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Hypervelocity impact penetration mechanics
International Journal of Impact Engineering 35 (2008) 1654–1660
Contents lists available at ScienceDirect
International Journal of Impact Engineering journal homepage: www.elsevier.com/locate/ijimpeng
Hypervelocity impact penetration mechanics C. McFarland a, *, P. Papados b, M. Giltrud c a
SAIC Inc., 4875 Eisenhower Avenue, Alexandria, VA 22304, USA Engineer Research and Development Center, U.S. Army Corps of Engineers, USACEGSL 7701 Telegraph Road, Alexandria, VA 22315, USA c Defense Threat Reduction Agency, 8725 John J. Kingman Road, Fort Belvoir, VA 22060, USA b
a r t i c l e i n f o
a b s t r a c t
Article history: Available online 13 August 2008
Inert dense metal penetrators having a mass and geometry capable of missile delivery offer significant potential for countering underground facilities at depths of tens of meters in hard rock. The proliferation of such facilities among countries whose support for terrorism and potential possession of Weapons of Mass Destruction (WMD) constitutes threats to world peace and U.S. Security. The Defense Threat Reduction Agency (DTRA), in cooperation with the U.S. Army Corps of Engineers, the Department of Energy National Laboratories and private sector R&D firms have pursued an aggressive research effort to explore the attributes of high velocity impact penetrators for countering such facilities. The penetration of crustal rocks with metal rods (such as tungsten or steel alloys) at high velocities involves complex wave propagation phenomena within the rod and inelastic response of both the penetrator and target material. In this paper we examine the sensitivity of penetration depth (for a fixed tungsten alloy mass impacting a limestone target) to impactor velocity, strength and geometry. Analyses are based upon a matrix of first principle finite difference calculations using the Sandia CTH (release 7.1) Shock Physics Code. Results indicate that impact velocity, penetrator yield strength and target yield strength strongly influence the penetration depth. Maximum penetration depth is achieved by a delicate trade off between penetrator kinetic energy and penetrator inelastic deformation (erosion). Numerical analyses for the parameter variations exercised in this study (impact velocities 1–3.5 km/s and penetrator yield strengths of 1–4 GPa) produced penetration depths of a tungsten alloy rod (length 200 cm, diameter 20 cm) which varied from 5.1 m to 28 m in a homogeneous limestone target. Ó 2008 Elsevier Ltd. All rights reserved.
Keywords: Penetration Tungsten Strength Erosion Limestone
1. Introduction The initial impact of the penetrator upon the target material produces a nearly instantaneous rise of stress in both materials, a reduction of the penetrator material velocity and a jump of material velocity of the target material. The resulting interface stress and particle velocity will be the same in both materials and the change in energy in the target material will be equally divided between internal and kinetic energies [1]. As time progresses, inelastic waves propagate into both materials, the penetrator shock wave encounters free surfaces producing relief waves and altering the initial one-dimensional flow (and state of uniaxial strain which exists at the shock front). We will first demonstrate that CTH correctly calculates initial impact conditions, i.e. impact stress and particle velocity and equally
* Corresponding author. E-mail address: [email protected] (C. McFarland). 0734-743X/$ – see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijimpeng.2008.07.080
partitions energy in the target material during the period of one-dimensional flow as required by the Hugoniot equations. Next we will examine the complex flow conditions that occur at late times during the penetration phase made possible through the sophisticated CTH post-processor editing options. Finally we will present a summary of parametric analyses in which a 1100 kg tungsten alloy penetrator (density 17.2 g/cm3) impacts a homogeneous limestone target (density 2.7 g/cm3) with penetration depths that vary from 5.1 m to 28 m depending upon penetrator geometry, yield stress, velocity and limestone target strength. 2. Impact Fig. 1 illustrates the empirically observed [2] shock condition arising from a tungsten flyer plate at 1.99 km/s impacting a limestone target. The tungsten is shocked to a point on its (reflected) Hugoniot from a condition which was initially stress free and at a velocity of 1.99 km/s to a normal stress of 28 GPa (280 kbars) and
C. McFarland et al. / International Journal of Impact Engineering 35 (2008) 1654–1660
Fig. 1. Initial impact conditions.
Fig. 2. Shock propagation at 11 ms and 41 ms.
Fig. 3. Interface particle velocity and stress.
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Fig. 6. Zoning (scaled calculations).
