Backman & Goldsmith - The Mechanics of Penetration of Projectiles Into Target

THE MECHANICS OF PENETRATION OF
PROJECTILES INTO TARGETS
MARVIN E. BACKMAN and WERNER GOLDSMITH

U.S. Naval Weapons Center, Code 3831,China Lake, CA 93555,U.S.A.
Abeact-The present survey is concerned with the terminal ballistics aspects and the penetration
mechanics viewpoint of the inIeraction of penetrators and targets. The fatter are categorized as semi-
infinite, thick, intermediate and thin. Initial velocity ranges are established for the partial purpose of
separating physical effects in various regimes. Target damage mechanisms are described and the concept of
a phase diagram distinguishing between the domains of embedment (or perforation) and ricochet is
presented. Methods for describing or observing the process are indicated: these include empirical relations,
analytical models for ballistic velocities based on rather simple damage mechanisms (primarily for targets,
but in some instances for the striker), hypervelocity descriptions, numerical techniques and experimental
methods. Projectile and target characteristics are described and the penetration into the targets is examined
for the various thickness categories. Finally, some unsolved problems in this area are indicated that require
further investigation.
TABLE OF CONTENTS
I. INTRODUCTION . . . . . . .
Definitions . . . . . . . . * . , . . . * . .
Classification of impacts . . . . . . ‘ . . . .
Velocity ranges . . . . . . . . . . . . . . . . . I
Target element phenomena . . . . , . . . . . .
Phase diagram . . , . . . .
Methods of obser;ation’and analysis . . . . I . * .
2. REVIEW OF PREVIOUS SUMMARIES INVOLVING TERMINAL BALLISTICS . , . , . .
3. DESCRIPTIVE OR OBSERVATIONAL METHODOLOGYFOR THE PENETRATION PROCESS

Empirical relations .............................
Analytical models at ordnance velocity ......................
Analytical models for hypervelocities ......................
Numerical evaluations ............................
Experimental methods ............................
4. CHARACTERISTICSOF PROJECTILES 21.

Classification . . . . . . . . 21.
Kinetic energy penetrators . . . 21.
Penetrators with warheads . . , 22.
Projectile parameters , . . . , . . 23
Projectile and trajectory orientation . . 24
Projectile deformations . . . . 2s
5. CHARACTERISTICS OF TARGETS . . . . . 29
Types of targets . . , . . , , . . . . 29
Target modeling hypotheses . , . . I . . . 30
Assumption of localized influence . . . . . . . . . . 30
Neglect of rigid body motions . . . . . ‘ . 30
Neglect of thermal effects . . , . . . * * . . 31
Thickness characterization . . . . . * . . . . ‘ . . . 31
Target element materials . . . . . . . . . . . . . . 32
6. SEMI-INFINITE TARGETS . . . . , I . . . . . . . . . 36
Empirical relations . . . . . . . . . . . . , . . . . . . 38
Analytical models . . . . . . . . . * . . . . . 1 .40
Numerical methods : . . . . , . . . . . . . . . . *
. 43
Experimental procedures . , . . , . * . * . . . . . . . .
.44
7. THIN PLATE PENETRATION AND PERFORATION . . . . . .44
Empirical relations . . . . . . . . . . . . . . . I . . . . . . 45
Analytical models . . . * . *. . . * ‘ , . .
. . .48
Single-force representation for normal impact of rigid projectiles . . . . . . . 49
Deformation without perforation , . . . . . ‘
. . . 49
Eiast~p~astic plates . . , . . : : : . . . _ . . . . . . . . . . 49
Elast~viscoplastic plates . . . . . . . . . . . . , . . . . . . . 50
UESVol. 16, No. I-A

1
M. E. BACKMAN and W. GOLDSMITH
Rigid blunt projectiles-failure by plugging 53

Sharp rigid projectiles 5.5
Hole initiation at the lip 55
Ductile hole enlargement . 55
Petaling . 59
Composite force functions . 62
Deforming projectiles 63
Single mechanism 63
Successive mechanisms 65
Target element failure limits . 70
Models based on hydrodynamic theories . 71
Multiple impact analyses . 74
Impact at obliquity and ricochet 75
Numerical solutions 76
8. INTERMEDIATE AND THICK TARGETS 77

Empirical equations . 78
Analytical models . 81
Quasi-empirical representations . 84
Hypervelocity impact . . 86
Comments on experimental work
Numerical solutions .
9. CONCLUSIONS . 91
1. INTRODUCTION
MOSTOFthis effort has been accomplished from the viewpoint of ballistics with emphasis on the
projectile and its trajectory, and terminal ballistic terminology such as “ballistic limit” and
“protection limit” has become the convention. The other perspective on penetration processes
is impact mechanics where equal weight is placed on the phenomenology of both penetrator and
target, and where there has been more interest in the overall mechanics of the process with
particular attention to transients.
During the last two decades, enormous strides have been made in an understanding of the
various mechanisms involved in the penetration process leading to significant improvements in
predictive capability. This paper will present a summary of the approaches used in these
investigations from both viewpoints and will also delineate current problems in the field.
Primary emphasis is placed on solid targets.
Definitions
Due to conflicting use of some terms in the literature, the following definitions will be
adopted in the present paper:
(a) Three generic terms for the object striking the target will be used; projectile will refer to a
device designed for ballistic performance and with substructure that serves specific ballistic
functions (examples are bullets, bombs, artillery shells); penetrator will refer to an object that,
in the context, has an exclusively terminal ballistic function; striker will refer to any object that
is considered to initiate an impact, and the term will not imply any functional restriction.
(b) The target will be defined as the smallest functionally and/or structurally independent
object whose performance is to be impaired by the projectile. (The target is the whole building
struck by an armor-piercing projectile and not the steel wall in which the projectile made a hole
and passed through.)
(c) Substructure is any single functional or operational unit of the target.
(d) A target element is that section of substructure affected by a single impact and which
constitutes the basic component to be specified in a model.
(e) Penetration will be defined as the entry of the penetrator into any region of the target,
i.e. the object struck by the penetrator and identified by structural and functional unity.
It should be emphasized that penetration of targets frequently occurs as the result of many
impacts against substructure. Penetration in this sense will involve perforation, embedment,
and ricochet in the course of the sequence of impacts. These terms will be reserved for
interaction with target elements and have the following meanings: perforation means that the
penetrator passed through the target element, embedment means that the penetrator is stopped
during contact with the target element, and ticochet means that the penetrator is deflected from
the target element without being stopped or perforating.
The mechanics of penetration of projectiles into targets 3
It is convenient to classify target elements by thickness, using the following operational

definitions:
(a) semi-infinite, if there is no influence of the distal boundary on the penetration process,
(b) thick, if there is influence of the distal boundary on the penetration process only after
substantial travel into the target element,
(c) intermediate, if the rear surface exerts considerable influence on the deformation
process during all (or nearly all) of the penetrator motion, and
(d) thin, if stress and deformation gradients throughout its thickness do not exist.
The term target damage will refer to any permanent change in the target. It is desirable to
distinguish this from dysfunction which will refer to the rendering inoperative of any target
substructure as the result of damage. In general, damage is directly correlated to penetration
processes, but dysfunction depends on the functional relation of the substructure to the target
as an operational entity.
Classification of impacts
The many phenomena of projectile penetration can be categorized by several schemes, for
example: (a) according to the angle of incidence of the impact; (b) the configuration and
material characteristics of the target; (c) the configuration and material characteristics of the
penetrator; (d) the range of initial velocities.
The last is, in many ways, the most fundamental because velocity so profoundly affects the
variety of phenomena that it overrides almost any other consideration.
There have been an increasing number of nonmilitary applications of penetration mechanics,
for example, mining and construction technology, the protective design of spacecraft, nuclear
reactors, and rotating machinery. But military interest has and will continue to be the major
motivation for development and research in this area. Military hardware is certainly the most
sophisticated form of controlled launching of penetrators and it is necessary to provide a brief
description of the capabilities of some of these devices in order to identify the means for
achieving certain velocity regimes.
Velocity ranges
The most common projection devices are conventional guns in which the propellant is
burned behind the projectile range in muzzle velocity from 500 to 13OOm/s, the nominal
ordnance range. The subordnance range of 25-5OOmls is achievable with pneumatic guns or
other laboratory apparatus. The lowest domain (O-25m/s) below the terminal velocity of freely
falling objects is obtained by drops or other experimental devices. The ultraordnance domain
from 1300 to 30OOmls is represented by warhead fragments and special purpose guns. Above
this upper limit is the hypervelocity range, involving either light-gas guns, shaped charges, or
meteors. Accelerators for this domain are described in [l, 21.
Target element phenomena

An alternative method of defining a velocity range involves the phenomenology of the
target. Elastic deformations of elements can be achieved only at extremely low striker
velocities that are employed in practice for a restricted range of laboratory experiments. The
limiting velocity L)El\of this domain in the case of normal contact of a plane-ended striker and
target is that required to produce the compressive yield stress uyc in either object, given by
_ ~Ychwm + PPCOPI
vEA - (PPGt)(PP%P)
(1.1)
where co = d/[(A + 2G)/pJ and co = q(E/p) are the dilatational and rod wave velocities, A and
G = E/2(1 + V) are the Lam6 constants, E is Young’s modulus, Y is Poisson’s ratio, p is the
mass density, and subscripts p and t refer to projectile and target, respectively[3,4]. For a
spherical projectile nose, this limit is much lower and must be evaluated frofn the Hertz law of
contact[5]. Permanent deformations may involve a variety of mechanical processes, either
singly or in combination. As shown in Fig. 1, nonperforating failure modes consist of two types
of transverse displacement of thin elements due to plastic deformation: (a) that in the contact
4 M. E. BACKMAN and W. GOLDSMITH
_+I@&
--___-___
\---------
---- CONFIGURATION AT
INITIAL CONTACT l-4
I I
1 I
Fig. 1. Permanent deformation of a thin target element showing bulging and dishing.
zone, called bulging, where the element conforms to the shape of the penetrator nose, and (b)
that induced by bending, termed dishing, which may extend to considerable distances from the
impact area. Striking velocities u, resulting in plastic deformations of the target are bounded by
eqn (1.1) and upA= q(crylp); here plastic stresses exceed hydrodynamic components. As the
target element thickness increases, these modes decrease and the deformation involves dis-
placements that tend to affect both the front and the rear surface so as to thicken the element
without significant deflection. This has been called cratering, a term that is appropriate as a
description of the effects in very thick plates and that is useful in referring to highly localized
deformations in target elements of any thickness.
In any of the velocity ranges, the target experiences a variety of phenomena including the
propagation of elastic, plastic, and/or hydrodynamic waves as well as frictional heating that
produces local and gross deformation. The hydrodynamic regime commences with up,,, and has
been described as terminating with velocities corresponding to those of the bulk modulus K in
the material, i.e. m 5 v,, 5 m = vH,,. Speeds in excess of this value are presumed to
cause shock waves due to the decrease of compressibility of solids in this domain. However,
this phenomenon is not usually treated separately in the penetration analysis of solid materials.
At sufficiently high, impact velocities, estimated to be above three VHA [3], comminution, phase
changes, vaporization, or even impact explosions may be expected[5-71.
Failure involving fracture results in the perforation of thin or intermediate targets. These
failures are due to the interaction of a variety of mechanisms with one predominating, depending
on material properties, geometric characteristics, and impact velocity. The most frequent types,
portrayed in Fig. 2, consist of fracture resulting from the initial compression wave, fracture in the
radial direction, spalling, scabbing, plugging, front or rear petaling, or fragmentation in the case of
brittle targets[8-121 and ductile hole enlargement.
Fracture due to an initial stress wave in excess of the ultimate compressive strength uuc
could conceivably occur in weak, low-density targets, while radial failure would be limited to
elements whose tensile strengths are substantially lower than their corresponding compressive
values, such as ceramics. Spalling is tensile material failure due to the reflection of the initial
compressive transient from the distal side of the target and is a commonplace phenomena under
explosive loading. Scabbing has a similar appearance, but the fracture is produced by defor-
mation and its surface is determined by local inhomogeneities and/or anisotropies such as the
rolling direction. Plugging develops as the result of a nearly cylindrical slug of approximately
the same diameter as the bullet being set in motion by the projectile. Failure occurs due to large
shears produced around the moving slug. Heat generated by the shear deformation is restricted
to a narrow annulus in which it decreases the material strength resulting in an instability; this is
called an adiabatic shearing process. Plugging is most frequently found when blunt penetrators
strike intermediate or thin, hard plates. Its presence is sensitive to velocity and the angle of
obliquity of pointed projectiles.
(ai FRACTURE DUE TO INITIAL (bl RADIAL FRACTURE BEHIND

STRESS WAVE INITIAL WAVE IN A BRITTLE
TARGET
(c) SPALL FAILURE (SCABBING! (dl PLUGGING
(ej PETALING (FRONTAL) (f) PETALING (REARWARD)
e n
tg) FRAGMENTATION Ihl DUCTILE
Fig. 2. Peroration mechanisms.

n
HOLE ENLARGEMENT
Petal& is produced by high radial and circumferential tensile stresses after passage of the
initial wave occurring near the lip of the penetrator. This deformation is the result of bending
moments created by the forward motion of the plate material being pushed ahead of the striker,
and by inhomogeneities or planes of weakness of the target. It is most frequently observed in
thin plates struck by ogival or conical bullets at relatively low impact velocities or by blunt
projectiles near the ballistic limit. Petaling is accompanied either by large plastic flows and/or
permanent flexure. As the material in the bulge on the back of the plate is further deforked by
the projectile, its tensile strength is eventually exceeded and a star-shaped crack develops
around the tip of the penetrator. The sectors-so formed-are then pushed back by the ensuing
motion of the bullet, forming petals. A combination of ductile failure and spalling seems to be
characteristic for the perforation of thick plates of medium or low hardness. In addition to
target element failure, projectile damage may also occur by plastic deformation, shattering[8],
bursting of shells, or component malfunction. Furthermore, the ejecta generated by failed
targets or projectiles must be considered as a penetrator when encountering any subsequent
target element. If intact, the bullet motion after emergence is, of course, of equal interest.
Figure 3 shows a sequence of normal steel-sphere impacts against 2024 aluminum alloy
plates at increasing velocities. The first picture (a) shows incipient plugging along approximately
spiral-shaped paths, as also noted elsewhere@, 131; (b) presents the completely formed plug
with substantially larger bulging of the target; (c) shows the removed plug and target scabbing;
(d) is an example of a combination of plugging and scabbing failure, while (e) shows a more
advanced stage of this process. The last two events are not frequently found, but show an
Fig. 3(a).
Fig. 3(b).
Fig. 3(c).
The mechanics of penetration of projectiles into targets
Fig. 3(d)
Fig. 3(e)
Fig. 3. Damagemechanisms:(a) Incipient plugging failure. (b) Completed plugging failure. (c) Incipient
pluggingand scabbing failures. (d) Completed plugging and well developedscabbingfailures.(e) Completed
pluggingand scabbingfailures.
interaction between scabbing and plugging. A possible application of ductile hole enlargement
hypothesis is presented in Fig. 4 for the entry of a conical-nosed projectile where slip lines
suggested by distortion of the rolling texture terminate away from the tip. This indicates that
plastic flow leads to rupture. Figure 5 shows two impacts of steel spheres striking at 30 and 60”,
respectively, from the normal to the plate. The first shows the formation of a plug, whereas the
second indicates a more hinge-like pattern.
Phase diagram
The phenomena described can be categorized by their dependence on variables other than
velocity. A useful concept for this purpose is that of a terminal ballistic phase diagram, which is
a representation of the intricate interrelationship between the principal parameters of an impact
system. The point of view is that of terminal ballistics with emphasis on the behavior of the
projectile. The central idea is that discrete final states of the projectile-target element system
Fig. 4. Internal deformations formed by penetration of a sharp projectile.
(b)
Fig. 5. (a) Plugging failure for an impact at 30” obiiquity. (b) Plugging failure for an impact at 45” obliquit
can be defined in terms of the final motion of the projectile relative to the target element-and
by the final condition of the projectile. The basis for identifying discrete final state of the
system in terms of projectile motion are the conventional ideas of perforation, embedment, and
ricochet. The basis for identifying discrete states in terms of projectile condition are such
concepts as projectile intact, projectile deformed, and projectile broken. Clearly, further
subdivision is feasible in terms of different modes of deformation and projectile breakup such
as distinctions between compressive and bending mode failure of the projectile.
An example of a phase diagram is shown in Fig. 6. It represents the behavior of a small
ogival-nosed projectile fired against a 0.635cm plate of 2024-T3 aluminum alloy. This diagram
has been constructed from a combination of experimental data and theoretical results involving
a particular model of the behavior of ogival projectiles fired against plates, using the methods of
[14]. The state variables in this representation are the impact velocity u. and the impact
obliquity 8. The diagram consists of a number of curves that represent the boundaries of
terminal ballistic phases. Any point on the boundary between states of perforation and either
ricochet or embedment is called a ballistic limit and the ensemble of points forms a ballistic
limit curve. Ballistic limit is a fundamental concept in evaluating projectile and armor per-
formance. This concept is formally defined in ordnance science; for example, the following
definition is given in [IS]:
“BaNistic Limit-The average of two striking velocities, one of which is the highest velocity giving a partial
penetration and the other of which is the lowest velocity giving a complete penetration. There are several
measures used in rating the resistance of armor or other materials to penetration, the three most widely
used criteria are: (I) the Army, (2) “protection”, and (3) the Navy ballistic limits. The essential difference
between these tests is the difference in the criterion employed to define a perforation as illustrated. In the
past, testing was performed using the Army or the Navy criterion for defining penetration, while the most
recent firings have emphasized the protection criterion.”
These concepts are shown in Fig. 7.

Other limit curves can be defined in an analogous way using criteria for projectile
deformation or failure in place of the criteria for perforation. The properties of phases that are
easy to investigate are best known, and phases that are difficult to observe and to rinalyze are
only poorly understood. Data and analyses on impacts at normal incidence and for velocities
less than 200 m/set are plentiful. Ballistic limit curves can be constructed for obliquities up to
50 or 60” with reasonable confidence. All other phase boundaries are, at best, speculative.
The individual phases of Fig. 6 correspond to appropriate correlations of initial and final
conditions. Thus, perforation intact corresponds to a correlation of initial speed and obliquity
to final speed and obliquity (where the obliquity is less than 90”); ricochet intact corresponds to
a correlation of initial speed and obliquity to final speed and obliquity (where the obliquities are
greater than 90”); while embedment corresponds to correlations of initial speed and obliquities
to penetration trajectories, which are usually characterized by penetration depth alone.
As with thermodynamics phase diagrams, any variable that defines the state of the system is
a state variable and could be used in these diagrams. Impact speed is virtually always one of the
variables because of its known importance. Impact speed and obliquity are appropriate to
represent the behavior of a specific projectile design against a specific target element. Impact
speed and the ratio of the thickness of the target element to projectile diameter have often been
used to represent the behavior of a specific projectile design against a family of target elements
in consideration of protective design. Early examples are given[l6], and others are scattered
throughout the military literature, usually without being identified as phase diagrams [ 171.
Figure 8 is a very crude phase diagram of the design-oriented type, portraying the behavior
of a blunt projectile in impacts against a single kind of target of variable thickness. The curve
that represents the variation of ballistic limit with the ratio of target thickness to projectile
diameter is adequately supported by data and analysis, but the rest of the limit curves are
hypothetical. These curves have been included because they illustrate considerations that are
important even if poorly understood.
The dependence of ballistic limit on target-element thickness has been extensively studied
and given simple empirical formulation. These tend to be piecise only for intermediate values
of hJD and are significantly less accurate for small or large extremes of this ratio. Here, h, is
the initial target thickness and D the projectile diameter. The predictions of a typical empirical
PERFORATES SHATTERED
1300 -
1200
IlOO-
-
PERFORATES BROKEN
T
:\
1000 -
900 -
800 -
700 -
600 -
PERFORATES INTACT
500 -
(BALLISTIC LIMIT CURVE)
400 -
300 - RICOCHETS INTACT
EMBEDS INTACT
200 -
(RICOCHET CURVE)
lOO-
O-
0
I
10
I
20
I
30
I
40
I
50
I
60
I
70
I
80
OBLIQUITY DEGREES
Fig. 6. Phase diagram for a 6.35-mm-dia. ogival-nosed projectile and a 6.35-mm aluminum alloy target
element.
ARMY PROTECTION NAVY

BALISTIC LIMIT BALLISTIC LIMIT BALLISTIC LIMIT
PEN1-j “IR; 1 PART9
WITNESS PLATE
Fig. 7. Definitions of perforation and partial penetration for defining the ballistic limit.
relation are depart from experimental data at low hAD and at high hJD. In the low range, the
reason for departure from experimental results is the occurrence of extensive “dishing” of the
target element, i.e. bending of the target element that extends 8-10 projectile diameters away
from the point of impact. For the blunt projectile, this phenomena interferes with the
mechanism of perforation and increases the ballistic limit beyond the trends described by the
PERFORATION
NO PERFORATION
SHATTER
SHATTER
PERFORATION
NO PERFORATION
INTACT
h,/D
Fig. 8. A phase diagram using the parameter hJD as a variable of state.
empirical equation. There are many observations that the dishing phenomena decrease with
velocity as the “dishing” limit curve suggests but the quantitative description of this depen-
dence is speculative. The identification of this phase is most significant in decisions of how
much of a complex target must be considered in providing an adequate dynamic model of the
impact. This will be discussed in Section 5.
A phase diagram will not have a ballistic limit curve if, for any reason, perforation is not a
terminal ballistic option. For example, in terminal ballistic studies of soils, it is appropriate to
consider vastly extended targets and to use the idealization of the “semi-infinite target” for
which no characterizatidn of target dimensions is needed. It is implicit in this idealization that
the dimensions of any target that is actually penetrated in the laboratory or in the field are large
enough so that it is indistinguishable from any larger target. The idea of a ballistic limit is not
applicable here by the very definition of a semi-infinite target element; however, the concept of
a ricochet limit is still meaningful, and a semi-infinite target could be defined as one for which
the ricochet limit curve depends only on the characteristics of the projectile and the target
material characteristics. Figure 9 shows the ricochet curve for sand[l8] compared to the
ricochet curve and ballistic limit curve for a comparatively thick aluminum plate. The ricochet
curves exhibit some of the same trends.
Methods of observation and analysis

Three general types of approaches for quantifying the interaction of penetrator and target
are :
(a) The experimental method either for the delineation of empirical equations (involving
primarily expressions for ballistic parameters in terms of their material properties, geometry
and initial velocity) or as a check and control on the predictions provided by theoretical
methods.
(b) Analytical procedures based on some relatively simple hypothesized model of system
response that then permits the stipulation of a forcing function acting on the penetrator, and
(c) Numerical solutions executed on high-speed digital computers and involving very
complicated programs that are based on conservation laws, compatibility relations, assume
constitutive equations, and failure criteria. Historically, the first known efforts to provide usable
information in the field of terminal ballistics date back about 230yr and involve the empirical
penetration relations attributed to Euler[19] and Robbins[201. While the advent of the 19th
century brought about the mathematical description of simple structural components such as
M. E. BACKMAN and W.GOLDSMITH
2800
1800
800
t
/
0 10 20 30 40
OBLIOUITY
50
DEGREES
60 70 80
! 90
Fig.
9.Phasediagram spheresimpacting
for6.35-mm 9.0-mmaluminum target
alloy elements andsand.
bars, beams and plates (first in terms of lumped-parameter and subsequently as continuum
representations), the first stipulated perforation process was the plugging model advanced by
Martel in 1897[21]. The initial proposal for concurrent multiple mechanisms producing per-
foration of plates was presented by Gabeaud in 1935[221.Fluid mechanical approaches for the
description of cratering in semi-infinite targets were initiated by iipik[23]. The closely related
theory of the Munroe jet, whose effect was first published in 1888[24] was not quantified until
1948[25]. Around l%O, various investigators, including Bjork[26] and Wilkins[27], developed
two-dimensional Eulerian and Lagrangian codes capable of analyzing such geometrically
limited interactions between projectile and target element with solutions obtained from high-
speed digital computers. Only recently has this field been extended to the three-dimensional
domain[28,29]. Current efforts are still being exerted in three disparate areas of observation
and analysis, employing increasing complex experimental equipment, composite penetration
models, and constitutive equations and failure criteria.
This paper will be concerned with a brief review of previous summaries covering the field of
terminal ballistics, a description of the methods employed for quantifying the events, a
description of both projectile and target characteristics, a summary of current knowledge for
penetration into semi-infinite target elements, thin and intermediate target elements, thick target
elements, and a discussion of important unsolved problems. A major use of this work is
believed to be the very extensive bibliography provided.
2,REVIEW OF PREVIOUS SUMMARIES INVOLVING TERMINAL BALLISTICS

While a number of books have been written on the subject of interior and exterior ballistics,
there are far fewer monographs devoted exclusively to terminal ballistics. The earliest com-
pilation in 18731301consists of an analysis of test results using an empirical relation known as

the Poncelet equation[31]. Shortly thereafter, the classical treatise of HeliC[32] devoted four
chapters to experimental data and their description by a variety of empirical formulas, followed
by a similar exposition for the perforation of armor plate[33]. Reference [34] summarizes
experimental results obtained earlier separating the targets according to the class of material
employed (concrete, steel, wood, earth and sand), and surveys proposed semi-empirical and
empirical force laws as well as mechanisms of penetration. Activities conducted in this area in
the United States during World War II were reported[l6]. In addition to the basic concepts and
variables involved in target failure, this source lists the relative effects of different projectile
types; that of normal versus oblique impact, and that of target parameters. The subject is
subdivided by target material (here separated into armor plate, concrete, plastic coverings, and
soil) with much of the information presented in qualitative, phenomenological, or empirical
form. Detailed references are given to individual investigations. Another monograph on the
entire subject of ballistics[35] contains a substantial section on terminal effects, primarily based
on empirical or quasi-empirical force-penetration relations that includes the topics of oblique
incidence and blunt and conical nose shapes. Force laws, post-perforation velocities and mass
loss predictions for frangible bullets were presented for various physical models. Some aspects
of the mechanics of penetration were also mentioned in two summary reports[36,37].
A contribution to the description of phenomena incident to the penetration of a projectile
into hard targets has been recently presented, based on semi-empirical and model
representations[38]. Empirical penetration data and correlating curves, as well as other terminal
ballistics information, is presented [39]; an extensive bibliography concerning this subject and
penetration mechanics is also available [40], and a compendium of quasi-empirical models of the
penetration process has been provided recently[41]. A compilation of analytical results
developed for the process of ballistic perforation of thin targets by rigid projectiles has been
published[l2], and a recent text emphasizing the fundamentals of interior, exterior, and
terminal ballistics of projectiles and warheads provides a description of the phenomena
encountered in these fields [42].
The subject of hypervelocity impact has received a great deal of individual attention,
particularly in the late 1950s and the 1960s. The Proceedings of the Hypervelocity Impact
Symposia (1955-1969) offer summaries of the topic and contain results of a host of analytical
and experimental investigations, the latter involving primarily light-gas gun data. Similar
information is found in the papers of the 1%9 AIAA Hypervelocity Impact Conference. A
collection of available knowledge of cratering and penetration under hypervelocity impact
conditions was compiled in 1.%1[43] and a review of the physical processes involved was
presented in 1%3[44]. A summary report containing substantial bibliographies in each section
on the study of hypervelocity impact phenomena categorizes the topic into meteoroid impact,
empirical approaches, theoretical approaches, and engineering considerations [45]. Another
useful bibliography dealing with the subject of meteoroid impact is contained in a companion
report[46]. Similarly, hypervelocity impact is also substantially covered in a bibliography
concerning aspects of the meteoroid hazard[47]. The special topic of the effects of hyper-
velocity jets and projectiles on rocks has received a separate review[48]. A collection of
articles presenting a combination of surveys and research information represents one of the
most recent contributions to the field[49].
Structural effects of impact are discussed in a series of papers[SO], while the subjects of
penetration mechanics, fracture, spallation, fragmentation and vulnerability are presented in a
symposium devoted to military ballistic problems [511. Considerable background information
and some discussion of specific ballistic situations are to be found in three monographs
covering the general topic of impact and wave propagation [lo, 52,531.
3. DESCRIPTIVE OR OBSERVATIONAL METHODOLOGY FOR

