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Analysis of jacketed rod penetration
International Journal of Impact Engineering 24 (2000) 891}905
Analysis of jacketed rod penetration夽 M. Lee* School of Mechanical & Aerospace Engineering, Sejong University, 98 Kwangjin-Gu, Kunja-Dong, Seoul, 143-747 South Korea Received 17 November 1999; received in revised form 18 February 2000
Abstract An analytical model for the cratering process in a metal target due to high-velocity penetration of a jacketed rod has been developed. The jacketed rod consists of a central high-density core surrounded by a low-density jacket. The analysis is based on the observation that two mechanisms are involved in the cavity growth by eroding rods: #ow of rod erosion products, which exerts radial stress on the target and opens a cavity, and radial momentum of the target as it #ows around the rod front nose. Hence, the mushrooming e!ects on the cavity growth are isolated. The model includes the centrifugal force exerted by the core and jacket, radial momentum of the target, and the strength of the target. A criterion for the onset of `bi-erosiona, where there exist two stagnation points such that the performance is di!erent signi"cantly from the homogeneous rods, is obtained from the model. Predicted results for the "nal cavity diameter are found to match well with the corresponding test values. Further numerical simulations are conducted and indicate that as long as `co-erosiona occurs the penetration performance of jacketed rods is governed by the aspect ratio not of the jacketed rod but of the high-density core rod. 2000 Elsevier Science Ltd. All rights reserved. Keywords: Cavity diameter; Jacketed rod; Mushrooming; Bi-erosion; Co-erosion
1. Introduction Recent trends in kinetic energy projectiles are toward higher aspect ratio (¸/D) and increased velocity [1], where ¸ is the length and D is the diameter of the projectile. For a given weapon system, however, the stability problem during acceleration and free-#ight phase becomes serious 夽
This article is an expanded and revised version of a paper that was "rst presented at the ASME Pressure Vessels and Piping Conference, 27}31 July 1997, Orlando, USA. * Tel.: 82-2-3408-3282; fax: 82-2-3408-3333. E-mail address: [email protected] (M. Lee). 0734-743X/00/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 7 3 4 - 7 4 3 X ( 0 0 ) 0 0 0 2 0 - 8
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M. Lee / International Journal of Impact Engineering 24 (2000) 891}905
for cylinders of extreme aspect ratio. The sti!ness in attack of thick targets is also one of the main concerns. To overcome these problems, a jacketed projectile that augments the sti!ness is usually suggested. The jacketed rod consists of a central high-density rod surrounded by a low-density jacket. Since two di!erent projectile materials are interacting with target material at the interface, two kinds of di!erent penetration processes are usually observed. If the outer diameter of the jacket is less than the diameter of the mushroom formed by the eroding core rod, it is known that a jacketed rod only penetrates as well as the core [2]. This condition results in `co-erosiona. If the relative thickness of the jacket exceeds a certain limit value, penetration performance is degraded because of the independent interaction of both core and jacket materials with the target. In this case, two di!erent stagnation points are observed. The jacket is decelerated much faster than the core since the density of jacket is lower than the core. This geometry is denoted `bi-erosiona [3]. Recently, a series of numerical simulations [4] and experiments [5] have been performed to examine the ballistic performance of the jacketed rod. Although the penetration performance of the jacketed rod is not diminished for co-erosion case, some reduction in the crater diameter compared with a homogeneous rod is measured at the identical aspect ratio [4]. This means that the jacketed rod is more sensitive to yawed penetration. One way to analyze a detailed high-velocity penetration process is to use computer predictions. However, this approach is computationally expensive. By using a simple analytical model, however, signi"cant parameters and how they vary over a wide velocity range can be obtained. Up to now, there are no detailed analytical models for the penetration mechanics of the jacketed rod and the critical value of D /D for the onset condition of `bi-erosiona, where D and D are the
diameter of the jacket and core materials. Hence, in this paper the previous model (two-stage cavity expansion model) for a homogeneous rod [6] is applied to the case of the jacketed rod. We "rst select a value of D /D less than the critical value, yielding `co-erosiona, and analytically
determine the cavity growth formed in a metal target. In addition, due to the two-step procedure in the theory the onset condition of the `bi-erosiona is obtained. For design concerns, these results are valuable.
2. Mushrooming and cavitation The radial motion of target material due to the penetration of eroding rods occurs in two stages as shown in the numerical simulation [7]. In the "rst stage, eroded rod elements play the dominant role in opening a cavity. In the second stage, the momentum in the target is responsible for further cavity expansion (cavitation e!ect) until the strength of the target forces it to come to rest. The term `cavitationa is borrowed from Hill's work [8] that deals with the attack of thick targets by non-deforming projectiles, where cavity growth is only due to the cavitation e!ect. When the projectile deforms, the entire physics is di!erent in kind. Actually, cavitation severely degrades the performance of anti-tank weapons because kinetic energy is absorbed by the enhanced cavity production at high velocity. Otherwise, it would go toward penetration. Simulations described here address the in#uence of the target momentum on the cavity growth (cavitation). In order to eliminate the e!ect of mushrooming on cavity production, hypothetical simulations, which is actually the same assumption made by Hill [8], were conducted using the
M. Lee / International Journal of Impact Engineering 24 (2000) 891}905
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Fig. 1. The e!ect of cavitation on cavity growth.
two-dimensional Eulerian wave propagation computer code, AUTODYN. A rigid long rod of a hemispherical nose impacts RHA steel target at 1.3 and 2.5 km/s. Fig. 1 shows the computed velocity "eld and cavity shape. At 1.3 km/s, the projectile essentially creates a cavity almost equal to its diameter. However, as the impact velocity increases there is some point at which the #ow separates from the projectile head, thus showing signi"cant cavity overshoot due to the target momentum. The target momentum has a great in#uence on the cavity growth, as does the mushrooming of the projectile. The analysis of cavity growth by a rigid projectile travelling through an in"nite medium at constant velocity was developed by Hill [8].
