Friction Law of Rock and Application to Modeling of Earthquake
Faulting
Naoyuki Kato
Earthquake Research Institute, University of Tokyo, 1-1-1 Yayoi, Bunkyo-ku, Tokyo 113-0032, Japan
e-mail: [email protected]
Abstract
Friction of rock has been experimentally investigated for many years to understand mechanics of
earthquakes. On the basis of the laboratory studies, some constitutive laws of rock friction were developed.
Among them, the rate- and state-dependent friction law well explains many observed characteristics of rock
friction at relatively low slip rates, such as the transition of slip mode from stick-slip to stable sliding,
preseismic sliding, and episodic aseismic slip events. Moreover, the friction law has been applied to
modeling earthquake cycles that includes interseismic periods of low slip rates, preseismic slip acceleration,
seismic slip, postseismic slow slip, and the fault healing process. Many numerical simulations that can be
compared with geophysical observation have been conducted to understand detailed slip processes and
frictional properties of faults or plate boundaries.
I review (1) derivation of the rate- and state-dependent friction law, (2) characteristics of the friction
law, and (3) the application to modeling of slip processes on plate boundaries.
The friction law incorporates important properties of rock friction. Frictional resistance increases with
shear loading rate, frictional strength increases with stationary contact time of sliding surfaces, and the
contact state evolves with slip distance. These properties are characterized by parameters a, b, and L,
respectively.
The steady-state friction coefficient ss(V) is defined as 0+(ab)ln(V/V*), where V is a slip velocity, 0
and V* are constants. When ab < 0, the steady-state friction decreases with an increase in V, possibly
leading to unstable slip. On the other hand, when ab > 0, the steady-state friction increases with V, and
aseismic sliding occurs. Even for ab < 0, a critical fault size is required to generate unstable slip. Aseismic
slip within the critical size fault is related to preseismic slip and episodic aseismic slip events.
Recent geodetic observations indicate that the coupling state of plate boundaries is spatially
nonuniform. In some regions, the plate boundaries are firmly coupled and strain energy is accumulated with
time, and large earthquakes are expected to occur in future. In the other regions, aseismic sliding always
takes place and little strain energy is accumulated. These observations are useful for evaluating the
potential of large earthquakes, and this behavior is successfully explained by modeling with the rate- and
state-dependent friction law.
1
Frictional properties of a fault in siliceous material at high slip velocities
Akito Tsutsumi
Department of Geology and Mineralogy, Division of Earth and Planetary Sciences
Graduate School of Science
Kyoto University
Kyoto, 606-8502, Japan
Understanding of the frictional properties of fault-zone material over a wide range of slip velocities is
of critical importance in earthquake mechanics. Previous experimental studies of rock friction performed
at high seismic slip velocities (to ~2 m/s) have indicated a dramatic decrease in friction (Di Toro et al.,
2011). Several models have been proposed to explain the fault weakening at high slip velocities,
including frictional melting (Tsutsumi and Shimamoto, 1997; Hirose and Shimamoto, 2005), thermal
decomposition of the fault material (Han et al., 2007, 2010), or silica-gel formation (Goldsby and Tullis,
2002; Di Toro et al., 2004). Among these processes, silica-gel formation may be distinguished from the
others because the weakening has occurred even at relatively low slip velocities (v = ~0.01 mm/s)
(Goldsby and Tullis, 2002; Di Toro et al., 2004), under which conditions transformation reactions are
unlikely to proceed because fault temperatures are expected to be low. Despite the general acceptance that
frictionally generated silica gel plays an important role in the weakening process of siliceous materials,
there exists little information on the frictionally-generated material; consequently it remains poorly
understood. To get a better understanding of fault zone process in siliceous material, we have performed
high-velocity friction experiments on quartz-rock samples and microstructural observation of the fault
surface material.
Friction experiments were performed on chert, a silica-rich sedimentary rock, at slip velocity of v =
104 mm/s and at low normal stress of 1.5 MPa. Fault weakening in chert samples occurred in association
with the formation of a 0.1-mm-thick fault gouge layer. SEM observations revealed that the fault surfaces
consisted of smooth and rough parts. On the smooth part of the surfaces, rod-shaped particles (with a
diameter < 0.5 mm), aligned perpendicular to the sliding direction, probably indicating that they were
rolled during the experiment. TEM observations revealed the following characteristics of the
experimentally generated fault surface material in chert: (1) the smooth fault surface consists of several
hundred-nm-thick amorphous silica layer, (2) Rolls exist on the smooth fault surface and are in contact
with the amorphous silica layer, (3) Rolls are made of amorphous silica, and (4) the gouge material is
composed of amorphous silica clasts. It follows from what we have observed that a thin hydrated
amorphous silica layer (silica gel layer) does form on fault surfaces in a siliceous rock during the high
velocity sliding.
