NPGNonlinear Processes in GeophysicsNPGNonlin. Processes Geophys.1607-7946Copernicus PublicationsGöttingen, Germany10.5194/npg-24-343-2017Generation and propagation of stick-slip waves over a fault with
rate-independent frictionKarachevtsevaIuliiajuliso22@gmail.comDyskinArcady V.https://orcid.org/0000-0001-5524-2566PasternakElenaSchool of Mechanical and Chemical Engineering, The University of
Western Australia, Crawley, AustraliaSchool of Civil and Resource Engineering, The University of Western
Australia, Crawley, AustraliaIuliia Karachevtseva (juliso22@gmail.com)11July201724334334921December201612January201712May201731May2017This work is licensed under the Creative Commons Attribution 3.0 Unported License. To view a copy of this licence, visit https://creativecommons.org/licenses/by/3.0/This article is available from https://npg.copernicus.org/articles/24/343/2017/npg-24-343-2017.htmlThe full text article is available as a PDF file from https://npg.copernicus.org/articles/24/343/2017/npg-24-343-2017.pdf
Stick-slip sliding is observed at various scales in fault
sliding and the accompanied seismic events. It is conventionally
assumed that the mechanism of stick-slip over geo-materials lies in the rate
dependence of friction. However, the movement resembling the
stick-slip could be associated with elastic oscillations of the rock around
the fault, which occurs irrespective of the rate properties of the
friction. In order to investigate this mechanism, two simple models
are considered in this paper: a mass-spring model of self-maintaining
oscillations and a one-dimensional (1-D) model of wave propagation through an
infinite elastic rod. The rod slides with friction over a stiff base. The
sliding is resisted by elastic shear springs. The results show that the
frictional sliding in the mass-spring model generates oscillations that
resemble the stick-slip motion. Furthermore, it was observed that the
stick-slip-like motion occurs even when the frictional coefficient is
constant. The 1-D wave propagation model predicts that despite the presence
of shear springs the frictional sliding waves move with the P wave velocity,
denoting the wave as intersonic. It was also observed that the amplitude of
sliding is decreased with time. This effect might provide an explanation to
the observed intersonic rupture propagation over faults.
Introduction
Earthquakes can lead to catastrophic structural failures and may trigger
tsunamis, landslides, and volcanic activities (Ghobarah et al., 2006; Bird
and Bommer, 2004). The earthquakes are generated at faults, and are either
produced by rapid (sometimes “supersonic”) propagation of shear
cracks/ruptures along the faults, or originated in the stick-slip sliding
over the fault. The velocity of rupture propagation is crucial for
estimating the earthquake damage. The rupture velocities can be classified
by comparing its speed with the speeds of stress waves in the rupturing
solid (Rosakis, 2002). There are several types of rupture propagation:
supersonic (V>VP), intersonic (VS<V<VP),
subsonic (V<VS), supershear (V>VS),
sub-shear (VR<V<VS), and sub-Rayleigh (V<VR). According to the data obtained from the seismic observation of
crustal earthquakes, most ruptures propagate with an average velocity that
is about 80 % of the shear wave velocity (Heaton, 1990). However, in some
cases, supershear propagation of earthquake-generating shear ruptures or
sliding is observed (Archuleta, 1984; Bouchon et al., 2000, 2001, 2010;
Dunham and Archuleta, 2004; Aagaard and Heaton, 2004). The above
observations introduced the concept of supershear crack propagation (e.g.
Bizzarri and Spudich, 2008; Lu at al., 2009; Bhat et al., 2007; Dunham,
2007; Vallee et al., 2008). However, due to the lack of strong motion recording, there are still
some debates regarding the data interpretation (Delouis et al., 2002; Bhat
et al., 2007). For instance, it was suggested that the 2002 Denali
earthquake was propagated at a supershear speed of about 40 km (Dunham and
Archuleta, 2004). However, the data were based on a single ground motion
record. The joint inversion of the combined data set provides a more robust
description of the rupture. The recent studies, which are aimed at deriving
the kinematic models for large earthquakes, have shown the importance of the
type of data used. It has been shown that slip maps for a given earthquake
may vary significantly (Cotton and Campillo, 1995; Cohee and Beroza, 1994).
