mechanisms of armour penetration
TRANSCRIPT
i
NIKO HOLKKO
MECHANISMS OF ARMOUR PENETRATION
Bachelor’s Thesis
Inspector: lecturer Risto Alanko
ABSCTRACT
NIKO HOLKKO: Mechanisms of armour penetration Tampere University of Technology Bachelor’s Thesis, 30 pages, 8 appendix pages March 2015 Bachelor’s Degree Programme in Mechanical Engineering Major: Machine Construction Examiner: Risto Alanko Keywords: armour penetration, armour, armour piercing, piercing
The ability of an armour piercing shell to penetrate armour depends on both the shell’s
and the armour’s geometries and their material properties. At the moment of impact, the
armour is perforated, or penetrated, with one of three perforation mechanisms. The ar-
mour can be damaged even in a failed penetration.
There are several types of shells, all of which have their unique properties and uses. The
impact behaviour differs as well between different types of shells. There are different
types of armours and armoured plates can be used to create multiple configurations that
impact the armour’s ability to resist penetration.
Predictive mathematical models can be created to different shells by using statistical data.
Using these models, the penetration capability of shells can be estimated as a function of
their type, calibre, mass and range of impact.
TABLE OF CONTENTS
1. INTRODUCTION .................................................................................................... 1
2. MECHANISMS OF ARMOUR PENETRATION ................................................... 2
2.1 Shell types ...................................................................................................... 7
2.2 Armour types .................................................................................................. 8
3. MATHEMATICAL PREDICTION MODEL ........................................................ 19
4. SUMMARY ............................................................................................................ 28
BIBLIOGRAPHY ........................................................................................................... 30
APPENDIX A: BALLISTIC PERFORMANCE INDEX
APPENDIX B: BALLISTIC PERFORMANCE MAPS
APPENDIX C: CONSTANTS OF THE SLOPE COEFFICIENT
APPENDIX D: PROPERTY TABLE OF AP-SHELLS
APPENDIX E: PROPERTY TABLES OF APC- AND APBC-SHELLS
APPENDIX F: PROPERTY TABLE OF APCBC-SHELLS
APPENDIX G: PROPERTY TABLE OF APCR-SHELLS
APPENDIX H: ARMOUR PIERCING SHELL TYPES
TABLE OF FIGURES
Kuva 1. Schematics of armour penetration. ................................................................ 2
Kuva 2. Nose shapes, conical and ogive. .................................................................... 3
Kuva 3. Fracture mechanisms. .................................................................................... 4
Kuva 4. Spalling (Rosenberg & Dekel, 2012 s. 40). .................................................... 5
Kuva 5. Penetrating sloped armour............................................................................. 9
Kuva 6. Slope coefficient as a function of the angle of impact when K = 1.3 and
0.4. ................................................................................................................ 10
Kuva 7. Slope coefficients for 76 mm and 90 mm APCR-shells as a function of
the angle if impact. ....................................................................................... 11
Kuva 8. Values for the BHN-coefficient as a function of caliber thickness for
different calibers, when the hardness value of the plate is 460 BHN. ......... 12
Kuva 9. Spaced Armour.. ........................................................................................... 14
Kuva 10. Single plates equivalent to spaced armour in different impact cases. ......... 16
Kuva 11. Comparison of layered and spaced armour. ................................................ 18
Kuva 12. Penetration of an AP-shell as a function of the kinetic energy
coefficient. .................................................................................................... 20
Kuva 13. Relational penetration of British AP-shells of different velocities as a
function of distance. ..................................................................................... 21
Kuva 14. Change in relational penetration with different calibers and masses. ........ 21
Kuva 15. Penetration and relational penetration for different shells types as a
function of distance. ..................................................................................... 25
ABBREVIATIONS AND NOTATIONS
AP Armour Piercing
APBC Armour Piercing Ballistic Capped
APC Armour Piercing Capped
APCBC Armour Piercing Capped Ballistic Capped
APCR Armour Piercing Composite Rigid
APHE Armour Piercing High Explosive
BHN Brinell Hardness Number
BPI Ballistic Performance Index
CHA Cast Homogeneous Armour
CRH Caliber-radius-heads
HEAT High Explosive Anti Tank
HVAP High Velocity Armour Piercing
RHA Rolled Homogeneous Armour
FHA Face-Hardened Armour
A shell and armour constant
BC ballistic coefficient
D diameter
E elastic modulus/Young’s modulus
KE kinetic energy
F force
H strain hardening rate
h thickness of molten layer
I ratio of mass and the cube of the diameter
i shape coefficient
K caliber thickness
L length of shell
L length of the nose of the shell
m mass
P penetration capability
R hardness
r distance
S sharpness
s radius
Tm melting temperature
t thickness
v velocity
vbl ballistic limit velocity
εr strain fracture
φ angle of oblique
ρ density
σm ultimate tensile strength
σr penetration strength
σspall spall strength
σε yield strength
µ coefficient of friction
γ Poisson’s constant
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1. INTRODUCTION
The term armour penetration is usually used to refer to the perforation of armoured plates
with varying ammunition in warfare. The goal of armour penetration is to destroy a target
protected by armoured plates, such a vehicle or its crew. Armour penetration has been a
phenomenon of interest of both civil and military engineers for nearly two hundred years.
The study of armour penetration first became important during the naval battles of the
19th century and at the advent of steel-protected war ships. As the first tanks appeared
during World War I, the science of armour penetration moved to study land targets as
well.
The first ammunition that was used against tanks and other armoured vehicles were made
of solid steel and shaped similar to bullets. Their penetration capability was based on their
kinetic energy. The shells were called armour piercing shells, or AP-shells. During the
2nd World War AP-shells were improved in multiple ways as the thickness of armoured
plates grew. In addition, the solid steel shots were designed to include parts made of other
materials than steel, such as tungsten. In the end, the traditional shells were replaced by
modern dart-shaped ammunition that were made completely out of heavy materials, such
as the aforementioned tungsten or depleted uranium. During World War II, other types of
shells appeared as well, such as chemical energy penetrators. These include shells such
as high explosive anti-tank shells (HEAT). HEAT-shells penetrate armour by firing a jet
of metal towards an armoured plate. The metal then penetrates through the armour
through its kinetic energy. In modern warfare, HEAT and dart-shaped ammunition are
the most commonly used ammunition types.
Armour piercing projectiles are fired out of a cannon. The projectile as a whole is com-
prised of an armour piercing shot and shell. The shot is the penetrating part and the shell
includes the primer and propellant. In general, the word shell can be used to refer to the
armour piercing part of to the whole combination. In this text, the word shell is used to
refer to the armour piercing projectile, or shot. The word projectile is also used.
This thesis focuses on the traditional kinetic AP-shells and the mechanisms of the event
of penetration. This work also studies how different ammunition and armour types affect
penetration. A mathematical model based on statistics is also derived. This model can be
used to predict the behaviour of different types of armour piercing shells.
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2. MECHANISMS OF ARMOUR PENETRATION
The penetration capability of a kinetic penetrator is based on its kinetic energy. The en-
ergy is maximised through the mass and velocity of the projectile. For this reason, the
projectiles are usually made out of heavy materials.
Kuva 1. Schematics of armour penetration.
In addition to its velocity and mass, the hardness values of the projectile (Ra) and armour
(Rp) also affect armour penetration (Bird & Livingston 2001, p. 21, 38). According to the
US Army Material Command (1963, p. 6-3) the general hardness value of an AP-shell is
653–722 BHN in the nose and 370–420 BHN everywhere else. AP-shells are usually
manufactured out of steel or steel alloys such as steel-molybdenum-chrome alloy. The
hardness of homogeneous armour plates is 220–300 BHN with an upper limit of 375
BHN. As the hardness increases over this value, the plates become brittle and their ability
to resist large diameter projectiles is reduced (Bird & Livingston 2001, p. 21). According
to Rosenberg & Dekel (2012) the ultimate tensile strength σm and yield strength σε also
affect armour’s ability to resist penetration. Figure 1 illustrates the physical properties
that affect penetration, when an AP-shell meets a homogeneous armour plate. Armour
plates are also manufactured out of steel and its alloys. The majority of armour plates are
homogeneous and rolled. Rolled homogeneous armour is usually denoted with RHA.
Modern American armour plates are manufactured according to standard MIL-DTL-
12560. Sometimes armour plates are manufactured out of aluminium. Even though alu-
minium resists penetration worse than steel, it is used in situations where light and thin
armour plates are needed.
The energy of the projectile is focused on a small area. This focus can be achieved with
the projectile’s diameter and the shape of the nose. The most common shapes are conical
and ogive. These shapes are illustrated in figure 2.
3
Kuva 2. Nose shapes, conical and ogive.
The radius s of a nose of an ogive projectile is 2–4 times the diameter D of the projectiles.
The ratio between the radius and the diameter is called calibre-radius-heads, or CRH. For
example, a projectile with s = 2D is a type 2CRH projectile. The sharpness S of a projec-
tile is defined as the ratio of the length l of the nose and the projectile’s diameter D.
(Rosenberg & Dekel 2012, s. 24.)
