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RESEARCH ARTICLE
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MAY 31 2022
An investigation of shock-induced phase transition in soda-
lime glass
Special Collection:
Shock Behavior of Materials
Akshay Joshi
;
Vatsa Gandhi
;
Suraj Ravindran
;
Guruswami Ravichandran
J. Appl. Phys.
131, 205902 (2022)
https://doi.org/10.1063/5.0086627
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Probing the properties and mechanisms of failure waves in soda-lime glass
J. Appl. Phys.
(May 2021)
06 October 2023 22:16:42
An investigation of shock-induced phase transition
in soda-lime glass
Cite as: J. Appl. Phys.
131
, 205902 (2022);
doi: 10.1063/5.0086627
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k
Submitted: 27 January 2022 · Accepted: 7 May 2022 ·
Published Online: 31 May 2022
Akshay Joshi,
1,2
,
a)
Vatsa Gandhi,
1
Suraj Ravindran,
1
and Guruswami Ravichandran
1
AFFILIATIONS
1
Graduate Aerospace Laboratories, California Institute of Technology, Pasadena, California 91125, USA
2
Delft University of Technology, 2628 CD Delft, The Netherlands
Note:
This paper is part of the Special Topic on Shock Behavior of Materials.
a)
Author to whom correspondence should be addressed:
ajoshi@caltech.edu
ABSTRACT
There exists a large body of evidence from experiments and molecular dynamics simulations to suggest the occurrence of phase transitions
in soda-lime glass (SLG) and other silica glasses subject to shock compression to pressures above 3 GPa. In light of these findings, the
current work investigated the existence of phase transition in SLG using shock and release experiments. The experiments employed symmet-
ric SLG
SLG impact to achieve complete unloading to zero stress after shock compression to stresses in the range of 3
7 GPa. The stress
strain response and the Lagrangian release wave speed behavior of SLG obtained from these experiments are seen to reveal a mismatch
between the loading and unloading paths of the pressure
strain curve for the material, which serves as compelling evidence for the occur-
rence of a shock-induced phase transition in the material at relatively low pressures. Furthermore, the release wave speed vs strain data
obtained from experiments were used to construct a methodology for modeling the shock and release behavior of SLG. This scheme imple-
mented in numerical simulations was able to capture the release behavior of shock compressed SLG, for which a robust and satisfactory
model was previously unavailable.
Published under an exclusive license by AIP Publishing.
https://doi.org/10.1063/5.0086627
I. INTRODUCTION
Shock compression and release experiments involve subjecting
a target material to high compressive stresses, using plate impact or
laser driven ablation, and releasing it to study its loading and
unloading response. This technique has been used to determine the
strength of a material
1
4
and its Grüneisen parameter
5
,
6
at high
pressures and strain rates. Shock and release experiments employ
optical velocimetry techniques such as velocity interferometer
system for any reflector (VISAR),
7
photon Doppler velocimetry
(PDV),
8
or embedded stress and strain gauges to infer the stress
strain loading and unloading history of the target material.
9
,
10
Further details of inferring the stress
strain history of the target
material from velocimetry data is provided in Sec.
II
.
Shock and release experiments have been conducted in previ-
ous works on soda-lime glass (SLG) and other silica glasses to
determine their Hugoniot elastic limit (HEL)
11
,
12
or to study onset
of phase transition in the material.
13
These studies on silica glasses
like SLG and fused quartz observed a progressively stiffer release
response for higher impact stresses. Possible causes for this
behavior are a gradual and irreversible phase transition occurring
in these glasses or a regular elastic release behavior in the material.
A more careful analysis of the release behavior of SLG and other
silica glasses is essential to verify the existence and properties of
phase transitions in the material. As will be seen in Sec.
IV
of this
work, such an analysis would have to involve the pressure
strain
curve of the material as opposed to just the stress
strain curve.
There are many anomalous properties associated with SLG
and other silica glasses in the 4
6 GPa pressure regime. These silica
glasses are known to undergo reduction in strength
14
,
15
and shear
modulus
16
with increase in pressure up to 5 GPa. For pressures
greater than 5 GPa, both strength and shear modulus increase with
pressure. Additionally, for impact stresses between 4 and 10 GPa,
SLG is observed to undergo a complete and sudden loss of spall
strength behind a failure front that travels significantly slower than
the compression wave. In the study by Joshi
et al.,
17
this failure
wave phenomenon was found to carry a significant densification
similar to what would be expected of a first-order phase transition.
5
It was speculated that localized densifications effected by phase
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transition in SLG were responsible for the observed failure wave
phenomenon, i.e, the failure wave was postulated to be a shock-
induced phase transition wave. Additional past studies, as discussed
next, indicate a possibility of the onset of phase transitions and
molecular rearrangements in silica glasses at pressures of 4
6GPa.
These phase transitions could be a cause for the anomalous proper-
ties of silica glasses observed at these pressures.
Silicon dioxide (SiO
2
, silica), which forms a major constituent
of SLG (

