# Shock-wave-induced cavitation of silicone oils

Justin Huneault  Department of Mechanical Engineering, McGill University, 817
Sherbrooke St. West, Montreal, QC H3A 0C3, Canada  Andrew Higgins
<EMAIL_ADDRESS>Department of Mechanical Engineering, McGill
University, 817 Sherbrooke St. West, Montreal, QC H3A 0C3, Canada

###### Abstract

The cavitation threshold of polydimethylsiloxane (silicone) oils was studied
using the planar impact of flyer plates to generate large transient negative
pressures within the liquids. The plate-impact experiments used a 64-mm-bore
gas-gun to launch thin sabot-supported flyer plates onto liquid capsule
targets in which a thin Mylar diaphragm formed a free surface at the back of
the sample. The shock wave driven into the target capsule by the flyer impact
placed the silicone oil in tension upon reflection from the rear free surface,
eventually causing the sample to cavitate. The spall strength, or critical
tension which cavitates the liquid, was determined by monitoring the free-
surface velocity using a photonic Doppler velocimetry system. This study
explored the effect of viscosity and loading strain rate on a system of three
silicone oils having vastly different viscosities
($4.8\text{\times}{10}^{-2}\text{\,}\mathrm{Pa}\text{\,}\mathrm{s}2.9\text{\times}{10}^{1}\text{\,}\mathrm{Pa}\text{\,}\mathrm{s}$),
but otherwise similar properties. The spall strength was found to remain
constant over the ranges of strain rate and viscosities probed in this work. A
comparison of the experimental results to models for the cavitation threshold
of liquids suggested that homogeneous nucleation of bubbles was the dominant
mechanism for tension relief at the onset of cavitation.

††preprint:

##  I INTRODUCTION

Liquids can be made to support significant tension despite the fact that they
are in a non-equilibrium state at pressures below their vapor pressure.
Trevena (1987) Tension in the liquid is relieved through the nucleation and
growth of vapor cavities in a process known as cavitation. Surface tension
creates an energy barrier which must be overcome for cavitation bubbles growth
to occur Fisher (1948), thus allowing liquids to exist in a metastable state
of tension equivalent to that of a superheated fluid. Cavitation bubbles
typically form from heterogeneous nucleation sites, including free bubbles and
bubbles attached to impurities within the liquid or at container walls.
Kedrinskii (2005) However, for sufficiently pure liquids or in cases where
tension is applied to the liquid on a sufficiently short timescale, cavitation
bubbles may nucleate from random molecular fluctuations in a process of
homogeneous nucleation. Fisher (1948); Zeldovich (1992); Carlson and Levine
(1975); Utkin and Sosikov (2009)

The reflection of a shock wave from a free surface can momentarily place a
liquid in a state of tension due to the interaction of the resulting reflected
expansion front with an expansion front behind the shock. This highly dynamic
phenomenon causes the liquid to be stretched until sufficient tension is
reached to cause rupture through the formation and growth of bubbles. The
reflection of a pressure pulse from a free surface has been used extensively
to study the behavior of liquids in tension Bull (1956a); Erlich, Wooten, and
Crewdson (1971); Carlson and Levine (1975); Trevena (1987); Utkin and Sosikov
(2009), and plays an important role in a number of physical processes,
including underwater explosions Kolsky  _et al._ (1949); Cole (1948);
Kedrinskii (2005), the fracture of kidney stones using shock wave lithotripsy
Zhu  _et al._ (2002), and the formation of jet-like instabilities during the
explosively driven dispersal of liquids. Milne  _et al._ (2017) Shock-wave-
induced cavitation is also of interest for a recently proposed Magnetized
Target Fusion concept in which a spherically imploding shock wave is used to
collapse a liquid metal cavity onto a plasma target in order to reach fusion
conditions. Laberge (2008, 2009); Suponitsky, Froese, and Barsky (2014);
Suponitsky  _et al._ (2017)

Bull Bull (1956a) was the first to develop an experimental apparatus that used
pressure wave reflection from a free surface to measure the maximum tension in
the liquid prior to cavitation. In these bullet-piston experiments, an impact-
driven pressure pulse is fed into a long vertical column of liquid with a free
surface in contact with a gas atmosphere near the top. Bull (1956a) The
incident pressure pulse reflects from the free surface as an expansion front
that generates significant transient tension in the liquid. Pressure sensors
mounted along the liquid column measure the incident pressure pulse that moves
up the column as well as the peak tension in the liquid caused by the
reflected tensile pulse. Trevena (1987) To access even greater loading rates,
Erlich et al. Erlich, Wooten, and Crewdson (1971) pioneered the use of spall
experiments, which had previously been used to study the dynamic fracture of
solids, to study cavitating liquids. In these experiments a planar shock wave
is driven into a target liquid, typically using the impact of a flyer plate,
and allowed to reflect from a free surface at the rear of the sample. As will
be described in Section II.2, the maximum tension within the liquid can be
inferred by observing the time evolution of the free-surface velocity. Antoun
_et al._ (2003) Such experiments determine the critical tension at which the
liquid begins to fail in tension under the loading conditions of the
experiment, a value typically referred to as spall strength. In spall tests,
the loading dynamics ensure that cavitation originates within the bulk liquid,
rather than at the walls of the container, which means that the maximum
tension is a true measurement of the limit of cohesion of the liquid. Utkin
and Sosikov (2009)

A number of studies have looked at the effect of viscosity and strain rate on
the cavitation threshold of liquids. Using a bullet-piston type experiment,
Bull Bull (1956a) found an empirical power law relationship
($P_{\mathrm{s}}\sim\eta^{0.2}$) between dynamic viscosity ($\eta$) and the
tension at the onset of cavitation ($P_{\mathrm{s}}$) for an assortment of
liquids spanning four orders of magnitude in viscosity. The observed increase
in the cavitation threshold with increasing viscosity was attributed to a
larger resistance to viscous growth of voids. Bull (1956a) Couzens and Trevena
Couzens and Trevena (1974) studied the relationship between spall strength and
viscosity for silicone oils over a range of three orders of magnitude using
bullet-piston experiments and found a similar, albeit less sensitive
relationship between cavitation threshold and viscosity
($P_{\mathrm{s}}\sim\eta^{0.06}$). A comparison of critical tension
measurements for glycerol obtained from the bullet-piston experiments of Bull
Bull (1956b) and the spall experiments of Erlich et al. Erlich, Wooten, and
Crewdson (1971), shows that the critical tension was seen to increase from
approximately $6\text{\,}\mathrm{MPa}$ at bullet-piston strain rates
($\dot{\epsilon}\approx$${10}^{2}\text{\,}{\mathrm{s}}^{-1}$) to
$23\text{\,}\mathrm{MPa}$ at spall experiment strain rates
($\dot{\epsilon}\approx$${10}^{5}\text{\,}{\mathrm{s}}^{-1}$). Grady developed
an energy-based spall strength model involving the viscous growth of pre-
existing bubbles in a liquid Grady (1988a), and showed relatively good
agreement with the data of Erlich et al. and Bull, indicating that the
observed increase in cavitation threshold with increasing strain rate may be
attributed to the greater viscous dissipation as the deformation rate is
increased.

