Chapter One

Introduction to Ballistics

Chapter1

2

CHAPTER ONE

Introduction to Ballistics

This chapter is only an introduction to the vast subject of ballistics. It

does not deal in advanced theory but does contain material in modest details.

In this chapter, ballistics is discussed in three separate phases:

a. Internal Ballistics: This concerns the events that take place from charge

ignition to the moment when the projectile leaves the muzzle.

b. External Ballistics: This deals with the motion of the projectile from the

moment of leaving the muzzle to the moment of impact or burst.

c. Terminal Ballistics: This deals with the motion of the projectile and parts

or fragments thereof, from the moment of impact or burst.

For a short period of time after the projectile leaves the muzzle, the projectile

is acted upon by the pressure of the emerging gas. This is known as the

transitional phase of ballistics (sometimes referred to as intermediate

ballistics). Its effect must be taken into account during the design and

construction of the equipment when such factors as the functioning of the

weapon, the steadiness and stability of the gun and the use of a muzzle brake

are considered.

Applications of Ballistics

Gunnery is the practical application of the science of ballistics to the

engagement of targets. It is applicable to the engagement of targets and to the

reduction of data obtained by shooting to a form suitable for the eventual re-

engagement of targets. While the appropriate data can be arrived at by use of a

Proforma through a rather mechanical process, knowledge of ballistics and

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3

ammunition characteristics will allow an understanding of the process being

followed.

1.1 Internal Ballistics

Internal ballistics is defined as the science that deals with the events that

take place in a gun from the moment the propellant charge is ignited until the

projectile leaves the muzzle. It deals with the complicated events during

burning of the propellant and the movement of the projectile, which in turn

depends on the design of the bore and the gas pressure. The task of the events

under consideration is to give the projectile the correct muzzle velocity and

the required rate of spin. In spite of numerous and detailed studies on the

subject, no exact scientific solution to the problem has yet been found. The

internal ballistics in this chapter relate to events as they pertain to the gun only

and no attempt has been made to relate these events to the mortar.

The evolution of heat and gases causes the temperature and pressure to

build up. Thermodynamic principles can be applied to find the relation

between the temperature and pressure, the physical properties of the products

of combustion, and the amount of heat released [1-2]. According to the first

Law of thermodynamics, the heat added, to a system is the sum (of the change

in its internal energy and the work done by the system

Q = ∆E + w (1-1)

Let the system consist of unit mass of propellant burned to its final products

in the free space of the gun chamber . The internal energy E can be given as

E = cvT0

0dT (1-2)

and if Q is the amount of heat related at constant volume (so that w = 0) ,

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Q = cvT0

0dT − cv

T

0dT = cv

T0

T0′ dT (1-3)

where T0′ is the temperature of the products corresponding to their initial state,

T0 the temperature of the products (the Adiabatic flame temperature), cv is

the specific heat at constant volume. The pressure can be obtained from the

equation of state of the product gases. Using a form of the Van der Waals

equation of state:

P(τ − b) = nRT , P = nRT/(τ − b) (1-4)

where τ = τc − τp it τc is chamber volume , τp is propellant volume, and

τp = c/ρis the charge mass divided by the propellant density, vc is the molar

volume at critical point and b=vc/3 .

The projectile is accelerated by the pressure acting on the base of the

Projectile AP1 = mdv

dt (1-5)

where P1 is the pressure at the base of the projectile, A is the projectile cross-

sectional area, m is the projectile mass, and v is the projectile velocity. If the

pressure were known as a function of the displacement of the projectile as

shown in Fig.1, the muzzle velocity could be calculated directly and the

central problem of interior ballistics would be solved. Unfortunately, interior

ballistics investigations have not found any explicit dependence of pressure on

projectile displacement, but rather show that the pressure depends on the

amount of propellant burned behind the projectile and on the volume of the

space behind the projectile. The relation between pressures, the amount of

charge burned, and the change in volume brought about by projectile

displacement is derived from conservation of energy. This relation is called

Resal's equation.

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1.1.1Resal's Equation

Let z be the fraction of the charge mass C that is burned at a given

instant; then (where J is the mechanical equivalent of heat)

E = JCz cvT0

0dT (1-6)

is the internal energy of the combustion products if there is no work done by

the system. Projectile displacement means that (1) the gases have expanded

and changed the internal energy and (2) the gases have done work. The

Fig. 1: Velocity and Pressure on the Base of a Projectile as a Function of

Distance along the Bore [Ref.3].

expansion of the gases lowers the temperature from T0 to T , and the internal

energy is E = JCz cvT

0dT (1-7)

The change in the internal energy is

∆E = JCz cvT

0dT − JCz cv

T0

0dT = −JCz cv

T0

TdT (1-8)

from Eq. 1-5 and 1-6, and the work of expansion is

w = A P1x

0dx (1-9)

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6

(It is implicit in the thermodynamic assumptions that, P is uniform over the

chamber. The fact that this is only approximately true will be considered later)

Using the first law of thermodynamics and Eq. 1-7 and 1-8,

−∆E = w thus JCz cvT0

TdT = A P1

x

0dx (1-10)

It is usually assumed that the variation in cv , over the range from T to T0 is

small so that cv can be replaced under the integral sign by its mean value cv ;

hence, Eq. 1-10 becomes

JCzcv T0 = JCzcv T + A P1x

0dx (1-11)

and T is replaced by using an equation of state

P τ − b = nRT (1-12)

so that JCzcv T0 =Jcv cP τ−b

nR + A P1

x

0dx (1-13)

Use the thermodynamic relation cP − cv = nR/J and the definition of the

force constant, F = nRT0 , which is common in internal ballistics, to obtain

FCz

γ−1=

PCz τ−b

γ−1+ A P1

x

0dx

Where cP is the specific heat at constant pressure , τ is the volume occupied

by the unit mass of gas , γ = cP/cv , Czτ is the volume of gas at the time

considered and consists of these parts: chamber volume (τc) plus volume of

bore behind the projectile (Ax) minus the volume of propellant burned at the

given time (C(1 − z)/ρp ). The initial volume of the free space is expressed

for analytical Convenience as added length of bore l0 ,

Al = τc − C/ρp

and the final result is Resal's equation

FCz

γ−1=

P

γ−1A x + l0 − Cz

b−1

ρp+ A P1

x

0dx (1-14)

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The quantity b − 1 /ρp is small and is frequently neglected to give a

simpler form

FCz

γ−1=

P

γ−1A x + l0 + A P1

x

0dx (1-15)

The term A P1x

0dx in Resal's equation includes all forms of energy changes

that are brought about by the pressure P . The kinetic energy of the projectile

is of most immediate interest and importance. A second major contribution is

the kinetic energy that the expanding gases in the chamber acquire as the

projectile is driven down the bore. An estimate of the magnitude of this term

can be made by using the Lagrange approximation. This estimate is based on

the assumption that the density of the gases and unburned propellant is

constant and that the velocity is linearly distributed between the breech and

the base of the projectile.

Ekg = A ρv2

2

x1

0dx for v = v1y

=Ax1

2ρv12

2 y21

0dy = Cv1

2/6 y = x/x1

The two major terms together are

mv2

2 +

Cv2

6= (m +

C

3)

v2

2

and can be lumped together by defining k2 = (1 +C

3m)

so that m +C

3

v2

2= k2

mv2

2 (1-16)

Other energy changes that are either neglected or included as apparent

changes in projectile mass are the energy of the recoil of the gun, the elastic

and heat energy absorbed by the gun, and the rotational energy imparted to the

projectile by rifling. Resal's equation becomes

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FCz

γ−1=

P

γ−1A x + l0 +

k2mv2

2 (1-17)

The equation of motion of the projectile has to be supplemented by

information on the dependence of the pressure at the base of the projectile on

the displacement of the projectile down the bore. Resal's equation partially

fulfills this need. It is a relation between projectile velocity, the pressure, the

amount of charge that has been burned, and the change in volume brought

about by projectile displacement. To complete the required information, the

dependence of burning on the pressure and temperature within the barrel must

be determined.

In the discussion of warheads the properties of the flame front of a

propellant will be compared to the properties of the detonation front of a high

explosive. In Anticipation of that comparison it should be noted that the flow

of material in the burning of propellant; is in the opposite direction to the

advance of the flame front. In detonation the motion of reacted material is in

the same direction as the advance of the detonation front.

