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Interaction between dynamic ruptures and off-fault yielding characterized by different rheologies

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Interaction between dynamic ruptures and off-fault yielding characterized by different rheologies

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Content Interaction between dynamic ruptures and off-fault yielding characterized by different rheologies by Shiqing Xu A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (Geological Sciences) August 2013 Copyright 2013 Shiqing Xu ii Acknowledgements First and the most, I will be forever grateful to my PhD advisor, Professor Yehuda Ben-Zion, for his inspiring, mentoring and encouragement during my PhD time at USC. I thank him for sharing with me his knowledge and way of thinking among many other skills in science. I am deeply grateful to Professor Jean-Paul Ampuero from Caltech and Doctor Vladimir Lyakhovsky from the Geological Survey of Israel, without whom this thesis would only be half complete. I thank both of them for sharing their knowledge and having fruitful discussion with me. I express my gratitude to Professor Nadia Lapusta from Caltech, for allowing me to take her class on fracture mechanics and earthquake faulting at Caltech. I thank her for teaching fundamental knowledge and introducing various research topics to me on dynamic fracture mechanics and its application to seismology. I would like to thank all the members of my thesis committee, who also served during my qualification exam, Professors Charles Sammis, Richard Thompson, John Platt and Frank Corsetti for their support and advice. I am also very thankful to many people in our department, especially Cindy Waite, John Yu and John McRaney, who helped me a lot in many ways. Special thanks go to my friends and colleagues. Zheqiang Shi and Wenzheng Yang helped me much during my initial stay in the US. Yoshihiro Kaneko shared with me many of his ideas in doing scientific research. Finally, I thank my girlfriend Lu Li for her support during my PhD study, and my mom for her constant support, patience and encouragement. iii Table of Contents Acknowledgements ii List of Tables vi List of Figures vii Abstract xx Introduction 1 Chapter 1: Properties of Inelastic Yielding Zones Generated by In-plane Dynamic Ruptures: I. Model Description and Basic Results 4 1.1 Summary 4 1.2 Introduction 5 1.3 Model setup 11 1.3.1 Friction laws 13 1.3.1.1 Slip-weakening friction 14 1.3.1.2 Rate- and state-dependent friction 16 1.3.2 Nucleation procedure 19 1.3.3 Normal stress response 21 1.3.4 Off-fault material response 22 1.3.5 Numerical method and parameters 26 1.4 Results 28 1.4.1 Crack- and pulse-like ruptures 28 1.4.2 Energy conservation and partition 30 1.4.3 Influence of background stress and rupture style 38 1.4.3.1 Predicted results in elastic medium 38 1.4.3.2 Simulations with generation of off-fault yielding 41 1.5 Discussion 49 Chapter 2: Properties of Inelastic Yielding Zones Generated by In-plane Dynamic Ruptures: II. Detailed Parameter-space Study 52 2.1 Summary 52 2.2 Introduction 53 2.3 Detailed parameter-space study 55 2.3.1 Influence of the S parameter 56 2.3.1.1 Yielding zone extent and decay form 59 2.3.1.2 Theoretical scaling relation 61 2.3.2 Influence of rock cohesion 65 2.3.3 Influence of material contrast across the fault 68 2.3.4 Ruptures at different depth sections 75 2.3.4.1 Without material contrast 76 2.3.4.2 With material contrast 81 iv 2.4 Discussion 85 2.4.1 On faulting process 86 2.4.2 Key signatures of PZM and RFM 88 2.4.3 Mechanisms for damage generation off a bimaterial fault 93 2.4.4 Limitations of the presented results and future improvements 96 Chapter 3: Numerical and Theoretical Analyses of In-plane Dynamic Rupture and Off-fault Yielding Patterns at Different Scales 99 3.1 Summary 99 3.2 Introduction 100 3.3 Model setup 102 3.3.1 Slip-weakening friction 103 3.3.2 Nucleation procedure 104 3.3.3 Off-fault material response 105 3.3.4 Numerical methods and parameters 106 3.4 Results 108 3.4.1 Basic properties of the yielding zones 108 3.4.1.1 Incremental yielding zone around the rupture tip 111 3.4.1.2 Stress analysis based on LEFM 112 3.4.1.3 Formation of cumulative yielding zones 117 3.4.2 Generation of large-scale off-fault shear fractures 119 3.4.2.1 A conceptual model 119 3.4.2.2 Numerical simulations with off-fault shear localization 122 3.4.3 Cases with fault heterogeneities 129 3.4.3.1 Rupture behavior around fault junction 130 3.4.3.2 Shear fractures around fault strength heterogeneities 132 3.5 Discussion 136 Chapter 4: Dynamic Ruptures on A Frictional Interface with Off-fault Brittle Damage: Feedback Mechanisms and Effects on Slip and Near-fault Motion 144 4.1 Summary 144 4.2 Introduction 145 4.3 Model description 147 4.3.1: Nucleation procedure, friction law, and normal stress regularization 147 4.3.2: Brittle damage rheology for the bulk 150 4.3.3: Numerical method and model parameterization 157 4.4 Results 159 4.4.1: Dynamic normal stress change along the fault 160 4.4.2: Development of a detached pulse front 162 4.4.3: Development of a train of pulses 169 4.4.4: Modulation of the rupture front 173 4.4.5: Effects on near-fault ground motion 175 4.5 Discussion 177 4.5.1: Mechanisms leading to pulse-like ruptures 177 v 4.5.2: Implication for tremor-like radiations 181 4.5.3: Relevance for fault zone structures 182 Bibliography 183 Appendices Appendix A: LEFM Stress Field Near a Propagating Crack Tip 201 Appendix B: Critical Conditions for Shear Localization 203 Appendix C: Mesh Dependence of Shear Localization 206 Appendix D: Numerical Tests on the Robustness of Slip Rate Features 213 vi List of Tables Table 1.1: Material properties and employed parameters for friction laws, nucleation procedure and viscoplasticity. 27 Table 1.2: Convention for converting physical quantities (x ) into dimensionless quantities ( ! x ) in 2-D. 28 Table 2.1: Key equations. 56 Table 2.2: Relation between the incremental rate of yielding zone thickness with along-strike distance from the hypocenter and rock cohesion c . 65 Table 2.3: Observable features of fault damage zones (in 2-D approximation). 95 Table 3.1: Parameter values of material properties used in the simulation. 107 Table 3.2: Convention between physical (x ) and normalized dimensionless ( ! x ) quantities. 108 Table 4.1 : Hessian matrix of U (multiplied by ρ ) ρ(∂ 2 U /∂ε ij e ∂ε kl e ) in 2-D. The original 3-D version can be found in Table 1 of Lyakhovsky et al. (1997). 152 Table 4.2: Parameter values used in the simulation. 158 Table 4.3: Convention between physical (x ) and normalized dimensionless ( ! x ) quantities. 158 vii List of Figures Figure 1.1: A 2-D model configuration of in-plane rupture along a planar frictional interface. The medium is loaded by a uniform background stress state with angle Ψ of the maximum compressive stress σ max relative to the fault. Symbols “C” and “T” represent the compressional and extensional quadrants in relation to the first motion of P waves from nucleation zone. In simulations incorporating a material contrast, medium-1 and -2 and the stiff and compliant side, respectively. 11 Figure 1.2: A linear slip-weakening friction law used to produce spontaneous crack-like ruptures outside the nucleation zone. D c is a characteristic slip distance for the reduction of the friction coefficient. 14 Figure 1.3: A rate- and state-dependent friction law used with velocity- weakening parameters (a<b ) to produce spontaneous pulse-like ruptures outside the nucleation zone. f ss (V) is the steady-state value of friction coefficient under a constant slip velocity V . 16 Figure 1.4: Effective slip-weakening distance D c eff for rate- and state- dependent friction with a step-like velocity jump from 0 to a constant value V (in steady state). Associated with an “aging law” for the evolution of the state variable θ , D c eff generally increases with the steady-state value of V . 18 Figure 1.5: Time-weakening friction law adopted to artificially nucleate the rupture. (Left) spatial distribution of friction coefficient along strike. (Right) time history of friction coefficient at a fixed point. L 0 is a characteristic length scale for the reduction of friction coefficient and v r is a prescribed outward-propagating rupture speed. 19 Figure 1.6: A schematic diagram of the off-fault Mohr-Coulomb yielding criterion, with φ being the internal friction angle and c denoting the rock cohesion. The solid circle described a trial stress state that may be measured along arbitrarily oriented surfaces and the blue dot (associated with σ xx and viii σ xy ) represents the stress on the plane normal to the x- coordinate of Fig. 1.1. The dashed circle and its interior represents the elastic regime preserving the current mean stress σ m . The Mohr-Coulomb yielding envelope is characterized by the thick black line. The yielding envelope in the tensile regime is represented by the red line, with T 0 being the yielding strength under tension (see text for more details). 22 Figure 1.7: Plastic strain distribution for (a) crack-like rupture and (b) pulse-like rupture under same loading conditions and nucleation procedure, but with different friction laws (see text for details). The intensity of the generated plastic strain is characterized by the scalar potency density ε 0 p = 2ε ij p ε ij p . The slip and slip velocity profiles for the crack and pulse ruptures are shown in (c) and (d). The red curve in (d) shows the maximum slip velocity profile for the pulse case. 29 Figure 1.8: Various energy components versus time for (a) the crack case and (b) the pulse case of Fig. 1.7. ΔE vol p : plastic energy dissipation, ΔE vol k : change of kinetic energy, ΔE sur f : frictional energy dissipation, ΔE vol e : released elastic strain energy, ΔE vol tot : energy mismatch (see text for details). The reference energy level E(t ref ) for each component is usually chosen as the one at the initial time t ref = 0 , but will be changed to the level right after the nucleation stage t ref =t nucl ≈ 59L 0 /c s when the normalized energy partition ΔE /(−E vol e ) is shown for the pulse case. 32 Figure 1.9: Energy rate versus time for (a) the crack case and (b) the pulse case of Fig. 1.7. 34 Figure 1.10: (a) Slip velocity as a function of space and time for the pulse case of Fig. 1.7b. (b) Snapshots of normalized shear traction, slip and slip velocity (only the right half is shown because of the symmetry). The propagation speed at the rupture front and the healing front are denoted by v r and v h , respectively and h denotes the pulse width. Except or the bounded maximum level, lim x→v r t Δ u∝1/ v r t−x behind the rupture front and lim x→v r t τ∝1/ x−v r t ahead of the rupture front. ix Around the healing front, ∂u /∂x is continuous, (∂u/∂x) x=v h t = 0 , and lim x→v h t Δ u∝ x−v h t ahead of the healing front. 36 Figure 1.11: Predicted pattern of off-fault mode-I tensile fractures (single black bar) and mode-II shear fractures (conjugate black bars: thick and thin for right-lateral and left-lateral shear fractures, respectively) for cracks expanding within an elastic medium. The value of v r indicates the instantaneous rupture speed at the time of the snapshot. Ψ=10° for (a) and (b), Ψ=45° for (c) and (d). 39 Figure 1.12: Similar to Fig. 1.11 for slip pulses under same initial stress conditions, nucleation procedure and rupture distance. 40 Figure 1.13: Distribution of (a) cumulative plastic strain for Ψ=10° , (b) same as (a) for Ψ=45° , (c) equivalent plastic strain increment for Ψ=10° , (d) same as (c) for Ψ=45° for expanding cracks similar to the cases in Fig. 1.11, but with off-fault yielding. 42 Figure 1.14: Inferred microfracture orientation (aligned to the direction of the maximum compressive stress ! σ max during failure) for the crack ruptures of Fig. 1.13 with (a) Ψ=10° and (b) Ψ=45° . Φ is defined as the acute angle between the inferred orientation of ! σ max and the fault plane. A positive value of Φ indicates that a local horizontal right-lateral slip can be promoted by ! σ max whereas a negative value of Φ favors a local horizontal left-lateral slip. The color scale represents the distance between the center of off-fault elements and the fault (see text for details). 45 Figure 1.15: Decay of potency density ε 0 p with fault normal distance d ⊥ for the crack cases of Fig. 1.13. (a) Ψ=10° , on the compressional side. (b) Ψ=45° , on the extensional side. Sampling locations are linearly mapped into a color scale, starting from X=60L 0 (blue) and ending at X=180L 0 (red) . T 180 represents the local thickness of off-fault yielding zone measured at the location X=180L 0 . More detailed are discussed in the text. 46 Figure 1.16: Distribution of (a) cumulative plastic strain for Ψ=10° , (b) same as (a) for Ψ=45° , (c) equivalent plastic strain x increment for Ψ=10° , (d) same as (c) for Ψ=45° for slip pulses similar to the cases in Fig. 1.12, but with off-fault yielding. 48 Figure 1.17: Inferred microfracture orientation for the pulse cases of Fig. 1.16. 49 Figure 2.1: Schematic diagram showing the migration of rupture tips along a planar fault, modified from Scholz et al. (1993). “C” and “T” represent the compressional and extensional quadrant(s), respectively. In the presence of a material contrast across the fault, the slip direction on the compliant side will be referred to as the positive direction, and the quadrants in the same or opposite directions will be distinguished by “+ /− ” signs. (b) Schematic diagram of a rough fault with geometric complexities, modified from Saucier et al. (1992) and Chester and Fletcher (1997). For both (a) and (b), the big arrows indicate the orientation of the far field background maximum compressive stress σ max while the small thin arrows represent the orientation and relative magnitude (indicated by the length) of the near-fault (dynamically or quasi-statically) perturbed maximum compressive stress ! σ max . 54 Figure 2.2: Plastic strain distribution for crack-like rupture with Ψ=45° and different values of S. The intensity of the plastic strain is quantified by the seismic potency density ε 0 p = 2ε ij p ε ij p . The background normal stress σ 0 differs to preserve comparable energy release from the nucleation zone. In (d), r and θ are polar coordinates with origin at the moving rupture tip, and ω is a conventional angle quantifying the incremental rate of yielding zone thickness with the along-strike distance from the hypocenter (i.e., tan(ω)≈ΔT /ΔX ). 58 Figure 2.3: Off-fault decay of ε 0 p versus fault normal distance d ⊥ for crack-like ruptures associated with different S and σ 0 values. The schematic diagram in (a) illustrates the employed mapping between sets of colors and distance from hypocenter. The inset in (b) shows the variation of ε 0 p from each side of the fault (or the summed value) along the strike. The inset plot in (d) reproduced the result in a double-linear scale. 60 xi Figure 2.4: Comparison of the numerical results and theoretical prediction of Eq. 2.5 (with assumed θ =−130° and v r =0.827c s ) on the scaling relation between T max /L and S. Numerical results are obtained by first calculating ΔT /ΔX based on the measurements in Fig. 2.3, and then converting it into angle ω to calculate T max /L through the relation T max /L=sin(ω)sin(θ )/sin(θ −ω) (see angle definition in Fig. 2.2d). 64 Figure 2.5: (a) Analytic prediction of close-to-fault microfracture orientation as a function of rupture speed v r with a fixed static friction coefficient f s =0.6 , based on the non-singular crack model of Poliakov et al. (2002). −R< x−v r t< 0, y≈ 0 − , with R being the size of process zone. (b) Variation of numerically inferred close-to-fault microfracture orientation and rupture speed along strike, for cases with different rock cohesion values. The inset shows the comparison between the numerical results in the dashed box and the analytic prediction in about the same selected range. 67 Figure 2.6: Plastic strain distribution for crack-like ruptures on a bimaterial interface with 20% contrast for (a) Ψ=10° and (b) Ψ=45° . For both cases, σ xy 0 =0.24σ c , σ yy 0 =−1.0σ c and c=0.2σ c . 68 Figure 2.7: Slip velocity profiles for the crack cases of Fig. 2.6, with the generated Rayleigh wave speed being c GR =0.825c s fast for γ =20% . 70 Figure 2.8: Inferred microfracture orientation or the crack cases of Fig. 2.6 (see Fig. 2.1a for quadrant notations). 72 Figure 2.9: Plastic strain distribution for pulse-like ruptures on a bimaterial interface with 20% contrast for a) Ψ=10° and (b) Ψ=45° . For both cases, σ xy 0 =1.04σ c , σ yy 0 =−4.0σ c and c=0 . 73 Figure 2.10: Slip velocity profiles for the crack cases of Fig. 2.9, with the generated Rayleigh wave speed being c GR =0.825c s fast for γ =20% . 74 xii Figure 2.11: Inferred microfracture orientation for the pulse cases of Fig. 2.9. 75 Figure 2.12: Plastic strain distribution for crack-like rupture at conditions representing different depth sections without material contrast across the fault. For all cases, τ 0 /(−σ 0 )= 0.24 or S=2.571. Parameters for different depth sections are: (a) σ 0 =−1.0σ c , c=0.2σ c ; (b) σ 0 =−1.62σ c , c=0.48σ c ; (c) σ 0 =−3.06σ c , c=1.2σ c . 76 Figure 2.13: (a)-(c) Inferred microfracture orientation for the crack cases of Fig. 2.12 at different depth sections. (d) Correlation between the close-to-fault microfracture orientation and rupture speed at three typical depth sections. 79 Figure 2.14: Plastic strain distribution for pulse-like rupture at conditions representing different depth sections without material contrast across the fault. For both cases, τ 0 /(−σ 0 )= 0.26 or S=2.125. Parameters for different depth sections are: (a) σ 0 =−4.0σ c , c=0.8σ c ; (b) σ 0 =−6.0σ c , c=2.4σ c . 80 Figure 2.15: Inferred microfracture orientation for the pulse cases of Fig. 2.14 at different depth sections. 81 Figure 2.16: Plastic strain distribution for crack-like rupture at conditions representing different depth sections on a bimaterial interface. Except for the variable degree of material contrast, all the other parameters are as in Fig. 2.12. Parameters for material contrast at different depth sections are: (a)γ =20%, c GR =0.825c s fast ; (b) γ =10% , c GR =0.873c s fast ; (c) γ =5% , c GR =0.896c s fast . 82 Figure 2.17: Inferred microfracture orientation for the crack cases of Fig. 2.16 at different depth sections. 83 Figure 2.18: Plastic strain distribution for pulse-like rupture at conditions representing different depth sections on a bimaterial interface. Except for the variable degree of material contrast, all the other parameters are as in Fig. 2.14. Parameters for material contrast at different depth sections are: (a)γ =20%, c GR =0.825c s fast ; (b) γ =10% , c GR =0.873c s fast . 84 Figure 2.19: Inferred microfracture orientation for the pulse cases of Fig. 2.18 at different depth sections. 84 xiii Figure 3.1: Model configuration for dynamic in-plane ruptures along a frictional interface (thick black line at the center) with off- fault plastic yielding. The red portion on the fault indicates the imposed nucleation zone with size L nucl . The medium is loaded by a uniform right-lateral background stress with angle Ψ between the background maximum compressive stress and the fault plane. Symbols “C” and “T” represent, respectively, the compressional and extensional quadrants, in relation to the first motion of P-waves from the nucleation zone. Because of symmetry, results in subsequent plots will be shown only for the right half. 102 Figure 3.2: Distribution of off-fault plastic strain generated by crack- like ruptures with (a) Ψ=10° and (b) Ψ=45° . The scale in the Y direction is exaggerated by a factor of 3.75. 109 Figure 3.3: Distribution of equivalent plastic strain increment near the rupture tip for the cases of Fig. 3.2 with (a) Ψ=10° and (b) Ψ=45° . Black bars represent local orientations of expected shear microfractures, with “thick” and “thin” bars used for right-lateral and left-lateral, respectively. 111 Figure 3.4: Calculated angular variation of the slip-induced incremental stress field Δσ αβ in a polar coordinate (based on LEFM) for the cases of Fig. 3.2 with (a) Ψ=10° and (b) Ψ=45° . The Coulomb failure stress change promoting left-lateral and right-lateral shear are denoted ΔCFS − (blue) and ΔCFS + (red), respectively. Locations of local peak values of ΔCFS ± are marked with numbers and letters, “L” for left-lateral and “R” for right-lateral. 114 Figure 3.5: Spatial representation in a polar coordinate system of the interaction between the background stress σ ij 0 and the slip- induced incremental stress Δσ ij for the cases of Fig. 3.2 with (a) Ψ=10° and (b) Ψ=45° . The results are based on the calculations shown in Fig. 3.4. 115 Figure 3.6: Schematic diagram illustrating the generation of off-fault large-scale shear fractures by propagating ruptures along a frictional fault. Panels (a) and (b) correspond, respectively, to the compressional side of Fig. 3.2a and extensional side of Fig. 3.2b. The grey lobes show the current failure zone around the rupture tip. The orientation of the transient xiv maximum compressive stress ! σ max near the rupture tip is indicated by a pair of arrows. 120 Figure 3.7: Distribution of off-fault shear bands with rate-independent plasticity generated by rupture with (a) Ψ=10° and (b) Ψ=45° . The listed values of frictional, stress and rupture parameters in this and other figures provide a balance between achieving numerical stability and producing prominent off-fault shear bands. The main features discussed in the text have been tested by a parameter space study and confirmed to be robust. Regions I-III highlight local features of the simulated shear bands at various locations. 124 Figure 3.8: Schematic diagram showing possible competition between synthetic (red) and antithetic (blue) shear bands. The solid and dashed lobes are constructed similar to Fig. 3.6, but with small finite shift and slight size increase corresponding to the changing rupture front configuration. Long solid curves represent already-developed paths of two conjugate shear bands (l s for synthetic and l a for antithetic), whereas short dashed curves indicate likely extension paths for the two bands. 127 Figure 3.9: Schematic diagram showing slip-induced ΔCFS lobes around a junction with (a) fault branch and (b) fault bend. The likely rupture evolutions are compared with laboratory experiments of Rousseau and Rosakis (2003, 2009). See text for more explanation. 130 Figure 3.10: Assumed space variations of the static friction coefficient f s (lower line) and corresponding evolution of normalized rupture speed averaged over 4L 0 (upper line) along the fault. 133 Figure 3.11: (a) Off-fault shear bands generated for the heterogeneous case of Fig. 3.10. The inset plot near the bottom shows (with scale at the right vertical axis) the plastic strain distribution along the fault. (b)-(d) close-up views of (a) in different locations associated with small upward and downward jumps of f s along he fault (Fig. 3.10). 134 Figure 3.12: Distribution of off-fault plastic yielding with a strong barrier ( f s >>1) located at X=60L 0 (a) with rate-independent and (b) rate-dependent plasticity. The inset plot in (a) xv schematically show how the high slip gradient near the barrier is partly compensated by the conjugate off-fault shear bands. Following King and Nábělek (1985), letters A, B and C denote different blocks separated by the main fault and shear bands, whereas the vectors on the right show the relative shear motion from a starting side to ending side. The maximum normalized value of ε 0 p in (a) is ~8000 and may correspond to a permanent strain of order 1 assuming σ c /µ ~10 −3 . The maximum value in (b) is about one order of magnitude less than that in (a). 136 Figure 3.13: An idealized Riedel shear structure containing various fracture elements, modified from Sylvester (1998) and Davis et al. (1999). The Y shear is parallel to the principal displacement zone (PDZ), whereas other fracture elements are inclined with characteristic acute or obtuse angles to the PDZ in relation to σ max and/or the internal friction angle. 139 Figure 4.1: Model configuration for dynamic rupture along a frictional interface (black line at the center) with possible generation of brittle damage in the surrounding bulk. The rupture is nucleated right-laterally with a prescribed speed over a small patch (red portion) until it can propagate spontaneously. The angle Ψ characterizes the relative orientation of the background maximum compressive stress to the fault. Symbols “C” and “T” denote the compressional and extensional quadrants, respectively, in relation to the first motion of P-waves radiated from the nucleation zone. Because of the symmetry, only the right half will be shown in later figures. 148 Figure 4.2: ξ−α phase diagram characterizing rock behaviors in different deformation regimes as a function of internal damage (in 2-D). From left to right: ξ =− 2 and ξ = 2 correspond to isotropic compaction and tension, respectively, while deformation has some non-zero shear component in between. The threshold value ξ 0 defines the level of ξ above which damage starts to accumulate. From bottom to top: α=0 corresponds to purely elastic continuum without internal damage, while α=1 defines the maximum damage level where convexity is lost at . Red and blue curves correspond to the conditions with loss of convexity in Eq. 4.11, and the green curve corresponds to the condition defined in Eq. 4.12, with all the conditions ξ 0 xvi properly normalized by µ 0 . The piece-wise black curve simply defines the common boundary between preserving (lower left) and losing convexity (upper right) for all the possibilities. The original 3-D version can be found in Figure 1 of Lyakhovsky et al. (1997). 154 Figure 4.3: Schematic diagram showing the path of damage evolution in ξ−α phase space, where the condition Λ 1 =0 dominates the determination of losing convexity (see the red curve in Fig. 4.2). The background color shows the normalized contour map (divided by µ 0 ) of Λ 1 (here simplified as Λ ) as a function of ξ and α . Damage starts to accumulate once ξ ≥ξ 0 , but will be damped by a “buffer zone” (the grey belt region) near the critical state (Λ=0 ), where a granular- related plasticity can significantly relax the strain. With a fixed value of C d , the width of the buffer zone will be predominantly controlled by the damping parameter q in Eq. 4.13, and the phase-transition parameter β in Eq. 4.14. 156 Figure 4.4: Snapshot distribution of the mapped shear wave speed (see the text for definition) for cases with (a) Ψ=14° and (b) Ψ=56° . For both plots, D c =2L 0 (it is set at 5.6L 0 for all the other cases), C d =0.25c s 0 /L 0 ≈1.2c s 0 /R 0 , C v =1/µ 0 , and C g =0 . Normal stress change Δσ along the fault is imposed in each plot, with a reference scale bar on the right. Positive and negative values indicate tensile and compressive change, respectively. The inset grey plots show the current failure zone defined by ξ ≥ξ 0 (Eq. 4.13) at the same time. As an illustration for the basic roles of damage, granular-related damping is turned off (q=0 ) and results are shown only for a limited rupture propagation distance. 160 Figure 4.5: Spatial distribution of (a) slip rate, (b) normal stress change, and (c) shear stress change for a case with Ψ=30° at three time steps. f d =0.1, C d =1.32c s 0 /R 0 , q= 0.66c s 0 /R 0 , C g = 0.53c s 0 /(µ 0 R 0 ) , C v =0 , and β=0.2 . The chosen times highlight the moments when the propagating rupture behaves like a crack (t 1 ), with a partially (t 2 ) or completely (t 3 ) detached pulse front. The black arrows connecting (a) and (b) at time t 3 indicate two notable tensile Δσ regimes behind the rupture front, both of which are associated with a local reduction in slip rate. 163 xvii Figure 4.6: Snapshot distribution of (a) damage and (b) its off-fault variation for the case of Fig. 4.5. The color bar in (b) indicates how the sampling locations (on the extensional side) are mapped to different colors: from X=60R 0 (blue) to X=150R 0 (red). Note the asymptotic saturation of damage with the proximity towards the fault, where a narrow waveguide within the overall LVZ may be defined. 165 Figure 4.7: Snapshot distribution in a zoom-in view of (a) mapped shear wave speed, (b) fault-normal particle velocity  u y , and (c) normal stress change Δσ yy for the case of Fig. 4.6. The distribution of slip rate at the same time is superimposed in each plot. 167 Figure 4.8: Snapshot distribution of slip rate for elastic calculations with a pre-existing finite-width (W ) LVZ (see inset in a). (a) W =5L 0 ≈ 0.4R 0 , (b) W =10L 0 ≈ 0.8R 0 and (c) W =40L 0 ≈3.2R 0 . For all cases, the P- and S- wave speed have a 30% reduction inside the LVZ compared to those of the country rocks. 168 Figure 4.9: Snapshot distribution of Δσ yy for the case of Fig. 4.8a (panel a) and Fig. 4.8b (panel b) at times when a detached pulse front has just formed. The solid black curve shows the slip rate distribution for each plot. The dashed black lines indicate the boundaries of the pre-existing LVZ. 168 Figure 4.10: Spatio-temporal distribution of slip rate for the case of Fig. 4.6. Various wave speeds of the intact or highly damaged rocks close to the fault (see Fig. 4.7a) are plotted for reference. 169 Figure 4.11: Similar to Fig. 4.5, but with slightly different model parameters: f d =0.2 , C d =13.2c s 0 /R 0 , q= 6.6c s 0 /R 0 , C g = 0.165c s 0 /(µ 0 R 0 ) , C v =0 , β=0.1. The chosen times highlight the moments when the propagating rupture behaves like a crack (t 1 ), with a partially detached pulse front ( t 2 ) or a completely detached pulse front followed by a train of pulses (t 3 ). 171 Figure 4.12: Spatial distribution of slip rate at different times for the case of Fig. 4.11. 172 xviii Figure 4.13: Snapshot distribution of (a) damage and (b) its off-fault variation for the case of Fig. 4.11. Slip rate at the same time is superimposed in (a). The color bar in (b) indicates how the sampling locations are mapped to different colors: from X=60R 0 (blue) to X=85R 0 (red). 172 Figure 4.14: Spatial variation of (a) rupture speed and (b) maximum slip rate for the case of Fig. 4.5. The blue bar in the zoom-in view roughly indicates the location where a detached pulse front has formed for the first time. 174 Figure 4.15: Snapshot distribution of amplitude of particle velocity at the same time as in Fig. 4.7. Two slip rate profiles are superimposed: the solid one corresponds to the same time as in Fig. 4.7, and the dashed one is associated with an earlier time when a completely detached pulse front has just formed. The green line indicates the location of receivers that will be used for analyzing seismograms. 175 Figure 4.16: Velocity seismograms recorded by 40 receivers across the fault (see Fig. 4.15 for reference) in the fault-normal component (panel a) and its zoom-in view (panel b). Receivers are arranged from the extensional side (bottom, starting at Y =−1.5R 0 ) to the compressional side (top, ending at Y =1.5R 0 ). The red and blue highlight the receivers nearest to the fault from the extensional and compressional side, respectively. 177 Figure 4.17: (a) Profiles of slip rate for cases with different Ψ values sampled at the same rupture front location (X /R 0 =131.5). The inset shows the zoom-in view near the rupture front (indicated by the dashed box) where the width of a completely or partially detached pulse front can be defined. For cases with a partially detached pulse front, the effective tail of the pulse is defined as the first local trough in slip rate behind the rupture front . (b) Diagram showing the pulse width as a function of Ψ (indicated by colors) and sampling location (indicated by marker shapes). Solid and open markers represent completely and partially detached pulse fronts, respectively, and the thick dashed line roughly characterizes the boundary between the two. The inset plot in the upper left corner shows schematically the configuration of the genereated LVZ with different Ψ values (for Ψ≥20° ) and rupture propagation distance. 178 xix Figure C1: Off-fault shear bands in simulations with square spectral elements with relatively large (a) and small (b) grids. The inset sketches illustrate the employed non-uniform distribution of internal nodes. Boxes I and II provide zoom- in view of features at relatively small and large propagation distance, respectively. The maximum plastic strain is rescaled in box I but remains unchanged in box II. 207 Figure C2: Off-fault shear bands in simulations with quadrilateral spectral elements with variable angle between the non- horizontal mesh edges and fault plane. The critical rupture length for transition from a possible antithetic to synthetic shear is very small in (a), intermediate in (b) and relatively large in (c). The mesh size is the same for all three cases. The ratio W s /W a in the dashed boxes roughly quantifies the mesh alignment effect (see text for more explanation). Boxes I and II provide zoom-in views of local features at different rupture propagation distances. 209 Figure D1: Numerical test on the resultions of slip rate (see Fig. 4.5 for reference) with two different mesh sizes. 213 Figure D2: Numerical test on the stability and resolution of oscillating slip rate (see Fig. 4.11 for reference) with various model parameters for the normal stress regularization. See the text for the definition of t * . 214 xx Abstract This dissertation mainly investigates mode-II in-plane dynamic ruptures along a frictional interface with off-fault yielding characterized by different rheology models. The goal is to provide improved understanding of off-fault yielding during earthquake ruptures and its contribution to earthquake physics. A detailed parameter-space study is performed to examine properties of the off-fault plasticity-type yielding under various conditions. The simulation results and theoretical analyses show the following features: (1) the off-fault yielding occurs primarily on the compressional and extensional side when the angle Ψ to the fault of the regional maximum compressive stress is low and high, respectively, regardless of the rupture mode; (2) the yielding zone thickness and the associated inelastic energy dissipation rate linearly increase with propagation distance for cracks while they approach constants for pulses in quasi-steady state; (3) the intensity of smoothly distributed plastic strain decays with fault normal distance, while for localized plastic strain in narrow bands the average geometric density of the bands decays with fault normal distance; (5) the local angle to the fault of inferred microcracks (aligned parallel to the mode-I type) is shallower and steeper than Ψ on the compressional and extensional sides, respectively, and increases with rupture speed (in the subshear regime) on the extensional side; (5) the average intensity of plastic strain increases while the yielding zone thickness decreases at greater depth; (6) off-fault yielding contributes to determining rupture directivity along a bimaterial fault, leading to reversed and preserved preferred direction when Ψ is low and xxi high, respectively; (7) during relatively smooth ruptures along typical strike-slip faults, the dominant large-scale secondary shear fractures (on the extensional side) are of the synthetic type, while the minor antithetic set can become prominent with sudden rupture deceleration or termination; on the other hand, abrupt rupture acceleration leads to relative lack of off-fault yielding in the vicinity of the transition regions. Comparisons to other numerical studies, laboratory experiments, and geological or seismological observations are also presented. The following work replaces plasticity by a more realistic brittle damage rheology that can account for temporal changes of elastic moduli inside the yielding zone, with the goal to explore additional feedback mechanisms between the generation of off-fault yielding and dynamic ruptures and their influence on ground motion. The numerical results reveal that off-fault brittle damage can produce dynamic normal stress change along the fault even within an initially homogeneous medium, leading to a possible rupture transition from crack-like in the early stage to a mode with a detached pulse front or a train of pulses in the late stage. Moreover, the reduction of elastic moduli inside the yielding zone creates a waveguide that allows the motion to propagate with little geometric attenuation, such as producing trapped wave signals. 1 Introduction Natural faults typically display an internal structure consisting a core region with one or several localized slip zones that accommodates most fault motion, and surrounding damage zone with fractured and crushed rock products. While the fault core plays the key role in the long-term deformation, characteristics of the broader damage zone contain important information on the stress conditions during failure and the corresponding faulting process. Many theoretical and numerical works have predicted and analyzed off-fault yielding by dynamic ruptures, owing to the increased off-fault stress with the rupture speed approaching its limit level (Poliakov et al., 2002; Rice et al., 2005). Moreover, laboratory experiments, field measurements, and seismological inversions have directly or indirectly shown the signatures of off-fault yielding during lab-based or natural dynamic ruptures. These include quasi-periodic array of tensile microfractures related to the tensile stress during dynamic failure (Griffith et al., 2009), pseudotachylyte injection vein related to shear heating under high slip rate (Di Toro et al., 2005), and temporal changes of rock elastic moduli related to damage generation and healing (Peng and Ben- Zion, 2006). On the other hand, another set of studies have focused on rupture propagation within heterogeneous elastic media, which environment can be possibly produced by the cumulative effects of rupture-induced off-fault yielding. Two end- member cases of such heterogeneities are the inclusion of a low-velocity zone surrounding the fault, and a bimaterial configuration with dissimilar rock properties on 2 different sides of the fault, which are found capable of influencing significantly rupture dynamics and ground motion (Ben-Zion, 2001). Therefore, it is reasonable and important to investigate the interaction between dynamic ruptures and the spontaneous generation of off-fault yielding, and their cumulative effects on the long-term properties of the fault zones and the occurrence of subsequent dynamic events. In this work, we incorporate off- fault yielding with two different types of rheologies into dynamic ruptures to numerically investigate their influences on various observable quantities. Our goal is to provide an improved understanding of earthquake physics and evolution of fault zone properties. In Chapter 1, we employ off-fault plasticity to study some basic properties of off- fault yielding zones, including the dependence of their location and extent on the rupture mode (e.g., crack-like versus pulse-like) and the background stress orientation. We also carried out simulations to explicitly show the energy partition as a function of time among different components and use some simple analytic expressions to explain the numerical results. Finally, we define several observable quantities or functional forms from our basic study on yielding zone properties, which will be used extensively in the following work. In Chapter 2, we follow and extend our procedures and definitions in Chapter 1 to perform a more detailed parameter-space study on the properties of off-fault yielding zones characterized by plasticity. Pre-existing bimaterial interface is also added to examine the competition between off-fault energy dissipation due to yielding and dynamic normal stress change on the influence of rupture directivity. We then combine different sets of parameters, with or without a bimaterial interface, to investigate the likely scenarios of yielding generation at different depth sections. Finally, we construct a 3 table summarizing key predictions on the observable features of yielding zones from different fault models, and provide comparison to laboratory, field and seismological observations. In Chapter 3, we still stick to off-fault plasticity but with a focus on the theoretical explanation of several yielding zone properties. They primarily include the angular pattern of incremental yielding zones around the rupture tip, and the location of cumulative yielding zones in the wake of propagating ruptures. In addition, we address an issue of off-fault synthetic versus antithetic shear (or forward versus backward branching) during dynamic ruptures, and find through our designed simulations that the preferred sense of shear depends strongly on the focused scale of fractures, and whether the main rupture is smoothly propagating or encountering heterogeneities. Our results particularly show prominent signals of large-scale antithetic fractures with a sudden rupture deceleration or arrest, in contrast to their weak signals during relatively smooth rupture propagation. We interpret this difference by using non-local properties of stress field in space at fixed time or in time with changing rupture front configuration. We also find some laboratory experiments and earthquake examples to support our numerical results and the provided theoretical explanation. Finally in Chapter 4, we switch to a more realistic brittle damage rheology allowing for changing elastic moduli to investigate additional feedback mechanisms between dynamic rupture and off-fault yielding. With a physical reduction of elastic moduli in the generated yielding zone (ignoring the post-healing effect), a local bimaterial effect and internal wave reflection can contribute to the determination of rupture mode and ground motion. 4 Chapter 1 Properties of Inelastic Yielding Zones Generated by In-plane Dynamic Ruptures: I. Model Description and Basic Results 1.1 SUMMARY We discuss results associated with 2-D numerical simulations of in-plane dynamic ruptures on a fault governed by slip-weakening and rate-and-state friction laws with off- fault yielding. The onset of yielding is determined by a Mohr-Coulomb type criterion while the subsequent inelastic response is described by a Duvaut-Lions type viscoplastic rheology. The study attempts to identify key parameters and conditions that control the spatial distribution and the intensity variation of off-fault yielding zones, the local orientation of the expected microfractures, and scaling relations or correlations among different quantities that can be used to characterize the yielding zones. In this chapter we present example results for crack and pulse ruptures, along with calculations of energy partition and characteristics of the simulated off-fault yielding zones. Chapter 2 provides a comprehensive parameter-space study of various examined features. In agreement with previous studies, the location and shape of the off-fault yielding zones depend strongly on the angle Ψ of the background maximum compressive stress relative to the fault and the crack vs. pulse mode of rupture. Following initial transients associated with nucleation of ruptures, the rate of various energy components (including off-fault dissipation) linearly increases with time for cracks, while approaching a constant level for pulse-like ruptures. 5 The local angle to the fault of the expected microfractures is generally shallower and steeper than Ψ in the compressional and extensional quadrants, respectively. The scalar seismic potency density decays logarithmically with increasing fault normal distance, with decay slope and maximum value that are influenced by the operating stress field. 1.2 INTRODUCTION Natural fault zones have an internal structure consisting of a core with one or several highly localized (e.g., cm-wide) slip zones that accommodate most of the fault motion, and surrounding damage zone with fractured and crushed rock products such as breccias and cataclasites (e.g., Ben-Zion and Sammis, 2003, and references therein). In large faults, the damage zone has typically significant fracture density over hundred to several hundred meters and it then tapers to properties of the host rocks (e.g., Chester et al., 1993; Caine et al., 1996; Faulkner et al., 2003; Dor et al., 2008). While the fault core plays the key role in the long-term deformation, characteristics of the broader damage zone contain important information on the stress conditions during failure and dynamic properties of earthquake ruptures (e.g., Wilson et al., 2003; Ben-Zion and Shi, 2005; Dor et al. 2006a; Templeton and Rice, 2008; Mitchell and Faulkner, 2009; Dunham et al. 2011a; Huang and Ampuero, 2011). Theoretical studies on off-fault yielding during rapid propagation of shear ruptures on a frictional fault have used analytical approaches and numerical simulations. Poliakov et al. (2002) and Rice et al. (2005) constructed, based on previous studies, non-singular crack and pulse models by balancing the stress intensity factor and the frictional resistance over a finite (process) zone behind the rupture tip. Both works examined properties of the elastically-predicted off-fault secondary fractures by adopting the Mohr- 6 Coulomb criterion. They found that the spatial pattern of potential yielding zones depends strongly on the orientation of the background stress field, the rupture speed, ratio of residual to peak stress, size of slipping patch and whether the rupture is in-plane or anti-plane. For mode-II in-plane ruptures, which are the focus of our study, the extent of the off-fault zones predicted to yield increases considerably with increasing rupture speed close to the limiting subsonic values. The inferred dynamic stress orientation inside the potential yielding zones can be significantly altered from the background and residual levels (with a possible reversed sign of shear component) when the ratio of residual to peak stress is small and the rupture speed approaches the Rayleigh wave speed. Yamashita (2000) and Dalguer et al. (2003) used, respectively, stress- and fracture- energy-based criteria to model the formation of discretely distributed off-fault tensile microfractures. Ando and Yamashita (2007) assumed a hoop shear maximization criterion to model the formation of shear branches nucleated by a propagating rupture at a series of prescribed points along the fault. Andrews (2005), Ben-Zion and Shi (2005), Duan (2008a and b), Ma and Andrews (2010) and others used Mohr-Coulomb or Drucker-Prager type criteria to model dynamic off-fault yielding as continuously distributed plastic strain over the surrounding bulk. These studies represented situations corresponding to large strike-slip faults by assuming that the angle between the background maximum compressive stress and the fault is ° = Ψ 45 , and found that the off-fault yielding is generated primarily on the extensional sides of the faults. Ben-Zion and Shi (2005) noted that the shape of the off-fault yielding zone for crack-like ruptures is triangular, while for steady-state pulse-like ruptures it is approximately constant, owing to the different stress concentrations in these two rupture styles: expanding cracks (or 7 expanding pulses) have increasing size of slipping zone and increasing stress concentration, while steady-state pulses have approximately constant slipping zone and stress concentrations. Templeton and Rice (2008) and Dunham et al. (2011a) performed systematic numerical simulations of off-fault plastic yielding for different values of Ψ and confirmed the theoretical expectations of Poliakov et al. (2002) and Rice et al. (2005): the plastic yielding zone is primarily in the compressional and extensional quadrants when ° < Ψ 20 and ° > Ψ 30 , respectively. The properties of distributed off-fault yielding zones have been examined in various field studies, believed to be closely related to propagation of dynamic ruptures along approximately planar surfaces, and laboratory experiments. Di Toro et al. (2005) documented quasi-periodic arrays of pseudotachylyte injection veins that are mainly on the extensional side, and at high angles (sometimes almost orthogonal), relative to the Gole Larghe fault in the Italian Alps. Rousseau and Rosakis (2003, 2009), Griffith et al. (2009) and Ngo et al. (2012) observed in laboratory experiments of dynamic ruptures along a glued interface between two Homalite samples arrays of tensile microfractures that are quasi-periodically distributed along the rupture path, at certain angles relative to the interface, depending on the loading conditions and rupture speed. The spatial distribution and local orientation of both pseudotachylyte injection veins in the field and tensile microfractures generated in labs can be well explained by theoretical analysis based on Linear Elastic Fracture Mechanics (LEFM) or other non-singular fault models emphasizing the dynamic effects of a rapidly propagating rupture near its tip (e.g., Di Toro et al., 2005; Ngo et al., 2012). 