Shell-Stripping Mechanism of Red Sandstone Under Hypervelocity Impact with Aluminum Spheres

Abstract

To investigate the size effect on fragmentation phenomena during hypervelocity impact, scaled experiments were conducted using a 30 mm smooth-bore ballistic range (DBR30) driven by a detonation-driven two-stage launching system. Unique stripping of sandstone target was observed, revealing that free-surface unloading waves govern peak pressure attenuation and fragmentation patterns. By establishing a shock wave attenuation model, the typical failure characteristics of different regions were distinguished, including jetting, crushing, and cracking. Parameter λ was defined to distinguish two forms of destruction, Class I (stripping-dominated) and Class II (cratering-dominated). Given the significant difference between the compressive and tensile strength of sandstone, the influence of the size effect on its failure characteristics was notable. This research also provides a valuable reference for understanding the evolution and formation mechanisms of binary asteroids.

1. Introduction

Near-Earth asteroids (NEAs) are defined as asteroids with heliocentric orbits intersecting Earth’s orbital vicinity. Those possessing a minimum orbital intersection distance (MOID) <0.05 AU with a diameter exceeding 140 m are designated as Potentially Hazardous Objects (PHOs), posing collision risks [1,2]. According to the Minor Planet Center (MPC), as of January 2018, there were more than 17,000 Near-Earth asteroids, of which nearly 1700 should be considered potentially hazardous [3]. The typical impact speed of asteroids is 15–20 km/s. Currently, there are 188 confirmed meteorite impact craters on Earth. The dimensions of these craters primarily depend on various factors, including the energy, size, composition, density, speed of the impactor, and so on [4]. Craters of various sizes exist all over the world, for example the Gosses Bluff crater [5], Chicxulub crater [6], Amguid crater [7], Barringer crater [8], Carancas crater [9], and Chelyabinsk crater [10], proving the existence of asteroid impact events directly and indirectly. Researchers worldwide have proposed a variety of strategies for asteroid defense to prevent the Earth from being hit. Among them, kinetic impact deflection of asteroids is the simplest and most effective approach [11]. In 2022, the DART (double asteroid redirection test) impactor struck Dimorphos, the secondary star of the double asteroid Didymos, at a speed of more than 6.2 km/s, successfully changing the asteroid’s orbit around Didymos and proving the feasibility of kinetic energy impact [12]. While momentum transfer quantification and asteroid property characterization require refinement, ground-based hypervelocity impact experiments have become a reliable method to help understand the mechanism of these phenomena.

Hypervelocity impact experiments, involving projectiles of varying sizes and targets, result in distinct debris and plume characteristics, thereby revealing novel fragmentation characteristic. George et al. [13] conducted sphere impact experiments with different pores and measured the velocity of the dust plume at about 1.6 km/s and the velocity of the large jet fragments at about 19.4 m/s. Meteorite ejecta has two stages of development: a high velocity, fine-grained ejecta cone, and high-velocity ejection of large ejecta fragments. These large ejecta fragments are the main cause of cratering recoil. For hard and thick target rocks such as anorthosite, cratering is the main feature after impact, and hyperbolic, parabolic, and power law equations can be used to evaluate crater excavation shape [14]. Fragmentation characteristics under hypervelocity impact affect crater formation efficiency [15]. Fragmentation accounts for more than 50% of the final crater volume, with quartzite targets exhibiting ~70% pellet fragmentation efficiency [16]. In contrast, sandstone projectile fragmentation under identical impact conditions exhibits an efficiency of ~50%, demonstrating dominant control over crater formation dynamics. Experimental observations [17] document large irregular rocks undergoing central pit formation and fragmentation along peripheral main cracks. Impactor size and material properties yield distinct, fragmentation-dependent phenomena. Conventional fragmentation modes primarily involve cratering and blocking, which are potentially determined by shock wave propagation at different stages.

Hypervelocity impact (HVI) on an asteroid, creating a crater, occurs in three stages: contact and compression, excavation, and modification. Each stage is dominated by a different set of physical mechanism [18]. Both strain rate and size have important effects on the formation of these craters [19]. The duration of dynamic fracture is very short. The energy accumulation and transfer speed at the crack tip are faster than the propagation speed of a new crack at the crack tip. Under a certain confining pressure, rock exhibits enhanced dynamic compressive strength and peak strain with increasing strain rate, demonstrating logarithmic scaling of compressive strength [20]. HVI-induced compression at the projectile–target interface launches shock waves along the compressed/uncompressed material boundary. Proximal to the impact site, the high pressure impact will induce vaporization, melting, and solid phase transformation. A strong impact material accompanied by a high-speed “jet” is ejected from the interface between the projectile and the target. Chen [21], Liu [22], and Johnson et al. [23] also explained this phenomenon through theory or simulation. The surface of the target, due to the interaction of the shock wave and unloading wave, will also produce tensile stress, resulting in spalling of the target [24]. Pressure pulse-induced rock deformation in small-scale crater experiments revealed enhanced dynamic strength under high strain rates, exhibiting significant size effects [25]. Shock loading and unloading strongly affect the crushing characteristics of the impact. Quantifying the shock transfer process is very important to clarify the crushing mechanism.

