††thanks: These authors contributed equally to this work.††thanks: These
authors contributed equally to this work.

# Three dimensional full-field velocity measurements in shock compression
experiments using stereo digital image correlation

Suraj Ravindran<EMAIL_ADDRESS>Aerosapce Engineering and Mechanics, University
of Minnesota, Minneapolis, Minnesota 55455, USA  Vatsa Gandhi  Division of
Engineering and Applied Science, California Institute of Technology, Pasadena,
California 91125, USA  Akshay Joshi  Division of Engineering and Applied
Science, California Institute of Technology, Pasadena, California 91125, USA
Guruswami Ravichandran  Division of Engineering and Applied Science,
California Institute of Technology, Pasadena, California 91125, USA  Jio
Institute, Ulwe, Navi Mumbai, Maharashtra 410206, India

###### Abstract

Shock compression plate impact experiments conventionally rely on point-wise
velocimetry measurements based on laser-based interferometric techniques. This
study presents an experimental methodology to measure the free surface full-
field particle velocity in shock compression experiments using high-speed
imaging and three-dimensional (3D) digital image correlation (DIC). The
experimental setup has a temporal resolution of $100$ ns with a spatial
resolution varying from $90$ to $200$ $\mu$m/pixel. Experiments were conducted
under three different plate impact configurations to measure spatially
resolved free surface velocity and validate the experimental technique. First,
a normal impact experiment was conducted on polycarbonate to measure the
macroscopic full-field normal free surface velocity. Second, an isentropic
compression experiment on Y-cut quartz-tungsten carbide assembly is performed
to measure the particle velocity for experiments involving ramp compression
waves. To explore the capability of the technique in multi-axial loading
conditions, a pressure shear plate impact experiment was conducted to measure
both the normal and transverse free surface velocities under combined normal
and shear loading. The velocities measured in the experiments using digital
image correlation are validated against previous data obtained from laser
interferometry. Numerical simulations were also performed using established
material models to compare and validate the experimental velocity profiles for
these different impact configurations. The novel ability of the employed
experimental setup to measure full-field free surface velocities with high
spatial resolutions in shock compression experiments is demonstrated for the
first time in this work.

Keywords: Shock Compression, Full-field measurements, High Speed imaging,
Stereo Digital image correlation

††preprint: AIP/123-QED

##  I Introduction

Plate impact experiments, generating shock compression, are employed to
understand the high pressure and high strain-rate behavior of broad classes of
materials, including metals, glasses, ceramics, and composites Meyers (1994).
In these experiments, the free surface velocities are typically measured,
which provide insights into various phenomena, such as equation of stateMeyers
(1994), inelastic material behavior Abou-Sayed  _et al._ (1976), polymorphic
phase transformations Bancroft  _et al._ (1956), and spall failureDandekar
(2004). Laser interferometric techniques are the commonly used method for
determining free-surface velocities due to their high sensitivity, accuracy,
time resolution, and non-intrusiveness towards propagating shock waves. In
these techniques, Doppler-shifted light reflected from the target free surface
is superposed with either a delayed version of itself (velocity
interferometer) or a reference light beam (displacement interferometer), and
the phase difference of the interfered beams is used to obtain the normal and
transverse free surface velocities. The most well-known interferometry
techniques that are employed for measuring point-wise normal particle velocity
are the Velocity Interferometer System for Any Reflector (VISAR) Barker and
Hollenbach (1972), and the Photon Doppler Velocimetry (PDV) Strand  _et al._
(2006); Mallick  _et al._ (2019a). The point-wise transverse particle
velocities are measured using a transverse displacement interferometer
(TDI)Kim  _et al._ (1977) or its corresponding PDV, a heterodyne transverse
velocimetry (HTV)Kettenbeil  _et al._ (2018) interferometer. Despite its high
temporal resolution ($\sim$ns), common laser interferometry techniques provide
only point-wise velocimetry and are unable to capture the spatial
heterogeneity in the shock wave. This limits the utility of such techniques in
the context of shock compression experiments on heterogeneous materials such
as composites, granular, and architected lattice materials, where
characterization of the material requires measurement of spatio-temporally
resolved free surface velocities. This issue can be partly addressed by
employing a variant of VISAR known as Optically Recording Velocity
Interferometer System (ORVIS) Bloomquist and Sheffield (1983), where velocity
measurements are conducted along a line to provide improved spatial resolution
Vogler  _et al._ (2008). ORVIS requires an elaborate arrangement of optics and
high-speed streak cameras in order to provide the desired horizontal scan of
the laser needed for the measurement Trott  _et al._ (2007). Additionally,
this technique only offers insight into the velocity variation along a line on
the free surface and not on the entire free surface. Such full-field free
surface velocity measurements can be achieved using the Digital Image
Correlation (DIC) technique Chu  _et al._ (1985); Sutton  _et al._ (2009).

Digital image correlation is well established as a full-field measurement
technique for both in-plane and out-of-plane displacement measurements Sutton
_et al._ (2009). This technique involves taking a series of images of a
deforming randomly speckled sample and is cross-correlated to extract
displacement fields across the entire sample surface Chu  _et al._ (1985);
Sutton  _et al._ (2009). Therefore, a single experiment yields thousands of
displacement measurements over the entire field of view, with sub-pixel
accuracy, which can be further used to obtain the velocity, acceleration, and
strain fields Pierron  _et al._ (2014); Rubino  _et al._ (2019). While DIC is
well-suited for quasi-static experiments, advances in high-speed imaging
capabilities have enabled DIC in the dynamic regime. For example, high-speed
imaging in conjunction with two-dimensional DIC was utilized to study the
dynamic micro- to macro-scale deformation behavior of materials Ravindran  _et
al._ (2016, 2017); Malhotra  _et al._ (2021); Kannan  _et al._ (2018);
Ravindran  _et al._ (2019). Recently, Rubino et al. Rubino  _et al._ (2019)
combined high-speed photography, and 2D DIC to characterize deformation fields
around dynamically propagating shear ruptures in laboratory earthquake
experiments. In addition to in-plane deformation fields, out-of-plane
deformation can be resolved using two cameras and high-speed three-dimensional
(stereo) digital image correlation, which is used in blast, ballistic and low
velocity impact applications. For instance, Tiwari et al. Tiwari  _et al._
(2009) conducted 3D DIC to measure the deformation of aluminum plates
subjected to blast loading, while work by Jannotti et al. Jannotti  _et al._
(2021) investigated anisotropic ballistic response of rolled AZ31B magnesium
plates. Additionally, Gupta et al. Gupta  _et al._ (2014) conducted 3D DIC to
understand the physical processes associated with the implosion of cylindrical
tubes submerged underwater. Pankow et al. Pankow  _et al._ (2010) implemented
stereo DIC with a single camera by using a series of mirrors to split the
image into two different views and examined the out-of-plane flexural response
of a dynamically loaded thin aluminum plate. In these studies, the authors use
the extracted full-field displacement and strain field to analyze the
deformation mechanisms at time scales ranging from $25-150$ $\mu$s. Overall,
previous impact studies using full-field measurements have focused on longer
time scale behavior of materials compared to the shock compression behavior
where the total event duration is typically smaller than $5$ $\mu$s.