8 Various studies examined the interaction between dynamic ruptures and properties of the bounding host rocks. During propagation of in-plane ruptures along a bimaterial interface separating different elastic solids, there is a coupling between slip and dynamic changes of normal stress Δσ that does not exist in a homogeneous solid (e.g., Weertman, 1980; Andrews and Ben-Zion, 1997; Ben-Zion, 2001). For subshear ruptures the change of σ at the tip propagating in the direction of particle motion in the compliant solid (referred to as the positive direction) is tensile, while the change at the tip propagating in the opposite direction is compressive. The amplitudes of the dynamic changes of Δσ near the rupture tips increase with propagation distance due to continual transfer of energy to shorter wavelengths (e.g., Adams, 1995; Ranjith and Rice, 2001; Ben-Zion and Huang, 2002; Rubin and Ampuero, 2007). The increase in the along-strike asymmetry of the dynamic bimaterial effects with propagation distance lead for various conditions (e.g., Ben-Zion and Andrews, 1998; Shi and Ben-Zion, 2006; Ampuero and Ben-Zion, 2008; Brietzke et al., 2009; Dalguer and Day, 2009) to the development of pulses propagating in the positive direction with strong reduction of normal stress near the propagating tip. Ben-Zion and Shi (2005) simulated dynamic ruptures on a bimaterial interface with constant friction coefficient, off-fault Coulomb plastic yielding and ° = Ψ 45 . In various cases associated with different sets of initial stress values, degree of material contrast and rock cohesion, ruptures evolved quickly to unilateral pulses in the positive direction and off fault yielding occurred primarily in a strip of approximately constant thickness in the extensional quadrant. They proposed that the cumulative effect of multiple such ruptures would produce a strong asymmetry in the distribution of off-fault yielding zones, with most yielding on the side with higher seismic velocity at seismogenic depth. Rubin and 9 Ampuero (2007) suggested from simulations of bimaterial ruptures with slip-weakening friction that asymmetric off-fault yielding may occur also for bilateral cracks, due to the strong asymmetry of the dynamic stress fields near the crack tips propagating in the opposite directions. Duan (2008b) obtained asymmetric yielding in simulations of bilateral crack ruptures on a bimaterial interface with slip-weakening friction, off-fault plasticity and ° = Ψ 45 . However, using in such cases low cohesion representing situations close to the free surface produced significant inelastic strain in both propagation directions. Ampuero and Ben-Zion (2008) showed that with velocity- dependent friction, the feedback between the asymmetric dynamic slip rates and stress drops near the different rupture tips leads to the development of macroscopically asymmetric ruptures. Dor et al. (2006a, 2006b, 2008), Wechsler et al. (2009) and Mitchell et al. (2011) observed with geological mapping and remote sensing data strongly asymmetric damage zones across sections of the San Andreas and San Jacinto faults in southern California, the North Anatolian fault in Turkey and the Arima-Takatsuki Tectonic Line in Japan. Lewis et al. (2005, 2007) observed with seismic trapped and head waves asymmetric damage zones across sections of the San Jacinto and San Andreas faults. These studies documented with multiple signals damage asymmetry in features of fault structures over length scales ranging from cm to several km. In all examined places, considerable more damage was present on the sides of the faults having, based on seismic imaging and laboratory measurements, faster seismic velocities at depth. DeDontney et al. (2011) performed detailed simulations of yielding patterns during propagation of in-plane bimaterial ruptures associated with different values of Ψ . They 10 found, in agreement with previous studies, that the pattern of the simulated yielding zones depends strongly on Ψ , and that for cases with sufficiently shallow Ψ , for which the yielding is primarily on the compressional side, the preferred rupture propagation is in the negative direction. Rudnicki and Rice (2006), Dunham and Rice (2008) and Viesca et al. (2008) considered interactions between dynamic ruptures and contrasts of poroelasticity or permeability inside and outside the fault zones. These fault zone ingredients are not considered in our work. In the present study we attempt to characterize various properties of yielding zones around faults generated spontaneously by dynamic in-plane ruptures with different frictional responses, variable initial stress conditions and rock cohesion values, possible existence of elasticity contrast across the fault, and conditions representing different depth sections. The results are presented in two related chapters. In the current chapter 1 we describe various components of the model and show example simulation results associated with crack- and pulse-type ruptures in a homogenous solid. The results are used to verify the consistency of the computations, through examination of the conservation and partition of energy during propagation of ruptures, and to define quantities that can be used to characterize off-fault yielding zones. These include the location, shape and intensity of the yielding patterns, the local orientation of expected microfractures, the decay of yielding density with normal distance from the fault, and possible scaling relations or correlations among different measurable quantities. In chapter 2 we present a systematic study of effects associated with various model parameters. The results of both chapters help to develop improved quantitative connections between mechanics and field observations of earthquake faults. 11 Figure 1.1: A 2-D model configuration of in-plane rupture along a planar frictional interface. The medium is loaded by a uniform background stress state with angle Ψ of the maximum compressive stress σ max relative to the fault. Symbols “C” and “T” represent the compressional and extensional quadrants in relation to the first motion of P waves from nucleation zone. In simulations incorporating a material contrast, medium-1 and -2 and the stiff and compliant side, respectively. 1.3 MODEL SETUP We aim to numerically simulate dynamic in-plane ruptures and the spontaneous generation of off-fault yielding along strike-slip faults. For a 3-D problem with plane strain assumption, the stress field is represented by ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ = zz yy yx xy xx ij σ σ σ σ σ σ 0 0 0 0 (1.1) with no motion or deformation along the z -axis ( 0 = z u , 0 / = ∂ ∂ z ). During the considered in-plane ruptures (Figure 1.1) the stress components of primary interest are xx σ , yx xy σ σ = and yy σ , while the stress component zz σ evolves with changing plane strain components ε xx and ε yy . According to Anderson's theory for strike-slip faulting, the intermediate principal stress int σ coincides with zz σ , while the maximum principal 12 stress max σ (compressive in the Earth's crust) and the minimum min σ (which is usually compressive but can become tensile) are in the y x− plane. The orientation and relative magnitude of max σ and min σ determine the partition of different stress components (or vice versa) (e.g., Scholz, 2002). In our 2-D simulations, right-lateral rupture is nucleated in a prescribed zone (marked with red color in Figure 1.1) and then is allowed to spontaneously propagate along the frictional fault (black solid line in Figure 1.1). The initial normal and shear stresses on the fault and surrounding medium are 0 0 yy σ σ = and 0 0 xy σ τ = . The relative strength S parameter, defined by d s S τ τ τ τ − − = 0 0 , (1.2) is often used to describe the relative closeness of the initial shear stress to the static yielding level (Andrews, 1976; Das and Aki, 1977). Here ) ( 0 σ τ − = s s f (negative sign for compressive normal stress) is the static shear strength (giving the peak shear stress level) and ) ( 0 σ τ − = d d f is the dynamic shear strength (giving the residual stress level under sliding), with s f and d f being the static and dynamic friction coefficient, respectively. In our all simulations, S is set at a relatively high level such that ruptures are in the subshear regime (e.g., for crack-like ruptures, 77 . 1 > S ). The acute angle between the maximum compressive stress and the fault plane is denoted Ψ . The initial stress state can be expressed in terms of Ψ , S and remote loading (Figure 1.1) as ( ) 0 0 1 yy d s xy S Sf f σ σ − + + = , (1.3) ( ) 0 0 0 0 2 tan 2 1 yy yy xy xx σ σ σ σ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ Ψ − = , (1.4) 13 With fixed friction coefficients 6 . 0 = s f and 1 . 0 = d f , we can change S and Ψ to assign initial values to 0 xy σ and 0 xx σ (e.g., Templeton and Rice, 2008). Once the rupture starts to propagate, the total stress field is the sum of the initial state and the slip-induced increment. A nominal dynamic value of Ψ can be inferred by: ) sgn( 4 ) ( arccos 2 1 2 2 xy xy xx yy xx yy σ σ σ σ σ σ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ + − − = Ψ , (1.5) where Ψ is still defined as the acute angle between the maximum compressive stress and fault plane, and () ⋅ sgn is the sign function accounting for a possibility that the sign of shear stress xy σ can be temporarily reversed by the dynamic effect of a rapidly propagating rupture (this is illustrated in the later Figures 1.11 and 1.12). To allow a possible material contrast across the fault, a subscript “i” is used to represent the Lamé parameters i λ , i µ and mass density i ρ in the lower stiff medium ( 1 = i ) and the upper compliant medium ( 2 = i ). Following the convention used by Ben- Zion and Shi (2005) and Shi and Ben-Zion (2006), we adopt a non-dimensional number 0 ≥ γ to quantify the degree of material contrast as 2 1 2 1 2 1 / / / 1 ρ ρ γ = = = + s s p p c c c c (with the same Poisson’s ratio ν for both sides), where ( ) i i i pi c ρ µ λ / 2 + = and i i si c ρ µ / = are the P- and S-wave speeds in the “i-th” medium, respectively. 1.3.1 Friction Laws Slip occurs when the on-fault shear stress reaches the frictional strength: ( ) σ τ − = f , where f is the friction coefficient and ( ) σ − is the effective normal stress. To study off- fault plastic response under different rupture styles, both slip-weakening friction (SWF) and rate-and-state friction (RSF) with fast weakening are employed to produce crack-like and pulse-like ruptures, respectively. 14 Figure 1.2: A linear slip-weakening friction law used to produce spontaneous crack-like ruptures outside the nucleation zone. D c is a characteristic slip distance for the reduction of the friction coefficient. 1.3.1.1 Linear slip-weakening friction (SWF) Slip-weakening friction laws (e.g., Ida, 1972; Palmer and Rice, 1973; Andrews 1976) have been widely applied to model a single earthquake rupture process as an expanding crack. The concept of a process zone where strength degradation spatially occurs, also called cohesive zone or breakdown zone (e.g., Ben-Zion, 2003), eliminates the stress singularity around the rupture tip and provides a spatial requirement for numerical resolution (e.g., Rice, 1980; Day et al., 2005). In our simulations, a simple form is adopted where the friction coefficient linearly decreases as a function of slip from its static value to the dynamic level: ( ) ⎩ ⎨ ⎧ > Δ ≤ Δ Δ − − = c d c c d s s D u f D u D u f f f f if if / , (1.6) where c D is the characteristic slip distance for the degradation of the friction coefficient (Figure 1.2). When the background shear stress is only slightly greater than the dynamic shear strength ( ) σ τ − ≈ d d f , the size of the process zone R can be estimated (Palmer and Rice, 1973) by ( ) r v f R R II 0 = . (1.7) 15 Here 0 R is the static value of R at zero rupture speed expressed as ( ) d s c f f D R − − = | | ) 1 ( 32 9 0 σ µ ν π , (1.8) and ( ) r v f II is a universal function of the rupture speed r v expressed as ( ) ] ) 1 ( 4 )[ 1 ( ) 1 ( 2 2 2 II s s p s s r v f α α α ν α α + − − − = , (1.9) with 2 2 / 1 p r p c v − = α and 2 2 / 1 s r s c v − = α . The function ( ) r v f II is often identified with the Lorentz contraction effect, since the size of the process zone R shrinks from 0 R to zero as r v increases from + 0 to the limiting speed. In a homogeneous solid, the latter is the Rayleigh wave speed R c for mode-II ruptures. When there is a material contrast across the fault, the limiting speed is (for the range of material contrast in our study) the generalized Rayleigh wave speed GR c (e.g., Weertman, 1980; Ben-Zion, 2001). The associated length scale Rʹ′ has been estimated by Rubin and Ampuero (2007) using the same friction law as τ µ π µ µ Δ = ʹ′ ʹ′ = ʹ′ 32 9 0 c D R R , (1.10) where µ is a function of the material properties and rupture speed r v (Weertman, 1980), 0 µ µ ≡ ʹ′ is an effective static modulus of the bimaterial solid (Rubin and Ampuero, 2007, Equation A8), 0 Rʹ′ is the static value of Rʹ′ , and τ Δ is the stress drop over Rʹ′ . 16 Figure 1.3: A rate- and state-dependent friction law used with velocity-weakening parameters (a<b ) to produce spontaneous pulse-like ruptures outside the nucleation zone. f ss (V) is the steady-state value of friction coefficient under a constant slip velocity V . 1.3.1.2 Rate- and state-dependent friction (RSF) Laboratory studies indicate that rock friction depends not only on slip, but also on slip rate and properties of the contact area (e.g., Dieterich, 1979, 1981; Ruina, 1983; Rice and Ruina, 1983; Marone, 1998; Bizzarri and Cocco, 2003). Moreover, seismic observations suggest earthquakes may rupture not necessarily as cracks but as narrow slip pulses (e.g., Heaton, 1990). Different mechanisms, with and without friction-dominant effects, have been proposed to produce pulse-like ruptures (Ben-Zion, 2001, and references therein). In our study we follow Ampuero and Ben-Zion (2008) and produce pulse-type ruptures using the following rate- and state- dependent friction law with fast weakening: c c s D b V V V a f f + − + + = θ θ , (1.11) 17 where c V and c D are characteristic slip velocity and slip distance (Figure 1.3). The response of the above friction to slip velocity consists of two competing effects. The a- term has a velocity-strengthening mechanism with an instantaneous (direct) response to the change of slip velocity V . The b-term has a velocity-weakening response to the change of V through an evolution (indirect) process, described by a state variable θ : c V τ θ θ − =  , (1.12) where c c c V D / = τ is a characteristic timescale over which friction evolves towards steady state in the current slip velocity regime ss V : c ss ss s ss V V V b a f f + − + = ) ( . (1.13) We set 0 ) ( < −b a so the friction in steady state is effectively velocity-weakening, which decays as 1/V towards a nominal dynamic level ) ( b a f f s d − + = for c V V >> . Due to the dependence of friction on V , the effective slip-weakening distance eff c D is not a constant. With a sudden jump of V from zero to a steady state value of ss V , eff c D increases with ss V (Figure 1.4). This trend is in general consistent with previous studies adopting an aging law for the state variable θ (e.g., Cocco and Bizzarri, 2002; Ampuero and Rubin, 2008), though the specific scaling relation between eff c D and ss V depends on details of the friction laws. 18 Figure 1.4: Effective slip-weakening distance D c eff for rate- and state-dependent friction with a step-like velocity jump from 0 to a constant value V (in steady state). Associated with an “aging law” for the evolution of the state variable θ , D c eff generally increases with the steady-state value of V . The above state variable used in our study (θ ) is related to that of Ampuero and Ben- Zion (2008) (θʹ′) through a unit conversion, c τ θ θ ʹ′ = . Therefore similar analysis on rupture-style transition can be performed by studying the characteristic healing timescale c τ : a rupture pulse is expected to be produced for θ θ τ  / << c (dominance of velocity- weakening behavior, c Vτ θ ≈ ) while a crack is expected for θ θ τ  / >> c (dominance of slip-weakening behavior, V ≈ θ  ). Since the behavior of the friction coefficient in steady state has the same dependence on slip velocity as in Ampuero and Ben-Zion (2008), we can follow the analysis in Appendix-A of that paper. This provides a critical wavelength to be well resolved within a homogeneous medium: 2 dyn ) / ( 1 1 | | ) )( 1 ( V V a b D c c cr + − − ≈ σ ν πµ λ , (1.14) with 19 a b a c a b V R − − − = 2 / | | ) )( 1 ( dyn µ σ ν . (1.15) For a bimaterial fault, the equivalent critical wavelength might be expressed as: 2 dyn ) / ( 1 1 | | ) ( V V a b D c c cr ʹ′ + ʹ′ − ʹ′ ≈ ʹ′ σ µ π λ , (1.16) where µ ʹ′ is the effective modulus which has been discussed for SWF, σʹ′ is the effective normal stress along the bimaterial interface and dyn V ʹ′ is the corresponding version of Equation (2.15) in Ampuero and Ben-Zion (2008) with the replacement of µ ν / ) 1 ( − by µʹ′ / 1 , σ by σʹ′ and R c by GR c ( GR c always exists in our study). Figure 1.5: Time-weakening friction law adopted to artificially nucleate the rupture. (Left) spatial distribution of friction coefficient along strike. (Right) time history of friction coefficient at a fixed point. L 0 is a characteristic length scale for the reduction of friction coefficient and v r is a prescribed outward-propagating rupture speed. 1.3.2 Nucleation Procedure We use a time-weakening friction (TWF) (e.g., Andrews, 1985; Bizzarri, 2010) with a uniform initial stress to artificially nucleate the rupture in a prescribed zone nucl I , during a certain time period ] , 0 [ nucl t : 20 ⎩ ⎨ ⎧ × ∈ ∀ = otherwise , ] , 0 [ } , , { }, , min{ PHY nucl nucl TWF PHY f t I t y x f f f . (1.17) The actual friction coefficient that governs the fault is chosen to be the minimum of the time-weakening friction TWF f and the physical friction PHY f during the nucleation time period ( nucl 0 t t≤ ≤ ) for regions inside the zone nucl I . For regions outside nucl I during the nucleation time period, and for the entire fault beyond that period, the physical friction PHY f determines how the friction coefficient evolves. The physical friction PHY f has the form defined by Eq. 1.6 ( SWF f ) or by Eq. 1.11 ( RSF f ). The prescribed time weakening friction TWF f within the nucleation time period is defined as: ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ − − − = s d r d s f f L r t v f f f f , , ) )( ( max min 0 0 TWF , (1.18) where ) /( 0 0 0 σ τ − = f , 2 0 2 0 ) ( ) ( y y x x r − + − = is the distance from the hypocenter } , { 0 0 y x and r v is the prescribed subshear rupture speed ( s r c v 75 . 0 = ). From the above definition, the friction coefficient linearly decreases at a fixed time from its static level s f at the rupture front to the dynamic level d f over a characteristic length scale 0 L behind the rupture front and is low-bounded by d f (Figure 1.5, left). At a fixed point, the friction coefficient weakens with time and is also low-bounded by d f (Figure 1.5, right). The size of the nucleation zone is determined by the prescribed rupture speed and desired time duration nucl t , which in practice is chosen to be large enough such that the subsequent spontaneous rupture can propagate over a long distance along the fault. 21 1.3.3 Normal Stress Response The Prakash-Clifton regularization of the normal stress response, as required for bimaterial rupture problems (e.g., Cochard and Rice, 2000; Ben-Zion and Huang, 2002), is used in the form proposed by Rubin and Ampuero (2007). Specifically, the fault strength } , 0 max{ * σ τ − ⋅ = f is proportional to a modified normal stress * σ (if it remains compressive, otherwise fault opening will be produced by setting 0 * = σ ), with some time delay in response to abrupt change of the actual fault normal stress σ : ) ( | | * * * σ σ δ σ − + = c V V  , (1.19) where * V is a reference slip velocity and c δ is a characteristic slip distance. As noted by Rubin and Ampuero (2007), the adopted form of * σ in Eq. 1.19 evolves with both time and slip. This can produce a prominent bimaterial effect if the evolution of * σ near the rupture front is rapid compared to that of the friction coefficient. In our study, * V is chosen to be comparable with the expected peak slip velocity near the rupture front and c δ is set to be a moderate fraction of c D (e.g., c c D 6 . 0 = δ ) such that the potential bimaterial effect is neither suppressed nor overly emphasized. 22 Figure 1.6: A schematic diagram of the off-fault Mohr-Coulomb yielding criterion, with φ being the internal friction angle and c denoting the rock cohesion. The solid circle described a trial stress state that may be measured along arbitrarily oriented surfaces and the blue dot (associated with σ xx and σ xy ) represents the stress on the plane normal to the x-coordinate of Fig. 1.1. The dashed circle and its interior represents the elastic regime preserving the current mean stress σ m . The Mohr-Coulomb yielding envelope is characterized by the thick black line. The yielding envelope in the tensile regime is represented by the red line, with T 0 being the yielding strength under tension (see text for more details). 1.3.4 Off-Fault Plastic Response Following Andrews (2005), Ben-Zion and Shi (2005) and later works, the Mohr- Coulomb criterion is adopted in our 2-D study for the onset of off-fault yielding. With this, yielding occurs when the maximum shear stress over all orientations 4 / ) ( 2 2 max yy xx xy σ σ σ τ − + = (1.20) exceeds a pressure-dependent yielding strength ) sin( ) cos( φ σ φ σ m Y c − = , (1.21) 23 where c is the rock cohesion, φ is the internal friction angle and 2 / ) ( yy xx m σ σ σ + = is the mean stress (Figure 1.6). The above criterion may be alternatively expressed by a yielding function Y f σ τ σ − = max ) ( , (1.22) with yielding starting when 0 ) ( = σ f . After the onset of yielding, the Duvaut-Lions type viscoplasticity (e.g., Simo and Hughes, 1998, chapter 2.7; Andrews, 2005; Duan and Day, 2008) is employed to describe the accumulation of plastic strain through the following non-associated flow rule: max max 2 τ τ µ σ τ ε ij v Y p ij T 〉 − 〈 =  . (1.23) Here v T is a viscoplastic timescale over which the stress is relaxed back to the elastic domain (bounded by Y σ ), 2 / |) | ( x x x + = 〉〈 is the ramp function (sometimes also called penalty function) defining the “driving force” for the plastic strain as the excess distance (in proper stress space) of the maximum shear stress max τ over the yielding strength Y σ , and ij kk ij ij δ σ σ τ ) 2 / 1 ( − = (with the Einstein summation convention assumed) are the deviatoric stress components in 2-D through which plastic strain is partitioned into different components (i.e. ij p ij τ ε ∝  ). This non-associated flow rule (with p ij ε  generally not proportional to ij f σ σ ∂ ∂ / ) ( ) also implies that the mean stress m σ does not change during each relaxation step and there is no volumetric change to the plastic strain over the bulk (i.e. 0 ≡ p kk ε ). Such rate-dependent rheology is often used as a regularization of plasticity to avoid or delay the occurrence of strain localization (such as shear band), that 24 is strongly mesh-dependent in numerical studies (e.g., Templeton and Rice, 2008; Dunham et al., 2011). In our study, the viscoplastic relaxation timescale v T is set to be the P-wave travel time over several grid points such that shear localization is effectively avoided (e.g., Andrews, 2005; Duan and Day, 2008). The magnitude of the accumulated plastic strain is quantified with a scalar quantity defined as p ij p ij p ε ε ε 2 0 = (e.g., Ben-Zion, 2003, 2008). This may be called the scalar seismic potency density per unit rupture length (or just potency density). We note that the plastic strain in our study is treated as a real strain tensor reflected by the factor 2 in the denominator of Eq. 1.23. This is similar to the well-known “transformation strain” of Eshelby (1957) that is used in seismology (e.g., Aki and Richards, 2002). Some other studies define the plastic strain using the engineering notation (e.g., Andrews, 2005; Duan, 2008b) without the above factor 2 in Eq. 1.23. A geometric interpretation for the employed yielding criterion and viscoplastic rheology is shown in Figure 1.6. A full description of the trial stress state is characterized by the solid Mohr circle, and is in particular described by the blue dot associated with a normal component xx σ and a shear component xy σ in the ) , ( τ σ stress space. As indicated by the fact that the solid Mohr circle is intersected by the yield envelope, the trial stress violated the yield strength and should be relaxed back towards the elastic domain. The external boundary of the elastic stress, keeping the current mean stress m σ unchanged, is described by the dashed Mohr circle. Depending on the viscoplastic timescale v T , the extent of stress relaxation can be different: for a perfect plasticity ( 0 = v T ), the trial stress will be relaxed exactly onto the boundary of the elastic domain, 25 while for a general rate-dependent viscoplasticity ( 0 > v T ) it will be relaxed to some transition state between its trial state and the fully relaxed state characterizing the external boundary of the elastic domain. Since the relaxation is always performed along the deviatoric stress direction, a specific relaxation path for the stress components xx σ and xy σ is given by the red arrow, starting from the blue dot and pointing to its closest projection onto the boundary of the associated elastic domain (the green dot on the dashed Mohr circle) (e.g., Simo and Hughes, 1998). The above employed Mohr-Coulomb yielding criterion is strictly appropriate only for rocks under absolute compression or partial tension (e.g., min σ may become tensile while max σ still remains compressive) loading conditions (e.g., Etheridge, 1983; Hancock, 1985; Jaeger et al., 2007), when the corresponding fracture can be treated as internal frictional sliding of mode-II shear type. Some field and laboratory observations have reported tensile microfractures (mode-I) or co-existence of both tensile and shear microfractures around the principal slip surface (e.g., Stanchits et al., 2006). This implies that it is more realistic to employ a yielding criterion that covers the compressive regime, the tensile regime, and the transition in between, with some attention to the fact that tensile-type yielding strength (e.g., 0 T in Fig. 1.6) is usually weaker than the shear-type yielding strength (e.g., Willson et al., 2007). However, due to the relative simplicity of the Mohr-Coulomb yielding criterion and the subsequent flow rule, we follow previous works on off-fault inelastic response and use these constitutive laws. As in previous studies, the results may be used to estimate the potential for inducing off-fault mode-I fractures (see section 1.4.3.1). 26 1.3.5 Numerical Method and Parameters We use the 2D spectral element code developed by Ampuero (SEM2DPACK-2.3.8, http://sourceforge.net/projects/sem2d/) to perform dynamic rupture simulations. The domain is discretized into square elements, with 5 Gauss-Lobatto-Legendre (GLL) nodes non-uniformly distributed per element edge. All physical quantities (constants or variables) in our numerical study are normalized so that scaling relations can be easily shown by their apparent values (e.g., Madariaga and Olsen, 2002). As examples, for a self-similar expanding crack, the slip u Δ is expected to scale with the stress drop τ Δ as: 2 2 ) / ( ~ x L u − ⋅ Δ Δ µ τ with µ being the shear modulus and L being the rupture distance measured from the hypocenter; the steady-state slip velocity ss u  Δ far behind the rupture tip is expected to scale with τ Δ as: s ss c u ⋅ Δ Δ ) / ( ~ µ τ  with s c being the S-wave speed (e.g., Scholz, 2002, page 194). Reference values of some fundamental parameters are summarized in Table 1.1. For convenience, we convert physical quantities into dimensionless quantities for plotting the results, as summarized in Table 1.2. To provide a fine resolution for simulations with and without material contrast, an average grid size 0 max min 0625 . 0 2 / ) ( L x x x = Δ + Δ = Δ is used (with x Δ being the distance between two neighbor nodes). This ensures that there are enough numerical cells to resolve the process zone size R (or Rʹ′ ) under SWF, the critical wavelength cr λ (or cr λʹ′ ) under RSF, and the characteristic length scale 0 L under TWF during the nucleation stage. The timestep is determined from the Courant-Friedrichs-Lewy (CFL) stability criterion: 27 min / CFL x t c p Δ Δ = , (1.24) with p c being the fastest P-wave speed and min x Δ being the minimum grid size. In all the simulations, CFL is fixed at 0.55. Table 1.1: Material properties and employed parameters for friction laws, nucleation procedure and viscoplasticity. Parameters Symbols Values Lamé parameters (medium-1 & -2) λ , µ 1, 1 P- and S- wave speeds (medium-1 & -2) p c , s c 1.732, 1 Mass density (medium-1 & -2) ρ 1 Poisson’s ratio ν 0.25 Static friction coefficient (SWF & RSF) s f 0.6 Dynamic friction coefficient (SWF) d f 0.1 Direct effect coefficient (RSF) a 0.001 Evolution effect coefficient (RSF) b 0.501 Dynamic friction coefficient (RSF) b a f f s d − + = 0.1 Characteristic slip distance (SWF & RSF) c D 1 Characteristic slip velocity (RSF) c V 1 Characteristic length scale for nucleation (TWF) 0 L 1 Internal friction angle φ ° 9638 . 30 Time scale for viscoplastic relaxation v T 0.075L 0 /c s = 0.075 Reference stress c σ 1 / 0 = L D c µ 28 Table 1.2: Convention for converting physical quantities (x ) into dimensionless quantities ( ! x ) in 2-D. Length Time Slip Slip rate Strain Energy Energy rate 0 L l l = ʹ′ 0 L tc t s = ʹ′ c D u u Δ = ʹ′ Δ s c c D L u u 0   Δ = ʹ′ Δ c σ εµ ε = ʹ′ 0 L D E E c c σ = ʹ′ s c c c D E E σ   = ʹ′ 1.4 RESULTS Before performing a detailed parameter-space study, we present basic simulations for crack- and pulse-like ruptures with off-fault yielding. The results are used to define various measures of quantifying the generated yielding patterns, and to examine the conservation and partition of energy during rupture propagation. An almost perfect balance among different energy components with off-fault elastic (not shown here but confirmed in our study) and elastoplastic response (section 1.4.2) demonstrates that the numerical code produces reliable results. 1.4.1 Crack- and Pulse-like Ruptures with Off-fault Yielding Figure 1.7 presents example results for an expanding crack (under SWF) and a slip pulse (under RSF). The following parameters are used for both examples: c yy xx σ σ σ 0 . 4 0 0 − = = , , 04 . 1 0 c xy σ σ = , 6 . 0 = s f , 1 . 0 = d f , 90 0 nucl L L = . 0 = c Additional parameters specific to the different friction laws used in the simulations are listed in Table 1.1. A Kelvin-Voigt type viscous damping layer, which is usually added around the fault to reduce numerical oscillations, is temporarily excluded to avoid interference with the energy calculations in section 1.4.2, but is included in later simulations. 29 Figure 1.7: Plastic strain distribution for (a) crack-like rupture and (b) pulse-like rupture under same loading conditions and nucleation procedure, but with different friction laws (see text for details). The intensity of the generated plastic strain is characterized by the scalar potency density ε 0 p = 2ε ij p ε ij p . The slip and slip velocity profiles for the crack and pulse ruptures are shown in (c) and (d). The red curve in (d) shows the maximum slip velocity profile for the pulse case. For the expanding crack in Figures 1.7a and 1.7c, both the off-fault plastic yielding zone and slip profiles follow generally a self-similar pattern with increasing rupture distance (or crack half-length) L . The off-fault yielding zone displays a triangular shape: its thickness T (extent in fault normal direction) scales linearly with L . All the slip profiles plotted at different timesteps with equal time intervals commonly show an overall elliptical shape, with the maximum slip max d in the center proportional to L . For the slip pulse case shown in Figures 1.7b and 1.7d, the growth of both the off- fault yielding extent and the envelope of slip profiles saturates beyond 0 45 ~ L X ± . This reflects a transition from an early crack-like nucleation phase to a spontaneous pulse-like 30 rupture, as more clearly shown by the inset for slip velocity in Figure 1.7d. The saturation for all dynamic quantities (i.e. off-fault yielding zone thickness, size of the slipping patch, maximum slip velocity and final slip at a local point) seems to approach a constant level, indicating that the slip pulse beyond the transition points has evolved into a quasi- steady state. The above results are generally similar to those obtained with off-fault yielding by Andrews (2005) for dynamic cracks and by Ben-Zion and Shi (2005) for pulse-type rupture, and follow up simulations of these types of ruptures. As noted, the key differences between the patterns of quantities generated for the crack and pulse ruptures stem from the fact that in the former class the stress concentration continues to grow with increasing rupture size, while in the latter class it remains (after initial transients) approximately constant (Ben-Zion and Shi, 2005). 1.4.2 Energy Conservation and Partition with Off-fault Yielding As part of the code verification, energy conservation and its partition during dynamic ruptures with spontaneous generation of off-fault yielding are investigated in this section. For the modeled faulting process, the total released elastic strain energy e E vol Δ should be balanced by the change of kinetic energy k E vol Δ , the dissipated plastic energy p E vol Δ and the dissipated frictional energy f E suf Δ : f p k e E E E E sur vol vol vol Δ + Δ + Δ = Δ − , (1.25) where subscripts “vol” or “sur” indicate energy stored in a volume or along the fault surface. For our 2-D in-plane cases, all calculated energy components should be 31 understood as energy per unit length along the anti-plane direction. Calculations for various energy components are performed at a series of timesteps and are stopped before the first P-wave reaches the nearest absorbing boundary, so there is no energy leakage outside the domain. The various energy components appearing in Eq. 1.25 are described in more detail below, with the understanding that all field values are functions of time. Elastic Strain Energy: ∫ Ω ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + = dxdy I I E e 2 2 1 vol 2 µ λ , (1.26) where λ and µ are Lamé parameters, and e kk I ε = 1 and e ij e ij I ε ε = 2 are the first and second elastic strain invariants. The released elastic strain energy e E vol Δ can be calculated by subtracting the value of the above defined quantity at an arbitrary time t from the value at the initial time 0 = t when rupture is initiated. Kinetic Energy: ∫ Ω = dxdy u u E i i k   ρ 2 1 vol , (1.27) where ρ is the mass density and i u  denotes the particle velocity vector. In our study, the medium is initially at quasi-static equilibrium (i.e. the reference kinetic energy k E vol is zero at time 0 = t ), so the change and absolute values of the kinetic energy at any time after rupture’s initiation are equal. 32 Dissipated Plastic Energy: ∫ ∫ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = Δ Ω t p ij ij p dt dxdy E 0 vol ε σ  , (1.28) where ij σ is the stress tensor and p ij ε  is the incremental rate of plastic strain tensor. Dissipated Frictional Energy: ∫ ∫ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ Δ = Δ Σ t i i f dt dx u T E 0 sur  , (1.29) where i T is the traction vector on the fault and i u  Δ is the slip rate vector across the fault. We do not separate between the on-fault fracture energy associated with frictional weakening and pure frictional heat, but simply calculate the total dissipated energy along the fault surface (e.g., Andrews, 2005; Kanamori and Rivera, 2006). Figure 1.8: Various energy components versus time for (a) the crack case and (b) the pulse case of Fig. 1.7. ΔE vol p : plastic energy dissipation, ΔE vol k : change of kinetic energy, ΔE sur f : frictional energy dissipation, ΔE vol e : released elastic strain energy, ΔE vol tot : energy mismatch (see text for details). The reference energy level E(t ref ) for each component is usually chosen as the one at the initial time t ref = 0 , but will be changed to the level right after the nucleation stage t ref =t nucl ≈ 59L 0 /c s when the normalized energy partition ΔE /(−E vol e ) is shown for the pulse case. 33 With the above definitions, we use the two examples of Figure 1.7 to test the conservation and partition of energy during the propagation of the simulated dynamic ruptures. Figures 1.8a and 1.8b present various energy components relative to their initial levels vs. time (solid curves) for the crack and pulse cases. For reference, the energy mismatch defined as ΔE vol tot =ΔE vol e +ΔE vol k +ΔE vol p +ΔE sur f is also plotted (solid purple). As seen, the purple curves in Figures 1.8a and 1.8b coincide closely with the background zero level (black dashed line), indicating that the total energy is almost perfectly conserved during the rupture propagation. The relative energy mismatch ) /( vol tot vol e E E Δ − Δ after the nucleation phase is less than 0.25% and 0.14% for the crack and the pulse cases, respectively. Further calculations including a viscous damping layer can produce a negative change of tot vol E Δ , because the energy loss due to damping is not included in the nominal energy balance equation. Nevertheless, with the assumed damping parameters that are used for later simulations, the results show no significant changes to ) /( vol tot vol e E E Δ − Δ or other relevant quantities compared to the cases without damping. The energy calculations indicate that the expanding crack generates larger kinetic energy than the slip pulse, in agreement with Shi et al. (2008, 2010). This can be seen from both the absolute value of the kinetic energy (solid green) over the entire time period and the relative energy ratio ) /( vol vo e k l E E Δ − Δ (dashed green in the inset) calculated with respect to a specified reference time ref t (28.5% for crack vs. 10.3% for pulse). All the energy components seem to evolve with time in a quadratic form ( ) 2 t E∝ Δ for the crack case, while they apparently evolve with time in a linear form ( ) t E∝ Δ for the pulse case after the initial nucleation phase. These results are further confirmed by plotting the 34 energy rate dt dE E / =  for each component against time in Figure 1.9. For the crack case (Figure 1.9a), E  linearly increases (with a proper sign) with time ( ) t E∝  , while for the pulse case (Figure 1.9b), E  approaches a constant level after the initial nucleation phase ( .) const ≈ E  . Figure 1.9: Energy rate versus time for (a) the crack case and (b) the pulse case of Fig. 1.7. The discussed energy characteristics reflect the differences in stress concentration and slipping zone size for the expanding crack and slip pulse noted earlier. Assuming the crack is expanding with a constant speed and most frictional energy dissipation along the fault goes to heat (both have been numerically confirmed in our study), we can obtain some explicit scaling relations. In such cases, the rate of dissipated frictional energy can be estimated from Eq. 1.29 as ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ Δ ⋅ = Δ = ∫ ∫ Σ t v d i i f r dx u dx u T E 0 sur 2    τ , (1.30) 35 where d τ is the residual shear stress (equals in magnitude to the dynamic frictional strength), and the factor 2 accounts for the fact that the crack is symmetrically expanding into two opposite directions in a 2-D in-plane configuration. Given the overall elliptical slip profile in Figure 7c, we can estimate the slip distribution as 2 2 1 ) ( x t v u r − Δ = Δ µ τ η , (1.31) where 1 η is a dimensionless constant of order 1.09 －1.5 in the sub-Rayleigh regime (Andrews, 2005). The slip velocity distribution can be estimated as 2 2 2 1 ) ( x t v t v u r r − Δ = Δ µ τ η  . (1.32) Putting expression 1.32 into Eq. 1.30, we find that for the crack case f E sur  and t are connected as t t v E r d f ∝ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ Δ ≈ 2 1 sur µ τ πτ η  , (1.33) where each quantity in the parentheses of 1.33 is approximately constant with time. In deriving Eqs. 1.31-1.33, we ignored the effects of the finite process zone and off- fault yielding, but these effects do not change our order of magnitude scaling estimate between f E sur  and t . Scaling relations between other energy rate components and time can be obtained from results of Andrews (2004, 2005) and Templeton (2009, chapter 4) by variable substitution between time t and crack half-length L through t v L r ≈ . 36 Figure 1.10: (a) Slip velocity as a function of space and time for the pulse case of Fig. 1.7b. (b) Snapshots of normalized shear traction, slip and slip velocity (only the right half is shown because of the symmetry). The propagation speed at the rupture front and the healing front are denoted by v r and v h , respectively and h denotes the pulse width. Except or the bounded maximum level, lim x→v r t Δ u∝1/ v r t−x behind the rupture front and lim x→v r t τ∝1/ x−v r t ahead of the rupture front. Around the healing front, ∂u /∂x is continuous, (∂u/∂x) x=v h t = 0 , and lim x→v h t Δ u∝ x−v h t ahead of the healing front. For the pulse case, since a quasi-steady state is implied by the distribution of plastic strain (Figure 1.7b) and the slip (or slip velocity) profiles (Figure 1.7d), one can naturally expect that each energy rate component is invariant under time translation after the nucleation stage, at least for the time period that is studied here. In particular, the features shown in Figure 1.10 (see figure caption for additional details) suggest that the slip velocity for the pulse case may be approximated by a Yoffe type function (Broberg, 1999; Nielsen and Madariaga, 2003): x t v t v h x c u r r s − − + Δ = Δ µ τ η 2  , (1.34) 37 where h is the pulse width (Figure 1.10b) and 2 η is a dimensionless function of propagation speeds at the rupture front and healing front. As implied by Figure 1.10a, both h and 2 η are expected to have constant values. Using Eq. 1.34, we can express the energy rate f E sur  for the pulse case as: ∫ ∫ − Σ − − + ⋅ ⋅ Δ = Δ = t v h t v r r s i i f r r dx x t v t v h x c dx u T E τ µ τ η 2 sur 2   . (1.35) As shown in Figure 1.10b, the shear traction τ has a relatively flat profile (compared to the profile for u  Δ ) over the major portion of the slipping zone (not close to the rupture front), suggesting that the effective value of τ in the integrand of Eq. (1.35) may be treated as a constant s d τ τ τ 26 . 0 } max{ 26 . 0 eff = ⋅ ≈ . With this, Eq. (1.35) can be approximately expressed as const. eff 2 sur = Δ ≈ µ τ τ π η s d f c h E  (1.36) The dynamic behavior of the examined steady-state pulse with elastoplastic response (and related earlier results) is different from that of the self-similar subshear pulse reported by Shi et al. (2008) with purely elastic response. Their pulse has triangular- shaped slip profile and increasing energy rate (i.e., a growing pulse), while ours has almost flat slip profile beyond the nucleation zone and approximately constant energy rate (i.e. a slip pulse in quasi steady-state). We note that both steady-state and growing pulses are possible rupture modes that can be produced (in addition to what is discussed above) by changing the stress condition or other model parameters (e.g., Dunham et al., 2011). The results in Figures 8 and 9 indicate higher kinetic energy ) /( vol vol e k E E Δ − Δ for 38 the two rupture modes than the values calculated by Shi et al. (2008). This stems from the relatively large stress drops in our simulations associated with dynamic friction coefficient of 0.1. Our calculated kinetic energy is also considerably higher than the about 5% or less estimates of radiated seismic energy based on seismological and laboratory data (e.g., McGarr, 1999; Fulton and Rathbun, 2011). This is related to the fact that the radiated seismic energy is estimated with certain observable (far field) quantities not reflecting the entire kinetic energy in the volume. 1.4.3 Influence of Background Stress Orientation Ψ and Rupture Style To illustrate the sensitivity of the results to some parameters and define quantities that can be useful for classifying and understanding the generated patterns, we consider cases with ° = Ψ 10 and ° 45 . 1.4.3.1 Predicted results in elastic medium Here we use results for a purely elastic solid (no off-fault yielding) to develop connections between two aspects of mode-I and II fractures: the overall spatial distribution and the local microfracture orientation. The inferred fractures of a certain type (mode-I or -II) are predicted based on an appropriate criterion. One of the advantages of using simulations in elastic solid is that it allows considerations of many physical possibilities associated with different yielding criteria and different background stresses. In addition, the predicted results in elastic solid provide a reference for results with generation of off-fault yielding. 