Illuminating the physics of dynamic friction through
laboratory earthquakes on thrust faults
Yuval Tal
a,b
, Vito Rubino
c
, Ares J. Rosakis
c,1
, and Nadia Lapusta
a,d
a
Seismological Laboratory, Division of Geological and Planetary Sciences, California Institute of Technology, Pasadena, CA 91125;
b
Department of Earth
and Environmental Sciences, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel;
c
Graduate Aerospace Laboratories, California Institute of
Technology, Pasadena, CA 91125; and
d
Division of Engineering and Applied Science, California Institute of Technology, Pasadena, CA 91125
Contributed by Ares J. Rosakis, April 15, 2020 (sent for review March 11, 2020; reviewed by Philippe H. Geubelle and Yonggang Huang)
Large, destructive earthquakes often propagate along thrust
faults including megathrusts. The asymmetric interaction of thrust
earthquake ruptures with the free surface leads to sudden varia-
tions in fault-normal stress, which affect fault friction. Here, we
present full-field experimental measurements of displacements,
particle velocities, and stresses that characterize the rupture inter-
action with the free surface, including the large normal stress reduc-
tions. We take advantage of these measurements to investigate the
dependence of dynamic friction on transient changes in normal
stress, demonstrate that the shear frictional resistance exhibits a
significant lag in response to such normal stress variations, and
identify a predictive frictional formulation that captures this effect.
Properly accounting for this delay is important for simulations of
fault slip, ground motion, and associated tsunami excitation.
dynamic friction
|
laboratory earthquakes
|
thrust faults
M
any large, destructive, tsunamigenic earthquakes occur
along thrust faults, such as the 2011 Mw 9.0 Tohoku
earthquake in Japan (1) and the 1999 Mw 7.7 Chi-Chi earth-
quake in Taiwan (2). An important feature of these two earth-
quakes is that the maximum slip occurred close to the free
surface (3
–
5) (Fig. 1), despite the notion that shallow portions of
the faults are supposed to be frictionally stable (e.g., ref. 6).
Numerical and theoretical studies have showed that the inter-
action of the ruptures with the free surface results in rapid var-
iations in the fault-normal stress,
σ
(7
–
11), which feed back into
the rupture process. The fault-normal stress affects the fault
frictional shear resistance, which controls the fault resistance to
sliding and, as a consequence, how far and how fast a fault slides.
For example, the unexpectedly large slip of 60 to 80 m in the
shallow portion of the 2011 Mw 9.0 Tohoku earthquake may
have been caused by this phenomenon (12, 13). While friction
traditionally is defined as being proportional to normal stress,
experimental studies (14
–
18) have shown that frictional shear
resistance does not change instantaneously in response to a step
changes in
σ
but rather evolves gradually with slip. This effect,
and its proper representation in friction formulations, is critically
important to understanding the dynamics of ruptures on thrust
and normal faults, as it affects near-fault shaking and tsunami
generation (7
–
10, 12, 13)
Recent advances in imaging spontaneous dynamic ruptures in
the laboratory (19
–
23) enable us to image and quantify the
normal stress reduction near the free surface and its effect on
friction for realistic fault-slip histories similar to natural thrust
earthquakes. We report and interpret measurements from our
laboratory earthquake experiments conducted in a setup, initially
designed in refs. 13 and 24 to mimic thrust earthquakes. The
original version of this setup used dynamic photoelasticity to
qualitatively visualize the rupture process while particle velo-
cimeters were employed to record the resulting ground motion
and to measure fault opening and slip at only a few discrete
locations on the fault and the free surface (13, 24, 25). The
current version, of this setup, described here, uses a newly de-
veloped, high-resolution, dynamic imaging technique based on a
combination of high-speed photography and digital image cor-
relation (DIC) (19
–
23). This method allows us to monitor the
entire rupture process within the field of view (FOV), to follow
individual ruptures as they approach and break the free surface
(Fig. 1
A
and
SI Appendix
, Fig. S1
), and to analyze this entire
process through, previously unavailable, full-field measurements
of displacements, velocities, strains, and stresses at time intervals
of 1
μ
s(
Materials and Methods
and
SI Appendix
). The analog
material used in the experiments results in critical length scales for
instability that are one to two orders of magnitude smaller than
those of rocks, allowing us to reproduce spontaneously propa-
gating dynamic ruptures in laboratory samples of manageable size.
