Impact Loading of Plates and Shells By

Pergamon Int. J. Impact Engno Vol. 18, No. 2, pp. 141-230, 1996

Copyright 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved

0734-743X(95)00023-2 0734-743X/96 $15.0 + 0.00

IMPACT LOADING OF PLATES AND SHELLS BY FREE-FLYING PROJECTILES: A REVIEW

G. G. CORBETT,I" S. R. REID:~ and W. JOHNSON 1"Department of Engineering, King's College, University of Aberdeen, Aberdeen AB9 2UE, U.K., ~/Department of Mechanical Engineering, UMIST, PO Box 88, Manchester M60 1QD, U.K. and

Ridge Hall, Chapel-en-le-Frith, via Stockport SK12 6UD, U.K.

(Received 15 February 1994; in revised form 24 February 1995)

Summary--This paper reviews recent research into the penetration and perforation of plates and cylinders by free-flying projectiles travelling at sub-ordnance velocities. It is shown that over the last few years there has been a significant amount of experimental research into a wide range of projectile-target configurations. Although most of this research has been concerned with the normal impact of monolithic metallic plates by non-deformable projectiles, valuable work has also been carried out on non-normal impact, impact by deformable projectiles, impact of non-homogeneous metallic and non-metallic targets (including laminated targets) and impact of pipes and tubes.

Recent analytical developments that enable the important characteristics of the penetration and perforation process to be modelled are reviewed. These include models that predict local deformations and failure, global deformations or both. It is shown that for some impact situations fairly simple analytical models are capable of predicting target response reasonably accurately, but for others, particularly when both local and global mechanisms contribute significantly to overall target response, more complicated models are required. The development of numerical codes that predict target response to projectile impact is briefly reviewed and the capabilities and limitations of current codes are discussed. The review also includes a section on the impact of soils and reinforced concrete structures.

NOTATION

Unless otherwise stated in the text the following notations apply:

A, B target plate dimensions C concrete aggregate size B1, 2 dashpot coefficients D tube or pipe diameter E material Young's modulus E c minimmn energy required for perforation E v energy absorbed in plate dishing Er minimum energy required for onset of fracture E k kinetic energy of projectile E m energy absorbed in projectile mushrooming E v energy absorbed in plastic deformation of target E, energy absorbed in shear E , energy absorbed in crack propagation F force F c compressive force component F i inertial torce component F, shear fo~rce component H length of conical section of frustum I total impulse g concrete constant L length of projectile M moment per unit length M o full plasl:ic yield moment per unit length N nose shape factor No full plastic yield membrane force per unit length R radius of target plate Ro radius of impulsive loading area S soil constant V projectile velocity

141

142 G.G. Corbett et al.

V r residual projectile velocity V~ experimentally obtained ballistic limit W weight of projectile

a,b constants dp diameter of projectile e width of shear zone f~ compressive strength of concrete h target thickness k mass ratio parameter I target span It, crack length mp mass of projectile mp~ mass of plug n work-hardening index p depth of penetration required for perforation of concrete r radius s depth of penetration of concrete required for scabbing t time w plate deflection wo central plate deflection x depth of penetration

Greek

~t work hardening coefficient fl half cone angle for conical projectile 7 visco-plastic constant for target material e direct strain 0 angle of obliquity of impact x curvature # mass density per unit area v Poisson's ratio p mass density a direct stress T shear stress ~j Johnson's damage number ~ modified damage number for circular plates ~q modified damage number for rectangular plates

Subscripts

c critical value i instability value n projectile nose 0 initial value p projectile r radial 0 circumferential y yield u ultimate

1. INTRODUCTION

The field of Impact Dynamics covers an extremely wide range of situations and is of interest to engineers from a number of different disciplines. For example, production engineers are interested in the subject in respect of its application to high speed blanking and hole-flanging processes; military scientists need to understand the subject in order to design structures that are more efficient at withstanding projectile impact or in order to design improved ballistic missiles; geologists use improved understanding of earth penetration processes to carry out remote seismic monitoring and surveying; and vehicle manufacturers use their understanding of the response of structures to impact loading to improve the performance and safety of their products. Indeed whenever two bodies collide or risk collision the subject of Impact Dynamics arises.

Impact loading of plates and shells by free-flying projectiles: a review 143

The genera]~ nature of the subject means that it incorporates a wide range of spatial, temporal and ~thermal scales, and also a wide range of materials. It also covers a large variety of loading situations such as hypervelocity impact, blast loading, jet impact, projectile penetration, dropped-object loading, structural crushing etc.

A useful non-dimensional parameter that is commonly used to classify the severity of projectile impact is Johnson's Damage number [1], given by

= (1) (Yd

where p is the target density, V0 is the impact velocity and ~d is the dynamic yield stress of the target material. This survey covers the field of "moderate velocity" projectile impact, i.e. impact of targets by projectiles travelling in the sub-ordance range (up to approximately 500 m/s). In this range (which corresponds to a Johnson damage number of up to approximately 5) the structural response of the target is often important and both local and global effects need to be taken into account. The survey concentrates primarily on the impact of finite thickness metallic targets from the viewpoint of investigating the penetration and perforation processes involved.

In 1978 Backman and Goldsmith [2] published a comprehensive survey of the mechanics of penetration of projectiles into targets which covered the major works that were published up to that date. This survey concentrates on work published since Backman and Goldsmith's paper, although it does include older works where these are thought necessary for complete- ness.

There has been considerable work carried out on the quasi-static piercing of metal sheets and this work has increased our understanding of the penetration and failure mechanisms that are present when a thin metal plate is subjected to this type of loading. Although this survey is predominantly concerned with dynamic loading of plates and shells, the information acquired from quasi-static investigations can be used to increase our understanding of the response of structures to dynamic loading. For further information on research that has been carried out on the quasi-static piercing of metal plates the reader is referred to papers by Johnson et al. [3,4]. In addition a brief review of research that has been carried out into the quasi-static and dynamic penetration of aluminium plates is given in [5].

A commonly used measure of a target's ability to withstand projectile impact is its ballistic limit and much work has been carried out by researchers to enable estimates of this parameter to be made. In general terms the ballistic limit of a structure is the greatest projectile vel~ocity the structure can withstand without perforation occurring. Precise definitions of this parameter vary depending on the interpretation of the term "perforation"--a more detailed discussion concerning these terms can be found in [2]. There are several different mechanisms by which a target can fail and these may occur singly or in combinations of two or more. For example, Backman and Goldsmith [2] identify the eight most commonly occurring types of failure of thin or intermediate targets as being those shown in Fig. 1. The large number of penetration and failure mechanisms present, coupled with the added complication of global structural response of the target, makes prediction of events arising from impact loading extremely difficult. In the past, engineers have had to rely exclusively on a few empirical or semi-empirical equations to obtain penetration data. The first formulae to be developed (see, for example Backman and Goldsmith [2] or Young [6]) predicted the penetration depths into semi-infinite targets (i.e. targets where the distal surface does not affect the penetration process) when struck normally by a projectile. The advent of battleship armour in the 19th century led to the development of equations predicting the depth of penetration of finite thickness armour plating [7]. Even to this day these formulae and others like them are extensively used by impact engineers.

In recent years appreciable advances have been made in the analytical approach to the problem of penetration with the models gradually becoming more sophisticated and more accurate. However, these too have relied heavily, and indeed still do, on experimental data to justify certain assumptions made and to supply various parameters for the models. The continuing importance of experimental data in improving the understanding of penetration


(a) Fracture due to initial stress wave

(c) Spatt failure (scabbing)

(b) Radial fracture behind initial wave in a brittle target

(d) Plugging

(e) Pefaling (frontal) (f) Pefating (rearwards)

(g) Fragmentation (h) Ouctite hole enlargement

Fig. 1. Perforation mechanisms [2].

and perforation processes is reflected in this survey by the substantial section allocated to it, Section 2.

Section 3 reviews the most commonly used empirical equations and recent work carried out on improving and up-dating them. Section 4 reveiws the main analytical advances in the field. These advances have been extremely important in identifying and explaining penetration and perforation phenomena and continue to constitute the main body of research.

Numerical methods have been relatively successful in predicting the response of targets to projectile impact. These methods are reviewed in depth in a number of surveys and are not dwelt upon in this paper. However, a brief discussion of some recent attempts at using such techniques to model the mechanics of penetration and perforation of thin-walled targets is given in Section 5.

The major part of this review concerns the impact of thin to intermediately thick metallic targets. A common method of guarding against possible impact damage when weight and space are not at a premium is to use a bulky shield to protect sensitive areas. The penetration of thick targets is therefore of great interest to impact engineers and a brief review of recent work on the impact loading of concrete and steel-concrete sandwich structures is given in Section 6.

2. EXPERIMENTAL INVESTIGATIONS

2.1. Flat plates

Whereas copious documented experimental evidence has been collected by the military, the number of non-classified reports on impact experiments is extremely limited. The experiments carried out at military establishments are also, quite often, not relevant to the


1800

.~ 1200

~ 900

~ 600 o/

~300

Ok' 0

A

0 Ix

I-1 i"1

500 1000 1500 2000 2500 Impact velocity (m/s)

Fig. 2. Projectile velocity drop through a target plate as a function of initial speed [8].

