#  Design of a low-velocity impact framework for evaluating space-grade
materials

Vineet Dawara  Department of Mechanical Engineering, Indian Institute of
Science, Bangalore  Ashok Bajantri  Department of Mechanical Engineering,
Indian Institute of Science, Bangalore  Harish Singh Dhami  Department of
Mechanical Engineering, Indian Institute of Science, Bangalore  SVS Narayana
Murty  Indian Space Research Organization, India  Koushik Viswanathan
<EMAIL_ADDRESS>Department of Mechanical Engineering, Indian Institute of
Science, Bangalore

###### Abstract

Material deformation and failure under impact loading is a subject of active
investigation in space science and often requires very specialized equipment
for testing. In this work, we present the design, operational analysis and
application of a low-velocity ($\sim 100$ m/s) projectile impact framework for
evaluating the deformation and failure of space-grade materials. The system is
designed to be modular and easily adaptable to various test geometries, while
enabling accurate quantitative evaluation of plastic flow. Using coupled
numerical methods and experimental techniques, we first establish an operating
procedure for the system. Following this, its performance in two complementary
impact configurations is demonstrated using numerical and experimental
analysis. In the first, a Taylor impact test is performed for predicting the
deformed shape of a cylindrical projectile impinging on a rigid substrate. In
the second, deformation of a plate struck by a rigid projectile is evaluated.
In both cases, physics-based models are used to interpret the resulting
fields. We present a discussion of how the system may be used both for
material property estimation (e.g., dynamic yield strength) as well as for
failure evaluation (e.g., perforation and fracture) in the same projectile
impact configuration.

#### Keywords -

Gas gun, Large strain deformation, projectile impact, low-velocity impact

## 1 Introduction

Failure of metallic materials under projectile impact is a subject of active
investigation in several sub-domains of space science. For instance,
mitigating satellite failure due to space debris requires materials and
structures that are resilient enough to withstand high strain-rate deformation
of $\sim 10^{3}-10^{5}$ s-1 [1, 2, 3]. While extensive work has been carried
out on hypervelocity ($>1$ km/s) impact events in the context of space debris
in low-earth orbit [4, 5], relatively lesser work has focused on low-velocity
($10-100$ m/s) impact events. However, these events are equally important to
understand—as an example, Whipple shields can be significantly less effective
at low velocity impact due to complete projectile penetration and lack of
fragmentation [6]. While hypervelocity impact is common in low earth orbit
applications, low-velocity events are expected to be more likely for deep-
space missions.

The performance of a complex structure is commonly obtained by piecing
together behaviour of representative samples under individual unit impact or
deformation events [7, 8]. These events can help drive the design of new
materials and/or structures with engineered internal features [9, 10].
Quantifying impact-induced failure is also vital for evaluating the
performance of more commonplace materials such as structural steels and glass
sandwich panels [11, 12] that are used for spacecraft and launch vehicle
applications.

There is hence a fundamental need for a well-controlled impact testing system
for materials performance evaluation, especially in the context of space
applications. Unfortunately, while many impact test systems exist in research
laboratories across the world (e.g., see Ref. [6]), with several research
results being continuously published, such a system is difficult to reproduce
from scratch, due to the paucity of available design information, operating
procedures and calibration data. This manuscript attempts to fill that gap by
outlining the design and performance of a projectile impact test-bed. We
provide the basic design, detailed performance evaluation, as well as
demonstrations of the use of this setup in two common impact configurations
[13]. In the first—the so-called Taylor impact test—a deformable projectile is
struck against a comparatively rigid target plate. This configuration is most
often used for material property evaluation—the yield strength of the
projectile material may be evaluated from its deformed shape [14]. The second
test, termed plate impact, is a complementary configuration, in which a
comparatively rigid projectile is impacted against a deforming target plate.
Here, the focus is on evaluating failure mechanisms in the target material. In
either case, specific microstructural mechanisms such as adiabatic shear
banding and crack bifurcation can be correlated with impact parameters and
kinematically measured deformation fields [15, 16, 17, 18, 19]. These
mechanisms are, in general, not amenable to direct real-time measurements,
necessitating the use of techniques such as ‘quick stop’ projectiles to infer
temporal evolution [20]. Results of these experiments often suggest the use of
structural and geometrical changes to either accentuate or mitigate the
occurrence of plastic failure in the projectile/target or both [21, 22, 23].

The primary kinematic/geometric parameters in a typical projectile impact test
in either configuration described above are the projectile velocity, size and
shape. The desired projectile velocity is achieved using a gas gun, wherein an
inert gas (such as He, Ar or N2) is compressed to a suitable initial pressure
and allowed to expand rapidly in order to accelerate the projectile. The
result is projectile exit velocities that can range from 10-250 m/s [24]. The
accelerated projectile is retained within a barrel and strikes a target plate
placed a fixed distance away. One of several impact models may then be
employed to evaluate failure in either the projectile or the target, as a
function of impact parameters [25, 26, 27].

While the setup described and analyzed in this manuscript is not unlike common
frameworks described before, we present a complete design-to-performance
description using a combination of quantitative mechanics models and
experimental observations for both impact configurations. The primary purpose
of the framework is to evaluate failure mechanisms in space-grade materials,
that are of specific interest to the Indian Space Research Organization. The
design does not focus on maximizing projectile velocity; rather, the emphasis
is on achieving versatility in conducting different projectile-target impact
experiments. The experimental configuration is also easily adaptable for
realizing larger target sizes and a wider variety of target shapes to simulate
actual parts. We envisage this manuscript as being the first in a series of
works that explore a systematic method for designing lightweight structural
materials for space applications.

The manuscript is organized as follows. The basic design and realization
details of the impact framework are discussed in Sec. 2. The main results are
presented in three parts. Firstly, coupled numerical simulations and
experimental velocity measurements are used to evaluate performance envelopes
in Sec. 3. Following this, we present investigations of the two impact
configurations described above—Taylor impact (Sec. 4) and plate deformation
(Sec. 5). In each of these tests, experimental results are accompanied by a
presentation of mechanics-based models to quantitatively understand the final
deformation. In all three parts, the findings are followed by a short
exposition on how the corresponding results may be used in a practical
setting. Finally, a broad overall discussion is presented in Sec. 6, along
with some concluding remarks.

## 2 Experimental configuration

The primary components of the impact framework are a gas gun, a barrel of
fixed length and a target chamber, housing the target, see Fig. 1(a). These
are mounted on dovetail guides fixed to a supporting structure. The overall
dimension of the setup is 2.5 m $\times$ 1 m $\times$ 1 m. Dovetail mounting
enables easy projectile loading, while also allowing changes to individual
components to be made quickly. The detailed description of individual
components, as well as the triggering mechanism and velocity measurement
system, are discussed below.

Figure 1: Operational drawings of the gas gun system. (a) Schematic of the
complete system showing the gun, barrel and target chamber mounted on
guideways for easy of disassembly. Panels (b) and (c) show the operation
mechanism using the two-valve system in sequence. The projectile is at the
breech end of the gun, and is fired by air escaping from the main chamber.

