速率和状态摩擦准则下断层滑动特征的数值模拟研究
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摘要
本文详细介绍了基于岩石摩擦实验提出的速率和状态摩擦准则,并研究了其在数值模拟震源参数变化、地震复杂性以及破裂动力学中的作用。断层滑动和地震破裂的物理机制主要受接触面上摩擦准则的控制,对岩石摩擦的认识是理解断层和地震行为的关键。因此本文首先详细回顾了速率和状态摩擦准则的发现过程及其基本特征,系统分析了该准则在简单的单自由度弹簧滑块模型下的稳定性问题,也简要介绍了最近十年来岩石高速摩擦下的最新实验室发现,以及相关的强弱化理论。总之对岩石摩擦的全面且系统的认识有助于理解断层和地震现象的本质。
观察震源参数随地震矩的变化一直是地震学家了解地震自相似性的手段之一。本文利用简单的单自由度弹簧滑块模型,数值模拟了速率和状态准则控制下的系统所表示的震源参数的变化。在该模型下,地震的大小主要由断层的尺度W(与模型中弹性刚度k成反比)控制。震源参数的变化受断层相对尺度(断层尺度W与成核尺度hc。的比值)的影响,对较小的地震,因为非震滑移和断裂能的比重比较大,震源参数有尺度效应；对较大的地震,非震滑移和断裂能可忽略,震源参数可视为自相似的。但是当考虑了热增压效应后,震源参数呈现非常明显的变化,视应力和应力降都随着地震矩的增加而明显增加。
本文发展了基于时间域牵引力边界积分的可同时模拟长时间尺度下断层滑动行为和极短时间尺度下同震过程的数值方法,并将其与准动态的数值模拟方法做了比较。尽管在同震过程准动态方法显然不能正确地模拟地震波效应,但是在均匀的或轻微非均匀的断层模型下,两者能给出大致相似的滑动模式(地震序列)和破裂风格(裂纹型或脉冲型)。本文同时利用准动态和新发展的动力学数值方法,研究了断层的相对尺度W/hc。对地震破裂特征和滑动复杂性的影响。发现当断层的相对尺度较小时,断层行为表现简单的周期性的裂纹型地震,当断层的相对尺度较大时,断层除了继续产生特征地震以外,会自发地产生更多自发停止的自愈合地震,这可能预示着均匀断层在相对尺度充分大的情况下,能自发地产生丰富的地震活动性。
总体上本文的研究显示速率和状态摩擦准则存在明显的尺度效应。断层相对尺度W/hc影响了断层和地震各方面的特征,地震的微观破裂特征和其宏观的活动复杂性在速率和状态摩擦准则下得到了统一的解释。
The laboratory-derived rate-and state-dependent friction law is introduced. We study how the friction law affects and controls the scaling of source parameters, seis-mic slip complexity, as well as rupture dynamics for a fault model obeying the law. The friction on the fault surface has strong influences on the natural fault behavior and earthquakes. Our understanding in the fault slip and earthquakes depends on how much we known about friction. Therefore, we first detailed review how the rate and state fric-tion law is proposed and developed, and then, give the stability analysis under a simple single-degree-of-freedom spring-slider model, and also summarize the latest friction facts from high-speed rock friction experiments over the past decade, as well as related thermal weakening theories. A comprehensive and systematic knowledge of the rock friction serve to understanding of the nature of faults and earthquakes.
The field observation of source parameters varying with the seismic moment is one way by which seismologists explore the self-similarity of earthquakes. In this study, a single-degree-of-freedom spring-slider model is used to numerical simulate the implied scaling relation of source parameters under the rate-and state-dependent friction law. In this case, earthquake size is controlled by the fault size W (being inversely propor-tional to the stiffness k) and source parameters are affected by the ratio of W and the nucleation size hc. The simulation result shows source parameters are scale-dependent for small events, but are almost self-similar for large events. The transition is attribut-ed to the relatively large proportion of both aseismic slip and fracture energy in small events. However, the model shows evident increase in stress drop and apparent stress if a thermal pressurization is included.
A numerical method that could simulate both the fault slip behavior in very long time-scale and co-seismic process in extremely short time-scale, based on the traction boundary integral equation method in time domain, is developed and compared with widely used quasi-dynamic method. The two methods produce generally similar slip patterns and rupture styles in an uniform or slightly nonuniform fault model, although the latter cannot represent the seismic wave effects correctly in the co-seismic process.
The effect of the relative fault size W/hc on the rupture style and the slip complexity is studied by both the quasi-dynamic and new developed dynamic methods. It is observed that the fault with a small W/hc produces regular, periodic crack-like characteristic event and the fault with a large W/hc produces the similar event, as well as more self-stopping and pulse-like events. It implies an uniform fault would spontaneously evolve into complex slip patterns with increasing W/hc.
This study indicates the rate-and state-friction law is scale-dependent. The rel-ative fault size W/hc affects all aspects of faults and earthquakes. The micro-scale rupture characteristics and the macroscopic seismic complexity hence have an unified interpretation under the friction law.
观察震源参数随地震矩的变化一直是地震学家了解地震自相似性的手段之一。本文利用简单的单自由度弹簧滑块模型,数值模拟了速率和状态准则控制下的系统所表示的震源参数的变化。在该模型下,地震的大小主要由断层的尺度W(与模型中弹性刚度k成反比)控制。震源参数的变化受断层相对尺度(断层尺度W与成核尺度hc。的比值)的影响,对较小的地震,因为非震滑移和断裂能的比重比较大,震源参数有尺度效应；对较大的地震,非震滑移和断裂能可忽略,震源参数可视为自相似的。但是当考虑了热增压效应后,震源参数呈现非常明显的变化,视应力和应力降都随着地震矩的增加而明显增加。
本文发展了基于时间域牵引力边界积分的可同时模拟长时间尺度下断层滑动行为和极短时间尺度下同震过程的数值方法,并将其与准动态的数值模拟方法做了比较。尽管在同震过程准动态方法显然不能正确地模拟地震波效应,但是在均匀的或轻微非均匀的断层模型下,两者能给出大致相似的滑动模式(地震序列)和破裂风格(裂纹型或脉冲型)。本文同时利用准动态和新发展的动力学数值方法,研究了断层的相对尺度W/hc。对地震破裂特征和滑动复杂性的影响。发现当断层的相对尺度较小时,断层行为表现简单的周期性的裂纹型地震,当断层的相对尺度较大时,断层除了继续产生特征地震以外,会自发地产生更多自发停止的自愈合地震,这可能预示着均匀断层在相对尺度充分大的情况下,能自发地产生丰富的地震活动性。
总体上本文的研究显示速率和状态摩擦准则存在明显的尺度效应。断层相对尺度W/hc影响了断层和地震各方面的特征,地震的微观破裂特征和其宏观的活动复杂性在速率和状态摩擦准则下得到了统一的解释。
The laboratory-derived rate-and state-dependent friction law is introduced. We study how the friction law affects and controls the scaling of source parameters, seis-mic slip complexity, as well as rupture dynamics for a fault model obeying the law. The friction on the fault surface has strong influences on the natural fault behavior and earthquakes. Our understanding in the fault slip and earthquakes depends on how much we known about friction. Therefore, we first detailed review how the rate and state fric-tion law is proposed and developed, and then, give the stability analysis under a simple single-degree-of-freedom spring-slider model, and also summarize the latest friction facts from high-speed rock friction experiments over the past decade, as well as related thermal weakening theories. A comprehensive and systematic knowledge of the rock friction serve to understanding of the nature of faults and earthquakes.
The field observation of source parameters varying with the seismic moment is one way by which seismologists explore the self-similarity of earthquakes. In this study, a single-degree-of-freedom spring-slider model is used to numerical simulate the implied scaling relation of source parameters under the rate-and state-dependent friction law. In this case, earthquake size is controlled by the fault size W (being inversely propor-tional to the stiffness k) and source parameters are affected by the ratio of W and the nucleation size hc. The simulation result shows source parameters are scale-dependent for small events, but are almost self-similar for large events. The transition is attribut-ed to the relatively large proportion of both aseismic slip and fracture energy in small events. However, the model shows evident increase in stress drop and apparent stress if a thermal pressurization is included.
A numerical method that could simulate both the fault slip behavior in very long time-scale and co-seismic process in extremely short time-scale, based on the traction boundary integral equation method in time domain, is developed and compared with widely used quasi-dynamic method. The two methods produce generally similar slip patterns and rupture styles in an uniform or slightly nonuniform fault model, although the latter cannot represent the seismic wave effects correctly in the co-seismic process.
The effect of the relative fault size W/hc on the rupture style and the slip complexity is studied by both the quasi-dynamic and new developed dynamic methods. It is observed that the fault with a small W/hc produces regular, periodic crack-like characteristic event and the fault with a large W/hc produces the similar event, as well as more self-stopping and pulse-like events. It implies an uniform fault would spontaneously evolve into complex slip patterns with increasing W/hc.
This study indicates the rate-and state-friction law is scale-dependent. The rel-ative fault size W/hc affects all aspects of faults and earthquakes. The micro-scale rupture characteristics and the macroscopic seismic complexity hence have an unified interpretation under the friction law.
引文
牛志仁,陈党民.1996.弹性摩擦系统奇怪吸引子的Lyapunov指数和维数[J].Acta Seismol. Sinica, 17(4):469-476.
吴忠良,陈运泰.2002.从弹簧滑块到地震预测：BK模型今昔谈[J].物理,31(11)：719-724.
刘秉正,彭建华.2004.非线性动力学[M].北京：高等教育出版社.
范天佑.2003.断裂理论基础[M].北京：科学出版社.
张海明.2004.半无限空间中平面断层的三维自发破裂传播的理论研究[D].北京：北京大学.
范天佑.2006.断裂动力学：原理与应用[M].北京：北京理工大学出版社.
张伟.2006.含起伏地形的三维非均匀介质中地震波传播的有限差分算法及其在强地面震动模拟中的应用[D].北京：北京大学.
周仕勇.2008.川西及邻近地区地震活动性模拟和断层间相互作用研究[J].物理,51(1)：165-174.
陈运泰.2009.地震预测：回顾与展望[J].中国科学D辑：地球科学,39(12)：1633-1658.
崔海霞,陈建敏,周惠娣.2010.奇妙的摩擦世界[M].北京：科学出版社.
尹祥础.2011.固体力学[M].北京：地震出版社.
ABERCROMBIE R E.1995. Earthquake source scaling relationships from -1 to 5 ML using seismograms recorded at 2.5-km depth[J]. J. Geophys. Res.,100(B12):24,015-24,036.
ABERCROMBIE R E, LEARY P.1993. Source parameters of small earthquakes recored at 2.5 km depth, Cajon Pass, southern California:implications for earthquake scaling[J]. Geophys. Res. Lett.,20:1511-1514.
ABERCROMBIE R E, RICE J R.2005. Can observation of earthquake scaling constrain slip weakening?[J]. Geophys. J. Int.,162:406-424.
AKI K, RICHARDS P G.2002. Quantitative seismology[M]. second edition. Sausalito, California:University Science Books.
ALLMANN B P, SHEARER P M.2009. Global variations of stress drop for moderate to large earthquakes[J]. J. Geophys. Res.,114(B01310):doi:10.1029/2008JB005821.
AMPUERO J P, BEN-ZION Y.2008. Cracks, pulses and macroscopic asymmetry of dynamic rupture on a bimaterial interface with velocity-weakening friciton[J]. Geophys. J. Int.,173:674-692.
AMPUERO J P, RUBIN A M.2008. Earthquake nucleation on rate and state faults-aging and slip laws[J]. J. Geophys. Res.,113, B01302, doi:10.1029/2007JB005082.
ANDREWS D J.1976a. Rupture propagation with finite stress in antiplane strain[J]. J. Geophys. Res., 81(20):3575-3582.
ANDREWS D J.1976b. Rupture velocity of plane strain shear cracks[J]. J. Geophys. Res.,81(32):5679-5687.
