Ballistic impact behaviour of woven fabric
composites: Parametric studies
N.K. Naik ∗ , P. Shrirao, B.C.K. Reddy
Aerospace Engineering Department, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India
Abstract
Polymer matrix composites undergo various loading conditions during their service life. For the effective use of such materials for high
performance applications, their behaviour under impact/ballistic impact loading should be clearly understood. This is because the polymer matrix
composites are susceptible to impact/ballistic impact loading. Ballistic impact behaviour of composite targets depends upon target geometrical
and material parameters as well as projectile parameters. In the present study, investigations on the ballistic impact behaviour of two-dimensional
woven fabric E-glass/epoxy composites are presented as a function of projectile and target parameters. The ballistic impact behaviour predictions
are based on the analytical method presented in our earlier work [N.K. Naik, P. Shrirao, Composite structures under ballistic impact, Compos.
Struct. 66 (2004) 579]. Firstly, perforation and partial penetration of the projectile into the target and cone formation on the exit side of the target
are studied. Further, effect of stress wave transmission factor is studied. The other parameters considered are: thickness of the target and mass and
diameter of the projectile. Ballistic limit, contact duration at ballistic limit and residual velocity are presented as a function of different target and
projectile parameters.
Keywords: Ballistic impact; Woven fabric composite; Prediction; Projectile parameters; Target parameters; Residual velocity
1. Introduction
Polymer matrix composites are finding increasing uses in
high performance applications. One of the important applications is that they can be used to provide effective protection for
ballistic impact. Such materials can absorb significant kinetic
energy of the projectile and are also characterized by high specific strength and specific stiffness.
In recent years textile composites are used for composites for
protection applications. This is because of their higher energy
absorption characteristics and through the thickness stiffness
and strength properties. Summary of investigations on ballistic
impact and an analytical model for the prediction of ballistic
impact behaviour of woven fabric composites are presented in
our earlier work [1]. Summary of the analytical model is presented in Appendix A.
There are three basic approaches to analyze ballistic impact
problems. They are:
(1) Empirical prediction models, which require lot of experimental tests and results.
(2) Prediction models, which require typical ballistic impact
experimental data as input.
(3) Analytical models, which take only mechanical and fracture
properties and geometry of the target and projectile parameters as input.
The analytical method presented in [1] is based on energy
transfer between the projectile and the target. The method
requires mechanical and fracture properties and geometry of
the target and projectile parameters as input.
Possible effects of various parameters on the ballistic impact
behaviour of the targets are presented in this study using the analytical method presented in [1] and summarized in Appendix A.
Firstly, ballistic impact behaviour with perforation and possible
partial penetration of the projectile into the target is studied.
Then, development of cone on the exit side of the target is
studied. Further, the effect of stress wave transmission factor,
target thickness and diameter and mass of the projectile are
studied.
105
Nomenclature
a
A
Adi
Aql
b
ce
cp
ct
d
dci
yarn width
cross-sectional area of fibre/yarn
damaged area at time (ti )
quasi-lemniscate area reduction factor
stress wave transmission factor
elastic wave velocity
plastic wave velocity
transverse wave velocity
projectile diameter
deceleration of the projectile during ith time interval
EDi
energy absorbed by deformation of secondary
yarns till time (ti )
EDLi
energy absorbed by delamination till time (ti )
EFi
energy absorbed by friction till time (ti )
energy absorbed by matrix cracking per unit volEmt
ume
EMCi
energy absorbed by matrix cracking till time (ti )
kinetic energy of the moving cone at time (ti )
EKEi
ESPi
energy absorbed by shear plugging till time (ti )
ETFi
energy absorbed by tension in primary yarns till
time (ti )
ETOTALi total energy absorbed by the target till time (ti )
contact force during ith time interval
Fi
GIIcd
critical dynamic strain energy release rate in mode
II
h
target thickness
hl
layer thickness
KEop
initial kinetic energy of the projectile
KEpi
kinetic energy of the projectile at time (ti )
mass of the projectile
mp
MCi
mass of the cone at time (ti )
N
number of layers being shear plugged in a time
interval
N0
number of layers
percent delaminating layers
Pd
Pm
percent matrix cracking
rdi
radius of the damage zone at time (ti )
rpi
distance covered by plastic wave till time (ti )
surface radius of the cone at time (ti ), distance
rti
covered by transverse wave till time (ti )
shear plugging strength
SSP
tc
contact duration of the projectile during ballistic
impact event
ti
ith instant of time
t
duration of time interval that tc is divided into
VBL , V50 ballistic limit velocity
fibre volume fraction
Vf
Vi
projectile velocity at time (ti )
VI , VO incident ballistic impact velocity
Vm
matrix volume fraction
VR
residual velocity
zi
height/depth of the cone at time (ti ), distance traveled by projectile at time (ti )
Greek letters
ε
strain
ε0
maximum strain in a yarn/fibre at any moment
εd
damage threshold strain
εi
maximum tensile strain in primary yarns at time
(ti )
εp
plastic strain
εpy
strain in primary yarns
εsy
strain in secondary yarns
ρ
density of the target
σ
stress
σp
plastic stress
σ sy
stress in secondary yarns
2. Damage mechanisms
Ballistic impact is normally a low-mass high-velocity impact
by a projectile propelled by a source onto a target. Since the
ballistic impact is a high velocity event, the effects on the target
can be only near the location of impact. During the ballistic
impact, energy transfer takes place from the projectile to the
target. Based on the target geometry, material properties and
projectile parameters the following are possible.