Dynamic Rupture Interaction with the Free Surface
The experiments illustrate in vivid detail how the thrust ruptures
interact with the free surface while dynamically capturing all of
the quantitative information needed to constrain fault-normal
stress variations and friction formulations. In a representative
experiment (experiment 1, Fig. 2), a rupture approaches the free
surface with a peak particle velocity magnitude of 4 m/s at a time
t
=
55
μ
s after nucleation. Note that the velocity map so far is
symmetric with respect to the fault line. The velocity field of the
propagating rupture shows a transition from a dominant fault-
normal motion ahead of the rupture front to a fault-parallel
motion behind it. As the rupture arrives at the free surface
(
t
=
62
μ
s), the symmetry in velocity field is broken, and the peak
particle velocities increase to 8 and 4.5 m/s in the hanging and
foot walls, respectively. At a later time (
t
=
75
μ
s), sliding
Significance
Our study explores a major challenge in earthquake science
—
the dynamics of thrust earthquake ruptures as they interact
with the Earth
’
s surface, which is relevant to some of the most
destructive earthquakes that have ever occurred. The work il-
lustrates key features of the complex dynamic behavior asso-
ciated with this interaction using an ultrahigh-speed imaging
technique. The interaction leads to large and rapid reductions
in normal stress. However, the frictional shear resistance does
not decrease instantaneously with normal stress, as typically
assumed, but experiences a significant delay. Such delay is
important for a range of earthquake source problems that in-
volve rapid normal stress variations.
Author contributions: Y.T., V.R., A.J.R., and N.L. designed research; Y.T., V.R., A.J.R., and
N.L. performed research; Y.T. analyzed data; and Y.T., V.R., A.J.R., and N.L. wrote
the paper.
Reviewers: P.H.G., University of Illinois at Urbana
–
Champaign; and Y.H., Northwestern
University.
The authors declare no competing interest.
Published under the
PNAS license
.
1
To whom correspondence may be addressed. Email: arosakis@caltech.edu.
This article contains supporting information online at
https://www.pnas.org/lookup/suppl/
doi:10.1073/pnas.2004590117/-/DCSupplemental
.
First published August 17, 2020.
www.pnas.org/cgi/doi/10.1073/pnas.2004590117
PNAS
|
September 1, 2020
|
vol. 117
|
no. 35
|
21095
–
21100
EARTH, ATMOSPHERIC,
AND PLANETARY SCIENCES
Downloaded at California Institute of Technology on September 3, 2020
continues with smaller levels of particle velocity. Note that the
motion of the hanging wall is mostly vertical, while the footwall
has a large horizontal component. The supershear rupture just
described is followed by another rupture, traveling at a sub-Rayleigh
speed. This sub-Rayleigh rupture originated the supershear one
(26), and subsequently trails behi
nd it (22, 24, 27). As the trailing-
Rayleigh signature arrives to the free surface (
t
=
101
μ
s), particle
velocities increase, with larger values at the hanging wall. The
particle velocities of the footwall are almost completely horizontal
near the free surface. Full-field images of the fault-parallel velocity,
shear stress,
τ
, and normal stress,
σ
(Fig. 3
A
), provide further insight
into the rupture process. As the rupture propagates through the
FOV,
τ
shows a symmetric pattern with respect to the fault. The
shear stress increases above the initial stress level (
τ
0
=
6.4 MPa) at
the rupture tip and decreases to smaller values behind it. The
normal stress and fault-parallel v
elocity show asymmetric pattern
with respect to the fault.
