigh range at x 80 mm, as defined by Rubinstein et al. (2004). The accelerating front agrees with experimental results from Rubinstein et al. (2004) however, the velocity is one order of magnitude smaller. The reason may lie in the interaction parameters that introducing additional correlation between stresses and slipping behavior. The influences of surface roughness on the initiation and propagation of the slip front nucleated at the left corner is very small, even though µ s of the left corner in the microroughness model (µ s 0.9) is higher than that in the homogeneous model. However, nucleation times at other locations are closely related to local friction coefficient. As depicted in Figure 4-8b, the differences in nucleation time and the variation in surface roughness have a positive relationship, and the most significant difference is seen around x 55 mm to x 75 mm where elements have relatively low friction coefficients. In addition to the rotation effect introduced by the shear loading, the low roughness in this region causes an early nucleation.

46 37 Figure 4-9. Propagation velocity of the slip front initiated from the left edge. The slip front starts as slow front and accelerates to sub-rayleigh front before it encounters other nucleation events. 4.4 Conclusion Simulations with different loading conditions and varying surface roughness are carried out in this study to investigate the onset of frictional slip. It was discussed in previous studies that at high normal pressure, the influence of friction coefficient on sliding behavior is little, and at low normal pressure, surface roughness becomes more influential (Byerlee, 1978). However in our simulations, the influences of surface roughness are minor regardless of loading conditions. With detailed comparisons, it was shown that the onset location is determined by stress conditions, and the location with highest apparent friction coefficient slips first. When multiple nucleations are involved, regions with low roughness are more readily to slip. The overall range of slip front propagation speed in first set of simulations is much slower than that in the second set, which indicates that the interaction algorithms introduce additional relationships between loading conditions and the propagation of slip fronts. Parameters designated for contact interactions (i.e., contact/tangential penalty) lead to this complexity but on the other hand, provide an approach to simulate realistic rock surfaces. Stress conditions monitored during the propagation of slip fronts indicated that stress conditions ahead of the slip tip are altered by the slip front, and may influence the propagation of the front

47 38 as described in Kammer et al. (2012). While the normal stress perpendicular to the sliding interface (σ yy ) and traction (τ) are controlled by the applied normal and shear loading conditions, the normal stress parallel to the sliding interface (σ xx ) shows a strong response to the propagation of the slip front. Several improvements need to be made before simulating sliding behavior of rock surfaces. First, as described above, contact interaction parameters need to be calibrated to simulate realistic range of slip front propagation speed. Second, the main body is modelled using conventional FEM in this study, and it would be interesting to invoke the DEM algorithm and examine crack breakage associated with slip nucleation. Moreover, FEM/DEM does not have the ability to simulate the relationship between the apparent contact area and real contact area, nor the relationship between contact time and healing of the sliding interface, which are essential to the sliding behavior. Last but not lease, in order to establish a stable simulation in a quasi-static condition, the time step is set to be 10-6 ms and shearing velocity is so slow that it takes extremely long for the slip front to initiate and propagate across the interface. Enabling parallel computing capability in FEM/DEM would help to mitigate this problem. The onset of frictional slip may be regarded as precursory phenomenon of an earthquake event, therefore, understanding this process may be the key to successful earthquake prediction (Ohnaka and Kuwahara, 1990). As depicted in stress profiles from both sets of simulations, the release of the normal stress that is perpendiculars to the fault normal (σ xx ) is a good indicator of the occurrence of slip events. However, to simulate a large scale fault zone, more factors need to be studied such as the influences of flow and the healing of the fault.

48 39 Chapter 5 Numerical study of AE in compression tests of granite samples under confinement 5 Numerical study of AE in compression tests of granite samples under confinement 5.1 Role of AE in the study of rock fractures Acoustic emission (AE) is defined as a transient elastic wave generated by the rapid release of energy within a material (Lockner, 1993), and AE (microseismic) monitoring techniques have been developed for decades. Early studies of AE were motivated by a desire to predict rock bursts and mine failure, however, this important goal has not been achieved (Lockner, 1993). Moreover, because the recording of the AE is essentially passive, it provides an ideal nondestructive method for studying crack growth. In our case, laboratory AE events are of interests. According to Lockner (1993), laboratory studies can be classified into four categories. First, simple counting of AE events based on which statistical analysis including AE rate, spatial distribution of AE events, and frequency-magnitude relation are examined. The second area of research involves the location of hypocenters of AE sources, and this technique requires precise arrival time measurements. In this dissertation however, is accomplish by an internal monitoring method proposed by Lisjak et al. (2013). A third area involves studies of waveform, this technique requires three-dimensional waveform data recording, and fault plane solutions of AE events can be produced. In a two-dimensional numerical simulation, this technique is not achievable; instead, failure modes discussed in Section 2.2 are examined. Last area is studies regarding P- and S-wave velocities and their spatial variations, which will not be considered here.

