Ballistic impact behaviour of thick composites: Parametric studies
N.K. Naik *, A.V. Doshi
Aerospace Engineering Department, Indian Institute of Technology Bombay, Powai, Mumbai 400 076, India
Abstract
Ballistic impact behaviour of typical woven fabric E-glass/epoxy thick composites is presented in this paper. Specifically, energy
absorbed by different mechanisms, ballistic limit velocity and contact duration are determined. The studies are carried out using the analytical method presented for the prediction of ballistic impact behaviour of thick composites in our earlier work [Naik NK, Doshi AV.
Ballistic impact behaviour of thick composites: analytical formulation. AIAA J 2005;43(7):1525–36.]. The analytical method is based on
wave theory and energy balance between the projectile and the target. The inputs required for the analytical method are: diameter, mass
and velocity of the projectile; thickness and material properties of the target. Analytical predictions are compared with typical experimental results. A good match between analytical predictions and experimental results is observed. Further, effect of incident ballistic
impact velocity on contact duration and residual velocity, effect of projectile diameter and mass on ballistic limit velocity and effect
of target thickness on ballistic limit velocity and contact duration are studied. It is observed that shear plugging is the major energy
absorbing mechanism.
Keywords: Ballistic impact; Thick composite; Parametric studies
1. Introduction
Composite materials are finding increasing uses in general engineering applications along with high performance
aerospace and defence applications during the last few decades because of their high specific strength and high specific
stiffness. This led to usage of more thick section composites
for different applications. Such composite structures
undergo different loading conditions during their service
life. Impact/ballistic impact is one of the typical loading
conditions. Thick composites behave differently compared
with thin composites under impact/ballistic impact loading
conditions.
When an impact load is applied to a body, instantaneous stresses are produced. But the stresses are not immediately transmitted to all parts of the body. The remote
portions of the body remain undisturbed for sometime.
The stresses progress in all directions through the body in
the form of disturbances of different types. In other words,
stresses (and their associated deformations or strains) travel through the body at specific velocities. These velocities
are functions of the material properties. Regardless of the
method of application of impact load, the disturbances
generated have identical properties based only on the target
material properties.
During an impact event the stress wave propagation
takes place in all the directions. Generally, this problem
is analyzed using 1D, 2D or 3D approaches. In 1D and
2D studies, the wave propagation through the thickness
direction is not considered [2–4]. When these approaches
are used to analyze structures, isotropic as well as orthotropic, it is assumed that the deformation behaviour along the
thickness direction of the target is the same along the entire
thickness. Such an assumption can be made for targets of
lower thickness or, in other words, can be used for thin
plates. If the thickness of the plate is increased the deformation and the induced stress behaviour of the plate would
be different at different locations along the thickness direc-
448
Nomenclature
A, Ap
Aql
d
dci
dh
E
Ebb
Ecf
Ecsy
Edl
Efr
Ehg
Emc
Emt
Ep
Erb
Esp
Etf
ETotal
F
Fc
Fi
G
GIIcd
h
hl
hlc
hp
K
KEp
KEp0
l
m
m0
nlfs, ns
nlft
nlsc
cross-sectional area of the projectile
quasi-lemniscate area reduction factor
diameter of the projectile
deceleration of the projectile during a given time
interval
diameter of the hole
energy/Young’s modulus
energy absorbed due to bulge formation on the
back face of the target
energy absorbed due to compression of the target directly below the projectile: Region 1
energy absorbed due to compression of the
yarns in the surrounding region of the impacted
zone: Region 2
energy absorbed due to delamination
energy absorbed due to friction
energy absorbed due to heat generated
energy absorbed due to matrix cracking
energy absorbed by matrix cracking per unit
volume
kinetic energy of the projectile at exit
energy absorbed due to reverse bulge formation
on the front face of the target
energy absorbed due to shear plugging of the
yarns
energy absorbed due to tension in the yarns in a
layer
total kinetic energy lost by the projectile
total force/contact force
compressive force
inertial force
shear modulus
critical strain energy release rate in Mode II
thickness of the target
thickness of each layer
thickness of a layer after compression
length of the plug
numerical constant (depends on the shape of the
projectile)
kinetic energy of the projectile at a particular
time interval
kinetic energy of the projectile: incident
length of the projectile
mass of the projectile
initial mass of the projectile
number of layers failed due to shear plugging
number of layers failed in tension
number of layers strained in compression
tion. For such cases the analysis is based on also considering the wave propagation along the thickness direction.
There are typical studies on ballistic impact behaviour of
thick composites [5–8].