(1) The projectile perforates the target and exits with a certain velocity. This indicates that the projectile initial kinetic
energy was more than the energy that the target can absorb.
(2) The projectile partially penetrates the target. This indicates
that the projectile initial kinetic energy was less than the
energy that the target can absorb. Based on the target material properties, the projectile can either be stuck within the
target or rebound.
(3) The projectile perforates the target completely with zero exit
velocity. For such a case, the initial velocity of the projectile of a given mass is referred to as the ballistic limit. For
this case the entire kinetic energy of the projectile is just
absorbed by the target.
For the complete understanding of the ballistic impact of
composites, different damage and energy absorbing mechanisms should be clearly understood. Possible energy absorbing mechanisms are [1]: cone formation on the back face of
the target, deformation of secondary yarns, tension in primary
yarns/fibres, delamination, matrix cracking, shear plugging and
friction between the projectile and the target. For different materials like carbon, glass or Kevlar, different mechanisms can
dominate. Also, the reinforcement architecture can influence the
energy absorbing mechanisms.
The total energy absorbed by the target till a particular time
interval is given in Appendix A.
ETOTALi = EKEi + ESPi + EDi + EDLi + EMCi + EFi
(1)
The following assumptions are made in the analytical model
used:
106
(4) Energy absorption due to primary yarn/fibre breakage and
deformation of the secondary yarns are treated independently.
(5) Longitudinal and transverse wave velocities are the same in
all the layers.
(6) Projectile is in contact with the target during the ballistic
impact event.
(7) Projectile displacement and the cone height formed are the
same at any instant of time.
The assumptions are valid for relatively thin plates.
The results presented are for a typical two-dimensional plain
weave fabric E-glass/epoxy composite. The input parameters are
given in Table 1.
3. Perforation and partial penetration of the projectile
into the target
Fig. 1. Energy absorbed by different mechanisms during ballistic impact event,
VI = 158 m/s, mp = 2.8 g, h = 2 mm, d = 5 mm, woven fabric E-glass/epoxy laminate.
(1) Projectile is perfectly rigid and remains un-deformed during
the ballistic impact.
(2) Projectile motion is uniform during penetration within each
time interval.
(3) Yarns/fibres in a layer act independently.
Figs. 1 and 2 present ballistic impact behaviour at incident
ballistic impact velocity (VI ) of 158 m/s, whereas Figs. 3 and 4
are with VI = 159 m/s. It may be noted that with VI = 158 m/s,
complete penetration is not taking place whereas, with
VI = 159 m/s, complete perforation is taking place with certain
residual velocity. Strictly speaking, ballistic limit is between
158 and 159 m/s. From the practical point of view, 158 m/s
can be taken as the ballistic limit. The main energy absorbing
mechanisms are: deformation of secondary yarns and fracture
of primary yarns. Significant amount of the kinetic energy of the
projectile is transferred as the kinetic energy of the moving cone
during ballistic impact event.
Fig. 2. Projectile velocity, strain and contact force variation during ballistic impact event, VI = 158 m/s, mp = 2.8 g, h = 2 mm, d = 5 mm, woven fabric E-glass/epoxy
laminate.