73%) and other silica glasses, has many crystalline
polymorphs such as
α
-quartz, coesite, and stishovite, all of which
are denser than amorphous silica. The temperature
pressure phase
diagram for SiO
2
18
indicates that
α
-quartz and coesite are the
thermodynamically favored crystalline structures for SiO
2
for pres-
sures of 4
6 GPa. The
α
-quartz to coesite phase transition, which
causes a volumetric densification of 9%
10% and occurs at
3
4 GPa pressure,
19
is known to be kinetically hindered and slow.
20
Crystallization of the
α
-quartz phase, from amorphous silica, was
not observed in previous shock compression experiments,
21
,
22
presumably because this transition is kinetically hindered at tem-
peratures achieved in these experiments. However, laser-driven
compression experiments on amorphous silica
21
seem to indicate
the onset of transition from the amorphous phase to the stishovite
phase at stresses of 4.7 GPa. More recent shock compression experi-
ments
23
were also able to obtain visual evidence for the shock-
induced nucleation of stishovite nanocrystals in soda-lime glass at
stresses of around 7 GPa. Another recent work involving quasistatic
compression of SLG nanopillars
24
also suggests the possibility of
SLG transforming to a stiffer stishovite phase at stresses of around
5 GPa. Past quasi-static compression experiments
25
and molecular
simulation studies
26
on amorphous SiO
2
also indicate the presence
of an ice-like first-order transition from low density amorphous
phase to high density amorphous phase at pressures of 3.6 GPa.
In their work involving molecular dynamics simulation studies,
Trachenko and Dove
27
attribute the observed anomalous change in
rigidity of fused silica (FS), in the 3
5 GPa pressure regime, to a
densification (increase in coordination number) in the amorphous
silica network, effected by molecular rearrangements.
The objectives of the present work on SLG, in context of the
aforementioned findings are as follows:
To study the release behavior of shock compressed SLG, with a
view toward discerning between the two possible mechanisms,
phase transition and regular ductile behavior, that explain the
stiffening of SLG
s release with increase in impact stress.
To develop a material model that adequately captures the release
behavior of shock compressed SLG. There are no robust and sat-
isfactory models available for this purpose yet.
The experiments in the current work are performed at impact
stresses of 4
7 GPa to probe the existence and onset of phase tran-
sition in SLG in a pressure regime where the material is known to
possess many anomalous properties. A salient feature of the
current shock and release experiments in comparison to previous
similar experiments on silica glasses
11
,
13
,
28
is that, for the first time,
complete unloading to zero stress is achieved using symmetric
SLG
SLG impact. This unloading to zero stress is critical to
unequivocally verify the existence of permanent densification in
shock compressed SLG, which was reported in earlier works
on SLG.
9
,
29
II. MATERIALS AND METHODS
The normal plate impact experiments conducted in this work
used either tungsten carbide (WC) or soda-lime glass (SLG) disks
to impact SLG disk targets as shown in
Fig. 1
. The SLG disks were
sourced from University Wafers, Inc., South Boston, MA, and had
densities of 2480
+
10 kg
=
m
3
. The target SLG disks were 5 mm
thick and 30 mm in diameter, while the impactor SLG disk was
3 mm thick and 30 mm in diameter. Both had an average surface
roughness of less than 1 nm. Aluminum rings were glued to the
disks as shown in
Fig. 1
to facilitate trigger upon impact. A 0.5
μ
m
thin aluminum layer was also deposited on the rear surface of the
SLG target to provide a reflective coating for velocimetry measure-
ments. The rear surface of the SLG was lightly scuffed with a 1200
grit sand-paper prior to aluminum deposition to obtain diffused
reflections from the surface. This is done to avoid any significant
loss in the intensity of light, received by the velocimetry probe, that
might occur due to shock-induced changes of the rear surface.
The WC impactor was of BC-00 grade and sourced from Basic
Carbide Corporation, Lowber, PA, and had a density of
15 480
+
100 kg
=
m
3
. The WC disk was 2 mm thick and 34 mm in
diameter. Lithium fluoride crystal (LiF[100]) disks were used as
windows in these experiments. These disks were 25.4 mm in diame-
ter, 6
:
32
+
0
:
01 mm in thickness, and were sourced from
ASPHERA, Inc., Santa Cruz, CA. The LiF crystals had densities of
2640 kg
=
m
3
