- Email: [email protected]
Experimental Investigation of Shock Wave Attenuation in Basalt
Icarus 156, 539–550 (2002) doi:10.1006/icar.2001.6729, available online at http://www.idealibrary.com on
Experimental Investigation of Shock Wave Attenuation in Basalt S. Nakazawa1 Institute of Space and Astronautical Science, Sagamihara 229-8510, Japan E-mail: [email protected]
S. Watanabe Department of Earth and Planetary Sciences, Nagoya University, Nagoya 464-8602, Japan
and Y. Iijima and M. Kato Institute of Space and Astronautical Science, Sagamihara 229-8510, Japan Received December 3, 1998; revised July 16, 2001
Shock compression experiments on Kinosaki basalt were carried out in the interest of studying collisional phenomena in the solar nebula. Shock waves of 7 and 31 GPa were generated using a thin flyer plate, and a shock wave of 16 GPa was generated using a thick cylindrical projectile. By employing in-material manganin and carbon pressure gauges, the shock wave attenuation was examined and the propagation velocities of the shock wave and rarefaction wave were measured. The attenuation mechanism consists of two effects: the rarefaction wave and geometrical expansion. The rarefaction effect includes the reflected wave and the edge wave. The efficiency of these mechanisms depends on the geometry of the projectile, initially induced pressure, and materials of the target and projectile. As a result of the experiments, a cylindrical impactor created an isobaric region of size almost equal to the projectile radius. The shock wave in the far field was attenuated similarly with the power of −1.7 to −1.8 of the propagation distance under our experimental conditions. The shock wave generated using a thin flyer plate was attenuated by the rarefaction wave generated on the back surface of the flyer plate and by geometrical expansion effects. The shock wave generated using a thick projectile was attenuated by edge-wave and by geometrical expansion effects. According to elastic theory, the rigidity of basalt at 7 and 31 GPa was calculated as 35 ± 7 and 0 ± 3 GPa, respectively, using the measured rarefaction wave velocities. The decayed shock pressure was related to the ejection velocity of the impact fragments, which were obtained in previous disruption experiments. The attenuation rates in previous experiments were consistent with ours. The previous impact scaling parameter called “nondimensional impact stress (PI)” has been improved. c 2002 Elsevier Science (USA)
Key Words: cratering; planetary formation; impact processes; planetesimals.
INTRODUCTION
It is widely accepted that planetesimals were formed by the repeated collisions of rocks with each other and accreted to grow into planetary bodies in the early solar nebula (e.g., Ohtsuki et al. 1993). Although the impact process of rocks consisting of polycrystals of some minerals is very complex, it must be studied to clarify the formation process of our planets. Basalt can be employed as one of the materials analogous to planetesimal and planetary bodies. Some impact disruption experiments with basalt have been performed by Fujiwara et al. (1977), Fujiwara and Tsukamoto et al. (1980), Takagi et al. (1984), Waza et al. (1985), and Nakamura and Fujiwara (1991). They collected impact fragments to measure their size distribution or took highspeed images to measure their ejection velocities. The disruption and fragment ejection are the results of shock waves which travel and decay through the target body. To explain their experimental results and to understand the physical process of impact phenomena, especially for the formation process of the planets, shock pressure attenuation must be investigated quantitatively. Some investigators have studied shock pressure attenuation observationally or theoretically. Dence et al. (1977) observationally examined the distribution of shock metamorphism in rocks below natural impact craters. They estimated the displacement of rocks in impact craters using the results of underground nuclear explosions and estimated the pressure attenuation as P ∝ x −2 – x −4.5 , where P and x are shock pressure and propagation distance, respectively. Dienes and Walsh (1970) and Ahrens and O’Keefe (1977) theoretically examined the regime of melting
1 Current address: National Space Development Agency of Japan, Tsukuba 305-8505, Japan. Fax: 81-298-68-2984. E-mail: [email protected].
539 0019-1035/02 $35.00 c 2002 Elsevier Science (USA) All rights reserved.
540
NAKAZAWA ET AL.
and vaporization and the crater formation for a spherical projectile impact using a hydrodynamics code. Their calculations showed the pressure–distance relation as P ∝ x−3 and P ∝ x −1.5 –x −3 , respectively. Based on their results, Melosh (1989), Melosh et al. (1992), and Asphaug (1997) employed the assumption of P ∝ x−2 in their hydrodynamics code simulation of impact fragmentation and impact cratering. Kieffer and Simonds (1980) considered that the energy was transferred from the projectile into the spherical region traversed by the shock wave and that the energy was distributed equally between kinetic energy and internal energy. Their analytical investigation resulted in P ∝ x −3 – x −3.6 . There is a large variation in the attenuation rates of these investigations. Natural crater observations showed some uncertainty in estimations of the displacement of rocks in the impact crater and of the pressure experienced. On the other hand, some of the theoretical studies discussed only the quantity of energy, without considering the mechanism of shock wave attenuation. Considering the mechanism of shock wave attenuation, it can be assumed that shock wave decay is influenced by the rarefaction wave and geometrical expansion (Fig. 1). Two rarefaction waves must be considered in the impact phenomena. One is the reflected rarefaction wave which is generated at the back surface of the projectile (Grady et al. 1975, Meyer 1994), and the other is the edge wave which is generated at the front edge of the projectile at the moment of impact. The second geometrical expansion effect means that the shock wave propagates hemispherically far from the impact point and decays while con-
FIG. 1. Schematic shock wave attenuation mechanisms: (1) rarefaction wave effect and (2) geometrical expansion effect. The rarefaction wave consists of two waves, the reflected rarefaction wave and the edge wave. In the far field, the shock wave propagates hemispherically and decays while conserving its wave energy.
serving its wave energy. This effect attenuates shock pressure (even higher than the Hugoniot elastic limit or HEL) with the power of −1 of the propagation distance (e.g., Melosh 1989), similar to the case of the spherical elastic wave (lower than the HEL). This effect is at work in the far field under any experimental condition. EXPERIMENTAL PROCEDURE
A single-stage powder gun with a 30-mm bore and a twostage light-gas gun with a 10-mm bore at Nagoya University were employed in the experiments. The experimental assembly is shown in Fig. 2. We measured shock pressure using an in-material shock pressure gauge with constant impact velocity and varying target thickness to investigate the mechanism of shock wave attenuation. Three types of copper impactors with cylindrical polycarbonate sabot were fired at the basalt target: a 3-mm-thick flyer plate with 28-mm diameter, a 1-mm one with 9-mm diameter, and a 9-mm one with 9-mm diameter. Each sabot was put in contact with a flyer plate. The aspect ratios of the two flyer plates were nearly the same. Although spherical projectiles are impacted in the natural phenomena, cylindrical projectiles were used in the experiments in order to distinguish the attenuation mechanism. A low-impedance manganin gauge (Mn10-0.05-FEP or Mn2-0.10-FEP; Dynasen Inc.), a 50- piezoresistive manganin gauge (Mn4-50-EK or Mn8-50-EK; Dynasen Inc.), or a 50- piezoresistive carbon gauge (C300-50-EKRTE; Dynasen Inc.) was sandwiched between two target plates to measure shock pressure. These gauges were used by Yoshida and Thadhani (1992), Sorrell and Kuo (1992), and Nakazawa et al. (1997). The calibration of the gauge, established by Dynasen Inc., showed only 5% error in shock pressure determination. Two gauges were used for most shots, and three or four gauges were used for thick targets in some 7-GPa experiments. Basalt from Kinosaki, Hyogo, Japan, was used for the targets in the experiments. Although the samples were quarried from a columnar rock, their physical properties varied from sample to sample. The sound velocities of each pre-shock sample were measured by the ultrasonic transmission method using a piezoelectric transducer. They ranged from 4.15 to 5.15 km s−1 for longitudinal waves and from 2.25 to 3.00 km s−1 for shear waves, which yield bulk sound velocities from 3.07 to 4.08 km s−1 . The measured initial bulk density of the basalt ranged from 2.63 to 2.74 g cm−3 , and the measured true density in crushed samples ranged from 2.85 to 2.89 g cm−3 , from which the porosity was calculated to be 4 to 8%. The target plates were polished parallel and flat to an accuracy of 0.03 mm, and the tilt of the target was set at under 0.3 degree. Target plates were cut large enough so that the wave reflected at the target side did not catch up with the shock wave. They were cut into rectangles of 50 × 80 mm for less than near field experiments (less than 50-mm distance) and into rectangles of about 200 × 200 mm for far field experiments (more than 50-mm distance).
SHOCK WAVE ATTENUATION IN BASALT
FIG. 2.
541
Illustration of attenuation experimental procedure. Three types of impactors and one or two shock pressure gauges were used in the experiment.
The HEL of basalt is 5 GPa according to Nakazawa et al. (1997). To investigate shock wave attenuation at pressures lower than the HEL, a 7-GPa shock wave was generated by firing a 3-mm-thick copper flyer plate with the impact velocity of 0.7 km s−1 . To investigate the attenuation at pressures higher than the HEL, a 31-GPa shock wave was generated by firing a 1-mmthick copper flyer plate with the impact velocity of 2.6 km s−1 . To investigate attenuation by a thick impactor, a 9-mm-thick copper projectile was fired with the impact velocity of 1.6 km s−1 to induce 16 GPa. The impact velocity was measured by the magnet flyer method or the wire cutting method. Because the
sabot is put into contact with the flyer plate, the reflected rarefaction wave does not release to zero pressure. However, it is weaker than the original shock wave and whether it has prevented the decay of the shock wave can be confirmed by checking the waveform. We also conducted experiments for measuring edge-wave velocity. The flyer plate and the experimental condition were the same as those in the 7-GPa experiments. Two gauges were sandwiched between two target plates; one was at the center of the target and the other at 7 mm off-center (Fig. 3). In the near field, the shock front is parallel to the target surface and the shock wave and reflected rarefaction wave should propagate through the target to reach the two gauges at the same moment. However, the edge wave should reach the off-center gauge before it reaches the center gauge. The edge-wave velocity was measured at the time when the shock pressure began to release at the off-center gauge. EXPERIMENTAL RESULTS
7-GPa and 31-GPa Thin Flyer Plate Experiments Experimental conditions and results are summarized in Table I. The initially induced shock pressure on the impact surface (P0 ) was estimated by the calculated particle velocity,
FIG. 3. Illustration of edge-wave experimental procedure. Two pressure gauges were sandwiched by two target plates to measure edge-wave velocity.
