NPGNonlinear Processes in GeophysicsNPGNonlin. Processes Geophys.1607-7946Copernicus PublicationsGöttingen, Germany10.5194/npg-24-467-2017Multistable slip of a one-degree-of-freedom spring-slider model in the
presence of thermal-pressurized slip-weakening friction and viscosityWangJeen-Hwajhwang@earth.sinica.edu.twInstitute of Earth Sciences, Academia Sinica,
P.O. Box 1–55, Nangang, Taipei, TaiwanJeen-Hwa Wang (jhwang@earth.sinica.edu.tw)11August201724346748025March201725April201723June201713July2017This 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/24/467/2017/npg-24-467-2017.htmlThe full text article is available as a PDF file from https://npg.copernicus.org/articles/24/467/2017/npg-24-467-2017.pdf
This study is focused on multistable slip of earthquakes based on a
one-degree-of-freedom spring-slider model in
the presence of thermal-pressurized slip-weakening friction and viscosity by
using the normalized equation of motion of the model. The major model
parameters are the normalized characteristic displacement, Uc, of
the friction law and the normalized viscosity coefficient, η, between
the slider and background plate. Analytic results at small slip suggest that
there is a solution regime for η and γ (=1/Uc) to
make the slider slip steadily. Numerical simulations exhibit that the time
variation in normalized velocity, V/Vmax (Vmax is the
maximum velocity), obviously depends on Uc and η. The
effect on the amplitude is stronger due to η than due to
Uc. In the phase portrait of V/Vmax versus the
normalized displacement, U/Umax (Umax is the maximum
displacement), there are two fixed points. The one at large
V/Vmax and large U/Umax is not an attractor, while
that at small V/Vmax and small U/Umax can be an
attractor for some values of η and Uc. When
Uc<0.55, unstable slip does not exist. When
Uc≥0.55, Uc and η divide the solution domain
into three regimes: stable, intermittent, and unstable (or chaotic) regimes.
For a certain Uc, the three regimes are controlled by a lower
bound, ηl, and an upper bound, ηu, of η.
The values of ηl, ηu, and
ηu-ηl all decrease with increasing
Uc, thus suggesting that it is easier to yield unstable slip for
larger Uc than for smaller Uc or for larger η
than for smaller η. When Uc<1, the Fourier spectra
calculated from simulation velocity waveforms exhibit several peaks, thus
suggesting the existence of nonlinear behavior of the system. When
Uc>1, the related Fourier spectra show only one peak,
thus suggesting linear behavior of the system.
Introduction
The earthquake ruptures consist of three steps: nucleation, dynamical
propagation, and arrest. Due to the lack of a comprehensive model, a set of
equations to completely describe fault dynamics has not yet been established,
because earthquake ruptures are very complicated. Nevertheless, some models,
for instance the crack model and dynamical lattice model, have been developed
to approach fault dynamics. Several factors will control earthquake ruptures
(see Wang, 2016b, and cited references herein), including at least
brittle-ductile fracture rheology, normal stress, re-distribution of stresses
after fracture, fault geometry, friction, seismic coupling, pore fluid
pressure, elastohydromechanic lubrication, thermal effect, thermal
pressurization, and metamorphic dehydration. A general review can be seen in
Bizzarri (2009). Among the factors, friction and viscosity are two important
ones in controlling faulting.
Burridge and Knopoff (1967) proposed a one-dimensional spring-slider model
(abbreviated as the 1-D BK model henceforth) to approach fault dynamics. Wang
(2000, 2012) extended this model to a two-dimensional version. The two models
and their modified versions have been long and widely applied to simulate the
occurrences of earthquakes (see Wang, 2008, 2012, and cited references
therein). In the following, the one-, two-, three-, few-, and many-body
models are used to represent the one-, two-, three-, few-, and
many-degree-of-freedom spring-slider models, respectively. The few-body
models have been long and widely used to approach faults (Turcotte, 1992).
Since the commonly used friction laws are nonlinear, the dynamical model
itself could behave nonlinearly. A nonlinear dynamical system can exhibit
chaotic behavior under some conditions
(Thompson and Stewart, 1986; Turcotte, 1992). This means that the system is
highly sensitive to initial conditions (SIC) and thus a small difference in
initial conditions, including those caused by rounding errors in numerical
computation, yields widely diverging outcomes. This indicates that long-term
prediction is impossible in general, even though the system is deterministic,
meaning that its future behavior is fully determined by their initial
conditions, without random elements. This behavior is known as
(deterministic) chaos (Lorenz, 1963).