Target thickness Projectile diameter Synabol Target material (ram) (ram)

O 4130 armour steel 6.35 6.35 A 1020 steel (large

grained) 6.35 6.35 2024-T4

[] aluminium 6.35 6.35 2024-T3 ahiminium 3.13 6.35

1020 steel (small grained) 1.57 6.35


O 2024-T3 aluminium 1.27 6.35

2024-0 ahiminium 1.27 6.35

needs of the civilian engineer whose aim is to highlight the relative importance of each parameter and to identify trends in target and projectile behaviour. Most of the experimental investigations reported here were carried out in order to validate existing theories or to highlight their short-comings and to point the way for further improvement, They are, however, also extremely useful in their own right as descriptions of the penetration process.

Goldsmith and Finnegan [8] reported a series of tests in which aluminium and mild steel plates were sltruck normally by hard steel spheres, travelling between 150 and 2,700 m/s, i.e. covering sub-ordnance (damage numbers less than 5), ordnance (damage numbers between 5 and 35) and ultra-ordnance ranges (damage numbers between 35 and 150). Data collected included the initial and final projectile velocities, radial and transverse strain histories at various positions from the impact point and crater and fragment dimensions. High-speed Kerr cell photographs of the impact process at pre-determined framing intervals were also taken in some of the tests. It was noted that the velocity drop experienced by the projectile decreases from the ballistic limit to a minimum value and then increases monotonically with initial velocity for all projectile-target systems (see Fig. 2). It was also observed that the amount of dishing of the plate is a maximum at the ballistic limit and then decreases as the impact velocity increases. The Kerr cell photographs revealed that the terminal plug and projectile velocities were very similar and post-perforation measurements indicated an inverse relationship between the final plug thickness and initial projectile velocity (Fig. 3).

Corran et al. [9] conducted an extensive experimental investigation of the impact of vroiectiles at sub-ordnance velocities a2ainst mild steel, stainless steel and aluminium alloy


oJt~ t "

.3

d : :

-~2

0 I I I I i 0 500 1000 1500 2000 2500 3000

Impact vetocity (mls)

Fig. 3. Measured central plug thickness as a function of impact velocity [8].

Target thickness Projectile diameter Symbol Target material (ram) (mm)

4130 armour steel 6.35 6.35 A 1020 steel (large

grained) 6.35 6.35 1020 steel (small

grained) 6.35 6.35 [] 2024-T4

aluminium 6.35 6.35 2024-T4

aluminium 6.35 9.53 2024-T3 aluminium 3.13 6.35

[q 1020 steel (small grained) 1.57 6.35


(> 2024-T3 aluminium 1.27 6.35

t 2024-0 aluminium 1.27 6.35

plates. The projectiles used were blunt silver steel cylinders with diameters of 12.5 mm and masses between 15 and 100 g. For most of the tests the plates were clamped with a transverse force of 40 tonnes acting on a circumference of 240 mm diameter. The plate thicknesses ranged from 1 to 10 mm. Figure 4 shows the variation of critical perforation energy and velocity with projectile mass, and compares the results of Corran et al. with the results from two previous investigations [10,11].

Extensive investigations into the quasi-static loading of thin metal plates have shown that the piercing load, the energy absorbed during perforation and the dominant failure mechanisms are all strongly dependent on the nose shape of the indenter [3,4,12,13]. For example Johnson etal. [13] compared the response of mild steel, brass and copper plates to quasi-static piercing by a variety of conical and ogival-ended punches. It was shown that the failure mode changed from being one of shear plugging to one of ductile hole flanging as the included angle of the conical-ended punches was decreased. Piercing load and energy required for perforation were also shown to be dependent on the nose shape with ogival- ended punches requiring significantly higher amounts of energy to perforate the plates than the conical-ended punches. This paper [13] also includes an interesting description of the hardness distribution in the vicinity of loadin2 followin2 oenetration from flat. conical and


200 oJ

o

aJ

a '~ 100

Shear. l Tensile Failure,,fo Failure

,o" I X #t

/ v S

0 I I I I I 0 t~ 8 12 16 20

11RxO.O1mm -I

ID

Fig. 5. Variation of perforation energy with projectile nose radius for 1.3 mm thick targets 19].

Cumulative error [] Plugging work + Membrane plastic work -+

50 \ ix Plastic bending o Elastic deformation

\ Mushrooming-soft projectiles ~ ", Mushrooming-hard projectiles

/,0 ~ f " i "~ . t.

,'- 7" ,, i~(e \ \ ~c nD

._E 30 ./" \~

.i.- /~ ', ,, ',, 0 /

~ 10 x ' ' ' x ' '~ ' '~O ..I

..o' . . . . t]"" ...... A.--- '-"" ' %

0 ""-.-a-"-'q , 0 2 t~ 6

Plate thickness (ram)

Fig. 6. Energy absorption mechanisms for plate perforation 19].

In these equations the value of yield stress, ay should be taken as the dynamic yield stress of the material, at a strain rate of approximately 50 s- 1. The derivation of Eqn (5) is given in [ 1], and details of the symbols used can be found in this reference.

Equations (2)-(5) are only approximate (to varying degrees) and have to be regarded as such, but they do allow an insight into the changing importance of each energy absorbing component as plate thickness increases. These relationships are plotted in Fig. 6 and exhibit the following trends:

(a) The proportion of membrane plastic work decreases with plate thickness. (b) The proportion of plastic bending work increases with plate thickness to a maximum

whereafter it falls away sharply.


100

E .b n 95 o >

-

o

~ 9O

85

0

, 'hO, ,

qo load ",,


15

~e 12.5

b~

lO

7-5 c 6J

s E

2-5

'~2024Optate~-~n~1127mmdia ~ 2024Opiate

o-~l',~_j27mm fhk "/ ~ I~ [ -~I".I.27mm fhk

H 50 100 150 200 250 300

Impact velocity imls)

Fig. 8. Permanent plate deflections following impact with a spherical steel projectile [15].

15

~12.5 t -

O 10

~_ 7.5 o

5

2'5

0 -40

~ ~i1"27mm /~,N~2.Tmm dia.-] T 2024-0ptate Projectile shape z ~ \ IV ~, 365mm dia.x for scales _~/ /~.f"~.~k \ -to -[7 1.27mm thk.

-30 -20 -10 0 10 20 30 Radius from impact point(mm)

t~0

Fig. 9. Transient plate profiles at various times after impact by a steel sphere [15].

retention) in thin metal plates following low velocity impact from flat-faced projectiles is given in [17]. Higher impact velocities produce a clean separation (Fig. 10e) between the plug and the target plate. The penetration process resulting from impact by hemispherically- tipped projectiles is significantly different from the thinning and shearing caused by a blunt or fiat-faced projectile (Fig. 11) and the star-shaped cracking caused by a cylindro-conical projectile (Fig. 12). Force-time histories obtained for different projectile profiles indicate that hemispherical and fiat-faced projectiles tend to produce shorter impact force duration times but higher peak loads than conically-ended projectiles (see, for example, [16]).

Leppin and Woodward [ 18] examined the effect of plate thickness and projectile geometry on the failure of thin titanium alloy targets. Square plates, 75 mm wide, with thicknesses of 1, 2 and 3 mm were struck with flat-faced and conically-ended cylindrical projectiles whilst clamped at their edges. Titanium alloy was chosen as the target material as it was known to be susceptible to failure by plugging. The projectiles were made of hardened steel, had diameters of 4.76 mm, weighed approximately 3 g and had included nose angles of between 45 and 180 (fiat-faced). The fiat-faced projectiles ejected plugs of equal diameter to the projectile on impact when the impact velocity was greater than or equal to the ballistic limit of the plate. The conically-ended projectiles in most cases ejected a plug with a diameter less than that of the projectile when struck with sufficient velocity. Leppin and Woodward [18] defined two critical energies for impact with conically-ended projectiles: one which is


(a)

(b)

Fig. 10. Penetration and perforation of a 1.27 mm thick aluminium plate when struck by a hemispherically nosed projectile at (a) 59 m/s, (b) 64 m/s, (c) 68 m/s, (d) 75 m/s, and (e) 127 m/s [16].

required for plug ejection and a higher one which is required for projectile passage through the plate. An example of the former occurrence is shown in Fig. 13(a), and an example of the latter in Fig. 13(b). The mechanism by which the plate allowed projectile passage once the initial plug was formed was seen to change as the plate thickness and/or nose cone angle were increased. The mechanism changed from one of plate dishing and local bending (with radial tearing, or l~talling being present) to one of hole enlargement through the formation of a secondary annular plug. The change in perforation mechanism with plate thickness is


(c)

(d)

Fig. 10 (continued)

shown schematically in Fig. 14. The plug diameter also increased with target thickness, and, for thin targets, is to a first approximation of the order of the target thickness.

