of 7
Evolution of d
ynamic shear strength of frictional
interfaces
during rapid normal stress variations
Vito
Rubino
1
*
,
Yuval
Tal
2
,3
,
Ares J.
Rosakis
1
,
and
Nadia
Lapusta
2
,
4
1
California Institute of Technology
,
Department of Aerospace (
Galcit)
, 91125
Pasadena
(
CA
), USA
2
California Institute of Technology
,
Division of Geological and Planetary Sciences
, 91125
Pasadena
(
CA
), USA
3
Ben
-
Gurion University of the Negev
,
Department of Earth and Environmental Sciences,
84105
Beer
-
Sheva
,
Israel
4
California Institute of Technology
,
Division of Engineering and Applied Science, 91125
Pasadena
(
CA
), USA
Abstract.
Pressure shear plate impact tests have revealed that when normal
stress changes rapidly enough, the frictional shear resistance is no longer
proportional to the normal stress but rather evolves with slip gradually.
Motivated by these findings, we focus on characterizing the dynamic shear
strength of frictional interface
s subject to rapid variations in normal stress.
To study this problem, we use laboratory experiments featuring dynamic
shear cracks
interacting with
a
free surface and resulting in pronounced and
rapid normal stress variations. As dynamic cracks tend to pr
opagate close to
the wave speeds of the material, capturing their behavior poses the
metrological challenge of resolving displacements on the order of microns
over timescales microseconds. Here we present our novel approach to
quantify the full
-
field behav
ior of dynamic shear ruptures and the evolution
of friction during sudden variations in normal stress, based on ultrahigh
-
speed photography (at 1
-
2 million frames/sec) combined with digital image
correlation. Our
measurements allow us to capture th
e evolution of
dy
namic
shear
cracks
during these short transients and enable us to decode the nature
of dynamic friction.
1
Introduction
The study of
dynamic
shear
cracks
along frictional interfaces
is relevant
to
a wide range of
problems, including fiber pull
-
out in the failure of composite materials, airplane actuators,
and earthquakes.
Friction formulations typically assume shear resistance to be proportional
to normal stress. However, when normal stress changes
rapidly
, frictional shear resistance no
longer obeys proportionality to the normal stress
. Instead
it
evolves with slip gradually
.
It is
important to
emphasize
the
highly
dynamic nature of this phenomenon as it occurs over
timescales
on the order
of
hundr
eds of
nanoseconds to microseconds.
The delayed response
*
Corresponding author:
vito.rubino@caltech.edu
© The Authors, published by EDP Sciences. This is an open access article distributed under the terms of the Creative Commons
Attribution License 4.0 (http://creativecommons.org/licenses/by/4.0/).
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of the shear resistance
to rapid normal stress variations was first
observ
ed
in experiments
performed using the pressure shear plate impact (PSPI)
test
by
Prakash and Clifton
[1]
(Figure 1)
.
This work was
developed
in
the context of high
-
speed machining applications and
was aimed at better characterizing the frictional response of the interface between tool and
workpiece to extend tool life and optimize machining costs.
Th
e proper
representation
of
these
effects
in fri
ction formulations is
key
for investigations of
a wider class of
engineering
and geophysics
problems
. Some examples
includ
e:
(i) slip on locally rough/nonplanar
interfaces
[2]
;
(ii) dynamic rupture on
bimaterial systems
[3
-
8]
;
and
(ii
i
) the dynamics of
ruptures on thrust and nor
mal faults near the Earth’s surface, which is important for near
-
fault shaking and tsunami generation, and which contains rapid changes in the fault
-
normal
stress due to the interaction of rupture with the free surface
[9
-
11]
.
Several different
formulations have been proposed that incorporate this effect
[1, 3, 12
-
14]
. Ignoring this effect
results in ill
-
posed problems and wrong results of numerical simulations
[e.g. 3, 4]
.
Recently,
we have developed an approach based on ultrahigh
-
speed digital image
correlation (DIC)
[15, 16]
that enables us to quantify the full
-
field behavior of dynamic
ruptures
[17, 18
]
and the evolution of dynamic friction
[19]
.
One key feature of our
experimental setup is that it is able to produce and record rapid normal stress variations
during dynamic rupture next to the intersection of the
interface
with the s
ide free surface
[11]
,
and therefore it is well suited to
study this problem
.
Previous versions of this
experimental
setup
ha
ve
been suc
cessfully used to reproduce a number of dynamic rupture phenomena,
including supershear transition, bimaterial effect, and pulse
-
like rupture propagation
[5, 20
-
22]
.
In
this study, we discuss our dynamic measurements and analysis aimed at understanding
the evolution of the shear resistance in response to rapid normal stress variations, based on
our recent advances
[23]
.
Fig. 1.
(a)
Experimental setup employed by Prakash and
Clifton (1993)
[1]
to produce sudden normal
stress variations.
(b) Effect of sudden normal stress change on shear stress which is equal to shear
resistance during slip: instead of changing immediately and proportionally, the shear stress
experiences gradua
l evolution. Modified from Prakash and Clifton (1993).