49 40 Previous studies (Mahabadi, Lisjak, et al., 2012; Mahabadi, Randall, et al., 2012; Lisjak et al., 2013) illustrated that by using accurate input properties and considering existing heterogeneity and microcracks, the mechanical response and failure mechanism of rock can be accurately reproduced in FEM/DEM. Meanwhile, Lisjak et al. (2013) elaborated the FEM/DEM based AE modeling algorithm which shows promising results. Based on these studies, a series of numerical simulations of compression test based on the granite model used in Lisjak et al. (2013) with a range of confinements is carried out. Quantitative analysis techniques are applied to study how AE events change as function of confining pressure. 5.2 Model description and simulation processes The same granite sample model from Lisjak et al. (2013) is used here and confining pressure conditions are applied in addition to the original set up. Detailed model calibration procedure and experimental results can be found in Mahabadi (2012), and a summarized description is provided as follows. The main body of the 2-D model consists of a diametrical vertical slice of a 54 mm 108 mm cylindrical sample which is meshed with a Delaunay triangle mesh with an average element size of 0.8 mm triangles and crack elements are established accordingly. The model is assigned with properties of the Stanstead Granite based on Mahabadi, Lisjak, et al. (2012) and Lisjak et al. (2013). The spatial heterogeneity of mineral phases was stochastically generated based on a discrete Poisson distribution of the rock mineral composition (Table 5-1). To simulate the weaker character of phase contacts, bounding strength among mineral grains, mode I fracture energy values, G Ic, for the boundaries between biotite and feldspar, biotite and quartz, and feldspar and quartz were reduced to 0.05, 0.05, and 0.6 J/m 2, respectively. Two rigid loading platens located at both ends of the rectangle are assigned with properties of steel, loading the sample at a constant velocity of m/s which corresponds to a strain rate of 2.31 s -1. The loading speed used, even thought is significantly higher than that used in an actual experiments, has been verified to ensure a quasi-static loading condition (Mahabadi et al., 2010;

50 41 Lisjak et al., 2013). AE events are recorded using the internal monitoring approach introduced by Lisjak et al. (2013). Table 5-1. Material properties of Stanstead Granite sample used in the simulation, from Lisjak et al. (2013). Parameter Feldspar Quartz Biotite Volume fraction (%) Young's modulus, E (GPa) Poisson's ratio, υ (-) Density, ρ (kg/m 3 ) Friction coefficient, µ (-) Internal cohesion, c (MPa) Tensile strength, f t (MPa) Mode I fracture energy, G Ic (J/m 2 ) Mode II fracture energy, G IIc (J/m 2 ) Note that dynamic friction is not used in this simulation, and friction coefficients are constants calculated using the internal friction angle (ϕ i ) from Lisjak et al. (2013) using tan( i ). Friction coefficients between steel platens and minerals are all assumed to be 0.1. The macroscopic behavior of rocks depends upon rock type and conditions such as temperature, strain rate, and confining pressure (Amitrano, 2003). Lisjak et al. (2013) studied AE events generated during a uniaxial compression test using FEM/DEM, simulation results were proved in good agreement with experimental values; however, the influences of confining pressure were not investigated. Taking uniaxial compression test as a special case of triaxial compression where 0, a set of conventional triaxial compression tests ( 1 2 3, Mogi (2007)) 2 3 with confining pressures ranging from 0 to 30 MPa are simulated in this study.