107
Table 1
Input parameters required for the analytical predictions of ballistic impact
behaviour
Projectile details (cylindrical)
Mass
Shape
Diameter
2.8 g
Flat ended
5 mm
Target details
Material
Vf
Thickness
Number of layers
Density
Woven E-glass/epoxy
50%
2 mm
6
1750 kg/m3
Other details
Stress-strain
Time-step
Quasi-lemniscate area reduction factor
Stress wave transmission factor
Delamination percent
Matrix crack percent
Shear plugging strength
Mode II dynamic critical strain energy release rate
Matrix cracking energy
Fibre failure energy
–
1 ␮s
0.9
0.825
100%
100%
–
1000 J/m2
0.9 MJ/m3
28 MJ/m3
With VI = 154 m/s as shown in Figs. 5 and 6, the maximum
strain in the bottom layer is lower than the ultimate strain limit.
Hence, the lower layers do not fail because of the ballistic impact.
The kinetic energy of the moving cone would be zero at the
end of ballistic impact event. Most of the energy is absorbed
because of the deformation of the secondary yarns. The energy
Fig. 3. Energy absorbed by different mechanisms during ballistic impact event,
VI = 159 m/s, mp = 2.8 g, h = 2 mm, d = 5 mm, woven fabric E-glass/epoxy laminate.
absorbed due to deformation of secondary yarns has two components: strain energy stored within the secondary yarns and
the energy absorbed because of possible matrix cracking and
delamination.
After the projectile velocity reaches to zero, there can be
rebounding of the projectile from the target. This is because of
Fig. 4. Projectile velocity, strain and contact force variation during ballistic impact event, VI = 159 m/s, mp = 2.8 g, h = 2 mm, d = 5 mm, woven fabric E-glass/epoxy
laminate.
108
Fig. 5. Energy absorbed by different mechanisms during ballistic impact event,
VI = 154 m/s, mp = 2.8 g, h = 2 mm, d = 5 mm, woven fabric E-glass/epoxy laminate.
the release of strain energy stored in the secondary yarns. This
energy would be converted into kinetic energy of the projectile.
Some energy is absorbed in the form of yarn/fibre tensile failure.
In this particular case, layers in the upper half have failed in the
form of fibre tensile failure. Layers in the lower half have not
failed. The energy absorbed due to matrix cracking and delamination is marginal.
Fig. 7. Energy absorbed by different mechanisms during ballistic impact event,
VI = 100 m/s, mp = 2.8 g, h = 2 mm, d = 5 mm, woven fabric E-glass/epoxy laminate.
Figs. 7 and 8 present the ballistic impact behaviour with
VI = 100 m/s. It may be noted that, in this case, yarns/fibres have
not failed anywhere throughout the thickness of the composite
plate. This is because the induced strain is lower than the ultimate strain limit. In this case the projectile would not penetrate
Fig. 6. Projectile velocity, strain and contact force variation during ballistic impact event, VI = 154 m/s, mp = 2.8 g, h = 2 mm, d = 5 mm, woven fabric E-glass/epoxy
laminate.
109
Fig. 8. Projectile velocity, strain and contact force variation during ballistic impact event, VI = 100 m/s, mp = 2.8 g, h = 2 mm, d = 5 mm, woven fabric E-glass/epoxy
laminate.
into the target. There can be marginal indentation because of
compression of the target.
4. Cone formation on exit side of the target
During the ballistic impact event, cone formation takes place
on the exit side of the target just below the point of impact
[1–3]. This is because of the transverse wave propagation. The
surface radius of the cone increases with time. Also, the cone
moves along with the projectile. With this, the height of the
cone increases. It may be noted that the projectile displacement
at any instant of time and the cone height formed would be the
same. Variation of cone surface radius with time is presented
in Fig. 9. Initially, transverse wave velocity increases significantly and then remains nearly constant during the remaining
period of ballistic impact event. Variation of cone surface radius
is nearly linear with respect to time. Cone height variation with
time is non-linear as shown in Fig. 9. The rate of increase of
Fig. 9. Cone surface radius, transverse wave velocity and projectile displacement variation during ballistic impact event, VI = 158 m/s, mp = 2.8 g, h = 2 mm, d = 5 mm,
woven fabric E-glass/epoxy laminate.
110
cone height/depth decreases with time. This depends upon the
velocity variation of the projectile during the ballistic impact
event.
5. Contact duration
The contact duration between the projectile and the target is
a function of incident ballistic impact velocity as presented in
Fig. 10. The contact duration can be defined as follows:
- Partial penetration: The time interval starting from the point
when the projectile just hits the target to the point when the
velocity of the projectile becomes zero.