Pd
Pm
Ssp
t
V
VBL
Vf
Vi
Vm
VR
Vzl
Vzt
xd
xl
xll, rp
xllc
xt
xtl, rt
z
zi
zl
zpl
zt
Dhlc
Dt
ecz
emax
et
etxl
c
m
q
rcz
rmax
rt
rtx
s
percent delaminating layers
percent matrix cracking
shear plugging strength
time/contact duration
incident impact velocity/velocity
ballistic limit velocity
fibre volume fraction
velocity of the projectile at the ith instant
matrix volume fraction
residual velocity
velocity of compressive stress wave in
z-direction
velocity of shear stress wave in z-direction
distance upto which damage has reached
distance the longitudinal wave has traveled in
x-direction
distance the longitudinal wave has traveled in
x-direction in a layer
change in length of the yarn after tension
distance the transverse wave has traveled in
x-direction
distance the transverse wave has traveled in
x-direction in a layer
total depth the projectile has penetrated/projectile displacement
Depth of projectile penetrated during a given
time interval
distance the compressive stress wave has
traveled in z-direction
distance by which a layer has moved in forward
direction from its original position
distance the shear stress wave has traveled in
z-direction
thickness by which a layer is compressed
increment in time interval
compressive strain along the thickness direction
ultimate strain
tensile strain along the radial direction
tensile strain in x-direction in a layer
shear strain
Poisson’s ratio
density of the target material
compressive stress along the thickness direction
ultimate stress
tensile stress along the radial direction
tensile stress along the radial direction
shear stress
Ballistic impact behaviour of typical woven fabric
E-glass/epoxy thick composites is presented in this paper.
First, typical experimental results are presented. Then, analytical results are presented based on the method presented
449
in [1]. Analytical studies are based on considering the projectile is rigid, cylindrical and flat-ended. The studies are
carried out to evaluate energy absorbed by different mechanisms, ballistic limit velocity, contact duration and damage shape and size. Analytical predictions are compared
with typical experimental results. The analytical method
[1] used is presented in brief in Appendix A.
2. Experimental studies
Typical experimental studies were carried out. The projectile and target information is given below:
• Projectile: Cylindrical, flat-ended hardened steel projectile of diameter, d = 6.33 mm; mass, m = 5.84 gm and
length, l = 24 mm.
• Target: Woven fabric E-glass/epoxy, unsupported area
of the target 125 mm · 125 mm, thickness 4–7 mm.
ballistic impact behaviour of composites used by Gellert
et al. [9] and Kumar and Bhat [10].
Compressive stress–strain data along the thickness
direction for woven fabric E-glass/epoxy composite at high
strain rate is given in Appendix B. Tensile stress–strain
data along the warp direction for woven fabric E-glass/
epoxy composite at high strain rate is given in Appendix C.
Details regarding load–displacement behaviour under
quasi-static loading for the study of frictional resistance
are provided in Appendix D. The plot obtained under
quasi-static loading is used for the calculation of frictional
energy during the ballistic impact event. The frictional
resistance is offered by the target for the movement of the
projectile during the later stage of the ballistic impact
event. During this stage, the velocity of the projectile is less.
Considering this, the plot obtained under quasi-static loading is used for the calculation of frictional energy during
the ballistic impact event.
The experimental ballistic limit velocity for the case of
5 mm thickness was 148 m/s and for the case of 4 mm
thickness it was 137 m/s. When the experiment was carried
out with the target thickness of 7 mm, complete penetration did not take place even with the incident ballistic
impact velocity of 173 m/s.
Gellert et al. [9] conducted ballistic impact experimental
studies on woven fabric E-glass/vinylester composites with
target thickness of 19, 14.5 and 9 mm using cylindrical, flatended steel projectiles. The experimental results are given
in Table 1.
3.1. Comparison of predicted and experimental results
3. Input data necessary for the analytical predictions of
ballistic impact behaviour
4. Ballistic impact behaviour of thick composites
Ballistic impact behaviour of typical woven fabric Eglass/epoxy composites is studied. Mechanical properties
of the target plate and the other details are given in Table
2 for a typical woven fabric E-glass/epoxy composite studied. The same set of mechanical properties is used in the
present study to predict the ballistic impact behaviour of
the woven fabric E-glass/epoxy laminates. Also, the same
set of mechanical properties is used for the prediction of
Ballistic limit velocity is predicted for different target
thicknesses and projectile parameters using the analytical
method and the mechanical and other properties presented in Table 2. The results are presented in Table 1
and compared with the experimental results. It can be
seen that there is a good match between the analytically
predicted and experimentally obtained ballistic limit
velocities.
Ballistic impact behaviour of thick composites is evaluated using the analytical method. The energy absorbed by
different energy absorbing mechanisms, projectile kinetic
energy, projectile velocity, contact force, distance traveled
by the projectile and longitudinal and shear waves along
the thickness direction are evaluated as a function of time.
Further, tensile strain at the failure of the layer is obtained
as a function of time and target thickness. Distances upto
which longitudinal and transverse waves have traveled
Table 1
Ballistic impact test results and analytical predictions for typical woven fabric E-glass/epoxy composites
Projectile mass,
m (gm)
Projectile diameter,
d (mm)
Target thickness,
h (mm)
Predicted
VBL (m/s)
Expt.
VBL (m/s)
Expt. results
Remarks
3.84
3.84
3.33
3.33
3.33
5.84
5.84
5.00
6.35
6.35
4.76
4.76
4.76
6.33
6.33
6.00
19
14.5
19
14
9
7
5
25
550.894
452.546
508.80
392.98
269.03
174.29
142.50
559.112
563
473
505
389
293
173
148
547
Penetrated
Penetrated
Penetrated
Penetrated
Penetrated
Not penetrated
Penetrated
Penetrated
Gellert et al. [9], 2000
Gellert et al. [9], 2000
Gellert et al. [9], 2000
Gellert et al. [9], 2000
Gellert et al. [9], 2000
Present study
Present study
Kumar and Bhat [10], 1998
Cylindrical projectiles with flat-end.