Systematic Understanding of the Friction Behavior in terms of the Interaction among Heat,
Fluid Pressure and Dilatancy in Natural Fault
Takehito Suzuki
Department of Physics and Mathematics, Aoyama Gakuin University
The presence of high-pressure fluid and frictional heating are considered to affect the process of
earthquake rupture as evidenced in field investigations [e.g., Sibson, 1975]. Inelastic porosity increase
with increasing fault slip (slip-induced dilatancy), revealed in laboratory experiments [e.g., Marone et
al., 1975], will also affect earthquake rupture significantly because of fluid pressure change with the
pore creation [Yamashita, 1999]. Recently, interactions among shear heating, fluid pressure, inelastic
pore creation and fault slip have attracted the attention of researchers [e.g., Segall et al., 2010;
Yamashita, 2013]. We have shown in a series of our papers [Suzuki and Yamashita, 2007; 2008; 2009;
2010; 2014] that shear heating, fluid pressure change and inelastic pore creation are keys for unified
understanding of earthquake source process. In particular, we considered these interactions in the
theoretical analysis and found three nondimensional controlling parameters Su , Su and Ta for
one-dimensional fault model. The parameter Su represents the relative dominance of the effect of
inelastic pore creation on the fluid pressure change over that of shear heating, while Su is associated
with the dominance of fluid flow effect over the effect of shear heating. Additionally, Ta denotes the
ratio of inelastic porosity increment c due to the characteristic slip by shear heating pressurization
of fluid to the upper limit  . Consideration of the parameters Su , Su and Ta and Ta will give
us a new insight into slip behavior.
Without the fluid flow ( Su  0 ), we can investigate the system behavior in an analytical way,
while the governing equations are nonlinear. In particular, our model simulates two types of slips; one
ceases spontaneously and the other one accelerates to the maximum allowable velocity in a single
framework. In the nondimensional parameter space, we obtained the clear boundary between these
two slip behaviors. We found the function G in terms of Su and Ta ; if G is positive, the
acceleration occurs, while the spontaneous slip cessation appears with negative G . Since our present
model has an advantage that its mathematical framework can be widely applied to phenomena other
than earthquakes, we discuss chemical reaction rate theory as an example for the application. In
particular, we will show transient state between an apparent stable point temporarily attracting orbits
but finally distracting them and an actual stable point attracting all the orbits.
Nanoscale studies of single asperities: Understanding the physics behind
rate-and-state friction
Robert W. Carpick1, Kaiwen Tian2, Nitya N. Gosvami1, David L. Goldsby3
1
Department of Mechanical Engineering and Applied Mechanics,
University of Pennsylvania, Philadelphia, PA 19104, USA
2
Department of Physics and Astronomy,
University of Pennsylvania, Philadelphia, PA 19104, USA
3
Department of Earth and Environmental Sciences,
University of Pennsylvania, Philadelphia, PA 19104, USA
Abstract:
Rate and state friction (RSF) laws are empirical relationships that describe the frictional
behavior of rocks and other materials in experiments reasonably well, and reproduce a wide
variety of observed natural behavior when employed in earthquake models. A pervasive
observation from rock friction experiments is frictional ‘ageing’: the increase of static friction
in a linear fashion with the log of contact time. Ageing is usually attributed to an increase in
real area of contact associated with asperity creep. However, recent atomic force microscopy
(AFM) experiments demonstrate that nanoscale silica-silica contacts exhibit ageing due to
progressive formation of interfacial chemical bonds in the absence of plastic deformation, in
a manner consistent with the multi-contact ageing behavior of rocks1. To further investigate
chemical bonding-induced ageing, we have examined the influence of normal load (and thus
contact normal stress) and contact time on ageing. We conducted experiments that mimic
slide-hold-slide rock friction experiments in the AFM for contact loads ranging from 23 to
393 nN and hold times from 0.1 s to 100 s, all in humid air (~50% RH) at room temperature.