### 2.1 Gas chambers and triggering mechanism

The gas gun has a total volume of 1.8 litres and is designed to withstand a
maximum pressure of 50 bar, see Fig. 1(b). The overall design and triggering
mechanism is based on that commonly used for split-Hopkinson pressure bars
(see, for instance, Ref. [28]). The gun has two internal chambers, termed
valve and main chambers. A two-valve arrangement regulates gas flow from an
external compressed air source (here a portable air compressor) to the gas
chamber. The projectile is typically supported by coaxial Teflon rings, called
sabots. One or two sabots are used, depending on the projectile length. Their
use is necessary because sabots minimize friction between the projectile and
interior barrel surface, while also helping align projectiles of arbitrary
diameter centrally with the barrel axis. The projectile is first loaded from
the target, through the barrel and close to the end connected to the gun
(termed the breech end). This is done by moving the target plate on the
dovetail and using a long enough tube to push the projectile through the
length of the barrel.

The actuation mechanism to accelerate the projectile is then operated as
follows, see Fig. 1(b) and (c). When valve 1 is opened and valve 2 closed,
pressurized gas from the external source flows into the valve chamber. This
gas then enters the main chamber via the clearance provided between the
sliding disc and interior surface of the valve chamber. O-rings are provided
on the sliding disc and support plate to prevent gas from flowing directly to
the barrel. Once the main chamber is filled to the required pressure (measured
using a pressure gauge), valve 1 is closed. The projectile placed at the
breech end of the barrel is now triggered by opening valve 2. When this is
done, pressure inside the valve chamber drops and high-pressure gas from the
main chamber flows back, pushing the sliding disc. It enters the barrel
immediately through the ports, see Fig. 1(c), providing the required driving
force for projectile motion.

This simple triggering system is inherently advantageous as it does not
require any operating skill, and can be easily reconfigured for subsequent
tests. Further, experiments have been performed to demonstrate that a
projectile velocity of $>100$ m/s can be easily achieved by simply using
compressed air, with higher velocities attainable using a light gas such as He
[29].

### 2.2 Barrel and velocity measurement system

Figure 2: Measurement methods for determining projectile exit velocities from
the barrel. (Left) Section near the barrel end showing vent holes and the
three independent measurement systems—a camera/light source combination, IR
sensors with a microcontroller and an oscilloscope for interrupt measurements.
(Right) Target chamber cross section showing the plate mounting details and
momentum trap.

The barrel housing the projectile has an inner diameter of 21 mm and total
length of 1.4 m. Five sets of four vent holes (10 mm diameter, 50 mm apart,
arranged at $90^{\circ}$ from each other) are provided along the length of the
barrel for allowing gas to escape, see Fig. 2. If this is not done, the
possibility of high gas pressure resulting in secondary impact may obscure the
results. The location of the first hole, at a distance of 740 mm from the
barrel breech end, is chosen to maximize projectile exit velocity, while
preventing secondary impacts. The velocity measurement system consists of IR
sensors and a microcontroller (Arduino Mega 2560), as shown in Fig. 2. An
oscilloscope is also used in parallel with the microcontroller for verifying
the measurements. A camera and light source arrangement may also be employed
across opposite lying ventholes for this purpose.

Projectile velocity within the barrel, and prior to entering the target
chamber, is measured using two IR sensors that read signals from the third and
fifth holes, spaced 100 mm apart, see Fig. 2. Each IR sensor consists of an
emitter and receiver placed at opposite facing vent holes on the barrel. The
sensor’s voltage output changes from low to high at the instant when the
projectile is at the vent hole, and is recorded by the microcontroller. The
projectile velocity is estimated the distance between these holes, divided by
the time difference between the two crossing events.

A second velocity measurement is performed using the voltage output of the IR
sensor placed at the third hole, using a digital oscilloscope (GW Instek GDS
11028). When the projectile passes through the hole, it obstructs the
receiver, and the voltage output changes from high to low and back. The width
of this signal gives the time taken by the projectile to pass through the
hole, and the velocity can be estimated by dividing the length of the
projectile by the obtained time interval.

Finally, a third velocity measurement, for comparison, is also performed by
capturing a high speed image sequence through the second vent hole using a
high-speed camera (Photron AX100). The camera and light source are placed
opposite each other, see Fig. 2. Initially, the camera records a bright image
that changes to dark as the projectile passes. By counting the number of dark
images $n$ and noting the frame rate (FPS) set for the camera, the velocity is
estimated as

$\displaystyle V_{0}=\frac{L_{0}}{n/\rm{FPS}}$ (2.1)

where $L_{0}$ is the length of the projectile. All three measurement methods
gave identical values (within $\pm 1\%$) for the projectile velocity and are
hence reported interchangably in the rest of the manuscript.

### 2.3 Target chamber

The fully enclosed target chamber consists of a circular mounting fixture with
holes for mounting the target plate, see Fig. 2. This circular fixture is
easily replaceable for mounting different sizes of targets, varying from
50-120 mm (square) and 150-200 mm (circular) dimensions. Appropriate spacer
lengths can be provided to change the distance between the barrel exit and the
target plate, to accommodate load cells for force measurement. Three viewing
windows—two parallel to each other and the other perpendicular, are provided
for imaging purposes. A momentum trap, made either of wood or silicone rubber
is placed at the end to arrest projectile motion in case of complete target
perforation.

## 3 Results I: Gas gun performance evaluation

The basic requirement of a single impact experiment is that a projectile of
mass $m$ be accelerated to a pre-determined velocity $V_{0}$. Since the gas
gun is pressure-controlled, we require a scheme for evaluating the projectile
exit velocity as a function of the filled chamber pressure $p_{0}$ and $m$.
This is obtained using numerical 1D gas dynamics calculations and results in
so-called velocity-pressure curves. These numerical results are then compared
with experimental measurements to provide an operation scheme for the gun.

### 3.1 Numerical 1D gas dynamics calculations

As discussed in Sec. 2.1, as gas exits the main chamber, it causes the
projectile to move forward. This is essentially effected via compression and
rarefaction waves that develop inside the main chamber, causing pressure
variations behind the projectile[29]. Since the pressure behind the projectile
determines the force moving it forward, these waves influence its velocity
during motion inside the barrel. Given the finite length of the main gas
chamber, reflected waves are soon set up and travel forward towards the
projectile, further lowering the pressure behind it. An analysis of these
pressure fluctuations as well as simulations of projectile velocity are
performed using the description in Ref. [29]. Only the final results are
presented in this section.

Figure 3: Gas dynamic calculations for dynamic gas pressure and exit velocity
of projectile. (Left) A space-time diagram showing characteristic curves for
rarefaction waves behind the projectile. Corresponding sections of the main
gas chamber and breech end of the barrel are shown in the middle schematic.
Solid (dashed) lines correspond to $u+a$ ($u-a$) characteristics (Right)
Variation of pressure and velocity with distance along the barrel. Projectile
mass $M=10$ g at initial pressure $p_{0}=5$ bar, $A_{b}$ is the cross-section
area of the barrel, and intial speed of sound $a_{0}=435$ m/s.