Many test; on the burning of propellants indicate that the rate of

advance of the propellant depends on the pressure. The form of the

mathematical relation that holds for many materials is

𝑑𝜉

𝑑𝑡= 𝑎𝑃0

𝑏 (1-18)

where 𝜉 is the displacement of the flame front, 𝑃0 is the pressure at the breech

where the propellant is burned, and 𝑎 and 𝑏 are empirical constants. This

equation is referred to as Vieille's law.

It is usually assumed that the reaction of the propellant is initiated on the

entire surface of all the propellant grains and advances into the grain parallel to

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the grain surface. The law that propellants burn parallel to the initial grain

surface surface is called Piobert's law. Although it may not be accurate in

minute detail, it is believed to be a reasonable description of the net effect of

burning. Piobert's law is the basis for analytical expressions of the relation

between the rate of chemical reaction and the grain shape and size.

It is convenient to express the burning rate in terms of the fraction 𝑓 of

a typical grain dimension (𝜉 = 𝑓𝑑𝑔 ), since this is a parameter that must

ultimately be considered.

Thus 𝑑𝜉

𝑑𝑡= 𝑑𝑔

𝑑𝑓

𝑑𝑡= 𝑎𝑃0

𝑏 (1-19)

A particular expression that is used frequently is

𝑧 = 1 − 𝑓 (1 + Ө𝑓) (1-20)

which is called the form function. Ө = Ө(𝑓), and the particular analytical

expression of Ө represents the effect of shape.

The following sections discuss three propellants (powder) grain

configurations: cylindrical, tubular, and multi-tubular [Fig. 2].

Cylindrical Grain

A very simple grain configuration is a cylindrical grain (Fig.2a). The

volume of the cylinder is

𝜏0 = 𝜋𝑑𝑔2𝑙/4 , 𝜏0 = 𝜋𝑑𝑔

3𝜆/4

where 𝑑𝑔 is the diameter of the cylinder and 𝑙 is the length of the cylinder

(let 𝑙 = 𝜆𝑑𝑔).

The dimensions of the cylinder are reduced by 𝑓𝑑𝑔 at a given time. Since

𝑧 =𝑚𝑔0−𝑚0

𝑚𝑔0=

𝜏0−𝜏

𝜏0

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10

and 𝜏 = 𝜋𝜆𝑓2𝑑𝑔 2 [𝑑𝑔 − 1 − 𝑓 𝑑𝑔]

then 𝑧 = 1 − 𝑓 [1 + (1 + 𝑓/𝜆)𝑓]

and Ө = 1 + 𝑓/𝜆

and for 𝜆 ≫ 1 Ө = 1

The burning surface of powder with this shape decreases as burning proceeds,

and the powder is called a digressive powder.

(a) Cylindrical grain (b) Tubular grain. (c) Multitubular grain

(d is the thickness of the wall)

Fig.2: Propellant Grains.

Tubular Grain

For a tubular grain [Fig.2b] the same types of considerations yield the

following equations for the initial and current volumes

𝜏0 = 2𝜋𝑑𝑔𝑟𝑙 , 𝜏0 = 2𝜋𝑑𝑔𝑟𝜆𝑑

and 𝑧 = 1 − 𝑓(𝜆 − 1 + 𝑓)/𝜆

= (1 − 𝑓)(1 + 𝑓/𝜆)

and = 1 − 𝑓 (if 𝜆 ≫ 1 and Ө = 0)

The burning surface of powder with this shape remains essentially constant as

burning proceeds, and the powder is called a neutral powder.

Multitubular Grain

Another common grain configuration is a cylinder with multiple

perforations [Fig.2c]. Ө for a seven-perforation grain is estimated at -0.172.

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The burning surface continually increases as burning proceeds, and the

powder is called a progressive powder.

1.1.2 Lagrange Correction

The pressure behind the projectile cannot be uniform since there is a

positive velocity gradient from the breech to the base of the projectile; so the

pressure gradient must be negative. The pressure at the base of the projectile is

estimated by using a factor called a Lagrange correction for converting breech

pressure to base pressure [Fig.3]. This correction is derived by assuming that

pressure will drop in proportion to the momentum acquired by the propellant

gases. Specifically it is assumed that P1

P0=

𝑚𝑣

(𝑚+𝐶

2)𝑣

(1-21)

where 𝐶𝑣/2 is the momentum of the propellant gases estimated by using the

same assumptions employed in estimating the energy of the gases in the

discussion of the corrections to Resal's equation. Thus,

P1 =P0

1+𝐶

2𝑚

= P0/k1 where k1 = 1 +𝐶

2𝑚 (1-22)

A mean value of pressure 𝑃 is used in the internal energy term of

Resal's equation. This is estimated by assuming that the ratio of the pressure at

the base of the projectile to the mean pressure is equal to the ratio of the

projectile kinetic energy to the sum of the projectile and gas kinetic energies

P1

p=

𝑚𝑣2/2

𝑚+𝐶

2 (𝑣2/2)

(1-23)

so that P = P1k2 where k2 is defined by Eq. 1-16. Equation 1-5 is expressed

in terms of the mean pressure as AP = k2mdv

dt (1-24)

The pressure at the breech is found by using Eq. 1-22 and 1-23 to give

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P0 = P(k1/k2) , and the burning rate, Eq. 1-19, becomes

dgdf

dt= −a(𝑃

k1

k2 )𝑏 (1-25)

Fig. 3: Pressure distribution between Breech and base of a Projectile [3].

The equation of motion, the burning rate equations, Resal's equation,

and the form function are mathematically sufficient information to determine

the motion of the projectile and the pressures in the barrel for a given the

propellant and barrel dimensions. These equations are expressed in terms of

mean pressure 𝑃 as follows:

𝐴𝑃 = k2mdv

dt (1-24)

FCz

γ−1=

P

γ−1A x + l0 − Cz

b−1

ρp+ A P1

x

0dx (1-14)

dgdf

dt= −a(𝑃

k1

k2 )𝑏 (1-25)

𝑧 = 1 − 𝑓 (1 + Ө𝑓) (1-20)

in which the constant k1 equals 1 + 𝐶/2𝑚 (the correction for obtaining base

pressure from beech pressure), the constant k2 equals 1 + 𝐶/3𝑚 (the

correction for Resal's equation), and p is the mean pressure in the chamber.

𝐏𝟏 =𝐏𝟎

𝟏 +𝑪

𝟐𝒎

Chapter1

13

These four equations are four separate and independent restrictions on

changes in pressure, velocity, and fraction of the charge that is burned as the

projectile moves down the barrel. The difficulties at this point are largely

mathematical the problem of extracting the required information in convenient

form.

Two of these equations can be simplified to give (Eqns. 1-17 and 1-19)

FcCz = PA 𝑥 + l0 (1-17a)

in which kinetic energy is deleted. and

dgdf

dt= −a𝑃

k1

k2 (1-19a)

which is the linear form, b = 1 . Equations 1-17a and 1-19a are reasonably

accurate in the early stages of projectile acceleration and are easier to solve.

Equations 1-24 and 1-19a are satisfied by

𝑣 = 𝐴dg(1 − 𝑓)/ak1m (1-26)

Equations 1-26. 1-17, and 1-17a imply

𝑑𝑥

𝑑𝑓=

A2dg 2 ( 𝑥+l0)

ma2 1+Ө𝑓 (1-27)

which has the solution

𝑥 + l0 = l0(1+Ө

1+Ө𝑓)G/Ө Ө ≠ 0 (1-28a)

and 𝑥 + 𝑙0 = 𝑙0exp[G 1 − f ] Ө = 0 (1-28b)

where G =A2dg

2 k2

mk1a2CFc (1-29)

The pressure is also found as a function of f from the above equations for

x + 1 and from Eq. 1-17a

P =Fc Ck1 1−𝑓 1+Ө𝑓

𝐴 l0k2(

1+𝑓

1+Ө)G/Ө Ө ≠ 0 (1-30a)

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14

P =Fc Ck1 1−𝑓 exp [−G 1−f ]

𝐴 l0k2 Ө = 0 (1-30b)

These equations are sufficient to calculate the pressure as a function of

projectile displacement. In general the pressure peaks before the propellant is

used up and the pressure curve has a rounded top, but this depends on the

constants of the system, and the propellant may be used up on the ascending

part of the curve (Fig. 4).