450
Table 2
Input parameters required for the analytical predictions of ballistic impact behaviour
Reference
Present study
Projectile details (cylindrical)
Mass (gm)
Shape
Diameter (mm)
5.84
Flat ended
6.33
Target details
Material
Vf (%)
Thickness (mm)
No. of layers
Density (kg/m3)
Tensile failure strain (%)
Compressive failure strain (%)
Shear plugging strength (MPa)
E-glass/epoxy
50
5
19
1850
3.5
13.5
90
Other details
Compressive stress–strain curve
Tensile stress–strain curve
Quasi-lemniscate factor
Delamination percentage
Matrix crack percentage
Strain energy release rate: mode II (J/m2)
Matrix cracking energy (MJ/m3)
Calculation of frictional energy
Fig. 20
Fig. 21
0.9
100
100
1000
0.9
Fig. 22
Fig. 1. Energy absorbed by different energy absorbing mechanisms during ballistic impact event, V = 550.894 m/s, m = 3.84 gm, d = 6.35 mm,
h = 19 mm, woven fabric E-glass/epoxy laminate.
along the radial direction until the failure of the layer and
damage variation along the thickness direction are also
presented.
4.1. Case I
Figs. 1–5 are with the following data:
451
Fig. 2. Energy absorbed by different energy absorbing mechanisms and tensile strain at failure of layer during ballistic impact event, V = 550.894 m/s,
m = 3.84 gm, d = 6.35 mm, h = 19 mm, woven fabric E-glass/epoxy laminate.
Fig. 3. Projectile velocity and contact force during ballistic impact event, V = 550.894 m/s, m = 3.84 gm, d = 6.35 mm, h = 19 mm, woven fabric
E-glass/epoxy laminate.
• Target material: Woven fabric E-glass/epoxy, h =
19 mm.
• Projectile parameters: d = 6.35 mm, m = 3.84 gm,
V = 550.894 m/s.
In this case, complete perforation of the target by the
projectile took place, i.e., the tip of the projectile was at
the back face of the target at the end of the ballistic impact
event. Hence, V = 550.894 m/s is the ballistic limit velocity.
452
Fig. 4. Distance traveled by projectile and waves along thickness direction during ballistic impact event, V = 550.894 m/s, m = 3.84 gm, d = 6.35 mm,
h = 19 mm, woven fabric E-glass/epoxy laminate.
Fig. 5. Distance upto which transverse and longitudinal waves have traveled along radial direction and damage variation along thickness direction during
ballistic impact event, V = 550.894 m/s, m = 3.84 gm, d = 6.35 mm, h = 19 mm, woven fabric E-glass/epoxy laminate.
The projectile kinetic energy decreases during the ballistic
impact event. This is because the energy is absorbed by
the target by different mechanisms. The ballistic impact
event can be subdivided into three stages as given in
Appendix A. During the first two stages, damage is taking
place upto the back face of the target and the energy is
453
absorbed by different mechanisms. As can be seen from
Fig. 1, for the case considered, the second stage ends at
8.1 ls. During the third stage, energy is absorbed only
because of the frictional resistance offered due to the movement of the projectile. For this case, at the end of the ballistic impact event, i.e., at time 359.43 ls, the velocity of the
projectile is zero (Fig. 3). The contact force increases during the first and second stages and decreases during the
third stage.
The major energy absorbing mechanism is shear plugging. Significant energy is absorbed by compression of
the target in the region directly below the projectile
(Region 1) and due to friction. Energy absorbed by matrix
cracking, delamination, tension in the yarns and compression in the surrounding region of the impacted zone
(Region 2) is relatively less (Fig. 2). Percentage energy
absorbed by different mechanisms is given in Table 3.
Tensile strain in the yarns at failure of different layers is
shown in Fig. 2. It can be seen that the tensile strain
exceeds the permissible tensile strain only in the last few
layers. It indicates that only last few layers fail in tension.
The other layers fail due to shear plugging. Fig. 2 presents
the time interval when a particular layer fails and it is plotted as a function of thickness. Among those layers which
fail in tension, it can be seen that lower layers fail earlier
than the layers above them. This is because of bulging effect
on the back face of the target.
Projectile displacement during the ballistic impact
event, i.e., distance traveled by the projectile is shown
in Fig. 4. At the end of the second stage the distance
traveled by the projectile is 3.1 mm. At this stage all
the layers have failed either due to tension in the
yarns/layers due to shear plugging or due to the combined effect of both the mechanisms. After this, during
the third stage, the projectile along with the plug formed
would be moving further towards the back face of the
target. Since the velocity of the projectile is less during
this stage, the time taken for the projectile to move further is significantly higher. The distance traveled by
through-the-thickness longitudinal and shear waves is
also shown in Fig. 4. This would be only upto the end
of second stage since the yarns/layers fail by the end of
second stage.
Distance upto which transverse and longitudinal waves
have traveled along the radial direction is shown in
Fig. 5. The radial distance indicates the distance up to
which the waves have reached when the corresponding layers fail either due to tension or due to shear plugging or due
to the combined affect of both the mechanisms. When the
induced stress exceeds the permissible strength, the fracture
of the yarns would take place. But much before that, clear
damage zone can be seen within the layers whenever the
induced stress exceeds the damage initiation threshold
stress. This damage is either due to delamination or matrix
cracking.