The velocity of an AP-shell is usually 600–900 m/s. The shell’s ability to maintain its
velocity can be expressed through a ballistic coefficient BC. Ballistic coefficient (Moss
et al. 1955, s. 86) can be expressed as
𝐵𝐶 =𝑚
𝐷2𝑖, (2.1)
where m is the projectile’s mass and i its shape coefficient. According to Cline (2002, p.
44) the shape coeffcient can be calcuated with
𝑖 =2
𝑠√4𝑠−1
𝑠. (2.2)
According to Masket (1949) and Rosenberg & Dekel (2012, p. 74) the amount of energy
required for penetration approximately the same for both projectile shapes. The penetra-
tion process can be made easier by increasing the sharpness of the projectile. Once the
sharpness reaches a value of 3, increasing its value no longer gives any more benefits
(Rosenberg & Dekel 2012, p. 75).
Caliber thickness K is defined as the ratio of the thickness of the armour plate and the
diameter (or caliber) of the projectile. Caliber thickness affects which fracture mechanism
the projectile uses to perforate an armour plate. The projectile perforates the armour with
smallest possible amount of energy. The fracture mechanisms (Rosenberg & Dekel 2012,
p. 121) can be roughly divided into three different main mechanisms: dishing, punching
and ductile hole enlargement. These fracture mechanisms are illustrated in figure 3.
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Kuva 3. Fracture mechanisms.
Both dishing and punching require a situation where caliber thickness K is less than 1.
This situation where the caliber of the projectile is bigger than the thickness of the armour
plate is called overmatching. Dishing is the dominant mechanism when caliber thickness
K is smaller than 1/3. In dishing, the caliber of the shell is much larger than the armour
plate, which leads to a situation where the plate is bent open. Thomson (1955) estimated
the energy needed for perforation for conical and ogive projectiles as following:
𝑊𝑝,𝑐𝑜𝑛𝑖𝑐𝑎𝑙 =1
4𝜋𝐷2𝑡 (
1
2𝜎𝜀 +
𝜌𝑝𝑣2𝐷2
4𝑙), (2.3)
𝑊𝑝,𝑜𝑔𝑖𝑣𝑒 =1
4𝜋𝐷2𝑡 (
1
2𝜎𝜀 + 1,86
𝜌𝑝𝑣2𝐷2
4𝑙), (2.4)
where v is the velocity of the projectile and t is the thickness of the armour plate and ρp
its density. Thomson noticed as well that the energy required for perforation is roughly
the same for both projectile shapes.
Punching is a special fracture mechanism and it requires a specific set of circumstances.
In addition to a small caliber thickness, it requires a blunt hit against the armour plate. A
blunt hit can be achieved if the nose of the projectile is flat or if the projectile hits the
armour plate with its edge. In punching, the force of the impact is so great that the shear
stress around the area of impact cuts a cylindrical section called a plug from the armour.
Punching can also occur in a situation where the nose of the projectile deforms into a flat
shape at the moment of impact. (Zener & Peterson 1943; Bird & Livingston 2001, p. 5).
If the caliber thickness is more than 1/3 and the circumstances for punching are not ful-
filled, the armour is perforated through ductile hole enlargement. In ductile hole enlarge-
ment the projectile pushes material away from itself, mainly in a radial direction. As the
projectiles travels through the armour plate, large amounts of friction is created (Thomson
1955). The friction causes the projectile to slow down. The heat from the frictional forces
causes the temperatures of the surfaces of the projectile and the hole to increase rapidly.
The increased temperature creates a layer of molten metal between the projectile and the
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hole. This molten metal acts as a lubricant which then reduces the friction. Both Zener &
Peterson (1943) and Rosenberg & Dekel (2012, p. 96) note that friction uses only a small
amount of the total kinetic energy of the projectile. The majority of the energy is used to
deform the armour plate and the projectile. According to Thomson (1955), the amount of
energy required to create the molten layer of metal during perforation is
𝑊𝑞 = 2𝜋𝜇𝑡𝑣 (𝜎𝜀𝐷𝑙
16𝑣+3𝜌𝑝𝐷
3𝑣
64𝑙), (2.5)
where µ is the coefficient of friction, which is roughly 0.02. Thomson also estimated that
the thickness h of the molten layer can be expressed with the equation
3
8𝜋𝐷2ℎ =
𝑊𝑞
285𝑇𝑚, (2.6)
where Tm is the required change in temperature to melt the material of the armour.
In a situation where the projectile does not penetrate the armour, it shatters against the
surface or bounces away. A failed penetration usually leaves a pit or a dent on the surface
of the armour plate and in some cases it can cause the inside layer of the armour plate to
spall. During impact, the pressure waves reflect from the back of the armour plate, which
causes tension on its surface. This tension can cause cleaving, chipping and fracturing,
which are often referred to as spalling. (Rosenberg & Dekel 2012, s. 39–42.)
Kuva 4. Spalling (Rosenberg & Dekel, 2012 s. 40).
The spalling of an aluminous plate caused by a glass ball can be seen in figure 5. Accord-
ing to Rosenberg & Dekel (2012, pp. 39–42) the spall strength of a material can be esti-
mated with
𝜎𝑠𝑝𝑎𝑙𝑙 =2𝜎𝜀
3[2 + ln (
𝐸
3(1−𝛾)𝜎𝜀)], (2.7)
where γ is the Poisson’s constant of the material and E its Young’s modulus. The formula
gives values close to real life empirical values according to Rosenberg & Dekel.
6
The ability of a material to resist penetration can be estimated through multiple ways. The
most common way is with the ballistic limit velocity. Ballistic limit velocity is the veloc-
ity of a projectile that it needs to penetrate an armour plate of certain thickness. In order
to define the ballistic limit velocity, the material parameters of both the projectile and
armour are needed. According to Rosenberg & Dekel (2012, pp. 117–120) ballistic limit
velocity can be calculated with
𝑣𝑏𝑙 = √2𝑡𝜎𝑟
𝜌𝑝𝐿, (2.8)
where L is the length of the projectile and σr is penetration strength. Penetration strength
characterises the armour’s ability to resist penetration. Penetration strength is dependent
on caliber thickness and it can be divided in to three different forms:
𝜎𝑟 =
{
(2
3+ 4𝐾)𝜎𝜀 , 𝐾 ≤
1
3
2𝜎𝜀 ,1
3< 𝐾 ≤ 1
(2 + 0,8 ln𝐾)𝜎𝜀 , 𝐾 > 1
. (2.9)
Rosenberg & Dekel (2012, s. 120) note, that the values for ballistic limit velocity calcu-
lated through the formula 2.8 differ from real life empirical values by ±2.5 %.
The suitability of a material as an armouring material can be measured by the ballistic
performance index BPI created by Srivathsa and Ramakrishnan (1997). BPI is a dimen-
sionless number and it can be used to compare different materials and different impact
velocities. BPI can be calculated with
Φ = [𝛼𝐼
2(1+𝑘𝑏)2 + 𝛼𝐼𝐼
(1+𝑘𝑒)2𝑘𝛾
2
2𝑘𝑗2 +
1
𝑘𝑗(1 +
1
𝑘𝑝) +
1
2𝑘𝑝2 +
1
2(1 +
1
𝑘𝑝)2
]. (2.10)
In the equation the first two terms describe the material’s elastic behaviour, the next two
its plastic behaviour and the last term includes the kinetic energy. Explaining the param-
eters ki is not necessary for this work and equation 2.10 is defined more in-depth in ap-
pendix A. However, the index is dependent on the mechanical properties of the material
and the impact velocity, so the index can be defined as a function in the form of
Φ = Φ(𝐸, 𝜌, 𝜎𝜀 , 𝜎𝑚, 𝜀𝑟 , 𝑣), (2.11)
where εr is fracture strain. Based on the BPI, Srivathsa & Ramakrishnan (1999) created
ballistic performance maps. The maps were created as a function of yield strength and
strain hardening rate. The strain hardening rate for a material can be derived from its other
material values and it can calculated with the following equation:
𝐻 =𝜎𝑚(1+𝜀𝑟)−𝜎𝜀
𝜀𝑟. (2.12)
7
Appendix B includes examples of ballistic performance maps for aluminium and steel
with different impact velocities. It is visible from the maps that aluminium suits better for
armour based on its ballistic properties when compared to steel of equal yield strength
and strain hardening rate. It is important to notice however, that the maps only indicate
the ballistic suitability of the material and they do not take into account the geometries of
the armour or the projectile (such as caliber, thickness or angle).
2.1 Shell types
AP-shells are the simplest type of ammunition used to penetrate armour. In addition to
these, there have been many variations that have had the aim to improve some of the
deficiencies of AP-shells. Different ammunition types are represented with different letter
combinations.
During World War II, it was noticed that projectiles often shattered as they hit armour
plates, especially in situations where they met face hardened armour plates. Face hard-
ened armour is inspected more closely in chapter 2.2. A face hardened plate has a bigger
hardness value than an RHA plate. This leads to a higher shatter probability in projectiles.
Due to this, a cap was added to the nose of AP-shells. The cap was made of softer material
than the rest of the projectile. The shells were called APC-shells (Armour Piercing
Capped). The aim for the soft cap was to absorb some of the impact energy by deforming
on impact. This reduced the strain on the actual penetrating part of the projectile, reducing
the probability of shattering. The soft cap is slightly blunter than the penetrating part of
the projectile, which leads to more rapid loss of velocity due to poorer aerodynamics.