Fig.4: Maximum Pressure Compared to Location of Burnout [3].

At f = 0 all of the propellant is burned, and the values of P , x and

are ; 𝑣b =𝐴dg

ak1m (1-31)

Pb =Fc Ck1

𝐴k2(1+Ө)G /Ө Ө ≠ 0 (1-32a)

Pb =Fc Ck1exp (G)

𝐴k2 Ө = 0 (1-32b)

𝑥b + l0 = l0(1 + Ө)G/Ө Ө ≠ 0 (1-33a)

𝑥b + l0 = l0exp(G) Ө = 0 (1-33b)

PMAX before Burnout

PMAX after Burnout

PMAX at Burnout

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1.1.3 Projectile Acceleration after Burning

After burning is completed, z is equal to 1, and only two of the original

four equations apply to the phenomena; thus for z = 1 . Eq. 1.17 is

Fc C

γ−1=

PA x+l0

γ+1+

k2m𝑣2

2 (1-34)

and using dv/dt = v(dv/dx) Eq. 1-24 is

k2m𝑣d𝑣

d𝑥= 𝐴𝑃 (1-35)

These equations can be solved by differentiating Eq. 1-34 and eliminating 𝑣

from the result, using Eq. 1-35, so that

𝑑𝑃𝐴 x + l0 + 𝑃𝐴𝑑𝑥 + (γ − 1)k2m𝑣𝑑𝑣 = 0

𝑑𝑃𝐴 x + l0 + 𝑃𝐴𝑑𝑥 + (γ − 1)𝐴𝑃𝑑𝑥 = 0 (1-36)

which has the solution 𝑃 = Pb(x+l0

𝑥b +l0)γ (1-37)

Resal's equation after burning is

Fc C

γ−1= PbA 𝑥b + l0 +

k2m𝑣b2

2= PA(𝑥 + l0)

k2m𝑣2

2 (1-38)

This equation can be combined with the previous result for 𝑃𝑏 and 𝑥𝑏 + 𝑙 to

determine 𝑣 as a function of 𝑥 and system parameters,

𝑣2 = 𝑣b2 +

2Fc C

mk 2(γ−1)[1 − (

x+l0

𝑥b +l0)1−γ] (1-39)

From the definition of 𝐺 and 𝑣𝑏

𝑣b2 =

GC Fc

mk2 (1-40)

and 𝑣2 =Fc C

mk2[G +

2

γ−1[1 − (

x+l0

𝑥b +l0)1−γ]] (1-41)

The general shapes of pressure/distance curves for different types of powders

are compared in Fig. 5. The advantages of a progressive powder are shown in

this figure. For the limitation placed on the pressure/distance curve by the

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strength of the barrel, the progressive powder may give the greatest muzzle

velocity. Small arms and cannon up to 20 mm usually use single-perforation

grains, and all larger guns use multi-perforated grains.

Fig. 5: Pressure/Distance Curves for Different Forms of Propellant [3].

An example of a specific pressure/distance curve is shown in Fig. 6.

Fig.7 compares a calculation of the pressure and velocity curves obtained for

instantaneous burning with that for the same powder burned gradually. This

comparison is interesting since the instantaneous burning is like the process of

energy release in a warhead. It will be noted that the instantaneous burning

gives the greater velocity (but at the price of very high pressures at the start of

the process). The velocity/distance curve for instantaneous burning [5] is

given by 𝑣 = 𝑣lim 1 − (x+l0

l0)1−γ (1-42)

which is Eqn. 39 with 𝑣𝑏 = 0 , 𝑥𝑏 = 0 and

𝑣lim = 2Fc C/(γ−1)

mk2 (1-43)

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Fig. 6: Pressure Distance Curve for a Fig. 7: Pressure/Distance Curve for

3.7-Inch Antiaircraft Gun [3] Instantaneous Burning [3]

Examples of muzzle velocities for, several weapons are given below [3]

Weapon

Muzzle velocity,

ft/sec

240-mm howitzer M1

155-mm gun M1

105.mm howitzer M2

105-mm howitzer M2A1

75-mm gun M3

1,50 to 2,300

2,745

650

1,235 to 1,550

2,030

240-mm howitzer M1 155-mm gun M1

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18

105.mm howitzer M2 105-mm howitzer M2A1

1.2 External Ballistics

External ballistics is the science that deals with the motion of a

projectile from the moment it leaves the muzzle of a gun to the moment of

impact or burst. The path followed by the projectile after ejection is called the

trajectory. Its form depends primarily on:

(i) muzzle velocity

(ii) angle of departure

(iii) Gravity

(iv) air resistance,

(v) weight and shape of the projectile,

(vi) spin of the projectile, and

(vii) Rotation of the earth.

Once the muzzle velocity and angle of departure have been decided, the two

main external influences on the projectile are gravity and air resistance. The

study of the motion of the projectile under the influence of gravity alone is

called motion in a vacuum and the results obtained are useful because of

certain similarities between the trajectory in air and the trajectory in a vacuum.

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1.2.1 Motion in a Vacuum

The motion of a projectile in a vacuum is free of the effects of air

resistance. Consider a projectile fired at a particular angle of elevation (ø) and

muzzle velocity. Newton's first law of motion states that a body in motion

continues to move in a straight line at a constant velocity unless acted upon by

an external force. This means that were it not for gravity, the projectile would

continue along the line of departure to infinity. This does not happen, of

course, because the vertical component of velocity of the projectile will be

retarded by gravity until it reaches zero velocity at the vertex, and then the

projectile will fall back to earth under the acceleration of gravity [3].

1.2.2 Vacuum Trajectories

The basic equations for a ballistic trajectory in a vacuum and a

gravitational field that is constant in magnitude and direction throughout the

trajectory are the following [3]:

𝑚𝑑2𝑥

𝑑𝑡 2= 0 , 𝑚

𝑑2𝑦

𝑑𝑡2= −𝑚𝑔 (1-44)

The initial conditions are

𝑥 = 0 , 𝑦 = 𝑦0

v𝑥 =𝑑𝑥

𝑑𝑡= v0𝑐𝑜𝑠𝜃 , v𝑦 =

𝑑𝑥

𝑑𝑡= v0𝑠𝑖𝑛𝜃

The vertical and horizontal motions are given by the equations

x = v0𝑐𝑜𝑠𝜃𝑡 (1-45)

and 𝑦 = v0𝑠𝑖𝑛𝜃𝑡 −𝑔𝑡2

2+ 𝑦0 (1-46)

and the time required for the projectile to reach the height 𝑦𝑓 is

𝑡 =v0𝑠𝑖𝑛𝜃 ± v0

2𝑠𝑖𝑛 2𝜃+2𝑔(𝑦𝑓−𝑦0)

𝑔 (1-47)

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20

The range of the projectile is

𝑥𝑓 =v0𝑐𝑜𝑠𝜃 (v0𝑠𝑖𝑛𝜃 + v0

2𝑠𝑖𝑛 2𝜃+2𝑔 𝑦𝑓−𝑦0 )

𝑔 (1-48)

=v0

2𝑠𝑖𝑛2𝜃

𝑔(1 ± 1 +

2𝑔 𝑦𝑓−𝑦0

v02𝑠𝑖𝑛 2𝜃

)

1.2.3 Trajectories with Air Resistance

The foregoing calculation of trajectories neglects the effect of the

atmosphere on the projectile. More accurate calculations include a resistance

to the motion of the projectile that is along the instantaneous direction of

motion. This resisting force is computed as [4];

𝐹𝐷 =𝐶𝐷𝜌𝐴v2

2 (1-49)

where 𝐶𝐷 is the drag coefficient, 𝜌 is the density of the atmosphere, and 𝐴 is

the cross-sectional area of the projectile. The equations of motion are

𝑚𝑑2𝑥

𝑑𝑡 2= −𝐹𝑑𝑐𝑜𝑠𝜃 (1-50)

and

𝑚 𝑑2𝑦

𝑑𝑡 2= −𝐹𝑑𝑠𝑖𝑛𝜃 − 𝑚𝑔 (1-51)

in which 𝜃 is the angle subtended between the tangent to the trajectory and the

horizontal. These equations cannot be solved in terms of simple algebraic

equations, but require either approximate solutions or solution by numerical

methods [5].