Temporal Evolution of Stresses and Slip Rate near the Free
Surface
The pronounced reduction in the fault-normal stress
σ
and the
associated response of shear resistance
τ
are captured by plots of
time histories of
τ
,
σ
, and the slip rate
V
at a point on the fault
close to the free surface (Fig. 3
B
). As the rupture arrives,
σ
initially increases from
σ
=
11.6 to
σ
∼
13.5 MPa (
t
=
59
μ
s), and
then decreases to
σ
∼
8 MPa over about 11
μ
s. At the arrival of
the trailing Rayleigh rupture (
t
=
102
μ
s), there is additional
reduction to a minimum of
σ
∼
5 MPa. The shear stress increases
to
τ
∼
7 MPa at the arrival of the supershear rupture, then de-
creases rapidly to
τ
∼
4 MPa over about 3
μ
s. At later stages (
t
>
70
μ
s), there are only small variations in
τ
, with a total decrease
of 1 MPa over 70
μ
s. The slip rate shows two peaks of
V
=
12 and
8 m/s, corresponding to the arrival of the supershear rupture and
the trailing Rayleigh signature, respectively.
Interestingly, supershear transition later in the rupture process
leads to a more intense near-surface rupture. An experiment
(experiment 2, Fig. 4
B
) under similar loading conditions to those
in experiment 1, but with a weaker nucleation process, have
resulted in a later supershear transition and a later arrival of the
rupture to the free surface (
t
∼
82
μ
s). Because of the later
transition, most of the rupture energy in experiment 2 is still
carried by the trailing Rayleigh signature rather than the (newly
formed) supershear rupture front. Thus, the peak in
V
and re-
duction of
σ
associated with the first arrival of the rupture to the
free surface are lower than in experiment 1. However, as the
supershear rupture is reflected, it interacts with the arriving
trailing Rayleigh, resulting in a significantly more intense trailing
Rayleigh in experiment 2 than both the supershear rupture front
and the trailing Rayleigh in experiment 1, with peak
V
of 16 m/s
and reduction of
σ
to
a minimum of 2 MPa.
To create different types of ruptures and gather a suite of data
suitable for constraining friction formulations, we have con-
ducted experiments at different initial levels of normal stress
σ
0
(
SI Appendix
, Table S1
). In the already presented experiment
1,
σ
0
=
11.6 MPa. The temporal evolution of
τ
,
σ
, and
V
near the
free surface in experiments under a prestress level of
σ
0
=
7.6
and 7.7 MPa (experiments 3 and 4, Fig. 4
C
and
SI Appendix
, Fig.
S2) is consistent with that in experiment 1, although with smaller
values of
σ
and
V
. Experiments under lower values of
σ
0
=
5.7
and 3.7 MPa (experiments 5 and 6, Fig. 3
C
and
SI Appendix
, Fig.
S2) are characterized by weaker ruptures with a delay of about
10
μ
s in the arrival of the rupture and even lower peaks in slip
rate
V
.
A
B
Fig. 1.
Experimental setup used to study the dynamics of thrust earth-
quakes. (
A
) Experimental setup that mimics thrust faults: Dynamic shear
ruptures evolve spontaneously along a frictional interface under resolved
shear and normal stresses. An ultrahigh-speed camera is used to take a series
of images (10
6
frames per s) that are analyzed by digital image correlation
(DIC). The use of an analog material enables us to produce spontaneously
propagating dynamic ruptures on a laboratory scale. (
B
) The setup allows to
explore the interaction of thrust earthquakes with the free surface, which
has resulted in unexpected recent observations, such as the 60- to 80-m slip
close to the free surface in the 2011 Mw 9.0 Tohoku earthquake. Adapted
from ref. 5, which is licensed under
CC BY 4.0
.
AB
CD
Fig. 2.
The dynamics of an experimental thrust rupture during interaction
with the free surface. (
A
–
D
) Snapshots of the particle velocity magnitude
(experiment 1), with overlaid vectors showing the direction and magnitude
of particle velocity near the interface and at the free surface. The interaction
breaks the symmetry in the velocity field, and the peak particle velocities
increase as the rupture front (
B
) and the trailing Raleigh (
D
) arrive at the free
surface, with larger velocity at the hanging wall compared to the foot wall.
The snapshots highlight a transition from a fault-parallel motion behind the
rupture tip, as it propagates upward (
A
) and breaks the free surface (
B
), to a
subvertical motion of the hanging wall and dominant horizontal motion of
the footwall at later stages (
C
and
D
).
21096
|
www.pnas.org/cgi/doi/10.1073/pnas.2004590117
Tal et al.