The present study extends the applications of high speed 3D DIC to plate
impact experiments for the first time to understand material behavior in the
shock regime (strain rates $>10^{6}$ s-1). In Section II, details of the
experimental setup, accuracy and resolution of measured data, and the
methodology of post-processing the acquired data are discussed. In Section
III, the feasibility of the developed experimental setup for measuring full-
field free surface velocity measurements and is first validated against point-
wise PDV measurements in normal and pressure-shear plate impact experiments.
In Section IV, concluding remarks are presented and potential future
directions are discussed.

##  II Materials and Methods

###  II.1 Experimental Setup

Figure 1: a) Schematic of the experimental setup with powder gun, camera
arrangements for 3D digital image correlation, b) image of the components
inside the vacuum chamber and the high-speed camera arrangement for stereo DIC
set up for imaging through the transparent window, c) representative speckle
patterns on the free surface of the sample.

In this study, plate impact experiments were conducted using a powder gun
facility at Caltech. Four materials, polycarbonate (PC), tungsten carbide
(WC), Y-cut quartz, and D2 tool steel, were investigated in different
experimental configurations to demonstrate the reliability, advantage, and
robustness of the high-speed stereo (3D) DIC for measuring full-field free
surface velocities in plate impact experiments. The schematic of the
experimental setup with high-speed imaging-based full-field diagnostics is
shown in Fig. 1a. Also, the images of the stereo high speed imaging DIC setup
and the components inside the vacuum chamber are shown in Fig. 1b. Samples
were shocked using the 38.7 mm slotted powder gun capable of launching
projectiles at velocities ranging from 0.2-1.5 km/s. Two Hyper Vision HPV-X2
high-speed cameras (Shimadzu, Kyoto, Japan) capable of capturing 5 million
frames/s at full resolution ($400\times 250$ pixel2) and 10 million frames/s
at half resolution ($400\times 250$ pixel2, where missing pixels are
interpolated) were used to capture the image of the free surface, see Fig. 1a.
These cameras were placed at a stereo angle between $15^{\circ}-16^{\circ}$
and were calibrated by capturing the image of the standard laser engraved
calibration grids (Correlated Solutions Inc., Columbia, SC), to facilitate
stereo DIC measurementSutton  _et al._ (2009). A total of 45 image pairs were
collected by tilting and rotating the calibration grid within the field of
view of both the cameras. It was ensured that the images were within the depth
of field of the imaging system to obtain accurate calibration of the stereo
camera system. The image of the camera arrangement and a typical calibration
grid used for calibration are shown in Fig. 1a. The images of the rear surface
of the sample were observed through a large mirror (50 mm × 100 mm) arranged
at $45^{\circ}$ with respect to the loading direction. A CAVILUX Smart laser
(Cavitar, Tampere, Finland), which provides high-speed incoherent laser
(wavelength, 640 nm) pulses as short as 10 ns, was used to illuminate the
sample during loading. The short duration laser pulses avoid blurring due to
the transients in the high-speed impact experiments. These laser pulses were
synchronized with the exposure of the camera sensor during the image
acquisition. An image of the complete target assembly, mirrors, and
illumination using laser inside the vacuum chamber is shown in the Fig. 1b.
Impact velocities of the experiments were measured using Photonic Doppler
velocimetry Strand  _et al._ (2006) (PDV). The PDV probe was attached to the
target holder and the optical return was checked during the target alignment.
In the pressure shear plate impact experiment, a heterodyne PDV was used to
simultaneously measure normal and transverse velocitiesKettenbeil  _et al._
(2018).

The materials used in this study were obtained from various commercial
vendors. Polycarbonate and D2 tool steel materials were procured from
McMaster-Carr (Los Angeles, CA) while the tungsten carbide was acquired from
Basic Carbide Corporation (Lowber, PA), ARMCO iron from AK Steel International
(Breda, The Netherlands), and Y-cut quartz from University Wafer (South
Boston, MA). The final dimensions of the flyers and target materials are shown
in Table 2.

The flyers and targets were lapped parallel to less than 10 $\mu$m and flat to
within 0.5 $\mu$m variation across the surface. Four encapsulated copper
shorting pins (tilt pins) were glued to the target and lapped flush to the
target surface; see inset in Fig. 1b. These pins were connected to a digital
circuit, which was connected to a high-speed digital oscilloscope sampling at
20 GigaSamples/s (MSO9404A, Keysight Technologies, Inc., Santa Rosa, CA) to
determine the contact time of the flyer with the target. The contact time of
each pin with the flyer was used to calculate the impact tilt in the
experiment. The flyer and target assemblies were aligned in the experiment
using an auto-collimator and an optically flat mirror assembly prior to the
shotKumar and Clifton (1977). This alignment ensures the impact tilt between
the flyer and target below 2 milli-radian in order to generate a plane wave in
the sample. A high contrast, random and isotropic speckle pattern was applied
on the free surface of the sample to facilitate DIC measurement. The speckles
in all the experiments were about $4-6$ pixels in size to avoid anti-aliasing
due to low sampling of the pixels Sutton  _et al._ (2009). They were created
using a fine tip pen (Pigma Micron, Sakura, Osaka, Japan) for the experiment
on polycarbonate. In other experiments, black speckles were applied using an
airbrush to obtain higher spatial resolution resulting in smaller speckles
(84-92 $\mu$m/pixel). The field of view and the images of the speckle patterns
used in the experiments are shown in Fig. 1c.

