into concrete targets by a hard

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1 Penetration projectile into concrete targets by a hard Q.M.Li and X. W.Chen Protective Technology Research Center School of Civil and Environmental Engineering Nanyang Technological Universi~, Singapore Abstract Two dimensionless numbers, termed impact function I and geometry function N, are introduced as dominant parameters to control the penetration depth into concrete targets subjected to a hard projectile impact [,2]. A definition of nose shape factor is defined quantitatively to eliminate the vagueness of the nose shape deftitions in empirical formulae. A unified formula on penetration depth into concrete target, proposed based on dimensional analysis and dynamic cavity expansion model in [,2] is applicable for a wide range of impact velocities, nose shapes and concrete strengths and reinforcements, The validity of the proposed penetration formula is verified through comparison with all major empirical formulae, e.g., modified NDRC, ACE, LJKAEA, and experimental results. There exists an upper limit of the rigid projectile assumption, beyond which, the penetration is dominated by semi-hydrodynamic regime. The proposed formula can also be extended to metallic and earth targets. Introduction Local impact effects on plain and reinforced concrete targets by a hard projectile have been investigated extensively for both civil and military applications. Based on large amount of laborato~ tests conducted in various countries [3-6], empirical formulae have been proposed to predict the local effects of a hard missile impact, i.e., penetration depth X, perforation thickness e and scabbing thickness hj. Empirical formulae on penetration depth, perforation thickness and scabbing

2 thickness in a thick concrete target had been reviewed by Kennedy [3], which covered most of the test data in U.S. and European till 970s. Slitcr [4] assessed empirical formulae on concrete impact based on a collection of test data for impact velocities between 27 and 32 rds. Further review and assessment on empirical formulae and published test data were given by Williams [5], Corbett, et al. [6], Yankelevsky [7], Dancygier [8] and Li and Chen [l]. Among the most commonly used empirical formulae are the Petry formula, the Army Corps of Engineers formula (ACE), the UKAEA (Bar) formula and the National Defence Research Committee (NDRC) formula [3-7]. These empirical formulae provide the most reliable, straightforward approach to design a concrete protective structure. The predictions of perforation and scabbing thickness in most of empirical formulae are based on the valid prediction of penetration depth, which, therefore. will be focused in the current research. In spite of their popularity in concrete penetration design, empirical formulae have several shortcomings that cotike their applications. Most of the empirical formulae are not dimensionally homogeneous, leading to the disadvantage of unit-dependency and difficulties to do the parametrical analysis. The definition of nose shape factor in the existing empirical formulae is not unique, e.g., the nose shape factor IV,vin NDRC is defined as 0.72 for flat nose, 0.84 for blunt nose,.0 for spherical nose and.4 for sharp nose while the nose shape coefficient N~ in Hughes [9] is chosen as.0..2,.26 and.39 for a flat, blunt, spherical and sharp nose: respectively. Thus, a unique definition of nose shape factor is necessary for eliminating its vagueness in different empirical penetration formulae. Furthermore, most of the published empirical formulae are valid for a small range of penetration depth in the low to medium impact velocity range. It was shown by Sliter [4] and Williams [5] that the modified NDRC formula and the Barr formula give good prediction of penetration depth oniy in the range of 0.6< X/d <2,0. Two important dimensionless numbers, i.e.. impact fimction and geometry fimction,v. are introduced in the present paper based on dimensional analysis and dynamic cavity expansion model [,2]. It shows that :hese two dimensionless numbers dominate other parameters to control the penetration process. Comparisons with various empirical formulae and experimental results on penetration will be presented for a wide range of impact velocities, nose shapes, projectile mass and diameter, concrete strengths and reinforcements. Thus, the shortcomings of existing empirical formulae may be avoid when the penetration test data are re-grouped and formulated using the impact and geometrical functions and IV.The limitation of rigid projectile assumption is also discussed. 2 Impact and geometry functions It is well known that Johnson s damage = p V; /o, [0], which relates directly to the impact velocity. is commonly used to classifi the severity