P0 = ρ0 flyer (C0 flyer + sflyer (vi − up0 basalt ))(vi − up0 basalt ) = ρ0 basalt (C0 basalt + sbasalt up0 basalt )up0 basalt ,
542
NAKAZAWA ET AL.
TABLE I Lists of Experimental Conditions and Results
Run No.
Propagation distance (x) (mm)
Normalized distance (x/r )
Bulk sound velocity (C0 ) (km s−1 )
Bulk density (ρ0 ) (g cm−3 )
Porosity (%)
Impact velocity (Vi ) (km s−1 )
Shock pressure (P) (GPa)
Initial pressure (P0 ) (GPa)
Normalized pressure (P/P0 )
7-GPa experiment 1 2 3 4 5 5 6 7 8 9 10 11 12 13 14 15 15 16 16 16 16
2.87 11.98 21.21 24.90 8.85 18.11 3.19 4.27 3.15 10.98 15.45 40.87 12.37 31.34 14.19 133.41 178.44 87.32 210.71 386.83 473.18
0.21 0.86 1.52 1.78 0.63 1.29 0.23 0.30 0.22 0.78 1.10 2.92 0.88 2.24 1.01 9.53 12.75 6.24 15.05 27.63 33.80
3.81 3.81 3.81 3.81 3.81 3.81 3.56 3.81 3.81 3.53 3.55 3.31 3.07 3.21 3.17 3.13 3.13 3.10 3.10 3.10 3.10
2.74 2.74 2.74 2.74 2.74 2.74 2.67 2.67 2.74 2.67 2.67 2.70 2.72 2.63 2.70 2.65 2.65 2.65 2.65 2.65 2.65
4 4 4 4 4 4 7 7 4 7 7 6 5 8 5 8 8 8 8 8 8
0.68 0.75 0.70 0.76 0.70 0.69 0.68 0.68 0.68 0.74 0.69 0.74 0.73 0.74 0.78 0.83 0.83 0.83 0.85 0.85 0.85
7.2 7.2 3.3 3.3 7.3 4.3 7.1 7.0 7.2 7.4 5.7 1.1 6.2 1.5 6.4 0.13 0.070 0.30 0.062 0.025 0.018
7.1 7.9 7.4 8.0 7.4 7.3 7.1 7.1 7.1 7.8 7.2 7.8 7.7 7.8 8.2 8.8 8.8 9.0 9.0 9.0 9.0
1.01 0.91 0.45 0.41 0.98 0.59 1.00 0.98 1.01 0.95 0.78 0.13 0.80 0.19 0.78 0.015 0.0080 0.034 0.0069 0.0027 0.0020
31-GPa experiment 17 17 18 18 19 19 20 20 21 21 22
3.00 12.87 8.52 22.55 1.05 11.05 5.36 33.46 8.16 10.12 4.48
0.67 2.86 1.89 5.01 0.23 2.46 1.19 7.43 1.81 2.25 1.00
3.42 3.53 3.24 3.25 3.61 3.35 3.49 3.09 3.41 3.35 3.10
2.74 2.74 2.74 2.74 2.74 2.74 2.74 2.74 2.74 2.74 2.74
4 4 4 4 4 4 4 4 4 4 4
2.70 2.70 2.58 2.58 2.68 2.68 2.36 2.36 2.66 2.66 2.58
31.0 4.0 6.7 1.4 32.5 5.0 16.0 0.75 10.5 5.0 21.0
32.5 32.5 30.6 30.6 32.2 32.2 27.3 27.3 31.9 31.9 30.6
0.95 0.12 0.22 0.046 1.01 0.16 0.59 0.028 0.33 0.16 0.69
16-GPa experiment 23 23 24 24 25 25
3.57 7.22 1.07 11.44 5.08 42.40
0.79 1.60 0.24 2.54 1.13 9.42
3.18 3.13 4.08 3.49 3.58 3.09
2.74 2.74 2.74 2.74 2.74 2.74
4 4 4 4 4 4
1.65 1.65 1.62 1.62 1.57 1.57
16.0 9.0 17.0 3.5 11.5 0.35
17.4 17.4 17.0 17.0 16.4 16.4
0.92 0.52 1.00 0.21 0.70 0.021
Note. r refers to the projectile radius. The initially induced pressure (P0 ) was estimated by the impedance matching method.
where ρ 0 , C0 , s, vi , and up0 are the mean initial bulk density, bulk sound velocity, coefficient of the Us − up relation, impact velocity, and initially induced particle velocity on the impact surface, respectively. In the calculation, the Hugoniot EOS of copper in Marsh (1980) was adopted for the impactor (C0 = 3.91 km s−1 , s = 1.51) and the Hugoniot EOS of basalt in Nakazawa et al. (1997) was adopted for the target (lower pressure: C0 = 4.3 km s−1 , s = 1.6; middle pressure: C0 = 4.9 km s−1 , s = 0.2; higher pressure: C0 = 3.0 km s−1 , s = 1.5). The initial particle velocity (up0 ) was calculated by the impedance matching method using
measured impact velocities and the Hugoniot equation of basalt. The error of the impact velocity was ±5%. Some examples of the shock profile are shown in Fig. 4. They are superimposed to fit the time of impact, which is calculated using the shock wave velocity estimated by the Us –P Hugoniot equation of basalt while taking into account the attenuation of the shock pressure (P) and the change of the shock velocity (Us ). The peak pressure decreased in the far field, and the shock profile appeared deltoid with very short shock duration time. Although a time resolution of the experimental system below 50 ns was confirmed, any
SHOCK WAVE ATTENUATION IN BASALT
543
FIG. 4. The observed shock profiles of the (a) 7-GPa, (b) 31-GPa, and (c) 16-GPa experiments. They are superimposed to fit the time of impact, which is calculated by using the Hugoniot equation of basalt.
steps at the HEL (5 GPa) were not identified in the shock profiles. The reported reason was that heterogeneity of basalt blurred the shock wave and the shock profile showed single-wave structures (Nakazawa et al. 1997). The measured shock pressure attenuation is shown in Fig. 5. The error in determining the normalized shock pressure (P/P0 ) is estimated at about ±10%, and at low pressures (<0.2 GPa)
it is estimated at about ±0.001 (= P/P0 ). The results of the 7-GPa experiment in Fig. 5a are divided into two regions; in region 1 (x < 11 mm), the pressure stays at maximum; in region 2 (x > 15 mm), the pressure attenuates with a power of −1.8 ± 0.2, where the value is called the “attenuation rate.” The transition of the attenuation rate suggests that the attenuation mechanism changed. Though there were fewer data, the data of
544
NAKAZAWA ET AL.
FIG. 5. The measured shock pressure attenuation of the (a) 7-GPa and (b) 31-GPa experiments. The results can be divided into two regions. The attenuation rates were nearly equal to each other.
the 31-GPa experiment in Fig. 5b also seem to be divided into two regions. The attenuation starting distance was at x = 3 mm. The attenuation rate is −1.7 ± 0.2, which is almost the same as in the 7-GPa experiment. The times when the pressure profile increased and then decreased indicate the arrival times of the shock wave and the rarefaction wave, respectively. The shock wave arrival time (tarrival ), the time when the shock pressure reached peak pressure (tpeak ), and the time when the shock pressure began to release (trelease ) versus propagation distance are plotted in Fig. 6 (x–t diagram). The time in the plots had error of ±0.1 µs. The time of impact (t0 ) was calculated by integrating the Us –x relation, which was estimated by the P–x relation and the Us –P Hugoniot taking into account the changing shock velocity (Us ). Because of large errors in estimating the time of impact, the data in the far field (>87.3 mm) were not plotted. The shock duration time, that is, the time lag between tpeak and trelease , became shorter in the near field (<14.2 mm) as the shock wave propagated. In these experiments, since the flyer plate was thin enough, i.e., one-ninth of the diameter, the reflected rarefaction wave caught up with the shock wave and decayed earlier than the edge wave. The reflected rarefaction wave caught up with the 7-GPa shock wave at about 12 mm (x/r = 0.86). In the 31-GPa experiment, there are only two data points in the near field (<3 mm) and it is impossible to estimate accurately the rarefaction wave velocity. However, the shock pressure at 3.0 mm (No. 17) was the same as the initial induced pressure (that is, the shock wave had not yet been attenuated) and the shock du-
ration time was short (that is, the reflected rarefaction wave already caught up with the shock wave). So, it can be concluded that the reflected rarefaction wave had just caught up with the shock wave at almost 3-mm distance. The distance at which the reflected rarefaction wave catches up with the shock wave depends on the thickness of the flyer plate and the shock wave velocity. The slope of trelease in Fig. 6 indicates the reflected rarefaction wave velocity. It is shown that the reflected rarefaction wave ran after the shock wave with higher velocity; velocities (Ur ) at 7 and 31 GPa were measured as 6.2 ± 0.3 and 6.7 ± 0.7 km s−1 , respectively, referred to Lagrangian coordinates. Edge-Wave Experiment and 16-GPa Cylindrical Thick-Projectile Experiment In the edge-wave experiments, two shock profiles arose at the same time and the release of the off-center gauge started earlier than that of the center gauge (Fig. 7). Since the stress gauge has a common problem, that the slope of the release profiles are not reproducible, only the release starting times were used in the analysis. The different release times indicate that the propagating direction of the rarefaction wave was not perpendicular to the gauge plane. Accordingly, the release on the off-center gauge was caused not by the rarefaction wave reflected at the back surface of the projectile but by the rarefaction wave generated at the front edge of the projectile. Assuming that the edge wave propagated spherically from the edge, the edge-wave velocity (Ue ) of 4.6 ± 0.2 km s−1 was measured in all edge-wave experiments
SHOCK WAVE ATTENUATION IN BASALT
545
FIG. 6. X–t diagram of the (a) 7-GPa and (b) 31-GPa experiments. The shock wave arrival time (t arrival ), the time when the shock pressure reached peak pressure (tpeak ), the time when the shock pressure began to release (t release ), and the calculated edge-wave arrival time are plotted. The errors are generally smaller than the symbol size. The slope of the curve indicates the wave velocity. The figure shows that the rarefaction wave runs after the shock wave with higher velocity and catches up to it.
