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,

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.

Figure 1 shows the one-body model whose equation of motion is

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

One-body spring-slider model. In the figure,

The viscosity coefficient,

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

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

The two types of viscous materials:

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,

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

The shear stress–slip function,

To conduct marginal analyses of the slip of the one-body model with friction,
Wang (2016b) used the friction law:

The variations in friction force with displacement for

Substituting the TP law and the linear viscous law into Eq. (1) leads to

The plot of

The time variation in

The time variation in

The time variation in

The time variation in

The term

The time variation in

Let

A phase portrait, denoted by

Numerical simulations for the time variation in

First, the cases excluding viscosity, i.e.,

Secondly, the cases including both friction and viscosity are studied.
Figure 7 is numerically made for four values of

Figure 8 is numerically made for four values of

Four examples for

Figure 13 exhibits the data points of

As mentioned above, the natural period of the one-body system at low
displacements is

The time variation in

The time variation in

The time variation in

The plot of

Based on the marginal analysis of the normalized equation of motion, i.e.,
Eq. (11), the plot of

Figure 5 exhibits the time variation in

Figure 6 exhibits the time variation in

Figure 7 exhibits the time variation in

Figure 8 shows the time variation in

Figures 9–12 show a variation from stable slip to intermittent slip and then
to unstable or chaotic slip when

Figure 13 exhibits the data points of

Values of

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.

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,

In this study no data are used, because only numerical simulations have been made.

The author declares that he has no conflict of interest.

The author would like to thank Richard Gloaguen (editor of