1.2.4 Aerodynamics of a Projectile

The aerodynamic forces on a projectile are due to the action of the

surrounding atmosphere as it is set in motion. The fluid exerts forces on the

Chapter1

21

projectile through frictional and fluid dynamic effects. It is convenient to

describe the motion of the atmosphere and projectile such that the projectile is

at rest and the fluid flows over it. A small element of the atmosphere might

flow over the projectile as is shown in Fig. 8. The rate of flow depends on the

velocity of the projectile, but since the projectile does not slow down rapidly,

the flow is approximately steady;

Fig.8: Flow of Fluid over a projectile

This means that although an element of the volume of the atmosphere will

change velocity as it goes from position 1 to position 6; all elements follow

the same path and go through the same changes of velocity. Thus, the velocity

of the element of atmosphere at a given position on the projectile is always the

same, and distribution of velocity is constant. The total force exerted on the

projectile by all such elements makes up the net aerodynamic force [6].

1.2.5 Steady Flow

Steady flow means that the paths of fluid elements through any point do

not change with time. Such paths are called streamlines. The shape of a

streamline is fixed, but the velocity of an element as it traverses the streamline

will vary. The points at which the velocity is zero, called stagnation points, are

of particular interest. In general, the velocity can be expressed as a function of

the time and the distance along the streamline, v = v(s, t) . The dynamics of

a fluid element can be analyzed by considering the dynamic equilibrium of the

forces on it. The equilibrium is between the contact forces due to the

Chapter1

22

hydrostatic pressure and viscosity of the atmosphere and the body forces due

to inertia and gravitation. A fluid has a hydrostatic pressure at any point, i.e., it

exerts a force per unit area normal to any given surface so that the total normal

force on a plane surface is [7];

𝐹p = 𝐴

pdA (1-52)

The hydrostatic pressure pushes in on all surfaces of the element of volume,

and it the pressure is uniform, there is no net force on the clement. The

pressure may vary from point to point and then a net force can develop due to

the difference in the pressures on opposite sides of the element. The element

accelerates along the streamline so that the force due to hydrostatic pressure

will come from the difference in pressure on the forward and rear surfaces of

the element:

𝐹p = p +dp

dsds − p dA =

dp

dsdAds (1-53)

The inertial force is

𝐹i = 𝑚𝑑v

𝑑𝑡 (1-54)

and since v = v(s, t)

𝑑v

𝑑𝑡=

𝜕v

𝜕𝑡+

𝜕v

𝜕𝑠

𝜕s

𝜕𝑡=

𝜕v

𝜕𝑡+

𝜕v

𝜕𝑠v (1-55)

Since the motion is steady, 𝜕v

𝜕𝑡= 0 so that

𝐹i = 𝑚v𝜕v

𝜕𝑠 (1-56)

The force due to gravity in this direction is

𝐹𝑔 = ρdl𝑔𝑐𝑜𝑠𝜃 (1-57)

= ρdl𝑔dy

ds

The equilibrium of forces is

Chapter1

23

𝐹i = 𝐹p + 𝐹𝑔 (1-58)

which, from Eq. 1-53, 1-56, and 1-57, is

ρv𝜕v

𝜕𝑠= −

dp

ds− ρ𝑔

dy

ds (1-59)

which can be integrated to give

ρv2

2+ p + ρ𝑔𝑦 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 (1-60)

thus the hydrostatic pressure is related to the flow rate and the height. The

equation is called Bernoulli's equation and has wide applicability to steady-

state fluid dynamic problems. The aerodynamic forces on a projectile are

important in regions of flow in which the gravitational effect is insignificant

so that [8];

v2/2 + p/ρ = constant (1-61)

is sufficient to relate flow rate and pressure. In Fig. 9 there is a stagnation

point at zero, and since v is zero, the pressure has a maximum value that is

equal to the constant for that streamline. At a large distance the velocity has its

maximum value equal to the projectile velocity and the pressure is a

minimum, the atmospheric pressure for undisturbed conditions. The theory of

incompressible flow has been developed to determine the properties of

streamlines from the contour of the objects over which flow occurs and from

these properties to determine the pressures and velocities at all points. The net

fluid dynamic force turns out to be zero for flow that follows the contour of

the projectile without regard to friction along the projectile surface or within

the fluid as viscosity. The friction between the projectile and the fluid causes

the fluid to slow down at the projectile surface. Shears develop in a thin layer,

and there is a shearing traction exerted on the projectile. The net force on the

Chapter1

24

projectile is still not enough to account for the fluid dynamic forces that are

known to exist [3].

Fig.9: Forces on a Fluid Element.

1.2.6 Effect of Viscosity

Fluid dynamic theory that includes viscosity of the fluid allows the

possibility that there will develop layers of intense shearing that are capable of

changing the character of the flow, for a projectile the shearing begins on the

ogival surface but separates from the projectile behind it and has the net effect

of setting up a low-pressure region behind the projectile. [Fig.10, Ref.[9]].

Fig.10: Flow with a Detached Boundary

Layer.

1.2.7 Lift, Drag, and Yawing Moments

A quasi-empirical method of representing the forces recognizes the

dynamic pressure 𝜌v2/2 as the dominant factor in the relation of the forces to

projectile motion. The aerodynamic forces are most conveniently expressed as

lift and drag forces acting at the center of mass in the direction normal to the

𝑭 = 𝒎𝒅𝐯

𝒅𝒕

𝐩 +𝐝𝐩

𝐝𝐬𝐝𝐬

Chapter1

25

projectile motion and parallel to it and a torque that acts about the center of

mass [10, 11].

Lift: 𝐹L =𝑐L𝜌Av 2

2 (1-62)

Drag: 𝐹D =𝑐D 𝜌Av 2

2 (1-63)

and Yawing moment 𝑀𝛿 =𝑐M 𝜌Alv 2

2 (1-64)

where l is the distance from the center of pressure to the center of mass. The

dimensionless coefficients cD , cL , and cM correct the dynamic pressure

𝜌v2/2 for the effects of the shape and orientation of the projectile and the

Mach number of the flow. The coefficients cl and cm can be defined to

express the approximately linear dependence on yaw at small yaw angles

cM = cm𝛿 (1-65)

and

cL = cl𝛿 (1-66)

The restoring torque tends to induce angular accelerations about the yaw axis

according to the relation

𝑚𝑅𝐼2 𝑑2𝛿

𝑑𝑡2=

Acm 𝜌lv 2𝛿𝑙

2 (1-67)

where RI is the radius of gyration of the projectile about the yaw axis. If the

center of mass of the projectile is ahead of the center of pressure, then the

moment tends to oppose the existing yaw, the right side of the equation is

negative, and the equation has the solution

𝛿 = 𝛿1cos(2v

ςt + ϵ) (1-68)

in which 𝛿1 and ϵ are constants of integration that depend upon the initial yaw

and yaw rates and

Chapter1

26

ς = 2π 2m𝑅𝐼

2

𝐴𝜌 lc m (1-69)

The quantity ς turns out to be the distance of projectile travel for one cycle of

oscillation. Fig.11 represents a trajectory with a yaw oscillation. The net lift

for such an oscillating projectile is zero, and the aerodynamic effects on the

trajectory depend only on the drag.

Fig.11: Section of a Trajectory with Yaw Oscillations.

1.2.8 Types of Drag

It is important to note that for stable behavior of projectiles the most

important part of the aerodynamic forces is the drag. There are three

contributions to the total drag: skin friction, base drag and wave drag [12,13].

Each of these contributes to the drag force through different processes and

depends on different aspects of projectile configuration. The skin friction is

the effect of the interaction between the flowing gases and the surface of the

projectile. This contribution to the total drag depends on the shape of the

projectile and in particular on the area of the base of the projectile, which

should be small for the least drag. Wave drag is the effect of the compression

and expansion of the air as it flows over the body. This contribution to the

total drag depends on projectile shape and is least for slender bodies, that is, it

decreases for an increase in the ratio of length to diameter. This contribution

to drag is only important as the Mach number Increases, and at subsonic

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27

speeds it is negligible. Wave drag is defined with reference to the cross-

sectional area.

The total drag coefficient is defined with reference to the cross-

sectional area; hence, it is related to skin friction and base and wave

components as

cD = cDw +cDw As

A+ cDb

Ab

A (1-70)

(wave drag) (skin friction) (base drag)

when As equals the wetted area of the body, Ab equals the effective area of the

base, and A equals the cross-sectional area of the body[fig.12].