Some APC-shells have an edge-like collar. The added cap reduces the penetration ability
of a projectile by roughly 14%. However, the shape of the nose helps against sloped ar-
mour, which will be inspected closer in chapter 2.2. (Bird & Livingston, 2001, pp. 16, 21
and 58.)
The loss of aerodynamics due to the soft cap was reduced by adding a ballistic cap on top
of the existing soft cap. This ammunition type was called APCBC-shells (Armour Pierc-
ing Capped Ballistic Capped). The ballistic cap can also be used to improve the aerody-
namics of an AP-shell, which then becomes an APBC-shell (Armour Piercing Ballistic
Capped). A ballistic cap also reduces the penetration capability of a projectile (Bird &
Livingston, 2001).
Often an AP-shell’s effectiveness is improved by adding explosives in to the projectile.
This ammunition type is called APHE (Armour Piercing High Explosive). The fuse is
connected to the nose of the shell so that at the moment of penetration the fuse sets of the
explosives and shatters the projectile on the other side of the armour plate. The main
purpose of the added explosive is to maximise the damage done to the target protected by
8
the armour plate, such as a vehicle or its crew. According to the US Army Material Com-
mand (1963, p. 6-4) the maximum proportional volume of the high explosive part is 5 %
of the total volume of the projectile. Increasing the size of the explosive filler further
weakens the structure of the projectile too much, causing it to shatter more easily. Bird &
Livingston (2001, p. 58) estimated that an explosive filler reduces the penetration capa-
bility of a projectile by 13 %. The explosive filler can also be added to APCBC-, APBC-
and APC-shells.
As armour grew thicker during World War II, the need for better AP-shells increased.
The increase in the penetration capability of traditional AP-shells could not be achieved
by increasing their velocity, as steel had the tendency to shatter at large velocities (a so
called shatter velocity). The problem was solved by adding a heavy metal core into AP-
shells. The high hardness and strength of the core allowed bigger impact velocities. In
addition, the stronger material offered a better penetration capability even at normal ve-
locities. The shells were called APCR-shells (Armour Piercing Composite Rigid). The
core of the APCR-shells is usually manufactured out of tungsten carbide. The hardness
value of APCR-shells is usually 760–800 BHN (Engineering Design Handbook - Ele-
ments of Terminal Ballistics, 1963, pp. 6-7–6-8). The velocity of APCR-shells is roughly
1200 m/s but their poor ballistic properties mean that they lose their velocity faster than
traditional AP-shells. APCR-shells are often shorter than their AP counterparts, resulting
in less mass. In American literature APCR-shells are often referred to as HVAP (High
Velocity Armour Piercing).
The most common shell types are illustrated in appendix H. Brown color denotes the base
of the shell, blue is the ballistic cap, grey the soft cap and green the heavy metal core. The
explosive filler is marked with red and the fuse with black.
2.2 Armour types
In chapter 2, the impact against a homogeneous vertical plate was discussed. By changing
the parameters of the armour plate, its ability to resist penetration can be improved, or in
some cases, worsened.
The most common way of improving penetration resistance is by changing the angle of
the armour plate, or sloping the armour. The slope causes the effective thickness of the
armour plate to increase so that the projectile has to travel a longer distance. An impact
against a sloped plate is illustrated in figure 5. The figure also illustrates the impact forces
affecting the projectile.
9
Kuva 5. Penetrating sloped armour.
The forces F1 and F2 that resist penetration create an asymmetrical pressure field against
the projectile. This asymmetry causes the projectile to be tilted away from the armour
plate. This causes the effective thickness of the plate to be more than just the geometrical
thickness. In the case of an APC-shell, the force F1 is smaller than force F2 due to the
blunt nose. This causes the shell to tilt slightly towards the normal which leads to a smaller
sloping effect than with AP-shells (Bird & Livingston, 2001, p. 16). The same effect can
be achieved even if the nose of the APC-shells isn’t blunt. At the moment of impact, the
softer metal spreads against the surface of the plate and “sticks” to it. According to Zener
& Peterson (1943) the projectile also tilts towards the normal when the penetration mech-
anism is punching. The creation of the plug reduces the force F1 which then creates a
pressure field that pushes the nose of the projectile downwards.
The angle of oblique φ is defined as the angle between the movement vector of the pro-
jectile and the normal of the armour plate. The effective thickness of the plate according
to its geometry would be
𝑡𝑒𝑓𝑓 =𝑡
cos𝜑. (2.2.1)
Like mentioned earlier, the effective thickness is in reality more than just the trigonomet-
rical result. Bird & Livingston (2001, p. 118) defined a slope coefficient that can be used
to calculate the true thickness of the armour plate. The slope coefficient can be calculated
with the equation
𝑠𝑙𝑜𝑝𝑒 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 = 𝑎𝐾𝑏, (2.2.2)
where a and b are empirical constants that depend both on the angle of oblique and the
type of the shells. Values for the constants can be found in the table of appendix C. The
true thickness of the armour can be calculated by multiplying the nominal thickness of
10
the armour with the coefficient. For example, let’s look at a situation where a 76 mm AP-
projectile impacts a 100 mm thick armour plate at an angle of 30°. This gives us a slope
coefficient of roughly 1.29 (K ≈ 1.316, a = 1.2195 and b = 0.19702). In this case, a 100
mm thick plate at an angle of 30° is equal to 129 mm vertical plate. By calculating with
just trigonometry, the effective thickness would be about 115 mm. The 30° angle of im-
pact increases the thickness of the armour plate by 14 % against the chosen AP-shell when
compared to the trigonometrical effective thickness.
Kuva 6. Slope coefficient as a function of the angle of impact when K = 1.3 and
0.4.
By inspecting the slope coefficient as a function of the angle of impact, the efficiency of
different shell types against sloped armour can be judged. From figure 6, it can be seen
that at small angles the type of the shell only has a miniscule impact on armour thickness.
When the caliber thickness is 1.3 and as the impact angle increases to 55°, the shells with
the soft cap (APCBC and APC) gain a superior advantage against sloped armour when
comparing to other shell types. As caliber thickness decreases, the difference between
shells types at large angles decreases as well. At a caliber thickness value of 0.4, it can be
seen that APBC- and AP-shells work better against sloped armour than APCPC- and
APC-shells regardless of the angle of impact. The limit value for this change, when the
projectiles with the soft cap perform better against sloped armour than the ones without,
can be estimated to be K ≈ 0.45.
According to Bird & Livingston (2001, p. 119) the effect that sloped armour has against
APCR-shells doesn’t depend on the caliber thickness but only on the angle of impact and
the caliber of the shells. Bird & Livingston defined the slope efficients for 90 mm ja 76
mm APCR-shells with the equations
0
2
4
6
8
10
12
14
10 15 20 25 30 35 40 45 50 55 60 65 70
Slo
pe
Co
effi
cien
t
Angle of Impact (°)
Slope Coefficient (K = 1.3)
APCBC/APC APBC AP
0
0,5
1
1,5
2
2,5
3
3,5
4
10 15 20 25 30 35 40 45 50 55 60 65 70
Slo
pe
Co
effi
cien
t
Angle of Impact (°)
Slope Coefficient (K = 0.4)
APCBC/APC APBC AP
11
𝑠𝑙𝑜𝑝𝑒 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡90 = {2,71828(𝜑
1,75∗0,000662), 𝑤ℎ𝑒𝑛 0° < 𝜑 ≤ 30°
0,9043 ∗ 2,71828(𝜑2,2∗0,001987), 𝑤ℎ𝑒𝑛 𝜑 > 30°
(2.2.3)
𝑠𝑙𝑜𝑝𝑒 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡76 = {2,71828(𝜑
2,2∗0,0001727), 𝑤ℎ𝑒𝑛 0° < 𝜑 ≤ 25°
0,7277 ∗ 2,71828(𝜑1,5∗0,003787), 𝑤ℎ𝑒𝑛 𝜑 > 25°
.(2.2.4)
The effect of caliber on the slope coefficient can be studied by plotting the slope coeffi-
cient values of the APCR-shells. Figure 7 has the plots of the slope coefficients both
above calibers as a function of angle of impact.
Kuva 7. Slope coefficients for 76 mm and 90 mm APCR-shells as a function of the
angle if impact.
It can be seen from the figure that at small angles of impact, the effect of the slope is
slightly smaller against the larger caliber. As the angle of impact rises to 55°, the smaller
caliber has the advantage of the larger one. Assuming that all APCR-shells follow the
form of the plots in figure 7, it can be stated that small caliber APCR-shells have an
advantage over large caliber APCR-shells when the angle of impact is larger than 55°.
The hardness of an armour plate can be improved greatly by face hardening it. Face hard-
ened armour plates are denoted with FHA. FHA-plates have a harder surface layer that
has a hardness value of 450–650 BHN. The depth of the hard layer is about 5–10 % of
the thickness of the whole plate. (Bird & Livingston, 2001, pp. 21–22). The aim of the
face hardening is to shatter projectiles that impact the plate and thus prevent penetration.