###  II.2 Postprocessing of images

The image sequence acquired using the HPV-X2 high-speed cameras was analyzed
to obtain the displacement field using a widely-used commercial software
Vic-3D (Correlated Solutions Inc., Columbia, SC). To obtain the stereo camera
arrangement calibration parameters, the bundle adjustment algorithm
implemented in Vic-3D was used Sutton  _et al._ (2009). This algorithm yields
the extrinsic and intrinsic parameters required for calculating all the
displacement components from the speckled images acquired during impact. Two
important parameters considered during post-processing of the images were the
subset size and step size. For a selected subset size, the algorithm
calculates all the displacement components at the center of the subset, while
the step size determines the distance between the centers of the subsets. For
instance, if one chooses step size of 3, the subset calculations will be
performed at a distance of 3 pixels from the previous subset and report the
data at the center of the subset. Therefore, a step size of 3 will skip three
points between the centers of the adjacent subsets. In this study, a
significant spatial gradient in displacement was not expected since the goal
of this study was to obtain the macroscopic velocity fields and the spatial
resolution chosen was much larger than the heterogeneities in the sample.
Therefore, subset size of $23\times 23$ pixel2 and step size of 7 was used for
post-processing. A summary of the experimental resolution and the parameters,
such as step and subset size, used for the analysis are shown in Table 1.

Table 1: Resolution and DIC postprocessing parameters Material |  | Resolution
---
[$\mu$m/pixel]
| Subset
---
[pixel$\times$pixel]
| Step size
---
[pixel]
Polycarbonate | 200 | $23\times 23$ | 7
Quartz-Tungsten Carbide | 95 | $23\times 23$ | 7
D2 Tool Steel | 93 | $23\times 23$ | 7

###  II.3 Measurement Noise

Figure 2: Mean and standard deviation of strain and velocity components from
the static images: a) in-plane strain ($\varepsilon_{xx}$), b) in-plane ($x$,
$y$) particle velocities, c) free surface (out of plane) $z-$velocity.

All the experiments utilized the half resolution high-speed mode in the camera
(Shimadzu HPV-X2) at 10 million frames/second. In the half-resolution mode,
every alternate pixel is unused during recording. The grayscale of the unused
pixel was interpolated from the neighboring pixels, which produces noise in
the measurement. Nine static images were analyzed to quantify the
uncertainties associated with the measurement in high-speed mode. The 3D image
correlation software, Vic-3D was used to post-process the images to calculate
the displacements and strains. Post-processing parameters for each experiment
are shown in Table 1. To quantify the velocity uncertainty as a result of
random noise in the displacement, the obtained displacement field is
numerically differentiated with time increment, $\Delta t=100$ ns, which is
the inter-frame time of the high speed camera at 10 million frames/s. Figure 2
shows the mean and standard deviation of full-field strain and velocity
components obtained by post-processing the static images. The highest
uncertainty in the strain was around 0.15% and the uncertainty in the in-plane
velocity components was approximately $8$ m/s. At the same time, the out-of-
plane velocity uncertainty ($dw/dt=$ $z-$velocity or normal free surface
particle velocity) was six times higher than the in-plane free surface
particle velocity owing to inherent lower sensitivity in the out-of-plane
displacement measurement Sutton  _et al._ (2009). This higher noise results in
5% uncertainty when measuring the free surface velocity of 1000 m/s. Note that
the uncertainty in velocity was calculated from the raw measurements, which
can be significantly reduced by the temporal and spatial filtering of the
dataRubino  _et al._ (2019). Optimal data filtering is employed to reduce the
uncertainties in computing free surface velocities and the procedure is
described in Appendix A.

###  II.4 Summary of experiments

The impact configurations, physical dimensions of the flyers, and samples for
all the experiments conducted in this study are shown in Table 2. A normal
impact experiment on a polycarbonate (PC) sample was performed to show the
accuracy and the robustness of the full-field DIC measurement technique in
plate impact experiments. The second experiment involved a dual-layer target,
tungsten carbide (WC) with a Y-cut Quartz (YQZ) driver, to show the capability
of the high-speed DIC measurement to accurately measure the free surface
velocity under isentropic compression. Next, a symmetric pressure shear plate
impact experiment (PSPI) was conducted on D2 tool steel to evaluate the use of
full-field measurements for resolving both normal and transverse full-field
velocities from the experiment.

Table 2: Impact configuration and dimensions of flyer and sample

Experiment |  | Flyer
---
material
| Target
---
material
| Flyer
---
thickness
[mm]
| Target
---
thickness
[mm]
| Impact
---
Velocity
[m/s]
| Normal
---
Stress
[GPa]
| Tilt
---
[mrad]
Normal Impact | Iron | Polycarbonate (PC) |  4.958$\pm$0.006 |  9.283$\pm$0.001 |  201$\pm$0.5 | 0.55 | 1.8
Normal Impact | WC | YQZ+WC |  4.465$\pm$0.002 |  | YQZ –
---
4.930$\pm$0.005
WC –
1.493$\pm$0.002
790$\pm$0.1 | 10.4 | 0.9
Pressure Shear Plate Impact | D2 tool steel | D2 tool steel |  5.630$\pm$0.004 |  3.983$\pm$0.003 |  530$\pm$0.9 | 10.3 | 1.2

  * •

**The diameter of the target in normal impact experiment on PC was 28 mm in
diameter, diameter of the target in the YQZ+WC was 30 mm, while in pressure
shear plate impact experiment, the diameter was 34 mm.

