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Theoretical considerations on the penetration of powdered metal jets
ARTICLE IN PRESS
International Journal of Impact Engineering 33 (2006) 316–325 www.elsevier.com/locate/ijimpeng
Theoretical considerations on the penetration of powdered metal jets Brenden Grove Schlumberger Reservoir Completions Center, 14910 Airline Road, Rosharon, TX 77583, USA Available online 13 November 2006
Abstract This paper explores some of the theoretical issues encountered when interpreting the penetration behavior of an oilwell perforating charge, whose jet forms from an unsintered powdered metal (PM) liner. Appropriate treatments of the jet’s porous compressible nature fill the gap between classical ‘‘continuous’’ and ‘‘fully particulated’’ jet penetration models. Within certain constraints, increasing a penetrator’s length (even if by distension) increases its hydrodynamic penetration depth, while reducing its impact pressure; and a porous penetrator penetrates deeper than a non-porous penetrator of the same density, length, and velocity. Dynamic target pressure considerations lead to the conclusion that highly distended, low-velocity, PM jets should penetrate moderate-strength geologic targets effectively. After demonstrating that initial transient shock pressures may be much higher than steady-state penetration pressures, we suggest that initial penetration rates may be higher than the steady-state rates. This, in conjunction with the well-known ‘‘residual penetration’’ phenomenon, indicates that a non-continuous jet’s penetration may be strongly influenced by transient effects. r 2006 Elsevier Ltd. All rights reserved. Keywords: Shaped charge; Perforate; Porous jet; Impact pressure
1. Introduction This paper explores the theoretical issues encountered when interpreting the penetration of a perforating charge whose jet forms from an unsintered powdered metal (PM) liner. Penetration model improvements are presented which take into account the jet’s porous compressible nature. Impact pressure and the role of transients will also be discussed. 2. Background Oilwell perforators are small-caliber (20–60 mm) shaped charges used to create holes into a hydrocarbonbearing subterranean reservoir, and the steel casing which lines a borehole connecting the reservoir to surface production facilities. These perforation tunnels, therefore, become part of the ultimate hydrocarbon production flowpath. Tel.: +1 281 285 5225; fax: +1 281 285 5453.
E-mail address: [email protected]. 0734-743X/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijimpeng.2006.09.034
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Since this flowpath must be ‘‘clean’’, it is desirable to employ liners which do not form competent slugs or large solid debris which may plug perforation tunnels. To this end, typical oilwell perforator liners are pressed to net shape from PM. Contrasted with conventional PM parts, oilwell perforator liners remain ‘‘green’’—i.e. they are not sintered to increase strength. Modest green strength results from particle interlocking and binderinduced adhesion. In a functioning charge, the liner collapses and jets in the usual way, but the formed jet exhibits no tensile strength; it is generally comprised of the constituent powder. Therefore, in the absence of any inward radial velocity, a stretching jet will distend to very low macroscopic densities; increased jet length results from lengthening gaps between powder particles. This calls into question any analysis based on solid, incompressible theory. 3. Jet penetration theory Standard penetration theory is first reviewed, as a foundation on which new concepts applicable to finely powdered jets are developed. Throughout this section, the terms ‘‘jet’’ and ‘‘penetrator’’ will be used interchangeably, referring conceptually to either a rod-type (constant velocity) penetrator, or an incremental element of what may be a stretching jet. Unless otherwise noted, none of the concepts discussed herein explicitly address the time-varying length of a real stretching jet. 3.1. Incompressible, hydrodynamic Hydrodynamic incompressible steady penetration theory relates penetration depth (PD) per unit jet length (L) to the square root of the jet-to-target density ratio [1]. A slight modification of this theory, motivated by momentum considerations [1,2] addresses particulated jets: sffiffiffiffiffiffiffi lrj PD ¼ , (1) L rt where 1plp2, depending on the extent of particulation, rj the average jet density including the gaps, and rt the target density. While allowing one to ‘‘bound’’ the solution by examining the limits, this analysis does not provide an obvious way of describing the extent of particulation (i.e., computing intermediate values of l). Also, it is noted that a fully particulated jet (l ¼ 2) is one in which the inter-particle gap length is much larger than the particle length, so that rj approaches zero. 3.2. Incompressible, target strength important Extending this hydrodynamic treatment to include target strength effects, the simplest models [1] add some characteristic ‘‘static’’ strength term to the dynamic pressure, resulting in the following PD per unit penetrator length: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi lrj PD 2s ¼ , (2) L rt rt ðV UÞ2 where V is jet velocity, U the penetration velocity, and s ¼ Y t Y j the net strength effect. Yt is generally related to some characteristic target material strength (tensile yield strength, for example), but the exact relationship has generally been empirical, and probably varies depending on material type (ductile metal vs. brittle ceramic, etc.). In reality, the operative resisting stresses are likely functions of strain, strain rate, and pressure. Contrasted with hydrodynamic penetration, Eq. (2) illustrates the following: – strength effects become more important as V decreases (approaching U), – jet strength and target strength tend to offset each other.