3. Cavity growth model 3.1. Assumptions It is assumed that the "nal cavity growth is equal to the sum of the cavity growth produced in each stage acting independently, regardless of the order of application of loads [6]. In other words,
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we use the principle of superposition for the total work done in each stage. The analysis includes the centrifugal force exerted by the projectile, radial momentum of the target, and the strength of the target. The following assumptions are made, many of which are motivated by the results of numerical simulations and experiments. (1) The #ow of eroded elements is considered as incompressible, steady, and inviscid relative to the stagnation region as shown in Fig. 2. r and r are the radius of the jacket and core material,
respectively. The eroded elements from each material arrive and exit with a velocity of
Fig. 2. Geometry of a jacketed rod penetration.
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Fig. 3. Force balance for the penetration of a jacketed rod.
of the stagnation region. From mass conservation in a coordinate system located at the penetration front, the mass of the eroded element is given by 2pr o q dl, (1a) 2pr o q dl. (1b)
Subscripts j and c denote jacket and core materials, respectively. r is the radius of the centerline of the eroded element, q is the thickness and dl is its arc length. The centrifugal acceleration of the #ow of each eroded element is given by (
where R(b) is the local radius of curvature of the centerline curve, and < is the impact velocity. From the de"nition of radius of curvature, 1 dr , R(b )"! (3a) sin b db 1 dr
. R(b )"! (3b)
sin b db
Each thickness of the eroded element is determined from Pappus's Theorem r q (b )" , (4a) 2r (b ) r (4b) q (b )" .
2r (b )
In this con"guration two centrifugal forces, one for the core and one for the surrounding jacket, must be balanced by the pressure resistance exerted by the target:
1 (
R(b )
(5)
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where R is the target resistance for radial cavity expansion and subscript t denotes the target material. The pressure resistance is acting normal to the #ow at radius r . If the eroded elements #ow together with a velocity of
2r db 2r From now on, we just use b instead of b . By introducing two constants indicating the ratios of dynamic to static pressures, Eq. (6) can then be written as
o o 2r dr "!2r # 1!
o o where
r S sin b db, r 1#¹ sin b
(7)
1/2o (
o L S sin b o db. a " r #2r # 1! r
o 1#¹ sin b o (9) This equation can be calculated by a numerical integration method. The strength of target is included in the model; otherwise Eq. (9) is not limited. To determine the initial condition r , now consider the case where the eroded element is inside the stagnation region in which resisting pressure is equal to 1/2o ;. We still apply the previous force balance argument applied to the outside stagnation region. In order to "nd the balancing forces exerted by the jacketed rod, it is necessary to consider two separate regions. For r 4r , there exists only one centrifugal force exerted by core element. For r 'r , however, there exist one centrifugal force exerted by the core and one hydrostatic force by the jacket. The jacket material is assumed to remain inside its stagnation region, such that the hydrostatic pressure is 1/2o ( , where, > is the dynamic yield strength of the jacket material. The
resistance by the target and the force by the jacket material are acting normal to the #ow at radius r . Again, Eq. (6) can be written by employing the radius of curvature and the thickness
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of eroded element,
r 1 dr \ R # o ;"! o ( ! o (
2 db 2r This equation is integrated for the interval from r (b"n)"r /2 to r [9]. After some algebraic manipulations (see the appendix), we "nd the radius of the stagnation region:
r "r
8S!3(1#¹) 1# , 4(N#¹!M)
(11)
where (R !> )
, N" (12a) R 1/2o (
. (12b) M" R Eqs. (9) and (11) give the size of cavity growth which is only due to the mushrooming e!ect. At this point, it is important to address that the expression for a jacketed rod becomes that of a homogeneous rod [6] if o /o "1 or c"1, where c"r /r .
In addition to the cavity expansion due to the centrifugal forces exerted by the eroded rod elements, the momentum imparted to the target is responsible for further cavity growth. This momentum is created by #ow in the target around a projectile head moving at constant velocity ;. In order to estimate the additional cavity expansion due to target momentum, we consider the target response to an `equivalent roda * a rigid body of the same shape of the eroded core head and penetration velocity [8]. We can then use the lumped energy method [6]. Since the rigid body is moving through the target with Poncelet resistance, the force (energy per unit target length) can be written as follows: F"A#B;,
(13)
where A"(pa )R , B"1/2(pa )C o . (14) Here C is the drag coe$cient. In this equation, A is the target resistance force associated with plastic work and B is the drag force that comes about due to the convective momentum e!ect. This momentum is responsible for the further cavity expansion from a to "nal crater radius a . We apply the lumped capacity method.