39 Figure 1.11: Predicted pattern of off-fault mode-I tensile fractures (single black bar) and mode-II shear fractures (conjugate black bars: thick and thin for right-lateral and left- lateral shear fractures, respectively) for cracks expanding within an elastic medium. The value of v r indicates the instantaneous rupture speed at the time of the snapshot. Ψ=10° for (a) and (b), Ψ=45° for (c) and (d). Following Poliakov et al. (2002) and Rice et al. (2005), we investigate properties of potential off-fault secondary fractures that may be induced by crack (Figure 1.11) and pulse-like (Figure 1.12) ruptures. The initial fault normal and shear stress conditions, and frictional parameters are the same as used to produce the crack and pulse cases of Figure 1.7. The simulations below are done for rupture size 0 200L L= , and the stress component 0 xx σ varies in relation to the associated Ψ (Eq. 1.5). The following criteria are used to designate the likely mode-I and mode-II off-fault fractures. If 0 min ≥ σ the yielding is assumed to be associated with mode-I fractures, while if Y σ τ ≥ max (with an assumption of zero rock cohesion) the yielding is designated as involving mode-II failures. Such assumptions in adopted yielding criteria might be appropriate for brittle rocks that are pre-damaged or for granular materials under low confining pressures. 40 Figure 1.12: Similar to Fig. 1.11 for slip pulses under same initial stress conditions, nucleation procedure and rupture distance. When comparing the predicted patterns for crack and pulse rupture styles, we first notice that the extent of the potential off-fault yielding zones (associated with either mode I or II) is larger for the expanding crack than for the slip pulse. This is again a consequence of the different stress concentrations produced by crack- and pulse-like ruptures (e.g., Ben-Zion and Shi, 2005). When focusing on the same rupture style, we find that the predicted off-fault mode-II yielding zones can extend to both sides of the fault, with partition depending strongly on Ψ . On the other hand, the predicted mode-I yielding zones are primarily located on the extensional side for both low and high angles of Ψ . On the extensional side where both yielding criteria are commonly exceeded, the predicted distributions of both modes I and II fractures have a similar pattern that may be characterized by the overall orientation of several yielding zone lobes, and by the magnitude decay of the strength excess with distance from the rupture tip inside each lobe. Based on this connection, we may utilize the information on the distributed off-fault mode-II shear fractures to estimate the spatial extent (the upper bound) and the off-fault density decay of the expected mode-I tensile fractures, at least for some locations on the extensional side of the rupture. 41 In addition to the overall relation of spatial distribution, modes I and II fractures may also be connected by their local orientations. A tensile fracture is usually oriented parallel to the direction of the local maximum compressive stress max σ . A pair of conjugate shear fractures are usually generated with an angle of ) 2 / 4 / ( φ π − ± relative to the local direction of max σ , where φ is the internal friction angle (e.g., Scholz, 2002; Jaeger et al., 2007). Therefore, with known or estimated information about the stress field, one can predict the orientation of the expected fractures. Conversely, with known information about the generated fractures, one can infer the transient stress field during the failure process. This should hold as long as the distributed fractures of a given type, tensile or shear, are individually negligible in size compared to the main fault and do not incoherently intersect (so the local stress field can be uniquely inferred) during the same failure process. Such fractures will be referred to as microfractures and are expected to be able to reflect the local stress field. Based on this second microscopic connection, we may use later equivalent mode-II shear microfractures, through simulated distributed plastic strain, to infer the transient maximum compressive stress orientation. This should be consistent with studies modeling mode-I tensile microfractures, as long as the rupture speed and other relevant dynamic properties of the main rupture remain similar. 1.4.3.2. Simulation with generation of off-fault yielding In this section we discuss and compare basic results with generation of off-fault yielding to the elastically-predicted results of the previous section. Except for the difference in off-fault response (elastoplastic vs. elastic), all the other conditions, procedures and parameters are similar to those of section 1.4.3.1. 42 Figure 1.13: Distribution of (a) cumulative plastic strain for Ψ=10° , (b) same as (a) for Ψ=45° , (c) equivalent plastic strain increment for Ψ=10° , (d) same as (c) for Ψ=45° for expanding cracks similar to the cases in Fig. 1.11, but with off-fault yielding. Figure 1.13 presents the distribution of the cumulative plastic strain and the equivalent plastic strain increment (through Eq. 1.23) for crack-like ruptures that correspond to the cases of Figure 1.11. As seen, the cumulative plastic strain (Figures 1.13a and 1.13b) is mainly located on the compressional or extensional side when Ψ is low (e.g., ° = Ψ 10 ) or intermediate to high (e.g., ° = Ψ 45 ), respectively, in agreement with previous studies (e.g., Templeton and Rice, 2008; Dunham et al., 2011a). The dependence of the location of off-fault plastic yielding zones on Ψ reflects the stress interaction between the slip-induced incremental stress field ij σ Δ (which does not vary too much for different cases based on LEFM) and the background stress field 0 ij σ (which varies considerably for a change of Ψ by ° 35 ). On the other hand, the triangular shape 43 of the plastic yielding zones in both cases indicates that the stress concentration near the crack tip increases with rupture size regardless of whether Ψ is low or high. Comparing the equivalent plastic strain increment (Figures 1.13c and 1.13d) with the predicted results for elastic medium (Figure 1.11), we find that the spatial pattern of the plastic activation zone is more localized than the elastically-predicted zone for cracks with a similar size. Moreover, the sense of horizontal shear can be reversed (i.e. from right-lateral to left-lateral) in the elastically-predicted yielding zone lobes ahead of the rupture tip, while a reversal can be hardly seen within actual off-fault yielding regions. Finally, during generation of off-fault plastic yielding with the employed parameters, the strength excess ) ( max Y σ τ − can only remain at a relatively low level (it will be exactly set at zero with a perfect rate-independent plasticity), while the elastically-predicted strength excess can stay at a higher level. These differences may be explained by variations in rupture speed that can affect the slip-induced incremental stress field (e.g., Poliakov et al., 2002), and the adopted rheology which determines how (e.g. the flow rule) and to what extent (e.g. the effective viscosity) the stress components exceeding the strength are relaxed. In addition to these results on the distribution of plastic strain along strike or near the rupture tip, we wish to investigate local orientations of the expected microfractures in relation to the simulated plastic strain. This requires establishing connections between discretely distributed microfractures and continuously distributed plastic strain. Our modeled plastic deformation represents brittle damage zones consisting of distributed shear microfractures, whose individual size and ability to concentrate stress are negligible compared to the main rupture. The simulated plastic strain at each location can be treated 44 as a test-particle response that reflects the transient stress field (with some modification) at the same location during the failure process. The reflected stress field is not necessarily the same as the state when plastic yielding just occurs, but should be understood as an average over the entire (short) period of the accumulation of plastic strain. The inferred stress field may be compared with inferences made with other modeling approaches. Following the above considerations, we re-examine the results in Figure 1.13 with a focus on the local orientation of the expected microfractures. To facilitate the comparison with other studies, instead of plotting pairs of conjugate shear microfractures at a point, we plot the average orientation of the inferred maximum compressive stress during the time when plastic strain is accumulated. The most-favored orientation for each expected shear microfracture can be obtained by rotating ° 30 ~ clockwise (for a left-lateral shear) or anti-clockwise (for a right-lateral shear) from the orientation of the inferred maximum compressive stress. We infer the principal stress orientation during the yielding process with a relation similar to Eq. 1.5 as follows. Based on Eq. 1.23, we assume the final accumulated plastic strain is proportional to the average deviatoric stress during the yielding process: ij kk ij ij p ij δ σ σ τ ε ) 2 / 1 ( − = ∝ . (1.37) Using Eq. 1.37 in Eq. 1.5 and applying the model assumption that there is no volumetric change for plastic strain, we get: ) sgn( ) ( ) ( arccos 2 1 2 2 p xy p xy p xx p xx ε ε ε ε ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ + − = Φ , (1.38) 45 where Φ is the inferred average orientation of the maximum compressive stress during the yielding process. In practice, to reduce possible numerical errors and displaying results, different plastic strain components are integrated over a spectral element (a 0 0 25 . 0 25 . 0 L L × square) and the integrated quantities are used to infer the stress orientation. A color scale is used to show the relative normal distance ⊥ d between the center of off-fault elements and the fault, starting from 0 375 . 0 L d = ⊥ (red) and ending at 0 875 . 4 L d = ⊥ (blue). Figure 1.14: Inferred microfracture orientation (aligned to the direction of the maximum compressive stress ! σ max during failure) for the crack ruptures of Fig. 1.13 with (a) Ψ=10° and (b) Ψ=45° . Φ is defined as the acute angle between the inferred orientation of ! σ max and the fault plane. A positive value of Φ indicates that a local horizontal right-lateral slip can be promoted by ! σ max whereas a negative value of Φ favors a local horizontal left-lateral slip. The color scale represents the distance between the center of off-fault elements and the fault (see text for details). Figure 1.14 shows the inferred microfracture orientation for the crack cases of Figure 1.13. We find that the inferred orientation strongly depends on the location with respect to the hypocenter. Φ is generally shallower than Ψ in the compressional quadrant for ° = Ψ 10 , and can become steeper than Ψ in the extensional quadrant for 46 both ° = Ψ 10 and ° = Ψ 45 . These results are generally consistent with previous model predictions emphasizing the role of a process zone around the rupture or fault tip (e.g., Scholz et al., 1993; Vermilye and Scholz, 1998), and also compatible with other studies on the generation of off-fault tensile microfractures (e.g., Yamashita, 2000; Griffith et al., 2009). This general tendency of Φ being shallow in the compressional quadrant and steep in the extensional quadrant may reflect the characteristics of the slip-induced incremental stress field. In particular, the strong asymmetry in the distribution of Φ on the different sides of the fault may be related with the anti-symmetric angular variation of the normal stress change across the fault (e.g., Freund, 1990). Figure 1.15: Decay of potency density ε 0 p with fault normal distance d ⊥ for the crack cases of Fig. 1.13. (a) Ψ=10° , on the compressional side. (b) Ψ=45° , on the extensional side. Sampling locations are linearly mapped into a color scale, starting from X=60L 0 (blue) and ending at X=180L 0 (red) . T 180 represents the local thickness of off-fault yielding zone measured at the location X=180L 0 . More detailed are discussed in the text. Another informative characteristic that has been measured in the field is the reduction of fracture density and other measures of damage intensity with distance from the fault (e.g., Wilson et al., 2003; Mitchell and Faulkner, 2009; Savage and Brodsky, 47 2011). To quantify this with our simulation results we examine the functional form that describes in different cases the relation between the potency density p 0 ε and normal distance from the fault. We assume p 0 ε is a good approximation of the fracture density evaluated in the field, because both quantities are correlated with the stress field around the fault. Figure 1.15 shows the off-fault decay of p 0 ε with ⊥ d for the crack cases of Figure 1.13. The color scale indicates the sampling location along strike starting from 0 60L X = (blue) and ending at 0 180L X = (red). The sampling locations are neither within the nucleation zone nor too close to the rupture tip at 0 200L X = . At each location (given color), data points are sampled normal to the fault on the side where plastic strain is primarily distributed. To avoid the singularity at 0 = ⊥ d in the semi-log scale of the plot, and to focus more on off-fault (rather than on-fault) properties, we choose 0 25 . 0 L d = ⊥ as the starting point for the fault-normal-distance. The results in the compressional quadrant for ° = Ψ 10 (Figure 1.15a) and in the extensional quadrant for ° = Ψ 45 (Figure 1.15b) suggest consistently that the potency density decays logarithmically with fault normal distance in regions not close to the edge of the off-fault yielding zones where p 0 ε tapers to zero. The obtained ) log( 0 ⊥ − ∝ d p ε form is in good agreement with other numerical simulations (e.g., Yamashita, 2000) and field observations at several locations (e.g., Vermilye and Scholz, 1998; Chester, et al., 2005; Mitchell et al., 2011). We note that Mitchell and Faulkner (2009) and Savage and Brodsky (2011) quantified observed fault-normal decays of fracture density using exponential and power law forms, respectively. However, these were not necessarily the best-fitting forms but rather preferred choices based on intended use (Mitchell and 48 Faulkner, 2009) or theoretical expectation (Savage and Brodsky, 2011). The decay slope in the simulated results seems to be independent of the sampling location along strike for a given rupture case (reflected by the sub-parallel color stripes), but can vary with the value of Ψ . The maximum value of p 0 ε for ° = Ψ 10 is higher than that for ° = Ψ 45 at the same location. This difference may be explained by the overprinting of the two yielding lobes in the compressional quadrant around the rupture tip for ° = Ψ 10 (Figure 1.13c), compared to the single yielding lobe in the extensional quadrant for ° = Ψ 45 (Figure 1.13d). The possible boundary of the overprinting region for ° = Ψ 10 is indicated by the dashed grey lines in the inset of Fig. 1.15a. Additional results related to the off- fault decay of p 0 ε are discussed in the follow up chapter 2. Figure 1.16: Distribution of (a) cumulative plastic strain for Ψ=10° , (b) same as (a) for Ψ=45° , (c) equivalent plastic strain increment for Ψ=10° , (d) same as (c) for Ψ=45° for slip pulses similar to the cases in Fig. 1.12, but with off-fault yielding. 49 Figure 1.16 shows the distribution of the cumulative plastic strain and the equivalent plastic strain increment for pulse-like ruptures that correspond to the pulse cases of Figure 1.12. As before, the cumulated plastic yielding zones have approximately constant thickness after the initial nucleation stage. The equivalent plastic activation zones (Figures 1.16c and 1.16d) are more confined to the rupture tip than in the crack case of Figures 1.13c and 1.13d. Figure 1.17 shows the inferred microfracture orientation for the pulse cases of Figure 1.16. Similar to the results for the crack cases (Figure 1.14), the average value of Φ is lower and higher than Ψ for locations at the compressional and extensional quadrant, respectively. Since the thickness of the off-fault yielding zone is relatively narrow for pulse-type ruptures, we do not present results on the decrease of p 0 ε with ⊥ d as done for the crack cases. Nevertheless, our trial functional fits to the simulated results suggest that ) log( 0 ⊥ − ∝ d p ε also for the pulse cases. Figure 1.17: Inferred microfracture orientation for the pulse cases of Fig. 1.16. 1.5 DISCUSSION We simulate in-plane dynamic ruptures on a frictional fault governed by slip- weakening and rate-and-state friction laws and Mohr-Coulomb off-fault yielding. The 50 model can be used to perform a detailed parameter-space study on various characteristics of the yielding zone generated by ruptures associated with different constitutive laws and parameters, different initial background stress, and different elastic moduli of the solids bounding the fault. Example simulations are used to demonstrate basic properties of crack and pulse ruptures and to define several measures that can help quantifying properties of off-fault yielding zones. These include the location, shape and intensity of the yielding zones, the expected orientation of modes I and II microfractures with respect to the fault, the functional decay form of yielding intensity with normal distance from the fault, and scaling relations or correlations among sets of quantities. The consistency of the numerical code is verified by the fact that energy conservation is satisfied to a very high degree. We confirm results of previous studies that the location and shape of the off-fault yielding zones depend strongly on Ψ and the crack vs. pulse mode of rupture. When Ψ is larger than about 35º, representing the likely situation for the San Andreas and other large strike-slip faults, the off-fault yielding zone produced by either crack or pulse rupture is primarily on the extensional side of the fault, in agreement with Andrews (2005), Ben-Zion and Shi (2005) and others. When Ψ is low (e.g., 15º), representing likely conditions for thrust faults, the off-fault yielding produced by either crack or pulse rupture is primarily on the compressional side, in agreement with Templeton and Rice (2008), Dunham et al. (2011a) and others. The spatial pattern of the yielding zone increment around the rupture tip and cumulative plastic strain along the fault depend strongly on the rupture style, reflecting differences in the stress concentration around the tips of cracks and pulses with propagation distance (Ben-Zion and Shi, 2005). 51 The energy partition among different components is influenced strongly by the rupture style under similar stress conditions and nucleation procedure. The energy rate E  for various components increases linearly with time for a self-similar crack expanding at a constant rupture speed, while it approaches a constant level for a slip pulse under quasi steady-state propagation. These energy characteristics are consistent with the generation of the yielding zones and slip (or slip velocity) profiles for the studied rupture cases (Figure 1.7). The relative energy partition ) /( vol e E E Δ − Δ for both rupture modes approaches in our study a constant value with time. The specific value of the constant depends on the rupture style (see the inset in Figure 1.8) and may depend also on other model parameters (Shi et al., 2008). As expected, a considerable portion of the initial elastic strain energy is transformed during ruptures into frictional heat and plastic dissipation (although less in our calculations than estimated in other studies as noted in section 1.4.2), and expanding cracks produce larger kinetic energy in the bulk than slip pulses. For both the crack- and pulse-like ruptures examined in this chapter, the angle Φ of the inferred transient maximum compressive stress during failure is lower and higher than Ψ on the compressional and extensional sides, respectively. The plastic potency density p 0 ε decays logarithmically with fault normal distance ⊥ d . These results are in good agreement with field observations (e.g., Vermilye and Scholz, 1998; Mitchell et al., 2011) and numerical simulations of off-fault tensile microfractures (e.g., Yamashita, 2000). A more complete characterization of fault yielding zones in the context of the adopted model requires a detailed parameter-space study and comparisons of predicted features with laboratory and field observations. This is done in the following chapter 2. 52 Chapter 2 Properties of Inelastic Yielding Zones Generated by In-plane Dynamic Ruptures: II. Detailed Parameter-space Study 2.1 SUMMARY We perform a detailed parameter-space study on properties of yielding zones generated by 2-D in-plane dynamic ruptures on a planar fault with different frictional laws and parameters, different initial stress conditions, different rock cohesion values, and different contrasts of elasticity and mass density across the fault. The focus is on cases corresponding to large strike-slip faults having high angle ( ° = Ψ 45 ) to the maximum compressive background stress. The simulations and analytical scaling results show that for crack-like ruptures (1) the maximum yielding zone thickness max T linearly increases with rupture distance L and the ratio L T / max is inversely proportional to 2 ) 1 ( S + with S being the relative strength parameter; (2) the potency density p 0 ε decays logarithmically with fault normal distance at a rate depending on the stress state and S; (3) increasing rock cohesion reduces L T / max , resulting in faster rupture speed and higher inclination angle Φ of expected microfractures on the extensional side of the fault. For slip pulses in quasi-steady state, T is approximately constant along strike with local values correlating with the maximum slip velocity (or final slip) at a location. For a bimaterial interface with ° = Ψ 45 , the energy dissipation to yielding contributes to developing macroscopically asymmetric rupture (at the scale of rupture length) with the 53 same preferred propagation direction predicted for purely elastic cases with Coulomb friction. When ° = Ψ 10 , representative for thrust faulting, the energy dissipation to yielding leads to opposite preferred rupture propagation. In all cases, Φ is higher on average on the compliant side. For both crack and pulse ruptures with ° = Ψ 45 , T decreases and p 0 ε increases for conditions representing greater depth. Significant damage asymmetry of the type observed across several large strike-slip faults likely implies persistent macroscopic rupture asymmetry (unilateral cracks, unilateral pulses, or asymmetric bilateral pulses). The results on various features of yielding zones expected from this and other studies are summarized in a table along with observations from the field and laboratory experiments. 2.2 INTRODUCTION The internal structure of fault zones reflects processes and conditions that have operated during the fault history, and can affect various aspects of earthquakes and seismic radiation in the future. It is therefore important to understand the relations between properties of yielding zones around faults and different types of fault motion (e.g. crack/pulse ruptures and aseismic failure on planar and rough surfaces). The goal of this work is to contribute toward such understanding with a systematic investigation of characteristics of yielding zones generated by dynamic ruptures on a planar frictional interface. Chapter 1 described the computational method and defined several measures that can be used to quantify properties of yielding zones using example results. In this chapter (2) we perform a detailed parameter-space study on various features of off-fault yielding generated by different types of dynamic ruptures associated with different 54 frictional laws and parameters, different initial conditions, and different elastic and cohesion parameters of the surrounding media. Figure 2.1: Schematic diagram showing the migration of rupture tips along a planar fault, modified from Scholz et al. (1993). “C” and “T” represent the compressional and extensional quadrant(s), respectively. In the presence of a material contrast across the fault, the slip direction on the compliant side will be referred to as the positive direction, and the quadrants in the same or opposite directions will be distinguished by “+ /− ” signs. (b) Schematic diagram of a rough fault with geometric complexities, modified from Saucier et al. (1992) and Chester and Fletcher (1997). For both (a) and (b), the big arrows indicate the orientation of the far field background maximum compressive stress σ max while the small thin arrows represent the orientation and relative magnitude (indicated by the length) of the near-fault (dynamically or quasi-statically) perturbed maximum compressive stress ! σ max . The example results in chapter 1 and previous studies (e.g., Ben-Zion and Shi, 2005; Templeton and Rice, 2008) demonstrated that the rupture style (crack vs. pulse) and angle Ψ of the background maximum compressive stress relative to the fault influence strongly the distribution of off-fault yielding zones. In section 2.3 of this chapter we investigate the roles of other parameters (relative strength parameter S, rock cohesion, contrast of elasticity), and combinations of parameters corresponding to certain physical situations, in controlling rupture dynamics and properties of yielding zones including 55 their location, extent, intensity, symmetry properties, microfracture orientations and decay with fault-normal distance. The obtained results are used to develop correlations and scaling relations among different manifestations of yielding zones. In section 2.4 the findings are discussed in relation to other models (e.g., fault motion on a rough surface) and observations from the field and the laboratory. The results provide improved criteria for interpreting various features of yielding zones around large strike-slip faults in terms of properties and conditions of earthquake ruptures on the faults. 2.3 Detailed Parameter-space Study Fig. 2.1a presents the geometry and several basic ingredients of the employed model. Fig. 2.1b illustrates aspects of a model with fault roughness that will be discussed in comparison with the obtained results. Since pulse-like ruptures are more sensitive than crack-like ruptures to small changes of nucleation procedure, initial stress state, fault frictional properties and other ingredients (e.g., Zheng and Rice, 1998; Ampuero and Ben-Zion, 2008; Shi et al., 2008; Dunham et al., 2011a), we mainly use crack ruptures to clarify the basic effects of various parameters. Results from both rupture styles will be presented only when additional distinct features are seen for rupture pulses. We are primarily interested in yielding zones associated with large strike-slip faults, so the angle Ψ will be generally fixed at ° 45 unless mentioned otherwise. For convenience, Table 2.1 summarizes key equations from chapter 1 that are used frequently in this work. The parameters specifying material properties, friction laws, nucleation procedure and viscoplasticity have same values as in chapter 1 (Table 1.1), except in cases of bimaterial ruptures where the elastic moduli and mass density of medium 2 are reduced. As in chapter 1, we provide normalized values of physical quantities. 56 Table 2.1: Key equations. d s S τ τ τ τ − − = 0 0 (2.T1) ( ) 0 0 1 yy d s xy S Sf f σ σ − + + = (2.T2) ( ) 0 0 0 0 2 tan 2 1 yy yy xy xx σ σ σ σ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ Ψ − = (2.T3) ) sgn( 4 ) ( arccos 2 1 2 2 xy xy xx yy xx yy σ σ σ σ σ σ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ + − − = Ψ (2.T4) ) sgn( ) ( ) ( arccos 2 1 2 2 p xy p xy p xx p xx ε ε ε ε ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ + − = Φ (2.T5) d s τ τ & : static & dynamic shear strength of the fault; 0 τ : initial shear stress. d s f f & : static & dynamic friction coefficient ( 6 . 0 = s f , 1 . 0 = d f ). Ψ : angle of the maximum compressive stress relative to the fault (based on the stress tensor ij σ ), specified as the angle for the background stress if using 0 ij σ . Φ : angle of the expected microfracture (aligned to the mode-I type) relative to the fault (based on the plastic strain tensor p ij ε ). 2.3.1 Influence of the S Parameter The relative strength parameter S, defined as the ratio between strength excess and dynamic stress drop (Eq. 2.T1), is a key quantity connecting the background initial stress with fault frictional properties (Andrews, 1976; Das and Aki, 1977). In natural fault settings the S parameter may vary with the ratio of differential stress to the confining pressure, the values of the static and dynamic friction coefficients, and changing strike of fault segments in an approximately uniform regional stress field (e.g., Lockner and 57 Beeler, 2002; Shi et al., 2008; Oglesby et al., 2008). As in chapter 1, we consider the range of values 77 . 1 > S known to generate ruptures with subshear speed, which is typical for most earthquakes. For the following investigation, the rock cohesion is set at 0 = c . To carefully investigate how the S parameter can influence the yielding zone properties, we choose the size of the applied nucleation zone nucl L somewhat larger than the critical value c L leading to dynamic instability (Palmer and Rice, 1973): 2 0 ) ( 3 16 d c c G L τ τ µ π − = , (2.1a) where µ is the shear modulus, c G is the fracture energy, and Poisson’s ratio is assumed to be 0.25. We then fix the ratio of the applied nucleation size nucl L to the critical size c L , so that comparable amount of energy is released in rupture nucleation for cases with different S values. The fracture energy evaluated from the evolution of stress as a function of slip during the breakdown process for slip-weakening friction (SWF) is c d s n c D f f G ) ( | | ) 2 / 1 ( − = σ , where c D is the slip-weakening distance. The critical size can be rewritten as 2 ) 1 ( ) ( | | 3 8 S f f D L d s n c c + − = σ µ π . (2.1b) As the S value increases, the magnitude of the background normal stress has to increase to retain a comparable amount of released energy over the same nucleation length. 58 Figure 2.2: Plastic strain distribution for crack-like rupture with Ψ=45° and different values of S. The intensity of the plastic strain is quantified by the seismic potency density ε 0 p = 2ε ij p ε ij p . The background normal stress σ 0 differs to preserve comparable energy release from the nucleation zone. In (d), r and θ are polar coordinates with origin at the moving rupture tip, and ω is a conventional angle quantifying the incremental rate of yielding zone thickness with the along-strike distance from the hypocenter (i.e., tan(ω)≈ΔT /ΔX ). 59 2.3.1.1 Yielding zone extent and decay form Figure 2.2 shows the distribution of plastic strain for crack-like rupture cases associated with different S values. As seen, the extent of the plastic yielding zone is relatively wide when the S value is relatively low but can be highly suppressed when S becomes high. This is consistent with results of Templeton and Rice (2008) for cases where off-fault plastic yielding primarily occurs on the same side (i.e., the extensional) of the fault. Templeton and Rice (2008) attributed this extent dependence on S partly to the closeness of initial stress state to yielding level ( 0 0 max / CF Y σ τ = ) as a function of S. We provide a quantitative explanation of the effect of S on the extent of the off-fault yielding zone in Section 2.3.1.2. Next we use the procedure described in chapter 1 to investigate the off-fault decay of the potency density p 0 ε . We examine the variation of p 0 ε in the fault normal direction and define the thickness T of the yielding zone as the distance from the fault where p 0 ε decreases to zero. In Figure 2.3 the sampling locations along the fault strike are mapped into different colors (see inset in Figure 2.3a): each color represents a trace normal to the fault strike, on the extensional side, starting from 0 100L X = (blue) and ending at 0 220L X = (red). The choice of the starting and ending points ensures that the selected range is neither within the nucleation zone nor too close to the rupture tip. As shown in Figures 2.3b-2.3d, there are three distance ranges where the off-fault variation of p 0 ε follows distinct patterns. In region (1) the yielding zone is affected strongly by the artificial nucleation procedure (see inset in Fig. 2.3b) and is therefore 60 excluded from detailed analysis. Once the rupture propagates away from that region p 0 ε may occur on both sides of the fault, but the summed value from both sides asymptotically approaches at zero distance a constant related to the breakdown process (inset in Fig. 2.3b). This feature is observed in all examined cases (a)-(d). Figure 2.3: Off-fault decay of ε 0 p versus fault normal distance d ⊥ for crack-like ruptures associated with different S and σ 0 values. The schematic diagram in (a) illustrates the employed mapping between sets of colors and distance from hypocenter. The inset in (b) shows the variation of ε 0 p from each side of the fault (or the summed value) along the strike. The inset plot in (d) reproduced the result in a double-linear scale. 61 In the region labeled (2) in Fig. 2.3c, representing most of the off-fault distance range where p 0 ε remains non-zero, there is a linear relation between p 0 ε and ) log( ⊥ d over the examined (one order of magnitude) range of ⊥ d . This logarithmic decay is observed in all presented cases (Figs. 2.3a-2.3d), consistent with the results of chapter 1 and Yamashita (2000). The slope of the logarithmic decay for each case does not depend on the sampling location (reflected by the sub-parallel color stripes), but it clearly varies for different rupture cases. The latter variation can be explained by the variable stress state which mainly determines the maximum value of p 0 ε close to the fault (the intercept with the vertical axis), and by the S parameter which controls the thickness of the yielding zone (the intercept with the horizontal axis). As ⊥ d continues to increase beyond some point, p 0 ε rapidly tapers to zero in a final third regime. We find a linear relation between the thickness of the yielding zone and the along-strike distance from the hypocenter (inset in Fig. 2.3d). This feature is consistent with the large scale view in Figure 2.2 (which is usually difficult to obtain in the field) that the off-fault yielding zones for the simulated crack cases have a triangular shape. 2.3.1.2 Theoretical scaling relation How far the activated off-fault yielding zone can extend depends on the interaction between the background stress field 0 ij σ and the slip-induced incremental stress field ij σ Δ at some transitional distance range from the rupture tip where the total stress just reaches the off-fault yielding strength. Following Ben-Zion and Ampuero (2009), we develop a scaling relation between yielding zone thickness and the S parameter with 62 order of magnitude quantities. For simplicity, we adopt here a singular crack model rather than a model with a finite cohesive zone. Differences between the two models are mentioned when necessary. Based on the singular crack model (e.g., Freund, 1990), the incremental stress field ij σ Δ in the vicinity of the crack tip can be described as: ) ( 0 0 ) , ( 2 II r O v r K r ij d ij + ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ Δ − Δ − + Σ = Δ τ τ θ π σ , (2.2) where r and θ are polar coordinates with the origin at the crack tip (Fig. 2.2d), d τ τ τ − = Δ 0 is the stress drop, L v k K r d π τ Δ = ) ( II II is the dynamic stress intensity factor with ) ( II r v k being a universal function of rupture speed r v , L denoting the half length of the crack (e.g., Broberg, 1999). At the farthest distance max r from the rupture tip where off-fault yielding can be activated, the total stress field ij ij ij σ σ σ Δ + = 0 is expected to just satisfy the yielding criterion for cohesionless rocks (Xu et al., 2012): ) sin( ) ( 5 . 0 4 / ) ( 2 2 φ σ σ σ σ σ yy xx yy xx xy + − = − + , (2.3) where φ is the internal friction angle. Writing 0 ij σ and τ Δ (normalized by 0 σ ) as functions of S and fault friction coefficients for ° = Ψ 45 , using Eq. (2.2) to express ij σ Δ as functions of ) ( II r v k , τ Δ , max /r L and ) , ( r ij v θ Σ , and using the obtained expression for ij σ in Eq. (2.3), we have 2 2 2 II max ) 1 ( ) ( S A v k L r r + ⋅ =η , (2.4) 63 where h is a factor of order 1 (to be determined numerically) and A is a dimensionless quantity depending on ) , ( r ij v θ Σ , friction coefficients and φ . Projecting max r on the fault normal direction to estimate the maximum yielding zone thickness max T , we get 2 2 2 II max ) 1 ( | ) sin( | ) ( S A v k L T r + ⋅ ⋅ = θ η . (2.5) For any r v , we can search for a value of θ that maximizes L T / max in Eq. (2.5). Assuming s r c v 827 . 0 = typical for our cases, we get ° − ≈ 130 θ . This is consistent with the simulation results (see the yielding pattern near the rupture tips in Fig. 2.2, noting the vertical exaggeration of factor 3.75). Figure 4 shows a comparison between the numerical simulation results and Eq. (2.5) with s r c v 827 . 0 = and ° − ≈ 130 θ . For simplicity, we also adopt 375 . 0 ) 827 . 0 ( II ≈ = s r c v k (Broberg, 1999, Fig. 6.9.8). The prediction with 1 = η (corresponding exactly to the truncated theoretical solution of Eq. 2.2) underestimates the ratio L T / max (dashed line in Fig. 2.4). This may be explained by the fact that the abrupt drop of shear stress in the singular solution (given by the second term of Eq. 2.2) leads to lower average shear stress behind the rupture tip (where major plastic yielding occurs in Fig. 2.2) compared with the non-singular model associated with a gradual shear stress drop over a finite length scale. Using a correction factor 72 . 1 = η provides a good agreement between the singular crack model with prescribed rupture process and the non-singular numerical results associated with spontaneous rupture and SWF. 64 Figure 2.4: Comparison of the numerical results and theoretical prediction of Eq. 2.5 (with assumed θ =−130° and v r =0.827c s ) on the scaling relation between T max /L and S. Numerical results are obtained by first calculating ΔT /ΔX based on the measurements in Fig. 2.3, and then converting it into angle ω to calculate T max /L through the relation T max /L=sin(ω)sin(θ )/sin(θ −ω) (see angle definition in Fig. 2.2d). A more detailed analysis may employ simulations with additional low values of S (before leading to supershear ruptures) or very high S values. At the former limit, additional terms of Eq. 2.2 (i.e., those included in ) ( r O ) may have to be considered, because when L T / max is expected to be fairly high (relatively) the above analyzed terms may not be sufficient to dominate the contribution to ij σ Δ . At the latter limit when L T / max is expected to be very low, Eq. 2.5 will reduce to 2 max 1 S L T ∝ . (2.6) 65 In this limit the finite size of the rupture tip cohesive zone, which is ignored in Eq. 2.2, should become important. In addition, this limit may approximate the case examined by Poliakov et al. (2002) for semi-infinite cracks ( ) ∞ → L with almost zero stress drop ( ∞ → S ). We note that the scaling relations given by Eqs. 2.5 and 2.6 do not apparently depend on the absolute value of the stress field or any length scale, but rather imply a dependence of one non-dimensional quantity on another. These scaling implications, along with the previously reported self-similar slip profiles in chapter 1, can help identify crack-like earthquake ruptures based on field data, and may be useful for jointly constraining (or inferring) the remote stress field and/or fault frictional properties. As noted by Ben-Zion and Ampuero (2009), theoretical estimates of W T / max for pulses of width W can be derived in analogous fashion, with an appropriate function ) , ( II h r v v f dependent also on the speed of the healing front h v . Table 2.2: Relation between the incremental rate of yielding zone thickness with along- strike distance from the hypocenter and rock cohesion c. c c σ 0 . 0 = c c σ 2 . 0 = c c σ 4 . 0 = c c σ 6 . 0 = X T Δ Δ / 0.0635 0.0364 0.0221 0.0139 2.3.2 Influence of Rock Cohesion Several studies showed that the assumed value of rock cohesion can influence rupture dynamics and off-fault yielding (e.g., Ben-Zion and Shi, 2005; Duan, 2008; Ma and Andrews, 2010). The effective value of this quantity usually varies with rock type 66 and confining pressure, and depends strongly on the initial rock damage (e.g., Scholz, 2002; Jaeger et al., 2007). Rock cohesion generally has two major effects that can influence the measurable properties associated with off-fault yielding. The first is directly related to the extent of the yielding zone. As illustrated in Table 2.2, with the same initial stress state and nucleation procedure, increasing rock cohesion can reduce the size of the yielding zone and decrease, for crack ruptures, the ratio of yielding zone thickness to rupture distance. This direct effect is naturally expected from the adopted yielding criterion and is generally consistent with the appearance of c in the closeness to failure parameter of Templeton and Rice (2008): φ φ σ σ σ σ σ cos sin ) )( 2 / 1 ( ) ( 4 / ) ( CF 0 0 2 0 2 0 0 xx c yy xx xy yy + + − + − = . (2.7) The second effect of rock cohesion is related to the microfracture orientation close to the fault. This effect operates through the evolving rupture speed that is correlated with the amount of off-fault yielding as a function of rock cohesion. To investigate the effect of rock cohesion on off-fault dynamic stresses, and hence yielding, we measure the expected microfracture orientation at a distance of 0 125 . 0 L (half spectral element) from the fault plane on the extensional side for ° = Ψ 45 and smooth the measurements over an along-strike length of 0 25 . 0 L (one spectral element). Figure 5a shows the predicted results for three friction coefficient ratios of SWF (with s f fixed at 0.6), based on the non-singular crack model of Poliakov et al. (2002). For all three cases, the angle Φ of the expected microfractures relative to the fault (aligned to mode I type) monotonically increases with the normalized rupture speed 67 s r c v / and asymptotically approaches ° 90 near the limit level s R c c / . Figure 2.5b shows the variation of the normalized rupture speed s r c v / (solid curves with scale at the left vertical axis) and the inferred angle Φ (discrete markers with scale at the right vertical axis) obtained in the numerical simulations for three c values. For a given rock cohesion the variations of Φ are positively correlated with those of s r c v / . When comparing different cases with the same rupture propagation distance, the angle Φ increases with the rupture speed (associated with increasing rock cohesion). Figure 2.5: (a) Analytic prediction of close-to-fault microfracture orientation as a function of rupture speed v r with a fixed static friction coefficient f s =0.6 , based on the non-singular crack model of Poliakov et al. (2002). −R< x−v r t< 0, y≈ 0 − , with R being the size of process zone. (b) Variation of numerically inferred close-to-fault microfracture orientation and rupture speed along strike, for cases with different rock cohesion values. The inset shows the comparison between the numerical results in the dashed box and the analytic prediction in about the same selected range. The inset plot in Figure 2.5b explicitly illustrates the correlation of Φ with s r c v / in the range indicated by the dashed box. The numerical results are generally in good agreement with the analytic prediction for 167 . 0 / = s d f f (which corresponds to 68 1 . 0 = d f and 6 . 0 = s f ). The systematic deviation of the numerically inferred angle Φ from the analytic prediction may be explained by the essential differences between the numerical and analytic models, e.g. with or without stress relaxation once reaching the yielding criterion. We note that as c becomes higher, reducing the influence of stress relaxation; the numerical results get closer to the analytic prediction. 2.3.3 Influence of Material Contrast across The Fault So far the properties of dynamic ruptures and off-fault yielding zones have been investigated in an isotropic homogeneous medium. However, large natural faults tend to separate different rock bodies (e.g., McGuire and Ben-Zion, 2005; Le Pichon et al., 2005; Thurber et al., 2006). In this section, we incorporate elasticity contrast across the fault into our numerical procedure (Ampuero and Ben-Zion, 2008) and investigate how the bimaterial effect can influence the generation and distribution of off-fault yielding. Figure 2.6: Plastic strain distribution for crack-like ruptures on a bimaterial interface with 20% contrast for (a) Ψ=10° and (b) Ψ=45° . For both cases, σ xy 0 =0.24σ c , σ yy 0 =−1.0σ c and c=0.2σ c . 69 As in Ben-Zion and Shi (2005) and later studies, we find that the locations of the off-fault yielding can be strongly affected by the existence of a bimaterial interface. In an isotropic homogeneous medium, although the off-fault yielding zone is usually asymmetrically distributed across the fault (for a single event), the partition pattern with respect to the hypocenter is symmetric. However, this latter symmetry can be broken by the presence of a material contrast across the fault. This and the different stress concentrations between crack and pulse type ruptures lead to various changes in yielding zones properties. Figure 2.6 shows the distribution of plastic strain with 20% material contrast (see chapter 1 for definition) for crack-like ruptures generated using SWF. When ° = Ψ 10 , typical for thrust faults, plastic strain is dominantly distributed on the compliant side in the positive direction, while it apparently extends to both sides of the fault in the negative direction, showing an asymmetric pattern with respect to the hypocenter. When ° = Ψ 45 , typical for strike-slip faults, plastic strain is mainly distributed with the employed parameters in the two extensional quadrants, generally following the pattern in an isotropic homogeneous medium. However, the off-fault extent of the yielding zone is wider on the stiff side (in the positive direction) than on the compliant side (in the negative direction), consistent with the numerical results of Duan (2008b). 70 Figure 2.7: Slip velocity profiles for the crack cases of Fig. 2.6, with the generated Rayleigh wave speed being c GR =0.825c s fast for γ =20% . Figure 2.7 shows the slip velocity profiles at different times and the estimated rupture speed beyond the nucleation zone for the crack cases of Figure 2.6. The results confirm that the rupture propagates as asymmetric bilateral crack for both cases, consistent with the expectation that slip-weakening friction with relatively large d s f f − and smooth nucleation procedures lead to asymmetric bilateral cracks on a bimaterial interface (e.g., Harris and Day, 1997; Shi and Ben-Zion, 2006; Rubin and Ampuero, 2007). In such cases, the direction with higher propagation speed and peak slip velocity depends, along with the generation of off-fault yielding, strongly on Ψ . When ° = Ψ 10 , the left propagating tip has faster rupture speed and higher peak slip velocity than the one propagating to the right (Figure 2.7a), in contrast to the prediction for a purely elastic model in the subshear regime (e.g., DeDontney et al., 2011). As Ψ increases to ° 45 , the tip propagating to the right has (Figure 2.7b) faster speed and higher peak slip velocity, consistent with the prediction for a purely elastic model (e.g., Shi and Ben-Zion, 2006; Brietzke et al., 2009). 