and less than 2 degree misalignment between the
<100> crystal axis and the disk axis. No anti-reflective coating was
deposited on the LiF disks as Fresnel reflections from the rear
surface of the LiF window do not interfere significantly with
photon Doppler velocimetry (PDV) measurements.
30
The velocity
time profile of the SLG
LiF[100] interface was
obtained using PDV,
8
which employs a 1550 nm wavelength light
FIG. 1.
Schematic of the plate impact experiment used to study the shock com-
pression and release behavior of SLG using photon Doppler velocimetry (PDV).
The Down
Barrel probe (DBP) measures the velocity of the impactor. The pres-
ence of the LiF[100] window ensures that the SLG remains under compression
as the PDV probe records the SLG
LiF interface velocity. A 0.5
μ
m thick alumi-
num coating provides a reflective surface for PDV measurement. The cavity
behind the impactor ensures that the stress releases to zero at the rear surface
of the impactor.
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J. Appl. Phys.
131,
205902 (2022); doi: 10.1063/5.0086627
131,
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to probe the interface. Due to the presence of the LiF[100] window,
appropriate optical and impedance mismatch corrections are
applied to the observed velocity profile to obtain the in-material
particle velocity using the procedure described by Joshi.
31
Furthermore, the in-material velocity profile thus obtained is used
to construct the stress
strain loading history of the SLG target for
which the following differential equations are integrated:
5
,
32
d
ε
¼
du
C
h
(
u
)
,
(1)
d
σ
¼
ρ
0
C
h
(
u
)
du
,
(2)
where
σ
and
ε
are the axial stress and strain, respectively,
C
h
(
u
)is
the wave speed in material frame and
u
is the particle velocity in
the target material, which is initially at rest. All compressive strains
are defined to be positive in this work.
C
h
(
u
), and
u
are obtained
from the in-material particle velocity data as illustrated in
Fig. 2
.
31
The time
t
0
, as shown in
Fig. 2
, corresponds to the arrival of the
compressive wave at the rear surface of the impactor. For the scope
of this work, the entire release fan is assumed to originate from this
point (
X
¼
0,
t
¼
t
0
) in the material position
time (
X
t
) diagram.
This is an approximation that provides accurate results for wave
profiles with small compressive fans. The accuracy of this method
can be verified using results from experiments involving impact
stresses much less than the HEL of the material.
The only significant source of uncertainty in the computed
stress
strain profile is the time of trigger,
33
which can be caused by
a tilt in impact. For the experiments conducted in this work, the
tilt was lower than 1.5 mrad (0.086 degrees). Further details about
computing the uncertainties in stress are outlined in the
Appendix
.
III. RESULTS AND DISCUSSION
The impact velocities, stresses, and other details corresponding
to each plate impact experiment are summarized in
Table I
. The
uncertainties in particle velocities were computed using the proce-
dure outlined by Dolan.
34
The procedure for computing the uncer-
tainties in stresses are outlined in the
Appendix
. The in-material
velocities are computed by correcting for the impedance mismatch
between SLG and LiF[100].
31
The shock and release experiments, which involved peak axial
stresses of 3
7 GPa, were conducted to probe the release behavior
of SLG at stresses corresponding to the onset of the failure wave
phenomenon, which is suspected to be caused by a localized phase
transition in the material.
17
,
35
A. Experiment No. WSL-1
Experiment No. WSL-1 involved impacting a 2 mm thick WC
disk onto a 5 mm thick SLG disk target. Since the timings of rever-
berations of the stress-waves in the WC impactor were accurately
observed by velocimetry of the WC
SLG interface in experiment
No. AJ-2 of a previous work,
17
the impactor thickness and velocity
were chosen to replicate that experiment, except now with the pres-
ence of the LiF window. These reverberation timings were then
used to determine the speeds of the release waves arriving at the
SLG
LiF interface. A plot of the observed and optically corrected
interface velocity alongside a material position-time diagram is
FIG. 2.
Schematic of procedure to evaluate the wave speed in material frame,
C
h
(
u
). When impactors other than SLG are used (such as WC), one must
account for the multiple reverberations in the impactor plate. L
I
and L
t
are the
thicknesses of the impactor and target, respectively.
t
is the arrival time of the