Downloaded at California Institute of Technology on September 3, 2020
In the experiments under lower initial normal stress,
σ
is
completely released, for a short period of time, as the trailing
Rayleigh signature arrives to the free surface (Fig. 3
C
and
SI
Appendix
, Fig. S2
). This indicates a potential local opening of the
fault. However, the shear resistance never goes to zero and the
normal stress returns to compressive values shortly after, indi-
cating that the actual opening may not have occurred at least for
the set of loading parameters and the rupture scenario of
supershear rupture approaching the free surface as reported
here. Note that fault opening near the point where rupture meets
the surface was suggested in previous experiments based on laser
velocimetry measurements of the fault opening velocity (13) but
under much lower initial compression resulting in a very differ-
ent rupture scenario than the one investigated here (i.e., purely
sub-Rayleigh rupture in ref. 13 vs. supershear rupture with sub-
Rayleigh signature here).
Frictional Response under Rapid Normal Stress Variations
Our experimental measurements indicate that the shear resis-
tance does not obey the traditionally assumed proportionality to
the normal stress but evolves gradually, as suggested in refs.
14
–
16 and 18. This delay is directly observed in plots of the ef-
fective friction,
τ
/
σ
, vs. slip near the free surface (Fig. 4). For
experiment 1, the effective friction initially increases to
τ
/
σ
∼
0.6
and then decreases with slip to
τ
/
σ
∼
0.35 at slip of about 25
μ
m.
At larger levels of slip, when the impinging rupture is reflected at
the free surface,
σ
decreases, and because of the delayed re-
sponse of the frictional shear resistance
τ
, the ratio
τ
/
σ
increases
back to a value of 0.6 at a slip of 120
μ
m. The friction
τ
/
σ
gradually decreases at larger slip, but as the trailing Rayleigh
arrives and
σ
temporarily decreases,
τ
/
σ
increases again to a peak
of 0.7, and later drops to 0.4.
The measurements of
τ
,
σ
, and
V
along the interface enable
testing different formulations of frictional shear resistance, as
well as constraining their parameters. We find that friction for-
mulations without the delayed evolution of shear stress in re-
sponse to normal stress changes cannot fit our experimental
measurements.
Friction Model 1.
In this model, we test a formulation featuring
rate-and-state friction with enhanced weakening but without
accounting for delayed shear stress response to variations in
normal stress. The formulation assumes that the frictional shear
resistance is proportional to
σ
as follows:
τ
=
f
σ
,
[1]
where
f
is the friction coefficient. Previous experiments per-
formed in the same setup (19) showed that, for constant
σ
,
the frictional behavior is consi
stent with a combined formula-
tion of rate-and-state (RS) friction (28, 29) with aging evolution
law, enhanced with flash-heating weakening (FH) (30
–
33) in the
form of:
f
=
f
w
+
[
f
p
+
a
ln
(
V
V
p
)
+
b
ln
(
V
p
θ
L
RS
)]
−
f
w
1
+
L
RS
θ
V
w
,
[2]
_
θ
=
1
−
θ
V
L
RS
,
[3]
where
f*
is the friction coefficient at the reference velocity
V
*,
a
and
b
are RS friction parameters,
L
RS
is the characteristic slip for
the state variable evolution,
V
w
is the weakening slip velocity,
and
f
w
is the residual friction coefficient. Note that, at steady
state,
L
RS
=
θ
→
V
. We discretize Eqs.
1
–
3
in time and search
for a set of frictional parameters that together with the measured
values of
V
(
t
)
and
σ
(
t
)
at each time
t
gives a time series of mod-
eled effective friction,
τ
=
σ
, that agrees with the measured one.
The values of
θ
t
are calculated as in ref. 34. To avoid constraining
a large set of frictional parameters, we use the following, previ-
ously measured, parameters of Homalite (e.g., refs. 19 and 35):
f*
=
0.58,
V
*
=
10
−
6
m/s,
a
=
0.011,
b
=
0.16,
V
w
=
1.1 m/s, and
f
w
=
0.27 and test different values of
L
RS
. We track the evolution
of
τ
=
σ
at a point on the interface near the free surface (location
marked in Fig. 3
A
) in experiment 1 and find that with
L
RS
=
2
μ
m, the friction formulation captures the reduction of
τ
=
σ
at slip
smaller than 25
μ
m. However, at larger slip, as
σ
decreases, the
modeled response is significantly below the observed response
because the formulation does not account for the delayed
response (Fig. 4
D
).