51 Stress-strain behavior and AE Four stages of mechanical behavior during a compression test are identified by Amitrano (2003) from experimental results: (1) linear, (2) non-linear prepeak, (3) non-linear postpeak, and (4) shearing. The simulation result from zero confining pressure produced a stress-strain curve where all four stages can been observed (Figure 5-1 & Figure 5-2). When modeling compression tests with FEM/DEM, the application of confining pressure changes the stress-strain response as follows: (i) An increase in Young's modulus is indicated by the slope change of the linear portion of curves with increasing confining pressure, which would result in AE wave speed increase; however, this effect is not studied in this work. Moreover, oscillations are introduced to the early stage of the linear portion when confinements are applied, which are caused by the stress wave induced by the instantaneous application of confining pressures at the initial step. Oscillations are stronger with larger confining pressure, and as simulations progress, these noises are damped out. (ii) A complete brittle-ductile transition is simulated at a confining pressure of approximately 5 MPa (Figure 5-1) which is much smaller than previous laboratory tests and numerical simulations. Even though a portion of the brittle-ductile transition was observed in experiment with confining pressure within 80 MPa, it was also stated that the behavior of granite will never become purely ductile (Amitrano, 2003). Moreover, confining pressure as high as 800 MPa has been tested, and the effect of the excessively high confining pressure on the mechanical behavior was reported minor (Brace et al., 1966; Amitrano, 2003). Brittle behavior of rocks is characterized by a sudden change of slope in the stress-strain curve near the yield point followed by a complete loss of cohesion or an appreciable drop in differential stress; meanwhile, ductile behavior is described as the deformation without any downward slope after the yield point (Mogi, 2007). The brittle-ductile transition depends on temperature, fluid pressure, strain rate, as well as rock type (Simpson, 1985). The Stanstead Granite simulated here is a relatively weak granite, and parameters used are verified against

52 43 laboratory tests by Mahabadi (2012). Further analyses are based on simulations with relatively low confining pressure from 0 MPa to 10 MPa, to avoid the influences of the transition. (iii) With confining pressure applied, for example 3 10MPa, the rock sample can endure a large permanent strain before a throughgoing fracture occurs (major stress drop). Mogi (2007) named this type of rocks A-type rocks. The large permanent strain before fracture occurs is because of homogeneous plastic deformation (Mogi, 2007). Although silicate rocks are usually not A-type (Mogi, 2007), the mechanical behavior of Stanstead Granite simulated here fits the A- type category. Figure 5-3 illustrates the locations and magnitudes of AE events monitored using the algorithm proposed by Lisjak et al. (2013) and their associated broken crack elements. These AE events are counted and displayed upon the points where axial stresses reach first-peak, and major failure planes begin to establish from these points. Figure 5-1. Stress-strain curves of the FEM/DEM compression tests with increasing confining pressure from 0 to 30 MPa. A transition from brittle to ductile starts at approximately 5 MPa, and the model becomes ductile after 10 MPa. With the application of confining pressures, samples endure large axial stains before the major stress drop, and in the ductile zone, strain hardening is observed.

53 44 Figure 5-2. Stress-strain curves of the compression tests with increasing confining pressure from 0 to 10 MPa. Red circles indicate occurrences of first peaks of axial stress on these curves, and these points are defined as main rupture points. Note that not all broken cracks radiate seismic energy because in Equation 7, a possible situation is that ( t) 0, for which the event energy is recorded as zero. These events may be referred E k to as asismic events. Moreover, in actual AE monitoring the AE recorded is limited by the resolution, synthetic AE vents with magnitude smaller than -10 were excluded from further investigations. Cumulative counting of AE events can be used to determine how much damage accumulates during loading and predict the failure (Lockner, 1993). Stress-displacement curves reach first peaks after approximately AE events occurred, and this fact is almost independent of the confining pressure. At lower confining pressures, it takes shorter time for similar total number of AE events to occur, which indicates more rapidly increasing AE rates. In another word, samples under low confining pressures fail more aggressively, while samples under high confinement stay intact for a relatively longer time.

54 45 Magnitude, M e Figure 5-3. Location and magnitude of AE events (color-coded dots) and their associated crack element breakages (red dashes) at peak axial stresses from simulations with increasing confining pressure. Only events with magnitudes larger than -10 (M e > -10) are considered and locations are corrected by taking lateral strain into consideration.

55 Peak Axial Stress Time (ms) Differential Stress (MPa) 46 The occurrences of first peaks of axial stresses are plotted against confining pressure in Figure 5-4a, and these points are defined as main rupture time of each simulation for further analyses purpose. Due to the brittle-ductile transition, first peaks are not well defined after 10 MPa. Since constant strain rate is applied, main rupture time points are equivalently represented by corresponding platen displacements (d). Figure 5-4b demonstrated the simulated strengths at increasing confining pressures. In the brittle region, strength increases nearly linearly at first, and when the curve approaches the brittleductile transition boundary the slope becomes gentler. This behavior agree with A-type rock behavior described by Mogi (2007), and the slope decreases near the transition pressure may be attributed to the gradual increase of local yielding due to heterogeneity (Mogi, 1966). a) b) Confining Pressure (MPa) Confining Pressure (MPa) Figure 5-4. The influence of confining pressure on peak axial stresses. a) Time points at which peak axial stresses occur; b) Strengths versus confining pressure curves. Further static analyses are based upon AE events prior to main ruptures (peak axial stresses). Large amount of low magnitude events are observed in the test without confinement, and many cracks are broken without radiating seismic energy, which indicates that without confining pressure, crack elements are more readily to break. With confining pressures applied, elements are more constrained, and most events are clustered around the vicinity of potential failure planes. Large magnitude AE events are mostly seen on these planes which are produced by the breakage of strong cracks and coalescence of microcracks.