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3.3. Compressible (porous) jet The foregoing restatement of broadly recognized basics allows us to next address jet compressibility, which has received much less attention. Flis and Crilly [3] developed a simple model for an incompressible jet penetrating a compressible target; we reverse their analysis here. This model describes a two-step process: (1) shock compaction of the distended jet to its maximum pore-free density, (2) penetration of the compacted jet, according to incompressible Bernoulli. Referring to Fig. 1, the total pressure in the target is 1 P2t ¼ Y t þ r0t U 2 , (3) 2 where Y t ¼ P0t , and r0t ¼ r2t ¼ rt . Next, we approximate the jet material’s compressibility as shown in Fig. 1, assume that the jet shock compresses just prior to impact (from state 0j to 1j), and apply the Rankine–Hugoniot shock jump conservation equations. Following Flis and Crilly [3] (leaving out the intermediate steps here for brevity), this gives the jet pressure: 1 P2j ¼ R þ r0 ðV UÞ2 ð1 þ fÞ, (4) 2 where r0 is the initial (distended) jet density, rs the solid (pore-free) jet density, f ¼ 1 r0 =rs the jet porosity, and R the jet compaction initial resistance (i.e., P1j). Equating P2t with P2j at the interface, penetration per unit jet length is given by sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi PD r0 2s ¼ ð1 þ fÞ . (5) L rt rt ðV UÞ2 For hydrodynamic penetration, this simplifies to rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi PD r0 ¼ ð1 þ fÞ. L rt
(6)
Noteworthy are the similarities between Eqs. (6) and (1), and (5) and (2), respectively; the only difference being (1+f) replaces (l). Eqs. (5) and (6), therefore, give the expected behavior at the limits—i.e. reproducing the previously shown ‘‘continuous’’ jet penetration for zero porosity, and the ‘‘fully particulated’’ jet penetration for 100% porosity. As mentioned before, however, jet density approaches zero as porosity approaches 100%.
Fig. 1. Compressible jet penetrating incompressible target [left]; Porous Jet Lockup Model [right] (following [3]).
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One useful implication is that, since f is known for a given r0, this analysis provides a method of determining l in the particulated jet model. Furthermore, this model indicates that hydrodynamic PD is greater for a porous jet than for a non-porous jet, given a length and initial density. 4. Jet length, density, and porosity: a summary Consider three hypothetical jets, each of identical mass, velocity, and cross-sectional area. Each varies in its density, length, and/or porosity as follows: Case A: rs, incompressible; length ¼ Ls
V
Case B: r0, compressible to rs, length ¼ L0
V
Case C: r0, incompressible, length ¼ L0
V
4.1. Hydrodynamic penetration depth The hydrodynamic PDs for each case are summarized as follows: rffiffiffiffiffi rs Case A PDA ¼ Ls . rt Case B
Case C
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r0 ð1 þ fÞ PDB ¼ L0 rt sffiffiffiffiffiffiffiffiffiffiffiffi 1þf ðalternative formÞ ¼ PDA 1f pffiffiffiffiffiffiffiffiffiffiffiffi ¼ PDC 1 þ f ðalternative formÞ. rffiffiffiffiffi r0 PDC ¼ L0 rt rffiffiffiffiffiffi L0 ¼ PDA Ls
ðalternative formÞ.
(7)
ð8Þ
ð9Þ
Note that PDB4PDC4PDA. Using the expression for f for case C (which exhibits no physical porosity, but for purposes of algebraic comparison), the PD ratios are sffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffi 1þf 1 : : 1. (10) PDB : PDC : PDA ¼ 1f 1f Eq. (10), represented graphically in Fig. 2, shows that increases in both length and porosity increase hydrodynamic PD; therefore, consideration of jet porosity is essential to an accurate PD calculation. 4.2. Dynamic jet pressure Considering the same three cases, the Bernoulli dynamic jet pressures are summarized as follows: Case A
1 PA ¼ rs ðV UÞ2 . 2
(11)
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5 Penetration Ratio
1.50
PDB:PDC
1.40 1.30 1.20 1.10
compressible low-density vs. fully-dense incompressible low-density vs. fully-dense
4 3 2 1
1.00 0
0.2
0.4 0.6 Porosity
0.8
1
0
0.2
0.4
0.6
0.8
1
1-ρ0/ρs
Fig. 2. Penetration ratios: compressible (case B) vs. incompressible long jet (case C) [left]; low-density long jets (cases B and C) vs. highdensity short jet (case A) [right].
Case B
1 r ðV UÞ2 ð1 f2 Þ 2 s 1 ¼ r0 ðV UÞ2 ð1 þ fÞ ðalternative formÞ. 2
PB ¼
1 PC ¼ r0 ðV UÞ2 . 2 U is not the same for each case, but varies with g: qffiffiffiffi 8 > gA ¼ rrt > > s > > ffi qffiffiffiffiffiffiffiffiffiffiffiffiffi rt gA V > qffiffiffiffi > > r gA > : gC ¼ r t ¼ pffiffiffiffiffiffiffi Case C
0
ð12Þ (13)
(14)
1f
Substitution of the appropriate forms of Eq. (14) into Eqs. (11)–(13) allows comparison among the impact pressures: qffiffiffiffiffiffiffiffiffiffiffiffiffiffi32 2 1 1= 1 þ gA = 1 f2 6 7 6 7 ð1 f2 Þ PA : PB : PC ¼ 1 : 4 5 1 1= 1 þ gA "
pffiffiffiffiffiffiffiffiffiffiffiffi#2 1 1= 1 þ gA = 1 f ð1 fÞ. : 1 1= 1 þ gA
ð15Þ
Fig. 3 shows the pressure ratios PB:PA, and PC:PA, for the illustrative case of gA ¼ 0:4. It can be shown that PA4PB4PC, for all gA and f. As with PD, accurate impact pressure calculation requires appropriate consideration of jet porosity. Finally, it is worth noting that case B yields the greatest hydrodynamic penetration, while case A produces the greatest impact pressure. 5. Impact pressure and target strength Consider some critical point of interest (Ucrit, Pcrit), which is observed late in penetration, when target strength effects are significant. Penetration velocity and jet density are expressed as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ðPcrit Y t Þ , (16) U crit ¼ rt
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Pressure Ratio
1 0.8 0.6 0.4
Compressible low-density vs. high-density
0.2
Incompressible low-density vs. high-density
0 0
0.2
0.4
0.6
0.8
1
1-ρ0 /ρs Fig. 3. Dynamic jet pressure ratios; ratio of cases C:A and B:A.