? 1 (pa )C o ;" 2prR dr"p(a!a )R . 2 ? Then, the solution to the "nal crater radius is determined by a "a
1/4o ; . 1# R
(15)
(16)
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The drag coe$cient is assumed to be 0.5 for a hemispherical nose. This equation coincides with Eq. (3.3) that is given by Hill [8] but with a replacing the projectile radius. This is because there is no mushrooming e!ect for the rigid body penetration. The extent of the compensation due to target momentum increases with impact velocity since the ; term in Eq. (16) becomes more important in the high impact velocity limit. 3.3. Target resistance and penetration velocity The theoretical R value determined by the classical spherical cavity expansion model as presented by Satapathy and Bless [10] is used in the present work, which gives for RHA steel the value R "5 GPa. Note that using a variable target resistance does not make any di$culty at all. For steady-state penetration the modi"ed Bernoulli equation applies: 1/2o ( "1/2o ;#R , (17) where > is the dynamic yield strength of the rod including the two-dimensional e!ect [11]. Then ; is given by Tate [12], )/o , (18) ;" (R !> ) where a"(o /o . Note that this equation is valid for materials of di!erent densities. Eq. (17) has become the standard reference for modeling homogeneous rod penetration of semi-in"nite targets. It has also been observed that this penetration velocity is valid for the co-erosion penetration of the jacketed rod [2]. However, it seems to be necessary to examine this observation. This aspect will be further discussed in the numerical simulation section. 3.4. Criteria for the onset of `bi-erosiona The limit of the ratio of jacket to core diameter for di!erent material combinations is valuable data to resolve design concerns. Analytical descriptions of this complex onset condition require some assumptions which may impose limits on the accuracy, but which also enforce a certain type of solution. The geometry of a jacketed rod at the onset condition of `bi-erosiona is displayed in Fig. 4. The key assumption made in this analysis is that bi-erosion occurs if the eroded core element is not able to #ow outside of the jacket material. In other words, if the diameter of the jacket is larger than the diameter of the mushroom formed by the eroded core rod, two stagnation points must be observed. Hence, determination of the extent of the eroded core #ow is the main subject. Again, the pressure resistance exerted by the target material is assumed to act normal to the centerline of the core material, and the eroded elements have the same centrifugal acceleration. These observations allow us to determine the trajectory of the eroded core element at the onset of bi-erosion. In this con"guration, one centrifugal force by the core plus pressure force (note that this is not a centrifugal force since it does not #ow) by the surrounding jacket must be balanced by the pressure force exerted by the target material. The case where the eroded core element is outside of the stagnation region is "rst considered. Using the local radius of curvature of the centerline curve
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Fig. 4. Geometry for the onset of bi-erosion.
and thickness of the eroded core element, the governing equation can be given by
dr \ 1 r 1 . R # o ; sin b" > # o (
2
2 db 2r Simplifying yields S sin b 2r dr "!2r db, N#(¹!M)sin b
(19)
(20)
where S and ¹ are de"ned by Eqs. (8a) and (8b) and N and M are de"ned by Eqs. (12a) and (12b). The cavity radius formed in the target material due to the mushrooming of the core can be determined by the integration of Eq. (20):
a " r #2r
p S sin b db. N#(¹!M)sin b
(21)
The previous procedure can also be applied to determine the inside stagnation region solution for r . Finally, by considering the momentum in the target the "nal cavity radius (a ) is obtained by using Eq. (16), while Eq. (21) is used to determine a . The onset condition of bi-erosion is de"ned as a /r "1. It is useful to examine the onset condition with impact velocity for di!erent jacket materials (di!erent properties). Fig. 5 shows the critical value of (r /r ) for the onset condition for two
di!erent jacket materials, one for an aluminum and the other for a titanium jacket. In both cases, the core rod material is a tungsten alloy. It is shown that the critical values for both materials increase with impact velocity. Each jacket material has a di!erent onset condition depending on the material properties, especially density and yield strength, such that the titanium jacket shows larger critical values than aluminum. The parameters used in the theoretical analysis are listed in Table 1.
4. Numerical simulations Numerical simulations of homogeneous and jacketed rods into in"nite homogeneous RHA steel targets were conducted using the AUTODYN "nite-di!erence Eulerian code. The objective of the simulation was to examine the penetration velocity of the steel-jacketed tungsten projectiles, since in the theory the penetration velocity for co-erosion case is assumed to be equal to that of
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Fig. 5. Onset condition of `bi-erosiona for jacketed rods.
Table 1 Material parameters Parameter
Projectile Core tungsten
Density (g/cm) Yield strength (GPa) Shear modulus (GPa)
17.4 2 145
Target Jacket
RHA
Aluminum
Titanium
2.77 0.4 27
4.57 2.0 55
7.86 1.2 80
a reference homogeneous rod. For the materials considered in the study, a constant shear modulus G and von-Mises yield surface with yield strength > is used. The Mie}Gruniesen equation of state (EOS) was used for each material. Two-dimensional axisymmetry calculations were conducted. The mesh used a regular cell size of 0.1;0.1 mm/cell for the rods, resulting in 150 cells across the length and 10 cells across the rod radius (150;10 mm). This mesh continues for three radii away from the rod. The total number of meshes is 450;110 for the computational domain of 450 mm;150 mm. For the jacketed rod, another 5 cells are "lled across the jacket. In this case the aspect ratio (¸/D) is reduced to 5, while that of core is not changed. In another case, 7 cells for the core and 3 cells for the jacket, is also considered. The dimensions of the homogeneous and jacketed rods simulated are summarized in Table 2. They are also shown in Fig. 6. In this study, the length of the projectile is held constant at
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Table 2 Simulation matrix showing projectile parameters Case
Length (mm)
Core (cells)
¸/D
Jacket (cells)
¸/D
1 2 3 4
15 15 15 15
10 10 7 7
7.5 7.5 10.7 10.7
0 5 3 0
N/A 5.0 7.5 N/A
(homog.) ( jacketed) ( jacketed) (homog.)
Fig. 6. Geometry of simulated homogeneous and jacketed rods.
150 mm. The thickness of the jacket is selected to ensure the co-erosion. The impact velocity considered is 1.8 km/s. Fig. 7 shows one of the co-erosion penetration processes of the jacketed rod, as shown in Fig. 2. Fig. 8 shows the depth formed in target during the penetration. The depth is calculated at the interface between the tungsten core and target. Regardless of the impact velocities considered, the results indicate that the jacketed rod does not detract from the rod's penetrability as long as the aspect ratio of the core remains constant (cases 1 and 2; cases 3 and 4). This matches the previously observed result for the co-erosion penetration velocity of the jacketed rod [4], and also con"rms the assumption made in the current model. For the identical aspect ratio (cases 1 and 3), while the aspect ratio of the core is reduced from 7.5 to 5, the change in penetration e$ciency (P/¸) is approximately 10%. This supports the well-known result that penetration e$ciency is greater for small ¸/D projectiles than for large ¸/D projectiles [13,14]. As shown in the "gure, we observe the longer steady-state phase for large value of ¸/D.