wave and
t
= 0 is the time of impact.
TABLE I.
Summary of shock compression and release experiments on soda-lime glass.
Experiment
No.
Impactor
Thickness
(mm)
Impact velocity
(m/s)
Target thickness
(mm)
Peak interface
velocity
(m/s) (optically
corrected)
Peak in-material
velocity (m/s)
Peak stress
(GPa)
WSL-1
WC
2
N/A
5
411.5 ± 0.51
424.1
5.68 ± 0.26
SSL-2
SLG
3
1266.1 ± 2.40
5
523.3 ± 0.03
577.3
7.27 ± 0.25
SSL-3
SLG
3
879.6 ± 0.79
5
393.8 ± 0.53
411.1
5.51 ± 0.19
SSL-4
SLG
3
462.7 ± 2.14
5
230.9 ± 0.98
231.2
3.14 ± 0.13
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shown in
Fig. 3
. The measured interface velocity can be seen to
consist of high frequency oscillations, which become more pro-
nounced after the first and second unloadings. This noise can be
attributed to a reduction in the intensity of the light reflected back
to the PDV and were excluded from the velocity measurements in
order to accurately capture the stress
strain response of SLG. The
optically corrected data were thus smoothened, by interpolation, to
remove these oscillations while retaining the important features in
the data such as the two-wave structures of the first and second
release. The stress
strain curve for this experiment, constructed
from the smoothened interface velocity using the procedure out-
lined in Sec.
II
[Eqs.
(1)
and
(2)
], is shown in
Fig. 4
. The peak com-
pressive stress is computed to be 5.68 GPa, after which the
unloading is observed to occur in multiple steps, each effected by
the reverberation of the release waves in the WC impactor. The
release wave speeds for the second release fan are computed using
point B (see
Fig. 3
) as the origin of the second release fan. The
time coordinate of points A and B are determined by using the
velocimetry results from Expt. No. AJ-2 in Joshi
et al.
,
17
which
used a WC impactor of almost identical thickness.
A significant observation is that the loading and unloading
paths do not coincide. The unloading occurs only partially, up to a
stress of around 2.8 GPa, due to the use of an impactor (WC) with
impedance higher than the SLG target. However, the second
unloading can be seen to proceed parallel to the loading curve. It
can, thus, be assumed that a complete unloading would have
resulted in a permanent densification in the material. It is very
unlikely that this small hysteresis is due to onset of inelasticity in
the material, as a small hysteresis would require a small yield
strength for the material at these stresses, which would further
entail an unlikely and abrupt increase in pressure due to a reduced
deviatoric stress. As will be discussed later (Sec.
IV
), this observed
hysteresis in the stress
strain curve is more likely due to a hysteresis
in the pressure loading and unloading effected by a gradual phase
transition occurring in the material.
The impact velocity could not be measured in this experiment
due to the lack of light reflected back to the Down
Barrel probe
(DBP). Although this impact velocity was not necessary to con-
struct the stress
strain curve for the experiment, a consequence of
not knowing the impact velocity is that the existence and extent of
densification due to the failure wave, expected in SLG at these
stresses, cannot be estimated. As was shown in Joshi
et al.,
17
the
densification due to the failure wave is associated with a difference
between the observed and expected peak velocity. Without knowl-
edge of the impact velocity, the expected peak velocity and hence
the deficit in velocity cannot be estimated. As will be discussed
later, numerical simulations of this experiment were able to match
the observed peak velocity by considering an impact velocity of
490 m/s.
Additionally, due to reflections from the rear surface of the
LiF window, the PDV probe was able to record the velocity
time
history of the rear surface of LiF as well. Velocity measurements of
the LiF-free surface indicate a spall occurring in the LiF material,
which could be a possible reason for the loss of signal at later times
from the SLG
LiF interface. The peak velocity for the LiF rear
FIG. 3.
Velocimetry data from experiment No. WSL-1 for SLG
LiF interface and
LiF[100] free surface. A failure wave velocity of 1.3 km/s was assumed in con-
structing the material position-time (
X
t
) diagram.
17
The points A and B corre-
spond to the arrival of the first and second reverberations of the release wave,
respectively, at the WC
SLG interface.
FIG. 4.
Stress
strain curve for SLG deduced from experiment No. WSL-1. The
uniaxial strain is also identical to the volumetric strain 1

