Granular Matter (2010) 12:337–344 DOI 10.1007/s10035-010-0186-7

Effects of density ratio and diameter ratio on critical incident angles of projectiles impacting granular media M. Nishida · M. Okumura · K. Tanaka

Received: 25 September 2009 / Published online: 16 April 2010 © Springer-Verlag 2010

Abstract The dynamic behavior of projectiles upon impact with granular media was recorded using two high-speed video cameras for capturing different angles. We used steel, brass, tungsten carbide spheres, and alumina ceramic spheres with diameters in the range of 6–20 mm as the projectiles and polystyrene beads (6 mm in diameter) and glass beads (1.7 mm in diameter) as the granular media. Upon impact, the projectiles penetrated the media, rebounded from the media, or were deflected such that their resulting motion was in a horizontal direction. Post-impact motion of the projectiles depended on the impact angles of the projectiles, the density ratio (bulk density/projectile density), and the diameter ratio (granular diameter/projectile diameter) and not on the impact velocity. The post-impact motion of the projectiles did not follow a clear trend in terms of the transient angle; instead, we observed the existence of a transient region. On the basis of the area of the transient regions, an empirical equation was derived for determining the critical angle of projectiles (the angle at which they can penetrate the granular media) as a function of the density ratio and the diameter ratio. Keywords Critical incident angles · Oblique impact · Projectile motion · Penetration · Rebound · Ricochet off

1 Introduction Over the last several decades, studying the mechanics of projectile impact on granular media has become increasingly important in the fields of physics, geophysics, particle technology, geology, and civil engineering. Although M. Nishida (B) · M. Okumura · K. Tanaka Nagoya Institute of Technology, Gokiso-cho, Showa-ku, Nagoya 466-8555, Japan e-mail: [email protected]

the mechanics of two colliding particles are now fairly well understood [1–5], the collision mechanics of granular media are not entirely clear. In particular, studying the impact behavior of granular media is difficult because the behavior is dependent on the compressibility and fluidity of the media. Prior to the year 2000, there had not been many researches on the formation of craters in granular media upon impact by projectiles and on the penetration behavior of projectiles upon impact with granular media. In 2003, Uehara et al. [6] and Walsh et al. [7] carried out experiments on the formation of impact craters by dropping balls onto dry, non-cohesive, granular media, and they formulated laws concerning the depth and diameter of the final crater. Newhall and Durian [8] investigated the effects of length, diameter, density, and tip shape of cylindrical projectiles on penetration depth. Pica Ciamarra et al. [9] conducted experimental and numerical investigations on projectiles penetrating a two-dimensional granular medium in order to study projectile deceleration and the probability distribution function of the forces acting on the grains of the medium. Goldman and Umbanhowar [10] directly measured the deceleration of spheres and disks upon impact with granular media and obtained scaling relations for the penetration depth and collision time. De Vet and de Bruyn [11] accurately measured the dimensions of impact crater surfaces using laser profilometry and showed that these impact craters are very nearly hyperbolic in profile. However, all the aforementioned researchers focused their attention on impact cratering and the penetration behavior of projectiles that were perpendicularly impacting granular media. In contrast, studies on oblique impacts of projectiles have been limited in number. Zheng et al. [12] examined the size and morphology of the impact craters formed when steel balls were made to obliquely impact granular media. From the viewpoint of aeolian sand transport, Rioual et al. [13,14]

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2 Experimental setup Figure 1 shows the experimental setup. We used three types of granular media, each consisting of spheres arranged randomly in a container made of polyvinyl chloride (PVC). The length, width, and height of the container were 300, 400, and 200 mm, respectively. The granular media used in the experiment comprised two types of polystyrene beads having diameters of 6.0 mm (Sekito Co., Ltd) and one type of glass

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Rubber band Tube

High-speed video camera

Projectile

Granular bed (Random packing)