Fig. 4. Target energy partitioning.
a particle velocity of 1.65 km/s. The Limestone is shocked from a stress free state to identical stress and velocity conditions. CTH computational results for the experiment above are presented in Figs. 2 and 3. Time sequenced plots show the tungsten projectile moving from the top at 1.99 km/s, the tungsten flyer plate (colored gray) is encased in a tungsten sabot, similarly the limestone target (light brown) is encased in a limestone sabot (the use of the sabot allows independent energy edits of the sabot and flyer plate/target materials so that energy edits within the flyer plate/ target can be made prior to the arrival of free surface effects). Isostress contours are superimposed upon the materials: red contours represent 100 kbars, yellow contours represent 10 kbars and blue contours represent 1 kbar. The red dots on the left hand side of the plots represent particle velocity. Dot density varies linearly around a mean value of 2 km/s.
Fig. 5. Rarefaction arrival.
Table 1 CTH computational matrix Velocity (km/s)
L/D
Rod segments
Penetrator yield strength, GPA (psi)
Target yield strength, GPA (psi)
Full scale depth of penetration (m)
3.5 3.5 3.5 3.5 3.5 3.5 3.5 3.0 3.0 3.0 3.0 3.0 3.0 3.0 2.5 2.5 2.5 2.5 2.5 2.0 2.0 2.0 2.0 2.0 2.0 2.0 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5
10 10 10 10 10 10 10 10 10 10 10 5 1 1 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 2 10 10 10
1 1 1 1 1 1 1 1 1 1 1 1 10 10a 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 5 1 1 1
4 3 2 2 1 1 1 4 3 2 1 3 3 3 4 3 2 1 3 4 3 2 1 4 3 2 4 3 2 1 4 3 2 3 1 2
0.1 0.1 0.1 0.2 0.1 0.06 0.4 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.06 0.1 0.1 0.1 0.1 0.06 0.06 0.06 0.1 0.1 0.1 0.1 0.24 0.06 0.06 0.02 0.06 0.1
12.0 11.0 9.0 8.5 6.8 6.8 6.0 17.0 11.0 9.3 7.0 7.8 9.1 12.2 23.0 12.5 11.0 7.8 15.0 28.0 19.5 11.5 8.7 27.0 21.0 19.0 26.5 19.0 12.0 9.1 20.0 22.0 11.0 15.0 11.5 10.5
(600,000) (450,000) (300,000) (300,000) (150,000) (150,000) (150,000) (600,000) (450,000) (300,000) (150,000) (450,000) (450,000) (450,000) (600,000) (450,000) (300,000) (150,000) (450,000) (600,000) (450,000) (300,000) (150,000) (600,000) (450,000) (300,000) (600,000) (450,000) (300,000) (150,000) (600,000) (450,000) (300,000) (450,000) (150,000) (300,000)
(15,000) (15,000) (15,000) (30,000) (15,000) (9000) (60,000) (15,000) (15,000) (15,000) (15,000) (15,000) (15,000) (15,000) (15,000) (15,000) (15,000) (15,000) (9000) (15,000) (15,000) (15,000) (15,000) (9000) (9000) (9000) (15,000) (15,000) (15,000) (15,000) (36,000) (9000) (9000) (30,000) (9000) (15,000)
C. McFarland et al. / International Journal of Impact Engineering 35 (2008) 1654–1660 Table 1 (continued ) Velocity (km/s)
L/D
1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.0 1.0 1.0 1.0 1.0 1.0 1.0
10 10 10 10 10 10 10 10 10 10 10 10 10 15 10 10 10
a
Rod segments
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
Penetrator yield strength, GPA (psi)
Target yield strength, GPA (psi)
Full scale depth of penetration (m)
1 4 2 3 2 3 1 2 2 4 4 4 3 3 2 3 1
0.1 0.1 0.2 0.1 0.06 0.4 0.2 0.2 0.06 0.2 0.2 0.1 0.1 0.1 0.1 0.2 0.1
9.2 23.0 7.5 21.0 20.0 12.5 8.2 9.0 15.0 21.0 18.5 25.0 16.0 19.0 9.0 10.1 5.1
(150,000) (600,000) (300,000) (450,000) (300,000) (450,000) (150,000) (300,000) (300,000) (600,000) (600,000) (600,000) (450,000) (450,000) (300,000) (450,000) (150,000)
(15,000) (15,000) (30,000) (15,000) (9000) (60,000) (30,000) (30,000) (9000) (30,000) (30,000) (15,000) (15,000) (15,000) (15,000) (30,000) (15,000)
sphere of same mass as cylinder with L/D ¼ 1.
The time history plots immediately above are for Lagrangian edit points at the leading edge of the impactor and the top center of the limestone. At time zero the tungsten flyer plate is separated from the limestone target by 2 cm and since the flyer plate velocity is 1.99 km/s impact is expected at approximately 10 ms. The 0.4 cm zone size accounts for the computational ‘‘noise’’ preceding impact. The stresses and particle velocities rapidly equilibrate to 280 kbars and 1.65 km/s, respectively, in near
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perfect agreement with the published results from Ahrens et al. [2]. As expected from the Hugoniot equations, Fig. 4 illustrates the equal partitioning of internal and kinetic energ in the limestone target as it is swept by the shock wave. As side rarefactions appear (approximately at 33 ms; see Fig. 5), the one-dimensional steady conditions no longer exist and equal partitioning is no longer expected (note that the target’s total energy at time zero reflects the random vibrational (i.e. internal) energy of the target which is approximately 5e12 ergs).
