Truncated Projectiles Into Concrete Targets

International Journal of Impact Engineering 30 (2004) 447–464

Penetration into concrete by truncated projectiles

Jan Arild Teland*, Henrik Sj^l

Norwegian Defence Research Establishment, P.O. Box 25, Kjeller N-2027, Norway

Received 30 October 2000; received in revised form 1 June 2002; accepted 23 May 2003

Abstract

In this paper, the penetration into semi-infinite concrete targets by rigid truncated projectiles is examined.The semi-analytical cavity expansion based penetration formula in Forrestal et al. (Int. J. Impact Eng. 15(4)(1994) 395) and Frew et al. (Int. J. Impact Eng. 21(6) (1998) 489) for ogive-nosed projectiles has earlier beenseen to give good agreement with experiments. In Lixin et al. (Int. J. Impact Eng. 24 (2000) 947), this modelwas extended to truncated-nosed projectiles by introducing two additional empirical constants. In thepresent paper, Forrestal’s approach is expanded to include a larger class of projectile nose geometries,especially flat and truncated noses. The new theory presented here introduces no new empirical constants,and is seen to agree very well with the available experimental data.r 2003 Elsevier Ltd. All rights reserved.

Keywords: Concrete; Penetration; Truncated nose

1. Introduction

The semi-analytical formula for penetration of rigid projectiles into semi-infinite concretetargets given in [1,2] has in various experiments been shown to give good results for differentconcrete qualities [3]. However, the formula is only applicable for projectiles with ogive noses. Anempirical approach at extending Forrestal’s formula to truncated ogive-nosed projectiles hasrecently been proposed by Lixin et al. [4]. This model, however, introduces two additionalempirical constants and is only valid when the truncated part of the nose is less than 1

3of the

original nose length.In this article, a purely analytical approach is used to extend Forrestal’s formula to other

projectile nose geometries, in particular all kinds of truncated noses. Typical projectiles are

ARTICLE IN PRESS

*Corresponding author. Fax: +47-63-80-75-09.

E-mail address: [email protected] (J.A. Teland).

0734-743X/$ - see front matter r 2003 Elsevier Ltd. All rights reserved.

doi:10.1016/S0734-743X(03)00073-3

depicted in Fig. 1. A direct consequence of such an approach is that no additional empiricalconstants need to be introduced, thus making the theory more widely applicable.

The physical assumptions made in this article are similar to those used by Forrestal. However,the cratering phase will be defined more generally as taking place when the projectile nose is notyet fully embedded in the concrete target. The tunneling phase then follows immediately

ARTICLE IN PRESS

Nomenclature

a surface area of projectileC; CF; CL; C0 non-dimensional proportionality constantsd diameter of projectiled 0 diameter of the flat part of the truncated nosef forceF non-dimensional force ð¼ f =scd

2ÞF0; F1; F2 non-dimensional forcesl length of truncated part of the noseL non-dimensional length of truncated part of the nose ð¼ l=dÞm mass of projectileM non-dimensional mass ð¼ m=d3rtÞN; N0; N2; Nb

2 nose factorspr radial stress at the surface of the projectile’s nosePr non-dimensional radial stress ð¼ pr=scÞR non-dimensional diameter ð¼ d 0=dÞs nose curvature radiusS non-dimensional material constant in Forrestal’s formulat timeT non-dimensional time ð¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðdsc=mÞ

ptÞ

U non-dimensional velocityv0 impact velocity of the projectilev velocity of the projectileV non-dimensional velocity ð¼ @X=@T ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðm=d3sc

pÞvÞ

V0 non-dimensional impact velocityV1 non-dimensional velocity when X ¼ X1

x penetration depthX non-dimensional penetration depth ð¼ x=dÞXp; XF

p ; XLp non-dimensional final penetration depths

X1 non-dimensional nose lengtha static material constantb dynamic material constantrt target densitysc concrete compressive strengthf;f0;f1 angles defining the projectile geometryc non-dimensional curvature radius (CRH) ð¼ s=dÞ

J.A. Teland, H. Sj^l / International Journal of Impact Engineering 30 (2004) 447–464448

afterwards. Cavity expansion theory is applied here to the complete projectile nose, including theflat part. This assumption is discussed more closely in Section 5 and shown to be reasonable. Thenew model is finally compared with various experimental data and discussed.