(Table II) by using the arrival time and the distances between the edge and the off-center gauge. Even when the error of the edgewave velocity was included, it was calculated that the edge wave could decay the shock wave at the center at distances farther than 25 mm (dashed line in Fig. 6a). Therefore, the edge wave could not decay the peak pressure in the 7-GPa experiment. The results of the 16-GPa experiment with a thick cylindrical projectile are shown in Fig. 8. The P–x diagram (Fig. 8a) is similar to the others (Fig. 5). The attenuation starting distance of 3.5 mm and the attenuation rate of −1.7 are almost the same as in the thin flyer plate experiment. Though the x–t diagram (Fig. 8b) also seems the same as the others, the attenuation mechanism was much different from the others. The projectile was so thick that the reflected rarefaction wave would take more than 2 µs to catch up to the shock wave, arriving later than the edge wave. Assuming the edge-wave velocity at 16 GPa is 6.5 km s−1 (a little smaller than the rarefaction wave velocity at 31 Gpa), the calculated arrival time of the edge wave is shown in Fig. 8b (dashed curve). Since the curve well matched the release line, it is confirmed that the edge wave, not the reflected rarefaction wave, with a propagation velocity of about 6.5 km s−1 attenuated the shock wave in the 16-GPa experiments. FIG. 7. The observed shock profiles of one of the edge-wave experiments. Two profiles arose at the same time, and the release of the off-center gauge started earlier than that of the center gauge. The edge-wave velocity of 4.6 km s−1 was measured by t release and the distance between the gauge and the impactor’s edge.
DISCUSSION AND CONCLUSIONS
No dominant attenuation in the shock pressure was detected for x < r (region 1 in Figs. 5a and 5b). This suggested that the
546
NAKAZAWA ET AL.
TABLE II Lists of Conditions and Results of Edge-Wave Experiments
Run No.
Propagation distance (x) (mm)
Normalized distance (x/r )
Bulk sound velocity (C0 ) (km s−1 )
Bulk density (ρ0 ) (g cm−3 )
Porosity (%)
Impact velocity (Vi ) (km s−1 )
Shock pressure (P) (GPa)
Initial pressure (P0 ) (GPa)
Edge-wave velocity (Ue ) (km s−1 )
8 9 10 12
3.15 10.98 15.45 12.37
0.23 0.78 1.10 0.88
3.81 3.53 3.55 3.07
2.74 2.67 2.67 2.72
4 7 7 5
0.68 0.74 0.69 0.73
7.3 6.5 4.1 4.8
7.1 7.1 7.2 7.5
4.75 4.64 4.43 4.61
Note. Edge-wave velocity (Ue ) was measured as the distance between the projectile front edge and the gauge divided by the edge-wave arrival time.
rarefaction wave did not catch up with the shock wave and that the geometrical expansion effect did not affect it in the near field. In the far field, the x–t diagram for the 7-GPa experiment (Fig. 6a) shows that the reflected rarefaction wave caught up with the shock front at 12 mm in propagation distance. Otherwise a spherical shock wave would have decayed in the far field (e.g., Melosh, 1989). Therefore, the attenuation in region 2 was caused by the sum of the geometrical expansion and the reflected rarefaction wave effects. There is a transition region between regions 1 and 2. However, we do not have enough data to characterize the mechanism. In the 31-GPa experiment, the reflected rarefaction wave caught up with the shock wave at 3 mm, and shock pressure started to decay beyond 3 mm distance. The geometrical expansion effect should also be included beyond the impactor radius as in the 7-GPa experiment. So, the attenuation farther than 3 mm
was caused by the sum of the geometrical expansion and the reflected rarefaction wave effects, which is the same mechanism as in the 7-GPa experiment. Because of the geometry (aspect ratio) of the impactor, the distance where the reflected rarefaction wave caught up with the shock wave was nearly equal to the impactor radius. Figure 8b shows that the shock wave generated by the thick impactor was attenuated by the edge wave. In the far field, the geometrical expansion effect is also at work. Accordingly, it is confirmed that the sum of the edge-wave and the geometrical expansion effects attenuate the shock wave. Though the impact velocities and the projectiles sizes were different in each experiment, the normalized shock pressures are shown in Fig. 9 for comparison. The distance (x) was normalized by the impactor radius (r ) and the measured shock pressure (P) was normalized by the initially induced shock
FIG. 8. (a) The measured shock pressure attenuation and (b) x–t diagram of the 16-GPa thick-projectile experiment. Though the features of the result are similar to those of the other thin flyer experiment, the sum of the edge-wave and geometrical effects attenuated the shock wave.
547
SHOCK WAVE ATTENUATION IN BASALT
FIG. 9. Normalized shock pressure vs normalized propagation distance for all experiments. Though the induced shock pressure and the attenuation mechanism were different, the shock pressures attenuate similarly, and the size of the all isobaric region was the same as the projectile radius.
pressure (P0 ), which was calculated by the impedance matching method. Although the data P/P0 have an error of ±10%, the figure shows that the size of the isobaric core is almost the same as the impactor size and that the attenuation rate in the far field is also the same even though the attenuation mechanisms are different. It is true that the size of the isobaric region typically depends on the projectile geometry and the impact velocity. If a thinner flyer plate were impacted, the size would be smaller by the reflected rarefaction wave effect. However, the edge-wave effect limits its upper size. In the case of a fully thick flyer plate, the edge wave catches up with the shock wave at first and the flyer plate thickness has little influence on the size. Under our experimental conditions, the upper size of the isobaric region would be seen, and it is nearly equal to the impactor radius. A spherical impactor, however, would cause another result. In the case of the spherical impactor, the edge wave can catch up with the original shock wave at the time of impact, in contrast to the result for a cylindrical impactor. And, it is uncertain whether the attenuation rate in our results could be applied to the case of a spherical impactor. According to our experimental results, the attenuation mechanism of a cylindrical impactor is as follows. The distance at which the geometrical expansion effect starts depends on the projectile radius, and that of the edge-wave effect also depends on the projectile radius. Also, the distance at which the reflected wave starts depends on projectile thickness. Of course, it also
depends on the rarefaction wave velocity, which is characterized by the material. When the projectile is thick enough, the reflected rarefaction wave reaches the shock wave after the edge wave and has no effect on peak pressure attenuation. In this case the shock wave is attenuated by the sum of the edge-wave and geometrical expansion effects. As a result of this mechanism, the shock pressure remains constant in the near field, and the shock pressure decays with a power of −1.7 of propagation distance in the far field. The attenuation rate in basalt is barely influenced by either the initial shock pressure-up to 31 Gpa or the projectile thickness. Although the impact condition dependency could not be identified in basalt, Arakawa et al. (1995) showed that the attenuation rate in ice depended on the impact velocity and that the rate was different from that of basalt. Basalt should also have such a dependency, but it is not seen in our experiments. Ice is more volatile than basalt, and so the effect is easily observed. The impact velocity dependency in basalt would appear if the induced shock pressure were high enough to cause melting and vaporization. The attenuation rate of −2 assumed in Melosh et al. (1992) is consistent with our result. Ahrens and O’Keefe (1977) showed some figures of peak pressure (P) versus distance (x) normalized by spherical projectile radius (r ) which had a much larger isobaric region. Pierazzo et al. (1997) showed a very similar feature for the attenuation. However, in their calculation, a spherical impactor was used and the shock pressure was much higher. Thus direct comparison and discussion are difficult. Mitani et al. (1995) constructed a shock wave propagation model in an elastic–plastic solid. To investigate shock attenuation, they assumed an elastic stage and a plastic stage, in both compression and release processes. They calculated elastic and plastic wave velocities at each pressure and concluded that, for the case of peak pressures lower than the HEL, the relation between peak pressure (P) and distance (x) in the plane-parallel wave case was P ∝ x−0.8 –x −1.1 , and in the spherical case it was P ∝ x −2.2 . These values are consistent with our result of P ∝ x−1.7 in the 7-GPa experiment. They also calculated the attenuation rate at higher pressure than the HEL; however, their result of P ∝ x−3 was incompatible with ours. When our results are applied to elastic theory, the highpressure behavior of basalt can be estimated. Nakazawa et al. (1997) calculated the pressure derivative of the bulk modulus (K ) to be 5 using the coefficient of Us –u p Hugoniot equation. Assuming that the bulk modulus (K ) is linearly related to the shock pressure (P), elastic theory shows that the bulk sound velocity (C) can be expressed as C=
K = ρ
K0 + K P , ρ
where K 0 and ρ are the mean bulk modulus at zero pressure and density, respectively. The typical bulk modulus of 35 ± 10 GPa at zero pressure (K 0 ) of Kinosaki basalt was theoretically estimated
548
NAKAZAWA ET AL.
by longitudinal and shear wave velocities measured before the experiments. The calculated bulk sound velocity (C) at 7 GPa of 4.8 ± 0.3 km s−1 is lower than the measured reflected rarefaction wave velocity (Ur ) at 7 GPa of 6.2 km s−1 in our experiments. The difference suggests that the rigidity (µ) at 7 GPa has a finite value. Since Ur is expressed as Ur =
K 0 + K P + 43 µ , ρ
a µ of 35 ± 7 GPa can be roughly estimated. A bulk sound velocity (C) at 31 GPa of 6.9 ± 0.2 km s−1 was calculated in the same way. The calculated velocity is close to the measured reflected rarefaction wave velocity (Ur ) of 6.7 km s−1 . This means that the rigidity (µ) at 31 GPa is almost 0 ± 3 GPa. The shear-wave velocity of basalt gives a rigidity at 0 GPa (µ0 ) of 19 ± 5 GPa (Fig. 10). Though there are only three data points, it is estimated that basalt has higher rigidity at 7 GPa than at 0 GPa and that it has almost no rigidity at 31 GPa. Indeed there were large errors; however, this behavior agrees with what one expects of elastic–plastic material (Graham 1993), which has maximum rigidity at the HEL and has no rigidity at high pressure. This behavior is important in estimating rarefaction velocity at high pressure and in calculating shock wave attenuation. According to the Rankine–Hugoniot equation, the shock pressure (P) is approximately proportional to the particle velocity (u p ) at low pressure. Arakawa et al. (1995) have reported that
FIG. 11. The previous ejection velocities are superimposed on our results. The ejection velocity (u ej ) is normalized by twice the initially induced particle velocity (up0 ), and the shock pressure (P) is normalized by the initially induced pressure (P0 ). Thick impactors were used in these experiments. All plots are located along a single curve.
the ejection velocity of impact fragments (u ej ) is almost equal to the free surface velocity (u fs ), which is known to be twice the particle velocity (u p ): P ∝ u p ≈ u fs /2 ≈ u ej /2. Therefore, P and u ej must decay at the same attenuation rate. When the initially induced shock pressure (P0 ) and the initially induced particle velocity (u p0 ) are constant, the following relation can be given at the distance x: P(x)/P0 = u ej (x)/2u p0 .