3. Penetration phase As will be demonstrated in results presented subsequently, the final calculated penetration depth is very sensitive to the initial impact conditions, (especially impact velocities which generate stresses which significantly exceed the penetrator yield strength) and the material properties of both the penetrator and geological target materials. The penetration phase poses more difficult computational challenges than those associated with initial impact, especially those dealing with mixed material cells along the penetrator target material. The Army Research Laboratory (ARL) has conducted a significant amount of testing of rod penetrators into metal and ceramic targets including tests used to evaluate CTH predictive capabilities. Ref. [3] provides detailed discussion of the CTH’s Boundary Layer Interface (BLINT) Model utilized in the results presented within this paper. The ARL calculated penetration depths and the transition velocity for rigid body to eroding body penetration were within the experimental data scatter.
Fig. 7. Axial, radial and yield stress, 300 ksi rod vs 600 ksi rod, 1.5 km/s.
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Fig. 8. Penetrator yield stress snapshots, 300 ksi (left), 600 ksi (right).
Fig. 9. Maximum penetration depth (Velocity ¼ 0, d(Yposition)/dt ¼ 0).
C. McFarland et al. / International Journal of Impact Engineering 35 (2008) 1654–1660
4. Computational program Parametric variations of impact velocity, penetrator yield strength (the material was assumed to be elastic-perfectly plastic), limestone target strength (Mohr–Coulomb strength model with Mises limit) and penetrator L/D were included. The BLINT algorithm was used in all calculations with a friction coefficient of 0.06. The matrix included both full scale ‘‘impactors’’ of 1100 kg and 1/10th scale (1.1 kg) variants. Comparisons between the full scale and 1/10th scale calculations showed that penetration depths and all edited parameters (e.g. ground kinetic energy, stress time histories, etc.) were virtually identical. The sole exception, as would be expected, was the material temperature which was slightly different between the full scale and 1/10th scale case. The motivation for varying target yield strength was primarily related to consideration of cyclic loading of the rock (either from a segmented rod single delivery or multiple delivery of unitary rods). It is well known (see, e.g. Ref. [4–7]) that cyclic loading of rocks generates micro fractures along grain boundaries thus reducing both strength and modulus.
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illustrates a subset for four values of rod strength (600 ksi, 450 ksi, 300 ksi, and 150 ksi) and a target (limestone) strength of 15 ksi. Aside from impact velocity, penetrator yield strength emerges as the calculation variable most strongly influencing penetration depth. Fig. 11 illustrates the dramatic difference in the penetration process for two extreme values of tungsten yield strength, 150 ksi vs 600 ksi. These calculations, with an impact velocity of 1 km/s are identical except for the tungsten yield strength. The 150 ksi rod (L/D ¼ 10) has completely eroded and dissipated most of its kinetic energy at a depth (full scale) of approximately 4 m whereas the 600 ksi penetrator has experienced no erosion and retains 50% of its initial kinetic energy at a depth of 4 m. The 600 ksi penetrator ultimately comes to rest intact at a depth of 20 m. Notice that the ‘‘crater radius’’ is more than twice as large for the 150 ksi case as opposed to the 600 ksi case for these snapshots in time. The maximum yield value used in the CTH analyses presented here was 4 GPa (or approximately 600 ksi). It has been observed [9,10] that tungsten materials exhibit substantial post-yield hardening characteristics with apparent yield strengths in excess of 3.4 GPa. It is recommended that additional research to be conducted to determine the appropriate values of strength for future analyses.