2. Existing theoretical models based on cavity expansion theory

It is convenient to introduce non-dimensional variables for the mathematical discussion. Todistinguish between quantities with and without dimension, capital letters will always denote non-dimensional quantities.

ARTICLE IN PRESS

Fig. 1. Projectile geometry (ogive and blunt noses).

J.A. Teland, H. Sj^l / International Journal of Impact Engineering 30 (2004) 447–464 449

2.1. Forrestal’s formula

In Forrestal’s original formulation, the penetration process is divided into a cratering phase anda tunneling phase. It was found that the crater depth usually was around two projectile diameters,suggesting a value of X ¼ 2 for the transition point between the two phases. In the crateringphase, when Xo2; the force F on the projectile was assumed proportional to the currentpenetration depth X ðTÞ;

F ¼ CFX ; Xo2; ð1Þ

where CF is determined later according to continuity conditions.In the tunneling phase, assuming no friction, the total force decelerating the projectile is

estimated according to the following integral over the projectile nose surface a:

F ¼1

d2

Za

PrðUÞ cosf da; U ¼ V cosf: ð2Þ

Given a particular target material model, either spherical or cylindrical cavity expansion theorycan be used to derive an analytical expression for pr; thereby enabling us to evaluate the integralexplicitly. In many cases, Pr is, at least approximately, found to take the following form:

PrðV Þ ¼ aþ bV2: ð3Þ

Thus one static and one dynamic (velocity-dependent) part, where the constants a and b dependon the chosen material model. An exact solution of the cavity expansion equations for a realisticconcrete model is very difficult to achieve. In [2], a and b were determined experimentally,resulting in

a ¼ S; b ¼1

M; S ¼ 82:6

sc

106

� ��0:544; ð4Þ

where sc must be given in Pascal. The material dependency of Eq. (4) is assumed to be valid for allprojectile geometries.

Performing the integration in Eq. (2) for an ogive nose gives the following expression for thetotal force on the projectile:

F ¼p4

S 1þN0

M

V2

S

� �; X > 2; ð5Þ

where N0 is a nose factor which only depends on the nose geometry. For an ogive nose it is givenby

N0 ¼8c� 1

24c2: ð6Þ

It should be noted that a sharp nose corresponds to a small value of the nose factor N0:Newton’s second law is used to calculate the complete penetration process, and the constant CF

is determined from the requirement that X ðTÞ; V ðTÞ and FðTÞ are all continuous at X ¼ 2:Forrestal’s result was

CF ¼ pS

81þ

N0

M

V21

S

� �;

V21

S¼

V20=S � p=2

1þ ðp=2ÞðN0=MÞð7Þ

ARTICLE IN PRESS


where V1 is the projectile velocity at the transition point between the cratering and tunnelingphases. The final penetration depth XF

p from Forrestal’s penetration formula thus becomes

XFp ¼

2

pM

N0ln

V20 =S þ M=N0

M=N0 þ p=2

� þ 2: ð8Þ

2.2. Lixin’s model

Lixin et al. [4] have extended Forrestal’s approach to truncated projectiles by introducingempirical constants C0 and XL in the force expression

F ¼C0CLðX þ LÞ; XoXL;

C0 p4

S 1þN0

M

V2

S

� �; X > XL:

8><>: ð9Þ

The parameter XL is interpreted as the crater depth, and is to be found experimentally. Theconstant CL is determined from continuity conditions, as described below, while C0 compensatesfor the increased force due to the truncated nose. According to Lixin, XL is in the range between1.5 and 2.5. In cases when no test reference is available, the value XL ¼ 2 is recommended. Theempirical constant C0 is assumed to depend on the diameter of the flat part of the nose in thefollowing way:

C0 ¼ 1 þ KR2: ð10Þ

By curve fitting to experimental data, Lixin found that K ¼ 1:5: The constant CL is thendetermined from continuity conditions at the transition point, and takes the following form:

CL ¼p

4ðXL þ LÞS 1 þ

N0

M

V21

S

� �V2

1

S¼

ðL þ XLÞV2

0

S� p

2C0XLðL þ 1

2XLÞ

L þ XL þ p2

C0 N0

MXLðL þ 1

2XLÞ

: ð11Þ

The final penetration depth in Lixin’s model is thus given by

XLp ¼

1

C0

2

pM

N0ln

V20

Sþ M

N0

MN0

þ p4 XLC0ðXLþ2L

XLþLÞ

0@

1Aþ XL: ð12Þ

3. New analytical model

Here we present an analytical attempt at generalizing Forrestal’s penetration formula (8) totruncated projectiles, thus removing the need to introduce additional empirical constants. Bothogive and blunt nose shapes will be considered. The various geometrical parameters of theprojectile are defined in Fig. 1.

Our modified penetration theory will be based completely on cavity expansion theory, whichwill therefore be applied to the flat part of the projectile nose as well. It is not immediately obviousthat cavity expansion theory is applicable to a flat nose geometry, so this point will be discussedmore closely in Section 5.

ARTICLE IN PRESS


The modification of Forrestal’s model is carried out in two steps. First by modifying thecratering phase and then by calculating the nose factor in the tunneling phase for the generalizedprojectile geometry.

3.1. The cratering phase

In Forrestal’s original calculation of the penetration depth, it was assumed that the crateringphase lasted until X ¼ 2; when the tunneling phase took over. In this paper, we instead proposethat the tunneling phase begins when the nose has completely entered the target, i.e. when X > X1;where X1 is the non-dimensional length of the nose. With this generalization, the new model isapplicable for nose geometries spanning from ogive to flat noses, i.e. no separate model for flat-nosed projectiles is needed. Simple geometry (see Fig. 1) shows X1 to be given by

X1 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifficð1� RÞ �

1

4ð1 � RÞ2

rðogive noseÞ; ð13Þ

X1 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffic2 �

R2

4

s�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffic2 �

1

4

rðblunt noseÞ: ð14Þ

In Fig. 2, we have plotted X1 from Eq. (13) as a function of c for R ¼ 0 and as a functionof R for different values of c: In the experiments described in [1,2], most of the projectileshad a nose length close to two projectile diameters, which is consistent with X1 ¼ 2 in ourapproach.

3.2. The force

In Forrestal’s original approach for ogive projectiles, the force is zero at impact as given byEq. (1). However, in our modified theory for truncated projectiles, a non-zero force F0 is assumedto act on the flat part of the projectile instantly after it has hit the target. This is also similar toLixin’s approach, as seen in Eq. (9), except that our method is purely analytical and therefore,does not require additional empirical constants to be introduced.

In principle, it should be possible to proceed by applying cavity expansion theory at every pointin time during the cratering phase, but this would involve a time-dependent nose factor, makingthe equations difficult to solve analytically. Instead, we assume similarly to Forrestal that theforce increases linearly with the penetration depth during the cratering phase. From varioushydrocode simulations, this assumption has been seen to be reasonable. In [5], deceleration–timemeasurements of projectiles during penetration were conducted, further indicating a roughlylinear behaviour in the cratering phase. The difference between our and Forrestal’s approach isthat the tunneling phase takes over at X ¼ X1; instead of X ¼ 2: The force can thus be written as

F ¼ F0 þ CX ; XoX1; ð15Þ

where C is a constant, similar to CF and CL; which later will be determined from continuityconditions at X ¼ X1:

In the tunneling phase, the force is calculated as the sum of the forces acting on theflat and ogive or blunt part of the nose, respectively. In mathematical terms, this can be

ARTICLE IN PRESS


written as

F ¼ F1 þ F2; X > X1; ð16Þ

where F1 is the force on the flat part, and F2 is the force on the ogive or blunt part of the nose. Theforce during the tunneling phase will in Section 3.2.4 be shown to be on the same form as inEq. (5), but with a more complicated expression for the nose factor.