In contrast to the results quoted above, which indicated that the cavitation
threshold of liquids may be determined by a heterogeneous nucleation
mechanism, previous studies performed at high strain rates
($\dot{\epsilon}>$${10}^{4}\text{\,}{\mathrm{s}}^{-1}$) have shown evidence
that the spall strength of liquids is determined primarily by a homogenous
cavitation mechanism. Carlson and Levine Carlson and Levine (1975) used an
electron beam to determine the spall strength of glycerol over a wide range of
temperatures (491 to 623 K), where the viscosity changes by five orders of
magnitude. The experiments showed a factor of eight decrease in spall strength
as temperature was increased. Carlson and Levine (1975) Utkin and Sosikov
Utkin and Sosikov (2009) used flyer-driven spall experiments to compare the
effect of varying strain rate on the spall strength of liquids having vastly
different viscosities: glycerol, hexane, methanol, and water. They observed
that the spall strength in hexane, methanol, and water was nearly independent
of strain rate over a range of approximately
${10}^{4}\text{\,}{\mathrm{s}}^{-1}{10}^{5}\text{\,}{\mathrm{s}}^{-1}$ Utkin
and Sosikov (2009), which was attributed to the weak dependence of the
cavitation threshold on the loading rate according to homogeneous nucleation
theory. Fisher (1948); Zeldovich (1992) In contrast, the spall strength of the
glycerol samples was seen to increase by a factor of 2.5 over a variation in
strain rate of
$1.5\text{\times}{10}^{4}\text{\,}{\mathrm{s}}^{-1}2.0\text{\times}{10}^{5}\text{\,}{\mathrm{s}}^{-1}$.
Both Carlson Carlson and Levine (1975) and Utkin Utkin and Sosikov (2009) used
models to show that the observed variation in the spall strength of glycerol
with temperature and strain rate can be attributed to the fact that the
loading rates in the experiments were on the same timescale as the relaxation
time needed to reach the steady-state homogeneous void nucleation rate, rather
than being caused by a heterogeneous viscous void growth mechanism. They
claimed that for nearly pure liquids placed in tension at sufficiently high
strain rates, the dominant mechanism which relieves tension and determines the
spall strength of the liquid is the homogeneous nucleation of bubbles. Carlson
and Levine (1975); Utkin and Sosikov (2009) From the theoretical work of
Fisher Fisher (1948) and Zeldovich Zeldovich (1992), the critical tension
required to cavitate a liquid should be strongly affected by its surface
tension, but only weakly related to its viscosity or the loading rate of the
experiment, provided that the relaxation time needed to reach the steady-state
void nucleation rate is much smaller than the loading rate. Carlson and Levine
(1975)

This paper, which builds upon the results of a previously reported preliminary
study Huneault  _et al._ (2018), will examine the effect of viscosity and
strain rate on the spall strength of a set of silicone oils which have vastly
different viscosities, but otherwise similar material properties. The
experiments will focus on observing the relationship between the measured
spall strength and the liquid properties or loading characteristics. The
results will be compared to existing models for the limiting tension of
liquids, in order to offer insight into the mechanism of cavitation
nucleation.

##  II MATERIALS AND METHODS

###  II.1 Materials

This study will examine the spall behavior of a set of three
polydimethylsiloxane (PDMS) silicone oils, sourced from Clearco Products Co.,
with dynamic viscosities ($\eta$) that span over two orders of magnitude, but
otherwise similar material properties (density ($\rho$), speed of sound
($c_{0}$), surface tension ($\sigma$), and bulk modulus ($K$)). The relevant
material properties for the three fluids can be seen in Tab. 1, where the oils
have been identified by their nominal kinematic viscosity in Stokes (St). As
can be seen from the molecular weight ($M$) values in Tab. 1, the difference
in viscosity between the oils is due to a variation in the average length of
the polymer chains, which affects the resistance of the fluid to shear.
Patterson (1998)

Table 1: Properties of the silicone oils used in the study. |  |  $\eta$111Taken from manufacturer data. |  $\rho_{0}$111Taken from manufacturer data. |  $c_{0}$222Taken from a study using a similar silicone oil.Leibacher, Reichert, and Dual (2015) |  $\sigma$111Taken from manufacturer data. |  $K$333Calculated from $K=c_{0}^{2}\rho_{0}$. |  $M$111Taken from manufacturer data.
---|---|---|---|---|---|---|---
Silicone Oil | Product No. | ($\text{\,}\mathrm{Pa}\text{\,}\mathrm{s}$) | ($\text{\,}\mathrm{kg}\text{\,}{\mathrm{m}}^{-3}$) | ($\text{\,}\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$) | ($\text{\,}\mathrm{N}\text{\,}{\mathrm{m}}^{-1}$) | ($\text{\,}\mathrm{GPa}$) | ($\text{\,}\mathrm{kg}\text{\,}{\mathrm{mol}}^{-1}$)
0.5 St | PSF-50cSt | $4.80\text{\times}{10}^{-2}$ | 960 | 1004 | $2.08\text{\times}{10}^{-2}$ | 0.97 | $3.78$
10 St | PSF-1,000cSt | $9.71\text{\times}{10}^{-1}$ | 971 | 1004 | $2.12\text{\times}{10}^{-2}$ | 0.98 | $2.80\text{\times}{10}^{1}$
300 St | PSF-30,000cSt | $2.93\text{\times}{10}^{1}$ | 976 | 1004 | $2.13\text{\times}{10}^{-2}$ | 0.98 | $9.17\text{\times}{10}^{2}$

A series of steady-state rheometry measurements were taken in order to verify
the viscosity of the silicone oils presented in Tab. 1 and determine their
shear rate sensitivity. The measurements were taken using an Anton Paar MCR
502 rotational rheometer with a concentric cylinder arrangement, which was
able to achieve shear rates of
$3\text{\times}{10}^{3}\text{\,}{\mathrm{s}}^{-1}$ for the low and
intermediate-viscosity oils and
$5\text{\times}{10}^{2}\text{\,}{\mathrm{s}}^{-1}$ for the high-viscosity oil.
Figure 1 shows the measured dynamic viscosity of the silicone oils as a
function of the shear rate. The zero-shear-rate viscosity of the fluids agreed
with the manufacturer provided values presented in Tab. 1. All three oils had
a constant viscosity at low shear rates, which was seen to transition to a
shear thinning behavior at higher shear rates for the intermediate and high-
viscosity oils. The measured zero-shear-rate viscosity ($\eta_{0}$) and
critical shear rate ($\dot{\gamma}_{c}$) for the onset of shear thinning are
presented in Tab. 2.