The failure mechanisms experienced by thin plates made of a ductile material when struck by various types of projectile was the subject of a study by Goldsmith [19]. Three different projectile profiles were used, namely flat, hemispherical and conical, and the types of failure causecl, when these projectiles were fired against aluminium alloy and mild steel plates at, or near, their respective ballistic limits were discussed. The flat-faced projectiles were seen to cause failure by plugging; the hemispherically-ended projectiles caused failure by plate


Fig. 10 (continued)

Fig. 11. Panetration and perforation mechanisms caused by a flat-faced projectile striking an sluminium plate at 54 m/s [16].

thinning followed by circumferential cracking leading to the formation of a spherical cap, with further

154 G.G. Corbettet al.

8~

e.i

Impact loading of plates and shells by froc-flying projectiles: a review 15 5

Fig. 13(a). Plug formation in a target where a plug of smaller diameter than the projectile has been ejected [18].

Fig. 13(b). Plug formation in a target where the projectile has passed through the target [18].

increase in the radial strain within the impact area with little increase, due to friction, in the circumferential strains. In [20] an analogy with the Forming Limit Diagram used in metal forming processes was made, in which the onset of necking is dependent on the ratio of the two principal in-plane strains.

Sangoy et al. [21] discussed the improvements in armour steels as ballistic protectors and believe that armour steels, although the oldest of armour materials, are still the most satisfactory material in dealing with ballistic protection due to continuing improvements in their performance. A basic requirement of armour steel is that it should have high hardness; but it was noted in [21] that there is no simple correlation between hardness and resistance to perforation, as measured by the structure's ballistic limit (Fig. 15). Three zones in the hardness-ballistic limit relationships were identified: (i) low hardness regime where resistance to perforation increases with hardness; (ii) medium hardness regime where the


~ct i le ~ ~ ~~.~entation t

C~'~'srnall diameter----'~ plogs

increasing target thickness Fig. 14. Schematic illustration of the effect of target thickness on failure mode of titanium alloy targets. At each thickness the top and bottom illustrations can be considered as either an early and late stage of a high velocity shot, or as a low velocity shot which does not allow perforation compared to a high velocity shot which allows perforation, respectively. Flow mechanisms are indicated rl 8].

u

n~

Plastic flow Armour adiabatic shearing fracture per frafi~l.~-

Hardness Fig. 15. Variation of ballistic limit with hardness for armour steel [21].

ballistic limit decreases due to the onset of adiabatic shear damage in the armour steel; (iii) high hardness regime where resistance increases again due to projectile break-up. The use of multi-layer armour steel was also discussed in [21] where it was noted that an efficient combination is a hard front face to break up the projectile and a ductile rear face to absorb the kinetic energy of the projectile.

Langseth and Larsen [22] carried out a detailed investigation into the plugging capacity of mild steel panels when subjected to central loading from dropped objects impinging normally with velocities up to 50 m/s. Similitude analysis was used to construct a quarter scale model of a typical offshore deck panel being struck by a falling drill collar. The indenters used in the tests were flat-faced and had masses ranging from 18 to 50 kg. The effects of target thickness, projectile mass and in-plane target stiffness on the energy required to perforate the panels was investigated. It was found that the critical plugging energy increased with plate thickness, decreased with increasing projectile mass over the velocity and mass range tested, and decreased with increasing in-plane panel stiffness. The effect of stringers attached to the free span on the critical plugging energy was investigated and found to be negligible. The result regarding the effect of varying the projectile mass, (i.e. that the critical plugging energy decreases with increasing mass, seemingly asymptotically approaching the quasi-static value) is contrary to the findings of previous investigations and seems to support the opinion of Corran et al. [9] that the variation of critical perforation energy with increasing projectile mass is not properly understood. This earlier work (Fig. 4) indicates that there is an increase in critical perforation energy with increasing projectile mass. However, these results were obtained from tests on thinner plates (1.3-2.5 mm) than those tested in [22] (4-10ram) and


300-0

250.0

200.0

i 150.0

.~ 100.0 o

u .

50-0

0.0

-50.0 0-0

Transient I GLobal mode phase phase I _ ~ " ' ~

Peak , . ~ %,

. . . . ' . . . . I . . . . . " " ' ' ' . . . . ' . . . . ' . . . . ' ' " - - "

I I I I I I I I

1.0 2.0 3.0 t~.0 5"0 6.0 7.0 8,0 9.0 Time, t (ms)

Fig. 16. Force-time history for a flat plate struck at 18 m/s with a 50kg mass 1-22].

the projectiles used were much lighter, increasing only up to 165 g. Other parameters such as free span width, support conditions, panel shape, target material, projectile shape (although all were classified as blunt) were also different. It is therefore apparent that the critical perforation energy of a target is not a simple function of projectile mass, and at present it is not clear which parameters affect the relationship and in what way.

Force-time pulses during the plugging process were obtained by Langseth and Larsen [22] by strain-gauging both the projectile and the target supports. The resulting traces from the projectiles were identified as containing two stages: a transient stage which is dominated by inertial effects and occurs before the supports register any loading; and a global stage which is dominated by the structural response of the system and occurs after the supports respond to tlhe loading (Fig. 16). Comparison with load-deflection data from quasi-static tests showed that the interface forces at perforation are approximately equal for both types of loading, and also that the load-displacement curves are also similar during the global-mode stage (Fig. 17). In Fig. 17 the dashed curve represents the static result shifted horizontally to a position such that the displacement at plugging coincides for the dynamic and static cases.

In another paper 123-1 Langseth and Larsen developed a methodology for designing steel plates to resist plugging failure following impact from a dropped object. The response of the plate was divided into three separate stages: a transition stage from elasto-plastic response to full plastic membrane response; a plastic membrane stretching stage; and a plate softening stage resulting from the effects of local indentation. The membrane stretching stage was seen to absorb most of the impact energy and the energy absorption characteristics of the plates were seen to be strongly influenced by the membrane stiffness and the plate plugging load. The analysis neglected dynamic effects and the effects of local indentation and used an

300"0

25(>0

20 0 A S 150'0 100.0 IL

" Supporf 50.0 I /~_~' - E acfivafed

0"0 . . . . , . . . . . . . . . , , " " , . . . . , . . . . . " ; : : '

-50.0 i i i i ~ I 0.0 10-0 20.0 3frO /,frO 50-0 60.0 70"0

DispLacemenf, w (ram)

Fig. 17. Comparison between static and dynamic force-displacement curves [22].

158 G.G. Corbettet al.

l

-1

[. 12.7mm -

I Projictile

D e p f ~

I... 1 Backup l [- hole diameter-

2S.~mm (a) before impact

Prjictile !

!indentert St0pper rinq [ I Target ' l Shear-band

specimen i, i_l ~ength

(b) after impact

Fig. 18. Controlld depth of penetration impact apparatus [24].

7 r (a) 7 - (b)

; / J7 ist- .12.7 ~ I I / | s o 9.S BHD(mm)

7. , 7 r ~ o 9.s

0.0 0-2 0.~, 0.6 0.8 1.0 1.2 1.~, 1.6 0.0 0.2 0.~, 0.6 0.8 1.0 1.2 1.4 1.6 Depth of penefrationlmm) Depth of penetration lmm)

7 6

E ES

e-

~3 g2 eft

1

0-0

(c)

BHO(mm) ~l 12.7 o 9.5 7 -9

I I I = i

0.2 0.l~ 0"6 0.8 1-0 1'2 1.4 1.6 Depth of penetration (ram)

Fig. 19. Shvar band lvngth vs depth of penetration of(a) pcarlitic steel, (b) martensitic steel tcmpared at 400 C, (c) martcnsitic steel tmpcrvd at 600 C for four diffvrcnt backup hole diameters (BHD).

Projcctil velocity -- 50 m/s [24].


empirically derived relationship to calculate the plate plugging load. In addition, data from experiments and finite element analyses were used to derive relationships that predict the "shift" in the load-deflection curve resulting from elasto-plastic effects. The analysis was seen to predict the energy absorbing capacity of steel plates to within 11%.

Chou et al. [24] carried out a detailed investigation into the formation of shear bands in steel when penetrated by a flat-faced projectile (dp/h o = 1). The plates were penetrated to a known depth using the apparatus shown in Fig. 18 and the effect of depth of penetration on shear band length was found for three different types of steel (Fig. 19). It was found that the steel with the highest flow stress formed shear bands at the smallest indentation depths and that the length of the shear band increased exponentially with depth of penetration for all of the types of steel tested. In these tests the plates were penetrated whilst standing on a "back-up ring" which had a diameter ranging from the indenter diameter to twice the indenter diameter, i.e. ranging from h o to 2he (Fig. 18). In Fig. 19 it is shown that the length of the shear band is strongly dependent on the back-up ring diameter, with smaller ring diameters producing longer shear bands for a given depth of penetration. This dependence of shear band length on support diameter to plate thickness ratio is important in predicting the onset of shea~r plugging in plates and indicates that further tests of this nature on plates with larger diameter to thickness ratios would be useful in increasing our understanding of the shear plugging process in larger span plates. A finite element simulation of adiabatic shear plugging was; also carried out in F24] with good agreement being obtained between the simulated response and the experimental results (Fig. 20).