THE PENETRATION PROCESS
This section will be concerned with the general methods of measurement, analysis and
interpretation of the processes involved in the impact of penetrators on targets. The subject will
be divided into, (a) empirical relations based on experimental results; (b) analytical models of
the penetration resistance and the failure mode of the target at ordnance velocities; (c)
14 M. E.BACKMAN andW. GOLDSMITH
analytical models for hypervelocity impact conditions; (d) numerical evaluation of the inter-
action; (e) experimental methods of data collection.
The vast majority of the investigations have been concerned with thin and intermediate
targets where defeat of the target by perforation or severe deformation, or else defeat of the
projectile by ricochet or embedment without dysfunction of the substructure was the desired
objective. Furthermore, the preponderance of analytical treatments and controlled experiments
are concerned with the impact in which both the direction of travel and the projectile axis are
normal to the target element. The basic reason for this is that it is much easier to analyze the
normal impact case than impact at obliquity with a genera1 flight orientation. A further aim is
that of achieving more comprehensive analyses. It is also more difficult to experimentally
control the initial conditions for such types of penetrations. Projectile damage tends to be
ignored in terminal ballistic analyses and when incorporated, is of an empirical nature.
Empirical relations
Empirical relations used in penetration mechanics are most useful when the number of
variables being correlated is small. Expansion of the number of correlated variables or the
range of the variables is usually at the expense of simplicity and statistical merit. The empirical
relations commonly used in penetration mechanics are based on experimental correlations of
such parameters as penetration depth P, crater volume rC, the ballistic limit u50(or the residual
velocity after perforation v,) with the dimensions of the projectile, mass m, the impact velocity
vo, and obliquity 6 and the type of target and its thickness ho. Examples are:
Morin: (1833) P/D = 2mv~/alnD3 (3.1)

Dideon: P/D = azp, In (1 + a3uz) (3.2)
HeliC: P/D = (4.608m/a4D3np,) loglo (1 + a5vz) (3.3)
de Marre: (1886) E. = a6D’.5ho’.4 (3.4)
Helie: r, = a7(1/2mvo). (3.5)
Here, D and E. are the diameter and required perforation energy of the penetrator, pr is the
target density, and ai are arbitrary constants. In this presentation all units are in the SI system.
These avoid the problem of incorporating a large number of variables by restricting the
correlation to a single material and thus eliminating the parameters of the material description
completely or reducing these to a minimum. When used with strict adherance to this limitation,
the empirical relation may be the best available prediction. This is an indication that penetration
mechanics is a complicated phenomena that depends on many parameters, some of which are
difficult to control.
Empirical relations are in themselves not the answer because not only do they fail to give
any insight into the processes described, but their acquisition for the many uses of ordnance
science is prohibitively costly, not only financially but even more in the time of the designers
and analysts that need the information.
Dimensionless parameters are frequently used in empirical relations as a convenience with
regard to units and as an expression of the feasibility of predicting the behavior of systems that
are scaled versions of a known system. Scaling normally means that the space and time
variables are changed by the same factor and that velocities are invariant. The boundaries on
the impact regimes are expressed in velocities so that the scaling of a system will have no
influence on the phenomena experienced unless there is some specific space or time depen-
dence. Possibilities for such dependencies are flaw dimensions which influence material
strengths, textural and granular characteristics of materials, and rate dependence of materials
which introduces a characteristic time. A few empirical relations are explicitly size-dependent.
The basis for the dependence will be discussed in the summary of those relations.
Analytical models at ordnance velocity

Analytical models provide correlations like those from empirical studies, but introduce
relations between parameters of the system on the basis of physical requirements. Thus, the
number of correlated variables is increased and the form of the correlation is improved without
a loss of tractability.
Simple representations may combine an empirical element with physical principles; for
example, the resistive force can be assigned an empirical formulation. The empirical force law
is then used to determine a correlation of some penetration parameter involving velocity, v, by
solving an equation of motion. The best known of these laws is the expression
F = a$ + a9v + a10v2 (3.6)
in which the terms on the right-hand side have been subsequently associated with the cohesive
resistance of the target, frictional effects, and the acceleration of target material in the
neighborhood of the projectile (virtual mass effect), respectively[54,55]. Use of the first and
third terms was proposed by Poncelet [31] in 1829 and has found wide application in the
literature.
Alternatively, dimensional analysis has been employed to provide some physical guidance
for the specification of otherwise purely empirical relations, particularly for penetration depth
and crater volume. Use of the Pi groups provides ratios of like-dimensional quantities, such as
impact to sound speed of target, or Mach number vJco, or the density ratio of projectile and
target p,lp,, whose exponents are arbitrary and, together with a coefficient, are determined
empirically from test data. Other combinations include the penetration ratio P/D, the Best (or
Metz) number prvo2/BHN, where BHN is the Brine11hardness number, the dimensionless yield
strength aY,/p,c&, the thermal parameter V’/Q,, where QVis the heat of melting or sublimation,
and QV,,/QIII. A number of texts and monographs deal with this topic, together with the more
general subject of modeling[%-581.
Usually, in order to solve the penetration problem simply and overcome gaps in knowledge
of the process, the application of the physical principles has to be accompanied by restrictive
assumptions and/or a priori specification of some aspect of the system behavior. Examples of
common restrictive assumptions or a priori specification of behavior are ideal rigidity of the
projectile, ideal plastic behavior of the target, and specified modes of deformation in the target
that lead to perforation.
In some instances these models require only a knowledge of initial conditions, impact
geometry and material properties of the system. They are then self-contained and completely
predictive. In most cases the complete analysis requires either the a posteriori measurement of
certain dimensions or else an a priori assumption. A force law has been derived for penetrator
entry into a semi-infinite medium incorporating the effects of target strength, medium friction,
equivalent fluid dynamic normal pressure, and shear drag integrated over the striker surface
with provisions for separation and target surface effects [59,60]. This model can most likely also
be applied to penetrator entry into thin target elements with replacement of free surface
manifestations by suitable damage mechanisms.
For thin plates, nonperforating damage mechanisms have been modeled in terms of the
bulge by which the target conforms to the projectile, or the dishing due to radial propagation of
strains to produce the deformation shown in Fig. 1[61]. Single processes involving piercing
have been described in terms of the shearing of a plug or the ductile enlargement of a hole, as
indicated in Fig. 2. In addition, momentum or energy balances between striker and target have
been successfully applied, but both of these methods required an a priori assumption concern-
ing the terminal position of the sectors ruptured and displaced by the penetrator. Multiple
perforation models have utilized either successive or concurrent mechanisms, such as initial
compression and terminal shearing of a plug, with an intermediate stage involving both
actions [62]. Other formulations have included: (a) target compression due to projectile motion,
the boring and hole expansion effect at its tip (causing plate chipping), the effect of friction and
target inertia[63], (b) only target resistance and inertial641, and (c) the combination of target
inertia, resistance to flow and friction[rll].
Analytical models for hypervelocities

A very significant break in the methodology occurs between ordnance and hypervelocity
impacts. The larger stresses occurring in hypervelocity impact permit neglect of rigidity and
16 M. E.BACKMAN andW.GOLDSMITH
compressibility of the striking bodies, and the impact is viewed as fluid flow. The description of
material properties is then greatly simplified. Subsequently, correction terms were added to the
original Bernoulli equation to account for target and projectile strength. Models which use the
Bernoulli equation, or a modification, as the dynamic basis for the interaction have been
employed to study the cratering effects of meteors, the penetration of hypervelocity
particles[23,65,66], Munroe jets or shaped charges[25,67-711, and to study wave propagation
in long rods simulating kinetic energy penetrators [72-741. As a result of this development, there
are two velocity ranges for which it is comparatively easy to define phenomena and to make
predictions: the low-ordnance range, and the hypervelocity range. In the first, deformation of
the penetrator is avoided and target deformation is treated by methods that use simplified
representations of strength. In the second, both bodies are treated by a simple theory of fluid
interaction with or without strength. The higher ordnance and ultraordnance range is a gap in
which the effect of projectile strength and magnitudes of projectile deformations introduce
considerable complexity. The success of the incompressible hydrodynamic theory and the
parallel success of applying finite difference methods and large computers to solve explosive
interactions with metals lead to the idea of improving the hydrodynamic theory of penetra-
tion by treating the materials as compressible fluids. The next step is to introduce equations
of state into elastoplastic constitutive equations so that analyses can be made in the low-
ordnancelultraordnance gap.
Numerical evaluations
Codes for predicting penetration encompass the elastic/plastic/hydrodynamic regimes of
material behavior and require equations of motion, continuity, energy, and compatibility. The
constitutive equations here given for homogeneous and isotropic materials provide the relations
between stress and a+ usually in terms of their deviatoric representations
Sij = Uij - Crsij eij = Eij - ESij

with
s,.=l i=i 1
0 = -a,$ (3.7)
’ 0 i#j 3
In addition, a yield criterion is provided, ordinarily chosen as that due to von Mises
(U, - fl$ + (v.1- u,)* + (uz - a,)* = 2& or sijsij = 20& (3.8)
or that of Tresca
UI - u3 = UyT (3.9)
where UyT is the yield stress in simple tension, and ul, u2, u3 are principal stresses; associated
flow rules for these criteria are used to obtain the plastic deformation. A hydrostatic pressure
variation is specified, along with the introduction of an artificial velocity. For the original
HEMP code, the governing relations in a single radial space variable, r, or the two-dimensional
case involving space variables x, y and 4 are given by
One-Dimensional Two-Dimensional
Lagrangian form %+%+@Y=p~ (3.10)
of equation ax aY Y X
of motion %Y+&5Y+s~~“-%=p,ii,
ax aY Y
Equation of V 1 a(?-‘~,) _!!=av,+?!!Y+*J!Y
ij=7 ar (3.11)
continuity V ax ay y
Energy B - V[s,i, + (Y - 1)&l B - V[s,i, + s&,, + s&&p
equation +(pH+j)V=O + uX&] + (pH + j) V = 0 (3.12)
Artificial * 2
j= j = B2p,, $ $ (3.13)
viscosity j 0
The mechanicsof penetrationof projectilesinto targets 17
Equation of
&=2G(+/V) s,=2G(i,+V)+C,
state for
the elastic
92=2G(1+V) s,,=2G(i,-firlV)+C, (3.14)
solid
S~=2G(+‘lV) &Y=G(&X~)+CX~
(Not all stresses are required, but are indicated to maintain analogy to the two-dimensional
case.)
For
a& &&Y
Compatibility’ en=--;
ax 8Y
+ Y = 2: e, = 0 * =%. &=$,av,
EYY ay 9 (3.15)
Y = 3: iz = i3 = 0 ax ay
Hydrostatic
pressure
Equation (3.16) is the Mie-Gtineisen equation of state. Here, u and u are particle displacement
and velocity, .??is the energy, V = pJp is the specific volume, Y is a constant with values. of 1,2
and 3 for plane, cylindrical, and spherical geometries, respectively, G is the shear modulus, and
C’iiare correction factors accounting for the rotation of the element in a fixed x-y coordinate
system[75], S = 1 or 0 according to whether the two-dimensional problem is axisymmetric or
plane, and a superposed dot represents differentiation with respect to time. An artificial
viscosity, j, linear for the one-dimensional case and quadratic for two dimensions is introduced
to stabilize the numerical solution; B, and B2 are constants, S* is the sound speed, and A is the
zone area. The hydrostatic pressure pH depends on the internal energy per unit original volume
e(im)and various material constants aI that must be chosen so that p~( V) + (2/3)am reproduces
the Hugoniot for shocks above the elastic limit and p(v) = -K In V below, with X as the
compressibility. A typical uniaxial stress-strain diagram is shown in Fig. lO[lll.
SLOPE: A + 2G
SLOPE K + 4/3G
VELOCITY c = JEZZ
P
SLOPE K = X + 2/3G
Fig. 10. A typical uniaxial stress-straindiagramfor representingmaterialbehavior from elastic through

hydrodynamicregimes.
IJIi.9 Vol. 16. No. I-B

Numerous variations of the constitutive assumptions, extensions to include discontinuous

phenomena and failure criteria have been incorporated. In particular, a variable yield strength
has been incorporated of the form
(3.17)
where x = p/p0 - 1 and e(in), is the reference energy. One of the earliest compressibility
relations, included for historical reasons, is independent of internal energy, and given by
B [eB,(l-lx+U"3)_ 11
(3.18)
p = (x + p
where B3 and B4 are constants.

A current alternate relation for the hydrostatic pressure employed for the HELP code is a
modified version of that proposed by Tillotson[76] to permit a smooth transition between
condensed states (subscript co) where p/p0 < 1, or cold states, where e(in)< e&,, and the
hydrostatic pressure is given by [77]
(3.19a)
and that for the expanded hot states (subscript ex), where p/p0 < 1 and e(,)> ez”,
+A2x
by an intermediate region where ez”, < e(in)< e[i,, and p/p, < 1, given by
I
e-%@d”-*l! e-Ad(PJp)-ll* (3.19b)
p = (Q) - &JP~,, + (eh- %dPm

(3.1%)
e;id - em
with pO,e(i,,),,e&, e;in),Ai and Bi as constants for a particular material. The failure criterion for
this numerical solution is represented by a minimum allowable compression ratio; when
violated, all stresses in the cell are set equal to zero. Other types of constitutive equations,
failure criteria, and energy relations are found in the various numerical schemes developed over
the years. The constants for any of these expressions are obtained by matching them with data
obtained under shock conditions: the so-called Hugoniot relations which have been widely
published [78’,791.
Most of the codes use finite difference methods. Finite element techniques are being
explored, especially for three-dimensional problems, and the method of characteristics has been
used in rare circumstances. Three-dimensional codes are beginning to be used on test
problems and will undoubtedly find wide application in the near future, but numerical methods
thus far have been primarily limited to two-dimensional situations due to problems of computer
capacity. The code construction may be Eulerian, where material motion is examined relative to
a fixed reference frame through which the material passes; Lagrangian, where the history and
position of specified particles are traced, or a combination of the two. The first widely
employed program was the Lagrangian scheme HEMP[27]. Commonly used Eulerian codes
include HELP [80,81] and METRIC [82]. Other codes involving these or combined categories
are titled VISTA[83,84] incorporating particle-in-cell methods, PICKWICK [83,851,
SHAPE[86], CSQ[87], STEEP[88], WAVE-L[89], PISCES[89,90], CRAM[l I], SHEP[911,
STRIDE[92], TOODY 1931,TOOREZI871, DORF[94], and SWISSMI[95], a structure-medium
interaction version of the SWIS three-dimensional Lagrangian code[%]. Each form of program
has its peculiar difficulties, Eulerian problems occur in accounting for material strength and
treating interfaces, and Lagrangian problems with severe grid distortion. As the programs have
been refined, these difficulties have been mitigated by various methods.
Experimental methods
Basic data for characterizing a terminal ballistic event are those that determine the initial
and final states of a system as defined by the speed, direction of travel, flight orientation and
configuration of the parts of the system. In the conventional reference system, the projectile is
initially in motion relative to a target element at rest and the required measurements are: (1)
changes in the trajectory of the projectile including the generation of new trajectories for
fragments of the projectile, (2) the motion imparted to the, target, both overall and the
trajectories of fragments, and (3) the changes in the configuration of projectile and target
element.
The trajectories of projectiles are measured by either high-speed photography or from the
measurement on the perforations of a series of thin paper or plastic sheets that are located
along the expected trajectory and which are assumed to have minimal effect on the motion of
the projectile. The position of the perforations on the sheets determines the projectile location
in the plane perpendicular to the trajectory and the shape of the perforation allows a
measurement of flight orientation.
Average values of speeds are determined from transit times over fixed distances. Arrival at
premeasured distances is detected by the breaking or making of electrical circuits in the form of
wire or printed circuit grids, the interruption of light beams monitored by photocells, elec-
tromagnetic sensors, or by synchronized photography of the projectile. The use of a given
sensing system must include consideration of the aeroballistic phenomena that occur in the
velocity range of the observed event. For example, shock waves at ultraordnance velocities
may result in premature breaking of circuits and ionization occurring in shocks can cause
spurious responses in electromagnetic detection systems.
It is frequently necessary to measure speed at a known distance from the point of impact.
The speed then needs to be corrected for losses due to atmospheric drag. In many cases, the
drag coefficient required for calculating the loss is not known and two separate speed
measurements are needed to estimate the deceleration due to the drag force.
The motion imparted to the target is ignored in many investigations. When measured, it is
usually by means of a ballistic pendulum that provides data in the form of the maximum
excursion where the target element is known to be part of the total mass. The momentum
transferred to the target is determined from the excursion by conservation of momentum. The
use of ballistic pendulum data has received renewed interest in recent investigations in which
small changes of projectile speed were measured[97,98].
The acquisition of data on the projectile or target element fragment systems requires a
combination of either photographic or X-ray records and the recovery of fragments by
procedures that inflict minimal damage to the fragments. The task of measuring individual
fragment speeds and directions of travel from films is formidable if there are many fragments
due to problems of identifying individual fragments from frame to frame. Ballistic pendulum
techniques can be used to measure the total momentum of a system and the velocities can be
estimated from separate dam on the total mass of the system obtained from fragment recovery.
The common practice in acquiring data on changes in configuration of the projectile and
target element is to measure the principal dimensions of the crater in the target element such as
depth, diameter and crater volume, and the final configuration of the projectile as measured by
its final length, diameter, nose shape, and mass of the projectile. Improved analytical models
and more comprehensive numerical calculations have resulted in more careful and precise
comparisons between the final configuration of input systems and predictions, but no standard
treatment currently exists to handle special details in the changes of configuration.
Metallurgical examination of cross sections of the projectile and target element reveal
details of internal changes of configuration such as deformations and changes of micro-
structure, the generation of internal fractures, and the development of adiabatic shears (intense
localized shearing); also, the development of fractures initiated by adiabatic shears that
represent the first phase of a plugging perforation. Figure 4 portrays internal deformation
through distortions of the grain pattern that are characteristic of sharp penetrators. In a few
instances, targets have been prepared by laminating materials with contrasting colors and
textures so that internal deformations can be even more explicitly revealed.
Photographic systems are a means for obtaining a two-dimensional representation of the
impact event. These data are either two-space dimensional representations of observations at
discrete times, or the representation by streak photography in which one-space dimension is
converted into a time coordinate by sweeping a one-space dimensional image of observations
over the frame at a known rate. The latter procedure requires that the time history over one
dimension of the observed field of view provides significant information on the system as, for
example, the displacements of a target element at ‘a given position or the motion of the
projectile along a straight path. The potential advantage of this procedure is that one-
dimensional motions are shown continuously and to the precision allowable with the optical
system and the photographic film. A combination streak and framing camera has been available,
but its use has not been fully documented.
A framing camera records a series of two-dimensional views of the impact system for a
sequence of time intervals. Clearly, the records provide a wealth of information on the spatial
relationship among the parts of the observed system. In the application of the task of
determining the trajectories of the projectile and fragments, the precision of the results depends
on the resolution of the system, exposure time, and the framing rate. Broad categories of
framing camera systems are moving film, rotating mirror, and electro-optical, including image
converter systems. Moving film systems allow long record durations but are limited by the
mechanical problems of obtaining precision film motion as the framing rate increases. Rotating
mirror systems are used at the higher framing rates. With this method, the film is stationary and
exposure occurs by means of a high-speed rotating mirror (as in a streak camera system) and a
set of individual lenses that produce the images on the film. The capacity for higher framing
rate is obtained at the cost of a limitation on the record duration. Electrooptical systems are
based on the use of an electrooptical shutter and have the advantage that the exposure time can
be very short (duration of the record is limited only by associated electronics), but these
systems are very limited in the number of frames that are obtained. The electrooptical shutters,
for example the Kerr-cell, also allow a small fraction of the light to pass through the system
when closed and this must be taken into account when the illumination for the event is so
intense and persists between frames.
Impact events are normally not self-luminous so that an essential part of the photographic
system is the light source. There are several techniques for providing illumination. The simplest
is the shadowgraph consisting of a very small but intense source of light; preferably so small
with respect to the object field that it may be regarded as a point source and that casts a
shadow of opaque objects on film behind the field of view. The source is either pulsed or
shuttered and can be used with a lens system in a “focused shadowgraph” variation. Other
methods involve simple diffuse backlighting which simply provides a uniform background for
observation by a camera and schlieren systems that are particularly sensitive to changes in
density of the air around the system and detect shocks and bow waves from fragment and
projectile motions. Details of these procedures are given in [!@I. Holography offers consider-
able promise in recording the motion of complex fragment patterns formed by impacts and a
few applications of this method have already been reported[ NO].
A number of techniques have been developed for measuring surface displacement-time
histories and interpreting these in terms of stress wave effects or other internal disturbances.
These techniques include optical lever methods, laser interferometry, pin contactor techniques,
application of condenser microphone principles and various kinds of strain gages, and
examples of these are described in [loll.
X-ray photography is an alternative means for observing an impact system in essentially a
shadowgraph mode and with the advantage that these radiations penetrate the debris that is
formed at the entrance and exit sides of target elements in perforating impacts in which there is
appreciable breakup of either the projectile or the target element. Pulsed X-ray systems have
been developed that make observations in orthogonal directions about the direction of travel
and thus provide data for determining trajectories in three dimensions [ 1021.
Extremely high-energy pulsed X-ray systems have been developed that are capable of
penetrating target elements and revealing the internal details of penetrating events such as the
configuration of the internal cavity. These techniques will also reveal disturbances within the
impacting bodies and thus provide additional information for determining stresses and strains
within the system by the application of shock dynamical theories that relate stress to shock
propagation rates.
4. CHARACTERISTICS OF PROJECTILES
Classification
Projectiles can be characterized according to the method of launching and the commodity
that is delivered. The means for setting them in motion are projection from guns (bullets, shells,
grenades), drops from a parent vehicle (bombs, mines), and self-projecting devices (missiles,
rockets, torpedoes). The commodity to be delivered by a projectile solely designed to penetrate is
its kinetic energy. It is an explosive or incendiary warhead for projectiles that are intended to
approach but not penetrate a target. It is a combination of both kinetic energy and warhead for
penetrating weapons that count on explosive or incendiary effects to contribute to the dysfunction
of the target.
Kinetic energy penetrators

Examples of projectiles that are designed for efficiency of penetration of targets are shown
in Fig. 11. Armor-piercing projectiles have a hard core to which other components are attached
that serve exterior and interior ballistic purposes, e.g. a rotating band for spin stabilization, a
windscreen for aerodynamic efficiency, and a tracer unit to make the trajectory visible. These
parts of the projectile increase its range and assure a favorable orientation at impact but tend to
degrade performance slightly compared to a bare core striking under the same conditions. The
windscreen, for example, increases the ballistic limit speed by about 5%. The usual practice is
to consider the core and the basic penetrator and to correct predictions for the effect of the
.30-CALIBER .50-CALIBER
CARBON
STEEL CORE
LEA0 ALLOY
FILLER
STEEL CORE
ARMOR-PIERCING CAPPED
ARMOR-PIERCING SOLID SHOT WITH BURSTING CHARGE
’ STEEL BODY
ROTATING BAND /ARMOR-PIERCING
TRACER CAVITY CAP
INCENDIARY FILLER
RETAINING DISK
PENETRATOR
ADAPTER f STARTER MIX I
ARMOR PlERCiNG INCENDIARY

FINS
FIN STABILIZED DISCARDABLE SABOT
Fig. I I. Examples of kinetic energy penetrators.

windscreen[l03]. The projectile parameters that are required to determine the ballistic limit in
regions of the phase diagram where the projectile remains intact are the projectile mass )n,
its dimensions of length LA, diameter D, and nose shape and length, LN, and the density of the
penetrator pp.This set of parameters constitutes a minimum description of the penetrator. In order
to determine the deformation and breakup limit curves, a further characterization of strength of the
projectile is needed.
Figure 12 shows typical deformation of projectiles after impacts against SAE 4130 steel
plates (230 BHN) of up to 6.35mm thickness and at obliquities up to 60”. Compressive
Fig. 12. Examples of deformations that result from impact.
deformations and shattering occurs near normal obliquity, while bending and associated
fracturing occurs at significant obliquity. Very simple failure criteria based on critical stresses
for compressive failure and critical bending moments for bending mode failure have been used
to establish the boundaries in the phase diagram for perforation-intact. For example, the
breakup of projectiles is predicted by the quantity[103]
x*= 16, V&sin [cos-’ (cos 0 cos 4)] (4.1)

?rBa,D3 cos 8
which estimates the ratio of bending moments developed during impact to the bending moment
required for failure so that impacts with x* = 1 correspond to flexural breakup. In this equation 0
is the obliquity, 4 the pitch, and B = 11.2- 0.022 BHN.
Reference 11041predicted projectile deformation at normal incidence using a continuum
mechanical model which uses equation of state data augmented by elastic properties E, v and
strengths Us, unr.
Penetrators with warheads