Fig. 12: Drag Coefficient as a Function of Mach number Typical

of Common Projectile Shapes[Ref.3].

1.2.9 Velocity

The higher the velocity of the projectile the greater the resistance of the

air will be. For a given increase in muzzle velocity the resulting increase in

range is greater at low muzzle velocities than those that are higher. For muzzle

velocities around the speed of sound (approximately 340 m/s) a small

variation in muzzle velocity causes a disproportionate variation in range. As

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28

the projectile travels through the air it is affected by three forces as follows:

(i).Nose Resistance: Nose resistance is a consequence of pressure drag at

subsonic velocities and compressive resistance (wave drag) at supersonic

velocities. Nose resistance increases as velocity increases until the velocity

reaches Mach 1. Nose resistance peaks near Mach 1 due to the development of

supersonic flow and shock waves around the nose [Fig.13]. Suffice to say that

a projectile with a long, slender nose will experience less pressure drag at

subsonic velocities and less compressive resistance at supersonic velocities

than a projectile with a blunt-shaped nose [14].

(ii).Tail/Base Drag: Tail drag arises from pressure in the base region of the

projectile that acts upon the base. Its effect can be very significant. The

stream-line flow around the projectile tends to break away at the base creating

a large cavity and wake. This causes the pressure at the base to be less than

ambient and results in tail drag. As projectile velocity increases towards Mach

1 the pressure at the base tends to zero [15].

Fig.13: Factors Affecting Projectile Velocity [5]

(iii).Skin Friction: A rough surface on the projectile will increase resistance,

thereby decreasing range. Its effect is normally the least of the three sources of

Chapter1

29

resistance but in the case of a long, thin projectile, such as a rocket body, it

can be relatively greater. In general, the effect of skin friction on a projectile is

small. The relationship between these three factors for a conventional

projectile is shown in Fig.13 [16].

1.2.10 Reynold's Number

The resistance encountered by the projectile as it passes through the

atmosphere is directly related to viscosity (the measure of the flow resistance

of the atmosphere gases) and is of particular interest to ballisticians.

Experiments have shown that if the ratio of the inertia forces to viscous forces

was preserved for similar shaped projectiles in different fluids, then the flow

about the projectiles would be identical. This ratio is known as Reynold's

number (Re) and is commonly shown as:

Re =Vκ

μ(1-71)

V = velocity , κ = length of projectile

μ = kinematic viscosity(1.45 × 10−5 M2/sec at sea level)

If Reynold's number is very large the inertia forces will dominate and if it is

very small, viscous forces will dominate. As the projectile moves through the

air, a velocity gradient forms between the undisturbed gases of the atmosphere

and those immediately adjacent to the projectile. This gradient grows in

thickness from the nose of the projectile; the degree of growth is dependent

upon the state of the boundary layer. Generally, when Reynold's number is

low the flow is orderly and steady (laminar flow). When Reynold's number is

high the flow is orderly but very energetic (turbulent flow) [17].

A Knowledge Reynold's number defines the state of the boundary layer and

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30

leads to one of the following:

(i) The effect that protuberance or roughness will have on skin friction;

(ii) The sensitivity of the flow field to body curvature; and

(iii) The extent of the cavity in the base of the projectile.

1.2.11 Effects Due to Velocity

Subsonic Velocities: If the projectile is travelling at less than the

velocity of sound (less than about 340 m/s) the compression at the nose is

transmitted away from the projectile in all directions and the resistance due to

the compression waves is negligible [Fig.14]. Most of the resistance at

velocities well below that of sound waves in air is due to:

(i) the formation of a wake behind the base of the projectile which is known

as tail drag (or base drag); and

(ii) Air sticking to the surface of the projectile, which is known as skin

friction.

For projectiles required to travel at subsonic speed, it is an advantage to

streamline the base in an effort to reduce tail drag [18].

Fig. 14: Compression Waves at Subsonic Velocities

Supersonic Velocity: At velocities above the speed of sound, large

pressure waves are suddenly generated, as from an explosion. These pressure

waves propagate into the atmosphere with an increasing velocity because they

Chapter1

31

are compressing the atmospheric gases, raising their temperature and hence

the local speed of sound.

The compression waves eventually come together and form a conical shock

wave at the nose of the projectile [Fig.15]. With a projectile that has a long

pointed nose, the shock wave forms at the nose, that is, is attached. With a

projectile having a blunt nose, the shock wave forms ahead of the projectile,

the blunter the nose the higher the wave drag and, therefore, the greater the

retardation. The angle of the shock cone decreases with the increase of the

Mach number and at hypersonic speeds (M>5) the shock wave almost follows

the shape of the body. Because the projectile is a finite length body travelling

supersonically, the disturbance generated must also be finite, and so a nose

shock wave is accompanied by a tail shock wave. The tail shock wave appears

to originate from the neck of the wake. This pair of shock waves constitutes

the sonic boom heard from bullets, projectiles or aircraft travelling at

supersonic velocities [19].

Transonic Velocities: The energy to overcome air resistance can only

come from the kinetic energy of the projectile. The projectile loses energy at a

greater rate when travelling supersonically than when travelling subsonically.

At the velocity of sound there is a condition of mixed subsonic and supersonic

motion. In this region, the projectile's behavior is unpredictable because the

air resistance is changing rapidly. This is known as the transonic zone. In this

zone, small changes in the projectile's velocity will cause marked changes in

resistance. Small variations in velocity due to wind, air or charge temperature

may contribute to these changes in resistance; however, this has not been

confirmed.

Chapter1

32

It is important that the projectile is steady when it enters the transonic zone.

Since it is common for the projectile to be unsteady just outside the muzzle of

the gun due to initial yaw, and the turning moment is strongest in the transonic

region, firing of projectiles in the area of the velocity of sound is avoided

when possible. Guns that have velocities in the area of the velocity of sound,

however, have given adequate accuracy. When the projectile is travelling at

the speed of sound, the compression waves and the projectile are travelling at

the same velocity [Fig.16] [20].

Fig. 15: Compression Waves at Supersonic Fig.16: Compression Waves at Transonic

Velocities Velocities

1.2.12 Projectile Design

Diameter: Two projectiles of identical shape but of different size will

not experience the same drag. For example, a larger projectile will offer a

larger area for the air to act upon; hence, its drag will be increased by this

factor. The drag of projectiles of the same shape is proportional to the square

of the diameter.

Base Design: A projectile with a boat-tail base (streamlined)

encounters less drag than one with a cylindrical base, especially at velocities

below the speed of sound. As the air moves over the cylindrical base

projectile, there is no external force to change its direction so the air flow

Chapter1

33

separates at the base and a vacuum forms, Fig.17 (1). A streamlined design

changes the direction of the air flow, thereby reduce the size of the vacuum

Fig.17 (2). A streamlined projectile also offers less base area, which further

reduces base drag. A considerable reduction in the amount of drag can be

achieved by the emission of a jet of gas from the base of the projectile. A

study was made in Sweden on the use of base bleed for increasing the range of

the 105 mm projectile. The results showed that at Mach 2, base drag was

reduced by about 50 percent and range improved by over 20 percent. This

principle is utilized in the Base Bleed option being developed with the

experimental 155 mm Extended Range Full Bore (ERFB) projectile. Drag can

also be reduced by spoilers, deliberately designed into the projectile near or at

the base. These spoilers break up the smooth and orderly (laminar) flow of air

and cause it to be turbulent, as shown in Fig.18. The turbulence tends to fill

the area behind the projectile with air, thereby reducing drag [21].

Nose/Head Design. The shape of the projectile nose is expressed in

terms of Caliber Radius Head (CRH) and is defined in terms of length and

radius of curvature of the ogive, both expressed in calibers. Fig.19 shows a

simple CRH of 2 calibers and a compound CRH with the same length of head

but the curvature is reduced by using a radius of 3 calibers [Ref.3].

Fig.17: Effects of Base Design on Air Flow Fig. 18: Air Turbulence Caused by Spoilers

Chapter1

34

Fig. 19: Caliber Radius Head

The optimum use of the length available for the ogival head [fig.20],

parallel portion and the base depends on the use for which the projectile was

designed. For projectiles that are fired at supersonic velocities, but may travel

at subsonic range, a streamlined base is required. Antiaircraft or anti-tank

projectiles are fired at supersonic velocities and are supersonic throughout

their operational range, hence, a streamlined base is not required and a

cylindrical base is used. A streamlined base is found on projectiles that travel

at subsonic velocities for a significant portion of their flight and is always

used when maximum range is a requirement.