Friction Model 2.
In this model, to account for the effects of rapid
normal stress variations, we test a formulation of RS friction with
enhanced-weakening featuring a delayed response of the shear
stress according to the Prakash
–
Clifton law. This model fits the
observed frictional response much better than model 1. In the
A
B
C
Fig. 3.
Experimental measurements of fault-parallel velocity, normal stress
reduction, and delay in shear resistance response as the rupture interacts
with the free surface. (
A
) Full-field images corresponding to the propagation
of the rupture upward through the FOV during experiment 1. (
B
) Local time
histories of
τ
(red),
σ
(black), and
V
(blue) near the free surface for experi-
ment 1 (location marked in
A
). The arrivals of the rupture front and trailing
Rayleigh lead to increases in
V
and decreases in
σ
. While
τ
initially decreases,
presumably due to velocity weakening, it barely responds to the variations in
σ
.(
C
) Local time histories of
τ
,
σ
, and
V
near the free surface for an exper-
iment performed under lower initial normal stress
σ
0
(experiment 5) show a
complete yet transient release of
σ
.
Tal et al.
PNAS
|
September 1, 2020
|
vol. 117
|
no. 35
|
21097
EARTH, ATMOSPHERIC,
AND PLANETARY SCIENCES
Downloaded at California Institute of Technology on September 3, 2020
Prakash and Clifton (PC) constitutive law (15, 16, 36), the shear
resistance is given by the following:
τ
=
f
ψ
.
[4]
The function
ψ
evolves exponentially with slip to the new value
of
σ
as
_
ψ
=−
V
L
PC
(
ψ
−
σ
)
,
[5]
where
L
PC
is another evolution distance. We discretize Eqs.
2
–
5
in time and search for the frictional parameters that together
with the measured values
V
(
t
)
and
σ
(
t
)
fit the observed frictional
response. Eqs.
4
and
5
require the values of
ψ
t
for each time,
which is obtained by discretizing Eq.
5
in time as follows:
ψ
t
=
(
ψ
t
−
1
−
σ
t
)
exp
(
−
V
t
L
PC
t
)
+
σ
t
,
[6]
where the initial value of
ψ
t
at the arrival of the rupture is
ψ
t
=
σ
0
. We use the frictional parameters as in the previous case
and estimate
L
PC
using a grid search, in which
L
PC
ranges be-
tween 0.1 (equivalent to a negligible delay) and 10,000
μ
m(a
large delay).
The inclusion of the delayed response in the formulation, with
L
RS
and
L
PC
as fitting parameters, enables to obtain a much
better fit between the experimental and modeled evolution of
τ
=
σ
in experiment 1 (Fig. 4
D
and
SI Appendix
, Fig. S6
). A value
of
L
PC
=
1,000
μ
m, which is three orders of magnitude larger
than the value of
L
RS
=
2
μ
m, the RS evolution distance, is
needed in order to capture the observed delay in shear resis-
tance. The root mean square (RMS) of the differences between
the observed and modeled responses (dashed black lines in
Fig. 5) decreases significantly from value of 0.17 to a value of
0.06 between
L
PC
=
10 and 1,000
μ
m, with most of the reduction
between
L
PC
=
30 and 600
μ
m, whereas, for
L
PC
>
1,000
μ
m,
there is only minor change in the RMS values. However, the
modeled values of
τ
=
σ
is consistently under the observed values,
suggesting that, in addition to the delayed response, the value of
f
w
in our experiment is slightly larger than that estimated in ref.
19. Note that the values of
V
w
=
1.1 m/s and
f
w
=
0.27 for the
weakening parameters were obtained in ref. 19 by fitting the
steady-state frictional behavior of Homalite at different slip
rates, using experimental data from experiments under different
initial normal stresses (
σ
0
)(
SI Appendix
, Fig. S3
).
Friction Model 3.
In this model, we improve the fit with the ob-
served response by considering a formulation of rate-and-state
friction with enhanced weakening and Prakash
–
Clifton law,
featuring weakening parameters that depend on normal stress.