ANDREWS D J.1985. Dynamic plane-strain shear rupture with a slip-weakening friction law calculated by a boundary integral method[J]. Bull. Seis. Soc. Am.,75(1):1-21.
ANDREWS D J.2002. A fault constitutive relation accounting for thermal pressurization of pore fluid[J]. J. Geophys. Res.,107, B12,2363, doi:10.1029/2002JB001942.
ANDREWS D J, BEN-ZION Y.1997. Wrinkle-like slip pulse on a fault between different materials[J]. J. Geophys. Res.,102(B1):553-571.
AOCHI H.2000. Theoretical studies dynamic rupture propagation along a 3D non-planar fault system[D].[S.l.]: The University of Tokyo.
AOCHI H, FUKUYAMA E, MATSUURA M.2000. Spontaneous rupture propagation on a non-planar fault in 3-D elastic medium[J]. Pure appl. geophys.,157:2003-2027.
BAK P, TANG C.1989. Earthquakes as a self-organized critical phenomenon[J]. J. Geophys. Res., 94(B11):15,635-16,637.
BAK P, TANG C, WIESENFELD K.1987. Self-organized criticality:an explanation of 1/f noise[J]. Phys. Rev. Lett.,59(4):381-384.
BAK P, TANG C, WIESENFELD K.1988. Self-organized criticality[J]. Phys. Rev. A,38(1):364-374.
BALTAY A, IDE S, PRIETO G, et al.2011. Variability in earthquake stress drop and apparent stress[J]. Geophys. Res. Lett.,38, L06303, doi:10.1029/2011GL046698.
BARBOT S, LAPUSTA N, AVOUAC J P.2012. Under the hood of the earthquake machine:toward predictive modeling of the seismic cycle[J]. Scinence,336:707-710.
Becker, T. W.1999. Deterministic chaos in two state-variable friction sliders and the effect of elastic interactions[J]. Geophysical Monograph,120:5-26.
BEELER N M.2001. Stress drop with constant, scale independent seismic efficiency and overshoot[J]. Geophys. Res. Lett.,28(17):3353-3356.
BEELER N M, TULLIS T E.1996. Self-healing slip pulses in dynamic rupture models due to velocity-dependence strength[J]. Bull. Seis. Soc. Am.,86(4):1130-1148.
BEELER N M, TULLIS T E, WEEKS J D.1994. The roles of time and displacement in the evolution effect in rock friction[J]. Geophys. Res. Let.,21(18):1987-1990.
BEELER N M, HICKMAN S H, WONG T F.2001a. Earthquake stress drop and laboratory-inferred interseismic strength recovery[J]. J. Geophys. Res.,106(B12):30,701-30,713.
BEELERN M, LOCKNER D L, HICKMAN S H.2001b. A simple stick-slip and creep-slip model for repeating earthquakes and its implication for microearthquakes at Parkfield[J]. Bull. Seis. Soc. Am.,91(6):1797-1804.
BEELER N M, WONG T F, HICKMAN S H.2003. On the expected relationships among apparent stress, static stress drop, effective shear fracture energy, and efficiency[J]. Bull. Seis. Soc. Am.,93(3):1381-1389.
BEELER N M, TULLIS T E, GOLDSBY D L.2008. Constitutive relationships and physical basis of fault strength due to flash heating[J]. J. Geophys. Res.,113, B01401, doi:10.1029/2007JB004988.
BELARDINELLI M E, BELARDINELLI E.1996. The quasi-static approximation of the spring-slider motion[J]. Nonlin. Processes Geophys.,3:143-149.
BEN-ZION Y.1996. Stress, slip, and earthquakes in models of complex single-fault systems incorporating brittle and creep deformations[J]. J. Geophys. Res.,101(B3):5677-5706.
BEN-ZION Y.2001. Dynamic ruptures in recent models of earthquake faults[J]. J. Mech. Phys. Solids, 49:2209-2244.
BEN-ZION Y.2008. Collective behavior of earthquakes and faults:continuum-discrete transitions, progressive evo-lutionary changes, and different dynamic regimes[J]. Rev. Geophys.,46, RG4006, doi:10.1029/2008RG000260.
BEN-ZION Y, RICE J R.1993. Earthquake failure sequences along a cellular fault zone in a three-dimensional elastic solid containing asperity and nonasperity regions[J]. J. Geophys. Res.,98(B8):14,109-14,131.
BEN-ZION Y, RICE J R.1995. Slip patterns and earthquake populations along different classes of faults in elastic solids[J]. J. Geophys. Res.,100(B7):12,959-12,983.
BEN-ZION Y, RICE J R.1997. Dynamic simulations of slip on a smooth fault in an elastic solid[J]. J. Geophys. Res.,102(B8):17,771-17,784.
BIZZARRI A.2009. What does control earthquake ruptures and dynamic faulting? A review of different competing mechanisms[J]. Pure appl. geophys.,166:741-776.
BIZZARRI A.2010a. How to promote earthquake ruptures:different nucleation strategies in a dynamic model with slip-weakening friction[J]. Bull. Seis. Soc. Am.,100(3):923-940.
BIZZARRI A.2010b. On the recurrence of earthquakes:role of wear in brittle faulting[J]. Geophys. Res. Lett.,37, L20315, doi:10.1029/2010GL045480.
BIZZARRI A.2010c. Pulse-like dynamic earthquake rupture propagation under rate, state-and temperature-dependent friction[J]. Geophys. Res. Lett.,37, L18307, doi:10.1029/2010GL044541.
BIZZARRI A.2011a. On the deterministic description of earthquakes[J]. Rev. Geophys., 49(RG3002):doi:10.1029/2011RG000356.
BIZZARRI A.2011b. Temperature variations of constitutive parameters can significantly affect the fault dynamics[J]. Earth Planet. Sci. Lett.,306(doi:10.1016/j.epsl.2011.04.009):272-278.
BIZZARRI A.2012. Modeling repeated slip failures on faults governed by slip-weakening friction[J]. Bull. Seis. Soc. Am.,102(2):812-821.
BIZZARRI A, COCCO M.2003. Slip-weakening behavior during the propagation of dynamic ruptures obeying rate-and state-dependent friction laws[J]. J. Geophys. Res.,108, B8.2373, doi:10.1029/2002JB002198.
BIZZARRI A, COCCO M.2005.3D dynamic simulations of spontaneous rupture propagation governed by different constitutive laws with rake rotation allowed[J]. Ann. Geophys.,48(2):279-299.
BIZZARRI A, COCCO M.2006. A thermal pressurization model for the spontaneous dynamic rupture propagation on a three-dimensional fault:1. methodological approach[J]. J. Geophys. Res., 111, B05303, doi:10.1029/2005JB003862.
BIZZARRI A, COCCO M, ANDREWS D J, et al.2001. Solving the dynamic rupture problem with different numerical approaches and constitutive laws[J]. Geophys. J. Int.,144:656-678.
BLANPIED M L, TULLIS T E.1986. The stability and behavior of a frictional system with a two state variable constitutive law[J]. Pure Appl. Geophys.,124(3):415-444.
BOATWRIGHT J.1980. A spectral theory for circular seismic sources, simple estimates of source dimension, dynamic stress drop, and radiated seismic energy[J]. Bull. Seis. Soc. Am.,70(1):1-27.
BOWDEN F P, TABOR D.1964. The friction and lubrication of solids:Part Ⅱ[M]. Oxford:Clarendon Press.
BRACE W F, BYERLEE J D.1966. Stick-slip as a mechanism for earthquakes[J]. Science,153(3739):990-992.
BRUNE J N.1970. Tectonic stress and the spectra of seismic shear waves from earthquakes[J]. J. Geophys. Res., 75(26):4997-5009.
BURRIDGE R.1969. The numerical solution of certain integral equations with non-integrable kernels arising in the theory of crack propagation and elastic wave diffraction[J]. Philos. Trans. R. Soc. London, Ser. A 265:353-381.
BURRIDGE R.1973. Admissible speeds for plane-strain self-similar shear cracks with friction but lacking cohe-sion[J]. Geophys. J. R. astr. Soc.,35:439-455.
BURRIDGE R, KNOPOFF.1969. Model and theoretical seismicity[J]. Bull. Seis. Soc. Am.,57(3):341-371.
BURRIDGE R, WILLIS J R.1969. The self-similar problem of the expanding elliptical crack in an anisotropic solid[J]. Mathematical Proceedings of the Cambridge Philosophical Society,66:443-468.
BYERLEE J D.1978. Friction of rocks[J]. Pure Appl. Geophys.,116:615-626.
CAO T, AKI K.1986. Seismicity simulation with a rate- and state-dependent friction law[J]. Pure App. Geophys., 124(3):487-513.
CARLSON J M, LANGER J S.1989. Mechanical model of an earthquake fault[J]. Phy. Rev. A,40(11):6470-6484.
CARPENTER B M, MARONE C, SAFFER D M.2011. Weakness of the San Andreas Fault revealed by samples from the active fault zone[J]. Nat. Geosci.,4:251-254.
CHEN T, LAPUSTA N.2009. Scaling of small repeating earthquakes explained by interaction of seismic and aseismic slip in a rate and state fault model[J]. J. Geophys. Res.,114(B01311):doi:10.1029/2008JB005749.
CHEN X, ZHANG H.2006. Modelling rupture dynamics of a planar fault in 3-D half space by boundary integral equation method:an overview[J]. Pure appl. geophys.,163:267-299.
CHOY G L, BOATWRIGHT J L.1995. Global patterns of radiated seismic energy and apparent stress[J]. J. Geophys. Res.,100(89):18,205-18,228.
COCCO M, BIZZARRI A.2002. On the slip-weakening behavior of rate-and state dependent constitutive Iaws[J]. Geophys. Res. Lett.,29,111516, doi:10.1029/2001GL013999.
COCHARD A, MADARIAGA R.1994. Dynamic faulting under rate-dependent friction law[J]. Pure Appl. Geo-phys.,142:419-445.
COCHARD A, MADARIAGA R.1996. Complexity of seismicity due to highly rate-dependent friction[J]. J. Geophys. Res.,101(B11):25,321-25,336.
COCHARD A, RICE J R.1997. A spectral method for numerical elastodynamic fracture analysis without spatial replication of the rupture event[J]. J. Mech. Phys. Solids,45(8):1393-1418.
COTTON F, ARCHULETA R, CAUSSE M.2013. What is sigma of the stress drop[J]. Seis. Res. Lett.,84(1):42-48.
DAHLEN F A.1974. On the ratio of P-wave to S-wave corner frequencies for shallow earthquake sources[J]. Bull. Seis. Soc. Am.,64(4):1159-1180.
DAHLEN F A.1977. The balance of energy in earthquake faulting[J]. Geophys. J. R. astr. Soc.,48:239-261.
DAS S.1976. A numerical study of rupture propagation and earthquake source mechanism[D]. [S.l.]:[s.n.].
DAS S, AKI K.1977. A numerical study of two-dimensional spontaneous rupture propagation[J]. Geophys. J. R. astr. Soc.,50:643-668.
DAY S M.1982. Three-dimensional finite difference simulation of fault dynamics:rectangular faults with fixed rupture velocity[J]. Bull. Seis. Soc. Am.,72(3):705-727.
DAY S M, D ALGUER L A, LAPUSTA N, et al.2005. Comparison of finite difference and boundary integral solutions to three-dimensional spontaneous rupture[J]. J. Geophys. Res.,110, B12307, doi:10.1029/2005JB003813.
De Paola N, HIROSE T, MITCHELL T, et al.2011. Fault lubrication and earthquake propagation in thermally unstable rocks[J]. Geology,39(1):35-38.
Di Toro G, GOLDSBY D L, TULLIS T E.2004a. Friction falls towards zero in quartz rock as slip velocity approaches seismic rates[J]. Nature,427:436-439.
Di Toro G, HIROSE T,NIELSEN S, et al.2004b. Natural and experimental evidence of melt lubrication of faults during earthquakes[J]. Science,311:647-649.
Di Toro G, HAN R, HIROSE T, et al.2011. Fault lubrication during earthquakes[J]. Nature,471:494-498.