The formation of failure cones in aluminium alloys through intensive plastic shear deformation was studied by Astanin et aL [25]. Flat-faced cylindrical steel projectiles with diameters of 14.5mm were fired at square (150mm x 150mm) aluminium plates with thicknesses of 30 and 60 mm (dp/h o = 0.24 and 0.48) at normal incidence. The plates were struck at various velocities and subsequently inspected for depth of penetration and evidence of cone formation. It was shown that the depth of penetration versus impact velocity curve was bi-linear with an increase in slope occurring at approximately 230 m/s for the 30 mm thick plates and 450 m/s for the 60 mm thick plates. The gradient of the initial part of the curve was shown to be only weakly dependent on plate thickness and was determined by the inertial and elasto-plastic properties of the target material.

~ IPROJECTILEI

1--

i - - - -

- - I - - - -

IT,,,l,,l,l,,lll,,,,l'llllrlt

(b)

Fig. 20. Comparison of (a) experimental observation of shear banding and (b) finite element simulation of shear banding [24].


2.2. Multi-layered tar#ets

The possibility of improving the penetration resistance of a target by layering it with materials having different properties has been known since the late 1800's when armour plating was first improved by hardening its surface. Since then the use of multi-layered armour plating has increased as the benefits of using such an approach have been realized. It has been shown that a hard impact surface layer to resist indentation backed-up by a tough, ductile inner layer to absorb the kinetic energy of the projectile is an efficient combination of layers to resist projectile impact I"1,21]. The possibility of using target plates made up of several separate, thinner plates has also been investigated. Marom and Bodner [26] carried out an experimental and analytical investigation into the response of multi-layered aluminium beams to projectile impact and found that multi-layered beams in contact showed greater resistance to penetration than equivalent weight monolithic beams, which in turn were more efficient than separated flat beams of equal weight. In all these tests the beams were struck with 5.6 mm diameter (0.22 in. calibre) projectiles travelling at velocities above the target ballistic limit and resistance was measured in terms of velocity drop by the projectile during the penetration process. Corran er al. [9] investigated the performance of multi-layered steel plates under projectile impact and found that layers placed in contact were superior to monolithic single layer plates if the adoption of multiple layers changed the response of the plates from being one dominated by plate bending and shearing to one dominated by membrane stretching.

The impact of bonded laminated metal targets was investigated by Woodward et al. [27]. The energy absorbed by the plates when struck at near their ballistic limits was separated into three components: delamination, bending and stretching. It was shown that, depending on the material properties of the plates, the laminated plates had higher or lower critical perforation energies than monolithic plates. The most effective laminated plates, in terms of impact resistance, were those that promoted high stretching energy absorption processes in the laminates near the non-impact face of the target plate.

Radin and Goldsmith [28] compared the impact resistance of monolithic and multi- layered aluminium targets and also a combination of aluminium and polycarbonate layers. In these tests both blunt and 60 conically-nosed projectiles were used. For the aluminium targets it was found that the ballistic resistance of adjacent layers of equal thickness was inferior to that of an equivalent single layer. Once again it was noted that spaced layers were less effective at impact resistance than layers in contact.

Hetherington and Rajagopalan 129] investigated the response of layered target plates consisting of a ceramic front layer and a fibre-reinforced plastic rear face. This combination was seen to be extremely effective at withstanding projectile penetration in terms of energy absorbed per unit areal density with the hard ceramic face serving to break the projectile up and spread the load on the ductile composite rear face. The analysis of Florence 130] which produced a model for the penetration of a two-layer (hard front face, ductile rear face) composite armour plate, was used to predict the energy absorbed by the plates to within ___ 32%. The combination of a ceramic impact face and a composite near face was also studied by Navarre et al. [31] and was shown to be particularly effective at withstanding high velocity impacts. Radin and Goldsmith [28] point out that the data on impact of multi- layered targets is very limited and patchy making direct comparisons between experimental results impossible. At present the benefits of replacing monolithic targets with multi-layered ones are not clear and further experimental work is needed to provide a reliable and adequate database.

2.3. Oblique impact of plates The majority of the investigations that have been carried out have concentrated on the

normal impact of rigid projectiles on targets. A few authors have adapted their analyses of the penetration process to accommodate oblique angles of impact (see, for example [32,33]), but readily available experimental data on this type of impact is extremely limited. Awerbuch and Bodner f34q investigated the velocitv drop experienced bv 5.5 mm calibre lead bullets when


1.0

0"9

0.8

~" 0.7

>~ 06

~0.s E

El.

P 0.4 "0

03

0"2

0-1

0 0 10" 20* 30* /,0" 50* 60* 700 80

Angle of impact

Fig. 21. Velocity drop vs angle of impact for aluminium plates struck by 0.22 in. calibre lead projectiles [34].

fired at aluminium plates at increasing angles of obliquity. There was found to be little change in the total w,'locity drop as this angle was increased to around 30 , after which the velocity drop rose at an increasing rate (Fig. 21). Similar results were obtained by Gupta and Madhu [35] who fired spinning hard core projectiles at approximately 820m/s against mild steel plates with thicknesses between 10 and 25 ram.

The influence of angle of obliquity on the failure mode of steel and aluminium plates was discussed by Woodward and Baldwin [36] and Goldsmith and Finnegan [37]. Woodward and Baldwin [36] found that the hardness at which adiabatic shear began to cause a reduction in the penetration resistance of metallic target plates (see Section 4.2.1) was higher for oblique impacts than for normal impacts. Goldsmith and Finnegan [37] carried out over 200 normal and oblique impact tests in which the velocity drop and change in angular orientation of projectiles striking soft aluminium, mild steel and medium carbon steel plates were determJined. Angles of obliquity ranged from 0 to 50 over the velocity range 20- 1025 m/s and the targets, which were damped on a diameter of approximately 115 mm, had thicknesses ranging from 1.25 to 25.4 mm. The projectiles used were made of either hardened steel (mass = 30 g) or soft aluminium (mass = 15 g), had diameters of approximately 12.7 mm, and were either flat-faced or conically-tipped. The dependence of velocity drop on the initial angle of obliquity was seen to be similar to that found by Awerbuch and Bodner [34] over the range of obliquities tested although both sets of results exhibited a large degree of scatter. However there was one significant difference in the findings of the two investigations. Goldsmith and Finnegan [37] noted a minimum value of velocity drop at approximately 20-30 obliquity for the higher velocity impact tests (over 300 m/s), whereas the tests of Awerbuch and Bodner [34], which involved impact velocities of approximately 400 m/s, exhibited a gradual increase in velocity drop with angle of obliquity with no local minimum value. A major difference between the two sets of tests (as well as differences in target material and geometry) was the type of projectile used: Goldsmith and Finnegan only report on the effect of angle of obliquity on velocity drop for hard-steel projectiles which, unlike the lead projectiles used by Awerbuch and Bodner, did not deform to any significant extent during most of the tests.


50

/+5

~,0

A 35

30 CD

'50.25 t-1

o 20

15

10

0 | 0

Plate thickness 3.175mm 6.35mm

o 122m/s x 160 m/s ,~ 30Sm/s o 610 m/s

910 mls

/Z / /

5 10 15 20 25 30 35 t~O t~S 50 Initial obliquity, 9i (deg)

Fig. 22. Final obliquity as a function of initial obliquity for 3.175 and 6.35mm thick 2024-0 aluminium targets struck by hard steel cylindro-eonieal projectiles. Solid lines, open symbols:

3.175 ram, Dashed lines, closed symbols: 6,35 mm [37],

An example of the effect of initial obliquity on the final angle of flight of the projectile following perforation is shown in Fig. 22. This figure shows the results of tests involving the impact of 3.175mm and 6.35mm thick aluminium alloy plates with hard steel cylindro- conical projectiles travelling at five different velocity levels. It can be seen that at the higher impact velocities there is little change in the angle of flight of the projectile during the penetration process; but at lower velocity levels the final angle of obliquity is always less than the initial angle, although as the initial angle of obliquity increases the difference between final and initial angle becomes less. A metallurgical examination of virgin and perforated plates was carried out in [37] and the effect of rolling direction on the initial cracking pattern was discussed.