An interesting question is the following. Can a simple few-body model with
total symmetry make significant predictions for fault behavior? Gu et
al. (1984) first found some chaotically bounded oscillations based on a
one-body model with rate- and state-dependent friction. Perez Pascual and
Lomnitz-Adler (1988) studied the chaotic motions of coupled relaxation
oscillators. Related studies have been made based on different spring-slider
models: (1) a one-body model with rate- and state-dependent friction (e.g.,
Gu et al., 1984; Belardinelli and Belardinelli, 1996; Ryabov and Ito, 2001;
Erickson et al., 2008, 2011; Kostić et al., 2013); (2) a one-body model
with velocity-weakening friction (e.g., Brun and Gomez, 1994); (3) a one-body
model with slip-weakening friction (e.g., Wang, 2016a, b); (4) a two-slider
model with simple static/dynamic friction (e.g., Nussbaum and Ruina, 1987;
Huang and Turcotte, 1990); (5) a two-body model with velocity-dependent
friction (e.g., Huang and Turcotte, 1992; de Sousa Vieira, 1999; Galvanetto,
2002); (6) a two-body model with rate- and state-dependent friction (e.g.,
Abe and Kato, 2013); (7) a two-body model with velocity-weakening friction
(Brun and Gomez, 1994); (8) a two-body model with slip-weakening friction
(e.g., Wang, 2017); (9) a many-body model with velocity-weakening friction
(e.g., Carlson and Langer, 1989; Wang, 1995, 1996); and (10) a one-body
quasi-static model with rate- and state-dependent friction (e.g., Shkoller
and Minster, 1997). Results suggest that predictions for fault behavior are
questionable due to the possible presence of chaotic slip.
The frictional effect on earthquake ruptures has been widely studied as
mentioned above. However, studies of the viscous effect on earthquake
ruptures are rare. The viscous effect mentioned in Rice et al. (2001) was
just an implicit factor which is included in the evolution effect of friction
law. In this work, I will investigate the effects of thermal pressurized
slip-weakening friction and viscosity on earthquake ruptures and the
generation of unstable (or chaotic) slip based on a one-body model.
ModelOne-body model
Figure 1 shows the one-body model whose equation of motion is
md2u/dt2=-K(u-uo)-F(u,v)-Φ(v),
where m is the mass of the slider, u and v (=du/dt)
are, respectively, the displacement and velocity of the slider,
uo is the equilibrium location of the slider, K is the spring
constant, F is the frictional force between the slider and the background
and a function of u or v, and Φ is the viscous force between the
slider and the background and a function of v. The slider is pulled by a
driving force FD due to the moving plate with a constant driving
velocity, vp, through a leaf spring of strength, K. Hence, the
driving force is FD=Kvpt, and thus
uo=vpt. When FD is slightly larger than the
static frictional force, Fo, friction changes from static
friction strength to a dynamic one, and thus the slider moves.
Viscosity
Jeffreys (1942) first emphasized the importance of viscosity on faulting.
Frictional melts in faults depend on temperature, pressure, water content,
etc. (Turcotte and Schubert, 1982), and can yield viscosity on the fault
plane (Byerlee, 1968). Rice et al. (2001) discussed that rate- and
state-dependent friction in thermally activated processes allows creep
slippage at asperity contacts on the fault plane. Scholz (1990) suggested
that the friction melts would present significant viscous resistance to shear
and thus inhibit continued slip. However, Spray (1993, 1995, 2005) stressed
that the frictional melts possessing low viscosity could generate a
sufficient melt volume to reduce the effective normal stress and thus act as
fault lubricants during co-seismic slip. His results show that viscosity
remarkably decreases with increasing temperature. For example, Wang (2011)
assumed that quartz plasticity could be formed in the fault zone when
T>300∘C after faulting and it would lubricate the
fault plane at higher T and yield viscous stresses to resist slip at lower
T. From numerical simulations, Wang (2007, 2016b, 2017) stressed the
viscous effect on faulting. Noted that several researchers (Knopoff et al.,
1973; Cohen, 1979; Xu and Knopoff, 1994; Knopoff and Ni, 2001; Dragoni and
Santini, 2015) took viscosity as a factor in causing seismic radiation to
reduce energy during faulting.
One-body spring-slider model. In the figure, u, K, η,
FD, N, and F denote, respectively, the displacement, the spring
constant, the viscosity coefficient, the driving force, the normal force,
and the frictional force.
The viscosity coefficient, υ, of rocks is mainly controlled by
temperature, T. An increase in T will yield partial melting of rocks and
thus the viscosity coefficient, υ, first is increased, then reaches
the largest value at a particular T, and finally decreases with increasing
T. The relation between υ and T can be described by the
following equation (e.g., Turcotte and Schubert, 1982): υ=υoexp(Eo+pVa/RT), where
υo is the largest viscosity at low ambient T of an area,
Eo is the activation energy per mole, p is the pressure,
Va is the activation volume per mole, and R is the universal
gas constant (Eo/R≈3×104 K). Obviously,
υ decreases with increasing T. This is particularly remarkable in
regions of high confining pressure. On the other hand, Diniega et al. (2013)
assume that υ exponentially depends on temperature: υ∼eβ(1-T*), where β is a constant and T*=(T-TC)/(TH-TC) is a dimensionless temperature
within a temperature range of TC to TH. The value of
υ increases with T* when T*<1 and decreases with
increasing T* when T*>1. Wang (2011) inferred that in the
major slip zone < 0.01 m of the 1999 Ms7.6 Chi-Chi,
Taiwan, earthquake, T(t) in the fault zone at a depth of 1111 m increased
from ambient temperature Ta≈45∘C at t=0 s to
peak temperature Tpeak=1135.1∘C at t=∼2.5 s.
T(t) began to decrease after t=2.5 s and dropped to 160 ∘C at
t=195 s. This yields a change in viscosity in the fault zone.