Figure 3(left) shows the gas dynamics calculation for one such situation when
the initial gas pressure is set to $5$ bar, the projectile mass is $10$ g,
$A_{b}$ is the cross-sectional area of the barrel, and $a_{0}=435$ m/s is the
initial speed of sound. The gas is assumed to be ideal with a specific heat
ratio of $1.4$. The horizontal (time) and vertical (distance along barrel)
axes are presented in non-dimensional form using $p_{0},M,a_{0}$ and $A_{b}$.
The transition (line OD) from the main chamber to the barrel is assumed to be
sudden in this analysis. The solid and dashed lines in this figure are the two
characteristic lines corresponding to isentropic one-dimensional gas flow,
given by

$\frac{D(u\pm\sigma)}{Dt}=0\;\;\text{along}\;\;u\pm a\;\;\text{characteristic
lines}$ (3.1)

where $u$ is the gas velocity, $a$ is the speed of sound in gas (in general,
not constant), $\sigma=\int dp/(a\rho)$ is the Riemann function, with $p$ and
$\rho$ the pressure and density of the gas, respectively. The $u-a$ ($u+a$)
lines, shown by dashed (solid) lines, correspond to disturbances moving away
from (towards) the projectile. Only a few characteristic lines are drawn,
along with the chamber and barrel schematic (middle). When the projectile
moves, it sends corresponding disturbance (rarefaction waves) towards the main
chamber, see Fig. 3(left). As these waves arrive at the transition between
barrel and main chamber, they are partly reflected towards the projectile
(compression waves) and partly transmitted (rarefaction waves) to the main
chamber, where they are further reflected (reflected waves) on arriving at the
other end. In this figure, OAB consists of only rarefaction waves, with no
effect of reflected waves, and is termed the simple wave region. When
reflected waves return to the transition, they again undergo partial
reflection/transmission away from/towards the projectile.

The characteristic gas equation, Eq. 3.1, is solved using a step-by-step
numerical calculation, simultaneously with Newton’s second law for tracking
the projectile’s trajectory. The result is a gas velocity $u$ and pressure $p$
immediately behind the projectile as it traverses the length of the barrel,
see Fig. 3(right). Here $u$ and $p$ are non-dimensionalized using the speed of
sound in static air $a_{0}$ and initial gas pressure $p_{0}$. The horizontal
distance $x$ along the barrel is again non-dimensionalized by a characteristic
length scale $Ma_{0}^{2}/p_{0}A_{b}$. It is clear from the $u$ vs. $x$ graph
that the projectile velocity tends to saturate after about $x\sim
0.1Ma_{0}^{2}/(A_{b}p_{0})$. At this length, the gauge pressure behind the
projectile also reduces significantly, resulting in negligible force. For a
given $p_{0}$, the projectile exit velocity $V_{0}$ is hence $u$ evaluated at
$x$ equal to the distance to the first vent hole.

### 3.2 Experimental measurement: velocity-pressure curve

The results of the 1D calculations provide the projectile exit velocity
$V_{0}$ as a function of filling pressure $p_{0}$ for various $m$. To
calibrate this prediction for the system described in Sec. 2, experimental
data was obtained for various projectiles, see Table 3.1. Friction between the
sabot and the (stainless steel) barrel was neglected since the corresponding
friction coefficient is very small (0.05–0.08, see Ref. [30]). Two sabots,
totalling one-third of the projectile length, were used with all projectiles,
except for the smallest one (mass 10 g). For this case, owing to length
constraints, a single sabot with one-third length was used instead.

Table 3.1: Blunt cylindrical projectile dimensions for velocity measurements Combined (projectile+sabot) mass (g) | 10 | 24 | 35 | 50
---|---|---|---|---
Material | Al | Al | SS | SS
Initial diameter $D_{0}$(mm) | $10.0$ | $14.0$ | $13.0$ | $13.7$
Initial length $L_{0}$ (mm) | $26.1$ | $50.0$ | $30.5$ | $44.0$

Before presenting the experimental measurements, the following distinctions
between the theoretical model and the actual configuration in Sec. 2 must be
considered. Firstly, since significant gas escape occurs when the projectile
crosses the first set of vent holes, the 1D gas analysis computation is only
performed until the first hole. Secondly, in the analysis, the main chamber is
assumed to be connected to the barrel by a sudden transition region, while the
actual design uses a more gradual transition. Hence, the gas chamber contains
$10\%$ lesser gas volume than considered in the model. As a first
approximation, this volume correction is incorporated by decreasing predicted
projectile kinetic energy by $10\%$. The final predicted relation between the
projectile exit velocity $V_{0}$ and the initial gas pressure $p_{0}$ is
summarized in Fig. 4 (left), for various values of (combined projectile $+$
sabot) mass $m$. This graph, termed a velocity-pressure curve, provides the
initial filling pressure $p_{0}$ required for accelerating a combined mass $m$
to the required $V_{0}$.

Figure 4: An operating velocity-pressure curve (left) and kinetic energy-
pressure curve (right) for the gas gun setup presented in this work.
Theoretical lines (solid lines) and experimental measurements (square markers)
are shown superimposed. The dashed lines are fits to the experimental data,
and shaded regions indicates theoretical bands as described in the text.

Experimental validation of this theoretical curve is performed using a total
of 8 to 10 velocity readings for a given $m$, using simultaneous independent
velocity measurements (_cf._ Sec. 2.2). The mean velocity (square marker) is
plotted over the predicted curves in Fig. 4 (left). Error bars represent
standard deviation of the measurements. From the data in this figure, it is
clear that projectiles can be propelled to $V_{0}\sim 100$ m/s at $p_{0}=5.5$
bar. While compressed air was used for these results, the velocity can be
nearly doubled if a light gas, such as He, is used instead [31].

A systematic discrepancy is clear in Fig. 4(left) between the predicted and
measured $V_{0}$ curves, primarily because the numerical model assumes an
idealized one-dimensional geometry, whereas flow from the main chamber to the
barrel is actually two-dimensional, as described in Sec. 2.1. To account for
this systematic deviation, we note the following primary observations from the
plot of kinetic energy ($=1/2mV_{0}^{2}$) of the projectile as a function of
initial pressure $p_{0}$, see Fig. 4(right). Firstly, all the theoretical
predictions (solid lines) fall within a very narrow band (shaded grey region,
labelled ‘prediction’) approximated by a single straight line
($E_{p}\approx\lambda p_{0}$) with slope $\lambda=20.9$ J/bar. Secondly, the
measured values are also found to lie on fitted straight lines (dashed lines)
for each projectile mass. These best-fit lines have slope lesser than
$\lambda$, indicating that the energy loss increases with $p_{0}$. As
mentioned earlier, except for the $10$ g projectiles, the slope of the other
$V_{0}-p_{0}$ experimental lines differ only marginally; thus, the theoretical
band can be scaled by factor $\lambda_{1}\simeq 0.71$ (shaded light orange,
labeled ‘2 sabots’) to coincide with the measurements. It should be recognized
that this factor $\lambda_{1}$ accounts for the energy losses caused by two-
dimensional flow from the main chamber to barrel, and backflow from the main
chamber to the atmosphere via the valve chamber. Thirdly, as aforementioned,
all the projectiles are supported by two sabot structures, except for the $10$
g projectile, which has only one. Moreover, with a single sabot, larger energy
loss occurs due to gas flow past the clearance between the sabot and the inner
barrel surface. This gas leak was also observed when measuring velocity using
high-speed imaging. To account for this, the theoretical band is further
scaled by factor $\lambda_{2}\simeq 0.64$ (shaded light cyan region, labeled
‘1 sabot’) to match experimental values for the $10$ g, single-sabot
projectile.