Virostek et aL [38] investigated the force-time histories produced by the penetration and perforation of thin aluminium and steel plates by cylindro-conical and hemispherically- ended projectiles at various angles of incidence. A brief review of the methods used to obtain the force-time pulses experienced by a projectile was made and a new method was described which allowed oblique impacts to be analysed. The force-time histories obtained from the impact of a 4.76 mm thick aluminium plate by a cylindro.conical projectile over the entire range of obliquities tested is shown in Fig. 23. Once again it was shown that there is little difference in behaviour over the lower range of obliquities (impact angle < 15 ) with the force-time curves being similar for these angles. For the higher angles of obliquity the test results indicated a higher force generated on impact followed by a sharp decrease in the force to zero, The difference in behaviour was also apparent in the post-perforated specimen with the mode of failure changing from one of symmetric petal formation to one of non-symmetric petal formation. It was also shown in [38] that for hemispherically-ended cylindrical projectiles, the peak forces tend to decrease as the angle of impact increases, contrary to the expected increase due to the increase in the effective target thickness--a feature apparent for the cylindro-conical projectiles. This decrease was explained by the change in failure mode from plugging or discing duc to a uniform circumferential tension to petalling caused by


32~.0

28'0

24.0

~20.0

~ 16"0 u~12. 0

8'0

lvO

0.0

f ~ e~ Vo(m/s) 0 144 / ] r ,~ ' ~ - - - - - 30 172

,~ ' - - '~ \ . . . . 45 170

0 100 200 300 400 500 Time {ps)

Fig, 23. Force histories for an instrumented cylindro-conical projectile striking 4.76 mm thick 2024-0 aluminium plates at various angles of obliquity, 0t [38],

radial cracking. Also, at higher angles of obliquity the presence of projectile "sliding" serves to delay the onset of the peak force. In general, for a given angle of obliquity, the peak force generated was seen to increase with impact velocity to the ballistic limit and thereafter remain constant. The results presented by Virostek et al. [38] reveal a number of interesting phenomena and it is clear that further research is needed before the effect of introducing obliquity into the penetration process is fully understood.

In Fig. 24 the variation of peak force with angle of obliquity for a cylindro-conical projectile striking aluminium plates is shown. The results are compared with a theoretical prediction atltributed to Virostek [39] and quoted in [38] as

g = ~mpV~ (6) 4[rp(2 + cos F) + ho/cOs 0]"

Whereas Eqn (6) gave a reasonable prediction of the peak force for aluminium plates, it over- predicted this parameter by about 30% for impact on steel plates. This was attributed to the assumption made in the derivation of Eqn (6) that the centre of the plate undergoes a total deflection of one projectile diameter during the impact process. A more detailed analysis of the global response of the plate is needed to improve the correlation between the theory

3O

25

z20

5

0

O

Q

e q u a t i o ~

U

4.76 mm A1 ) 3,18mm A1 I Experimental data

I I I 10 20 30

Angle of incidence (deg)

U

13

I 40 5O

Fig. 24. Peak forcevs angle ofinciden~ for a cylindro-conicalprojectile striking 3.18ram thick and 4.76 ram thick 2024-0 aluminium targets [38"1.


and experiment. A proposed model for hemispherically-ended projectiles overestimated the peak force obtained, although the shape of the force-time curves showed fairly good agreement.

The effect of projectile yaw was discussed by Zukas [40] in a review of investigations into the penetration and perforation of solids. The results of Graberek [41] were quoted which indicate that small angles of yaw (less than 5 ) have little effect on the ballistic limit of a target, but as the angle increases the ballistic limit is increased significantly. Yaw angles of 10% were quoted as increasing the ballistic limit of a metallic target plate by 12%. For high yaw angles projectile break up is a distinct possibility. Bless et al. [42] used the method of reversed impact to investigate the effect of yaw angle on the ballistic limit and crater geometry of armour plates when struck at high velocities. The plates were launched against a stationary target rod at approximately 2.15 km/s with the rods set at various angles of yaw to the plate velocity vector. It was found that at these high speeds the rods did not substantially rotate during penetration. It was shown that there is a relationship between the critical yaw angle required for perforation of the plate at a given impact velocity and the plate thickness- projectile diameter ratio.

Crouch et al. [17] developed a simple analysis to predict the effect of yawing of blunt projectiles on the impact energy absorbing characteristics of metallic plates. The work done (neglecting plate dishing) in perforating a plate with a yawed projectile was predicted to be

)l Ey = arh o L ~ + (L sin 4~ + dp cos 40 + dp~f (7) where ~ is the angle of yaw. However, in impact tests that were carried out on thin aluminium alloy plates it was shown that the presence of yaw in a blunt projectile may alter the mode of failure from plug shearing to plate tearing, especially when the ho/d p ratio is very small. When this change in failure mode occurred higher exit velocities were measured following perforation, implying a reduction in the ballistic limit of the target.

2.4. Moving targets

The normal impact of moving plates with a blunt projectile was investigated experimentally and analytically by Wu and Goldsmith [43,44]. In these tests plates were struck whilst moving at a velocity of 40 m/s in a direction perpendicular to the line of flight of the projectile. It was found that the motion of the target increased the ballistic limit from 24 to 29% over that of the corresponding stationary target. This was attributed to the addition of a transverse velocity component for the target increasing the contact duration between the projectile and target over that which could have been the case for a stationary target. The test results also indicate that there is considerable "smearing" present when a projectile strikes a moving target. This "smearing" is caused by the action of the projectile being spread over a greater area than would be the case for a stationary target thus resulting in the effects of the impact being less concentrated and involving a larger amount of target material, a phenomenon discussed in [45].

2.5. Impact loading of pipes and tubes Interest in the response of cylindrical shells to local impact loading has increased over the

last few years. Initial investigations were concerned with response of pipes and tubular members to quasi-static loading, with a variety of tests being carried out in which cylindrical shells were either pierced (for example [46-48]) or locally indented by punches with various nose shapes (for example [49-51]). Rigid-plastic analysis has been used to predict the load-deflection behaviour of tubular beams [52,53]. When compared with experimental data, it was shown that rigid-plastic analysis provides a very useful first approximation to the large deformation of tubular beams. Reid and Goudie [54] extended the rigid-plastic analysis to take into account elastic effects in a semi-empirical model.

The residual strength of damaged cyclindrical shells is an area of research of particular interest to the offshore oil and gas industry where there is concern about the damage caused


250

~200 E

o 150 to

E " 100

(a)

50

Fig. 25.

ho:l.2mm

~ s.~.~'~ "~

LI 0 O 8

i I I

p,,=O Pn=O'09 Pn=0-19

250

~ 200 150

- 1ool

'~b) .~.~ ~ tho =2'Omm

I 50 I I p,=0 p~= 0-16 ~=0.31

Results of impact tests on mild steel tubes. (a) Tubes with wall thickness, ho = 1.2 mm, (b) tubes with wall thickness, h o = 2.0 mm [20].

to tubular m,~mbers on offshore jacket structures resulting from a ship collision (i.e. low velocity-high mass impact) and the effect this damage has on the residual strength of the members [55-59].

The response of cylindrical shells to impact from free-flying projectiles has been studied by relatively few researchers. Palomby and Stronge [20] conducted an experimental investigation into blunt missiles of different nose radii fired at mild steel plates and tubes. The effect of nose radius on the ballistic limit of 50.8 mm O.D. tubes in the as-received (V.H.N. = 178) and annealed condition (V.H.N. = 98) was investigated and the results are shown in Fig. 25 for two different wall thicknesses. The difference in the response of the two different types of tube to changing nose radius is clearly illustrated. The dimensionless energy parameter,

J = mpV2 (8) ayhorZp

was used to highlight the effect of annealing and nose radius further. J is plotted against p = h/r, (where r, is the projectile nose radius) for both types of tube, and compared with plate impact data in Fig. 26. For the annealed specimens the non-dimensional perforation energy, J increases with p until it reaches a maximum and then decreases. This is similar to the behaviour of mild steel plates as reported by Corran et al. [9] (see Fig. 5). For the as-received tvbes, J varied with p to a much lesser extent than for the annealed tubes. The difference in response of the two different types of tube to changing nose radius is attributed to the higher tensile strain capacity of the annealed specimens allowing more energy to be dissipated during indentation for the rounder projectiles than for the flat-faced projectiles. A decrease in the projectiles nose radius resulted in less indentation, a change in mode of failure from one of tensile tearing to one of shear plugging and a decrease in the critical perforation energy. The as-received tubes with their lower strain to failure failed through shear for all projectile nose radii and consequently were less sensitive to changes in this parameter.

Microscopic observations of sections cut from the annealed tubes at velocities near the ballistic limit illustrate clearly the changing failure modes. The 45 distortion of the grains in Fig. 27(a) indicates the shearing mode of failure associated with flat-faced projectile impact, whereas the tensile necking shown in Fig. 27(b) is indicative of discing failure caused by round-nosed projectiles. A combination of the two failure modes, found when the tube is struck by a l)rojectile with a nose radius near to the critical value (p = 0.16) is shown in Fig. 27(c).

Ma and Stronge [60] also investigated experimentally the indentation, rupture and perforation of thin-walled cold-drawn mild steel tubes struck by spherical missiles travelling at velocities near to the ballistic limit. The penetration and failure process observed in


No %

I!

/,0- mild steel tubes

o as received 36 cylindrical missiles

mild steel tubes annealed

32 j cylindrical missiles

mild steel tubes 28 A as received

spherical missile [86]

2/, mild steel plates cylindrical missiles[9]

16-

12-r o

8-

4 -

0 u i i 0 v. u 0 0"1 0'2 0'3 0.t~ 5 0'6 0"7

Pn=h0/rn

Fig. 26. Plot of non-dimensional impact energy at perforation vs non-dimensional missile nose curvature [20].

a typical test can be seen in Fig. 28. In general, the first fracture occurs on the distal surface of the tube at projectile speeds 50-75% of the ballistic limit at a radius smaller than the projectile radius.