The description of the physical models of viscosity can be found in several
articles (Jaeger and Cook, 1977; Cohen, 1979; Hudson, 1980; Wang, 2016b). A
brief description is given below. For many deformed materials, there are
elastic and viscous components. The viscous component can be modeled as a
dashpot such that the stress–strain rate relationship is σ=υ(dε/dt) where σ and ε are
the stress and the strain, respectively. Two simple models (shown in Fig. 2)
commonly used to describe the viscous materials are the Maxwell model and the
Kelvin–Voigt model (or the Voigt model). The first one can be represented by
a purely viscous damper (denoted by “D”) and a purely elastic spring
(denoted by “S”) connected in series. Its constitution equation is
dε/dt=dεD/dt+dεS/dt=σ/υ+E-1dσ/dt where E is the elastic modulus and σ=Eε.
The constitutive relation of the second model is σ(t)=Eε(t)+υdε(t)/dt.
Under a constant tensile stress, the strain will increase, without a upper
limit, with time for the Maxwell model, while the strain will increase, with
an upper limit, with time for the Kelvin–Voigt model. Wang (2016b) assumed
that the latter is more appropriate than the former to be applied to the
seismological problems as suggested by Hudson (1980). Hence, the
Kelvin–Voigt model is taken in this study. To simplify the problem, only a
constant viscosity coefficient is considered in a numerical simulation as
given below. The viscous stress at the slider is represented by υv.
The two types of viscous materials: (a) for the
Kelvin–Voigt model and (b) for the Maxwell model. (κ= spring constant and υ= coefficient of viscosity.)
However, it is not easy to directly implement viscosity in a dynamical
system as used in this study. Wang (2016b) represented the viscosity
coefficient in an alternative way. Viscosity leads to the damping of
oscillations of a body in viscous fluids. The damping coefficient, η,
depends on the viscosity coefficient, υ, and the linear dimension,
R, of the body in a viscous fluid. According to Stokes' law, the η of
a sphere of radius R in a viscous fluid of υ is η=6πRυ (cf. Kittel et al., 1968). In order to simplify the problem,
the damping coefficient is taken in this study. Hence, the viscous force is
Φ=ηv. Noted that the unit of η is N(m/s)-1.
Friction caused by thermal pressurization
Numerous factors can affect friction (see Wang, 2009, 2016b, and cited
references herein). When fluids are present and temperature changes in
faults, thermal pressurization will yield resistance on the fault plane and
thus play a significant role in earthquake rupture (Sibson, 1973;
Lachenbruch, 1980; Chester and Higgs, 1992; Fialko, 2004; Fialko and Khzan,
2005; Bizzari and Cocco, 2006a, b; Rice, 2006; Wang, 2000, 2006, 2009, 2011,
2013, 2016b, 2017; Bizzarri, 2010, 2011a, b, c).
Rice (2006) proposed two end-member models for thermal pressurization: the
adiabatic-undrained-deformation (AUD) model and the slip-on-a-plane (SOP)
model. He also obtained the shear stress–slip functions caused by the two
models. The first model corresponds to a homogeneous simple shear strain
ε at a constant normal stress σn on a spatial
scale of the sheared layer that is broad enough to effectively preclude heat
or fluid transfer. The second model shows that all sliding is on the plane
with τ(0)=f(σn-po) where po is
the pore fluid pressure on the sliding plane (y=0). For this second model,
heat is transferred outwards from the fault plane. Although the stress τsop(u) also shows slip weakening (Wang, 2009), the SOP model is
not appropriate in this study because of the request of a constant velocity
for this model.
The shear stress–slip function, τ(u), caused by the AUD model is
τaud(u)=f(σn-po)exp(-u/uc).
The parameters uc are the characteristic displacements associated
with the thickness and some physical properties of the fault zone. The stress
τaud(u) displays exponentially with u and thus exhibits
slip-weakening friction. Based on the AUD model, Wang (2009) proposed a
simplified slip-weakening friction law (denoted by the TP law hereafter),
F(u)=Foexp(-u/uc), where Fo is the static
frictional force, to study seismic efficiency. Wang (2016b, 2017) applied the
law to simulate slip of one-body and two-body spring-slider models. Figure 3
exhibits F(u) versus u for five values of uc, i.e., 0.1, 0.3,
0.5, 0.7, and 0.9 m. The friction force decreases with increasing u and it
decreases faster for smaller uc than for larger uc.
Meanwhile, the force drop decreases with increasing uc. For small
u, exp(-u/uc) can be approximated by 1-u/uc (Wang,
2016a, b, 2017). The parameter uc-1 is almost the decreasing
rate, γ, of friction force with slip at small u. Small (large)
uc is related to large (small) γ.
Predominant frequency and period of the system
To conduct marginal analyses of the slip of the one-body model with friction,
Wang (2016b) used the friction law: F(u)=Fo-γu. His results
show that the natural periods are To=2π/(K/m)1/2 when
friction and viscosity are excluded and
Tn=To/[1-To2(η2+4mγ)/(4πm)2]1/2
when friction and viscosity are included. Clearly, Tn is longer than
To. Equation (4) shows that Tn increases with η and γ, thus indicating that friction and viscosity both lengthen
the natural period of the system.
The variations in friction force with displacement for
F(u)=exp(-u/uc) when uc=0.1, 0.3, 0.5, 0.7, and
0.9 m (following Wang, 2016b).