In order to study the size effect in hypervelocity impact, scaled specimen experiments were conducted in this paper. A special phenomenon of stripping effect on sandstone was observed during hypervelocity impact. Shock wave attenuation during the impact process is closely related to the crushing effect of the target. Especially, the influence of the unloading wave needs to be considered. Therefore, the relation between shock wave propagation and the broken region is established, explaining the shell-stripping model. The parameter λ emerges as the quantitative threshold defining sandstone’s size effect-driven failure mode transitions. A critical λ = 1 distinguishes Class I (stripping-dominated) and Class II (cratering-dominated) failure modes.

2. Experimental Methods

In this work the projectiles were 8 mm diameter Al-spheres (6061-T6 Al) (Yasite company, Jinhua, China), and a red sandstone cube with a size of 100 mm served as the impact target. Red sandstone impact tests were carried out using a 30 mm smooth-bore ballistic (DBR30) driven by gaseous detonation from the Institute of Mechanics, Chinese Academy of Sciences [26,27], as shown in Figure 1. The main body of the bombing driver of the secondary light cannon balloon (DBR30) mainly includes the hypervelocity transmitter main body (explosion tube and compression pipe), test section, and recycling section, as shown in Figure 2a. A sabot of low-density high-pressure polyethylene was utilized in the launch system. Nitrogen was pressurized in the test section to achieve projectile–sabot separation. The ballistic target uses a stimulating imaging system, and the impact process of the shooting target was recorded by a high-speed camera (Kirana 5 (Pitstone, UK)), as shown in Figure 2b. The camera sampling frame rate was 75,000 Hz, and the speed of the incident projectile was measured by laser velocimetry system. The optical path was visualized using a schlieren imaging system, as shown in Figure 2c. The sandstone specimen was wrapped by insulating tape to prevent the rock from collapsing. The sandstone was suspended in the target chamber with two steel wire ropes, configured as a ballistic pendulum, as shown in Figure 2d. A polyvinylidene fluoride (PVDF) sensor was employed to capture the impact response of the center point behind the sandstone. In order to preserve the high-frequency characteristics of the signal, the sampling frequency of the oscilloscope was set to 10 MHz. The sampling time was not less than 2 ms, and the trigger voltage was 210 mV.

3. Experimental Results

Ballistic impact tests of an aluminum ball on red sandstone specimens at different velocities were conducted, and the results were summarized in

Table 1. Experimental parameters and results.

vp (m/s)	ρp (g/cm3)	ρt (g/cm3)	mp (g)	mr (g)	mq (g)
2891	2.700	2.518	19.0	1942	647
3154	2.700	2.504	19.0	1738	611
3509	2.700	2.498	19.0	1629	619
4453	2.700	2.498	19.0	-	485

Note: v_p is the speed of the projectile; ρ_p is the density of the projectile; ρ_t is the density of the target; m_p is the mass of the projectile (including the cartridge); m_r is the total remaining mass of the target (including stripping ball); m_q is the mass of the stripping ball.

. It can be inferred that, with an increase in impact velocity, the mass of the remaining targets decreases. When the velocity exceeded 4453 m/s, the target was ejected, and the catastrophic fragmentation rendered the residual mass measurement unfeasible. The final impact morphology was illustrated in Figure 3. Centrally-located spherical craters surrounded by petaloid fractures were observed. Notably, spheres of the sandstone target, with a mass of m_q, were left after impact across all tested velocities, as circled in Figure 3. Moreover, the fracture morphology on the rear face of the target was velocity-dependent. At an impact velocity of 2891 m/s, there was no detectable fracture on the back of the target. When the velocity reaches 3154 m/s, a single crack could be observed. And at an impact velocity of 3509 m/s, a cross crack was introduced, with the target being divided into four fragments.