The analytical (e.g. Burridge, 1973) and numerical (e.g. Das and Aki,
1977) research in fracture dynamics indicate that only the Mode II rupture
(shear-induced slip occurring in the direction perpendicular to the crack
front) can propagate with intersonic velocity (VS<V<VP) for short durations, as long as the prestress of the fault is high
compared to both failure and residual stresses (Dunham, 2007). Intersonic
Mode II crack propagation was first confirmed in laboratory by Rosakis et
al. (1999).
Sliding over pre-existing fractures and interfaces is one of the forms of
instability in geo-materials. It is often accompanied by stick-slip – a
spontaneous jerking motion between two contacting bodies sliding over each
over. It is assumed that the mechanism of stick-slip lies in intermittent
change between static and kinetic friction and the rate dependence of the
frictional coefficient (Popp and Rudolph, 2004).
The investigation of the friction law on geological faults is the key
element in the modelling of earthquakes. Rate- and state-dependent friction
laws proposed by Dieterich, Ruina, and Rice (Dieterich, 1978; Ruina, 1983;
Rice, 1983) have successfully modelled frictional sliding and earthquake
phenomena. There are two types of frictional sliding between surfaces that
include the tectonic plates. The first type occurs when two surfaces slip
steadily (V=V0 condition, where V is relative velocity and V0 is
the load point velocity) and is analogous to the fault creep (Byerlee and
Summers, 1975). In the stable state, the sliding over discontinuities
(faults and fractures) is prevented by friction. Modelling of the frictional
sliding is an important tool for understanding the initiation and the
development of rupture, and also, the healing of the faults. Many models and
numerical methods are developed to describe seismic activities and the
supershear fracture/rupture propagation (Noda and Lapusta, 2013; Lapusta and
Rice, 2003; Lu at al., 2009; Lapusta et al., 2000; Sobolev, 2011; Bak and
Tang, 1989; Harris and Day, 1993).
The faults are continuously subjected to variations in both shear and normal
stresses, and can produce sliding over initially stable fractures or
interfaces (Boettcher and Marone, 2004). In the Earth's crust, the increase
in shear stress is an obvious consequence of tectonic movement, while
oscillations in the normal stress can be associated with the tidal stresses
or seismic waves generated by other seismic events. These can generate the
second dynamic state when the sliding occurs jerkily (slip, stick, and then
slip again). This type of sliding is called “stick-slip” sliding, which
exhibit cyclic behaviour. Brace and Byerlee (1966) supposed that the stick-slip
instabilities in the tectonic plates are associated with the appearance of
earthquakes. Both types of sliding are usually
investigated using a spring-block model introduced by Burridge and Knopoff
(BK)
in 1967 (Turcotte, 1992). The BK model consists of an assembly of blocks,
where each block is connected via the elastic springs to the next block and
to the moving plate.
In the present paper, we first simulate a single element block model,
which is one block undergoing frictional sliding on a stiff base. The
movement is caused by a spring attached to the block. The other end of the
spring moves with a constant velocity. The paper begins with considering
stick-slip-like movement occurring under rate-independent friction due to
the eigenoscillations of the fault faces and the associated wave
propagation. This demonstrates that the rate dependence of friction is not
necessarily a controlling phenomenon. We also analyse a simple mechanism of
unusually high shear fracture or sliding zone propagation, also referred as
the P sonic propagation of sliding area over a frictional fault. The
analysis is based on the fact that accumulation of elastic energy in the
sliding plates on both sides of the fault can produce oscillations in the
velocity of sliding even if the frictional coefficient is constant. We note
that Walker and Shearer (2009) found evidence of the intersonic rupture
speeds close to the local P wave velocity by analysing the Kokoxili and
Denali earthquakes seismic data. This paper considers a highly simplified
one-dimensional (1-D) rod model where many properties of a real fault system have been
neglected. (Considerable fault geometry simplification is in use in
analysing intersonic ruptures; see, e.g., Bouchon et al., 2010.)