Face hardening increases efficiency against small caliber (K > 1) AP-shells. If the AP-
shell has a soft cap, the face hardened layer makes the armour weaker. Part of the energy
0
1
2
3
4
5
6
7
8
9
10
10 15 20 25 30 35 40 45 50 55 60 65 70
Slo
pe
Co
effi
cien
t
Angle of Impact (°)
Slope Coefficient
APCR 76mm APCR 90mm
12
is absorbed by the soft cap which prevents the projectile from shattering. The armour
plate however can’t absorb large amounts of energy due to the surface layer of increased
hardness which leads to brittle behaviour of the armour plate. According to Bird & Liv-
ingston (2001, p. 24) the effect of face hardening on an armour plate can be estimated
with the equation
𝐵𝐻𝑁 − 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 = 0,01 ∗ 977,07 ∗ 𝐷0,06111 ∗ 𝐾0,2821 ∗ 𝐵𝐻𝑁−0,4363, (2.2.5)
where BHN is the Brinell hardness of the armour plate and D is the diameter of the shells
in millimetres. By multiplying the thickness of the FHA-plate with the BHN-coefficient,
the thickness of an equivalent RHA-plate can be evaluated. Figure 8 illustrates the values
for the BHN-coefficient as a function of caliber thickness for three different calibers.
Kuva 8. Values for the BHN-coefficient as a function of caliber thickness for differ-
ent calibers, when the hardness value of the plate is 460 BHN.
It can be seen from the figure that with the chosen hardness value (460 BHN) and shell
calibers, FHA-plates are stronger than equal RHA-plates once caliber thickness increases
beyond the value of 1.5. According to Bird & Livingston (2001, p. 23) the slope coeffi-
cient for an FHA-plate can be calculated the same way as the coefficients for an RHA-
plate. FHA-plates are weak against APCR-shells. An APCR-shell will penetrate roughly
1.1–1.3 thicker FHA-plate than RHA-plate (Bird & Livingston, 2001, p. 24). Rosenberg
& Dekel (2012, p. 261) noticed however that even a relatively thin face hardened plate
(K < 0.3) is capable of shattering an APCR-shell during penetration. This leads to a situ-
ation where the post-penetration damage are less than in a regular penetration.
Even though armour plates are usually rolled, they can also be cast. Cast armour plates
are denoted with CHA (Cast Homogeneous Armor). The hardness value for CHA-plates
is usually the same as for RHA (220–330 BHN). When making rolled armour plates, the
manufacturing process removes impurities and flaws from the material and the grain
0
0,2
0,4
0,6
0,8
1
1,2
1,4
0,1 0,5 0,9 1,3 1,7 2,1 2,5 2,9
BH
N-c
oef
fici
ent
Caliber Thickness
BHN-coefficient (460 BHN)
40 mm 76 mm 122 mm
13
structure of the material is made stronger. If the plates are manufactured through casting,
this doesn’t happen which leads to cast plates being weaker than rolled ones. As a general
rule, cast armour is roughly 15 % weaker than rolled armour. When caliber thickness is
extremely big (K > 2.5), the differences between cast and rolled armour are minimal. Bird
& Livingston (2001, p. 26) created an equation to estimate the effect of casting. The equa-
tion is in the form of
𝑐𝑎𝑠𝑡 𝑐𝑜𝑒𝑓𝑓𝑐𝑖𝑒𝑛𝑡 = 0,8063 + 0,001238𝑡 − 0,0002628𝐷 + 0,02706𝐾, (2.2.6)
where t and D are the thickness of the armour plate and the diameter of the shell in milli-
metres. The maximum value for the cast coefficient is 1, which means that cast armour is
never stronger than rolled armour. The cast coefficient is used like the BHN- or sloped
coefficient. The sloped coefficients for cast armour is the same as for rolled armour.
The ability for armour to resist penetration also depends on its quality. During production,
several flaws can form in the armour plates. These flaws include impurities, cracks and
flaws in the grain structure of the material. The effect of a flaw is directly proportional to
the caliber thickness. As caliber thickness decreases, the effect of the flaw increases (Bird
& Livingston, 2001, pp. 28–29). Any damage inflicted on the armour also decreases its
ability to resist penetration. Usually the damage is caused by projectiles that haven’t pen-
etrated the armour. The non-penetrating hits often create cracks on the armour’s surface.
In addition to this, they enlarge the existing cracks of the armour through fatigue.
By having two armour plates separate from each other, spaced armour is created. Usually,
spaced armour is used to protect from shaped charges (HEAT) but they can also bring
protection against traditional armour piercing shells if certain conditions are met. Figure
9 illustrates the principle of spaced armour.
14
Kuva 9. Spaced Armour..
According to Bird & Livingston (2001, p. 36) a single plate that is equivalent to a certain
spaced armour combination can be calculated with Okun’s equation
𝑡𝑒𝑓𝑓 = [(1,15𝑡1)1,4 + 𝐴1,4𝑡2
1,4]1
1,4, (2.2.6)
where t1 and t2 are the thickness of the primary and secondary plates and A is a constant
that is dependent on the type of shell and armour. A is 1 if the shell type is APC, APBC
or APCBC. If the shell is an AP-shell, A is 1.05. If the primary plate is face hardened and
the secondary plate is homogeneous, A is 1.10. By looking at the equation, it can be seen
that regardless of the value of A, the primary plate has a larger impact on the effective
thickness of the plate. If the impacts against the primary and secondary plates are not
perpendicular, the thickness of the single plate can be estimated with the slope coeffi-
cients of equation 2.2.2. Once the angle have been taken into account, the effective thick-
ness can be calculated with equation 2.2.6.
Figure 10 illustrates the contour curves of equation 2.2.6 as a function of plate thicknesses
in all three cases. The figure also includes the contours of the unified thickness of the
plates (𝑡1 + 𝑡2). It can be seen from the figure that spaced armour is slightly better than a
single plate if the primary plate is noticeably thicker than the secondary plate. For exam-
ple, a primary plate of 36.0 mm and a secondary plate of 2.0 mm, would equal to a single
plate of 41.8 mm, 40.2 mm or 40.3 mm depending on the value of A. In the case of both
the AP-shells and the face hardened primary plate, the effective thickness can be made
stronger than the unified thickness when the primary plate is noticeably thinner than the
secondary plate. In all the cases where the spaced armour combination has better effective
thickness than a single plate of their unified thickness, the difference in these thicknesses
15
is very small. If one the plates is very thin, it is possible that the fracture mechanism is
dishing or punching. In these cases the plate resists the penetration worse than predicted,
as Okun’s equation assumes that both plates are perforated trough ductile hole enlarge-
ment. It can be stated that spaced armour is almost always worse that a single plate that
has the same thickness as the unified thickness of the primary and secondary plates.
The distance between the plates doesn’t affect the resistance against penetration. The abil-
ity for spaced armour to resist kinetic penetrator is based on the reducing its kinetic energy
during the penetration of the primary plate. In addition, the deforming of the nose of the
projectile also weakens its ability to penetrate the second plate. The projectile might also
change its flight direction or tumble or roll after penetration.
16
Kuva 10. Single plates equivalent to spaced armour in different impact cases.
Two plates that are attached to each other are called layered armour. A layered armour
resists penetration less than a single plate of equal thickness. Layered armour is often a
temporary solution or a field modification (so called appliqué armour) that is used to
17
strengthen an already existing armour plate. Bird & Livingston (2001, pp. 38–39) define
three methods of calculating the effective thickness of layered armour. The first method
is based on tests made by the US Navy and the statistical analysis of their results. The
effective thickness of layered armour is then
𝑡𝑒𝑓𝑓,𝑠𝑡𝑎𝑡 = (𝑡1 + 𝑡2) [0,3129 (𝑡1
𝑡2)0,02527
∗ (𝑚𝑎𝑥(𝑡1, 𝑡2))0,2439
]. (2.2.7)
The equation has a both minimum and maximum value. The minimum and maximum
values for the equation are
𝑡𝑒𝑓𝑓,𝑠𝑡𝑎𝑡,𝑚𝑖𝑛 = 0,3 ∗ min(𝑡1, 𝑡2) + max (𝑡1, 𝑡2) (2.2.8)
𝑡𝑒𝑓𝑓,𝑠𝑡𝑎𝑡,𝑚𝑎𝑥 = 0,96(𝑡1 + 𝑡2). (2.2.9)
In the equations the function max(x1,x2) evaluates as the larger number inside the paren-
theses and min(x1,x2) evaluates as the smaller of the values. The second way of calculating
the effective is through the navy rule of thumb, which is
𝑡𝑒𝑓𝑓,𝑛𝑎𝑣𝑦 = 0,7𝑡1 + 𝑡2. (2.2.10)
The third way is to use Nathan Okun’s equation. Okun’s layered armour equation is based
on the average of the spaced armour equation and the unified thickness of the plates.
Okun’s layered armour equation is in the form of
𝑡𝑒𝑓𝑓.𝑂𝑘𝑢𝑛 = 0,5 ∗ [(𝑡1 + 𝑡2) + (𝑡11,4 + 𝑡2
1,4)1
1,4]. (2.2.11)
It is important to notice that Okun’s equation doesn’t take into account which of the plates
is thicker. For example, a 40 mm primary plate and a 20 mm secondary plate get an ef-
fective thickness of 47 mm through the statistical method, 48 mm through the navy rule
of thumb and 55 mm through Okun’s equation. The same plates in the reverse order would
get thicknesses of 45 mm, 54 mm and 55 mm respectively. The exact effective thickness
is difficult to evaluate but it can be stated that the effective thickness is between the uni-
fied thickness of the plates and the thickness of the thicker plate.