- Complete perforation: The time interval starting from the point
when the projectile just hits the target to the point when the
projectile exits from the back surface of the target.
At ballistic limit, the projectile would be exiting the target
with zero velocity. From Fig. 10, it is seen that as the incident ballistic impact velocity increases, the contact duration decreases. It
may be noted that at ballistic limit and above, the drop in contact
duration is significant. Contact duration at ballistic limit velocity
(VBL ) as a function of target thickness is presented in Fig. 11.
Generally, it is observed that as the target thickness increases,
the contact duration at ballistic limit decreases.
6. Effect of stress wave transmission factor
During the ballistic impact event, as the longitudinal stress
wave travels through the yarns, stress wave attenuation takes
place. The stress wave transmission factor indicates the fraction
of the stress wave that is transmitted as the longitudinal stress
wave propagates further. Higher the transmission factor, there
would be less stress attenuation, in turn, more stress transmission
[1,10].
Fig. 10. Contact duration variation as a function of incident ballistic impact
velocity, mp = 2.8 g, h = 2 mm, d = 5 mm, woven fabric E-glass/epoxy laminate.
Fig. 11. Contact duration at ballistic limit velocity as a function of target thickness, mp = 2.8 g, d = 5 mm, woven fabric E-glass/epoxy laminates.
If the stress wave transmission factor is higher, corresponding stress and, in turn, strain would be higher. This would lead
to possible larger damage area and hence, higher energy absorption. This would lead to possible higher VBL . Fig. 12 presents
ballistic limit velocity as a function of stress wave transmission factor. Higher the stress wave transmission factor, higher is
VBL .
For woven fabric composites, stress wave transmission factor
is a function of the geometry of woven fabric composite as well
as the material properties. For the given material properties of the
yarns of the woven fabric composite, stress wave transmission
Fig. 12. Ballistic limit velocity variation as a function of stress wave transmission factor, mp = 2.8 g, h = 2 mm, d = 5 mm, woven fabric E-glass/epoxy
laminate.
111
Fig. 13. Ballistic limit velocity variation as a function of projectile diameter,
mp = 2.8 g, h = 2 mm, woven fabric E-glass/epoxy laminate.
Fig. 14. Ballistic limit velocity variation as a function of projectile mass,
h = 2 mm, d = 5 mm, woven fabric E-glass/epoxy laminate.
factor can be controlled by controlling the geometry of the woven
fabric composite.
Parameter “b” used in this study is based on stress wave attenuation characteristics for each composite [10]. It is a material
property and depends upon geometry of the fabric as well as
mechanical and physical properties of the reinforcing material
and the matrix. This parameter is determined for each material
separately. In the present case, stress wave transmission factor,
b = 0.825 (Appendix A).
159 m/s, 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 158 m/s. But with incident ballistic impact
velocity of 159 m/s, complete perforation takes place with the
residual velocity of 54 m/s.
7. Effect of projectile and target parameters
Variation of VBL as a function of projectile diameter is presented in Fig. 13 for the same mass of the projectile. As the
projectile diameter increases, VBL increases. It may be noted
that VBL increase is not significant in this case.
Variation of VBL as a function of projectile mass is presented
in Fig. 14 for the same diameter of the projectile. As the mass
increases, VBL decreases and the behaviour is nearly linear.
Variation of VBL as a function of target thickness with the
same projectile mass and diameter is presented in Fig. 15. As the
target thickness increases, VBL also increases and the behaviour
is nearly linear. At higher incident impact velocities, contact
duration would decrease. This effect can be seen in Fig. 11.
8. Residual velocity
Residual velocity (VR ) of the projectile as a function of incident ballistic impact velocity is presented in Fig. 16. This plot
is for the case with target thickness, h = 2 mm; projectile mass,
mp = 2.8 g and projectile diameter, d = 5 mm. For this case ballistic limit is 158 m/s. This indicates that complete perforation
would not take place up to the incident ballistic impact velocity
of 158 m/s. If the projectile velocity is equal to or more than
Fig. 15. Ballistic limit velocity variation as a function of target thickness,
mp = 2.8 g, d = 5 mm, woven fabric E-glass/epoxy laminates.
112
tile parameters has been investigated. The specific observations
are:
Fig. 16. Residual velocity variation as a function of incident ballistic impact
velocity, mp = 2.8 g, d = 5 mm, h = 2 mm, woven fabric E-glass/epoxy laminate.