DIETERICH J H.1972. Time-dependent firction in rocks[J]. J. Geophys. Res.,77(20):3690-3697.
DIETERICH J H.1978. Time dependent firction and the mechanics of stick slip[J]. Pure Appl. Geophys., 116:790-806.
DIETERICH J H.1979a. Modelling of rock friction:1. Experimental results and constitutive equations[J]. J. Geophys. Res.,84(85):2161-2168.
DIETERICH J H.1979b. Modelling of rock friction:2. Simulation of preseismic slip[J]. J. Geophys. Res., 84(85):2169-2175.
DIETERICH J H.1992. Earhtquake nucleation on faults with rate- and state-dependent strength[J]. Tectonophysics, 211.
DIETERICH J H.1994. A constitutive law for rate of earthquake production and its application to earthquake clsutering[J]. J. Geophys. Res.,99(B2):2601-2618.
DIETERICH J H, RICHARDS-DINGER K B.2010. Earthquakes recurrence in simulated fault systems[J]. Pure Appl. Geophys.,167.
DUBLANCHET P, BERNARD P, FAVREAU P.2013. Interactions and triggering in a 3-D rate-and-state asperity model[J]. J. Geophys. Res.,118:1-21.
ELLSWORTH W L, BEROZA G C.1995. Seismic evidence for an earthquake nucleation phase[J]. Science, 268(5212):851-855.
ERICKSON B, BIRNIR B, LAVALLEE D.2008. A model for aperiodicity in earthquakes[J]. Nonlin. Processes Geophys.,15:1-12.
ERICKSON B, BIRNIR B, LAVALLEE D.2011. Periodicity, chaos and localization in a Burridge-Knopoff model of an earthquake with rate-and-state friction[J]. Geophys. J. Int.,187:178-198.
ERICKSON B A.2010. Complexity in the nonlinear Dieterich-Ruina law[D].[S.l.]:University of California, Santa Barbara.
ESHELBY J D.1957. The determination of the elastic field of an ellipsoidal inclusion, and related problems[J]. Proc. R. Soc. Lond. A,241:376-396.
ESHELBY J D.1969. The elastic field of a crack extending non-uniformly under general anti-plane Ioading[J]. J. Mech. Phys. Solids,17:177-199.
FOSSUM A F, FREUND L B.1975. Nonuniformly moving shear crack model of a shallow focus earthquake mechanism[J]. J. Geophys. Res.,80(23):3343-3347.
FREUND L B.1972a. Crack propagation in an elastic solid subjected to general loading-1. constant rate of exten-sion[J]. J. Mech. Phys. Solids,20:129-140.
FREUND L B.1972b. Crack propagation in an elastic solid subjected to general loading-II. non-uniform rate of extension[J]. J. Mech. Phys. Solids,20:141-152.
FREUND L B.1972c. Energy flux into the tip of an extending crack in an elastic solid[J]. J. Elasticity,2(4):341-349.
FREUND L B.1990. Dynamic fracture mechanics[M]. The Edinburgh Building, Shaftesbury Road, Cambridge CB2 2RU:Cambridge University Press.
FUKUYAMA E, MADARIAGA R.1998. Rupture dynamics of a planar fault in a 3-D elastic medium:rate- and slip-weakening friction[J]. Bull. Seis. Soc. Am.,88:1-17.
GABRIEL A A, AMPUERO J P, DALGUER L A, et al.2012. The transition of dynamic rupture styles in elastic media under velocity-weakening friction[J]. J. Geophys. Res.,117, B09311, doi:10.1029/2012JB009468.
GELLER R J.1997. Earthquake prediction:a critical review[J]. Geophys. J. Int.,131:425-450.
GEUBELLE P H, RICE J R.1995. A spectral method for three-dimensional elastodynamic fracture problems[J]. J. Mech. Phys. Solids,43(11):1791-1824.
GOLDSBY D L, TULLIS T.2011. Flash heating leads to low fricional strength of crustal rocks at earthquake slip rates[J]. Science,334:216-218.
GU J C, RICE J R, RUINA A L, et al.1984. Slip motion and stability of a single degree of freedom elastic system with rate and state dependent friction[J]. J. Mech. Phys. Solids,32(3):167-196.
GU Y J, WONG T F.1991. Effects of loading velocity, stiffness, and inertia on the dynamics of a single degree of freedom spring-slider system[J]. J. Geophys. Res.,96(B13):21,677-21,691.
HASSARD B D.1981. Theory and application of Hopf bifurcation[M]. New York, NY10022, USA:Cambridge Unversity Press.
HE C, WONG T F, BEELER N M.2003. Scaling of stress drop with recurrence interval and loading velocity for laboratory-derived fault strength relations[J]. J. Geophys. Res.,108(B1,2037):doi:10.1029/2002JB001890.
HEATON T H.1990. Evidence for and implications of self-healing pulses of slip in earthquake rupture[J]. Phys. Earth Plantet. Int.,64:1-20.
HILLERS G, BEN-ZION Y, MAI P M.2006. Seismicity on a fault controlled by rate- and state-dependent friction with spatial variations of the critical slip distance[J]. J. Geophys. Res., 111, B01403, doi:10.1029/2005JB003859.
HOROWITZ F G.1988. Mixed state variable friction laws:some implications for experiments and a stability analysis[J]. Geophys. Res. Lett.,15(11):1243-1246.
HOROWITZ F G.1989. Slip patterns in a spatially homogeneous fault model[J]. J. Geophys. Res., 94(B8):10,279-10,298.
HUANG J, TURCOTTE D L.1990. Evidence for chaotic fault interactions in the seismicity of the San Andreas fault and Nankai trough[J]. Nature,348:234-236.
IDA Y.1972. Cohesive force across the tip of a longitudinal-shear crack and Griffith's specific surface energyfJ]. J. Geophys. Res.,77(20):3796-3805.
IDE S.2002. Estimation of radiated energy of finite-source earthquake models[J]. Bull. Seis. Soc. Am., 92(8):2994-3005.
IDE S, BEROZA G C.2001. Does apparent stress vary with earthquake size[J]. Geophys. Res. Lett., 28(17):3349-3352.
IZUTANI Y.2005. Radiated energy from the mid Niigata, Japan, earthquake of October 23,2004, and its after-shocks[J]. Geophys. Res. Lett.,32:L21313, doi:10.1029/2005GL024113.
KANAMORI H, HEATON T H.2000. Microscopic and macroscopic physics of earthquakes[C]//. RUNDLE J, TURCOTTE D L, KLEIN W. Geocomplexity and the physics of earthquakes. Washington, D. C.:AGU, 120:147-161.
KANAMORI H, RIVERA L.2006. Energy partitioning during an earthquake[C]//. ABERCROMBIE R, MCGARR A, KANAMORI H, et al. Earthquakes:radiated energy and the physics of faulting. AGU, Washington, D. C., doi:10.1029/GM170:[s.n.]. Geophys. Monogr. Ser., vol.170.
KANAMORI H, MORI J, HAUKSSON E, et al.1993. Determination of earthquake energy release and ML using TERRAscope[J]. Bull. Seis. Soc. Am.,83(2):330-346.
KANEKO Y, LAPUSTA N, AMPUERO J P.2008. Spectral element modeling of spontaneous earthquake rup-ture on rate and state faults:effects of velocity-strengthening friction at shallow depths[J]. J. Geophys. Res., 113,B09317,doi:10.1029/2007JB005553.
KANEKO Y, FIALKO Y, SANDWELL D T, et al.2013. Interseismic deformation and creep along the central section of the North Anatalian Fault (Turkey):InSAR observations and implications for rate-and-state friction properties[J]. J. Geophys. Res.,118:316-331, doi:10.1029/2012JB009661.
KATO N.2003. Repeating slip events at a circular asperity:numerical simulation with a rate- and state-dependent friction law[J]. Bull. Earthq. Res. Inst. Univ. Tokyo,78:151-166.
KATO N.2004. Interaction of slip on asperities:numerical simulation of seismic cycles on a two-dimensional planar fault with nonuniform frictional property[J]. J. Geophys. Res.,109,B12306,doi:10.1029/2004JB003001.
KATO N.2007. How frictional properties lead to either rupture-front focusing or cracklike behavior[J]. Bull. Seis. Soc. Am.,97(6):2182-2189.
KATO N.2012. Dependence of earthquake stress drop on critical slip-weakening distance [J]. J. Geophys. Res., 117,B01301,doi:10.1029/2011JB008359.
KATO N, TULLIS T E.2001. A composite rate-and state-dependent law for rock friction[J]. Geophys. Res. Lett., 28(6):1103-1106.
KATO N, TULLIS T E.2003. Numerical simulation of seismic cycles with a composite rate- and state-dependent friction law[J]. Bull. Seis. Soc. Am.,93(2):841-853.
KATO N, LEI X, WEN X.2007. A synthetic seismicity model for the Xianshuihe fault, southwestern China: simulation using a rate- and state-dependent friction law[J]. Geophys. J. Int.,169:286-300.
KEILIS-BOROK V I.1959. On estimation of the displacement in an earthquake source and of source dimensions[J]. Ann. Geofis.,12:205-214.
KNOPOFF L.1958. Energy release in earthquakes[J]. Geophys. J. Roy. Astron. Soc., 1(1):44-52.
KOSLOFF D D, LIU H P.1980. Reformulation and discussion of mechanical behavior of the velocity-dependent friction law proposed by Dieterich[J]. Geophys. Res. Lett.,7(11):913-916.
KOSTROV B V.1964. Self-similar problems of propagation of shear cracks[J]. J.appl. Math. Mech.,28:1077-1087.
KOSTROV B V.1966. Unsteady propagation of longitudinal shear cracks[J]. J. appl. Math. Mech.,30:1042-1049.
KOSTROV B V.1974a. Crack propagation at variable velocity[J]. J. appl. Math. Mech.,38(3):551-560.
KOSTROV B V.1974b. Seismic moment and energy of earthquakes, and seismic flow of rock[J]. Izv. Earth Phys., 1:23-40.
LAPUSTA N, LIU Y.2009. Three-dimensional boundary integral modeling of spontaneous earthquake sequences and aseismic slip[J]. J. Geophys. Res.,114, B09303, doi:10.1029/2008JB005934.
LAPUSTA N, RICE J R.2003. Nucleation and early seismic propagation of small and large events in a crustal earthquake model[J]. J. Geophys. Res.,108, B4.2205, doi:10.1029/2001JB000793.
LAPUSTA N, RICE J R, BEN-ZION Y, et al.2000. Elastodynamic analysis for slow tectonic loading with spontaneous rupture episodes on fault with rate- and state-dependent friction[J]. J. Geophys. Res., 105(B10):23,765-23,789.
LINKER M F, DIETERICH J H.1992. Effects of variable normal stress on rock friction:observations and constitutive equations[J]. J. Geophys. Res.,97(B4):4923-4940.
LIU Y.2013. Numerical simulations on megathrust rupture stailized under strong dilatancy strenghening in slow slip region[J]. Geophys. Res. Lett.,40:1311-1316.
LIU Y, RICE J R.2005. Aseismic slip transients emerge spontaneously in three-dimensional rate and state modeling of subduction earthquake sequences[J]. J. Geophys. Res.,110,B08307,doi:10.1029/2004JB003424.
LIU Y, RICE J R.2007. Spontaneous and triggered aseismic deformation transients in a subduction fault model[J]. J. Geophys. Res.,112,B09404,doi:10.1029/2007JB004930.
LIU Y, MCGUIRE J J, BEHN M D.2012. Frictional behavior of oceanic transform faults and its influence on earthquake characteristics[J]. J. Geophys. Res.,117,B04315,doi:10.1029/2011JB009025.
LOCKNER D A, MORROW C, MOORE D, et al.2011. Low strength of deep San Andreas fault gouge from SAFOD core[J]. Nature,472:82-85.
MADARIAGA R.1976. Dynamics of an expanding circular fault[J]. Bull. Seis. Soc. Am.,66(3):639-666.