ðincompressible; 1plp2Þ rj ¼
ðcompressible; following½3Þ
rt U 2crit þ 2s , lðV U crit Þ2
rj ¼ rs ð1 fÞ;
(17) sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r U 2 þ 2s . f ¼ 1 t crit rs ðV U crit Þ2
(18)
When jet density can vary, therefore, Pcrit does not imply a fixed Vcrit. Rather, V varies inversely with rj on a Pcrit isobar. Fig. 4 shows such isobars (in {rj, V} space) for four different targets, given the values shown in Table 1. The compressible jet model approaches continuous jet behavior at high densities, and particulated jet behavior at low densities; both trends are as expected. For comparison, Fig. 4 also includes curves based on compressible Bernoulli theory, the details of which are omitted here. The good agreement between both treatments of jet compressibility makes either equally suitable for comparison against incompressible treatments. Fig. 5 shows the isobars for all four target materials of Table 1, for compressible jet impact. Mild steel exhibits a lower critical jet velocity than aluminum, indicating that more of a jet would be consumed in penetrating mild steel; while initially counterintuitive, this is consistent with experimental observations [4], and can be explained by the fact that the mild steel and aluminum considered have identical strengths, but the dynamic impact pressure in the steel is higher due to its higher density. Also of interest—and this hinges on the appropriateness of UCS (unconfined compressive strength) as a benchmark—even highly distended PM jets (density o2 g/cc) should penetrate concrete or sandstone effectively at velocities as low as 500 m/s. 6. Transients and L/D effects Two assumptions in the forgoing are (1) continuous penetration and (2) L/D independence. For very long rods and continuous jets, this is reasonable, since the initial transients and terminal effects are operative during very small portions of the overall penetration, and aspect ratio does not generally play a significant role. However, a highly distended jet, comprised of millions of equi-axed particles and large inter-particle spacing, calls into question these simplifying assumptions. Penetrator aspect ratio effects have been studied at some length [5–7]; short penetrators are observed to penetrate deeper, per unit length, than long penetrators; even exceeding the hydrodynamic limit as L/D-1+ at sufficiently high velocities. This is the rationale for the segmented rod penetrator concept [1]. The increased penetration effectiveness is likely due largely to residual or inertial penetration (‘‘afterflow’’) [1]; but may also be due in part to the initial transient pressures, which will now be demonstrated to exceed the steady-state Bernoulli pressures. Applying the Rankine–Hugoniot equation for momentum conservation across a shock for 1D impact (which, admittedly, does not accurately describe the particle impact in question), we have Px ¼ rx0 U xs U xp ,
(19)
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Mild Steel
Aluminum
18 incomp. particulated
14
comp. Bernoulli
12
comp. F&C
10 8 6 4 2
Jet Density rho (g/cc)
incomp. solid
16 Jet Density rho (g/cc)
18 16 14 12 10 8
incomp. solid
6
incomp. particulated
4
comp. Bernoulli
2
comp. F&C
0 2.0E+04 4.0E+04 6.0E+04 8.0E+04 1.0E+05 1.2E+05
0 2.0E+04 4.0E+04 6.0E+04 8.0E+04 1.0E+05 1.2E+05
Jet Velocity V (cm/s)
Jet Velocity V (cm/s)
Sandstone
Concrete 18
18 incomp. solid
14
comp. Bernoulli
12
incomp. solid
16
incomp. particulated comp. F&C
10 8 6 4
Jet Density rho (g/cc)
Jet Density rho (g/cc)
16
incomp. particulated
14
comp. Bernoulli
12
comp. F&C
10 8 6 4 2
2 0 2.0E+04 4.0E+04 6.0E+04 8.0E+04 1.0E+05 1.2E+05
0 2.0E+04 4.0E+04 6.0E+04 8.0E+04 1.0E+05 1.2E+05
Jet Velocity V (cm/s)
Jet Velocity V (cm/s)
Fig. 4. Jet density vs. velocity on Pcrit isobars (various targets). Table 1 Target parameters for jet density-vs.-velocity (on isobar) calculation Target material
rt (g/cc)
Yt (kbar)
Pcrit
rs (g/cc)
R ¼ Yj (kbar)
Mild steel Aluminum Concrete Sandstone
7.86 2.7 2.25 2.25
3 (UTSa) 3 (UTS) 0.3 (UCSa) 0.6 (UCS)
2Yt 2Yt 2Yt 2Yt
16 16 16 16
0 0 0 0
UTS ¼ ultimate tensile strength, UCS ¼ unconfined compressive strength. 12
Jet Density ρ(g/cc)
a
Al Fe sandstone concrete
8
4
0 0.0E+00
5.0E+04 1.0E+05 Jet Velocity V (cm/s)
1.5E+05
Fig. 5. Jet density vs. velocity on Pcrit isobars (various targets).