Similar observation was made by Zhu et al. [12], Jenq et
al. [13], Potti and Sun [14] and Jenq et al. [15] during their
experimental studies. They observed that, as the incident ballistic impact velocity is increased beyond the ballistic limit, the
corresponding residual velocity of the projectile increased. But
the increase was very steep just above the ballistic limit.
This can be explained considering the projectile displacement and the increase in cone surface radius during the ballistic
impact event. As shown in Fig. 6, at certain incident ballistic
impact velocity, the upper layers would be failing whereas the
lower layers may not be failing. The strain in the lower layers would be increasing initially and then start decreasing. This
is because of the geometry of the cone formed, which is governed by projectile displacement, i.e., change in cone height and
the cone surface radius at any instant of time. The complete
perforation would take place if the induced strain exceeds the
permissible strain before it starts decreasing as shown in Fig. 6.
If complete perforation does not take place at this incident ballistic impact velocity, e.g., for the present case it is 158 m/s, the
incident ballistic impact velocity will have to be increased for
complete perforation. The necessary increase in incident ballistic impact velocity above 158 m/s could be so small that in a
practical sense 158 m/s can be taken as the ballistic limit. The
geometry of the cone formed, and in turn, the induced strain is
governed by projectile displacement, i.e., change in cone height
(zi ) and the cone surface radius (rti ) at any instant of time. The
trend given in Fig. 16 is because of the combined effect of rti
and zi .
9. Conclusions
Studies have been carried out on ballistic impact behaviour of
a typical plain weave fabric E-glass/epoxy composite. Effect of
various target geometrical and material parameters and projec-
- During the ballistic impact event the kinetic energy of the
projectile is transferred to the target. Significant amount of
this energy is transferred as the kinetic energy of the moving
cone during ballistic impact event.
- At the end of ballistic impact event, major energy absorbing
mechanisms are: deformation of secondary yarns and fracture
of primary yarns.
- Increase in cone surface radius is nearly linear with respect to
time.
- The rate of increase of cone depth/height decreases with time.
- The contact duration between the projectile and the target
decreases as the incident ballistic impact velocity increases.
Beyond the ballistic limit velocity the decreases in contact
duration is significant.
- As the stress wave transmission factor increases, the ballistic
limit velocity increases.
- For the same mass of the projectile, as the diameter of the
projectile increases, the ballistic limit velocity increases.
- For the same diameter of the projectile, as the mass of the
projectile increases, the ballistic limit velocity decreases.
- For the same mass and diameter of the projectile, as the target
thickness increases, the ballistic limit velocity increases.
- 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 the ballistic limit.
Appendix A. Analytical formulation
On striking a target, energy of the projectile is absorbed by
various mechanisms like kinetic energy of the moving cone EKE ,
shear plugging ESP , deformation of secondary yarns ED , tension in primary yarns ETF , delamination EDL , matrix cracking
EMC , frictional energy EF along with other energy dissipation
mechanisms. Here, an analytical model is presented for the
above-mentioned mechanisms, which dominate during ballistic impact of two-dimensional woven fabric composites [1].
A.1. Analytical formulation
Cone formation was observed on the back face of the target as
shown in Fig. 17 when struck by a projectile [2,3]. Shear plugging on the front face and cone formation on the back face start
taking place depending on the target material properties during
the ballistic impact event. It is assumed that the velocity of the
moving cone is the same as the velocity of the projectile. Initially, the moving cone has velocity equal to that of the projectile
and has zero mass. As time progresses, the mass of the moving
cone increases and velocity of the cone decreases. As the cone
formation takes place, the yarns/fibres deform and absorb some
energy. The primary yarns, which provide the resistive force to
the projectile motion, are strained the most, thus leading to their
failure. When all the primary yarns fail, the projectile exits the
target. Tensile failure of the yarns thus absorbs some energy
113
The innermost longitudinal wavelet, called the plastic wave,
propagates at velocity [4],
1 dσ
(A.3)
cp =
ρ dε ε=εp
Here, εp denotes the strain at which plastic region starts.
As the strain wavelets pass a given point on the yarn, material of the filament flows inward towards the impact point. The
material in the wake of the plastic wave front forms itself into
a transverse wave, shaped like an inverted tent with the impact
point at its vertex. In a fixed coordinate system, the base of
the tent spreads outward with the transverse wave velocity. The
transverse wave velocity is given by [4],
εp (1 + εp )σp
1 dσ
ct =
dε
(A.4)
−
ρ dε
ρ
0
If the complete impact event is divided into a number of small
instants, then at ith instant, the time is given by ti . By that time
the transverse wave has traveled to a distance rti and the plastic
wave has traveled to a distance of rpi . The projectile has moved
through a distance zi .