MADARIAGA R, COCHARD A.1996. Dynamic friction and the origin of the complexity of earthquake sources[J]. Proc. Natl. Acad. Sci.,93:3819-3824.
MADARIAGA R, OLSEN K B.2000. Criticality of rupture dynamics in 3-D[J]. Pure appl. geophys.,157:1981-2001.
MADARIAGA R, OLSEN K, ARCHULETA R.1998. Modeling dynamic rupture in a 3D earthquake fault model[J]. Bull. Seis. Soc. Am.,88(5):1182-1197.
MALAGNINI L, NIELSEN S, MAYEDA K, et al.2010. Energy radiation from intermediate- to large-magnitude earthquakes:implications for dynamic fault weakening[J]. J. Geophys. Res., 115, B06319, doi:10.1029/2009JB006786.
MARONE C.1998. Laboratory-derived friction laws and their application to seismic faulting[J]. Annu. Rev. Earth Planet. Sci.,26:643-695.
MAYEDA K, GOK R, WALTER W R, et al.2005. Evidence for non-constant energy/moment scaling from coda-derived source spectra[J]. Geophys. Res. Lett.,32, L10306, doi:10.1029/2005GL022405.
MCGARR A.1999. On relating apparent stress to the stress causing earthquake faults slip[J]. J. Geophys. Res., 104(B2):3003-3011.
MCGARR A, FLETCHER J B.2000. A method for mapping apparent stress and energy radiation applied to the 1994 Northridge earthquake fault zone[J]. Geophys. Res. Lett.,27(13):1953-1956.
MCGARR A, FLETCHER J B.2001. A method for mapping apparent stress and energy radiation applied to the 1994 Northridge earthquake fault zone-revisited[J]. Geophys. Res. Lett.,28(18):3529-3532.
MITSUI Y, HIRAHARA K.2009. Coseismic thermal pressurization can notably prolong earthquake recurrence intervals on weak rate and state friction faults:numerical experiments using different constitutive equations[J]. J. Geophys. Res.,114, B09304, doi:10.1029/2008JB006220.
MITSUI Y, HIRAHARA K.2011. Fault instability on a finite and planar fault related to early phase of nucleation[J]. J. Geophys. Res.,116, B06301, doi:10.1029/2010JB007974.
MORI J, ABERCROMBIE R E, KANAMORI H.2003. Stress drops and radiated energies of aftershocks of the 1994 Northridge, California, earthquake[J]. J. Geophys. Res.,108, B112545, doi:10.1029/2001JB000474.
NADEAU R M, JOHNSON L R.1998. Seismological studies at Parkfield Ⅵ:moment release rates and estimates of source paramters for small repeating earthquakes[J]. Bull. Seis. Soc. Am.,88(3):790-814.
NAGATA K, NAKATANI M, YOSHIDA S.2012. A revised rate- and state-dependent friction law obtained by constraining constitutive and evolution laws separately with laboratory data[J]. J. Geophys. Res.,117, B02314, doi:10.1029/2011JB008818.
NODA H, LAPUSTA N.2010. Three-dimensional earthquake sequences simulations with evolving temperature and pore pressure due to shear heating:effect of heterogeneous hydraulic diffusivity[J]. J. Geophys. Res.,115, B12314, doi:10.1029/2010JB007780.
NODA H, LAPUSTA N.2013. Stable creeping fault segments can become destructive as a result of dynamic weakening[J]. Nature,493:518-523.
NODA H, DUNHAM E M, RICE J R.2009. Earthquake ruptures with thermal weakening and the operation of major faults at low overall stress levels[J]. J. Geophys. Res.,114, B07302, doi:10.1029/2008JB006143.
OANCEA V G, LAURSEN T A.1997. Stability analysis of state dependent dynamic frictional sliding[J]. Int. J. Non-Linear Mechanics,32(5):837-853.
OHNAKA M.2003. A constitutive scaling law and a unified comprehension for frictional slip failure, shear fracture of intack rock, and earthquake rupture[J]. J. Geophys. Res.,108, B2,2080, doi:10.1029/2000JB000123.
OKADA Y.1985. Surface deformation due to shear and tensile faults in a half-space[J]. Bull. Seis. Soc. Am., 75(4):1135-1154.
OKADA Y.1992. Internal deformation due to shear and tensile faults in a half-space[J]. Bull. Seis. Soc. Am., 82(2):1018-1040.
OKUBRO P G.1989. Dynamic rupture modeling with laboratory-derived constitutive relations[J]. J. Geophys. Res., 94(B9):12,321-12,335.
PALMER A C, RICE J R.1973. The growth of slip surfaces in the progressive failure of over-consolidated clay[J]. Proc. Roy. Soc. Lond. A.,332:527-548.
PERRIN G, RICE J R, ZHENG G.1995. Self-healing slip pulse on a frictional surface[J]. J. Mech. Phys. Solids, 43(9):1461-1495.
PRESS W H, TEUKOLSKY S A, VETTERLING W T, et al.1992. Numerical Recipes in Fortran 77[M].2nd edition. New York:Cambridge University Press.
RANJITH K, RICE J R.1999. Stability of quasi-static slip in a single degree of freedom elastic system with rate and state dependent friction[J]. J. Mech. Phys. Solds,47:1207-1218.
REMPEL A W, RICE J R.2006. Thermal pressurization and onset of melting in fault zones[J]. J. Geophys. Res., 111, B09314, doi:10.1029/2006JB004314.
RICE J R.1980. The Mechanics of Earthquake Rupture[M]//. DZIEWONSKI A, BOSCHI E. Physics of the Earth's Interior. North Holland:Amsterdam.
RICE J R.1983. Constitutive relations for fault slip and earthquake instabilities[J]. Pure Appl. Geophys., 121(3):443-475.
RICE J R.1993. Spatio-temporal complexity of slip on a fault[J]. J. Geophys. Res.,98(B6):9885-9907.
RICE J R.2006. Heating and weakening of faults during earthquake slip[J]. J. Geophys. Res., 111(B05311):doi:10.1029/2005JB004006.
RICE J R.BEN-ZION Y.1996. Slip complexity in earthquake fault models[J]. Proc. Natl. Acad. Sci.,93:3811-3818.
RICE J R, GU J C.1983. Earthquake aftereffects and triggered seismic phenomena[J]. Pure Appl. Geophys., 121(2):187-219.
RICE J R, RUINA A L.1983. Stability of steady frictional slipping[J]. J. Appl. Mech.,50:343-349.
RICE J R, TSE S T.1986. Dynamic motion of a single degree of freedom system following a rate and state dependent friction law[J]. J. Geophys. Res.,91(B1):521-530.
RICE J R, LAPUSTA N, RANJITH K.2001. Rate and state dependent friction and the stability of sliding between elastically deformable solids[J]. J. Mech. Phys. Solids,49:1865-1898.
RICHARDS-DINGERK, DIETERICH JH.2012. RSQSim earthquake simulator[J]. Seis. Res. Lett.,83(6):983-990.
RIVERA L, KANAMORI H.2005. Representations of the radiated energy in earthquakes[J]. Geophys. J. Int., 162:148-155.
ROBINSON R.2004. Potential earthquake triggering in a complex fault network:the northern South Island, New Zealand[J]. Geophys. J. Int.,159.
ROBINSON R, BENITES R.1995. Synthetic seismicity models of multiple interacting faults[J]. J. Geophys. Res., 100(B9):18,229-18,238.
ROBINSON R, BENITES R.1996. Synthetic seismicity models for the Wellington Region, New Zealand:Implica-tions for the temporal distribution of large events[J]. J. Geophys. Res.,101(B12):27,833-27,844.
ROBINSON R, BENITES R.2001. Upgrading a synthetic seismicity models for more realistic fault ruptures[J]. Geophys. Res. Lett.,28(9):1843-1846.
ROBINSON R, NICOL A, J WALSH J, et al. 2009. Features of earthquake occurence in a complex normal fault networks:results from a synthetic seismicity model of the Taupo Rift, New Zealand[J]. J. Geophys. Res.,114, B12306, doi:10.1029/2008JB006231.
ROBINSON R, Van Dissen R, LITCHFIELD N.2011. Using synthetic seismicity to evaluate seismic hazard in the Wellington region, New Zealand[J]. Geophys. J. Int.,187.
ROJAS O, DUNHAM E M, DAY S M, et al.2009. Finite difference modelling of rupture propagation with strong velocity-weakening friction[J]. Geophys. J. Int.,179(3):1831-1858.
ROY M, MARONE C.1996. Earthquake nucleation on model faults with rate-and state-dependent friction:effects of inertia[J]. J. Geophys. Res.,101(B6):13,919-13,932.
RUBIN A M, AMPUERO J P.2005. Earthquake nucleation on (aging) rate and state faults[J]. J. Geophys. Res., 110, B11312, doi:10.1029/2005JB003686.
RUDNICKI J W, FREUND L B.1981. On energy radiation from seismic sources[J]. Bull. Seis. Soc. Am., 71(3):583-595.
RUINA A L.1980. Friction laws and instabilities:A quasistatic analysis of some dry frictional behavior[D]. Provi-dence, R. I.:Brown Univ.
RUINA A L.1983. Slip instability and state variable friction laws[J]. J. Geophys. Res.,88(B12):10,359-10,370.
RUNDLE P B, RUNDLE J B, TIAMPO K F, et al.2006. Virtual California:fault model, frictional parameters, applications[J]. Pure Appl. Geophys.,163:1819-1846.
SACHS M K, HEIEN E M, TURCOTTE D L, et al.2012. Virtual California Earthquake Simulator[J]. Seis. Res. Lett.,83(6):973-978.
SAVAGE J C, WOOD M D.1983. The relations between apparent stress and stress drop[J]. Bull. Seis. Soc. Am., 61(5):1381-1388.
SCHOLZ C H.1988. The critical slip distance for seismic faulting[J]. Nature,336:761-763.
SCHOLZ C H.1997. Whatever happened to earthquake prediction[J]. Geotimes,42(3):16-19.
SCHOLZ C H.1998. Earthquakes and friction laws[J]. Nature,391:37-42.
SCHOLZ C H.2002. The mechanics of earthquakes and faulting[M].2nd.[S.l.]:Cambridge University Press.
SCHOLZ C H, ENGELDER J T.1976. Role of asperity indentation and ploughing in rock friction[J]. Int. J. Rock Mech. Min. Sci. and Geomech.,13:149-154.
SCHOLZ C H, MOLNAR P, JOHNSON T.1972. Detailed studies of frictional sliding of granite and implicatins for the earthquake mechanism[J]. J. Geophys. Res.,77(32):6392-6406.
SCHOLZ C H, AVILES C A, WESNOUSKY S G.1986. Scaling differences between large interplate and intraplate earthquakes[J]. Bull. Seis. Soc. Am.,76(1):65-70.
SHAW B E.2009. Constant stress drop from small to great earthquakes in magnitude-area scaling[J]. Bull. Seis. Soc. Am.,99(2A):871-875.
SHAW B E, RICE J R.2000. Existence of continuum complexity in the elastodynamics of repeated fault ruptures[J]. J. Geophys. Res.,105(B10):23,791-23,810.
SHKOLLER S, MINSTER J B.1997. Reduction of Dieterich-Ruina attractors to unimodal maps[J]. Nonlin. Pro-cesses Geophys.,4:63-69.
STUART W D.1988. Forecast model for great earthquakes at the Nankai Trough subduction zone[J]. Pure Appl. Geophys.,126(2-4):619-641.
STUART W D, TULLIS T E.1995. Fault model for preseismic deformation at Parkfield, California[J]. J. Geophys. Res.,100(B12):24,079-24,099.
TAKAHASHIT, SATO H, OHTAKE M, et al.2005. Scale dependence of apparent stress for earthquakes along the subducting Pacific plate in northeastern Honshu, Japan[J]. Bull. Seis. Soc. Am.,95(4):1334-1345.