Examples of designs of projectiles that are intended to deliver a warhead with significant
penetration into a target are shown in Fig. 13. The body of these projectiles and their high
explosive content are the principal elements of the penetrator, but the fuze is the vital
functioning element of the warhead. The penetrator must achieve some degree of entry without
dysfunction of the warhead through structural failure, fuze malfunctioning, or premature
initiation of the explosive system. Clearly, the design of penetrating projectiles with warheads
must embody a compromise between penetration and warhead performance. This compromise
has been expressed by determining the minimum thickness h, for the warhead case such that
the case will not fail in perforation of plates of known thickness h [105]. For flat-ended
projectiles, the thickness h, is estimated as
h, = h(a&r~p)Wn) (4.2)
where Sr is the mass of the projectile to the rear of section at which failure occurs which is a
region just beyond the bat nose section, a: is the dynamic shear strength of the target element,
and orrp is the strength yield of the projectile body.
The phase diagram for a high-explosive projectile will exhibit the main features of the
diagram for an armor-piercing projectile but, in addition, should have special phases for fuze
operation and dysfunctioning. The explosive filling of these projectiles decreases the average
density and the resistance to compressive and bending mode failure. As a result, the ballistic
limit is increased and boundaries on the phase for intact perforation are compressed.
EXPLOSIVE
COMPONENTS
PROJECTILE BODY /
HIGH EXPLOSIVE PROJECTILE
FUZING AND
EXPLOSIVE
COMPL)NENTS
WARHEAD CASE
GUIDED MISSILE WARHEAD
Fig. 13. Examples of projectiles with warheads.
Projectile parameters
Penetrator shape is significant in determining the mode of perforation and the resistance of a
target element to penetration. Pointed penetrators exhibit a piercing mode of perforation in
which target failure centers about the projectile axis. Blunt shapes, on the other hand, exhibit a
plugging mode of perforation in which target failure occurs over either a cylindrical or a
roughly conical surface. The transition depends on penetrator shape, but the relationship
between configuration and material properties that is needed to define that transition is only
known approximately. This correlation of shape to mode of perforation for target elements of
intermediate thicknesses has been investigated both experimentally and by analyses that
estimate the compressive, inertial, and frictional forces on a projectile[lO6]. For conical
projectiles, sharp and blunt shapes are defined as follows:
sharp.. . . .for nose half-angles. . . . .14”

blunt. . . . .for nose half-angles. . . . .!W
and definitions of “pseudo-sharp” and “pseudo-blunt” are given between these definitions in
terms of a critical angle that depends on the ratio h/d and varies from 30 to 50”. The concept of
a criterion for “sharp” and “blunt” shapes has been more simply stated in [103] by requiring
that LN/D 2 1 for “sharp” and LdD C 1 for “blunt”. The flight orientation c$,the angle between
the velocity vector and projectile axis, is an additional parameter that influences the mode of
perforation and this is treated as another aspect of shape[l03]. An impact by a sharp projectile
with 4 I IO”becomes blunt, i.e. it is assumed that perforation occurred in the plugging mode.
Empirical expressions for the ballistic limit will be presented in Section 7 that express the
difference between sharp and blunt projectiles by assigning slightly different forms for the
ballistic limit expression. These tend to attribute less resistance to the blunt shape. The
comparative penetration resistance of thin target elements to five penetrator shapes has
recently been studied experimentally[97] and numerically[l07], leading to the conclusion that
sharp shapes encounter less resistance than blunt shapes. It is evident, on the basis of these
experimental data, that the comparative efficiency of a sharp or blunt shape depends on the
relative thickness of the target element h/D. The details of this dependence is properly the
subject for Section 7, but the important point is that the recognition of the effect of sharp and
blunt shapes plays a key role in establishing the penetration resistance. Equations (3.1) and (3.3)
can be rewritten as
(4.3)’
(4.4)
to introduce the parameter [4m/ppnD3] which is the ratio of the length to diameter of a cylinder
of the same mass and diameter as a given projectile. This parameter has the property that it
specifies the ratio of major dimensions in a form that is independent of nose shape. It is clear
from these rewritten eqns (4.3) and (4.4), that it is desirable for a penetrator to be long and
dense in order to make the penetration large. Deformation of the projectile increases the
diameter of the penetrator and therefore decreases this same coefficient. Thus, the penetrator
should ideally be difficult to deform. Steels, forms of tungsten carbide and depleted uranium are
all attractive materials for use as penetrators. The latter substances have the advantage of high
density, but brittleness puts definite limits on the range of impact conditions for which these
heavy materials can be used. Steel is reasonably dense and can be given considerable
toughness; its comparative cheapness and availability are distinct advantages.
It is desirable that a penetrator be long, but two disadvantages appear with increased length:
(1) increased susceptibility to bending mode failures, and (2) exterior ballistic instability for
spin-stabilized projectiles. Projectile designs that use aerodynamic stabilization by tail fins (as
in Fig. 11, bottom projectile) avoid the latter problem. Many projectiles that are used to
penetrate soils are bombs for which fin-stabilization and length of the projectile tend to
overcome the inherent instability problems (as is discussed in Section 6).
Projectile and trajectory orientation

The well-established convention for designating the direction of approach by a projectile to
a target element is based on the choice of motion perpendicular to the front surface of the
target element in zero obliquity. Thus, the obliquity 6 of a projectile is the angle between its
velocity vector and the normal into the front surface. This convention and the following
relationships hold for all obliquities.
(1) A projectile that passes through the target element without being deflected has no
change of obliquity.
(2) A projectile that emerges normal to a parallel back surface of a plate has zero obliquity.
(3) A ricochet has final obliquity greater than 90“.
It is a common (and mildly confusing) practice to draw illustrations with the outward normal
as the reference vector for ease of presentation. Ricochet obliquities, both initial and final, are
sometimes measured as the angle between the velocity vector and the front surface. In that
convention, the initial angle of incidence is the complement of the angle defined above and the
final angle is the angle defined above less 90”.
The flight orientation of a projectile is the angular displacement between the axis of
symmetry and the instantaneous line of flight. There is no consensus on a convention for the
components of the flight orientation and, unfortunately, the convention for aircraft cannot be
applied because of the axial symmetry of projectiles and lack of significance of the gravitational
force. The angle of obliquity is an important parameter in specifying terminal ballistic
interactions and since it is defined as the angle subtended by the velocity vector and the normal
to the surface of the target element, it is convenient to choose a reference system for terminal
ballistics that has two directions in the plane defined by these vectors with the remaining
direction normal to this plane. The direction perpendicular to this plane is the pitch axis, with
rotations about this axis designated as pitch. The direction of flight is chosen as one of the axes
in the designated plane. The axis perpendicular to both the direction of flight and the pitch axis
”
NORMAL TO
t TARGET ELEMENT
Fig. 14. Kinematic and geometric properties of the projectile and target element used in defining obliquity and
flight orientation.
is the yaw axis and rotations about this axis are defined as yaw. These conventions for the
direction of impact and angular orientations of a projectile relative to a target element are
shown in Fig. 14.
Projectile deformations
Projectile deformations is clearly affected by the corresponding deformations in the target
element and cannot realistically be assumed to occur as an independent phenomenon; never-
theless, analyses that neglect target deformation have been used to estimate projectile yield
strength or mass loss. The earliest and simplest of these considered the impact of a cylinder at
normal incidence against a semi-infinite target element. The material behavior of the projectile
was idealized as rigid/perfectly plastic [ 1081 or rigid/work-hardening[109] while the target
element was considered completely rigid so that only projectile deformation could occur. A
later effort included elastic effects and a concave downward stress-strain law in the plastic
domain for the striker[l lo]. These analyses are uniaxial, incompressible, and neglect lateral
inertia. In tire case of the first two approaches, the portion of the striker not yet reached by the
plastic vave front travels toward the target as a rigid body with velocity a,. Comparison with
experimental results [ 11l] indicated that a rigid/work-hardening model provides good correlation
with observed terminal projectile shapes. The perfectly-plastic analysis predicts a series of
steps rather than the smooth contour calculated by the other techniques and observed
experimentally. The yield strength of the perfectly-plastic projectile was estimated from the
expression
(4.5)
with
$ = exp K- PPU~(V,+ 2c~N2&1 (4.6)

0
where J?+ was the final total striker length and J$’ the final length of its undeformed rear
portion.
This approach has been extended to include impact of cylinders against nonrigid plates of
finite thickness[ll2]. It utilizes two mechanisms, depending on whether the impact velocity u,
is below or above the plastic wave velocity of the rod cp = d(l/p)(do/de) which, for the
assumed bilinear stress-strain curve in accord with observation for steel, is given by CP=
Y&& where E,, is the plastic modulus of the projectile.
Two target deformation models are examined: (a) a deformable half-space, as shown in Fig.
15, that neglects wave motion and specifies a constant value of interface velocity uI deter-
mined by impedance matching with the striking rod, and (b) a rigid plate plus removed from a
plate of thickness ho, as shown in Fig. 16, where ur is time-dependent and the mass sheared
from the deformed cylinder is that beyond 1.25& in accordance with experimental data[ll3].
When the second model predicts more deformation than (a), it is replaced by the former.
Velocities are measured relative to a coordinate system in which the target is motionless before
impact.
Consider first the case of uI 5 cp. Here, cylinder mass m is constant and plastic deformation
dZ z
cp -v
ELASTIC WAVE
Co-V-
----- _
p
PLASTIC WAVE
w
- VI
V
ORIGINAL CYLINDER /TARGET

INTERFACE
Fig. 15. An idealized representation of the deformation of a projectile striking a deformable half space.
L; UNDEFORMED
LOST DURING
ATE PERFORATION
ACE
LOST THROUGH
STANDING SHOCK
Fig. 16. An idealized representation of the deformation of a projectile striking a target element that fails by
plugging.
occurs only within increment
dz = (cP -{v - or})dt = (CP- UL)dt. (4.7)
Conservation of mass requires that
where oL = v - uI,
Let L* = L, - c,t be the length of the part of the striker that is still elastic; then this portion has
mass mE = p,,A&*, and the decelerating force is F = -v&A0 = mE(du/dt) so that the in-
stantaneous relative velocity is
VL = v, + -$$ln(*-F)-VI.
P 0
For (a), pressure balance is obtained at the interface if
VO
with cH = m
v1= 1 f (p&/ppcP)
while, for (b), momentum conseryation with negligible impulse transferred to the target yields
m&, = mfiv + (m, - mE + m&l

or (4.11)
01= (m& - mEv)/(h - mE - mq)
28 M.E. BACKMAN and W.GOLDSMITH
where m4 is the plug mass M, =&+(1.25Q,),,)*h,.The length of the undeformed portion of the
cylinder is
for (a) LT = Lo exp (- u,p,cdZ*u~~) (4.12)

and (b) LT = L, exp (- u,p,cd&!s)[md(m, + h)l
where the impedance factor Z*= 1+(p,cdp,~,). Integration of eqn (4.7) yields z = f(t).
Equation (4.8) can then be combined with eqn (4.7) and implicitly solved for t = &A/A,), so
that A/A, can be related to z. This yields for the two cases
f=t+(l-$+{I
* +AdA~ew(~[l--f&I) (4.13a)
where
and
5E+ 5
;=Lo 1.54
K1-E>In1- **CPdL +t_+lnX*
CP?
1
+O.%~*~-[&*ln~*+~~j~O]ln[l+LO(~~_l)] (4.13b)
with x* = 1+ (m,,/m,,). Equation (4.13b) utilizes the approximation in the variable of integration
[(In T)/( 1- q)] = [(ln 7)/1.54] - O.%, which is satisfactory for Z < 0.1 where the cylinder is
nearly disintegrated. Also
$=(z+F)/[(?+$t(l-z)ln(t-E)-(?)$I (4.14)
and combination of eqns (4.13b) and (4.14) yields z = g,(A/A,,) at specified values of time.
For UI> cp, the plastic wave cannot propagate and a standing shock wave is located before
the interface which erodes the cylinder material passing by; when 2rr is less than cP and
approaches cp from below the scheme previously detailed applies and the problem can be
solved in two steps. First, it is necessary to determine the residual length L, when uI
approaches cp. This shortened cylinder of mass ml = (L,/L,)m, and initial velocity v* = Z*cP
is then treated by eqns (4.6)-(4.11) to ascertain its terminal shape as shown in Fig. 16. During
subsequent plate perforation, the laterally expanded portion of the mass may be sheared off and
this will also be considered.
Neglecting the distance between the shock wave and the impact interface and utilizing the
equation of motion yields upon integration for the half-space
(4. IS)
and for the plate, with mJm, neglected (i.e. a thin target)
(4.16)
with v, = cp.
In thin plate perforation, a ring of material is often separated from the deformed cylinder as
shown in Fig. 16. Test data indicate that the final deformed rod diameter is III = 1.250,, when
A/A, = 1.56. The time t is computed when the cylinder is deformed to this diameter, and
Z(Q)= z, can then be calculated. The residual mass of the cylinder after losses due to erosion
and/or shear is
mfrS)= p&L~ - Wf + 1.56zf) (4.17)
where LCrS,is the remaining cylinder length when erosion has just stopped. The predictions of
this method of analysis are in quite satisfactory agreement with the results of 0.22 caliber steel
fragments striking steel plates at velocities up to 1175m/s.
Other models combining striker and target deformation will be discussed in Section 7.
The stresses in a projectile that impacts a target element have been calculated as a function
of obliquity [ 1141by assuming hydrodynamic loading of the nose of the projectile over the part
of its surface that is immersed in the target element. The response of the projectile is modeled
by a rotary beam column that includes the effect of rotatory inertia neglecting the effects of
shear deformations, and by a shear column to compute the shear stresses. The forces applied
are assumed to be of the form
Fs = uybA tan 8 (4.18)
for the shear force and
F, = aycA = ppC,vy (4.19)
for the normal force where vy is the particle velocity corresponding to the compressive yield
stress. The maximum normal stress is found to be
u,,,, = 0.28~~~ tan 8, (4.20)
the maximum shear stress is found to be
US, = gyc tan t? (4.21)
and the maximum octahedral stress CQ~~)

is estimated as
%a) --~[2+1.12tanB+6.15tan2$l’“. (4.22)
5. CHARACTERISTICS OF TARGETS
Penetration mechanics is properly concerned with interactions between penetrators and
target elements. The analytical process for reducing targets into appropriate elements requires
many methods and data that are not a part of penetration mechanics; however, the criteria for
the size ‘of the target element and the parameters that are needed to describe it depend directly
on penetration-mechanical considerations. This section will survey types of targets and the
methods used to determine target elements. The assumptions and physical bases for charac-
terizing target elements will then be discussed.
Types of tatgets
Vehicular passages such as roads, runways, railroad tracks, locks, bridges and docks are
particularly simple targets in that the number of substructural components are small and
dysfunction is directly related to damage. In general, these target elements are either thick or
semi-infinite and the basis for characterizing target elements is fairly straightforward.
Munition and fuel storage facilities likewise involve comparatively few substructural com-
ponents. Target elements will vary from thin for storage containers located above ground to
semi-infinite for storage below ground. Dysfunction of these targets is not all commensurate
with damage, but the basis for evaluating dysfunction such as kills and tires is comparatively
simple.
Structures such as buildings, factories, bunkers, silos, and hangars may have considerable
substructural complexity, and target elements vary from thin to thick. If any significant part of
the structure is subterranean, that part of the target could be essentially semi-infinite. In
anything as complex as a factory, the relation between dysfunction and damage is clearly
complicated and requires separate analysis.
Vehicles such as aircraft, armored personnel carriers, missiles and ships involve extensive
substructure, and have many functional capabilities for which dysfunction needs to be deter-
mined. Target elements vary from thin to thick. The relation between dysfunction and damage
is complicated by the great number of substructural components to be considered, and the
number of functions that exist.
The penetration of a multicomponent target such as an aircraft or armored vehicle is usually
analyzed by following the line of flight of the penetrator through the target, and identifying
target elements along the way. In the simplest schemes, the trajectory is a single straight line. In
more sophisticated analyses, deflection of the line of flight is introduced at each element. In
either case, the properties of the target element are those at its intersection with the line of
flight. The curved surfaces of components are recognized only through the effect on the
obliquity of impact. Penetration of the target is reduced to a series of interactions with a
sequence of target elements that are characterized by the material properties at the point of
intersection with the substructure. Substructural components are usually classed as vulnerable
or nonvulnerable. If the component is nonvulnerable the impact is analyzed to determine the
residual motion of the penetrator. If the component is vulnerable the impact is analyzed for
residual motion and for dysfunction of the component. Reference [ 1131provides a summary of
the techniques employed.
This process can be carried out by an analyst working from target drawings and data on the
functional interrelationship among substructural elements or the process can be carried out by
computer programs that have automated the target descriptions, e.g. MAGIC [116], and the data
acquisition along the shot line, e.g. SHOTGEN [117], and the vulnerability analysis, e.g.
VAREA[118].
Target modeling hypotheses

There are several assumptions that are implicit in this approach to target characteristics:
(1) Localized influence. The assumption that only a region of the target with dimensions
comparable to the projectile diameter is affected by the impact and that the characteristics of
the substructure beyond that point are not relevant for the outcome of the impact.
(2) Neglect of rigid body motion of the substructure.
(3) Neglect of all thermal phenomena. This includes frictional effects with a variable
coefficient of friction and significant effects from the heating due to deformation and friction
such as changes in material behavior and phase changes.
(4) A corollary of the tirst assumption is that all target elements have planar surfaces.
(5) Initially stress-free target elements.
The employment of all but the last of these hypotheses greatly simplifies the analyses of
target behavior.
Assumption of localized influence

Observations on many kinds of projectiles and target elements show that cratering defor-
mation extends only two or three projectile diameters and, therefore, the assumption of
localized influence is adequately justified where this kind of deformation is dominant. The
assumption is much more questionable for impact velocities near the ballistic limit and for thin
plates that exhibit the plate-bending modes of deformation called dishing. The few in-
vestigations of this dishing process[61, 119, 1201,of impacts by flat-ended cylinders against thin
plates indicates substantial plastic deformation well outside the “cratering” region. For exam-
ple, permanent deflections for aluminum alloy plates struck at speeds up to the ballistic limit
were observed extending 10-12 radii from the impact axis[61]. Further details will be found in
Section 7.
Neglect of rigid body motions

It is characteristic of the ballistic viewpoint that the target merely represents an environ-
ment for the projectile. The temptation inherent in this viewpoint is to assume that each part of
that environment is fixed in space, i.e. that there are no rigid body motions of any of the target
elements. Thus, the characteristics of the target that are needed to determine its interaction
with the penetrator are those of a small target element, by the assumption of localized
influence, but the inertia of the element is that of the whole target by this assumption of
negligible rigid body motion. However, the assumption can result in significant error even if the
target mass is significantly larger than the mass of the projectile, and if small local motion
occurs[l21]. Clearly, if the target and the projectile masses are comparable and the projectile
embeds in the target, then conservation of momentum will require that the combined masses
possess the initial momentum and the final rigid body motion wiil depend on the ratio of the
projectile mass to the combined masses.
It is unlikely that a projectile will have mass comparable to the target. It is far more likely
that a target will have substructure with mass that is comparable to that of the projectile and
that this is attached to the rest of the target by mounting devices with some given strength. The
analysis for this system can be made using a scheme analogous to that employed for the
analysis of perforation by plugging[l22, 1231. The projectile striking a substructure uses a
certain amount of energy in separating it from the target and in deforming it. As in [122], the
energy used in deforming the substructure is just the right amount to assure, from a combined
energy and momentum analysis, that the separated unit and projectile will have the same
velocity after impact. The expression for the final velocity has the same form as that of a plug
sheared by a blunt striker.
Neglect of thermal effects

The neglect of thermal effects is standard practice in much of mechanics. This is justified by
the small dissipation of heat in much of dynamic elasticity and fluid flow, by the insensitivity of
many of the material properties to temperature and by the rapidity of the mechanical events.
Penetration processes are highly dissipative and, moreover, this dissipation occurs in com-
paratively small regions. Furthermore, both the rate of dissipation increases and the region of
dissipation decreases with increased penetration rate so that while material properties can be
stated without attention to thermal properties for armor penetration problems, hypervelocity
impact requires equation of state data and the accounting for changes of temperature[l24].
Some theoretical studies of iinpacts in the ordnance range have used such equations of state
and considered thermal effectsIll, 68, 104)].
Thickness characterization
Definitions of thin, intermediate, and thick targets were given earlier. These definitions can
be related simply to target and projectile dimensions through considerations of the length of
time required to communicate effects from the back surface. Assuming that these effects are
propagated at elastic stress wave rates (which is realistic at the lowest velocities), the number
of traversals of the target thickness during a single traverse of the projectile is
n= 3 L
(>O
CP 7i
(5.1)
or, expressed as the dimensionless ratio h/D
(5.2)
Thin target elements are those with n greater than 5; this number is chosen as a somewhat
arbitrary estimate of the requirement for stability of stress in the region immediately ahead of
the projectile. Intermediate target elements are those with 1-e n < 5 that are influenced by the
stress-free rear surface but (by this arbitrary criterion) not enough to approach equilibrium.
Thick target elements are those with n < 1, where the distal surface reflections return no faster
than those initially traversing the projectile. These choices of traversal numbers, for which the
exact numerical values are stipulated a priori do establish rough ranges of relative target and
projectile dimensions for which it is safest to compare results from experiments to theory.
There is experimental evidence that dishing is not confined to the thin target element range,
but dishing is the dominant process in that range. Dishing may extend out to h/D values
corresponding to n = 2, but with substantially lower magnitude and minimal below the value
!I= 2.
Target element materials

An integral part of penetration mechanics is the delineation of the material characteristics of
target elements. This can be accomplished qualitatively by the distinction between natural
substances such as soils, rock, wood and water, or manufactured materials such as metals (and
alloys), plastics, ceramics, and miscellaneous compounds like rubber. It is also convenient to
consider materials according to weight categories. Lightweight material, with specific gravities
up to about 3, include wood, plastics, ceramics, soil, many rocks and low-density metals and
alloys. Intermediate range weight material with specific gravities up to about 8 encompass most
of the common metallic compounds such as steel, copper, brass and tin. Heavy materials with
specific gravity above 8 include lead, tungsten and uranium. Target resistance can be opera-
tionally described in terms of penetrability of a standard projectile fired under identical impact
conditions. However, the analysis, prediction, and interpretation of penetration phenomena
requires a specification of the physical properties of such materials that must include initial
density, a complete constitutive description, and fundamental damage properties; that is,
yielding and failure limits.
Penetrability of substances is a quantitative measure of particular interest to ballisticians. A
comparison of performance of a single projectile type against a spectrum of materials will
establish some type of order among them. For example, a monobloc round-ended steel
projectile with’ a length-to-diameter ratio of 8, striking normally at 150m/s, will penetrate
representative media as follows:
Penetration
Material (dia.)
Wet mud[125] 2200

Sand[lZS] 350
Concrete (2500psi strength)[l26] 36
Concrete (5OOOpsistrength)il261 25
Aluminum allo; 2024-T3Ildj] _ 1.5
Steel (BHN) 100[103] 0.6
Steel (BHN) 350[103] 0.3
Comparative penetrability is the basis for a very crude classification of target materials as
low resistant (mainly soils), moderately resistant such as concrete, masonry and low-strength
metal alloys, and highly resistant, including the high-strength metals, alloys and ceramics. These
classes tend to correspond to different penetration phenomena, the use of different investigative
approaches, and correspond very roughly to the present classification of targets by relative
thickness and by density.
The low resistant materials are almost exclusively soils. These tend to be porous and
particulate. Further, many such as clays are highly anisotropic and inhomogeneous. Consider-
able variation in their constituents and in the size of inclusions will be found from one sample
to another. Soils are rheologically very different from structural materials so that it is seldom
possible to compare the characteristics of these two classes of materials. Semi-infinite targets
are primarily composed of low-resistant substances.
At the other extreme, highly resistant materials approximate homogeneous, isotropic
behavior and their properties are reproducible and consistent. Target elements are treated as
structural entities that’ permit explicit examination of their transient response. These materials
are of primary interest in the case of thin and intermediate target elements.
The characteristics of moderately resistant substances is a mixture of the homogeneity and
isotropy of metals, the inhomogeneity of concrete due to its aggregate, and the anisotropy of
reinforced concrete and wood. Rheologically, these are treated with reasonable accuracy by the
same models as metals. Moderately resistant materials are primarily used for thick target
elements. In all cases, the target material must be properly identified in terms of its physical
characteristics.
The behavior of a specified material is usually represented in terms of a mode1 that depends
on its state and character. Most targets are characterized as solids that may be regarded as
either a continuum or a particulate system. Gases are excluded from present considerations, but
in some instances the substance acts like a fluid, as for example the behavior of solids under
the extreme pressures produced by hypervelocity impact. Target character, particularly for
solids, is determined by its homogeneity (relative to pulse length for dynamic conditions),
isotropy, and composite nature including initial solid/fluid combinations. Solid models include
one or more of the domains of elastic, plastic, viscous, or hydrodynamic behavior. Fluids are
described as incompressible or compressible and either viscous or inviscid. Permanent defor-
mation of a solid occurs when stresses attain a critical value, or yield limit, and is described by
means of a flow rule, while rupture (or fracture) is characterized by an ultimate strength.
Stresses, in turn, are designated by their action as either tensile or compressive, shear, or a
combination such as bending. Failure of a liquid results in cavitation when the tensile strength
of the substance is exceeded.
An accurate specification of material response of any solid is complicated by the extreme
range of stresses to which an object is subjected during penetration and by the diversity of
deformation and failure patterns that occur. Furthermore, some of the proposed models of
material behavior incorporate extrapolations from limited experimental information into
domains of doubtful or, at best, unverified validity. For example, while three-dimensional
behavior in the elastic region of solids can be accurately predicted by extensions of one-
dimensional data and the magnitude of the elastic constants are found to be identical under both
static and dynamic conditions, these conclusions do not have comparable validity in the plastic
domain. Here, no reliable data is available for dynamic yielding, especially under multiaxial
stress conditions. Behavioral descriptions in this regime are specified to be identical to
idealizations of static deformations such as eqns (3.8) or (3.9) and their associated flow rules.
Hydrostatic pressure variations such as eqns (3.16) or (3.18-21) are inferred from one-
dimensional mechanical Hugoniot measurements with internal energy variations either deduced
from a specified model or else assumed a prior?. Some questions remain concerning the
applicability of this information to three-dimensional stress situations or its extension to other
polytropic thermodynamical processes. In fact, one of the largest sources of inaccuracy in the
numerical solutions employing the various types of codes is the uncertainty in the specification
of material behavior.
The materials of common target elements exhibit differences in behavior that ideally should
be incorporated in constitutive descriptions. Metals and plastics approximate the homogeneous
and isotropic class of solids to which the most well known constitutive relations apply. Their
description is mathematically simplified because their comparable or identical response under
tension or compression, and their cohesiveness. In contrast, fiber-embedded matrices and many
rocks are distinctive as anisotropic materials; ceramics are characterized by drastically different
strengths under tension and compression; earth materials are typically inhomogeneous; and
particulate media such as sands and soils are inhomogeneous and noncohesive. Constitutive
relations for these materials are not well developed and infrequently applied.
The material descriptions of metals and alloys merit special attention because of their
frequent occurrence as target elements. The isotropic elastic behavior that is appropriate for
metals and alloys requires the specification of two constants, Young’s modulus E and Poisson’s
ratio v (or alternatively, the Lame constants A and G), and the density of the material. The limit
of this elastic domain is characterized by a yield strength uy, ordinarily secured from uniaxial
stress-strain curves in tension or compression. A multidimensional yield surface is constructed
using this value on the basis that it is a function only of the stress invariants. One of the most
common hypothesis is that uy depends only on the second invariant ;Sijsii as, for example,
described by eqn (3.8), while another relates this value to the maximum principal stress difference
(Tresca criterion). Associated flow rules specify the direction of the strain vector as normal to the
yield surface.
Either perfectly plastic solids, where uy is invariant, or work-hardening materials where gy
is a function of prior plastic work performed, are utilized for stress analysis [ 1271.In some cases
the yield stress has been found to be dependent on strain rate[128]. Numerous models also
superpose a viscous deformation component on plastic strain[l29,130].
A universal fracture criterion, such as the history-dependent yield surface, in stress space,
does not exist at present. On the other hand, simple uniaxial failure limits have been specified
by an ultimate tensile strength uvT and/or a failure strain luT. In addition, the total energy in
IJESVol. 16.No. I-C