4.2. Case II
Results are presented in Figs. 6–10 for the same target
and projectile parameters (Table 2) with V = 600 m/s. This
velocity is above the ballistic limit velocity. In this case, the
projectile is exiting with velocity equal to 303.9 m/s. In
other words, the entire kinetic energy of the projectile is
not absorbed by the target. In this case, the second stage
of the ballistic impact event is over at 7.1 ls. The projectile
exits the target at 59.1 ls (Fig. 6). In this case also, only a
few layers fail in tension near the back face of the target.
The other layers fail in shear plugging. The contact force
increases during the first and second stages and decreases
during the third stage. Projectile displacement is 3.4 mm
at the end of the second stage. Further movement of the
projectile would be during the third stage. The total contact
duration is less in this case as compared to case I. This is
because of the higher velocity of the projectile. Damage
shape is nearly identical to that for case I.
4.3. Case III
Results are presented in Figs. 11–15 for the same target
and projectile parameters (Table 2) with V = 500 m/s. This
velocity is below the ballistic limit velocity. In this case, the
projectile did not completely penetrate the target. The
velocity of the projectile was zero at time equal to 7.1 ls.
During the ballistic impact event, shear wave has traveled
along the thickness direction to a distance of 12 mm. From
Fig. 12, it can be seen that all the layers upto a distance of
Table 3
Percentage energy absorbed by different mechanisms with different incident ballistic impact velocities, m = 3.84 gm, d = 6.35 mm, h = 19 mm, woven
fabric E-glass/epoxy laminate
Incident ballistic impact velocity (m/s)
600
550.894
500
Energy (%)
Total kinetic energy of projectile, KEpo
Energy absorbed due to compression: Region 1, Ecf
Energy absorbed due to compression: Region 2, Ecsy
Energy absorbed due to tension in yarns, Etf
Energy absorbed due to shear plugging, Esp
Energy absorbed due to matrix cracking, Emc
Energy absorbed due to delamination, Edl
Energy absorbed due to friction, Efr
Kinetic energy of projectile at exit, Ep
100.0
7.1
2.2
1.4
58.0
3.0
0.4
6.0
22.1
100.0
8.4
2.5
2.3
76.4
2.9
0.5
7.3
0
100.0
8.0
3.1
0.8
87.0
0.7
0.1
0
0
454
Fig. 6. Energy absorbed by different energy absorbing mechanisms during ballistic impact event, V = 600 m/s, m = 3.84 gm, d = 6.35 mm, h = 19 mm,
woven fabric E-glass/epoxy laminate.
Fig. 7. Energy absorbed by different energy absorbing mechanisms and tensile strain at failure of layer during ballistic impact event, V = 600 m/s,
m = 3.84 gm, d = 6.35 mm, h = 19 mm, woven fabric E-glass/epoxy laminate.
12 mm have failed due to shear plugging. The remaining
layers have not failed. Projectile displacement is 2.4 mm
when the velocity of the projectile reaches to zero. Since
complete failure has not taken place, and the projectile
velocity is zero at a distance of 2.4 mm, projectile does
not move further. Hence, frictional energy is not present
455
Fig. 8. Projectile velocity and contact force during ballistic impact event, V = 600 m/s, m = 3.84 gm, d = 6.35 mm, h = 19 mm, woven fabric E-glass/
epoxy laminate.
Fig. 9. Distance traveled by projectile and waves along thickness direction during ballistic impact event, V = 600 m/s, m = 3.84 gm, d = 6.35 mm,
h = 19 mm, woven fabric E-glass/epoxy laminate.
456
Fig. 10. Distance upto which transverse and longitudinal waves have traveled along radial direction and damage variation along thickness direction
during ballistic impact event, V = 600 m/s, m = 3.84 gm, d = 6.35 mm, h = 19 mm, woven fabric E-glass/epoxy laminate.
Fig. 11. Energy absorbed by different energy absorbing mechanisms during ballistic impact event, V = 500 m/s, m = 3.84 gm, d = 6.35 mm, h = 19 mm,
woven fabric E-glass/epoxy laminate.
in this case. The contact force builds up at the end of the
ballistic impact event when the velocity of the projectile
reaches to zero. This can lead to delamination at that par-
ticular interface upto which shear plugging has taken place.
The damage shape is similar to those obtained for the cases
I and II, but the size is smaller.
457
Fig. 14. Distance traveled by projectile and waves along thickness
direction during ballistic impact event, V = 500 m/s, m = 3.84 gm,
d = 6.35 mm, h = 19 mm, woven fabric E-glass/epoxy laminate.
Fig. 12. Tensile strain at failure of layer during ballistic impact event,
V = 500 m/s, m = 3.84 gm, d = 6.35 mm, h = 19 mm, woven fabric
E-glass/epoxy laminate.
tact force is higher at velocities lower than ballistic limit
velocity.
5. Parametric studies
5.1. Incident ballistic impact velocity
Contact duration between the projectile and the target
as a function of incident ballistic impact velocity is presented in Fig. 16. The contact duration can be defined as
follows:
• Partial penetration: The time interval starting from
when the projectile just hits the target to when the velocity of the projectile becomes zero.