TINTI E, BIZZARRI A, PIATANESI A, et al.2004. Estimates of slip weakening distance for different dynamic rupture models[J]. Geophys. Res. Lett.,31, L02611, doi:10.1029/2003GL018811.
TSE S T, RICE J R.1986. Crustal earthquake instability in relation to the depth variation of frictional slip properties[J]. J. Geophys. Res.,91(B9):9452-9472.
TULLIS T E.1988. Rock friction constitutive behavior from laboratory experiments and its implications for an earthquake prediction field monitoring program[J]. Pure Appl. Geophys.,126(2-4):555-588.
VERAART A J, FAASSEN E J, DAKOS V, et al.2011. Recovery rates reflect distance to a tipping point in a living system[J]. Nature,481:357-359.
VIEGAS G, ABERCROMBIE R E, KIM W Y.2010. The 2002 M5 Au Sable Forks, NY, earthquake sequence: source scaling relashionships and energy budget[J]. J. Geophys. Res.,115, B07310, doi:10.1029/2009JB006799.
WALTER W R, MAYEDA K, GOK R.2006. The scaling of seismic energy with moment:simple models compared with observations[C]//. ABERCROMBIE R, MCGARR A, KANAMORIH, et al. Earthquakes:radiated energy and the physics of faulting. AGU, Washington, D. C., doi:10.1029/GM170:[s.n.]. Geophys. Monogr. Ser., vol. 170.
WANG J H.2002. A dynamic study of two one-state-varible, rate-dependent, and state-dependent friction laws[J]. Bull. Seis. Soc. Am.,2:687-694.
WANG J H.2008. One-dimensional dynamical modeling of earthquakes:a review[J]. Terr. Atmos. Ocean Sci., 19(3):183-203.
WANG J H.2012. Some intrinsic properties of the two-dimensional dynamical spring-slider model of earthquake faults[J]. Bull. Seis. Soc. Am.,102(2):822-835.
WARD S N.1991. A synthetic seismicity model for the Middle America trench[J]. J. Geophys. Res., 96(B13):21,433-21,442.
WARD S N.2000. San Francisco Bay area earthquake simulations:a step toward a standard physical earthquake model[J]. Bull. Seis. Soc. Am.,90(2):370-386.
WARD S N.2012. ALLCAL earthquake simulator[J]. Seis. Res. Lett.,83(6):964-972.
WIGGINS S.2003. Introduction to applied nonlinear dynamical systems and chaos[M].2nd ed.175 Fifth Avenue, New York, NY 10010, USA:Springer-Verlag New York, Inc.
WILLIS J R.1973. Self-similar problems in elastodynamics[J]. Phil. Trans. R. Soc. Lond. A,274:434-491.
YAMADA T, MORI J J, IDE S, et al.2007. Stress drops and radiated seismic energies of microearthquakes in a South African gold mine[J]. J. Geophys. Res.,112, B03305, doi:10.1029/2006JB004553.
YAMASHITA T.1976. On the dynamical process of fault motion in the presence of friction and inhomogeneous initial stress:Part Ⅰ. rupture propagation[J]. J. Phys. Earth,24:417-444.
YANG H, LIU Y, LIN J.2012. Effects of subducted seamounts on megathrust earthquake nucleation and rupture propagation[J]. Geophys. Res. Lett.,39, L24302, doi:10.1029/2012GL053892.
ZHANG H, CHEN X.2006a. Dynamic rupture on a planar fault in three-dimensional half space-Ⅰ. Theory[J]. Geophys. J. Int.,164:633-652.
ZHANG H, CHEN X.2006b. Dynamic rupture on a planar fault in three-dimensional half space-Ⅱ. Validations and numerical experiments[J]. Geophys. J. Int.,167:917-932.
ZHANG W, ZHANG Z, CHEN X.2012. Three-dimensional elastic wave numerical modelling in the presence of surface topography by a collocated-grid finite-difference method on curvilinear grids[J]. Geophys. J. Int., 190:358-378.
ZHANG Z, ZHANG W, LI H, et al.2013. Stable discontinuous grid implementation for collocated-grid finite-difference seismic wave modelling[J]. Geophys. J. Int.,192(3):1179-1188.
ZHENG G, RICE J R.1998. Conditions under which velocity-weakening friction allows a self-healing versus a cracklike mode of rupture[J]. Bull. Seis. Soc. Am.,88(6):1466-1483.
ZZIV A, COCHARD A.2006. Quasi-dynamic modeling of seismicity on a fault with depth-variable rate- and state-dependent friction[J]. J. Geophys. Res., 111, B08310, doi:10.1029/2005JB004189.
ZIV A, RUBIN A M.2003. Implications of rate-and-state friction for properties of aftershock sequences:quasi-static inherently discrete simulations[J]. J. Geophys. Res.,108, B1.2051, doi:10.1029/2001JB001219.
吴忠良,陈运泰.2002.从弹簧滑块到地震预测：BK模型今昔谈[J].物理,31(11)：719-724.
刘秉正,彭建华.2004.非线性动力学[M].北京：高等教育出版社.
范天佑.2003.断裂理论基础[M].北京：科学出版社.
张海明.2004.半无限空间中平面断层的三维自发破裂传播的理论研究[D].北京：北京大学.
范天佑.2006.断裂动力学：原理与应用[M].北京：北京理工大学出版社.
张伟.2006.含起伏地形的三维非均匀介质中地震波传播的有限差分算法及其在强地面震动模拟中的应用[D].北京：北京大学.
周仕勇.2008.川西及邻近地区地震活动性模拟和断层间相互作用研究[J].物理,51(1)：165-174.
陈运泰.2009.地震预测：回顾与展望[J].中国科学D辑：地球科学,39(12)：1633-1658.
崔海霞,陈建敏,周惠娣.2010.奇妙的摩擦世界[M].北京：科学出版社.
尹祥础.2011.固体力学[M].北京：地震出版社.
ABERCROMBIE R E.1995. Earthquake source scaling relationships from -1 to 5 ML using seismograms recorded at 2.5-km depth[J]. J. Geophys. Res.,100(B12):24,015-24,036.
ABERCROMBIE R E, LEARY P.1993. Source parameters of small earthquakes recored at 2.5 km depth, Cajon Pass, southern California:implications for earthquake scaling[J]. Geophys. Res. Lett.,20:1511-1514.
ABERCROMBIE R E, RICE J R.2005. Can observation of earthquake scaling constrain slip weakening?[J]. Geophys. J. Int.,162:406-424.
AKI K, RICHARDS P G.2002. Quantitative seismology[M]. second edition. Sausalito, California:University Science Books.
ALLMANN B P, SHEARER P M.2009. Global variations of stress drop for moderate to large earthquakes[J]. J. Geophys. Res.,114(B01310):doi:10.1029/2008JB005821.
AMPUERO J P, BEN-ZION Y.2008. Cracks, pulses and macroscopic asymmetry of dynamic rupture on a bimaterial interface with velocity-weakening friciton[J]. Geophys. J. Int.,173:674-692.
AMPUERO J P, RUBIN A M.2008. Earthquake nucleation on rate and state faults-aging and slip laws[J]. J. Geophys. Res.,113, B01302, doi:10.1029/2007JB005082.
ANDREWS D J.1976a. Rupture propagation with finite stress in antiplane strain[J]. J. Geophys. Res., 81(20):3575-3582.
ANDREWS D J.1976b. Rupture velocity of plane strain shear cracks[J]. J. Geophys. Res.,81(32):5679-5687.
ANDREWS D J.1985. Dynamic plane-strain shear rupture with a slip-weakening friction law calculated by a boundary integral method[J]. Bull. Seis. Soc. Am.,75(1):1-21.
ANDREWS D J.2002. A fault constitutive relation accounting for thermal pressurization of pore fluid[J]. J. Geophys. Res.,107, B12,2363, doi:10.1029/2002JB001942.
ANDREWS D J, BEN-ZION Y.1997. Wrinkle-like slip pulse on a fault between different materials[J]. J. Geophys. Res.,102(B1):553-571.
AOCHI H.2000. Theoretical studies dynamic rupture propagation along a 3D non-planar fault system[D].[S.l.]: The University of Tokyo.
AOCHI H, FUKUYAMA E, MATSUURA M.2000. Spontaneous rupture propagation on a non-planar fault in 3-D elastic medium[J]. Pure appl. geophys.,157:2003-2027.
BAK P, TANG C.1989. Earthquakes as a self-organized critical phenomenon[J]. J. Geophys. Res., 94(B11):15,635-16,637.
BAK P, TANG C, WIESENFELD K.1987. Self-organized criticality:an explanation of 1/f noise[J]. Phys. Rev. Lett.,59(4):381-384.
BAK P, TANG C, WIESENFELD K.1988. Self-organized criticality[J]. Phys. Rev. A,38(1):364-374.
BALTAY A, IDE S, PRIETO G, et al.2011. Variability in earthquake stress drop and apparent stress[J]. Geophys. Res. Lett.,38, L06303, doi:10.1029/2011GL046698.
BARBOT S, LAPUSTA N, AVOUAC J P.2012. Under the hood of the earthquake machine:toward predictive modeling of the seismic cycle[J]. Scinence,336:707-710.
Becker, T. W.1999. Deterministic chaos in two state-variable friction sliders and the effect of elastic interactions[J]. Geophysical Monograph,120:5-26.
BEELER N M.2001. Stress drop with constant, scale independent seismic efficiency and overshoot[J]. Geophys. Res. Lett.,28(17):3353-3356.
BEELER N M, TULLIS T E.1996. Self-healing slip pulses in dynamic rupture models due to velocity-dependence strength[J]. Bull. Seis. Soc. Am.,86(4):1130-1148.
BEELER N M, TULLIS T E, WEEKS J D.1994. The roles of time and displacement in the evolution effect in rock friction[J]. Geophys. Res. Let.,21(18):1987-1990.
BEELER N M, HICKMAN S H, WONG T F.2001a. Earthquake stress drop and laboratory-inferred interseismic strength recovery[J]. J. Geophys. Res.,106(B12):30,701-30,713.
BEELERN M, LOCKNER D L, HICKMAN S H.2001b. A simple stick-slip and creep-slip model for repeating earthquakes and its implication for microearthquakes at Parkfield[J]. Bull. Seis. Soc. Am.,91(6):1797-1804.
BEELER N M, WONG T F, HICKMAN S H.2003. On the expected relationships among apparent stress, static stress drop, effective shear fracture energy, and efficiency[J]. Bull. Seis. Soc. Am.,93(3):1381-1389.
BEELER N M, TULLIS T E, GOLDSBY D L.2008. Constitutive relationships and physical basis of fault strength due to flash heating[J]. J. Geophys. Res.,113, B01401, doi:10.1029/2007JB004988.
BELARDINELLI M E, BELARDINELLI E.1996. The quasi-static approximation of the spring-slider motion[J]. Nonlin. Processes Geophys.,3:143-149.
BEN-ZION Y.1996. Stress, slip, and earthquakes in models of complex single-fault systems incorporating brittle and creep deformations[J]. J. Geophys. Res.,101(B3):5677-5706.
BEN-ZION Y.2001. Dynamic ruptures in recent models of earthquake faults[J]. J. Mech. Phys. Solids, 49:2209-2244.
BEN-ZION Y.2008. Collective behavior of earthquakes and faults:continuum-discrete transitions, progressive evo-lutionary changes, and different dynamic regimes[J]. Rev. Geophys.,46, RG4006, doi:10.1029/2008RG000260.
BEN-ZION Y, RICE J R.1993. Earthquake failure sequences along a cellular fault zone in a three-dimensional elastic solid containing asperity and nonasperity regions[J]. J. Geophys. Res.,98(B8):14,109-14,131.
BEN-ZION Y, RICE J R.1995. Slip patterns and earthquake populations along different classes of faults in elastic solids[J]. J. Geophys. Res.,100(B7):12,959-12,983.
BEN-ZION Y, RICE J R.1997. Dynamic simulations of slip on a smooth fault in an elastic solid[J]. J. Geophys. Res.,102(B8):17,771-17,784.