34 M.E. BACKMAN andW.GOLDSMITH
the stressed region must decrease if fracture is to proceed. The hardness of the material is
characterized by some measure of static or dynamic resistance to indentation such as that
involved in ball penetration tests where the crater diameter may be expressed as a Brine11
hardness number (NW). This quantity can be expressed approximately by [43]
l3HN = 3.61~~~. (5.3)
Metallurgical processes or conditions, though vital for material behavior, cannot be considered
here due to space limitations.
The simple failure criteria cited are frequently employed only because it is not possible to
make exact predictions of such failures as spallation, where the applied stress state (and
frequently its history), the size of the stressed region, the critical fracture stress value, and the
effective surface energy of the material to be cracked would have to be known
beforehand[ 1311.A cumulative damage criterion that has found recent popularity[l32] yields a
stress functional f[u(t)] such that
f[o(t)]=-&,-U)b’ and o(,k)5oso, (5.4)
where A1 and bl are constants, o(fch)is the theoretical cohesive strength, and a, is a measure of
the long time fracture stress oR.This relation was found to be in good agreement with observations
for 6061-T6 aluminum. The cumulative rate of void formation leads to the expression for the
fracture time tR
I:” f[u(t)] dt = 1 (5.5)
where an example of the functional is given by eqn (5.4). The above expression can be directly
related to plate spa11experiments where oR is measured as a function of stress pulse duration.
Other failure modes require the specification of different criteria; for example, the observed
presence of adiabatic shear in plugging has led to the construction of a computational model for
shear bands[1331. An extensive discussion of fracture criteria for homogeneous materials is
given in [134].
The enormous scope of theoretical descriptions and experimental investigations of material
behavior permit citation of only a few isolated examples of actual data. Table 1[135,136]
presents static constitutive and failure information for three typical homogeneous and isotropic
metallic substances often used as target elements. Linear work-hardening can be employed to
approximate the plastic regime when yield and ultimate strengths are significantly different. For
a .number of metals, the uniaxial stress-strain curve (including yield and failure limits) has
been found to be significantly strain-rate dependent[l37,138]. In general, aluminum alloys are
relatively insensitive, with maximum deviations of about 15 percent, and none in some cases.
Titanium and many steels exhibit substantially larger variation with this parameter. Several
monographs have been devoted to the examination of dynamic material behavior [139,140].
Temperature elevation generally reduces the magnitude of moduli and values of limiting
strength. For 2024T4 aluminum a reduction of 25% in E has been found at 425°C relative to
room temperature[l41]. To a much smaller extent, the same trend is noted with increasing
specimen size. A value of 1000s-’ and a range of 50-400 s-’ for the viscoplastic relaxation
time y has been employed for aluminum and various mild steels, respectively, to characterize
this type of behavior [119,1421.
In the hydrodynamic regime, density and compressibility are the primary variables depicting
material behavior; strength considerations are secondary. A hydrodynamic model of behavior
of a solid must provide a pressure-density relation, with or without variation of internal energy,
such as eqn (3.16) or comparable equations involving shock velocity U and particle velocity.
The information necessary for construction of a uniaxial stress-strain curve incorporating
compressibility, such as shown in Fig. 10, can be obtained from a Hugoniot curve representing
shock front data. Figure 17 exhibits the experimental results from three metals that can be used
Fig. 17. Hugoniotsfor three metals.
either directly or with mixing theory mod~~tions (that are not expected to produce si~ificant
numerical differences) to represent the hydrod~amic behavior of the substances in Table 1.
Both adiabats and isotherms in the neighborhood of the curves of Fig. 17 can be calculated
from them by appropriate thermodynamic relations [783.
Information on other metals is listed in a host of sources[78, 13.5, 1361. A hysteretic
elastic/ideally-plastic solid using a nonassociated flow rule has been utilized to represent
earth[143], while a cap model has been employed to portray rock[W]. Mechanical properties
of rocks and soils are summarized[l45-141, and those for plastics may be found in [148].
Light-armor material data has also been coflected[l49] and information on ceramics is given in
{150].
A large class of “homogeneous” target mate&s is composed of two or more diffusely
Table 1. Mechanicalpropertiesof three typicalmetallictarget materials
E, GPa a, (0.2%)MPa u,, MPa Reduction

P G Elongation, in area,
Material gIcm3 T C GPa v BHN T C S T C S %in2in. eUT %
2024Al-T4 2.77 74.2 j2.8 27.6 0.33 1’20 324 324 114t 438 283 20 0.010
Ti-W-2.5 Sn 4.46 110 110 48.2 0.34 286 827 8% 45s 861 793 758 15 0.011 40
(millannealed)
SAE, AfSI-4130 7.85 200 200 77.5 0.29 305 910 IONI 500 1034 655 20 0.010 60
oilquenched
at 857°Cand
temperedat 538°C
tTaken as 55%cryr
distributed components that may be either solid, liquid, or gaseous. Examples include light-
weight armor consisting of fiber-reinforced matrices, honeycombs, plastic foams and all
biological systems. A variety of methods have been developed for the constitutive description
of these as well as “nonhomogeneous” systems such as a filled fluid conduit. Each of the latter
has been treated on an individual basis. The scope of this subject is too broad to be covered
here; numerous references can be found in texts and journals in the fields of material science
and mechanics.
Soils, masonry and concrete are multiphase media requiring consideration of the interaction
of+at least two components, including the special case of solids and voids. The interaction of
these phases has been analyzed[l51] by the use of a model involving a hollow sphere of
incompressible material obeying Coulomb’s law of failure to obtain the constitutive relations
for the volumetric response of porous rocks. Figures 18 and I9 show a comparison of
loading and unloading predictions with data for sandstone and tuff.
P
Kbars
V(porousmaterial)
Lx=
Vlmatrix material)
Fig. 18. A theoretical constitutive relation for sandstone compared to experimental data.
6. SEMI-INFINITE TARGETS
Semi-infinite targets have been defined as those for which there is no meaning either to a
distal surface or to a ballistic limit. Examples are massive structures such as dams or buried
facilities that are attacked by a penetrating projectile with a warhead and for which some
degree of penetration is needed in order for the warhead to achieve acceptable amounts of
damage.
The phase diagram for a semi-infinite target has no ballistic limit curve and the ricochet limit
curve is determined solely by the properties of the material and the characteristics of the
projectile. Projectile integrity requirements are often extremely important and concerned with
comparatively small amounts of damage since considerations of fuze and warhead malfunc-
tioning can hinge on levels of deformation that would be considered nondeforming for a kinetic
energy penetrator. Thus the phase boundaries to be determined for semi-infinite targets are the
ricochet limit curve and a projectile deformation curve for a small specified degree of
deformation. Figure 20 is an estimate of the terminal ballistic phase for a hard soil (such as a
silty sand) struck by an explosive-filled, ogival-nosed projectile at zero degrees pitch and yaw.
The ricochet limit curve is estimated using the same basic procedure employed in determining
Fig. 6[14], but with a linear dependence on velocity. The deformation limit is here arbitrarily
40
30 -
P
20 -
KbXS
10 -
0
0 1.1 1.2 1.3 1.4 1.5 1.6 1.
V(porous materml)
(1=
Vlmatrlx material)
Fig. 19. A theoretical constitutive relation for tuff compared to experimental data.
800
PENETRATION
PROJECTILE
DEFORMED
/
/
PENETRATION
PROJECTILE
UNDEFORMED
I I I I /I I 1 I
10 20 30 40 50 60 70 80 90
OBLIQUITY DEGREES
Fig. 20. Phase diagram for an ogival-nosed projectile and a moderately resistant soil
38 M.E.BACKMAN andW.GOLDSMITH
chosen as the value of the axial force corres~nd~~ to an impact velocity of 550 m/s and 45
degrees obliquity. These conditions are arbitrary, but are in the regions where failures have
been observed.
Underground structures, runways, and roadways are often attacked by low-flying aircraft
delivering bombs on the target. In this tactical situation, ricochet is of interest not only because
it is a failure to achieve the conditions required for target dysfunction, but it is also a danger to
the attacking aircraft. In the past, the dynamics of this problem has received little attention and
there are few analyses and data on the ricochet curve. Several of the analyses have used media
that are easy to define and describe such as water and sand[l52-1561. Reference [IO31contains
an analysis based on data for metallic targets, but the form of the final relations could apply to
hard soils since the development is in no way dependent on the relative thickness of the target
element. The analytical method[l4] has been applied to ogival projectiles to predict the curve
shown in Fig. 20 by using constants that are appropriate for a m~erately resistant soil.
An analysis for ricochet from water, which also applies to sand[154,1.56], considers the net
effect on the trajectory of a sphere by the lift force due to the flow of water over the immersed
surface and the gravitational force and provides the expression
2 4&t
v=1 -‘- P **2
(6.f)
lQP*
as an overestimate of the ricochet curve where B* is the complement of the obliquity angle in
radians. The experimental ricochet limit curve for sand is shown in Fig. 9 and compared to the
ricochet curve for a thick ~uminum alloy piate (141for which there is ex~rimen~l verification
from 3o-60”. The curve for sand has the same trends except for the abrupt drop at 61” that
apparently coincides with the sound speed.
The determination of ricochet curves for soils and soft rock that are penetrated by
projectiles of standard pointed shapes is complicated by the common observation that these
projectiles have unstable flight orientations and trajectories. This type of projectile usually
pitches or yav+s to the point of tumbIing and trajectories develop severe deflections that are
descriptively called “1-ing”.
The tumbling and deflecting effect is greater for pointed than for blunt shapes and is reduced
by large length-to-diameter ratios. The shapes of the tunnels produced in soils suggest that the
flow of soil around a sharp-nosed projectile undergoes a separation from its body, as is also
inferred from the selective scraping pattern on projectile bodies. These flow separations are
capable of causing the observed instabilities because these can result in the relocation of the
center of pressure to a point ahead of the center of mass. The motion of the projectile is then
unstable in that any pitch or yaw will tend to grow rather than be damped. The trajectories of
the projectile undergo :arge deflections because of increasing magnitudes of the components of
force acting away from the axis of symmetry. The immediate implication of this kind of
unstable behavior is that ricochet limits for pointed projectiles are strongly dependent on flight
orientation since small values of pitch can result in a significant change in trajectory. In most
cases this will mean some degree of indeterminacy in the ricochet limit since small deviations
of flight orientation cannot be detected.
The establishment of the depth of penetration as a function of impact speed, obliquity, and
flight orientation is of primary interest. A number of empirical relations will be discussed and
several analytical models will be summarized.
Empirical Relations
Empirical relations that have been used to compute the depth of penetration from projectile
and target characteristics are
Poncelet
(6.2)
Resal
(6.3)
Petry
P=a,*yog(l+&) (6.4)
Young[ 1661
P = 13.4o,&dm us<61 m/s

(6.5)
P = (0.257)a& y( D- 30.5) u, 2 61 m/s
J--
where uI3and Cpare material and shape factors, respectively. These all assume a straight trajectory,
and if impact is at obliquity, it is assumed that there has been no deflection due to the
unsymmetrical loading of the projectile during entry into the target element. Table 2[42] gives the
range of values of a,@, as, alo and a,, for several common soils as designation of their trafficability.
It is obviously helpful to have some externally observable characteristic of the medium, such as
trafficability, in order to establish at least a crude classification of the soil.
The Young equations are distinctive in the square root form in the relationship of mg/A and
the consequent implication that the penetration scales as
This conclusion on scaling is based on experimental firings into soils with projectiles that range
from 100 to 460 mm in diameter[l66]. Many dynamic systems are such that if all spatial and
time dimensions of the system are scaled by a common factor and the same material is used in
Table 2. Penetration resistance constants for three soils
Historical
(a,,@) for long, Density values General
slender a,, for Petry tratlicability
weapons equation g/cm3 a, aI0 estimate
I. Soft:
Partially saturated
clays 0.017-0.028 1.6-2.0 Too soft for
Wet tide flat muds 40 0.078 1.1-1s ordinary
Soft wet clays 50 I s-2.0 wheeled
Typical soft soil vehicles
(sand, 30%; silt,
30%; clay, 40%) 3.34 17.8
2. Medium: Passable for
Loose, moist sand 6.5 0.0072-0.010 I.6 wheeled vehicle
Loess (loam) with varying
Loose, dry, sandy degrees of
alluvium difficulty
Moist, sandy, silty
clay
Typical loam
(sand, 50%; silt,
35%; clay, 15%) 7.31 20.7
3. Hard: -
Dry, silty sand 2.5 0.0035-0.0039 2.7 Passable for all
Frozen clayey silt 3.8 0.0042-0.0046 wheeled vehicles
Glacier ice 4.2 0.0057 1.0
Hard, dense, clayey
silt 5.2 2.6
Typical sand
(sand, 95%
silt, 5%) 15.7 24.7
both the scaled and reference systems, then common phenomena wifl occur at comparable
points of space and time[%]. This does not hold if the material properties are in any way
explicitly dependent on a size dimension or on time. The possible dependence of soil
parameters on the comparatively large size of soil granules and inclusions would make the
applicability of strict scaling questionabIe. Certainly, if materials are expliWy size-dependent,
,as for Figs. 18 and 19, then scaling would include scaling of the media.
Equations (6.2) and (6.3) can be obtained by integration of the equation of motion with an
axial resisting force of the form of eqn (3.6). Clearly, the constants in eqn (3.6) can be
interpreted as lumped parameter represen~tions of all relevant phenomena that are, respec-
tively, independent, linearly dependent, and quadratically dependent on velocity.
Equation (3.6) has been used to correlate experimental data obtained for sand [54,55]. The
form with a9 = 0 is found to fit the data provided that two velocity regimes are identified, which
appear to be the subsonic and supersonic domains for the medium and that have different
mechanical responses.
Equation (3.6) has been interpreted much the same way [157] but explicitly identifying ato as
pressure drag’, u9u as viscous drag, and us as plastic flow resistance so that
ACod
UlOV = -
2 (6.7)
aqv = 6~~~(~2)v (6.8)

as = Auy. (6.9)
A similar conclusion was reached using a force law that contained a power series in both
velocity and penetration, yielding{1581
rn~=Al+A2~+A32:+A~~+AI~~+**. (6.10)
when coefficients Ai were statistically correlated with experimental data for which the diameter
D and nose length LN were varied, the equations of motion were found to be
(6.11)
Analy~jcal models
One approach to the problem of deep penetration of semi-infinite target elements has been
to modify static analyses of the expansion of cavities with spherical and cylindrical shapes [ 1591
to account for dynamic expansion and work-h~dening by employing idea~~ations of material
behavior that simplify the analyses but approximate real materials[l60]. These analyses were
originally concerned with the penetration of spheres into metallic target elements that were
regarded as semi-infinite in that perforation and the effects of distal surfaces were excluded.
More recently, these same procedures have been applied to compressible materials 11611that are
more common as semi-infinite target elements, such as soils and concrete[l62,163].
The deep penetration theories first considered the entries of a rigid sphere into an isotropic
elastic-plastic medium. A theory of cavity expansion idealized material behavior to have certain
“locking” properties. The material exhibits ideal locking behavior in dilatation and bilinear
elastic-plastic strain-hardening behavior in shear. The elastic locking condition requires that on
application of a hydrostatic pressure to the cavity, the material around the cavity obey the
relation for dilatation
e,+2e.$=& (6.12)
where iE is the elastic locking strain, while in shear the stress-strain relatipn is
vi+- ur = ~G(Q, - c,). (6.13)

When the yield stress is reached, the material experiences plastic locking stresses in dilatation,
but plastic behavior in shear so that
&+2E$+=BP (6.14)
where iP is the plastic locking strain, and
2
a,+- a, = uyc + 3Ep(eb - c.,). (6.15)
Figure 21 shows properties of the idealized material and the two regions around an expanding
cavity where they apply. The expanding medium exerts on the cavity a pressure that depends
Fig. 21. Stress-strain diagrams for locking-idealizations of material behavior and regions.
on the cavity radius. The exact relation is-obtained from an equation of motion for the
expansion of the medium, the stress-strain relation for the material, and conservation of mass
and momentum applied to the expanding boundaries of the locked elastic and locked plastic
regions. The pressure has a static and a dynamic part given, respectively, by
(6.16)
and
pD = p$(B,ri: + B2?) (6.17)
where
r is the Eulerian radial coordinate

PQ is the elastic locking density
p$ is the locking density
A, = (&2E) - (~g3)
EE is elastic locking strain
42 hf. E. BACKMAN and W. GOLDSMITH
EP is the tangent modulus for linear strain hardening.
The rate of expansion of the cavity is equated to the velocity of the rigid sphere and a
cosine variation of pressure over this sphere is assumed in order to determine the force
resisting penetration. The equation of motion for the rigid sphere is then
(6.18)
and the penetration P is
(6.19)
A variant of the cavity expansion model has been used[162] to determine the axial resisting
force for deep soil penetration. The axial stress is composed of a radial stress CT~
and a shear
stress us, so the equation of motion is
(6.20)
where a, and OS are assumed to be of the form
a, = f 1+ sin 11f V$)C+~~, (6.21)

crs = (1 + sin r))ascCe,e? (6.22)
The quantity (1 + sin ~1)is a factor evaluated from the shape of the projectile by
(6.23)
for a conical nose, and

(4LID) (6.24)
sin?-/
=(4LQLlp+1
for an ogival nose, and the parameter JI is a form of the Best number, given by
(6.25)
The parameters crti,) and ctsfCe)are values of the normal and shear stress obtained from cavity
expansion theory modified to make compressibility effects consistent with an analysis[l61]
when reduced to the incompressible case. These have the form
(6.26)
4 sinIF
(6.28)
k=3( 1 f sin[F)
where a(&) is the cohesion, tj? is the internal friction angle, and f is the rigidity index.
Table 3. Material properties for use in eqn (6.26)t
Soil type (g,cPm’) %Hh (dk.) I
Clay 2.8 0.419 x IO5 41 13

Gypsite deposits 1.92 2.78x 16 46 60
Selenite deposits 1.94 2.37 x IO’ 42 60
Clayey silt I .54 1.15 x 16 33.4 2%135t
tReference [163).
SVaried with depth.
Table 3[163] lists the constants approximating the material description of several specific
soils and [162] gives further details on the relationship of these parameters to material behavior.
The models discussed above assume that only an axial force acts on the projectile and that
the projectile follows a straight trajectory. Another approach[59] is based on a relation similar
to eqn (3.6), but with specific interpretations of the components in terms of media parameters,
and generalized to compute the force on the projectile for unsymmetrical contact with the
resisting medium. In this description, the expression for stress at any point on the projectile is
given by
cr={pcu
exp(-a,t[H(t- t,)]+ a&l +f)+ C,fpv2+ Gipv2 (6.29)
where parameters c14, aIs, C,, C,, and f characterize the medium and are determined from small
caliber experiments; H(t - t,) is the Heaviside step function_ The total force on any projectile is
determined by a two-dimensional code that integrates the stresses over all area increments. This
total force is then used to determine the three-dimensional rigid body motions.of the projectile.
Experimental inputs are needed to characterize a given medium that, once characterized,
applies to any axially symmetric projectile interacting with that substance. The term
pcu exp (-a,& represents shock phenomena, aI5 the static resistive properties of the medium,
(1 +fl frictional effects, and C, and C, the dynamic pressure due to inertial effects. Figure 22
presents the predicted and measured flight orientation of a 20-mm projectile in soft soil when
striking at an obliquity of 45” and an initial velocity of 280m/s, that are in reasonable
agreement. The lack of precise knowledge of the mechanical response of the highly variable
medium represented by natural terrain might readily lead to significant differences in calculated
and observed trajectories, particularly since small variations in material properties can have
large influences on the unstable flight path of the penetrator.
Numerical methods
Numerical methods have been applied to soil penetrations using the relations (3.7-3.12) and
Prandtl-Reuss descriptions of material behavior[l64] to match given properties. The codes
used for this work were WAVE-L and PISCES-DL2[89,90].
The final results indicated that after the nondeforming projectile has been fully embedded in
a given soil layer, the penetration process is steady and the transient responses, for which
numerical methods are.most appropriate, are a minor part of the event. Disturbances of the soil
do not extend far from the projectile but are appreciable within a projectile diameter and result
in separation of the soil from the projectile along the nose.
Deformations of the projectile are extremely small compared to those in the resisting media
so that it is appropriate to model the behavior of the projectile as rigid. Since the resistance
offered by the medium is low, the trajectories are of the order of hundreds of projectile
diameters in length. On the other hand, if the entry phase is to be treated in any detail, the mesh
size for the calculation must be a reasonably small fraction of the nose length and the conditions
for stable computational procedures will demand a small time increment, so that the number of
calculations will be large. The number of steps required for following deformations of the
projectile will be still larger since even sinaller cell sizes must be used to characterize the
deforming projectile.
44 M. E. BA~KMAN and W. GOLDSMITH
I f I 1 I I I I
Z
V 1 980 FT/SEC IN PLANE TRAJECTORY

SURFACE
2--
-4 -
PREDICTEO TRAJECTOR
AND ORIENTATION
12 -
OUT OF PLANE TRAJECTORY
0 30Q
I ~~
I I I I I I
4 8 12 16 20 24 28 32 36
Fig. 22. Predictions and measurementsof a curved trajectory and changes of flight orientation for a 20-mm
projectile penetrating sand.
Experimental procedures
Earth penetrators are intended to deliver a warhead whose detonation either damages a
subterranean structure or massive target, or else causes extensive cratering to a runway, road,
or other vehicular passage. For this task the projectile should be of the order of half a meter in
diameter. The experimental basis for conclusions on the penetration of semi-infinite targets
consists of some small scale measurements with projectiles of the order of centimeters in
diameter, but the bulk of studies are at full scale. This has been true because of the ~nding~~57]
that modeiing by geometric similarity is not reliable. Full-scale testing is expensive because
even the simplest measurement, the penetration depth, requires the handling of massive targets
or excavating with heavy equipment. The large size of earth penetrators has the advantage that
it is feasible to instrument the projectile in order to obtain records of the projectile response to
the,penetration process.
Accelerometers have been mounted in projectiles and the readings telemetered to a
receiving station by a trailing antenna[l65,166]. These records have correlated with such details
of the target structure as the layers of differing resistance to penetration. A recorded sharp
increase in deceleration at the end of the trajectory is attributed to the change from a cavitating
flow that limits contact to the nose section to a Aow that envelops the entire projectile.
A simple accelerometer uses the inden~tion made by hard spheres that are driven by a fixed
mass into plate. The device has been calibrated beforehand so as to record the maximum
acceleration of the projectile[W]. The measurement method correlated to the damage ob-
served in projectiles of the warhead type. The location of the damage was at the regions of
greatest change of thickness and agreed with that designated in [IOS].
7. THIN PLATE PENETRATION AND PERFORATION
Incontrast to the semi-infinite medium where neither bulk target deformation nor the exit of
the striker on the distal side is considered, projectile impact on a thin element will produce the
plate deformation of bulging and dishing, and various modes of perforation. The target damage
patterns shown in Figs. 1 and 2 occur, particularly those of 2(b), 2(d), 2(f), and 2(h). These are
produced by elastic, plastic and hydrodynamic wave propagation involving normal, bending and
shear stresses, by frictional effects, by the initiation, propagation and arrest of cracks leading to
both microscopic and macroscopic fracturing and changes of phase at ultra-high speeds. The
analyses of these processes is simplest for the significant range of impact conditions in which
the projectile does not deform. However, the projectile is subject to the same failure
mechanisms, although these may be manifested in a different fashion. Elastic effects in the
target elements by themselves may be neglected [ 168,169], but even when generated by impacts
in remote regions[l70], these could become significant in the presence of other loads.
The following treatment of these target elements will be subdivided into empirical relations,
analytical models of target damage produced by rigid projectiles, models that include the effect
of a deformable projectile, hypervelocity characterizations, and the application of numerical
methods to the general continuum mechanical treatment of thin and intermediate target
elements. Experimental investigations of thin target elements are voluminous and closely tied to
models; they will not be separately cited. Normal incidence is usually chosen to simplify
experimental and analytical procedures with the tacit assumption that the conclusions for this
case are fundamental and can be logically extended to account for obliquity. Impact at obliquity
and a nonzero flight orientation are discussed to broaden the scope of this treatment and to
assess this assumption.
Empirical relations
The phase diagram shown in Fig. 6 is typical of systems for which deformations of the distal
surface of the target element are important in governing their dynamics. There is a phase for
embedment of the projectile, but since the ballistic limit is low, this phase is restricted to slow
speeds and the phase narrows to insignificance with increasing obliquity. Predictions of
penetration per se are of limited interest. The ballistic limit and the residual velocity are the
quantities required in order to assess the protective capability of the system.
Reference [ 1031presents a collection of equations consisting of either empirical relations or
simple formulas derived from analytical models that specifically apply to target elements of thin
and intermediate thicknesses. These equations have been expressed in terms of dimensionless
parameters and were chosen to provide the information for a simplified phase diagram using the
parameters describing common military projectiles, warhead fragments, and target elements.
Figure 23 is an example of such a simplified phase,diagram. It will be noted that an embedment
phase is ignored and all nonperforating impacts are regarded as ricochets. The equations
1000
PROJECTILE RICOCHETS INTACT
800
z
E
t? PROJECTILE PERFORATES
; 600 WITH CORE BROKEN
0’
,4 PROJECTILE RICOCHETS
z 400 -
10 20 30 40 50 60 70 80 90
OBLIQUITY DEGREES
Fig. 23. A simplifiedphase diagramfor a 7.62~mmdia.projectileand a 1.27-mm-thickaluminumalloy plate.