The entire impact process was recorded using high-speed photography with a frame size of 512 × 320 px and a sampling rate of 75,000 Hz. High-speed photography is triggered by the output signal of the laser velocimetry system. Figure 4 shows the projectile perforation process of the projectile hitting that target, as captured by a high-speed camera. After the impact, the central region of the target fragmented, developing an outward-propagating conical jet structure. At the end of the collision, large debris was ejected from the target center.

4. Simulation Verification

4.1. Simulation Method

We used low-density spherical aluminum projectiles to represent spacecraft equivalently. The projectile perforation process was simulated with the Smoothed Particle Hydrodynamics (SPH) method implemented in LS-DYNA. The Johnson–Cook (JC) constitutive model [28] coupled with the Mie–Gruneisen equation of state (EOS) [29] was utilized to characterize the thermo-mechanical response of the 6061-Al alloy projectile, and the mechanical behavior of the red sandstone specimen was described by the Holmquist–Johnson–Cook (HJC) model [30]. Critical parameters are listed in

Table 2. Parameters of the JC 6061-T6 model [29].

ρ0 (kg·m−3)	µ	G (GPa)	σs (MPa)	Nq (MPa)
2773	0.33	27.6	290	203
Ne	Q	Spall type	D0	D2–5
0.35	0.011	3	1	0

,

Table 3. Parameters of the Mie–Gruneisen EOS [29].

C (m·s−1)	S1	γ
5386	1.339	1.997

and

Table 4. Parameters of the red sandstone with HJC model [30,31].

ρ0 (kg·m−3)	fc (MPa)	A	B	C0
2500	20	0.79	2.2	0.007
Smax	G (Gpa)	T (MPa)	D01	D02
7	4.1	0.4	0.03	1
Pcrush (MPa)	µcrush	Plock (GPa)	µlock	K1 (GPa)
370	0.06	1.0	0.15	15
K2 (GPa)	K3 (GPa)	EFmin	N	FS
50	0	0.01	0.80	0.004

. A free boundary condition was employed in the SPH modeling.

4.2. Simulation Results Analysis

The simulation results are compared with the experimental results under the same circumstances to verify their accuracy, as shown in Figure 5. The simulation and experimental results have equal scaling. At 130 μs, the simulated debris cloud morphology matches experimental observations, exhibiting identical funnel-shaped geometry. The hypervelocity impact formed a crater and debris ejected from the target. The peripheral deformation of the target exhibits velocity-dependent flaring under varying impact velocities. The debris cloud morphology obtained from simulations and experiments is generally consistent.

The SPH-simulated maximum principal stress fields in the central section of the target under varied impact velocities are shown in Figure 6. The red region indicates propagation of the unloading wave, and the central green region corresponds to the stripping area, which is consistent with experimental observations. The energy variation during the hypervelocity impact process was also obtained through numerical simulation, as shown in Figure 7. The ratio of kinetic energy to internal energy can be calculated according to the particle velocity during the contact and compression stage (Stage A). It is found that 50% of the kinetic energy of the projectile was transferred to the target, with a portion converted into internal energy to perform work during expansion, thereby inducing a phase transition. The kinetic energy of the target peaks at the end of the contact and compression stage. The initial decline is attributed to reduced particle velocities during shock wave expansion, but this decline continues as excavation flow velocity decreases. In the subsequent stage (after about 10τ, τ = d/V_i; d is the diameter of the projectile; V_i is the speed of the projectile), the ratio of internal energy to total energy of the projectile stabilizes, with most initial energy transferred to the target.

5. The Shell-Stripping Model

During the impact experiment, regardless of the velocity, large stripping spheres were detached, and their formation process was recorded with an ultra-high-speed camera, as shown in Figure 8b. According to Adriano’s [32] research, asteroid shapes observed in rubble-pile asteroids result from stochastic low-velocity collisions during the impact process. Mass, fragment shapes, and angular momenta collectively govern the stochastic emergence of distinct configurations within each system. Contact binary asteroids may form through gravitational regrouping, and their largest fragments need not occupy the central region of the rubble-pile structure. This phenomenon warrants further investigation to establish a plausible explanation.

5.1. Shock Wave Attenuation Model

The shock wave accumulated in the initial contact and compression stages of the hypervelocity impact, while the subsequent attenuation stage affects the fragmentation characteristics of materials. An effective shock wave attenuation model was developed by Gault et al. [33]. However, the assumption of spherical (or hemispherical) symmetry inadequately represents near-surface conditions and neglects the effects of the unloading wave from free surfaces, particularly when considering the impact size. This limitation underscores the need for a more accurate attenuation model through simulation. In order to model shock wave theoretical propagation in space, it is assumed that the shock wave exhibits spherically symmetric attenuation originating from a source at depth (p), and r₀ represents the radial distance between the uniformly shocked volume and the peak pressure zone. r₀ defines a sphere centered at depth p.