  * •

**The flyer in all experiments was 34 mm in diameter

###  II.5 Numerical simulation

Three-dimensional finite element simulations of the experiments were conducted
using the commercial software ABAQUS/ExplicitHibbitt  _et al._ (2001)
(Dassault Systemes, Providence, RI) employing previously calibrated material
models for PC, Y-cut quartz, WC, and D2 tool steel. In all the simulations, an
element size of $50$ $\mu$m was used based on mesh convergence studies.
Further details of the simulations are presented in the respective sections of
the experiments. The material models in the simulations include both the
equation of state (EOS) and strength models taken from previous studies. The
volumetric part of the deformation (Pressure-Volume Hugoniot relation) was
modeled using linear Grüniesen equation of stateMeyers (1994):

$P_{H}=\frac{\rho_{0}C_{0}\upsilon}{(1-S\upsilon)^{2}}$ (1)

where, $C_{0}$ is the bulk sound speed, $S$ is the slope of linear shock
velocity ($U_{s}$) - particle velocity ($u_{p}$) relation,
$U_{s}=C_{0}+Su_{p}$, $u_{p}$ is the normal particle velocity, $\upsilon$ is
the volumetric strain, $\upsilon=1-\frac{\rho_{0}}{\rho}=\frac{u_{p}}{U_{s}}$,
where $\rho_{0}$ and $\rho$ are the initial and final densities, respectively.
In the simulations, parameters ($C_{0}$, $S$) of the linear equation of state
were taken from past studies and are shown in Table 3.

Table 3: Equation of state parameters for the materials in the experiments. Material |  | Density, $\rho_{0}$
---
[kg/m3]
| Bulk Sound
---
Speed, $C_{0}$
[m/s]
$S$
IronBarker and Hollenbach (1974) | 7800 | 4460 | 1.72
Polycarbonate Dwivedi  _et al._ (2012); Carter and Marsh (1995) | 1197 | 2330 | 1.57
Tungsten CarbideDandekar and Grady (2002) | 15600 | 4930 | 1.30
D2 Tool Steel Ravindran  _et al._ (2021) | 7900 | 4590 | 1.42

Two different models were used to capture the strength behavior of the
materials used in the experiments. The flow strength $(Y)$ of polycarbonate
(PC) was modeled using the Johnson-Cook material model with the calibration
parameters obtained from the Dwivedi et al. Dwivedi  _et al._ (2012) and Rai
et al.Rai  _et al._ (2020), respectively.

$Y=\left(A+B\varepsilon_{p}^{n}\right)\left(1+C\ln\frac{\dot{\varepsilon}}{\dot{\varepsilon}_{ref}}\right)\left(1-\left(\frac{T-T_{R}}{T_{m}-T_{R}}\right)^{m}\right)$
(2)

where $A,B$, and $n$ are the parameters for the power-law strain hardening,
$\varepsilon_{p}$ is the plastic strain, and $C$ is the constant for the
strain rate hardening, $\dot{\varepsilon}_{ref}$ is the reference strain rate,
and $m$ is the exponent for thermal softening. Tungsten carbide and D2 tool
steel were modeled using a modified Johnson-Cook model, in which the
logarithmic strain rate hardening term is replaced with the Cowper-Symmonds
power-law strain rate hardening:

$Y=\left(A+B\varepsilon_{p}^{n}\right)\left(1+\left(\frac{\dot{\varepsilon}_{p}}{D\dot{\varepsilon}_{ref}}\right)^{p}\right)\left(1-\left(\frac{T-T_{R}}{T_{m}-T_{R}}\right)^{m}\right)$
(3)

where $p$ and $D$ are parameters associated with the strain-rate hardening
term. Parameters for these models were obtained from the past studies and
listed in Table 4. Additionally, thermal softening was not considered in the
simulations as the estimated temperatures under the loading conditions
considered here is only a small fraction of the melting temperature.

Table 4: Parameters of the strength model used in simulations for Polycarbonate (PC), Tungsten Carbide (WC), Iron, and D2 tool Steel.  |  |  Model parameters (Eqs. (2) and (3))
---|---|---
Material |  | Shear Modulus
---
$G$ [GPa]
| $A$
---
[MPa]
| $B$
---
[MPa]
$n$ | $C$ | $m$ | $p$ |  | $D$
---
($\times 10^{3}$)
$\dot{\varepsilon}_{ref}$
PCDwivedi  _et al._ (2012) | 0.93 | 80 | 70 | 2.00 | 0.052 | 0.548 | - | - | 1
WCRavindran  _et al._ (2021) | 278 | 4300 | 9200 | 0.38 | - | - | 2 | 1500 | 1
D2 tool steel Ravindran  _et al._ (2021) | 78 | 850 | 150 | 0.24 | - | - | 2 | 500 | 1
IronSadjadpour  _et al._ (2015) | 78 | 32.6 | 430 | 0.1 | 0.28 | 0.55 | - | - | 355

Unlike other materials, Y-cut quartz (YQZ) was modeled using the anisotropic
elastic model since the pressures that we are interested in the study do not
create appreciable plastic strain in the samples. Anisotropic elastic
constants and the density of the YQZ were obtained from Johnson et al.Johnson
(1971) and Heyliger et al. Heyliger  _et al._ (2003) and listed in Table 5.

Table 5: Density and elastic constants of Y-cut quartz Johnson (1971); Heyliger  _et al._ (2003) | Density
---
[kg/m3]
| $C_{11}$
---
[GPa]
| $C_{12}$
---
[GPa]
| $C_{13}$
---
[GPa]
| $C_{14}$
---
[GPa]
| $C_{33}$
---
[GPa]
| $C_{44}$
---
[GPa]
2650 | 86.8 | 7.0 | 11.9 | -18.0 | 105.8 | 58.2

##  III RESULTS AND DISCUSSION

###  III.1 Normal impact experiment on polycarbonate

Figure 3: a) Variation of the free surface displacement $(w)$ from experiment
and simulation in a normal plate impact on polycarbonate at different
Lagrangian axial coordinates along the line shown in the inset. The inset
figure shows the full-field particle displacement at $t=4.2$ $\mu$m, b)
evolution of free surface displacement from simulation and experiment at the
center of the sample, where the inset shows the displacement from the
simulation, c) normal free surface particle velocity $(dw/dt)$ from simulation
and experiment at the center of the target, d) full-field free surface
particle velocity at different times.