A difference in response between thick-walled and thin-walled tubing was also observed by Ma and Stronge [60]. For thin-walled tubes higher impact speeds resulted in an increase in dishing leading eventually to a tensile type of failure. Tubes with thicker walls responded to an increase in impact velocity by being penetrated further leading to a blister being pushed out on the distal side and ultimately a plug being sheared off. For steel tubes the transition between the two types of failure occurred at dp/h ratios of 3. The response of the tubes to impact was compared with that of plates of similar thickness in [60], with the tubes proving to be stiffer than their plate counterparts.

The effect of the presence of a filling medium (sand or water) was also investigated by Ma and Stronge [60]. An increase in the stiffness of the tube was seen to occur as well as an increase in its ballistic limit. Hence the ballistic limit of a given tube increased with wall thickness and density of the filling medium, but decreased with increasing missile diameter. Neilson et al. I-61] compared the strain histories of empty and pressurized water-filled pipes in a series of tests in which nominal 150mm bore, 7.6 nun wall thickness pipes were struck with fiat-faced projectiles with diameters of 25 and 60mm. The water-filled pipes were pressurized to values that gave circumferential wall pre-stresses up to 25% of the material yield stress. It was shown that there was a noticeable difference in the strains measured in the two types of pipe, with the largest difference being in the circumferential direction. For impact on empty pipes the circumferential strains at the 0 location (i.e. in the same vertical plane as the point of impact) were all compressive whereas for the water-filled pipes tensile circumferential strains were recorded. This was explained in [61] by the fact that the water


Fig. 27(a). Tube hit by flat-nosed projectile. Top: cross-section of tube for impact near the ballistic limit; Bottom: magnification of area where shear fracture has occurred [20].

tended to spread the loading of the projectile more uniformly around the circumference of the pipe than would be the case if the water was not there. The incompressibility of the water resulted in the compressive effect of projectile indentation being countered by circumferential tensile forces required to maintain a constant flow area. Neilson et al. [-61] noted that the perforation energy of pipes decreased when the pipes were water-filled as opposed to being empty: an observation contrary to that of Ma and Stronge [-60]. The main difference between the two investigations was the type of projectile used: Ma and Stronge used lightweight (< 10 g), sphe:rical missiles in their tests, whereas Neilson et al. used heavy (4 kg), flat-faced projectiles. Another difference was the pipe wall thickness. Although the dp/h o and, dp/D ratios were :fimilar for both series of tests the overall scale of the Neilson tests were approximately 3-4 times larger than that used in the Ma and Stronge tests and hence the wall thicknesses were larger. Neilson et al. [,,61] (like Ma and Stronge [-60]) noted that the presence of the water had far more of an effect on the critical perforation energy than the pressure it was at. Howe~ver, it was found that altering the water pressure may alter the form of fracture produced by the projectile.

Further work is deafly needed in this area in order to gain a better understanding of the influence of filling medium on pipe response. The presence of an incompressible filling


Fig. 27(b). Tube hit by round-nosed projectile. Top: cross-section for impact near the ballistic; Bottom: magnification of area where tensile necking has occurred [20].

medium has been shown to strongly influence the strain histories in the pipe wall when subjected to projectile impact, and these strain histories will determine the fracture modes and the energy absorbing capacity of the pipes. What is not fully understood at present is how the development of the strain fields are dependent on filling medium density, pressure etc and how these strains determine the final fracture mode.

Corbett et al. [62] conducted an experimental investigation into the impact of steel tubes by round nosed projectiles concentrating on the effect that the projectile mass, projectile nose shape, the type of tube support and the tube material properties had on the ballistic limit of the tubes. In [62] the critical perforation energies of the tubes were shown to be dependent on projectile mass in a manner similar to that observed in flat plates [9-11]. The dependence of critical perforation energy on the material properties of the tube and the projectile nose shape was seen to be similar to that described by Palomby and Stronge [20] with there being a particular nose radius for which the critical perforation energy was a maximum. A comparison was also made in [62] between the response of steel tubes to dynamic and quasi-static local loading. It was shown that dynamic loading results in a more localized deformation zone than static loading; material properties have less of an effect on global response and energy absorbing capacity of tubes under dynamic loading than under static loading; and supporting conditions have less of an effect on tube response to dynamic loading than static loading.

2.6. Material properties at high rates of strain

The plastic stress-strain relationship of many materials is strain-rate sensitive and this dependercy needs to be taken into account if dynamic structural behaviour is to be modelled


Fig. 27(c). Tube hit by round nosed projectile. Top: cross-section for impact near the ballistic limit; Bottom: magnification of impact area showing tensile fracture on the distal side [27].

accurately. For example, it has been shown that the lower yield stress of mild steel increases by 170% and the ultimate tensile strength increases by 40% when the strain rate is increased from 10- 6 to 103 s- 1 [63]. It is generally thought that, as a rule of thumb, for the mean strain rates involved in sub-ordnance impacts an enhancement of 40% to the yield stress is a reasonable factor [64].

In recent years there has been an extensive amount of experimental and analytical research into the characterization of dynamic plastic behaviour of materials and although a discussion of this research is beyond the scope of this paper, excellent reviews can be found in [65-67].

2.7. Effects of material constraint

Another effect that makes accurate representation of material strength difficult is that of lateral constraint. It is a well known fact that metallic blocks, when indented by blunt punches, require pressures well in excess of the yield stress of the material in simple tension or compression in order to cause plastic deformation [68]. This apparent enhancement of strength, often characterized as a constraint factor, K, is due to the prevention oflateral flow by the undeformed surrounding material giving rise to a hydrostatic stress component. Woodward and de Morton [69] showed that for quasi-static indentation of steel plates with flat-faced cylindrical punches the constraint factor varies from approximately 2 at yield to values greater than 3 at high strains. As indentation proceeds, the amount of constraint increases due to increased contact area between the punch and the plate and hence the loading pressure required for further plastic flow also increases. The constraint factor for any particular situation is strongly dependent on the loading and target parameters that define that situation.


meridional deformation axial deformat ion

Vo = 85 mls

Vo = 116 mls first distal fracture

Vo = 195 IxlUislic

Vo = 259 mJs

Fig. 28. Cross-section and axial section of thin steel tube (diameter = 50 mm, wall thickness = 2.1 mm) when struck by a 12.7 mm diameter sphere travelling at near the tubes ballistic limit [60].

Liss et al. [70] investigated the effect of material constraint on the material properties of aluminium 2024-0 under dynamic loading using the split Hopkinson bar tech- nique. It was found that an average constraint factor of 2 needs to be incorporated when estimating the yield stress of constrained aluminium (specimen thickness-bar diameter ratio = 0.5) under static loading, and a factor of approximately 1.75 when loaded dynamically.

2.8. Friction

Krafft [71] investigated the importance of surface friction in the penetration process. His experiment shows that, at the most, sliding friction accounts for 3% of the total striking energy of the projectile. However, if "ploughing" of the target plate occurs during the impact process sliding friction may well become an important factor.

3. EMPIRICAL PREDICTIONS OF MINIMUM PERFORATION ENERGY

3.1. Introduction

The first studies of penetration and perforation processes were of an experimental nature. Test data were used in conjuction with analytical and dimensional considerations to define relationships between the various parameters. For example, Robins [72] found experimentally that "if bullets of the same diameter and density impinge on the same solid substance with different velocities they will penetrate that substance to different depths, which will be in the duplicate ratio of those velocities nearly, and the resistance of solid substances to the penetration of bullets is uniform". This discovery gave rise to the well-known Robins-Euler formula predicting the approximate depth of penetration,

mv V 2 x = (9)

2a

where a is a measure of the resistance to penetration, assumed to be constant and found from experimentation. Since the work of Robins a number of similar empirical equations predicting such parameters as the depth of penetration or the energy required to perforate a structure have been proposed. They are well-documented (for example see [1,2]) and are still widely used today.


3.2. Similitude theory and scaling Similitude ~theory (or dimensional analysis) is a powerful tool in analysing experimental

data and is often used in the derivation of empirical formulae. An early example of the use of dimensional analysis to model the effects of explosive loading of structures was that due to Hopkinson in 1915 as reported by Christopherson 173]. More recently Buckingham's Pi theorem was applied to projectile penetration of solids to identify 16 dimensionless groups, 14 of which are applicable to sub-ordnance impact [74]. The number of these groups is less than the number of variables and allows insight into how certain groups are related. The use of similitude theory to obtain dimensionless groups also allows scaling to be carried out. Scaled models can be tested at far less cost than prototypes and by removing dimensions from experimental groups the laws of scaling can be used to apply results from tests on a model to a prototype. However, for scaling laws to be applied with confidence experimental evidence is needed to confirm its applicability and accuracy when applied to a given situation: this is not available for all circumstances.