5. Comparison of theory and experiments The results of the ratio of crater diameter to rod outer diameter for homogeneous and jacketed rods are summarized in Fig. 9. It is shown that the results obtained from the present theory match Cullis and Lynch's experimental results [4], which is the only data available now, quite well in the velocity range of the test for a homogeneous rod and a jacketed rod of same mass and outer
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Fig. 7. Simulated co-erosion penetration of a jacketed rod.
Fig. 8. Penetration depth versus time after impact for unitary and jacketed rods.
diameter. RHA was used as the target material. In this test, a 10.6 mm diameter by 45.6 mm long tungsten alloy projectile (¸/r "8.8) was used to obtain the performance of a homogeneous rod. By employing a steel jacket (EN24) the tungsten core is allowed to increase to 73.5 mm (¸/r "13.8),
while maintaining the same diameter. Thus the ratio of r /r is 1.767, which results in co-erosion.
As shown in Fig. 9, the jacketed projectile does not maintain the crater diameter of the homogeneous rod, such that the jacketed rods become much more sensitive to yawed penetration [15]. This is due to the lower mass per unit length of the jacketed rod that co-erodes.
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Fig. 9. Ratio of crater diameter to homogeneous and jacketed rod diameter. Tungsten core, steel jacket and RHA target, experimental data from Ref. [4].
6. Conclusions An analytical model for the cratering process in a thick steel target by high-velocity penetration of jacketed rods has been developed. The model allows us to determine the onset condition of bi-erosion where the penetration e$ciency is signi"cantly diminished compared with co-erosion. An estimation of the amount of the mushrooming with respect to impact velocity is also a valuable contribution of the present analysis. Maximum penetration for an available kinetic energy can occur with a reduction of the mushrooming. It has also been demonstrated that the theoretical predictions for the cavity diameter are in good agreement with experimental data. Due to the low-density jacket, a smaller crater diameter is observed compared with the homogeneous rod of identical aspect ratio. In terms of the cavity size, the jacketed rod is more sensitive to yawed penetration. From a computational study for the co-erosion case, the penetration velocity of a jacketed rod is shown to be identical to that of a homogeneous rod. That is, the penetration performance of the jacketed rod does not detract from that of homogeneous rod as long as the aspect ratio of the core remains constant.
Acknowledgements This work was supported by Korea Research Foundation Grant (KRF-99-E00015).
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Appendix A. Stagnation region solution This appendix describes the derivation of the extent of the stagnation region, r . Rearranging Eq. (10b) yields S sin b db, r )r 2r dr "!2r 1#¹
(A.1)
S "!2r sin b db, r 'r , N#¹!M
(A.2)
where S and ¹ are de"ned in Eqs. (8a) and (8b) and N and M are de"ned in Eqs. (12a) and (12b). Integration of Eq. (A.1) from an initial condition r /2 to r yields 3(1#¹)!8S cos b" . (A.3) 8S Again, integration of Eq. (A.2) from r
to r yields
2S cos b . r "r 1! ¹#N!M
(A.4)
The stagnation region is "nally determined by combining Eqs. (A.3) and (A.4): r "r
8S!3(1#¹) 1# . 4(¹#N!M)
(A.5)
References [1] Sorensen BR, Kimsey KD, Zukas JA, Frank K. Numerical analysis and modeling of jacketed rod penetration. Int J Impact Engng 1999;22:71}91. [2] Orphal DL, Miller CW, McKay WL, Borden WF, Larson SA, Kennedy CM. Hypervelocity impact and penetration of concrete by high ¸/D projectiles. Sixth International Symposium on Ballistics, 1982. [3] Orphal DL, Miller CW. Hypervelocity impact of high L/D penetrators. 1986 Hypervelocity Impact Symposium, DARPA-TIO-87-62, 1987. [4] Cullis IG, Lynch NJ. Hydrocode and experimental analysis of scale size jacketed KE projectiles. 14th International Symposium on Ballistics; Vol. TB-7, Quebec, Canada, 1994. p. 271}80. [5] Lehr HF, Wollman E, Koerber G. Experiments with jacketed rods of high "neness ratio. Int J Impact Engng 1995;17:517}26. [6] Lee M, Bless JS. Cavity models for solid and hollow projectiles. Int J Impact Engng 1998;21:881}94. [7] Kivity Y, Hirsch E. Penetration cuto! velocity for ideal jets. 10th International Symposium on Ballistics, San Diego, Ca, 1987. [8] Hill R. Cavitation and the in#uence of headshape in attack of thick targets by non-deforming projectiles. J Mech Phys Solids 1980;28:249}63. [9] Miller CW. Two-dimensional engineering model of jet penetration. 15th International Symposium on Ballistics, Israel, 1995. p. 8}21. [10] Satapathy S, Bless JS. Quasi-static penetration tests of PMMA: analysis of strength and crack morphology. Mech Mater 1996;23:323.
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[11] Tate A. Long rod penetration models * Part II. Extensions to the hydrodynamic theory of penetration. Int J Engng Sci 1986;8:599}612. [12] Tate A. Further results in the theory of long rod penetration. J Mech Phys Solids 1969;17:141}50. [13] Anderson CE, Walker JD, Bless SJ, Partom Y. On the ¸/D e!ect for long-rod penetrators. Int J Impact Engng 1996;18:247}64. [14] Hohler V, Strip AJ. Hypervelocity impact of rod projectiles with ¸/D from 1 to 32. Int J Impact Engng 1987;5:323}31. [15] Lee M. An engineering impact model for yawed projectiles. Int J Impact Engng 2000;24:797}807.