Radii rti and rpi after time ti = i t is given by,
Fig. 17. Conical deformation during ballistic impact on the back face of the
composite target.
rti =
n=i
ctn t
(A.5)
n=0
of the projectile. Even before the failure of the primary yarns,
there would be some energy absorption because of tension in
the yarns. During the ballistic impact event, delamination and
matrix cracking take place in the laminate area, which forms the
cone. The total kinetic energy of the projectile that is lost during
ballistic impact is the total energy that is absorbed by the target
till that time interval and is given by,
ETOTALi = EKEi + ESPi + EDi
+ ETFi + EDLi + EMCi + EFi
(A.1)
An analytical model to predict the ballistic impact behaviour
is presented below. Firstly, modelling of a single yarn is presented. The model is further extended to the woven fabric composites. The analytical model is for the case when the projectile
impacts the target normally.
A.2. Modelling of a single yarn subjected to transverse
ballistic impact
When a yarn is impacted by a projectile transversely, longitudinal strain wavelets propagate outward along the filament
[4–6]. The outermost wavelet, called the elastic wave, propagates at velocity,
1 dσ
ce =
ρ dε ε=0
(A.2)
rpi =
n=i
cpn t
(A.6)
n=0
A.3. Modelling of woven fabric composites
Woven fabric composites consist of warp and fill yarns, interlaced in a regular sequence [7–9]. As the projectile impacts
on to the woven fabric composite, there can be many yarns
beneath the projectile. In the present analysis, all the warp and fill
yarns are treated separately. Behaviour of each yarn is analyzed
as explained in the previous section. Overall behaviour of the
woven fabric composite is presented considering the combined
effect of all the individual yarns within a layer.
A.4. Strain and deceleration history: single yarn of a
woven fabric composite
When a projectile impacts a woven fabric composite, a cone
would be formed on the back face of the target. The shape of
the cross-section of the cone formed on the surface of the target plate would be quasi-lemninscate. Quasi-lemniscate shape
is because of difference in elastic properties and critical strain
energy release rates along warp/fill directions and the other
directions. For simplicity, the cone formed can be considered
to be of circular cross-section, rather than quasi-lemniscate. The
base of the cone spreads with a transverse wave velocity as given
in Eq. (A.4). The plastic wave velocity is given in Eq. (A.3).
After time ti the transverse wave travels to a distance rti and
the plastic wave travels to a distance of rpi . The projectile moves
114
through a distance zi . Radii rti and rpi after time ti = it are given
in Eqs. (A.5) and (A.6), respectively.
Because of stress wave attenuation, stress in the yarn and
hence, strain in the yarn varies with the distance from the
point of impact. Stress and in turn, strain is maximum at the
point of impact and decreases along the length of the yarn. At
ith time interval, the strain at the point of impact is given by
[10],


 (d/2) + (rti − (d/2))2 + z2i + (rpi − rti ) − rpi 
εi =


b(rpi /a) − 1
×
ln b
a
(A.7)
Studies have been carried out on stress wave attenuation of
longitudinal waves during ballistic impact on typical woven
fabric composites based on wave reflection/transmission at the
interfaces [10]. Normalized stress can be represented as,
y = f (x) = bx/a
(A.8)
where b is a constant of magnitude less than 1, x the distance, a
the yarn size and y the normalized stress. It may be noted that
the constant b represents the transmitted component of the stress
wave. Hence, b is referred to as stress wave transmission factor.
Parameter “b” is obtained from stress wave attenuation studies for each composite. It is a material property and depends
upon geometry of the fabric as well as mechanical and physical
properties of the reinforcing material and the matrix. For the
target analyzed, b = 0.825.
As a result of stress wave attenuation from the point of impact
to the point up to which the longitudinal stress wave has reached,
there would be strain variation also. The maximum strain would
be at the point of impact. The strain would be decreasing with
distance from the point of impact to the point up to which longitudinal stress wave has reached.
Once, the strain variation in a yarn/fibre is known, the energy
absorbed by various mechanisms such as kinetic energy of the
moving cone, deformation energy absorbed by the secondary
yarns, tension in primary yarns, delamination and matrix cracking can be calculated.