Single degree of freedom frictional oscillator
We start with the self-excited oscillations, which resembles the
stick-slip-like motion, but occur under constant friction. A single
degree-of-freedom block-spring model is used for this purpose. A block
sliding on a rigid horizontal surface is driven by a spring, whose other end
is attached to a driver moving with a constant velocity (Fig. 1). All
variables and constants used in the equations are listed in Table 1.
The list of variables and constants.
SymbolMeaningSymbolMeaningV0load point velocityτshear stressVrelative velocity of blockτffriction stressk1single spring stiffnessEYoung's modulusmblock masscvelocity of longitudinal wave (p= wave)Ngravity forceωeigenfrequencyTshear forcek2the spring stiffness relating stress and displacement discontinuity (the difference between the rod displacement and the zero displacement of the base)μfriction coefficientJ0Bessel function of the order of 0ω0eigenfrequencyJ0′derivative of Bessel functionttimeiimaginary uniththickness of an infinite rodξindependent variableρvolumetric rod densityzintegration variableσNuniform compressive loadf,garbitrary functionsσlongitudinal stress
Friction is assumed to be cohesionless: Tcr=μN, where Tcr is
the force at which sliding starts.
The single block model.
The system of equations representing the motion of the block reads
mV˙=f(T,μN)T˙=k1(V0-V).
The appearance of the f(T,μN) function in the system of equations
represents the fact that V≥0.
The function f(T,μN) is defined as
f(T,μN)=T-μN,T>μN and V>00,T<μN or V<0
In order to represent the system of Eq. (1) in dimensionless form, it
is convenient to introduce a dimensionless time t∗:
t∗=tω0,ω02=k1m,
where ω0 is the eigenfrequency of the block-spring system, m is
the block mass and k1 is the spring stiffness.
The governing system of equations in dimensionless form is defined as
V˙=f(T∗,μN∗)T˙=1-V∗,
where the dot represents the derivative with respect to dimensionless time
t∗, and V*,T*, and N* are the dimensionless velocity, shear force, and
gravity force respectively.
V∗=VV0,T∗=TmV0ω0,N∗=NmV0ω0
Behaviour of the system
In order to demonstrate the behaviour of the system at stick-slip-type
regime, we consider the block sliding under the following set of initial
conditions:
V(0)=0,T˙(0)=0.
Figure 2 represents the corresponding behaviour of the system (dimensionless
velocity vs. dimensionless time).
Block sliding with constant friction
coefficient.
It is observed that the system exhibits self-excited oscillations even with
constant friction coefficient, which somewhat resemble the stick-slip-type
sliding. Furthermore, the energy in the system does not change with time,
obviously due to the constant energy influx by velocity V0, where the
excess of the V0 is dissipated by friction.
A detailed investigation of the behaviour of a system described in a Sect. 2
was undertaken in our previous work (Karachevtseva et al., 2014, 2015). It should also be noted that similar
oscillation-type movements were observed in laboratory experiments with the
sliding of two granite blocks under biaxial compression (Sobolev et al.,
2016).
Stress wave propagation in frictional sliding (generalization 1-D
solid)
In the previous section, we showed the stick-slip-like motion occurring even
when the friction coefficient is constant. In this section we will expand
our understanding to incorporate the slide over a fault where a stick-slip
phenomenon is traditionally flagged as a mechanism of earthquakes. We shall
keep assuming the constant friction law, which will permit us to obtain an
analytical solution. For this purpose, following Nikitin (1998), we consider
the simplest possible 1-D model of fault sliding, which takes into account
the rock elastic response and the associated dynamic behaviour. The model is
shown in Fig. 3. It consists of an infinite elastic rod of height
(thickness) h, and of unit length in the direction normal to the plane of
drawing in Fig. 3. The linear density is ρ and the rod is assumed to
be able slide over a stiff surface. The sliding is resisted by friction. The
stiff surface can be described as a symmetry line such that instead of the
(horizontal) fault, only the upper half of the line is considered. The rod
is connected to a stiff layer moving with a constant velocity V0. The
connection is achieved through a series of elastic shear springs. Both the
elastic rod and the elastic springs describe the model of the elasticity of
the rock around the fault, as shown in Fig. 3. We assume that the system
is subjected to a uniform compressive load σN such that the
friction stress is kept constant, which is assumed equal to τf=μσN=const.