ρ
0
ρ

.
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surface can be seen to be approximately twice the optically cor-
rected peak interface velocity, thus lending further credibility to the
optical corrections applied to the observed interface velocity.
B. Experiment No. SSL-2
Experiment No. SSL-2 involved symmetric impact between a
3 mm thick SLG disk onto a 5 mm thick SLG disk target at an
impact velocity of 1266 m/s. The SLG target is backed by a 6.5 mm
thick LiF[100] window. A plot of the observed and optically cor-
rected SLG
LiF interface velocity alongside a material position
time diagram is shown in
Fig. 5
, which is used to construct the
stress
strain curve shown in
Fig. 6
. The peak compressive stress is
computed to be 7.27 GPa. It can also be observed that complete
unloading of stress to 0 GPa is achieved in this experiment, with a
major part of the unloading curve parallel to the loading curve. It
is, thus, observed that SLG retains a permanent (residual) volumet-
ric strain of around 2%, which is higher than the permanent strain
observed in experiment No. WSL-1.
The observed peak velocity for the LiF-free surface can again
be seen to be approximately twice the optically corrected peak
value for the interface velocity, thus lending further credibility to
the optical corrections applied to the observed data. A significant
observation can be made in the SLG
LiF interface velocity data
after they have been corrected for optical effects and impedance
mismatch between SLG and LiF. The resultant in-material particle
velocity can be seen to have a peak value of 577.3 m/s. This is
around 56 m/s smaller than 633 m/s, which would be the expected
peak velocity for symmetric SLG
SLG impact. This deficit occurs
due to a fast traveling release wave that arrives at the SLG
LiF
interface at 1.5
μ
s, thereby quenching/attenuating any compression
wave that travels slower than 3.3 km/s. These slower traveling waves
can be due to the inelastic (plastic) behavior of the SLG, which
gives rise to a slower plastic shock wave, or due to the failure wave
traveling at 1.3 km/s.
17
Thus, depending on the speed of this
second wave and the observed deficit in peak velocity, an additional
1.8%
4.6%
17
of volume densification/strain will have to be consid-
ered in the stress
strain curve. For further illustrations and calcula-
tions in this work, the additional densification will be assumed to
be 4.6% in magnitude and taken to be caused by the failure wave.
The stress
strain curve incorporating this densification is also
shown in
Fig. 6
.
C. Experiment No. SSL-3
This experiment involved impacting a 5 mm thick SLG disk
with a 3 mm thick SLG disk at 880 m/s impact velocity. The LiF
[100] window used was 8 mm thick. The velocimetry data for the
SLG
LiF interface and the LiF-free surface, obtained in this experi-
ment, are shown in
Fig. 7
. The optically corrected SLG
LiF inter-
face velocity is also shown in
Fig. 7
. Furthermore, the in-material
particle velocity for SLG is calculated from the SLG
LiF interface
velocity. These velocities are listed in
Table I
. A significant
observation is that the SLG in-material particle velocity, which is
expected to be exactly half of the impact velocity for symmetric
impact, is around 30 m/s lower than expected. As discussed in
Joshi
et al.
,
17
this deficit in velocity can be attributed to a phase
FIG. 5.
Velocimetry data of experiment No. SSL-2 for SLG
LiF interface and
LiF-free surface. A failure wave velocity of 1.3 km/s was assumed in the material
position
time (
X
t
) diagram.
FIG. 6.
Stress
strain curve for SLG material inferred from experiment No.
SSL-2 not accounting and accounting for densification due to the failure wave.
The uniaxial strain is also identical to the volumetric strain 1