### 3.3 Implications

The operating regime for carrying out impact experiments using a given
projectile can be selected based on Fig. 4. For a given projectile of mass $m$
and required exit velocity $V_{0}$, the required initial operating pressure is
obtained from the abscissa of the corresponding curve in this figure.

The pneumatic triggering mechanism used here allows projectile velocities of
$\sim$100 m/s with just compressed air, resulting in much higher strain rates
than in conventional configurations such as Izod-Charpy or drop-weight
tests[13]. This velocity range is suitable for evaluating deformation at
strain rates of $10^{3}-10^{5}$/s. The present design does not emphasize
optimizing velocity, considering the complexity involved in achieving higher
velocities, such as the use of multiple stages, but instead focuses on
conducting versatile laboratory-scale low-velocity impact experiments.
However, as evident from the 1D gas dynamics analysis in Sec. 3.1, the maximum
projectile velocity in the current gas gun setup can be further increased by
increasing the initial $p_{0}$ or by using helium [31]. It is also worth
mentioning that such analyses are usually integrated with the design phase of
the gun in order to estimate the various dimensions of the gas chamber and
barrel for the desired velocity range.

In summary, for a given initial pressure $p_{0}$ and projectile mass $m$, the
theoretical kinetic energy can be estimated as $E_{p}=\lambda p_{0}$, then the
output kinetic energy of the projectile in our setup is obtained as either
$E_{m}=\lambda_{1}E_{p}$ if two sabots are used, or
$E_{m}=\lambda_{2}\lambda_{1}E_{p}$ if single sabot is used.

## 4 Results II: Taylor Impact test

Having established a method to determine the operating parameters, we now
proceed to discuss two complementary impact tests. The first is the Taylor
impact test where the projectile is deformable and the target plate is
comparatively rigid. The converse test is discussed in Sec. 5.

The Taylor impact test uses a deformable cylindrical projectile to strike a
stationary plate. Compressive elastic and plastic waves start propagating
towards the rear end of the projectile immediately after impact. Elastic
waves, being much faster, reach the rear end first and are reflected as
tensile waves, which then undergo subsequent reflection from compressive waves
arising from the elastic-plastic boundary. This cyclic process results in
projectile deceleration until a finite plastic region is developed at the
impact end, see schematic in Fig. 5. We now present theoretical analysis of
this process, followed by experimental evaluation using our setup.

### 4.1 Theoretical analysis

In his pioneering work, Taylor proposed a one-dimensional simplified model[14]
to estimate the dynamic yield strength in terms of density, impact velocity,
and initial/final lengths of the projectile. Several modifications to this
model have been proposed subsequently, to incorporate more complex
constitutive material properties and for matching with the experimental
observations [32, 33, 34]. However, most of these models are one-dimensional
and rely on measurements of the final dimensions of the impacted specimen
[35].

To predict the final projectile geometry, we use an axisymmetric model in
which the plastic zone geometry is approximated as the frustum of a cone [36].
This model considers incompressible plastic deformation that causes bulging at
the impact end and incorporates spatio-temporal variations of strain and
strain rate, thus also predicting deformation history. Given the constitutive
equation of the material, the deformation of the impacting projectile can be
obtained by solving a set of ordinary differential equations (ODEs) in time,
derived purely based on physical arguments. We use this formulation to predict
the influence of the impact velocity $V_{0}$ on the mechanics of projectile
deformation. The operating parameters required to determine the impact
velocity are to be estimated from the velocity-pressure relationship curve
shown (_cf._ Fig. 4).

Figure 5: Schematic of Taylor impact showing the meaning of various symbols.
The definitions of quantities $\beta$ and $\gamma$ are also displayed.

We have made two minor modifications to the model presented in Ref. [36].
Firstly, the effective plastic strain and strain rates are determined by
averaging the strain rate components along the elastic-plastic boundary since
the unyielded part of the projectile is subjected to stress waves generated at
this boundary. The corresponding expressions for strain rate components are
(_cf._ Ref. [36]):

$\displaystyle\dot{\epsilon}_{xx}$
$\displaystyle=-\frac{\beta^{2}vx}{D_{0}(\gamma-1-\log\gamma)(D_{0}+\beta x)}$
$\displaystyle\dot{\epsilon}_{rr}=\dot{\epsilon}_{\theta\theta}$
$\displaystyle=\frac{\beta^{2}vx}{2D_{0}(\gamma-1-\log\gamma)(D_{0}+\beta x)}$
$\displaystyle\dot{\epsilon}_{xr}$
$\displaystyle=-\frac{\beta^{2}vr}{2(\gamma-1-\log\gamma)(D_{0}+\beta x)^{2}}$

Here, the initial diameter, length, and impact velocity of the projectile are
denoted by $D_{0}$, $L_{0}$, and $V_{0}$, respectively. During impact, $D$ and
$v$ represent the bulge end diameter and projectile velocity, see schematic in
Fig. 5. Also depicted here is the axial shortening $h$, undeformed length $l$,
and plastic zone size $s$ that occur during impact, as well as the
dimesionless ratios $\beta$ and $\gamma$. The coordinate system is located at
the center of the elastic-plastic boundary (see bottom row, Fig. 5). At this
boundary ($x=0$), the average strain rates are obtained by integrating over
the cross-sectional area,

$\displaystyle\dot{\bar{\epsilon}}_{xr}$ $\displaystyle=\frac{4}{\pi
D_{0}^{2}}\int_{0}^{2\pi}\int_{0}^{D_{0}/2}\dot{\epsilon}_{xr}\Big{|}_{x=0}rdrd\theta=\frac{\beta^{2}v}{6D_{0}(\gamma-1-\log\gamma)}$

and the other components are identically zero
($\dot{\bar{\epsilon}}_{xx}=\dot{\bar{\epsilon}}_{rr}=\dot{\bar{\epsilon}}_{\theta\theta}=0$).
The effective plastic strain rate can now be obtained by using the von-Mises
criterion:

$\displaystyle\dot{\bar{\epsilon}}_{p}=\sqrt{\frac{2}{3}\dot{\epsilon}_{ii}}=\frac{\beta^{2}v}{3\sqrt{3}D_{0}(\gamma-1-\log\gamma)}$
(4.1)

The effective plastic strain can be determined by time integration of this
equation. Additionally, the yield stress ($\sigma_{y}$) is determined assuming
a Johnson-Cook constitutive relation, and ignoring thermal effects

$\displaystyle\sigma_{y}=\left[A+B\bar{\epsilon}_{p}^{n}\right]\left[1+C\ln\left(\frac{\dot{\bar{\epsilon}}_{p}}{\dot{\epsilon}_{0}}\right)\right]$
(4.2)

where, $A,B,n,C$ and $\dot{\epsilon}_{0}$ are material constants. Combining
these relations, the following set of coupled non-linear ODEs are obtained

$\displaystyle\frac{dl}{dt}$ $\displaystyle=-c_{p}$ (4.3)
$\displaystyle\frac{dh}{dt}$ $\displaystyle=v$ (4.4)
$\displaystyle\frac{dv}{dt}$ $\displaystyle=-\frac{c_{e}(c_{e}+c_{p})}{El}$
(4.5) $\displaystyle\frac{dD}{dt}$ $\displaystyle=\frac{\beta
v(\gamma-1)}{2(\gamma-1-\ln\gamma)}$ (4.6)

along with the expressions (4.1) for strain rate and the constitutive equation
(4.2). The symbols $E$, $c_{e}$, and $c_{p}$ denote the Young’s modulus, the
elastic wave speed, $c_{e}=\sqrt{E/\rho}$, and the plastic wave speed
$c_{p}=\sqrt{\sigma_{y}/\rho}$, respectively. As pointed out in Ref. [36],
numerical instability prevails at the beginning of time integration due to the
presence of $s$ in the denominator of the expression for $\beta=(D-D_{0})/s$.
Time integration is hence performed by setting constant values of $\beta$ and
$\gamma$.

The simulation steps are as follows—we begin with an initial guess for $\beta$
and $\gamma$ for the first iteration and obtain the final deformed geometry.
The diameter after bulging ($D_{f}$) is determined using the final undeformed
length $l_{f}$ and plastic zone size $s_{f}$ from the previous iteration,
using the incompressibility condition. Further, we update $\beta$ and $\gamma$
equal to the average values $\beta_{av}(=0.5\beta_{f})$ and
$\gamma_{av}(=0.5(1+\gamma_{f}))$, respectively for the subsequent iterations,
until the tolerance for these quantities is below $10^{-3}$.

Figure 6: Taylor specimens Al 6061-T6 after impact. The initial parameters are
tabulated beneath the image. The model predicted deformed geometry (in orange)
is overlaid over the deformed specimens. Figure 7: Numerical results for
specimen ’c’ ($D_{0}=10$ mm, $L_{0}=25$ mm, and $V_{0}=105$ m/s). Figure
(left) shows the time history (left) of the bulge diameter $D$ and length $L$
of the projectile. Figure (right) shows the spatial effective strain field
distribution ($\dot{\bar{\epsilon}}_{p}^{*}$) (right) for three distinct time
instances.

### 4.2 Experimental measurements

We prepared cylindrical specimens of as-received aluminum 6061 with initial
diameter $D_{0}$ and length $L_{0}$ impacting an EN19 plate (20 mm thickness)
with different velocities $V_{0}$, see Fig. 6. The specimens are labeled with
small alphabets (‘a’ to ‘e’), and their initial parameters are tabulated in
the figure. The sabots used for the test consists of one Teflon ring of length
2/3 times the projectile length, placed at the free end. The combined sabot
plus projectile mass for all projectiles is in the range of 12-15 g, and the
corresponding operating pressure is 7 bar. Using scaling factors for single
sabots (see Sec. 3.2), the estimated velocities are again consistent with
measured values during the test. The deformed specimens are shown in Fig. 6,
along with the predicted shapes overlaid (orange dashed lines) from the model.
The close agreement between the final experimentally observed shapes and the
prediction is clear from this figure. For the model computation, Johnson-Cook
material constants for aluminum 6061 were taken from Ref. [37] as: $A=334$
MPa, $B=114$ MPa, $n=0.42$, $C=0.002$ and $\dot{\epsilon}_{0}$ = 1. We also
observed minor buckling in the deformed specimens, alongside the cone-shaped
bulging of the plastic zone.

The strain rate varies from a maximum at the beginning to zero at the end of
the impact. The model additionally predicts deformation history during impact,
which is otherwise challenging to obtain experimentally, see Fig. 7. Temporal
variation of the frustum diameter $D$ and length $L$ of the projectile
(corresponding to specimen ‘c’) is shown in Fig. 7(left). Corresponding
spatial variation of the effective plastic strain rate
($\dot{\bar{\epsilon}}_{p}^{*}$) for three different time instances is shown
in Fig. 7(right). All quantities are reported here in non-dimensional form,
time $t^{*}=tV_{0}/L_{0}$, velocity $v^{*}=v/V_{0}$ and strain-rate
($\dot{\bar{\epsilon}}_{p}^{*}$) is expressed in terms of $t^{*}$. These
results clearly predict the large instantaneous strain-rate that the bulge
region near the impact end is subjected to, making it particularly susceptible
to premature damage. It is expected, especially given the close match in final
shapes seen in Fig. 6, that the spatio-temporal evolution matches that in the
experiments, as may perhaps be confirmed by careful high-speed _in situ_
imaging. This, however, is beyond the scope of the present manuscript.

Another interesting prediction of the model is the deformation path traced
over the yield surface, see Fig. 8(left). Effective plastic strain
($\bar{\epsilon}_{p}$), strain rate ($\dot{\bar{\epsilon}}_{p}$), and yield
stress ($\sigma_{y}^{*}$) are all computed at the elastic-plastic boundary.
Note that the non-dimensional yield stress is defined as
$\sigma_{y}^{*}=(\sigma_{y}-A)/A$ using the Johnson-Cook parameter $A$. This
deformation history is useful for semi-quantitatively predicting the final
microstructure in the deformed projectile—given this information, one could
easily use a multi-scale (e.g., crystal-plasticity) model to estimate grain
shape/size evolution and potentially predict the onset of fracture. The varied
paths of two very similar specimens ‘a’ and ‘b’ is also noteworthy in this
figure.

As first-attempt experimental validation, the microstructure of specimen ‘c’,
and a schematic showing corresponding locations, is reproduced in Fig.
8(right). The microstructure is obtained by mechanical polishing with emery
papers of various grades, followed by cloth polishing using diamond paste, and
then etched by submerging it for 1-3 minutes in Keller’s reagent. The change
in grain size is indistinguishable as the specimen is only subjected to 15%
plastic strain and 18% more stress than the static yield stress as inferred
from the deformation path of specimen ‘c’in Fig. 8(left).

Figure 8: The deformation path (left) traced over the yield surface by all
Taylor specimens ’a’ to ’e’. The microstructure images (right) of a small
section cut at two different locations indicated by coloured squares in the
schematics on the longitudinal cross-section of specimen ’c’.