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ðpressure continuityÞ Pp ¼ Pt , ðvelocity continuityÞ
323
(20)
U pp ¼ V 0 U tp ,
(21)
ðassumed linear UsUp relationshipÞ U xs ¼ C x0 þ Sx U xp ,
(22)
where x ¼ p or t (indicating projectile or target), P is the pressure, r0 is the initial density, C0 is the bulk sound velocity, S is an empirical constant, and Us, Up, and V0 are the shock, particle, and impact velocities, respectively. 6.1. Particle velocity Eqs. (19)–(22) can be combined into the well-known quadratic equation in Utp: 8 rt0 t > > S Sp a ¼ > > > rp0 > < t aðU tp Þ2 þ bðU tp Þ þ c ¼ 0 b ¼ r0 C t þ C p þ 2V Sp 0 > p 0 > r0 0 > > > > : c ¼ ðSp V 2 þ C p V Þ 0
0
(23)
0
For like-on-like impact, the expression for Up simplifies to 1 U tp ¼ U pp ¼ V 0 , 2 which is identical to the steady-state penetration velocity: sffiffiffiffiffi rp0 g U ¼V . ; where : g ¼ 1þg rt0
(24)
(25)
For the general case of dissimilar material impact, however, initial particle velocities and steady penetration velocities are not equal. 6.2. Pressure Utp, once found, can be used to give the shock pressure in the target: Pt ¼ rt0 ðC t0 U tp þ St U tp U tp Þ.
(26)
Comparing this with the steady-state Bernoulli pressure 1 Pt ¼ rt U 2 2 gives the following ratio: t Pshock C ¼ 2 0 þ St . Psteadystate U
(27)
(28)
Considering realistic values of C0, U, and S, this ratio is always greater than unity i.e. the initial shock pressure is always higher than the steady-state impact pressure. Fig. 6 shows particle velocity and pressure as functions of impact velocity, under both initial shock and steady-state conditions, for the illustrative case of tungsten impacting aluminum; Fig. 7 shows the ratio of shock pressure to steady-state pressure. At moderateto-low impact velocities, shock pressure can be an order of magnitude greater than the Bernoulli pressure.
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8.E+05 Up (shock) Up (steady)
Up (cm/sec)
6.E+05 4.E+05 2.E+05 0.E+00 0.0E+00
2.5E+05
5.0E+05
7.5E+05
1.0E+06
Impact Velocity (cm/sec) 3.E+12
P (dyn/cm2)
P (shock) P (steady)
2.E+12
1.E+12
0.E+00 0.0E+00
2.5E+05
5.0E+05
7.5E+05
1.0E+06
Impact Velocity (cm/s)
Fig. 6. Transient shock vs. steady-state penetration; target particle velocity [top] and impact pressure [bottom].
30
Pressure Ratio
25 20 15 10 5 0 0.0E+00
2.5E+05
5.0E+05
7.5E+05
1.0E+06
Impact Velocity (cm/s)
Fig. 7. Ratio of transient shock pressure/steady-state bernoulli pressure.
7. Conclusions We developed a treatment of jet compressibility, based on the method of Flis and Crilly [3]. At the limits, this treatment reduces to the well-known expressions for penetration of fully solid and fully particulated jets, respectively. For a given mass and diameter, a porous (compressible) penetrator was shown to penetrate (hydrodynamically) deeper than an incompressible penetrator of the same length and density; and also deeper than a short incompressible penetrator of higher density. The distended penetrator produces impact pressures intermediate between those produced by the comparable long low-density and short high-density solid penetrators.
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Impact pressures were analyzed in the context of isobars in (rj, V) coordinates, indicating loci of jet properties which can produce specified target pressures. Highly distended (o2 g/cc) slow-moving (500 m/s) jets were determined capable of penetrating moderate strength (UCS 300–600 bar) geologic targets. Finally, we proposed that non-steady effects (initial shocks, terminal residual penetration) may be significant with such non-continuous jets. This would somewhat preclude the application of any continuum steady-state treatment, including those discussed herein. Acknowledgments The author gratefully acknowledges Schlumberger, for supporting this work and its publication. Particular appreciation is extended to Dr. Ian Walton, for his many insights, contributions, and critical review of this work. References [1] Walters WP, Zukas JA. Fundamentals of shaped charges. New York: Wiley; 1989 reprinted 1998 by CMC Press, Baltimore, MD. [2] Evans W. The hollow charge effect. Bull Inst Min Metall 1950. [3] Flis WJ, Crilly MG. Hypervelocity jet penetration of porous materials. In: Proceedings of the 18th international symposium on ballistics, vol. 2, 15–19 November 1999, San Antonio, TX. p. 869–76. [4] Grove B. Determination of the fundamental characteristics of a perforating charge. Schlumberger Intern Rep 2003. [5] Hohler V, Stilp A. Influence of the length-to-diameter ratio in the range from 1 to 32 on the penetration performance of rod projectiles. In: Proceedings of the eighth international. symposium on ballistics, 23–25 October 1984, Orlando, FL. p. IB13–IB19. [6] Anderson CE, Walker JD, Bless SJ, Partom Y. On the L/D effect for long-rod penetrators. Int J Impact Eng 1996;18(3):247–64. [7] Anderson CE, Walker JD, Bless SJ, Sharron TR. On the velocity dependence of the L/D effect for long-rod penetrators. Int J Impact Eng 1995;17:13–24.