• Complete penetration: The time interval starting from
when the projectile just hits the target to when the projectile tip reaches to the back face of the target.
Fig. 13. Projectile velocity and contact force during ballistic impact event,
V = 500 m/s, m = 3.84 gm, d = 6.35 mm, h = 19 mm, woven fabric
E-glass/epoxy laminate.
At ballistic limit velocity, when the projectile tip just
reaches to the back face of the target, the velocity of the
projectile would be zero. For the case considered
(Fig. 16), the ballistic limit velocity is 550.894 m/s. The
contact duration behaviour as given in Fig. 16 and Table
4 can be divided into two parts:
Part 1: Incident ballistic impact velocity is less than the
ballistic limit velocity.
Comparison of ballistic impact parameters with different
incident ballistic impact velocities is presented in Table 4. It
can be noted that the total contact duration is more at ballistic limit velocity compared to other velocities. The con-
As the incident ballistic impact velocity increases the
contact duration increases. At lower velocities, the energy
of the projectile would be lower and the time taken for
the projectile to reach to zero velocity would be lower.
458
Fig. 15. Distance upto which transverse and longitudinal waves have traveled along radial direction and damage variation along thickness direction
during ballistic impact event, V = 500 m/s, m = 3.84 gm, d = 6.35 mm, h = 19 mm, woven fabric E-glass/epoxy laminate.
Table 4
Comparison of ballistic impact parameters with different incident ballistic
impact velocities, m = 3.84 gm, d = 6.35 mm, h = 19 mm, woven fabric
E-glass/epoxy laminate
Incident ballistic impact velocity, V (m/s)
500
550.894
600
Ballistic impact parameters
Contact duration, t (ls)
Upto 2nd stage
Total
–
7.1
8.1
359
7.1
59
Peak contact force, F (N)
708
356
179
Projectile displacement, z (mm)
Upto 2nd stage
2.4
3.1
3.4
Residual velocity, VR (m/s)
–
0
304
As the velocity increases, the time required for the projectile to reach to zero velocity would be more. Also, at higher
velocities some energy is absorbed due to friction. During
this phase, the deceleration of the projectile is very slow
leading to larger contact duration. The maximum contact
duration is with ballistic limit velocity.
Part 2: Incident ballistic impact velocity is more than
the ballistic limit velocity.
As the incident ballistic impact velocity increases the
contact duration decreases. This is because, as the incident
ballistic impact velocity increases, the exit velocity of the
projectile increases. This also means that the velocity of
the projectile within the target would be higher with higher
incident ballistic impact velocity.
Residual velocity, VR of the projectile as a function of
incident ballistic impact velocity is presented in Fig. 17.
This plot is for the case with h = 19 mm, projectile mass
3.84 gm and projectile diameter 6.35 mm. For this case,
ballistic limit is 550.894 m/s. This indicates that complete
perforation would not take place upto the incident ballistic
impact velocity of 550.89 m/s. If the projectile velocity is
more than ballistic limit velocity, there would be complete
perforation and the projectile would be exiting with certain
velocity.
It is interesting to note the plot of residual velocity versus incident ballistic impact velocity. As the incident ballistic impact velocity is increased beyond the ballistic limit,
the corresponding residual velocity of the projectile also
increases. But the increase is very steep just above the ballistic limit. Just to give an example, complete perforation
does not take place with incident ballistic impact velocity
of 550.89 m/s. But with incident ballistic impact velocity
of 552 m/s, complete perforation takes place with the residual velocity of 67 m/s.
Similar observations were made by Zhu et al. [11], Jenq
et al. [12] and Potti and Sun [13] during their experimental
studies. The residual velocity of the projectile is increased
as the incident ballistic impact velocity is increased above
the ballistic limit. It was clearly observed that the increase
in residual velocity was very steep immediately after the
ballistic limit velocity was reached.
459
Fig. 16. Contact duration as a function of incident ballistic impact
velocity, m = 3.84 gm, d = 6.35 mm, h = 19 mm, woven fabric E-glass/
epoxy laminate.
Fig. 18. Ballistic limit velocity as a function of projectile mass and
projectile diameter, h = 19 mm, woven fabric E-glass/epoxy laminate.
tionship is nearly linear. Further, with the same diameter of
the projectile, as the mass of the projectile increases, ballistic
limit velocity decreases. But as the mass increases, the rate
of decrease in ballistic limit velocity increases.
5.3. Target thickness
Ballistic limit velocity and contact duration at ballistic
limit velocity as a function of target thickness are shown
in Fig. 19. The plots are for the case of same mass and same
Fig. 17. Residual velocity as a function of incident ballistic impact
velocity, m = 3.84 gm, d = 6.35 mm, h = 19 mm, woven fabric E-glass/
epoxy laminate.
5.2. Projectile diameter and mass
Effect of projectile diameter and mass on ballistic limit
velocity is presented in Fig. 18. With the same mass of the
projectile, as the diameter of the projectile increases, ballistic limit velocity increases. For the case considered the rela-
Fig. 19. Ballistic limit velocity and contact duration at ballistic limit
velocity as a function of target thickness, m = 3.84 gm, d = 6.35 mm,
woven fabric E-glass/epoxy laminate.