ARTICLE IN PRESS

Fig. 2. Nose length X1 as a function of nose curvature c and truncation diameter R:


3.2.1. Flat part

To calculate F1; we use cavity expansion theory and Eq. (2) with U ¼ V (i.e. the same stress allover the surface), and integrate only over the flat part of the projectile:

F1 ¼p4

S 1þ1

M

V2

S

� �R2: ð17Þ

It is seen that the flat part of the nose corresponds to a nose factor N0 ¼ 1:The instant force at impact, F0; can be calculated by using cavity expansion theory on the flat

part of the projectile, just as above. The only difference is that the impact velocity V0 must be usedinstead of a general velocity V : The instant force F0 thus becomes

F0 ¼ F1ðV0Þ ¼p4

S 1þ1

M

V20

S

� �R2: ð18Þ

For flat projectiles, the nose length is X1 ¼ 0; and cavity expansion theory is applied for thecomplete penetration process. In this case, the force is given by Eq. (17) with R ¼ 1:

3.2.2. Ogive part

The force F2 on the ogive part of the projectile is found from cavity expansion, but integrationis now only carried out over the ogive part of the surface, as illustrated in Fig. 1a.

This amounts to only integrating from f1 to p=2 instead of from f0 to p=2; where f1 and f0 aredefined in Fig. 1. Some simple geometry shows these angles to be given by

sin f0 ¼2c� 1

2c; sin f1 ¼

2c� 1þ R

2c: ð19Þ

The expression for the force can thus be written as

F2 ¼ 2pc2

Z p=2

f1

PrðUÞðsin f� sinf0Þ cosf df; U ¼ V cosf: ð20Þ

The actual calculations are now straightforward, although slightly cumbersome. Eventually, thefollowing result is obtained:

F2 ¼p4

S ð1 � R2Þ þN2

M

V2

S

� ; ð21Þ

N2 ¼ N0 � R2 6ð4c� 1Þ þ 8Rð1 � 2cÞ � 3R2

24c2

� : ð22Þ

3.2.3. Blunt part

For a blunt projectile, the force in the tunneling phase is given by

F2 ¼ 2pc2

Z f0

f1

PrðUÞ sin f cosf df; U ¼ V cosf ð23Þ

ARTICLE IN PRESS


which after integration can be written as

F2 ¼p4

S ð1 � R2Þ þNb

2

M

V2

S

� ; ð24Þ

Nb2 ¼ ð1� R2Þ 1 �

1 þ R2

8c2

� �� : ð25Þ

3.2.4. Total force

As anticipated earlier, on adding the two force contributions from F1 and F2 together, the newtotal force expression is seen to be identical to the old force of Eq. (5), except for a new nose factorN in the dynamic part

F ¼ F1 þ F2 ¼p4

S 1þN

M

V2

S

� �; ð26Þ

N ¼N0 þ R2 6ð2c� 1Þ2 þ 8Rð2c� 1Þ þ 3R2

24c2

� ; ogive nose;

1 �1 � R4

8c2

� ; blunt nose:

8>>><>>>:

ð27Þ

This means that the static part of the force is unaffected by the nose shape. The nose factor N as afunction of c and R; respectively, is shown in Fig. 3.

3.3. Comparison to Lixin’s model

In Fig. 4, we have plotted the penetration depth and force as a function of time for Lixin’s semi-empirical approach and our analytical model for an impact velocity of 400 m=s: The values ofprojectile and target parameters correspond to Lixin’s experiments with 0:282 kg projectiles [4]. Itis seen that in the cratering phase the FFI-model predicts a larger decelerating force than theLixin-model, while the Lixin force is stronger in the tunneling phase. However, for the finalpenetration depth the difference is less than 5%.

4. Penetration depth

The final penetration depth Xp is calculated in two stages. First, the penetration depth duringthe tunneling phase is calculated as a function of a so far undetermined transition velocity V1:This velocity is then determined from conditions of continuity at the transition point.