International Journal of Impact Engineering 33 (2006) 316–325 www.elsevier.com/locate/ijimpeng
Theoretical considerations on the penetration of powdered metal jets Brenden Grove Schlumberger Reservoir Completions Center, 14910 Airline Road, Rosharon, TX 77583, USA Available online 13 November 2006
Abstract This paper explores some of the theoretical issues encountered when interpreting the penetration behavior of an oilwell perforating charge, whose jet forms from an unsintered powdered metal (PM) liner. Appropriate treatments of the jet’s porous compressible nature fill the gap between classical ‘‘continuous’’ and ‘‘fully particulated’’ jet penetration models. Within certain constraints, increasing a penetrator’s length (even if by distension) increases its hydrodynamic penetration depth, while reducing its impact pressure; and a porous penetrator penetrates deeper than a non-porous penetrator of the same density, length, and velocity. Dynamic target pressure considerations lead to the conclusion that highly distended, low-velocity, PM jets should penetrate moderate-strength geologic targets effectively. After demonstrating that initial transient shock pressures may be much higher than steady-state penetration pressures, we suggest that initial penetration rates may be higher than the steady-state rates. This, in conjunction with the well-known ‘‘residual penetration’’ phenomenon, indicates that a non-continuous jet’s penetration may be strongly influenced by transient effects. r 2006 Elsevier Ltd. All rights reserved. Keywords: Shaped charge; Perforate; Porous jet; Impact pressure
1. Introduction This paper explores the theoretical issues encountered when interpreting the penetration of a perforating charge whose jet forms from an unsintered powdered metal (PM) liner. Penetration model improvements are presented which take into account the jet’s porous compressible nature. Impact pressure and the role of transients will also be discussed. 2. Background Oilwell perforators are small-caliber (20–60 mm) shaped charges used to create holes into a hydrocarbonbearing subterranean reservoir, and the steel casing which lines a borehole connecting the reservoir to surface production facilities. These perforation tunnels, therefore, become part of the ultimate hydrocarbon production flowpath. Tel.: +1 281 285 5225; fax: +1 281 285 5453.
E-mail address: [email protected]. 0734-743X/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijimpeng.2006.09.034
ARTICLE IN PRESS B. Grove / International Journal of Impact Engineering 33 (2006) 316–325
317
Since this flowpath must be ‘‘clean’’, it is desirable to employ liners which do not form competent slugs or large solid debris which may plug perforation tunnels. To this end, typical oilwell perforator liners are pressed to net shape from PM. Contrasted with conventional PM parts, oilwell perforator liners remain ‘‘green’’—i.e. they are not sintered to increase strength. Modest green strength results from particle interlocking and binderinduced adhesion. In a functioning charge, the liner collapses and jets in the usual way, but the formed jet exhibits no tensile strength; it is generally comprised of the constituent powder. Therefore, in the absence of any inward radial velocity, a stretching jet will distend to very low macroscopic densities; increased jet length results from lengthening gaps between powder particles. This calls into question any analysis based on solid, incompressible theory. 3. Jet penetration theory Standard penetration theory is first reviewed, as a foundation on which new concepts applicable to finely powdered jets are developed. Throughout this section, the terms ‘‘jet’’ and ‘‘penetrator’’ will be used interchangeably, referring conceptually to either a rod-type (constant velocity) penetrator, or an incremental element of what may be a stretching jet. Unless otherwise noted, none of the concepts discussed herein explicitly address the time-varying length of a real stretching jet. 3.1. Incompressible, hydrodynamic Hydrodynamic incompressible steady penetration theory relates penetration depth (PD) per unit jet length (L) to the square root of the jet-to-target density ratio [1]. A slight modification of this theory, motivated by momentum considerations [1,2] addresses particulated jets: sffiffiffiffiffiffiffi lrj PD ¼ , (1) L rt where 1plp2, depending on the extent of particulation, rj the average jet density including the gaps, and rt the target density. While allowing one to ‘‘bound’’ the solution by examining the limits, this analysis does not provide an obvious way of describing the extent of particulation (i.e., computing intermediate values of l). Also, it is noted that a fully particulated jet (l ¼ 2) is one in which the inter-particle gap length is much larger than the particle length, so that rj approaches zero. 3.2. Incompressible, target strength important Extending this hydrodynamic treatment to include target strength effects, the simplest models [1] add some characteristic ‘‘static’’ strength term to the dynamic pressure, resulting in the following PD per unit penetrator length: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi lrj PD 2s ¼ , (2) L rt rt ðV UÞ2 where V is jet velocity, U the penetration velocity, and s ¼ Y t Y j the net strength effect. Yt is generally related to some characteristic target material strength (tensile yield strength, for example), but the exact relationship has generally been empirical, and probably varies depending on material type (ductile metal vs. brittle ceramic, etc.). In reality, the operative resisting stresses are likely functions of strain, strain rate, and pressure. Contrasted with hydrodynamic penetration, Eq. (2) illustrates the following: – strength effects become more important as V decreases (approaching U), – jet strength and target strength tend to offset each other.