2
Experimental setup
and data processing
The lab
oratory
setup features a dynamic rupture along
the
frictional interface
, inclined by
an angle
a
,
formed by two quadrilateral sections of Homalite
-
100
(Figu
re
2
).
The applied
compressive load
P
results in normal and shear prestresses on the interface given by
!
=
cos
"
and
!
=
sin
cos
, respectively.
Ruptures are
initiated
by the small burst of a Ni
-
Cr wire placed across the interface. A flash light (Cordin 640)
is
triggered 200
μ
s
prior
to the
wire detonation in order to ramp up in intensity and
guarantee
adequate illumination during
(a)
(b)
2
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the transient. The imaged area of the s
pecimen is covered by a black and white speckle pattern
to provide a characteristic pattern for image matching.
2.1
Full
-
field measurements
with
traction continuity across the interface
Evolving ruptures are captured by an ultrahigh
-
speed camera (Shimadzu HPV
-
X) at 1
-
2
million frames/sec, depending on the experiment, and with an exposure time of 200 ns
(Figure 2)
. The sequence of images is subsequently analyzed with a local digital image
correlation algorithm (VIC
-
2D, Correlated Solutions Inc.). The analysis is
performed
independently over the two domains separated by the interface
,
in order to
avoid smoothing
of the displacement discontinuity across the interface
.
Accurate measurements of
displacements and
stresses near the interface are very important to study
the dynamics of
shear cracks
. However, DIC algorithms involve small errors that can lead to non
-
physical
discontinuities in the stress field across the interface.
In order to enforce traction continuity
across the interface, we employ an algorithm we have
developed
to
locally adjusts the
displacements computed by DIC
[24]
. The procedure is based on extrapolating the
displacements near the interface using polynomials constructed u
sing a constrained
inversion, such that (i) traction continuity is satisfied at the interface and (ii) displacements
produced by DIC are matched at the center of the pixel where the measurements are accurate.
Particle velocity
and strain
fields are
subsequ
ently
obtained by temporal
and spatial
differentiation
,
respectively
,
of the displacement fields
[15]
.
2.2
Ca
pturing the evolution of f
riction
al shear resistance
S
tress
maps
are
computed
from strain
fields
using linear elastic constitutive properties, with
the dynamic Young’s modulus of Homalite
E
= 5.3 GPa and Poisson’s ratio
n
= 0.35
[15]
.
The
se
measurements provide the stress changes associated with dynamic rupture
propagation. The total levels of
normal and shear
stress
es
are obtained by summing the stress
change to the prestress levels
!
and
!
, respectively.
Friction
time history
is compute
d
as
the ratio of
shear to normal stress along the interface and it
is
tracked
along
with
that
of slip
(relative displacement along the interface) and its rate.
Fig
.
2.
Schematic of the experimental configuration used in this study featuring a dynamic r
upture
along a frictional interface approaching the free surface and an ultrahigh
-
speed camera capturing the
transient event at 1 million frames/sec.
3
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Fig
.
3
.
F
ull
-
field
of
measurements
capturing
rupture propagation near the free surface
with the
associated
normal stress reduction and shear stress evolution.
Full
-
field
interface
-
parallel velocity
(left), shear stress (middle), and
interface
-
normal stress (right) at different stages of a supershear
crack
-
like rupture for
a test
with
a
= 29
o
,
P
= 15 MPa, and field of view of
18.7
x 11
.7
mm. The first
four rows correspond to stages when the rupture arrives
to
and
is
reflected from the free surface,
while the last
row
corresponds to the arrival of the
trailing Rayleigh
rupture
.
3
Dynamic measurements of full
-
field quantities and friction
Our recently developed
full
-
field imaging technique
[15, 24]
enables
us
to image the
particle
velocities
and stress changes within a field of view (FOV) close to the free surface (Figure
3).
This approach
allows us to both
produce full
-
field maps capturing
the dynamic
s
of the
phenomenon
and to quantify
the evolution of
interface
-
normal stress, shear stress,
and hence
friction along the interface close to the free surface
(Figures 4a and 4b)
. These measurements
allow us
to
study how friction evolves under the conditions of rapid normal stress variations
and to distinguish between different proposed formulatio
ns.
The
f
ull
-
field
interface
-
parallel
velocity, shear stress, and interface
-
normal stress
are shown
in Figure 4
at different stages of
a supershear crack
-
like rupture for a test
performed
with
a
= 29
o
and
P
= 15 MPa, and
with
a
field of view of 18.7 x 11.7
mm
[23]
. The first
four rows correspond to stages when the
rupture arrives to and is reflected from the free surface, while the last row corresponds to the
arrival of the trailing Rayleigh rupture.