A pure iron flyer impacted a polycarbonate target at a velocity of $201\pm
0.5$ m/s in a normal plate impact experiment. The stress associated with the
impact calculated using impedance matching Meyers (1994) was $0.58$ GPa. For
these calculations, linear equation of state (EOS) of iron and polycarbonate
was used, see Table 3. Images of the speckled free surface taken using a high-
speed camera during the experiment were used to calculate the full-field free
surface displacement of the sample. Figure 3a (inset) shows the full field
out-of-plane (free surface) displacement ($w$) at $t=4.2$ $\mu$s, with $t=0$
being the time of impact. It is noted that the displacement at the left edge
is about 60 $\mu$m higher than at the right edge of the sample. To visualize
the variation of the displacement at a given time, out of plane displacement
at four different times, $t=3.8$, 4.2, 4.6, and 5.2 $\mu$s, along a 9 mm line
that passes through the center of the sample is plotted in Fig. 3a. The
observed spatial displacement variation along the line is associated with the
impact tilt in the experiment. The average tilt measured using the shorting
pins was $1.8$ mrad. Numerical simulations were performed to evaluate the
accuracy of the free surface displacement measurement and other derived
quantities. In the 3D simulation, the flyer was tilted in the same orientation
as in the experiment, which was measured using tilt pins, to simulate the
exact impact conditions of the experiment. Material strength and the pressure-
volume behavior were modeled using the Johnson-cook material model and the
linear equation of state $\left(U_{s}-u_{p}\right)$, respectively. Parameters
for the linear equation state and the Johnson-cook model are shown in Tables 3
and 4 respectively. It is noted that the spatial variation of the displacement
in the experiment (solid lines) and simulation (symbols) closely match with
each other, including the higher displacement at the left edge, as shown in
Fig. 3a. Also, the temporal evolution of the free surface displacement at the
center of the sample matches closely with the simulation, see Fig. 3b. The
inset in Fig. 3b shows the section at the center of the sample from the
simulation at $t=4.2$ $\mu$s, which indicates that the displacement at the
center is in close agreement to the displacement measured ($0.25$ mm) in the
experiment.

The particle velocity was calculated by numerically differentiating the
displacement data using a central difference formula. Figure 3c shows the
particle velocity calculated from the raw displacement data, filtered
displacement data, and simulation. Calculation of the velocity from noisy
displacement data shows unrealistically large oscillations with magnitude
close to $60$ $m/s$. Hence, the random noise in the displacement data was
filtered using the method described in Appendix A. Consequently, the velocity
calculated from the filtered data shows nearly constant steady-state free
surface velocity, which is close to the simulated and theoretically calculated
particle velocity. Filtering reduces the uncertainty in the out-of-plane
velocity down to $20$ m/s. However, the rise time of the experiment is
difficult to capture due to the reduced temporal resolution from filtering.
The evolution of full-field free surface particle velocity calculated from the
full-field displacement is shown in Fig. 3d. These particle velocity contours
also show the effect of tilt on the spatial distribution; at $t=3.8$ $\mu$s,
the velocity at the left edge is higher than the sample’s right edge. However,
after $t=4.0$ $\mu$s, the free surface shows a relatively homogeneous velocity
field. It indicates that the flyer takes about 200 ns to close the gap with
the target, which corroborates with the tilt measured in the experiment.

Shock wave velocity was also calculated using the known thickness of the
sample and shock wave transit time in the sample. Shock transit time was
calculated by finding the time difference between the contact time of the
flyer with the sample and the wave arrival time at the free surface. The
contact time was obtained from the copper shorting pins, while the arrival
time of the shock at the free surface was determined by inspecting the
temporal evolution of the out-of-plane displacement or particle velocity.
Shock arrival time at the free surface of the sample was $3.5\pm 0.1$ $\mu$s.
Thus, the shock velocity in the sample was calculated (sample thickness/time
of shock arrival) to be $2620\pm 75$ m/s, which is in accord with the
previously reported shock velocity of 2553 m/s at similar pressuresYe  _et
al._ (2019).

The normal stress ($\sigma$) from the shock wave velocity and particle
velocity can be calculated usingMeyers (1994),

$\displaystyle\sigma=\rho_{0}U_{s}u_{p}$ (4)

where, $\rho_{0}$ is the initial density, $U_{s}$ is the shock velocity, which
is calculated from the linear equation of state, $U_{s}=C_{0}+Su_{p}$ of the
sample, $S$ is the slope of the linear $U_{s}-u_{p}$ relation, $C_{0}$ is the
bulk sound velocity of the sample, and $u_{p}$ is the particle velocity, which
is half the measured normal free surface particle velocity ($dw/dt$). The
shock stress calculated from the unfiltered data using Eq. (4) was $0.58\pm
0.12$ GPa, where the uncertainty is associated with both the particle velocity
and shock wave velocity and is around $20\%$. The uncertainty in the particle
velocity can be reduced by removing the random noise in the measured
displacement, as discussed in the Appendix. After filtering the displacement,
the stress in the sample was calculated to be $0.58\pm 0.06$ GPa, with the
uncertainty reduced to $10\%$.

The experiment was also designed to measure the spall strength Meyers (1994).
Free surface velocity histories obtained at the center of the sample is used
to deduce the spall strength of the material,

$\displaystyle\sigma_{\text{spall }}=\rho_{0}C_{L}\Delta
u\frac{1}{1+\frac{C_{L}}{C_{B}}}$ (5)

where $C_{L}$ and $C_{B}$ are the longitudinal wave speed and bulk sound
speed, respectively, and $\Delta u$ is the pullback velocity determined from
the free surface velocity measurement. The values of the various parameters
used in the calculation of spall strength using Eq. (5) are, $C_{L}=2190$ m/s,
$C_{B}=1930$ m/s, $\Delta u=V_{a}-V_{b}=110$ m/s, see Fig. 3c. The spall
strength of polycarbonate estimated from this experiment at a normal stress of
$0.58$ GPa and a strain rate of $0.12\times 10^{5}$ s-1 is $0.14$ GPa. The
spall strength reported for polycarbonate at a similar shock stress (at a
higher strain rate) is $0.16$ GPa Majewski  _et al._ (1986). This is $12.5$ %
higher than the measured value in this study and the discrepancy could be
attributed to strain rate effects and measurement uncertainty.