46 M. E.BACKMAN and W.GOLDSMITH
contained in the collection are of two kinds. The first defines the boundaries of phases by the
ballistic limit curve which is given by expressions of the form
(7.1)
and.by the projectile breakup limit curve expressed in the form
Bm,,D3
VA =
J
m sin [cos-’ (cos 8 cos &)] cos 8 (7.2)
which is derived from eqn (4.1) and in which the projectile parameters D, m, and uuP are those
of the core of the projectile; for fragments, the deformation limit curve is given by the
empirical equation
vi = Bdl + cos B/(B#dLd] + Be)] (7.3)
where Ld is the length of the fragment.

The other kind of equation defines the changes in the state of motion and changes in the
shape of the projectile in a given phase. For example, in any perforating phase, the residual
speed of blunt projectiles and fragments is given by the equation
(7.4)
which will be discussed under plugging perforations. The change of mass of a deforming
fragment is predicted by
where m* is a fixed mass of 0.031 kg. This expression was obtained from the results of a large
series of experimental firings[l71]. The set of equations was selected to achieve uniformity in
the calculations performed by different organizations of vulnerability of aircraft targets. The set
is internally consistent, based on available experimental data and includes as an integral part
deformations and breakup of the striker.
Another form of the ballistic limit equation similar to eqn (7.1) that implies size dependence
is given by
B,( hlD)o.7
‘.50 = Do.os(mg/D3)o.s (7.6)
where the constant 8, depends on target material and projectile shape. Equation (7.1) with
BZ = 0 is nearly the same as that derived from an analytical model for a conical-nosed
projectile[41]. Yet another equation derivable from an analytical model[41] which predicts
effects similar to eqn (7.1) is
>3
38.2( mg/ D)3 -- c2 ln f + ~~50sin B
h/D = (7.7a)
(Ktg#2(sin B + f cos B) “‘O sin B C2
where
(7.7b)
Other relations correlate the specific limit energy, mg(&)/D3 to h/D as given in [391.
The ballistic limit velocity unsohas also been given as the sum of the velocities required to
produce the various mechanisms that are activated, i.e. lateral displacements, plug shear,
petalling, bulging, dishing and penetrator deformation. Fire 24 represents the data for the
ballistic limit of hardened steel fragments of various sizes and a blunt cylinder with a hardness
of 230 BHN. The cylinder impact is represented by equations of the form of 7.1, given here by
uSso= 1100(h/D)“~7s
m/s. (7.8)
The fragment data illustrates the cumulative effects of plugging, bulging, dishing, lateral
expansion and penetrator deformation [ 1721.
.The effect of nose shape has been empirically reported as the ratio of the ballistic limit for a
blunt projectile to the ballistic limit for a sharp projectile[411
U”5WrW= e-b,‘ND’+ 1.14~1 _ e-b5W’)) (7.9)

b5GXsharp)
in which bg and b5 are dimensionless constants dependent on plate properties. Corrections of

these limits to account for various degrees of nose sharppess are also given.
In the hypervelocity regime, the plate thickness required to stop a particle has been cited in
terms of the kinetic energy To and Young’s modulus of the target Et [1731as
h = 3(2)“3($* (7.10)
PLATE
MIL 12560 230 BHN
7.6 WAL FSP
1200 -
12.7 WAL FSP
20 MM WAL FSP
1100 - A BLUNT RIGID COMPACT CYL.

BHN 230
1000 -
900 -
BLUNT RIGID PENETRATOR

-PENETRATION- ADDED COMPONENT
BOO -
DUE TO PENETRATOR
DEFORMATION
= 700 -
e
-z
,& BLUNT RIGID PENETRATOR
5:
-SIMPLE PLUGGING-
’ 600 -
TRANSITION FROM PLUGGING TO

PENETRATION
BLUNT RIGID PENETRATOR (AVrjop)
“5On = “SOr + A”500 + Ai&, + AVsw
200
100 ADDED COMPONENT DUE TO DISHING & BULGING(D.“~(M)
0.6 0.8 1.0 1.2
T/d
Fig. 24. Contributions to the ballistic limit for rigid and deforming blunt compact fragments striking steel
plates.
48 M. E.BACKMAN and W. GOLDSMITH
where C, = 6.9 GPa; the hole diameter is generally substantially larger than the fragment except
for very thin plates. The hole diameter DCfor thin plate perforation has been described by the
relation [ 1741
(7.11)
where coefficient Cz was evaluated as 2.6 from theoretical considerations, but a value of 3.2 was
found to better fit data for Pyrex spheres fired into thin aluminum targets at a velocity of
1.5mm/ps. Alternatively, eqn (7.11) can be replaced by a relation developed from an empirical
expression based on shear strength[175] and developed from a host of experimental in-
vestigations, and is given by
(7.12j
Threshold penetration information for 2024 aluminum targets struck by nylon andcadmium
spheres in the range from 3.0-U mm/ps indicated the applicability of either of the relations
proposed earlier[176], with run as the percent elongation in 2 in.
(7.13a)
or
(7.13b)
Steel rods fired at rolled homogeneous armor, normally and at 60” obliquity, were first fitted to
an empirical expression for the ballistic limit given by[177]
(7.14)
with B, and b, are constants. This was simplified for a specific projectile-target configuration,
yielding
(7.15)
with constants 0 s Bq s 1 and b3> 1 determined from a least squares data regression analysis;
for LID= 5, Bq=0.74, bg=4.4, and vnSO=720 m/s. Variations of the ballistic limit with
obliquity, residual mass expressions and distribution of the fragments behind the target are also
empirically described [ 1771.
Analytical models
An impact produces a contest between target element and projectile for the occupancy of a
region of space that initially belongs to the target element. The idealization of the projectile
behavior as rigid greatly simplifies the analysis of this process, and it is, therefore, convenient
to consider two kinds of models: (1) those that assume a rigid projectile, and (2) those that
account for the deformation of the projectile.
From the definition of rigidity, the only permissible effect of impact on a rigid projectile is a
change in its motion, and the deformable target element must undergo local displacements that
accommodates its shape to that of the intruding projectile. These displacements result in forces
on the projectile that, in turn, change its motion. An analytical model of penetration by a rigid
projectile must delineate these kinematic changes and their influence. An essential part of the
specification of displacements is the internal separations that permit perforation.
The mechanics of penetration of projectiles into tar@s 49
Salient aspects of any model of penetration by a rigid penetrator must include the following:
(1) an assumption about the local displacements of the target element; (2) a choice of the
internal separation mechanism required for perforation (which is dependent on the shape of the
projectile); (3) a principle for associating these kinematic details with changes of projectile
motion; (4) criteria for the termination of the impact.
Single force representations for the normal impact of rigid projectiles

Deformation without perforation. Plate deformations consist of bulging-where the plate
conforms to the shape of the projectile-and dishing-where bending of the plate extends
outward from the contact and between projectile and plate. Bulging may be treated by the static
and quasi-static methods of analysis used in metal processing problems[178, 1791.Dishing, on
the other hand, requires some type of dynamical theory. The modeling of dishing has
considered elastoplastic deformation of both membranes and plates. The impulsive loading of
elasto-plastic membranes is the simplest case involving two stress components. Only one space
variable is employed for axial symmetry, and this system might approximate the dynamic
deformation of sufficiently flexible and thin target elements[52]. The central normal impact of a
pointed projectile on a circular membrane at constant velocity has been depicted in terms of the
variable rlt, presuming dynamic similarity, a tool likely to be of considerable importance for
penetration mechanics. This has the advantage that the phenomena can be described by
ordinary rather than partial differential equations. The same similarity hypothesis was employed
to depict the impact of a cone on an infinite elastic-perfectly plastic, or work-hardening
membrane[l80] utilizing the analogy between the behavior of thin plates and wires.
The dynamic response of thin plates, either infinite in extent or with various circular
supports, and composed of rigid/plastic, viscoplastic or elasto-viscoplastic materials has been
studied for impulsive loading over all or much of their surfaces[52, 181, 1821.These conditions
correspond to blast loading rather than projectile impact [52], but introduce the important
concept of regions separated by stationary or outward traveling yield hinges. In subsequent
formulations, reasonable correlations with experimental results were obtained only when both
bending and membrane stresses were incorporated[l83,184].
Efasto-plastic plates. The governing equations for these cases[l85, 1861also permit large
deformations and strains and may serve as the starting point in a description of permanent plate
deformation accompanied by perforation. The maximum central deflection of a perfectly plastic
plate subjected to projectile impact has been calculated from an energy balance based on an
assumed permanent deflection shape of the target[lIV]; its extension to work-hardening
materials is sketched below [119]. For the usual thin circular plate of radius R, plane stress and
symmetry conditions, the total plastic strain energy I$ is
(7.16)
where E is strain, r and I$ refer to radial and circumferential directions, respectively, and
integration is performed over the total volume P. Neglecting c, and the contribution of the
second term of eqn (7.16) and using the von Mises yield condition for a linear work-hardening
solid defined in terms of an initial yield stress uy,, and a constant a*, given by
a,’ - up, + 0; = a; =(cry0+ff*e,y (7.17)

eqn (7.16) becomes
uyoe, + ia*e:
I
r dr.
For large deqections, the .radial stretching strain for small radial displacements can be
approximated by E, = 1/2(aw/ar)* so that eqn (7.18) becomes
(7.19)
DE.5Vol. 16,No. I-D

For an assumed permanent transverse deflection profile w(r) = w&r), where w, is the per-
manent deflection at the origin, equating the loss of the striker kinetic energy (AT), to relation
(7.19) yields
[I,“(~~4rdr]w~+[~R(~)‘~d~][~]-4(1~~~~”z(AT), =o (7.20)
which may be solved for w when f(r) is known. Experiments indicated that the profile could
empirically be represented by the relation f(r) = e-‘, yielding for eqn (7.20)
“‘;-;; “)(AT& = 0. (7.21)
This equation was in excellent accord with data obtained from the impact of a 12.7-mm-
diameter steel sphere on a l.27-mm-thick 2024-Oaluminum plate for impact velocities up to
120m/s[119].
Elasto-uiscoplastic plates. Mathematical difficulties noted in solutions of perfectly plastic
and work-hardening plates subjected to point loading have been obviated by the inclusion of
rate dependence. A number of investigations have been performed concerning the normal
central impact of projectiles on viscoelastic circular plates[120, 142, 183, 187-1891, using a
uniaxial constitutive equation of a viscoplastic solid given by
(7.22)
where y is the inverse viscoplastic relaxation time.

The motion of a viscoplastic plate of radius. R, and perfectly plastic yield moment per unit
length MY subjected to central projectile impact based on small deflections and only flexural.
action is
t+h$+-$$lrGR,tps)=O, OCrGR, (7.23a)
3’tiy
-p,h$=O, R,<r<R, (7.23b)
v4v + 2hMy
where ps is the static pressure distribution’ corresponding to dynamic loading p, amounting to

47rMylfl for a concentrated load at r = 0. Then
(7.24)
replaces eqn (7.23a). This system is solved using appropriate boundary conditions. An approxi-
mate solution is given by, with r* = r/R,
x(&,(r*)exp{-A$/t}-$I-
VT r*2(1-2111 r*)l (7.25)
with
6 = 3flyptRtlZM,,
I\ = 16rp,hR: Ir p,hR:
1 m ( *--B -) m
and
8phR:
n = 2,3, . . . ,m. (7.26)
=5$i-
Motion ceases at time t, where v(r*, t) = 0, given by
m -I+,
(7.27)
p,hR: > I
and the final permanent deflection is given by
II . (7.28)
This result is expressed in terms of the static collapse load F,” and the initial central velocity of
the first plate mode fiOwhere
(7.29)
for the present case. Good agreement has been obtained between the predictions of eqn (7.28)
and experimental results for 6.35-mm-thick mild steel targets of 101.6-mmnominal diameter out
to the clamped edge when struck by mild steel projectiles of 10.52-mm-diameter weighing log
over a velocity range from 91.5 to 595 m/s and bullets of the same diameter weighing 31 g
over a velocity range from 204 to 292 m/s[189]. The static collapse load was estimated to be
between 24,500 and 26,700 N and the viscoplastic relaxation time was chosen as 2.5 ms.
For plates considerably thinner than those.employed in the tests above, eqn (7.25), which is
the first term of a series, does not represent a suitable approximation. Consequently, an
alternative method of solution for the boundary value problem of very thin plates was sought.
This was achieved by means of the Laplace transform technique for the*velocity which then
permitted explicit evaluation of the displacement at the origin, yielding
w(O,t)=
(vo-s >[ +e b*Z’
erfc(b*P)+ ;(;lQ+]+~l-$$f;P
(7.30)
with
b* = 16prm:
mS
where erfc is the complementary error function. The velocity and acceleration of the impact
point can be obtained by direct differentiation of eqn (7.30). The permanent central plate
deflection is calculated at time tf when ~(0, tf) = 0; this is given by
41rMy
ebo2’ferfc (b*t;‘2) + *[l-26*e)1’2]=o. (7.31)
vo-t/jmb*2 >
Comparison of eqn (7.30) evaluated at tf with experimental data for thin aluminum and steel
plates showed a predicted value higher by a factor of three than measured. This is attributed to
the neglect in the analysis of membrane action which is expected to be significant for such
impact situations[120].
An approach to this problem that considers both large bending and membrane deformation
without penetration is sketched in Fig. 25[61]. The governing equation for the transverse plate
deflection, w, is given by
(7.32)
Fig. 25. Model of plastic bending and membrane deformation of a plate due to impact by a blunt projectile.
where N, is the in-plane radial force per unit circumferential length; the maximum value of M,
for total yielding in bending is the fully plastic moment MY = $‘ay and the maximum value of
N, is IraY for a perfectly plastic material of yield stress oy. The waves produced in the target
by the impact are regarded as dividing the plate into the five regions indicated: (1) the most
remote from the impact point, r > rc = R, + c,,f is beyond the range of compressional elastic
waves traveling with velocity co and thus totally stress-free; (2) covers the range rs =
Rp + cst G r c rc of elastic in plane compression producing no transverse deflection, with cs as
the shear wave velocity; in domain (3) where rB 5 R,, + c,t =s r c rs, only elastic bending occurs
such as described in [lo] with c, = A/&&; in the annulus (4) with $D S r G rB, plastic
deformation obtains, as well as in region (5) r 6 R, that travels with the bullet velocity.
Impacts produced by undeformable projectiles of 12.7-mm-diameter with masses of 15-1OOg
at initial velocities ranging from 25.4-102 m/s on several types of aluminum alloy plates with
thicknesses of 0.64-4.8 mm indicate that the slope awlar lies between 0.1 and 0.2; here, the
radial load N, predominates over bending effects unless the bending moment distribution
changes almost discontinuously. Thus, from eqn (7.32) with the moments omitted, the plastic
zone assumed to have completely yielded throughout its thickness is governed by
la aw 1 a2w
(7.33)
;Z ( ‘r > ‘ZW?F
involving primarily membrane forces, with bending insignificant. The initial and boundary
conditions including the joint motion of projectile and plate in region (5) are given by
w(r, t) = 0
r> rB
$(r, t) = 0 I
2?rR,ayh $ = (m+vR&h)$, r=R,
w(r,O)=O, r>R, (7.34)

A finite difference solution of this set of relations was found to be in excellent accord with
experimental data, but for thick plates or low values of oY a modification of the first boundary
condition will be required to incorporate the neglected elastic bending effects. The interaction
of plastic bending and membrane action in other domains also requires further investigation.
Ballistic effects on textile structures are described in [191], where wave propagation
resulting from transverse fiber impact is presented. This type of approach is an extension of the
work on cables[l92], and of the membrane treatment of elasto-plastic plates by projectiles[lSO]
employing a finite element numerical procedure.
Rigid blunt projectiles-failure by plugging

A blunt projectile induces displacement of the target element in the direction of travel
through shearing in the periphery of the impact region. Flat cylinders are the ideal for this
purpose and cause such displacement over the whole front surface. Spherical, ogival and
large-angle conical projectiles induce significant axial displacements over a major part of the
contact surface. The internal separations that eventually lead to perforation initiate in the
periphery of the region of contact-at the edge of flat-ended cylinders and at angles of 45” and
greater from the axis of the projectile for ogival and spherical-ended cylinders.
The first model of plugging[21] in which the resistive force was equated to the product of a
constant yield stress in shear and the instantaneous area of contact between a separating plug
and the remainder of the target element and which neglected inertial effects was abandoned due
to lack of correlation with data. A very early model[193] assumes that the plug and projectile
come to a common velocity vW instantaneously and then conservation of momentum gives
Lao
(7.35)
‘*=D+h,
with L, the original length of the projectile and h,, the original target thickness. When the target
element is modeled as a fluid, the resisting force,F is given by
F = Awp,v2. (7.36)
The process terminates when the projectile traverses the thickness of the target element so that
the terminal velocity is
0, = v,[exp (- pAwhJm)l (7.37)
where Aw is the presented area of the projectile. Figure 26 shows that eqns (7.35) and (7.37)
bracket the largest observed fragment velocity produced by the normal impact of I6 mm x
7.1 mm projectiles impacting mild steel plates of various thicknesses at a velocity of 1470m/s.
This model provides force-time histories but will not predict a ballistic limit. A later
model[122] considers the plugging process as an exchange between kinetic energy T, of the
projectile and energy of deformation of the plug partitioned into two energies Es and &. Es is
0 5 IO 15 20
h PLATE THICKNESS cm
Fig. 26. Predictions from eqns (7.35) and (7.37) compared IO experimental data
the energy expended at the periphery of the crater in those deformation processes that separate
the plug from the target element. Eq is the energy expended throughout the plug in plastic
deformations that account for the projectile and plug reaching a common velocity. The energy
E4 is obtained from an energy and momentum balance applied to the plug-projectile system
with the assumption that ES is constant for a given target element and projectile. This energy is
expressed in terms of the ballistic limit velocity vsoas
Es =;m& (7.38)
whereas
(7.39)
The energy balance becomes
i mui = k(m + m,)vf+ Eq + Es. (7.40)
The projectile and plug diameters, DP and Dq are not necessarily assumed to be identical
here. With the hypothesis that Es is constant, the final velocity is
v, = .&(v:- v:o)“* = [ 1 + E ($)*($)]-‘(v: - v:o)“*. (7.41)
For oblique impact of such blunt cylinders, at an angle of obliquity 0, eqn (7.41) has been
modified to read
Vf =
cos e*cva
- v;o)“*= cos o*[ 1+2(2&-&) j-‘&t- v;o)“* (7.42)
1 + (m/m,)
where the change in direction of travel, B*, is obtained from an expression developed in the
Section on obliquity[l22].
A viscoplastic model of the target has been used to analyze the uniform shearing defor-
mation of a plug in normal impact. The model approximates the case of high impact velocities
and plate thicknesses small compared to the projectile diameter, but since the shear stresses
must vanish on the free surface, the theory is in error near these boundaries, and, thus, the
plate cannot be too thin.
A Bingham-type equation featuring a dynamic viscosity coefficient g
(7.43)
has been used as a constitutive equation for the model[194-l%] although the dynamic yield
stress has also been neglected relative to the strain rate term[197]. Here, E, = awlar and
au/at-CO;hence, u& is negative. Other constitutive relations have been employed for this
problem involving perfectly plastic material [ 1981, a linear work-hardening solid and elasto-
viscoplastic solid [ 199,200] or an empirical relation from experimental data[9]. The equation of
motion is
(7.44)
and the proper initial and boundary conditions are
=O, r>R,
vk 0) (7.45a)
= vo, r = R,,
and
~(03,t) = 0 for an infinite plate
(7.45b)
o(&, t) = 0 for a plate of radius R,
and
(m + m,)ti = 2mRph,u, with the plug mass mg = np,Rih,, at r = R,. (7.46)
The first solution for plugging assumed a constant projectile velocity, neglected the plug
mass relative to the projectile mass and dealt with a plate of finite radius with zero plate
velocity at the edge[l94]. Numerical results have been derived for more realistic conditions
using the methods of characteristics and the Laplace transform [ 198,200].
A solution of the boundary value problem with the strength term neglected but using the
proper shear conditions under the projectile; (eqn 7.46), was obtained in closed form by the use
of the Laplace transform using asymptotic expansions and thus valid for short time
intervals[l95]. The initial velocity employed in eqn (7.45a) for this case was that of the
projectile-plug combination, ugf,given from conservation of momentum by
(7.47)
A similar technique using eqn (7.47), also applicable only during a brief initial interval, yielded a
series solution for the deflection involving the complementary error function of a combination
of dimensionless radius and time[200,201].
A number of experimental results were found to be in reasonable correspondence with the
predictions of viscoplastic models [ 1971.
Sharp rigid projectiles

Hole initiation at the tip. Sharp projectiles are those that displace a significant amount of
target material radially as ideally represented by small-angle conical projectiles. Ogival pro-
jectiles are usually in this class, but may have significant axial displacements near the tip for
blunter ogives. The internal separations required for perforation initiate at the tip. The principal
alternatives for further development of separations are: (I) confinement to a small region near
the tip, but continuously generated so that a hole in the target element is enlarged along the
trajectory of the projectile, and (2) the development of the radial fractures that travel outward
from the point of initiation to form petals. In the first case, the material of the target element is
free to expand over the projectile in a process of ductile hole enlargement. In the second case,
the fractures relieve the stress at the base of the petals as they propagate away from the
projectile nose.
Models that represent the penetration of sharp projectiles into thin and intermediate target
elements differ as to the mode of separation and displacement. The earliest treatment of ductile
hole enlargement used elasto-plastic theory to determine the work done, and conservation of
energy is then sufficient to specify the change of motion of the projectile. The initial model of
petaling assumes that strength can be neglected and that the momentum of the petals is the
dominant contribution to the change of projectile motion that can then be determined solely
from conservation of momentum. Other theories that do not specify the kinematics of the
process, except that the hole initiation is at the tip, have correlated the forces acting on the
projectile due to inertial, compressive and frictional phenomena to the displacements and rates
of displacement.
Ductile hole enlargement. A quasi-static analysis of the completely symmetrical enlargement
of the hole that develops at the moving point of a sharp projectile was given in a classical paper
for a thin infinite elastic-perfectly plastic sheet[202]. This description was improved by a more
precise stress analysis in the region of significant target thickening[203]. The work W required
to expand such a hole to a given radius r, is
W = I .33arfh,,uyT. (7.48)
Subsequent development improved the dynamic representations of the previous

models[204] and considered another process where the hole enlarged first at constant velocity
u,, and at constant acceleration after a certain time. Here, a dynamical similarity solution was
obtained in terms of radial particle velocity ti, given by
(7.49)
(7.50)
that depend only on variable r/t.

Experiments conducted on the normal penetration of lead sheets by conical steel rods
rotated during the enlargement process[203] indicated the possibility of producing two modes
of cratering, a symmetric pattern (for R/h, = 2.7) which is specified as the requisite geometry
for the above analyses and an antisymmetric pattern (for R/h, = 10). In the latter case, which is
the configuration generally observed in perforation at ballistic speeds, the material in the
tubular crater expands under the action of the tangential stress CT+ = on, while the normal
stress uz vanishes.
For large holes (R/h,,% l), the radial stress can be considered to be negligible and the stress
state is approximately uniaxial. The tubular deformation pattern is then shown in Fig. 27 where
Fig. 27. Deformation pattern for perforation of a thin target element by ductile hole enlargement process
h is the local crater thickness, H is the crater height, and z* is the distance from the crater tip
to the section in question. In view of incompressibility
$= (g= (zJ”2 (7.51)
H
rrR2h, = 2vR h d.z* = 2?rRh, dz* (7.52)
so that
H=aR and (7.53)
The plastic work per unit volume d Ws for the deformation of an elemental ring from the
original position at r to the crater lip is given by
(7.54)
so that the total static work required to produce the crater is
(7.55)
This estimate for the unsymmetrical case is substantially smaller than for the symmetrical case
(eqn 7.52). This may explain why the unsymmetrical crater is observed far more frequently,
particularly under conditions of high-speed impact [203].
Treatment of this modeI[205,206] on the basis of an energy approach yields for the
accelerating force on the plate generated by the penetrator and the mass m* displaced at time t
d*r dm*dr
F = m*-&‘+--& with m*= Irp,h,ri (7.56)
so that the dynamic work WD is given by
WD = np,h, ‘d”) ?$dr + 27rp,h, 16”“’ r[$r dr (7.57)

I0
where rr is the time when the final hole size is attained. For the conical and ogival nose shapes
shown in Fig. 28, defined by
z= 2 tanp (Cone) (7.58a)
and
(Ogive) (7.58b)
the expressions for the striker velocity u = dddr are given by
** = b:v; cmb:
(7.59a)
(bI+z?tan*/3)2-2~,~a~2/3+2p, tan2/?(bI+z2tan4&
where
and
sin -% dz = 0 (7.59b)
LN
where
$= T4R4
4 4
32L; 56, +s)‘