18
Kuva 11. Comparison of layered and spaced armour.
Figure 11 has a comparison between the effective thicknesses of spaced armour (round
lines) and layered armour (polylines). It can be noted that in the case of thin plates, spaced
armour is more effective than layered armour. With thick plates, the situation is opposite.
Layered armour is better against traditional kinetic energy penetrators when the desired
effective thickness is more than 120 mm.
19
3. MATHEMATICAL PREDICTION MODEL
Often when studying different types of ammunition, the main point of interest is finding
out how much a certain projectile can penetrate. Most current models are based on statis-
tical analysis and require a reference case to be used. One of the most common ways to
estimate a projectile penetration capability is through DeMarre’s equation. DeMarre
equation can be used to estimate penetration against RHA-plates if the penetration for a
projectile of the same type is known. DeMarre’s equation can be written as
𝑃 = 𝑃𝑟𝑒𝑓𝑓 (𝑣
𝑣𝑟𝑒𝑓𝑓)1.4283
(𝐷
𝐷𝑟𝑒𝑓𝑓)1.0714
(𝑚
𝐷3𝑚𝑟𝑒𝑓𝑓
𝐷𝑟𝑒𝑓𝑓3
)
0.7143
, (3.1)
where P is the penetration of the projectile. The index reff indicates the values of a known
projectile. Penetration against FHA-plates can be estimated with the use of the Krupp
equation, which is based on the DeMarre equation (Bird & Livingston, 2001, p. 78). The
equation requires that the reference values are also against face hardened armour. By us-
ing the denotation 𝐼 =𝑚
𝐷3, we can write the equations of DeMarre and Krupp ass
𝑃𝑅𝐻𝐴 = 𝑃𝑟𝑒𝑓𝑓 (𝑣
𝑣𝑟𝑒𝑓𝑓)1.4283
(𝐷
𝐷𝑟𝑒𝑓𝑓)1.0714
(𝐼
𝐼𝑟𝑒𝑓𝑓)0.7143
(3.2)
𝑃𝐹𝐻𝐴 = 𝑃𝑟𝑒𝑓𝑓 (𝑣
𝑣𝑟𝑒𝑓𝑓)1.250
(𝐷
𝐷𝑟𝑒𝑓𝑓)1.250
(𝐼
𝐼𝑟𝑒𝑓𝑓)0.625
. (3.3)
There are no equations to approximate general penetration values. Based on the theory in
chapter 2, it can be stated that penetration depends on the kinetic energy of the projectile.
A bigger kinetic energy gives a better penetration capability in an ideal situation, where
the projectile doesn’t shatter and both the projectile and the armour are flawless. The
energy of the projectile, and thus its mass, is concentrated on a small area. Based on this,
the caliber of the projectile affects the penetration as well. The smaller the area that the
energy is concentrated on, the better the penetration. From this, we can assume that pen-
etration is in the form of 𝑃 = 𝑃 (𝐾𝐸
𝐷), where KE is the projectile’s kinetic energy. By
using the statistics offered by Bird & Livingston (2001), Koll (2009), Honner (1999),
Boyd (2015) and Ankerstjern (2015), a series of property tables can be created for differ-
ent projectiles. Appendix D has the properties for different AP-shells. The properties in-
clude the cannon that the projectile was fired with and the diameter, velocity and pene-
tration of the shell at different ranges. The penetration is measured against a vertical
20
RHA-plate. The penetration value at 100 m can be assumed to be the maximum penetra-
tion of the said projectile. A kinetic energy coefficient was calculated for all of the shells
using the equation
𝑘𝑖𝑛𝑒𝑡𝑖𝑐 𝑒𝑛𝑒𝑟𝑔𝑦 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 =𝑚𝑣2
𝐷∗104 (3.4)
The kinetic energy coefficient represents how the kinetic energy is distributed in relation
to the projectile’s diameter.
Kuva 12. Penetration of an AP-shell as a function of the kinetic energy coefficient.
Figure 12 illustrates the penetration values of different projectiles as a function of their
kinetic energy coefficient. It can be seen from the figure that as the kinetic energy coef-
ficient increases, so does the penetration. By fitting a curve into the data points, the pen-
etration of an AP-shell can be estimated with an equation of
𝑃0 = 62.804138 (𝑚𝑣2
𝐷∗104)0.477171
, (3.5)
where the unit of mass m is kilogrammes, unit of velocity v is m/s and the unit of diameter
D is mm. The values given by the equation differ on average by 8,29 % from the real life
values, which makes the equation suitable for preliminary evaluation.
The projectile’s penetration decreases as a function of distance, as the projectile slows
down due to drag, thus reducing its kinetic energy. Figure 13 has a comparison between
the relational penetration values 𝑃𝑟
𝑃0 of different British 57 mm shells of different velocities
as a function of distance r.
0
50
100
150
200
250
0 2 4 6 8 10 12 14
Pen
etra
tio
n (
mm
)
Kinetic Energy Coefficient (J/mm)
Penetration of an AP-shell
21
Kuva 13. Relational penetration of British AP-shells of different velocities as a func-
tion of distance.
It can be seen from the figure that the impact of the velocity to the rate at which the
projectile loses its penetration is miniscule. The same phenomenon can be seen with 37
mm American AP-shells and 87.6 mm British AP-shells. The effect that the velocity has
on relational penetration is at most 2 %-units. Based on this it can be assumed that the
relational penetration is dependent only on the caliber and mass of the projectile.
Kuva 14. Change in relational penetration with different calibers and masses.
By studying AP-shells with the same velocity but different mass and caliber, we get figure
14. Based on chapters 2 and 2.1 it can be stated that the relational penetration is dependent
on the ballistic coefficient BC. Based on the curves of figure 14, the relational penetration
is in the form of 𝑎𝑒𝑏𝑟. Table 1 has ballistic coefficients of the projectiles of figure 14 and
0
0,2
0,4
0,6
0,8
1
1,2
0 1000 2000 3000 4000
Pen
etra
tio
n/o
rigi
nal
pen
etra
tio
n
Distance (m)
Change in relational penetration
6 pounder L45(822,96 m/s)
6 pounder L45(862,58 m/s)
6 pounder L52(899,16 m/s)
0
0,2
0,4
0,6
0,8
1
1,2
0 500 1000 1500 2000 2500 3000 3500
Pen
etra
tio
n/o
rigi
nal
pen
etra
tio
n
Distance r (m)
Change in relational penetration(v = 792,48 m/s)
37 mm Gun M3(American)
2 pounder (British)
3-inch Gun M5(American)
85L52 52-K (Soviet)
122L43 D-25T (Soviet)
22
the values for the constants a and b that fit their curves. When calculating the ballistic
coefficients, it was assumed that all projectiles have the same coefficient of form (i = 1).
Taulukko 1. Ballistic coefficients and constants of different AP-shells.
Cannon Caliber
(mm)
Mass (kg) BC m/D2
(kg/mm2)
a b
37 mm
Gun M3
37 0.87 0.000636 1.066213 -0.000643
2 pounder 40 1.08 0.000675 1.058075 -0.000633
3-inch
Gun M5
76.2 6.8 0.001171 1.040843 -0.000401
85L52
52-K
85 9.2 0.001273 1.031434 -0.000314
122L43
D-25T
122 25 0.001680 1.024319 -0.000236
Based on figure 14 and table 1, a projectile loses penetration slower when it has a bigger
ballistic coefficient. By making a similar analysis on all the shells of appendix D and
fitting the constants a and b as a function of the ballistic coefficient, we get equations
𝑎 = 0.808933𝐵𝐶−0.037164 (3.6)
𝑏 = 0.000356 ln(𝐵𝐶) + 0.002019 (3.7)
As we know that relational penetration is in the form of 𝑎𝑒𝑏𝑟, we get
𝑃𝑟
𝑃0= 0.808933𝐵𝐶−0.037164𝑒(0.000356 ln(𝐵𝐶)+0.002019)𝑟 (3.8)
Equation 3.8 can be used in situations where the distance r is more than 100 m. Otherwise
it can be assumed that the penetration is already at its maximum. Table 2 has a comparison
between equation 3.8 and real life values with r = 2000 m.
23
Taulukko 2. Functionality of equation 3.8, when r = 2000 m.
Gun BC (kg/mm2) P0 (mm) Predicted
(mm)
Real value
(mm)
Error
2 cm KwK
38 L/55
0.00037 45 10 11 9.1 %
5 cm KwK
39 L/60
0.000824 100 38 33 15 %
57L73 ZiS-
2
0.000982… 134 57 54 5.6 %
17 pounder 0.001326… 200 100 105 4.8 %
152L28
ML-20
0.002112… 165 120 111 8.1 %
The most probable reason for large errors in equation 3.8 is the assumption that all pro-
jectiles have a coefficient of form of 1. In reality, the projectiles have different shapes and
if these were taken into account, the results would be more accurate.