FIG. 10. The rigidity (µ) of basalt is calculated by the reflected rarefaction wave velocity with assuming that basalt behaves as an elastic–plastic material.
Fujiwara and Tsukamoto (1980) and Takagi et al. (1992) measured ejection velocities of fragments in the target antipode of basalt. The normalized shock pressure (P/P0 ) and the normalized ejection velocity (u ej /2u p0 ) are compared in Fig. 11. Thick cylindrical projectiles were used in all these experiments. The propagation distance was normalized by the projectile radius to compensate for the difference in the projectile radius. The shock waves in the three experiments were attenuated by the sum of the edge-wave effect and the geometrical expansion effect. All plots are located along a single curve. The results show that the ejection velocity of the impact fragments from the target antipode is attenuated with the power of the target size and that it could be estimated by the shock pressure attenuation rate. To apply the experimental results to natural impact phenomena, Mizutani et al. (1990) introduced the “isobaric spherical
SHOCK WAVE ATTENUATION IN BASALT
549
core” in which the volume was equal to the projectile volume, and they derived the “nondimensional impact stress (PI )” as a scaling parameter of the impact phenomena as PI =
P0 3 a L L , Yt p t
where a, L t , L p , and Yt are the mean attenuation rate, target size, projectile size, and fracture strength, respectively. They assumed an attenuation rate of −3 and scaled the experimental largest fragment mass with the parameterPI . However, the attenuation rate should be estimated by considering each influential attenuation effect. Especially near the isobaric core, the attenuation rate should not be −3 and the scaling parameter PI must be improved. The improved parameter PI with the largest fragment mass is shown in Fig. 12. The original data were extracted from the experimental results on basalt by Takagi et al. (1984), which satisfied the condition 1 < L t /r < 10, where r is the projectile radius, and the improved parameter PI was calculated with the attenuation rate of −1.7. The plots with the improved parameter show a more concentrated distribution, which shows that our attenuation rate is appropriate for improving the impact scaling law. Mizutani et al. (1990) also showed from the experimental results on basalt by Takagi et al. (1984) that the slope of the size distribution curves of fragments in each regime, γ 1 , γ 2 ,
FIG. 13. The slope of the size distribution curve of fragments in each regime vs previous and renewed PI . The distribution of the renewed plots was concentrated and shifted.
FIG. 12. The largest fragment mass vs the previous parameter PI and the improved parameter. The original data are from Takagi et al. (1984). The renewed parameter was calculated using the attenuation rate of −1.7. The renewed parameter can concentrate the data and can improve the impact scaling law.
and γ 3 , could be given as a function of PI calculated by the attenuation rate of −3. However, our results showed that PI should be calculated using the attenuation rate of −1.7 and the function of the slope should be renewed, which is shown in Fig. 13. The distributions of the plots have been concentrated and have shifted. To estimate the volume of impact melt for 14 terrestrial impact craters, Grieve and Cintala (1992) proposed a shock wave attenuation model. According to their model, the attenuation rate was −2 when the shock pressure was higher than the HEL, and it decreased to −1 when the pressure was lower than the HEL. Although the material in their model was not specified
550
NAKAZAWA ET AL.
and a detailed comparison was difficult, our experimental results are consistent with their model for high pressure. However, our experimental attenuation rate for pressures lower than the HEL is still close to −2 and no transition was detected. Their estimated shock pressure in the far field was higher. However, since it is in the high pressure field, i.e., in the near field, that the melt deposit occurs, the estimated volume of impact melt is barely changed. SUMMARY
We performed impact experiments using cylindrical impactors and showed constant attenuation rates from near to far field, up to 30 times the projectile radius. The results could be explained by two attenuation effects: the rarefaction wave and geometrical expansion effects. The measured attenuation rate of basalt was constant within the experimental condition, up to 31 GPa of the initial shock pressure. The experimental results show that basalt behaves as an elastic–plastic material. According to our results, the scaling law of nondimensional impact stress (PI ) has been improved. ACKNOWLEDGMENTS We are deeply grateful to Dr. M. Arakawa of Hokkaido University for useful discussions and his encouragement. We acknowledge Dr. A. Fujiwara of ISAS and Dr. T. Kadono of Nagoya University for useful advice in developing the two-stage light-gas gun. We also greatly appreciate the technical help and suggestions of Mr. T. Masuda, Mr. K. Suzuki, and Ms. C. Miwa at the Instrument Development Center of Nagoya University. We thank Dr. M. Higa, Mr. K. Shirai, Mr. T. Kiyono, and Mr. S. Fujinami for their help with the experiments.
REFERENCES Ahrens, T. J., and J. D. O’Keefe 1977. Equation of state and impact-induced shock-wave attenuation on the moon. In Impact and Explosion Cratering (D. J. Roddy, R. O. Pepin, and R. B. Merrill, Ed.), pp. 639–656. Pergamon, New York. Arakawa, M., N. Maeno, M. Higa, Y. Iijima, and M. Kato 1995. Ejection velocity of ice impact fragments. Icarus 118, 341–354. Asphaug, E. 1997. Impact origin of the Vesta family. Meteorit. Planet. Sci. 32, 965–980. Dence, M. R., R. A. F. Grieve, and P. B. Robertson 1977. Terrestrial impact structures: Principal characteristics and energy considerations. In Impact and Explosion Cratering (D. J. Roddy, R. O. Pepin, and R. B. Merrill, Ed.), pp. 247–275. Pergamon, New York. Dienes, M. R., and J. M. Walsh 1970. Theory of impact: Some general principles and the methods of Eulerian codes. In High Velocity Impact Phenomena (R. Kinslow, Ed.), pp. 45–104, Academic, San Diego. Fujiwara, A., and A. Tsukamoto 1980. Experimental study on the velocity of fragments in collisional breakup. Icarus 44, 142–153.
Fujiwara, A., G. Kamimoto, and A. Tsukamoto 1977. Destruction of basaltic bodies by high-velocity impact. Icarus 31, 277–288. Grady, D. E., W. J. Murri, and P. S. De Carli 1975. Hugoniot sound velocity and phase transformations in two silicates. J. Geophys. Res. 80, 4857–4861. Graham, R. A. 1993. In Solid under High-Pressure Shock Compression, pp. 31–36. Springer-Verlag, Berlin. Grieve, R. A. F., and M. J. Cintala 1992. An analysis of differential impact meltcrater scaling and implications for the terrestrial impact record. Meteoritics 27, 526–538. Kieffer, S. W., and C. H. Simonds 1980. The role of volatiles and lithology in the impact cratering process. Rev. Geophys. Space Phys. 18, 143–181. Marsh, S. P. 1980. In LASL Shock Hugoniot Data (S. P. Marsh, Ed.), pp. 57–60. Univ. of California Press, Berkeley. Melosh, H. J. 1989. Cratering mechanics: Excavation stage. In Impact Cratering, pp. 60–68. Oxford Univ. Press, New York. Melosh, H. J., E. V. Ryan, and E. Asphaug 1992. Dynamic fragmentation in impacts: Hydrocode simulation of laboratory impacts. J. Geophys. Res. 97, 14735–14759. Meyer, M. A. 1994. Shock wave attenuation, interaction, and reflection. In Dynamic Behavior of Materials, pp. 179–201. Wiley-Interscience, New York. Mitani, N., S. Ida, and S. Watanabe 1995. Numerical simulation of shock attenuation in solids. In Proc. 28th ISAS Lunar and Planetary Symp. (M. Shimizu and H. Mizutani, Eds.). Inst. Space and Aeronautical Science, pp. 98–101. Sagamihara. Mizutani, H., Y. Takagi, and S. Kawakami 1990. New scaling laws on impact fragmentation events. Icarus 87, 307–326. Nakamura, A., and A. Fujiwara 1991. Velocity distribution of fragments formed in a simulated collisional disruption. Icarus 92, 132–146. Nakazawa, S., K. Shirai, S. Watanabe, M. Kato, Y. Iijima, T. Kobayashi, and T. Sekine 1997. Hugoniot equation of state of basalt. Planet. Space Sci. 45, 1489–1492. Ohtsuki, K., S. Ida, Y. Nakagawa, and K. Nakazawa 1993. Planetary accretion in the solar gravitational field. In Protostars and Planets 3 (E. H. Levy and J. I. Lunine Eds.), pp. 1089–1108. Univ. of Arizona Press, Tucson. Pierazzo, E., A. M. Vickery, and H. J. Melosh 1997. A reevaluation of impact melt production. Icarus 127, 408–423. Sorrell, F. Y., and T.-M. Kuo 1992. Dynamic response of moist soil to shock loading. In Shock Compression of Condensed Matter 1991 (S. C. Schmidt, R. D. Dick, J. W. Forbes, and D. G. Tasker, Eds.), pp. 505–508. Elsevier, Amsterdam. Takagi, Y., H. Mizutani, and S. Kawakami 1984. Impact fragmentation experiments of basalts and pyrophyllites. Icarus 59, 462–477. Takagi, Y., M. Kato, and H. Mizutani 1992. Velocity distribution of fragments of catastrophic impacts. In Asteroids, Comets, Meteors 1991 (A. W. Harris and E. Bowell, Eds.), pp. 597–600. Lunar Planet. Inst., Houston. Waza, T., T. Matsui, and K. Kani 1985. Laboratory simulation of planetesimal collision 2. Ejecta velocity distribution. J. Geophys. Res. 90, 1995–2011. Yoshida, M., and N. N. Thadhani 1992. Study of shock induced solid state reactions by recovery experiments and measurements of Hugoniot and sound velocity. In Shock Compression of Condensed Matter 1991 (S. C. Schmidt, R. D. Dick, J. W. Forbes, and D. G. Tasker, Eds.), pp. 585–592. Elsevier, Amsterdam.