Normalization of equation of motion
Substituting the TP law and the linear viscous law into Eq. (1) leads to
md2u/dt2=-K(u-uo)-Foexp(-u/uc)-ηv.
To simplify numerical computations, Eq. (4) is normalized based on the
following normalization parameters: Do=Fo/K, ωo=(K/m)1/2, τ=ωot, U=u/Do,
Uc=uc/Do, and
ΓD=FD/K. This gives
du/dt=[Fo/(mK)1/2]dU/dτ,
d2u/dt2=(Fo/mK)d2U/dτ2. The
driving velocity becomes
Vp=vp/Doωo Hence, the
normalized acceleration and velocity are, respectively,
A=d2U/dτ2 and V=dU/dτ. The phase
ωt is replaced by Ωτ, where Ω=ω/ωo is the dimensionless angular frequency. Note that η/(mK)1/2 is simply denoted by η below. Clearly, all normalization
parameters are dimensionless. Hence, Eq. (4) becomes
d2U/dτ2=-U-ηdU/dτ-exp(-U/Uc)+ΓD.
When FD=vpt or ΓD=Vpτ, Eq. (5) is
transformed to a set of three first-order differential equations by defining
x=U/Uc, y=V/Vp, and z=-U+Vpτ-ηVpyτ (yt=dy/dτ):
xτ=(Vp/Uc)y,yτ=(z-e-x)/Vp,zτ=Vp(1-y-ηyτ).
As x≪1, e-x≈1-x and thus Eq. (6b) can be approximated by
yτ≈(z-1+x)/Vp. The condition of x≪1 shows
U/Uc≪1. The differential of this equation leads to yττ≈ (zτ+xτ)/Vp, where yττ=d2y/dτ2. Substituting Eqs. (6a) and (6c) into this
equation gives
yττ+ηyτ+(1-1/Uc)y=1.
The homogeneous equation of Eq. (7) is
yττ+ηyτ+(1-1/Uc)y=0.
Let the general solution be y∼eλτ. This leads to [λ2+ηλ+(1-/Uc)]y=0 or
λ2+ηλ+(1-/Uc)=0.
The solutions of Eq. (9) are
λ±=-η/2±[η2-4(1-1/Uc)]1/2/2.
The plot of η versus 1/Uc exhibits the phase portrait
and root structure of the system. The solid line displays the function:
D(η,1/Uc)=η2-4(1-1/Uc)=0. The solid circle,
open circle, and solid square represent, respectively, a stable inflected
node with D=0, a stable spiral with D<0, and a stable node with
D>0.
The time variation in V/Vmax, the phase portrait of
V/Vmax versus U/Umax, and the power spectrum for four
values of Uc: (a) for Uc=0.1; (b)
for Uc=0.4; (c) for Uc=0.7; and (d)
for Uc=0.9 for the TP law of F(U)=exp(-U/Uc) when η=0.
The time variation in V/Vmax, the phase portrait of
V/Vmax versus U/Umax, and the power spectrum for four
values of Uc: (a) for Uc=1.00; (b)
for Uc=1.01; (c) for Uc=1.15; and
(d) for Uc=2.00 for the TP law of F(U)=exp(-U/Uc)
when η=0.
The time variation in V/Vmax, the phase portrait of
V/Vmax versus U/Umax, and the power spectrum for four
values of η: (a) for η=0.20; (b) for η=0.50; (c) for η=0.87; and (d) for η=0.90 when
Uc=0.20 for the TP law of F(U)=exp(-U/Uc).
The time variation in V/Vmax, the phase portrait of
V/Vmax versus U/Umax, and the power spectrum for four
values of η: (a) for η=0.43; (b) for η=0.47; (c) for η=0.98; and (d) for η=0.99 when
Uc=055 for the TP law of F(U)=exp(-U/Uc).
The term -η/2 of Eq. (10) leads to e-λ/2 which yields
attenuation of y. Define D(η, 1/Uc) to be η2-4(1-1/Uc). As mentioned above, Uc-1 is the
normalized decreasing rate of friction, Γ, at U=0. Figure 4 shows
the plot of η versus 1/Uc and thus exhibits the root
structure of the system. Because η>0 and
Uc>0, only the plot in the first quadrant is present
in Fig. 4. The solid line displays the function: D(η,
1/Uc)=η2-4(1-1/Uc)=0. Along the line, we have
η2=4(1-1/Uc), and thus λ±=-η/2. In other
words, the roots are equal and real, and thus the solution is a stable
inflected node displayed by a solid circle in Fig. 4. As D(η,
1/Uc)>0 or η2>4(1-1/Uc), the roots are both real and negative. The solution
shows no oscillation and thus is a stable node shown by a solid square in
Fig. 4. As D(η, 1/Uc)<0 or η2<4(1-1/Uc), the roots are complex with a negative real part. This
results in oscillations of exponentially decaying amplitude. The solution is
a stable spiral or a stable focus shown by an open circle in Fig. 4.
The time variation in V/Vmax, the phase portrait of
V/Vmax versus U/Umax, and the power spectrum for four
values of η: (a) for η=0.39; (b) for η=0.83; (c) for η=0.84; and (d) for η=0.85 when
Uc=0.6 for the TP law of F(U)=exp(-U/Uc).