By combining the equations 3.8 and 3.5, we get an equation that can be used to estimate
the penetration value of an AP-shell as a function of its caliber, mass, velocity and dis-
tance. The equation can be written as
𝑃𝐴𝑃 = 0.808933𝐵𝐶−0.037164𝑒(0.000356 ln(𝐵𝐶)+0.002019)𝑟 ∗ 62.804138(
𝑚𝑣2
𝐷∗104)0.477171
, (3.9)
which then becomes
𝑃𝐴𝑃 = 50.804340 (𝑚𝑣2
𝐷∗104)0.477171
𝐵𝐶−0.037164𝑒(0.000356 ln(𝐵𝐶)+0.002019)𝑟, (3.10)
where unit for caliber D is mm, the unit for mass m is kg, the unit for velocity v is m/s
and the unit for distance r is m and it has a minimum value of 100 m.
Appendix E has the properties for APC- and APBC-shells. The same property tables for
APCBC- and APCR-shells can be found in appendices F and G respectively. By perform-
ing the same analysis for these shell types, their penetration behaviour can be predicted
as well.
For APC-shells we get the equations:
24
𝑃0 = 48.844680 (𝑚𝑣2
𝐷∗104)0.627189
(3.11)
𝑃𝑟
𝑃0= 0.817308𝐵𝐶−0.035818𝑒(0.000349 ln(𝐵𝐶)+0.001964)𝑟 (3.12)
𝑃𝐴𝑃𝐶 = 39.921148 (𝑚𝑣2
𝐷∗104)0.627189
𝐵𝐶−0.035818𝑒(0.000349 ln(𝐵𝐶)+0.001964)𝑟. (3.13)
For APBC-shells we get the equations:
𝑃0 = 42.980939 (𝑚𝑣2
𝐷∗104)0.596242
(3.14)
𝑃𝑟
𝑃0= 0.703895𝐵𝐶−0.055240𝑒(0.000267 ln(𝐵𝐶)+0.001522)𝑟 (3.15)
𝑃𝐴𝑃𝐵𝐶 = 30.254068 (𝑚𝑣2
𝐷∗104)0.596242
𝐵𝐶−0.055240𝑒(0.000267 ln(𝐵𝐶)+0.001522)𝑟. (3.16)
When looking at APC- and APBC-shells, it should be noticed that their property tables
only include a few different projectiles. This may lead to great difference between the
behaviour of these equations and their real life counterparts.
For APCBC-shells we get the equations:
𝑃0 = 47.338655 (𝑚𝑣2
𝐷∗104)0.620892
(3.17)
𝑃𝑟
𝑃0= 0.908771𝐵𝐶−0.017257𝑒(0.000189 ln(𝐵𝐶)+0.001061)𝑟 (3.18)
𝑃𝐴𝑃𝐶𝐵𝐶 = 43.019997 (𝑚𝑣2
𝐷∗104)0.620892
𝐵𝐶−0.017257𝑒(0.000189 ln(𝐵𝐶)+0.001061)𝑟. (3.19)
And for APCR-shells we get the equations:
𝑃0 = 88.951277 (𝑚𝑣2
𝐷∗104)0.482321
(3.20)
𝑃𝑟
𝑃0= 0.533666𝐵𝐶−0.092555𝑒(0.000966 ln(𝐵𝐶)+0.006577)𝑟 (3.21)
𝑃𝐴𝑃𝐶𝑅 = 47.470272 (𝑚𝑣2
𝐷∗104)0.482321
𝐵𝐶−0.092555𝑒(0.000966 ln(𝐵𝐶)+0.006577)𝑟. (3.22)
Figure 15 includes the penetration values and relational penetrations of different shell
types based on the previous equations. The shell was given the following parameters: D
= 75 mm, m = 6.5 kg, v = 700 m/s. For the APCR-shell the mass was 4.0 kg and the
velocity 1000 m/s.
25
Kuva 15. Penetration and relational penetration for different shells types as a func-
tion of distance.
From the relational penetration of figure 15, it can be seen that the models take into ac-
count the faster penetration drop of APCR-shells. APC-shells should lose penetration
faster than AP-shells, which isn’t visible in the derived models. This could be due to the
small amount of data available for the APC-shells, as mentioned earlier. Shells with a
ballistic cap lose their penetration slower than other shells types, which is consistent with
the observations in chapter 2.1.
By looking at the absolute penetration values, it can be seen that APCR-shells have the
highest penetration value. Shells that have a ballistic cap, a soft cap or both have a worse
penetration that regular AP-shells. The models are consistent with the observations in
chapter 2.1 when it comes to absolute maximum penetration.
26
Taulukko 3. 5 cm KwK 39 L/60 (BC = 8.24 * 10-4 kg/mm2 for AP and APCBC, 3.7*10-4
kg/mm2 for APCR), predicted penetration and true values against different armour con-
figurations.
Projectile
Armour
AP (mm) Real AP
(mm)
APCBC
(mm)
APCR
(mm)
Real APCR
(mm)
RHA (100 m) 104 100 91 137 149
RHA (2000 m) 40 33 53 18 32
FHA (460 BHN,
100 m)
100 97 90 164 179
CHA (100 m) 106 103 95 137 149
Sloped RHA
(φ = 30°, 100 m)
78 76 70 94 105
Spaced Armour
(100 m)
60 + 55 60 + 50 60 + 40 60 + 97 60 + 111
Layered Armour
(100 m)
60 + 62 60 + 58 60 + 46 60 + 88 60 + 98
By using the models derived in chapter 3 and the information from chapter 2.2, the pen-
etration capability of a projectile can be studeied. Table 3 has different penetration values
for a German 5 cm KwK 39 L/60 cannon and compares the theoretical values to real life
values.
The true values against RHA-plates in table 3 come from Bird & Livingston (2001). The
predictions against RHA-plates were done by using equations 3.10, 3.19 and 3.22. The
FHA-plates for AP- APCBC-shells were calculated through equation 2.2.5. In this spe-
cific situation (K ≈ 2) it can be seen that the FHA-plate is better against the AP-shell than
an RHA-plate. In the case of the APCBC-shell the FHA-plate is better as well even though
the said shell type is designed to be better against face hardened armour. In both of the
cases, this is due to the large caliber thickness. In the case of the APCR-shells, the thick-
ness of the FHA-plate is 1.2 times of the RHA-plate, as mentioned in chapter 2.2. As
mentioned in chapter 2.2, cast armour is worse than rolled armour. The exception in the
table is the APCR-shells, for which the cast and rolled armour are equal. This is due to
27
the large caliber thickness. The values for CHA-plates were calculated with equation
2.2.6.
Against a sloped armour of 30°, all shells were roughly 15 % weaker. The sloped armour
was calculated using equation 2.2.2. For the APCR-shell the coefficient was estimated
through the values of the 76 mm and 90 mm shells. Spaced armour was calculated with
the equation 2.2.6. In the case of the APCR-shell, it was assumed that it behaves like
APC-, APBC- and APCBC-shells, meaning that after the primary plate is penetrated, the
projectile hasn’t suffered deformations. The layered armour was calculated with the equa-
tion 2.2.7
28
4. SUMMARY
An armour piercing shell penetrates armour through its kinetic energy. Perforation is af-
fected by the mass, density, velocity, diameter, hardness and sharpness of the projectile
and the hardness, density, ultimate tensile strength and yield strength of the armour. Pen-
etration can be achieved through three different mechanisms: dishing, punching or ductile
hole enlargement. The dominant fracture mechanism depends mainly on the caliber thick-
ness. If the caliber thickness is small, the mechanism is dishing. In dishing, the armour
plate is bent open. Punching requires a blunt impact and a small caliber thickness. During
punching a plug is detached from the armour due to the shear tension of the impact. In
other cases the mechanism is ductile hole enlargement. In ductile hole enlargement the
projectile digs in to the armour causes the material to flow away from the projectile. If
the armour doesn’t penetrate the armour, spalling may occur.
The ability of armour to resist penetration can be measured in different ways, the most
common of which is the ballistic limit velocity. Ballistic limit velocity is the velocity
required for a certain projectile to penetrate an armour plate of certain thickness. The
suitability of armour material can be measured with the ballistic performance index.
Basic shell types can be divided into groups based on their properties. The shell types are
AP – armour piercing
APHE – armour piercing high explosive
APC – armour piercing capped
APBC – armour piercing ballistic capped
APCBC – armour piercing capped ballistic capped
APCR – armour piercing composite rigid.
Different shell types apply for different situations. Shells with high explosives are de-
signed to explode after penetration and maximised the damage to the target behind the
armour. Adding a soft cap reduces the probability of the shattering of the projectile and
improve its efficiency against sloped armour. A ballistic cap increases the aerodynamics
of a projectile. A projectile with a rigid core are an improved version of the traditional
armour piercing shells. These projectiles have their mass concentrated on a smaller area,
which increases their penetration.
Armour plates can be divided into three groups based on the manufacturing method:
rolled homogeneous armour (RHA), face hardened armour (FHA) and cast homogeneous
armour (CHA). RHA-plates are the most common. Face hardened plates have a harder
surface layer. Face hardened plates work well against AP-shells when caliber thickness
29
is big. Cast plates are weaker than RHA- or FHA-plates. When manufacturing armour
plates, any flaws in their structure will weaken them.