460
diameter of the projectile. As the target thickness increases,
the ballistic limit velocity increases. But as the thickness of
the target increases, the rate of increase in ballistic limit
velocity decreases. Also, as the target thickness increases,
contact duration increases. As the thickness of the target
increases, the increase in contact duration is nearly linear
initially. But the rate of increase decreases later.
6. Conclusions
Studies have been carried out on ballistic impact behaviour of typical woven fabric E-glass/epoxy thick composites. Effects of different target and projectile parameters
are investigated. Specific observations are:
• The major energy absorbing mechanism is shear plugging. Compression of the region directly below the projectile (Region 1) and friction between the projectile
and the target also absorb significant amount of
energy. Energy absorbed by matrix cracking, delamination, tension in the yarns and compression in the surrounding region of the impacted zone (Region 2) is
not significant.
• Contact duration depends upon incident ballistic impact
velocity.
– For the case when incident ballistic impact velocity is
less than ballistic limit velocity, contact duration
increases with the increase in incident ballistic impact
velocity.
– The maximum contact duration is obtained with ballistic limit velocity.
– For the case when incident ballistic impact velocity is
more than ballistic limit velocity, contact duration
decreases with the increase in incident ballistic impact
velocity.
• Beyond the ballistic limit, as the incident ballistic impact
velocity increases, the residual velocity of the projectile
increases. But the rate of increase in residual velocity
is very significant just above ballistic limit.
• For the same mass of the projectile, as the diameter of
the projectile increases, ballistic limit velocity increases.
The behaviour is nearly linear.
• For the same diameter of the projectile, as the mass of
the projectile increases, ballistic limit velocity decreases.
But as the mass increases, the rate of decrease in ballistic
limit velocity increases.
• For the same mass and diameter of the projectile, as
the thickness of the target increases, ballistic limit
velocity increases. But as the thickness of the target
increases, the rate of increase in ballistic limit velocity
decreases.
• For the same mass and diameter of the projectile, as the
thickness of the target increases, contact duration
increases. As the thickness of the target increases, the
increase in contact duration is nearly linear initially.
But the rate of increase decreases later.
Appendix A. Analytical formulation
Stress waves are generated within the target when it is
impacted by a projectile. Due to transverse impact, compressive wave and shear wave propagate along the thickness direction and tensile wave and shear wave propagate
along the radial direction. The projectile applies forces on
to the target. The forces acting are the compressive force,
inertial force and frictional force. These forces act at various stages of the impact event.
Different damage and energy absorbing mechanisms
during ballistic impact are: compression of the target
directly below the projectile (Region 1), possible reverse
bulge formation on the front face, compression in the surrounding region of the impacted zone (Region 2), tension
in the yarns, shear plugging, bulge formation on the back
face, delamination and matrix cracking, friction between
the target and the projectile and heat generation due to
impact. Impacted materials can fail in a variety of ways.
The actual mechanisms depend on such variables as projectile size, shape, mass and velocity, and target material
properties and geometry and relative dimensions of the
projectile and the target.
Generally, the ballistic impact event can be sub-divided
into three stages. During the first stage, the projectile
strikes to the target and compression of the target takes
place directly below the projectile face. As the compression
progresses, the material would flow predominantly along
the thickness direction. Material flow can also be in the
radial direction as well as towards the front face of the target. This stage would continue until through-the-thickness
compressive wave reaches to the back face of the target.
During this stage, compressive stresses are generated within
the target directly below the projectile. As a result of this
the surrounding region would be under tension along the
radial direction.
Additionally, because of the impact force, shear stresses
are generated within the target around the periphery of the
projectile. Any of these stresses could lead to failure of the
target. As the projectile moves further, the yarns in Region
2 of the upper layers exert pressure on the yarns in Region
2 of the lower layers. In other words, compressive deformation of the yarns takes place in Region 2 also.
Because of the compression of the layers, as well as possible failure of the target by different modes, the projectile
moves further. This leads to bulge formation on the back
face. This is the second stage of impact. Along with bulge
formation on the back face, failure of the yarns/layers
would take place by different mechanisms in the upper layers of the target. This process continues and the projectile
moves further. A clear plug formation can take place in
front of the projectile. Also, the yarns/layers can fail in tension on the back face because of bulge formation. During
the entire ballistic impact event, inplane matrix cracking
and delamination between the layers can also take place.
During the third stage, the projectile moves further and
the plug and the projectile exit from the back face of the
461
target. As the projectile penetrates into the target and starts
moving further, frictional forces act between the projectile
and the target. Heat can also be generated during this
process.
The total kinetic energy of the projectile lost during the
ballistic impact event is equal to the total energy absorbed
by the target till that time interval. It is given by the following relation:
ETotali ¼ Ecfi þ Erbi þ Ecsyi þ Etfi þ Espi þ Ebbi þ Edli
þ Emci þ Efri þ Ehgi :
ð1Þ
The following assumptions are made in the analytical
formulation:
• Projectile impact is normal to the surface of the target.
• The projectile is cylindrical with a flat-end and perfectly
rigid.
• The compressive stress is experienced along the thickness direction only within those layers through which
the compressive wave has traveled. Also, compressive
strain is uniform within those layers.
• The shear plugging stress is experienced along the thickness direction only within those layers through which
the shear wave has traveled. Also, shear plugging stress
is uniform within those layers.