While

$$p~2({\displaystyle \frac{2r_{0m}}{U_{s,\mathrm{p}}}})u_t$$

(1)

U_s,p is the shock wave velocity of the projectile; u_t is the target particle velocity; r_0m is the radius of the projectile; and the initial radius of energy deposition in the target is denoted as r_0t.

The shock wave attenuation model has several basic assumptions. Shock wave accumulation is completed at a specific depth. The energy transferred to the target during the shock stage is uniformly distributed in a compressed sphere behind the shock wave at any time. Without considering the real representation of near-surface conditions, the unloading wave influence from the free surface is not considered.

Then, the total energy in the system E_T is the sum of the kinetic and internal energies:

$$E_{\tau }={\displaystyle \frac{4}{3}}\pi r^{3}P\left[1-{\left({\displaystyle \frac{Pn}{K_0}}+1\right)}^{-1/n}\right]$$

(2)

The differential energy dE_T for a small radius change dr is

$${\displaystyle \frac{d{E}_T}{dr}}=4\pi r^2\left[1-{\left({\displaystyle \frac{Pn}{K_0}}+1\right)}^{-1/n}\right]\left[P+{\displaystyle \frac{r}{3}}{\displaystyle \frac{dP}{dr}}\right]+{\displaystyle \frac{4}{3}}\pi r^3{\displaystyle \frac{P}{K_0}}\left[{\left({\displaystyle \frac{Pn}{K_0}}+1\right)}^{-1-(1/n)}\right]{\displaystyle \frac{dP}{dr}}$$

(3)

The internal energy regained during isentropic expansion is given under a release adiabat, with specific waste heat ΔEw defined as

$$∆E_w={\displaystyle \frac{1}{2}}P(V_0-V)-{\displaystyle {\int }_{V_0}^{\nu }P}dV$$

(4)

Then,

$${\displaystyle {\int }_{V_0}^{\nu }P}dV={\displaystyle \frac{K_0}{n}}{\displaystyle \frac{1}{(1-n)}}[V_0\left\{1-{\left({\displaystyle \frac{Pn}{K_0}}+1\right)}^{1-(1/n)}\right\}+V_0(1-n){\left({\displaystyle \frac{Pn}{K_0}}+1\right)}^{-1/n}-(1-n)V_0]$$

(5)

Therefore

$$∆E_w={\displaystyle \frac{1}{2}}\left[P{V}_0-{\displaystyle \frac{2K_0{V}_0}{n}}\right]\left[1-{\left({\displaystyle \frac{Pn}{K_0}}+1\right)}^{-1/n}\right]+{\displaystyle \frac{K_0{V}_0}{n(1-n)}}\left[1-{\left({\displaystyle \frac{Pn}{K_0}}+1\right)}^{1-(1/n)}\right]$$

(6)

$${\displaystyle \frac{d{E}_w}{dr}}=4\pi {\rho }_0{r}^2\left\{{\displaystyle \frac{1}{2}}\left[P{V}_0-{\displaystyle \frac{2K_0}{n}}{V}_0\right]\left[1-{\left({\displaystyle \frac{Pn}{K_0}}+1\right)}^{-1/n}\right]\right.+\left.{\displaystyle \frac{K_0{V}_0}{n(1-n)}}\left[1-{\left({\displaystyle \frac{Pn}{K_0}}+1\right)}^{1-(1/n)}\right]\right\}$$

(7)

Equating (3) and (7):

$$\begin{array}{l}{\displaystyle \frac{dX}{dR}}=\left\{-{\displaystyle \frac{3}{2}}X+{\displaystyle \frac{3}{2}}X{(Xn+1)}^{-1/n}+{\displaystyle \frac{1}{n}}-{\displaystyle \frac{1}{n}}{(Xn+1)}^{-1/n}+{\displaystyle \frac{1}{n(1-n)}}[{(nX+1)}^{1-(1/n)}-1]\right\}\\ \cdot {\left\{{\displaystyle \frac{R}{3}}[1-{(nX+1)}^{-1/n}+X{(nX+1)}^{-1-(1/n)}]\right\}}^{-1}\end{array}$$

(8)

Specify the initial value X = P/K₀ (representing X₀) at any initial value of R= r/r₀ (R₀). K₀ is adiabatic bulk modulus, and n is adiabatic bulk modulus derivative. P₀ is the initial pressure, calculated by Hugoniot curve, and V₀ = 1/ρ₀. Here all properties refer to target properties, so the subscript t, as in ρ_t, is dropped.