4.1. The tunneling phase

The projectile velocity V1 after the cratering phase is used as initial condition for the tunnelingphase. Using Newton’s second law and Eq. (26) eventually yields the following relationship

ARTICLE IN PRESS


between Xp and V1:

Xp ¼2

pM

Nln 1þ

N

M

V21

S

� �þ X1: ð28Þ

Now, only an expression for V1 as a function of the impact velocity V0 is required. This is derivedin the next section.

ARTICLE IN PRESS

Fig. 3. Nose factor N as a function of nose curvature c and truncation diameter R:


4.2. Cratering phase

Using Newton’s second law on the cratering phase gives the following equation:

@2X

@T2þ CX ¼ �

p4

S 1 þ1

MV2

0

� �R2 ¼ �F0; XoX1 ð29Þ

ARTICLE IN PRESS

Fig. 4. Penetration depth x and force f as a function of time for the FFI and Lixin models.


with initial conditions X ð0Þ ¼ 0 and Vð0Þ ¼ V0: The solution of Eq. (29) is

X ðTÞ ¼V0ffiffiffiffi

Cp sin

ffiffiffiffiC

pT þ

F0

Cðcos

ffiffiffiffiC

pT � 1Þ: ð30Þ

The result from the cratering and tunneling phase must be continuous at X ¼ X1 in X ; V and F :These conditions eventually yield the following result:

C ¼1

X 21

ðV20 � V2

1 Þ �2F0

X1;

V21

S¼

½1 � p4

R2

MX1�

V20

S� p

4X1 � p

4R2X1

1þ NM

p4

X1

: ð31Þ

This is a generalization of the original Eq. (7) by Forrestal. Inserting the expression for V1 intoEq. (28), we obtain the final penetration depth Xp as a function of the impact velocity V0

Xp ¼2

pM

Nln

½1 � p4

R2

MX1�

V20

Sþ M

N� p

4R2X1

MNþ p

4X1

" #þ X1: ð32Þ

We note that for the special cases R ¼ 0 and X1 ¼ 2; Forrestal’s original formula (8) is retrievedwith N ¼ N0: The penetration depth as a function of scaled impact velocity for different values ofR are shown in Fig. 5 for M ¼ 5; 20 and 100, respectively. The nose shape corresponds to c ¼ 3 inthis figure. It is seen that for large values of M; roughly corresponding to long slim projectiles,there is only a little difference in penetration depth whether the nose is truncated or not. However,for small values of M; nose shape has a large effect on penetration depth, which seems intuitivelyreasonable.

5. The validity of cavity expansion theory applied to flat nose projectiles

A potential problem with the new penetration model is that it relies on cavity expansion theorybeing applied to the flat part of the projectile nose. It may be questioned whether cavity expansionreally is applicable in such a case.

To investigate this point, it would be most convenient to compare the predictions of theanalytical theory with the exact solution of the penetration problem. However, such a solution isnot available, which is indeed why simplified penetration theories based on cavity expansion havebeen developed. However, for particular values of the various physical parameters, thepenetration problem can be solved numerically using Autodyn [6] or similar hydrocodes,enabling a comparison between the analytical solution and the numerical solution (given thosespecific input parameter values).

It would have been convenient to use the empirical concrete model of Eq. (4) in such acomparison, but unfortunately this model cannot be applied directly in an Autodyn simulation.However, this should not be necessary, since it is the applicability of the framework of cavityexpansion theory to penetration of projectiles with flat noses, which we are interested in testing.Our conclusion in this matter should be independent of the target material model, i.e. it should beof no importance whether the target is concrete, aluminium, steel or any other material, as longas the fundamental assumptions of the analytical penetration theory based on cavity expansionare satisfied.

ARTICLE IN PRESS


In our chosen example, we look at a compressible Mises elastic–plastic target materialmodel with a bulk modulus 10:5 GPa; shear modulus 7:9 GPa; density 2300 kg=m3 andyield strength 200 MPa: Despite the apparent simplicity of this material model, it was convenientto derive a numerical expression for Pr of Eq. (3) instead of a very complicated analyticalexpression.