56 47 Examining the failure mode (Figure 2-2) of these events, the number and percentage of each failure mode are listed in Table 5-2. A clear trend of decreasing tensile mode (mode I) and increasing shear mode (mode II) is shown, which indicates that with increasing confining pressure, crack elements are more difficult to open, instead, shear events contributes more to the total damage of the rock. The way these failure events contributes to the deformation varies with confining pressure, as discussed in J. Tullis and Yund (1977). At low confining pressure, there is only a small amount of strain before the microcracks link up to form a throughgoing fracture. At high confinement, failed crack elements form grain-scale microcracks and deformation bands, and there is substantial sample shortening before a through-going fault is formed (J. Tullis and Yund, 1977). Table 5-2. The variation of failure mode associated with increasing confining pressure. Confinement (MPa) Mode I Mode II Mode I-II (81%) 59 (14.7%) 17 (4.2%) (76.8%) 61 (17.5%) 20 (5.7%) (66.7%) 90 (24%) 35 (9.3%) (58%) 115 (29.8%) 47 (12.2%) (45.2%) 163 (39.2%) 65 (15.6%) (44.3%) 132 (38.7%) 58 (17%) 5.4 Temporal evolution and spatial clustering of AE Seismic processes are shown to have a stochastic self-similarity in time (Scholz, 1968a), space (Hirata et al., 1987), and magnitude (Gutenberg and Richter, 1956) domains, and this selfsimilarity manifests itself in power laws. It was also found that the self-similarity holds true over a wide range of magnitudes, and can be adopted to microfracturing studies (Hirata, 1987). AE during rock fracture experiments follows a power law evolution in the time domain, and this phenomenon is typically described by the Omori's law. The generalized form of the Omori's law is expressed as (Utsu, 1962):

57 48 dn dt K p ( c t) (28) where dn/dt is the aftershock rate, t, is the time after the mainshock of earthquakes, and K, c, and p are empirical fitting parameters. In case of AE events preceding the main rupture, t represents the reverse time from the main rupture. In logarithmic scale, this relation becomes linear and the slope of the best fitting curve represents the p value. As displayed in Figure 5-5, p value decreases with increasing confining pressure which indicates a lower AE rate acceleration. Hirata (1987) reported a typical range of Omori's exponent p values from 0.8 to 2.3 from laboratory tests, and the simulated range of p values ( ) fits into this range. Figure 5-5. Logarithmic scale plot of the rates of acoustic events as function of the time before main ruptures and their best fitting lines. Omori's exponent values, p, decrease with increasing confining pressure. Note that quality of linear regressions are very poor, and two reasons may be attributed to: abnormally high AE rate at 0.4 ms ahead of the main rupture, and extremely low AE rate approximately 1 ms before the main rupture. Besides, samples under different confining pressures deform over different time span, even though the general trend of AE rate agree with previous studies, no further plausible information can be obtained. Eliminate low magnitude

58 49 events ( 10 M 9) and apply linear regression only for linear portion of the AE rate could e improve the analysis quality. Hirata et al. (1987) used the correlation integral C(r) for the AE source distributions to examine whether or not the spatial distribution of microfracturing is a fractal, which gave positive results. Here this method is used to study the spatial variation of AE events distribution under different confining pressures. The correlation integral is given by Hirata et al. (1987) as 2 C( r) Nr ( R r), (29) N( N 1) where N r ( R r) is the number of AE source pairs separated by a distance smaller than r, and N is numbers of AE events reported in Figure 5-5 for different confining pressures. If the distribution has a fractal structure, C(r) is expressed by D C( r) r, (30) where D is the fractal dimension of the distribution. Similar to the Omori s law, in log-log scale, this relation appears to be linear and the fractal dimension D is approximated by the slope of the best fitting line (Figure 5-6). In two dimensions, D 2 indicates complete randomness in the source location distribution, while lower values suggest the presence of clustering. However, D- value does not carry any information about the shape of the spatial distribution, and the fractal analysis must be accompanied with a visual inspection of the actual source pattern. AE events from the model without confinement have the largest D-value of 1.61, and as the confining pressure increases, D-value decreases to Generally, the larger confining pressure applied the smaller D-value. Visual interpretations from Figure 5-3 agree with the D-value variation, decrease of the D-value indicates a spatial localization of the fracture process, more specifically; AE events are more concentrated around potential failure planes and randomly distributed low magnitude events are restricted when confining pressures are applied.