Our experimental measurements indicate that the shear resistance does not obe
y the
traditionally assumed proportionality to the normal stress but evolves gradually. This delay
is directly observed in plots of the effective friction,
τ
/
σ
, vs. slip near the free surface (Figure
4
)
[23]
. For experiment 1
in Figure 4
, the effective friction initially increases t
o
τ
/
σ
0.6 and
4
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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
response 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
formulations without the delayed evolution of shear stress in response
to normal stress
changes cannot fit our experimental measurements. In model 1, we test a formulation
featuring rate
-
and
-
state
(RS)
friction
[25, 26]
enhanced
with flash heating (FH)
weakening
[27, 28]
but without accounting for delayed shear stress response to variations in normal
stress
, so that
=
푓휎
and the frictio
n coefficient
f
evolves according to
=
#
+
$
%
&
'()
*
"
"
+
&
,()
-
"
#
$
%&
.
/
0
%
'
1
&
$
%&
#
"
'
(1)
̇
=
1
23
4
%&
,
(
2
)
where
f*
is the friction coefficient at the reference velocity
V
*,
a
and
b
are RS friction
parameters,
67
is the characteristic slip for the state variable evolution,
V
w
is the weakeni
ng
slip velocity, and
f
w
is the residual friction coefficient.
We track the evolution of
τ
/
σ
at a point
on the interface near the free surface (location marked in Figure
3, bottom right panel
) in
experiment 1 and find that 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 a
ccount for the delayed
response (Figure 4
c
). In model 2, to account for the effects of rapid normal stress variations,
we test a formulation of rate
-
and
-
state friction with enhanced
-
weakening featuring a delayed
response of the shear stress according to th
e Prakash
Clifton law
[1, 3, 29]
:
=
푓휓
, where
the function
evolves with slip as:
̇
=
!
"
!"
(
)
,
(
3
)
where
L
PC
is
a characteristic slip scale. This formulation was proposed in the context of
regularizing the ill
-
posed problem of sliding along a bimaterial interface.
This model fits the
observed frictional response much better than model 1 (Figure 4d). In model 3, we improve
the fit with the observed response by considering a formulation of ra
te
-
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 experiments
performed under different normal loads and showed that
V
w
[30]
and
f
w
[31]
decrease with
σ
in the form of power laws. This formulation does fit better the observed behavior in
experiment 1. Once identified a formulation and its parameters that capture the measurements
of experiment 1, we
examine
its predictive value. Remarkably, we find that th
e
same
formulation
and
parameters allows prediction of the friction evolution near the free surface
in
various
other experiments
.
A comparison with a selected experiment (labelled as
experiment
3) is shown in Figure 4
. Accounting for the normal stress depe
ndence of the
enhanced
-
weakening friction parameters (
V
w
and
f
w
) in the PC law mostly affect experiments
performed under lower
σ
0
than experiment 1, and consequently experienced smaller
reductions in the friction coefficient.
5
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Fig. 4
.
Experimental
measurements showing the delayed response of the frictional shear resistance to
rapid normal stress variations and fitting with friction models that capture this phenomenon (modified
after Tal et al., 2020).
(
a
b
)
Time histories
of
τ
(red),
σ
(black), and
V
(blue) near the free surface (
for
the
location
marked
in the bottom right panel of
Fig
ure
3) for experiments
labelled as
1
and
3. (
c
)
Fitting of the measured effective friction
τ
/
σ
(experiment 1) for three models
discussed in the text.
The experimental d
ata 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. (
d
) Comparison of the
measured and predicted values of
τ
/
σ
for experiment
3 and friction
model 3. The parameters
of the
friction model, which have been
constrained
using
the data
of
experiment 1
, coupled with the
measured velocity time history of experiment 3,
allow us to predict the nontrivial friction evolution in
this
experiment, as well as
for various other
experiments
not show here.
4
Conclusions
In this study we have shown the behavior of dynamic shear cracks approaching and
interacting with a free surface
, using ultrahigh
-
speed photography coupled with digital image
correlation
. The full
-
field measurements show in vivid detail the
particle
motion
characterizing the
rupture
dynamics
and the associated rapid normal stress reductions
,
occurring over timescales of microseconds.
Our findings clearly demonstrate
a significant
delay be
tween normal stress changes and the corresponding changes in
the shear
resistance
.
These conclusions,
based on a
laboratory
configuration featuring the
interaction of a shear
rupture with the free surface,
have
important implications for the dynamics of th
rust
earthquakes near the free surface.
In particular, our results indicate that the delay in shear
resistance response to variations in normal stress is associated with an evolution distance that
is two to three orders of magnitude larger than that of rat
e
-
and
-
state friction. Such delay is
also
important in
many engineering and geophysics
problems that involve rapid normal stress
variations, such as slip on nonplanar
interfaces and
bimaterial
systems.
We acknowledge the following sources of support:
NSF
(Grant EAR
-
1651235), US Geological Survey
(USGS) (Grant G20AP00037), Caltech/ Mechanical and Civil Engineering Big Idea Fund (2019),
Caltech’s Division of Geological and Planetary Sciences, and the Southern California Earthquake
Center (SCEC) (Contributio
n 19093). SCEC is funded by NSF Cooperative Agreement EAR
-
1033462
and USGS Cooperative Agreement G12AC20038.
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EPJ Web of Conferences
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, 01016 (2021)
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