300 mm

Impact angle

200 mm

investigated grains obliquely impacting a two-dimensional granular bed at velocities ranging from 6 to 22 m/s. Using the discrete element method (DEM), Oger et al. [15] numerically investigated the ejection process (splash function) during the oblique impact of a sphere with two- and three-dimensional packings; the projectiles were of the same material as the granular packings. Studies on the ejection process are still being carried out by their research group at the University of Rennes [16,17]. However, other than from the viewpoint of aeolian sand transport, the behavior of projectiles obliquely impacting granular media has not yet been fully understood. Many studies have been carried out in the fields of geophysics and planetary science [18,19] to understand impact cratering on Earth and other planets. However, most of these studies have focused on the amount of ejecta and the sizes and shapes of craters formed from perpendicular impacts. In several studies, the effects of oblique impact of projectiles on the amount of ejecta have been taken into account [20–22]. After carrying out pre-launch investigations, the Japan Aerospace Exploration Agency (JAXA) had launched the Hayabusa Project (asteroid sample return project) in 2007; the explorer is currently on its return journey to the Earth. When projectiles perpendicularly impact granular media comprising particles whose densities and diameters are smaller than those of the projectiles, they penetrate deep into the media and easily form craters on the surface of the media. When the projectiles impact granular media at an angle close to 0◦ relative to the horizontal plane, the projectiles tend to ricochet off the media. Soliman et al. [25] examined the ricochet angles when three types of spheres (steel, duralumin, and lead) with the same diameters (9.52 mm) impacted granular media comprising a particular type of sand (density 2.7 g/cm3 , diameter 1 mm) and obtained an equation to determine the ricochet angles. However, the effect of ricochet angle on the density and diameter ratios has not yet been fully understood. We examined the ricochet angles and penetration angles for eight types of projectiles that were made to obliquely impact three types of granular media at impact velocities less than 20 m/s. We proposed an equation to determine the critical angle of projectiles (the angle at which they penetrate the granular media).

M. Nishida et al.

400 mm

Polyvinyl chloride (PVC)

High-speed video camera Fig. 1 Experimental setup for studying projectile impacts on granular media

beads with mean diameters of 1.7 mm (FGB-10, Fuji Manufacturing Co., Ltd). The characteristics of the granular media (diameter, density, volume fraction, and repose angle) are listed in Table 1. Light and heavy polystyrene beads (G6SL and G6SH, respectively) were prepared by changing the particle densities of the beads by the addition of different amounts of calcium carbonate. The as-prepared polystyrene beads were used as BB bullets. Therefore, the diameters of the polystyrene beads had a very narrow distribution. In contrast, the diameter distribution of the glass beads was wider. The volume fraction of the granular media was 61– 63%, which is close to the well-accepted value for random close packing of monodisperse spheres [26]. The granular media were prepared by randomly spreading the beads in the container. Individual projectiles were made to obliquely impact the top-center of the granular media with velocities less than 20 m/s. The projectiles were loaded into an aluminum tube and were slung using a rubber band as a slingshot. The impact angle was varied by varying the angle made by the aluminum tube with the horizontal plane of the container. We did not use an air gun because the granular media are affected by the air released from the gun upon firing. We used eight types of projectiles for impact tests, as listed in Table 2: alumina ceramic spheres, brass spheres, and tungsten carbide spheres of diameters 11.1 mm and steel spheres of diameters 6.0, 9.0, 11.1, 12.0, and 20.0 mm (Nakano Steel Ball Co., Ltd). The diameters and densities of the projectiles and granular media were selected such that the density and diameter ratios were large. The ratio between the densities of the granular media and the projectile was in the range of 0.055–0.39, and the ratio between the diameters of the granular media and the projectile was in the range of 0.085–0.67. We carried out 17 sets of experiments (Table 3). The motions of the projectiles and granular media just before and after impact were recorded using two syn-

Effects of density ratio and diameter ratio on critical incident angles

339

Table 1 Granular media used in the experiments Granular beds

Diameter Dt ( mm)

Particle density (kg/m3 )

Volume fraction

Bulk density ρ t (kg/m3 )

Repose angle (◦ )

Polystyrene beads (G6SL)

6.0

1.06 × 103

0.61

0.64 × 103

23

Polystyrene beads (G6SH)

6.0

2.21 × 103

0.63

1.39 × 103

23

1.7

2.50 ×

0.61

1.52 ×

24

Glass beads (G2G)

Table 2 Projectiles used in the impact tests

103

103

Projectiles

Diameter D p ( mm)

Density ρ p (kg/m3 )

(P06S)

Steel sphere

6.0

7.81 × 103

(P09S)

Steel sphere

9.0

7.81 × 103

(P11S)

Steel sphere

11.1

7.81 × 103

(P12S)

Steel sphere

12.0

7.81 × 103

(P20S)

Steel sphere

20.0

7.81 × 103

(P11A)

Alumina ceramic sphere

11.1

3.94 × 103

(P11B)

Brass sphere

11.1

8.45 × 103

(P11W)

Tungsten carbide sphere

11.1

14.7 × 103

Table 3 Results of all the experiments conducted in the study Experiment no.