The model of infinitive elastic rod driven by elastic
shear spring.
Equation of movement of the rod reads
∂σ∂x+1h(τ-τf)=ρ∂V∂t,
where σ is the longitudinal (normal) stress in the rod, τ is
the contact shear stress, τf is the frictional stress, V0
is the load point velocity, and V(x,t) is the velocity of point x of the rod at
time t, as shown in Fig. 3.
According to the Hooke's law:
σ=E∂u∂x,
where u(x,t) is the displacement and E is the Young's modulus of the rod. After
differentiating, we have
∂σ∂t=E∂V∂x.
The elastic reaction of the shear springs is expressed as
∂τ∂t=k2(V-V0),
where k2 is the spring stiffness relating stress and displacement
discontinuity (the difference between the rod displacement and the zero
displacement of the base).
Defining ΔV=V-V0 and solving the system of Eqs. (7)–(10),
we get the following wave equation:
∂2ΔV∂t2=c2∂2ΔV∂x2-ω2ΔV,
where c=EhEhρρ is the velocity of the longitudinal wave
(P wave) and ω=k2k2(hρ)(hρ) is regarded as
eigenfrequency of the system consisting as a unit length of the rod
considered as a lamp mass on the shear springs.
Propagation of initial sliding in the form of a triangular
function f(z) of zero
area.
Propagation of initial sliding with different initial
conditions.
It is observed that despite the presence of shear springs and friction
between the rod and the stiff surface, the waves propagate with the P wave
velocity determined by the Young's modulus and density of the rod.
Therefore, according to the terminology described in the Introduction, the
wave should be named p-sonic wave. It should be highlighted that while such waves look
like the shear waves, they are in fact compressive waves propagation along
the rod, hence denoted as the P wave velocity.
In order to analyse the way the pulse propagates, Eq. (11) is
complemented by the initial conditions as
ΔV(x,t)=f0(x);dΔVdt=F0(x).
The solution of the wave in Eq. (11) can be found by using the Riemann method
(e.g. Koshlyakov, 1964).
ΔV(x,t)=12[f(x-ct)+g(x+ct)]+12∫x-ctx+ctΦ(x,t,z)dz,
where
Φ(x,t,z)=1c2t2-(z-x)2φ(x,t,z).
The integral from Eq. (13) can be found by using the Chebyshev–Gauss method
I(x,t)=∫x-ctx+ctΦ(x,t,z)dz≈πn∑j=1nφ(x,t,x+ζjat),ξj=cos2j-12nπ,
where
φ(x,t,z)=1cF(z)J0ωcic2t2-(z-x)2c2t2-(z-x)2+ωtf(z)1iJ0′iωcc2t2-(z-x)2.
Propagation of an initial sliding
Figures 4–5 represent the propagation of initial sliding under the different
initial conditions. Particularly, a triangular velocity impulse, Eq. (17)
and zero acceleration were used as initial conditions for Fig. 4. As
shown in Fig. 5, linear and harmonic functions are used for velocity and
acceleration as initial conditions.
f(x;a,b,c)=maxminx-ab-a,c-xc-b,0,
where x
is the vector, a, b, and c are scalar parameters.
It is seen that the initial sliding (impulse) propagating with P wave
velocity keeps its width but the amplitude reduces with time. It is also
observed that as the impulse propagates, it loses energy that goes to
increase the energy of shear springs.