Numerical simulations
Let y1=U and thus y2=dU/dτ. Equation (5) can be re-written as
two first-order differential equations:
dy1/dτ=y2,dy2/dτ=-y1-ηy2-exp(-y1/Uc)+ΓD.
Equation (11) will be numerically solved using the fourth-order Runge–Kutta
method (Press et al., 1986). To simplify the following computations, the
value of ΓD is set to be a small constant of 10-5,
which can continuously force the slider to move.
A phase portrait, denoted by y=f(x), is a plot of a physical quantity
versus another of an object in a dynamical system (Thompson and Stewart,
1986). The intersection point of the bisection line, i.e., y=x, and f(x)
is called the fixed point, that is, f(x)=x. If the function f(x) is
continuously differentiable in an open domain near a fixed point xf and
|f′(xf)|<1, attraction is generated. In other
words, an attractive fixed point is a fixed point xf of a function
f(x)
such that for any value of x in the domain that is close enough to xf,
the iterated function sequences, i.e., x, f(x), f2(x),
f3(x), …, converges to xf. An attractive fixed point
is a special case of a wider mathematical concept of attractors. Chaos can
be generated at some attractors. The details can be seen in Thompson and
Stewart (1986) or other nonlinear literature. In the following plots, the
intersection points of the bisection line (denoted by a thin solid line)
with the phase portrait of V/Vmax versus U/Umax are the fixed
points. To explore nonlinear behavior of a system, the Fourier spectrum
F[V(Ωk)], where Ωk=k/δτ is the
dimensionless angular frequency at k=0, …, N-1, is calculated for the
simulation velocity waveform through the fast Fourier transform (Press et
al., 1986). The bifurcation from a predominant period to others will be seen
in the Fourier spectra.
Numerical simulations for the time variation in V/Vmax, the phase
portrait of V/Vmax versus U/Umax, and the Fourier
spectrum based on different values of model parameters are displayed in
Figs. 5–12. In the figures, Vmax and Umax are,
respectively, the maximum velocity and displacement for case (a) of each
figure, because the maximum values of U and V decrease from case (a) to
case (d) in this study.
First, the cases excluding viscosity, i.e., η=0, are explored. Figure 5 is numerically made for four values of Uc: (a) for Uc=0.1; (b)
for Uc=0.4; (c) for Uc=0.7; and (d) for Uc=0.9 when
η=0. Figure 6 is numerically made for four values of Uc: (a) for
Uc=1.00; (b) for Uc=1.01; (c) for Uc=1.15; and (d) for
Uc=2.00 when η=0. A comparison between Figs. 5 and 6
suggests that Uc=1 is a transition value of the friction law between
two modes of slip as displayed in Fig. 4. Only Uc<1 is
considered below.
Secondly, the cases including both friction and viscosity are studied.
Figure 7 is numerically made for four values of η: (a) for η=0.20;
(b) for η=0.50; (c) for η=0.87; and (d) for η=0.90
when Uc=0.20. Obviously, the time variation in V/Vmax exhibits
cyclic oscillations with a particular period when η<ηl=0.86 and has intermittent slip with shorter periods when η>ηl. Such a phenomenon holds also for η<5.5.
Figure 8 is numerically made for four values of η: (a) for η=0.46; (b) for η=0.47; (c) for η=0.98; and (d) for η=0.99
when Uc=0.55. The Fourier spectrum is not calculated for case
(d), because the velocity becomes an abnormally large negative value at a
certain time and the phase portrait also displays unstable or chaotic slip at
small V and U. This exhibits unstable slip of the system. In other words,
the problem becomes ill-posed in this parameter regime. The time variation in
V/Vmax exhibits cyclic oscillations specified with three main
frequencies when η<ηl=0.47. There is
intermittency slip with shorter periods when ηl<η<ηu=0.98. There are unstable slips when η>ηu. This phenomenon holds also when 0.55<Uc<1.
Four examples for η varying from η <ηu
to η >ηu for different values of
Uc are displayed in Figs. 9–12. Figure 9 is made for four values
of η: (a) for η=0.39; (b) for η=0.83; (c) for η=0.84;
and (d) for η=0.85 when Uc=0.6. Figure 10 is made for four
values of η: (a) for η=0.34; (b) for η=0.71; (c) for η=0.72; and (d) for η=0.73 when Uc=0.7. Figure 11 is made
for four values of η: (a) for η=0.25; (b) for η=0.53; (c)
for η=0.54; and (d) for η=0.55 when Uc=0.8. Figure 12
is made for four values of η: (a) for η=0.14; (b) for η=0.35; (c) for η=0.36; and (d) for η=0.37 when
Uc=0.9. The Fourier spectrum is not calculated for case (d) in
each example, because the velocity becomes negative infinity at a certain
time.
Figure 13 exhibits the data points of ηl (with a solid square)
and that of ηu (with a solid circle) for several values
Uc. The values of ηl and ηu for
several values of Uc are given in Table 1. The figure exhibits a
stable regime when η≤ηl, an intermittency or transition
regime when ηl<η≤ηu, and an
unstable regime when η>ηu.