71 The above differences in rupture and slip velocities may be explained by the generation of off-fault yielding, whose distribution depends strongly on Ψ and on the material contrast. When ° = Ψ 10 , more plastic strain is distributed (Fig. 2.6a) on the compliant side (in the positive direction) than on the stiff side (in the negative direction). Therefore, although the right propagating rupture may be encouraged by the tensile change of normal stress right behind the rupture front, the energy absorbed by the larger off-fault yielding leads to lower rupture and peak slip velocities. When ° = Ψ 45 , the left propagating rupture generates larger plastic strain (Fig. 2.6b) and compressive dynamic change of normal stress. Both effects lead to higher rupture and slip velocities in the right (positive) direction. These features are generally consistent with the numerical results of DeDontney et al. (2011) with off-fault elasto-plastic response. We note that DeDontney et al. (2011) reported failed ruptures under ° = Ψ 45 and unilateral ruptures in the positive direction under ° = Ψ 35 , in contrast to our asymmetric bilateral crack under ° = Ψ 45 . These small differences may be due to details associated with the employed material contrast, S parameter, and the nucleation procedure. It should be mentioned that the bimaterial interface in our study separates as in, e.g., Ben-Zion and Shi (2005) and DeDontney et al. (2011), materials of different elastic moduli and mass densities. The lower density on the compliant side contributes to generation of higher plastic strain on that side compared to the stiff side. This, in turn, leads to reversed preferred rupture direction under low Ψ values. Duan (2008b) used same mass density across the fault in his studies of bimaterial ruptures and obtained higher magnitude of plastic strain on the stiff side in contrast to our results and to those of DeDontney et al. (2011). 72 Figure 2.8: Inferred microfracture orientation or the crack cases of Fig. 2.6 (see Fig. 2.1a for quadrant notations). In addition to the asymmetry of off-fault yielding zones with respect to the hypocenter, we wish to find more signatures that may help identify preferred rupture direction or reflect the existence of a bimaterial fault. Among all the quantities that have been investigated before, we find that off-fault microfracture orientation may be a good indicator. Figure 2.8 shows the inferred results for the crack cases of Figure 2.6. When ° = Ψ 10 , the inferred angle Φ on the compliant side has a higher average value than the one on the stiff side over the same off-fault distance range (see “C + ” vs. “C - ” and “T - ” vs. “T + ” in Fig. 8a). Moreover, an interesting feature with a reversed sign of Φ is observed in “C - ”, probably reflecting the interaction between the slip-induced stress change and the local dynamic change of normal stress. When ° = Ψ 45 , although the major plastic yielding switches to the extensional quadrants, the inferred angle Φ still has a higher average value on the compliant side (“T - ”) than on the stiff side (“T + ”). The results hold for off-fault yielding produced by single asymmetric bilateral ruptures. The cumulative effect of multiple such ruptures with different hypocenter locations is expected to 73 produce off-fault yielding on both sides of the fault, but still with a higher average angle of Φ on the compliant side than on the stiff one. Figure 2.9: Plastic strain distribution for pulse-like ruptures on a bimaterial interface with 20% contrast for a) Ψ=10° and (b) Ψ=45° . For both cases, σ xy 0 =1.04σ c , σ yy 0 =−4.0σ c and c=0 . Next we briefly discuss results for pulse-like bimaterial ruptures, whose dynamic behavior is more sensitive to the generation of off-fault yielding. When ° = Ψ 10 , the rupture propagates as an asymmetric bilateral pulse with higher rupture and slip velocities to the left (Figs. 2.9a and 2.10a). In contrast, as Ψ increases to ° 45 , the left propagating pulse is arrested after some distance and the rupture eventually becomes a unilateral pulse propagating to the right (Figs. 2.9b and 2.10b). For both ° = Ψ 10 and ° = Ψ 45 , the local T value correlates with the maximum slip velocity and final slip (not shown here) at the same location (Figs. 2.9 and 2.10), consistent with the earlier results of Ben-Zion and Shi (2005). The correlation with final slip can be related to the analysis in section 2.3.1.2 and the previous work by Ben-Zion and Ampuero (2009) that both the final slip and the yielding zone thickness are expected to scale with the pulse width. The inferred angle Φ 74 at a given propagation distance has higher average value on the compliant side than on the stiff side for both cases (Fig. 2.11). Figure 2.10: Slip velocity profiles for the crack cases of Fig. 2.9, with the generated Rayleigh wave speed being c GR =0.825c s fast for γ =20% . The above dynamic features can again be explained by the interaction between the generation of off-fault yielding and the bimaterial effect through the angle Ψ . However, we emphasize that the dynamic behavior of a propagating pulse is more sensitive to off- fault yielding than that of a crack. With the same bimaterial fault interface, the examined cracks continue to propagate bilaterally (asymmetrically) with and without off-fault yielding, while the studied pulses can switch from being asymmetrically bilateral to unilateral, depending on the amount of off-fault yielding. The latter may be adjusted by changing the value of rock cohesion and/or S parameter. This implies that, with a material contrast across the fault, the asymmetry of the dynamic behavior and the associated off- fault yielding zones for pulse type ruptures can be more prominent than that for cracks. In particular, the cumulative effect of the case shown in Fig. 2.9b (with many hypocenter locations) is expected to produce much more extensive off-fault yielding zones on the 75 stiff side than on the compliant side, in agreement with the numerical results of Ben-Zion and Shi (2005) based on Coulomb type friction and various geological and seismological observations (e.g., Dor et al., 2006a, 2008; Lewis et al., 2005, 2007; Mitchell et al., 2011). Figure 2.11: Inferred microfracture orientation for the pulse cases of Fig. 2.9. 2.3.4 Ruptures at Different Depth Sections So far we investigated properties of rupture-induced inelastic yielding zones by individually varying several controlling parameters including the background stress state (in chapter 1), S parameter, rock cohesion and a possible contrast of rock elasticity and density across the fault. Here we explore how combinations of these parameters representing different depth sections can influence the generation and properties of the inelastic yielding zones, with and without material contrast across the fault. In particular, we define three typical depth sections, referred to as “shallow”, “intermediate” and “deep”, considering that the on-fault initial shear and normal stresses increase with depth. We also assume that the value of rock cohesion increases with depth, in agreement with 76 laboratory observations that damage healing and cohesion increase with Δσ (e.g., Jaeger et al., 2007; Johnson and Jia, 2005). Figure: 2.12: Plastic strain distribution for crack-like rupture at conditions representing different depth sections without material contrast across the fault. For all cases, τ 0 /(−σ 0 )= 0.24 or S=2.571. Parameters for different depth sections are: (a) σ 0 =−1.0σ c , c=0.2σ c ; (b) σ 0 =−1.62σ c , c=0.48σ c ; (c) σ 0 =−3.06σ c , c=1.2σ c . 2.3.4.1 Without material contrast across the fault We first quantify the effect of depth on crack-like ruptures (see caption of Figure 2.12). The angle Ψ is fixed at ° 45 , consistent with our focus on large strike-slip faults, 77 and the shear to normal stress ratio ) /( 0 0 σ τ − is fixed at 0.24, associated with S = 2.571. To allow comparable amount of released energy to nucleate the rupture, we also fix the ratio of the applied nucleation size nucl L to the critical size c L for different cases, leading to smaller nucl L with increasing depth ( 0 σ ). Figure 2.12 shows the plastic strain distribution for crack ruptures at the three depth sections. As expected, the off-fault yielding zone for all three cases displays a triangular shape that is mainly distributed in the extensional quadrant (only the right half is shown due to the symmetry). However, the off-fault extent and magnitude of the plastic strain vary with depth. The thickness of the yielding zone becomes progressively narrower for conditions representing greater depth due to the increasing rock cohesion. The magnitude of the plastic strain consistently increases with depth due to the increasing background stress and dynamic stress drop. These features are consistent overall with the early 2-D results of Ben-Zion and Shi (2005) and Rice et al. (2005), the more recent 3-D simulation results of Ma (2008), Ma and Andrews (2010) and Kaneko and Fialko (2011), and geological and seismological observations of “flower-like” fault zone structure with depth (e.g., Rockwell and Ben-Zion, 2007, and references therein). In particular, seismic trapped waves and related studies imply that low velocity fault zone layers with considerable thickness are generally limited to the top several kilometers of the crust (e.g., Ben-Zion et al., 2003; Lewis et al., 2005; Yang and Zhu, 2010; Lewis and Ben- Zion, 2010; Yang et al., 2011). We note that, as in Ben-Zion and Shi (2005) and later studies, we used a fairly low cohesion value for shallow depth and relatively high c for deeper sections. It appears that 78 the assumed value of rock cohesion at different depth sections plays a more important role in controlling the yielding zone extent than other depth-dependent conditions such as the normal stress. This is anticipated by the scaling relation of Eq. 2.5, indicating that when c is negligible, T is similar for depth ranges with similar S parameter and rupture length, similar angle θ near the rupture tip (Fig. 2.2d), and comparable rupture speeds. To verify this expectation we performed simulations with same rock cohesion for different depth sections and found no prominent variation of yielding zone thickness with depth, as long as the assumed c value remains a small fraction (possibly zero) of s n at the shallowest examined depth. This is consistent with the 3-D numerical simulation by Ma and Andrews (2010) of crack-like rupture with constant non-zero rock cohesion. As mentioned, laboratory data indicate that rock cohesion should generally increase with depth (e.g., Jaeger et al., 2007). Figure 2.13 shows the expected microfracture orientation over certain off-fault distance range for the crack cases of Fig. 2.12. The results generally display a similar pattern for all three depth sections: Φ has a maximum value approaching ° 80 at a fault normal distance 0 4 . 0 L d ≈ ⊥ and it gradually decreases to about ° 60 at 0 5L d ≈ ⊥ (Figs. 2.13a-2.13c). This similarity may be explained by the fact that the dynamic stress drop is proportional to the background stress, and the total stress in Eq. 2.T4 and plastic strain in Eq. (2.T5) increase linearly with depth. Despite the overall similarity, the difference in the inclination angle Φ can still be observed by investigating close-to-fault microfractures ( 0 125 . 0 L d = ⊥ ) due to rupture speed variation with depth (Fig. 2.13d). 79 Figure 2.13: (a)-(c) Inferred microfracture orientation for the crack cases of Fig. 2.12 at different depth sections. (d) Correlation between the close-to-fault microfracture orientation and rupture speed at three typical depth sections. Figure 2.14 shows the plastic strain distribution for pulse type ruptures for conditions representing two depth sections. Since the dynamic behavior of rupture pulses is more sensitive to the generation of off-fault yielding (and many other factors), only two depth sections, referred to as “shallow” and “intermediate”, are used. The assumed values for S and c are slightly different from those used for the crack ruptures but are self-consistent with variable depth conditions. In contrast to the crack cases of Fig. 2.12, the yielding zone thickness for the pulse cases only slightly decreases with depth and approaches a constant value along strike after its initial growth. The mild depth variation might be explained by the tendency of the ruptures generated with the employed rate- 80 and-state friction law to evolve from steady-state to self-similar pulse behavior with increasing dynamic stress drop while keeping all other constitutive parameters unchanged (e.g., Zheng and Rice, 1998; Nielsen and Carlson, 2000; Ampuero and Ben-Zion, 2008). This is related to the fact that self-similar (growing) ruptures usually produce broader yielding zones than pulse-like ruptures that are approximately in steady-state (Ben-Zion and Shi, 2005; Xu et al., 2012a). The case representing “intermediate” depth is associated with higher stress than the one representing “shallow” section. As a result, the more energetic rupture pulse at greater depth may partially counter or even overcome the effect of the increased c with depth. Figure 2.14: Plastic strain distribution for pulse-like rupture at conditions representing different depth sections without material contrast across the fault. For both cases, τ 0 /(−σ 0 )= 0.26 or S=2.125. Parameters for different depth sections are: (a) σ 0 =−4.0σ c , c=0.8σ c ; (b) σ 0 =−6.0σ c , c=2.4σ c . 81 Figure 2.15 shows the expected microfracture orientation for the pulse cases of Fig. 2.14. The inferred angle Φ has an average value around ° 70 at both depth sections, and is slightly higher at the intermediate depth probably due to the faster rupture speed at that depth (e.g., Rice et al., 2005). Another interesting phenomenon is that the angle Φ is approaching a constant as a function of fault normal distance (characterized by the color) beyond 0 200L X = at shallow depth (Fig. 2.15a). This and the constant yielding zone thickness beyond 0 200L X = in Fig. 2.14a imply that the rupture pulse approaches quasi- steady propagation conditions around 0 200L X = . Figure 2.15: Inferred microfracture orientation for the pulse cases of Fig. 2.14 at different depth sections. 2.3.4.2 With material contrast Here various degrees of material contrast across the fault are added to represent different depth ranges. Following Ben-Zion and Shi (2005), the degree of material contrast γ decreases with depth: % 20 = γ for “shallow”, % 10 = γ for “intermediate” 82 and % 5 = γ for “deep” sections (see chapter 1 for definition). Except for the variable value of γ , all other parameters are the same as in section 2.3.4.1. Figure 2.16: Plastic strain distribution for crack-like rupture at conditions representing different depth sections on a bimaterial interface. Except for the variable degree of material contrast, all the other parameters are as in Fig. 2.12. Parameters for material contrast at different depth sections are: (a) γ =20% , c GR =0.825c s fast ; (b) γ =10% , c GR =0.873c s fast ; (c) γ =5%, c GR =0.896c s fast . Starting with crack type ruptures, we find that with variable γ the slip velocity profiles still show some asymmetry with respect to the epicenter (similar to the results in Fig. 2.7b), while the slip profiles and overall distribution of off-fault yielding zone have a weak asymmetry with respect to the epicenter. The yielding zone thickness linearly scales with the rupture distance at each depth, with a mildly higher growth rate in the positive direction, and it systematically decreases with depth for each propagation direction (Figure 2.16). The inferred angle Φ has a slightly higher average value on the compliant 83 side than on the stiff side over the same fault normal distance range; this asymmetry becomes weaker with depth (Figure 2.17). Therefore, such cases are not expected to produce clear asymmetric signals (co-seismic slip profiles and generated yielding zones) that may be observed in geological studies. One may expect that the cumulative effect of such cases with different hypocenter locations will produce damage zones with a weak asymmetry across the fault at shallow depth, and with almost symmetric distribution at deeper sections. Figure 2.17: Inferred microfracture orientation for the crack cases of Fig. 2.16 at different depth sections. 84 Figure 2.18: Plastic strain distribution for pulse-like rupture at conditions representing different depth sections on a bimaterial interface. Except for the variable degree of material contrast, all the other parameters are as in Fig. 2.14. Parameters for material contrast at different depth sections are: (a) γ =20% , c GR =0.825c s fast ; (b) γ =10% , c GR =0.873c s fast . Figure 2.19: Inferred microfracture orientation for the pulse cases of Fig. 2.18 at different depth sections. The corresponding results for pulse cases exhibit stronger differences for conditions representing different depth sections of bimaterial faults. As shown in Figure 2.18a, although the pulse rupture manages to propagate bilaterally for the shallow depth case (compare with the reference case in Fig. 2.9b), the asymmetry of the distributed yielding 85 zone with respect to the hypocenter is still strong. In the positive direction the pulse is associated with faster rupture speed and growing yielding zone thickness, while in the negative direction it has slower rupture speed and more localized off-fault yielding zone with an approximately constant thickness. The difference in yielding zone thickness at 0 220L X ± = can be up to a factor two and seems to continue to increase with propagation distance. Similarly, the asymmetry in the local microfracture orientation with respect to the hypocenter is also prominent. The inferred angle Φ has a higher average value and a smaller standard deviation on the more compliant side than on the stiffer side (Figure 2.19a). For conditions representing intermediate depth, the asymmetries of the yielding zone thickness and local Φ are highly reduced (Figures 2.18b and 2.19b). The cumulative effect of multiple pulse type ruptures on different depth sections of a bimaterial fault is expected to produce a relatively wide-spread damage zone at shallow depth mainly on the stiff side, and a highly localized relatively symmetric damage zone at greater depth. This is generally consistent with the simulation results of Ben-Zion and Shi (2005). 2.4 DISCUSSION As discussed and reviewed by various authors (e.g., Ben-Zion and Sammis, 2003; Wilson et al., 2003; Mitchell and Faulkner, 2009; Yamashita, 2009), there are many proposed fault models for the formation and development of fault zone structures. Although these models generally agree on the overall structure and elements of active fault zones (e.g., principal plane, fault core, damage zone), they may be distinguished at different spatio-temporal scales (e.g., width and functional form where inelastic deformation is non-zero, single vs. many ruptures, co-seismic vs. inter-seismic periods) 86 and by some damage features that are specific to certain mechanical processes. In the following subsections we attempt to synthesize (Table 2.3) the findings from our parameter-space study on different characteristics of yielding zones with theoretical and observational results of others. 2.4.1 On Faulting Processes The mechanism of generating off-fault yielding zones in our study, by rapid progression of an earthquake rupture tip along a pre-existing fault plane, is referred to as the “fifth model” by Mitchell and Faulkner (2009). This is the dynamic counterpart to their “third model” where off-fault damage is produced by the formation and migration of a “process zone” around the tips of a quasi-statically growing fault. We generally do not differentiate between these two models because they share many predictions such as asymmetrically distorted stress field around the rupture or fault tip (e.g., Vermilye and Scholz, 1998; Poliakov et al., 2002). These models may be distinguished based on features (not simulated here) likely to be specific to dynamic ruptures such as pseudotachylytes (e.g., Wenk et al., 2000; Di Toro et al., 2005) or pulverized rocks (e.g., Dor et al., 2006; Mitchell et al., 2011; Doan and Billi, 2011). More subtle features that depend on rupture speed, such as the degree of stress distortion near rupture tip reflected by the microfracture orientation, and magnitude of stress drop reflected by the off-fault extent of damage zone (e.g., Andrews, 1976; Rice, 1980; Sibson, 1989) are highly non- unique. For simplicity, we will call both models of damage production around propagating rupture and/or fault tip the “process zone” model (PZM) and assume that the fault plane generally remains planar. 87 Another fault model (referred to by Mitchell and Faulkner (2009) as the “fourth” model) emphasizes the roughness of the fault surface and suggests that the off-fault damage is due to the stress interaction and cycling over fault irregularities during displacement (e.g., Scholz, 1987; Chester and Chester, 2000; Wilson, et al., 2003; Dieterich and Smith, 2009). Depending on the fault surface model (e.g., roughness, frictional properties), the resulting off-fault damage can have different attributes at different scales. For convenience, we will call this model the rough fault model (RFM). One outcome of this model is the prediction that the perturbed principal stress ' max σ can have higher magnitude and higher inclination angle relative to local fault surface around a restraining bend than around a releasing bend (Fig. 2.1b), assuming the two bends have comparable size (e.g., Chester and Fletcher, 1997; Griffith et al., 2010). Additional mechanisms for generation of fault zone damage, referred to by Mitchell and Faulkner (2009) as the “first” and “second” models are, respectively, processes related to fault initiation and interaction between different faults. These two processes are not considered in our following discussion. Table 2.3 summarizes results on signatures of fault damage from our parameter- space study, along with expectations from other studies involving quasi-static PZM and fault motion associated with the RFM. The items listed at the top of Table 2.3 are various features that may be used to characterize different controlling mechanisms and the associated damage zone structure. The items listed vertically on the left give additional specifications for each fault model. The angle Ψ is assumed to be moderate to high, consistent with our focus on large strike-slip faults. A single dominant principal displacement surface in 2-D approximation is typically assumed. Damage structures 88 involving multiple fault cores, intersection between different faults, and various 3-D effects may have features associated with superposition of the discussed entries and additional interactions not considered in this work. Various quantitative connections between faulting processes and damage structure are discussed further below. 2.4.2 Key Signatures of PZM and RFM in Yielding Zones With our assumption on relatively high Ψ and failure criteria that depend on normal stress, the PZM generally predicts for both quasi-static process and a single dynamic rupture that the off-fault yielding zone is more prominent on the extensional side of the fault (e.g., Yamashita, 2000; Poliakov et al., 2002; Rice et al., 2005; Willson et al., 2007). This prediction is consistent with observations of asymmetrically distributed tensile cracks near the tips of shear fractures or pre-existing cuts in analogue experiments (Misra et al., 2009a), tensile cracks along the two extensional quadrants of a small fault (Lim, 1998), and tensile cracks along a frictional interface sustaining dynamic rupture in laboratory experiments (Griffith et al., 2009). Coupling this asymmetry with an assumed preferred propagation direction of bimaterial ruptures was used to explain the prominent damage asymmetry observed across several large strike-slip faults, with significantly more damage on the sides with faster seismic velocity at depth (e.g., Dor et al., 2006, 2008; Wechsler et al., 2009; Mitchell et al., 2011). There have been suggestions that damage asymmetry may be expected also for macroscopically symmetric bilateral ruptures (Rubin and Ampuero, 2007; Duan, 2008b). However, the results of DeDontney et al. (2011) and our study indicate that significant damage asymmetry is unlikely to be generated by macroscopically symmetric bilateral 89 cracks for conditions representing shallow depth where the damage asymmetry has been observed. As mentioned in the introduction of chapter 1, this is consistent also with results of Duan (2008b) with low cohesion. Considerable damage asymmetry consistent with observations seems to require macroscopic rupture asymmetry in the form of unilateral cracks (e.g., Fig. 5a of DeDontney et al. 2011), asymmetric bilateral pulses (e.g., Fig. 2.19a), or unilateral pulses (e.g., Fig. 5a of Ben-Zion and Shi 2005; Fig. 2.9b). This is discussed further in section 2.4.3. The PZM also predicts that microfractures are asymmetrically oriented around the rupture or fault tips, with lower and higher angles on the compressional and extensional side, respectively. The predicted sets of microfractures for small and immature faults (related to designation “A” in Table 2.3) have been observed in the field (Vermilye and Scholz, 1998). For relatively large and mature faults that are seismically active (related to designation “B” in Table 2.3), a mixture of predicted sets of microfractures is likely to be observed due to overprinting involving multiple rupture events nucleated at different locations. The dynamic effects of rapidly propagating ruptures, for both crack and pulse types, may modify the microfracture orientation by promoting higher inclination angle on the extensional side at higher rupture speed in the subshear regime (e.g., Poliakov et al., 2002; Rice et al., 2005). This has been observed in laboratory experiments for the rupture speed range R r R c v c 85 . 0 7 . 0 < < (Ngo et al., 2012), but the effect may be too subtle to observe in the field. When considering different rupture types (crack vs. pulse) of PZM, specific signatures are additionally predicted by the scaling relation between the macroscopic properties of the yielding zone (e.g., shape, thickness) and kinematic properties of the 90 rupture (e.g. rupture length, slip and slip velocity). As mentioned, self-similar ruptures (either cracks or pulses) are expected to produce for single events triangular-shape damage zones (e.g., Andrews, 2005; Templeton and Rice, 2008), while nearly-steady pulse-like ruptures produce more localized damage zones with approximately constant or a slightly growing thickness (e.g., Ben-Zion and Shi, 2005; Dunham et al., 2011a). A triangular-shape damage zone may also be produced by quasi-statically expanding cracks, and it has been observed at various positions along the trace of the Gozo fault in Maltese islands (Kim et al., 2004, Fig. 7a). The dynamic and quasi-static crack type faulting processes may be distinguished by the ratio of maximum damage zone thickness to crack length, if the stress drop during dynamic ruptures is usually less than that during quasi-static processes involving fracturing intact rocks (Sibson, 1989). Observations of more localized damage generated by nearly steady pulse ruptures may provide information on rupture speed, stress drop and pulse width (Ben-Zion and Ampuero, 2009), along with the maximum slip velocity at a position having a given yielding thickness (Ben-Zion and Shi, 2005). The general signatures of the RFM-promoted fault damage zones may be characterized by “irregular” distribution of yielding zone properties along directions parallel or normal to the general fault strike. Nevertheless, at certain sections of the rough fault, some yielding zone properties may show correlations with the local geometric or kinematic properties of the fault. Assuming small quasi-static displacement and small amplitude-to-wavelength ratio, simple model calculations for a wavy fault with a single spectral component predict off-fault yielding that is symmetric with respect to the local fault plane and local damage extent that scales with the wavelength of the fault surface 91 (Chester and Chester, 2000). The local microfracture orientation in potential yielding zones depends on the type of fault bend, background stress state, and fault frictional properties. For relatively weak faults with moderate to high Ψ , the perturbed stress field induced by fault roughness may produce high-angle microfractures associated with high- magnitude fault normal stress around restraining bends (designated by “C~” in Figure 2.1b and Table 2.3), low-angle microfractures associated with possible fault opening around releasing bends (designated by “T~” in Figure 2.1b and Table 2.3), and/or strong rotation in principal stress axes (see Figure 2.1b for definition) near the convex corners of the fault (Saucier et al., 1992; Chester and Chester, 2000; Griffith et al., 2010). More realistic quasi-static calculations incorporating fault roughness at many scales are expected to produce complex distribution of yielding zones along the entire fault, but not to change the basic features produced by recent failure events around each spectral component at its characteristic scale. The forgoing predictions related to fault roughness have been applied to explain the reversed sense of shear stress near the SAF in the Cajon Pass area (Saucier et al., 1992) and the observed fault-normal and fault-parallel microfractures along the Punchbowl fault in southern California (Wilson et al., 2003). With increasing fault displacement, large-scale geometrical asperities are expected to become involved and rocks may undergo stress cycling by the juxtaposition of different irregularities of various scales and types during the change of fault configuration. This may lead to two long-term cumulative features. (1) The average yielding zone thickness is expected to scale with the total displacement and largest wavelength of fault roughness along the examined fault segment (Chester and Chester, 92 2000; Savage and Brodsky, 2011). (2) An overprinting of microfractures is expected to produce a mixture of low- and high-angle microfractures at an arbitrary location inside the yielding zone, and the maximum microfracture density is expected to correlate with the fault displacement (Wilson et al., 2003; Mitchell and Faulkner, 2009). All models predict a decay of microfracture density with distance from the principal fault surface. The results of this study indicate that the functional form of the decay is logarithmic over most of the yielding zone extent (e.g., Figure 2.3), consistent with the numerical study of Yamashita (2000). In contrast, Dieterich and Smith (2009) suggested based on quasi-static simulations of slip on a rough fault a power law decay of damage with distance from the fault. Some studies on quasi-static deformation of a wavy fault had an exponential component in the solution for the decay of the perturbed stress field from the fault (Saucier et al., 1992; Chester and Chester, 2000), suggesting exponential damage decay in locations where this component dominates the full solution. The functional forms used to fit field observations vary in different studies. Vermilye and Scholz (1998) fitted microfracture density decay for several small faults in New York State with a logarithmic form. Mitchell et al. (2011) used the logarithmic form to fit the reduction of rock pulverization intensity with distance from the main slip surface of the Arima-Takatsuki Tectonic line in Japan. Savage and Brodsky (2011) used a power law to fit fracture density decay for small faults near Santa Cruz, California. Mitchell and Faulkner (2009) used an exponential decay form for the Atacama fault zone in northern Chile. The difference in fitting functions may stem from a preference related to assumed models or intended use, or it may reflect actual differences in the key operating processes associated with the different examined locations. 93 2.4.3 Mechanisms for Damage Generation Off a Bimaterial Fault Several effects can contribute to asymmetric off fault damage distribution produced by multiple earthquake ruptures on a planar bimaterial fault. The opposite senses of normal stress changes near the rupture tips propagating in the opposite directions can lead to the development unidirectional pulse or bilateral rupture with asymmetric slip and rupture velocities (e.g., Shi and Ben-Zion, 2006; Ampuero and Ben-Zion, 2008; Brietzke et al., 2009). Since damage is promoted on the extensional side for cases with high Ψ representing large strike slip faults, ruptures that are unilateral or more pronounced in one direction, will produce asymmetric damage across the fault. As shown in Figures 2.6-2.7 and 2.9-2.10 and by DeDontney et al. (2011), in cases with high Ψ the coupling of off-fault energy dissipation due to yielding contributes to the asymmetry of bimaterial ruptures compared to purely elastic cases. (In cases with low Ψ representing thrust faults, the off-fault energy dissipation can produce opposite rupture asymmetry). The combined result is expected to produce for subshear ruptures on large strike-slip faults a statistically preferred propagation in the positive direction (Fig. 2.1a), associated with more prominent damage on the stiff side. The type of damage generated may be shear, tensile, or a mixture of both, depending on the adopted criteria and competition between different failure modes. We note that although in some studies the elastically-predicted or modeled plastic strain is a shear type failure, its generation in some regions may be associated with tensile stress or normal stress close to tensile (e.g., Andrews and Ben-Zion, 1997; Ben-Zion and Huang, 2002; Dalguer and Day, 2009). Therefore this type of damage may be used as an indicator for potential tensile stress, at 94 least for some locations on the extensional side and along the fault right behind the rupture tip in the positive direction (Duan, 2008b). Some damage asymmetry across a bimaterial interface can also be produced under quasi-static loadings. This has been invoked to explain asymmetric tensile micro- cracking near grain boundaries separating different minerals (e.g., Dey and Wang, 1981; Kranz, 1983). However, in such cases involving approximately planar boundaries the damage is confined to the immediate vicinity of the interface. In cases of large bimaterial faults with roughness, more extensive damage with potential tensile cracking may occur quasi-statically during the interseismic period around releasing bends (e.g., Chester and Chester, 2000). However, this is expected to be dominated by the roughness and hence produce approximately symmetric damage. A related topic is the observation of pulverized rocks along several large strike-slip faults (e.g., Dor et al., 2006a, 2008; Mitchell et al., 2011). The pulverized rocks were found to be strongly asymmetric with respect to the main slip zone, with most pulverized rock bodies on the side with faster seismic velocity at depth. The damage asymmetry documented in these studies extends over hundreds of meters, and has been observed in places to be accompanied by across-fault asymmetry over scales of km involving various geomorphic features (e.g., Dor et al., 2008; Wechsler et al., 2009). Several signatures of the observed pulverized rocks suggest that they are produced under conditions associated with tensile stress (see Mitchell et al., 2011, and references therein). Experimental studies suggest that generation of pulverized rocks require high dynamic strain rates (Doan and Gary, 2009; Doan and Billi, 2011). 95 Table 2.3: Observable features of fault damage zones (in 2D approximation) Location Spatialpatternandextent Microfractureorientation † Densitydecay Scalingrelationorcorrelation NotesandReferences Quasi-static in“C”(I d &II,minor) T increasesfromtheinitiationpoint(s) Φ≈ 20 o in“C”andΦ≈ 70 o logarithmic,expotential dmax∝ L;d0∝ L;Tmax∝ L; Crack (A&B) &“T”(I t &II,major) towardsfaultgrowingdirection in“T” orpower-lawdecay ρmax≈ const.,indep. ofL CowieandScholz(1992a),Scholzetal. (1993), (triangularshape) VermilyeandScholz(1998),MitchellandFaulkner(2009), SavageandBrodsky(2011) A:prominentlyin“T” A:T increasesfromthehypocenter A:Φ <Ψin“C”;Φ >Ψin logarithmicdecay A:dmax∝ L;Tmax/L∝ S −2 Crack (I t &II),mightalso towardsrupturepropagationdirection “T”,Φ(in“T”)increaseswith (A&B) (S isrelativelylarge); Cowie&Scholz(1992b),Scholzetal. (1993), in“C”(I d &II) (triangularshape),usuallynarrowerthan increasingvr (∼ 90 o whenvr ρmax≈ const.,indep. ofL; Yamashita(2000),Poliakovetal. (2002),Andrews(2005), B:onbothsides thatforaquasi-staticallygrowingfault approachesvlim)ordecreasing B:ρmax growswith#ofEQs TempletonandRice(2008),Ben-ZionandAmpuero(2009), Dynamic B:T ≈ const.(moderatetowide) τd/τs;senseoffault-parallel (overprint 1 ),mightbeupper MitchellandFaulkner(2009),Xuetal. (2012a,2012b) shearaheadofrupturetipcan bounded A:T ≈ const.(narrow),canincrease bereversedathighvr A:d&T correlateswithvmax Pulse towardsrupturepropagationdirection B:twoclusteredsets:Φ <Ψ andpulsewidth,usuallyapproach Riceetal. (2005),Ben-ZionandAmpuero(2009), PZM B:T ≈ const.(narrowtomoderate) (minor)andΦ >Ψ(major) const. butcangrowwithL Dunhametal. (2011a),Xuetal. (2012a,2012b) B:overprint 1 ofρmax A:similartothecase similartothecasew/omaterial A:Φ − 2 >Φ + 1 >Ψin“T”, logarithmicdecaywith A:d ± max ∝ L ± ;T ± max ∝ L ± ; Crack w/omaterialcontrast contrast(A&B),slightly-to-moderately Φ <Ψin“C” asymmetricslopes asymmetryofT &dwrt. the Duan(2008b),DeDontneyetal. (2011), Dynamic B:onbothsides morepronouncedin“+”direction(A) B:similartothecasew/o wrt. thehypocenter(A) hypocentercorrelateswithγ; Xuetal. (2012b) + (“1”: stiffside,“2”: oronthestiffside(B) materialcontrast;forthe oracrossthefault(B) B:T1/T2 correlateswithγ bimaterial compliantside;“+/–” A:T + ≈ const.(narrowtomoderate), majority:Φ2 >Φ1 >Ψ A:d ± &T ± correlatewithv ± max ; Pulse forparticlemotion cangrowwithL + ;T − depends, d + ≈ const.,mightgrowwithL + ; Ben-ZionandShi(2005),AmpueroandBen-Zion(2008), directiononside2/1) usuallyhighlylocalized; d − depends,usuallyrapidlytapers Xuetal. (2012b) B:T1 (moderate)$ T2 (localized) B:T1/T2 correlateswithγ onbothsides(A&B) A:T fluctuatesalongstrike,maybe irregular: fromfaultparallel exponentialor A:dmax∝ β −2 ,indep. ofL(Lis Crack moreextensivearound“T∼”than tofaultnormalwhenf is power-lawdecay(A&B) large),dmax∝ L(Lissmall); Saucieretal(1992),ChesterandFletcher(1997), Quasi-static around“C∼” notveryhigh;senseoffault- T ∝ λ,ρmax∝ λ −α (α> 0) ChesterandChester(2000),Wilsonetal. (2003), B:T ≈ const. parallelshearnearconvex B:T ∝ d,alsodependsonλmax; DieterichandSmith(2009),Griffithetal. (2010) cornercanbereversedwith ρmax growswithd(overprint 2 ), RFM lowf (A&B) mightbeupperbounded A:theoveralllocation A:T fluctuatesalongstrike,may severalclusteredsets,which N.A.(A&B) A:lackofmacroscopicscaling Crack/Pulse mayagreewellwith showpatternsofPZMalongportions areexpctedtoreflectthe relationsatthepresenttime; DuanandDay(2008),Dumhametal. (2011b), Dynamic thatpredictedbyPZM withlowβ;damagemaybeabsentinthe competitionbetweendynamic vr anti-correlateswiththeslope ShiandDay(2011) B:onbothsides immediatevincinityof“T∼” effectandlocalroughfault offaultprofile B:T ≈ const.withfluctuation surface(A&B) B:N.A. PZM=ProcessZoneModel(withapproximatelyplanarfaultinterface),RFM=RoughFaultModel. Asingleprincipalfaultsurfaceisassumed. TheangleΨofthebackgroundmaximumcompressivestressrelativetothefaultisassumedtobemoderateto high,representivefor(butnotlimitedto)largestrike-slipfaults. †TheorientationisdescribedbytheangleΦofthemeasured(orinferred)microfractures(alignedtothemode-Itype)relativetothefault. A: in a short-term process (e.g. during a single seismic event or with small amount of displacement), B: in a long-term process (e.g. after multiple seismic cycles with variable hypocenter locations or with large cumulative displacement); “C”&“T”: compressional and extensional quadrants, respectively; “C∼”&“T∼”: restraining and releasing bends, respectively; I d : dilatant microfractures (under compression), I t : tensile microfractures (under transient tension);II: mode-II shear microfractures;T: damage zone thickness;d: displacement or seismic slip;d0: characteristic displacement on the trailing edge of process zone;L: fault half length or rupture distance;ρ: microfracture density or its equivalence (e.g. seismic potency density);vr: rupture speed;τd: dynamicshearstrength ofthefault;τs: staticshearstrengthofthefault;S: relativestrengthS parameter(seethetext);vmax: maximumslipvelocity;γ: degreeofmaterialcontrast;f: frictioncoefficient;β: roughness (e.g. rmsslope)offault profile;λ: wavelengthoffaultprofile. Subscript“max”isusedtospecifythemaximumvalueofaquantity(e.g.dmax,Tmax,ρmax). 96 The wide extent of the observed pulverized rocks, their observations in the context of large bimaterial strike-slip faults, and their existence primarily on the side with faster velocity at depth suggest that they are likely produced by repeated predominantly unilateral or strongly asymmetric bilateral ruptures. Based on the results of sections 2.3.3 and 2.3.4.2 that weakly asymmetric ruptures are not expected to produce significant damage asymmetry in low cohesion materials typical of shallow depths and previous related studies, we may conclude that strongly asymmetric fault zone damage that includes pulverized rocks is likely generated by pulse-type ruptures with statistically- preferred propagation direction, although we cannot exclude the unilateral or strongly asymmetric crack-type ruptures (see discussion in section 2.4.2). 2.4.4 Limitations of The Presented Results and Potential Future Improvements We have used 2-D simulations to explore changes of fault zone damage with depth that is likely generated by dynamic ruptures. Our 2-D simulations with the adopted Mohr- Coulomb criterion do not account for the stress/strength gradient with depth (Ma and Andrews, 2010), the effect of the intermediate principal stress in influencing rock damage (Lockner and Beeler, 2002), the finite width of the seismogenic zone and the free surface (Day, 1982; Ma and Andrews, 2010). In particular, the finite seismogenic depth implies an upper limit on self-similar rupture growth (Day, 1982). Concerning our results about the evolution of the plastic zone thickness for self-similar and quasi-steady ruptures, this suggests that the extent of the damage zone scales linearly with fault length up to a value proportional to the seismogenic depth. More precisely, our results for ° = Ψ 45 (Fig. 2.4) imply that the fault zone thickness saturates at a value of a few percent the seismogenic depth, which is a few hundred meters. This provides a possible explanation for the 97 observation that fault zone thickness for faults with large cumulated slip is typically a few hundred meters, independently of fault length (Mitchell and Faulkner, 2009; Savage and Brodsky, 2010). In addition our 2-D simulations do not properly reflect the sensitivity of the Green’s function to a local disturbance in 3-D (Evans, 2000, chapter 2.4). Moreover, we have adopted a simple setting of model parameters without introducing saturation of the effective normal stress at some depth or implied scaling relation of the slip-weakening distance with final slip (e.g., Rice, 1993; Abercrombie and Rice, 2005), nor did we consider effects associated with pre-existing low velocity fault zone layer (e.g., Harris and Day, 1997; Ben-Zion and Huang, 2002; Huang and Ampuero, 2011). These limitations could accordingly affect our evaluation of the 3-D structure of fault damage zones. Given that many adopted yielding criteria in both 2-D and 3-D are pressure- dependent (e.g., Templeton and Rice, 2008; Ma and Andrews, 2010) and the size of potential off-fault failure zone may scale with the slip-weakening distance (Rice et al., 2005), our results should be augmented by a future related parameter-space study using 3-D simulations of crack- and pulse-like ruptures. Only limited numerical simulations of dynamic ruptures along rough faults have been performed so far, mainly with a focus on high-frequency radiation and basic properties of off-fault yielding (e.g., Dunham et al., 2011b; Shi and Day, 2011). As indicated in Table 2.3 for this category, there are many yielding zone properties that are not covered or explored by these studies, such as the competition between properties of dynamic ruptures and generated yielding zones. Some results of dynamic rupture models show no or little damage in the immediate vicinity of fault releasing bends (Dunham et al., 2011b, 98 Fig. 3c) or fault kink that is oriented into the extensional quadrant (Duan and Day, 2008, Fig. 13), in contrast to the quasi-static expectations for a wavy fault (Chester and Chester, 2000). These and other issues should be clarified by future simulations of ruptures on rough faults. The simulations done in this study and related earlier works used off-fault yielding in the form of plastic strain, rather than brittle damage as observed geologically (e.g., Wilson et al., 2003; Dor et al., 2008; Mitchell and Faulkner, 2009) and seismologically (e.g., Lewis et al., 2005; Allam and Ben-Zion, 2012) in the structure of natural faults. Brittle damage is associated with permanent volumetric changes (e.g., Jaeger et al., 2007), and the reduction of elastic moduli in the damage zones can lead to significant motion amplification (e.g., Spudich and Olsen, 2001; Peng and Ben-Zion, 2006) and additional dynamic feedback mechanisms not accounted for by plasticity. Examining the effects of such mechanisms, and producing clearer predictions on damage products that may be compared with in-situ observations, require simulations that incorporate brittle damage. This will be done in a follow up work. 99 Chapter 3 Numerical and Theoretical Analyses of In-plane Dynamic Rupture and Off-fault Yielding Patterns at Different Scales 3.1 SUMMARY We perform numerical simulations of in-plane ruptures with spontaneous Mohr- Coulomb yielding in the bulk and analyze properties of the ruptures and yielding zones at different scales. Using a polar coordinate system, we show that the overall shapes and patterns of the simulated yielding zones can be well explained by combining the slip- induced Coulomb stress change and the background stress. While there is no apparent mechanism for preferring synthetic vs. antithetic shearing at a scale much smaller than the yielding zone size, this is not the case at larger scales. For shallow angles Ψ between the maximum background compressive stress and the fault, representing thrust faulting, large-scale off-fault synthetic fractures are dominant but there are two conjugate sets of fractures with a typical size comparable to the yielding zone thickness. For smooth rupture propagation with moderate-to-high Ψ values representing large strike-slip faults, most of the off-fault fractures that grow across the entire yielding zone are of the synthetic type. The less preferred antithetic set may become more pronounced for rupture propagation encountering fault heterogeneities. In particular, a strong fault barrier promotes antithetic fractures with a comparable size to those of the synthetic type around the barrier, where very high permanent strain is also observed. A consideration of non- 100 local properties of the stress field in space or time can explain the above differences. Our results provide an alternative way of understanding Riedel shear structures and the potentially preferred synthetic shear fractures suggested in previous studies. The examined dynamic processes may be distinguished from quasi-static patterns by the timing, location, and inclination angle of characteristic fracture elements. In agreement with other studies, we propose that backward or orthogonally inclined antithetic shear fractures on strike-slip faults and very high permanent strain could be used as signals that reflect abrupt rupture deceleration. On the other hand, relative lack of off-fault yielding at given locations may indicate abrupt rupture acceleration. 