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3.3. Compressible (porous) jet The foregoing restatement of broadly recognized basics allows us to next address jet compressibility, which has received much less attention. Flis and Crilly [3] developed a simple model for an incompressible jet penetrating a compressible target; we reverse their analysis here. This model describes a two-step process: (1) shock compaction of the distended jet to its maximum pore-free density, (2) penetration of the compacted jet, according to incompressible Bernoulli. Referring to Fig. 1, the total pressure in the target is 1 P2t ¼ Y t þ r0t U 2 , (3) 2 where Y t ¼ P0t , and r0t ¼ r2t ¼ rt . Next, we approximate the jet material’s compressibility as shown in Fig. 1, assume that the jet shock compresses just prior to impact (from state 0j to 1j), and apply the Rankine–Hugoniot shock jump conservation equations. Following Flis and Crilly [3] (leaving out the intermediate steps here for brevity), this gives the jet pressure: 1 P2j ¼ R þ r0 ðV UÞ2 ð1 þ fÞ, (4) 2 where r0 is the initial (distended) jet density, rs the solid (pore-free) jet density, f ¼ 1 r0 =rs the jet porosity, and R the jet compaction initial resistance (i.e., P1j). Equating P2t with P2j at the interface, penetration per unit jet length is given by sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi PD r0 2s ¼ ð1 þ fÞ . (5) L rt rt ðV UÞ2 For hydrodynamic penetration, this simplifies to rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi PD r0 ¼ ð1 þ fÞ. L rt
(6)
Noteworthy are the similarities between Eqs. (6) and (1), and (5) and (2), respectively; the only difference being (1+f) replaces (l). Eqs. (5) and (6), therefore, give the expected behavior at the limits—i.e. reproducing the previously shown ‘‘continuous’’ jet penetration for zero porosity, and the ‘‘fully particulated’’ jet penetration for 100% porosity. As mentioned before, however, jet density approaches zero as porosity approaches 100%.
Fig. 1. Compressible jet penetrating incompressible target [left]; Porous Jet Lockup Model [right] (following [3]).
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One useful implication is that, since f is known for a given r0, this analysis provides a method of determining l in the particulated jet model. Furthermore, this model indicates that hydrodynamic PD is greater for a porous jet than for a non-porous jet, given a length and initial density. 4. Jet length, density, and porosity: a summary Consider three hypothetical jets, each of identical mass, velocity, and cross-sectional area. Each varies in its density, length, and/or porosity as follows: Case A: rs, incompressible; length ¼ Ls
V
Case B: r0, compressible to rs, length ¼ L0
V
Case C: r0, incompressible, length ¼ L0
V
4.1. Hydrodynamic penetration depth The hydrodynamic PDs for each case are summarized as follows: rffiffiffiffiffi rs Case A PDA ¼ Ls . rt Case B
Case C
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r0 ð1 þ fÞ PDB ¼ L0 rt sffiffiffiffiffiffiffiffiffiffiffiffi 1þf ðalternative formÞ ¼ PDA 1f pffiffiffiffiffiffiffiffiffiffiffiffi ¼ PDC 1 þ f ðalternative formÞ. rffiffiffiffiffi r0 PDC ¼ L0 rt rffiffiffiffiffiffi L0 ¼ PDA Ls
ðalternative formÞ.
(7)
ð8Þ
ð9Þ
Note that PDB4PDC4PDA. Using the expression for f for case C (which exhibits no physical porosity, but for purposes of algebraic comparison), the PD ratios are sffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffi 1þf 1 : : 1. (10) PDB : PDC : PDA ¼ 1f 1f Eq. (10), represented graphically in Fig. 2, shows that increases in both length and porosity increase hydrodynamic PD; therefore, consideration of jet porosity is essential to an accurate PD calculation. 4.2. Dynamic jet pressure Considering the same three cases, the Bernoulli dynamic jet pressures are summarized as follows: Case A
1 PA ¼ rs ðV UÞ2 . 2
(11)
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5 Penetration Ratio
1.50
PDB:PDC
1.40 1.30 1.20 1.10
compressible low-density vs. fully-dense incompressible low-density vs. fully-dense
4 3 2 1
1.00 0
0.2
0.4 0.6 Porosity
0.8
1
0
0.2
0.4
0.6
0.8
1
1-ρ0/ρs
Fig. 2. Penetration ratios: compressible (case B) vs. incompressible long jet (case C) [left]; low-density long jets (cases B and C) vs. highdensity short jet (case A) [right].
Case B
1 r ðV UÞ2 ð1 f2 Þ 2 s 1 ¼ r0 ðV UÞ2 ð1 þ fÞ ðalternative formÞ. 2
PB ¼
1 PC ¼ r0 ðV UÞ2 . 2 U is not the same for each case, but varies with g: qffiffiffiffi 8 > gA ¼ rrt > > s > > ffi qffiffiffiffiffiffiffiffiffiffiffiffiffi rt gA V
0
ð12Þ (13)
(14)
1f
Substitution of the appropriate forms of Eq. (14) into Eqs. (11)–(13) allows comparison among the impact pressures: qffiffiffiffiffiffiffiffiffiffiffiffiffiffi32 2 1 1= 1 þ gA = 1 f2 6 7 6 7 ð1 f2 Þ PA : PB : PC ¼ 1 : 4 5 1 1= 1 þ gA "
pffiffiffiffiffiffiffiffiffiffiffiffi#2 1 1= 1 þ gA = 1 f ð1 fÞ. : 1 1= 1 þ gA
ð15Þ
Fig. 3 shows the pressure ratios PB:PA, and PC:PA, for the illustrative case of gA ¼ 0:4. It can be shown that PA4PB4PC, for all gA and f. As with PD, accurate impact pressure calculation requires appropriate consideration of jet porosity. Finally, it is worth noting that case B yields the greatest hydrodynamic penetration, while case A produces the greatest impact pressure. 5. Impact pressure and target strength Consider some critical point of interest (Ucrit, Pcrit), which is observed late in penetration, when target strength effects are significant. Penetration velocity and jet density are expressed as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ðPcrit Y t Þ , (16) U crit ¼ rt
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Pressure Ratio
1 0.8 0.6 0.4
Compressible low-density vs. high-density
0.2
Incompressible low-density vs. high-density
0 0
0.2
0.4
0.6
0.8
1
1-ρ0 /ρs Fig. 3. Dynamic jet pressure ratios; ratio of cases C:A and B:A.
