To investigate the cratering effects of hypervelocity rod projectile impacting on rocks, a two-stage light gas gun was used to carry out 10 groups of small-scale experiments, whose velocity ranges from 1.5 km/s to 4.1 km/s. After each experiment, the morphology and size of the hypervelocity impacting crater were accurately obtained by using a device for image scanning. According to the morphology of the final crater, the impact crater can be divided into crushing area, spallation area, and radial crack area. Based on the experimental results of steel projectile vertical impacting on granite targets, the relationship between the depth and the diameter of the crater is analyzed, i.e.,
The problem of hypervelocity impact on rocky media originated from the study of Geology and Planetary Science [
Generally, hypervelocity impact is a high-temperature, high-pressure, and high strain-rate process caused by the rapid release of energy. It is convenient to divide cratering processes into two stages, i.e., a relatively short initial high-pressure stage and a longer cratering flow stage [
This cratering problem can be conveniently divided into three regimes. In the “early stage,” the impact velocity and the impedances of the two materials determine the shock propagation speed. In the “intermediate stage,” the pressures are still large compared to either material strength, viscosity, or gravity-induced stresses so that the response is still much like that of an inviscid, compressible fluid. In the “late stage,” the pressures decay down to the kilo bar level or less, where depending upon the problem, material strength, viscosity, or gravity forces act to retard and ultimately stop the crater growth at its final configuration. An interesting phenomenon was discovered that both small-scale impact craters in the laboratory and bowl-shaped craters which are less than 5 km in diameter on the Earth are controlled by strength (of rocks) [
At present, there are lots of investigations on the mechanism and propagation process of hypervelocity impact cratering, including results of small-scale experiments in the laboratory and large-scale meteorite impact events in the nature. However, the physical model and dimensional analysis were always established on the spherical projectile, and a lot of studies did not consider cratering caused by the rod projectile. Actually, hypervelocity kinetic energy weapons are always rod projectiles. Besides, it is nearly impossible to build a theoretical model to calculate hypervelocity rod projectile impact cratering with simplistic constitutive equations. Therefore, it is important to investigate the cratering effects caused by the hypervelocity rod projectile, which should be based on small-scale experiments and dimensional analysis.
A two-stage light gas gun (Figure
Schematic diagram of the experimental system for a two-stage light gas gun.
The projectile has an ogive nose shape whose caliber radius head (CRH) equals to 3, and the diameter and the length are equal to 7.2 mm and 36 mm, respectively (i.e., the length-diameter ratio is 5). The material is 30CrMnSiNiA, the mass of the projectile is 9.67 g, and the density is 7850 kg/m3. The projectile is mounted in a polycarbonate sabot with a diameter of 18 mm. The sabot is formed as a three-lobed structure and a mechanical shelling device (Figure
Photographs of the projectile with sabots.
The target material of all experiments is granite. The parameters of granite measured before the experiment are density, elastic P-wave velocity, uniaxial compressive strength, ultimate shear strength, shear modulus, and Poisson’s ratio. As shown in Figure
Photograph of targets.
The target is made of granite with a density (
There were 10 groups of hypervelocity impact experiments conducted, whose velocity range was 1.5 km/s∼4.1 km/s. At the end of each experiment, the recovery of the projectile and the measurement of the target were carried out. In all the experiments, no residual projectile was found; therefore, erosion damage of projectiles occurred. There were no tensile cracks on the edge of the rock target, which indicated that the target was large enough to be considered as a semi-infinite target without being affected by the boundary reflection wave.
The concepts of the transient crater and final crater are very important for brittle media such as rocks. Because the tensile strength of the brittle medium is far lower than the compressive strength, the reflected tensile wave on the surface of the target will cause irregular fragmentation of the target, which is called “spalling.” If the zone of the crater caused by spalling is excluded, a bowl-shaped crater will be obtained. This part of the crater is formed entirely by the flow of the medium driven by the shock compression wave, which can be called as the “transient crater.” Because of the existence of spalling, the diameter and volume of the crater will be greatly increased, and the crater as a whole will present a very rough and irregular shallow dish shape on the inner surface. The final morphology of the crater affected by shock compression and spalling is called as the “final crater.”