Using this material model in both Autodyn and the analytical penetration model, calculationsand simulations were performed for three different types of rigid projectiles, each having adiameter of 75 mm and total mass of 6:26 kg: The projectiles had different nose shapes,corresponding to N ¼ 0:22 (sharp nose), N ¼ 0:5 (spherical nose) and N ¼ 1:0 (flat nose). Ifthe application of cavity expansion theory to penetration of projectiles with flat noses is invalid,one would expect a significant difference between analytical and numerical results in the flatnose case.

ARTICLE IN PRESS

Fig. 5. Penetration depth Xp as a function of impact velocity V=S1=2 for various values of M and R ðc ¼ 3Þ:


In Fig. 6, we have plotted the analytical and numerical results for final penetration depth as afunction of impact velocity. There is generally seen to be a very good correspondence between thetwo as the deviations are never larger than a few percent. Interestingly, the flat nose and thespherical nose appear to give the best agreement, whereas for the sharp nose the cavity expansionmodel underpredicts the penetration depth by a few percent at low velocities.

Comparisons have been performed for a variety of target material models with similar outcome.It appears that cavity expansion gives very good predictions and that applying it to flat noses iswell justified. It should be mentioned, though, that a physical effect which is not captured by thecavity expansion approach to flat noses, is that right after impact a large peak in the force as afunction of time is generated. This effect, however, lasts for a very short time, after which cavityexpansion seems to be valid. The total deceleration in this period of time is usually negligible dueto the short duration of this phase, except if the target is very thin. However, this case is irrelevanthere since we are working with semi-infinite targets.

The good agreement for flat noses might possibly be explained by the geometrical shape of theplastic zone around the projectile during penetration. Even though the projectile nose was flat,this zone was always seen to take on a spherical form in our numerical simulations. However,making an explicit mathematical proof of the applicability of cavity expansion to flat nosepenetration remains for further study. The same can be said about the application of cavityexpansion theory to the penetration of ogive projectiles (including spherical), which suffers fromthe same conceptional shortcomings despite being generally accepted.

We conclude that the framework of cavity expansion seems to work well for a variety ofprojectile geometries. Combining the general nose geometry with the material parameters defined

ARTICLE IN PRESS

Fig. 6. Penetration depth of a 75 mm projectile as a function of impact velocity calculated by cavity expansion theory

and Autodyn for different nose shapes.


by Forrestal in Eq. (4) then results in our new Formula (32), for penetration into concrete.Recently, without being aware of the present work, a similar result has also been reached by Liand Chen [7]. In the next chapter, the new formula will be compared to experimental data fordifferent projectile geometries.

6. Comparison to experimental data

In this section, we make comparisons with experimental data taken from Lixin et al. [4] andsome experiments performed at FFI [8,9]. In Fig. 7, a sketch of the projectiles used in theseexperiments are shown.

6.1. Experimental data with truncated noses (Lixin et al. [4])

Lixin et al. performed several penetration experiments to validate their theory. To compare ouranalytical theory with Lixin’s data it is, however, necessary to know the concrete density, which isnot given in [4]. The density for a typical C-30 concrete will usually lie somewhere in the range of2000–2500 kg=m3: Assuming a density of 2300 kg=m3 and using K ¼ 1:5; we have in Fig. 8plotted the penetration depth as a function of impact velocity for Lixin’s 0:282 kg projectiles.These projectiles had a non-dimensional truncated diameter given by R ¼ 0:5: Agreement betweenour theory and the experiments is seen to be very good, especially considering that no additionalempirical constants have been introduced.

Further, we note that our model is not very sensitive to the concrete density as the predictedpenetration depth given by Eq. (32) only deviates by about 5% within the density range givenabove.

ARTICLE IN PRESS

Fig. 7. The projectiles used in the experiments described in [4,8,9].