3.2 INTRODUCTION Various studies have used plasticity in recent years to model off-fault yielding during propagation of dynamic ruptures on frictional faults (e.g., Andrews, 2005; Ben- Zion and Shi, 2005; Templeton and Rice, 2008; Dunham et al., 2011a). There is no length scale or specific orientation in plasticity described by stress invariants in a continuum, so the smoothly distributed plastic strain can only reflect certain yielding zone properties. These include the overall shape of the yielding zone and local orientations of two possible conjugate microfractures. However, plasticity allowing for shear localization (Templeton and Rice, 2008) or direct modeling of discrete shear branches (Ando and Yamashita, 2007) can provide some insights on the potential large-scale fractures inside the yielding zone. The properties of the generated yielding zones reflect, in addition to the assumed yielding rheology, the background stress operating on the fault and the dynamic stress generated during rupture propagation. Improved understanding of the conditions leading to different sets of yielding properties at different scales can help 101 inferring from field observations information on the occurrence and properties of dynamic ruptures. Examinations of the dominant stress field at various scales and comparisons with expectations from Linear Elastic Fracture Mechanics (LEFM) or modified models in the near field, and the background stress in the far field, provide fundamental tools for understanding the yielding zone properties (e.g., Poliakov et al., 2002; Rice et al., 2005). However, previous analyses of yielding features either did not consider fully effects resulting from superposition of space-time varying transient failure zone lobes (Poliakov et al., 2002), or considered effects of changing rupture front configuration but presumed the type of shear branches from two possibilities (Ando and Yamashita, 2007). In addition, most previous studies on the topic did not explore the relation between the evolving stress field and the smoothness of rupture process. In this chapter we consider the above mentioned effects explicitly and show that analyses of non-local properties of the stress field, in space at fixed time and in time with changing rupture front configuration, along with the smoothness of rupture propagation, lead to a refined understanding of off-fault yielding characteristics on several scales. The results highlight the roles of incremental yielding zones, the existence of outer and possible internal envelopes of cumulative yielding, and competition between two possible sets of conjugate shear fractures during smooth and non-smooth rupture processes. The presented results, together with those of previous studies, help to develop better connections between failure processes on a fault and yielding zone properties. 102 Figure 3.1: Model configuration for dynamic in-plane ruptures along a frictional interface (thick black line at the center) with off-fault plastic yielding. The red portion on the fault indicates the imposed nucleation zone with size L nucl . The medium is loaded by a uniform right-lateral background stress with angle Ψ between the background maximum compressive stress and the fault plane. Symbols “C” and “T” represent, respectively, the compressional and extensional quadrants, in relation to the first motion of P-waves from the nucleation zone. Because of symmetry, results in subsequent plots will be shown only for the right half. 3.3 MODEL SETUP We consider 2-D in-plane ruptures and off-fault yielding under plane strain conditions. The relevant stress components are illustrated in Figure 3.1. A right-lateral rupture is initiated in a prescribed zone (red bar in Figure 3.1) and is then allowed to propagate spontaneously along the frictional fault (solid black line). The first motion of the radiated P-waves defines four quadrants relative to the hypocenter with “C” and “T” denoting compressional and extensional quadrants, respectively. The initial normal and shear stresses on the fault are 0 0 yy σ σ = and 0 0 xy σ τ = , respectively, and Ψ represents the acute angle between the background maximum compressive stress max σ and the fault. A 103 relative strength parameter ) /( ) ( 0 0 d s S τ τ τ τ − − = represents the relation between the initial shear stress, static strength ) ( 0 σ τ − = s s f and dynamic shear strength ) ( 0 σ τ − = d d f , with s f and d f being the static and dynamic friction coefficient. It is well known that increasing the initial shear stress on a frictional fault toward the static strength leads to a transition from subshear to supershear rupture propagation (Burridge, 1973; Andrews, 1976; Das and Aki, 1977; Day, 1982). In this study we set the value of S to produce subshear ruptures, which is the typical situation for most earthquakes (e.g., Ben-Zion, 2003, and references therein). 3.3.1 Slip-weakening Friction We adopt a linear slip-weakening friction (SWF) to describe the breakdown process along the fault outside the nucleation zone. Specifically, the frictional strength τ has the following dependence (e.g., Ida, 1972; Palmer and Rice, 1973; Andrews, 1976) on slip u Δ : ⎩ ⎨ ⎧ > Δ ≤ Δ Δ − − = c d c c d s s D u D u D u if if / ) ( τ τ τ τ τ (3.1) where c D is a characteristic slip distance over which shear strength reduces from s τ to d τ . When the background shear stress is only slightly higher than d τ , the size of the spatial region associated with the strength reduction, referred to as the process zone, can be estimated (e.g., Rice, 1980) by ) ( II 0 r v f R R= , (3.2a) 104 where 0 R is the static value of R at zero rupture speed. For Poissonian solids this is given by ) ( 8 3 0 d s c D R τ τ µ π − = , (3.2b) where µ is shear modulus and ) ( II r v f is a monotonic function of rupture speed r v that increases from unity at r v = 0 to infinity at the limiting Rayleigh wave speed. For proper resolution in numerical simulations, we discretize 0 R with multiple numerical cells (19 or more) and check the simulation results to ensure that the process zone R is well resolved. 3.3.2 Nucleation Procedure We follow the procedure of Xu et al. (2012a) with a time-weakening friction (TWF) to artificially trigger the rupture within the nucleation zone. The rupture front during the nucleation stage is enforced to propagate outward with a constant subshear speed and the frictional strength at locations reached by the rupture front linearly weakens with time up to the dynamic level d τ : ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ − − − = s 0 0 , , ) )( ( max min τ τ τ τ τ τ d r d s L r t v , (3.3) where r is the along-strike distance from the hypocenter and 0 L is a spatial scale for the strength reduction (similar to the length scale R for SWF). We usually use 0 L as a reference for other length scales since its value is fixed. The size of the nucleation zone is determined by the prescribed r v and a desired time span. In practice, we choose the 105 effective frictional strength within the nucleation zone to be the minimum of those determined by the TWF and SWF, and make the nucleation zone large enough to produce a sustained rupture outside the nucleation zone under SWF. 3.3.3 Off-fault Material Response The off-fault yielding during dynamic ruptures is governed in most cases by the Mohr-Coulomb plasticity. The onset of yielding is described by a yield function of stress invariants: Y F σ τ − = max , (3.4) where 2 2 max 4 / ) ( xy yy xx σ σ σ τ + − = is the maximum shear stress and ) cos( ) sin( ) ( 2 / 1 φ φ σ σ σ c yy xx Y + + − = is the yielding strength, with φ being the internal friction angle and c being the rock cohesion. Yielding occurs when 0 ≥ F . After the onset of yielding, plastic flow is partitioned into different components through a plastic flow potential max τ = M assuming zero inelastic volumetric deformation over a characteristic time scale v T : ij v p ij M T F σ µ ε ∂ ∂ 〉〈 =  , (3.5) where p ij ε  is the rate of plastic strain and 2 / |) | ( x x x + = 〉〈 is the ramp function. An appropriate finite value of v T for stress relaxation (rate-dependent visco-plasticity) can help reducing shear localization that usually has a strong dependence on the numerical mesh. A rate-dependent (rather than instantaneous) response may also reflect the actual 106 physical process of microcracking growth under high strain rates (see Simo and Ju, 1987, and references therein). In some of the simulations we adopt this strategy to obtain smoothly distributed plastic strain. In other cases we use a perfect rate-independent plasticity by setting 0 = v T (e.g., Loret and Prevost, 1990; Prevost and Loret, 1990) to focus on physical implications of the shear localization features. Relevant theoretical background for shear localization is reviewed when appropriate. To facilitate our theoretical stress analysis, we do not consider post-failure weakening or hardening mechanisms in this study. Various weakening or hardening mechanisms provide more realistic yielding behaviors for given circumstances (e.g., de Borst, 1988; Leroy and Ortiz, 1989; Ando and Yamashita, 2007; Shi et al., 2010; Lyakhovsky et al., 2011) and should be explored in future studies. In all cases we use the scalar seismic potency defined as p ij p ij p ε ε ε 2 0 = to measure the intensity of the plastic strain (e.g., Ben-Zion, 2008). More details on the adopted plasticity and related energy balance verification can be found in Xu et al. (2012a). 3.3.4 Numerical Methods and Parameters The simulations are done with the 2D spectral element code developed by Ampuero (SEM2DPACK-2.3.8, http://sourceforge.net/projects/sem2d/). The calculation domain is typically discretized into square elements with 5 Gauss-Lobatto-Legendre nodes non- uniformly distributed per element edge. Occasionally we use obliquely oriented spectral elements to investigate the mesh alignment effect on the pattern of dynamic shear bands (see Appendix C). A typical traction-at-split-node technique has been implemented in the code to solve dynamic rupture problems (Andrews, 1999; Kaneko et al., 2008). 107 Absorbing boundary conditions are assumed around the calculation domain, which is set large enough (much larger than that shown in figures) to ensure that the rupture and generated off-fault yielding patterns do not interact with the absorbing boundaries. A visco-elastic layer of the Kelvin-Voigt type is added near the fault to damp the high frequency numerical oscillations near the fault. Additional details on the code can be found in the SEM2DPACK-2.3.8 user’s guide. Table 3.1: Parameter values of material properties used in the simulations. Parameters Values Lamé parameters λ , µ 1, 1 P- and S- wave speeds p c , s c 1.732, 1 Mass density ρ 1 Poisson’s ratio ν 0.25 Characteristic slip distance (SWF) c D 1 Characteristic length scale (TWF) 0 L 1 Internal friction angle φ ° 9638 . 30 Rock cohesion c 0 Time scale for stress relaxation v T 075 . 0 / 075 . 0 0 = s c L (rate-dependent) 0 (rate-independent) Reference stress c σ 1 / 0 = L D c µ 108 Table 3.2: Convention between physical ( ) and normalized dimensionless ( ) quantities. Length Time Stress Strain 0 L l l = ʹ′ 0 L tc t s = ʹ′ ! σ = σ σ c c σ εµ ε = ʹ′ Similar to Xu et al. (2012a), the simulated quantities are normalized by reference parameters. The values of some quantities corresponding to conditions for natural faults are estimated in some cases. Table 3.1 lists the values of various parameters that are fixed in this work. The quantities most relevant for the focus of our study are the length scale and intensity of the off-fault yielding. We therefore use 0 L and µ σ / c (Table 3.1) to provide reference length and strain values, respectively. Table 3.2 summarizes the conversions between the physical and normalized quantities. We typically use an average grid size (between two adjacent nodes) of 16 / 0 L x≈ Δ for the numerical simulations; refined meshing is used when studying the mesh size effect on the pattern of dynamic shear bands (see Appendix C). The relation between x Δ and the estimated value of 0 R for ruptures under various stress and strength conditions are also presented. 3.4 RESULTS 3.4.1 Basic Properties of The Yielding Zones Previous works have shown that the distribution of off-fault plastic yielding generated by in-plane ruptures depends strongly on the angle Ψ (e.g., Poliakov et al., 2002 and later studies). Yielding occurs primarily on the compressional side when Ψ is less than x ! x 109 about 15º and progressively switches to the extensional side when Ψ is larger than about 30º. The strong dependence on Ψ holds regardless of whether the rupture mode is crack- like (e.g., Andrews, 2005; Templeton and Rice, 2008; Xu et al., 2012a) or pulse-like (e.g., Ben-Zion and Shi, 2005; Rice et al., 2005; Dunham et al., 2011a). This is illustrated in Figure 3.2 for crack-like ruptures with 2 values of Ψ characterizing faults stressed with low (  10 = Ψ ) and high (  45 = Ψ ) angles. The assumed angles represent approximately 2-D configurations corresponding, respectively, to thrust faults (in a cross- section view) and large strike-slip faults (in a map view), although other factors may also influence the faulting style. Figure 3.2: Distribution of off-fault plastic strain generated by crack-like ruptures with (a) Ψ=10° and (b) Ψ=45° . The scale in the Y direction is exaggerated by a factor of 3.75. 110 Poliakov et al. (2002) analyzed the conditions favoring large-scale synthetic shear (i.e. with the same sense of shear as the main fault) along off-fault paths and showed that the optimal orientation is located on the compressional and extensional sides when Ψ is low and high, respectively. They also suggested where to search for the potential failure plane by analyzing through Mohr circles the stress field including the background stress and slip-induced stress change in front of the rupture tip. Dunham et al. (2011a) calculated based on the non-singular model proposed by Poliakov et al. (2002) the stress evolution with respect to the yielding level to estimate where off-fault yielding is likely to occur under relatively low and high values of Ψ . These studies explain the general location dependence of the off-fault yielding zone on Ψ . However, recent results of Xu et al. (2012a) show a characteristic angular distribution pattern of the transient stress field around the rupture tip that may lead to several distinct failure zone lobes. The orientation and relative size of different lobes provide additional information about the dynamic stress field that can be used to improve the understanding of off-fault yielding. In the following we first discuss briefly key results of Xu et al. (2012a) on the angular failure zone pattern around the rupture tip (section 3.4.1.1). Then we provide a simple theoretical analysis to explain the simulated angular pattern (section 3.4.1.2), and describe how the cumulative yielding zone forms in the wake of a propagating rupture (section 3.4.1.3). 111 Figure 3.3: Distribution of equivalent plastic strain increment near the rupture tip for the cases of Fig. 3.2 with (a) Ψ=10° and (b) Ψ=45° . Black bars represent local orientations of expected shear microfractures, with “thick” and “thin” bars used for right- lateral and left-lateral, respectively. 3.4.1.1 Incremental yielding zone around the rupture tip We follow the procedure described by Xu et al. (2012a) to show the equivalent incremental plastic yielding zone around the rupture tip. To achieve the goal, we plot the normalized Coulomb stress | | / ) ( 0 max m Y σ σ τ − around the rupture tip in Figure 3.3 at the same time step used for Figure 3.2, where 0 m σ is the initial mean stress. The thick and thin black bars represent, respectively, the local orientation of expected right-lateral (synthetic) and left-lateral (antithetic) shear fractures inside the current yielding zone at the examined time step. According to the employed visco-plasticity in Eq. 3.5 and the assumed non-zero value for v T (Table 3.1), if the stress field exceeds the yielding strength it will be relaxed gradually while staying above the yielding level within the same time step. Therefore the stress-based plots of Figure 3.3 have similar patterns to the incremental plastic strain. As seen, in addition to the overall location-dependence of the 112 plastic yielding zone on Ψ , the expected plastic strain increment displays an angular distribution pattern around the rupture tip with several distinct lobes (marked by letters). Some lobes are oriented forward while others are oriented backward and may dominate the contribution to off-fault yielding (e.g., lobe-A in Fig. 3b). The angular distribution of yielding zone increment, and especially the emergence of the backward-oriented lobes, has not been explicitly explained by previous studies (e.g., Poliakov et al., 2002) and is analyzed in more detail below. 3.4.1.2 Stress analysis based on LEFM With the assumed constant values for rock cohesion and internal friction angle, the distribution of off-fault plastic yielding could be determined by the total stress field (Eq. 3.4). In our study, this is the sum of the background stress field 0 ij σ , which is given as a model input, and the slip-induced incremental stress field ij σ Δ . For basic analysis, we adopt the singular crack model (Freund, 1990) to investigate the angular variation of ij σ Δ with respect to the crack tip. In this case, ) 1 ( ) , ( 2 II II O v r K r ij d ij + Σ = Δ θ π σ , (3.6) where d K II is the instantaneous dynamic stress intensity factor, 2 2 y x r + = and ) / ( tan 1 x y − = θ are the transformed polar coordinates with origin at the crack tip, II ij Σ are dimensionless functions of θ and rupture speed r v , and ) 1 ( O denotes higher order terms that are bounded by constants as + →0 r . The expressions for the different components of II ij Σ and the dynamic stress intensity factor d K II are given in Appendix A. 113 With the expressed quantities in a Cartesian coordinate system (Eq. A1), the normal and shear stress change along arbitrarily oriented planes around the crack tip can be written as θ σ θ σ σ σ θ σ θ σ θ σ σ θ θθ 2 cos 2 / ] 2 sin ) [( 2 sin cos sin 2 2 xy xx yy r xy yy xx Δ + Δ − Δ = Δ Δ − Δ + Δ = Δ , (3.7) where θ is the inclination angle of the chosen plane with respect to the fault (see inset in Figure 3.4b). The angular variations of the leading terms of θθ σ Δ and r θ σ Δ normalized by r K d π 2 / II are plotted in Figure 4, using s r c v 87 . 0 = for ° = Ψ 10 and s r c v 83 . 0 = for ° = Ψ 45 , with c s being the S-wave speed. The different assumed r v values are based on the simulations results of Figure 3.2. As seen, r θ σ Δ is an even function of θ (solid black curve) and θθ σ Δ is an odd function of θ (dashed black curve). Two sets of the Coulomb Failure Stress change θθ θ σ σ Δ + Δ − = Δ − 6 . 0 CFS r (blue) and θθ θ σ σ Δ + Δ = Δ + 6 . 0 CFS r (red) are plotted to help evaluating the efficiency of ij σ Δ in promoting non-local (i.e. over some distance r from the crack tip) left-lateral and right-lateral shear fractures. Local maxima of CFS Δ are marked with numbers and letters to indicate the preferred sense of shear with respect to the rupture tip. We have checked that each local maximum is associated with a local peak value of CFS Δ and a non-zero shear component r θ σ Δ of a proper sign (e.g. a local maxima of CFS Δ with very low r θ σ Δ will not be identified as a favored shear fracture; rather it indicates a potential tensile fracture). 114 Figure 3.4: Calculated angular variation of the slip-induced incremental stress field Δσ αβ in a polar coordinate (based on LEFM) for the cases of Fig. 3.2 with (a) Ψ=10° and (b) Ψ=45° . The Coulomb failure stress change promoting left-lateral and right-lateral shear are denoted ΔCFS − (blue) and ΔCFS + (red), respectively. Locations of local peak values of ΔCFS ± are marked with numbers and letters, “L” for left-lateral and “R” for right- lateral. The incremental stress field induced by slip is expected to determine whether off-fault yielding can be activated near the rupture tip (e.g., Poliakov et al., 2002). Determining the ability of activated yielding zone to extend to larger scales requires consideration of the stress field at those scales. We follow these ideas to examine the orientations and extent where the two stress fields 0 ij σ and ij σ Δ interact positively to promote Coulomb-type yielding. As pointed out by Poliakov et al. (2002), there are two (constant) stress fields 1 ij σ and 0 ij σ , differing only by the shear component, that affect the total stress at intermediate and far distance ranges from the rupture tip, respectively. In our study the shear stress components in 1 ij σ and 0 ij σ are a small fraction of s τ . We therefore do not distinguish between 1 ij σ and 0 ij σ , and adopt 0 ij σ as the (constant) dominating stress field at large scales. The resolved maximum compressive stress orientation obtained using 1 ij σ or 0 ij σ differs only by about ° 5 , which does not affect significantly our analysis. 115 Figure 3.5: Spatial representation in a polar coordinate system of the interaction between the background stress σ ij 0 and the slip-induced incremental stress Δσ ij for the cases of Fig. 3.2 with (a) Ψ=10° and (b) Ψ=45° . The results are based on the calculations shown in Fig. 3.4. Figure 3.5 illustrates the spatial interaction between 0 ij σ and ij σ Δ in a conventional polar coordinate system. Four dashed quarter circles represent the quadrants favoring left- lateral shear (dashed blue) or right-lateral shear (dashed red) by the background stress field 0 ij σ . The polar diagrams in the center display the angular variation of the positive part of ± ΔCFS (defined in Figure 3.4), showing where left-lateral (solid blue) or right- lateral (solid red) shear is encouraged by the leading term of the incremental stress field ij σ Δ near the rupture tip. Where off-fault yielding can be activated near the rupture tip depends on the location of local maxima of ± ΔCFS (in terms of ij σ Δ ). On the other hand, whether the activated yielding zone can extend to larger scales depends on the spatial (angular) interaction of ± ΔCFS with 0 ij σ . This is indicated conveniently in Figure 3.5 by whether or not the solid lobes are located within a dashed quadrant with the same color. 116 As shown in Figure 3.5a for ° = Ψ 10 , two ij σ Δ -promoted local maxima in the upper half plane ] 180 , 0 [ ° ° ∈ θ , corresponding to “#3-R” and “#4-L” in Figure 3.4a, are located in the right-lateral ] 80 , 10 [ ° ° − ∈ θ and left-lateral ] 170 , 80 [ ° ° ∈ θ quadrants, respectively. Therefore, the two stress fields are positively interacting in these orientations, consistent with the well-developed “lobe-B” and “lobe-C” in Figure 3.3a. However, in the lower half plane ] 0 , 180 [ ° ° − ∈ θ , two ij σ Δ -promoted local maxima, corresponding to “#1-L” and “#2-R” in Figure 3.4a, are located in the right-lateral and left-lateral quadrants, indicating a competition between the two stress fields. The only exception is near ° − = 100 θ , where either right-lateral or left-lateral shear is encouraged by both 0 ij σ and ij σ Δ (probably due to the maximum tensile stress change along this orientation; see Figure 3.4a). However, as this yielding zone extends further, the background maximum compressive stress max σ , operating as the normal stress along the inclined plane ° − = 100 θ , starts to dominate the stress field and is expected to suppress its further development. This prediction is consistent with the activated but only weakly developed “lobe-A” in Figure 3.3a. Similarly, as shown in Figure 3.5b for ° = Ψ 45 , among all the three ij σ Δ -promoted local maxima of Figure 3.4b, only “#1-L” oriented to ° − ≈ 125 θ is located in a stress quadrant with the same sense of shear promoted by 0 ij σ . Therefore only in the vicinity of this orientation can the activated yielding zone extend to a larger scale from the rupture tip, in agreement with the well-developed “lobe-A” in Figure 3.3b. Although none of the ij σ Δ -promoted local maxima is located right ahead of the rupture tip (e.g., 117 ] 45 , 45 [ ° ° − ∈ θ ), the two stress fields ij σ Δ and 0 ij σ are still positively interacting and promote together right-lateral shear. This can produce a weakly-developed yielding zone or at least a close-to-failure stress zone right ahead of the rupture tip, corresponding to the weak “lobe-B” in Figure 3.3b. For such a case where the examined feature has a size comparable to or smaller than the process zone, the singular crack model may not work very well. We note that considering the interaction between the remote and slip-induced stress fields in the employed polar coordinate system is merely a convenient way of illustrating a tendency for promoting left-lateral or right-lateral shear with respect to the rupture tip. On the other hand, the constitutive law for accumulating plastic strain at each point depends on the local stress field only through Eq. 3.5, with no connection to the overall non-local pattern of the stress field and no preference to a particular coordinate system for describing the stress field. 3.4.1.3 Formation of cumulative yielding zones We can now describe the process producing cumulative plastic yielding zones of the type simulated in Figure 3.2. As the rupture propagates, a series of incremental yielding zone lobes are successively produced along strike. Each of these is expected to have fixed shape soon after their activation due to the short timescale v T for stress relaxation and the low residual stress level left behind the rupture tip. This implies that the envelope(s) of the cumulative yielding zone will be constructed by the farthest point(s) of each set of the incremental yielding lobes, and that the overall partition of the final yielding zone onto different sides of the fault will generally follow the pattern of the incremental yielding 118 lobes. Considering the yielding zones generated during the rupture propagation stage along these lines leads to several expectations on observable properties of the cumulative final yielding zone. One observable is the shape of the final yielding zone. Although an exact prediction is not possible due to the non-linearity of the problem, several end-member cases with relatively simple rupture history can be addressed. For example, a self-similarly expanding rupture is expected to produce a triangular-shaped yielding zone, whose thickness linearly increases in the rupture propagation direction (Figure 3.2). Another end-member case is a slip pulse in a quasi-steady state, which is expected to generate a yielding zone with approximately constant thickness along fault strike. These end- member cases have been demonstrated earlier by Andrews (2005) and Ben-Zion and Shi (2005), respectively, and were explored further by Xu et al. (2012a, 2012b). A second possible observable is an overprinting feature in the envelope of the yielding zone. As shown in Fig. 3.2a for ° = Ψ 10 , in addition to the outermost yielding zone envelope (blue) that extends on the compressional (top) side beyond the rupture tip, there is another internal envelope with more intense yielding (red to green) that only approaches the rupture tip from behind. This is not surprising if we recognize that the former and latter envelopes form by the progression of the forward- and backward- oriented incremental yielding zone lobes, respectively (Fig. 3.3a). The observation that the more intense internal yielding zone (red to green in Fig. 2a) approaches the rupture tip from the behind can be explained by the fact that this region is swept commonly by both the backward- and forward-oriented incremental yielding zone lobes. Naturally, we expect a prominent overprinting feature if there are several distinct yielding zone lobes 119 on the same side of the fault and each lobe is with a considerable size. For comparison, the case ° = Ψ 45 in Fig. 3.2b does not have an internal envelope with a prominent overprinting feature on the extensional side, since there is only one dominant yielding zone lobe on that side (Fig. 3.3b). As a result, the maximum plastic strain for ° = Ψ 45 is lower than that for ° = Ψ 10 . This indicates that in addition to influencing the off-fault yielding zone location, the angle Ψ also affects the maximum level of the permanent off- fault strain. 3.4.2 Generation of Large-scale Off-fault Shear Fractures Here we explore how large-scale shear fractures, whose size may be comparable to or larger than the thickness of the yielding zone, may be induced by dynamic ruptures. The interaction between large-scale off-fault shear fractures and the stress failure zone associated with the moving rupture tip is shown to produce, on a fault with uniform properties, a preference for generating synthetic off-fault fractures over antithetic ones. 3.4.2.1 A conceptual model The analysis done so far has no mechanism for preferring either left-lateral or right- lateral microfractures with scale much smaller than the size of the yielding zone. Therefore, we assume that the generated yielding zones are filled with many conjugate sets of microfractures (Figure 3.6). In contrast, the formation of large-scale shear fractures requires that the large-scale stress field should be consistently constructed in some non-local way in space or time. A non-locality in space means that the stress field should coherently promote the activation and growth of potential shear fractures along certain orientations over a considerable distance range. A non-locality in time means that 120 the potential large-scale fractures should continuously be subjected over a considerable time span, with changing configuration of the rupture tip, to a stress field of similar sign. Figure 3.6: Schematic diagram illustrating the generation of off-fault large-scale shear fractures by propagating ruptures along a frictional fault. Panels (a) and (b) correspond, respectively, to the compressional side of Fig. 3.2a and extensional side of Fig. 3.2b. The gray lobes show the current failure zone around the rupture tip. The orientation of the transient maximum compressive stress ! σ max near the rupture tip is indicated by a pair of arrows. Following the above reasoning, we may expect that the slip-induced incremental stress field near the rupture-tip is responsible for the activation and early growth of potential large-scale fractures, approximately along the orientations given by the local maxima of the “driving force” (e.g., CFS Δ in Figure 3.4). We note that there may be 121 different criteria to predict the initiation orientation of such fractures with respect to the rupture tip. After the initial activation, whether or not the factures can continue to grow will depend on the stress field at large scales and/or the time period of being loaded coherently from the moving rupture tip. Figure 3.6a and Figure 3.6b illustrates schematically two scenarios corresponding to the compressional side of Figure 3.2a and the extensional side of Figure 3.2b. The grey lobes and their interiors represent the current stress failure zone defined by the Mohr- Coulomb criterion. The overall orientation and relative size of each lobe depend on the angular pattern of CFS Δ near the rupture tip (Figure 3.4) and its spatial interaction with 0 ij σ at large scales (Figure 3.5). The orientation of the transient maximum compressive stress max σʹ′ is indicated by a pair of arrows based on Fig. 3.3. As mentioned, the seed of each potential large-scale shear fracture is likely activated by the stress field near the rupture tip, presumably following the orientation given by one of the local maxima of CFS Δ . As the rupture front moves forward the activated seeds will be located behind the rupture tip. The seeds associated with antithetic shear have growing direction opposite to the rupture propagation direction. Therefore, they spend relatively short time within any failure zone lobe associated with the moving rupture tip and are likely to stop growing, unless other mechanisms can promote their further growth. In contrast, the seeds associated with a synthetic shear tend to grow in the rupture direction so they spend longer time inside the moving rupture-tip failure zones. As illustrated in Fig. 6, the synthetic shear fractures are promoted over some ranges of space and time by the stress field inside the updated failure zone lobes (see the relative 122 orientation of max σʹ′ ), so they can continue to grow in the forward direction. As the rupture tip moves sufficiently forward, so that the synthetic shear fractures are no longer promoted by any failure zone lobe, they will be arrested unless other mechanisms exist. Since we are using the concept of failure zone lobes around the propagating rupture tip to construct the formation and extension of discrete large-scale shear fractures, the overall distribution of the large-scale fractures is expected to display a self-similar pattern along strike as indicated by the dashed black lines in Figure 3.6. However, we note that the above discussion assumes that the synthetic and antithetic fractures are equally nucleated at a series of locations along the fault and their subsequent growth is not affected by overlapping with other fractures. We also have not considered the feedback between the generated off-fault fractures and the main rupture, and the interaction between neighboring off-fault fractures. Moreover, when a rupture reaches a critical length that depends on the ambient stress environment, it is likely to continue to grow spontaneously further over some distance (see e.g. Fig. 12 of Ando and Yamashita, 2007 and Figure 13 of Ben-Zion, 2008). This provides a possibility for the occurrence of dynamic instabilities at the tips of shear fractures that are long enough, which may generate a hierarchical structure of off-fault shear fractures with branches and bifurcations over several length scales. These complexities are partly discussed in the following section. 3.4.2.2 Numerical simulations with off-fault shear localization Many observational and theoretical studies analyzed shear localization features under various loading conditions (e.g., see the review by Hobbs et al., 1990 and Rosakis and 123 Ravichandran, 2000). While numerically simulated localization could have a strong dependence on the employed mesh (e.g., Needleman, 1989; McKinnon and Garrido de la Barra, 1998), they have been carefully used to successfully explain observed localization features in the field and in laboratory experiments (e.g., Poliakov and Herrmann, 1994; Lecomte et al., 2012; Li et al., 2002). Templeton and Rice (2008) showed numerically that analysis based on the bifurcation theory for localization under quasi-static deformation could also be applied for dynamic rupture problems. Here we follow and extend their work to investigate deformation localization onto shear bands for yielding characterized by the Mohr-Coulomb type plasticity. We use perfect rate-independent plasticity ( 0 = v T ), with no hardening or softening and no volumetric change. The relevant background material is summarized in Appendix B. Issues related to the reliability of numerically simulated localization during dynamic ruptures are discussed in Appendix C. From the results in Appendix B, it can be shown that for our numerical implementation with the above plasticity, the following double-side inequality is satisfied: max min 0 h h h < = < , where min h and max h are the critical minimum and maximum hardening moduli related to localization, and h is the specified hardening modulus. Therefore, shear localization could occur and the critical angle corresponding to max h is given by ° ≈ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = 5 . 37 sin 2 1 arccos 2 1 φ θ c . This value is larger than the Mohr- Coulomb angle ° ≈ − = 5 . 29 2 4 M φ π θ (Jaeger et al., 2007), smaller than the approximated Roscoe angle ° = − = 45 2 4 ψ π θ R where ψ is the dilatancy angle assuming that elastic 124 strains are negligible compared to plastic strains (Roscoe, 1970), and is approximately equal to the angle suggested by Arthur et al. (1977) ° ≈ − − = 3 . 37 4 4 4 A ψ φ π θ . Figure 3.7: Distribution of off-fault shear bands with rate-independent plasticity generated by rupture with (a) Ψ=10° and (b) Ψ=45° . The listed values of frictional, stress and rupture parameters in this and other figures provide a balance between achieving numerical stability and producing prominent off-fault shear bands. The main features discussed in the text have been tested by a parameter space study and confirmed to be robust. Regions I-III highlight local features of the simulated shear bands at various locations. 125 Figure 3.7a shows simulation results for ° = Ψ 10 characterizing low angle thrust faulting. As seen, the overall pattern of the generated shear bands is similar to the conceptual model illustrated in Figure 3.6a. The off-fault yielding zone filled by shear bands displays a triangular shape as expected for self-similar growing pattern. There are clearly two sets of shear bands, with the longer (synthetic) and shorter (antithetic) sets oriented forward and backward, respectively. The relatively high angles of these off-fault shear bands imply that some amount of the total potency is partitioned onto the fault normal direction. The envelope formed by the longer shear bands characterizes the outmost boundary of the entire yielding zone with an overall self-similar pattern, and the internal envelope formed by the shorter shear bands also shows a general self-similarity. Although the two band sets are mainly developed in different locations characterized by the forward- and backward- oriented lobes (see Fig. 3.6a and Fig. 3.7a), the acute angle between the two sets is generally in a good agreement with the theoretical prediction c θ 2 , implying that the transient stress orientations within the two lobes are similar. Additional details may be seen in close-up views of the results at various locations. In location-I in Fig. 3.7a with a limited rupture distance, shear bands are hardly generated at the very beginning and the developed shear bands are primarily of the synthetic type. The overprinting feature as discussed in section 3.4.1.3 mainly involves at this stage re- activation of the synthetic shear bands. In location-II with an intermediate propagation distance, both synthetic and antithetic shear bands are clearly observed and develop in a regular way, reflected by the quasi-linear envelope and almost uniform spacing between adjacent bands. In contrast, the results in location-III with long propagation distance show strong non-linear features such as fluctuations in shear band length and spacing. 126 The apparent slope of the envelope formed by the longest shear bands also deviate from the previous slope where self-similarity is well retained. Figure 3.7b shows corresponding results for ° = Ψ 45 characterizing large strike-slip faults. In contrast to the conceptual model in Figure 3.6b with two sets of shear bands, the simulations show only one predominant set of synthetic shear bands, similar to previous results of Templeton and Rice (2008). Additional numerical simulations with various meshing strategies (Appendix C) and investigation of the nucleation procedure confirm that a single dominant set of shear bands as shown in Figure 3.7b is a robust feature at relatively large scales comparable to the rupture propagation distance (for the examined cases with homogeneous fault properties). The predicted antithetic shear bands may be observed in the very beginning of the rupture, near the edge of nucleation zone (location- I), as well as at a local small scale close to the fault (location-II). However, the most likely situation when rupture entered the spontaneous propagation stage is that the antithetic shear bands are not activated at all, or soon stop growing, due to the extension of earlier activated synthetic shear bands. The acute angle between two conjugate shear bands at locations where they co-exist is about ° 75 , also in a good agreement with the theoretical prediction c θ 2 . At larger propagation distance (location-III), clear second- order bifurcation feature indicates that the generated synthetic bands develop their own branches, with strong fluctuation in band orientation and spacing between adjacent bands. The bifurcation feature may be explained as resulting from increasing stress field between two adjacent well-developed synthetic bands with increasing rupture distance, so secondary branches need to be triggered to relax the high stresses. 127 Figure 3.8: Schematic diagram showing possible competition between synthetic (red) and antithetic (blue) shear bands. The solid and dashed lobes are constructed similar to Fig. 3.6, but with small finite shift and slight size increase corresponding to the changing rupture front configuration. Long solid curves represent already-developed paths of two conjugate shear bands (l s for synthetic and l a for antithetic), whereas short dashed curves indicate likely extension paths for the two bands. It is interesting to ask why the simulation for ° = Ψ 45 shows only one dominant set of shear bands, in contrast to the results for ° = Ψ 10 with two well-developed sets. A careful examination of the stress field surrounding the moving rupture front suggests that this may be explained by a combination of several effects. The first is related to the off- fault locations where the two sets of shear bands tend to grow. For ° = Ψ 10 , the activated conjugate shear bands tend to grow in different locations associated with distinct stress lobes (Figs. 3.6a and 3.7a). Although later activated antithetic bands may have to pass through earlier formed synthetic sets, the antithetic bands usually do not intersect at the tip region of a currently growing synthetic band thus may still extend outward. For 128 ° = Ψ 45 , however, the two sets attempt to grow in the same location at the same time (inside the backward-oriented stress lobe), so there is a clear competition between the antithetic and synthetic bands (Figs. 3.6b and 3.7b). Therefore, the earlier development of the synthetic set could suppress the growth of the antithetic set at the same location. Figure 3.8 illustrates schematically a second effect related to competition between generations of the two sets of shear bands. The grey stress lobes are constructed in a similar way to Fig. 3.6, with solid lobes corresponding to a reference state at time t and dashed lobes to an updated state at time t t Δ + . The rupture advance x Δ is assumed to be small enough (compared to the size of the backward-inclined lobe) such that the transient stress fields around the rupture front at t and t t Δ + are similar (they will be exactly the same for a propagating steady-state pulse regardless of the value of x Δ ). The total Coulomb stress at a given time generally decays with distance from the rupture front inside each characteristic lobe (Fig. 3.3b) in agreement with the r / 1 term in ij σ Δ (Eq. 3.6). This is also generally true for a Lagrangian description (e.g. the stress field ahead of a virtually growing shear band along ˆ r or ! ˆ r ) if the change of rupture configuration is small compared to the characteristic size of the stress lobe (as assumed above). In contrast, the Lagrangian-type stress variation along the transverse direction of the lobe (e.g., ˆ θ or ! ˆ θ ) is probably small because the examined path more or less follows the translation direction from the reference state to the updated state. Another effect is related to the orientation of the stress field. As shown in Figure 3.3b and examined more quantitatively by Xu et al. (2012b), there is an anti-clockwise rotation of the principal stress orientation with increasing distance from the fault. This 129 can result in bending of the shear band growing path, moving initially backward-inclined antithetic bands towards the fault-normal direction and initially forward-inclined synthetic bands towards the fault-parallel direction. Combining the two effects, the antithetic set tends to grow in a direction with relative large stress decay gradient and/or relatively strong rotation of stress orientation, while the corresponding stress variations along the growing path of the synthetic set are likely to be relatively small. Therefore, the synthetic shear bands are expected to be more promoted and their subsequent development may further suppress the growth of the less promoted antithetic set. The above analysis assumes that the rupture-induced stress field is dominant and the expected yielding zone size is relatively large. Therefore, it may not be appropriate for explaining the situation in the very beginning of a rupture. As mentioned in section 3.4.2.1, the synthetic bands also benefit from the moving stress lobes over longer duration than the antithetic shear bands even. All these effects explain the dominant generation of large- scale synthetic shear bands (or fractures) by dynamic ruptures along larger strike-slip faults. 3.4.3 Cases With Fault Heterogeneities So far the main fault was assumed to have homogeneous initial stress and frictional properties. However, natural faults are associated with various geometrical and rheological heterogeneities (e.g., Ben-Zion and Rice, 1993). In this section we analyze two simple end-member cases of fault heterogeneities. 130 Figure 3.9: Schematic diagram showing slip-induced ΔCFS lobes around a junction with (a) fault branch and (b) fault bend. The likely rupture evolutions are compared with laboratory experiments of Rousseau and Rosakis (2003, 2009). See text for more explanation. 