ρ
0
ρ

.
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transition/densification in SLG, occurring at stresses of around
5 GPa. This densification causes an abrupt change in the slope of
the stress
strain equation of state (EOS) of SLG and is observed as
a slower traveling secondary compression wave. Joshi
et al.
17
observed that, for SLG, this secondary wave, commonly referred to
as the
failure wave,
travels at a speed of around 1.3 km/s. As
shown in the position
time (
X
t
) diagram in
Fig. 7
, this failure
wave is too slow to be observed at the SLG
LiF interface, being
attenuated by the longitudinal release wave from the rear surface of
the SLG impactor.
The observed SLG
LiF interface velocity, after appropriate
optical and impedance mismatch corrections, and analysis, reveals
the peak stress attained in the SLG to be around 5.5 GPa, similar to
what is observed in experiment No. WSL-1. The stress loading and
unloading paths, inferred from the optically corrected SLG
LiF
interface velocity, is shown in
Fig. 8
. The slope of and strain-change
across the
densification
section of the loading
unloading plot are
inferred by assuming the second wave
s speed to be 1.3 km/s
17
(same as the failure wave). The existence and extent of this densifi-
cation could not be verified in experiment No. WSL-1, as the
impactor velocity was not known in that experiment.
The unloading can be seen to proceed along a slightly steeper
path as compared to the loading path. A resultant mismatch
between loading
unloading paths and permanent densification of
over 2% can be observed in SLG.
D. Experiment No. SSL-4
This experiment involved impacting a 5 mm thick SLG disk
with a 3 mm thick SLG impactor with an impact velocity of 463 m/s.
The LiF window used was 8 mm thick. The velocimetry data for the
SLG
LiF interface and the LiF-free surface recorded for this
experiment are shown in
Fig. 9
. Optical and impedance mismatch
corrections are applied to the observed SLG
LiF interface velocity.
The optically corrected SLG
LiF interface velocity is shown in
Fig. 9
, and the impedance corrected peak velocity is shown in
Table I
. A significant observation is that the impedance corrected
(in-material) velocity is almost exactly half the impact velocity, as
expected in a symmetric impact experiments. This observation
serves as a verification of the densification inferred from the
velocity deficit in experiment No. SSL-3 and also validates the
optical correction scheme used in this work.
30
,
31
The SLG
LiF interface velocity, corrected for optical path
length change and impedance mismatch, was used to extract the
stress
strain history of the SLG. The loading
unloading history
for SLG is shown in
Fig. 10
, which suggests that the material
behavior is nearly elastic. The peak compressive stress attained in
the SLG target is computed using Eq.
(2)
to be 3.14 GPa. A minor
mismatch between the loading and unloading paths, with a result-
ing minor permanent densification of around 0.3% can also be
seen in
Fig. 10
.
E. Summary of experimental results
Figure 11
provides, in summary, a plot of SLG
s stress
strain
response for experiment Nos. WSL-1, SSL-2, SSL-3, SSL-4, AT-3,
and AT-4. The interface
velocity data for experiments Nos.
AT-3 and AT-4 were taken from Alexander
et al.,
28
corrected for
SLG
LiF impedance mismatch, and then processed to obtain the
stress
strain curve shown in the plot. Experiment Nos. AT-3 and
FIG. 7.
Velocimetry data from experiment No. SSL-3 for the SLG
LiF interface
and the LiF-free surface. A failure wave velocity of 1.3 km/s was assumed in the
material position
time (
X
t
) diagram. Impact velocity is 879.6 m/s.
FIG. 8.
Plot of stress
strain curve corresponding to experiment No. SSL-3. The
uniaxial strain is also identical to the volumetric strain 1