Experiments were conducted by sequentially sliding the AFM tip on the sample at a velocity
of 0.5 𝜇m/s, setting the velocity to zero and holding the tip stationary for a given time, and
finally resuming sliding at 0.5 𝜇m/s to yield a peak in the friction value followed by a drop to
the sliding friction value. We demonstrate that chemical bonding-induced ageing, as
measured by the peak friction minus the sliding friction after a hold, increases approximately
linearly with the product of the normal load and the log of the hold time. Theoretical studies
of the roles of reaction energy barriers in nanoscale ageing indicate that frictional ageing
depends on the total number of reaction sites and the hold time2. We combine chemical
kinetics analyses with contact mechanics models to explain our experimental results, and
develop a new approach for curve fitting ageing vs. load data, which allows us to show that
the friction drop data points fall onto one master curve. The analysis yields physically
reasonable values for the activation energy and activation volume of the chemical bonding
process. Our study of the role of normal load in ageing of nanoscale silica contacts provides a
basis to hypothesize that the kinetic processes do not depend strongly on the normal load. We
will discuss the implications of these findings for interpreting macroscopic RSF behavior.
1.
2.
Li, Q., Tullis, T.E., Goldsby, D. and Carpick, R.W. Frictional Ageing from Interfacial
Bonding and the Origins of Rate and State Friction. Nature 480, 233-236 (2011).
Liu, Y. and Szlufarska, I. Chemical Origins of Frictional Aging. Phys. Rev. Lett. 109,
186102/1-4 (2012).
Takahiro Hatano, University of Tokyo
"Atomistic Origin of Rate and State Friction Law "
A theoretical account is given of the microscopic basis of the rate- and state- dependent
friction (RSF) law. Although the RSF law is commonly used to model earthquake-related
phenomena, the theoretical basis has not been very clear. As macroscopic friction force
is the sum of traction at many microscopic asperities, macroscopic friction laws should
be derived by assuming the physical processes at the microscopic asperities. Here we derive
the RSF law starting from constitutive laws for the microscopic asperities, and give the
atomistic expressions for the empirical RSF parameters.
In particular, we show that both the critical slip distance and the state variable are
the 0th weighted power means of the corresponding microscopic
quantities: a linear dimension of asperities and its contact duration. Evolution laws for
the state variable are derived and the approximations behind two major evolution laws (the
aging and the slip laws) are clarified.
Adhesion-Detachment Motions in Friction of Gels"
1
2
Tetsuo Yamaguchi , Shmuel Rubinstein , and Yoshinori Sawae
[email protected]
1
3
Department of Mechanical Engineering, Kyushu University, Fukuoka 819-0395, Japan
2
School of Engineering and Applied Sciences, Harvard University, USA
We report our studies on stick-slip friction experiments between a Plexiglass block and a Silicone
Gel plate driven at a constant plate velocity[1]. In order to study the effects of the normal stress and
its gradient on the adhesion-detachment motions (Schallamach waves) in sheared dissimilar bodies,
we controlled the normal force and the initial inclination angle of the upper block. As a result, we
successfully observed a variety of slip events and size statistics. In small normal force and small
inclination angle conditions, slip events were dominated by slow events which occur uniformly at the
block-plate interface, and the size statistics obeyed the power law with cutoff. In intermediate force
and angle conditions, coexistence of small slow events in “deep” (large normal stress) regions and
giant events rupturing “shallower” regions were clearly seen and the size distributions obeyed powerlaw plus a bump at large sizes. In larger force and angle conditions, however, the giant slip events
disappeared and only small and intermediate slip events were observed. Interestingly, when we did
experiments at the boundary between the latter two regimes, very complex stick-slip cycles which
exhibit switching of the latter 2 dynamical modes were seen. Based on our visualization of 2D contact
fields at the frictional interface, we analyzed stick-slip dynamics at the mesoscopic scale. We found
that giant event was well characterized by the stress level and the areal fraction of detached regions.
References
[1] T. Yamaguchi, S. Rubinstein, Y. Sawae, Dynamics of adhesion and detachment in sliding friction
of soft elastic solids, in preparation.
Modeling of frictional sliding and slip precursors, analogies
between friction and fracture
J.F. Molinari
Civil Engineering Institute, Institute of Materials Science
Ecole Polytechnique Fédérale de Lausanne (EPFL), Switzerland
Recent experimental observations [1] show that the propagation speed of the slip front varies along its
path and is coupled to the local shear to normal stress ratio. We simulate these laboratory-earthquake
experiments with the finite-element method and rate-and-state friction laws. Numerical results reveal
that the slip front speed varies with a changing stress state along the interface, which is coherent with
experimental observation. However, a static stress criterion does not seem to be sufficient to fully
characterize the propagation speed of the interface rupture. Instead, we show that a dynamic energetic
criterion, which relates the slip front speed with the relative rise of the energy density at the slip tip,
captures all acquired data [2]. We also discuss the transition from sticking to sliding at the frictional
interface, which is marked by the occurrence of local slip events, called slip precursors. These initiate
at shear levels much below the global static friction coefficient threshold. These precursors stop
before propagating over the entire interface, and their length increases with increasing shear force,
which can be fully predicted by linear elastic fracture mechanics [3].