3 Strwtures [. ndev.wock and lmpmt I 93 of a projectile impact, i.e., <, the global response is more important than local effect; -, both the local effect and global (structural) response are important; (3) Q ~ >>, the local phenomenon is more important than the global one. In empirical formulae, local effects measured by perforation thickness and scabbing thickness of the target are determined by penetration depth, which, however, cannot be uniquely determined by the damage number OJ, but by two dimensionless numbers, i.e., impact finction and geometrical fimction N [,2]. 2. Dimensional analysis For a rigid projectile penetrating into concrete target, the final penetration depth is generally determined by X= f(m, VO,d, N*; pc, fc,fi, E.; a, F,,%) () where PC, EC and fc and A are the density, the Young s modulus, the unconfined uniaxial compressive strength and uniaxial tensile strength of the concrete target, respectively. a is the characteristic size of aggregate, r is the average amount of reinforcement in percentage (o/o) and p., is the representative sliding friction coefficient between projectile and concrete during penetration. M and P Oare the mass and the initial impact velocity of a projectile, d is the projectile shank diameter and N*is a nose factor to describe the characteristics of nose geometry, which will be discussed later. It has been shown that influences of parameters, J, l?,, a, r and pm on penetration are secondary comparing with other parameters [], and therefore, a dimensional analysis based on Eq.( ) gives in which, impact factor is defined by m : I*= (3) d3f. which has been used by Chang [], Hughes [9] and Haldar and Hamieh [2] in their empirical formulae, and M?.= (4) P ~3 represents the ratio between the missile section pressure ivf/d and a characteristic target area density pcd (2) for an infinite thick target. Missile section pressure and target area density have been widely used in penetration and perforation mechanics [7, 3]. However, both of them have dimensions, and therefore, empirical formulae based on them are unit-dependent. Johnson s Damage Number [0], can be related to dimensionless numbers Ix

4 94.Strucrt(res( trder.yhoch Lztxi lmp~ic[j and d = /,4. (5) 2.2 Dynamic cavity expansion model Forrestal, et al. [4, 5] s analytical penetration model for concrete target (density p and unconfined compressive strength j C) impacted by a rigid ogival missile (mass M and diameter d) with normal incidence velocity VO is based on the cavity expansion theory. This rnodei is generalized and extended to any nondeformable projectile with arbitrary nose (y=y(x), as shown in Fig. ) in []. It assumes that the axial resistant force on the projectile nose is F=c.x for x<kd (6a) F= (SfC mi 2 +N*pV2) for x > kd (6b) 4 where x and V are the instantaneous depth and velocity of the projectile. Parameter c is a constant depending on concrete strength and density, projectile nose shape, diameter and mass []. Dimensionless crater depth k is determined by slip line tieid according to [] k= $, (7) [) where A is the nose height and k reflects the sharpness of the projectile nose. Specially, k = for a flat nose, k =.207 for a hemi-spherical nose and k = and 2.77 for ogive-noses with CRH=3 and CRH=4.5, respectively. S in Eq.(6b) is a dimensionless material constant depending on the uncotilned compressive strength [,4], i.e.,.s= 82.6j C-0 s44 or S = 72.0j_c:O-s( fc in MPa). (8a,b) where Eq.(8a) was recommended and used in [4-7]. Equation (8b), suggested in [], is almost the same as Eq.(3a) but is comparable to a term in empirical formulae. e.g., modified NDRC formula, on penetration depth. The dimensionless number N* in Eqs.(,2.6b) is termed as a nose factor in [], which is defined by 3 N = $J;+ (9a) according to Fig.. Expressions of N= for various nose shapes are N-. J-- 3Y 24W for ogive nose where y = r/d is the caliber-radius-head (CRH) and r is the radius of the ogive nose,.v = -J--- l+4y2 (9b) and,v- = ~v? (9c,d)