### 4.3 Implications

The physics-based model described here provides a direct method to predict the
deformation geometry given $m,V_{0}$. It also suggests two inverse problems.
The first one, as originally suggested by Taylor, is to deduce the material
parameters $A,B,n$ for an unknown material, from measurements of the final
shape and the impact velocity $V_{0}$. In such a case, the inverse problem is
posed in which the material parameters are perturbed in the model in such a
way that the predicted deformation best fits the experimental measurements,
see, for instance, Refs. [36, 38]. Since the Taylor impact configuration is
simple to set up, a number of experiments can be easily carried out using the
framework presented here.

A second inverse problem is to determine the impact velocity $V_{0}$, given
the material properties and the final deformed shape. This is of critical
importance in applications where determining impact conditions are necessary
as a _post mortem_ procedure, as for e.g., in failure of spacecraft
structures. One could even go a step further and envisage predicting $V_{0}$
based on microstructural features in the final projectile. This will be
necessary in practical situations where, perhaps part of the projectile is
lost due to subseqeuent impact events and deformation history must be
reconstructed from a very small portion of the final projectile. For this
purpose, the relations presented in Fig. 8, showing the deformation history of
the specimen, may be exploited to obtain the initial $V_{0}$. This route must
be taken with care, since it is quite likely that unique solutions cannot be
guaranteed in general.

## 5 Results III: Plate deformation experiments

We now discuss a configuration complementary to that of the Taylor
experiment—impact of a rigid projectile against a deformable plate. Upon
impact, the projectile will induce plastic deformation that may lead to
fracture, and potentially, resulting in perforation [27]. For a given
projectile-target geometry, a range of impact velocities can result in either
mere initiation of plastic flow to complete perforation [39, 40]. We deal here
only with the non-perforating case to estimate plastic deformation in the
plate.

### 5.1 Theoretical analysis

Upon impact of the rigid projectile, elastic and plastic waves begin to
propagate within the plate, causing transverse deflections in the direction of
projectile motion. Two plastic deformation regions may be delineated—bulging
where the target conforms to the projectile shape, and bending-induced
dishing, which extends over a considerable distance from the impact zone.

To estimate the amount of deformation as a function of impact parameters
$V_{0},m$, we adopt the model proposed by Beynet and Plunkett[39], assuming a
rigid blunt projectile and a perfectly-plastic target material with no elastic
effects. The plate bends as the bulge region moves with the projectile. The
plate subsequently yields due to the developed radial stress, resulting in
plastic strain in the transverse direction. Assuming that the deformation is
dominated by radial stress, the transverse displacement $w$ is governed by

$\displaystyle\frac{\partial^{2}w}{\partial\bar{r}^{2}}+\frac{1}{\bar{r}}\frac{\partial
w}{\partial\bar{r}}=\frac{\partial^{2}w}{\partial\bar{t}^{2}}\;\;\text{for}\;\;1<\bar{r}<\infty,\bar{t}\geq
0$ (5.1)

where $\bar{r}=r/r_{0}$, and $\bar{t}=c_{p}t/r_{0}$ are the nondimensionalized
radial distance and time, respectively. Here, $r_{0}$ is the projectile radius
and $c_{p}=\sqrt{\sigma_{y}/\rho}$ is the plastic wave speed. Equation 5.1
describes dishing as a radial propagation of plastic strains with speed
$c_{p}$. To determine the initial and boundary conditions for solving Eq. 5.1,
we note that the radial stress near the bulge region retards the velocity of
an equivalent body consisting of projectile and bulge region, which serves as
a boundary condition, and can be written as

$\displaystyle\alpha\frac{\partial
w}{\partial\bar{r}}=\frac{\partial^{2}w}{\partial\bar{t}^{2}},\;\;\text{at}\;\;\bar{r}=1,\bar{t}\geq
0$ (5.2)

with $\alpha=2\Delta M/(M+\Delta M)$, $M$ and $\Delta M$ are masses of
projectile and the bulge region, respectively. Initially, the transverse
displacement is zero everywhere, and only the bulge region moves, which forms
the initial conditions given by:

$\displaystyle w(\bar{r},0)$ $\displaystyle=0$ $\displaystyle\frac{\partial
w(\bar{r},0)}{\partial t}$ $\displaystyle=\begin{cases}V_{0}\frac{M}{M+\Delta
M}\frac{r_{0}}{c_{p}}&\bar{r}=1\\\ 0&\bar{r}>1\end{cases}$ (5.3)

with $V_{0}$ the projectile velocity at the onset of impact. Equation (5.1) is
then numerically solved using a finite-difference scheme, subject to the
initial (Eq. (5.3)) and boundary (Eq. (5.2)) conditions to obtain the final
shape of the plate.

### 5.2 Experimental measurements

For obtaining experimental measurements to evaluate the predicted temporal
target deformation, we impact a stainless steel projectile of diameter 10.2 mm
and length 45.8 mm with $V_{0}=65$ m/s against an aluminum 6061 (as-received)
target plate of dimensions 50 mm $\times$ 50 mm $\times$ 3 mm. The resulting
shape is shown in Fig. 9(left), with bulging and dishing regions demarcated.
The corresponding microstructure (Fig. 9(right)) shows clearly distinguishable
grains in these two regions. The occurrence of locally compressed grains
within the bulge region is evident in this figure, and this can be related to
the projectile impact velocity $V_{0}$ using grain-size analysis.

Figure 9: Deformed plate of Al 6061 (left) along with microstructure (right)
of the small section near the bulge region marked in red.

Since the model predicts transverse displacement, we obtain equivalent
experimental data by measuring the deformed surface profile of the plate.
While this profile can be accurately reconstructed using laser scanning, we
use a more primitive, yet easily implementable, method instead. This is done
by tracing grid points embossed _a priori_ onto the plate’s surface during
preparation, see supplementary material. Using the simple fact that a static
liquid surface will form a perfectly horizontal surface, we recover $x,y,z$
coordinates of points on the surface of the deformed plate. A more
comprehensive description of this method is provided in the supplementary
material. The final transverse displacement and developed surface strain
fields in the plate can be estimated from the deformed profile, and by
correlating initial and final grid locations, respectively.

Figure 10: Results of the measurement using the camera-ink setup for the
deformed aluminum plate. Figure (left) shows the transverse deflection $w$ as
a function of radial distance $r$ from the center of the projectile impact
predicted from the model compared with the experiment. The strain field
component $\epsilon_{rr}$ using camera-ink setup is shown in Figure (right).

A radial section of the obtained profile gives the deflection $w$ of the
points in the deformed plate plotted against the predicted deflection from the
model, see Fig. 10(left). For comparison, $\sigma_{y}=334$ MPa and $\rho=2700$
kg/m3 were chosen as representative for Al6061. The model assumes the plate to
be infinite, without accounting for edge effects from clamping. In contrast,
the considered plate in the test is finite; hence, the predicted deformation
is much smaller than that reconstructed from the final shape. We expect the
discrepancy between the two results to reduce for larger in-plane plate
dimensions.