At the beginning of the first time interval of impact, entire
energy is in the form of kinetic energy of the projectile. Later,
this energy is dissipated into energy absorbed by various damage mechanisms and the kinetic energy of moving cone and
projectile. Considering the energy balance at the end of ith time
interval,
(A.9)
Rearranging the terms in the above equation,
1
2
2 mp V 0
− Ei−1 = 21 (mp + MCi )Vi2
Ei−1 = ESP(i−1) + ED(i−1) + ETF(i−1)
+ EDL(i−1) + EMC(i−1) + EF(i−1)
(A.11)
The terms on the right hand side of Eq. (A.11) are known at
(i−1)th instant of time. From this, the velocity of the projectile
at the end of ith time interval can be obtained as,
1/2(mp V02 − Ei−1 )
(A.12)
Vi =
1/2(mp + MCi )
If the projectile velocity is known at the beginning and the
end of ith time interval, then the deceleration of the projectile
during that time interval can be found out. It is given by,
dci =
Vi−1 − Vi
t
(A.13)
It may be noted that the velocity at the beginning of the ith
time interval is the same as the velocity at the end of (i−1)th
time interval. Distance traveled by the projectile zi up to ith time
interval is given by,
zi =
n=i
zn
(A.14)
n=0
1
zi = Vi−1 t − dci (t)2
2
(A.15)
Utilizing dci , the deceleration of the projectile during ith time
interval, the force resisting the projectile motion can be calculated. It is given by,
Fi = mp dci
(A.16)
The magnitude of the force on the projectile is the same as
the magnitude of the force applied on the target, by the projectile. Hence, this force can be used to determine whether shear
plugging takes place or not.
The above process is repeated until all the primary yarns in the
target fail, i.e., the complete perforation takes place. The velocity
at the end of time interval, during which all the yarns are broken,
is the residual velocity of the projectile. On the other hand, if
the numerator in Eq. (A.12) becomes zero, then the projectile
does not penetrate the target completely with the given initial
velocity. Thus by repetition of the above procedure with various
velocities so as to get complete perforation with zero residual
velocity, the ballistic limit of the target laminate can be obtained.
A.5. Kinetic energy of the moving cone formed
KEP0 = KEpi + EKE + ESP(i−1) + ED(i−1)
+ ETF + EDL(i−1) + EMC(i−1) + EF(i−1)
where
(A.10)
The cone formed on the back face of the target absorbs some
energy. By the end of ith time interval the surface radius of the
cone formed is given by Eq. (A.5).
Mass of the cone formed is,
MCi = πrti2 hρ
(A.17)
115
The velocity of the cone formed is equal to Vi , the velocity
of the projectile at the end of ith time interval. So the energy of
the cone formed at the end of ith time interval is,
EKEi = 21 MCi Vi2
(A.18)
(A.20)
ESPn
n=0
A.7. Energy absorbed due to deformation of secondary
yarns
The secondary yarns experience different strains depending
on their position. The yarns, which are close to the point of
impact experience a strain, equal to the strain in the outermost
primary yarn, whereas those yarns, which are away from the
impact point, experience less strain.
The energy absorbed in the deformation of all the secondary
yarns can be obtained by the following integration [10],
rti εsyi
EDi =
σsy (εsy )dεsy
√
d/ 2
0
× h 2πr − 8r sin
(A.22)
ε=0
where ε0 is the ultimate strain limit. If during ith time interval N
numbers of yarns/fibres are failing, then the right hand side of
the above expression is multiplied by N.
It may be noted that during the movement of the cone formed,
kinetic energy is converted into strain energy within yarns up
to the point of failure. It retards the projectile. At the point of
failure of the strained yarns, the strain energy stored in the yarns
is dissipated.
A.9. Energy absorbed due to delamination and matrix
cracking
(A.19)
where N indicates the number of layers shear plugged during ith
time interval and SSP denotes shear plugging strength.
The energy absorbed by shear plugging by the end of ith time
interval is given by,
n=i
ε=ε0 b
σ(ε) dε dx
0
When the target material is impacted by the projectile, shear
plugging stress in the material near projectile periphery rises.