Experimental Investigation of Shock Wave Attenuation in Basalt S. Nakazawa1 Institute of Space and Astronautical Science, Sagamihara 229-8510, Japan E-mail: [email protected]
S. Watanabe Department of Earth and Planetary Sciences, Nagoya University, Nagoya 464-8602, Japan
and Y. Iijima and M. Kato Institute of Space and Astronautical Science, Sagamihara 229-8510, Japan Received December 3, 1998; revised July 16, 2001
Shock compression experiments on Kinosaki basalt were carried out in the interest of studying collisional phenomena in the solar nebula. Shock waves of 7 and 31 GPa were generated using a thin flyer plate, and a shock wave of 16 GPa was generated using a thick cylindrical projectile. By employing in-material manganin and carbon pressure gauges, the shock wave attenuation was examined and the propagation velocities of the shock wave and rarefaction wave were measured. The attenuation mechanism consists of two effects: the rarefaction wave and geometrical expansion. The rarefaction effect includes the reflected wave and the edge wave. The efficiency of these mechanisms depends on the geometry of the projectile, initially induced pressure, and materials of the target and projectile. As a result of the experiments, a cylindrical impactor created an isobaric region of size almost equal to the projectile radius. The shock wave in the far field was attenuated similarly with the power of −1.7 to −1.8 of the propagation distance under our experimental conditions. The shock wave generated using a thin flyer plate was attenuated by the rarefaction wave generated on the back surface of the flyer plate and by geometrical expansion effects. The shock wave generated using a thick projectile was attenuated by edge-wave and by geometrical expansion effects. According to elastic theory, the rigidity of basalt at 7 and 31 GPa was calculated as 35 ± 7 and 0 ± 3 GPa, respectively, using the measured rarefaction wave velocities. The decayed shock pressure was related to the ejection velocity of the impact fragments, which were obtained in previous disruption experiments. The attenuation rates in previous experiments were consistent with ours. The previous impact scaling parameter called “nondimensional impact stress (PI)” has been improved. c 2002 Elsevier Science (USA)
Key Words: cratering; planetary formation; impact processes; planetesimals.
INTRODUCTION
It is widely accepted that planetesimals were formed by the repeated collisions of rocks with each other and accreted to grow into planetary bodies in the early solar nebula (e.g., Ohtsuki et al. 1993). Although the impact process of rocks consisting of polycrystals of some minerals is very complex, it must be studied to clarify the formation process of our planets. Basalt can be employed as one of the materials analogous to planetesimal and planetary bodies. Some impact disruption experiments with basalt have been performed by Fujiwara et al. (1977), Fujiwara and Tsukamoto et al. (1980), Takagi et al. (1984), Waza et al. (1985), and Nakamura and Fujiwara (1991). They collected impact fragments to measure their size distribution or took highspeed images to measure their ejection velocities. The disruption and fragment ejection are the results of shock waves which travel and decay through the target body. To explain their experimental results and to understand the physical process of impact phenomena, especially for the formation process of the planets, shock pressure attenuation must be investigated quantitatively. Some investigators have studied shock pressure attenuation observationally or theoretically. Dence et al. (1977) observationally examined the distribution of shock metamorphism in rocks below natural impact craters. They estimated the displacement of rocks in impact craters using the results of underground nuclear explosions and estimated the pressure attenuation as P ∝ x −2 – x −4.5 , where P and x are shock pressure and propagation distance, respectively. Dienes and Walsh (1970) and Ahrens and O’Keefe (1977) theoretically examined the regime of melting
1 Current address: National Space Development Agency of Japan, Tsukuba 305-8505, Japan. Fax: 81-298-68-2984. E-mail: [email protected].
539 0019-1035/02 $35.00 c 2002 Elsevier Science (USA) All rights reserved.
540
NAKAZAWA ET AL.
and vaporization and the crater formation for a spherical projectile impact using a hydrodynamics code. Their calculations showed the pressure–distance relation as P ∝ x−3 and P ∝ x −1.5 –x −3 , respectively. Based on their results, Melosh (1989), Melosh et al. (1992), and Asphaug (1997) employed the assumption of P ∝ x−2 in their hydrodynamics code simulation of impact fragmentation and impact cratering. Kieffer and Simonds (1980) considered that the energy was transferred from the projectile into the spherical region traversed by the shock wave and that the energy was distributed equally between kinetic energy and internal energy. Their analytical investigation resulted in P ∝ x −3 – x −3.6 . There is a large variation in the attenuation rates of these investigations. Natural crater observations showed some uncertainty in estimations of the displacement of rocks in the impact crater and of the pressure experienced. On the other hand, some of the theoretical studies discussed only the quantity of energy, without considering the mechanism of shock wave attenuation. Considering the mechanism of shock wave attenuation, it can be assumed that shock wave decay is influenced by the rarefaction wave and geometrical expansion (Fig. 1). Two rarefaction waves must be considered in the impact phenomena. One is the reflected rarefaction wave which is generated at the back surface of the projectile (Grady et al. 1975, Meyer 1994), and the other is the edge wave which is generated at the front edge of the projectile at the moment of impact. The second geometrical expansion effect means that the shock wave propagates hemispherically far from the impact point and decays while con-
FIG. 1. Schematic shock wave attenuation mechanisms: (1) rarefaction wave effect and (2) geometrical expansion effect. The rarefaction wave consists of two waves, the reflected rarefaction wave and the edge wave. In the far field, the shock wave propagates hemispherically and decays while conserving its wave energy.
serving its wave energy. This effect attenuates shock pressure (even higher than the Hugoniot elastic limit or HEL) with the power of −1 of the propagation distance (e.g., Melosh 1989), similar to the case of the spherical elastic wave (lower than the HEL). This effect is at work in the far field under any experimental condition. EXPERIMENTAL PROCEDURE
A single-stage powder gun with a 30-mm bore and a twostage light-gas gun with a 10-mm bore at Nagoya University were employed in the experiments. The experimental assembly is shown in Fig. 2. We measured shock pressure using an in-material shock pressure gauge with constant impact velocity and varying target thickness to investigate the mechanism of shock wave attenuation. Three types of copper impactors with cylindrical polycarbonate sabot were fired at the basalt target: a 3-mm-thick flyer plate with 28-mm diameter, a 1-mm one with 9-mm diameter, and a 9-mm one with 9-mm diameter. Each sabot was put in contact with a flyer plate. The aspect ratios of the two flyer plates were nearly the same. Although spherical projectiles are impacted in the natural phenomena, cylindrical projectiles were used in the experiments in order to distinguish the attenuation mechanism. A low-impedance manganin gauge (Mn10-0.05-FEP or Mn2-0.10-FEP; Dynasen Inc.), a 50- piezoresistive manganin gauge (Mn4-50-EK or Mn8-50-EK; Dynasen Inc.), or a 50- piezoresistive carbon gauge (C300-50-EKRTE; Dynasen Inc.) was sandwiched between two target plates to measure shock pressure. These gauges were used by Yoshida and Thadhani (1992), Sorrell and Kuo (1992), and Nakazawa et al. (1997). The calibration of the gauge, established by Dynasen Inc., showed only 5% error in shock pressure determination. Two gauges were used for most shots, and three or four gauges were used for thick targets in some 7-GPa experiments. Basalt from Kinosaki, Hyogo, Japan, was used for the targets in the experiments. Although the samples were quarried from a columnar rock, their physical properties varied from sample to sample. The sound velocities of each pre-shock sample were measured by the ultrasonic transmission method using a piezoelectric transducer. They ranged from 4.15 to 5.15 km s−1 for longitudinal waves and from 2.25 to 3.00 km s−1 for shear waves, which yield bulk sound velocities from 3.07 to 4.08 km s−1 . The measured initial bulk density of the basalt ranged from 2.63 to 2.74 g cm−3 , and the measured true density in crushed samples ranged from 2.85 to 2.89 g cm−3 , from which the porosity was calculated to be 4 to 8%. The target plates were polished parallel and flat to an accuracy of 0.03 mm, and the tilt of the target was set at under 0.3 degree. Target plates were cut large enough so that the wave reflected at the target side did not catch up with the shock wave. They were cut into rectangles of 50 × 80 mm for less than near field experiments (less than 50-mm distance) and into rectangles of about 200 × 200 mm for far field experiments (more than 50-mm distance).
SHOCK WAVE ATTENUATION IN BASALT
FIG. 2.
541
Illustration of attenuation experimental procedure. Three types of impactors and one or two shock pressure gauges were used in the experiment.
The HEL of basalt is 5 GPa according to Nakazawa et al. (1997). To investigate shock wave attenuation at pressures lower than the HEL, a 7-GPa shock wave was generated by firing a 3-mm-thick copper flyer plate with the impact velocity of 0.7 km s−1 . To investigate the attenuation at pressures higher than the HEL, a 31-GPa shock wave was generated by firing a 1-mmthick copper flyer plate with the impact velocity of 2.6 km s−1 . To investigate attenuation by a thick impactor, a 9-mm-thick copper projectile was fired with the impact velocity of 1.6 km s−1 to induce 16 GPa. The impact velocity was measured by the magnet flyer method or the wire cutting method. Because the
sabot is put into contact with the flyer plate, the reflected rarefaction wave does not release to zero pressure. However, it is weaker than the original shock wave and whether it has prevented the decay of the shock wave can be confirmed by checking the waveform. We also conducted experiments for measuring edge-wave velocity. The flyer plate and the experimental condition were the same as those in the 7-GPa experiments. Two gauges were sandwiched between two target plates; one was at the center of the target and the other at 7 mm off-center (Fig. 3). In the near field, the shock front is parallel to the target surface and the shock wave and reflected rarefaction wave should propagate through the target to reach the two gauges at the same moment. However, the edge wave should reach the off-center gauge before it reaches the center gauge. The edge-wave velocity was measured at the time when the shock pressure began to release at the off-center gauge. EXPERIMENTAL RESULTS
7-GPa and 31-GPa Thin Flyer Plate Experiments Experimental conditions and results are summarized in Table I. The initially induced shock pressure on the impact surface (P0 ) was estimated by the calculated particle velocity,
FIG. 3. Illustration of edge-wave experimental procedure. Two pressure gauges were sandwiched by two target plates to measure edge-wave velocity.