BIZZARRI A.2009. What does control earthquake ruptures and dynamic faulting? A review of different competing mechanisms[J]. Pure appl. geophys.,166:741-776.
BIZZARRI A.2010a. How to promote earthquake ruptures:different nucleation strategies in a dynamic model with slip-weakening friction[J]. Bull. Seis. Soc. Am.,100(3):923-940.
BIZZARRI A.2010b. On the recurrence of earthquakes:role of wear in brittle faulting[J]. Geophys. Res. Lett.,37, L20315, doi:10.1029/2010GL045480.
BIZZARRI A.2010c. Pulse-like dynamic earthquake rupture propagation under rate, state-and temperature-dependent friction[J]. Geophys. Res. Lett.,37, L18307, doi:10.1029/2010GL044541.
BIZZARRI A.2011a. On the deterministic description of earthquakes[J]. Rev. Geophys., 49(RG3002):doi:10.1029/2011RG000356.
BIZZARRI A.2011b. Temperature variations of constitutive parameters can significantly affect the fault dynamics[J]. Earth Planet. Sci. Lett.,306(doi:10.1016/j.epsl.2011.04.009):272-278.
BIZZARRI A.2012. Modeling repeated slip failures on faults governed by slip-weakening friction[J]. Bull. Seis. Soc. Am.,102(2):812-821.
BIZZARRI A, COCCO M.2003. Slip-weakening behavior during the propagation of dynamic ruptures obeying rate-and state-dependent friction laws[J]. J. Geophys. Res.,108, B8.2373, doi:10.1029/2002JB002198.
BIZZARRI A, COCCO M.2005.3D dynamic simulations of spontaneous rupture propagation governed by different constitutive laws with rake rotation allowed[J]. Ann. Geophys.,48(2):279-299.
BIZZARRI A, COCCO M.2006. A thermal pressurization model for the spontaneous dynamic rupture propagation on a three-dimensional fault:1. methodological approach[J]. J. Geophys. Res., 111, B05303, doi:10.1029/2005JB003862.
BIZZARRI A, COCCO M, ANDREWS D J, et al.2001. Solving the dynamic rupture problem with different numerical approaches and constitutive laws[J]. Geophys. J. Int.,144:656-678.
BLANPIED M L, TULLIS T E.1986. The stability and behavior of a frictional system with a two state variable constitutive law[J]. Pure Appl. Geophys.,124(3):415-444.
BOATWRIGHT J.1980. A spectral theory for circular seismic sources, simple estimates of source dimension, dynamic stress drop, and radiated seismic energy[J]. Bull. Seis. Soc. Am.,70(1):1-27.
BOWDEN F P, TABOR D.1964. The friction and lubrication of solids:Part Ⅱ[M]. Oxford:Clarendon Press.
BRACE W F, BYERLEE J D.1966. Stick-slip as a mechanism for earthquakes[J]. Science,153(3739):990-992.
BRUNE J N.1970. Tectonic stress and the spectra of seismic shear waves from earthquakes[J]. J. Geophys. Res., 75(26):4997-5009.
BURRIDGE R.1969. The numerical solution of certain integral equations with non-integrable kernels arising in the theory of crack propagation and elastic wave diffraction[J]. Philos. Trans. R. Soc. London, Ser. A 265:353-381.
BURRIDGE R.1973. Admissible speeds for plane-strain self-similar shear cracks with friction but lacking cohe-sion[J]. Geophys. J. R. astr. Soc.,35:439-455.
BURRIDGE R, KNOPOFF.1969. Model and theoretical seismicity[J]. Bull. Seis. Soc. Am.,57(3):341-371.
BURRIDGE R, WILLIS J R.1969. The self-similar problem of the expanding elliptical crack in an anisotropic solid[J]. Mathematical Proceedings of the Cambridge Philosophical Society,66:443-468.
BYERLEE J D.1978. Friction of rocks[J]. Pure Appl. Geophys.,116:615-626.
CAO T, AKI K.1986. Seismicity simulation with a rate- and state-dependent friction law[J]. Pure App. Geophys., 124(3):487-513.
CARLSON J M, LANGER J S.1989. Mechanical model of an earthquake fault[J]. Phy. Rev. A,40(11):6470-6484.
CARPENTER B M, MARONE C, SAFFER D M.2011. Weakness of the San Andreas Fault revealed by samples from the active fault zone[J]. Nat. Geosci.,4:251-254.
CHEN T, LAPUSTA N.2009. Scaling of small repeating earthquakes explained by interaction of seismic and aseismic slip in a rate and state fault model[J]. J. Geophys. Res.,114(B01311):doi:10.1029/2008JB005749.
CHEN X, ZHANG H.2006. Modelling rupture dynamics of a planar fault in 3-D half space by boundary integral equation method:an overview[J]. Pure appl. geophys.,163:267-299.
CHOY G L, BOATWRIGHT J L.1995. Global patterns of radiated seismic energy and apparent stress[J]. J. Geophys. Res.,100(89):18,205-18,228.
COCCO M, BIZZARRI A.2002. On the slip-weakening behavior of rate-and state dependent constitutive Iaws[J]. Geophys. Res. Lett.,29,111516, doi:10.1029/2001GL013999.
COCHARD A, MADARIAGA R.1994. Dynamic faulting under rate-dependent friction law[J]. Pure Appl. Geo-phys.,142:419-445.
COCHARD A, MADARIAGA R.1996. Complexity of seismicity due to highly rate-dependent friction[J]. J. Geophys. Res.,101(B11):25,321-25,336.
COCHARD A, RICE J R.1997. A spectral method for numerical elastodynamic fracture analysis without spatial replication of the rupture event[J]. J. Mech. Phys. Solids,45(8):1393-1418.
COTTON F, ARCHULETA R, CAUSSE M.2013. What is sigma of the stress drop[J]. Seis. Res. Lett.,84(1):42-48.
DAHLEN F A.1974. On the ratio of P-wave to S-wave corner frequencies for shallow earthquake sources[J]. Bull. Seis. Soc. Am.,64(4):1159-1180.
DAHLEN F A.1977. The balance of energy in earthquake faulting[J]. Geophys. J. R. astr. Soc.,48:239-261.
DAS S.1976. A numerical study of rupture propagation and earthquake source mechanism[D]. [S.l.]:[s.n.].
DAS S, AKI K.1977. A numerical study of two-dimensional spontaneous rupture propagation[J]. Geophys. J. R. astr. Soc.,50:643-668.
DAY S M.1982. Three-dimensional finite difference simulation of fault dynamics:rectangular faults with fixed rupture velocity[J]. Bull. Seis. Soc. Am.,72(3):705-727.
DAY S M, D ALGUER L A, LAPUSTA N, et al.2005. Comparison of finite difference and boundary integral solutions to three-dimensional spontaneous rupture[J]. J. Geophys. Res.,110, B12307, doi:10.1029/2005JB003813.
De Paola N, HIROSE T, MITCHELL T, et al.2011. Fault lubrication and earthquake propagation in thermally unstable rocks[J]. Geology,39(1):35-38.
Di Toro G, GOLDSBY D L, TULLIS T E.2004a. Friction falls towards zero in quartz rock as slip velocity approaches seismic rates[J]. Nature,427:436-439.
Di Toro G, HIROSE T,NIELSEN S, et al.2004b. Natural and experimental evidence of melt lubrication of faults during earthquakes[J]. Science,311:647-649.
Di Toro G, HAN R, HIROSE T, et al.2011. Fault lubrication during earthquakes[J]. Nature,471:494-498.
DIETERICH J H.1972. Time-dependent firction in rocks[J]. J. Geophys. Res.,77(20):3690-3697.
DIETERICH J H.1978. Time dependent firction and the mechanics of stick slip[J]. Pure Appl. Geophys., 116:790-806.
DIETERICH J H.1979a. Modelling of rock friction:1. Experimental results and constitutive equations[J]. J. Geophys. Res.,84(85):2161-2168.
DIETERICH J H.1979b. Modelling of rock friction:2. Simulation of preseismic slip[J]. J. Geophys. Res., 84(85):2169-2175.
DIETERICH J H.1992. Earhtquake nucleation on faults with rate- and state-dependent strength[J]. Tectonophysics, 211.
DIETERICH J H.1994. A constitutive law for rate of earthquake production and its application to earthquake clsutering[J]. J. Geophys. Res.,99(B2):2601-2618.
DIETERICH J H, RICHARDS-DINGER K B.2010. Earthquakes recurrence in simulated fault systems[J]. Pure Appl. Geophys.,167.
DUBLANCHET P, BERNARD P, FAVREAU P.2013. Interactions and triggering in a 3-D rate-and-state asperity model[J]. J. Geophys. Res.,118:1-21.
ELLSWORTH W L, BEROZA G C.1995. Seismic evidence for an earthquake nucleation phase[J]. Science, 268(5212):851-855.
ERICKSON B, BIRNIR B, LAVALLEE D.2008. A model for aperiodicity in earthquakes[J]. Nonlin. Processes Geophys.,15:1-12.
ERICKSON B, BIRNIR B, LAVALLEE D.2011. Periodicity, chaos and localization in a Burridge-Knopoff model of an earthquake with rate-and-state friction[J]. Geophys. J. Int.,187:178-198.
ERICKSON B A.2010. Complexity in the nonlinear Dieterich-Ruina law[D].[S.l.]:University of California, Santa Barbara.
ESHELBY J D.1957. The determination of the elastic field of an ellipsoidal inclusion, and related problems[J]. Proc. R. Soc. Lond. A,241:376-396.
ESHELBY J D.1969. The elastic field of a crack extending non-uniformly under general anti-plane Ioading[J]. J. Mech. Phys. Solids,17:177-199.
FOSSUM A F, FREUND L B.1975. Nonuniformly moving shear crack model of a shallow focus earthquake mechanism[J]. J. Geophys. Res.,80(23):3343-3347.
FREUND L B.1972a. Crack propagation in an elastic solid subjected to general loading-1. constant rate of exten-sion[J]. J. Mech. Phys. Solids,20:129-140.
FREUND L B.1972b. Crack propagation in an elastic solid subjected to general loading-II. non-uniform rate of extension[J]. J. Mech. Phys. Solids,20:141-152.
FREUND L B.1972c. Energy flux into the tip of an extending crack in an elastic solid[J]. J. Elasticity,2(4):341-349.
FREUND L B.1990. Dynamic fracture mechanics[M]. The Edinburgh Building, Shaftesbury Road, Cambridge CB2 2RU:Cambridge University Press.
FUKUYAMA E, MADARIAGA R.1998. Rupture dynamics of a planar fault in a 3-D elastic medium:rate- and slip-weakening friction[J]. Bull. Seis. Soc. Am.,88:1-17.
GABRIEL A A, AMPUERO J P, DALGUER L A, et al.2012. The transition of dynamic rupture styles in elastic media under velocity-weakening friction[J]. J. Geophys. Res.,117, B09311, doi:10.1029/2012JB009468.
GELLER R J.1997. Earthquake prediction:a critical review[J]. Geophys. J. Int.,131:425-450.
GEUBELLE P H, RICE J R.1995. A spectral method for three-dimensional elastodynamic fracture problems[J]. J. Mech. Phys. Solids,43(11):1791-1824.
GOLDSBY D L, TULLIS T.2011. Flash heating leads to low fricional strength of crustal rocks at earthquake slip rates[J]. Science,334:216-218.
GU J C, RICE J R, RUINA A L, et al.1984. Slip motion and stability of a single degree of freedom elastic system with rate and state dependent friction[J]. J. Mech. Phys. Solids,32(3):167-196.
GU Y J, WONG T F.1991. Effects of loading velocity, stiffness, and inertia on the dynamics of a single degree of freedom spring-slider system[J]. J. Geophys. Res.,96(B13):21,677-21,691.
HASSARD B D.1981. Theory and application of Hopf bifurcation[M]. New York, NY10022, USA:Cambridge Unversity Press.
HE C, WONG T F, BEELER N M.2003. Scaling of stress drop with recurrence interval and loading velocity for laboratory-derived fault strength relations[J]. J. Geophys. Res.,108(B1,2037):doi:10.1029/2002JB001890.