A projectile with a tapered base to overcome tail drag and a long

pointed head to overcome nose resistance would seem ideal. A long thin

projectile, however, does not stand up well to the stresses in the bore and

requires a high rate of spin, or fins, to keep it stable in flight. A tapered base

allows the propellant gases to escape at shot ejection in a way that may cause

instability and is the reason why high velocity projectiles normally have a

cylindrical base. Also, a long head on a projectile of fixed overall length

Chapter1

35

results in the projectile being inadequately supported in the bore (because of

the short walls). This would cause high initial yaw and loss of stability.

Where a projectile has a secondary role, i.e. anti-tank or illuminating, separate

firing tables are provided. The projectile could be designed to match the

primary round but this is not desirable as separate firing tables would still be

required and in some cases, performance would be degraded. Current design

philosophy is to use the same CRH and base design on the complete family of

projectiles [22].

Fig.20: Long Nose ogive of the 6.18 Calibers

1.2.13 Ballistics Coefficient

Because of their physical characteristics, some projectiles are more

effective in penetrating air than others. Those projectiles that perform well in

air can be said to have a better carrying power. If two different projectiles are

projected at the same velocity under the same conditions, the projectile with

the better carrying power will travel further. To illustrate the point, it is

common knowledge that a baseball will travel further than a tennis ball when

thrown in the same direction with the same velocity.

The energy available to overcome the resistance of the air is derived from the

kinetic energy of the projectile, that is, from the mass and velocity of the

Chapter1

36

projectile. For a given velocity and type of atmosphere, the carrying power

must depend on the following physical characteristics of the projectile:

(i) Mass (M). The greater the mass, the greater the energy and the greater the

carrying power.

(ii) Diameter (d). The greater the size of the hole the projectile must bore

through the air, the greater the resistance and the less the carrying power.

The size of the hole varies as d2.

(iii) Shape. This is deduced by Κ . In fact, it has been found impossible to

separate the effect of this factor from a steadiness factor, 𝜍 (sigma). Their

product Κ𝜍 is used as a single factor and is determined by trial firings.

Κ ∗is 1 for the standard shape for which the drag law has been obtained

but any degradation of this shape, for example, by a fuze profile that is less

sharp than the standard, will give a value of a Κ ∗greater than 1. The

greater the value of Κ ∗, the less the carrying power.

In summary, for a given velocity and atmosphere, the carrying power of a

projectile is proportional to its mass M and is inversely proportional to Κ ∗ 𝑑2.

That is to say:

Carrying power =M

Κ∗𝑑2 (1-72)

This ratio is called the Standard Ballistic Coefficient (c𝜍 ):

c𝜍 =M

Κ∗𝑑2

Where c𝜍 is the ballistic coefficient in a standard atmosphere, M is the

projectile weight,d is the projectile diameter,Κ ∗ are two coefficients that

represent shape and degree of stabilization.

The numerical value of c𝜍 depends on the units of measurement. Defined in

this way, c𝜍 is a universal measure of ballistic performance. No matter what

Chapter1

37

the projectiles are, if they have the same value of c𝜍 they will behave

identically in flight in the same atmosphere.

A projectile becomes considerably heavier with only a small increase in

diameter d, since for a given material the weight varies as 𝑑2. Thus a large

projectile will normally have a larger (better) value of c𝜍 than a smaller

projectile. However, if a small projectile is made of dense material so that its

mass M is comparatively large, this projectile may have a c𝜍 as good as, or

better than, the c𝜍 of a larger projectile of normal density. This is an important

consideration in connection with modern composite armour-piercing shot

[23].

1.3 Terminal Ballistics

Terminal ballistics is that part of the science of ballistics that relates to

the interaction between a projectile and target. The target could be a tank,

truck, command post, aircraft, missile, ship, etc., so that the possible

interactions could depend on a myriad of structural details. Fortunately,

terminal ballistics is simplified by the common observation that a projectile

traveling at ordnance velocities produces the hole through which the projectile

passes and extremely localized material failure immediately adjacent to the

hole. The only relevant properties of the target are those along the path of the

projectile and a few projectile diameters to the side. Thus, the overall

interaction of a projectile with a complex target is simplified to the impact of

the projectile with an array of barriers. The way these barriers are

interconnected and supported is not very important. In designing weapons and

ammunition, maximum terminal effect is the desired objective. A proper

balance among many factors is essential to accomplish this purpose. The most

Chapter1

38

important of these factors are; (i) terminal velocity; (ii) shape, weight and

material used in the projectile; (iii) type and weight of explosive charge;

and(iv) fuzing system.

1.3.1 Examples of Targets

A simple example of a target is a building that stores munitions

[24].The building is usually a reinforced concrete structure with soil pushed

up against its walls. The concrete and soil barriers protect the vulnerable

munitions that are stored inside. The thicknesses and obliquities that are

presented to a projectile depend on the trajectory of the projectile. If the

projectile still has velocity after perforating the barriers, it may either cause

mechanical damage or detonate the munitions. A tank is a much more

sophisticated target. Its military value lies in its mobile firepower. Anything

that decreases its mobility or impairs its firepower destroys its military value.

The components of greatest value are the ammunition, engine parts, personnel,

fuel, and the Suns and fire-control system. These are protected by the armor

plate of the turret, sides, and bottom of the vehicle [Fig.21]. The terminal

interaction of a projectile with this target is a series of perforations through

these barriers that impacts against vulnerable components. The weight that a

tank designer can allot to armor is limited by the power of the engine, and

therefore the armor material has to be very tough.

An aircraft is still another target that derives its military value from its

mobility and firepower. The vulnerable components of an aircraft are

comparable to those of the tank. Weight limitations on aircraft prevent the use

of thick protective barriers. In fact, there are few protective barriers as such,

Chapter1

39

but the structural elements to comprise the airframe must also serve as

protective barriers.

Fig.21: Distribution of Armor Thicknesses, Typical World War II Tank.

These and many other examples show that the terminal ballistics of a

projectile can almost always be viewed as one or more interactions with

barriers that serve to deny the projectile access to vulnerable components at

velocities that could destroy the vulnerable components. A barrier succeeds by

(1) simply decelerating the projectile, (2) breaking the projectile into

fragments and decelerating the fragments, or (3) deforming the projectile and

decelerating the projectile in its less efficient shape. It is assumed that the

projectile is properly designed so that the barrier decelerates the projectile but

never succeeds in resisting the projectile with sufficient force to cause it to

shatter or deform. This is actually the assumption of a majority of the theories

of armor penetration. It is at least approximately correct for many real

situations and vastly simplifies analysis.

1.3.2 Examples of Projectiles

Figure 22 shows different types of projectiles and the features of internal

design that are used to make the projectile non-deforming during impact.

25mm

37 mm

50 mm

75 mm

Chapter1

40

(a) Bullets (ball)

(b) Bullets (armor-piercing)

(c) Projectiles

Fig.22: Examples of non-deforming projectiles.

Fig.22 (a) shows the cross sections of .30- and .50-calilber bullets that are

intended to attack lightly protected personnel and equipment. These consist of

a lead alloy or mild steel core and a jacket. These will be non-deforming only

in the attack of light armor and extended barriers that are not very resistant to

penetration. Fig.22 (b) shows bullets of the same calibers that are intended to

attack armor. The cores of these armor-piercing bullets are hard, tough steels

or a material like tungsten carbide. The nose shape of the core is chosen to

attain reasonable performance at obliquities. The jacket includes a windshield

that gives the bullet optimum aerodynamic properties. As long as the core

material has an advantage in hardness over the barrier material, the projectile

will be essentially non-deforming. Fig.22(c) shows larger-caliber projectiles,

one that is solid steel, and one that has a cap on the nose of the core to assist in

armor penetration.

Chapter1

41

1.3.3 Mechanism of Penetration

It is assumed that the projectile keeps its shape during ballistic impact.

The barrier material must be pushed aside or forward through a combination

of elastic, plastic, and brittle failures in order to accommodate the barrier to

the moving contour imposed by the projectile. In so doing the barrier material

exerts its resisting force on the projectile and changes its motion. One aspect

of the mechanism of penetration is the specific combinations of deformations

that occur within a given material.