Discussion
As mentioned above, the natural period of the one-body system at low
displacements is To=2π/ωo=2π(m/K)1/2 in the absence of friction and viscosity and Tn=2π/ωn=To/[1-To2(η2+4mγ)/(4πm)2]1/2 in the presence of friction and viscosity. Due
to γ=1/uc at u=0, Tn increases with decreasing
uc. Obviously, Tn is longer than To and increases
with η and γ, thus indicating that friction and
viscosity both lengthen the natural period of the system.
The time variation in V/Vmax, the phase portrait of
V/Vmax versus U/Umax, and the power spectrum for four
values of η: (a) for η=0.34; (b) for η=0.71; (c) for η=0.72; and (d) for η=0.73 when
Uc=0.7 for the TP law of F(U)=exp(-U/Uc).
The time variation in V/Vmax, the phase portrait of
V/Vmax versus U/Umax, and the power spectrum for four
values of η: (a) for η=0.25; (b) for η=0.54; (c) for η=0.55; and (d) for η=0.56 when
Uc=0.8 for the TP law of F(U)=exp(-U/Uc).
The time variation in V/Vmax, the phase portrait of
V/Vmax versus U/Umax, and the power spectrum for four
values of η: (a) for η=0.14; (b) for η=0.36; (c) for η=0.37; and (d) for η=0.38 when
Uc=0.9 for the TP law of F(U)=exp(-U/Uc).
The plot of ηl (with a solid square) and ηu
(with a solid circle) versus Uc.
Based on the marginal analysis of the normalized equation of motion, i.e.,
Eq. (11), the plot of η versus 1/Uc is displayed in Fig. 4
which exhibits the phase portrait and root structure of the system. Since
η and Uc are both positive, only the plot of η versus
1/Uc in the first quadrant is displayed. In Fig. 4, the solid
line displays the function: D(η,1/Uc)=η2-4(1-1/Uc)=0. Along the line, the solution
λ±=-η/2 and thus exp(λt)=exp(-η/2). In other
words, the roots are equal and real, and, thus, the phase portrait is a
stable inflected node displayed by a solid circle in Fig. 4. Because of η≥0, we have 1/Uc≤1. As D(η,1/Uc)>0 or η2>4(1-1/Uc), the roots are both real and negative. The solution
shows no oscillation and thus the phase portrait is a stable node shown by a
solid square in Fig. 4. Because of η≥0, we have 1/Uc≤1. As D(η,1/Uc)<0 or η2<4(1-1/Uc), the roots are complex with a negative real part. This
results in oscillations with exponentially decaying amplitude. The phase
portrait is a stable spiral or a stable focus shown by an open circle in
Fig. 4.
Figure 5 exhibits the time variation in V/Vmax, the phase
portrait of V/Vmax versus U/Umax, and the Fourier
spectrum for four values of Uc: (a) for Uc=0.1; (b)
for Uc=0.4; (c) for Uc=0.7; and (d) for
Uc=0.9 when η=0. In the first panels, the time variation in
V/Vmax exhibits cyclic behavior, the amplitude of
V/Vmax decreases, and the predominant period of signal increases
with increasing Uc. This is consistent with Eq. (3) in which
Tn increases with Uc. Although the four phase
portraits are almost similar, their size decreases with increasing
Uc. The second panels exhibit two fixed points: one at V=0 and
U=0 and the second one at larger V and larger U. The slope values at
the first fixed points decrease with increasing Uc, thus
suggesting that the fixed point is not an attractor for small Uc
and can be an attractor for larger Uc. The slope values at the
fixed points for smaller Uc are greater than 1, and thus they
cannot be an attractor. The third panel for each case displays the Fourier
spectrum. Fourier spectra show that, in addition to the peak related to the
predominant frequency, there are numerous peaks associated with higher
frequencies. This shows nonlinear behavior caused by nonlinear friction. The
frequency related to the first peak decreases with increasing Uc.
The amplitude of a peak decreases with increasing Uc. The
amplitude of a peak decreases with increasing Ω for small
Uc, while it first increases up to the maximum and then decreases
with increasing Ω for large Uc. The amplitude of a peak
becomes very small when Ω>0.25.
Figure 6 exhibits the time variation in V/Vmax, the phase
portrait of V/Vmax versus U/Umax, and the Fourier
spectrum for four values of Uc: (a) for Uc=1.00; (b)
for Uc=1.01; (c) for Uc=1.15; and (d) for
Uc=2.0 when η=0. In the first panels, the time variation
in V/Vmax exhibits cyclic behavior and the amplitude of
V/Vmax remarkably decreases with increasing Uc when
Uc>1. In the second panels, the size of the phase
portrait decreases with increasing Uc and there are two fixed
points: the first one at V=0 and U=0 and the second one at larger V
and larger V. With comparison to the phase portrait of Uc=1.0,
the phase portrait becomes very small when Uc≥1.15. In
contrast to Fig. 5, the absolute values of slope at the fixed points in
Fig. 6 increase with Uc. Hence, the fixed points cannot be an
attractor for Uc≥1. In the third panels, Fourier spectra
exhibit that except for Uc=1, there is only one peak and the
predominant frequency increases or the predominant period decreases with
increasing Uc. This is consistent with Eq. (3). Results show that
nonlinear behavior disappears when Uc>1. In addition,
the amplitude of a peak decreases with increasing Uc when
Uc>1. Obviously, Uc=1 is the critical
value of the friction law as displayed in Fig. 4.
