1
Supplementary
Information
for
Illuminating the physics of dynamic friction through laboratory
earthquakes on thrust faults
Yuval Tal, Vito Rubino, Ares J. Rosakis, and Nadia Lapusta.
Ares J. Rosakis
Email:
arosakis@caltech.edu
This PDF file
includes:
Supplementary
t
ext
Figures
S1
to
S
8
Tables S1
SI
References
www.pnas.org/cgi/doi/10.1073/pnas.2004590117
2
Supplementary
Information Text
S1.
Laboratory setup
The laboratory setup provides full field measurements of displacements, velocities, strains,
and stresses associated with dynamic shear ruptures on preexisting inclined frictional
interfaces (Fig. S1). Two Homalite
-
100 quadrilateral plates with a frictiona
l interface
inclined at a dip angle
b
are loaded under uniaxial compression
P
(Fig.
S1), resulting in
initial shear and normal stresses on the interface of
t
0
=
P
sin(
b
)cos(
b
) and
s
0
=
P
sin
2
(
b
),
respectively. The values of
b
and
s
0
for the six experiments presented in the paper are
shown in table S1. The rupture is nucleated by a local pressure release provided by a rapid
expansion of a NiCr wire filament due to an electrical discharge of a high
-
voltage capacitor
(Cordin 640). A key
aspect of the setup is that the low shear modulus of Homalite enables
to produce well
-
developed dynamic ruptures in samples of tens of centimeters. Once the
rupture initiates, a target area coated with a random black
-
speckle pattern is monitored near
the
free
-
surface using an ultrahigh
-
speed camera system (Shimadzu HPV
-
X), capable of
recording up to 10 million frames per second, and a high
-
speed white light source system
with two light heads (Cordin 605) (Fig. S1). In the experiments reported here, the cam
era
records a sequence of 128 images of the patterns distorted by the propagating rupture with
a resolution of 400 x 250 pixels
2
, at temporal sampling of 1 million frames/second and
exposure time of 200 ns.
S
2
.
Full
-
fields analysis
We employ the methodolo
gy in
(1, 2)
to obtain full
-
field maps of displacements, velocities,
and stresses from the sequence of images acquired with the ultrahigh
-
speed
camera. We
analyze the images with the local digital image correlation (DIC) software Vic
-
2D
(Correlation Solutions Inc.) to produce evolving displacement maps, computed with
respect to a selected reference configuration. The 2D
-
DIC algorithms provide the
two in
-
plane displacement components at each subset center. In order to capture the discontinuous
displacement field across the interface, the correlation is performed separately for the
domains above and below the interface. While standard local DIC appr
oaches are able to
produce the displacement map up to half a subset away from the interface, the “Fill
boundary” algorithm of Vic
-
2D uses affine transformation functions to extrapolate the
displacements from the center of the subset up to the interface. We
filter the high
-
frequency
noise from the displacement fields using a non
-
local
-
means (NL
-
means) filter
(3
–
5)
and
have developed a post processing algorithm
(6)
that locally adjust the displacements
computed by DIC near the interface to ensure continuity of tractions across the interface.
We employ a frame system x
1
-
x
2
parallel to the interface and calculate the strains
from the filtered displacement fields using a finite difference approximation, with central
difference scheme for pixels away from the boundaries and second
-
order backward and
forward difference schemes
at the pixels immediately above and below the interface,
respectively
(1, 2)
. The stress changes with respect to the reference configuration (b
efore
3
rupture) are computed from the strain fields using the standard plane
-
stress linear elastic
constitutive equations. Because Homalite
-
100 is a strain
-
rate sensitive material at the strain
rate levels developed during the dynamic ruptures, we use the d
ynamic values of the elastic
constants to compute the stress changes
(1, 2)
. The actual stresses are obtained from the
stress changes by adding
the initial stresses. The full
-
field analysis enables to observe
displacements, velocities, and stresses close to the interface and study how the normal (
s
)
and shear (
t
) tractions, slip (
U
), and slip rate (
V
) evolve at any point along the interface. A
mo
re detailed description of the laboratory setup and full
-
fields analysis is given in
(1, 2)
.