( N
The ballistic limit for the conical projectile is obtained from eqn (7.59a) as
(7.60)
a) CONE bi OGIVE
Fig. 28. Conical and ogival projectiles perforating thin plates.
The corresponding total mechanical work, W = Ws + WD is
(Cone) (7.61a)
and
(Ogive) (7.61b)
when the velocity u is assumed to have the constant value u,. The frictional heating effect could
be incorporated in the energy relation, but this quantity is small compared with the plastic
deformation effects, especially at relatively low ballistic velocities, and may be neglected.
However, the energy contained in waves of permanent deformation and in fracture mechanisms
is not readily calculable and may be a significant fraction of the total energy.
The partial perforation of a thin metallic plate by a rigid, truncated cylindro-conical
projectile under conditions of normal impact has been analyzed by a quasi-dynamic energy
method similar to that of Ref. [205]; no attempt was made to conserve momentum[207]. It is
assumed that deformation of the target occurs as shown in Fig. 29 with the shearing of a cap of
radius R, equal to the radius of the frontal flat of the projectile. The crater consists of a
cylindrical section with a radius R,, and a conical section at its base of half-cone angle 5 and
maximum radius R,. It is further hypothesized that only the circumferential stress u4 = oy is
effective, that shear and deformation in the cylindrical region of the crater occur with constant
axial velocity u,, which must be determined separately and that in the conical region occurs
with axial velocity u(R, - r)/(R, - Rp). The work of plastic deformation W,” is that required to
form the two parts of the crater. The dynamic work W,” involves the axial motion of the plug
region of the target, the axial and radial motion in the cylindrical crater domain and the axial
Fig. 29. The deformation of a thin target element under impact by a truncated cylindroconical projectile.
motion in the conical segment. The shear energy W, = ?rR&aus, where ous is the ultimate
shear strength. For partial perforation, the sum of the work terms must equal the initial kinetic
energy To of the projectile, so that
To=~mv:=~vhocry
[
Ri-R:-2R:In?+Z(R:-R,$lnl
4 1
++rp,h,of R,,+3R,)(R,,-R,,)+ I2(R;--;R;+(R;-R:) ( RpLRqr)] + nR,h,‘ut/s.
(7.62)
&ruling. The previous solutions do not take into account certain mechanisms that might
represent a significant fraction of the total energy. A class of analyses postulate the presence of
fractures that initiate at the projectile tip and propagate radially. These neglect the strength of
the target and require an assumed deformation pattern of the plate as well as the prescription of
a zone of radius r,, outside of which the target is not affected by the impact of the
striker[208-2101. The process is sketched in Fig. 30; within the circle r,,, the principal effect is
plate fracture and plastic deformation. The validity of the solution is clearly dependent on the
accuracy of the stipulated geometrical model. The terminal velocity at any obliquity for this
case is given by [ 1221
Uf = (of - I&p*. (7.63)
When plugging is absent, a momentum balance in the direction of projectile motion z yields
(7.64)
where m, is an effective target mass. and w,(r,,. z) is the axial displacement of a deformed plate
tiLAI
DEFORMED PLATE
a) SKETCH OF PROCESS u PLATE
bl CROSS SECTION AND NOMENCLATURE
l-------------i
cl CONICAL PROJECTILE
Fig. 30. The deformation process for sharp projectiles perforating thin plates.
element initially at r,. An exact solution of this equation can be obtained if the deformation
pattern is known; otherwise this deformation can usually be approximated with reasonable
accuracy.
For the conical projectile with total angle 28 shown in Fig. 30(c), it will be assumed that the
plate is distorted to conform to the shape of the nose; this implies that the target strength is
neglected relative to inertial effects. With no radial stretching of the petals-as confirmed by
test-w, is given by
w, = (z tan /3 - rO)cos p (7.65)

yielding a velocity drop
Au = (u,?rp,h[z tan /3]*sin /3)/m. (7.66)
For small values of Au, u may be replaced by its initial value u, so that
rp,hR,2
[Au]~= u, - ur = u0 sin p (7.67)
m
where the final projectile velocity is assumed to occur at the instant when z tan p becomes
equal to the radius R, of the cylindrical section of the bullet. Figure 31 shows a comparison of
the predictions of eqn (7.67) with corresponding experimental data and also with the results
from the energy analysis (eqn 7.59a). Two values of o vT were selected for the latter so as to
afford the best fit with the experimental curves. The energy approach appears to provide better
correlation at low impact velocities, while the momentum solution seems to agree more nearly
%=FE ho MATERIAL LA mm 2Rp m Jo
ENERGY METHOD u,, = 407 M Pa
ENERGY METHOD “Y = 1586 M Pa
MOMENTUM METHOD
0 200 400 6CO 800 1000 1200

V, INITIALVELOCITY
Fig. 31. A comparison of the predictions from a momentum analysis and an energy analysis of perforation
of thin plates with experimental data.
at higher initial speeds. Thus, the e#ect of target strength should probably be incorporored for
impact velocities below 500 m/s.
Although eqn (7.66) cannot be extended to the case of blunt-nosed projectile impact by
taking the limit /3+:7r in view of the a priori assumed deformation pattern, such a procedure
will probably yield a reasonable upper limit for the event. However, the neglect of target
strength introduces a successively greater error with increasing /3. Thus a different petaling
hypothesis for blunt-nosed impact has been constructed where the projectile first plastically
deforms the plate into a dish before fracture begins at a time when the bullet speed is reduced
to a value of u, [ I I]. This may apparently occur in thin, ductile targets at initial velocities close
to the ballistic limit. A deformation pattern is hypothesized consisting of a cylindrical crater
terminating in a yield hinge that permits an initially circular section of the target to open into a
conical shape. The velocity drop subsequent to initial fracture is given by
4pr h, (7.68)
cl-t+=- 3 LJp L,O’
)
and the total velocity drop is thus
(7.69)
The execution of a force analysis for the simplified conical model requires the assumption
that the contact pressure between projectile and petal vanishes. Thus? the force F exerted on
each of the n petals acts at their base and is uniformly distributed from symmetry con-
siderations. The total axial force, per unit circumferential length, FX = F,/Z?rztan /3 is approxi-
mately
FZ = hp,uztan /3 sin f3 (7.70)
while the radial load per unit length is evaluated as
= p,hoz[ 1- sin B] tan’ /3. (7.71)
Thus, the total line load per unit length, F*, becomes
F* = IF*,* + Ft*]“* = (p,hui tan* /3)(2[1- sin /3])ln (7.72)
which is inclined at an angle of a?r - $3 to the projectile axis.

When a flat (meplat) exists on the tip of the projectile, a slug is usually punched out that
travels with a velocity B2c0 where B2 depends on plate thickness and initial projectile
velocity(2091. Experiments indicate that B2 decreases with plate thickness and increases with
velocity, with a lower theoretical limit of unity and an upper range of perhaps l-1/2 for normal
ballistic speeds and thin metallic targets. The strength effect could again be reasonably
neglected at sufficiently high velocities. Inclusion of the plug momentum yields a velocity drop
in lieu of eqn (7.64) given by
Ao = &m,u, + m,(.W (7.73)

m
For a small value of Au, the total velocity drop is given by
Au = %vO[B2R$+ (R*- Rk) sin PI (7.74)
where RN is the radius of the frontal flat and R that of the cylindrical projectile body.
62 M. E.BACKMAN and W. GOLDSMITH
Exclusive of the plug punching effect, the line loads are those given by eqns (7.70-7.72).
Corresponding results have also been developed for an ogival-nosed bullet.
Composite force functions. Several models for sharp projectiles that assume initial failure at
the projectile tip consider a number of mechanisms. One accounts for (1) compression and (2)
distortion of the target, (3) inertial effects, and (4) frictional effects that depend on the
displacements and displacement rates. A composite forcing function covering these effects in
the order cited is stipulated as
F(z) = rp,[r,2(2) - I,‘(2- h)] + 2rppxI:_, r(z)[g(Z)li dz
+ 4ahR:Pru22z(LN +$I-z)+ 7ff(p,[J(z-h)-.l(z)]+px[rz(z)-r,2(z-h)]} (7.75)

3(%+9)
where
LP
J(w) = 2
I
I
r(z) dz for z G L,
and
r(z) = 0 for z d 0
J(w)=0 for z>LP r(z) = Rp for z 3 LN (7.76)
and the notation is that shown in Fig. 30. This expression depends upon two “dynamic”
material parameters, p1 = pH(l+ a tan’ p)“‘, with pH as the hydrostatic pressure. The other is
the distortional pressure component px which accounts for the creation of new surfaces in a
manner similar to that of a surface tension and whose corresponding force component is active
only in a quasi-static hole enlargement. Although known to be variable throughout the
perforation process, both quantities may be assumed constant for convenience. This is
reasonable for the hydrostatic pressure variation, but not for distortion where the pressure
vanishes in zones of spalling. The coefficient of friction f is also presumed to be invariant.
The process described neglects the deformation of the projectile, the flexure of the target,
shattering and spalling effects and further approximates plastic deformation in.a crude manner.
The model assumes a process where the target material is accelerated to a mean speed equal to
one-third that of the projectile, somewhat akin to the concept of virtual mass. This assumption
is not unreasonable since the kinetic energy acquired by the plate in the range up to
v, - 1000m/s is only about S-10% of .the initial energy. It is easy to criticize this development
on the basis of the neglect of phenomena that quite evidently do occur, and the use of grossly
oversimplified assumption (for example, a solid friction representation is employed instead of a
fluid zone around the projectile periphery where frictional effects would be velocity-depen-
dent), and because of its quasi-empirical approach that requires two arbitrary dynamic pressure
values. However, the model appears to correlate well with measurements involving steel
projectiles striking armor plate at normal and oblique incidence.
For the latter case, two linear and one angular component of acceleration must be
considered and calculations are performed in steps. It is also at least conceivable that the
empirical pressures can be related to more fundamental material properties derivable from
independent tests, and that their variation throughout the target thickness might be accounted
for in a reasonably rational manner. A modified version of eqn (7.75) has been given in
differential form[211,212] as
dF = dFc + dFx + dFF + dF,
ZG
b,nrPayT(tan @)ViGZ + 2neUltan @ sec*Wcos @- rK)
[ (
+ 4b2p,r,, tan m [r, tan ati + v(tan’ a -c-b)]}{ I+ sin (o lz fi + 5Fj)] dz (7.77)
where @= tan-’ (dr,/dw,) is the local slope of the ogive at the point rD(wt)with curvature
K = - cos @(d@/dS) and e,, is the specific surface energy of the target. Constants b, = 1 and
b2 = 2.5, and 5~ is the friction angle which diminishes rapidly with increasing u as the result of
temperature rise. Frictional effects account for less than three percent of the total energy[213];
the effects of plate distortion are also considered negligible. The ballistic limit associated with
this and the previous model is
~~=21rR:h(P,[l+f~]+PX[2~LNr(r){~)d~+f])
(7.78)
m -W$d~)
Another representation also valid for thicker plates considered only the motion acquired by
the target and the resistance to plastic indentation that is described in terms of Brine11hardness
BHN. Friction and tearing are neglected, and separation of the plate material from the ogival
nose of the striker could occur. The force acting on the striker whose nose displacement is z
and whose velocity is u is proposed as[22,214].
F = n?(r) [ 6BHN ( 1 -h)2-4BHN(I-;)l+p,u2].

z (7.79)
A number of expressions proposed for the force acting on the projectile have the same form
as the empirical Poncelet eqn (31); scientific respectability was acquired by associating each of
the terms with a hypothetical mechanism. Thus, a differential force law per unit area dA of the
surfaces in contact, incorporating static resistance and the effect of virtual target mass has been
proposed for tests on aluminum targets as [64]
$= [pS+ Au2 sin’ /I] sin /3. (7.80)
The form of this law is applicable to both blunt- and sharp-nosed projectiles. Here /Yz, the
average contact pressure required for perforation under quasistatic conditions was empirically
determined from static tests for the aluminum employed as
ds= 5.29h MPa (h in mm). (7.81)
This quantity is probably linked to the average pressure, pS under a very slowly moving
indenter, which can be ascertained from a slip-line solution of the problem and shows a value of
pW- 3arr for a Mises type of yield criterion[4]. Integration of eqn (7.80) is carried out over
three stages of the displacement w of a cyclindro-conical bullet: (1) initial piercing where
0~ w c h, (2) partial protrusion where h < w < LN, and (3) completed hole enlargement where
LN < w <h + LN. For a rigid projectile, the terminal velocity ur is evaluated as
~f=exp(-[2~~R~hsinBl/m)[u~-(h~~28)]{exp([2?rplR~hsin2~]/m)-l}. (7.82)
From eqn (7.82), the ballistic limit is given by
1)~)= pt
c
$,f2
{exp
B(12mR:h sin PI/m) - I}]. (7.83)
Experiments indicate that perforation occurred by plug removal and permanent projectile
deformation was observed in varying degrees of severity. The predictions of eqn (7.82) were in
fair agreement with test data when the bullet distortion was not too great[64]. Test results of
other investigators have also been correlated with this theory[215].
Deforming projectiles
Single mechanism. Deformation of the projectile complicates its interaction with a target
element by allowing ‘variations in its shape as well as in its motion. A model of these
interactions must postulate projectile deformations or in some way allow for them in consider-
ing the compromise that must develop between the displacements of the projectile and the
target element.
A model for a plugging mechanism has been proposed for the perforation of an elastic-
plastic-hydrodynamic target of thickness h at speeds ranging from the ballistic limit to about
1200m/s by a cylindrical projectile of time-dependent length L[ 123). The impact is expected to
produce a high pressure p at the projectile interface that creates a fluid zone in this region. The
plug ahead and the bullet behind. are regarded as incompressible. The penetration P up to plate
failure at time f1 when plug motion commences, is resisted by an additional stress (T/ given by
uj = ay,(h - P)/R for 0 < t < ffi (7.84)
This stress is produced at the plug periphery and acts on the projectile face. The initial pressure
generated by the imvact is
PKHIPPCH~ ”
PO= PtCHl”I = &cHp(", - “I) =
hCHt + PflHp > ”
with
p&H”,
VI = (7.85b)
PrCHf + PpCHp
where shock velocity U has been replaced by hydrodynamic sound speed cH and or is the
velocity of the interface, which is governed by fluid behavior. It is further hypothesized that the
RSiStiVe pressure p is a step pUk of magnitude p0 acting for the I, = 2h/(cHt + v,) when
P, = vItr = 2ht$/(cHt + 0,); for 1, SOS 1,. p is assumed to vanish.
When spreading is neglected, the equation of motion of the projectile is
dv
- &tp + UJ) = - rR;p - ~R~q,(h - P) (7.86)
m;lT=
with
m = np,RzL.
Further, dZ../dt= -(v - v,) = -(o, - Y,) so that L = L, - (0, - v,)t. With the conditions P = 0
for v = v, and P = h for v = v,, integration of eqn (7.86) using eqn (7.85) yields an expression
for IQ.In simplified form for a constant value of L beyond time rR this velocity is given by
)I
‘$‘&i,CHph _ ‘%(PrCHr + PPCHP)
(7.87)
PP~CH&L ’
ppcHp”o
PtcHf + PpcHt
This permits evaluation of us0from the relation
uwh*
vfo = (7.88)
&H&P
The exit velocity may be approximated as
“,_$JT. (7.89)
However, this analysis has been questioned in view of the large number of assumptions and the
violation of mass conservation.
Successive mechanisms.The representations in this section are equally appropriate for thin
targets as well as those of intermediate thickness. A recent descriptive model uses an analysis
of the perforation process based on a sequence of three interconnected stages as shown in Fig.
32[216). The first phase comprises indentation and compression of the target to an empirically
ascertained depth h, - h,, when incipient plug shear occurs. The second stage involves further
continued compression and shearing of the plug and terminates when plug and projectile move
at the same velocity. The third domain commences at this instant and continues until plug
ejection-only shearing forces act during this interval. All other effects such as friction and
heating, plate flexure and wave propagation, are neglected.
In the first stage, the material ahead of the projectile is compressed to the ultimate strength
auc and a portion of the target concurrently acquires momentum transferred by the bullet. The
equation of motion is then given by
do
F,(t) = - ;C,g+A,d- u”=Aw=~Awu2+(m,+~Awl)udz forOazSh,-h,. (7.90)
Here m, and AW are the initial mass and projected area of the projectile, /r, the initial target
thickness, and h, the thickness remaining when the compression phase ends (this is sub-
stantially the cap or plug thickness so that /I, = h,). C, is a constant depending on the geometry
of the striker that accounts for the virtual mass of the target, with values of l/2, 1 and cos2p
computed for a spherical, cylindrical, and conical-nosed bullet, respectively. A value of unity
might even be appropriate for any striker shape under conditions of severe flattening or
breakup of the impact surface so that a more nearly planar surface is presented to the target.
ADDED TARGET MASS
r
1
I- h
(al FIRST STAGE Ib) SECOND STAGE
c- Z-h,-
IcJ END OF SECOND STAGE fdl THIRD STAGE
Fii 32. Three stags of an rwuad perforath process.
lJR3VdM.Ne.l-S
The projectile velocity during this phase is obtained upon integration of eqn (7.90) as
(7.91)
v= [~+~,~~~~~,J~~~cl-~,~~~c.,l”’ o~r~hl
and a further integration yields the displacement in quadratures as
(7.92)
The force history of eqn (7.90) can be evaluated numerically in conjunction with eqns (7.91) and
(7.92).
In the second stage inertial, compressive, and shear forces act concurrently. The first of
these acts on a plug area A, which in many instances can be assumed to be identical to the
projected area of the bullet. If the plug exhibits a significantly different base diameter, a linear
variation of D with z is a good approximation. The value of C, is here chosen as 0.5, as
non-armor-piercing projectiles tend to assume a spherical shape during penetration. The
resultant discontinuity in force at the end of stage 1 is very small. The compression uuAq at this
instant is reduced to zero at the end of stage 2, and its variation will be assumed to be parabolic.
The shear force acting on the plug periphery is assumed to be of the form of eqn (7.43) with the
shear strain rate C, = dAr.s, where Ars, termed the radial clearance, is the width of the shear
zone. This rate is a function only of the target material at high deformation rates and is readily
derivable experimentally or analytically[200,217]. The equation of motion for stage 2 is thus
with It, - h d z s h, (7.93)
while the velocity during this interval may be evaluated from the relation
duz(z)
dz= [_(I+~C,)~~u’-u~~~z-~+~~~~-~~~
Ill[(m,
+PA&I.
z-(ho-h) ’
+ &dz(ho - h) - uucAq h
(7.94)
>
The effective mass at the end of the first stage is meI = m, + p,A,(h, - h,). The duration for
stage 2 is t2 = &,, (l/u) dz.
In the third stage, projectile and plug with a total mass m,2 = m, + p,&, move together
under shear stress a, acting on area & of average cavity diameter’s developed during the
second stage. The equation for stage 3 is then
d2z*
m.2-p = F3 = -anA: = - (uR+ p&)A; (7.95)
where z* = z - h , = r,Ars and the cylindrical surface area of the plug is A: = d&h,. The small
shear during stage 3 is neglected.
The solution of eqn (7.95) is
(7.96)
with u2f as the velocity at the end of the second phase. The force in domain 3 is thus
The mechanics of pmwation of proj~~tik~
into targets 61
(7.97)
acting during a time t3 until z* = 27. The time required for the plug to leave the target is
(h, - zt)/u, and the total time for plug ejection is thus
t=t,+t*+t3+-_
Vf
This interval is followed by the ejection of fragments representing the effective mass added in
stage 1, and then by the projectile itself.
A typical calculation involving this model is shown in Fig. 33; the values of ~1were taken
both from ballistic tests and direct measurement[62], while Ars and i& were measured. The
value of a was found to be well approximated by the average of 4 and the diameter D, of the
ejected plug. Tests have also shown that the values of && and hrlh, for a given striker-target
arrangement are nearly constant over the velocity range of interest and can be determined from
a small number of experiments. Earlier analyses of the type cited involving only the tirst and
third deformation phases[216,218-2201 lead to a sudden drop in force at time 1, followed by a
gradual decay to zero force at the end of perforation.
Another model for plate perforation by blunt projectiles first considers symmetrical com-
pression under lateral constraint producing the deformation pattern shown in Fig. 34[221]. This
is modified by shear represented by. the depicted hypothetical force-deformation history to
produce the actual plugging scheme shown in Figs. 34(b) and 34(c). The initial phase also
involves plug acceleration and the second stage must account for frictional work done between
striker and target. The target material obeys Hooke’s law in the elastic range and the empirical
relation Q = a,, ebl in the plastic range, where a, = EbJ&’ and b1 is an empirical constant.
Constraint to side flow is expressed by a factor C, which has a theoretical value of 2 and an
experimentally determined range extending considerably beyond that value, with a mean value
of CZ= 2.7 chosen for aluminum. The confined plastic stress-strain relation and wave velocity
cp are thus
0 = C*U&bl (7.99a)
CP = ~C~b,uoc”+, (7.Wb)
and velocity UIcorresponding to strain lI is then given by
-=-
(7.100)
5.59 MM DIAMETER LEAD BULLET ON A

50 MM THICK Al ALLOY TARGET AT
400 MIS
I,,,,,/- STAGE 2 i’iiG’ j
Fii. 33. Cakdationl from a three-stqc model of perforation.

FORCE 2n Rpa, h,
I J -7
1 ‘h lh, - hq)
0
0 DISPLACEMENT
~-j-q-q=---p-~~~~~~-~
Ia) Ibl IC)
UNIFORM SYMMETRICAL PLUG COMPRESSION

COMPRESSION; NO SHEAR F.ND SHEAR
Fig. 34. The dependence of assumed deformations and calculated forces on displacement.
The force on the plug at a uniform compressive strain E is
F = ~R;C~IY&~~ (7.101)
and the work done Wc, in moving the plug a distance z = Lk is given by
4 Lh
WC,=
I
0
F dr = aR;Czu,
I
0
lbldt (7.102)
where e = l(ALh) and varies from zero to eI. The kinetic energy T4 of the plug which is
included in eqn (7.102) is
T, = $rfiR;h,o: (7.103)
where the plug thickness hd after compression is given by hd = h, exp (- l I).

The energy balance required to achieve the conditions shown in Fig. 34(c) must include the
shear work that will bring the plug from the position of Fig. 34(a), as well as that of friction.
For uniform acceleration of the plug to velocity ul and concurrent uniform deceleration of the
projectile interface from u0 to ul, it follows that
L,=$, Lh=;(uo+~~)f. Lh-Lp=h,-h=;v,t. (7.104)
The shear work is the area under the curve of Fig. 34(c), given by
Ws, = %lh,(h, - A,) + L,(2h, - LJI (7.105)
where the factor \/5 arises from the use of the Mises criterion. Sticking friction is assumed to
act over the average contact length {L at the same yield stress, so that its work is
WF, = 7rV&L2ti. (7.106)

The mechanics of penetration of projcctites into targets 69
The total work WC, + WSI + WFI during this first stage of plug acceleration is evaluated in
increments of strain l until the projectile velocity has been reduced to that attained by the plug.
After this phase, projectile and plug move together; the work of shear and friction
W.. = rRp,,(h, - Ld2/t/? (7.107a)

Wn = ?rR,u,(h,t - La/t/j (7.107b)
respectively, reduce the velocity of the plug to the terminal ejection velocity 0,.
A still different approximation intended to apply to the projectile perforation of a porous
medium involves a quasi-static analysis featuring a cylindro-conical plug as shown in Fig.
35[222,223]. This consists of a compressed cylinder of the same diameter as the normally
impinging blunt-nosed striker prefaced by a conical frustrum with a larger diameter 4 at the
distal side of the target. Such plugs have been experimentally produced in Styrofoam by blunt
steel bullets with masses of 0.341 and 4.40 kg corresponding to diameters of 15.88 and
31.75 mm, respectively, for initial velocities ranging from 0.6 to 6.1 m/s, and target thicknesses
of 2 < h,,/R, ~88. Conical-ogival plugs were found either when the plate thickness was
increased or conical-nosed bullets were employed.
Fii. 35. The forces and plugging failures postulated for the plugging perforation of a porous medium.
For the model of Fig. 35, conservation of momentum and energy yield
t Ps ups
-mob
2
=
I
0
]a,(~, i)l[21raRP(PS - z)] dz +
I0
nu,,(z, i)R; dz (7.108)
for zero terminal projectile velocity, with a as the compression ratio of collapsed plug height to
shear penetration Ps. Assuming stress variations of the form
E
a,, = -z
ho
(7.109b)
yields
(7. I IO)
At incipient perforation, the frustrum-considered rigid-is held by tensile and shear stresses
assumed constant over the conical area. The equation of equilibrium in the vertical direction
neglecting inertial effects relative to contact stresses and some crude assumptions concerning
the stresses yields the expression
(7.111)
where 7 = R&h, - Ps). Rupture is assumed to occur when the tensile stress normal to the slant
surface reaches a critical value ati = a. so that the corresponding angle /3 will yield the
minimum equilibrium impact force u,rrI?i. Substitution of the maximum compressive stress
during penetration, a, = (E/h,)Ps, in eqn (7.111) and elimination of Ps provides the expression
for the critical impact energy,
(7.112)
where f* is * for PP/ap = 0 from which the critical impact velocity can be obtained by
iteration.
The velocity range and target material do not immediately appear to be applicable to the
problem of ordinary terminal ballistics. However, the geometry of the model is noteworthy in
that a number of plugs produced in the impact of projectiles on metallic targets at ballistic
speeds frequently exhibit shapes like those observed here rather than being cylinders with
constant diameters. Consequently, the analysis of Ref. [222] which utilizes unrealistic stress
distributions and ignores what may be important effects might be improved and expanded to
accommodate situations of primary interest in the defeat of armor plate.
Target-element failure limits

Crude estimates have been provided for the limit expressions for some of the failure modes
sketched in Fig. 2 that are induced by a blunt-nosed projectile[9]. For compressive failure on
the distal side of the plate due to the initial dilatation wave produced by the impact, the value of
vAis given by
vA=($$)~[l+($~]“2 (7.113)
with
2 * (hcar)(P,coJ(Ptcol+ &Cop)
based on the hypothesis that failure occurs when the maximum shear stress developed is equal
to the yield value. Failure resulting from the tensile stress in the initial wave is calculated from
the condition that the radial stress at the back surface is equal to the yield value in tension;
thus,
(7.114)
In a similar manner spalling failure due to the reflected tensile wave corresponds to the limit
velocity
(7.115)
For plugging and petaling, the resisting forces were taken as the maximum shear stress
u,&,/R,) and the maximum normal stress 4urr(hJ&), integrated across the plate, yielding
v* = [y(*)]“2[exp (@$) - 1]“2 for plugging (7.116a)
VA= 2[F(*)]“2[exp (%) - l]“2 for petaling (7.116b)
respectively.
A simplification of the viscoplastic shear analysis of [200, 2011 has been suggested which
stipulates the occurrence of failure when, simultaneously, the strain equals or exceeds a critical
value and the strain rate equals or falls below some critical value. For steel and aluminum,
these are taken as 0.02 and 1.0, respectively[224].
While expression (7.113X7.1 16) are very appealing in view of their simplicity, considerable
caution should be employed in their application. These values have not been extensively
checked against test data and, moreover, are also suspect on theoretical grounds.
Models based on hydrodynamic theories

Models have been developed for the impact/of short, blunt projectiles traveling at hyper-
velocities where shear resistance is minimal and impact phenomena are amenable to treatment
by hydrodynamic theory. The phenomena for thin targets, in particular the formation of a
system of tiny particles on both sides of the target element, has been examined from the
viewpoint of shock wave propagation, such as is shown in Fig. 36(a) and (b)[225-2271. In the
first diagram two shock waves SI and Sz have propagated outward from the interface I and
rarefaction (tensile) waves generated at the edge of the striker cause rearward material ejection
while traveling towards the center. The pattern after reflection of S, as rarefaction R3 from the
distal side of the target is shown in Fig. 36(b). This process causes the formation of a bubble of
debris that grows due to addition of material from projectile and target.
PROJECTILE
“0
I .
<?
I ;’ I
:< EJECTED DEBRIS
fJ;.,
0
‘l.,, t
2.
?*
oi. S2 .d’
‘?o:
0;.
‘co.
‘do
PROJECTILE
BUBBLE OF DEBRIS
Fig. 36. Shock propagation and the ejection of debris in hypervelocity impact of a cylinder against a thin
plate.
Fracture occurs when the net tensile stress exceeds a critical value, creating other waves in
order to satisfy the boundary conditions of zero stress at any free surface. Transient stresses
associated with the fracture may produce additional failures. The phenomenon can thus be
depicted as a series of spalls starting at the initial free surfaces. Each material element is
shocked by the non-isentropic Hugoniot process to some pressure pI and returned adiabatically
to ambient pressure by the rarefaction waves. This involves shock heating which can lead to
fusion, liquefaction and vaporization at successively increasing values of pressure. Incipient
melting, complete liquefaction, and vaporization for a series of metals was found to occur in the
ranges of 30-200, 35-200, and IOO-2OOGPa,respectively. The shock pressure pI and particle
velocity UIbetween the shocks SI and S, (whose velocities are UI and UZ)are derived from the
one-dimensional conservation equations across the shock front (also known as the Hugoniot
relations). The result is
ul=uo l-t pt,(l -PI&P,) I’* -’

[ { PW(1-P&,1) 11
u1 = (%-~)v*/(~- 1) (7.119a)
u2 = d(l -P&t*) (7.119b)
where the subscript o denotes the reference state.