3.4.3.1 Rupture behavior around fault junction We consider a configuration where multiple piece-wise planar fault segments are connected at a junction point. In particular, we show how the ideas presented in section 3.2 can be applied to explain the experimental results by Rousseau and Rosakis (2003, 2009) on fault bend and branch problems. To be consistent with the impact loading and the material properties used in their laboratory experiments, we only consider the stress field induced by an incoming mode-II crack and assume an equivalent internal friction coefficient of 0.577 for the surrounding bulk. The model configuration is described by a fault branch or bend path that is oriented at ° − = 100 θ with respect to the main fault segment (Figure 3.9). For the fault branch problem an extended horizontal path remains beyond the junction point (Figure 3.9a), while for the fault bend problem it is excluded (Figure 3.9b). 131 Based on the stress analysis of section 3.4.1, we plot the normalized stress field (the leading term) in the form of ± ΔCFS (“+” and “–” sign for synthetic and antithetic rupture triggering, respectively) around the rupture tip (currently located at the junction point), with an instantaneous rupture speed R r c v 85 . 0 = where c R ≈ 0.92c s denotes the Rayleigh wave speed. We do not include a finite cohesion in our consideration of rupture triggering, but refer to the clarification by Rousseau and Rosakis (2003) that in the experiments fault interfaces are bonded with an ability to sustain shear ruptures even under a transient tensile stress. For the fault branch problem (Figure 3.9a), since the backward branch is unfavorably oriented for a direct rupture jump (because of the large change in segment direction), the incoming rupture is likely to stay on the horizontal path, although part of its induced stress field (solid blue lobe) attempts to trigger an antithetic shear rupture along the branch path. As the rupture tip rapidly moves forward along the horizontal path, the induced stress field is also removed quickly from the backward-oriented fault branch. Unless a critical state has been reached along the branch within this short time period, the potential antithetic shear rupture will not be triggered. Because some energy may have been consumed in the triggering attempt, the main rupture is expected to experience deceleration or even get arrested after some distance beyond the junction point. For the fault bend problem (Figure 3.9b), the expectation may be different. Unless some breakouts are created in the bulk around the junction point, most of the energy of the incoming rupture will be transferred more efficiently to the weak bend path. Still, due to the unfavorable orientation of the bend path for a rupture jump, one may expect that the incoming rupture and its induced stress field can be temporarily halted (with a strong 132 deceleration) around the junction point. Since the increased transient dynamic stress field associated with the reduced rupture speed tends to stay around the junction point and in particular along the bend path for longer time duration than for the fault branch problem, a new rupture will have more opportunities to be triggered following the stress distribution along the bend path. This may most likely be an antithetic shear rupture with a reversed sense of slip compared to the incoming rupture. If the bend path is not favorably oriented for a rupture transfer (e.g. ] 120 , 60 [ ° ° ∈ θ ), the energy of the incoming rupture will probably be absorbed by breaking into the surrounding bulk through off-fault damage process. We note that in contrast to the theoretical analysis of Rousseau and Rosakis (2003, 2009) on the absolute “driving stress”, our analysis with separate considerations of CFS Δ for synthetic and antithetic shears, and the time duration effect, has more explicit connections to observations. We also note that it is more realistic to consider in general the triggered secondary rupture along a kink path to be mixed mode-I and mode-II, in particular without remote compressive stress. Indeed, an “opening mode” was presumably observed by Rousseau and Rosakis (2003, Figure 13c) for a continuing intersonic rupture along a fault bend path with ° − = 56 θ relative to the previous fault segment. 3.4.3.2 Off-fault shear fractures around fault strength heterogeneities The result in section 3.4.2.2 that there is a relatively clear antithetic shear around the edge of the nucleation zone (location-I in Fig. 3.7b), suggests that abrupt changes of rupture behavior can promote the generation of off-fault antithetic shear fractures for 133 moderate-to-high Ψ values. This is examined here with simulations assuming heterogeneous static friction coefficient along the fault. We assume that the pattern of co- seismically generated shear bands indicate triggering potential of subsequent rupture if secondary faults exist along the optimal orientation of the shear bands, with the efficiency of triggering more or less proportional to the length of the shear bands. Figure 3.10: Assumed space variations of the static friction coefficient f s (lower line) and corresponding evolution of normalized rupture speed averaged over 4L 0 (upper line) along the fault. Figure 3.10 shows a fault configuration with periodic spatial variations of s f between a reference level and a slightly perturbed level, along with the evolution of rupture speed for this case. The generated distribution of off-fault shear bands is given in Figure 3.11. Within each piece-wise domain with constant s f , the evolution of rupture speed is relatively smooth and there are no prominent antithetic shear bands, similar to the previous results. However, at locations with abrupt jumps of s f , the rupture speed 134 changes rapidly (Figure 3.10) with opposite sign to that of the strength change ( s f Δ ). This results in locally pronounced antithetic shear bands near places with rapid rupture deceleration and a seeming vacancy of antithetic band near places with rapid rupture acceleration (Figure 3.11). Since the rupture can still propagate through the assumed weak fault heterogeneities, the size of the activated antithetic shear bands is small. Figure 3.11: (a) Off-fault shear bands generated for the heterogeneous case of Fig. 3.10. The inset plot near the bottom shows (with scale at the right vertical axis) the plastic strain distribution along the fault. (b)-(d) close-up views of (a) in different locations associated with small upward and downward jumps of f s along he fault (Fig. 3.10). Figure 3.12 shows the distribution of shear bands with a strong fault barrier located at X=60L 0 where s f abruptly jumps from 0.55 to a very high level that arrests the rupture. The rapid deceleration and eventual rupture arrest produces considerable off-fault yielding around the strong barrier (Fig. 3.12a). In this case there are several distinct shear 135 band sets, including a well-developed antithetic band that is decoupled from the forward- oriented synthetic bands (i.e., the synthetic and antithetic bands do not overlap around the strong barrier, so they can grow approximately independently of each other). A corresponding simulation with a finite relaxation timescale produces similar shear band patterns around the strong barrier (Fig. 3.12b), suggesting that the decoupled shear bands are general features. Another general feature is the very high plastic strain around the strong barrier (see the estimated values in the caption of Fig. 3.12). The generated large-scale off-fault synthetic and antithetic shear bands around the strong barrier, together with the arrested rupture, form a pattern similar to the proposed diagram for non-conservative barriers of King and Nábělek (1985, Fig. 2). For our case, the finite motion resulted by the arrested rupture along the main fault, if constrained as a plane strain problem, will be partly compensated by two conjugate sets of off-fault shear-type yielding around the barrier (see insets in Fig. 3.12a and explanation in the caption). A permanent volumetric strain in the form of topographic features related to abrupt rupture arrest may also be generated around strong barriers (e.g., Ben-Zion et al., 2012). We note that some off-fault yielding also occurs between the two dominant shear band sets around the strong barrier (on the extensional side). This is likely triggered by the radiated waves associated with the strong stopping phase and is presumably related to a potential tensile stress at that location. The relatively large size of the antithetic shear band in Figure 3.12 suggests that if a secondary fault path exists along the orientation θ∈[−120°,−100°] , a potential antithetic shear rupture would have a high probability to 136 be triggered, similar to the fault bend problem (Figure 3.9b) but with the presence of background stress. Figure 3.12: Distribution of off-fault plastic yielding with a strong barrier ( f s >>1) located at X=60L 0 (a) with rate-independent and (b) rate-dependent plasticity. The inset plot in (a) schematically show how the high slip gradient near the barrier is partly compensated by the conjugate off-fault shear bands. Following King and Nábělek (1985), letters A, B and C denote different blocks separated by the main fault and shear bands, whereas the vectors on the right show the relative shear motion from a starting side to ending side. The maximum normalized value of ε 0 p in (a) is ~8000 and may correspond to a permanent strain of order 1 assuming σ c /µ ~10 −3 . The maximum value in (b) is about one order of magnitude less than that in (a). 3.5 DISCUSSION The main goal of this chapter is to develop improved connections between characteristics of off-fault yielding at different scales (microfractures, intense inner 137 yielding zone, outer yielding zone) and dynamic ruptures on earthquake faults. Previous works on the topic focused on properties of stationary stress fields with respect to given rupture tip locations (e.g. Poliakov et al., 2002), or considered a changing rupture tip configuration but ignored the possible generation of antithetic type fractures in non- smooth rupture processes (e.g. Ando and Yamashita, 2007). Here we show that explicit considerations of the cumulative effects of transient features of the dynamic stress field (e.g., Figs. 3.4-3.6, 3.8-3.9), stemming from interactions between successive positions of the crack tip and the background stress field, provide refined understanding of several yielding features. These include the competition between large-scale synthetic and antithetic shear bands off the main fault, the existence of well-developed (decoupled) conjugate sets with both synthetic and antithetic bands in some circumstances, and the recognition of an internal intense yielding zone with overprinting of deformation fractures (produced by different yielding lobes) and an outer yielding zone generated by one dominant yielding lobe. Our considerations of stress over extended regions around the rupture tip at given times and extended time intervals at given locations, plus the degree of smoothness of the evolving stress field, help to clarify the type of large-scale shear bands generated during the initiation, propagation and arrest of ruptures. As shown in sections 3.4.2-3.4.3, these ideas explain consistently the relative generation of synthetic and antithetic shear bands for faults with homogeneous frictional properties under different background stress fields ( ° = Ψ 10 and ° = Ψ 45 ), as well as examples with heterogeneous fault friction. We also explain results on rupture along a conjugate backward-oriented fault in the laboratory 138 experiments of Rousseau and Rosakis (2003, 2009), and illustrate the generation of pronounced antithetic fractures by abrupt rupture arrest. The results imply that both the stress amplitude (average or maximum level) and duration of a “favorable” stress field (e.g., positive Coulomb stress change) are important for rupture branching or damage generation, in particular at high angle or backward orientation to the main fault. The duration factor helps to explain (better than a mere consideration of “stress shadow effects”) frustrated failure attempts during rupture propagation with rapidly changing stress field, and the more encouraged attempt during strong rupture deceleration or arrest with longer operation of the stress field. Positive contributions from both the stress amplitude and duration are consistent with favored failure scenarios in laboratory experiments (Rousseau and Rosakis, 2003), numerical simulations (Kase and Kuge, 2001) and natural earthquake examples (e.g., Hudnut et al., 1989). The generation of microfractures by a single rupture at a scale much smaller than the yielding zone size is likely to be accomplished rapidly, and hence depends essentially only on the local stress field. Therefore, unless other mechanisms exist, there is no preference for either synthetic or antithetic shear microfractures. On the other hand, repeated earthquakes typically occur along the same fault segment, so distributed microfractures generated by earlier events could be reactivated by later events that can also increase the microfracture density. If a zone with high density of microfractures of a certain type (synthetic or antithetic) is aligned favorably with the stress generated by a new event, the microfractures can coalesce and form large-scale fractures. Since the length scale separating dense microfractures is small, the coalescence of microfractures 139 into large scales could occur much faster than developing large-scale fractures in intact rocks. In such cases, well-developed antithetic large-scale fractures may develop during rupture propagation along a strike-slip fault (i.e., the relative importance of changing rupture configuration is reduced for rapid coalescence of pre-existing microfractures). Figure 3.13: An idealized Riedel shear structure containing various fracture elements, modified from Sylvester (1998) and Davis et al. (1999). The Y shear is parallel to the principal displacement zone (PDZ), whereas other fracture elements are inclined with characteristic acute or obtuse angles to the PDZ in relation to σ max and/or the internal friction angle. Another important issue is how to distinguish between deformation features generated by quasi-static faulting vs. dynamic rupture. A comparison between classic Riedel shear structure and the simulated results with off-fault shear bands for typical strike-slip faults suggests several distinguishable features which may be used jointly. The classic Riedel shear system (Fig. 3.13) is formed in a quasi-static process associated with a transition 140 from distributed deformation to localized deformation (e.g., Tchalenko, 1970; Ben-Zion and Sammis, 2003, and references therein). The initially developed characteristic fracture elements (R, R’, etc) are generally quasi-symmetrically distributed with respect to the later formed Principal Displacement Zone (PDZ). Each set of fracture elements are usually regularly spaced along the general strike (referred to as en échelon), but may show more complex patterns towards increasing displacement direction in a single system (Ahlgren et al., 2001). The average angle with respect to the PDZ, under small background shear strain, is ° 15 ~ for the synthetic R-shear and ° 75 ~ for the antithetic R’-shear (Davis et al., 1999). Cases with very high-angle or backward-oriented R’-shear, produced by large background shear strain (Wilcox et al., 1973), significant volumetric compaction (Ahlgren, 2001), or rigid block rotation (Nicholson et al., 1986) are not treated as representative quasi-static results in this chapter, but can be otherwise excluded by the recognition of sigmoidal form (Ramsay and Huber, 1983), systematically re- oriented fracture elements of various types (Lecomte et al., 2012, Fig. 1), or severe fragmentations (Brosch and Kurz, 2008). On the other hand, the dynamic counter part of Riedel shear usually requires a pre- existing localized fault to accommodate the rupture, and the off-fault fractures are produced by the stress concentration around the propagating rupture tip (e.g., Ando and Yamashita, 2007). In typical strike-slip faults, the rupture-induced off-fault shear fractures or bands are mainly distributed on one (extensional) side of the fault, with likely increasing fracture length and/or complexity (e.g. hierarchical patterns and variations in fracture separation and length) towards the rupture propagation direction. In cases of expanding ruptures this is the direction of decreasing slip, leading to opposite expectation 141 on the correlation between slip and complexity compared to quasi-static deformation (Ahlgren et al., 2001). Also, the average angle between the dynamically generated synthetic or antithetic shear bands and the fault plane is typically steeper than the corresponding quasi-static value, due to the re-orientation of the maximum compressive stress in the dynamic process (Xu et al., 2012b). In particular, if the antithetic shear bands emerge, they are usually oriented backward (with an obtuse angle) or almost orthogonally to the background shear direction. Despite the above differences, our results for smoothly propagating rupture (e.g., Fig. 3.7b) agree with the quasi-static model that there may be a statistical preference for the synthetic Riedel shear over the antithetic shear (e.g., Schmocker et al., 2003; Katz et al., 2006; Misra et al., 2009b). In cases with strong fault heterogeneities, there could be strong fluctuations of rupture speed or even rupture arrest. When rupture experiences a sudden deceleration, either due to encountering a fault strength barrier (Figs. 3.11-3.12) or due to un-favored changing direction of fault path (Figs. 7 and 13 of Duan and Day, 2008), prominent signals of antithetic shear fractures (or bands) associated with high permanent strain are observed, with the intensity of the signal generally proportional to the abruptness of rupture deceleration. This suggests that backward or almost- orthogonally inclined antithetic shear fractures, primarily on one (the extensional) side of a large strike-slip fault, may be used as a signal that reflects abrupt deceleration of rupture process. On the other hand, instances of abrupt rupture acceleration produce relative lack of off-fault yielding in the vicinity of the corresponding locations. Examples include the locations with reduced frictional strength (marked blue locations in Fig. 3.11), fault releasing bends (Fig. 3 of Dunham et al., 2011b), fault kink where the continued 142 path is favorably oriented (Fig. 13 of Duan and Day, 2008), and locations where a subshear rupture rapidly jumps to supershear regime associated with a low S value (Fig. 13 of Templeton and Rice, 2008). Although the listed examples include both rupture acceleration within the subshear regime (Figs. 3.10 and 3.11; Dunham et al., 2011b) and more abrupt rupture transition from subshear to supershear (Duan and Day, 2008; Templeton and Rice, 2008), The similarity in producing relative lack of damage in these diverse examples likely stems from the fact that a fault portion over which the rupture accelerates has less chance to be subjected to high (transient) stress concentration. A comparison between our model predictions and observations is best done in the context of damage zones well below the surface, since our 2-D calculations with the adopted Mohr-Coulomb criterion do not account for free surface effects (Ma and Andrews, 2010) or the competition between tensile and shear type failures under low confining pressures (Bourne and Willemse, 2001). The predicted high-angle antithetic shear pattern by abrupt rupture deceleration is well supported by the indirect but strong evidence of aftershock distribution and/or conjugate earthquake triggering around known rupture termination ends. Numerous examples show clearly two groups of aftershock clusters around mainshocks termination ends: one is located in front of earthquake terminus while the other forms a high angle (roughly along the conjugate direction) to the main fault plane mainly on the extensional side (King et al., 1994, and references therein). The overall lineaments of two distinct aftershock clusters around the same earthquake terminus, and sometimes the emergence of one dominant aftershock cluster, or triggered big event(s) along the orthogonal or even backward direction near the termination end(s) of a prior event (e.g., Aktar et al., 2007; Chi and Hauksson, 2006; Das, 143 1992; Hudnut et al., 1989) are in good agreement with our expectation without or with (preferred) off-fault plane of weakness (see section 3.4.3). More examples of conjugate earthquake triggering consistent (or at least partially consistent) with our expectations can be found in Das and Henry (2003), Jones and Hough (1995), and the recent off-Sumatra earthquake sequence (Meng et al., 2012; Yue et al., 2012). The above discussion on triggered antithetic shear near the tip of a master fault or earthquake terminus enriches the list of mechanisms for producing near orthogonal conjugate faulting, which also includes post-rotation of two conjugate faults or weak strength-dependence on pressure (Thatcher and Hill, 1991, and references therein), block rotation of antithetic shear between two master faults (Kilb and Rubin, 2002), and re-activation of a rifting system in a contemporary obliquely-oriented compressive stress field (Hamburger et al., 2011, and references therein). Several important aspects of natural brittle deformation have not been considered in this work. These include effects associated with rough faults (e.g., Dunham et al., 2011b), free-surface effects (e.g., Ma and Andrews, 2010), volumetric deformation (e.g., Rudnicki and Rice, 1975; Viesca et al., 2008), post-failure frictional weakening (Ando and Yamashita, 2007), and changes of elastic moduli in the yielding zones (e.g., Lyakhovsky et al., 1997, 2011). These effects should be considered in future studies. 144 Chapter 4 Dynamic Ruptures on A Frictional Interface with Off-fault Brittle Damage: Feedback Mechanisms and Effects on Slip and Near-fault Motion 4.1 SUMMARY The spontaneous generation of brittle rock damage near and behind the tip of a propagating rupture can produce dynamic feedback mechanisms that modify significantly the rupture properties, seismic radiation and generated fault zone structure. In this work we study such feedback mechanisms during a single event and their consequences for earthquake physics and various possible observations near the fault and in the far field. This is done through numerical simulations of in-plane dynamic ruptures on a frictional fault with bulk behavior governed by a brittle damage rheology that incorporates reduction of elastic moduli in off-fault yielding regions. The model simulations produce several features that modify key properties of the ruptures and local wave propagation. These include (1) dynamic changes of normal stress on the fault, (2) rupture transition from crack-like to a detached pulse, (3) emergence of a rupture mode consisting of a train of pulses, (4) quasi-periodic modulation of slip rate on the fault, and (5) asymmetric near- fault ground motion with higher amplitude and longer duration on the side with fresh 145 damage generation. The results can have significant implications for multiple topics ranging from rupture directivity and near-fault seismic motion to tremor-like radiations. 4.2 INTRODUCTION The high stress concentration at the front of dynamic ruptures in the seismogenic zone is expected to produce brittle rock damage (reduction of elastic moduli) in the material surrounding the fault. The associated energy absorption can reduce the amplitude of ground motions. However, damage process can contribute to the radiated wavefield (Ben-Zion and Ampuero, 2009) and the generated damage zone with reduced elastic moduli may behave as a waveguide that can amplify near-fault motion. Waves reflected from the boundaries of the generated damage zone may influence the subsequent rupture properties as previously shown in simulations with a pre-existing low velocity fault zone (e.g., Ben-Zion and Huang, 2002; Huang and Ampuero, 2011). Moreover, if the damage zone is asymmetrically distributed around the fault, the spontaneously generated bimaterial interface can lead to coupling between slip and dynamic change of normal stress that can change significantly the mode and other properties of ruptures (e.g., Weertman, 1980; Andrews and Ben-Zion, 1997; Ampuero and Ben-Zion, 2008). There has been considerable research in recent years on simulations of dynamic ruptures that incorporate off-fault plastic yielding (e.g., Andrews, 2005; Ben-Zion and Shi, 2005; Duan and Day, 2008; Templeton and Rice, 2008; Ma and Andrews, 2010; Dunham et al., 2011a; Kaneko and Fialko, 2011; Xu and Ben-Zion, 2013; Gabriel et al., 2013). These studies provided insights on various connections between properties and 146 processes of deformation on and off the main faults. However, models with off-fault plasticity keep the elastic moduli in the yielding region unchanged, so they do not account for potentially important feedback mechanisms between generation of brittle rock damage and properties of dynamic ruptures and ground motion. Laboratory experiments (e.g., Lockner and Byerlee, 1980; Stanchits et al., 2006) and seismological observations (e.g., Peng and Ben-Zion, 2006; Wu et al., 2009) show that brittle failures are accompanied by significant temporal changes of elastic moduli. To study the consequences of dynamic reduction of elastic moduli near propagating ruptures, we incorporate in this work brittle off-fault damage in simulations of dynamic in-plane ruptures during a singe event. Healing effect and damage evolution over multi-cycles are not considered in this work. The simulations employ a continuum visco-elastic damage model with co-evolution of inelastic strain and elastic moduli in off-fault regions where the stress reaches the elastic limit (e.g., Lyakhovsky et al., 1997, 2011; Lyakhovsky and Ben-Zion, 2009, 2012). The results illustrate the richness of new dynamical features that can arise from the interaction between ruptures on main faults and dynamically evolving elastic moduli in the yielding damage zones. In particular, we find under some conditions transitions of the rupture mode to different types of pulses and oscillatory slip rate that may produce tremor-like seismic radiation away from the fault. The results also show an apparent supershear rupture propagation with respect to the generated damage zone and an off- fault yielding pattern that is different in some respects from that produced by plasticity under similar stress conditions. 147 4.3 MODEL DESCRIPTION We consider 2-D in-plane ruptures and off-fault brittle damage allowing dynamic changes of elastic moduli under plane strain conditions. The relevant stress components operating on the fault and the surrounding medium are listed in Figure 4.1. A right-lateral rupture is nucleated with a prescribed speed over a small patch (red portion in Fig. 4.1) until it can propagate spontaneously along the frictional fault (solid black line in the center). The first motion of P-waves radiated from the hypocenter defines four quadrants, with “C” and “T” denoting compressional and extensional quadrants, respectively. The initial normal and shear stresses on the fault are σ 0 =σ yy 0 and τ 0 =σ xy 0 , respectively, and Ψ represents the angle between background maximum compressive stress σ max and the fault. A relative strength parameter S= (τ s −τ 0 )/(τ 0 −τ d ) is used to quantify the relation between initial shear stress, static strength τ s = f s (−σ 0 ) and dynamic strength τ d = f d (−σ 0 ) , with f s and f d being the static and dynamic friction coefficient, respectively. It is known that increasing the initial shear stress level towards the static strength can induce a transition to supershear ruptures (Burridge, 1973; Andrews, 1976; Das and Aki, 1977; Day, 1982). In this chapter, we choose the value of S and rupture propagation distance range to produce subshear ruptures relative to the shear wave speed of the intact medium. We will later discuss the relation between rupture speed and the wave speed of the spontaneously generated damaged material. 4.3.1 Nucleation Procedure, Friction Law, and Normal Stress Regularization We follow the procedure of Xu et al. (2012a) with a time-weakening friction (TWF) to artificially trigger the rupture within the nucleation zone. The rupture front during the 148 nucleation stage is enforced to propagate outward with a constant subshear speed. The frictional strength at locations reached by the prescribed rupture front linearly weakens with time up to the dynamic level τ d . An associated length scale R TWF can be defined over which the fault strength spatially reduces from τ s to τ d , and is well resolved by the employed numerical cells. The actual time duration for the nucleation process depends on when the prescribed rupture can propagate spontaneously under a physical friction law that is described below. Figure 4.1: Model configuration for dynamic ruptures along a frictional interface (black line at the center) with possible generation of brittle damage in the surrounding bulk. The rupture is nucleated right-laterally with a prescribed speed over a small patch (red portion) until it can propagate spontaneously. The angle Ψ characterizes the relative orientation of the background maximum compressive stress to the fault. Symbols “C” and “T” denote the compressional and extensional quadrants, respectively, in relation to the first motion of P-waves radiated from the nucleation zone. Because of the symmetry, only the right half will be shown in later plots. We adopt a linear slip-weakening friction (SWF) to physically describe the breakdown process along the fault outside the nucleation zone. Specifically, the frictional 149 strength τ has the following dependence (e.g., Ida, 1972; Palmer and Rice, 1973; Andrews, 1976) on slip Δu : τ = τ s − (τ s −τ d )Δu /D c if Δu≤D c τ d if Δu>D c $ % & ' & , (4.1) where D c is a characteristic slip distance over which the fault strength reduces from τ s to τ d . When the background shear stress is only slightly higher than τ d , the size of the spatial region associated with the strength reduction, referred to as the process zone, can be estimated (e.g., Rice, 1980) by R= R 0 f II (v r ) , (4.2a) where f II (v r ) is a monotonic function of rupture speed v r that increases from unity at v r =0 to infinity at the limiting Rayleigh wave speed, and R 0 is the static value of R at zero rupture speed. For Poissonian solids this is given by R 0 = 3π 8 µD c (τ s −τ d ) , (4.2b) where µ is the shear modulus. To provide proper numerical resolution for this study with evolving rock elastic properties, we discretize R 0 (by using the µ value of intact rocks) with multiple numerical cells, and check its dynamic value during rupture propagation with evolving µ value to ensure that R is always well resolved. 150 A related consideration for this study is the fault frictional response under abrupt changes of normal stress, owing to the possible damage-related material contrast across the fault. To ensure the numerical stability in such scenarios, a Prakash-Clifton normal stress regularization of the type proposed by Rubin and Ampuero (2007) is added to describe the evolution of a modified normal stress σ * :  σ * = V +V * δ c σ −σ * ( ) , (4.3) where V * and δ c are characteristic slip rate and slip distance, respectively, and V is the slip rate. Similar to Xu et al. (2012a), we choose the values for V * and δ c so that normal stress regularization is only moderately but not overly emphasized. A discussion on the influence of wave-induced normal stress change at a local bimaterial interface further behind the rupture front can be found in Appendix D. 4.3.2 Brittle Damage Rheology for The Bulk We follow the model formulation of Lyakhovsky et al. (2011) and related works (e.g., Lyakhovsky et al., 1997; Hamiel et al., 2004; Ben-Zion and Lyakhovsky, 2006; Lyakhovsky and Ben-Zion, 2008) to describe material response under brittle deformation. The elastic potential energy per unit mass accounting for internal damage is expressed as: U ε ij e ,α ( ) = 1 ρ λ 2 I 1 2 +µI 2 −γI 1 I 2 " # $ % & ' (4.4) where ρ is the mass density, λ and µ are the Lamé parameters, and γ is an additional modulus that vanishes in a damage-free solid. The elastic strain tensors ε ij e and a damage 151 scalar α are treated as state variables, with α being interpreted as a non-dimensional measure of the density of non-interacting microcracks in a macroscopically representative volume (Lyakhovsky et al., 1997). In the field, α may be measured to be proportional to the weighted number of microcracks per unit volume, while in our study it is scaled to have an explicit connection to elastic moduli (see Eq. 4.5; also see Budiansky and O’Connell, 1976). The first and second strain invariants are described by I 1 =ε kk e and I 2 =ε ij e ε ij e . The elastic moduli in (4.4) are assumed to have the following dependence on the damage variable α : λ=λ 0 µ=µ 0 +ξ 0 γ r α γ =γ r α (4.5) where λ 0 and µ 0 are the Lamé parameters of the intact solid, ξ 0 is a material parameter (with a negative value) related to the onset of damage generation and γ r is a scaling factor that maps damage into elastic moduli and sets the maximum damage level at one. The parameter ξ 0 can be connected to the internal friction angle φ : ξ 0 = − 2 1+(λ 0 /µ 0 +1) 2 sin 2 φ . (4.6) Eq. (4.6) is the 2-D version of the relation derived by Lyakhovsky et al. (1997). From the above expressions, a non-linear stress-strain relation can be readily derived as: σ ij =ρ ∂U ∂ε ij e = λI 1 −γ I 2 ( ) δ ij + 2µ−γI 1 / I 2 ( ) ε ij e . (4.7) 152 For deformations with non-zero I 1 , the above equation may be re-written using effective moduli depending on ξ = I 1 / I 2 : σ ij = λ−γ /ξ ( ) I 1 δ ij + 2µ−γξ ( ) ε ij e . (4.8) Depending on the level of internal damage, the elastic potential energy U may lose its convexity under different types of loadings (see Lyakhovsky et al., 1997, 2011 for a detailed discussion on this topic). Mathematically the loss of convexity leads to non- uniqueness of the solutions to quasi-static problems, while physically it signifies brittle instability and change in the state of the material. In the context of our work, a loss of convexity of the energy function can lead to severe numerical instabilities and thus needs to be carefully regularized. Table 4.1: Hessian matrix of U (multiplied by ρ ) ρ(∂ 2 U /∂ε ij e ∂ε kl e ) in 2-D. The original 3-D version can be found in Table 1 of Lyakhovsky et al. (1997). ε 11 e ε 22 e ε 12 e ε 11 e λ+2µ−γξ+λξe 1 2 −2γe 1 λ−γ(e 1 +e 2 )+γe 1 e 2 0 ε 22 e λ−γ(e 1 +e 2 )+γe 1 e 2 λ+2µ−γξ+λξe 2 2 −2γe 2 0 ε 12 e 0 0 2µ−γξ Notation e 1 =ε 1 e / I 2 , e 2 =ε 2 e / I 2 , where ε 1 e and ε 2 e are principal strains 153 The requirement of convexity of U is equivalent to requiring (Lyakhovsky et al., 1997, 2011) that all the eigenvalues of the Hessian matrix ρ(∂ 2 U /∂ε ij e ∂ε kl e ) are positive (see Table 4.1; note that 0 33 23 13 = = = e e e ε ε ε for adopted plane strain conditions). The first and second eigenvalues can be calculated as the roots of a quadratic equation x 2 +bx+c=0 , (4.9) which is associated with the 2×2 sub-matrix in the upper left of Table 4.1: b=−(2λ+4µ−3γξ), c=(2µ−γξ) 2 +(2µ−γξ)(2λ−γξ)+(λγξ−γ 2 )(2−ξ 2 ). (4.10) With the above expressions, the corresponding conditions for the loss of convexity are: Λ 1 =(−b− b 2 −4c)/2= 0, Λ 2 =(−b+ b 2 −4c)/2= 0. (4.11) The third eigenvalue is given by the isolated principal minor in the lower right of Table 4.1, and the corresponding condition for the loss of convexity is: Λ 3 =2µ−γξ =0. (4.12) Each condition in Eqs. (4.11) and (4.12) defines a characteristic curve in the ξ−α phase space, which can be used to distinguish materials in different regimes. Figure 4.2 shows such a phase diagram within the range of the current 2-D study (− 2 ≤ξ ≤ 2, 0≤α≤1 ). Eqs. (4.11) and (4.12) normalized by µ 0 are plotted with different colors to separate material phases with or without losing convexity of different types, and the superposed black curve defines the common boundary for all the possibilities. We note that Λ 1 =0 is 154 the strongest condition in the compressive (ξ <0 ) or slightly tensile (ξ >0 ) regime, while Λ 3 =0 dominates in the vicinity of 2-D extension (ξ→ 2 ). For our focus on earthquake-related deformation, we mainly use Λ 1 =0 to discuss the issue of convexity. Figure 4.2: ξ−α phase diagram characterizing rock behaviors in different deformation regimes as a function of internal damage (in 2-D). From left to right: ξ =− 2 and ξ = 2 correspond to isotropic compaction and tension, respectively, while deformation has some non-zero shear component in between. The threshold value ξ 0 defines the level of ξ above which damage starts to accumulate. From bottom to top: α=0 corresponds to purely elastic continuum without internal damage, while α=1 defines the maximum damage level where convexity is lost at . Red and blue curves correspond to the conditions with loss of convexity in Eq. 4.11, and the green curve corresponds to the condition defined in Eq. 4.12, with all the conditions properly normalized by µ 0 . The piece-wise black curve simply defines the common boundary between preserving (lower left) and losing convexity (upper right) for all the possibilities. The original 3-D version can be found in Figure 1 of Lyakhovsky et al. (1997). During damage accumulation the modulus γ increases and the shear modulus µ decreases. This leads to material evolution from linear elastic solid (α=0 ) to ξ 0 155 macroscopic brittle instability at a critical damage level (α cr ) leading to loss of convexity. Lyakhovsky et al. (1997, 2011) derived an evolution equation for the damage state variable using energy and entropy balance equations and by requiring non-negative entropy production. Previous studies with the damage model have shown that constant remote loading leads to an accelerated damage accumulation once exceeding the threshold ξ 0 for damage onset (e.g., Ben-Zion and Lyakhovsky, 2002). This would produce inevitable loss of convexity unless the local stress is relaxed rapidly. To regularize the damage evolution close to the critical state leading to loss of convexity, we simplify the model developments of Lyakhovsky et al. (2011) by adding a damping term that becomes effective when the critical state is approached. The resulting equation for damage evolution is  α= I 2 C d (ξ −ξ 0 )− q (Λ−Λ cr ) 2 # $ % & ' ( if ξ ≥ξ 0 0 if ξ <ξ 0 * + , , - , , (4.13) where C d is a model parameter controlling the rate of damage growth, the q -term is added to damp the damage growth near the critical state, and Λ−Λ cr is the distance quantifying the closeness to the critical state Λ cr =0 . Note that originally has a unit of elastic modulus (see Eqs. 4.10-4.12). To facilitate the understanding of Λ−Λ cr , we normalize its value by µ 0 in the numerical implementation such that its distribution in terms of a non-dimensional contour map can be easily shown in the ξ−α phase space (Fig. 4.3). Λ 156 Figure 4.3: Schematic diagram showing the path of damage evolution in ξ−α phase space, where the condition Λ 1 =0 dominates the determination of losing convexity (see the red curve in Fig. 4.2). The background color shows the normalized contour map (divided by µ 0 ) of Λ 1 (here simplified as Λ ) as a function of ξ and α . Damage starts to accumulate once ξ ≥ξ 0 , but will be damped by a “buffer zone” (the grey belt region) near the critical state (Λ=0 ), where a granular-related plasticity can significantly relax the strain. With a fixed value of C d , the width of the buffer zone will be predominantly controlled by the damping parameter q in Eq. 4.13, and the phase-transition parameter β in Eq. 4.14. Following the interpretation of Lyakhovsky et al. (2011) and Lyakhovsky and Ben- Zion (2013), Λ=0 separates between solid-like (Λ>0 ) and granular-like (Λ<0 ) states of material in the compressive and partially tensile regime. This is illustrated in Figure 4.3 with a gradual phase transition between solid-like and granular-like phases of a material under brittle deformation. The dependence of the critical state on a mechanical variable (ξ ) and an entropy-related variable (α ) is analogous to the dependence of the boiling point of water on pressure and temperature. A probability function describing how solid-like a material element is may be defined as: 157 P(Λ)= 1 exp −(Λ−Λ cr )/β [ ] +1 , (4.14) where β is a parameter quantifying the width of the phase transition zone between solid- like and granular-like states. Two end-member cases relevant for this model formulation are: P Λ−Λ cr β >>1 # $ % & ' ( →1 for solid-like phase far away from the critical state and P Λ−Λ cr β →0 $ % & ' ( ) →1/2 when the critical state is approached. Two separate terms can contribute to the accumulation of plastic strain during failure (Lyakhovsky et al., 2011):  ε ij p = C g (1−P)+C v  α " # $ % τ ij . (4.15) The first term arises from the intrinsic viscous behavior of the granular phase, which scales with the probability of the granular state 1−P (Lyakhovsky et al., 2011). The second term was originally proposed by Hamiel et al. (2004) and represents the ordinary generation of damage-induced plastic deformation. Similar to previous studies, we assume that plastic strain is partitioned to different components according to the deviatoric stress τ ij =σ ij −1/2σ kk δ ij (in 2-D). 4.3.3 Numerical Method and Model Parameterization The simulations are performed using the 2-D spectral element code (Ampuero, 2002; SEM2DPACK-2.3.8, available at http://sourceforge.net/projects/sem2d/), with implementation of the above presented brittle damage rheology. We extended the initial work by Ampuero et al. (2008) to incorporate the control on loss of convexity (Eqs. 4.13- 4.15). Absorbing boundary conditions are assumed around the calculation domain, which 158 Table 4.2: Parameter values used in the simulation. Parameters Value Lamé parameters for intact rocks λ 0 , µ 0 1, 1 P- and S- wave speeds for intact rocks c p 0 , c s 0 1.732, 1 Mass density ρ 1 Reference unit length scale L 0 1 Characteristic length scale (TWF) R TWF 4L 0 Characteristic slip distance (SWF) D c 2L 0 or 5.6L 0 Static value of process zone (SWF) R 0 variable Critical elastic strain invariant for the onset of rock damage ξ 0 -0.98 Table 4.3: Convention between physical (x ) and normalized dimensionless ( ! x ) quantities. Length Time Slip rate Normal stress change Shear stress change ! l = l R 0 ! t = tc s 0 R 0 Δ" u = Δ uR 0 D c c s 0 Δ " σ = Δσ σ 0 Δ " τ = Δτ (τ 0 −τ d ) is set large enough to ensure no interference with the propagating rupture and the generated off-fault damage. A visco-elastic layer of the Kelvin-Voigt type is added surrounding the fault to damp the high-frequency numerical noise. The implementation of a non-linear bulk rheology is described by Lyakhovsky et al. (2009). The accuracy of this code has been validated by various studies (Ampuero, 2002; Lyakhovsky et al., 2009; 159 Huang and Ampuero, 2011; Gabriel et al., 2013; Xu et al., 2012a; see also the cited references in the user’s guide of SEM2DPACK-2.3.8). Similar to our previous studies with off-fault plasticity (Xu et al., 2012a, 2012b), the calculated quantities are normalized by reference parameters. Table 4.2 lists values of some basic parameters that are fixed, unless mentioned otherwise. Table 4.3 summarizes the conversion between the physical and normalized quantities that are frequently discussed in this study. We typically use an average grid (spectral node) size of Δx≈ L 0 /4 (or Δx≈ R 0 /53 ) to perform the simulations, with L 0 being a reference unit length scale (see Table 4.2). For cases requiring extremely large domains, we switch to a doubled grid size but check that the key features are not influenced. The relation between Δx and estimated value of R 0 are also presented. 4.4 RESULTS In principle, we could perform a detailed parameter-space study to investigate how different model parameters can influence properties of the generated damage zones and their interaction with fault motion. However, since the adopted damage model shares some similarities with plasticity, such as the dependence of the location and extent of damage zones on the background stress and rupture mode (Ampuero et al., 2008), we mainly present additional features that are produced with the damage model and can be relevant to the interpretation of various observations. 160 Figure 4.4: Snapshot distribution of the mapped shear wave speed (see the text for definition) for cases with (a) Ψ=14° and (b) Ψ=56° . For both plots, D c =2L 0 (it is set at 5.6L 0 for all the other cases), C d =0.25c s 0 /L 0 ≈1.2c s 0 /R 0 , C v =1/µ 0 , and C g =0 . Normal stress change Δσ along the fault is imposed in each plot, with a reference scale bar on the right. Positive and negative values indicate tensile and compressive change, respectively. The inset grey plots show the current failure zone defined by ξ ≥ξ 0 (Eq. 4.13) at the same time. As an illustration for the basic roles of damage, granular-related damping is turned off (q=0 ) and results are shown only for a limited rupture propagation distance. 4.4.1 Dynamic Normal Stress Change along The Fault One fundamental feature produced by off-fault brittle damage is coupling between slip and dynamic changes of the normal stress Δσ on the fault. It is well known that during propagation of mode-II ruptures in a homogeneous linear elastic medium, the normal stress does not change on the fault because of the symmetry in the fault-normal displacement field. However, a contrast of elastic properties across the fault breaks the symmetry and leads to coupling between Δσ and slip. For a bimaterial frictional fault, 161 the sign and amplitude of Δσ depend on the sense of slip, degree of material contrast, rupture propagation direction, position relative to the rupture front and rupture speed regime (e.g., Weertman, 1980, 2002; Ben-Zion, 2001). Here we show that the anticipated properties of Δσ can also be observed with the spontaneous generation of a material contrast across the fault, owing to an asymmetric pattern of damage (through Eq. 4.5), although the introduced non-linear stress-strain relation for a damaged medium (Eqs. 4.7 and 4.8) may also contribute to normal stress changes even under a pure shear deformation ( I 1 =0 ). Figures 4.4a and b show, at a selected time, the spatial distributions of the effective shear wave speed calculated by µ(α)/ρ . The readers are directed to the work by Hamiel et al. (2009) for a more detailed discussion on the stress- and damage-induced seismic wave anisotropy in spite of the scalar nature of the damage variable. They show that the speed of seismic wave depend not only on the damage level, but also on the state of deformation. Therefore the speed depends on the normal stress change on the fault for cases with background stress orientations Ψ=14° and 56°, respectively. The low and high Ψ values are chosen as representative of thrust and large strike-slip faults, respectively. The Ψ values were shown to be important for determining the location of dynamically generated off-fault yielding (e.g., Templeton and Rice, 2008; Ampuero et al., 2008; Xu et al., 2012a, 2012b). More specifically, the strain-based prediction of the current failure zone in Fig. 4.4 (the inset grey plots) could also be well explained by the stress-based theoretical analysis of Xu and Ben-Zion (2013). When evaluating the resultant normal stress change, we observe a tensile normal stress change right behind the rupture front for Ψ=14° with reduced shear wave speed mainly on the compressional 162 side, and a compressive normal stress change in the corresponding location for Ψ=56° with reduced shear wave speed on the extensional side. These normal stress changes and their dependence on the location of off-fault damage (compliant) zone are generally consistent with results associated with a pre-existing material contrast across the fault (e.g., Andrews and Ben-Zion, 1997; Ben-Zion and Andrews, 1998; Shi and Ben-Zion, 2006). On the other hand, the amplitude of the normal stress change may be modified by plasticity-related stress relaxation (Duan, 2008b; DeDontney et al., 2011), as well as the details of the regularization of normal stress changes (Rubin and Ampuero, 2007; Ampuero and Ben-Zion, 2008). 