5 Fig : General nose shape of a hard projectile. respectively, for conical nose where y = h/d bluntisphencal nose where ~ = r/d Same value of N* Fig 2: Dependence of X/d on I and N based on Eqs.(l Oa,b). and h is the nose length and for and r is the radius of the sphere. gives the same resistant force on the projectile nose during penetration [] when friction is neglected, Specially, N = for flat nose and N* = /2 for hemi-spherical nose. In general, the nose factor N* must be in the range of O<N* < ; and the smaller the nose factor N*, the sharper the nose. The introduction of nose factor N* makes it possible to describe projectile s nose shape quantitatively, which is one of the shortcomings in empirical formulae on penetration depth, perforation thickness and scabbing thickness. The motion of the rigid projectile is controlled by the penetration resistance in Eqs.(6a,b), which, together with initial conditions, lead to dimensionless penetration depth [], (loa) (lob) Y 4c(-sin El,cosO) n=!i-, ~(cose,sin@) e, m(x) o d> x h where two dimensionless numbers termed impact function,, and geometrical function, N, were introduced [], i.e.,, kfl~~ M =~= and N=~=- ( la,b) S Sd3f, N*pd3 It implies that dimensionless numbers and N are dominant parameters to determine the final dimensionless penetration depth in a concrete target when subjected to hard projectile impact. Fig.2 shows the variation of dimensionless penetration depth X/d in a plane of and IV of possible impact velocity, projectile geometry and target characteristics in practical applications. A larger

6 96. itpl,crw~,~ ( ridershock and [Inpuc[ I [ N corresponds to a slender and sharper projectile. is the ratio between impact energy and energy dissipation capability of the target. When N is large enough and l/n<<, Eqs.(lOa,b) can be simplified into where a single dimensionless number can be used to predict penetration depth. 3 Comparison with empirical formulae Many empirical formulae for shallow to medium penetration into concrete targets were published in the past half-century. Kennedy [3] and Yankelevsky [7] presented a comparison between several empirical formulae proposed to predict the penetration depth in a thick target and the perforation thickness. Take their example of a missile having weight of 45.45kg, diameter of 5.24cm and nose shape factor of N*=, Fig. 3 shows the calculated penetration depths according to various empirical formulae and the prediction by Eqs. ( 0a,b) for a hemispherical nose projectile. Sliter [4] compiled an experimental data set from both US and European tests to assess the empirical formulae on concrete impact. Figs. 4 shows the comparison among Sliter s test data collection, modified NDRC, Barr s formula, Eqs.( 0a,b) and Eqs.( 2a,b) in respect to impact function. It should be noted that measurement of small penetration depth is far less accurate than the measurement in deep penetration and the uncertainties of the exact nose shape exist for each test. However, Eqs.( 0a,b) or Eqs.( 2a,b), which are more physically meaningfid than modified NDRC and Barr s formulae and are able to consider the exact nose shape effect on penetration depth give reasonable V.lUCI>nwcc. k. P.. / /-f., J Fig 3: Penetration depth calculated by various empirical formulae (from [7]). / + THt EJ+ x Barr r A ~q~,(l Oa,b) h (h,b} 0 o, 4 6 Fig 4: Dependence of X.id on I for X/ dup to 6.0.

7 4 Comparison with experimental results When l/n <, a simple formula is suggested.wwctzws i rider Shock md Impact [l 97 Xld=I2. (3) All the test data in Forrestal et al.[4, 5] and Frew et al.[6] for deep penetration into concrete target by ogive nose projectile are re-grouped using and N in [] and Fig. 5 demonstrates the variation of X/d with for different N. In spite of the great differences on projectiles, concrete targets and impact velocities, the test data are very well represented by the proposed two dimensionless numbers. Because most of the test data satisfi l/n< 0.5, X/d increases with almost linearly and a little dependence on N is observed. Current model gives a simple and direct method to treat more complicated nose shapes, e.g., truncated ogive and stepped-nose projectile [2,7,8]. Fig. 6 demonstrates the test data from Qian et al.[7] for concrete target and the predictions from proposed formulae. Again, good agreement is observed. Ref.[ 9] extends the current model to predict the penetration depth of a high strength concrete (HSC) and fiber reinforced concrete targets under a hard projectile impact, as shown as Figs.7-8, In spite of other empirical formulae underestimating the penetration depth of HSC targets, theoretical prediction shows good agreement with the published penetration tests [20,2 ]. The experiments of deep penetration into various al-alloy targets with different projectile nose shapes have been systemically conducted by Forrestal, et al. [22-26], which are compared with the current predictions in Fig. 9 [2]. While, Fig. 0 shows the comparison between the penetration prediction and the test results [27] for soil target [2]. 5 Discussion and conclusions Interesting phenomenon was observed recently in [25,26] that there exists a transition point of impact velocity, beyond which the penetration depth reduces dramatically. This observation is contrast to most of other experimental observations on long-rod projectile penetration where the final penetration depth increases steadily with increasing impact velocity. This phenomenon was also noticed in [28] when they studied the behind armour effects of penetration. Thus, it is believed that the penetration mechanics can be generally classified into three basic regimes in a broad range of impact velocities varying from subsonic velocity in the order of 0zm/s to hyper-velocities in the order of 03rrds. The first regime is the non-deformable projectile penetration, which is valid in the sub-ordnance velocity range (less than.0-.5 kmls), in which projectile has negligible deformation and conventional material strength parameters of target, such as strength, stiffness, hardness and toughness, govern the extent of the penetration. The second is the semi-hydrodynamic regime defined by the valid application range of Alekseevskii-Tate model [29,30], which has been widely accepted for long-rod penetration into thick target where the projectile erodes as it penetrates the target. The impact velocity for the validity of