Discussion
This paper introduced the notion that the frictional movement resembling the
stick-slip sliding, which are often observed and usually attributed to the
rate dependence of friction, can be obtained with constant friction by
taking into account the elasticity of the surrounding and its
self-oscillations. This understanding is applied to propagation of slip over
infinitely long fault leads to a simple model that predicts that the slip
will propagate with P wave velocity. This conclusion is made under the
assumption of constant (rate-independent) friction. Relaxing this
assumption, which is taking into account that
τf=τf(∂ΔV∂t), leads to the
following equation replacing Eq. (11):
1+1ρhdτfdΔVt′∂2ΔV∂t2=c2∂2ΔV∂x2-ω2ΔV,ΔV′=∂ΔV∂t.
It is seen that when the sliding rate changes slowly, the propagation speed
of rupture c1 can be approximated as
c12≈c21+1ρhdτfdΔVt′-1.
Furthermore, it is observed that when the friction increases with the
sliding rate, c1 becomes smaller than P wave velocity. If the rate
dependence of friction is lowered further, the slip propagation can become
intersonic.
Conclusions
In this paper, it is shown that the accumulation of elastic energy in the
sliding plates on both sides of the fault can produce oscillations in the
velocity of sliding even when the friction is constant. These oscillations
resemble stick-slip movements, but they manifest themselves in terms of
sliding velocity rather than displacement. The sliding exhibits wave-like
propagation over long faults. Furthermore, the 1-D model shows that the zones
of sliding propagate along the fault with the velocity of P wave (the
propagation speed can however be lower if the rate dependence of friction is
taken into account). The mechanism of such fast wave propagation is the
normal (tensile/compressive) stresses in the neighbouring elements (normal
stresses on the planes normal to the fault surface) causing a P wave to
propagate along the fault rather than the shear stress controlling the
sliding. This manifests itself as a P sonic propagation of an apparent shear
rupture.
No data sets were used in this article.
The authors declare that they have no conflict of interest.
This article is part of the special issue “Waves in media with pre-existing or emerging inhomogeneities and dissipation”. It is a result of the EGU General Assembly 2014,
Vienna, Austria, 27 April–2 May 2014.
Edited by: Sergey Turuntaev
Reviewed by: two anonymous referees
ReferencesAagaard, B. T. and Heaton T. H.: Near-source ground motions from
simulations of sustained intersonic and supersonic fault ruptures, B.
Seismol. Soc. Am., 94, 2064–2078, 10.1785/0120030249, 2004.Archuleta, R. I.: A faulting model for the 1979 Imperial Valley earthquake,
J. Geophys. Res., 89, 4559–4585, 10.1029/JB089iB06p04559, 1984.Bak, P. and Tang, S.: Earthquakes as self-organized critical phenomenon, J.