Analysis of jacketed rod penetration夽 M. Lee* School of Mechanical & Aerospace Engineering, Sejong University, 98 Kwangjin-Gu, Kunja-Dong, Seoul, 143-747 South Korea Received 17 November 1999; received in revised form 18 February 2000
Abstract An analytical model for the cratering process in a metal target due to high-velocity penetration of a jacketed rod has been developed. The jacketed rod consists of a central high-density core surrounded by a low-density jacket. The analysis is based on the observation that two mechanisms are involved in the cavity growth by eroding rods: #ow of rod erosion products, which exerts radial stress on the target and opens a cavity, and radial momentum of the target as it #ows around the rod front nose. Hence, the mushrooming e!ects on the cavity growth are isolated. The model includes the centrifugal force exerted by the core and jacket, radial momentum of the target, and the strength of the target. A criterion for the onset of `bi-erosiona, where there exist two stagnation points such that the performance is di!erent signi"cantly from the homogeneous rods, is obtained from the model. Predicted results for the "nal cavity diameter are found to match well with the corresponding test values. Further numerical simulations are conducted and indicate that as long as `co-erosiona occurs the penetration performance of jacketed rods is governed by the aspect ratio not of the jacketed rod but of the high-density core rod. 2000 Elsevier Science Ltd. All rights reserved. Keywords: Cavity diameter; Jacketed rod; Mushrooming; Bi-erosion; Co-erosion
1. Introduction Recent trends in kinetic energy projectiles are toward higher aspect ratio (¸/D) and increased velocity [1], where ¸ is the length and D is the diameter of the projectile. For a given weapon system, however, the stability problem during acceleration and free-#ight phase becomes serious 夽
This article is an expanded and revised version of a paper that was "rst presented at the ASME Pressure Vessels and Piping Conference, 27}31 July 1997, Orlando, USA. * Tel.: 82-2-3408-3282; fax: 82-2-3408-3333. E-mail address: [email protected] (M. Lee). 0734-743X/00/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 7 3 4 - 7 4 3 X ( 0 0 ) 0 0 0 2 0 - 8
892
M. Lee / International Journal of Impact Engineering 24 (2000) 891}905
for cylinders of extreme aspect ratio. The sti!ness in attack of thick targets is also one of the main concerns. To overcome these problems, a jacketed projectile that augments the sti!ness is usually suggested. The jacketed rod consists of a central high-density rod surrounded by a low-density jacket. Since two di!erent projectile materials are interacting with target material at the interface, two kinds of di!erent penetration processes are usually observed. If the outer diameter of the jacket is less than the diameter of the mushroom formed by the eroding core rod, it is known that a jacketed rod only penetrates as well as the core [2]. This condition results in `co-erosiona. If the relative thickness of the jacket exceeds a certain limit value, penetration performance is degraded because of the independent interaction of both core and jacket materials with the target. In this case, two di!erent stagnation points are observed. The jacket is decelerated much faster than the core since the density of jacket is lower than the core. This geometry is denoted `bi-erosiona [3]. Recently, a series of numerical simulations [4] and experiments [5] have been performed to examine the ballistic performance of the jacketed rod. Although the penetration performance of the jacketed rod is not diminished for co-erosion case, some reduction in the crater diameter compared with a homogeneous rod is measured at the identical aspect ratio [4]. This means that the jacketed rod is more sensitive to yawed penetration. One way to analyze a detailed high-velocity penetration process is to use computer predictions. However, this approach is computationally expensive. By using a simple analytical model, however, signi"cant parameters and how they vary over a wide velocity range can be obtained. Up to now, there are no detailed analytical models for the penetration mechanics of the jacketed rod and the critical value of D /D for the onset condition of `bi-erosiona, where D and D are the
diameter of the jacket and core materials. Hence, in this paper the previous model (two-stage cavity expansion model) for a homogeneous rod [6] is applied to the case of the jacketed rod. We "rst select a value of D /D less than the critical value, yielding `co-erosiona, and analytically
determine the cavity growth formed in a metal target. In addition, due to the two-step procedure in the theory the onset condition of the `bi-erosiona is obtained. For design concerns, these results are valuable.
2. Mushrooming and cavitation The radial motion of target material due to the penetration of eroding rods occurs in two stages as shown in the numerical simulation [7]. In the "rst stage, eroded rod elements play the dominant role in opening a cavity. In the second stage, the momentum in the target is responsible for further cavity expansion (cavitation e!ect) until the strength of the target forces it to come to rest. The term `cavitationa is borrowed from Hill's work [8] that deals with the attack of thick targets by non-deforming projectiles, where cavity growth is only due to the cavitation e!ect. When the projectile deforms, the entire physics is di!erent in kind. Actually, cavitation severely degrades the performance of anti-tank weapons because kinetic energy is absorbed by the enhanced cavity production at high velocity. Otherwise, it would go toward penetration. Simulations described here address the in#uence of the target momentum on the cavity growth (cavitation). In order to eliminate the e!ect of mushrooming on cavity production, hypothetical simulations, which is actually the same assumption made by Hill [8], were conducted using the
M. Lee / International Journal of Impact Engineering 24 (2000) 891}905
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Fig. 1. The e!ect of cavitation on cavity growth.
two-dimensional Eulerian wave propagation computer code, AUTODYN. A rigid long rod of a hemispherical nose impacts RHA steel target at 1.3 and 2.5 km/s. Fig. 1 shows the computed velocity "eld and cavity shape. At 1.3 km/s, the projectile essentially creates a cavity almost equal to its diameter. However, as the impact velocity increases there is some point at which the #ow separates from the projectile head, thus showing signi"cant cavity overshoot due to the target momentum. The target momentum has a great in#uence on the cavity growth, as does the mushrooming of the projectile. The analysis of cavity growth by a rigid projectile travelling through an in"nite medium at constant velocity was developed by Hill [8].