S3.
Determining combinations of the parameters
a
v
,
b
v
,
a
f
, and
b
f
that agree with the
experimental data in ref.
(1)
In
model 3
of
the frictional response, we test a formulation of
rate
-
and
-
state friction
with
e
nhanced
weakening
and
Prakash
-
Clifton law
,
featuring
weakening parameters
V
w
and
f
w
that
decrease with
normal stress
in the form of a power law.
We consider only combinations
of the
power law
parameters
a
v
,
b
v
,
a
f
, and
b
f
that agree with the experimental data in
(1)
.
The experiments in
(1)
were performed with the same experimental setup, monitoring a
region close to the center of the specimen
. Because of the distance from the free surface,
the normal stress was nearly constant
a
nd equal to the initial normal stress (
s
0
).
We find the allowable combinations of
the parameters
a
v
,
b
v
,
a
f
, and
b
f
by fitting
results from
(1)
for the experiments under
s
0
= 10 MPa, within a certain tolerance, and
then narrow the sets of parameters by fit
ting the results from
(1)
for
s
0
= 5.7 and 17.6 MPa.
As a result, we obtain 23 combinatio
ns of the parameters
a
v
,
b
v
,
a
f
, and
b
f
, from which we
search for the best fit for Exp. #1.
Lets us describe the procedure for finding the allowable combinations of the
parameters
a
v
,
b
v
,
a
f
, and
b
f
in more details. For
s
0
= 10 MPa (
휎
"#
)
, there are two sets of
measurements, at
푉
&
"#
,
(#
.
*
= 0.2 m/s and
푉
&
"#
,
(+
= 3 m/s (Fig. S3). At steady
-
state,
퐿
-.
/
휃
→
푉
, thus equations 2 and 3 give
푓
&
"#
(
푉
)
=
푓
5
,
&
"#
+
7
89
(
:
)
;
7
<
,
=
>?
"
@
A
A
<
,
=
>?
,
(S1)
where
푓
-.
(
푉
)
is the steady state values of RS friction. The curve in equation
S1
should go
through values of friction
푓
&
"#
(
푉
&
"#
,
(#
.
*
)
and
푓
&
"#
(
푉
&
"#
,
(+
)
that are within the experimental
data. That can be achieved by choosing the enhanced weakening parameters
V
w
,σ
10
and
f
w,σ
10
at
s
0
= 10 MPa such that they satisfy:
푓
5
,
&
"#
+
7
89
C
:
=
>?
,
D?
.
E
F
;
7
<
,
=
>?
"
@
A
=
>?
,
D?
.
E
A
<
,
=
>?
=
푓
&
"#
C
푉
&
"#
,
(#
.
*
F
(S2)
and
푓
5
,
&
"#
+
7
89
C
:
=
>?
,
DG
F
;
7
<
,
=
>?
"
@
A
=
>?
,
DG
A
<
,
=
>?
=
푓
&
"#
C
푉
&
"#
,
(+
F
,
(S3)
4
which give
푉
5
,
&
"#
=
H
푓
&
"#
C
푉
&
"#
,
(#
.
*
F
−
푓
&
"#
C
푉
&
"#
,
(+
F
J
/
K
푓
&
"#
C
푉
&
"#
,
(#
.
*
F
−
푓
-.
C
푉
&
"#
,
(#
.
*
F
푉
&
"#
,
(#
.
*
−
푓
&
"#
C
푉
&
"#
,
(+
F
−
푓
-.
C
푉
&
"#
,
(+
F
푉
&
"#
,
(+
L
(S4)
and
푓
5
,
&
"#
=
푓
&
"#
C
푉
&
"#
,
(#
.
*
F
+
:
<
,
=
>?
:
=
>?
,
D?
.
E
H
푓
&
"#
C
푉
&
"#
,
(#
.
*
F
−
푓
-.