4.1. Numerical results 6. Conclusions
5. Discussion Table 1 shows a wide disparity in penetration depth (from 5.1 m to 28 m for a tungsten mass of 1100 kg impacting Limestone). Subsets of the results do, however, show very distinct trends. Fig. 10
First principle analyses using the Sandia National Laboratory CTH Shock Physics code indicate that substantial variations in penetration depth for high velocity/hypervelocity impact penetrators will result depending upon nominal variations in key parameters. Full scale and 1/10th scale calculations were performed for a range of impact velocities, geometric configurations and penetrator/target material properties. Maximum penetration occurs when the combination of impact velocity and material properties prevents or minimizes penetrator erosion. For the tungsten alloy penetrator (1100 kg full scale), penetration depths varied from 5.1 m to 28 m in a homogeneous limestone target. Aside from impact velocity, which was varied from 1 km/s to 3.5 km/s, the penetrator strength emerged as the dominant variable. Values of penetrator strength were parametrically varied from 150 ksi to 600 ksi, the 150 ksi penetrator experienced significant erosion at all velocities whereas the 600 ksi penetrator experienced minimal erosion below impact velocities of 2 km/s. The maximum penetration depth occurred between 1.5 km/s and 2 km/s impact velocities for all parametric sets. The appropriate dynamic yield
PENETRATION DEPTH VS VELOCITY VARIABLE ROD STRENGTH, LIMESTONE STRENGTH=15 KSI
PENETRATION DEPTH (M)
Fig. 6 illustrates zoning for the 1/10th scale calculations. The rod dimensions are 2 cm diameter by 20 cm length with 40 cells across the diameter and 200 cells along the rod length. The tungsten rod is modeled as an elastic-perfectly plastic material with yield strength varied parametrically from 150 ksi to 600 ksi. The limestone target material (density 2.7 g/cm3) was modeled as a Mohr–Coulomb material, the Mises limit was 15 ksi for most cases. Several excursions to both higher and lower Mises limits were also included. The complete computational matrix (Table 1) includes variation of velocity, tungsten alloy yield strength, limestone yield strength and penetrator geometry. Fig. 7 (impact velocity of 1.5 km/s) presents detailed output of rod radial stress, axial stress and a standard CTH edit of a function of the second invariant of the deviatoric stress tensor (sqrt3j2p) commonly used as a failure or yield criteria (Ref. [8]), in this case a maximum value of 300 ksi or 600 ksi. The actual CTH yield values of 2 GPa (20e9 dynes/cm2) and 4 GPa (40e9 dynes/cm2), have been rounded off to 300 ksi and 600 ksi in the text inserts. Both rods in this case experience some plastic flow (sqrt[3j2p] has a limiting, i.e. yield, value of 20 kbars (w300 ksi) for the blue curves and 40 kbars (w600 ksi) for the red curves). As indicated by Fig. 7, however, the associated plastic strains for the 600 ksi material are negligible. Spatially resolved values of the yield stress at two identical points in time are shown in Fig. 8 below. The color palettes of the yield stresses are adjusted so that red constitutes yield in both cases (300 ksi rod on left, 600 ksi rod on right). The evidence of plastic flow (erosion and mushroom tip) is readily apparent for the 300 ksi rod. A summary of case parameters and final penetration depths has been provided in Table 1. Penetration depths were determined from post-processor editing of penetrator Lagrangian tracers. Fig. 9 illustrates the 1/10th scale edits for the case of 1.5 km/s, L/D ¼ 10, Yield (penetrator) ¼ 450,000 psi and Yield (target) ¼ 60,000 psi which indicate a (full scale) penetration depth of 12.5 m.
600 KSI
30
450 KSI
300 KSI
150 KS
25 20 15 10 5 0
1
1.5
2
2.5
IMPACT VELOCITY (KM/SEC) Fig. 10. Penetration depth vs velocity.
3
3.5
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Fig. 11. 150 ksi penetrator vs 600 ksi penetrator.
strength for applications such as these are currently subjective and warrants additional investigation. References [1] Zel’dovich YB, Raizer YP. Physics of shock waves and high-temperature hydrodynamic phenomena. New York: Academic Press; 1967. [2] Ahrens TJ, Anderson WW, Zhao Y. Shock propagation in crustal rocks: sandstone, limestone and shale. California Institute of Technology, CIT. Report No. 64649. [3] Segletes SG. Analysis of the noneroding penetration of tungsten alloy long rods into aluminum targets. Army Research Laboratory. Technical Report 3075; September 2003. [4] Haimson BC. Effect of cyclic loading on rock. ASTM STP 654. In: Dynamic geotechnical testing. American Society for Testing and Materials; 1978. p. 228–45.
[5] Haimson BC. Mechanical behavior of rock under cyclic loading. In: Advances in rock mechanics. Washington DC: National Academy of Sciences; 1974. [6] Rajaram V. Mechanical behavior of Berea sandstone and Westerly granite under cyclic compression, a thesis submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy. University of WisconsinMadison; 1978. [7] Touloukian YS. Physical properties of rocks and minerals, McGraw-Hill/CINDAS data series on material properties. vol. II-2; 1981. [8] Malvern LE. Introduction to the mechanics of a continuous medium. New York: Prentice-Hall; 1969. [9] Scheffler DR. Modeling the effect of penetration nose shape on threshold velocity for thick aluminum targets. ARL-TR-1417. Maryland: Aberdeen Proving Ground; July 1997. [10] Dandekar D, Grady D. Shock equation of state and dynamic strength of tungsten carbide. American Institute of Physics, Conference Proceedings 2002;620(1):783–6.