8 00 75 N=25.7i?4=78.35 X/d=O.5 XJd f A ++_ Test:N= Test-N= 43.4 A Test-N=200 ~ + Test-N= ~ Fig 5: Dependence of X/d on I for concrete target by various ogive nose projectiles. u o Fig 6: Dependence of Wd on I for concrete target by truncatedogive nose projectiles. 6 A 0 0 ~ Fig 7: Dependence of X/d on I for high strength concrete by conical nose projectile with N=334. ~?.5 I / / Analysis W RC-H l RC-R - A SF-H A SF-R SX-H o SX-R :.:*<* o Fig 8: Dependence of X/d on I for high strength concrete by conical no~e projectile with N= ~ - f 60YXld=O O Ref [22] X/d X Ref [23] x 20 A Ref [24] +Ref [25] Ref [26] o --- o Fig 9: Dependence of X/d on I for aluminum alloy targets by various projectile. 80 / X/d=O.5 ~ 60 \..- & / 40 8 / X Id 20 Eq. (lob) /- Test o I o 40, Fig 0: Dependence of Xid on I for soil targets by ogival nose projectile with N=l 0.2.

9 .Sftvlctllw.s [ tder.silm k ad II?ip(lcr [ 99 Aiekseevskli-Tate model ranges from,0 krds to 3.0 km/s depending on the target and projectile material properties. The third is the hydrodynamics regime of penetration. The impact velocity is usually greater than 3,0 km/s, where the strengths of projectile and target materials are negligible and the impact can be characterised as a fluid-fluid interaction, governed by the laws of steady-state fluid. The penetration mechanism transition between first and second regime is responsible for the dramatic drop of penetration depth with increasing impact velocity. It has great theoretical and practical interest to study the transition between these two regimes in order to get the best performance of a projectile. The proposed two dimensionless numbers, i.e., impact function and geometry function N, may be adopted as dominant parameters to control the penetration depth into various targets subjected to a hard projectile impact. A new quantitative defhition of nose shape factor can eliminate the vagueness of the nose shape definitions in empirical formulae. A unified formula on penetration depth, proposed based on dimensional analysis and dynamic cavity expansion model is applied to a wide range of impact velocities, nose shapes, concrete strengths and reinforcements and other mediums. The validity of the proposed penetration formula is verified by comparison with all major empirical formulae and published experimental results. A possible transition between rigid projectile regime and semi-hydrodynamic regime is pointed out. References [] Li QM, Chen XW. Dimensionless formulae for penetration depth of concrete target impacted by a non-deformable projectile. To Int J Impact Engng, 200. [2] Chen XW, Li QM. Deep penetration of a non-deformable projectile with different geometrical characteristics. To Int J Impact Engng, 200. [3] Kennedy I@. A review of procedures for the analysis and design of concrete structures to resist missile impact effects. Nucl Eng Des 976;37: [4] Sliter GE. Assessment of empirical concrete impact formulas. J Struct Div- ASCE 980; 06(ST5): [5] Williams MS. Modeling of local impact effects on plain and reinforced concrete. ACI Structural Journal 994;9(2): [6] Corbett GG, Reid SR and Johnson W. Impact loading of plates and shells by free-flying projectiles: A review. Int J Impact Engng 996; 8: [7] Yankelevsky DZ. Local response of concrete slabs to low velocity missile impact. Int J Impact Engng 997;9(4): [8] Dancygier AN. Scaling of non-proportional non-deforming projectiles impacting reinforced concrete barriers. Int J Impact Engng 2000; 24: [9] Hughes G. Hard missile impact on reinforced concrete. Nucl Engng Design 984;77: [0] Johnson W. Impact Strength of Materials. Edward Arnold, 972. [] Chang WS. Impact of solid missiles on concrete barriers. J Struct Div- ASCE 98; 07(ST2):257-27I [2] Haldar A, Hamieh H. Local effect of solid missiles on concrete structures. J