• The plug formed is freely moving forward, i.e., resistance is not offered by the target except for the frictional
resistance.
• The peak tensile strain within the yarns is near the
periphery of the projectile. Hence, the tensile failure of
the yarns would take place near the periphery of the
projectile.
• The deceleration of the projectile remains constant during each time interval.
The analytical formulation for the three stages is presented below: The total force acting on the projectile is
given by
F ¼ F i þ F c;
ð2Þ
where, inertial force,
1
F i ¼ qKAV 2
2
ð3Þ
and, compressive force,
F c ¼ rcz A:
ð4Þ
For the cylindrical projectile with a flat-end, K = 1
[14].
These forces are acting on the effective mass of the projectile. The effective mass of the projectile includes the
material of the target displaced by the projectile moving
with it. The effective mass of the projectile used in the equation is, therefore, m0 + qAz. This is based on considering
that the projectile has moved by a distance z. The equation
of motion for the penetration process is
d
ðmV Þ ¼ F i þ F c
dt
d
dm
dV
1
ðmV Þ ¼ V
þm
¼ KqAV 2 þ rcz A
dt
dt
dt
2
ð5Þ
The rate of change of effective mass of the projectile is
dm
dz
¼ qA ¼ qAV
dt
dt
ð6Þ
dV
dV dz
dV
¼
¼V
dt
dz dt
dz
Substituting Eq. (6) into Eq. (5), following relation is
obtained:
dV
1
¼ KqAV 2 þ rcz A:
qAV 2 þ ðm0 þ qAzÞV
ð7Þ
dz 2
The above equation is solved by separation of variables
to calculate the velocity [14] and the expression for velocity
is obtained as
V ðzÞ ¼
rcz ðzÞ
V þ
qð1 þ 0:5KÞ
2
i
!1=2
2þK
m0 =qA
rcz ðzÞ
:
ðm0 =qAÞ þ z
qð1 þ 0:5KÞ
ð8Þ
The time required for the projectile to penetrate distance
z is calculated by the following expression:
t¼
Z
z
0
¼
Z
0
z
1
dz
V ðzÞ
V 2i þ
!1=2
2þK
rcz ðzÞ
m0 =qA
rcz ðzÞ
dz:
qð1 þ 0:5KÞ ðm0 =qAÞ þ z
qð1 þ 0:5KÞ
ð9Þ
The velocity calculated by this method is used only for
the first time interval. After the calculation of the velocity,
the energy absorbed by different energy absorbing mechanisms during the first time interval is calculated. Knowing
the initial kinetic energy of the projectile and the energy
absorbed during the first time interval, the velocity of the
projectile for the next time interval is calculated. By knowing the velocity, various parameters such as displacement
of the projectile, strain, contact force and energy absorbed
by different mechanisms are calculated for the given time
interval. This procedure is continued until the compressive
wave reaches to the back face of the target. The shear wave
follows the compressive wave along the thickness direction.
The wave velocities along the thickness direction are
given by
sffiffiffiffiffiffiffiffiffiffiffiffiffi
1 drcz
V zl ¼ k
;
ð10Þ
q decz
sffiffiffiffiffiffiffiffiffi
1 ds
;
ð11Þ
V zt ¼ k
q dc
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð1 tÞ
k¼
:
ð12Þ
ð1 þ tÞð1 2tÞ
For the elastic waves, E and G are used instead of the
local slopes as given in Eqs. (10) and (11). The distance
traveled by compressive wave and shear wave along the
thickness direction at any instant of time is
462
zl ¼ V zl t;
zt ¼ V zt t:
ð13Þ
ð14Þ
The number of layers through which the compressive
wave has traveled is
zl
nlsc ¼ :
ð15Þ
hl
As the projectile is impacted onto the target, tensile and
shear waves are also generated along the radial direction in
the target. The wave velocities can be calculated using Eqs.
(10) and (11). But, in this case, the material properties are
with respect to radial direction.
The compressive strain in each layer at any instant of
time is given by
z
ecz ¼ :
ð16Þ
zl
The calculation of tensile strain in a particular yarn/
layer is given in [1].
Considering the strain variation as linear with maximum
strain at the periphery of the projectile and zero at a distance upto which the longitudinal radial wave has reached
at that particular time, the maximum strain is calculated as
below:
emax
txl ¼ 2etxl :
ð17Þ
In the beginning of the ballistic impact event, all the
energy is in the form of kinetic energy of the projectile.
During the impact event, the target absorbs some of the
energy. The energy balance at the end of ith time interval
is obtain as
KEp0 ¼ KEpi þ Ecfði1Þ þ Erbði1Þ þ Ecsyði1Þ þ Etfði1Þ
þ Espði1Þ þ Ebbði1Þ þ Edlði1Þ þ Emcði1Þ
þ Efrði1Þ þ Ehgði1Þ :
ð18Þ
Rearranging the terms in the above equation
i1
X
1
1
m0 V 2 Ei1 ¼ mi V 2i
2
2
i¼1
or
V i1 V i
:
Dt
ð22Þ
The distance traveled by the projectile during ith time
interval is obtained as
1
zi ¼ V i1 Dt dci ðDtÞ2 :
ð23Þ
2
The total distance traveled by the projectile is equal to
the summation of the distance the projectile has traveled
in each time interval. The force resisted by the target
against the motion of the projectile is given by
F i ¼ mi dci :
ð24Þ
This force is used to determine if the shear plugging is
taking place or not. The above-mentioned procedure is
continued during all the three stages of the ballistic impact
event.