Contents lists available at ScienceDirect
International Journal of Impact Engineering journal homepage: www.elsevier.com/locate/ijimpeng
Hypervelocity impact penetration mechanics C. McFarland a, *, P. Papados b, M. Giltrud c a
SAIC Inc., 4875 Eisenhower Avenue, Alexandria, VA 22304, USA Engineer Research and Development Center, U.S. Army Corps of Engineers, USACEGSL 7701 Telegraph Road, Alexandria, VA 22315, USA c Defense Threat Reduction Agency, 8725 John J. Kingman Road, Fort Belvoir, VA 22060, USA b
a r t i c l e i n f o
a b s t r a c t
Article history: Available online 13 August 2008
Inert dense metal penetrators having a mass and geometry capable of missile delivery offer significant potential for countering underground facilities at depths of tens of meters in hard rock. The proliferation of such facilities among countries whose support for terrorism and potential possession of Weapons of Mass Destruction (WMD) constitutes threats to world peace and U.S. Security. The Defense Threat Reduction Agency (DTRA), in cooperation with the U.S. Army Corps of Engineers, the Department of Energy National Laboratories and private sector R&D firms have pursued an aggressive research effort to explore the attributes of high velocity impact penetrators for countering such facilities. The penetration of crustal rocks with metal rods (such as tungsten or steel alloys) at high velocities involves complex wave propagation phenomena within the rod and inelastic response of both the penetrator and target material. In this paper we examine the sensitivity of penetration depth (for a fixed tungsten alloy mass impacting a limestone target) to impactor velocity, strength and geometry. Analyses are based upon a matrix of first principle finite difference calculations using the Sandia CTH (release 7.1) Shock Physics Code. Results indicate that impact velocity, penetrator yield strength and target yield strength strongly influence the penetration depth. Maximum penetration depth is achieved by a delicate trade off between penetrator kinetic energy and penetrator inelastic deformation (erosion). Numerical analyses for the parameter variations exercised in this study (impact velocities 1–3.5 km/s and penetrator yield strengths of 1–4 GPa) produced penetration depths of a tungsten alloy rod (length 200 cm, diameter 20 cm) which varied from 5.1 m to 28 m in a homogeneous limestone target. Ó 2008 Elsevier Ltd. All rights reserved.
Keywords: Penetration Tungsten Strength Erosion Limestone
1. Introduction The initial impact of the penetrator upon the target material produces a nearly instantaneous rise of stress in both materials, a reduction of the penetrator material velocity and a jump of material velocity of the target material. The resulting interface stress and particle velocity will be the same in both materials and the change in energy in the target material will be equally divided between internal and kinetic energies [1]. As time progresses, inelastic waves propagate into both materials, the penetrator shock wave encounters free surfaces producing relief waves and altering the initial one-dimensional flow (and state of uniaxial strain which exists at the shock front). We will first demonstrate that CTH correctly calculates initial impact conditions, i.e. impact stress and particle velocity and equally
* Corresponding author. E-mail address: [email protected] (C. McFarland). 0734-743X/$ – see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijimpeng.2008.07.080
partitions energy in the target material during the period of one-dimensional flow as required by the Hugoniot equations. Next we will examine the complex flow conditions that occur at late times during the penetration phase made possible through the sophisticated CTH post-processor editing options. Finally we will present a summary of parametric analyses in which a 1100 kg tungsten alloy penetrator (density 17.2 g/cm3) impacts a homogeneous limestone target (density 2.7 g/cm3) with penetration depths that vary from 5.1 m to 28 m depending upon penetrator geometry, yield stress, velocity and limestone target strength. 2. Impact Fig. 1 illustrates the empirically observed [2] shock condition arising from a tungsten flyer plate at 1.99 km/s impacting a limestone target. The tungsten is shocked to a point on its (reflected) Hugoniot from a condition which was initially stress free and at a velocity of 1.99 km/s to a normal stress of 28 GPa (280 kbars) and
C. McFarland et al. / International Journal of Impact Engineering 35 (2008) 1654–1660
Fig. 1. Initial impact conditions.
Fig. 2. Shock propagation at 11 ms and 41 ms.
Fig. 3. Interface particle velocity and stress.
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Fig. 6. Zoning (scaled calculations).