C
푉
&
"#
,
(#
.
*
F
J
.
(S5)
Considering that the experimental data may include errors and that the interfaces of
different experiments may slightly change, we allow some flexibility fitting to the
experimental data of
(1)
. We use a threshold value of
e
f
= 0.03 above and below the
experimental data and allow the values of
푓
&
"#
(
푉
&
"#
,
(#
.
*
)
and
푓
&
"#
(
푉
&
"#
,
(+
)
to range
between
f
σ
10,
V
0.2,min
-
e
f
and
f
σ
10,
V
0.2,max
+
e
f
and between
f
σ
10,
V
3,min
-
e
f
and
f
σ
10,
V
3,max
+
e
f
,
respectively (Fig. S4). Note that
, at
푉
&
"#
,
(+
= 3 m/s, the curve generated with
V
w
and
f
w
that
are independent of
s
(
V
w
= 1.1 m/s and
f
w
= 0.27) is lower than the experimental data by
more than 0.03. The values of
V
w,σ
10
and
f
w,σ
10
for a trial pair of
푓
&
"#
(
푉
&
"#
,
(#
.
*
)
and
푓
&
"#
(
푉
&
"#
,
(+
)
give two constraints for the parameters
a
v
,
b
v
,
a
f
, and
b
f
:
푉
5
,
&
"#
=
푏
N
휎
"#
O
P
⇒
푏
N
=
푉
5
,
&
"#
/
휎
"#
O
P
(S6)
and
푓
5
,
&
"#
=
푏
7
휎
"#
O
R
⇒
푏
7
=
푓
5
,
&
"#
/
휎
"#
O
R
.
(S7)
We use here
휎
"#
instead of
휓
(
휎
)
because each of the experiments in
(1)
was performed
under constant value of
s
=
s
0
.
Then, we use the frictional data for
s
0
= 5.7 MPa (
휎
T
.
U
)
and
s
0
= 17.6 MPa (
휎
"U
.
V
)
to provide additional constraints for the choices of the parameters
a
v
,
b
v
,
a
f
, and
b
f
for each
trial pair of
V
w,σ
10
and
f
w,σ
10
. To avoid repetition, we show in the following only the
development of the constraint for
s
0
= 5.7 MPa in details. The values of
V
w,σ
5.7
(
a
v
,
b
v
) and
f
w,σ
5.7
(
a
f
,
b
f
) should generate a curve of
푓
&
T
.
U
(V) that goes through a trial value
푓
&
T
.
U
(
푉
&
T
.
U
)
betwe
en
푓
&
T
.
U
,
WXY
−
ε
7
and
푓
&
T
.
U
,
W[\
+
ε
7
, where
푉
&
T
.
U
= 0.9 m/s is the average slip rate for
the frictional data of
휎
T
.
U
(Fig. S3). That gives the following condition:
푓
5
,
&
T
.
U
+
7
89
(
:
=
]
.
^
)
;
7
<
,
=
]
.
^
"
@
A
=
]
.
^
A
<
,
=
]
.
^
=
푓
&
T
.
U
(
푉
&
T
.
U
)
,
(S8)
where
푓
-.
(
푉
&
T
.
U
)
is the steady state values of RS friction for
푉
&
T
.
U
. Substituting
푉
5
,
&
T
.
U
=
푏
N
휎
T
.
U
O
P
and
푓
5
,
&
T
.
U
=
푏
7
휎
T
.
U
O
R
gives
푏
7
휎
T
.
U
O
R
+
푓
-.
(
푉
&
T
.
U
)
−
푏
7
휎
T
.
U
O
R
1
+
푉
&
T
.
U
푏
N
휎
T
.
U
O
P
=
푓
&
T
.
U
(
푉
&
T
.
U
)
,
(S9)
5
which together with the constraints in equations
S6
and
S7
becomes:
푓
5
,
&
"#
`
휎
T
.
U
휎
"#
a
O
R
+
푓
-.
(
푉
&
T
.