Geophys. Res.-Sol. Ea., 94, 15635–15637 10.1029/JB094iB11p15635,
1989.Bhat, H. S., Dmowska, R., King, G. C. P., and Klinger, Y.: Off-fault damage
patterns due to supershear ruptures with application to the 2001 Mw 8.1
Kokoxili (Kunlun) Tibet earthquake, J. Geophys. Res., 112, B06301, 10.1029/2006JB004425, 2007.Bird, J. F. and Bommer, J. J.: Earthquake losses due to ground failure,
Eng. Geol., 75, 147–179, 10.1016/j.enggeo.2004.05.006, 2004.Bizzarri, A. and Spudich, P.: Effects of supershear rupture speed on the
high-frequency content of Swaves investigated using spontaneous dynamic
rupture models and isochrone theory, J. Geophys. Res., 113, B05304, 10.1029/2007JB005146, 2008.Boettcher, M. S. and Marone, C.: Effects of normal stress variation on the
strength and stability of creeping faults, J. Geophys. Res., 109, B03406,
10.1029/2003JB002824, 2004.Bouchon, M., Toksoz, N., Karabulut, H., Bouin, M. P., Dieterich, M., Aktar,
M., and Edie, M.: Seismic imaging of the 1999 Izmit (Turkey) rupture
inferred from the near-fault recordings, Geophys. Res. Lett., 27, 3013–3016,
10.1029/2000GL011761, 2000.Bouchon, M., Bouin, M. P., Karabulut, H., Toksoz, M. N., Dietrich, M., and
Rosakis, A. J.: How fast is rupture during an earthquake? New insights from
the 1999 Turkey earthquakes, Geophys. Res. Lett., 28, 2723–2726, 10.1029/2001GL013112, 2001.Bouchon, M., Karabulut, H., Bouin, M. P., Schmittbuhl, J., Vallee, M.,
Archuleta, R., Das, S., Renard, F., and Marsan, D.: Faulting characteristics of
supershear earthquakes, Tectonophysics, 493, 244–253, 10.1016/j.tecto.2010.06.011,
2010.Brace, W. F. and Byerlee, J. D.: Stick-slip as a mechanism for earthquakes,
Science, 153, 990–992, 10.1126/science.153.3739.990, 1966.Burridge, R.: Admissible speeds for plane-strain self-similar shear cracks
with friction but lacking cohesion, Geophys. J. Roy. Astr. S., 35,
439–455, 10.1111/j.1365-246X.1973.tb00608.x, 1973.Byerlee, J. D. and Summers, R.: Stable sliding preceding stick-slip on
fault surfaces in granite at high pressure, Pure Appl. Geophys., 113, 63–68,
10.1007/BF01592899, 1975.Cohee, B. P. and Beroza, G. C.: Slip distribution of the 1992 Landers
earthquake and its implications for earthquake source mechanism, B.
Seismol. Soc. Am., 84, 692–712, 10.1016/0148-9062(95)94486-9,
1994.Cotton, F. and Campillo, M.: Frequency domain inversion of strong motions:
application to the 1992 Landers earthquake, J. Geophys. Res., 100, 3961–3975, 10.1029/94JB02121,
1995.Das, S. and Aki, K.: A numerical study of two-dimensional spontaneous rupture
propagation, Geophys. J. Roy. Astr. S., 50, 643–668, 10.1111/j.1365-246X.1977.tb01339.x, 1977.Delouis, B., Giardini, D., Lundgren, P., and Salichon, J.: Joint Inversion
of InSAR, GPS, Teleseismic, and Strong-Motion Data for the Spatial and
Temporal Distribution of Earthquake Slip: Application to the 1999 Izmit
Mainshock, B. Seismol. Soc. Am., 92, 278–299, 10.1785/0120000806,
2002.Dieterich, J. H.: Time-dependent friction and the mechanics of stick-slip, J.
Geophys. Res., 77, 790–806, 10.1007/BF00876539, 1978.Dunham, E. M.: Conditions governing the occurrence of supershear ruptures
under slip-weakening friction, J. Geophys. Res., 112, B07302, 10.1029/2006JB004717, 2007.Dunham, E. M. and Archuleta, J. R.: Evidence for a Supershear Transient during
the 2002 Denali Fault Earthquake, B. Seismol. Soc. Am., 94,
S256–S268, 10.1785/0120040616, 2004.Ghobarah, A., Saatcioglu, M., and Nistor, I.: The impact of the 26 December 2004
earthquake and tsunami on structures and infrastructure, Eng. Struct., 28,
312–326, 10.1016/j.engstruct.2005.09.028, 2006.Harris, R. A. and Day, S. M.: Dynamics of fault interaction: parallel
strike-slip faults, J. Geophys. Res.-Sol. Ea., 98, 4461–4472, 10.1029/92JB02272, 1993.Heaton, T. H.: Evidence for and implications of self-healing pulses of slip
in earthquake rupture, Phys. Earth Planet. In., 64, 10–20, 10.1016/0031-9201(90)90002-F, 1990.Karachevtseva, I., Dyskin, A. V., and Pasternak, E.: The cyclic loading as a
result of the stick-slip motion, Adv. Mat. Res., 891–892, 878–883,
10.4028/www.scientific.net/AMR.891-892.878, 2014.Karachevtseva, I., Dyskin, A. V., and Pasternak, E.: Stick-slip motion and the
associated frictional instability caused by vertical oscillations,
Bifurcation and Degradation of Geomaterials in the New Millennium, Springer
Series in Geomechanics and Geoengineering, 135–141, 10.1007/978-3-319-13506-9_20,
2015.Koshlyakov, N. S., Smirnov, M. M., and Gliner, E. B.: Differential equations
of mathematical physics, Moscow, 701, 1964 (in Russian).Lapusta, N. and Rice, J. R.: Nucleation and early seismic propagation of
small and large events in a crustal earthquake model, J. Geophys.