References:
[1] O. Ben-David, G. Cohen, and J. Fineberg (2010) Science.
[2] D.S. Kammer, V.A. Yastrebov, P.Spijker, and J.F. Molinari (2012) Tribolology Letters
[3] D.S. Kammer, M. Radiguet, J.P. Ampuero, J.F. Molinari (2015) Tribology Letters
Numerical and experimental aspects of the onset of sliding
J. Scheibert1, R. Sahli1, J.K. Trømborg1,2, A. Malthe-Sørenssen2, H.A. Sveinsson2, K. Thøgersen2, G.
Pallares1
1
Laboratoire de Tribologie et Dynamique des Systèmes, CNRS / Ecole Centrale de Lyon, Ecully,
France
2
Department of Physics, University of Oslo, 0316 Oslo, Norway
The contact between macroscopic rough solids is made of many micrometric asperities in contact.
Under increasing shear force, the rupture of this so-called multi-contact occurs through the
propagation of a micro-slip front along the interface. Gross sliding of the solids only begins when
the front has spanned the whole interface.
I will first introduce a multi-scale model [1,2] capable of reproducing the space and time features of
micro-slip fronts observed in recent experiments [3,4], in particular the existence of anomalously
slow fronts. I will show how the macroscale behavior depends on the assumptions made for the
microscopic behavior of individual junctions at a multicontact interface.
I will then show how the individual rupture behavior of micro-contacts can be accessed
experimentally and give an example of how to tune this behavior through adequate preparation of
the interface. I will finally attempt an upscaling from the micro- to the macroscale and test it against
experiments (similar to that of [5]) on a full multicontact interface.
References:
[1] J.K. Trømborg, H.A. Sveinsson, J. Scheibert, K. Thøgersen, D.S. Amundsen, A.
Malthe-Sørenssen, PNAS 111, 8764 (2014)
[2] J.K. Trømborg, H.A. Sveinsson, K. Thøgersen, J. Scheibert, A. Malthe-Sørenssen, Phys. Rev. E
(in press)
[3] S.M. Rubinstein, G. Cohen, J. Fineberg, Nature 430, 1005 (2004)
[4] O. Ben-David, S.M. Rubinstein, J. Fineberg. Nature 463, 76 (2010)
[5] A. Prevost, J. Scheibert, G. Debrégeas, Eur. Phys. J. E 36, 17 (2013)
Atomic-Scale Exfoliation and Adhesion of Nano-Carbon
K. Miura1, M. Ishikawa1, N. Sasaki2
1
Department of Physics, Aichi University of Education, Hirosawa 1, Igayacho, Kariya-shi, Aichi
448-8542, Japan
2
Department of Engineering Science, University of Electro-Communications, Chofu, 182-8585
Tokyo, Japan
Exfoliation is a daily ordinary phenomenon. Detachment and exfoliation experiments are expected
to provide information on the adhesion forces and energies of solid surfaces in contact with each
other. However, it was not easy to scientifically solve exfoliation and fracture, because we could not
approach them at the atomic scale. On the other hand, developments of adhesive materials that easily
provide an exfoliation are desirable from eco-society. Recently, it has been reported that carbon
nanotube (CNT) arrays with curved entangled tops exhibit a macroscopic adhesive force of
approximately 100 N/cm2, almost 10 times as large as that of a gecko foot, and a shear force much
stronger than the normal adhesion force. It is well-known that a gecko sticks on a wall and can walk
easily on it. However, until now, it has not known that an origin of the sticking force comes from van
der Waals force acting between many setae on gecko’s feet and a wall. It is now interesting to note
that a decrease of exfoliation and adhesion forces is closely related to low frictional force and leads to
a realization of nanomachine or biomolecular motor. Therefore, we focus our attention on the
elementary processes involved in the exfoliation of a CNT and a graphene on a substrate [1][2], and
perform exfoliation experiments using a CNT and a graphene [3].