By observation of the final crater, it can be found that the impact crater can be divided into three typical zones after the hypervelocity penetration. Each zone owns its obvious characteristics, which are as follows: The first zone is the crushing area. As it is shown in Figure
Crushing area at the bottom of the crater.
The second zone is the spallation area. As shown in Figure
Spallation area around the center of the crater.
The third zone is the radial crack area. When the impact speed is relatively high, radial cracks are distributed radially on the target. As shown in Figure
Radial cracks on the free surface.
The influence of stripping on the shape of the crater.
A device for image scanning of surface morphology was used to measure the crater size, and the scanned image and profile of the crater are shown in Figure
Cratering test on granite under hypervelocity projectile impact. (a) Photograph of a crater. (b) 3D scanned image of the crater. (c) Crater profile.
As brittle material can be destroyed and peeled on the free surface by the tension wave, the crater shape is very irregular. Therefore, each target is scanned by three independent cross sections to obtain the maximum depth of the crater and the average diameter of the crater, and the crater volume can be calculated by numerical calculation. The statistics in Table
Experimental results of the final crater.
No. | Parameter of the projectile | Size of the final crater | ||||||
---|---|---|---|---|---|---|---|---|
1 | 7.85 | 0.72 | 9.67 | 1.5550 | 16.7 | 4.770 | 372 | 3.50 |
2 | 7.85 | 0.72 | 9.67 | 1.8294 | 18.5 | 4.468 | 503 | 4.14 |
3 | 7.85 | 0.72 | 9.67 | 2.2310 | 27.5 | 4.606 | 652 | 5.97 |
4 | 7.85 | 0.72 | 9.67 | 2.8069 | 28.5 | 5.137 | 715 | 5.55 |
5 | 7.85 | 0.72 | 9.67 | 2.8782 | 33.5 | 6.260 | 1575 | 5.35 |
6 | 7.85 | 0.72 | 9.67 | 3.1478 | 40.5 | 6.068 | 1391 | 6.67 |
7 | 7.85 | 0.72 | 9.67 | 3.1996 | 38.7 | 5.979 | 1716 | 6.47 |
8 | 7.85 | 0.72 | 9.67 | 3.5421 | 47.0 | 6.222 | 2863 | 7.55 |
9 | 7.85 | 0.72 | 9.67 | 3.5584 | 46.1 | 6.584 | 2761 | 7.00 |
10 | 7.85 | 0.72 | 9.67 | 4.1356 | 59.7 | 6.565 | 6299 | 9.09 |
According to the experimental results, the relationship between the depth and the diameter of the crater is analyzed, i.e.,
Relation between the dimensionless diameter and the dimensionless depth.
Some interesting conclusions can be drawn from Figure
Due to the limitation of observation and measurement, little is known about the actual situation of transient crater formation. In recent years, MEMIN, a German research team, had carried out a series of hypervelocity impact experiments of tuff, sandstone, and quartzite, which measured the process and morphology of transient cratering. Kenkmann et al. [
Parabolic fitting of transient craters.
The parabolic equation for fitting the transient crater in Figure
The volume of the transient crater (
Experimental results of the transient crater.
No. | ||||||
---|---|---|---|---|---|---|
1 | 1555.0 | 372 | 105 | 7.6 | 0.28 | 0.46 |
2 | 1829.4 | 503 | 182 | 10.4 | 0.36 | 0.56 |
3 | 2231.0 | 652 | 207 | 10.9 | 0.32 | 0.40 |
4 | 2806.9 | 715 | 288 | 12.6 | 0.40 | 0.44 |
5 | 2878.2 | 1575 | 248 | 10.1 | 0.16 | 0.30 |
6 | 3147.8 | 1391 | 376 | 12.7 | 0.27 | 0.31 |
7 | 3199.6 | 1716 | 320 | 12.5 | 0.19 | 0.32 |
8 | 3542.1 | 2863 | 1775 | 28.3 | 0.62 | 0.60 |
9 | 3558.4 | 2761 | 997 | 20.0 | 0.36 | 0.44 |
10 | 4135.6 | 6299 | — | — | — | — |
Combining Tables
The volumes of the final and transient crater change with the kinetic energy of the projectile.