Brittles Penetration: The process of brittle penetration illustrated in Fig.24

can in principle at least be continued until the failure zone reaches the exit

surface and fractures run to that surface.

Ductile Penetration: In ductile materials (materials that undergo large

amounts of deformation before fracture or rupture) the point of a conical or

ogival projectile concentrates stresses at the tip and results in intense

deformations along the axis of the crater. Extensive plastic deformation

culminates in fractures on the axis. The projectile opens a hole in the barrier

along the projectile axis, and the hole is enlarged as the projectile passes

through. This mode of perforation is characteristic of extremely ductile

materials [Fig. 24].

Radial Fractures: Projectiles of a wide variety of shapes may cause failure

on the exit side of a ductile barrier that occurs in a star pattern about the

projectile axis. In thin barriers the star fractures extend through the entire

barrier thickness and, if sufficiently ductile, the angular piece between the

cracks folds back into petals. This is called petalling. In thick targets the

fractures extend only partway through the barrier and combine with other

modes of failure to form fragments [Fig. 24]

Chapter1

42

Plugging: Cylinders and blunt projectiles in general, knock out a plug that is

roughly cylindrical in shape. Even sharp projectiles may do this under the

proper conditions. The essentia1 feature is failure on cylindrical or conical

surfaces around the projectile axis. This is called plugging [Fig.24].

Spalling and Scabbing: Two other modes of failure that may be associated

with high-velocity impact are spalling and scabbing. Spalling has been

intensely studied for explosive loading and is failure caused by the interaction

of intense stress waves. The configuration of the fracture surface is

determined by the impact configuration and stress wave transmission and

reflection. Scabbing may have the same general causes but the fracture surface

is determined by patterns of in-homogeneities and anisotropy in the barrier

materials. Under certain circumstances these failure processes could

contribute to the formation of a single fragment and resemble plugging.

Fig.24: Primary Modes of Perforation Failure.

Fragmenting: The above processes of ductile perforation, radial fractures,

and plugging may occur in pure form at lower velocities. At higher velocities

Chapter1

43

these combine with spalling and scabbing or with secondary ductile and brittle

failure processes to produce many fragments [Fig.24].

1.3.4 Application of Conservation Laws

The final results of the interaction of a projectile and a tough barrier are

new rigid body motions of the projectile and barrier, changes of shape for

these bodies, and changes of energy. Conclusions, about the interrelation of

initial and final motions and changes of energy can be reached by applying

conservation of energy and momentum to, the initial and final states with little

or no concern for the details of the intervening events.

Conservation of energy and momentum are first applied to the initial

and final states of rigid body motion (Fig.25 of the projectile/barrier system

with clean perforation of the barrier, i.e., perforations by ductile or petalling

modes. Conservation of momentum requires that

𝑚1𝑣0 = 𝑚1𝑣1 + 𝑚2𝑣2 line-of-flight momentum (1-73)

0 = 𝑚1𝑤1 + 𝑚2𝑤2 off-line-of-flight momentum (1-74)

Conservation of energy requires that

𝑚1𝑣0

2

2=

𝑚1 ( 𝑣12+𝑤1

2)

2+

𝑚2 ( 𝑣22+𝑤2

2)

2+ 𝐸0 (1-75)

where 𝐸0 is the amount of energy that is converted from kinetic to non-kinetic

form. This can occur by many processes, in dissipative processes in elastic

waves or in the permanent deformations of plasticity and fracture, for

example.

The components of off-line-of-flight momentum depend on the deflection

angles Ω and 𝜓 𝑤1 = 𝑣1𝑡𝑎𝑛Ω (1-76)

𝑤2 = 𝑣2𝑡𝑎𝑛𝜓 (1-77)

Chapter1

44

and thus 𝜓 and Ω are interrelated

𝑡𝑎𝑛𝜓 =𝑚1𝑣1

𝑚2𝑣2𝑡𝑎𝑛Ω (1-78)

Since 𝑚2𝑤2 = 𝑚1𝑤1

= 𝑚1𝑣1𝑡𝑎𝑛Ω (1-79)

𝑤2 =𝑚1𝑣1

𝑚2tanΩ (1-80)

Fig.25: Perforation at obliquity With Deflection of the Projectile

Conservation of energy is expressed as

𝑚1𝑣02 = 𝑚1𝑣1

2 + 𝑚2𝑣22𝑡𝑎𝑛2Ω +

𝑚12

𝑚2 𝑣1

2𝑡𝑎𝑛2Ω + 2𝐸0 (1-81)

The equation for momentum and energy conservation can be solved for v1 and

v2;

v1 = (R1v0 ± R2 𝑣0 2 −

2𝐸0

R2m1+ 𝑡𝑎𝑛2Ω (R1 − R2) 𝑣0

2 +2𝐸0

m1R1 )𝑐𝑜𝑠2Ω

(1-82)

v2 =𝑚1𝑣0

𝑚2 1 + R1𝑐𝑜𝑠2Ω ±

Chapter1

45

R1( 𝑣0 2 −

2𝐸0

R2m1+ 𝑡𝑎𝑛2Ω (R1 − R2)v0 +

2𝐸0

m1R1 )𝑐𝑜𝑠2Ω (1-83)

Where R1 =m1

m1+m2 , R2 =

m2

m1+m2

The quantities 𝐸0 and Ω account for the individual properties of the given

impact system, such is the projectile shape, the mechanical properties of the

barrier, and the mode of failure of the barrier. Special cases of the above

equations make it easier to understand the, consequences of the conservation

laws and the role of the parameters 𝐸0 and Ω .

1.3.5 Simple Terminal Ballistics Theories

Several theories of the interaction between a projectile and barrier at

zero obliquity calculate measures of the defeat of a projectile in tractable form

by using simple assumptions about the response of the barrier to impact [26].

These all result in calculations of the energy E0 that is changed from kinetic to

non-kinetic form. The earlier theories calculate the work done in penetration

using specific assumptions on the nature of the contact forces resisting

penetration. More recent theories explicit determine the changes in energy that

occur as penetration proceeds. In addition, assumptions are made often

implicitly on the motion of entry and exit boundaries and on the failure

process that terminates resistance or initiates changes in the form of

resistance.

Penetration resistance has been estimated by simple expressions that use

empirical constants for each barrier of interest. The Euler-Robinson form is

𝐹𝑅 = 𝑐1𝐴 (1-84)

Chapter1

46

where 𝐴 is the cross-sectional area of the projectile at the current stage of

embedment, and 𝑐1 is a constant for the particular projectile and barrier. This

is equivalent to assuming that the barrier material has been loaded until it has

reached a limit 𝜍𝑌 beyond which it cannot go. The constant 𝑐1 is then the yield

limit 𝜍𝑌 .

A slightly more complicated form for the penetration resistance is the Poncelet

form that was introduced in the discussion of soil penetration

𝐹𝑅 = (𝑐1 + 𝑐3𝑣2)𝐴 (1-85)

where 𝑐1 and 𝑐3 are empirical constants. This can be interpreted as an

extension of the Euler-Robinson form by consideration of inertial effects

through an adaptation of the methods of exterior ballistics. Thus the Euler-

Robinson form of penetration resistance can be combined with the assumption

that the entry and exit surfaces have negligible motion, and that the mode of

perforation is ductile (the barrier material opens at the tip of the projectile and

expands about it). The force on the projectile at any penetration is 𝐹𝑅 = 𝑘𝐴

and the energy 𝐸0 that is dissipated in the penetration is 𝐸0 = 𝑘𝐴𝑑𝑝 = 𝑘𝜏𝑃

0,

where 𝑘 shape factor and 𝜏 is the volume of the indentation in the barrier.