Fig. 4. Target energy partitioning.
a particle velocity of 1.65 km/s. The Limestone is shocked from a stress free state to identical stress and velocity conditions. CTH computational results for the experiment above are presented in Figs. 2 and 3. Time sequenced plots show the tungsten projectile moving from the top at 1.99 km/s, the tungsten flyer plate (colored gray) is encased in a tungsten sabot, similarly the limestone target (light brown) is encased in a limestone sabot (the use of the sabot allows independent energy edits of the sabot and flyer plate/target materials so that energy edits within the flyer plate/ target can be made prior to the arrival of free surface effects). Isostress contours are superimposed upon the materials: red contours represent 100 kbars, yellow contours represent 10 kbars and blue contours represent 1 kbar. The red dots on the left hand side of the plots represent particle velocity. Dot density varies linearly around a mean value of 2 km/s.
Fig. 5. Rarefaction arrival.
Table 1 CTH computational matrix Velocity (km/s)
L/D
Rod segments
Penetrator yield strength, GPA (psi)
Target yield strength, GPA (psi)
Full scale depth of penetration (m)
3.5 3.5 3.5 3.5 3.5 3.5 3.5 3.0 3.0 3.0 3.0 3.0 3.0 3.0 2.5 2.5 2.5 2.5 2.5 2.0 2.0 2.0 2.0 2.0 2.0 2.0 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5
10 10 10 10 10 10 10 10 10 10 10 5 1 1 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 2 10 10 10
1 1 1 1 1 1 1 1 1 1 1 1 10 10a 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 5 1 1 1
4 3 2 2 1 1 1 4 3 2 1 3 3 3 4 3 2 1 3 4 3 2 1 4 3 2 4 3 2 1 4 3 2 3 1 2
0.1 0.1 0.1 0.2 0.1 0.06 0.4 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.06 0.1 0.1 0.1 0.1 0.06 0.06 0.06 0.1 0.1 0.1 0.1 0.24 0.06 0.06 0.02 0.06 0.1
12.0 11.0 9.0 8.5 6.8 6.8 6.0 17.0 11.0 9.3 7.0 7.8 9.1 12.2 23.0 12.5 11.0 7.8 15.0 28.0 19.5 11.5 8.7 27.0 21.0 19.0 26.5 19.0 12.0 9.1 20.0 22.0 11.0 15.0 11.5 10.5
(600,000) (450,000) (300,000) (300,000) (150,000) (150,000) (150,000) (600,000) (450,000) (300,000) (150,000) (450,000) (450,000) (450,000) (600,000) (450,000) (300,000) (150,000) (450,000) (600,000) (450,000) (300,000) (150,000) (600,000) (450,000) (300,000) (600,000) (450,000) (300,000) (150,000) (600,000) (450,000) (300,000) (450,000) (150,000) (300,000)
(15,000) (15,000) (15,000) (30,000) (15,000) (9000) (60,000) (15,000) (15,000) (15,000) (15,000) (15,000) (15,000) (15,000) (15,000) (15,000) (15,000) (15,000) (9000) (15,000) (15,000) (15,000) (15,000) (9000) (9000) (9000) (15,000) (15,000) (15,000) (15,000) (36,000) (9000) (9000) (30,000) (9000) (15,000)
C. McFarland et al. / International Journal of Impact Engineering 35 (2008) 1654–1660 Table 1 (continued ) Velocity (km/s)
L/D
1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.0 1.0 1.0 1.0 1.0 1.0 1.0
10 10 10 10 10 10 10 10 10 10 10 10 10 15 10 10 10
a
Rod segments
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
Penetrator yield strength, GPA (psi)
Target yield strength, GPA (psi)
Full scale depth of penetration (m)
1 4 2 3 2 3 1 2 2 4 4 4 3 3 2 3 1
0.1 0.1 0.2 0.1 0.06 0.4 0.2 0.2 0.06 0.2 0.2 0.1 0.1 0.1 0.1 0.2 0.1
9.2 23.0 7.5 21.0 20.0 12.5 8.2 9.0 15.0 21.0 18.5 25.0 16.0 19.0 9.0 10.1 5.1
(150,000) (600,000) (300,000) (450,000) (300,000) (450,000) (150,000) (300,000) (300,000) (600,000) (600,000) (600,000) (450,000) (450,000) (300,000) (450,000) (150,000)
(15,000) (15,000) (30,000) (15,000) (9000) (60,000) (30,000) (30,000) (9000) (30,000) (30,000) (15,000) (15,000) (15,000) (15,000) (30,000) (15,000)
sphere of same mass as cylinder with L/D ¼ 1.
The time history plots immediately above are for Lagrangian edit points at the leading edge of the impactor and the top center of the limestone. At time zero the tungsten flyer plate is separated from the limestone target by 2 cm and since the flyer plate velocity is 1.99 km/s impact is expected at approximately 10 ms. The 0.4 cm zone size accounts for the computational ‘‘noise’’ preceding impact. The stresses and particle velocities rapidly equilibrate to 280 kbars and 1.65 km/s, respectively, in near
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perfect agreement with the published results from Ahrens et al. [2]. As expected from the Hugoniot equations, Fig. 4 illustrates the equal partitioning of internal and kinetic energ in the limestone target as it is swept by the shock wave. As side rarefactions appear (approximately at 33 ms; see Fig. 5), the one-dimensional steady conditions no longer exist and equal partitioning is no longer expected (note that the target’s total energy at time zero reflects the random vibrational (i.e. internal) energy of the target which is approximately 5e12 ergs).