P0 = ρ0 flyer (C0 flyer + sflyer (vi − up0 basalt ))(vi − up0 basalt ) = ρ0 basalt (C0 basalt + sbasalt up0 basalt )up0 basalt ,
542
NAKAZAWA ET AL.
TABLE I Lists of Experimental Conditions and Results
Run No.
Propagation distance (x) (mm)
Normalized distance (x/r )
Bulk sound velocity (C0 ) (km s−1 )
Bulk density (ρ0 ) (g cm−3 )
Porosity (%)
Impact velocity (Vi ) (km s−1 )
Shock pressure (P) (GPa)
Initial pressure (P0 ) (GPa)
Normalized pressure (P/P0 )
7-GPa experiment 1 2 3 4 5 5 6 7 8 9 10 11 12 13 14 15 15 16 16 16 16
2.87 11.98 21.21 24.90 8.85 18.11 3.19 4.27 3.15 10.98 15.45 40.87 12.37 31.34 14.19 133.41 178.44 87.32 210.71 386.83 473.18
0.21 0.86 1.52 1.78 0.63 1.29 0.23 0.30 0.22 0.78 1.10 2.92 0.88 2.24 1.01 9.53 12.75 6.24 15.05 27.63 33.80
3.81 3.81 3.81 3.81 3.81 3.81 3.56 3.81 3.81 3.53 3.55 3.31 3.07 3.21 3.17 3.13 3.13 3.10 3.10 3.10 3.10
2.74 2.74 2.74 2.74 2.74 2.74 2.67 2.67 2.74 2.67 2.67 2.70 2.72 2.63 2.70 2.65 2.65 2.65 2.65 2.65 2.65
4 4 4 4 4 4 7 7 4 7 7 6 5 8 5 8 8 8 8 8 8
0.68 0.75 0.70 0.76 0.70 0.69 0.68 0.68 0.68 0.74 0.69 0.74 0.73 0.74 0.78 0.83 0.83 0.83 0.85 0.85 0.85
7.2 7.2 3.3 3.3 7.3 4.3 7.1 7.0 7.2 7.4 5.7 1.1 6.2 1.5 6.4 0.13 0.070 0.30 0.062 0.025 0.018
7.1 7.9 7.4 8.0 7.4 7.3 7.1 7.1 7.1 7.8 7.2 7.8 7.7 7.8 8.2 8.8 8.8 9.0 9.0 9.0 9.0
1.01 0.91 0.45 0.41 0.98 0.59 1.00 0.98 1.01 0.95 0.78 0.13 0.80 0.19 0.78 0.015 0.0080 0.034 0.0069 0.0027 0.0020
31-GPa experiment 17 17 18 18 19 19 20 20 21 21 22
3.00 12.87 8.52 22.55 1.05 11.05 5.36 33.46 8.16 10.12 4.48
0.67 2.86 1.89 5.01 0.23 2.46 1.19 7.43 1.81 2.25 1.00
3.42 3.53 3.24 3.25 3.61 3.35 3.49 3.09 3.41 3.35 3.10
2.74 2.74 2.74 2.74 2.74 2.74 2.74 2.74 2.74 2.74 2.74
4 4 4 4 4 4 4 4 4 4 4
2.70 2.70 2.58 2.58 2.68 2.68 2.36 2.36 2.66 2.66 2.58
31.0 4.0 6.7 1.4 32.5 5.0 16.0 0.75 10.5 5.0 21.0
32.5 32.5 30.6 30.6 32.2 32.2 27.3 27.3 31.9 31.9 30.6
0.95 0.12 0.22 0.046 1.01 0.16 0.59 0.028 0.33 0.16 0.69
16-GPa experiment 23 23 24 24 25 25
3.57 7.22 1.07 11.44 5.08 42.40
0.79 1.60 0.24 2.54 1.13 9.42
3.18 3.13 4.08 3.49 3.58 3.09
2.74 2.74 2.74 2.74 2.74 2.74
4 4 4 4 4 4
1.65 1.65 1.62 1.62 1.57 1.57
16.0 9.0 17.0 3.5 11.5 0.35
17.4 17.4 17.0 17.0 16.4 16.4
0.92 0.52 1.00 0.21 0.70 0.021
Note. r refers to the projectile radius. The initially induced pressure (P0 ) was estimated by the impedance matching method.
where ρ 0 , C0 , s, vi , and up0 are the mean initial bulk density, bulk sound velocity, coefficient of the Us − up relation, impact velocity, and initially induced particle velocity on the impact surface, respectively. In the calculation, the Hugoniot EOS of copper in Marsh (1980) was adopted for the impactor (C0 = 3.91 km s−1 , s = 1.51) and the Hugoniot EOS of basalt in Nakazawa et al. (1997) was adopted for the target (lower pressure: C0 = 4.3 km s−1 , s = 1.6; middle pressure: C0 = 4.9 km s−1 , s = 0.2; higher pressure: C0 = 3.0 km s−1 , s = 1.5). The initial particle velocity (up0 ) was calculated by the impedance matching method using
measured impact velocities and the Hugoniot equation of basalt. The error of the impact velocity was ±5%. Some examples of the shock profile are shown in Fig. 4. They are superimposed to fit the time of impact, which is calculated using the shock wave velocity estimated by the Us –P Hugoniot equation of basalt while taking into account the attenuation of the shock pressure (P) and the change of the shock velocity (Us ). The peak pressure decreased in the far field, and the shock profile appeared deltoid with very short shock duration time. Although a time resolution of the experimental system below 50 ns was confirmed, any
SHOCK WAVE ATTENUATION IN BASALT
543
FIG. 4. The observed shock profiles of the (a) 7-GPa, (b) 31-GPa, and (c) 16-GPa experiments. They are superimposed to fit the time of impact, which is calculated by using the Hugoniot equation of basalt.
steps at the HEL (5 GPa) were not identified in the shock profiles. The reported reason was that heterogeneity of basalt blurred the shock wave and the shock profile showed single-wave structures (Nakazawa et al. 1997). The measured shock pressure attenuation is shown in Fig. 5. The error in determining the normalized shock pressure (P/P0 ) is estimated at about ±10%, and at low pressures (<0.2 GPa)
it is estimated at about ±0.001 (= P/P0 ). The results of the 7-GPa experiment in Fig. 5a are divided into two regions; in region 1 (x < 11 mm), the pressure stays at maximum; in region 2 (x > 15 mm), the pressure attenuates with a power of −1.8 ± 0.2, where the value is called the “attenuation rate.” The transition of the attenuation rate suggests that the attenuation mechanism changed. Though there were fewer data, the data of
544
NAKAZAWA ET AL.
FIG. 5. The measured shock pressure attenuation of the (a) 7-GPa and (b) 31-GPa experiments. The results can be divided into two regions. The attenuation rates were nearly equal to each other.
the 31-GPa experiment in Fig. 5b also seem to be divided into two regions. The attenuation starting distance was at x = 3 mm. The attenuation rate is −1.7 ± 0.2, which is almost the same as in the 7-GPa experiment. The times when the pressure profile increased and then decreased indicate the arrival times of the shock wave and the rarefaction wave, respectively. The shock wave arrival time (tarrival ), the time when the shock pressure reached peak pressure (tpeak ), and the time when the shock pressure began to release (trelease ) versus propagation distance are plotted in Fig. 6 (x–t diagram). The time in the plots had error of ±0.1 µs. The time of impact (t0 ) was calculated by integrating the Us –x relation, which was estimated by the P–x relation and the Us –P Hugoniot taking into account the changing shock velocity (Us ). Because of large errors in estimating the time of impact, the data in the far field (>87.3 mm) were not plotted. The shock duration time, that is, the time lag between tpeak and trelease , became shorter in the near field (<14.2 mm) as the shock wave propagated. In these experiments, since the flyer plate was thin enough, i.e., one-ninth of the diameter, the reflected rarefaction wave caught up with the shock wave and decayed earlier than the edge wave. The reflected rarefaction wave caught up with the 7-GPa shock wave at about 12 mm (x/r = 0.86). In the 31-GPa experiment, there are only two data points in the near field (<3 mm) and it is impossible to estimate accurately the rarefaction wave velocity. However, the shock pressure at 3.0 mm (No. 17) was the same as the initial induced pressure (that is, the shock wave had not yet been attenuated) and the shock du-
ration time was short (that is, the reflected rarefaction wave already caught up with the shock wave). So, it can be concluded that the reflected rarefaction wave had just caught up with the shock wave at almost 3-mm distance. The distance at which the reflected rarefaction wave catches up with the shock wave depends on the thickness of the flyer plate and the shock wave velocity. The slope of trelease in Fig. 6 indicates the reflected rarefaction wave velocity. It is shown that the reflected rarefaction wave ran after the shock wave with higher velocity; velocities (Ur ) at 7 and 31 GPa were measured as 6.2 ± 0.3 and 6.7 ± 0.7 km s−1 , respectively, referred to Lagrangian coordinates. Edge-Wave Experiment and 16-GPa Cylindrical Thick-Projectile Experiment In the edge-wave experiments, two shock profiles arose at the same time and the release of the off-center gauge started earlier than that of the center gauge (Fig. 7). Since the stress gauge has a common problem, that the slope of the release profiles are not reproducible, only the release starting times were used in the analysis. The different release times indicate that the propagating direction of the rarefaction wave was not perpendicular to the gauge plane. Accordingly, the release on the off-center gauge was caused not by the rarefaction wave reflected at the back surface of the projectile but by the rarefaction wave generated at the front edge of the projectile. Assuming that the edge wave propagated spherically from the edge, the edge-wave velocity (Ue ) of 4.6 ± 0.2 km s−1 was measured in all edge-wave experiments
SHOCK WAVE ATTENUATION IN BASALT
545
FIG. 6. X–t diagram of the (a) 7-GPa and (b) 31-GPa experiments. The shock wave arrival time (t arrival ), the time when the shock pressure reached peak pressure (tpeak ), the time when the shock pressure began to release (t release ), and the calculated edge-wave arrival time are plotted. The errors are generally smaller than the symbol size. The slope of the curve indicates the wave velocity. The figure shows that the rarefaction wave runs after the shock wave with higher velocity and catches up to it.