U
)
−
푓
5
,
&
"#
H
휎
T
.
U
휎
"#
J
O
R
1
+
푉
&
T
.
U
푉
5
,
&
"#
H
휎
T
.
U
휎
"#
J
O
P
=
푓
&
T
.
U
(
푉
&
T
.
U
)
.
(S10)
With some algebra:
`
휎
T
.
U
휎
"#
a
O
R
=
푓
&
T
.
U
(
푉
&
T
.
U
)
`
1
+
푉
&
T
.
U
푉
5
,
&
"#
H
휎
T
.
U
휎
"#
J
O
P
a
(
푉
&
T
.
U
)
−
푓
-.
(
푉
&
T
.
U
)
푉
&
T
.
U
푉
5
,
&
"#
H
휎
T
.
U
휎
"#
J
O
P
(S11)
and an expression for
a
f
is obtained:
푎
7
=
log
f
푓
&
T
.
U
(
푉
&
T
.
U
)
`
1
+
푉
&
T
.
U
푉
5
,
&
"#
H
휎
T
.
U
휎
"#
J
O
P
a
(
푉
&
T
.
U
)
−
푓
-.
(
푉
&
T
.
U
)
푉
&
T
.
U
푉
5
,
&
"#
H
휎
T
.
U
휎
"#
J
O
P
g
/
log
`
휎
T
.
U
휎
"#
a
(S12)
Similarly, for
s
0
= 17.6 MPa, the values of
V
w,σ
17.6
(
a
v
,
b
v
) and
f
w,σ
17.6
(
a
f
,
b
f
) should
generate a curve of
푓
&
"U
.
V
(V) that goes through a trial value
푓
&
"U
.
V
(
푉
&
"U
.
V
)
between
푓
&
"U
.
V
,
WXY
−
ε
7
and
푓
&
"U
.
V
,
W[\
+
ε
7
. That provides another expression for
a
f
:
푎
7
=
log
f
푓
&
"U
.
V
(
푉
&
"U
.
V
)
`
1
+
푉
&
"U
.
V
푉
5
,
&
"#
H
휎
"U
.
V
휎
"#
J
O
P
a
(
푉
&
"U
.
V
)
−
푓
-.
(
푉
&
"U
.
V
)
푉
&
"U
.
V
푉
5
,
&
"#
H
휎
"U
.
V
휎
"#
J
O
P
g
/
log
`
휎
"U
.
V
휎
"#
a
.
(S13)
For a given combination of trial values
푓
&
"#
(
푉
&
"#
,
(#
.
*
)
,
푓
&
"#
(
푉
&
"#
,
(+
)
,
푓
&
T
.
U
(
푉
&
T
.
U
)
,
and
푓
&
"U
.
V
(
푉
&
"U
.
V
)
, we use equations
S4
,
S5
,
S12
, and
S13
and solve numerically for
푎
N
and
푎
7
, using Matlab function “
fzero
”. Note that not all the combinations of trial values
provide valid solutions for the parameters
푎
N
and
푎
7
. In cases where a solution exists and
푎
N
<
0
and
푎
7
<
0
, we use equations
S6
and
S7
to obtain
푏
N
and
푏
7
. We test 5 values of
푓
&
"#
(
푉
&
"#
,
(#
.
*
)
between
f
σ
10,
V
0.2,min
-
e
f
and
f
σ
10,
V
0.2,max
+
e
f
, 4 values of
푓
&
"#
(
푉
&
"#
,
(+
)
between
f
σ
10,
V
3,min
-
e
f
and
f
σ
10,
V
3,max
+
e
f
, 7 values of
푓
&
T
.
U
(
푉
&
T
.
U
)
between
푓
&
T
.
U
,
WXY
−
ε
7
and
푓
&
T
.
U
,
W[\
+
ε
7
, and 4 values of
푓
&
"U
.
V
(
푉
&
"U
.
V
)
between
푓
&
"U
.
V
,
WXY
−
ε
7
and
푓
&
"U
.