ρ
0
ρ

.
Journal of
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AT-4 were chosen for analysis here as they too used symmetric
SLG
SLG plate impact with LiF[100] windows for shock compres-
sion and release of SLG. Both Nos. AT-3 and AT-4 used 6 mm
thick SLG impactors and targets. Experiment No. AT-3 was con-
ducted at an impact velocity of 1.99 km/s, while experiment
No. AT-4 was at an impact velocity of 2.38 km/s. Complete unload-
ing to zero stress were not achieved in these experiments due to the
use of a thick impactor, which delays the unloading wave traveling
into the target from the impact surface. The use of an SLG impac-
tor makes the velocity data more reliable for the purpose of extract-
ing the stress
strain response. Due to the complete unloading to
zero stress achieved in experiment Nos. SSL-2, SSL-3, and SSL-4, it
can be observed that the loading
unloading hysteresis and perma-
nent densification starts in SLG at an impact stress of around
5 GPa and continues to increase as the impact stress increases. The
failure wave phenomenon, which is observed to occur in SLG that
is shock compressed to stresses between 4.7 GPa
36
,
37
and
10.8 GPa,
17
seems to be correlated with the onset of the aforemen-
tioned loading
unloading hysteresis at 5 GPa. It was further
observed in Joshi
et al.
17
that failure waves carried an additional
densification making them more reminiscent of phase transition
waves.
35
,
38
Thus, the permanent densification and stress
strain hys-
teresis observed in the current work could be, in part or wholly,
due to a phase transition occurring in SLG.
Figure 11
shows a plot of the observed stress
strain behavior
of SLG alongside the pressure isotherms of
α
-quartz
39
and
coesite.
18
Pressure isotherms of
α
-quartz and coesite were chosen
for comparison, as opposed to stress Hugoniots for the respective
materials, for a better approximation of the release (unloading)
curve of the materials, which is generally isentropic and largely par-
allel to the pressure
strain curve.
5
For a material undergoing
gradual, irreversible phase transition to a stiffer phase, the release
FIG. 9.
Velocimetry data from experiment No. SSL-4 for the SLG
LiF interface
and LiF-free surface. Impact velocity is 463 m/s.
FIG. 11.
Summary of stress
strain response of SLG and pressure vs strain
curves (isotherms) for polymorphs of silica (SiO
2
),
α
-quartz, and coesite.
SLG
LiF interface data for experiment Nos. AT-3 and AT-4 were taken from
Ref.
28
and processed to obtain the shown stress
strain plots. EOS-1 is the
equation of state for SLG used in Joshi
et al.
17
Equations of state for
α
-quartz
and coesite were taken from McWhan
39
and Hemley
et al.
,
18
respectively. The
strains were computed using 1