The application of scaling laws to dynamic loading of inelastic structures was discussed by Jones [75] who concluded that the laws of geometric scaling cannot, with confidence, be applied to impact loaded targets due to two factors: strain-rate effects and the existence of ductile-brittle transitions which may occur in either the model or the prototype tests. Dallard and Miles [76] and Booth et al. [77] also investigated the applicability of geometric scaling to impact situations and reached a similar conclusion. The latter paper includes an interesting appendix by Calladine who discussed both qualitatively and quantitatively the effect of strain-rate on the scaling laws and showed that the form of the load-deflection curve is important in determining the appropriate scaling system. In [77] Calladine's reasoning was seen to account for some, but not all, of the discrepancy between the predicted and actual behaviour of scaled models of plated steel structures impact loaded by dropped objects. It is clear that inertia also plays an important role in these problems, inhibiting the onset of bending dominated modes of deformation which will also disturb scalability.

Scaling theory was applied to low velocity projectile impact of plates by Duffey et al. [78] and produced compatible results for full and half-sized models to within 10%. Anderson et al. [79] carried out a computational study to quantify the effects of scaling on the penetration and perforation processes present in high velocity impacts (V o = 1.5 km/s). It was found that the impact resistance of small scale targets was slightly greater than for the full size models clue to strain rate effects, although the differences were extremely small (typically 5% for scaling factors greater than 10) and difficult to separate from experimental scatter. Further work is needed to quantify the effect of strain-rate in the sub-ordnance impact range on the scaling process before it can be applied with confidence to dynamic penetration and perforation ]processes. Furthermore, material failure complicates the scaling process as the ductile-brittle transition problem is exacerbated during fracture and this phenomenon needs to be investigated further.

3.3. Empirical predictions of critical impact energy for steel plates The use of analytical methods to predict the response of plates to projectile impact has

increased significantly over recent years. However, the use of empirical formulae to predict the energy required to perforate target plates continues to be an important tool for the impact engineer. As the importance of impact mechanics has increased so has the development of empirical formulae to cover a wide range of impact situations. Some of these formulae were developed many years ago [for example Eqn (9), 1742] and others are the product of recent research. The most well-known and widely used empirical formulae predicting the minimum energy required to perforate a plate are

(i) De Marre (1886): (i) De Marre (1886): E_ _..,1.5,.1.4 velocity range not stated. (10) c - - U Up n 0 ,

Here a is a constarit found from experimental evidence.


i0 s

5

t _

oJ

(I/

10 4 L

Q .

- SRI t - - - De Marre ~o- - - ' '9 . BRt oL 1o "

', x \T T

x Perforation - m[racks or small hote - - * Deformation onty - -

C] ED El> 90 [ ] :> 60*

Fig. 29. Impact test results on 7mm thick steel target plates. Circles: non-perforating impact; triangles: partial perforation; crosses: full perforation [80].

(ii) Stanford Research Institute Formula (SRI) (1963):

audp 2 E, = ~ [42.7h o + lho] (11)

validity range: 0.1 < hold p < 0.6; 0.002 < ho/l < 0.05; 10 < L/dp < 50; 5 < lid v < 8; l/h o < 100; 21 < V o < 122m/s.

(iii) BRL formula (1968):

E c = 1.44 109(hodp) 1"5, validity range not stated. (12)

The parameters in all these formulae [Eqns (10)-(12)] must be expressed in SI units. Ohte et al. [80] performed a series of experiments in which flat-faced, hemispherically-ended and conically-nosed cylindrical projectiles were fired against carbon steel target plates 7-38 mm thick in order to investigate the validity of these formulae. The projectiles had diameters ranging from 66 to 160 mm and masses ranging from 3 to 50 kg. The test results for the 7 mm thick plates (Fig. 29) show that under these test conditions for flat-faced and hemispherically- ended projectiles the perforation energy is similar and reasonably well predicted by these formulae, with the De Marre formula being closest. However for conically-nosed projectiles the required energy for perforation drops sharply as the nose angle is decreased. This decrease in the critical energy required was attributed to the decrease in effective contact area for the conically-nosed projectiles by Ohte et al. [80] who investigated the change in this contact area with target thickness and nose angle and developed an improved estimate of the perforation energy:

E c = 3.0 x 108ho~Sd~ 5 (13)

where de is the effective nose diameter, d e = ho{ 1 + 2.9(tan fl)2.1} for conically-nosed projectiles provided that d e < dp. If d e is calculated as being greater than dp, then the value of dp is taken as being the effective projectile diameter. In this formula the perforation energy is given in joules when h o and d e are in metres. These relationships were obtained from tests on SGV49 carbon steel plates 7-38mm thick, w/ho>39, and 25< Vo< 180m/s and were shown to predict critical perforation energy reasonably well.

Neilson [81] also investigated the applicability of the SRI and BRL formulae. Dimen- sional analysis was used to condense the results into a more manageable form and the data correlated for long projectiles, in the following form:

/h , ,1.7/ l \0.6 E =Aaudap("l | - - | (14) \d,,/ \d.]

for parameter ranges O.14


=~1=

10

o Haximum energy giving non-perforation x Hinimurn energy giving perforation

L/dp >13

....=.. +30%

. . . . . .o~ -30% o

Ec 1.7 o.6

o.ud, = 1"/* (d~l (d~/

100 1 i i , i , i , , I

10 t/dp

Fig. 30. Comparison of test data with proposed empirical formula for mean perforation energy [Eqn (14)] [81].

calculation of the minimum perforation energy. Figure 30 shows a comparison between Eqn (14) and test results obtained from flat-faced cylindrical projectiles with diameters ranging from 32 to 85 mm and masses from 1 to 20kg striking target plates with thickness ranging from 1 to 25 rnm.

The perforation energy for long penetrators appears to be independent of panel width for w/dr, ratios greater than 22. There is not such a consistent relationship for the short and intermediate length penetrators. An explanation for this can be found in Fig. 4 [9]. For mild steel plates t]Oe perforation energy increases with increasing projectile mass (and hence increasing projectile length) to a plateau. Before this plateau the perforation energy is highly dependent on the L/dr, ratio but once the plateau is reached the energy remains approximately constant, whatever the L/dr, ratio. In 1986 Jowett [82] assembled data from various sources and provided a bi-functional relationship for the perforation energy for shorter projectiles:

/h \1.74/ l \o.61 E c = 1.32a, da | " | | - - | for 0.1 < ho/d p < 0.25 (15a)

"\dr,) \dr,J

/h \o.s4/ I \o.61 Ec = 0.38a, dva |"---| / - - | for 0.25 < ho/dr, < 0.64 (15b)

\dr,] \dr,,!

having a validity range: 2 < L/dr, < 8, 315 < a, < 483 MPa, 40 < Vo < 200 m/s, l/dr, < 12. For l/dr, ratios > 1L2 the l/dr, term in Eqns (15) should be replaced by unity. Once again SI units are used in these equations. A comparison between these relationships, named the AEA short missile equations and the assembled experimental evidence [along with equation (14)] is shown in Fig. 31.

The importance of only applying the empirical formula to impact situations that lie within their stated ranges of applicability was highlighted by Wen and Jones [83]. Low velocity ( < 20 m/s) impact tests with blunt and flat-faced penetrators were carried out on circular mild steel plates with D/h o ratios between 25 and 100. It was shown that the Neilson [Eqn (14)], SRI [Eqn (11)] and AEA short missile [Eqns (15)] formulae all over-predicted the critical impact energy required for perforation of the plates. It was also noted that the test parameters lay outside the stated ranges of applicability for all these equations. The BRL formula [Eqn (12)] gave the best prediction for critical impact energy for these tests. In [83] a new empirical formula was ]proposed which was obtained by dividing the energy absorbing mechanisms of the plate inte two components: a local component consisting of indentation and shear; and a global component consisting of membrane stretching. Dimensional analysis was used to


10

J ,.ol I- o.81 I e 0-6 I

0'4 0"3

0"2

0-1

I /

/ /

/ /

/ 0 /

/ /

/ /

i i I ~ '~ I -- O / -

/ ~o" x /

,/~""~ o Zaid 1861 , " Corran [9] I Lefhaby[99]

~[~" x Neitson [811

i i i i i i 0-2 0-3 0'40"5 0"7 1-0

ho/dp Neitson's long projectite equafion [Bli

E----.L = 1"32(h0 ) " ( [ )'"0.1 ~h*


SRI (hemispherically-ended projectile)

o-ud 3 E c = ~3 [42.7ho2 + lho] , (17a)

Neilson (hemispherically-ended projectile)

/h \1 .7 / i ,,0.6 3 0 E c=0.9cr~d / - -~ [ - - | . (17b) "\d,,/ \d,,/

These equations were shown to be valid over the ranges 0.2


4. ANALYTICAL SOLUTIONS

4.1. Ener#y and momentum methods

The first attempt to investigate analytically the mechanics of penetration has been attributed to Bethe who carried out a static analysis of the penetration process (see for example 1-86]). The analysis was subsequently improved upon by Taylor C87] who evaluated the work required to expand a hole in the target plate to the radius of the bullet. Whereas Bethe tried to utilize a relationship between the state of stress in the plate and its deformation, Taylor realized that, due to the variation in the ratios of the principal stresses during the penetration process, the only relationship that is valid is between stress and strain increments. This relationship was used along with the Mises yield criterion to establish the stress distribution in the plate and the work required for perforation, giving

E = 1.33rcr~hoay. (21)

This is the work required to produce a symmetrical mode of deformation (Fig. 32) and is lower than Bethe's value of

E c = 2nr~h0ay. (22)

The ratio 2/ho, where 2 is the height of the crater resulting from projectile perforation, is a useful parameter in defining the geometry of the plastically deformed portion of the plate: this ratio is known as the shape factor. Taylor's analysis of symmetrical plate deformation gave a shape factor of 2.66, considerably higher than Bethe's resultof 2. The work required for

Fig. 32. Ductile hole formation failure associated with the penetration of mild steel plate by a cylindro-conical projectile [91].


unsymmetric deformation to failure (i.e. petalling) was given [87] as

Eo = 0.5nr2vhotry. (23)

The latter mode of deformation occurs more commonly in thin ductile plates (ho/d p < 1) than symmetric deformation, which tends to occur in thicker ductile plates (hold p > 1) [3]. Taylor's analysis was based on quasi-static considerations with no account taken of inertial effects. Later authors [88,89] have extended Taylor's theory to include these effects. Thompson [9,0] arrived at the same expression for the energy required for unsymmetrical plastic hole erdargement as Taylor but via a different and much simpler path.