This formulation is consistent with high-speed friction experi-
ments on gabbro that were performed under different normal
loads and showed that
V
w
(37) and
f
w
(38) decrease with
σ
in the
form of power laws:
V
w
=
b
v
σ
a
v
,
[7]
and
A
B
C
D
E
F
Fig. 4.
Experimental evidence, fitting, and subsequent prediction of pronounced delay in shear resistance response to rapid normal stress variations. (
A
–
C
)
Temporal evolution of
τ
(red),
σ
(black), and
V
(blue) near the free surface (location marked in Fig. 3
A
) for experiments 1 to 3. Although experiments 1 and 2
were performed under similar loading conditions, a later supershear transition in experiment 2 leads to a more intense near-surface rupture with larger peak
in
V
and larger reduction of
σ
.(
D
) Fitting of the measured effective friction
τ
/
σ
(experiment 1) for three models: 1) enhanced-weakening RS friction without
account for a delayed response to variations in
σ
(blue); 2) enhanced-weakening RS friction that accounts for the delayed response by the Prakash
–
Clifton law
(purple); and 3) enhanced-weakening rate-and-state friction with the Prakash
–
Clifton law and weakening parameters that depend on normal stress (red). The
experimental data are best fit by friction model 3 with the Prakash
–
Clifton evolution distance that is two to three orders of magnitude larger than that of
rate-and-state friction. (
E
and
F
) Comparison of the measured and predicted values of
τ
/
σ
for experiments 2 and 3 and friction model 3. The parameters
constrained with the data in experiment 1 allow us to predict the nontrivi
al friction evolution in experiments 2 and 3, as well as experiments 4 to 6
(
SI Appendix
,Fig.S7
).
21098
|
www.pnas.org/cgi/doi/10.1073/pnas.2004590117
Tal et al.
Downloaded at California Institute of Technology on September 3, 2020
f
w
=
b
f
σ
a
f
,
[8]
where
a
v
,
b
v
,
a
f
, and
b
f
are the power laws coefficients. Each of
the experiments in refs. 37 and 38 examined the steady-state
frictional behavior under constant sliding velocity and normal
stress. To account for rapid stress variations observed in our
experiments, we modify Eqs.
7
and
8
such that
V
w
and
f
w
evolve
gradually in response to a change in the normal stress as
V
w
=
b
v
ψ
(
σ
)
a
v
[9]
and
f
w
=
b
f
ψ
(
σ
)
a
f
.
[10]
We assume here that the same form of delay
(
ψ
)
affects all of the
frictional properties that vary with
σ
; thus, we avoid using addi-
tional fitting parameters. We discretize Eqs.
2
–
5
,
9
, and
10
in
time and search for a set of parameters
a
v
,
b
v
,
a
f
,
b
f
, and
L
PC
, that
together with the RS friction parameters above and the mea-
sured values
V
(
t
)
and
σ
(
t
)
fit the observed frictional response,
similarly to the analysis carried out for the previous two models.
As before, we estimate
L
PC
using a grid search, in which
L
PC
ranges between 0.1 (equivalent to a negligible delay) and 10,000
μ
m (a large delay). We do not simply perform a grid search over
the parameters
a
v
,
b
v
,
a
f
, and
b
f
, but consider only combinations
of the parameters that agree with the experimental data in ref. 19
(
SI Appendix
).
This formulation does fit better the observed behavior in ex-
periment 1, with the best fit obtained for evolution distances of
L
PC
=
370
μ
m and
L
RS
=
2
μ
m and power law parameters of
a
f
=
−
0.34 and
b
f
=
85 (Figs. 4
D
and 5). With these values,
f
w
varies between 0.33 and 0.36 (Eq.
10
), and the modeled effective
friction fits the observed friction at slip of 25 to 200
μ
m better
than model 2, although the additional improvement of the fit
from model 2 to model 3 is significantly smaller than that from
model 1 to model 2. Note that the differences between model 2
and model 3 should increase for the experiments performed
under lower value of
σ
0
(Eqs.
9
and
10
). Plots of the RMS of vs.