3. Penetration phase As will be demonstrated in results presented subsequently, the final calculated penetration depth is very sensitive to the initial impact conditions, (especially impact velocities which generate stresses which significantly exceed the penetrator yield strength) and the material properties of both the penetrator and geological target materials. The penetration phase poses more difficult computational challenges than those associated with initial impact, especially those dealing with mixed material cells along the penetrator target material. The Army Research Laboratory (ARL) has conducted a significant amount of testing of rod penetrators into metal and ceramic targets including tests used to evaluate CTH predictive capabilities. Ref. [3] provides detailed discussion of the CTH’s Boundary Layer Interface (BLINT) Model utilized in the results presented within this paper. The ARL calculated penetration depths and the transition velocity for rigid body to eroding body penetration were within the experimental data scatter.
Fig. 7. Axial, radial and yield stress, 300 ksi rod vs 600 ksi rod, 1.5 km/s.
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Fig. 8. Penetrator yield stress snapshots, 300 ksi (left), 600 ksi (right).
Fig. 9. Maximum penetration depth (Velocity ¼ 0, d(Yposition)/dt ¼ 0).
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4. Computational program Parametric variations of impact velocity, penetrator yield strength (the material was assumed to be elastic-perfectly plastic), limestone target strength (Mohr–Coulomb strength model with Mises limit) and penetrator L/D were included. The BLINT algorithm was used in all calculations with a friction coefficient of 0.06. The matrix included both full scale ‘‘impactors’’ of 1100 kg and 1/10th scale (1.1 kg) variants. Comparisons between the full scale and 1/10th scale calculations showed that penetration depths and all edited parameters (e.g. ground kinetic energy, stress time histories, etc.) were virtually identical. The sole exception, as would be expected, was the material temperature which was slightly different between the full scale and 1/10th scale case. The motivation for varying target yield strength was primarily related to consideration of cyclic loading of the rock (either from a segmented rod single delivery or multiple delivery of unitary rods). It is well known (see, e.g. Ref. [4–7]) that cyclic loading of rocks generates micro fractures along grain boundaries thus reducing both strength and modulus.
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illustrates a subset for four values of rod strength (600 ksi, 450 ksi, 300 ksi, and 150 ksi) and a target (limestone) strength of 15 ksi. Aside from impact velocity, penetrator yield strength emerges as the calculation variable most strongly influencing penetration depth. Fig. 11 illustrates the dramatic difference in the penetration process for two extreme values of tungsten yield strength, 150 ksi vs 600 ksi. These calculations, with an impact velocity of 1 km/s are identical except for the tungsten yield strength. The 150 ksi rod (L/D ¼ 10) has completely eroded and dissipated most of its kinetic energy at a depth (full scale) of approximately 4 m whereas the 600 ksi penetrator has experienced no erosion and retains 50% of its initial kinetic energy at a depth of 4 m. The 600 ksi penetrator ultimately comes to rest intact at a depth of 20 m. Notice that the ‘‘crater radius’’ is more than twice as large for the 150 ksi case as opposed to the 600 ksi case for these snapshots in time. The maximum yield value used in the CTH analyses presented here was 4 GPa (or approximately 600 ksi). It has been observed [9,10] that tungsten materials exhibit substantial post-yield hardening characteristics with apparent yield strengths in excess of 3.4 GPa. It is recommended that additional research to be conducted to determine the appropriate values of strength for future analyses.