The in-plane strain field components for a finite deformation are computed by
using the displacement mapping function from the original to the deformed
profile. The radial component $\epsilon_{rr}$ is only shown here, see Fig.
10(right), estimated in the red region marked in Supplementary Figure 1. The
other components $\epsilon_{\theta\theta}$ and $\epsilon_{r\theta}$ are found
to be less than 10% of the maximum value of $\epsilon_{rr}$, which _a
posteriori_ justifies the radial symmetry assumption in the model.

### 5.3 Implications

Despite being based on simplified theoretical arguments, the analysis
presented here provides predictions of material response in typical non-
perforating impact. It should be noted that any material can typically exhibit
a wide variety of failure mechanisms in a plate impact test [27]. Some common
possibilities include fracture due to initial compressive waves, spalling
caused due to the reflection of compressive wave from the distal boundary of
the plate, plugging resulting from highly localized shear zone formation[18,
41, 20] and even fragmentation [42, 43]. These failure types predominantly
depend on material properties, geometrical characteristics, and impact
velocity. As a result, and given its practical consequences for space
applications, plate deformation has attracted significant research effort for
decades [44, 45, 46].

Considering the broad variability in failure response, a predictive model
relating the impact velocity with the deformation mechanics is essential in
setting the test operating parameters in an experiment. As per test
configuration and material heterogeneities (for instance, porous solids,
composite materials _etc._), one needs to resort to more involved analyses to
predict the deformation [47, 48].

## 6 Discussion and Summary

In this work, we’ve described a compact laboratory-scale gas gun setup for
studying low-velocity impact events on both projectiles and targets. Some
noteworthy features of the configuration are that it is easy to setup and
adapt to multiple test modalities (e.g., Taylor impact, perforation) and
realize different final target geometries (e.g., multi-layer shields, curved
structures). Our work here has focused on determining an operating scheme for
the test setup, as well as calibrating theoretical velocity-pressure curves
with those obtained experimentally. using only compressed air, the system was
demonstrated to maximum projectile velocities of $\sim$100 m/s, making it
ideal for low-velocity impact studies in a laboratory setting.

The capabilities of the presented design have been demonstrated for two
standard impact test configurations _viz._ Taylor and plate impact. The former
has been performed by impacting an aluminum specimen against a rigid alloyed
steel plate and using an axisymmetric model for predicting deformation
geometry. Close agreement between experiments and the model predictions were
noted, in addition to the determination of time history for the deformation
fields, which are extremely challenging to obtain experimentally. In the
second impact configuration, an aluminum plate is deformed by a stainless
steel projectile. A physics-based model is presented to relate the plate’s
transverse deflection with the impact velocity. The final displacement and
strain fields after impact were measured using a visual grid mapping
technique, and agrees reasonably well with the numerical prediction. While
accounting for the disparities between the simplified models and the actual
experiments, the numerical models help estimate the effect of impact velocity
on the material deformation behaviour. The derived impact velocity will then
be suitably used to estimate the operating parameters for the test by
referring to the produced velocity-pressure relationship curve.

We believe that the framework described in this work is easily replicable and
will be useful for groups that are exploring the use of novel structural
designs for space applications, such as metamaterials or microarchitected
internal features [52, 53], for impact energy absorption applications. We also
believe that it will help elucidate the impact performance of emerging classes
of metallic materials such as high-entropy alloys [50, 51] that represent an
area of active research interest. The question of of how these multi-component
alloys fail and how such potentially catastrophic mechanisms can be mitigated
either by diverse microarchitecting, or additional alloying or a combination
of both, represent interesting areas for future research for space-grade
materials. We are presently investigating some of these questions and hope to
communicate our results in due course.