Res., 108, 2205, 10.1029/2001JB000793, 2003.Lapusta, N., Rice, J. R., Ben-Zion, Y., and Zheng, G.: Elastodynamic
analysis for slow tectonic loading with spontaneous rupture episodes on
faults with rate- and state-dependent friction, J. Geophys. Res.- Sol. Ea., 105, 23765–23789,
10.1029/2000JB900250, 2000.Lu, X., Lapusta, N., and Rosakis, A. J.: Analysis of supershear transition
regimes in rupture experiments: the effect of nucleation conditions and
friction parameters, Geophys. J. Int., 177, 717–732, 10.1111/j.1365-246X.2009.04091.x, 2009.Nikitin, L. V.: Statics and dynamics of solids with an external dry
friction, Moscow Lyceum,Moscow, 272, 1998 (in Russian).Noda, H. and Lapusta, N.: Stable creeping fault segments can become
destructive as a result of dynamic weakening, Nature, 518–523, 10.1038/nature11703,
2013.Popp, K. and Rudolph, M.: Vibration Control to Avoid Stick-Slip Motion, J.
Vib. Control, 10, 1585–1600, 10.1177/1077546304042026, 2004.Rice, J. R.: Constitutive relations for fault slip and earthquake
instabilities, Pure Appl. Geophys., 121, 443–475, 10.1007/BF02590151, 1983.
Rosakis, A. J.: Intersonic shear cracks and fault ruptures, Adv. Phys.,
51, 1189–1257, 10.1080/00018730210122328, 2002.Rosakis, A. J., Samudrala, O., and Coker, D.: Cracks faster than the shear
wave speed, Science, 284, 1337–1340, 10.1126/science.284.5418.1337,
1999.Ruina, A.: Slip instability and state variable friction laws, J. Geophys. Res., 88, 10359–10370, 10.1029/JB088iB12p10359, 1983.Sobolev, G. A.: Seismicity dynamics and earthquake predictability, Nat. Hazards Earth Syst. Sci., 11, 445–458, 10.5194/nhess-11-445-2011, 2011.Sobolev, G. A., Ponomarev A. V., and Maibuk, Yu. Ya.: Initiation of unstable
slips-microearthquakes by elastic impulses, Izvestiya, Phys. Sol.
Earth, 52, 674–691, 10.1134/S106935131605013X, 2016.Turcotte, D. L.: Fractals and chaos in geology and geophysics, Cambridge
University Press, 221, 10.1002/gj.3350280216, 1992.Vallee, M. M., Landes, M., Shapiro, N. M., and Klinger, Y.: The 14 November 2001
Kokoxili (Tibet) earthquake: High-frequency seismic radiation originating
from the transitions between sub-Rayleigh and supershear rupture velocity
regimes, J. Geophys. Res., 113, B07305, 10.1029/2007JB005520,
2008.Walker, K. T. and Shearer, P. M.: Illuminating the near-sonic rupture
velocities of the intracontinental Kokoxili Mw 7.8 and Denali fault Mw 7.9
strike-slip earthquakes with global P wave back projection imaging, J.
Geophys. Res., 114, B02304, 10.1029/2008JB005738, 2009.