HEATON T H.1990. Evidence for and implications of self-healing pulses of slip in earthquake rupture[J]. Phys. Earth Plantet. Int.,64:1-20.
HILLERS G, BEN-ZION Y, MAI P M.2006. Seismicity on a fault controlled by rate- and state-dependent friction with spatial variations of the critical slip distance[J]. J. Geophys. Res., 111, B01403, doi:10.1029/2005JB003859.
HOROWITZ F G.1988. Mixed state variable friction laws:some implications for experiments and a stability analysis[J]. Geophys. Res. Lett.,15(11):1243-1246.
HOROWITZ F G.1989. Slip patterns in a spatially homogeneous fault model[J]. J. Geophys. Res., 94(B8):10,279-10,298.
HUANG J, TURCOTTE D L.1990. Evidence for chaotic fault interactions in the seismicity of the San Andreas fault and Nankai trough[J]. Nature,348:234-236.
IDA Y.1972. Cohesive force across the tip of a longitudinal-shear crack and Griffith's specific surface energyfJ]. J. Geophys. Res.,77(20):3796-3805.
IDE S.2002. Estimation of radiated energy of finite-source earthquake models[J]. Bull. Seis. Soc. Am., 92(8):2994-3005.
IDE S, BEROZA G C.2001. Does apparent stress vary with earthquake size[J]. Geophys. Res. Lett., 28(17):3349-3352.
IZUTANI Y.2005. Radiated energy from the mid Niigata, Japan, earthquake of October 23,2004, and its after-shocks[J]. Geophys. Res. Lett.,32:L21313, doi:10.1029/2005GL024113.
KANAMORI H, HEATON T H.2000. Microscopic and macroscopic physics of earthquakes[C]//. RUNDLE J, TURCOTTE D L, KLEIN W. Geocomplexity and the physics of earthquakes. Washington, D. C.:AGU, 120:147-161.
KANAMORI H, RIVERA L.2006. Energy partitioning during an earthquake[C]//. ABERCROMBIE R, MCGARR A, KANAMORI H, et al. Earthquakes:radiated energy and the physics of faulting. AGU, Washington, D. C., doi:10.1029/GM170:[s.n.]. Geophys. Monogr. Ser., vol.170.
KANAMORI H, MORI J, HAUKSSON E, et al.1993. Determination of earthquake energy release and ML using TERRAscope[J]. Bull. Seis. Soc. Am.,83(2):330-346.
KANEKO Y, LAPUSTA N, AMPUERO J P.2008. Spectral element modeling of spontaneous earthquake rup-ture on rate and state faults:effects of velocity-strengthening friction at shallow depths[J]. J. Geophys. Res., 113,B09317,doi:10.1029/2007JB005553.
KANEKO Y, FIALKO Y, SANDWELL D T, et al.2013. Interseismic deformation and creep along the central section of the North Anatalian Fault (Turkey):InSAR observations and implications for rate-and-state friction properties[J]. J. Geophys. Res.,118:316-331, doi:10.1029/2012JB009661.
KATO N.2003. Repeating slip events at a circular asperity:numerical simulation with a rate- and state-dependent friction law[J]. Bull. Earthq. Res. Inst. Univ. Tokyo,78:151-166.
KATO N.2004. Interaction of slip on asperities:numerical simulation of seismic cycles on a two-dimensional planar fault with nonuniform frictional property[J]. J. Geophys. Res.,109,B12306,doi:10.1029/2004JB003001.
KATO N.2007. How frictional properties lead to either rupture-front focusing or cracklike behavior[J]. Bull. Seis. Soc. Am.,97(6):2182-2189.
KATO N.2012. Dependence of earthquake stress drop on critical slip-weakening distance [J]. J. Geophys. Res., 117,B01301,doi:10.1029/2011JB008359.
KATO N, TULLIS T E.2001. A composite rate-and state-dependent law for rock friction[J]. Geophys. Res. Lett., 28(6):1103-1106.
KATO N, TULLIS T E.2003. Numerical simulation of seismic cycles with a composite rate- and state-dependent friction law[J]. Bull. Seis. Soc. Am.,93(2):841-853.
KATO N, LEI X, WEN X.2007. A synthetic seismicity model for the Xianshuihe fault, southwestern China: simulation using a rate- and state-dependent friction law[J]. Geophys. J. Int.,169:286-300.
KEILIS-BOROK V I.1959. On estimation of the displacement in an earthquake source and of source dimensions[J]. Ann. Geofis.,12:205-214.
KNOPOFF L.1958. Energy release in earthquakes[J]. Geophys. J. Roy. Astron. Soc., 1(1):44-52.
KOSLOFF D D, LIU H P.1980. Reformulation and discussion of mechanical behavior of the velocity-dependent friction law proposed by Dieterich[J]. Geophys. Res. Lett.,7(11):913-916.
KOSTROV B V.1964. Self-similar problems of propagation of shear cracks[J]. J.appl. Math. Mech.,28:1077-1087.
KOSTROV B V.1966. Unsteady propagation of longitudinal shear cracks[J]. J. appl. Math. Mech.,30:1042-1049.
KOSTROV B V.1974a. Crack propagation at variable velocity[J]. J. appl. Math. Mech.,38(3):551-560.
KOSTROV B V.1974b. Seismic moment and energy of earthquakes, and seismic flow of rock[J]. Izv. Earth Phys., 1:23-40.
LAPUSTA N, LIU Y.2009. Three-dimensional boundary integral modeling of spontaneous earthquake sequences and aseismic slip[J]. J. Geophys. Res.,114, B09303, doi:10.1029/2008JB005934.
LAPUSTA N, RICE J R.2003. Nucleation and early seismic propagation of small and large events in a crustal earthquake model[J]. J. Geophys. Res.,108, B4.2205, doi:10.1029/2001JB000793.
LAPUSTA N, RICE J R, BEN-ZION Y, et al.2000. Elastodynamic analysis for slow tectonic loading with spontaneous rupture episodes on fault with rate- and state-dependent friction[J]. J. Geophys. Res., 105(B10):23,765-23,789.
LINKER M F, DIETERICH J H.1992. Effects of variable normal stress on rock friction:observations and constitutive equations[J]. J. Geophys. Res.,97(B4):4923-4940.
LIU Y.2013. Numerical simulations on megathrust rupture stailized under strong dilatancy strenghening in slow slip region[J]. Geophys. Res. Lett.,40:1311-1316.
LIU Y, RICE J R.2005. Aseismic slip transients emerge spontaneously in three-dimensional rate and state modeling of subduction earthquake sequences[J]. J. Geophys. Res.,110,B08307,doi:10.1029/2004JB003424.
LIU Y, RICE J R.2007. Spontaneous and triggered aseismic deformation transients in a subduction fault model[J]. J. Geophys. Res.,112,B09404,doi:10.1029/2007JB004930.
LIU Y, MCGUIRE J J, BEHN M D.2012. Frictional behavior of oceanic transform faults and its influence on earthquake characteristics[J]. J. Geophys. Res.,117,B04315,doi:10.1029/2011JB009025.
LOCKNER D A, MORROW C, MOORE D, et al.2011. Low strength of deep San Andreas fault gouge from SAFOD core[J]. Nature,472:82-85.
MADARIAGA R.1976. Dynamics of an expanding circular fault[J]. Bull. Seis. Soc. Am.,66(3):639-666.
MADARIAGA R, COCHARD A.1996. Dynamic friction and the origin of the complexity of earthquake sources[J]. Proc. Natl. Acad. Sci.,93:3819-3824.
MADARIAGA R, OLSEN K B.2000. Criticality of rupture dynamics in 3-D[J]. Pure appl. geophys.,157:1981-2001.
MADARIAGA R, OLSEN K, ARCHULETA R.1998. Modeling dynamic rupture in a 3D earthquake fault model[J]. Bull. Seis. Soc. Am.,88(5):1182-1197.
MALAGNINI L, NIELSEN S, MAYEDA K, et al.2010. Energy radiation from intermediate- to large-magnitude earthquakes:implications for dynamic fault weakening[J]. J. Geophys. Res., 115, B06319, doi:10.1029/2009JB006786.
MARONE C.1998. Laboratory-derived friction laws and their application to seismic faulting[J]. Annu. Rev. Earth Planet. Sci.,26:643-695.
MAYEDA K, GOK R, WALTER W R, et al.2005. Evidence for non-constant energy/moment scaling from coda-derived source spectra[J]. Geophys. Res. Lett.,32, L10306, doi:10.1029/2005GL022405.
MCGARR A.1999. On relating apparent stress to the stress causing earthquake faults slip[J]. J. Geophys. Res., 104(B2):3003-3011.
MCGARR A, FLETCHER J B.2000. A method for mapping apparent stress and energy radiation applied to the 1994 Northridge earthquake fault zone[J]. Geophys. Res. Lett.,27(13):1953-1956.
MCGARR A, FLETCHER J B.2001. A method for mapping apparent stress and energy radiation applied to the 1994 Northridge earthquake fault zone-revisited[J]. Geophys. Res. Lett.,28(18):3529-3532.
MITSUI Y, HIRAHARA K.2009. Coseismic thermal pressurization can notably prolong earthquake recurrence intervals on weak rate and state friction faults:numerical experiments using different constitutive equations[J]. J. Geophys. Res.,114, B09304, doi:10.1029/2008JB006220.
MITSUI Y, HIRAHARA K.2011. Fault instability on a finite and planar fault related to early phase of nucleation[J]. J. Geophys. Res.,116, B06301, doi:10.1029/2010JB007974.
MORI J, ABERCROMBIE R E, KANAMORI H.2003. Stress drops and radiated energies of aftershocks of the 1994 Northridge, California, earthquake[J]. J. Geophys. Res.,108, B112545, doi:10.1029/2001JB000474.
NADEAU R M, JOHNSON L R.1998. Seismological studies at Parkfield Ⅵ:moment release rates and estimates of source paramters for small repeating earthquakes[J]. Bull. Seis. Soc. Am.,88(3):790-814.
NAGATA K, NAKATANI M, YOSHIDA S.2012. A revised rate- and state-dependent friction law obtained by constraining constitutive and evolution laws separately with laboratory data[J]. J. Geophys. Res.,117, B02314, doi:10.1029/2011JB008818.
NODA H, LAPUSTA N.2010. Three-dimensional earthquake sequences simulations with evolving temperature and pore pressure due to shear heating:effect of heterogeneous hydraulic diffusivity[J]. J. Geophys. Res.,115, B12314, doi:10.1029/2010JB007780.
NODA H, LAPUSTA N.2013. Stable creeping fault segments can become destructive as a result of dynamic weakening[J]. Nature,493:518-523.
NODA H, DUNHAM E M, RICE J R.2009. Earthquake ruptures with thermal weakening and the operation of major faults at low overall stress levels[J]. J. Geophys. Res.,114, B07302, doi:10.1029/2008JB006143.
OANCEA V G, LAURSEN T A.1997. Stability analysis of state dependent dynamic frictional sliding[J]. Int. J. Non-Linear Mechanics,32(5):837-853.
OHNAKA M.2003. A constitutive scaling law and a unified comprehension for frictional slip failure, shear fracture of intack rock, and earthquake rupture[J]. J. Geophys. Res.,108, B2,2080, doi:10.1029/2000JB000123.
OKADA Y.1985. Surface deformation due to shear and tensile faults in a half-space[J]. Bull. Seis. Soc. Am., 75(4):1135-1154.
OKADA Y.1992. Internal deformation due to shear and tensile faults in a half-space[J]. Bull. Seis. Soc. Am., 82(2):1018-1040.
OKUBRO P G.1989. Dynamic rupture modeling with laboratory-derived constitutive relations[J]. J. Geophys. Res., 94(B9):12,321-12,335.
PALMER A C, RICE J R.1973. The growth of slip surfaces in the progressive failure of over-consolidated clay[J]. Proc. Roy. Soc. Lond. A.,332:527-548.