V
,
W[\
+
ε
7
,
all at intervals of
D
f
= 0.01. That gives a total of 560 combinations. However, only 23
combinations provide an allowable set of the parameters
a
v
,
b
v
,
a
f
, and
b
f
that generate three
curves
푓
(
푉
,
휎
#
)
, for
휎
#
= 5.7, 10, and 17.6 MPa, that are all within the defined limit of 0.
03
from experimental data in
(1)
(Figs. S5A and S5B). The parameters
a
v
and
a
f
decrease
l
inearly with the logarithms of
b
v
and
b
f
, respectively (Fig. S5C), while there is no clear
relationship between
a
v
and
a
f
(Fig. S5D).
6
Fig. S1
. Schematics of the laboratory setup. Dynamic shear ruptures evolve spontaneously
along a frictional interface inclined at a dip angle
b
between two Homalite plates under a
compressional load
P
. Ruptures are initiated by the small burst of a NiCr wire plac
ed across
the interface and connected to a capacitor bank. The white light produced by a flash source
is reflected by the specimen’s surface and captured by a low
-
noise high
-
speed camera at 1
million frames/sec. A 19 x 12 mm
2
region to be imaged is coated
by a flat white paint and
decorated by a characteristic speckle pattern used for image correlation.
Frame 1
Frame n
P
18 cm
25 cm
Homalite
β
Ultrahigh speed camera
1 Million frames/s
128 frames
High voltage trigger
p
ulser (
Cordin
640)
Light Source
(
Cordin
605)
NiCr wire
F
oot wall
Hanging
wall
19 mm
12
DIC
I
nterface
-
parallel
displacement (
μ
m)
20
-
20
0
4
-
4
0
I
nterface
-
normal
displacement (
μ
m)
7
Fig. S2.
Local behavior of the ruptures near the free surface in all six experiments.
(
A
to
F
) The temporal evolution of
t
(red),
s
(black), and
V
(blue) at about 1.5 mm from the
free surface for six experiments performed under different values of
b
and
s
0
.
The curves
are generated using a temporal moving average, with a width of three data points, on the
local data for each quantity.
50 70 90 110 130
0
4
8
12
16
0
4
8
12
16
20
50 70 90 110 130
0
4
8
12
16
0
4
8
12
16
20
50 70 90 110 130
0
4
8
12
16
0
4
8
12
16
20
50 70 90 110 130
0
4
8
12
16
0
4
8
12
16
20
50 70 90 110 130
0
4
8
12
16
0
4
8
12
16
20
50 70 90 110 130
0
4
8
12
16
0
4
8
12
16
20
Exp #1:
β
=
61
o
σ
0
= 11.6 MPa
t (
μ
s
)
t (
μ
s
)
τ
and
σ
(MPa)
Exp #2
β
=
62
o
σ
0
= 11.5 MPa
Exp #3:
β
=
61
o
σ
0
= 7.6 MPa
Exp #4:
β
=
62
o
σ
0
= 7.7 MPa
Exp #6:
β
=
61
o
σ
0
= 3.7 MPa
Exp #5:
β
=
61
o
σ
0
= 5.7 MPa
V (m/s)
τ
σ
V
Supershear
Trailing Rayleigh
Late
supershear
Fault opening
Fault opening
A
C
E
F
D
B
8
Fig.
S3
. The steady
-
state values of
f
vs.
V
in the experiments in
(19)
that were monitored
with a small field of view and had slip rates larger than 0.1 m/s, as well as two low
-
velocity
measurements that were obtained in a different set of experiments in
(35)
. The data was
fitted in
(19)
with friction model 1, that is, a combined formulation of RS friction enhanced
by flash heating with the weakening parameters
V
w
= 1.1 m/s, and
f
w
= 0.27 (black curve).
-8-6-4-20
0.2
0.3
0.4
0.5
0.6
0.7
σ
0
= 10 MPa
σ
0
= 5.7 MPa
σ
0
= 17.6 MPa
low velocity experiments
f
log
10
(V) (m/s)