4.4.2 Development of A Detached Pulse Front According to the basic results in section 4.4.1, there is a compressive normal stress change behind the rupture front for Ψ=56° . If this is the only feedback mechanism produced by off-fault brittle damage, the dynamic strength drop would be reduced and the propagating rupture would be somewhat suppressed compared to cases with zero or tensile normal stress change. In other words, this is the most unfavorable direction for rupture propagation in a bimaterial interface. Moreover, the rupture (if sustained) would generally remain crack-like with the adopted nucleation approach and friction law (Duan, 2008b). On the other hand, the generated damage zone is expected to modify the rupture properties through interactions between waves reflected from the boundaries of the damage zone and the propagating rupture (e.g., Harris and Day, 1997; Ben-Zion and Huang, 2002; Huang and Ampuero, 2011, 2013). However, in contrast to previous studies of bimaterial ruptures with a low velocity zone (LVZ), the damage zone bounding the fault here is evolving, leading to more complex responses. Below we discuss a case 163 where a detached pulse is produced by the interactions between the rupture and asymmetrically generated off-fault damage zone. Figure 4.5: Spatial distribution of (a) slip rate, (b) normal stress change, and (c) shear stress change for a case with Ψ=30° at three time steps. f d =0.1 , C d =1.32c s 0 /R 0 , q= 0.66c s 0 /R 0 , C g = 0.53c s 0 /(µ 0 R 0 ), C v =0 , and β=0.2 . The chosen times highlight the moments when the propagating rupture behaves like a crack (t 1 ), with a partially ( t 2 ) or completely (t 3 ) detached pulse front. The black arrows connecting (a) and (b) at time t 3 indicate two notable tensile Δσ regimes behind the rupture front, both of which are associated with a local reduction in slip rate. Figure 4.5 shows the slip rate (panel a), normal stress change (panel b) and shear stress change (panel c) for a simulation similar to that producing Fig. 4.4b, but with Ψ=30° and much longer propagation distance. At the early stage, the slip rate profile still displays a classic crack style (time t 1 ). However, with increasing rupture propagation distance and damage generation, there are additional tensile type changes of normal stress further behind the rupture front. These lead to a local reduction of slip rate at time t 2 and 164 finally produce a completely detached pulse front at time t 3 . As a result of the tensile change of normal stress, there is an additional shear stress drop at the corresponding location (see the zoom-in view in Fig. 4.5c). When a detached pulse front has formed, there is a sharp shear stress trough in the region between the pulse front and the remaining slipping patch, which resembles what is observed in elastic homogeneous calculations with a triggered supershear pulse front (Fig. 1b of Festa & Vilotte 2006). Since there is no rate-dependent healing in the employed friction law, the detached pulse front is followed by a long tail of freely slipping patch under dynamic friction, where a second pulse may be further developed (see the left arrow between Figs 4.5a and 4.5b). However, if we allow a quick healing on the frictional fault once the slip rate is below a certain threshold value (as in Huang and Ampuero, 2011), a pure pulse-type rupture could be generated after the transition. Next we investigate and discuss details of the transition process to the detached pulse front. As mentioned, reflected waves within the LVZ are most likely responsible for producing the slip arrest. To confirm this expectation in the context of spontaneously evolving LVZ, we first plot the damage distribution and its off-fault variation in Figure 4.6. As seen, within the overall damage zone, there is an internal region near the fault that contains a high damage level with relatively gradual spatial variation. This internal region behaves more efficiently as a waveguide than the overall damage zone, although both zones have growing width towards rupture propagation direction. 165 Figure 4.6: Snapshot distribution of (a) damage and (b) its off-fault variation for the case of Fig. 4.5. The color bar in (b) indicates how the sampling locations (on the extensional side) are mapped to different colors: from X=60R 0 (blue) to X=150R 0 (red). Note the asymptotic saturation of damage with the proximity towards the fault, where a narrow waveguide within the overall LVZ may be defined. Figure 4.7 displays zoom-in views of the mapped effective shear wave speed, the associated fault-normal particle velocity  u y and normal stress change Δσ yy near the rupture front. As clearly shown in Fig. 4.7c, there are series of alternating normal stress changes behind the rupture front, generally with opposite signs on different sides of the fault. The position of the first tensile normal stress change regime behind the rupture front (on the fault and to the compressional side) more or less coincides with the location where a fresh wedge-shape LVZ has just formed (Fig. 4.7a). The precise boundary where waves are reflected to heal the rupture is difficult to discern from the current particle 166 velocity distribution (Fig. 4.7b), because of the complex geometry of the LVZ near the rupture front and its evolving shape. For comparison, Figures 4.8 and 4.9 show corresponding results generating single or multiple detached pulse front(s) and alternating normal stress changes (Figure 4.9) behind the rupture front in elastic calculations with a pre-existing asymmetric finite-width LVZ. The similarity between those results and the (somewhat more complex) fields shown in Figure 4.7 with evolving off-fault damage suggests that wave reflection contributes to the rupture healing process. The operating mechanism may also include the contribution of head waves (e.g., Ben-Zion, 1989, 1990; Huang and Ampuero, 2013) that are backward propagating along the boundaries of the LVZ, and an asymmetric “Mach front” only on the extensional side with a speed faster than the shear wave speed of the LVZ (~ 0.7c s 0 ) that remains behind (Figure 4.10). These mechanisms are suspected because there are linear features in the particle velocity distribution (Fig. 4.7b) around X=159R 0 , which are usually observed for head waves or a Mach front (as well as its reflected counterpart within a LVZ). In addition to the complex wave signals around the rupture front, waves are also emitted from the tail of the detached pulse front and propagate backward (Figs. 4.7b and c). Some of these waves become internally reflected or trapped inside the well- established LVZ further behind the rupture front. When such waves reach the fault, they may result in oscillations of slip rate in the freely slipping patch of the fault, similar to what have been illustrated in the results and Appendix of Ben-Zion and Huang (2002) for the interaction with the rupture front, see also Huang and Ampuero (2011, 2013). The generation of trapped-type waves is enhanced by the following two conditions that exist further behind the rupture front: (1) a well-established LVZ with minor additional shape 167 evolution to allow constructive interference of internal wave reflections, and (2) smooth distribution of background particle velocity with a relatively low amplitude. The effects of these trapped waves on near-fault ground motion and potential tremor-like signals are discussed in subsequent subsections. Figure 4.7: Snapshot distribution in a zoom-in view of (a) mapped shear wave speed, (b) fault-normal particle velocity  u y , and (c) normal stress change Δσ yy for the case of Fig. 4.6. The distribution of slip rate at the same time is superimposed in each plot. 168 Figure 4.8: Snapshot distribution of slip rate for elastic calculations with a pre-existing finite-width (W ) LVZ (see inset in a). (a) W =5L 0 ≈ 0.4R 0 , (b) W =10L 0 ≈ 0.8R 0 and (c) W =40L 0 ≈3.2R 0 . For all cases, the P- and S- wave speed have a 30% reduction inside the LVZ compared to those of the country rocks. Figure 4.9: Snapshot distribution of Δσ yy for the case of Fig. 4.8a (panel a) and Fig. 4.8b (panel b) at times when a detached pulse front has just formed. The solid black curve shows the slip rate distribution for each plot. The dashed black lines indicate the boundaries of the pre-existing LVZ. 169 Figure 4.10: Spatio-temporal distribution of slip rate for the case of Fig. 4.6. Various wave speeds of the intact or highly damaged rocks close to the fault (see Fig. 4.7a) are plotted for reference. 4.4.3 Development of A Train of Pulses Since internal wave reflection can provide additional feedback mechanism between off-fault brittle damage and rupture dynamics, we may use analytical model results with a pre-existing LVZ (e.g., Ben-Zion, 1998) to investigate the competition between different mechanisms (velocity contrast inside and outside the damage zone, rupture propagation distance, width of the damage zone, damping effects) in influencing the efficiency of generating internal wave reflection and its interaction with the propagating rupture. Here we mainly focus on one feature generated by our numerical simulations, the development of a train of pulses under certain conditions. 170 Figure 4.11 shows snapshots of slip rate (a), normal stress changes (b) and shear stress change (c) for a case with Ψ=30° and slightly different values of other model parameters (see caption) from those leading to Fig. 4.5, generating a narrower damage zone. Similar to the results in Fig. 4.5, the rupture initially behaves as a crack (time t 1 ), and later has an abrupt reduction in slip rate behind the rupture front (time t 2 ). In contrast to the previous results, however, at a later stage (time t 3 ) the rupture produces in addition to a relatively large detached front pulse a train of pulses further behind. As shown in Appendix D, the oscillations of simulated fields are well resolved numerically so they reflect a genuine physical effect. A further investigation of slip rate at different times in Figure 4.12 reveals that the oscillations actually migrate towards rupture propagation direction, with a somewhat expanding size, which is consistent with our expectation that they are triggered by a moving source (i.e., waves emitted from a propagating pulse front). The strong oscillations in slip rate and the long duration of the process imply that the anticipated internal wave reflections in the LVZ are more efficiently interacting with the frictional fault. To more explicitly see this, we plot the damage distribution and its off-fault variation (Figure 4.13). The overall damage zone follows a similar pattern to that shown for the previous case in Fig. 6, but the near-fault damage zone structures are quite different in the two cases. The current case has a more localized internal damage zone with almost constant width along strike and almost constant damage level. This is indicated by the dashed line in Fig. 4.13a and the overlap of different damage decay profiles close to the fault in Fig. 4.13b. The current case develops a train of pulses during the duration of the simulation for the following reasons: (1) Its generated narrower damage zone near the fault enhances the development of trapped waves (Ben-Zion, 1998) 171 because the number of reflections per unit length parallel to the fault increases with decreasing damage zone width if other factors remain about the same. (2) Its generated near-fault damage zone has approximately a constant width along strike, which can construct trapped waves more coherently than with a spatially varying width (e.g., Igel et al., 1997). On the other hand, we notice that the train of pulses appears only after the rupture has propagated over a certain distance range (Figs. 4.11 and 4.12), and that it emerges with some distance behind the detached pulse front. These may be explained by the fact that the construction of internal wave reflections requires some propagation distance to produce strong enough disturbance over the fault frictional strength. Figure 4.11: Similar to Fig. 4.5, but with slightly different model parameters: f d =0.2 , C d =13.2c s 0 /R 0 , q= 6.6c s 0 /R 0 , C g = 0.165c s 0 /(µ 0 R 0 ) , C v =0 , β=0.1. The chosen times highlight the moments when the propagating rupture behaves like a crack (t 1 ), with a partially detached pulse front (t 2 ) or a completely detached pulse front followed by a train of pulses (t 3 ). 172 Figure 4.12: Spatial distribution of slip rate at different times for the case of Fig. 4.11. Figure 4.13: Snapshot distribution of (a) damage and (b) its off-fault variation for the case of Fig. 4.11. Slip rate at the same time is superimposed in (a). The color bar in (b) indicates how the sampling locations are mapped to different colors: from X=60R 0 (blue) to X=85R 0 (red). 173 Our simulations indicate another scenario where the detached pulse front itself can become a train of pulses, which is usually associated with an extremely narrow and highly damaged zone close to the fault. However, in such scenario we also find that high- intensity damage can localize, as shown also for plasticity (Duan, 2008a), onto mesh edges close to the boundary of the generated (or pre-existing) narrow compliant zone in off-fault regions. This localization produces a mesh-dependent problem for the simulation results, so the robustness of this scenario and its implication for the observed boundary Y-shears at the interfaces between fault gouge and country rocks (Gu and Wong, 1994) should be more carefully investigated by our future work. 4.4.4 Modulation of The Rupture Front From the previous results it is seen that radiated waves and their reflections within a LVZ can interact with the fault portion behind the rupture front. We also expect that the waves interact with the rupture front itself. Indeed, with a pre-existing constant-width LVZ, a simple geometric relation between the rupture speed (in the subshear regime of the LVZ) and properties of the LVZ (width, wave speed) can be readily established (Ben- Zion and Huang, 2002). However, in the current study with evolving geometrical and material properties of the LVZ the results are more complex. The radiated waves in our case may initially propagate with the speed of the intact material (there is no LVZ in some region around the rupture front), get later trapped inside an evolving LVZ and finally interact with the rupture front at its new position. Since the backward-propagating waves and their reflections continuously produce healing signals behind the rupture front, the stress concentration and energy release rate 174 will be influenced by the fluctuation of the pulse width, leading to local acceleration or deceleration of the rupture speed. Figure 4.14 shows the evolution of rupture speed v r and maximum slip rate v max for the case leading to Fig. 4.5. As seen, beyond the location X ~45R 0 , both v r and v max are locally modulated by quasi-periodic oscillations. Again, the oscillations are well resolved numerically and reflect a genuine physical outcome of the examined case. The coherence of the oscillating signals becomes stronger beyond X ~92R 0 (marked blue bar in Fig. 4.14), which corresponds to the location where a detached pulse front has just formed. These types of oscillations are observed for all the cases in our study where a detached pulse front is developed. It should be mentioned that the whole process is essentially non-linear because the perturbed rupture front determines the updated wave radiation and off-fault damage generation, which can influence the interference with the rupture front in a further step. Figure 4.14: Spatial variation of (a) rupture speed and (b) maximum slip rate for the case of Fig. 4.5. The blue bar in the zoom-in view roughly indicates the location where a detached pulse front has formed for the first time. 175 4.4.5 Effects on Near-Fault Ground Motion So far we have focused on the interplay between the propagating rupture and radiated waves through the generated off-fault damage zone. Here we consider in more detail effects of the brittle damage zone on the near-fault ground motion. Figure 4.15 shows the amplitude of particle velocity  u = ( u x ) 2 +( u y ) 2 for the case of Fig. 4.7. As seen, the pattern is almost symmetric across the fault ahead of the rupture front, because damage has not yet been generated at those locations. In contrast, there is a strong asymmetry in the distribution of particle velocity right behind the rupture front, with higher amplitude over a wider off-fault distance range on the extensional side. This asymmetry can be well understood as due to the reduced seismic impedance and the possible multiple reflections of waves within the damage zone generated at that location. It is interesting to notice that the latter effect is directly connected to the wedge shape of the fresh damage zone near the rupture front (see Fig. 4.7a), which resembles the effect of a dipping fault and free surface on the amplified ground motion on the hanging wall (Oglesby et al., 1998; Shi and Brune, 2005). Figure 4.15: Snapshot distribution of amplitude of particle velocity at the same time as in Fig. 4.7. Two slip rate profiles are superimposed: the solid one corresponds to the same time as in Fig. 4.7, and the dashed one is associated with an earlier time when a completely detached pulse front has just formed. The green line indicates the location of receivers that will be used for analyzing seismograms. 176 Another asymmetry can be found in the ground motion further behind the rupture front. As discussed in section 4.4.2, waves can be more coherently trapped inside a well- established damage zone, but will propagate away in a rather homogeneous medium. Therefore, receivers located on the side with damage generation are expected to record longer ground shaking with higher amplitude in the later part of seismograms than those on the other side. This is explicitly shown in fault-normal velocity seismograms in Figure 4.16 for a linear array of receivers that is symmetrically distributed across the fault at X=106.1R 0 (marked by a green line in Fig. 4.15). As seen, after the passage of the rupture front (t>126R 0 /c s 0 ), receivers on the extensional side (red and below) continue to record oscillatory signals with large amplitude, low frequency and long duration. In contrast, receivers on the compressional side (blue and above) only record smooth motion that gradually reduces towards zero. This contrast is more pronounced in the fault-normal component than in the fault-parallel component (not shown here), similar to the previous study with pre-existing LVZ surrounding the fault (Duan, 2008a). Detailed investigation of seismograms on the extensional side also indicates that the first several receivers from the bottom are outside the effective waveguide, due to the lack of long-lived signal oscillations compared to those closer to the fault. This is confirmed by checking the mapped effective shear wave speed distribution where the internal LVZ (within the overall LVZ) has a thickness less than 1.5R 0 around X=106.1R 0 . 177 Figure 4.16: Velocity seismograms recorded by 40 receivers across the fault (see Fig. 4.15 for reference) in the fault-normal component (panel a) and its zoom-in view (panel b). Receivers are arranged from the extensional side (bottom, starting at Y =−1.5R 0 ) to the compressional side (top, ending at Y =1.5R 0 ). The red and blue highlight the receivers nearest to the fault from the extensional and compressional side, respectively. 4.5 DISCUSSION 4.5.1 Mechanisms Leading to Pulse-like Ruptures Various mechanisms have been proposed to produce pulse-like ruptures that can explain the seismological observation of earthquakes with a short rise time (Heaton, 1990) and the field observation of a lack of frictional heating along strike-slip faults (Brune et al., 1969). These include rate-dependent friction (Zheng and Rice, 1998), finite- width seismogenic zone (Day, 1982), heterogeneities of fault strength or initial stress (Beroza and Mikumo, 1996), rupture on a bimaterial interface (Andrews and Ben-Zion, 1997), pre-existing symmetric or asymmetric LVZ (Harris and Day, 1997; Ben-Zion and 178 Huang, 2002; Huang and Ampuero, 2011), and generation of off-fault yielding by bimaterial ruptures (Ben-Zion and Shi, 2005; Xu et al., 2012b). Figure 4.17: (a) Profiles of slip rate for cases with different Ψ values sampled at the same rupture front location (X /R 0 =131.5 ). The inset shows the zoom-in view near the rupture front (indicated by the dashed box) where the width of a completely or partially detached pulse front can be defined. For cases with a partially detached pulse front, the effective tail of the pulse is defined as the first local trough in slip rate behind the rupture front . (b) Diagram showing the pulse width as a function of Ψ (indicated by colors) and sampling location (indicated by marker shapes). Solid and open markers represent completely and partially detached pulse fronts, respectively, and the thick dashed line roughly characterizes the boundary between the two. The inset plot in the upper left corner shows schematically the configuration of the genereated LVZ with different Ψ values (for Ψ≥20° ) and rupture propagation distance. 179 In cases where multiple mechanisms co-exist, whether pulse-like ruptures can still be produced and their robustness depends strongly on competitive effects. In this study, we show that the spontaneous generation of off-fault brittle damage accounting for changing elastic moduli can also lead to the development of a pulse front. More specifically, we investigate the competition between a negative bimaterial effect (with a moderate-to-high Ψ value) and internal wave reflection by a finite-width LVZ (section 4.4.2), and show that the latter can overcome the former under certain conditions. These conditions include a slowly growing or approximately constant-width damage zone, relatively large velocity contrast across the effective boundary of the damage-related LVZ, and relatively weak fault frictional strength behind the rupture front. Connecting the first condition to our model parameters, cases with 20°≤Ψ≤35° while keeping other parameters (including the initial fault normal and shear stress components) the same as in Fig. 4.5 can successfully produce a detached pulse front within the range of X<152R 0 (Figure 4.17a). Detailed investigation reveals the following features (Figure 4.17b): (1) the critical rupture propagation distance producing a detached pulse front for the first time and the pulse width (see figure caption for definition) sampled at the same location increase with the Ψ value for Ψ≥20° , which is associated with an increasing ratio of damage zone thickness to rupture propagation distance; (2) the pulse width with a fixed Ψ value increases with rupture propagation distance, which is associated with an increase of damage zone thickness. Similar correlation between pulse width and LVZ thickness can also be observed in elastic calculations with symmetric or asymmetric pre- existing LVZ (Figure 8 and results of Huang & Ampuero 2011, 2013). This provides 180 additional evidence for our expectation (section 4.4.2) that the healing signal comes from the trapped waves inside the LVZ. The result summarized in Figure 4.17 may also explain why Yamashita (2000) did not observe a detached pulse front in his numerical study by modeling off-fault tensile microcracks. In that case the rupture either did not propagate over a long enough distance or the growing rate of LVZ thickness is too fast, although other factors such as material anisotropy and degree of velocity contrast may also be important for this topic. In addition to a single detached pulse front, our results also produce a rupture mode with a train of pulses (section 4.4.3). This type of rupture mode has been seen in previous numerical studies under certain conditions. These include cases with pre-existing LVZ bounding the fault (Harris and Day, 1997; Huang and Ampuero, 2013), along with ruptures that are triggered by impact loadings (Coker et al., 2005; Shi et al., 2010) and highly energetic frictional nucleation procedure (Shi et al., 2008). The mechanism leading to trains of pulses in this work shares naturally similarities with the mechanisms in previous studies with pre-existing LVZ, although there are also some differences. In Harris and Day (1997) trains of pulses were observed with asymmetric LVZ primarily in the positive direction defined by a local bimaterial interface (Fig. 9a of their paper), while in Huang and Ampuero (2013) it was produced by a friction law with fast healing inside a symmetric finite-width LVZ. Here multiple pulses are observed with pre-existing or dynamically generated LVZ (see Figs. 4.8 and 4.11) on the extensional side, which corresponds to the negative direction of a bimaterial interface. The different LVZ configurations that produce trains of pulses suggest that internal wave reflections are an important mechanism that can dominate other effects over ranges of conditions. 181 4.5.2 Implication for Tremor-like Radiations The strong persisting oscillations of slip rate in Fig. 4.11 provide a possible mechanism for producing tremor-like signals. The mirror image of Fig. 4.11 from right to left represents a waveform recorded by a near-fault receiver, which is characterized (like tremor) by non-impulsive shape, long duration and relatively low amplitude. The conditions leading to persisting oscillatory slip rate in our study share some commonalities with those producing non-volcanic tremor and/or slow slip events in some natural settings. These include relatively deep fault sections (with presumably very thin LVZ), high P-to-S wave speed ratio, an overall low fault strength (see reviews by Peng and Gomberg, 2010; Rubinstein et al., 2010). Compared to the commonly suggested physical models for explaining natural tremors, such as involvement of fluids and fault heterogeneities with rate-and-state friction (Rubinstein et al., 2010, and references therein), the tremor-like signals in our study are produced by waves that are radiated from the tail of a detached pulse front and then get trapped inside a narrow LVZ. As mentioned in section 4.4.3, the efficiency of this mechanism increases for a LVZ with approximately constant and narrow width, which may exist primarily at the bottom of or below the seismogenic zone. Our suggested mechanism may explain why in most cases tremors are rarely observed at shallow depths , since fault damage zones typically display a flower- like structure (Ben-Zion and Sammis, 2003; Rockwell and Ben-Zion, 2007; Ma and Andrews, 2010). In addition, the geometry of the generated LVZ must introduce a length scale to the tremor-like radiation pattern, which also resembles the suggested mechanism of fluid conduits in generating volcanic or non-volcanic tremors (Rubinstein et al., 2010). 182 4.5.3 Relevance for Fault Zone Structures The brittle damage model used in this study can provide, with additional simulations, multiple signals for interpretation of different elements of fault zone structures. As discussed in section 7 of Ben-Zion (2008) and studied by Lyakhovsky et al. (2011) and Lyakhovsky and Ben-Zion (2013), a brittle instability leading to dynamic rupture may correspond to a phase transition from a solid-like to a granular-like state of rocks. This corresponds physically to a transition from the fault damage zone where the rock volume still maintains cohesion to the slip zone filled with rock particles (gouge, cataclasite, etc). The model suggests increasing rock damage density with proximity towards the fault core, followed by a possible saturation where rocks can no longer accommodate additional fracturing but are crushed into finer grains, as has been documented by some field observations (e.g., Mitchell and Faulkner, 2009). This general feature of off-fault damage variation can be simulated with the adopted damage rheology during both dynamic ruptures (Fig. 4.6 and Fig. 4.13) and under long-term quasi-static loadings (Lyakhovsky and Ben-Zion, 2009; Finzi et al., 2009), with an explicit granular-related mechanism that can physically damp the damage growth (Lyakhovsky et al., 2011). An interesting topic for a future study is multi-cycle simulation of fault zone evolution, accounting for damage accumulation and healing, as well as possible phase transitions in the states of rocks. 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Xu, S., and Ben-Zion, Y., 2013, Numerical and theoretical analyses of in-plane dynamic rupture on a frictional interface and off-fault yielding patterns at different scales: Geophysical Journal International, v. 193, p. 304–320, doi:10.1093/gji/ggs105. Yamashita, T., 2000, Generation of microcracks by dynamic shear rupture and its effects on rupture growth and elastic wave radiation: Geophysical Journal Internation, v. 143, p. 395–406. Yamashita, T, 2009, Chapter 8: Rupture Dynamics on Bimaterial Faults and Nonlinear Off-Fault Damage: in Fault-Zone Properties and Earthquake Rupture Dynamics, International Geophysics, v. 94, edited by Fukuyama, E., p. 187–215, Academic Press. Yang, H., and Zhu, L., 2010, Shallow low-velocity zone of the San Jacinto fault from local earthquake waveform modeling: Geophysical Journal International, v. 183, p. 421–432. 200 Yang, H., Zhu, L., and Cochran, E.S., 2011, Seismic structures of the Calico fault zone inferred from local earthquake travel time modeling: Geophysical Journal International, v. 186, p. 760-770. Yue, H., Lay, T., and Koper, K.D., 2012, En échelon and orthogonal fault ruptures of the 11 April 2012 great intraplate earthquakes: Nature, v. 490, p. 245–249, doi:10.1038/nature11492. Zheng, G., and Rice, J.R., 1998, Conditions under which velocity-weakening friction allows a self-healing versus a crack-like mode of rupture: Bulletin of the Seismological Society of America, v. 88, p. 1466–1483. 201 Appendix A LEFM Stress Field Near a Propagating Crack Tip The expressions for the different components of II ij Σ of Eq. (3.6) in a Cartesian coordinate system, subjected to 1 ) , 0 ( II = Σ r xy v , are given by (e.g., Freund, 1990) ⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ − + = Σ ⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ + − = Σ ⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ + − − + − = Σ s s d d s s yy s s s d d d s xy s s s d d s d s xx D D D γ θ γ θ α α γ θ α γ θ α α γ θ α γ θ α α α 2 1 2 1 2 II 2 1 2 2 2 1 II 2 1 2 2 1 2 2 II sin sin ) 1 ( 2 cos ) 1 ( cos 4 1 sin ) 1 ( sin ) 2 1 ( 2 , (A1) where 2 2 / 1 d r d c v − = α with c d being the dilatational (P-) wave speed, 2 2 / 1 s r s c v − = α with c s being the shear (S-) wave speed, 2 ) / sin ( 1 d r d c v θ γ − = , 2 ) / sin ( 1 s r s c v θ γ − = , ) tan ( tan 1 θ α θ d d − = , ) tan ( tan 1 θ α θ s s − = , and 2 2 ) 1 ( 4 s s d D α α α + − = . The dynamic stress intensity factor d K II can be further expressed as a product of the static stress intensity s K II and a universal function of rupture speed ) ( II r v k : s r d K v k K II II II ) ( = , (A2) 202 where L K s π τ Δ ∝ II , with τ Δ being the stress drop and L being the half length of the crack, and s r R r r c v c v v k / 1 / ) / 1 ( ) ( II − − ≈ that monotonically decreases from 1 at + ≈0 r v towards 0 as r v approaches the Rayleigh wave speed R c (Freund, 1990). 203 Appendix B Critical Conditions for Shear Localization Rice (1976) showed that for elastically isotropic materials and deformation processes where rotational effects on stress rate can be neglected, the critical value of plastic hardening modulus c h can be described as h c 2µ = 2n⋅P⋅Q⋅n−(n⋅P⋅n)(n⋅Q⋅n)−P :Q− ν 1−ν (n⋅P⋅n−trP)(n⋅Q⋅n−trQ), (B1) where n is a unit vector representing the normal direction of shear band, M σ ∇ ∝ P and F σ ∇ ∝ Q are the normalized gradients with respect to the stress tensor ij σ of the plastic flow potential M and the yield function F , and ν is the Poisson's ratio. To make Eq. (B1) applicable, c h must be larger than a minimum admissible value min h , which corresponds to a loading requirement that the scalar plastic strain rate (sometime also called the rate of equivalent plastic strain) should never be negative (e.g., Rice and Rudnicki, 1980). From Bardet (1990), this condition can be more explicitly expressed as: Q P Q P : tr tr 2 1 2 min − − − = ν ν µ h . (B2) For our employed Mohr-Coulomb type yielding criterion and non-associated flow rule without volumetric change, we can follow the procedure of Bardet (1990) to re- 204 express the entries on the right side of (B1) and (B2) in the principal stress coordinate system as: ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − = 2 / 1 0 0 2 / 1 ij P , ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ + + + + − = ) sin 1 ( 2 sin 1 0 0 ) sin 1 ( 2 sin 1 2 2 φ φ φ φ ij Q , (B3) T ] cos sin [ θ θ = i n where θ is the angle between shear band and the maximum compressive stress. Compared to the original expressions in Bardet (1990), each component of ij P or ij Q in Eq. (B3) is associated with an opposite sign due to the treatment of compressive normal stress with negative values in our study. However, this difference in sign-convention does not affect the evaluation of Eq. (B1). Using the expressions in Eq. (B3) into (B1), we have: φ ν φ θ φ µ 2 2 2 sin 1 ) 1 ( 8 ) sin 2 cos 2 ( sin 2 + − − − = c h . (B4) The maximum value of c h that satisfies Eq. (B4) is given by φ ν φ µ 2 2 max sin 1 ) 1 ( 8 sin 2 + − = h , (B5) 205 which is usually treated as the criterion for predicting the occurrence of shear bands for the first time. And the corresponding angle θ satisfies: φ θ sin 2 1 2 cos = . (B6) 206 Appendix C Mesh Dependence of Shear Localization It is well documented that certain features of numerically modeled shear bands (the emergence, width, two possible conjugate sets, etc) can have a strong dependence on the adopted numerical meshing in both quasi-static and dynamic processes (e.g., Needleman, 1989; McKinnon and Garrido de la Barra, 1998; Templeton and Rice, 2008). To get a confidence on the physical implications of the modeled shear bands, one should try different meshing strategies to ensure that the key discussed features are robust. If a particular mesh structure is designed to mimic material properties with some pre-existing preference (anisotropy, internal texture, etc), which is also believed to be the dominant mechanism leading to localization, the potentially-biased results may still be considered acceptable. Here we mainly discuss effects involving the size and orientation of numerical meshes on the generated shear bands. Figure C1 shows generated shear bands with square elements of two different sizes. Consistent with the study by Templeton and Rice (2008), the average shear band width decreases with refining the mesh size (e.g., see the zoom-in view in box II). This mesh- size dependence has been explained by the expectation that, in the absence of physical parameters specifying shear band width, the mesh size provides a minimum length scale that is numerically allowable to optimize shear localization (e.g., Prevost and Loret, 1990). In the spectral method used in our study the internal nodes within each element are 207 not uniformly distributed, but have narrower and wider spacing towards the end and the center of an element, respectively (see the structure of the inset square). Therefore, the minimum possible width of the simulated shear bands in our study is not only controlled by the element size, but also depends on the node partition within the element. Moreover, the introduced non-uniformity by the internal nodes may contribute to shear band triggering near the fault. Figure C1: Off-fault shear bands in simulations with square spectral elements with relatively large (a) and small (b) grids. The inset sketches illustrate the employed non- uniform distribution of internal nodes. Boxes I and II provide zoom-in view of features at relatively small and large propagation distance, respectively. The maximum plastic strain is rescaled in box I but remains unchanged in box II. 208 Despite the above size-dependent features, some properties of the generated shear bands are preserved with varying the mesh size. (1) Within a limited distance range, there is a transition from antithetic shear bands to synthetic ones, and the angle between the two sets in the transition region has a good agreement with the theoretical prediction (see box I). (2) After the initial transition, the synthetic shear bands in the uniform fault problem always dominate the off-fault yielding zone and can develop their own branches at larger propagation distances (see box II). (3) The intensity distribution of plastic strain within each well-developed shear band is quite uniform (ignoring the edge and tip effects), although the absolute value depends on the mesh size. (4) The average area density not occupied by shear bands (but still within the overall yielding zone) seems to increase with off-fault distance. It is interesting to note that the combination of (3) and (4), when compared to Fig. 3.2, may imply some equivalence of patterns between distributed and localized deformation (with a similar overall yielding zone shape) during dynamic ruptures. For distributed deformation, the geometric measure is macroscopically continuous (uniform) but the potency density decays with off-fault distance, while for localized deformation the potency density within each discrete band is uniform but the geometric measure of band density decays (on average) with off-fault distance. It may be interesting to explore in a future work whether a generalized deformation density, accounting for both yielding intensity in a failed element and geometric measure of failed elements can be used to describe uniformly distributed and localized deformation patterns. 209 Figure C2: Off-fault shear bands in simulations with quadrilateral spectral elements with variable angle between the non-horizontal mesh edges and fault plane. The critical rupture length for transition from a possible antithetic to synthetic shear is very small in (a), intermediate in (b) and relatively large in (c). The mesh size is the same for all three cases. The ratio W s /W a in the dashed boxes roughly quantifies the mesh alignment effect (see text for more explanation). Boxes I and II provide zoom-in views of local features at different rupture propagation distances. 210 Templeton and Rice (2008) suggested based on numerical simulations with small propagation distance (~10R 0 ) a characteristic shear band spacing that may physically scale with the current size of process zone (e.g., 0.2−0.4R ). According to our results and Duan and Day (2008), these suggested features may hold in the very beginning of ruptures but appear to gradually lose their validity at larger propagation distance (e.g., the spacing between adjacent well-developed shear bands fluctuates and sometimes is even larger than the static size of process zone). This implies that a statistical description of shear band spacing is needed and the possible scaling coefficient with process zone may evolve with propagation distance. The numerically simulated shear bands may also have a strong dependence on the mesh orientation. The effective width of shear bands can vary with the relative angle between shear bands and mesh edges. The minimum possible width can only be obtained when the corresponding mesh edges are aligned parallel to the band orientation (this is referred to as the mesh-alignment effect). For our study of two possible conjugate shear band sets, it is important to check the effect of mesh orientation on the relation between the generated shear bands. If quadrilateral meshes are used, the two pairs of mesh edges may be aligned parallel to the two principal stress axes such that any potential conjugate shear bands will be equally biased by the mesh-alignment effect (McKinnon and Garrido de la Barra, 1998). For our dynamic problem, instead of performing a rigid rotation of the entire array of previously used square meshes, we keep a pair of mesh edges parallel to the fault but vary the angle between the other pair and the fault plane (i.e., changing the square mesh to a more general quadrilateral mesh). Other approaches for reducing the 211 mesh-alignment and/or mesh-size effect can be found in related works (e.g., McKinnon and Garrido de la Barra, 1998; Li et al., 2001; Bažant and Jirásek, 2002). Figure C2 shows the generated shear bands with a variable angle δ between the inclined mesh edges and the fault plane. Clear difference can be found in the early stage of ruptures right after the emergence of shear bands; the dominant set is of the synthetic type in Fig. C2a while it is of the antithetic type in Figs. C2b and C2c (see box I). This difference may be explained by the contrast in resolvable shear band widths (see the conventionally defined s W and a W in the dashed box), of which the set with a narrower width is numerically more encouraged. However, with increasing rupture propagation distance the synthetic shear bands continue to be dominant in Fig. C2a, while there is a clear transition of the dominant set from antithetic to synthetic in Figs. C2b and C2c. The critical rupture propagation distance leading to this transition seems to increase with the value of a s W W / , and may never be reached in the explored distance range if a s W W / is too high (not shown here but confirmed by a test with inclined mesh edges almost parallel to the expected antithetic shear bands). Finally, at relatively large propagation distances, the synthetic shear bands always dominates in all three test cases, while the possible existence of minor antithetic bands near the fault depends strongly on the mesh alignment (see box II). These results imply that some physical mechanism (like the one in section 3.2.2), rather than a numerical artifact, is involved in the generation of significant synthetic shear bands (in the examined cases without pre-existing favorable failure orientations off the main fault). 212 Our examined test cases with approximately equal values for s W and a W , along with the implied features insensitive to mesh size, support our conclusion on large-scale shear-type yielding generated by earthquake ruptures in large strike-slip faults (see section 3.2.2). End-member cases with a s W W / strongly deviating from one are severely affected by the mesh-alignment effect, and may not produce representative results that are useful in the context of this study. 213 Appendix D Numerical Tests on the Robustness of Slip Rate Features Figure D1: Numerical test on the resultions of slip rate (see Fig. 4.5 for reference) with two different mesh sizes. To demonstrate the validity of results with oscillating fields, we conduct numerical tests on the robustness of the detached pulse front and train of pulses. Two different mesh sizes are used, with possible modification of fault normal stress regularization, to test the accuracy and stability of the resulting ruptures. Fig. D1 shows simulated profiles of slip rate for the case of Fig. 4.5 with two different mesh sizes. Although there are slight differences in the onset of slip and the peak values, the overall profiles match well, including the resolution of the healing signal behind the rupture front (Fig. D1b). The overall match is retained after the development of a completely detached pulse front (not shown here but confirmed). Therefore, we conclude that the detached pulse front is a robust feature. 214 Figure D2: Numerical test on the stability and resolution of oscillating slip rate (see Fig. 4.11 for reference) with various model parameters for the normal stress regularization. See the text for the definition of t * . Fig. D2 shows test results on the resolved oscillating slip rate for the case of Fig. 4.11 by varying parameters of the normal stress regularization. We use an estimated time scale t * = δ c V * + V to quantify the effect of normal stress regularization, with extremely small and large t * values corresponding to almost instantaneous and little response of fault strength to fault normal stress changes, respectively. As seen, the resolved pulse front profiles with three tested t * values overlap well with each other. This is because the off- fault energy dissipation due to damage and plasticity-related stress relaxation near the rupture front can help stabilize the slip rate (Ben-Zion and Shi, 2005), so that the details of performing normal stress relaxation become less important at that location (we remark that the absolute value of each tested t * is still considered small compared to the travel time of shear wave through R 0 ). However, at locations further behind the rupture front 215 where the generated damage is frozen, the assumed t * value can significantly affect the rupture behavior. The two cases with relatively small t * values (green and red) preserve the feature of trains of pulses with about the same “wavelength” (see the inset plot) while the rest case (blue) can only produce a smooth slip rate profile. One may argue that the resultant trains of pulses with small t * values are produced by the numerical instability for bimaterial problems (Cochard and Rice, 2000). However, they do not diverge with time, although they can grow in the very beginning (Fig. 4.12). Moreover, we notice that the slip rate oscillations are quite monochromatic, suggesting a process characterized by a dominant scale (e.g., like the thickness of the internal waveguide, see Fig. 4.13). More specifically, it could involve a physical instability between oscillating shear stress and fault strength (the product of a constant friction coefficient and oscillating normal stress) that are contributed by series of wave reflections at a local bimaterial interface. These features distinguish the operating mechanism from the ones near the rupture front with evolving but not highly oscillating normal stress change in previous bimaterial studies, and we believe that such process could exist in natural settings for situations under overall low fault strength (e.g., with the presence of migrating fluids through a narrow fault zone). Our extended test with refined meshes produces the similar results if t * remains small. Therefore, we think the reported trains of pulses are likely features, although their robustness in nature needs to be validated by more observations.