ðincompressible; 1plp2Þ rj ¼
ðcompressible; following½3Þ
rt U 2crit þ 2s , lðV U crit Þ2
rj ¼ rs ð1 fÞ;
(17) sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r U 2 þ 2s . f ¼ 1 t crit rs ðV U crit Þ2
(18)
When jet density can vary, therefore, Pcrit does not imply a fixed Vcrit. Rather, V varies inversely with rj on a Pcrit isobar. Fig. 4 shows such isobars (in {rj, V} space) for four different targets, given the values shown in Table 1. The compressible jet model approaches continuous jet behavior at high densities, and particulated jet behavior at low densities; both trends are as expected. For comparison, Fig. 4 also includes curves based on compressible Bernoulli theory, the details of which are omitted here. The good agreement between both treatments of jet compressibility makes either equally suitable for comparison against incompressible treatments. Fig. 5 shows the isobars for all four target materials of Table 1, for compressible jet impact. Mild steel exhibits a lower critical jet velocity than aluminum, indicating that more of a jet would be consumed in penetrating mild steel; while initially counterintuitive, this is consistent with experimental observations [4], and can be explained by the fact that the mild steel and aluminum considered have identical strengths, but the dynamic impact pressure in the steel is higher due to its higher density. Also of interest—and this hinges on the appropriateness of UCS (unconfined compressive strength) as a benchmark—even highly distended PM jets (density o2 g/cc) should penetrate concrete or sandstone effectively at velocities as low as 500 m/s. 6. Transients and L/D effects Two assumptions in the forgoing are (1) continuous penetration and (2) L/D independence. For very long rods and continuous jets, this is reasonable, since the initial transients and terminal effects are operative during very small portions of the overall penetration, and aspect ratio does not generally play a significant role. However, a highly distended jet, comprised of millions of equi-axed particles and large inter-particle spacing, calls into question these simplifying assumptions. Penetrator aspect ratio effects have been studied at some length [5–7]; short penetrators are observed to penetrate deeper, per unit length, than long penetrators; even exceeding the hydrodynamic limit as L/D-1+ at sufficiently high velocities. This is the rationale for the segmented rod penetrator concept [1]. The increased penetration effectiveness is likely due largely to residual or inertial penetration (‘‘afterflow’’) [1]; but may also be due in part to the initial transient pressures, which will now be demonstrated to exceed the steady-state Bernoulli pressures. Applying the Rankine–Hugoniot equation for momentum conservation across a shock for 1D impact (which, admittedly, does not accurately describe the particle impact in question), we have Px ¼ rx0 U xs U xp ,
(19)
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Mild Steel
Aluminum
18 incomp. particulated
14
comp. Bernoulli
12
comp. F&C
10 8 6 4 2
Jet Density rho (g/cc)
incomp. solid
16 Jet Density rho (g/cc)
18 16 14 12 10 8
incomp. solid
6
incomp. particulated
4
comp. Bernoulli
2
comp. F&C
0 2.0E+04 4.0E+04 6.0E+04 8.0E+04 1.0E+05 1.2E+05
0 2.0E+04 4.0E+04 6.0E+04 8.0E+04 1.0E+05 1.2E+05
Jet Velocity V (cm/s)
Jet Velocity V (cm/s)
Sandstone
Concrete 18
18 incomp. solid
14
comp. Bernoulli
12
incomp. solid
16
incomp. particulated comp. F&C
10 8 6 4
Jet Density rho (g/cc)
Jet Density rho (g/cc)
16
incomp. particulated
14
comp. Bernoulli
12
comp. F&C
10 8 6 4 2
2 0 2.0E+04 4.0E+04 6.0E+04 8.0E+04 1.0E+05 1.2E+05
0 2.0E+04 4.0E+04 6.0E+04 8.0E+04 1.0E+05 1.2E+05
Jet Velocity V (cm/s)
Jet Velocity V (cm/s)
Fig. 4. Jet density vs. velocity on Pcrit isobars (various targets). Table 1 Target parameters for jet density-vs.-velocity (on isobar) calculation Target material
rt (g/cc)
Yt (kbar)
Pcrit
rs (g/cc)
R ¼ Yj (kbar)
Mild steel Aluminum Concrete Sandstone
7.86 2.7 2.25 2.25
3 (UTSa) 3 (UTS) 0.3 (UCSa) 0.6 (UCS)
2Yt 2Yt 2Yt 2Yt
16 16 16 16
0 0 0 0
UTS ¼ ultimate tensile strength, UCS ¼ unconfined compressive strength. 12
Jet Density ρ(g/cc)
a
Al Fe sandstone concrete
8
4
0 0.0E+00
5.0E+04 1.0E+05 Jet Velocity V (cm/s)
1.5E+05
Fig. 5. Jet density vs. velocity on Pcrit isobars (various targets).