Figure 4:  a) Experimental configuration with $\mathrm{Y}$-cut quartz-tungsten
carbide $(\mathrm{YQZ}+\mathrm{WC})$ target assembly, b) distance-time $(t-Z)$
diagram of the experimental configuration, c) free surface particle
displacement $(w)$ and free surface particle velocity $(dw/dt)$ at $t=2.0$
$\mu$s, $3.0$ $\mu$s, and $4.0$ $\mu$s, d) free surface particle velocity
evolution at different points marked on the free surface of the WC.

###  III.2 Normal impact experiment on Quartz-WC assembly

The high-pressure properties of condensed matter under isentropic compression
is important in many dynamic applications. In such loading, the rise time is
relatively large compared to loading conditions that generate strong shocks. A
model experiment was conducted to demonstrate the capability of high-speed DIC
experiments for measuring the free surface velocities with long rise times. To
resolve the longer rise times ($>300$ ns), measurement diagnostics with a time
resolution of 100 ns is deemed sufficient. Y-cut quartz (YQZ)-tungsten carbide
(WC) dual-layer assembly was used as a target in this experiment, see Fig. 4a.
The driver was the YQZ crystal of approximately 5 mm in thickness and 30 mm in
diameter, and the measurement was performed on the rear side of the WC of the
target, see Fig. 4a. The flyer material was also made of tungsten carbide.
Figure 4b plots the distance-time $(t-Z)$ diagram of the experiment, which
shows the wave reverberation within the thin WC attached to the YQZ. Three
recompression signals can be recorded for a total duration of $1.89$ $\mu$s
before the lateral relief wave arrives at the center of the sample. This long
rise time of $1.89$ $\mu$s allows to capture 18 images for the displacement
measurement; therefore, it was relatively easy to resolve the temporal
features of the free surface velocity history.

Figure 4c shows the full-field free surface displacement and the free surface
particle velocity at different times $t=$$2.0\ \mu$s, $3.0\ \mu$s, and $4.0\
\mu$s. The displacement at various times is relatively homogeneous, with the
peak displacements of $410\ \mu$m, $1190\ \mu$m, and $2190\ \mu$m at $t=$$2.0\
\mu$s, $3.0\ \mu$s, and $4.0\ \mu$s, respectively. The peak out-of-plane
displacement (or normal displacement, $w$) was observed to be near the center
of the WC as a result of the uniaxial strain condition in shock compression.
The normal free surface velocity $(dw/dt)$ contours determined from the full-
field displacement is also plotted in Fig. 4c, showing a homogeneous field.
The small heterogeneity in the velocity is mainly associated with the noise in
the measurement, similar to the observation in the normal impact experiments
on polycarbonate described earlier in Section III.1. To visualize the free
surface velocity variation as a function of time, the data was plotted at
randomly selected points shown in the inset of Fig. 4d. The velocity evolution
at different points on the free surface indicates that the magnitude of free
surface velocities is very close to each other since the microscale
heterogeneity in WC is at much smaller length scales. The temporal evolution
of the particle velocity shows step-like features at $t=1.01$ $\mu$s, 1.45
$\mu$s, and 1.89 $\mu$s, see Fig. 4d. This timing corresponds to the arrival
of the reflected waves from the YQZ-WC interface, depicted in Fig. 4b,
indicating that these step-like features are due to wave reverberations with
the WC target.

Numerical simulations were conducted using the material models discussed in
Section III.1, and the resulting free surface velocity is plotted in Fig. 4d.
The Y-cut quartz was modeled using an anisotropic elastic constitutive
description, and the material parameters are shown in Table 5. WC was modeled
using the linear equation of state and the strength model with constitutive
description in Eq. (3). The strength model was calibrated in our previous
study using the data from symmetric pressure shear plate impact (PSPI)
experimentsRavindran  _et al._ (2021), and the parameters are shown in Table
4. Figure 4d shows the normal particle velocity calculated from the numerical
simulations and the experiments. Notably, the step-like features resolved in
the experiments were also reproduced by the simulations, indicating the
confidence of the measurement technique developed in this study. Hence, the
temporal resolution used in the study is sufficient to capture the longer rise
time measurements with different temporal features seen in Fig. 4.

###  III.3 Pressure shear plate impact experiment on D2 tool steel

Figure 5: a) Pressure shear plate impact (PSPI) configuration, b) free surface
normal and transverse particle velocity profiles for D2 tool steel from
PDV/HTV and high-speed DIC measurements, c) contour plots of transverse
velocity $(dv/dt)$ at $t=2.2$ $\mu$s, $2.4$ $\mu$s, and $2.6$ $\mu$s.

Pressure shear plate impact (PSPI) experiments are advantageous in measuring
the shear strength of materials at high pressures and strain ratesAbou-Sayed
_et al._ (1976); Ravindran  _et al._ (2021); Kettenbeil  _et al._ (2020);
Klopp  _et al._ (1985). The difference between PSPI and normal plate impact
experiments is the skewness $(\theta)$ of the flyer plate and target plate
configuration, as shown in Fig. 5a. The inclined impact generates both
longitudinal and shear waves within the flyer and the target assembly. In
these experiments, normal velocities are typically measured using photonic
Doppler velocimetry (PDV) and transverse velocity using heterodyne transverse
velocimeterKettenbeil  _et al._ (2018) (HTV), transverse displacement
interferometerKim  _et al._ (1977) (TDI), or modified PDV Mallick  _et al._
(2019b). This is done by depositing a diffraction grating on the free surface
of the target where the diffracted $0^{\textnormal{th}}$ order beam is
interfered with a reference signal to extract the normal free surface velocity
and the diffracted $\pm 1^{\textnormal{st}}$ order beams are interfered with
each other to determine the transverse velocity. Detailed descriptions of PSPI
experiments and the velocity measurement techniques can be found elsewhere
Kettenbeil  _et al._ (2018, 2020); Klopp  _et al._ (1985); Kim  _et al._
(1977); Zuanetti  _et al._ (2017). For PSPI experiments, in addition to the
fabrication of the diffraction grating, the main difficulty arises from the
precise alignments required to ensure that (1) the diffraction plane is
perfectly aligned with the shearing direction, and (2) the fiber optic probes
are precisely aligned to maximize the light input in the transverse probes.
Also, the transverse signals are highly sensitive to impact tilt which could
result in light loss during an experiment. On the other hand, full-field DIC
does not require precise alignment and it is insensitive to impact tilt,
therefore, using DIC relaxes the tight constraints required in conventional
(PDV, HTV) transverse velocity measurements. Thus, digital image correlation
can be a powerful free surface velocity measurement technique for PSPI
experiments. To explore its feasibility, a PSPI experiment was conducted on D2
tool steel with an impact skew angle of $\theta=18^{\circ}$ and the full-field
normal and transverse velocities were measured using high-speed digital image
correlation. This measurement was then compared with our previous PSPI
experimentRavindran  _et al._ (2020) with similar loading conditions that
utilized laser interferometry (PDV/HTV) to extract the normal and transverse
free surface velocities at a single point, i.e., the center of the rear free
surface.