Abstract (if available)

Abstract This dissertation mainly investigates mode-II in-plane dynamic ruptures along a frictional interface with off-fault yielding characterized by different rheology models. The goal is to provide improved understanding of off-fault yielding during earthquake ruptures and its contribution to earthquake physics. ❧ A detailed parameter-space study is performed to examine properties of the off-fault plasticity-type yielding under various conditions. The simulation results and theoretical analyses show the following features: (1) the off-fault yielding occurs primarily on the compressional and extensional side when the angle Ψ to the fault of the regional maximum compressive stress is low and high, respectively, regardless of the rupture mode

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Creator Xu, Shiqing (author)

Core Title Interaction between dynamic ruptures and off-fault yielding characterized by different rheologies

School College of Letters, Arts and Sciences

Degree Doctor of Philosophy

Degree Program Geological Sciences

Publication Date 07/23/2013

Defense Date 04/30/2013

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag dynamics and mechanics of faulting,elasticity and anelasticity,fractures and faults,mechanics,microstructures,modelling,OAI-PMH Harvest,rheology and friction of fault zones,theory

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Advisor Ben-Zion, Yehuda (committee chair), Corsetti, Frank A. (committee member), Platt, John (committee member), Sammis, Charles G. (committee member), Thompson, Richard S. (committee member)

Creator Email shiqingx@usc.edu,stiller.xu@gmail.com

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