Figure 7 exhibits the time variation in V/Vmax, the phase
portrait of V/Vmax versus U/Umax, and the Fourier
spectrum for four values of η: (a) for η=0.20; (b) for η=0.50; (c) for η=0.87; and (d) for η=0.90 when
Uc=0.20. In the first panels, the time variation in
V/Vmax exhibits cyclic behavior and the amplitude of
V/Vmax decreases with increasing η. The predominant period
of the signal only slightly increases with increasing η, because η changes in a small range. In the second panels, the size of the phase
portrait decreases with increasing Uc and there are two fixed
points: the first one at V=0 and U=0 and the second one at larger V and
larger U. Since the slope values of fixed points are clearly all
higher than 1, they are not an attractor. In the third panels, the Fourier
spectra exhibit that in addition to the peak related to the predominant
frequency, there are numerous peaks associated with higher Ω. This
shows nonlinear behavior, mainly caused by nonlinear friction, of the model.
The highest peak for case (a) appears at the second frequency. When η<0.9, the amplitude of a peak decreases with increasing η.
The frequencies related to the peaks do not change remarkably, because η varies in a small range. Except for case (a), the amplitude of a peak
decreases with increasing Ω. The third peak amplitude disappears when
η>0.5. The amplitude of a peak becomes very small when
Ω>0.25. Except for Uc=0.1, the frequencies
related to the peaks in Fig. 7 are different from and slightly smaller than
those in Fig. 5. Note that when Uc<0.55 the simulation
results in Fig. 5 are similar to those in Fig. 6.
Figure 8 shows the time variation in V/Vmax, the phase portrait
of V/Vmax versus U/Umax, and the Fourier spectrum for
four values of η: (a) for η=0.46; (b) for η=0.47; (c) for
η=0.98; and (d) for η=0.99 when Uc=0.55. When η≤0.47, the time variation in V/Vmax exhibits cyclic
oscillations specified with different main angular frequencies. When η>0.47 (for example, η=0.98 in the figure), in addition to
cyclic behavior there is a small intermittent slip with shorter periods. This
phenomenon also exists when ηl<η<ηu=0.98. There are unstable (or chaotic) slips when η>ηu. Hence, the phase portraits in the second
panels display unstable slip at small V and U when
ηl<η≤ηu=0.98. When η=0.99,
the velocity becomes an abnormally large negative value at a certain time and
the phase portrait also displays unstable or chaotic slip at small V and
U. This exhibits unstable slip of the system. In other words, the problem
becomes ill-posed in this parameter regime. Since the slope values of fixed
points at large V and U are clearly higher than 1, they are not an
attractor. Due to the appearance of infinity velocity when η=0.99, the
Fourier spectrum is not calculated for η=0.99. The Fourier spectra
exhibit that when η<0.47, in addition to the peak related to
the predominant frequency, there are numerous peaks associated with higher
Ω. This shows nonlinear behavior of the model caused by nonlinear
friction. The amplitude of a peak decreases with increasing Uc
and the peak amplitude decreases with increasing Ω. When η=0.98, the amplitude of the highest peak is much larger than others. For the
first three cases, the amplitude of a peak becomes very small when Ω>0.25. The frequencies related to the peaks in Fig. 8 are
different from and slightly smaller than those in Fig. 7.
Figures 9–12 show a variation from stable slip to intermittent slip and then
to unstable or chaotic slip when η increases from a smaller value to a
larger one for Uc=0.6, 0.7, 0.8, and 0.9. The values of
ηu for Uc=0.20–0.95 with a unit difference of
0.05 are given in Table 1. Like Fig. 8, when η≤ηl, the
time variation in V/Vmax exhibits only cyclic oscillations
specified with different frequencies. When ηl<η≤ηu, small intermittent displacements appear in the cyclic
oscillations. Hence, the phase portraits display that unstable slip at small
V and U when ηl<η≤ηu. When
η>ηu, the velocity becomes an abnormally large
negative value at a certain time and the phase portrait also displays an
unstable or chaotic slip at small V and U. This exhibits unstable slip of
the system. In other words, the problem becomes ill-posed in this parameter
regime. Due to the appearance of abnormally large negative velocity, the
Fourier spectrum is not calculated for η>ηu.
When η<ηl, in addition to the peak related to the
predominant frequency, there are numerous peaks related to higher Ω.
This shows nonlinear behavior, mainly caused by nonlinear friction, of the
model. The amplitude of a peak decreases with increasing Uc and
the amplitude of a peak decreases with increasing Ω. For the first
three cases, the amplitude of a peak becomes very small when Ω>0.25. Figures 7–12 show that the frequencies related to the
peaks slightly decrease with increasing Uc and the decreasing
rate decreases with increasing Uc. In other words, the
frequencies related to the peaks for large Uc are almost similar.
The number of higher peaks and the amplitudes of peaks at higher Ω
both decrease with increasing η. This indicates that viscosity makes a
stronger effect on higher-frequency waves than on lower ones, and the effect
increases with η.