(Table II) by using the arrival time and the distances between the edge and the off-center gauge. Even when the error of the edgewave velocity was included, it was calculated that the edge wave could decay the shock wave at the center at distances farther than 25 mm (dashed line in Fig. 6a). Therefore, the edge wave could not decay the peak pressure in the 7-GPa experiment. The results of the 16-GPa experiment with a thick cylindrical projectile are shown in Fig. 8. The P–x diagram (Fig. 8a) is similar to the others (Fig. 5). The attenuation starting distance of 3.5 mm and the attenuation rate of −1.7 are almost the same as in the thin flyer plate experiment. Though the x–t diagram (Fig. 8b) also seems the same as the others, the attenuation mechanism was much different from the others. The projectile was so thick that the reflected rarefaction wave would take more than 2 µs to catch up to the shock wave, arriving later than the edge wave. Assuming the edge-wave velocity at 16 GPa is 6.5 km s−1 (a little smaller than the rarefaction wave velocity at 31 Gpa), the calculated arrival time of the edge wave is shown in Fig. 8b (dashed curve). Since the curve well matched the release line, it is confirmed that the edge wave, not the reflected rarefaction wave, with a propagation velocity of about 6.5 km s−1 attenuated the shock wave in the 16-GPa experiments. FIG. 7. The observed shock profiles of one of the edge-wave experiments. Two profiles arose at the same time, and the release of the off-center gauge started earlier than that of the center gauge. The edge-wave velocity of 4.6 km s−1 was measured by t release and the distance between the gauge and the impactor’s edge.
DISCUSSION AND CONCLUSIONS
No dominant attenuation in the shock pressure was detected for x < r (region 1 in Figs. 5a and 5b). This suggested that the
546
NAKAZAWA ET AL.
TABLE II Lists of Conditions and Results of Edge-Wave Experiments
Run No.
Propagation distance (x) (mm)
Normalized distance (x/r )
Bulk sound velocity (C0 ) (km s−1 )
Bulk density (ρ0 ) (g cm−3 )
Porosity (%)
Impact velocity (Vi ) (km s−1 )
Shock pressure (P) (GPa)
Initial pressure (P0 ) (GPa)
Edge-wave velocity (Ue ) (km s−1 )
8 9 10 12
3.15 10.98 15.45 12.37
0.23 0.78 1.10 0.88
3.81 3.53 3.55 3.07
2.74 2.67 2.67 2.72
4 7 7 5
0.68 0.74 0.69 0.73
7.3 6.5 4.1 4.8
7.1 7.1 7.2 7.5
4.75 4.64 4.43 4.61
Note. Edge-wave velocity (Ue ) was measured as the distance between the projectile front edge and the gauge divided by the edge-wave arrival time.
rarefaction wave did not catch up with the shock wave and that the geometrical expansion effect did not affect it in the near field. In the far field, the x–t diagram for the 7-GPa experiment (Fig. 6a) shows that the reflected rarefaction wave caught up with the shock front at 12 mm in propagation distance. Otherwise a spherical shock wave would have decayed in the far field (e.g., Melosh, 1989). Therefore, the attenuation in region 2 was caused by the sum of the geometrical expansion and the reflected rarefaction wave effects. There is a transition region between regions 1 and 2. However, we do not have enough data to characterize the mechanism. In the 31-GPa experiment, the reflected rarefaction wave caught up with the shock wave at 3 mm, and shock pressure started to decay beyond 3 mm distance. The geometrical expansion effect should also be included beyond the impactor radius as in the 7-GPa experiment. So, the attenuation farther than 3 mm
was caused by the sum of the geometrical expansion and the reflected rarefaction wave effects, which is the same mechanism as in the 7-GPa experiment. Because of the geometry (aspect ratio) of the impactor, the distance where the reflected rarefaction wave caught up with the shock wave was nearly equal to the impactor radius. Figure 8b shows that the shock wave generated by the thick impactor was attenuated by the edge wave. In the far field, the geometrical expansion effect is also at work. Accordingly, it is confirmed that the sum of the edge-wave and the geometrical expansion effects attenuate the shock wave. Though the impact velocities and the projectiles sizes were different in each experiment, the normalized shock pressures are shown in Fig. 9 for comparison. The distance (x) was normalized by the impactor radius (r ) and the measured shock pressure (P) was normalized by the initially induced shock
FIG. 8. (a) The measured shock pressure attenuation and (b) x–t diagram of the 16-GPa thick-projectile experiment. Though the features of the result are similar to those of the other thin flyer experiment, the sum of the edge-wave and geometrical effects attenuated the shock wave.
547
SHOCK WAVE ATTENUATION IN BASALT
FIG. 9. Normalized shock pressure vs normalized propagation distance for all experiments. Though the induced shock pressure and the attenuation mechanism were different, the shock pressures attenuate similarly, and the size of the all isobaric region was the same as the projectile radius.
pressure (P0 ), which was calculated by the impedance matching method. Although the data P/P0 have an error of ±10%, the figure shows that the size of the isobaric core is almost the same as the impactor size and that the attenuation rate in the far field is also the same even though the attenuation mechanisms are different. It is true that the size of the isobaric region typically depends on the projectile geometry and the impact velocity. If a thinner flyer plate were impacted, the size would be smaller by the reflected rarefaction wave effect. However, the edge-wave effect limits its upper size. In the case of a fully thick flyer plate, the edge wave catches up with the shock wave at first and the flyer plate thickness has little influence on the size. Under our experimental conditions, the upper size of the isobaric region would be seen, and it is nearly equal to the impactor radius. A spherical impactor, however, would cause another result. In the case of the spherical impactor, the edge wave can catch up with the original shock wave at the time of impact, in contrast to the result for a cylindrical impactor. And, it is uncertain whether the attenuation rate in our results could be applied to the case of a spherical impactor. According to our experimental results, the attenuation mechanism of a cylindrical impactor is as follows. The distance at which the geometrical expansion effect starts depends on the projectile radius, and that of the edge-wave effect also depends on the projectile radius. Also, the distance at which the reflected wave starts depends on projectile thickness. Of course, it also
depends on the rarefaction wave velocity, which is characterized by the material. When the projectile is thick enough, the reflected rarefaction wave reaches the shock wave after the edge wave and has no effect on peak pressure attenuation. In this case the shock wave is attenuated by the sum of the edge-wave and geometrical expansion effects. As a result of this mechanism, the shock pressure remains constant in the near field, and the shock pressure decays with a power of −1.7 of propagation distance in the far field. The attenuation rate in basalt is barely influenced by either the initial shock pressure-up to 31 Gpa or the projectile thickness. Although the impact condition dependency could not be identified in basalt, Arakawa et al. (1995) showed that the attenuation rate in ice depended on the impact velocity and that the rate was different from that of basalt. Basalt should also have such a dependency, but it is not seen in our experiments. Ice is more volatile than basalt, and so the effect is easily observed. The impact velocity dependency in basalt would appear if the induced shock pressure were high enough to cause melting and vaporization. The attenuation rate of −2 assumed in Melosh et al. (1992) is consistent with our result. Ahrens and O’Keefe (1977) showed some figures of peak pressure (P) versus distance (x) normalized by spherical projectile radius (r ) which had a much larger isobaric region. Pierazzo et al. (1997) showed a very similar feature for the attenuation. However, in their calculation, a spherical impactor was used and the shock pressure was much higher. Thus direct comparison and discussion are difficult. Mitani et al. (1995) constructed a shock wave propagation model in an elastic–plastic solid. To investigate shock attenuation, they assumed an elastic stage and a plastic stage, in both compression and release processes. They calculated elastic and plastic wave velocities at each pressure and concluded that, for the case of peak pressures lower than the HEL, the relation between peak pressure (P) and distance (x) in the plane-parallel wave case was P ∝ x−0.8 –x −1.1 , and in the spherical case it was P ∝ x −2.2 . These values are consistent with our result of P ∝ x−1.7 in the 7-GPa experiment. They also calculated the attenuation rate at higher pressure than the HEL; however, their result of P ∝ x−3 was incompatible with ours. When our results are applied to elastic theory, the highpressure behavior of basalt can be estimated. Nakazawa et al. (1997) calculated the pressure derivative of the bulk modulus (K ) to be 5 using the coefficient of Us –u p Hugoniot equation. Assuming that the bulk modulus (K ) is linearly related to the shock pressure (P), elastic theory shows that the bulk sound velocity (C) can be expressed as C=
K = ρ
K0 + K P , ρ
where K 0 and ρ are the mean bulk modulus at zero pressure and density, respectively. The typical bulk modulus of 35 ± 10 GPa at zero pressure (K 0 ) of Kinosaki basalt was theoretically estimated
548
NAKAZAWA ET AL.
by longitudinal and shear wave velocities measured before the experiments. The calculated bulk sound velocity (C) at 7 GPa of 4.8 ± 0.3 km s−1 is lower than the measured reflected rarefaction wave velocity (Ur ) at 7 GPa of 6.2 km s−1 in our experiments. The difference suggests that the rigidity (µ) at 7 GPa has a finite value. Since Ur is expressed as Ur =
K 0 + K P + 43 µ , ρ
a µ of 35 ± 7 GPa can be roughly estimated. A bulk sound velocity (C) at 31 GPa of 6.9 ± 0.2 km s−1 was calculated in the same way. The calculated velocity is close to the measured reflected rarefaction wave velocity (Ur ) of 6.7 km s−1 . This means that the rigidity (µ) at 31 GPa is almost 0 ± 3 GPa. The shear-wave velocity of basalt gives a rigidity at 0 GPa (µ0 ) of 19 ± 5 GPa (Fig. 10). Though there are only three data points, it is estimated that basalt has higher rigidity at 7 GPa than at 0 GPa and that it has almost no rigidity at 31 GPa. Indeed there were large errors; however, this behavior agrees with what one expects of elastic–plastic material (Graham 1993), which has maximum rigidity at the HEL and has no rigidity at high pressure. This behavior is important in estimating rarefaction velocity at high pressure and in calculating shock wave attenuation. According to the Rankine–Hugoniot equation, the shock pressure (P) is approximately proportional to the particle velocity (u p ) at low pressure. Arakawa et al. (1995) have reported that
FIG. 11. The previous ejection velocities are superimposed on our results. The ejection velocity (u ej ) is normalized by twice the initially induced particle velocity (up0 ), and the shock pressure (P) is normalized by the initially induced pressure (P0 ). Thick impactors were used in these experiments. All plots are located along a single curve.
the ejection velocity of impact fragments (u ej ) is almost equal to the free surface velocity (u fs ), which is known to be twice the particle velocity (u p ): P ∝ u p ≈ u fs /2 ≈ u ej /2. Therefore, P and u ej must decay at the same attenuation rate. When the initially induced shock pressure (P0 ) and the initially induced particle velocity (u p0 ) are constant, the following relation can be given at the distance x: P(x)/P0 = u ej (x)/2u p0 .