59 50 Figure 5-6. Correlation integral versus source distance and best fitting lines for their linear portion. Correlation integral, C(r), and source distance, r, follow a linear relationship on a log-log plot when r 50 mm, and a linear regression is applied to the linear descending branch for each distribution. 5.5 Dependence of b-value on confining pressure Acoustic events magnitude distribution during compression tests has been shown to obey a power law relationship (Lockner, 1993; Amitrano, 2003): b N( A) aa, (31) where A is the maximum amplitude of AE, N(>A) is the number of events with amplitude larger than A, and a and b are constant. This corresponds to the well-known Gutenberg-Richter b-value relation for earthquakes (Gutenberg and Richter, 1954): log N( M) a bm, (32) where N is the number of AE events with magnitude larger than M, and a and b are constants. In this linear expression, b-value is given by the slope of the frequency-magnitude distribution curve. A systematic decrease of b-value from 1.63 to 1.06 was observed with increasing confining pressure in numerical simulations by Amitrano (2003). It was also stated that b-value variation is

60 51 related to the macroscopic behavior (brittle-ductile transition) which is affected by confining pressure (Amitrano, 2003; Jaeger et al., 2007). As depicted in Figure 5-7, b-value drops continuously from 1.14 to 0.93 when confining pressures are applied. The increasing confining pressure is equivalent to increasing depth in the field, and the decreasing trend, as well as the overall range, of b-value agrees with previous studies such as in Gerstenberger et al. (2001). Figure 5-7. Frequency-magnitude plots for each simulation. b-values are calculated based on the linear portion of each distribution, and only events with magnitude larger than -10 are considered. A systematical decrease of b-value is simulated when confining pressures are applied. Lisjak et al. (2013) stated that during a uniaxial compression test, low b-values are due to fast fracture growth and crack coalescence accompanied by high energy release. The relatively low b-value of the test without confinement agrees this interpretation. When confining pressures are applied, crack elements are less readily to break and fractures grow slower, therefore models fail more progressively than in a uniaxial compression test. However, mechanical behavior of models with confining pressures consistent among themselves.

61 52 Events with largest magnitudes in each simulation with different confinement are clustered from -6 to -7 magnitude range, and even though the decreasing b-value suggests a potential of larger magnitude events at high confining pressure, not a clear trend is observed here. However, a spatial distribution pattern is obviously demonstrated by these large magnitude events which are mostly located on the potential failure plane with an angle of conjugate planes (Figure 5-8). 60 to the horizontal or its According to the Mohr-Coulomb failure criterion, angles (β) of two possible planes of shear failure can be calculated from angle of internal friction (ϕ i ) using (Jaeger et al., 2007): 1 45 i. (33) 2 Two directions indicated by Equation 30 are referred to as conjugate directions of failure (Jaeger et al., 2007). ϕ i is used uniformly as failure planes are oriented at 51.8 for all the element properties; thus, two potential 70 and 20 to the horizontal. The simulated failure plane inclined 60 has a good agreement with the theoretical prediction, and the difference may be related to heterogeneity introduced by the arbitrary distribution of mineral phases. Note that the largest event in simulation with 0 MPa is generated by same crack element as in simulation with 2 MPa (x=-6.4, y=-7.8), and the largest events in simulations with 8 MPa and 10 MPa are from identical crack element (x=-18.6, y=-24.9). This fact suggests that irrespective of confining conditions, high stresses build up at these locations and generate large magnitude events, which may be induced by local heterogeneity. Mineral phases of crack elements associated with AE events are further investigated in Section 5.6. In FEM/DEM, only newly developed cracks are simulated, influences of pre-existing cracks are not captured. Since pre-existing cracks are of great importance to AE source mechanism and energy emission, this shortage should be overcome in future studies. In natural earthquakes, shearing of pre-existing faults is the main mechanism, especially at large depth where geostatic pressure is high. The increasing role of shearing events listed in Table 5-2 confirmed this fact.