Figure 13 exhibits the data points of ηl (with a solid square)
and that of ηu (with a solid circle) for several values
Uc. The values of ηl and ηu for
several values of Uc are given in Table 1. The figure exhibits a
stable regime when η≤ηl, an intermittency (or
transition) regime when ηl<η≤ηu,
and an unstable (or chaotic) regime when η>ηu.
When Uc<0.55, there is no ηl; in other
words, unstable slip does not exist. Clearly, ηl,
ηu, and their difference ηu-ηl
all decrease with increasing Uc. This means that it is easier to
yield unstable slip for larger Uc than for smaller
Uc. Since smaller Uc is associated with a larger
γ of decreasing rate of friction force with slip, it is easier to
yield unstable slip from smaller γ than from larger γ.
Huang and Turcotte (1990, 1992) observed intermittent phases in the
displacements based on a two-body model. In other words, some major events
are preceded by numerous small events. Those small events could be
foreshocks. They also claimed that earthquakes are an example of
deterministic chaos. Ryabov and Ito (2001) also found intermittent phase
transitions in a two-dimensional one-body model with velocity-weakening
friction. Their simulations exhibit that intermittent phases appear before
large ruptures. From numerical simulations of earthquake ruptures using a
one-body model with a rate- and state-friction law, Erickson et al. (2008)
found that the system undergoes a Hopf bifurcation to a periodic orbit. This
periodic orbit then undergoes a period doubling cascade into a strange
attractor, recognized as broadband noise in the power spectrum. From
numerical simulations of earthquake ruptures using a two-body model with a
rate- and state-friction law, Abe and Kato (2013) observed various slip
patterns, including the periodic recurrence of seismic and aseismic slip
events, and several types of chaotic behavior. The system exhibits typical
period-doubling sequences for some parameter ranges, and attains chaotic
motion. Their results also suggest that the simulated slip behavior is
deterministic chaos and time variations of cumulative slip in chaotic slip
patterns can be well approximated by a time-predictable model. In some cases,
both seismic and aseismic slip events occur at a slider, and aseismic slip
events complicate the earthquake recurrence patterns. The present results
seem to be comparable with those made by the previous authors, even though
viscosity was not included in their studies. This suggests that nonlinear
friction and viscosity play the first and second roles, respectively, in the
intermittent phases. The intermittent phases could be considered foreshocks
of the mainshock which is associated with the main rupture. Simulation
results exhibit that foreshocks happen for some mainshocks and not for
others.
Conclusions
In this work, the multistable slip of earthquakes caused by slip-weakening
friction and viscosity has been studied based on the normalized equation of
motion of a one-degree-of-freedom spring-slider model in the presence of the
two factors. The friction is caused by thermal pressurization and decays
exponentially with displacement. The major model parameters are the
normalized characteristic distance, Uc, for friction and the
normalized viscosity coefficient, η, between the slider and the
background moving plate, which exerts a driving force on the former. Analytic
results at small U suggest that there is a solution regime for η and
γ (=1/Uc) to make the slider slip steadily. Numerical
simulations lead to the time variation in V/Vmax, the phase
portrait of V/Vmax versus U/Umax, and the Fourier
spectrum. Results show that the time variation in V/Vmax
obviously depends on Uc and η. The effect on the amplitude
is stronger from η than from Uc. When
Uc>1, the time variation of V/Vmax
exhibits cyclic oscillations with a single period and the amplitude of
V/Vmax remarkably decreases with increasing Uc. When
Uc<1, the slip changes from stable motion to
intermittency and then to unstable motion when η increases. For a
certain Uc, the three regimes are controlled by a lower bound,
ηl, and an upper bound, ηu, of η. When
Uc<0.55, ηu is absent and thus unstable
or chaotic slip does not exist. When Uc≥0.55, the plots of
ηl and ηu versus Uc exhibit a
stable regime when η≤ηl, an intermittency (or
transition) regime when ηl<η≤ηu,
and an unstable (or chaotic) regime when η>ηu.
The values of ηl, ηu, and
ηu-ηl all decrease with increasing
Uc, thus suggesting that it is easier to yield unstable slip for
larger Uc than for smaller Uc or larger η than
for smaller η. The phase portraits of V/Vmax versus
U/Umax exhibit that there are two fixed points: the first one at
large V/Vmax and large U/Umax is not an attractor for
all cases under study, while the second one at small V/Vmax and
small U/Umax can be an attractor for some values of
Uc and η. When Uc<1, the Fourier
spectra calculated from simulation velocity waveforms exhibit several peaks
rather than one, thus suggesting the existence of nonlinear behavior of the
system. When Uc>1, the related Fourier spectra show
only one peak, thus suggesting linear behavior of the system.
In this study no data are used, because only numerical simulations have been made.
The author declares that he has no conflict of interest.
Acknowledgements
The author would like to thank Richard Gloaguen (editor of Nonlinear Processes in Geophysics), J. G. Spray, and one anonymous reviewer for their
valuable comments and suggestions to substantially improve this article. The
study was financially supported by Academia Sinica, the Ministry of Science
and Technology (grant nos.: MOST-105-2116-M-001-007 and
MOST-106-2116-M-001-005), and the Central Weather Bureau (grant no.:
MOTC-CWB-106-E-02). Edited by: Richard
Gloaguen Reviewed by: John G. Spray and one anonymous referee
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