ρ
0
ρ

, with
ρ
0
¼
2480 kg
=
m
3
.
FIG. 10.
Stress
strain plot for experiment No. SSL-4. The uniaxial strain is also
identical to the volumetric strain 1

ρ
0
ρ

.
Journal of
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J. Appl. Phys.
131,
205902 (2022); doi: 10.1063/5.0086627
131,
205902-7
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06 October 2023 22:16:42
response is expected to become significantly stiffer with increasing
peak compressive stresses. This can be partly attributed to the
residual stiffer phase. It was postulated by Lacks,
26
through molecu-
lar dynamics simulations, that fused silica undergoes a transition to
a stiffer high density amorphous phase at pressures of 3
5 GPa. It
can also be noted that the unloading paths of SLG for experiment
Nos. AT-3 and AT-4 closely resemble the pressure isotherm for
α
-quartz. With progressive stiffening, the release response of SLG
can be expected to closely resembl
ethecoesiteisotherm.Itis,
therefore, very likely that SLG undergoes a localized transition
17
to one or more of these higher density phases at different stress
levels. A more compelling evidence for shock-induced phase tran-
sition in SLG is discussed in Sec.
IV
. Additionally, from
Fig. 11
it
is evident that the hysteresis observed for Expt. No. SSL-4 is
almost negligible, thereby validating the accuracy of the procedure
employed to infer the stress
strain history of SLG from the veloc-
imetry data.
Figure 12
provides a plot of the Lagrangian wave speeds of
compression and release waves observed in SLG for experiment
Nos. WSL-1, SSL-2, SSL-3, SSL-4, AT-3, and AT-4. The experimen-
tal observations for Lagrangian wave speeds in the present work
seem to be in very good agreement with results of previous shock
compression and release data for SLG.
11
From
Fig. 12
, it can be
seen that for SLG, the onset of hysteresis occurs at a strain of 0.04,
beyond which the compression and release wave speeds differ, and
the strain corresponding to the HEL is around 0.1, beyond which
the compression wave speeds increases with strain.
IV. PHASE TRANSITION IN SLG
Since, in general, plasticity is associated with the deviatoric
stresses in the material, the pressure
volume equation of state
(EOS) of the material can be assumed to be unaffected by inelastic
behavior. Thus, for the case of a regular ductile material without
phase transitions, neglecting the differences between the
shock-Hugoniot and the release isentrope, the loading and unload-
ing paths in the pressure
volume EOS can be assumed to be the
FIG. 13.
Schematic stress
strain and pressure
strain diagrams of shock
loading and release for materials with (a) regular inelastic response and (b)
phase transition. The major difference in the two diagrams is the significant mis-
match of the pressure vs strain loading and unloading paths for the latter case.
The pressure
strain slope at release
dp
d
ε

release
can be seen to be higher com-
pared to the slope at peak compression
dp
d
ε

loading
. In both cases, material is
assumed to lack strain hardening.
FIG. 12.
Plot of Lagrangian wave speeds of compression and release waves in
SLG. The data for experiments AT-3 and AT-4 are taken from Alexander
et al.
28
The compression (blue line) and release (yellow line) wave speeds for SLG pro-
vided as a function of peak particle velocity in Alexander
et al.
11
are plotted
here as functions of strains (refer to the Appendix in Joshi
et al.
17
for the proce-
dure). The onset of hysteresis is the strain (0.04) at which the release and com-
pression wave speeds start to differ significantly. The strain (
ε
) is computed as
1

ρ
0
ρ

.
Journal of
Applied Physics
ARTICLE
scitation.org/journal/jap
J. Appl. Phys.
131,
205902 (2022); doi: 10.1063/5.0086627
131,
205902-8
Published under an exclusive license by AIP Publishing
06 October 2023 22:16:42