PERRIN G, RICE J R, ZHENG G.1995. Self-healing slip pulse on a frictional surface[J]. J. Mech. Phys. Solids, 43(9):1461-1495.
PRESS W H, TEUKOLSKY S A, VETTERLING W T, et al.1992. Numerical Recipes in Fortran 77[M].2nd edition. New York:Cambridge University Press.
RANJITH K, RICE J R.1999. Stability of quasi-static slip in a single degree of freedom elastic system with rate and state dependent friction[J]. J. Mech. Phys. Solds,47:1207-1218.
REMPEL A W, RICE J R.2006. Thermal pressurization and onset of melting in fault zones[J]. J. Geophys. Res., 111, B09314, doi:10.1029/2006JB004314.
RICE J R.1980. The Mechanics of Earthquake Rupture[M]//. DZIEWONSKI A, BOSCHI E. Physics of the Earth's Interior. North Holland:Amsterdam.
RICE J R.1983. Constitutive relations for fault slip and earthquake instabilities[J]. Pure Appl. Geophys., 121(3):443-475.
RICE J R.1993. Spatio-temporal complexity of slip on a fault[J]. J. Geophys. Res.,98(B6):9885-9907.
RICE J R.2006. Heating and weakening of faults during earthquake slip[J]. J. Geophys. Res., 111(B05311):doi:10.1029/2005JB004006.
RICE J R.BEN-ZION Y.1996. Slip complexity in earthquake fault models[J]. Proc. Natl. Acad. Sci.,93:3811-3818.
RICE J R, GU J C.1983. Earthquake aftereffects and triggered seismic phenomena[J]. Pure Appl. Geophys., 121(2):187-219.
RICE J R, RUINA A L.1983. Stability of steady frictional slipping[J]. J. Appl. Mech.,50:343-349.
RICE J R, TSE S T.1986. Dynamic motion of a single degree of freedom system following a rate and state dependent friction law[J]. J. Geophys. Res.,91(B1):521-530.
RICE J R, LAPUSTA N, RANJITH K.2001. Rate and state dependent friction and the stability of sliding between elastically deformable solids[J]. J. Mech. Phys. Solids,49:1865-1898.
RICHARDS-DINGERK, DIETERICH JH.2012. RSQSim earthquake simulator[J]. Seis. Res. Lett.,83(6):983-990.
RIVERA L, KANAMORI H.2005. Representations of the radiated energy in earthquakes[J]. Geophys. J. Int., 162:148-155.
ROBINSON R.2004. Potential earthquake triggering in a complex fault network:the northern South Island, New Zealand[J]. Geophys. J. Int.,159.
ROBINSON R, BENITES R.1995. Synthetic seismicity models of multiple interacting faults[J]. J. Geophys. Res., 100(B9):18,229-18,238.
ROBINSON R, BENITES R.1996. Synthetic seismicity models for the Wellington Region, New Zealand:Implica-tions for the temporal distribution of large events[J]. J. Geophys. Res.,101(B12):27,833-27,844.
ROBINSON R, BENITES R.2001. Upgrading a synthetic seismicity models for more realistic fault ruptures[J]. Geophys. Res. Lett.,28(9):1843-1846.
ROBINSON R, NICOL A, J WALSH J, et al. 2009. Features of earthquake occurence in a complex normal fault networks:results from a synthetic seismicity model of the Taupo Rift, New Zealand[J]. J. Geophys. Res.,114, B12306, doi:10.1029/2008JB006231.
ROBINSON R, Van Dissen R, LITCHFIELD N.2011. Using synthetic seismicity to evaluate seismic hazard in the Wellington region, New Zealand[J]. Geophys. J. Int.,187.
ROJAS O, DUNHAM E M, DAY S M, et al.2009. Finite difference modelling of rupture propagation with strong velocity-weakening friction[J]. Geophys. J. Int.,179(3):1831-1858.
ROY M, MARONE C.1996. Earthquake nucleation on model faults with rate-and state-dependent friction:effects of inertia[J]. J. Geophys. Res.,101(B6):13,919-13,932.
RUBIN A M, AMPUERO J P.2005. Earthquake nucleation on (aging) rate and state faults[J]. J. Geophys. Res., 110, B11312, doi:10.1029/2005JB003686.
RUDNICKI J W, FREUND L B.1981. On energy radiation from seismic sources[J]. Bull. Seis. Soc. Am., 71(3):583-595.
RUINA A L.1980. Friction laws and instabilities:A quasistatic analysis of some dry frictional behavior[D]. Provi-dence, R. I.:Brown Univ.
RUINA A L.1983. Slip instability and state variable friction laws[J]. J. Geophys. Res.,88(B12):10,359-10,370.
RUNDLE P B, RUNDLE J B, TIAMPO K F, et al.2006. Virtual California:fault model, frictional parameters, applications[J]. Pure Appl. Geophys.,163:1819-1846.
SACHS M K, HEIEN E M, TURCOTTE D L, et al.2012. Virtual California Earthquake Simulator[J]. Seis. Res. Lett.,83(6):973-978.
SAVAGE J C, WOOD M D.1983. The relations between apparent stress and stress drop[J]. Bull. Seis. Soc. Am., 61(5):1381-1388.
SCHOLZ C H.1988. The critical slip distance for seismic faulting[J]. Nature,336:761-763.
SCHOLZ C H.1997. Whatever happened to earthquake prediction[J]. Geotimes,42(3):16-19.
SCHOLZ C H.1998. Earthquakes and friction laws[J]. Nature,391:37-42.
SCHOLZ C H.2002. The mechanics of earthquakes and faulting[M].2nd.[S.l.]:Cambridge University Press.
SCHOLZ C H, ENGELDER J T.1976. Role of asperity indentation and ploughing in rock friction[J]. Int. J. Rock Mech. Min. Sci. and Geomech.,13:149-154.
SCHOLZ C H, MOLNAR P, JOHNSON T.1972. Detailed studies of frictional sliding of granite and implicatins for the earthquake mechanism[J]. J. Geophys. Res.,77(32):6392-6406.
SCHOLZ C H, AVILES C A, WESNOUSKY S G.1986. Scaling differences between large interplate and intraplate earthquakes[J]. Bull. Seis. Soc. Am.,76(1):65-70.
SHAW B E.2009. Constant stress drop from small to great earthquakes in magnitude-area scaling[J]. Bull. Seis. Soc. Am.,99(2A):871-875.
SHAW B E, RICE J R.2000. Existence of continuum complexity in the elastodynamics of repeated fault ruptures[J]. J. Geophys. Res.,105(B10):23,791-23,810.
SHKOLLER S, MINSTER J B.1997. Reduction of Dieterich-Ruina attractors to unimodal maps[J]. Nonlin. Pro-cesses Geophys.,4:63-69.
STUART W D.1988. Forecast model for great earthquakes at the Nankai Trough subduction zone[J]. Pure Appl. Geophys.,126(2-4):619-641.
STUART W D, TULLIS T E.1995. Fault model for preseismic deformation at Parkfield, California[J]. J. Geophys. Res.,100(B12):24,079-24,099.
TAKAHASHIT, SATO H, OHTAKE M, et al.2005. Scale dependence of apparent stress for earthquakes along the subducting Pacific plate in northeastern Honshu, Japan[J]. Bull. Seis. Soc. Am.,95(4):1334-1345.
TINTI E, BIZZARRI A, PIATANESI A, et al.2004. Estimates of slip weakening distance for different dynamic rupture models[J]. Geophys. Res. Lett.,31, L02611, doi:10.1029/2003GL018811.
TSE S T, RICE J R.1986. Crustal earthquake instability in relation to the depth variation of frictional slip properties[J]. J. Geophys. Res.,91(B9):9452-9472.
TULLIS T E.1988. Rock friction constitutive behavior from laboratory experiments and its implications for an earthquake prediction field monitoring program[J]. Pure Appl. Geophys.,126(2-4):555-588.
VERAART A J, FAASSEN E J, DAKOS V, et al.2011. Recovery rates reflect distance to a tipping point in a living system[J]. Nature,481:357-359.
VIEGAS G, ABERCROMBIE R E, KIM W Y.2010. The 2002 M5 Au Sable Forks, NY, earthquake sequence: source scaling relashionships and energy budget[J]. J. Geophys. Res.,115, B07310, doi:10.1029/2009JB006799.
WALTER W R, MAYEDA K, GOK R.2006. The scaling of seismic energy with moment:simple models compared with observations[C]//. ABERCROMBIE R, MCGARR A, KANAMORIH, et al. Earthquakes:radiated energy and the physics of faulting. AGU, Washington, D. C., doi:10.1029/GM170:[s.n.]. Geophys. Monogr. Ser., vol. 170.
WANG J H.2002. A dynamic study of two one-state-varible, rate-dependent, and state-dependent friction laws[J]. Bull. Seis. Soc. Am.,2:687-694.
WANG J H.2008. One-dimensional dynamical modeling of earthquakes:a review[J]. Terr. Atmos. Ocean Sci., 19(3):183-203.
WANG J H.2012. Some intrinsic properties of the two-dimensional dynamical spring-slider model of earthquake faults[J]. Bull. Seis. Soc. Am.,102(2):822-835.
WARD S N.1991. A synthetic seismicity model for the Middle America trench[J]. J. Geophys. Res., 96(B13):21,433-21,442.
WARD S N.2000. San Francisco Bay area earthquake simulations:a step toward a standard physical earthquake model[J]. Bull. Seis. Soc. Am.,90(2):370-386.
WARD S N.2012. ALLCAL earthquake simulator[J]. Seis. Res. Lett.,83(6):964-972.
WIGGINS S.2003. Introduction to applied nonlinear dynamical systems and chaos[M].2nd ed.175 Fifth Avenue, New York, NY 10010, USA:Springer-Verlag New York, Inc.
WILLIS J R.1973. Self-similar problems in elastodynamics[J]. Phil. Trans. R. Soc. Lond. A,274:434-491.
YAMADA T, MORI J J, IDE S, et al.2007. Stress drops and radiated seismic energies of microearthquakes in a South African gold mine[J]. J. Geophys. Res.,112, B03305, doi:10.1029/2006JB004553.
YAMASHITA T.1976. On the dynamical process of fault motion in the presence of friction and inhomogeneous initial stress:Part Ⅰ. rupture propagation[J]. J. Phys. Earth,24:417-444.
YANG H, LIU Y, LIN J.2012. Effects of subducted seamounts on megathrust earthquake nucleation and rupture propagation[J]. Geophys. Res. Lett.,39, L24302, doi:10.1029/2012GL053892.
ZHANG H, CHEN X.2006a. Dynamic rupture on a planar fault in three-dimensional half space-Ⅰ. Theory[J]. Geophys. J. Int.,164:633-652.
ZHANG H, CHEN X.2006b. Dynamic rupture on a planar fault in three-dimensional half space-Ⅱ. Validations and numerical experiments[J]. Geophys. J. Int.,167:917-932.
ZHANG W, ZHANG Z, CHEN X.2012. Three-dimensional elastic wave numerical modelling in the presence of surface topography by a collocated-grid finite-difference method on curvilinear grids[J]. Geophys. J. Int., 190:358-378.
ZHANG Z, ZHANG W, LI H, et al.2013. Stable discontinuous grid implementation for collocated-grid finite-difference seismic wave modelling[J]. Geophys. J. Int.,192(3):1179-1188.
ZHENG G, RICE J R.1998. Conditions under which velocity-weakening friction allows a self-healing versus a cracklike mode of rupture[J]. Bull. Seis. Soc. Am.,88(6):1466-1483.
ZZIV A, COCHARD A.2006. Quasi-dynamic modeling of seismicity on a fault with depth-variable rate- and state-dependent friction[J]. J. Geophys. Res., 111, B08310, doi:10.1029/2005JB004189.
ZIV A, RUBIN A M.2003. Implications of rate-and-state friction for properties of aftershock sequences:quasi-static inherently discrete simulations[J]. J. Geophys. Res.,108, B1.2051, doi:10.1029/2001JB001219.
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