If the projectile doesn’t hit the armour perpendicularly, it is a case of sloped armour. The
thickness of a sloped armour plate can be calculated through a slope coefficient. The slope
coefficient is dependent on the shell type, angle of impact and caliber thickness. For
APCR-shells, the caliber thickness doesn’t affect the slope coefficient. By having two
plates separated from each other, spaced armour can be created. If the plates are in contact
with each other, it is called layered armour. In most cases, the combinations of two armour
plates are worse than a single plate of equal unified thickness. When comparing armour
of two combined plates, we notice that spaced armour is better if the plates are thin. In
the case of thick plates, layered armour is better.
The penetration capability of different shell types can be estimated with the following
equations:
𝑃𝐴𝑃 = 50.804340 (𝑚𝑣2
𝐷∗104)0.477171
𝐵𝐶−0.037164𝑒(0.000356 ln(𝐵𝐶)+0.002019)𝑟 (4.1)
𝑃𝐴𝑃𝐶 = 39.921148 (𝑚𝑣2
𝐷∗104)0.627189
𝐵𝐶−0.035818𝑒(0.000349 ln(𝐵𝐶)+0.001964)𝑟 (4.2)
𝑃𝐴𝑃𝐵𝐶 = 30.254068 (𝑚𝑣2
𝐷∗104)0.596242
𝐵𝐶−0.055240𝑒(0.000267 ln(𝐵𝐶)+0.001522)𝑟 (4.3)
𝑃𝐴𝑃𝐶𝐵𝐶 = 43.019997 (𝑚𝑣2
𝐷∗104)0.620892
𝐵𝐶−0.017257𝑒(0.000189 ln(𝐵𝐶)+0.001061)𝑟 (4.4)
𝑃𝐴𝑃𝐶𝑅 = 47.470272 (𝑚𝑣2
𝐷∗104)0.482321
𝐵𝐶−0.092555𝑒(0.000966 ln(𝐵𝐶)+0.006577)𝑟 (4.5)
In the equations, the unit for the mass m is kg, the unit for the velocity v is m/s, the unit
for the diameter D of the shell is mm and the distance r is m. BC is the ballistic coefficient
of the projectile. The efficient can be calculated with equation 2.1, assuming that the co-
efficient of form i = 1. The minimum value for the distance is 100 m. The equation can
be used to calculate a preliminary estimate of a projectile’s penetration. The equation
follow projectile properties (AP- and APCR-shells have the highest penetration, but lose
it the fastest). From the equations, the equations for APC- and APC-shells can be seen as
unreliable. The reason for this was the small amount of data available when creating the
model. If the different ballistic coefficient of the different projectiles were taken into ac-
count during the creation of these models, they would be more accurate.
30
BIBLIOGRAPHY
Bird, L.R. & Livingston, R.D. (2001). World War II Ballistics: Armor and Gunnery. 2nd
ed. Albany, New York, and Woodbridge, Connecticut, U.S.A, Overmatch Press.
British Equipment of the Second World War (2015), David Boyd, web page. Available
(referenced 7.3.2015): http://www.wwiiequipment.com/
Cline, Donna (2002). Exterior Ballistics Explained, Trajectories, Part 3 "Atmosphere".
Lattie Stone Ballistics.
Engineering Design Handbook - Elements of Terminal Ballistics, Parts One and Two:
(AMCP 706-160, 706-161). (1963). U.S. Army Materiel Command.
Guns vs Armour 1939 to 1945 (1999), David Michael Honner, web page. Available (ref-
erenced 7.3.2015): http://amizaur.prv.pl/www.wargamer.org/GvA/index.html
Masket, A.V. (1949). The Measurement of Forces Resisting Armor Penetration. Journal
of Applied Physics 20, 2, pp. 132–140.
Moss G.M., Leeming D.W. & Farrar C.L. (1995). Brassey's Land Warfare Series: Mili-
tary Ballistics. Royal Military College of Science, Shrivenham, UK.
Panzerworld, Christian Ankerstjerne (2015), web page. Available (referenced 7.3.2015):
http://www.panzerworld.com/
Rosenberg, Z. & Dekel, E. (2012). Terminal Ballistics. 14th ed. Springer Berlin Heidel-
berg.
Srivathsa, B. & Ramakrishnan, N. (1997). On the ballistic performance of metallic mate-
rials. Bulletin of Materials Science 20, 1, pp. 111–123.
Srivathsa, B. & Ramakrishnan, N. (1999). Ballistic performance maps for thick metallic
armour. Journal of Materials Processing Technology 96, 1–3, pp. 81–91.
The Russian Ammunition Page (2009), Christian Koll, web page. Available (referenced
7.3.2015): http://russianammo.org/index.html
Thomson, W.T. (1955). An Approximate Theory of Armor Penetration. Journal of Ap-
plied Physics 26, 1, pp. 80–82.
Zener, C. & Peterson, R.E. (1943). Mechanism of Armor Penetration. Watertown, Mas-
sachusetts, Watertown Arsenal. 710/492.
APPENDIX A: BALLISTIC PERFORMANCE INDEX
When inspecting penetration, the armour is divided into two sectors in the direction of
the projectile and into three sector in a radial direction. In sector I the material flows into
radially and in sector II the armour bulges in the direction of movement of the projectile.
The radial sector are divided into the projectile’s diameter i, the plastic region ii and the
elastic region iii. The parameters of equation 2.10 are defined as:
𝑘𝛾 = √1−𝛾
(1−2𝛾)(1+𝛾),
𝑘𝑒 =𝑣𝑟
𝑘𝛾√𝜌
𝐸,
𝑘𝑗 =𝜌𝑣𝑟
2
𝜎𝜀,
𝑘𝑏 = 𝑣𝑟√𝜌
𝐶, 𝑤ℎ𝑒𝑟𝑒 𝐶 =
𝐸
3(1−2𝛾) ,
𝑘𝑝 = 𝑣𝑟√𝜌
𝐸𝑝, 𝑤ℎ𝑒𝑟𝑒 𝐸𝑝 =
𝜎𝑚(1+𝜀𝑟)−𝜎𝜀
𝜀𝑟,
𝛼𝐼 = 1 − 𝛼𝐼𝐼 = 1 − √𝑣⊥
𝑣, 𝑤ℎ𝑒𝑟𝑒 𝑣⊥ =
−𝑘𝛾√𝜌𝐸+√𝑘𝛾2𝐸𝜌+10,4𝜌𝜎𝜀
2𝜌,
𝑣𝑟 =𝑣
1,85,
where γ is the Poisson’s constant of the material, ρ its density, E Young’s modulus, σε
yield strength, σm ultimate tensile strength, εr fracture strain ja v impact velocity. (Shri-
vathsa & Ramakrishnan, 1999.)
APPENDIX B: BALLISTIC PERFORMANCE MAPS
The impact velocity in figures marked with (a) was 400 m/s, and in figures marked with
(b) 800 m/s. (Shrivathsa & Ramakrishnan, 1999.)
APPENDIX C: CONSTANTS OF THE SLOPE EFFICIENT
φ APCBC/APC
a/b
APBC
a/b
AP
a/b
10 1.0243/0.0225 1.039/0.01555 0.98297/0.0637
15 1.0532/0.0327 1.055/0.02315 1.00066/0.0969
20 1.1039/0.0454 1.077/0.03448 1.0361/0.13561
25 1.1741/0.0549 1.108/0.05134 1.1116/0.16164
30 1.2667/0.0655 1.155/0.07710 1.2195/0.19702
35 1.3925/0.0993 1.217/0.11384 1.3771/0.22546
40 1.5642/0.1388 1.313/0.16952 1.6263/0.26313
45 1.7933/0.1655 1.441/0.24604 2.0033/0.34717
50 2.1053/0.2035 1.682/0.37910 2.6447/0.57353
55 2.5368/0.2427 2.110/0.56444 3.2310/0.69075
60 3.0796/0.2450 3.497/1.07411 4.0708/0.81826
65 4.0041/0.3353 5.335/1.46188 6.2644/0.91920
70 5.0803/0.3478 9.477/1.181520 8.6492/1.00539
75 6.7445/0.3831 20.22/2.19155 13.751/1.074
80 9.0598/0.4131 56.20/2.56210 21.8713/1.17973
85 12.8207/0.4550 221.3/2.93265 34.4862/1.28631
If the angle of impact is not in the table, the values for the constants can be calculated
through interpolation. (Bird & Livingston, 2001, p. 118.)