3. Cavity growth model 3.1. Assumptions It is assumed that the "nal cavity growth is equal to the sum of the cavity growth produced in each stage acting independently, regardless of the order of application of loads [6]. In other words,
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we use the principle of superposition for the total work done in each stage. The analysis includes the centrifugal force exerted by the projectile, radial momentum of the target, and the strength of the target. The following assumptions are made, many of which are motivated by the results of numerical simulations and experiments. (1) The #ow of eroded elements is considered as incompressible, steady, and inviscid relative to the stagnation region as shown in Fig. 2. r and r are the radius of the jacket and core material,
respectively. The eroded elements from each material arrive and exit with a velocity of
Fig. 2. Geometry of a jacketed rod penetration.
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Fig. 3. Force balance for the penetration of a jacketed rod.
of the stagnation region. From mass conservation in a coordinate system located at the penetration front, the mass of the eroded element is given by 2pr o q dl, (1a) 2pr o q dl. (1b)
Subscripts j and c denote jacket and core materials, respectively. r is the radius of the centerline of the eroded element, q is the thickness and dl is its arc length. The centrifugal acceleration of the #ow of each eroded element is given by (
where R(b) is the local radius of curvature of the centerline curve, and < is the impact velocity. From the de"nition of radius of curvature, 1 dr , R(b )"! (3a) sin b db 1 dr
. R(b )"! (3b)
sin b db
Each thickness of the eroded element is determined from Pappus's Theorem r q (b )" , (4a) 2r (b ) r (4b) q (b )" .
2r (b )
In this con"guration two centrifugal forces, one for the core and one for the surrounding jacket, must be balanced by the pressure resistance exerted by the target:
1 (
R(b )
(5)
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where R is the target resistance for radial cavity expansion and subscript t denotes the target material. The pressure resistance is acting normal to the #ow at radius r . If the eroded elements #ow together with a velocity of
2r db 2r From now on, we just use b instead of b . By introducing two constants indicating the ratios of dynamic to static pressures, Eq. (6) can then be written as
o o 2r dr "!2r # 1!
o o where
r S sin b db, r 1#¹ sin b
(7)
1/2o (
o L S sin b o db. a " r #2r # 1! r
o 1#¹ sin b o (9) This equation can be calculated by a numerical integration method. The strength of target is included in the model; otherwise Eq. (9) is not limited. To determine the initial condition r , now consider the case where the eroded element is inside the stagnation region in which resisting pressure is equal to 1/2o ;. We still apply the previous force balance argument applied to the outside stagnation region. In order to "nd the balancing forces exerted by the jacketed rod, it is necessary to consider two separate regions. For r 4r , there exists only one centrifugal force exerted by core element. For r 'r , however, there exist one centrifugal force exerted by the core and one hydrostatic force by the jacket. The jacket material is assumed to remain inside its stagnation region, such that the hydrostatic pressure is 1/2o ( , where, > is the dynamic yield strength of the jacket material. The
resistance by the target and the force by the jacket material are acting normal to the #ow at radius r . Again, Eq. (6) can be written by employing the radius of curvature and the thickness
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of eroded element,
r 1 dr \ R # o ;"! o ( ! o (
2 db 2r This equation is integrated for the interval from r (b"n)"r /2 to r [9]. After some algebraic manipulations (see the appendix), we "nd the radius of the stagnation region:
r "r
8S!3(1#¹) 1# , 4(N#¹!M)
(11)
where (R !> )
, N" (12a) R 1/2o (
. (12b) M" R Eqs. (9) and (11) give the size of cavity growth which is only due to the mushrooming e!ect. At this point, it is important to address that the expression for a jacketed rod becomes that of a homogeneous rod [6] if o /o "1 or c"1, where c"r /r .
In addition to the cavity expansion due to the centrifugal forces exerted by the eroded rod elements, the momentum imparted to the target is responsible for further cavity growth. This momentum is created by #ow in the target around a projectile head moving at constant velocity ;. In order to estimate the additional cavity expansion due to target momentum, we consider the target response to an `equivalent roda * a rigid body of the same shape of the eroded core head and penetration velocity [8]. We can then use the lumped energy method [6]. Since the rigid body is moving through the target with Poncelet resistance, the force (energy per unit target length) can be written as follows: F"A#B;,
(13)
where A"(pa )R , B"1/2(pa )C o . (14) Here C is the drag coe$cient. In this equation, A is the target resistance force associated with plastic work and B is the drag force that comes about due to the convective momentum e!ect. This momentum is responsible for the further cavity expansion from a to "nal crater radius a . We apply the lumped capacity method.