Figure 1: Steady-state measurements of viscosity as a function of shear rate for the silicone oils used in this study. Table 2: Steady-state measurements of silicone oil viscosity as a function of strain rate. | $\eta_{0}$ | $\dot{\gamma}_{c}$
---|---|---
| ($\text{\,}\mathrm{Pa}\text{\,}\mathrm{s}$) | ($\text{\,}{\mathrm{s}}^{-1}$)
0.5 St oil | $5.17\text{\times}{10}^{-2}$ | $2\text{\times}{10}^{2}$
10 St oil | $1.05$ | $2\text{\times}{10}^{3}$
300 St oil | $3.08\text{\times}{10}^{1}$ | -

The shear thinning behavior discussed above has been observed previously in
PDMS oils. Barlow  _et al._ (1964); Ghannam and Esmail (1998); Carré and Woehl
(2006); Vázquez-Quesada  _et al._ (2017) These oils are known to behave as
Newtonian fluids at low shear rates, then begin to show shear thinning
behavior at a critical shear rate which increases with decreasing zero-shear-
rate viscosity. Ghannam and Esmail (1998); Carré and Woehl (2006); Vázquez-
Quesada  _et al._ (2017) The onset of shear thinning indicates that at these
deformation rates, the viscoelastic properties of the PDMS oils, which result
from interactions of the polymer molecules, begin to affect the flow. The high
strain rates encountered in the experiments presented in this study ($\approx
10^{4}\,\mathrm{s}^{-1}$), will result in a very dynamic cavitation process
with large bubble growth rates that are likely to induce viscoelastic material
behavior.

###  II.2 Plate-impact experiments

In this study, the planar impact of flyer plates onto liquid samples was used
to determine the spall threshold of the PDMS oils. Poly(methyl methacrylate)
(PMMA) or aluminum (Al 6061) flyer plates, held by a sabot, were launched from
a 64 mm-bore single-stage gas gun, and impacted the liquid target assembly. A
magnet was embedded in the sabot in order to monitor the impact velocity of
the flyer plate using a series of magnetic coil gauges located at the end of
the projectile launch tube. The front face of the target assembly consisted of
a 3-mm-thick Polycarbonate driver plate which transmitted the shock wave into
the test liquid upon impact. The liquid was contained within an aluminum ring
which had fill ports through which the liquid was injected via Luer-Lok
fittings. Prior to being injected in the capsule, the oils were degassed under
vacuum (approximately 100 Pa) until free bubbles no longer appeared. The rear
surface of the liquid was held in place by a 51-$\upmu$m-thick aluminized
polyester sheet (Mylar), which formed a free surface through which the
incident shock wave reflected to place the sample in tension. Although there
was a slight impedance mismatch between the liquid sample and the Mylar, the
short acoustic travel time across the film (on the order of
$30\text{\,}\mathrm{ns}$) ensured the film had no significant effect on the
dynamics of the experiment. The rearmost portion of the target assembly held
the collimating optical probe used to track the velocity of the Mylar surface.
A sealed air gap between the Mylar surface and the probe allowed the liquid
and Mylar to move freely during the experiment, which lasts on the order of
$1\text{\,}\mathrm{\SIUnitSymbolMicro s}$. The dimensions of the target
assembly were chosen such that waves caused by the interaction of the impact-
driven shock wave with the boundaries of the liquid did not reach the central
axis in time to interfere with the spall measurement, thus ensuring that the
strain remained uniaxial. The target assembly was fastened in a manner which
ensured that the Mylar sheet was taut and free of wrinkles and was mounted
onto a flange on the end of the gas gun launch tube in order to ensure a
planar impact. Face-seal o-rings and stopcock valves were used to seal the
target assembly, which allowed the target chamber to be evacuated to a 200 Pa
Helium atmosphere before the shot, while maintaining a 1 atm environment
within the target assembly. A labelled schematic of the projectile and target
assembly is shown in Fig. 2(a), while a labeled picture of the target assembly
mounted onto the gas gun can be seen in Fig. 2(b).

Figure 2: (a) Schematic of the liquid capsule assembly, and (b) labelled
picture of the assembly mounted onto the launch tube of the gas gun.

Plate-impact experiments rely on measuring the time evolution of the free-
surface velocity to determine the spall strength of the material being
studied. Antoun  _et al._ (2003) The use of aluminized Mylar, which is
reflective to infrared radiation, allowed the free-surface velocity to be
monitored by a photonic Doppler velocimetry (PDV) system. Strand  _et al._
(2006) PDV measures surface velocity by observing the variation in beat
frequency between a laser beam reflected off a moving surface and a reference
beam. The system has a trade-off between time resolution and velocity
accuracy, due to the fact that the beat frequency is obtained by performing a
sliding window fast Fourier transform (FFT). Dolan (2010) A window sample size
of 2000 to 3000 samples was used during the sliding FFT analysis, resulting in
an estimated velocity uncertainty of 2 m/s and a time resolution of
approximately 10 ns. Figure 3, shows a typical spectrogram produced by
performing a sliding window FFT on the PDV data, where the high intensity
contours represent the velocity of the free surface. The extracted
velocity–time curve for the experiment is also shown in black. It should be
noted that previously reported experiments Huneault  _et al._ (2018) related
to this study used significantly thinner aluminized Mylar diaphragms
($4\text{\,}\mathrm{\SIUnitSymbolMicro m}$). The
$4\text{\,}\mathrm{\SIUnitSymbolMicro m}$ diaphragms had a lower reflectivity
and a tendency to wrinkle which affected the PDV return signal, sometimes
leading to noisy data or lost experiments. Increasing the Mylar thickness to
$51\text{\,}\mathrm{\SIUnitSymbolMicro m}$ improved the quality and
consistency of the velocimetry data, and was not seen to affect the spall
strength measurements.

Figure 3: Typical spectrogram obtained from performing a sliding window FFT on
the PDV data. The black curve is the evolution in the free-surface velocity as
a function of time, which is obtained from the analysis.