For non-deforming projectiles the crater volumes are easily estimated

from the nose shape and penetration, 𝜏 = 𝐴(𝑝)𝑑𝑝𝑃

0 where 𝐴(𝑝) is the cross-

sectional area of the projectile as a function of penetration. Examples of 𝐴(𝑝)

are

Spherical nose 𝐴 𝑝 = 𝜋(𝑑𝑝 − 𝑝2) (1-86)

Conical nose (cone angle a) 𝐴 𝑝 = 𝜋𝑡𝑎𝑛2(𝑑/2)𝑝2 (1-87)

Flat-ended nose (and penetration 𝐴 𝑝 = 𝜋𝑑2/4 (1-88)

beyond complete nose embedding)

Chapter1Chapter1Chapter1Chapter1

47

An example of the application of these results to penetration data is shown in

Fig.26. The data are penetrations of conically ended steel projectiles into brass

targets. The targets were plates 9 inches on a side and 1/4 inch thick. The

0.875-inch-long projectiles were 1/4 inch in diameter with a cone angle of 90

degrees. The values of the mass ratios R� and R� are 0.002 and 0.009,

respectively, so that special case 3 is appropriate. The penetration of the

projectile up to the shaft is given by

���

� = k �� �(/2)������ = k ���

� (1-89)

or � = ��������

� (1-90)

Fig. 26: Penetration Data for a Conical Ended Projectile [3].

and beyond the nose

���

� = k�� + �!�" (p − ��) (1-91)

Region of in-deformation

of the conical nose

%& = '(

& + )*(+) − +()),-.)&

%/& = /+)/0

Chapter1

48

or p = 𝑝0 +2𝑚(𝑣2−𝑣0

2)

𝐾𝜋𝑑2 (1-92)

where 𝜏0 is the volume of the nose and 𝑝0 is the length of the nose. The

constant k was chosen to fit the data [fig.26].

1.3.6 Anti-Tank Projectile

There are two main types of anti-tank projectiles which, by virtue of

their action, can be classified as either shot or shell.

Shot. Shot defeats armour by means of kinetic energy imparted by velocity

and weight. As shot is fired at high velocity it is necessary that it has a

good ballistic shape so that in flight loss of velocity is kept to a minimum.

The use of high velocities is a user advantage in that corrections for

moving targets are normally small.

Shell. HEAT and HEP, defeat armour by virtue of the action of their

explosive content. The highest possible velocities are required in order to

improve the chance of a hit with both kinetic energy (KE) and chemical

energy (CE) projectiles. The performance against armour of both HEAT

and HEP projectiles is independent of range [5].

1.3.7 The behavior of materials under impact and explosive loading

The properties of a given material are described by the relation between

the stresses and strains within the material, that is between the internal forces

per unit area and the relative elongations or contractions and shearing of the

material. The elongation of a rod in a simple testing machine is an example.

The machine applies a pulling force on the rod, and allows measurement of

the elongation of the rod along its axis and the amount of contraction of the

Chapter1

49

rod in cross-sectional area. From these data we have the following stresses and

strains;

A tensile stress 𝜎1 = 𝐹/𝐴 (1-93)

A tensile strain (from the extension along the axis) 𝜖1 = ∆𝑙/𝑙 (1-94)

A compressive strain (from the contraction perpendicular to the axis of the

rod) 𝜖1 = ∆𝑙/𝑙 (1-95)

A test is usually carried out to the point of final failure of the rod. If the stress

and strain data are plotted directly using the initial cross-sectional area A0 in

the computation of stress, and ∆𝑙/𝑙0 for the strain, the curve representing the

data looks like the curve of Fig.27, and is called an engineering stress/strain

curve.

Fig.27: Engineering Stress/Strain Data [3]

1.4 Glossaries Related to Ballistics

Target: The target is a specified point at which fire is directed.

Point of Impact: The point of impact is the point where the projectile first

strikes an object.

Point of Burst: The point of burst is the point at which a projectile actually

bursts. It may occur before, at, or beyond the point of impact.

Chapter1

50

Inclination of the Trajectory: The inclination of the trajectory is the acute

angle measured from the horizontal plane passing through a given point on

the trajectory to the oriented tangent to the trajectory at this point.

Angle of fall: The angle of fall is the inclination of the trajectory at the level

point, the sign being positive.

Line of Impact: The line of impact is a line tangent to the trajectory at the

point of impact or burst.

Angle of Impact: The angle of impact is the acute angle, at the point of

impact, between the line of impact and a plane tangent to the surface struck.

This term should not be confused with the term angle of fall. They are the

same only when the point of impact is at the level point.

Angle of Incidence: The angle of incidence is the acute angle between the

normal to the surface struck and the line of impact.

Guns and Howitzers: There are no sharp distinguishing features between

guns and howitzers. Generally, guns produce higher muzzle velocities, fire

low angle and have fewer charges than howitzers that fire at both high and

low angles. Although certain differences may be noted, both have the

following properties:

(i)They give projectiles specified initial velocity and direction of motion.

(ii)There is a rapid burning of a propellant charge in a chamber that produces

gas under pressure which forces the projectile to move along the barrel.

Mortars: Mortars are usually small, light and easily handled equipments that

propel projectiles at high angles of elevation. They are usually loaded through

the muzzle whereas guns and howitzers are loaded through the breech.

Mortars usually have smooth bores, but can have rifled bores.

Chapter1

51

Rockets: Rockets are weapons consisting essentially of a warhead and a tube

filled with propellant. Rockets depend for flight on the reaction set up by a jet

of rapidly expanding gases released by the propellant.

Recoilless Guns: Recoilless guns reduce or eliminate recoil forces on the

carriage by creating an opposing force that is normally achieved by venting a

portion of the propellant gases. Lighter carriages can thus be used.

Rifling: Rifling is the set of twisted grooves cut along the interior of the bore,

leaving raised ribs or lands between them.

Caliber: This is the (standard) diameter of the bore, excluding the depth of

the rifling grooves. It is measured from land to land.

Weapon: The term weapon refers to the trunnions, the axis about which the

barrel rotates during elevation or depression and which is at right angles to the

weapon axis.

Weapon Axis: The weapon axis is the axis of the bore taken at the breech and

it is a straight line. This axis will not go through the weapon if the trunnions

are offset from the centre line of the bore.

Axis of the Bore: The axis of the bore is the line passing along the centre of

the barrel .This may, owing to drop, be slightly curved. In this manual, the

axis of the bore will be assumed to be a straight line from the weapon axis to

the muzzle axis.

Muzzle Axis: The muzzle axis is the axis taken at the bore and it is a straight

line.

Droop: Droop is the vertical angle between the axis at the breech and the

muzzle axis. Droop varies with barrel length and/or temperature.

The Breech Clinometers Plate: This is an accurately machined plane surface

Chapter1

52

on top of the breech ring parallel to the weapon axis. Most angular

measurements and adjustments made on the gun are based on this plane.

Muzzle Brake: Muzzle brakes reduce the recoil forces by deflecting a portion

of the propellant gases rearward at the muzzle, thus creating an opposing

force. They are used to increase the stability of the carriage on firing.

Muzzle Velocity: muzzle velocity is the velocity of the projectile at the

muzzle.

The Gun: A gun is a weapon that ejects its projectile by the action of a

burning propelling charge. In a closed chamber a propellant charge burns

more vigorously under pressure .The gun provides the chamber in which the

charge burns.

Projectile: A projectile is an elongated object, such as a bullet, that is

propelled from a gun by a rapidly burning, low explosive propelling charge. It

is fitted with a soft metal rotating (driving) band or bands near its base which

is designed:

(i) To engage with the rifling of the barrel causing spin to be imparted to the

projectile as it moves along the bore;

(ii) To prevent the escape of gases forward past the projectile;

(iii) To offer a certain initial resistance to movement that has the effect of

allowing an initial pressure rise which contributes to the regularity of

burning of the propellant charge and hence regularity in muzzle velocity;

(iv) To assist in centering the projectile in the bore. This is particularly

evident when two driving bands are fitted, one well forward of the other;

(v) For equipments using separate loading ammunition, to hold the projectile

in position when rammed, and to prevent slip-back when the gun is

elevated.

Chapter1

53

Propellant Charge: This is a rapidly burning composition of low explosive

that is burned in a gun to propel the projectile. When suitably ignited, the

propellant charge has an extremely rapid rate of burning, producing many

times its own volume of gases at a high temperature and pressure. No outside

agent, for example oxygen, is necessary for its burning. The rate at which the

contained propellant burns increases with, and is approximately proportional

to, the pressure developed. The higher the pressure, the faster the rate of

burning ; the lower the pressure, the slower the rate of burning.

The total effect of all interior ballistic factors determines the muzzle velocity,

which is expressed in meter per second (m/s) or feet per second (ft/s).

More literature on Ballistics can be found in Refs. [26-33].