ARTICLE IN PRESS B. Grove / International Journal of Impact Engineering 33 (2006) 316–325
ðpressure continuityÞ Pp ¼ Pt , ðvelocity continuityÞ
323
(20)
U pp ¼ V 0 U tp ,
(21)
ðassumed linear UsUp relationshipÞ U xs ¼ C x0 þ Sx U xp ,
(22)
where x ¼ p or t (indicating projectile or target), P is the pressure, r0 is the initial density, C0 is the bulk sound velocity, S is an empirical constant, and Us, Up, and V0 are the shock, particle, and impact velocities, respectively. 6.1. Particle velocity Eqs. (19)–(22) can be combined into the well-known quadratic equation in Utp: 8 rt0 t > > S Sp a ¼ > > > rp0 > < t aðU tp Þ2 þ bðU tp Þ þ c ¼ 0 b ¼ r0 C t þ C p þ 2V Sp 0 > p 0 > r0 0 > > > > : c ¼ ðSp V 2 þ C p V Þ 0
0
(23)
0
For like-on-like impact, the expression for Up simplifies to 1 U tp ¼ U pp ¼ V 0 , 2 which is identical to the steady-state penetration velocity: sffiffiffiffiffi rp0 g U ¼V . ; where : g ¼ 1þg rt0
(24)
(25)
For the general case of dissimilar material impact, however, initial particle velocities and steady penetration velocities are not equal. 6.2. Pressure Utp, once found, can be used to give the shock pressure in the target: Pt ¼ rt0 ðC t0 U tp þ St U tp U tp Þ.
(26)
Comparing this with the steady-state Bernoulli pressure 1 Pt ¼ rt U 2 2 gives the following ratio: t Pshock C ¼ 2 0 þ St . Psteadystate U
(27)
(28)
Considering realistic values of C0, U, and S, this ratio is always greater than unity i.e. the initial shock pressure is always higher than the steady-state impact pressure. Fig. 6 shows particle velocity and pressure as functions of impact velocity, under both initial shock and steady-state conditions, for the illustrative case of tungsten impacting aluminum; Fig. 7 shows the ratio of shock pressure to steady-state pressure. At moderateto-low impact velocities, shock pressure can be an order of magnitude greater than the Bernoulli pressure.
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8.E+05 Up (shock) Up (steady)
Up (cm/sec)
6.E+05 4.E+05 2.E+05 0.E+00 0.0E+00
2.5E+05
5.0E+05
7.5E+05
1.0E+06
Impact Velocity (cm/sec) 3.E+12
P (dyn/cm2)
P (shock) P (steady)
2.E+12
1.E+12
0.E+00 0.0E+00
2.5E+05
5.0E+05
7.5E+05
1.0E+06
Impact Velocity (cm/s)
Fig. 6. Transient shock vs. steady-state penetration; target particle velocity [top] and impact pressure [bottom].
30
Pressure Ratio
25 20 15 10 5 0 0.0E+00
2.5E+05
5.0E+05
7.5E+05
1.0E+06
Impact Velocity (cm/s)
Fig. 7. Ratio of transient shock pressure/steady-state bernoulli pressure.
7. Conclusions We developed a treatment of jet compressibility, based on the method of Flis and Crilly [3]. At the limits, this treatment reduces to the well-known expressions for penetration of fully solid and fully particulated jets, respectively. For a given mass and diameter, a porous (compressible) penetrator was shown to penetrate (hydrodynamically) deeper than an incompressible penetrator of the same length and density; and also deeper than a short incompressible penetrator of higher density. The distended penetrator produces impact pressures intermediate between those produced by the comparable long low-density and short high-density solid penetrators.
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Impact pressures were analyzed in the context of isobars in (rj, V) coordinates, indicating loci of jet properties which can produce specified target pressures. Highly distended (o2 g/cc) slow-moving (500 m/s) jets were determined capable of penetrating moderate strength (UCS 300–600 bar) geologic targets. Finally, we proposed that non-steady effects (initial shocks, terminal residual penetration) may be significant with such non-continuous jets. This would somewhat preclude the application of any continuum steady-state treatment, including those discussed herein. Acknowledgments The author gratefully acknowledges Schlumberger, for supporting this work and its publication. Particular appreciation is extended to Dr. Ian Walton, for his many insights, contributions, and critical review of this work. References [1] Walters WP, Zukas JA. Fundamentals of shaped charges. New York: Wiley; 1989 reprinted 1998 by CMC Press, Baltimore, MD. [2] Evans W. The hollow charge effect. Bull Inst Min Metall 1950. [3] Flis WJ, Crilly MG. Hypervelocity jet penetration of porous materials. In: Proceedings of the 18th international symposium on ballistics, vol. 2, 15–19 November 1999, San Antonio, TX. p. 869–76. [4] Grove B. Determination of the fundamental characteristics of a perforating charge. Schlumberger Intern Rep 2003. [5] Hohler V, Stilp A. Influence of the length-to-diameter ratio in the range from 1 to 32 on the penetration performance of rod projectiles. In: Proceedings of the eighth international. symposium on ballistics, 23–25 October 1984, Orlando, FL. p. IB13–IB19. [6] Anderson CE, Walker JD, Bless SJ, Partom Y. On the L/D effect for long-rod penetrators. Int J Impact Eng 1996;18(3):247–64. [7] Anderson CE, Walker JD, Bless SJ, Sharron TR. On the velocity dependence of the L/D effect for long-rod penetrators. Int J Impact Eng 1995;17:13–24.