As and when the shear plugging stress exceeds shear plugging
strength, shear plugging failure occurs. As a result, plug formation takes place. This phenomenon is generally observed for
carbon/epoxy composites. If at the beginning of the ith time
interval, shear plugging stress exceeds shear plugging strength,
then the energy absorbed by shear plugging during that time
interval is given by the product of distance sheared, shear plugging strength and the area over which shear plugging stress is
applied. It is given by,
ESPi =
x
ETF = A
A.6. Energy absorbed due to shear plugging
ESPi = Nhl SSP π dh
this phenomenon is stress wave attenuation. When the strain in
yarns/fibres exceeds failure strain, it fails and some energy is
absorbed due to tensile failure. For a yarn/fibre of cross-section
area A it is given by,
x/a
−1
d
2r
dr
(A.21)
Delamination and matrix cracking absorb some part of the
initial kinetic energy of the projectile. A part of the conical
area undergoes delamination and matrix cracking. The extent
to which composite has delaminated and the matrix has cracked
till (i + 1)th time instant can be calculated on the basis of strain
profile in the composite at that time interval. From the results
derived in the earlier section it can be observed that the strain at
the impact point is the highest and it decreases along the length of
the yarn. The area in which strain is more than the damage initiation threshold strain εd , undergoes damage in the form of matrix
cracking and delamination. However, complete matrix cracking
may not take place. Evidence for this phenomenon is provided
by the fact that after ballistic impact, matrix is still attached to
the fibres and does not separate from the reinforcement completely. Due to matrix cracking, the interlaminar strength of the
composite decreases. As a result, further loading and deformation causes delamination. This delamination is of mode II type.
Again, delamination may not occur at all the lamina interfaces.
Towards the end of ballistic impact event, when only a few nondelaminated layers are left, these non-delaminated layers are
more likely to bend rather than delaminate.
The area undergoing delamination and matrix cracking in the
conical region is of quasi-lemniscate shape, which is taken to be
Aql percent of the corresponding circular area. During (i + 1)th
time interval the area of delamination and matrix cracking is
given by,
A.8. Energy absorbed due to tension in primary yarns
2
2
π(rd(i+1)
− rdi
)Aql
The yarns directly below the projectile, known as the primary
yarns, fail in direct tension. All the primary yarns within one
layer do not fail at one instant of time. As and when the strain of
a particular yarn reaches the dynamic failure strain in tension,
the yarn fails. It may be noted that the length of yarns/fibres
failing in tension is twice the distance covered by the longitudinal wave. Also, the complete length of a primary yarn is not
strained to the same extent as explained earlier. The reason for
where rdi indicates the radius up to which the damage has propagated until ith time interval. So the respective energies absorbed
by delamination and matrix cracking during this time interval
are given by,
(A.23)
2
2
− rdi
)Aql GIIcd (N0 − 1)
EDLi = Pd π(rd(i+1)
(A.24)
2
2
EMCi = Pm π(rd(i+1)
− rdi
)Aql Emt hVm
(A.25)
116
The factors Pd and Pm stand for percentage delamination
and percentage matrix cracking. Hundred percent delamination
indicates that complete delamination has taken place along all
the interfaces within the damaged area. Hundred percent matrix
cracking indicates that entire matrix within the damaged area
has cracked.
Energy absorbed till the ith time interval is given by,
EDLi =
n=i
EDLn
(A.26)
n=1
EMCi =
n=i
EMCn
(A.27)
n=1
Effect of matrix properties has been taken into account by
considering matrix cracking energy and the energy absorbed by
delamination.
A.11. Calculation of contact duration
The impact event starts when the projectile touches the front
face of the target. End of the impact event is taken to be when the
entire primary yarns fail or when the velocity of the projectile
becomes zero. Total time from the start of the impact event till
the end of the event is called as contact duration. If the end of the
event occurs during nth time interval, then the contact duration
is obtained as,
tc = n t
(A.28)
A.10. Other possible energy absorbing mechanisms
References
Other possible energy absorbing mechanisms are: bending
strain around the hinges at the edge of the contact patch, bending
strain around the hinges at the edge of the cone formed and radial
compression around the penetrating projectile.
Target penetration takes place when, either all the fibres fail
due to tension or all the layers fail due to shear plugging or due to
the combined effect of both the mechanisms. Even after tensile
failure of all the yarns or shear plug formation, the projectile has
to overcome frictional resistance provided by the damaged laminate [11]. In cases when the projectile has just enough energy
to fracture all the yarns but not enough energy to overcome the
frictional resistance, it may get stuck up in the target. The frictional resistance depends on the type of fit between the projectile
and the damaged target. And accordingly, it can result in local
temperature rise.
The present method is mainly for relatively thin and flexible
target plates. Hence, the energy absorbing mechanisms as given
in this section are not considered.
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