4.1. Numerical results 6. Conclusions
5. Discussion Table 1 shows a wide disparity in penetration depth (from 5.1 m to 28 m for a tungsten mass of 1100 kg impacting Limestone). Subsets of the results do, however, show very distinct trends. Fig. 10
First principle analyses using the Sandia National Laboratory CTH Shock Physics code indicate that substantial variations in penetration depth for high velocity/hypervelocity impact penetrators will result depending upon nominal variations in key parameters. Full scale and 1/10th scale calculations were performed for a range of impact velocities, geometric configurations and penetrator/target material properties. Maximum penetration occurs when the combination of impact velocity and material properties prevents or minimizes penetrator erosion. For the tungsten alloy penetrator (1100 kg full scale), penetration depths varied from 5.1 m to 28 m in a homogeneous limestone target. Aside from impact velocity, which was varied from 1 km/s to 3.5 km/s, the penetrator strength emerged as the dominant variable. Values of penetrator strength were parametrically varied from 150 ksi to 600 ksi, the 150 ksi penetrator experienced significant erosion at all velocities whereas the 600 ksi penetrator experienced minimal erosion below impact velocities of 2 km/s. The maximum penetration depth occurred between 1.5 km/s and 2 km/s impact velocities for all parametric sets. The appropriate dynamic yield
PENETRATION DEPTH VS VELOCITY VARIABLE ROD STRENGTH, LIMESTONE STRENGTH=15 KSI
PENETRATION DEPTH (M)
Fig. 6 illustrates zoning for the 1/10th scale calculations. The rod dimensions are 2 cm diameter by 20 cm length with 40 cells across the diameter and 200 cells along the rod length. The tungsten rod is modeled as an elastic-perfectly plastic material with yield strength varied parametrically from 150 ksi to 600 ksi. The limestone target material (density 2.7 g/cm3) was modeled as a Mohr–Coulomb material, the Mises limit was 15 ksi for most cases. Several excursions to both higher and lower Mises limits were also included. The complete computational matrix (Table 1) includes variation of velocity, tungsten alloy yield strength, limestone yield strength and penetrator geometry. Fig. 7 (impact velocity of 1.5 km/s) presents detailed output of rod radial stress, axial stress and a standard CTH edit of a function of the second invariant of the deviatoric stress tensor (sqrt3j2p) commonly used as a failure or yield criteria (Ref. [8]), in this case a maximum value of 300 ksi or 600 ksi. The actual CTH yield values of 2 GPa (20e9 dynes/cm2) and 4 GPa (40e9 dynes/cm2), have been rounded off to 300 ksi and 600 ksi in the text inserts. Both rods in this case experience some plastic flow (sqrt[3j2p] has a limiting, i.e. yield, value of 20 kbars (w300 ksi) for the blue curves and 40 kbars (w600 ksi) for the red curves). As indicated by Fig. 7, however, the associated plastic strains for the 600 ksi material are negligible. Spatially resolved values of the yield stress at two identical points in time are shown in Fig. 8 below. The color palettes of the yield stresses are adjusted so that red constitutes yield in both cases (300 ksi rod on left, 600 ksi rod on right). The evidence of plastic flow (erosion and mushroom tip) is readily apparent for the 300 ksi rod. A summary of case parameters and final penetration depths has been provided in Table 1. Penetration depths were determined from post-processor editing of penetrator Lagrangian tracers. Fig. 9 illustrates the 1/10th scale edits for the case of 1.5 km/s, L/D ¼ 10, Yield (penetrator) ¼ 450,000 psi and Yield (target) ¼ 60,000 psi which indicate a (full scale) penetration depth of 12.5 m.
600 KSI
30
450 KSI
300 KSI
150 KS
25 20 15 10 5 0
1
1.5
2
2.5
IMPACT VELOCITY (KM/SEC) Fig. 10. Penetration depth vs velocity.
3
3.5
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Fig. 11. 150 ksi penetrator vs 600 ksi penetrator.
strength for applications such as these are currently subjective and warrants additional investigation. References [1] Zel’dovich YB, Raizer YP. Physics of shock waves and high-temperature hydrodynamic phenomena. New York: Academic Press; 1967. [2] Ahrens TJ, Anderson WW, Zhao Y. Shock propagation in crustal rocks: sandstone, limestone and shale. California Institute of Technology, CIT. Report No. 64649. [3] Segletes SG. Analysis of the noneroding penetration of tungsten alloy long rods into aluminum targets. Army Research Laboratory. Technical Report 3075; September 2003. [4] Haimson BC. Effect of cyclic loading on rock. ASTM STP 654. In: Dynamic geotechnical testing. American Society for Testing and Materials; 1978. p. 228–45.
[5] Haimson BC. Mechanical behavior of rock under cyclic loading. In: Advances in rock mechanics. Washington DC: National Academy of Sciences; 1974. [6] Rajaram V. Mechanical behavior of Berea sandstone and Westerly granite under cyclic compression, a thesis submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy. University of WisconsinMadison; 1978. [7] Touloukian YS. Physical properties of rocks and minerals, McGraw-Hill/CINDAS data series on material properties. vol. II-2; 1981. [8] Malvern LE. Introduction to the mechanics of a continuous medium. New York: Prentice-Hall; 1969. [9] Scheffler DR. Modeling the effect of penetration nose shape on threshold velocity for thick aluminum targets. ARL-TR-1417. Maryland: Aberdeen Proving Ground; July 1997. [10] Dandekar D, Grady D. Shock equation of state and dynamic strength of tungsten carbide. American Institute of Physics, Conference Proceedings 2002;620(1):783–6.