Relation between the kinetic energy of the projectile and the volume of the final crater.
Relation between the kinetic energy of the projectile and the diameter of the final crater.
Figure
In Figure
Dimensional analysis is a method to determine the similarity criteria by using dimension theory. It is a common method to determine the conditions of cratering in studies [
Therefore, for any given materials, note parameter
According to the point source theory and power law, equation (
Equation (
Then, equation (
The experiments [
There are several models to determine material strength. Housen and Holsapple [
The relationship of dimensionless
In Figure
Fitting curves of the impact crater caused by the spherical projectile (i.e., results of Polanskey and Ahrens) and rod projectile (i.e., results of this article) are both in the power-law form as equations (
According to the slope of equations (
As stated, the only point source limit that exists is for
In the problem of cratering caused by the hypervelocity impact of the rod projectile, the “fundamental” independent variables which will affect the dimensional analysis process and the scaling law should be carefully considered. When considering, a projectile of a given material (density
A simple case occurs for a final crater geometry measure such as the volume
In a length-force-time system, the independent and dependent variables were developed according to dimensional analysis [
Equation (
Small-scale craters are strength-dominated, whereas sufficiently large craters are gravity-dominated. Therefore, these are strength-dominated craters, and it is reasonable to assume no gravity or atmosphere dependence. In addition, the dimensionless terms
According to the qualitative relationship between the crater efficiency and the dimensionless strength, the lower the strength of the material is, the greater the final crater volume is. Therefore, there must be such conditions: For Referring to the studies done by Wang et al. and Li and Chen [ For From equation (
Therefore, no matter what the
Certainly, equation (
The same as the above, there must be such conditions: For where For
Therefore, no matter what the
Generally, equation (
The relation between energy scale and momentum scale can be built up as follows:
According to equation (
The relation between dimensionless Π
The relation between dimensionless
From Figure
From Figure
Combining Figures
According to the morphology of the crater, the impact crater can be divided into crushing area, spallation area, and radial crack area. When a hypervelocity rod projectile impacts on a granite target, spallation occurs on the free surface under the action of shock wave propagation and reflection, which also lead to an irregular shape of the impact crater. It shows that the depth of the crater is much smaller than the diameter of the crater, and the crater seems to be a shallow dish. According to the experimental results, the relationship between the depth and the diameter of the crater is analyzed, i.e., With the increase of the projectile kinetic energy, it is uncertain whether the depth of the crater increases, but the volume of the crater will increase. This is because the increase of the kinetic energy leads to much more increase of the diameter of the crater. The volume of the transient and final crater increases with the increase of the projectile kinetic energy, and the contribution of spallation to the volume is growing more rapidly. When calculating the relationship between dimensionless crater efficiency and dimensionless strength by the dimensional analysis method, the point source solution cannot be used to analyze the problem of cratering caused by the hypervelocity rod projectile. Dimensional analysis was redesigned, and the similarity law was re-established. Another interesting and reasonable conclusion had been proved by experimental investigation in which it is equivalent to calculate the final crater by using the energy scale or the momentum scale.
The data used to support the findings of this study are included within the article.
The authors declare that there are no conflicts of interest regarding the publication of this paper.
The authors acknowledge the financial support received from the Natural Science Foundation of China (Grant nos. 51808552, 51808553, and 11602303), the China Postdoctoral Science Foundation (Grant nos. 2017M621752 and 2018M643853), and the Natural Science Foundation of Jiangsu Province (Grant no. BK20190570).