? 1 (pa )C o ;" 2prR dr"p(a!a )R . 2 ? Then, the solution to the "nal crater radius is determined by a "a
1/4o ; . 1# R
(15)
(16)
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The drag coe$cient is assumed to be 0.5 for a hemispherical nose. This equation coincides with Eq. (3.3) that is given by Hill [8] but with a replacing the projectile radius. This is because there is no mushrooming e!ect for the rigid body penetration. The extent of the compensation due to target momentum increases with impact velocity since the ; term in Eq. (16) becomes more important in the high impact velocity limit. 3.3. Target resistance and penetration velocity The theoretical R value determined by the classical spherical cavity expansion model as presented by Satapathy and Bless [10] is used in the present work, which gives for RHA steel the value R "5 GPa. Note that using a variable target resistance does not make any di$culty at all. For steady-state penetration the modi"ed Bernoulli equation applies: 1/2o ( "1/2o ;#R , (17) where > is the dynamic yield strength of the rod including the two-dimensional e!ect [11]. Then ; is given by Tate [12], )/o , (18) ;" (R !> ) where a"(o /o . Note that this equation is valid for materials of di!erent densities. Eq. (17) has become the standard reference for modeling homogeneous rod penetration of semi-in"nite targets. It has also been observed that this penetration velocity is valid for the co-erosion penetration of the jacketed rod [2]. However, it seems to be necessary to examine this observation. This aspect will be further discussed in the numerical simulation section. 3.4. Criteria for the onset of `bi-erosiona The limit of the ratio of jacket to core diameter for di!erent material combinations is valuable data to resolve design concerns. Analytical descriptions of this complex onset condition require some assumptions which may impose limits on the accuracy, but which also enforce a certain type of solution. The geometry of a jacketed rod at the onset condition of `bi-erosiona is displayed in Fig. 4. The key assumption made in this analysis is that bi-erosion occurs if the eroded core element is not able to #ow outside of the jacket material. In other words, if the diameter of the jacket is larger than the diameter of the mushroom formed by the eroded core rod, two stagnation points must be observed. Hence, determination of the extent of the eroded core #ow is the main subject. Again, the pressure resistance exerted by the target material is assumed to act normal to the centerline of the core material, and the eroded elements have the same centrifugal acceleration. These observations allow us to determine the trajectory of the eroded core element at the onset of bi-erosion. In this con"guration, one centrifugal force by the core plus pressure force (note that this is not a centrifugal force since it does not #ow) by the surrounding jacket must be balanced by the pressure force exerted by the target material. The case where the eroded core element is outside of the stagnation region is "rst considered. Using the local radius of curvature of the centerline curve
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Fig. 4. Geometry for the onset of bi-erosion.
and thickness of the eroded core element, the governing equation can be given by
dr \ 1 r 1 . R # o ; sin b" > # o (
2
2 db 2r Simplifying yields S sin b 2r dr "!2r db, N#(¹!M)sin b
(19)
(20)
where S and ¹ are de"ned by Eqs. (8a) and (8b) and N and M are de"ned by Eqs. (12a) and (12b). The cavity radius formed in the target material due to the mushrooming of the core can be determined by the integration of Eq. (20):
a " r #2r
p S sin b db. N#(¹!M)sin b
(21)
The previous procedure can also be applied to determine the inside stagnation region solution for r . Finally, by considering the momentum in the target the "nal cavity radius (a ) is obtained by using Eq. (16), while Eq. (21) is used to determine a . The onset condition of bi-erosion is de"ned as a /r "1. It is useful to examine the onset condition with impact velocity for di!erent jacket materials (di!erent properties). Fig. 5 shows the critical value of (r /r ) for the onset condition for two
di!erent jacket materials, one for an aluminum and the other for a titanium jacket. In both cases, the core rod material is a tungsten alloy. It is shown that the critical values for both materials increase with impact velocity. Each jacket material has a di!erent onset condition depending on the material properties, especially density and yield strength, such that the titanium jacket shows larger critical values than aluminum. The parameters used in the theoretical analysis are listed in Table 1.
4. Numerical simulations Numerical simulations of homogeneous and jacketed rods into in"nite homogeneous RHA steel targets were conducted using the AUTODYN "nite-di!erence Eulerian code. The objective of the simulation was to examine the penetration velocity of the steel-jacketed tungsten projectiles, since in the theory the penetration velocity for co-erosion case is assumed to be equal to that of
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Fig. 5. Onset condition of `bi-erosiona for jacketed rods.
Table 1 Material parameters Parameter
Projectile Core tungsten
Density (g/cm) Yield strength (GPa) Shear modulus (GPa)
17.4 2 145
Target Jacket
RHA
Aluminum
Titanium
2.77 0.4 27
4.57 2.0 55
7.86 1.2 80
a reference homogeneous rod. For the materials considered in the study, a constant shear modulus G and von-Mises yield surface with yield strength > is used. The Mie}Gruniesen equation of state (EOS) was used for each material. Two-dimensional axisymmetry calculations were conducted. The mesh used a regular cell size of 0.1;0.1 mm/cell for the rods, resulting in 150 cells across the length and 10 cells across the rod radius (150;10 mm). This mesh continues for three radii away from the rod. The total number of meshes is 450;110 for the computational domain of 450 mm;150 mm. For the jacketed rod, another 5 cells are "lled across the jacket. In this case the aspect ratio (¸/D) is reduced to 5, while that of core is not changed. In another case, 7 cells for the core and 3 cells for the jacket, is also considered. The dimensions of the homogeneous and jacketed rods simulated are summarized in Table 2. They are also shown in Fig. 6. In this study, the length of the projectile is held constant at
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Table 2 Simulation matrix showing projectile parameters Case
Length (mm)
Core (cells)
¸/D
Jacket (cells)
¸/D
1 2 3 4
15 15 15 15
10 10 7 7
7.5 7.5 10.7 10.7
0 5 3 0
N/A 5.0 7.5 N/A
(homog.) ( jacketed) ( jacketed) (homog.)
Fig. 6. Geometry of simulated homogeneous and jacketed rods.
150 mm. The thickness of the jacket is selected to ensure the co-erosion. The impact velocity considered is 1.8 km/s. Fig. 7 shows one of the co-erosion penetration processes of the jacketed rod, as shown in Fig. 2. Fig. 8 shows the depth formed in target during the penetration. The depth is calculated at the interface between the tungsten core and target. Regardless of the impact velocities considered, the results indicate that the jacketed rod does not detract from the rod's penetrability as long as the aspect ratio of the core remains constant (cases 1 and 2; cases 3 and 4). This matches the previously observed result for the co-erosion penetration velocity of the jacketed rod [4], and also con"rms the assumption made in the current model. For the identical aspect ratio (cases 1 and 3), while the aspect ratio of the core is reduced from 7.5 to 5, the change in penetration e$ciency (P/¸) is approximately 10%. This supports the well-known result that penetration e$ciency is greater for small ¸/D projectiles than for large ¸/D projectiles [13,14]. As shown in the "gure, we observe the longer steady-state phase for large value of ¸/D.