6.2. Experimental data with flat nosed projectiles

Small scale experiments with 12 mm flat nosed projectiles against concrete targets withcompressive strength of 35 MPa were performed at FFI [8,9]. This is an extreme case of atruncated projectile, and should therefore be another good test of the applicability of cavityexpansion theory to flat noses. Projectiles of three different masses were used, namely 20.5, 65.8and 122:8 g: The 20:5 g projectiles were made of steel, while the others were tungsten projectiles.According to the theory discussed above, these projectiles correspond to R ¼ 1 and X1 ¼ 0: Theexperimental results are in Fig. 9 compared with the new FFI model and Forrestal’s originalmodel.

There is seen to be very good agreement between the experimental data and the modifiedpenetration formula given by Eq. (8), especially for the 20.5 and 122:8 g projectiles. Theexperiments with the 65:8 g projectiles seem, however, to be slightly above the FFI-model.The reason for this is unclear at the moment. We have not compared these experiments withLixin’s model because, according to Lixin et al. [4], this model is not valid for such projectiles. Theforce expression in Eq. (9) will be too large due to the factor C0; which takes the value of 2.5 forflat projectiles. The final penetration depth will therefore be too small compared to theexperiments.

6.3. Predrilled cavities

For completeness, we note that by using F1 ¼ 0 (i.e. F ¼ F2) and N ¼ N2 from either Eq. (22)or (25), depending on whether the projectile is ogive or blunt, a model for penetration into targetswith predrilled cavities is obtained. This topic has been further explored in Teland [10].

ARTICLE IN PRESS

Fig. 8. Comparison between analytical theory and experimental data in [4].


7. Conclusions

We have modified Forrestal’s semi-analytical penetration formula to obtain a new modelwhich is valid for a large range of projectiles. The physical assumptions in the new modelare to a large part similar to those made in Forrestal’s original work [1,2], and which aregenerally accepted as valid. New features include a redefinition of the transition point betweenthe cratering and tunneling phase according to a more physical criterion. Further, we haveargued that cavity expansion theory can be applied to penetration of flat nose projectiles and usedthis observation to develop a penetration model that is valid for a large range of projectilegeometries.

The new model does not depend on any new empirical factors, which means that it does nothave to be calibrated to new experiments in each case. The formula was compared with differentexperimental data and good agreement was found.

ARTICLE IN PRESS

Fig. 9. Comparison between analytical theory and experimental FFI-data [8,9].


References

[1] Forrestal MJ, Altman BS, Cargile JD, Hanchak SJ. An empirical equation for penetration depth of ogive nose

projectiles into concrete targets. Int J Impact Eng 1994;15(4):395–405.

[2] Frew DJ, Hanchak SJ, Green ML, Forrestal MJ. Penetration of concrete targets with ogive-nosed steel rods.

Int J Impact Eng 1998;21(6):489–97.

[3] Sj^l H, Teland JA. Prediction of concrete penetration using Forrestal’s formula. FFI/RAPPORT-99/04415.

[4] Lixin Q, Yunbin Y, Tong L. A semi-analytical model for truncated-ogive-nose projectiles penetration into

semi-infinite concrete targets. Int J Impact Eng 2000;24:947–55.

[5] Forrestal MJ, Frew DJ, Hickerson JP, Rohwer TA. Penetration of concrete targets with deceleration-time

measurements. Int J Impact Eng 2003;28(5):479–97.

[6] Autodyn theory manual. Horsham, UK: Century Dynamics Ltd.

[7] Li QM, Chen XW. Deep penetration of a non-deformable projectile with different geometrical characteristics.

Int J Impact Eng 2002;27(6):619–37.

[8] Sj^l H, Teland JA, Kaldheim Ø: Penetration into concrete with 12 mm projectiles. FFI/NOTAT-98/04392

(in Norwegian).

[9] Sj^l H, Teland JA, Kaldheim Ø: Penetration into concrete—analysis of experiments with 12 mm projectiles.

FFI/RAPPORT-2000/04414.

[10] Teland JA. Cavity expansion theory applied to penetration of targets with predrilled cavities. International

Ballistic Symposium, Interlaken, Switzerland, 7–11 May 2001.

ARTICLE IN PRESS


Documents

Truncated Projectiles Into Concrete Targets