62 Values 53 Figure 5-8. Magnitudes and locations of five largest events in each simulation with different confinements. Solid stars indicate events with the largest magnitude in each simulation, and they are aligned to the potential failure plane with a dipping angle of 60 degrees. Hollow circles represent the rest large events which are mostly clustered near the potential failure plane or on its conjugate planes b-value p-value D-value Confining Pressure (MPa) Figure 5-9. b-value, p-value, and D-value plotted against confining pressure. A decreasing trend is observed for all these values when confining pressures are applied. p-values have large uncertainties indicated by error bars; meanwhile standard errors of D-values are within 0.007, and standard errors of b-values are all less than therefore, are not displayed.

63 54 To summarize the influences of confinement on a compression test, the variation of frequencymagnitude distribution (b-value), Omori's law (p-value), and fractal dimension (D-value) are plotted against confining pressures in Figure 5-9. Although not a clear correlation among them is discussed in previous studies, a general decreasing trend with increasing confining pressure holds true for these values. In another word, with higher confining pressure, AE events are more spatially concentrated, but temporally more separated. b-value and D-value are positively correlated through the mechanical behavior of rock under different confinements, in another word, diffused damage is associated with high b-value and localized damage is associated with low b-value. This correlation between damage pattern and b- value is different from Amitrano (2003), in which a negative correlation between b-value and D- value is observed. Nevertheless, these results do not conflict since the controlling factors of damage patterns are different. In Amitrano (2003), internal friction angle controls the damage pattern of the rock, meanwhile in our simulations, external confinement determines the damage pattern. Further investigations are needed to examine which factor is more dominant. Amitrano (2003) stated that the variation of b-value is associated with the type of macroscopic behavior (brittle/ductile), rather than with confining pressure. However, confining pressure is one of the most important controlling factors for the brittle-ductile transition, along with rock properties, and temperature. We can draw a more general conclusion that b-value is related to macroscopic behavior which is influenced by the environment (confining pressure, temperature, etc.) and properties of the rock (internal friction angle). Despite the scale difference, persuasive similarities between laboratory and natural seismicity have been reported by recent studies (Goebel et al., 2012; Goebel et al., 2013). The spatial distribution of b-values has been formulated to forecast earthquakes (Wiemer and Wyss, 1997; Schorlemmer and Wiemer, 2005), while the relationship between earthquake and the temporal distribution of b-value is still controversial (Goebel et al., 2013). For both spatial and temporal changes in b-value, stress condition plays an important role.

64 Influences of crack element inclination and mineral phase The heterogeneity and microstructural features of the rock influence the stress distribution and consequently the fracture initiation and growth (Mahabadi, 2012). As demonstrated in Figure 5-3, at mesoscopic scale, large amount of events are clustered in the vicinity of a potential failure plane dipping 60 to the horizontal direction and its conjugate planes, and coalescence of these events establishes major fractures; meanwhile, the breakage of crack elements which represent microscopic fractures shows preferred inclinations as well. Although similar degree of inclination is observed, orientations of microscopic breakages and mesoscopic fractures are not directly related. a) σ 3 = 0 MPa, Total events = 401 b) σ 3 = 2 MPa, Total events = 349 c) σ 3 = 4 MPa, Total events = 375 d) σ 3 = 6 MPa, Total events = 386 e) σ 3 = 8 MPa, Total events = 416 f) σ 3 = 10 MPa, Total events = 341 Figure Frequency count of the inclination of broken crack elements before the main rupture time. The inclination is calculated with the reference of the positive direction of the x axis, counterclockwise.

65 56 (I) d = [0, ] mm, Total events=341 Mode I: 44.3%, Mode II: 38.7%, Mode I-II: 17% (II) d = [0.5845, 1.5] mm, Total events =3750 Mode I: 30.3%, Mode II: 59.8%, Mode I-II: 9.9% (III) d = [1.5, 2.5] mm, Total events=2553 Mode I: 53.3%, Mode II: 37.3%, Mode I-II: 9.4% Figure Frequency distribution of the inclination of broken crack elements at different phases and associated mechanical behavior in the simulation with σ 3 =10 MPa. Percentages of failure modes are calculated for each phase as well. Right column shows the spatial distributions of failed cracks at the last frame of each simulation. Figure 5-10 quantitatively assesses the inclination of broken crack elements before the main rupture of each simulation under different confinements. The inclination of a crack element is approximated by the orientation of one of the edges on which the crack element is established, and is calculated by referring to the positive (right) direction of the x axis, counterclockwise.