Tracing of the shock and rarefaction waves is most effectively accomplished by numerical
procedures. A full discussion of the subject of shock waves in solids is given in [228,229].
A one-dimensional Lagrange diagram of the flow for the regions shown in Fig. 36, not yet
reached by rarefactions RI and R2, is shown in Fig. 37[225]. The leading edge of rarefaction R3
produced by reflection of SZat the distal target face travels with adiabatic sound speed co,), for
pressure pI relative to the target which moves with velocity vi. Upon reaching interface Z, this
wave generates transmitted rarefaction wave R4 traveling with adiabatic sound speed c(,,)~
relative to the projectile as well as a reflected wave. No reflected wave is generated for identical
projectile and target materials. For a sufficiently thin shield, rarefaction wave R4 can overtake
the shock at a distance L, from the face of the uncompressed projectile, given by
LI
-=
L y
*!3z
c cwm
PI0 C(m), -
+ u2-
Ul - 01
“l][$q
C(SO)I
“1:,“3
(7.120)
where y* is the ratio of the mass per unit area of target to projectile; it weakens the shock
beyond this point. Although the sound speeds c o0) cannot be computed from Hugoniot data, a
reasonable approximation for metallic materials below 0.5 TPa is given by
CC,,)= $X49+ (y)2]“2. (7.121)
Here, U is the shock velocity corresponding to pressure p and u is the particle velocity behind
the shock, both measured relative to the undisturbed material, considered at rest. The
attenuation of shock wave SI due to R:, is given in [230] based on hydrodynamic, compressible
inviscid flow under isentropic conditions. Furthermore, rarefaction RI meets shock SI at the
t
-h _
i- 0 I
Fig. 37. Lagrangediagramfor shocksand flow during the hypervelocityimpact of a cylinder againsta thin
plate.
axis of symmetry of the projectile at distance Li, and RZ meets SZ on this line in the target at
L.,,, where
L,, = 0.720, and L,, = 0.720, (7.122)
Each of these distances is measured from the original position of the interface in the
uncompressed configuration. The analysis presented is valid only up to the instant of such
intersection. If h CL,,, the side rarefaction will never meet the axial segment of the shock
before its reflection from the distal target surface. The spray angle of the debris cloud was
estimated to be 45 degrees, based on these one-dimensional considerations.
An alternative theory of the debris cloud generated during thin target perforation[231]
assumes the punching of a plug with dimensions equal to the presented area of the projectile.
This yields momentum and energy balances given by
mph = (mp+ m&f (7.123)
m,vo2
= m,+mq vf + he(*) (7.124)
2 2
where de(*) is the increase in specific internal energy of projectile and target. It is now assumed
that projectile and target plug shatter forming a sphere of uniform mass distribution expanding
at a uniform rate vCu)and whose center of gravity moves with velocity fi, as shown in Fig. 38.
This leads to the result
(7.125)
5, = tan-’ (7.126)
where g* and & are the spray angles of the front and rear hemisphere, respectively. The fastest
and slowest fragments will travel with velocity
VW= Vf + V(,)
v, = Uf - %.x).
Although significant differences have been found between these relations and experimental
evidence, this treatment is the only model of the process developed thus far. A more exact
description of the behavior of the debris necessitates a two- or even three-dimensional analysis
which will probably necessitate the use of numerical codes.
Fig. 38. A model of the pattern of debris and its motion from hypervelocity impact of a projectile against a
thin plate.
Multiple impact analyses

Plugging analyses have been further developed to encompass the inelastic impact of a
warhead fragment pattern of nd fragments per unit area of the target[232]. Each plug removed
is assumed to have the same uniform cross section as the striker. Conservation of momentum in
the absence of forces at the boundary yields
mdud = (m;- m& + (md + m&f (7.127)
where u, is the impulsively acquired plate velocity and rn; = &h/nd is the plate element mass
accelerated. For initial velocities less than the ballistic limit n&t ur = 0; thus the value of ulsoas
a function of adSOis given as
(7.128)
The energy balance, which accounts for target motion outside the regions of perforation,
also includes a loss J$ due to plug acceleration by the plastic impact of the striker and, upon
plug separation, a further loss Z?o assumed independent of t&. Thus
1
-m&j0 = $f~o+~(m:-m,)u:+~(md+m,)uf (7.129)
2
if t& 2 r&o. If the plate is infinite in extent, uf reduces t0 equation (7.41). The plate VdOCity
magnitude depends upon whether or not perforation occurs, i.e. 5 s uJu450> 1 or < 1, respec-
tively, and is given by
VI_ m
--m+,;F(b)
u
(7.130)
with
(7.131)
If a fraction al of the fragment attaches itself to the rear of the plate, the value of F(5) for 5 > 1
becomes instead of eqn (7.131)[233]
F(n=rC1-Jlc(l-I:,(m~m;) *
( >I
l-i
(7.132)
The warhead is assumed to have a Gaussian probability density P(5) for the fragment
velocities expressed in terms of the normalized fragment velocity 5 and standard deviation
W*=g/ed as
P(l)= -exp 1
a*V!%
[-%I (7.133)
where the standard deviation u * &, and a bar over a symbol denotes the mean value. The
expected impulsive plate velocity 5, is given by
(7.134)
with Z(E a”) = Jo”F(l)P(l) d&

It is assumed that large velocity gradients imparted to the plate, which responds as a
membrane, may produce rupture in zones of large strain e. This occurs near the plate supports
Themechanicsof penetrationof projectilesinto targets 7s
or where large density changes in the fragment pattern exist, such as near its periphery. Based
upon a momentum analysis of the target whose geometry is characterized by a traveling yield
hinge, the mean plate velocity is given by
(7.135a)
or
(7.135b)
and its critical value at rupture & corresponds to the ultimate strain CU.The approximate value
of eqn (7.135b) for the range of practical interest, eu G0.2, is
(7.136)
and when this is combined with eqn (7.134) it leads to the inequality
(7.137)
as a fracture criterion for the plate. For plate rupture to occur due to a fragment pattern with
m& < p,h it is necessary that (&R/l)dM)61(~ a*)/2. Other fragment distributions and some
associated vulnerability information are described in [35] and [233]. Maximum fragment
velocities and associated masses from bursting cylinders are cited in [234], together with
suggested criteria and methods for shielding targets from ballistic damage.
Impact at obliquity and ricochet

The shapes of perforation craters show that the trajectory of the projectile begins with
increases of obliquity as if for a ricochet, but this trend is reversed somewhere along the
trajectory so that the projectile emerges from the target element with no more than the initial
obliquity, and often with a marked decrease. For the geometrically simple shape of a sphere,
the direction of these initial deflections for a perforation and the deflections for a ricochet are
readily associated with the direction of the force due to contact with the target element.
Analyses of ricochet that identify the deflecting force with a lift component of the flow pressure
on the surface of the projectile were discussed in Section 6[154-1561. The reversal of this
ricochet trend in a perforating trajectory of a sphere has recently been associated with the
development of failures in thin and intermediate target elements and specifically attributed to
the redistribution of stress on the sphere so as to change its direction of motion[ 141.
The simple analytical model of [14] deals primarily with the determination of the contact
surface from the location of the sphere relative to the surface of the target element and
assumes that contact is limited by the formation of a crater resulting from permanent changes
of shape. This model predicts that final deflections for speeds below the ballistic limit tend to
increase up to 180”. Near the ballistic limit, perforations are close to zero degree and then
increase up to the initial obliquity. These predictions are in reasonable agreement with
experimental data and indicate that the assumption that analyses at normal incidence can be
applied to obliquity by increasing the thickness of the target element is approximately correct
for velocities sufficiently far above the ballistic limit.
The residual velocity for the oblique impact of blunt projectiles was given in eqn (7.42)[ 1221
where the problem is formulated as the calculation of the change in the obliquity B* imposed on
the trajectory of the projectile by the condition of oblique incidence. This angle is estimated by
a consideration of the momentum imparted to the projectile during perforation. The impulse
that is responsible for the momentum change of the deflection is assumed to result only from
the stresses acting on the projectile over its cylindrical surface and to depend only on the
dynamic strength in this region. Thus, the force component of the impulse is independent of the
velocity, but the time component of the impulse depends on the duration of the contact with
the target element and is, therefore, roughly inversely proportional to the velocity. The final
expression for the estimate of 8* is
. ^
=(-g+;&
,y (7.138)
sin2e*
where e^is to be determined experimentally and for blunt projectiles is found to be e^= 8 since
blunt projectiles tend to emerge with zero obliquity at the ballistic limit.
Equation (7.42) can be used to determine the normalized velocity values x = o,,/ux,to check
the validity of the assumption that oblique perforations can be estimated by using the presented
thickness in an expression for perforation at normal incidence. For example, if x = 2, the
deflection 0* is approximately 5”. For a thin target, it is more important to account for bulk
plate deformations like dishing.
The momentum theory of the perforation of thin plates [208,209] by truncated conical-nosed
projectiles was specifically formulated for oblique impact(2101 and is one of the earliest
analyses to consider obliquity; however, the objective was to obtain the appropriate change in
velocity of this case. Deflections were not explicitly derived, but a procedure for determining
deflections was outlined. The theory when modified for obliquity recognized three regimes
according to the changes in target response. At low obliquities, there is no rearward petaling. At
high obliquities, petaling is of the conventional form over those parts of the cone with the
greatest obliquity of its surface with respect to the target element, but the petaling is reversed
(protrudes from the front surface of the target element). At very high obliquities, the petaling
again has a rearward part, but contact with the plate begins along the side of the projectile. The
final expression for the change in velocity is given by
m Au B,n
(7.139)
p,h& tan p c = tan fi cos 8 +[(+jL]G’B.e,
w!tere G(& e) is an integral that represents the axial momentum of the petals. The parameter BI
is an experimental constant.
Numerical solutions
The computer codes based on eqns (3.10-3.16), or similar representations of fundamental
mechanical principles, have been applied to a wide variety of penetration problems-including
thin plate perforation. Representative problems solved by this .technique are discussed below.
One of the earliest solutions using an axisymmetric particle-in-cell code VISTA concerned
the normal impact of solid cylindrical projectiles against laminated plates at speeds of 15 mm/@
and that of hollow aluminum cylinders against the same target at speeds up to that value[235,
2361. Shock focusing along the axis generates both a forward moving jet behind the penetrated
plate and another jet moving backward from the impact surface. Two problems of a steel
cylinder with an aspect ratio of unity striking thin steel or aluminum plates at velocities of
0.61 km/s were treated by the CRAM code and indicated a plugging type of failure11 1I. The
plate velocity and calculated failure time were in good agreement with the predictions of [ 1221.
A computer solution for the deformation of clamped circular plates under explosive loading,
using the structural elastic/plastic program DEPROSS were found to compare satisfactorily
with measured deflections and strains(l871. The results of this investigation have been
frequently compared to those involving projectile impact. An Eulerian code, STEEP, was
employed to study the ejecta characteristics behind thin plates of aluminum, copper, and
cadmium struck by spheres at 7.5 and 15 mm/F. The expansion of the debris cloud was traced
and the mass, axial momentum and kinetic energy of the ejecta was evaluated as a function of
solid angle[84]. The SHAPE program which uses either a particle-in-cell or Eulerian code for
the initial hydrodynamic process and a Lagrangean method for the later elastic/plastic
phenomena has been employed to evaluate ballistic limits on laminated multi-material targets
struck normally by steel cylinders at velocities of 7 mm/j&s[237]. Incipient and perforation
computations were performed using a generalized maximum plastic strain criterion that leads to
plugging and petaling failure, the former pccurring in thinner targets. Further use of CRAM
involved the penetration of equal weight, equal base diameter steel cylinders, truncated cones,
and ogival projectiles into steel and aluminum targets at velocities up to 1 km/s and into
epoxy-steel slabs up to 130m/s [238]. The effects of oblique layup geometries and bond strength
were investigated here.
The monograph on High-Velocity Impact Phenomena[49] contains several papers on general
computational applicability as well as a description of the CAMEO calculations for the
behavior of debris generated upon hypervelocity perforation of thin targets and shields[227].
Substantial momentum multiplication was found in the cases of cadmium-cadmium, lead-lead
and low density impacts at velocities ranging from 3 to 7.5 mmlps. A number of calculations
using the VISTA code are also included in this volume[83]; in particular, the response of
laminated meteor bumpers at a, = 15 mmlps are detailed.
Reference [51] contains a section on penetration mechanics with several useful references to
numerical aspects of thin plate penetration. The impact of blunt and pointed tungsten carbide
and steel projectiles on steel armor and aluminum plates at velocities of l-l.3 mmlps was
examined by means of the SHEP code; penetration, plugging and hole formation were explicitly
treated[91]. Use of the HELP code in a variety of problems, some previously published, is
described in [239]. In particular, applications to a numerical parameter study of hypervelocity
impact, plugging failure in thin plates. occurring during ballistic impact, shaped charge jet
formation, and fragmenting munitions are described with reference to design considerations.
The HELP code was employed to study the effects of five different projectile geometries
upon impact on three aluminum targets with differing thicknesses and one steel target at
velocities of about 0.5 mm/ps[107]. The solutions were found to be in excellent agreement with
experimental results, but deviated from the predictions of simple perforation models[64, 122,
2051that did not provide adequate residual energies. The same code was employed in a study of
iron-on-iron and iron-on-aluminum impacts in the ballistic range[81]. It was also used to
examine the plugging process generated by the propagation of slip discontinuities in the
direction of maximum shear stress upon attainment of a critical value of the specific plastic
work for the material ahead of the discontinuity[240]. Another application considered the
diverse effects on thin plate perforation and the generation of debris of kinetic energy
penetrators, steel cylinders, and shaped charge jets[241].
The two-dimensional codes HELP, CSQ and TOODY have been used to treat oblique
impacts by approximating this three-dimensional problem by a two-dimensional plane strain
configuration rather than by axial symmetry[242, 2431.The results obtained from these codes
are not presented as quantitatively valid, but are assumed to give qualitative indications of the
results for the real system. The codes predict the deformations in the plates and the projectile
and the changes in the orientation and direction of travel of projectile.
The EPIC code which is a Lagrangean finite element formulation of the fundamental
mechanical equations has a three-dimensional capability [244-2461. The code has predicted
deformations of a moderately thick target element struck by a sphere that is in reasonable
agreement with experimental data[l4].
8. INTERMEDIATE AND THICK TARGETS

Intermediate and thick target elements are those for which the existence of a free rear
surface has an effect on the interaction between the projectile and target element. Perforation is
usually the design objective of the projectile, although in some instance? a warhead may cause
damage without perforation. In contrast to the thin target element, the phase diagram of an
intermediate or thick target element has a significant embedment phase. These penetrations, in
which the projectile comes to a complete stop, represent tactical failure of kinetic energy
penetrators and for explosive projectiles, the depth of penetration determines the capacity to
cause damage. Thus, the correlation of depth of penetration to impact parameters is as essential
for characterizing the impact system as the ballistic limit and residual velocity. Many of the
results for thin or semi-infinite targets have also been applied to this domain. This may be
acceptable in some instance; however, considerable caution must be exercised in interpreting
such conclusions. Modifications may be required to account for the consequences of either the
virtual absence or distinct presence of constraints at the rear of the target incorporated in the
two other cases, respectively, that do not occur here.
Empirical equations
The relations cited in Sections 3 and 7 may also be applicable for thick targets, particularly
if the empirical coefficients are appropriately modified. In general, such equations reflect
concepts of force or energy in the form of dimensionless ratios. An extensive discussion of this
subject is to be found[3] which is relatively inaccessible; some portions will be repeated here.
All expressions given should be considered to be applicable only within the ranges of material
properties, velocities, and geometries used in their construction.
Penetration and the ballistic limit velocity have frequently been expressed by products of
the independent variables of projectile mass, target thickness, and initial velocity such as
P = y,mpc’u,oC, (8.la)
and
v.~ = yzm>hC4R> (8.lb)
where yl and Ci are material and geometric constants. The postulated constant resistance to
penetration[20] due to cohesive strength was mod&d to account for free surface effects on the
basis that material near the free surface offers less resistance to lateral displacement than that
near the midplane. This yields an expression for the ballistic limit
” u)=
n
[
- WC,) In
2rryzR:(h
m, 1 (8.2)
where 6, is an experimental constant associated with free surface effects. A similar expression
for a deforming projectile including target cohesive inertial resistance and free-surface effects
was given as [43]
(8.3)
where the projectile radius Rpz was determined for u0 = 762 m/s from the empirical relation
Rp = R,[I + ys(u - 762)/762]“2. (8.4)
For projectiles not shattered by the impact, eqn (8.lb) has been applied to different projectile-
target conf&u-ations yielding the following combinations of empirical parameters
(a)’ y2 = 3.37, C, = -0.5, C-4= 1, c5 = 0.5

(b) y2=0.21, C3= -0.33, Cr = 0.67, C5= 0.33
(c) y2 = 0.064, C3= -0.5, Cq= 0.65, C, = 0.75
(d) yz is variable, CS= -0.75, C, = 0.75, C5= 1.5 with units of m-kg-s.
For obliquity at angIe 0, an expression was proposed as
(8.5)
Penetration relations for normal impact may also be deduced from hypothesized force laws.
If the resistance is considered to be only proportional to the presented area Aw, the penetration
can be expressed as [43]
PI D = y,ppo: + 62 (8.6)
with yi as an empirical constant; b2 has been added to account for free-surface effects. If the
phenomenon is assumed to be dependent either on the tirst or the second power of the
penetration, eqn (8.1) is modified to
AD = y&u, (8.7)
or
PID = y&3’v,o.66 (8.8)
the latter being similar to that proposed by [247]. Alternatively, the force on the projectile may
be assumed to depend upon some power of the velocity, i.e.
F = - a,dv; F = - ag’v; F = a&v; and F = - a,.z21ppv2. (8.9)
All of these force laws may also be expressed solely as some function of entry distance I,
namely
F = - aszap,; F = - asz51dpp; F = - a7(p,#z)‘n and F = - ae(@z)‘n (8.10)
respectively, with ai as constants, that yield the corresponding penetrations
AD = yloppv,,; PID = y,,pj’3v~‘3; AD = y12p$$~‘2 and PID = y,grv!‘. (8.11)
If the force is expressed as the sum of an inertial and a resistance term apd(aloa~c + p,v2) in
the manner of [31], also employed in the theories of [202,205], the resulting penetration relation
is
PID=% (8.12)
1 k9++&w
which is of the same general form as the result based on an analytical model which will be
described subsequently[248]. Alternatively, albayc may be replaced by af&YN. At low
velocities the strength effect is dominant: it may be neglected only when pro:%-aloayc, i.e. in
the hypervelocity regime.
The force law suggested for the partial entry of conically-nosed penetrators into hard
metallic targets at ballistic velocities has been of the form
(8.13)
where the crushing pressure pa is a function of temperature 8, and AN is the base area of the
embedded cone (2491.
Normal penetration depth relations, which may also apply for targets of finite thickness
providing the effects of the distal surface are unimportant, have also been proposed in a variety
of forms, with particular reference to high- and hypervelocities. A popular functional represen-
tation, particularly for the impact of cylinders and spheres at normal incidence based on
dimensional analysis, has been used in the form
pID= YM(~)"(~)"(~)" (8.14)
with D = DI as the diameter of an equivalent spherical projectile, if a sphere was not used and
with the constants fitted to the particular experimental conditions. Typical values have included
Yl4 c6 G cs
:; 2.28
l.% 213
112 213 0 [247,250,251]
[2521
(c) arbitrary l/3 0 l/3 1451
(d) arbitrary l/3 l/3 0 ]451

(e) arbitrary 213 0 l/3 ]2531
(f) 2.5 0 0 l/4 for lSu/coS2 ]2541
(g) 0.395 0.56 0.625 - 1 for two types of aluminum ]2551
(h) 0.482 0.537 0.576 - 0.235 I2561
(i) 0.311 0.167 0 0.282 with uyT replaced by uus,. w71
Other expressions of this type include
P/D = y,@2’3 1” (1 + $!l;)

(8.15)
YIS = 0.6 ~,~==4 [43,253,254]
found to be applicable over a wide range of velocities for impacts involving ductile projectiles,
but requires a specific knowledge of yIs and aI3 for each combination of metallic materials used.
The relation
P/D = y,~(~)2’3~hU:,BHN,)1’1 (8.16)
was found to be applicable for many different metallic targets in the limited range from 2.5 to
5 km/s [253]. Extrapolation beyond this range is unwarranted. Also, for conservative estimates
of penetration [ 1731
P/D = l.3($3(~)1)[6_9~log~ (8.17)
where E, is the Young’s modulus of the target in GPa. A penetration relation has also been
proposed based on momentum per unit presented area corrected for a strength term[250,258],
p = YdPp~oLp - 63). (8.18)
The penetration resistance for metals has also been empirically expressed in terms of Brine11
hardness by
P = ~,sm~‘3u,C/BZfNC7 (8.19)
where Cs is either l/3 or 2/3, and C, either l/4 or l/3, respectively.

A number of empirical relations for the crater volume PC,in the ultrahigh and hypervelocity
regions have been proposed. Two initial hypotheses stipulated that this quantity should be
proportional to initial kinetic striker energy T, [8] or else the product ptT,Qd2601 where Qe is
the heat of fusion. Most of the subsequent expressions developed for crater volume are
accompanied by corresponding crater depth relations. These include the following, based on
actual or equivalent spherical projectile volume Prp or initial energy T,
T’JP,, = 8.8(~)(~)‘(&~& in MPa (eqn 8.17)[164] (8.20)
I’,./P,, = 34ey(z)2 (eqn 8.14a)[247] (8.21)
PC/P, = 30.25(zy(;)2 (eqn 8.14b)]252] (8.22)
PJl-+, = 46.8($($)*y$ [261] (8.23)
rdrpr = o.l2e)(er (eqn 8.143)[257] (8.24)

rJrpe = 1- e-+9To
WI (8.25)
l-c/T, = ~,z(co@J [2631 (8.26)
rdT, = 100 3.1- 0.6 ; BHN m2/kg[43) (BHN in dyne/cm*) (8.27)

c + V)m I-’
where yus is a parameter related to the ultimate strength. Other expressions may be found in
1711and [264]. The crater diameter DC has been empirically represented by
DAD = 4.S(p,/p,)(v~c~)*” (for hemispherical craters)[251] (8.28)
DAD = (2.6[(pdp,)(v~~~)]‘.~(hlD)~.~] + 1 (for thin plates)]1741 (8.29)
D~D=(pp/p,)o~osr[(~)(~)]o’o’(~)~3+ 1 [257]. (8.30)
Analytical models
The classical model of penetration by ductile hole enlargement cited in Section 7 has a form
for thick targets. This model depicts the expansion of a uniform hole of instantaneous radius R
to its final size Rf in an infinite plate subject to conditions of plane strain, i.e. where
h, > D(43,202]. The model incorporates a yield circle of radius ry separating an outer elastic
region from an inner, perfectly-plastic regime at yield stress uy; only inplane stresses are
considered. The boundary ry is determined as a first approximation from the quasi-static
solution of the expanding cylinder problem as
(8.3 1)
ry = R
The stresses in the elastic and plastic regions, governed by the Tresca criterion (eqn 3.9) are given
1 rY 2 (8.32a)
ur=--ar 2 0r ’
. r 3 ry
ad = icry(rJr)“, (8.32b)
and
0,=-u, ln[rdr]++ , (8.33a)
( >
rSry
Ud = a, (8.33b)
and the displacement is given by
l+v R2
W’=a- r> ry.
Equation (8.32) is modified by use of the dynamic equilibrium equation
d2wr=*+ur--OQ (8.35)
‘ai at r
and the hole expansion history, given for an ogival projectile of nose length LN, penetrating at
constant velocity u,, approximately by
t<o
o<t+ (8.36)
t==Lfgfvlu,
to yield a more accurate value of w, and ry.
UBSVd. 16 No. I-P