The impact velocity of the D2 tool steel flyer plate in the experiment was
measured to be $530\pm 0.9$ m/s , which was close to a previous PSPI
experiment ($539\pm 0.8$ m/s) with PDV and HTV based single point velocity
measurementsRavindran  _et al._ (2020). Figure 5b shows the normal and
transverse free surface velocity at the center of the sample from high-speed
DIC measurement along with the PDV-HTV measurements from an earlier
studyRavindran  _et al._ (2021). As expected, the normal and transverse
components of the impact velocity match the measured free surface velocities
of both the DIC and the laser interferometry results. Comparing the DIC data
to the PDV measurements, the normal velocity arrival time and the final
Hugoniot state of the material were close, considering the uncertainty
associated with the out-of-plane velocity measurement (Fig. 2) and the
temporal resolution of the experiment. Also, the high-speed DIC measurement
captures the shock wave rise time relatively well, however, resolving the
Hugoniot elastic limit was difficult due to the low temporal resolution in the
DIC measurement. The peak transverse velocity that was measured in the
previous experiment, approximately 45 m/s, closely matched the transverse
velocity profile measured in high-speed DIC measurements. However, the DIC
transverse velocity profile shows oscillatory behavior, which is associated
with uncertainty due to numerical differentiation of the noisy transverse
displacement. The shear wave dissipates when traveling through the plastically
deformed medium due to the forerunning normal shock which reduces the
transverse displacement magnitude on the backside of the target. The
displacement resolution required is high to resolve such small displacements,
which can be increased by using higher magnification optics.

Figure 5c shows the full-field transverse particle velocity from the
experiment. Note that the transverse velocity at the edge of the sample is
higher than $70$ m/s due to the lateral deformation at the circumference of
the target sample. At the center of the sample, the velocity is very close to
the previously measured transverse velocity in this material in the PSPI
configuration using single point laser interferometry.

##  IV Summary and conclusion

An experimental methodology has been developed for the first time to measure
the full-field particle velocity in shock compression experiments using high-
speed imaging and stereo 3D digital image correlation (DIC). Since DIC cross-
correlates speckle images to determine displacement fields, computing the
derivative from the data with noise to obtain velocities is challenging. To
overcome this limitation, an optimal smoothing method (See Appendix A) was
adopted to filter the data such that the roughness of the data is minimized
without oversmoothing the key features in particle velocity. Three different
types of experiments were conducted to validate the measurement technique to
resolve free-surface velocities in experimental configurations involving shock
compression, which are summarized below.

  1. 1.

A normal impact experiment on polycarbonate at low spatial resolution
demonstrated that the full-field free surface velocity under shock compression
can be measured using high-speed digital image correlation. The results from
numerical simulation closely matched the filtered free surface velocity using
the optimal spline smoothing of the data. The temporal resolution of the
experiment was critical to resolve the shock rise time, which can be
challenging in strong shock experiments, where the shock thicknesses are
relatively small.

  2. 2.

In the normal impact experiment on a Y-cut quartz-tungsten carbide (YQZ-WC)
target assembly, the long rise time experimental profile associated with
isentropic compression was captured remarkably well and matched the numerical
simulation. In this experiment, the spatial resolution used was around 94
$\mu$m, a factor of two smaller than in the polycarbonate experiment. Hence,
the velocity uncertainty was reduced to 30 m/s as compared to 45 m/s
uncertainty for the polycarbonate experiment.

  3. 3.

A symmetric pressure shear plate impact (PSPI) experiment was conducted on D2
tool steel to show the capability of the technique in measuring both normal
and transverse velocities under combined normal and shear loading arising from
an oblique impact. The high-speed DIC measurement closely matched the
previously measured normal and transverse velocity using point-wise
measurement using PDV and HTV interferometric techniques.

The newly developed technique offers many advantages for both normal and
oblique plate impact experiments. DIC requires cameras and a speckled sample,
which circumvents traditional interferometer setups involving complicated
optical arrangements, laser alignments, and for oblique impacts, diffraction
gratings Kettenbeil  _et al._ (2018). The high-speed DIC technique is
convenient for measuring the spatially resolved data. With the current state-
of-the-art cameras, the best time resolution that can be used for accurate 3D
digital image correlation measurement is $100$ ns, which is sufficient to
resolve rise time as short as 300 ns. The spatial resolution can be achieved
as small as $0.2$ $\mu$m/pixel, enabling the measurement of the local particle
velocities in heterogeneous materials with grain sizes as small as $5$ $\mu$m.
This capability allows one to probe material behavior at mesoscale under shock
compression, which remains a great challenge with conventional optics based
techniques. It is expected that the framing rate of cameras will increase in
the future, enabling one to measure the full field particle velocity in strong
shock regimes where the rise time are below 10 ns.

###### Acknowledgements.

The research reported here was supported by the DOE/NNSA (Award No. DE-
NA0003957), which is gratefully acknowledged. The authors acknowledge the
support from the Office of Naval Research (Award No. N00014-16-1-2839) for
developing the high-pressure PSPI capability, and the Army Research Laboratory
(Cooperative Agreement Number W911NF-12-2-0022) for the acquisition of the
high-speed cameras.