###  II.3 Analysis methods

This section will outline the methods used to analyze the free-surface
velocity of the target liquid in order to determine the maximum pressure in
the liquid prior to unloading, the rate at which tension increases in the
liquid prior to cavitation, and the maximum tension in the liquid (i.e., spall
strength). A schematic of the wave diagram for the plate-impact experiment is
shown in Fig. 4(a). Initially, the impact drives a shock wave into both the
driver plate and the flyer plate. The shock in the driver is transmitted into
the liquid sample, while the shock in the relatively thin flyer plate rapidly
reflects from its free surface as an expansion front, which is then
transmitted into the driver and liquid. Nonlinearity allows the head of the
expansion front from the flyer free surface to overtake the shock wave for
sufficiently long travel distances. The thickness of the flyer plate and
target capsule in these experiments have been chosen such that the shock is
unsupported by the time it reaches the free surface (i.e., the shock is
directly followed by an expansion front). The shock in the liquid reflects
from the rear free surface as an expansion front. The two expansion fronts
then interact with each other and generate a rapidly increasing tension in the
liquid, which is eventually relieved by cavitation at a critical tension which
corresponds to the spall strength of the liquid. The resulting increase in
pressure caused by cavitation forms a compressive wave, known as a spall
pulse, which travels towards the rear free surface.

Figure 4: (a) Position-time wave diagram of a plate-impact experiment which
demonstrates the mechanism for shock induced cavitation. (b) Schematic of the
evolution in the particle velocity and pressure within the silicone oil target
during a plate-impact experiment. The key points in the loading path are
labelled in part (a) of the figure.

The process described above is illustrated in the pressure–particle velocity
plot in Fig. 4(b), while a representative example of the time evolution of the
liquid free-surface velocity is shown in Fig. 5, where the main features of
the signal have been labeled. The arrival of the shock wave increases the
pressure and particle velocity in the liquid to state (1), which transitions
to state (2) as the reflected expansion front relieves the pressure and
doubles the particle velocity. On the free-surface velocity trace, the shock
wave manifests itself as a sharp increase in pressure, with the peak velocity
($u_{\mathrm{max}}$) corresponding to state (2). As can be seen in Fig. 5,
wave reverberations in the Mylar sheet cause a slight delay in reaching the
peak velocity behind the shock. The interaction of both expansion fronts
further reduces the pressure into the tensile region (state (2)-(3)), which
manifests itself on the free-surface velocity trace as a nearly linear
decrease in velocity. This velocity pullback signal has a slope which is
proportional to the rate at which tension develops in the sample. The state of
maximum tension ($P_{\mathrm{s}}$, state (3)), corresponds to the point where
tension is relieved by cavitation in the liquid, while state (4) corresponds
to the minimum free-surface velocity ($u_{\mathrm{min}}$) observed in Fig. 5,
which occurs at the arrival of the cavitation-induced spall pulse.

Figure 5: Labeled schematic of the evolution in the silicone oil target free-
surface velocity as a function of time for a plate-impact experiment.

From Fig. 4(a), it can be seen that cavitation is expected to initiate at the
intersection of the tail of the rear free-surface expansion front (line
(2)-(3)) with the forward-moving characteristic where the critical tension is
first reached (line (3)-(4)-(5)). Therefore, in the acoustic approximation,
the spall strength can be evaluated from the velocity pullback ($\Delta
u_{\mathrm{fs}}=u_{\mathrm{max}}-u_{\mathrm{min}}$) recorded on the free
surface-velocity trace using the following expression Antoun  _et al._ (2003)

$P_{\mathrm{s}}=\frac{1}{2}\rho_{0}c_{0}\Delta u_{\mathrm{fs}},$ (1)

where $\rho_{0}$ and $c_{0}$ are the ambient density and speed of sound of the
liquid. Similarly, the strain rate is inferred by fitting a linear slope to
the free-surface velocity pullback signal ($\dot{u}_{\mathrm{fs}}$) Luo  _et
al._ (2009)

$\dot{\epsilon}=\frac{-\dot{u}_{\mathrm{fs}}}{2c_{0}}.$ (2)

It is also of interest to estimate the maximum pressure experienced by the
liquid target as it is shock compressed prior to unloading. The particle
velocity behind the shock wave ($u_{\mathrm{p}}$) can be estimated from the
maximum free-surface velocity during shock-up using the following relation
Walsh and Christian (1955)

$u_{\mathrm{p}}\approx\frac{1}{2}u_{\mathrm{max}}.$ (3)

From the conservation of mass and momentum across a shock wave, the post-shock
pressure ($P_{\mathrm{h}}$) can be expressed as

$P_{\mathrm{h}}=\rho_{\mathrm{0}}U_{\mathrm{s}}u_{\mathrm{p}},$ (4)

where $U_{\mathrm{s}}$ is the shock velocity. It is well known that most
materials exhibit a linear relationship between shock velocity and particle
velocity, known as the $U_{\mathrm{s}}-u_{\mathrm{p}}$ Hugoniot. Cooper (2002)
Although this empirical relationship has not been determined for PDMS oils, it
is possible to estimate the shock velocity from the ambient speed of sound of
the liquid using the so-called universal Hugoniot relationship proposed by
Woolfolk et al. Woolfolk, Cowperthwaite, and Shaw (1973)

$U_{\mathrm{s}}=1.37c_{0}-0.37c_{0}\mathrm{exp}\left(\frac{-2u_{\mathrm{p}}}{c_{0}}\right)+1.62u_{\mathrm{p}}.$
(5)

It has been shown that this relationship agrees well with a wide variety of
liquids as long as the post-shock pressure remains below the bulk modulus of
the liquid Garrett, Chhabildas, and Reinhart (2006), which is the case in this
study.

##  III RESULTS

The test parameters, including the target liquid thickness ($w_{\mathrm{l}}$),
the flyer plate thickness ($w_{\mathrm{f}}$), and the flyer plate impact
velocity ($v_{\mathrm{i}}$), as well as the resulting maximum particle
velocity ($u_{\mathrm{max}}$) at the free surface, calculated post-shock
pressure ($P_{\mathrm{h}}$), pullback velocity ($\Delta u_{\mathrm{fs}}$),
strain rate ($\dot{\epsilon}$), and measured spall strength ($P_{\mathrm{s}}$)
for 21 plate-impact experiments performed on the PDMS oils are presented in
Tab. 3. The free-surface velocity profiles for the experiments are presented
in Fig. 6, where the curves have been arbitrarily shifted on the time axis in
order to display them on the same plot. A spall event, identified by an arrow
in Fig. 6, was recorded for each test. As expected, an increase in the impact
velocity resulted in greater shock pressures and peak free surface velocities.
Similarly, experiments with aluminum flyers generated greater shock pressures
than those with PMMA flyers at equivalent impact velocities due to the
difference in the shock impedances of the materials. In general, greater
impact velocities also led to greater strain rates.