Energy absorbed due to compression of the target in
Region 1
Z ecz
Ecf ¼ Ap
rcz ðecz Þde zl :
ð25Þ
ec ¼0
Energy absorbed due to compression of the yarns in the
surrounding region of impacted zone in Region 2
nlsc Z xt Z e¼ecz
X
rcz ðecz Þde x dx:
ð26Þ
Ecsy ¼ 2ph
j¼nlf
d=2
e¼0
Energy absorbed due to tension in yarns
nlsc Z xl Z etxl
X
Etf ¼ Ay
rtx ðetxl Þde dx:
j¼nlf
0
ð19Þ
Here,
ð27Þ
e¼0
The total energy absorbed due to tension is sum of the
energy absorbed by different yarns.
Energy absorbed by shear plugging during a time interval is given by the product of thickness of the target that is
sheared, shear plugging strength and the area over which
shear stress is acting. It is given by
DEspi ¼ ns hl S sp p dh:
1
1
mi1 V 2i1 Ei1 ¼ mi V 2i
2
2
ð28Þ
Energy absorbed by shear plugging by the end of ith
time interval is given by
i
X
Esp ¼
DEspi :
ð29Þ
n¼1
Eði1Þ ¼ Ecfði1Þ þ Erbði1Þ þ Ecsyði1Þ þ Etfði1Þ
Energy absorbed by delamination and matrix cracking
during ith time interval is given by
þ Espði1Þ þ Ebbði1Þ þ Edlði1Þ þ Emcði1Þ
þ Efrði1Þ þ Ehgði1Þ :
dci ¼
ð20Þ
The terms on the right-hand side of Eq. (20) can be calculated for each time interval. Hence, the velocity of the
projectile for the next time interval is calculated as
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
ffi
m V 2 Eði1Þ
2 i1 i1
Vi ¼
:
ð21Þ
1
m
2 i
The deceleration of the projectile during ith time interval
is obtained as
Edli ¼ P d px2d Aql GIIcd ðN 0 1Þ;
Emci ¼ P m px2d Aql Emt hðV m Þ
ð30Þ
Frictional resistance offered by the target towards the
movement of the projectile is a material property. It
depends upon projectile diameter, length and surface condition. It also depends upon the diameter of the hole
formed within the target due to ballistic impact, hole surface condition and the target material properties. Frictional resistance is determined experimentally.
463
Fig. 20. Compressive stress–strain curve along thickness direction for
woven fabric E-glass/epoxy laminates at high strain rate.
Appendix B. Stress–strain data at high strain rates:
compressive loading
The compressive stress–strain curve along the thickness
direction for the woven fabric E-glass/epoxy composite at
high strain rate is given in Fig. 20. Typical parameters
for this stress–strain curve are as follows:
•
•
•
•
emax = 13.5%.
rmax = 1350 MPa.
Area under the stress strain curve = 127.57 MPa.
The compressive stress–strain curve is represented by the
equation,
y ¼ 0:0496x4 þ 2:0004x3 31:956x2 þ 288:86x:
Fig. 21. Tensile stress–strain curve along radial direction for woven fabric
E-glass/epoxy laminates at high strain rate.
Here, y indicates stress in MPa and x indicates strain in
percentage.
Appendix C. Stress–strain data at high strain rates: tensile
loading
The tensile stress–strain curve along the radial direction
for the woven fabric E-glass/epoxy composite at high strain
Fig. 22. Load–displacement plot under quasi-static loading: penetration of a cylindrical projectile into a composite target with a hole, h = 19 mm,
d = 8 mm, dh = 8 mm, woven fabric E-glass/epoxy laminate.
464
rate is given in Fig. 21. Typical parameters for this stress–
strain curve are as follows:
•
•
•
•
emax = 3.5%.
rmax = 560 MPa.
Area under the stress strain curve = 13.72 MPa.
The tensile stress–strain curve is represented by the
equation,
y ¼ 1:0884x4 þ 16:003x3 106:09x2 þ 381:9x:
• Here, y indicates stress in MPa and x indicates strain in
percentage
Appendix D. Load–displacement plot under quasi-static
loading for the study of frictional resistance
Typical experiments were conducted to study the frictional resistance offered during the movement of the projectile within the target with a hole. Load–displacement
plot under quasi-static loading condition to study the penetration behaviour of a cylindrical, flat-ended hardened
steel projectile into a composite target with a hole is presented in Fig. 22. The figure shows the distance moved
by the projectile within the hole and the corresponding
load. Based on this, work done to move the projectile over
a distance and the corresponding energy absorbed due to
friction are calculated. This information is made use for
the calculating the frictional energy during the ballistic
impact event.
It can be seen from the figure that the behaviour is nonlinear and slope of the curve decreases as the projectile
moves further. The behaviour depends upon the surface
condition of the hole formed during the ballistic impact
event, diameter of the hole, thickness of the target, diameter of the projectile and the material of the target. For the
present study, the plot given in Fig. 22 is used for calculating frictional energy absorbed during ballistic impact event
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