In the plastic regime, the equation to be satisfied is
2
&J=~-"' (8.37)
r
which yields an approximate solution for ry
(8.38)
The radial stress at the hole, t+(R), is given by
where Si and Ci are the sine and cosine integrals and
(8.40a)
(8.40b)
with
The total resistance to penetration is
W=-h, ‘t?rRu,(R) dR
= ?rR;h, 3.030~ + 0.455p, (8.41)
where the last quantity in the bracket is the dynamic term. An average yield stress, en given by
(8.42)
)
is recommended where u’t corresponds to strain rate
i* = (Jyqy)(2k)
and y* is a numerical constant with values between 0.1 and 0.2 for steel.
Another penetration model[248] specified the combination of the static and dynamic
components of the force on a rigid projectile to be of the form
F = ~R2[BHN”* + (1/2q~j,)"*~
I* (8.43)
where shape factor Y was given the values l/3, 213, and 1 for pointed projectiles, spheres, or
micrometeorites, respectively. Integration of eqn (8.43) gives for the penetration P
(8.44a)
or
(8.44b)
for a spherical projectile. This expression was the only theoretical prediction that seemed to
agree well with the results of experiments, including ductile penetrators over a wide range of
initial velocities.
The application of the energy balance technique employed in conjunction with eqn (7.40)
requires a modification for thick plates in that the initial plug mass mg may no longer be equal
to the plug ejected from the plate, m+ Thus, eqn (7.41) should be revised to read
(8.45)
for normal impact of blunt cylinders. For pointed projectiles, no plug is formed, and the
appropriate relation for any angle of obliquity becomes
(Uf/VSO)
=[ (2) - l]“*. (8.46)
This assumes that the work of displacing the material laterally is constant, a hypothesis that
seems to be reasonably well satisfied experimentally when uJvs,,> 1.5. For thick plates, at
obliquity 8, the emerging plug mass has been represented by [41].
vtD*ho (8.47)
mqf= 4cos 9.
An extension of the hole enlargement analysis for sharp projectiles embodied by eqn (7.80)
to thick plates considers five geometrically different stages of penetration, as shown in Fig. 39,
in each of which the force acting on the rigid projectile is different [265]. The initial conditions
for each phase are the terminal displacement and velocity for the previous regime. The total
retarding force on the penetrator is the axial component of the sum of the normal hole
enlargement force and tangential friction force yielding for each stage, with v = i
Stage I: O<ZSLN, F(z) = -(Ali*+ B,)z*
with
AI = $ *pl tan p sin* @(tan@+ f D,
and
B1 = ?T(Tyctan /3(tan /l+ f D,
Stage II: L,<zsL, F(z)=-_[A,i2+C2(z-LN)+B2]

with
A2 = A,L$,
C2 = ruycf DD
B2= B,Lk
Stage III: L<z<h, F(z)=-[Azi*+C2(L-L,)+B,]
Stage IV: h<z5h+LN, F(~) = _ [Ade(-2A,/m)(Z-L)+ B4(z _ h)2e(-2A1/3m)(x-h)3]
where
Aq = A&,- Cz(L - LN) - I32
Bq=-A,v;,,+B,
Stage V:
z>h+LN, F(z)=-Cz(h+L-2) (8.48)
where the velocities with Roman subscripts denote their values at the end of the respective
phases. Test data involving penetration of blunt, conical, hemispherical, and ogival-tipped
STAGE -I
;Tt____ +L;_____--------I-
III
IV
’ L
-----________
r
-z
A
i”t _______
LN
__
Fig. 39. A five-stage penetration process for a sharp penetrator and a thick target.
cylinders of 50 caliber, all weighing 40 g, into Lexan at velocities ranging from 467 to 767 m/s
correlated reasonably well with predictions using values of fD = 0.25 and ~YC= 75.8 MPa,
except at the end points.
Quasi-empirical representations
A mode1 of the perforation of a target by a rigid sphere has been proposed that incorporates
a strain-rate independent elastic-plastic strength term and an inertia term acting normal to the
leading striker surface as well as a tangentially acting frictional force, whose coefficient
decreases with increasing velocity. This is claimed to be valid in the initial velocity region of
several hundred m/s, where the penetration time exceeds UC,, but the velocity is not sufficiently
high to deform the penetrator. Since these stresses are constant at any instant, the total force
can be obtained by integration. The front and rear surface effects are described by two factors
&(z) and t,,(z) with values between zero and one. The retardation force is given by [266].
F = &&k%z)(~~ + a2+f%w:c+ b2M2) (8.49)

where
&(z) = 1 - eeJBID, &(z) = 1 _ e-(z-l-B,DW,D)
a(D-z)z, Osz10/2
al = at = (8.50)
z>o/2
with
(=cos_‘(l-21/0)
fD = f[ 1+ e(z-BdB~]-‘. (8.51)
The material parameters Bi and 62, low-velocity friction coefficient f and dynamic yield
strength a& must be determined from specially designed experiments. The value of 3.5 was
assigned to y, but a more appropriate magnitude would lie in the range from 2.6 to 2.9
corresponding to the ratio of punch indentation resistance to yield stress[267]. Values of the
parameters determined from special experiments for two types of aluminum alloys and mild
steel are given in Table 4.
Table 4. Values of material parameters for the theory of 12661
Property Parameter
P E =YC cut BHN Gc f b, B, BI B, B, B,

g/cm’ GPa MPa MPa MPa m/s m/s
SIS 4007-14 2.7 70 70 90 25 80 0.95 0.29 0.5 I.15 0.7 510 120
Al
SIS 42 12-6 2.7 70 260 Hx) 115 181 0.92 0.29 0.5 I.15 0.7 570 210
Al
Mild steel 7.85 205 692 530 0.52 0.29 0.5 1.15 0.7 2000 100
Good agreement was found between the prediction of eqns (8.18), (8.19) and results from
impact tests of an 8 mm hard-steel sphere striking the two types of aluminum targets, 100 and
25-mm-thick, respectively, in the range from 500 to 1500m/s.
It has been suggested that the model be extended to include an arbitrary nose shape defined
by the relation[268]
r = f,(z*), 0< z*< LN

(8.52)
r=U2, z*> LN
where .z* is the axial distance from the nose tip. The penetration force is now given by
F(z, i) = 27r[yc$c+ bzp,i2] I;; c(z*)&(z*)Mz*)[~ +P(i)]

(8.53a)
for{;;;;aLN(8.53b)
and12=;N
The factors describing the edge effects, assumed to be roughly the same as for a spherical
penetrator, are given by
&(r*)= I- exp
(8.54)
-z*+z+l+B,D
&*> = 1-exp
2&f,@*) I-
Reasonable agreement of the predictions of this theory with data for conical-nosed projectiles
was found.
Hyperuelocity impact
The hypervelocity domain is of concern to the ballistician because of the application of
shaped charges and of the comparable effects produced by high velocity long-rod penetrators. It
is also of fundamental scientific interest in the study of impacts produced by meteors and
meteorites; it permits the understanding of system behavior in a regime not ordinarily
experienced. In general, the analysis of phenomena consists of (1) a purely incompressible
hydrodynamic approach, (2) shock wave propagation and intersection for compressible materi-
als, (3) viscoplastic theories that attempt to take into account inertial, strength, and viscous
effects or (4) the elastoplastic-hydrodynamic characterization described by eqns (3.10)-(3.16)
that can only be evaluated by extensive computational codes. Because of the vast literature on
the subject and its limited applicability to penetration mechanics, only a few of the more
important developments will be described in this paper. A recent brief survey of the subject is
given in 12691.
Both empirical equations and analytical models show that the principal means for increasing
penetrator efficiency are: (1) to increase its velocity, (2) to increase its relative length L/D, and
(3) to increase its density. The shaped charge is an explosive device that produces, on
detonation, a slender column of comparatively dense material (such as copper) that travels at
speeds in the hypervelocity regime, and thus exploits all three means for achieving increases of
penetration. These penetrators have proven to be efficient and are used to attack thick tough
armor. The speeds at which these impact target elements result in inertial forces greatly in
excess of material strength so that the mechanical principles that govern penetration are
essentially those of hydrodynamics. A theory of the penetration of long rods into thick targets
has been developed from hydrodynamic principles that predict the essential elements of shaped
charge penetration[12,25,65,67,69-711.
This model, shown in Fig. 40, consists of a jet of material of density pj and length Li that
travels with velocity u(z) and penetrates a target of density pf to a distance P(z) at a rate u(t).
To represent standard forms of shaped charges the leading element of the jet has a velocity z’,
greater than the trailing element with velocity z]h.Ls is the standoff distance, i.e. the distance
from the target element at the completion of the formation of the jet, which is significant since
it represents the distance over which the jet can increase in length due to the velocity gradient.
If 5 denotes the initial z coordinate of the element that arrives at the stagnation point PZ at the
TARGET
I
Fig. 40. The penetration of a thick target by a high-speed jet.

time t and if o&) is presumed to be invariant prior to impact, the displacement of the element
at time t is
fdf) = ZW + P(f). (8.54)
For steady-state conditions, a modified form of Bernoulli’s equation governs
(8.55)
that accounts for a dynamic yield strength of the target, given by a:, and the yield stress of the
striker a&. If these strengths are neglected, the penetration P of the jet can be expressed in the
form [70]
If, further, the velocity of the jet is uniform over its length (so that the rod is rigid), eqn (8.56)
reduces to [25]
p = LJ
0 “2.
f
If the dynamic yield strength of the target is included, the penetration for a uniform jet is given
by
The change in the behavior of the penetrator from that of a rigid rod to the jet inversion
shown in Fig. 40 occurs at or in the neighborhood of a critical velocity u, that depends on the
state of cavitation around the striker and has been experimentally found to lie between 1000
and 2000 m/s [38]. In contrast to the assumption for the shaped charge that leads to eqn (8.55)
for a velocity u, just above the uX,the crater bottom is not moving supersonically either with
respect to striker or target, and therefore there is a second critical speed ok above which such
supersonic behavior is attained. This velocity can be determined from eqn (8.55) by replacing
uJ -u with the sound velocity of the striker, and amounts to twice this value for identical
striker and target materials. Between the two limits, the rod inverts, but an additional force is
transferred to the rod that effects the penetration velocity u.
The behavior of a rod with initial length Lo,but varying with time L(t) is obtained by the
use of eqn (8.55) and an equation of motion for the incoming projectile acted upon by the
braking force at the base of the crater and given by [74]
du, where
dL(t)
-=-(u,-u)=dt-u.
dz
P,LW~ = - a$ (8.59)
dt
The value of u is obtained by assuming the strength difference & - & to be small compared
to the dynamic pressures, yielding
+pu-~ B*u2
where
D
0 1 &-ay,
and B*= 7 (8.60)
0 DO
A*‘l+V& UG'
With the definitions
d 1 and f(x)= exp _ *2)]

it follows that
L(f) = LoxB*‘c’f(X); (8.61a)
~~*‘~‘f(x)dx (8.61b)
(8.61c)
The crater radius R, is obtained from a balance of the energy of deformation of the target and
that lost by the projectile and is given by
(8.62)
where
and e* is the work necessary to displace unit volume of the material. The validity of the
equations decreases as v0+ v: since the shock then travels slowly toward the projectile rear,
and the braking force cannot be considered continuous, as indicated by eqn (8.59). It is
interesting to note that the perforation of a target by a bullet traveling initially at a velocity
v, < v, takes place with a significant drop in velocity, but no decrease in mass; for a shaped
charge with a velocity v, > v:, this takes place without a drop in velocity but with a decrease in
mass; in between, the slender rod with a velocity v, 5 v S v: loses both mass and speed.
The first and also one of the most significant effects on the target in hypervelocity impact is
the propagation of shock waves and the motion of the material behind such fronts, both of
which are treated by hydrodynamical equations. This topic is described in numerous papers
and monographs[228, 229, 270, 2711. Hypervelocity penetration by rod projectiles or Munroe
jets involves three additional stages[68, 711: (a) a primary penetration region during which a
constant target-projectile interface velocity exists, with the relatively short duration
(8.63)
when strength is neglected, and governed by eqns (8.55)-(8.58); (b) a secondary penetration
domain (or cavitation) when the projectile has been completely deformed, but the shock wave
continues to expand along with the crater cavity; and (c) a recovery stage when the energy
density at the cavity wall no longer can overcome the inherent material resistance, and
elastic-plastic or brittle restitution ensues resulting in crater contraction, occasionally to a size
smaller than at the end of stage (a), or else brittle failure. Based on shaped charge analysis, the
total penetration at the end of process (b) is given by the sum of eqn (8.57) and half the crater
diameter, in reasonable agreement with experimental data for steels and aluminum in the range
from 2 to 6.7 mmlps. However, a better correlation was found by using the expression
P = (L, -D)($z+o.13($*[&]“2 (8.64)
where E, is the initial kinetic energy in joules and BHNf is the Brine11hardness of the material
after impact. Shock and particle velocity were found to be linearly related in the regimes
below and above phase transitions for the materials investigated. Viscous effects on shock
wave propagation are discussed in [2721.
A semi-empirical model for this regime considers two stages of the motion of a rigid
projectile in such a target. First, for velocities above Mach 5 the target is considered as a
compressible fluid exerting a drag force with a coefficient CD = 1, yielding an equation of
The mechanics of penetration of projectiles into targets
motion for a spherical striker[273]
d’p, d( y2)_ 1 lrD2

- - 2 4h02 (8.65)
12 dr
with a penetration P at Mach 5 given by
WQ(Mach 4&]nU”
5) = - (8.W
3 PI 5CP
where cp is the plastic wave velocity. Below this speed, the penetration was empirically
specified, yielding a total penetration of
(8.67)
where h = S(CT~/CT~~of copper) in MPa and cp was taken as 305 m/s. The neglect of target
melting was justified on the basis that mechanical energy in the penetration process is
transferred much more rapidly than heat.
Comments on experimental work

Almost all analyses of the penetration of target elements have either been accompanied by
experimental data or have referenced experimental work. The empirical expressions, in
particular, are representations of numerous experimental results that were obtained in general
by the experimental procedures discussed in Section 3. Some experimental work will be
discussed next that represents either significant departures from standard procedures or that
illustrates particular methods.
Examples of the transmission of information from a projectile concerning the conditions
that exist during impact were described in Section 6. This technique is much more difficult to
apply to impacts against thin, intermediate and thick target elements, but has been accom-
plished by avoiding the complete instrumentation of the projectile. The strains in a projectile
were measured by strain gages mounted on the projectile for impacts against target elements
with hJD of from 0.25 to 12[274]. The problem of data transmittal was solved by the expedient
of firing the target elements at the stationary projectile with the electrical connections leading
directly to the recording instrumentation. The target element was placed in the nose of hollow
projectiles which were 5.8 times larger in diameter than the instrumented projectile. Obliquities
were simulated by fixing the target element at the required orientation with respect to the
projectile axis.
The force that the projectile exerts on a plate during normal impact has also been measured
by strain gages mounted on a load cell which is attached to the rear of the plate to record the
force exerted on it[275]. The output of the strain gages was calibrated as force by applying
static loads of known magnitude to the apparatus. The instantaneous penetrations of the
projectile were measured concurrently with the force by means of a coaxial capacitor that
sensed the displacements of the rear end of the projectile. The output of the capacitor was
calibrated as penetration by static displacements of a projectile. The prediction of force from
the acceleration derived from the measurements of penetration as a function of time were
compared to the direct measurements of force and found to agree within 7%.
Measurements obtained by this scheme indicate that the maximum force attained with a
given target does not change significantly for flat-ended, hemispherical, or ogival nose shapes.
The location of the peak force during the penetration process does reflect the effect of nose
shape and the energy required to perforate a given plate thickness is shown to be least for the
flat-ended shape and the greatest for the ogive. These differences in energy are attributed to the
differences in the way that material of the plate is displaced for each projectile shape and which
results in significant differences in the conditions required for the final breakout of the
projectile.
In Section 7 the observation was made that perforation at obliquity begins with deflections
that increase the obliquity of the trajectory of the projectile and that this trend is reversed as
penetration proceeds. High-speed photographic studies 12761record these trends for target
elements of intermediate thickness and projectile with ogival nose shape. Reference [276] also
includes X-ray pictures of the penetration of long copper and steel into plastic and aluminum
alloy targets and show the inversion process illustrated in Fig. 41.
Fig. 41. Cross section of a thick-laminated target element showing the inversion of the penetrating rod.
Numerical solutions
Many of the numerical procedures described in Chapter 7 either apply equally well or may
be readily converted to the case of thick targets, although, here, the failure processes are far
more complicated and the proper criteria less well known. Design investigations to determine
the best profile of five different heavy-penetrator nose shapes fired against thick targets were
conducted with the aid of the HELP program[107]. The failure criterion was based on a
minimum allowable compression ratio of the material. This occurred in regions where adiabatic
shear failure was observed in corresponding experiments. Measured crater depths were found
to be somewhat shallower than predicted. The spa11threshold for the hard alloy-steel penetra-
tors exhibited values of 3.65 and 0.91 GPa, respectively.
Investigations of long-rod penetration at normal and oblique incidence, including a
parametric study of material property variation, were also carried out by means of this code[82,
242, 2771. The results were in good correspondence with those obtained by other numerical
methods for comparable problems. The HEMP code was further used to determine the
response of steel plates struck normally by steel cylinders with diameters comparable to plate
thickness at velocities of 0.8 mm/ps[278]. Stress contours were generated in both objects that
indicated a difference in spa11velocity depending upon target hardness; results were compared
to corresponding experiments.
Numerical studies involving the performance of steel-jacketed kinetic energy penetrators

employing the same code indicated that a core of high-strength tungsten did not yield any
improvement over one of ordinary tungsten [ 1041.Furthermore, it was found that a conical nose
shape was not efficient for penetration of thicker targets. A variety of thick targets were
investigated by the use of two-dimensional codes such as VISTA, PICKWICK, CRAM,
DEPROSS, or a combination, with respect to penetration by low-density projectiles simulating
a meteoroid [83].
9. CONCLUSIONS
The previous chapters have surveyed the subject of penetration mechanics by listing the
various methods employed for the interpretation and categorization of the phenomena. It is
evident that ranges of impact velocity, thickness, obliquity, and material characteristics are the
basis for establishing a reasonable degree of order among the many approaches. It is also
evident that there is a great diversity among these approaches, which range in sophistication
from simple descriptions of test results to intricate mechanical theories. The diversity of these
approaches is due in part to several aspects of the contributing phenomena. The range of
impact speeds that is easily attained result in mechanical phenomena whose descriptions are
nonlinear and involve significant changes of material behavior. Several distinct types of
material behavior are exhibited by common target elements. The diversity of these approaches
is also the result of the varying interests of the sources of support for the development of
penetration mechanics. These sources, which are largely concerned with the effectiveness of
ordnance, are often satisfied with specific solutions to highly restricted problems and may
change the direction and level of interest for reasons unrelated to progress in the field. A period
of activity in a given area followed by neglect does not necessarily mean that the area was
brought to ‘completion or even to a point of diminishing results. For example, studies of
dynamic material properties were strongly pursued during the 1950s and early l%Os and then
greatly reduced. Hypervelocity impact was also intensively investigated in connection with
problems such as the meteroid hazard to space vehicles, and when a better perspective was
achieved on this problem the subject was virtually abandoned.
There is considerable diversity in the various approaches, and these are also not necessarily
consistent. This is to be expected in an active field of research, but there are few guidelines for
choosing from among these approaches the one most appropriate for a given system and given
conditions. At the present time the impediment to establishing such guidelines is not so much a
lack of means for either observing or interpreting observations as a lack of a synthesis of
observations and interpretations into a comprehensive theory that encompasses the specific
theories of particular systems and particular impact conditions.
Sets of differential equations that represent the fundamental conservation laws and con-
stitutive relations certainly promise to provide the required theory, and there are many
indications that in the long run numerical solutions of these equations will provide predictions
of the theory. The difficulty at the present is not just the time, expense and complexity of this
approach, but also that the numerical solutions tend to hide significant trends in a welter of
detail and in inaccessible steps of computation. This criticism does not mean that the numerical
approach has an inherent weakness as a predictive tool, but rather that there is a difficulty in
stating problems so that the generality and capacity for detail of the numerical approach can be
used effectively.
Analytical models are potentially the means for arriving at the appropriate problem
statements because these derive conclusions from fundamental concepts through idealizations
and simplifications that explicitly isolate trends and at the same time put the expressions in a
tractable form.
The following subjects are considered to merit investigation as contributions to
phenomenological modeling:
1. The development of more rational bases of specifying modeling parameters. In several
successful penetration models certain parameters are specified beforehand to make the models
completely predictive, such as ballistic limits, or failure loci that determine plug masses and
changes of particle velocities in perforation processes. A second order of analytical studies
should be initiated to better define these phenomena and permit the determination of these
parameters by less empirical procedures. The next two topics can contribute to such studies but
have value in themselves.
2. The improved description of interface phenomena. There is a continuing need for
improvement in the modeling of the formation of the crater in the target element and the
expansion of the adjacent anelastic region and for verification by experimental measurement of
spatial and temporal pressure variation and deformation history of the crater. The need for
modeling these phenomena is strongest for interface phenomena involving oblique impact,
nonzero flight orientations, and deforming projectiles.
3. The improved analytical modeling of force contributions. Current specification for the
resistive force acting on the projectile includes inertia, target compression and shear, and
occasionally a simple friction representation. There is a need to extend existing modeling
concepts to thicker targets. There is also a need to account for the dishing and bulging of thin
target elements by considering the bulk deformation of the element, for example, due to
transferred shearing forces by a plug. Inclusion of this phenomenon is particularly important in
the vicinity of the ballistic limit. The plug displacement can then be given relative to that of the
target; methods of deriving the latter using large deformation theory (for thin targets) are
suggested in [80,83,180].
4. Eflects of obliquity. Most phenomenological models of penetration and perforation are
applicable only to the case of normal impact, although some recent developments have
specifically attacked the question of the effects of obliquity[l4]. Further efforts in this direction
are needed, particularly for the case of intermediate and thick target configurations.
5. Efects of flight orientation of the projectile. Virtually all analytical and most experimental
configurations do not include considerations of yaw or pitch of the projectile prior to impact
(and occasionally not even spin). Such initial motions should be included in the theoretical
development, and the terminal kinematics of the projectile, if intact, should incorporate the
possibility of such rotations.
6. Improved constitutive representations. The vast variety of possible targets is reflected in
the present catalog of possible behavior patterns of the materials. More general rheological
models may need to be investigated for some substances, and the effect of viscosity may need
to be given greater prominence. As an example of relative ignorance of the response of
materials, there is virtually no reasonable bank of information concerning ice and snow under
impact conditions.
7. Thermo-mechanical coupling and heat dissipation. Present models generally ignore the
effects of temperature changes on stresses and material properties, excepting possibly their
influence on yield and ultimate strength. Coupled thermo-mechanical fields should be examined
to delineate various zones of material behavior.
8. Failure critetia. Analytical and experimental investigations should be further directed to
provide better information on multi-axial failure criteria and their dependence on strain rate
and/or temperature. This should be applied to the determination of failure conditions in both
target and projectile.
Acknowledgements-The authors gratefully acknowledge the assistance of William J. Stronge, Robert G. S. Sewell and
John Pearson who offered critical suggestions; Stephen A. Finnegan who performed special studies and computations; and
Toni Goff and JoAnn Santelman who prepared the manuscript.
NOMENCLATURE PRIMARY
r volume
A dilatation
8 temperature
A dimensionless variable
@ slope
Y dimensionless variable
\y shape factor
fl solid angle
a material property constant
/3 half-cone angle
7 material property constant
6, Kromecker delta = A :i:
strain
; angle
9 variable of integration
6 obliquity
curvature
TrLame constant, eigenvalue when subscripted
CL coefficient of viscosity
Poisson’s ratio
; general variable
P density
* stress, standard deviation
7 time interval
a acceleration, constant when numerically subscripted, variable radial coordinate
b constant when numerically subscripted
C wave velocity
specific energy, deviatoric strain
; coefficient of friction 1
gravitational constant
h” target element thickness
i artificial viscosity
I length variable
m mass, projectile mass
n integer, number of objects
P pressure
4 specific heat energy
r radial coordinate
s stress deviator
t time
u particle velocity
u velocity
w displacement
x
Cartesian coordinates or rigid body displacements-
Y
I z in direction perpendicular to a target element
i area, constant when numerically subscripted
B constant when numerically subscripted
BHN Brine11hardness
c aerodynamic coefficient, constant when numerically subscripted
D projectile diameter
E Young’s modulus
B energy
force
G” shear modulus
H height of bulge or crater
I linear impulse, linear momentum, rigidity index in material description
K bulk modulus
L length
M moment/unit length
N force/unit length
P penetration, probability distribution
heat energy
2 radius, radial dimension
S entropy, standoff
T kinetic energy
u shock wave velocity
V specific volume
w work
z impedance factor
Subscripts
a axial coordinate of projectile
c effective
n melting or sublimation
K critical
A A limiting
CL maximum
Y minimum
6 embedment
v perforation
p ricochet
7 time
4 coordinate
0 fusion
(ad) additional
(ce) cavity expansion

(ch) cohesive
(co) condensed
(ex) expanded
(in) internal
(re) reflected
(rr) transmitted
(so) sound
(SO) ballistic limit
A total
B bending
C compression
D dilatational
E elastic
F friction
G inertial
H hydrodynamic
I interface
J jet
K plate
M moment
N nose
0 rod
P plastic
Q separation
R fracture
S shear
T tension
U ultimate
V viscoplastic
W presented
X distortion
Y yield
Z hydrostatic
(I acceleration
b blunt
c crater
d fragment
e equivalent
f final
g front
h back
i
indeces
: t
m mass
n normal
0 initial
p projectile
4 plug
rcoordinate
ssharp
t target
u surface
w membrane
X
Y coordinates
L
i dimensionless variable
4 Right orientation, component coordinate
IJ eigen function
o angular variable
Superscripks
. time derivative
S static
D dynamic
,._
*, I special designation
- over a symbol = average (mean)
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Backman & Goldsmith - The Mechanics of Penetration of Projectiles Into Target

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THE MECHANICS OF PENETRATION OF

PROJECTILES INTO TARGETS

MARVIN E. BACKMAN and WERNER GOLDSMITH

Target element phenomena . . . . , . . . . . .

Methods of obser;ation’and analysis . . . . I . * .

2. REVIEW OF PREVIOUS SUMMARIES INVOLVING TERMINAL BALLISTICS . , . , . .

3. DESCRIPTIVE OR OBSERVATIONAL METHODOLOGYFOR THE PENETRATION PROCESS

4. CHARACTERISTICSOF PROJECTILES 21.

UESVol. 16, No. I-A

Rigid blunt projectiles-failure by plugging 53

8. INTERMEDIATE AND THICK TARGETS 77

It is convenient to classify target elements by thickness, using the following operational

Target element phenomena

(ai FRACTURE DUE TO INITIAL (bl RADIAL FRACTURE BEHIND

(c) SPALL FAILURE (SCABBING! (dl PLUGGING

(ej PETALING (FRONTAL) (f) PETALING (REARWARD)

Fig. 2. Peroration mechanisms.

Fig. 4. Internal deformations formed by penetration of a sharp projectile.

These concepts are shown in Fig. 7.

300 - RICOCHETS INTACT

ARMY PROTECTION NAVY

PEN1-j “IR; 1 PART9

Fig. 8. A phase diagram using the parameter hJD as a variable of state.

Methods of observation and analysis

2,REVIEW OF PREVIOUS SUMMARIES INVOLVING TERMINAL BALLISTICS

pilation in 18731301consists of an analysis of test results using an empirical relation known as

3. DESCRIPTIVE OR OBSERVATIONAL METHODOLOGY FOR

Morin: (1833) P/D = 2mv~/alnD3 (3.1)

Analytical models at ordnance velocity

F = a$ + a9v + a10v2 (3.6)

Analytical models for hypervelocities

Sij = Uij - Crsij eij = Eij - ESij

Fig. 10. A typical uniaxial stress-straindiagramfor representingmaterialbehavior from elastic through

IJIi.9 Vol. 16. No. I-B

Numerous variations of the constitutive assumptions, extensions to include discontinuous

where B3 and B4 are constants.

p = (Q) - &JP~,, + (eh- %dPm

Kinetic energy penetrators

ADAPTER f STARTER MIX I

ARMOR PlERCiNG INCENDIARY

FIN STABILIZED DISCARDABLE SABOT

Fig. I I. Examples of kinetic energy penetrators.

Fig. 12. Examples of deformations that result from impact.

x*= 16, V&sin [cos-’ (cos 0 cos 4)] (4.1)

Penetrators with warheads

HIGH EXPLOSIVE PROJECTILE

GUIDED MISSILE WARHEAD

Fig. 13. Examples of projectiles with warheads.

sharp.. . . .for nose half-angles. . . . .14”

Projectile and trajectory orientation

$ = exp K- PPU~(V,+ 2c~N2&1 (4.6)

ORIGINAL CYLINDER /TARGET

occurs only within increment

dz = (cP -{v - or})dt = (CP- UL)dt. (4.7)

Conservation of mass requires that

For (a), pressure balance is obtained at the interface if

m&, = mfiv + (m, - mE + m&l

for (a) LT = Lo exp (- u,p,cdZ*u~~) (4.12)

u,,,, = 0.28~~~ tan 8, (4.20)

v, = .&(v:- v:o)“* = [ 1 + E ($)($)]-‘(v: - v:o)“. (7.41)