Table 3: Summary of plate-impact experiments. Shot | Target | $w_{\mathrm{l}}$ | Flyer | $w_{\mathrm{f}}$ | $v_{\mathrm{i}}$ | $u_{\mathrm{max}}$ | $P_{\mathrm{h}}$ | $\Delta u_{\mathrm{fs}}$ | $\dot{\epsilon}$ | $P_{\mathrm{s}}$
---|---|---|---|---|---|---|---|---|---|---
No. | Liquid | (mm) | Material | (mm) | ($\text{\,}\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$) | ($\text{\,}\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$) | ($\text{\,}\mathrm{MPa}$) | ($\text{\,}\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$) | ($10^{4}$$\text{\,}{\mathrm{s}}^{-1}$) | ($\text{\,}\mathrm{MPa}$)
1 | 0.5 St | 13 | Al 6061 | 4.8 | 124 | 203 | 120 | 54 | 1.1 | 26
2 | 0.5 St | 14 | PMMA | 1.6 | 445 | 265 | 166 | 53 | 2.4 | 25
3 | 0.5 St | 9 | PMMA | 1.6 | 628 | 439 | 314 | 39 | 3.7 | 19
4 | 0.5 St | 17 | Al 6061 | 6.3 | 395 | 540 | 413 | 57 | 2.4 | 27
5 | 0.5 St | 13 | Al 6061 | 4.8 | 402 | 574 | 450 | 48 | 3.0 | 23
6 | 0.5 St | 9 | Al 6061 | 4.1 | 387 | 598 | 475 | 46 | 2.3 | 22
7 | 0.5 St | 17 | Al 6061 | 6.3 | 592 | 705 | 596 | 47 | 0.6 | 23
8 | 0.5 St | 17 | Al 6061 | 6.3 | 581111Projectile velocity estimated from experiments with the same projectile and launcher parameters. | 731 | 627 | 48 | 3.2 | 23
9 | 0.5 St | 14 | Al 6061 | 4.6 | 545 111Projectile velocity estimated from experiments with the same projectile and launcher parameters. | 741 | 639 | 39 | 3.2 | 19
10 | 0.5 St | 9 | Al 6061 | 4.1 | 550 | 800 | 713 | 39 | 4.1 | 19
11 | 0.5 St | 9 | Al 6061 | 4.1 | 573 | 842 | 767 | 49 | 4.1 | 24
12 | 10 St | 13 | Al 6061 | 4.8 | 120 | 191 | 114 | 44 | 1.1 | 22
13 | 10 St | 9 | PMMA | 1.6 | 622 | 432 | 311 | 48 | 4.4 | 23
14 | 10 St | 14 | Al 6061 | 4.6 | 545 | 710 | 609 | 43 | 3.4 | 21
15 | 10 St | 17 | Al 6061 | 6.3 | 570 | 760 | 671 | 48 | 3.8 | 23
16 | 10 St | 9 | Al 6061 | 4.1 | 564 | 826 | 754 | 49 | 4.1 | 24
17 | 300 St | 13 | Al 6061 | 4.8 | 123 | 199 | 119 | 49 | 1.0 | 24
18 | 300 St | 14 | PMMA | 1.6 | 442 | 274 | 176 | 45 | 2.6 | 22
19 | 300 St | 9 | PMMA | 1.6 | 605 | 417 | 297 | 49 | 4.5 | 24
20 | 300 St | 17 | Al 6061 | 6.3 | 379 | 531 | 409 | 47 | 2.4 | 23
21 | 300 St | 13 | Al 6061 | 4.8 | 572 | 796 | 716 | 44 | 4.4 | 21
Figure 6: Time-velocity plot of the free surface velocities for the plate-
impact experiments. The time axis has been arbitrarily shifted to fit all the
data on the same plot. Each curve is labeled by its shot number and the
arrival of the spall pulse is marked by an arrow.

The measured spall strength for the plate-impact experiments is plotted as a
function of strain rate ($\dot{\epsilon}$) in Fig. 7 for the three types of
PDMS oils. Also included in Fig. 7 are curves based on homogeneous and
heterogeneous nucleation models described in Section IV. The experimental data
does not show a strain rate dependency for either of the three silicone oils.
The measured spall strength is also plotted as a function of the shock
pressure ($P_{\mathrm{h}}$) in Fig. 8. As can be seen, the measured spall
strength is unaffected by the wide variation in shock pressure. In Fig. 9, the
measured spall strength is plotted as a function of the zero-shear-rate
viscosity for the three PDMS oils. Curves based on homogeneous and
heterogeneous cavitation models described in Section IV are also shown, as
well as the data and correlations between critical tension and viscosity
obtained from bullet-piston experiments of Bull Bull (1956a) and Couzens and
Trevena. Couzens and Trevena (1974) A second set of power-law correlations
based on the results of Bull Bull (1956a) and Couzens and Trevena Couzens and
Trevena (1974) ($P_{\mathrm{s}}\sim\eta^{0.2}$ and
$P_{\mathrm{s}}\sim\eta^{0.06}$, respectively) are also plotted such that they
intercept the average spall strength value of $23\text{\,}\mathrm{MPa}$ at
$9.71\text{\times}{10}^{-1}\text{\,}\mathrm{Pa}\text{\,}\mathrm{s}$, which
corresponds to the viscosity of the 10 St oil. As can be seen, the measured
spall strength in this work showed no clear correlation with the viscosity of
the oils. The average spall strength was 22.7 MPa for the 0.5 St silicone oil,
22.6 MPa for the 10 St oil, and 22.8 MPa for the 300 St oil.

Figure 7: Spall strength as a function of strain rate for the plate-impact
experiments. Also shown is the expected sensitivity of spall strength to
strain rate based on homogeneous and heterogeneous cavitation models. Figure
8: Spall strength as a function of shock pressure for the plate-impact
experiments. Figure 9: Measured spall strength from the plate-impact
experiments as a function of the zero-shear-rate viscosity ($\eta_{0}$) of the
PDMS oils. Also shown is the expected sensitivity of spall strength to
viscosity based on homogeneous and heterogeneous cavitation models, and
previously published data and viscosity correlations from Bull Bull (1956a)
and Couzens and Trevena. Couzens and Trevena (1974)