## References

  * [1] Akahoshi Y, Nakamura T, Fukushige S, Furusawa N, Kusunoki S, Machida Y, et al.  Influence of space debris impact on solar array under power generation.  International Journal of Impact Engineering. 2008;35(12):1678–1682.
  * [2] Krag H, Serrano M, Braun V, Kuchynka P, Catania M, Siminski J, et al.  A 1 cm space debris impact onto the sentinel-1a solar array.  Acta Astronautica. 2017;137:434–443.
  * [3] Liou JC, Johnson NL.  Risks in space from orbiting debris.  Science. 2006;311(5759):340–341.
  * [4] Silnikov M, Guk I, Nechunaev A, Smirnov N.  Numerical simulation of hypervelocity impact problem for spacecraft shielding elements.  Acta Astronautica. 2018;150:56–62.
  * [5] Poniaev S, Kurakin R, Sedov A, Bobashev S, Zhukov B, Nechunaev A.  Hypervelocity impact of mm-size plastic projectile on thin aluminum plate.  Acta Astronautica. 2017;135:26–33.
  * [6] Yasaka T, Hanada T, Hirayama H.  Low-velocity projectile impact on spacecraft.  Acta Astronautica. 2000;47(10):763–770.
  * [7] Nayan N, Gurao NP, Murty SN, Jha AK, Pant B, Sharma SC, et al.  Microstructure and micro-texture evolution during large strain deformation of an aluminium–copper–lithium alloy AA 2195.  Materials and Design. 2015;65:862–868.
  * [8] Shockey DA, Simons JW, Brown CS, Kobayashi T.  Shear failure of Inconel 718 under dynamic loads.  Experimental Mechanics. 2007;47(6):723–732.
  * [9] Selivanov VV, Silnikov MV, Markov VA, Popov YV, Pusev VI.  Using highly porous aluminum alloys and honeycomb structures in spacecraft landing gear.  Acta Astronautica. 2021;180:105–109.
  * [10] Dudziak R, Tuttle S, Barraclough S.  Harpoon technology development for the active removal of space debris.  Advances in Space Research. 2015;56(3):509–527.
  * [11] Chen S, Zang M, Wang D, Yoshimura S, Yamada T.  Numerical analysis of impact failure of automotive laminated glass: A review.  Composites Part B: Engineering. 2017;122:47–60.
  * [12] Kim SB, Huh H, Bok HH, Moon MB.  Forming limit diagram of auto-body steel sheets for high-speed sheet metal forming.  Journal of Materials Processing Technology. 2011;211(5):851–862.
  * [13] Ruan D, Kariem MA, Crouch IG.  High strain rate and specialised testing.  The Science of Armour Materials. 2017;p. 581–637.
  * [14] Taylor GI.  The use of flat-ended projectiles for determining dynamic yield stress I. Theoretical considerations.  Proceedings of the Royal Society of London Series A Mathematical and Physical Sciences. 1948;194(1038):289–299.
  * [15] Rogers HC.  Adiabatic plastic deformation.  Annual Review of Materials Science. 1979;9(1):283–311.
  * [16] YL B, Dodd B.  Adiabatic shear localization: occurrence, theories, and applications.  Pergamon Press; 1992.
  * [17] Meyers MA.  Dynamic behaviour of materials.  John Wiley and Sons; 1994.
  * [18] Viswanathan K, Yadav S, Sagapuram D.  Shear bands in materials processing: Understanding the mechanics of flow localization from Zener’s time to the present.  Applied Mechanics Reviews. 2020;72(6).
  * [19] Zhou M, Rosakis AJ, Ravichandran G.  Dynamically propagating shear bands in impact-loaded prenotched plates—I. Experimental investigations of temperature signatures and propagation speed.  Journal of the Mechanics and Physics of Solids. 1996;44(6):981–1006.
  * [20] Wingrove AL.  The influence of projectile geometry on adiabatic shear and target failure.  Metallurgical Transactions. 1973;4(8):1829–1833.
  * [21] Sankar HR, Parameswaran V.  Effect of multiple holes on dynamic buckling of stubby shells: An experimental and numerical investigation.  International Journal of Impact Engineering. 2016;96:129–145.
  * [22] Faye A, Parmeswaran V, Basu S.  Mechanics of dynamic fracture in notched polycarbonate.  Journal of the Mechanics and Physics of Solids. 2015;77:43–60.
  * [23] Rittel D, Lee S, Ravichandran G.  A shear-compression specimen for large strain testing.  Experimental Mechanics. 2002;42(1):58–64.
  * [24] Swift HF.  Light-gas gun technology: a historical perspective.  In: High-Pressure Shock Compression of Solids VIII. Springer; 2005. p. 1–35.
  * [25] Anderson Jr CE, Bodner SR.  Ballistic impact: the status of analytical and numerical modeling.  International Journal of Impact Engineering. 1988;7(1):9–35.
  * [26] Zhang GM, Batra RC.  Modified smoothed particle hydrodynamics method and its application to transient problems.  Computational Mechanics. 2004;34(2):137–146.
  * [27] Backman ME, Goldsmith W.  The mechanics of penetration of projectiles into targets.  International Journal of Engineering Science. 1978;16(1):1–99.
  * [28] Sharma S, Chavan VM, Agrawal RG, Patel RJ, Kapoor R, Chakravartty JK.  Split-Hopkinson pressure bar: an experimental technique for high strain rate tests.  Bhabha Atomic Research Centre; 2011.
  * [29] Seigel AE.  The theory of high speed guns.  Advisory Group for Aerospace Research and Development Neuilly-sur-seine (France); 1965.
  * [30] Constantinou M, Caccese J, Harris HG.  Frictional characteristics of Teflon–steel interfaces under dynamic conditions.  Earthquake engineering & structural dynamics. 1987;15(6):751–759.
  * [31] Hutchings IM, Winter RE.  A simple small-bore laboratory gas-gun.  Journal of Physics E: Scientific Instruments. 1975;8(2):84.
  * [32] Hawkyard JB.  A theory for the mushrooming of flat-ended projectiles impinging on a flat rigid anvil, using energy considerations.  International Journal of Mechanical Sciences. 1969;11(3):313–333.
  * [33] Jones SE, Gillis PP, Foster Jr JC.  On the equation of motion of the undeformed section of a Taylor impact specimen.  Journal of applied physics. 1987;61(2):499–502.
  * [34] Jones SE, Maudlin PJ, Foster Jr JC.  An engineering analysis of plastic wave propagation in the Taylor test.  International Journal of Impact Engineering. 1997;19(2):95–106.
  * [35] Jones SE, Drinkard JA, Rule WK, Wilson LL.  An elementary theory for the Taylor impact test.  International Journal of Impact Engineering. 1998;21(1-2):1–13.
  * [36] Chakraborty S, Shaw A, Banerjee B.  An axisymmetric model for Taylor impact test and estimation of metal plasticity.  Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. 2015;471(2174):20140556.
  * [37] Lesuer DR, Kay GJ, LeBlanc MM.  Modeling large-strain, high-rate deformation in metals.  Lawrence Livermore National Lab.(LLNL), Livermore, CA (United States); 2001.
  * [38] Sen S, Banerjee B, Shaw A.  Taylor impact test revisited: Determination of plasticity parameters for metals at high strain rate.  International Journal of Solids and Structures. 2020;193:357–374.
  * [39] Beynet P, Plunkett R.  Plate impact and plastic deformation by projectiles.  Experimental Mechanics. 1971;11(2):64–70.
  * [40] Calder CA, Goldsmith W.  Plastic deformation and perforation of thin plates resulting from projectile impact.  International Journal of Solids and Structures. 1971;7(7):863–881.
  * [41] Rittel D, Wang ZG.  Thermo-mechanical aspects of adiabatic shear failure of AM50 and Ti6Al4V alloys.  Mechanics of materials. 2008;40(8):629–635.
  * [42] Kooij S, van Dalen G, Molinari JF, Bonn D.  Explosive fragmentation of Prince Rupert’s drops leads to well-defined fragment sizes.  Nature Communications. 2021;12(1):2521.
  * [43] Chaudhri MM.  Dynamic fracture of inorganic glasses by hard spherical and conical projectiles.  Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences. 2015;373(2038):20140135.
  * [44] Prosser WH, Gorman MR, Humes DH.  Acoustic emission signals in thin plates produced by impact damage.  Journal of Acoustic emission. 1999;17(1-2).
  * [45] Børvik T, Langseth M, Hopperstad O, Malo K.  Perforation of 12 mm thick steel plates by 20 mm diameter projectiles with flat, hemispherical and conical noses: Part I: Experimental study.  International journal of impact engineering. 2002;27(1):19–35.
  * [46] Kaboglu C, Mohagheghian I, Zhou J, Guan Z, Cantwell W, John S, et al.  High-velocity impact deformation and perforation of fibre metal laminates.  Journal of materials science. 2018;53(6):4209–4228.
  * [47] Anderson Jr CE.  Analytical models for penetration mechanics: a review.  International Journal of Impact Engineering. 2017;108:3–26.
  * [48] Umanzor ME, Batra RC, Williams CB, Druschitz AP.  Penetration Resistance of Cast Metal- Ceramic Composite Lattice Structures.  Advanced Engineering Materials. 2021;23(12):2100577.
  * [49] Kadono T, Arakawa M.  Crack propagation in thin glass plates caused by high velocity impact.  Physical Review E. 2002;65(3):035107.
  * [50] Sonkusare R, Jain R, Biswas K, Parameswaran V, Gurao NP.  High strain rate compression behaviour of single phase CoCuFeMnNi high entropy alloy.  Journal of Alloys and Compounds. 2020;823:153763.
  * [51] George EP, Raabe D, Ritchie RO.  High-entropy alloys.  Nature Reviews Materials. 2019;4(8):515–534.
  * [52] Kadic M, Milton GW, van Hecke M, Wegener M.  3D metamaterials.  Nature Reviews Physics. 2019;1(3):198–210.
  * [53] Fleck NA, Deshpande VS, Ashby MF.  Micro-architectured materials: past, present and future.  Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. 2010;466(2121):2495–2516.