Projectiles

Granular medium

Diameter ratio, Dt /D p

Density ratio, ρ t/ ρ p

Transient region, θ E (◦ )

E1

P09S

G6SL

0.67

0.082

41–55

E2

P12S

G6SL

0.50

0.082

32–38

E3

P11S

G6SL

0.55

0.082

32–42

E4

P11A

G6SL

0.55

0.16

51–66

E5

P11B

G6SL

0.55

0.076

36–47

E6

P11W

G6SL

0.55

0.041

21–26

E7

P09S

G6SH

0.67

0.18

55–90

E8

P12S

G6SH

0.50

0.18

45–61

E9

P20S

G6SH

0.30

0.18

35–53

E10

P11S

G6SH

0.55

0.18

50–71

E11

P11A

G6SH

0.55

0.35

68–90

E12

P11B

G6SH

0.55

0.16

46–64

E13

P06S

G2G

0.28

0.19

38–45

E14

P09S

G2G

0.19

0.19

29–31

E15

P20S

G2G

0.085

0.19

16–18

E16

P11S

G2G

0.15

0.19

27–29

E17

P11A

G2G

0.15

0.39

37–44

chronized high-speed video cameras (IDT, MotionPro X-3 and MotionScope PCI 2000S) positioned to capture different angles; the framing rate was 500 frame/s and the shutter speed was 1/10,000 s. The actual impact velocity and impact angle were calculated from the images of the high-speed video cameras. Observation of images recorded by the high-speed video cameras showed few rotations of the projectiles before impact.

3 Experimental results 3.1 Classification of projectile motion Figure 2 shows images recorded by the high-speed video cameras just after the oblique impact of 11.1- mm diametral steel projectile (P11S) on the granular media comprising G6SL. From Fig. 2a, we could infer the following: the impact

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Top view

Side view

50 mm

50 mm

(a) Top view

Side view

Projectile Projectile

50 mm

50 mm

(b) Top view

Side view

Projectile

50 mm

50 mm

(c)

Projectile

Fig. 2 Motion of the projectile and granular media after the impact of 11.1- mm diametral steel projectiles (P11S) on 6- mm diametral light polystyrene beads (G6SL); the images were taken using a high-speed video camera (a) Penetration, Velocity 10.2 m/s, Angle 69◦ (100 ms after impact) (b) Horizontal movement, Velocity 7.9 m/s, Angle 34◦ (100 ms after impact) (c) Rebound, Velocity 10.1 m/s, Angle 22◦ (42 ms after impact)

velocity was 10.2 m/s and the impact angle was 69◦ , the projectile penetrated deep into the granular surface, a few of the beads were uniformly displaced in the upward direction, and a circular crater was observed on the granular surface. In Fig. 2b, we can see that when the impact velocity was 7.9 m/s and the impact angle was 34◦ , the projectiles were deflected such that their resulting motion was in a horizontal direction (hereafter referred to as the “horizontal motion”), and a number of beads were greatly displaced in the upward direction. The number of beads scattered in the direction of motion of the projectile was found to be greater than that scattered in the opposite direction. The circular crater spread toward the direction of projectile motion and became elliptical. In Fig. 2c, we can see that when the impact velocity was the same as that in Fig. 2a and the impact angle was 22◦ , the projectile rebounded from the granular surface (right-hand side photograph) and the number of beads scattered in the

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Fig. 3 Projectile motion after the impact of 11.1- mm diametral steel projectiles (P11S) on 6- mm diametral light polystyrene beads (G6SL) (penetration black triangles; rebound white triangles; motion in a direction parallel to the horizontal plane of the granular media: white circles)

direction of motion of the projectile was found to be greater than that scattered in the opposite direction. We believe that the motion of the projectiles and beads resembles that of stones skipping across water; in other words, the mechanics underlying the skipping of stones is similar to that underlying oblique impacts and ricochets of solid projectiles against a liquid surface [27,28]. Figure 3 shows the experimental results of projectiles impacting the media at different velocities and angles. Each symbol in Fig. 3 represents the result for a particular experimental condition. Upon impact at velocities ranging from 3 to 20 m/s and at angles ranging from 32◦ to 90◦ , the projectile penetrated deep into the surface of the packed particles (indicated by black triangles in the figure). At impact angles ranging from 32◦ to 42◦ , the projectile penetrated deep into the surface (black triangles), showed horizontal motion (white circles), or rebounded from the granular media (white triangles). At impact angles below 32◦ , the projectiles did not penetrate the granular media. The changes in the motion of the projectiles upon impact depended only on the impact angle and not on the impact velocity. In addition, the change was not rapid, and instead of a clear-cut transient angle, a relatively wide range of transient angles was evident. We called this range the transient region. However, classification of projectile motion was sometimes difficult. We could also observe ambiguous motions that showed similarities to both penetration and horizontal motion. In such cases, when the resting position of the projectile after impact was close to the impact position, we classified the motion as penetration (black triangles in Fig. 3), regardless of the penetration depth of the projectiles. However, when the resting position of the projectile after impact was far from the impact position and close to the crater rim, we classified the motion as motion parallel to the horizontal plane (white circles in Fig. 3). We then calculated the probability of penetration