##  IV CAVITATION MODELS

The spall strength obtained from plate-impact experiments on liquid targets is
a measure of the critical tension required to induce cavitation bubble growth.
A liquid in tension is in a metastable state, where bubble nucleation and
growth, the mechanism by which tension is relieved, is stabilized by the
surface energy required to grow these voids. Bubbles in a liquid in tension
($P$) must be larger than a certain critical radius
($R_{\mathrm{c}}=2\sigma/P$), before it is energetically favorable for them to
grow. The vapor pressure inside the bubble has been neglected in the critical
radius expression above due to its small contribution compared to $P$, which
is on the order of $10\text{\,}\mathrm{MPa}$ in this work. $R_{\mathrm{c}}$
corresponds to the radius at which the work of formation of the bubble is
maximum ($W_{\mathrm{max}}$). As was discussed in Section I, the cavitation
bubbles which grow to relieve tension in the liquid may originate from pre-
existing flaws, such as free bubbles or bubbles attached to contaminant
particles, but bubbles may also nucleate from random molecular fluctuations in
a homogeneous cavitation process. A further consideration in spall experiments
is that tension is applied on a timescale which can be comparable to the rate
of cavitation bubble growth. This can allow the liquid to remain in a state of
tension despite the growth of bubbles if their density or growth rate is
insufficient to relieve the tension that is accumulating in the sample. Utkin
(1997); Utkin, Sosikov, and Fortov (2006) The rate of void growth in a
spalling material is typically characterized by the damage rate
($\dot{V}_{\mathrm{v}}$), which is a measure of the time rate of change of the
specific volume of voids in the target material. For plate-impact experiments,
it has been shown that the damage rate must exceed the strain rate by a factor
of four or greater ($\dot{V}_{\mathrm{v}}>4\dot{\epsilon}/\rho_{0}$) in order
to relieve tension in the sample and form the spall pulse on the free-surface
velocity signal. Antoun  _et al._ (2003) The spall strength of a liquid may
therefore be affected by its physical properties, the distribution of pre-
existing flaws, and the rate of loading. This section will present models
which can be used to predict the spall strength and its dependence on strain
rate and viscosity for both homogeneous and heterogeneous cavitation
nucleation.

###  IV.1 Homogeneous nucleation

In a homogenous nucleation process, random molecular fluctuations allow
bubbles to overcome the energy barrier created by surface tension in order to
reach the critical radius and continue to grow. According to the theory of
nucleation, the rate at which bubbles of a critical radius form is
proportional to $\mathrm{exp}(-W_{\mathrm{max}}/kT)$, where $k$ is the
Boltzmann constant, and $T$ is the temperature. Fisher (1948); Zeldovich
(1992); Carlson and Levine (1975) The work of formation needed to reach the
critical radius therefore provides a barrier to cavitation and stabilizes the
liquid phase. Following the approach of Fisher Fisher (1948) and Zeldovich
Zeldovich (1992), an expression can be obtained which expresses the volumetric
rate of formation of critical bubbles as a function of the tension in the
fluid Bogach and Utkin (2000),

$J=\frac{N_{0}\sigma}{\eta}\left(\frac{\sigma}{kT}\right)^{\frac{1}{2}}\mathrm{exp}\left(\frac{-16\pi\sigma^{3}}{3P^{2}kT}\right),$
(6)

where $N_{0}$ is the number of molecules per unit volume. As noted by Fisher
and Zeldovich, Eq. 6 can be used to determine the critical tension
($P=P_{\mathrm{s}}$) for cavitation by substituting a reasonable value for the
nucleation rate ($J$). Indeed, Fisher Fisher (1948) showed that varying the
nucleation rate over a range of ${10}^{-15}$ to ${10}^{18}$ only changes the
value of critical tension obtained in the calculation by a factor of 1.58. The
exponential factor is the dominant term in Eq. 6, and it should therefore be
expected that the critical tension is relatively insensitive to the viscosity
of the fluid or the rate at which tension is applied. Bogach and Utkin Bogach
and Utkin (2000) noted that from Eq. 6, the relationship between the critical
tension and strain rate should follow the empirical expression below

$P_{\mathrm{s}}\approx\frac{A}{\left[\mathrm{ln}\left(\frac{B}{\dot{\epsilon}}\right)\right]^{\frac{1}{2}}},$
(7)

where $A$ and $B$ are constants. Model curves based on the homogeneous
nucleation theory presented above have been included in Fig. 7 (Eq. 7, $A$ =
$1.24\text{\times}{10}^{8}$, $B$ = $1.00\text{\times}{10}^{17}$) and Fig. 9
(Eq. 6, $J$ = $1\text{\,}{\mathrm{m}}^{-3}\text{\,}\mathrm{s}$). As can be
seen, according to the steady-state homogeneous nucleation theory presented
above, the spall strength should be expected to marginally increase with
increasing strain rate and viscosity.

###  IV.2 Heterogeneous nucleation

Any real fluid will contain a number of heterogeneities from which cavitation
can nucleate. Kedrinskii (2005) A cavitation process dominated by the growth
of pre-existing bubbles should be expected to be strongly influenced by the
dynamics of the bubbles. The time dependent evolution in the radius ($R$) of a
single bubble in an effectively infinite liquid is described by the Rayleigh-
Plesset equation. Plesset and Prosperetti (1977) Below is the Rayleigh-Plesset
equation for a Newtonian fluid where the liquid is under a far-field tension
($P$) and the internal bubble pressure and mass diffusion across the bubble
interface are negligible

$R\ddot{R}+\frac{3}{2}(\dot{R})^{2}+\frac{4\eta}{\rho_{0}R}\dot{R}+\frac{2\sigma}{\rho
R}=\frac{P}{\rho_{0}}.$ (8)

It has been noted that for the case of a system subjected to a large sudden
increase in tension, such as that in a spall experiment, the initial
normalized growth rate of a bubble that is significantly larger than the
critical radius can be expressed as the following Bull (1956a); Erlich,
Wooten, and Crewdson (1971); Bogach and Utkin (2000)

$\frac{\dot{R}}{R}=\frac{P}{4\eta}.$ (9)

The volumetric growth rate of a single bubble ($\dot{V}_{\mathrm{b}}$), can
then be expressed by the simple expression below Bogach and Utkin (2000)

$\dot{V}_{\mathrm{b}}=\frac{\pi R^{3}P}{\eta}.$ (10)

From Eq. 10, it can be seen that heterogeneous cavitation bubble growth should
be expected to strongly depend on the viscosity of the fluid. Obtaining an
estimate of the spall strength from the approach described above poses a
number of challenges, notably determining the distribution of pre-existing
bubbles, and considering the effect of neighboring bubbles in a multi-bubble
system.