##  Appendix A Optimal data filtering

Figure 6: a) True function ($f$) and noisy data ($y_{i}$) as a function of
time ($t$) with a signal-to-noise ratio (SNR) of 25 dB, b) velocity calculated
from the noisy data and the true function. Figure 7: a) Roughness vs Error in
log-log scale, roughness and error are calculated based on 6 and 7, b)
velocity calculated from filtered data.

Evaluating particle velocity from the displacement measurements is essential
to understanding the physics of materials subjected to shock compression .
However, calculating the derivatives from the experimentally measured
displacement is nontrivial due to noise. The error in the derivative is
typically an order of magnitude higher than the noise in the measurements and
is highly susceptible to small errors in the data. Thus, to accurately infer
the particle velocity (first time-derivative of displacement), a smoothing
spline is generally fit to the displacement vs. time data. The smoothing
spline, characterized by its error in relation to the original data and by its
roughness, is a piece-wise polynomial fit to the data that can capture any
functional form Epps (2010); De Boor (1978); Silverman (1985). The error, $E$,
and roughness, $R$, associated with a smoothing spline, $s(t)$, and dataset,
($t_{i},y_{i}$), are given by:

$\displaystyle E(s)$
$\displaystyle=\sum_{i=1}^{N}\left[y_{i}-s\left(t_{i}\right)\right]^{2}$ (6)
$\displaystyle R(s)$
$\displaystyle=\sum_{i=1}^{N}\left[\frac{d^{3}s\left(t_{i}\right)}{dt^{3}}\right]^{2}$
(7)

Determining the optimal smoothing spline fit involves a trade off between the
total error of the spline, $E(s)$, and the roughness of the spline, $R(s)$.
For example, a spline with zero error would pass through all the (noisy) data
points and thus have the highest roughness. A spline with zero roughness would
be a straight line of regression through all the data points and thus involve
the highest error.

The majority of literature employs the ‘de Boor formulation’ De Boor (1978)
(terminology adopted from Epps Epps (2010)) to determine the optimum smoothing
spline for the data. The de Boor formulation involves minimizing the weighted
sum of the error and roughness measures by an appropriate choice for the
weight (smoothing parameter). An equivalent approach can be found in the
‘Reinsch formulation’Reinsch (1971) (terminology adopted from Epps Epps
(2010)), which involves finding the smoothing spline with the least roughness
but with its error below a chosen threshold. The Reinsch formulation, as
implemented in Epps Epps (2010), is said to overcome smoothing inadequacies
that can result from the de Boor formulation. Epps Epps (2010) used the
Reinsch formulation to construct a plot of minimum roughness vs. critical
error tolerance, which was then used to identify the optimal smoothing spline
for the data. This procedure is best suited for extracting the velocity
history for the region of interest as it is amenable to automation and can
operate upon a large number of datasets. This procedure was adopted in the
current work and is illustrated next.

Consider a general sine-cosine signal:
$f(t)=\mathrm{A}\sin(t)+\mathrm{B}\cos(t)$ with an added noise
$\tilde{\epsilon}(t)$, whose Signal-to-Noise-Ratio (SNR) $=25\mathrm{~{}dB}$
and is sampled at $N$ data points. We thus obtain a test dataset
$\left(t_{i},y_{i}\right)$, where
$y\left(t_{i}\right)=f\left(t_{i}\right)+\tilde{\epsilon}\left(t_{i}\right)$.
The noisy data $y(t_{i})$ and the true function $f(t)$ are illustrated in Fig.
6a, and their corresponding derivatives $\left(df/dt\textnormal{ and
}dy_{i}/dt\right)$ are shown in Fig. 6b. It is noted that that the noise in
the derivative is significantly larger than the noise in the original data. An
outline of the procedure Epps (2010); Reinsch (1971) to obtain the smoothing
spline to best approximate the function $f(t)$ is described next. The MATLAB
noa (2021) function spaps implements the Reisnch formulation (discussed above)
to output a minimum roughness smoothing spline for a given noisy dataset and
specified maximum error tolerance. The order of the smoothing spline (cubic,
quintic, etc.) can also be specified in the function. The noisy dataset
$\left(t_{i},y_{i}\right)$ and a certain maximum error tolerance $E$ are
provided to the spaps function, which then outputs the smoothed spline
datapoints $s\left(t_{i}\right)$. The smoothing spline $s\left(t_{i}\right)$
has the minimum possible roughness $R(s)$ for the given error tolerance of
$E(s)$. $E(s)$ and $R(s)$ can be computed using Eqs. (6) and (7),
respectively. By varying the $E(s)$, which is provided to the spaps function,
various smoothing splines with different $R(s)$ can be obtained. The log-log
plot of $R(s)$ vs. $E(s)$, as shown in Fig. 7a, is then analyzed to obtain the
critical error tolerance $E_{\textnormal{critical}}$, which is the solution to
the de Boor formulation for the optimal smoothing spline. The critical point
in the $R(s)$ vs. $E(s)$ plot is the point corresponding to the sudden change
in curvature. A plot of the first derivatives of smoothing splines
corresponding to error tolerances of $E_{\textnormal{low}}$,
$E_{\textnormal{critical}}$ and $E_{\textnormal{high}}$ are shown in Fig. 7b
alongside the derivative of the true function, $f(t)$. It can be seen in Fig.
7b that the minimum roughness smoothing spline for the dataset, $y({t_{i}})$,
with a specified error tolerance of $E_{\textnormal{critical }}$ provides an
excellent estimate of the first derivative, which closely matches that of the
true function. When $E=E_{\text{high}}$, the data is over smoothed, while for
$E=E_{\text{low}}$, the data is under smoothed, which will introduce large
error while calculating the derivative. This procedure would work only for
datasets with low to moderate noise. For datasets with high level of noise, a
clear kink in the $R(s)$ vs. $E(s)$ cannot be observed. Further details
regarding the procedure and theory used here can be found in Epps Epps (2010).

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