5. Comparison of theory and experiments The results of the ratio of crater diameter to rod outer diameter for homogeneous and jacketed rods are summarized in Fig. 9. It is shown that the results obtained from the present theory match Cullis and Lynch's experimental results [4], which is the only data available now, quite well in the velocity range of the test for a homogeneous rod and a jacketed rod of same mass and outer
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Fig. 7. Simulated co-erosion penetration of a jacketed rod.
Fig. 8. Penetration depth versus time after impact for unitary and jacketed rods.
diameter. RHA was used as the target material. In this test, a 10.6 mm diameter by 45.6 mm long tungsten alloy projectile (¸/r "8.8) was used to obtain the performance of a homogeneous rod. By employing a steel jacket (EN24) the tungsten core is allowed to increase to 73.5 mm (¸/r "13.8),
while maintaining the same diameter. Thus the ratio of r /r is 1.767, which results in co-erosion.
As shown in Fig. 9, the jacketed projectile does not maintain the crater diameter of the homogeneous rod, such that the jacketed rods become much more sensitive to yawed penetration [15]. This is due to the lower mass per unit length of the jacketed rod that co-erodes.
M. Lee / International Journal of Impact Engineering 24 (2000) 891}905
903
Fig. 9. Ratio of crater diameter to homogeneous and jacketed rod diameter. Tungsten core, steel jacket and RHA target, experimental data from Ref. [4].
6. Conclusions An analytical model for the cratering process in a thick steel target by high-velocity penetration of jacketed rods has been developed. The model allows us to determine the onset condition of bi-erosion where the penetration e$ciency is signi"cantly diminished compared with co-erosion. An estimation of the amount of the mushrooming with respect to impact velocity is also a valuable contribution of the present analysis. Maximum penetration for an available kinetic energy can occur with a reduction of the mushrooming. It has also been demonstrated that the theoretical predictions for the cavity diameter are in good agreement with experimental data. Due to the low-density jacket, a smaller crater diameter is observed compared with the homogeneous rod of identical aspect ratio. In terms of the cavity size, the jacketed rod is more sensitive to yawed penetration. From a computational study for the co-erosion case, the penetration velocity of a jacketed rod is shown to be identical to that of a homogeneous rod. That is, the penetration performance of the jacketed rod does not detract from that of homogeneous rod as long as the aspect ratio of the core remains constant.
Acknowledgements This work was supported by Korea Research Foundation Grant (KRF-99-E00015).
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Appendix A. Stagnation region solution This appendix describes the derivation of the extent of the stagnation region, r . Rearranging Eq. (10b) yields S sin b db, r )r 2r dr "!2r 1#¹
(A.1)
S "!2r sin b db, r 'r , N#¹!M
(A.2)
where S and ¹ are de"ned in Eqs. (8a) and (8b) and N and M are de"ned in Eqs. (12a) and (12b). Integration of Eq. (A.1) from an initial condition r /2 to r yields 3(1#¹)!8S cos b" . (A.3) 8S Again, integration of Eq. (A.2) from r
to r yields
2S cos b . r "r 1! ¹#N!M
(A.4)
The stagnation region is "nally determined by combining Eqs. (A.3) and (A.4): r "r
8S!3(1#¹) 1# . 4(¹#N!M)
(A.5)
References [1] Sorensen BR, Kimsey KD, Zukas JA, Frank K. Numerical analysis and modeling of jacketed rod penetration. Int J Impact Engng 1999;22:71}91. [2] Orphal DL, Miller CW, McKay WL, Borden WF, Larson SA, Kennedy CM. Hypervelocity impact and penetration of concrete by high ¸/D projectiles. Sixth International Symposium on Ballistics, 1982. [3] Orphal DL, Miller CW. Hypervelocity impact of high L/D penetrators. 1986 Hypervelocity Impact Symposium, DARPA-TIO-87-62, 1987. [4] Cullis IG, Lynch NJ. Hydrocode and experimental analysis of scale size jacketed KE projectiles. 14th International Symposium on Ballistics; Vol. TB-7, Quebec, Canada, 1994. p. 271}80. [5] Lehr HF, Wollman E, Koerber G. Experiments with jacketed rods of high "neness ratio. Int J Impact Engng 1995;17:517}26. [6] Lee M, Bless JS. Cavity models for solid and hollow projectiles. Int J Impact Engng 1998;21:881}94. [7] Kivity Y, Hirsch E. Penetration cuto! velocity for ideal jets. 10th International Symposium on Ballistics, San Diego, Ca, 1987. [8] Hill R. Cavitation and the in#uence of headshape in attack of thick targets by non-deforming projectiles. J Mech Phys Solids 1980;28:249}63. [9] Miller CW. Two-dimensional engineering model of jet penetration. 15th International Symposium on Ballistics, Israel, 1995. p. 8}21. [10] Satapathy S, Bless JS. Quasi-static penetration tests of PMMA: analysis of strength and crack morphology. Mech Mater 1996;23:323.
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[11] Tate A. Long rod penetration models * Part II. Extensions to the hydrodynamic theory of penetration. Int J Engng Sci 1986;8:599}612. [12] Tate A. Further results in the theory of long rod penetration. J Mech Phys Solids 1969;17:141}50. [13] Anderson CE, Walker JD, Bless SJ, Partom Y. On the ¸/D e!ect for long-rod penetrators. Int J Impact Engng 1996;18:247}64. [14] Hohler V, Strip AJ. Hypervelocity impact of rod projectiles with ¸/D from 1 to 32. Int J Impact Engng 1987;5:323}31. [15] Lee M. An engineering impact model for yawed projectiles. Int J Impact Engng 2000;24:797}807.