Grady Grady (1988a) proposed a unique approach to estimating the spall
strength of dynamically stretched materials, based upon the criteria that the
onset of fracture should be expected to occur when it is energetically
favorable for the material to break apart. More precisely, failure in a
dynamically stretched liquid is assumed to occur when the stored energy
(elastic and kinetic) in the stretching liquid is equal to the viscous
dissipation and surface energy produced in the fracture (cavitation) process.
Grady (1988a) Inherently, this approach assumes that the material is favorably
disposed to fracture, which for a liquid means that there are pre-existing
bubbles from which cavitation can nucleate. Grady (1988a) The following
expression, which relates the spall strength to the strain rate for the
experiment ($\dot{\epsilon}$) and the surface tension ($\sigma$), viscosity
($\eta$), and speed of sound ($c_{0}$) of the liquid, can be obtained from the
energy based approach Grady (1988a)

$P_{\mathrm{s}}^{3}-2\eta\rho
c_{0}^{2}\dot{\epsilon}P_{\mathrm{s}}-6\rho^{2}c_{0}^{3}\sigma\dot{\epsilon}=0.$
(11)

The second and third term of Eq. 11 are the viscous dissipation and surface
energy contributions, respectively. As can be seen, the viscous dissipation
term is more sensitive to strain rate that the surface energy term
($P_{\mathrm{s}}\sim\eta^{1/2}$ versus $P_{\mathrm{s}}\sim\eta^{1/3}$), which
means that as the strain rate is increased, the spall strength transitions
from being dominated by the contribution of surface energy to that of viscous
dissipation. A transition strain rate ($\dot{\epsilon}_{\mathrm{t}}$), can be
defined as the rate at which the viscous and surface energy are equal Grady
(1988a)

$\dot{\epsilon}_{\mathrm{t}}=\frac{9}{2}\frac{\rho\sigma^{2}}{\eta^{3}}.$ (12)

The transition strain rate corresponds to $1.7\text{\times}{10}^{4}$, $2.2$,
and $7.9\text{\times}{10}^{-5}\text{\,}{\mathrm{s}}^{-1}$ for the low,
intermediate, and high-viscosity silicone oils, respectively. For the strain
rates encountered in experiments
($\dot{\epsilon}\approx$${10}^{4}\text{\,}{\mathrm{s}}^{-1}$), the oils,
particularly the intermediate and high-viscosity samples, can be considered to
be in the viscous dissipation dominated regime, where the spall strength can
be expressed as

$P_{\mathrm{s}}=(2\eta\rho c_{0}^{2}\dot{\epsilon})^{\frac{1}{2}}.$ (13)

In this regime, the spall strength is seen to have a strong dependence on both
strain rate and viscosity ($P_{\mathrm{s}}\sim\dot{\epsilon}^{1/2}$,
$P_{\mathrm{s}}\sim\eta^{1/2}$). The dependence of spall strength on strain
rate based on the viscous limit of the Grady heterogeneous nucleation model is
shown in Fig. 7, while the relationship between the spall strength and the
viscosity of the silicone oils predicted by Eq. 11 is shown in Fig. 9. As can
be seen, the heterogeneous nucleation theory presented above predicts a
significant variation in spall strength with increasing strain rate and
viscosity.

##  V DISCUSSION

The spall strength values reported in Section III are comparable to
measurements reported in previous plate-impact studies performed on similar
liquids, where the critical tension was also found to be on the order of
$10\text{\,}\mathrm{MPa}100\text{\,}\mathrm{MPa}$. Erlich, Wooten, and
Crewdson (1971); Carlson and Levine (1975); Bogach and Utkin (2000); Utkin and
Sosikov (2009) The spall strength of the three PDMS oils remained relatively
constant over the range of strain rates and shock pressures probed in this
work. As can be seen in Fig. 7, the low sensitivity to strain rate is
consistent with the homogeneous cavitation model but does not agree with the
heterogeneous cavitation model of Eq. 11. Utkin and Sosikov Utkin and Sosikov
(2009) noted a similar lack of strain rate dependence in plate-impact
experiments performed with a range of liquids including water, methanol, and
hexane, which was attributed to homogeneous cavitation nucleation. The absence
of a correlation between spall strength and shock pressure further suggests
that pre-existing bubbles do not play an important role in determining the
critical tension at which the oils cavitate for the conditions probed in this
work. The significant variation in shock pressure would be expected to affect
the size of the voids due to their collapse prior to tensile loading. The
results also showed that the spall strength of the PDMS oils was unaffected by
their zero-shear-rate viscosity, with all three oils having an equivalent
average spall strength. A comparison of the plate-impact data with the model
curves for $P_{\mathrm{s}}$ as a function of viscosity in Fig. 9, shows that
the insensitivity of the spall strength to viscosity is captured by the
homogeneous cavitation model, whereas the heterogeneous cavitation model
suggests that the variation in the viscosity of the silicone oils should
result in an order of magnitude increase in spall strength. The agreement
between the spall strength measurements and Eq. 6, which has no free
parameters, again suggests that the dominant cavitation mechanism in these
experiments is homogeneous nucleation of bubbles via molecular fluctuations.

The relationship between spall strength and viscosity observed in this work
does not agree with previously published results from bullet-piston
experiments Bull (1956a); Couzens and Trevena (1974), where a power-law
correlation was seen between viscosity and critical tension. Indeed, Couzens
and Trevena Couzens and Trevena (1974) studied a similar system of silicone
oils with nominal kinematic viscosities ranging from 0.01 St to 10 St and
observed a power law dependence between the critical tension and viscosity
($P_{\mathrm{s}}\sim\eta^{0.06}$). It is interesting to note that the critical
tension measured during plate-impact experiments is typically an order of
magnitude greater than values obtained using bullet-piston experiments. This
can be seen in Tab. 4, where the measured critical tensions obtained from
bullet-piston and plate-impact experiments are compared for a number of
liquids. This comparison indicates that the critical tension which induces
cavitation increases significantly as the strain rate is increased from
${10}^{2}\text{\,}{\mathrm{s}}^{-1}$ in the bullet-piston studies Grady
(1988a) to ${10}^{4}\text{\,}{\mathrm{s}}^{-1}$ for plate-impact experiments.
However, the results presented in this paper show that a sensitivity to
loading rate is no longer observed at the strain rates encountered in plate-
impact experiments. The lack of correlation between viscosity, strain rate,
and spall strength observed in the results presented in this paper indicates a
possible transition in the mechanism of cavitation nucleation as the rate at
which tension is applied is increased from
${10}^{2}\text{\,}{\mathrm{s}}^{-1}$ (bullet-piston) to
${10}^{4}\text{\,}{\mathrm{s}}^{-1}$ (plate-impact).