Effects of density ratio and diameter ratio on critical incident angles

Fig. 4 Probability distribution function of penetration when 11.1- mm diametral steel projectiles (P11S) were made to impact 6- mm diametral light polystyrene beads (G6SL)

Fig. 5 Projectile motion after the impact of 12.0- mm diametral steel projectiles (P12S) on 6- mm diametral light polystyrene beads (G6SL) (penetration black triangles; rebound white triangles; motion in a direction parallel to the horizontal plane of the granular media: white circles)

from the experimental results for impact angles less than 4◦ and obtained the penetration distribution function shown in Fig. 4. As the impact angle increased, the probability of penetration gradually increased. The impact angles for which the probability of penetration was 0 and 1 were averaged, and the probability of penetration for these averaged angles was 0.5. Figure 5 shows the experimental results obtained for projectiles with large diameters (12 mm, P12S). For large impact angles in the range of 32 − 90◦ , the projectiles penetrated the granular media. The transient region became narrower. As shown in Fig. 6, when we used the projectiles of 11.1mm diamtral alumina ceramic spheres (P11A) and G6SH, the transient region became wider because the difference in diameters and densities decreased. When the difference in diameters and densities between the projectiles and beads (P11S and G2G) was considerably large, a very narrow transient region (almost a line, as shown in Fig. 7) was observed. The post-impact motion of the projectiles suddenly changed

341

Fig. 6 Projectile motion after the impact of 11.1- mm diametral alumina ceramic projectiles (P11A) on 6- mm diametral heavy polystyrene beads (G6SH) (penetration black triangles; rebound white triangles; motion in a direction parallel to the horizontal plane of the granular media: white circles)

Fig. 7 Projectile motion after the impact of 11.1- mm diametral steel projectiles (P11S) on 1.7- mm diametral glass beads (G2G) (penetration black triangles; rebound white triangles; motion in a direction parallel to the horizontal plane of the granular media: white circles)

from penetration to rebound at a certain impact angle. The expansion and contraction of the transient region as a function of the diameter and density ratios was within the predicted range. However, in the case of oblique impact, we inferred that the vertical component of impact velocity is important. Therefore, prior to the experiment, we predicted that the lines that indicate a constant vertical component of impact velocity, and hence, the boundaries of penetration, would run diagonal in the graph of impact angle versus impact velocity. In other words, we expected the transient region to depend on the impact velocity as well as the impact angle. However, contrary to our expectation, the impact experiments revealed that the penetration of the projectile into the granular media did not depend on the impact velocity under the conditions adopted in the present study.

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3.2 Equation for critical incident angle of projectiles We proposed an equation to determine the critical angle of the projectiles between penetration and not penetration. In all our experiments, when the impact angle was close to 90◦ , the projectiles almost penetrated the media. As the impact angle decreased, the projectiles started to move in a direction parallel to the horizontal plane of the granular media. When the impact angle was close to 0◦ , the projectiles always rebounded. The impact angles at which penetration occurred depended on the density ratio and diameter ratio between the projectiles and the granular media. In all the experiments, the impact angles at which penetration into the granular medium ceased to occur were not clear, and transient regions (regions showing a combination of rebound, penetration, and motion parallel to the horizontal plane of the granular media) were always visible. Therefore, we examined the minimum impact angle at which the projectile penetrated the granular media and the maximum angle at which the projectile did not penetrate the granular media (the maximum angle for motion parallel to the horizontal plane of the granular media). The results of calculation are presented in Table 3 and are shown by dashed lines in Figs. 3, 5, 6, and 7. Soliman et al. [25] proposed that the ricochet angle for dry sand, θ , is given by  ρs o . (1) θ = 82 ρp Here, ρs and ρ p are the densities of dry sand and the projectile, respectively. On the basis of Soliman’s equation and our experimental results, we assumed that the incident critical angle θ E can be described as   β  Dt γ ρt θE = α . (2) ρp Dp Here, ρ t is the bulk density of the granular media, Dt is the particle diameter of the granular media, and D p is the projectile diameter; α, β, and γ are variables. We added the diameter term to Soliman’s equation. We wanted to verify the density dependency of the critical angle when the diameter ratio was small, because the projectile diameter was 10 times larger than the particle diameter of sand in the case of the study of Soliman et al. In addition, they considered the specific gravity of sand (2.7), whereas we considered the bulk density of the granular media in Eq. 2 in order to examine the effect of their volume fraction on the critical incident angle. As in their study, we did not add the term of impact velocity to Eq. 2 because the experimental results did not depend on the impact velocity under the conditions adopted in our study. On the other hand, we assumed that the angle of repose (or frictional characteristics), the shape factor of the granular media, and the shape of the projectile (length and tip shape) were very important. As is well known, when using