References:
[1] M. Ishikawa, M. Kato, R. Harada, N. Sasaki, and K. Miura, Visualization of nanoscale peeling
of carbon nanotube on graphite, Appl. Phys. Lett., 93, 083122, 2008.
[2] M. Ishikawa, R. Harada, N. Sasaki, and K. Miura, Adhesion and peeling forces of carbon
nanotubes on a substrate, Phys. Rev. B, 80, 193406, 2009.
[3] M. Ishikawa, M. Ichikawa, H. Okamoto, N. Itamura, N. Sasaki, Kouji Miura, Atomic-scale
Peeling of Graphene, Applied Physics Express Vol. 5 (2012) 065102.
Fingerprint of Atomic Species in Friction at the Atomic Level
Denis Damiron, Pierre Alain, Dai Kobayashi, and Hideki Kawakatsu
Institute of Industrial Science, University of Tokyo, Japan
In ‘non-contact’ mode lateral force microscopy, the frequency shift curve
exhibits in UHV, though not confirmed on all samples, a W shaped profile.
When the tip sample distance is regulated to maintain the local minima
further away from the sample during raster scanning, atomic resolution is
obtained. For the case of Si(111), the frequency shift curve was of a typical
W shape, and mapping of the minimum frequency of oscillation showed
atomic resolution with 1 angstroem sized sites. The atomic features
exhibited different dfmin values from site to site. Although compared to
the deflection mode, further study is needed to explain what is being
mapped, the technique has the ability to detect differences in the tip
sample interaction in the lateral direction at the atomic level. We will
introduce the technique, and compare the technique with on-the-fly morse
parameter mapping in the deflection mode.
Nano-scale Control of Friction and Adhesion at
Surfaces and Interfaces
N. Sasaki1*, M. Suzuki1, K. Miura2, and H. Fujita3
1
Department of Engineering Science, The University of Electro-Communications
2
Department of Physics, Aichi University of Education
3
CIRMM, Institute of Industrial Science, The University of Tokyo
*e-mail: [email protected]
Control of friction and adhesion is one of the most practical problems in our daily life, which covers
almost all the areas from basic science to applied engineerings. In this talk we discuss our recent
numerical and experimental studies on atomic-scale control of friction and energy dissipation of
carbon and Si nanocontacts and interfaces.
First, in order to control friction, or reduce friction of nanocarbon interfaces, we have
experimentally developed the fullerene molecular bearings [1,2] and numerically evaluated the
ultralow friction of the graphene/C60 and graphene/graphene interfaces [3,4]. Anisotropy of friction
of graphite/C60 interface is found, where the maximum and minimum friction appears along the
commensurate and other scan directions, respectively. In order to control adhesion, we have
measured and simulated the peeling process of graphene [5,6]. Similar anisotropy is also found
during the peeling process at graphene/graphene interface. These results show that crystalline
anisotropy can be used for controlling magnitude of friction and its related adhesion. How the
mechanism of energy dissipation is included in the lateral force curve is also discussed.
Next the shear-fracture process of Si nanocontact is studied by molecular dynamics simulation
[7]. First amorphous contact region formed under the high loading condition of several GPa, slides
at Si/Si interface. Then it becomes thinner and longer to become nanowire, which elongates until it
breaks. The simulated contact diameter and angle are in good agreement with experimental one
observed by MEMS in TEM [7]. The shear fracture driven by amorphous deformation for Si
nanocontact is different from that driven by stick-slip process for carbon- and metal-nanocontact,
which can give us clue to understand one of the mechanism of dynamics of the single real contact.
References:
[1] K. Miura, S. Kamiya and N. Sasaki, Phys. Rev. Lett. 90, 055509 (2003).
[2] K. Miura, D. Tsuda, S. Kamiya and N. Sasaki, e-J. Surf. Sci. Nanotech. 3, 21 (2005).
[3] N. Itamura, K. Miura, and N. Sasaki, Jpn. J. Appl. Phys. 48, 060207(R) (2009).
[4] N. Sasaki, N. Itamura, H. Asawa, D. Tsuda, and K. Miura, Tribol. Online 7, 96 (2012).
[5] N. Sasaki, H. Okamoto, N. Itamura and K. Miura, e-J. Surf. Sci. Nanotech. 8, 105 (2010).
[6] M. Ishikawa, N. Sasaki, and K. Miura et al., Appl. Phys. Exp. 5, 065102 (2012).
[7] T. Ishida, T. Sato, T. Ishikawa, M. Oguma, N. Itamura, K. Goda, N. Sasaki and H. Fujita, Nano
Lett. 15, 1476 (2015).