Table 4: Comparison of critical tension measurements from bullet-piston and plate-impact experiments. Liquid |  $P_{\mathrm{s}}$ (Bullet-Piston) |  $P_{\mathrm{s}}$ (Plate-Impact)
---|---|---
| ($\text{\,}\mathrm{MPa}$) | ($\text{\,}\mathrm{MPa}$)
PDMS oil (10 St) | 2.3 Couzens and Trevena (1974) | 21-24111Data presented in this work.
Water | 0.9-1.5 Couzens and Trevena (1969) | 32 Huneault  _et al._ (2018)
| 1.8 Bull (1956b) | 22.5-46.0 Bogach and Utkin (2000)
Glycerol | 6.3 Bull (1956b) | 22-24 Erlich, Wooten, and Crewdson (1971)
|  | 57-142 Utkin, Sosikov, and Bogach (2003)

The mechanisms of spall failure in liquids bears resemblance to ductile spall
failure in solids, which also proceeds via a mechanism of void nucleation,
growth, and coalescence. Grady (1988a) In metals, failure sites are known to
nucleate at grain boundaries, inclusions, and second phase particles. Curran,
Seaman, and Shockey (1987); Antoun  _et al._ (2003) Experiments performed on
single crystal samples show a significant increase in spall strength Razorenov
_et al._ (2006, 2007) due to the fact that failure sites must now nucleate
within the lattice at sub-microscopic heterogeneities such as dislocation
pileups. Antoun  _et al._ (2003); Curran, Seaman, and Shockey (1987) As the
strain rate is increased, the spall strength of polycrystalline materials can
approach that of single crystals due to the fact that the loading pulse length
approaches the scale of the distribution of favorable nucleation sites. Antoun
_et al._ (2003); Wilkerson and Ramesh (2016) This behavior is indicative of a
transition from energy-limited spall, where there are a large number of
failure nucleation sites that allow the sample to fail when it is
energetically favorable, to that of flaw-limited spall, where significant
elastic energy can be stored within the body before failure sites nucleate.
Grady (1988b) The concept of energy and flaw-limited spall in solids is
directly analogous to the concepts of homogeneous and heterogeneous cavitation
nucleation in liquids. The comparison of the results from this work and
similar studies to those performed at lower strain rates in bullet-piston
experiments suggests a transition from energy-limited spall (heterogeneous
nucleation) to flaw-limited spall (homogeneous nucleation) as strain rate is
increased. It is important to note that this work has focused on the maximum
tension supported by the liquid at the onset of cavitation, which represents a
small portion of the shock induced cavitation process. A shock wave reflecting
from a liquid free surface can produce large cavitation particle clouds and a
series of spall layers which break up into a bubbly spray. Kolsky  _et al._
(1949); Kedrinskii (2005) These processes, which occur on much longer
timescales than those considered in this work (tens of microseconds to
milliseconds), should be expected to be influenced by heterogeneities in the
fluid Kedrinskii (2005), regardless of the strain rate at the onset of
cavitation.

The heterogeneous cavitation theory presented in Section IV.2 and the
discussion above were based on the assumption that the viscosity of the oils
during cavitation bubble growth could be adequately represented by the zero-
shear-rate viscosity. As was discussed in Section II.1, the bubble growth
rates required to relieve the tension during plate-impact experiments are
likely to induce non-Newtonian behavior in the PDMS fluids, particularly for
the intermediate and high-viscosity oils. At shear rates above the critical
shear rate ($\dot{\gamma}_{\mathrm{c}}$), PDMS oils behave as a pseudo-plastic
material: their viscosity is shear-rate dependent and they have a resistance
to shear deformation (non-zero shear modulus). Chhabra (2010) While the
elasticity of the oils during high strain rate deformation should be expected
to increase resistance to bubble growth Warnez and Johnsen (2015), the
opposite is true of their shear thinning properties, which reduce the apparent
viscosity of the fluid. The viscous resistance to bubble growth arises from
the extensional flow field as the fluid near the bubble is displaced. Pearson
and Middleman (1977) In a viscoelastic material, the Newtonian relationship
between extensional and shear viscosity
($\eta_{\mathrm{E}}(\dot{\epsilon})=3\eta(\dot{\gamma})$) is no longer
applicable and the ratio between the two values can be much greater than
three. Jones, Walters, and Williams (1987) Although data on the extensional
shear rate dependence of PDMS oils is limited, published results of capillary
rheometry experiments showed a decay from
$20\text{\,}\mathrm{Pa}\text{\,}\mathrm{s}12\text{\,}\mathrm{Pa}\text{\,}\mathrm{s}$
as $\dot{\epsilon}$ was increased from
${10}^{2}\text{\,}{\mathrm{s}}^{-1}{10}^{4}\text{\,}{\mathrm{s}}^{-1}$ for a
10 St oil, and a decay from
$125\text{\,}\mathrm{Pa}\text{\,}\mathrm{s}25\text{\,}\mathrm{Pa}\text{\,}\mathrm{s}$
for a 50 St oil over the same range of strain rates. Day  _et al._ (2008);
Williams  _et al._ (2010) These results indicate that the high strain rates
studied in this work likely affected the viscosity of the high-viscosity
silicone oil and reduced the overall difference in the viscosity of the three
oils. Nonetheless, it seems reasonable to assume that even at the strain rates
experienced in plate-impact experiments (${10}^{4}\text{\,}{\mathrm{s}}^{-1}$)
there remains a large difference (approximately two orders of magnitude) in
the viscosity and resulting resistance to void growth of the three oils
considered in this work.

##  VI CONCLUSION

The mechanism by which cavitation nucleates in systems where rupture of the
liquid is induced by shock or pressure waves is of significant scientific and
engineering interest. The spall strength of three PDMS oils having vastly
different viscosities was evaluated over a range of loading conditions using
plate-impact experiments. The results showed that the spall strength was
unaffected by the variation in the viscosity of the oils or the rate at which
tension was applied. These findings were shown to be consistent with a
homogeneous cavitation model where the principal mechanism for tension relief
is the formation of voids via random molecular fluctuations. The results
suggest that cavitation originating from heterogeneities (pre-existing
bubbles) does not have a significant effect on the spall strength of the PDMS
oils for the conditions probed in this work. Comparison with previously
published data obtained using bullet-piston experiments, where the loading
rates were two orders of magnitude less than those presented in this work,
suggests a transition from homogeneous to heterogeneous cavitation nucleation
as the strain rate is increased to that of plate-impact experiments.

###### Acknowledgements.

The authors would like to thank Jihane Kamil for her help in developing the
experiment, Charles Dubois and Asher Bechimol from the Department of Chemical
Engineering at École Polytechnique de Montréal for their assistance with the
rheometry measurements, Hansen Liu, Zhuo Fan Bao, and Hin Fung Ng for their
help in performing experiments, and David Plant and Victoria Suponitsky for
their guidance with the experiments. This work was supported by General Fusion
and the Natural Sciences and Engineering Research Council of Canada (NSERC)
under Collaborative Research and Development Grant 477617-14.

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