FIG. 10. The rigidity (µ) of basalt is calculated by the reflected rarefaction wave velocity with assuming that basalt behaves as an elastic–plastic material.
Fujiwara and Tsukamoto (1980) and Takagi et al. (1992) measured ejection velocities of fragments in the target antipode of basalt. The normalized shock pressure (P/P0 ) and the normalized ejection velocity (u ej /2u p0 ) are compared in Fig. 11. Thick cylindrical projectiles were used in all these experiments. The propagation distance was normalized by the projectile radius to compensate for the difference in the projectile radius. The shock waves in the three experiments were attenuated by the sum of the edge-wave effect and the geometrical expansion effect. All plots are located along a single curve. The results show that the ejection velocity of the impact fragments from the target antipode is attenuated with the power of the target size and that it could be estimated by the shock pressure attenuation rate. To apply the experimental results to natural impact phenomena, Mizutani et al. (1990) introduced the “isobaric spherical
SHOCK WAVE ATTENUATION IN BASALT
549
core” in which the volume was equal to the projectile volume, and they derived the “nondimensional impact stress (PI )” as a scaling parameter of the impact phenomena as PI =
P0 3 a L L , Yt p t
where a, L t , L p , and Yt are the mean attenuation rate, target size, projectile size, and fracture strength, respectively. They assumed an attenuation rate of −3 and scaled the experimental largest fragment mass with the parameterPI . However, the attenuation rate should be estimated by considering each influential attenuation effect. Especially near the isobaric core, the attenuation rate should not be −3 and the scaling parameter PI must be improved. The improved parameter PI with the largest fragment mass is shown in Fig. 12. The original data were extracted from the experimental results on basalt by Takagi et al. (1984), which satisfied the condition 1 < L t /r < 10, where r is the projectile radius, and the improved parameter PI was calculated with the attenuation rate of −1.7. The plots with the improved parameter show a more concentrated distribution, which shows that our attenuation rate is appropriate for improving the impact scaling law. Mizutani et al. (1990) also showed from the experimental results on basalt by Takagi et al. (1984) that the slope of the size distribution curves of fragments in each regime, γ 1 , γ 2 ,
FIG. 13. The slope of the size distribution curve of fragments in each regime vs previous and renewed PI . The distribution of the renewed plots was concentrated and shifted.
FIG. 12. The largest fragment mass vs the previous parameter PI and the improved parameter. The original data are from Takagi et al. (1984). The renewed parameter was calculated using the attenuation rate of −1.7. The renewed parameter can concentrate the data and can improve the impact scaling law.
and γ 3 , could be given as a function of PI calculated by the attenuation rate of −3. However, our results showed that PI should be calculated using the attenuation rate of −1.7 and the function of the slope should be renewed, which is shown in Fig. 13. The distributions of the plots have been concentrated and have shifted. To estimate the volume of impact melt for 14 terrestrial impact craters, Grieve and Cintala (1992) proposed a shock wave attenuation model. According to their model, the attenuation rate was −2 when the shock pressure was higher than the HEL, and it decreased to −1 when the pressure was lower than the HEL. Although the material in their model was not specified
550
NAKAZAWA ET AL.
and a detailed comparison was difficult, our experimental results are consistent with their model for high pressure. However, our experimental attenuation rate for pressures lower than the HEL is still close to −2 and no transition was detected. Their estimated shock pressure in the far field was higher. However, since it is in the high pressure field, i.e., in the near field, that the melt deposit occurs, the estimated volume of impact melt is barely changed. SUMMARY
We performed impact experiments using cylindrical impactors and showed constant attenuation rates from near to far field, up to 30 times the projectile radius. The results could be explained by two attenuation effects: the rarefaction wave and geometrical expansion effects. The measured attenuation rate of basalt was constant within the experimental condition, up to 31 GPa of the initial shock pressure. The experimental results show that basalt behaves as an elastic–plastic material. According to our results, the scaling law of nondimensional impact stress (PI ) has been improved. ACKNOWLEDGMENTS We are deeply grateful to Dr. M. Arakawa of Hokkaido University for useful discussions and his encouragement. We acknowledge Dr. A. Fujiwara of ISAS and Dr. T. Kadono of Nagoya University for useful advice in developing the two-stage light-gas gun. We also greatly appreciate the technical help and suggestions of Mr. T. Masuda, Mr. K. Suzuki, and Ms. C. Miwa at the Instrument Development Center of Nagoya University. We thank Dr. M. Higa, Mr. K. Shirai, Mr. T. Kiyono, and Mr. S. Fujinami for their help with the experiments.
REFERENCES Ahrens, T. J., and J. D. O’Keefe 1977. Equation of state and impact-induced shock-wave attenuation on the moon. In Impact and Explosion Cratering (D. J. Roddy, R. O. Pepin, and R. B. Merrill, Ed.), pp. 639–656. Pergamon, New York. Arakawa, M., N. Maeno, M. Higa, Y. Iijima, and M. Kato 1995. Ejection velocity of ice impact fragments. Icarus 118, 341–354. Asphaug, E. 1997. Impact origin of the Vesta family. Meteorit. Planet. Sci. 32, 965–980. Dence, M. R., R. A. F. Grieve, and P. B. Robertson 1977. Terrestrial impact structures: Principal characteristics and energy considerations. In Impact and Explosion Cratering (D. J. Roddy, R. O. Pepin, and R. B. Merrill, Ed.), pp. 247–275. Pergamon, New York. Dienes, M. R., and J. M. Walsh 1970. Theory of impact: Some general principles and the methods of Eulerian codes. In High Velocity Impact Phenomena (R. Kinslow, Ed.), pp. 45–104, Academic, San Diego. Fujiwara, A., and A. Tsukamoto 1980. Experimental study on the velocity of fragments in collisional breakup. Icarus 44, 142–153.
Fujiwara, A., G. Kamimoto, and A. Tsukamoto 1977. Destruction of basaltic bodies by high-velocity impact. Icarus 31, 277–288. Grady, D. E., W. J. Murri, and P. S. De Carli 1975. Hugoniot sound velocity and phase transformations in two silicates. J. Geophys. Res. 80, 4857–4861. Graham, R. A. 1993. In Solid under High-Pressure Shock Compression, pp. 31–36. Springer-Verlag, Berlin. Grieve, R. A. F., and M. J. Cintala 1992. An analysis of differential impact meltcrater scaling and implications for the terrestrial impact record. Meteoritics 27, 526–538. Kieffer, S. W., and C. H. Simonds 1980. The role of volatiles and lithology in the impact cratering process. Rev. Geophys. Space Phys. 18, 143–181. Marsh, S. P. 1980. In LASL Shock Hugoniot Data (S. P. Marsh, Ed.), pp. 57–60. Univ. of California Press, Berkeley. Melosh, H. J. 1989. Cratering mechanics: Excavation stage. In Impact Cratering, pp. 60–68. Oxford Univ. Press, New York. Melosh, H. J., E. V. Ryan, and E. Asphaug 1992. Dynamic fragmentation in impacts: Hydrocode simulation of laboratory impacts. J. Geophys. Res. 97, 14735–14759. Meyer, M. A. 1994. Shock wave attenuation, interaction, and reflection. In Dynamic Behavior of Materials, pp. 179–201. Wiley-Interscience, New York. Mitani, N., S. Ida, and S. Watanabe 1995. Numerical simulation of shock attenuation in solids. In Proc. 28th ISAS Lunar and Planetary Symp. (M. Shimizu and H. Mizutani, Eds.). Inst. Space and Aeronautical Science, pp. 98–101. Sagamihara. Mizutani, H., Y. Takagi, and S. Kawakami 1990. New scaling laws on impact fragmentation events. Icarus 87, 307–326. Nakamura, A., and A. Fujiwara 1991. Velocity distribution of fragments formed in a simulated collisional disruption. Icarus 92, 132–146. Nakazawa, S., K. Shirai, S. Watanabe, M. Kato, Y. Iijima, T. Kobayashi, and T. Sekine 1997. Hugoniot equation of state of basalt. Planet. Space Sci. 45, 1489–1492. Ohtsuki, K., S. Ida, Y. Nakagawa, and K. Nakazawa 1993. Planetary accretion in the solar gravitational field. In Protostars and Planets 3 (E. H. Levy and J. I. Lunine Eds.), pp. 1089–1108. Univ. of Arizona Press, Tucson. Pierazzo, E., A. M. Vickery, and H. J. Melosh 1997. A reevaluation of impact melt production. Icarus 127, 408–423. Sorrell, F. Y., and T.-M. Kuo 1992. Dynamic response of moist soil to shock loading. In Shock Compression of Condensed Matter 1991 (S. C. Schmidt, R. D. Dick, J. W. Forbes, and D. G. Tasker, Eds.), pp. 505–508. Elsevier, Amsterdam. Takagi, Y., H. Mizutani, and S. Kawakami 1984. Impact fragmentation experiments of basalts and pyrophyllites. Icarus 59, 462–477. Takagi, Y., M. Kato, and H. Mizutani 1992. Velocity distribution of fragments of catastrophic impacts. In Asteroids, Comets, Meteors 1991 (A. W. Harris and E. Bowell, Eds.), pp. 597–600. Lunar Planet. Inst., Houston. Waza, T., T. Matsui, and K. Kani 1985. Laboratory simulation of planetesimal collision 2. Ejecta velocity distribution. J. Geophys. Res. 90, 1995–2011. Yoshida, M., and N. N. Thadhani 1992. Study of shock induced solid state reactions by recovery experiments and measurements of Hugoniot and sound velocity. In Shock Compression of Condensed Matter 1991 (S. C. Schmidt, R. D. Dick, J. W. Forbes, and D. G. Tasker, Eds.), pp. 585–592. Elsevier, Amsterdam.