Experiments were performed by Woodward [91] in which conically-tipped cylindrical missiles perforated steel and aluminium plates and the results were compared with an improved version of Taylor's expression [Eqn (21)] obtained by a more accurate numerical integration [92], namely

E c = 1.921tr2hoay (24)

for symmetrical mode of deformations, giving a shape factor of 3.835. For the work done in unsymmetdc perforation, Woodward [91] improved Eqn (23) (which he termed the Thompson model) by including bending effects and the contribution of the ductile hole enlargement element of the symmetric mode that occurs when the plate thickness is over 1.8 times the projectile radius, giving

E=2rphotry(rp+2ho)+l.42~zryho(1-~81) 2. (25)

Woodward proposed that a realistic yield stress should be used to take into account strain hardening effects. A value of the yield stress at a natural strain of 1.0 was found to give good correlation with the experimental results. A comparison between these models and the experimental results is shown in Table 1. It can be seen that the theories, once the appropriate modifications are made, give reasonable, if a little low, estimates of the critical projectile velocities. It was suggested in [91] that the discrepancy between the observed shape factors (2/ho) and the constant predicted value of 3.835 is due to work hardening effects.

By considering dynamic effects, Thomson [90] also analysed the penetration and perfor- mation of a plate by projectiles of various shapes giving an estimate of the residual projectile velocity, V r

r2h 2 V2 41t p OFry .a_ PVo] v, 2= o - mp

(26a)

which can be reduced to:

V, = (V~ - V~) 1/2 if mp >> mpv (28)

Zaid and Paul [94] used momentum balance, extending the method used by Thompson [90], to determine the residual velocity of a perforating projectile following normal impact of

and an estimate of the critical energy,

2 Vorp 2

where the constant .4 - 1 for a conically-tipped projectile and A = 1.86 for an ogival-headed projectile. Sodha and Jain [93] subsequently corrected the analysis for the ogival-headed projectile, giving a new value of A -- 0.62.

Recht and Ipson [32] used momentum considerations along with an energy balance to analyse the mechanics of penetration by projectiles. The analysis yields the following expression for residual projectile velocity, Vr, following a normal impact at a velocity, V o above its ballistic limit, Vx

Vr ---~ mp (V 2 -- V2) t/2 (27) mp+ ?lqpl 0


0

_=

~2

0

e~

e~

0

~ I .~-~ ~

..~ ~ I~ 0

O~ ~

O~ I I

a~

.r. a

k~

e~

e~

8

-I--


45 to 202t~.0 plate o. . " 360ram d iax

Vo L \ L - - " 1'27mm thick "~ 30 o~ 12"7~m dia !

o E ;;;; meota resu.,

equatlon~291 0 - " -- J I

0 60 120 180 2/,0 300 Impact velocity (m/s)

Fig. 33. Experimental velocity drop and comparison with theory for perforation by a 12.7mm diameter cylindro-conical projectile [15].

a thin plate to give the relationship

Vr ~ mp VO mp+ 2nphor~ sin/~" (29)

In [952, Paul and Zaid apply the method to various truncated conical and ogival nose shapes and compare the results with experimental data. In [332, the analysis is extended to account for oblique impact.

All of these approaches to estimating the energy required for perforation and the residual velocity of the projectile neglect any bending, stretching or dynamic effects beyond the zone of impact. This implies that, for the analyses to be valid, the impact velocity must be appreciably larger than the ballistic limit. Various studies (e.g. [86,962) have confirmed that the energy and momentum approaches give a good prediction of residual velocity when the impact velocity is suitably high, but tend to over-predict as the impact velocity approaches the ballistic limit. Calder and Goldsmith [15] plotted the results of their experiments along various analytical predictions. Whereas all three theories give good prediction of velocity drop at high impact velocity, only Eqn (27) with its experimentally obtained value of V x gives good agreement at velocities near to the ballistic limit (Fig. 33).

4.2. Analysis of failure mechanisms

4.2.1. Failure by plugging The energy and momentum methods of analysis outlined in the previous section are useful

in predicting the velocity drop experienced by a projectile striking a target plate at velocities appreciably higher than the ballistic limit of the target. They are not suitable for instances where the response of the target material at and away from the point of impact significantly affects the penetration process, i.e, impact velocities near to the ballistic limit. Also they fail to provide any insight into the actual process of perforation. These perforation processes (see Fig. 1) are numerous and extremely complicated, occurring either singly or as a combination of two or more.

Ductile hole enlargement has been discussed in Section 4.1. It occurs when conically- tipped projectiles perforate ductile plates of all thicknesses, although it is most commonly found in intermediate to thick plates [Fig. 34(a)2. Failure by plugging occurs when blunt projectiles strike intermediate to thick targets: a band of high shear strain is produced at a radius close to the projectile radius and this results in a plug being sheared off [see Fig. 34(b)2. Catastrophic shear results from an interplay between thermal softening and work hardening of the plate material within the shear bands (see, for example Recht [97]). When


(a)

(b)

Fig. 34. Internal deformations and fracture formed by penetration of (a) a sharp projectile and (b) a blunt projectile l-2].

the local increase in temperature has a negative effect on the strength of the material which is equal to or greater than the positive effect of strain hardening, catastrophic shear failure will occur. The importance of the temperature effect on failure by shear has been the concern of a number of authors, the majority of whom have been interested in the metal forming aspects (punching and blanking) of the process. Recently however, its importance in the role of projectile impact has been realized. Stock and Thompson [98] show that for aluminium alloys these bands of intense shear generate enough heat to raise the temperature of material within these bands to melting point, and in steel to give changes of phase. These effects are commented on by Johnson [1]. Lethaby and Skidmore [99] showed the plugging process to be a slow one under threshold conditions, and this means that for thin plates much of the energy is expended on global target deformation before plugging is completed. This implies that any analysis of thin plates which assumes failure by shear but does not take into account global target deforrhation is only applicable for high velocity impacts. It was also shown that, for plugging to occur, far greater energies are required at impact velocities near the ballistic limit than at impact velocities appreciably higher than this.

Woodward [100] used Taylor's equation for ductile hole enlargement [Eqn (21)] to predict the change in failure mode from ductile enlargement to adiabatic shearing. It was


shown that the mode of failure will change from one of ductile hole enlargement to one of plugging when the depth of plate to be penetrated diminishes to below x/3rp, and that the total work done in adiabatic shearing is

E = 2rcr 2 try(h o - x/~rp) + 2rtr~,try 3 tan ff

(30)

In the analysis it was assumed that, due to thermal softening, the work required to eject the plug, once formed, is negligible. Equation (30) also includes a term for the work required for indentation.

For thin targets (h 0 < x/3rp) it was assumed that no work is done in radial hole expansion and that the work done in cone indentation is given by:

2ntryh3 (31) Ei = X//'3 tan fl"

These equations are plotted in [100] for various plate thicknesses and compared with experimental values obtained from impact tests on three different types of plate: (i) 5083 Aluminium, (oy = 452 MPa); (ii) IMI Titanium 125, (try = 1167 MPa); (iii) IMI Titanium 318, (try = 1685 MPa). The value for try was obtained by fitting the stress-strain curve obtained from a quasi-static uniaxial compression test to a function of the form tr = trye', n being the work-hardening index and try the flow stress at a natural strain of 1.0. No attempt was made to incorporate strain-rate or temperature effects into the estimate of dynamic yield stress and hence the value of try obtained using this method is somewhat arbitrary and will only be valid for low velocilly impacts.

In Woodward's tests 50 mm square panels were struck by cylindro-conical projectiles with various cone angles, diameters of 4.76 mm and a mass of approximately 3 g. A more detailed experimental investigation into plugging behaviour of thin titanium alloy plates when struck by this type of projectile [18] has been described earlier (Section 1.2). The aluminium and titanium 125 plates were seen to fail by ductile hole enlargement with a plug being form

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