10 00.$FH[c[[(ws [ rfder Shock a)lci /)72pLzcf I Struct Div-ASCE 984; 0(5): [3] Smith PD, Hetherington JG. Blast and bailistic loading of structures, Butterworth-Heinemann Ltd, Oxford, 994. [4] Forrestal MJ, Altman BS, Cargile JD, Hanchak SJ. An empirical equation for penetration depth of ogive-nose projectiles into concrete targets. Int J Impact Engng 994; 5(4): [5] Forrestal MJ, Frew DJ, Hanchak SJ, Brar NS, Penetration of grout and concrete targets with ogive-nose steel projectiles. Int J Impact Engng 996; 8(5): [6] Frew DJ, Hanchak SJ, Green ML, Forrestal MJ. Penetration of concrete targets with ogive-nose steel rods. Int J Impact Engng 998;2 : [7 Qian LX, Yang YB, Liu T. A semi-analytical model for truncated-ogivenose projectiles penetration into semi-fmite concrete targets, Int J Impact Engng 2000; 24: [8 Chen XW, Li QM. Dee penetration of truncated-ogive-nose projectile into concrete target. Proc. 4? Int. Symp. on Impact Engng., Kumamoto, 200. [9 Chen XW, Li QM. Local impact effects on hi h strength concrete by a nondeformable projectile. Proceedings of the 4!. Asia-Pacific Conference on Shock & Impact Loads On Structures (SILOS/4), Singapore, Nov 200. [20 Luo X, Sun W, Chan YN. Characteristics of high-performance steel fiberreinforced concrete subject to high velocity impact. Cement and Concrete Research 2000; 30: [2] Dancygier AN, YankeIevsky DZ. High strength concrete response to hard projectile impact. Int J Impact Engng 996; 8(6): [22] Forrestal MJ, Okajima K, Luk VK. Penetration of 606-T65 Aluminum Targets with rigid long rods. ASME J Appl imech 988;55: [23] Forrestal MJ, Brar NS, Luk VK. Penetration of strain-hardening targets with rigid spherical-nose rods. ASME J Appl Mech 99;58(): 7-0. [24] Forrestal MJ, Luk VK. Penetration of7075-t65 aluminum targets with ogival-nose rods. Int J Solids Struct 992;29: [25] Piekutowski, AJ, Forrestal h4j, Poormon KL, Warren TL. Penetration of 606 -T65 aluminium targets by ogive-nose steel projectiles with striking velocities between 0.5 and 3.0 ids. Int J Impact Engng 999;23:? [26] Forrcstal MJ, Piekutowski, AJ. Penetration experiments with 606-T65 aluminium targets and spherical-nose steel projectiles at striking velocities between 0.5 and 3.Okrn/s.Int J Impact Engng 2000:24: [27] Forrestal MJ, Luk VK. Penetration into soil targets. Int J Impact Engng 992; 2: [28] Hazell PJ, Fellows NA, Hetherington JG. A note on the behind armour effects from perforated alumintialuminium targets. Int J Impact Engng 998;22: [29] Aiekseevskii VP, Penetration of a rod into a target at high velocity. Combustion, explosions and shock waves.(translated from the Russian). New York:Faraday Press, 966;2:63-6. [30] Tate,4.,4 theory for the deceleration of long rods atler impact. J Mech Phys Solids 967; 5:387-99,