L
PC
for different values of
a
f
and
a
v
(Fig. 5
A
and
B
, respectively)
provide more insight into the differences between the models
and the effect of using different model parameters. The RMS
decreases from 0.18 for model 1 to 0.06 for model 2 with (
L
PC
≥
1,000
μ
m) and further to 0.045 for model 3, with a large effect of
the parameters
a
f
,
b
f
, and
L
PC
(Fig. 5
A
) and negligible effect of
the parameters
a
v
and
b
v
(Fig. 5
B
). The RMS value of 0.045 is
obtained for four combinations of the parameters
a
v
,
b
v
,
a
f
, and
b
f
, with the following parameter ranges:
a
f
=
−
0.346 to
−
0.334,
b
f
=
72 to 95,
a
v
=
−
1.55 to
−
0.14, and
b
v
=
7to8
×
10
9
. The
values of
L
PC
that give the smallest RMS value for each of the
four combination ranges between 350 and 415
μ
m.
Now that we have identified a formulation and its parameters
that capture the measurements of experiment 1, we verify its
predictive value by applying it to experiments 2 to 6, without any
changes in the parameters. Remarkably, we find that the for-
mulation in model 3, together with the parameters constrained
with the data in experiment 1, allows prediction of the friction
evolution near the free surface in the other experiments (Fig. 4
and
SI Appendix
, Fig. S7
). Accounting for the normal stress de-
pendence of the enhanced-weakening friction parameters (
V
w
and
f
w
through Eqs.
9
and
10
, respectively) in the PC law mostly
affect experiments 3 to 6 (
SI Appendix
, Figs. S7 and S8
), which
were performed under lower
σ
0
than experiment 1, and conse-
quently experienced smaller reductions in the friction coefficient.
For experiments 1, 2, 3, and 5, the decreases in RMS of the
differences between the observed and modeled frictional re-
sponse between models 1 and 2 is significantly larger than that
between models 2 and 3, while for experiments 4 and 6, the
difference between the RMS obtained with models 1 and 2
is comparable to or smaller than that between models 2 and 3
(
SI Appendix
,Fig.S8
).
Conclusions
Our study focuses on the conceptual frictional behavior of thrust
faults for cases with rapid variations in normal stress. We present
experimental measurements featuring rapid normal stress reduc-
tion as the ruptures interact with th
e free surface, with a temporary
complete release for experiments under the initial compressive
stresses of less than 5.7 MPa. Larger reduction in normal stress and
larger slip rates for a rupture with l
ate supershear transition com-
pared to that with early transition suggest that the former may be
more destructive in the case of thru
st faults, in contrast with current
assumptions that well-develope
d supershear ruptures are more
damaging. Our findings clearly
demonstrate the delay between
normal stress changes and the corresponding changes in frictional
resistance for the laboratory setting in an analog material, with
important implications for the dynamics of thrust earthquakes near
the free surface. In particular, our results indicate that the delay in
shear resistance response to variati
ons in normal stress is associated
with an evolution distance that is two to three orders of magnitude
larger than that of rate-and-state friction. Such delay is important
in other earthquake source problems that involve rapid normal
stress variations, such as seism
ic slip on nonplanar faults, on
bimaterial faults, and in the presence of shear-heating
–
induced
pressurization of pore fluids. Our past studies have demonstrated
that similar conceptual features uncovered in our experiments are
highly relevant to real faults in
rocks. However, to establish the
connection to real earthquakes, inc
luding the time delay, one would
B
A
Fig. 5.
Comparison of the observed response in experiment 1 between friction models 1 to 3. Root mean square (RMS) of the differences between the
observed and the modeled frictional response vs.
L
PC
for different values of
a
f
(
A
) and
a
v
(
B
). The colored curves represent different values of
a
f
and
a
v
used in
model 3, while the dashed black line represents the RMS for model 2 with weakening parameters
V
w
=
1.1 m/s and
f
w
=
0.27 that do not vary with
σ
. Note
that, as
L
PC
decreases, model 2 converges to model 1.
Tal et al.
PNAS
|
September 1, 2020
|
vol. 117
|
no. 35
|
21099
EARTH, ATMOSPHERIC,
AND PLANETARY SCIENCES
Downloaded at California Institute of Technology on September 3, 2020