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Fig. 8 Density dependency of projectile motion after impact (diameter ratio 0.55)

extremely loose packing, jet formation and void collapse of granular media can be observed just after projectile impact [29–32]. Newhall and Durian [8] discussed the importance of projectile shape on impact craters in loose granular media. In order to establish an empirical equation that considers these factors, further experiments employing various types of projectiles and granular media are required. We first determined the density dependency β by using the results obtained for a constant diameter ratio. Figure 8 shows seven experimental results wherein the diameter ratio was 0.55 (E3–E6 and E10–E12 in Table 3). On the graph, the transient region is shown using error bars. This means that the minimum impact angle of penetration is at the lower end of the graph and the maximum impact angle at which the projectiles did not penetrate the granular media (motion parallel to the horizontal plane of the granular media) is at the higher end. Please note that this is a double logarithmic graph. By fitting β to the experimental data in Fig. 8, it was found that when the value of β was close to 0.5, we could draw lines connecting the values obtained from all the seven experiments. We used the β value of density dependency as evaluated by Soliman et al. (0.5). Next, we calculated the normalized value (ρt /ρ p )1/2 and plotted it on the vertical axis in Fig. 9 in order to determine the diameter dependency γ and the variable α. When the diameter ratio was close to 0.043, the transient region was narrow. By fitting α and γ to the experimental data in Fig. 9, it was found that the line at α = 210 and γ = 2/3 could be drawn to connect the values obtained from all the 13 experiments. We derived the following equation to determine the critical incident angle of the projectiles. 

ρt θ E = 210 ρp

1/2 

Dt Dp

2/3 .

(3)

We performed many experiments with projectiles whose densities and diameters were larger than those of the granular

Effects of density ratio and diameter ratio on critical incident angles

343 supported in part by a Grant-in-Aid for Scientific Research (C), KAKENHI (19560089), from the Japan Society for the Promotion of Science (JSPS).

References

Fig. 9 Diameter dependency of projectile motion after impact

media. When the densities and diameters of the projectiles were smaller than those of the granular media (the regions in Figs. 8 and 9 wherein ρt /ρ p and Dt /D p are greater than 1.0, i.e., x-axis values greater than 1.0), the projectiles did not penetrate the granular media even at the impact angle of 90◦ ; they could only penetrate under special conditions such as high-velocity impact, which led to large plastic deformation of the projectiles and the granular media. In fact, when 9.0- mm diametral steel spheres (P09S) and P11A impacted G6SH, we observed that the projectiles did not penetrate the granular medium; instead, they moved in a direction parallel to the horizontal plane of the granular media, even at an impact angle of 90◦ . As predicted, under such experimental conditions, the impact point of the projectile on a particle of the granular medium plays a very important role in determining the post-impact motion of the projectiles.

4 Summary By using two high-speed video cameras, we examined in detail the post-impact motion of projectiles upon impact with granular media. Upon impact, the projectiles either penetrated the media, rebounded from the media, or moved in a direction parallel to the horizontal plane of the media. The post-impact motion of the projectiles depended on the impact angle. This impact angle was affected by the density and diameter ratios. When the projectile diameter was close to the diameter of the particles composing the granular media, the critical angle for penetration did not show a remarkable, sudden cutoff. Instead, we observed transient regions for the post-impact motion of the projectiles. By using the transient regions, we derived an empirical equation for determining the critical incident angle of projectiles. Acknowledgements The authors greatly appreciate Mr. Ken Fukushima and Mr. Koshi Ando, IDT Japan, for their help in taking photographs with the high-speed video cameras. This work was

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