APPENDIX D: PROPERTY TABLE OF AP-SHELLS
Gu
nD
(m
m)
m (
kg)
v (m
/s)
100
250
500
750
1000
1250
1500
2000
2500
3000
2 cm
Kw
K 3
8 L/
55 (
Ge
rman
)20
0,14
875
9,86
6445
4033
2823
1915
117
5
3,7
cm P
ak L
/45
(Ge
rman
)37
0,68
573
9,74
9664
5952
4540
3530
2318
13
37 m
m G
un
M3
(Am
eri
can
)37
0,87
792,
4876
6959
5043
3631
2216
12
37 m
m G
un
M3
(Am
eri
can
)37
0,87
883,
9289
8169
5950
4337
2719
14
2 p
ou
nd
er
(Bri
tish
)40
1,08
792,
4882
7463
5446
3934
2418
13
5 cm
Kw
K 3
8 L/
42 (
Ge
rman
)50
2,06
684,
8856
7668
5849
4135
2921
1511
5 cm
Kw
K 3
9 L/
60 (
Ge
rman
)50
2,06
834,
5424
100
9279
6960
5245
3325
18
6 p
ou
nd
er
L45
(Bri
tish
)57
2,86
822,
9611
710
997
8777
6861
4838
30
6 p
ou
nd
er
L45
(Bri
tish
)57
2,86
862,
584
128
119
105
9383
7365
5140
32
6 p
ou
nd
er
L52
(Bri
tish
)57
2,86
899,
1613
512
611
210
089
7970
5544
35
57L7
3 Zi
S-2
(So
vie
t)57
3,19
398
9,68
5613
412
511
198
8777
6954
4233
75 m
m G
un
M2
(Am
eri
can
)75
6,32
563,
8895
9081
7366
6054
4536
30
75 m
m G
un
M3
(Am
eri
can
)75
6,32
618,
744
109
102
9284
7668
6251
4134
17 p
ou
nd
er
(Bri
tish
)76
,27,
788
3,92
200
190
175
160
147
135
124
105
8874
3-in
ch G
un
M5
(Am
eri
can
)76
,26,
879
2,48
154
145
131
119
107
9788
7259
48
85L5
2 52
-K (
Sovi
et)
859,
279
2,48
142
135
125
116
107
9992
7867
57
25 p
ou
nd
er
(Bri
tish
)87
,69,
147
2,44
7873
6659
5348
4335
2823
25 p
ou
nd
er
(Bri
tish
)87
,69,
157
8,20
5610
396
8677
6961
5544
3528
90 m
m G
un
M3
(Am
eri
can
)90
10,6
182
2,96
188
179
163
150
137
125
115
9681
68
90 m
m G
un
M3
(Am
eri
can
)90
10,6
185
3,44
206
201
193
185
178
170
164
150
139
128
100L
52 B
S-3
(So
vie
t)10
014
889,
7112
208
200
188
176
164
154
144
126
111
97
122L
43 D
-25T
(So
vie
t)12
225
792,
4819
618
917
916
815
815
014
112
511
199
152L
28 M
L-20
(So
vie
t)15
248
,859
9,84
6416
516
015
214
513
713
012
411
110
090
APPENDIX E: PROPERTY TABLES FOR APC- AND APBC-
SHELLS
Gu
nD
(m
m)
m (
kg)
v (m
/s)
100
250
500
750
1000
1250
1500
2000
2500
3000
5 cm
Kw
K 3
8 L/
42 (
Ge
rman
)50
2,06
684,
8856
7367
5951
4539
3426
2015
5 cm
Kw
K 3
9 L/
60 (
Ge
rman
)50
2,06
834,
5424
9689
7970
6255
4938
3023
12,8
cm
Pak
80
L/55
(G
erm
an)
128
26,3
587
9,65
2828
227
025
123
321
720
218
716
214
012
1
12,8
cm
Pak
80
L/55
(G
erm
an)
128
26,3
585
9,84
0826
425
423
722
120
719
318
015
713
712
0
Gu
nD
(m
m)
m (
kg)
v (m
/s)
100
250
500
750
1000
1250
1500
2000
2500
3000
57L7
3 Zi
S-2
(So
vie
t)57
3,19
398
9,68
5611
911
410
698
9185
7868
5850
85L5
2 52
-K (
Sovi
et)
859,
279
2,48
139
133
123
114
105
9891
8173
65
100L
52 B
S-3
(So
vie
t)10
014
914,
423
522
621
119
718
517
216
114
112
310
8
122L
43 D
-25T
(So
vie
t)12
225
792,
4820
119
418
317
216
215
214
412
911
810
9
152L
28 M
L-20
(So
vie
t)15
246
,559
9,84
6413
513
112
812
311
911
611
411
010
610
2
APPENDIX F: PROPERTY TABLE FOR APCBC-SHELLS
Gu
nD
(m
m)
m (
kg)
v (m
/s)
100
250
500
750
1000
1250
1500
2000
2500
3000
37 m
m G
un
M3
(Am
eri
can
)37
0,87
792,
4866
6358
5450
4643
3732
27
37 m
m G
un
M3
(Am
eri
can
)37
0,87
883,
9278
7469
5950
4337
2719
14
2 p
ou
nd
er
(Bri
tish
)40
1,22
822,
9673
7065
6157
5349
4337
33
6 p
ou
nd
er
L45
(Bri
tish
)57
3,23
792,
4810
710
396
9084
7873
6456
49
6 p
ou
nd
er
L52
(Bri
tish
)57
3,23
830,
5811
511
010
396
9084
7868
6052
57 m
m G
un
M1
(Am
eri
can
)57
3,3
822,
9611
010
598
9185
7973
6455
48
7,5
cm K
wK
37
L/24
(G
erm
an)
756,
838
4,96
2454
5350
4846
4442
3835
32
7,5
cm K
wK
40
L/43
(G
erm
an)
756,
873
9,74
9613
312
812
111
410
710
195
8575
67
7,5
cm P
ak 4
0 L/
46 (
Ge
rman
)75
6,8
792,
4814
614
113
312
511
811
110
593
8273
7,5
cm K
wK
40
L/48
(G
erm
an)
756,
874
9,80
813
513
012
311
610
910
397
8676
68
7,5
cm K
wK
42
L/70
(G
erm
an)
756,
893
5,12
6418
517
916
815
814
914
013
211
610
391
75 m
m G
un
M2
(Am
eri
can
)75
6,32
563,
8878
7667
5952
4540
3124
19
75 m
m G
un
M3
(Am
eri
can
)75
6,32
618,
744
8885
8177
7369
6559
5347
76 m
m g
un
M1A
1 (A
me
rica
n)
767
792,
4812
512
111
611
110
610
197
8981
74
17 p
ou
nd
er
(Bri
tish
)76
,27,
788
3,92
174
170
163
156
150
143
137
126
116
107
77m
m H
V (
Bri
tish
)76
,27,
778
4,86
147
143
137
131
126
121
116
106
9890
7,62
cm
Pak
36
L/51
(G
erm
an)
76,2
7,6
709,
8792
133
128
121
115
108
102
9786
7769
3-in
ch G
un
M5
(Am
eri
can
)76
,27
792,
4812
412
111
510
910
398
9384
7668
8,8
cm K
wK
36
L/56
(G
erm
an)
8810
779,
6784
162
158
151
144
138
132
126
116
106
97
8,8
cm K
wK
43
L/71
(G
erm
an)
889,
8799
9,74
423
222
721
921
120
419
619
017
616
415
3
90 m
m G
un
M3
(Am
eri
can
)90
10,9
480
7,72
164
156
150
143
137
131
125
114
104
92
90 m
m G
un
M3
(Am
eri
can
)90
10,9
485
3,44
169
168
164
157
151
144
138
127
115
104
12,8
cm
Pak
80
L/55
(G
erm
an)
128
28,3
844,
9056
267
262
253
245
237
230
222
208
195
182
APPENDIX G: PROPERTY TABLE FOR APCR-SHELLS
Gu
nD
(m
m)
m (
kg)
v (m
/s)
100
250
500
750
1000
1250
1500
2000
2500
3000
2 cm
Kw
K 3
8 L/
55 (
Ge
rman
)20
0,1
1005
,84
6345
2615
85
31
00
3,7
cm P
ak L
/45
(Ge
rman
)37
0,36
894
4,88
9071
4832
220
00
00
2 p
ou
nd
er
Litt
lejo
hn
(B
riti
sh)
400,
5711
88,7
212
912
110
998
8879
7157
4637
5 cm
Kw
K 3
8 L/
42 (
Ge
rman
)50
0,92
510
49,7
3113
011
594
7763
5142
2819
12
5 cm
Kw
K 3
9 L/
60 (
Ge
rman
)50
0,92
511
49,7
0614
913
210
888
7259
4832
2114
57L7
3 Zi
S-2
(So
vie
t)57
1,55
511
99,6
9318
316
914
712
811
197
8464
4836
7,5
cm K
wK
40
L/43
(G
erm
an)
754,
191
9,88
6417
316
415
113
912
711
710
891
7765
7,5
cm P
ak 4
0 L/
46 (
Ge
rman
)75
4,1
989,
6856
195
186
170
157
144
132
121
102
8673
7,5
cm K
wK
40
L/48
(G
erm
an)
754,
192
9,64
176
167
154
141
130
119
109
9278
66
7,5
cm K
wK
42
L/70
(G
erm
an)
754,
7511
29,5
8926
525
323
421
619
918
417
014
512
410
5
76 m
m g
un
M1A
1 (A
me
rica
n)
764,
2610
36,3
223
922
720
819
117
516
014
712
410
488
76,2
L41,
5 Zi
S-3
(So
vie
t)76
,23
954,
6336
130
114
9275
6049
3926
1711
85L5
2 52
-K (
Sovi
et)
854,
9910
49,7
3117
515
913
611
710
085
7354
3929
8,8
cm K
wK
36
L/56
(G
erm
an)
887,
392
9,64
219
212
200
190
179
170
160
143
128
115
8,8
cm K
wK
43
L/71
(G
erm
an)
887,
311
29,5
8930
429
628
226
925
724
523
421
319
417
7
90 m
m G
un
M3
(Am
eri
can
)90
910
18,0
3230
629
527
826
224
623
221
819
317
115
1
APPENDIX H: ARMOUR PIERCING SHELL TYPES