Then the equation can be numerically integrated to obtain the dimensionless attenuation law during the impact process, as shown in Figure 9a. However, due to the size effect of the target, the shock wave produces an unloading wave that catches up with the shock surface during the unloading process. It is necessary to determine the unloading pressure, considering the sparse wave produced under the simulation conditions. The unloading pressure was simulated by taking one SPH particle every ten particles in the impact direction and connecting its pressure into a curve. It can be seen that, at R = 5 to the left, the shock wave attenuation pressure calculated theoretically is greater than that of SPH simulation and then tends to be stable when R > 5.

The temporal shock wave attenuation in the impact direction obtained by simulation is shown in Figure 10. According to the peak pressure decay profile, the diameter of the stripping spheres can be roughly estimated as illustrated in Appendix A—Figure A2. The dimensionless mass/size of stripping balls obtained by experiments/simulations is shown in Figure 11. It is indicated that impact velocity correlates inversely with residual spall diameter. Such an unloading of shock waves is an important cause of stripping failure.

5.2. Shock Wave Stripping Model

During HVI penetration, the contact zone experienced extreme temperature and pressure upon the initial collision. An intense shockwave was generated at the projectile–target interface, propagating through both materials. Regions ② and ③ in Figure 12 represent shock waves. HVI-induced shock pressures, ranging from 21.95 GPa to 39.62 GPa, were derived from Hugoniot curve analysis during the initial impact stages and far exceed the yield strengths of the projectile and the target. The unloading wave reflected at the lateral boundary of the projectile–target contact interface induced jet ejection. The unloading wave is represented by purple arrows in Figure 12. Shock attenuation modeling initiated when the projectile reached depth p (defined in the approximate model), with initial conditions established at this position, whereas the SPH simulations were carried out along the impact path directly. The area above point p is the cavity-crushing area. Spatial pressure decay in the shock wave is quantified by simulation based on the shock wave attenuation model. Appendix A—Figure A2 shows the concentric contours characterizing the spatial distribution of shock waves calculated from shock attenuation positions at different velocities of the target.

Due to the limited size of the target, the shock attenuation transmitted to the lateral surface and the lower surface will produce an unloading wave. The dynamic compressive strength of sandstone is generally 60–200 MPa, and the tensile strength is generally about 1.5 MPa [34,35,36]. Region ② is a high-compression area, and the target is broken and debris splashes from here. The results indicate that the shock pressure in region ③ remains subcritical to sandstone’s compressive strength, while the lower surface edge transfer of shock pressure to an unloading wave can generally reach the tensile strength of sandstone. The interaction of the compression wave and the unloading wave leads to a sphere-stripping effect in region ③. At the same time, tensile cross cracks will also occur in target region ④, as shown in Figure 3.

To investigate the scaling effects of projectile/target dimensions on failure characteristics, the HVI of different projectile body and target sizes were simulated at 3000 m/s. λ is defined as the ratio of the peak shock pressure at the target’s backside center (red dot in Figure 13b) to the tensile strength of the target. λ = Pα/f_T, where Pα is the peak shock pressure at the center of the back of the target, and f_T is the tensile strength of the target. The peak pressure at this position corresponds to the pressure rise from the shock wave’s initial arrival at the interface, with subsequent unloading initiating at this peak. It is the obvious difference between the compressive strength and the tensile strength of sandstone that induces size-dependent fracture mechanisms in HVI failure. Variation of ratio λ with the size ratio is obtained, as shown in Figure 13. λ decreases as the size ratio reduces, falling below 1 beyond a critical threshold. This indicates that shock unloading at such ratios will not lead to tensile failure generation, consistent with the semi-infinite target impact response. λ greater than 1 will produce damage in region ③, in the pattern shown in Figure 12. The damage to the target can be classified into Class I or Class II damage, according to variation in λ. Class I failure features the stripping effect, with retention of large, spherical fragments. Class II failure features traditional cratering and crushing, which generally occurs in the case of large target size and is consistent with the impact effect of a semi-infinite target.

6. Conclusions

In this paper, experiments and simulations of HVI of aluminum spheres on sandstone were conducted, and a stripping phenomenon under specific dimensional impacts was found. The spatial shock pressure attenuation model clarified distinct regional failure mechanisms in the target, incorporating unloading wave effects on peak pressure. The significant disparity between sandstone’s compressive and tensile strengths drives the combined effect of shock compression and unloading wave tension, inducing a regional shell-stripping phenomenon. We establish a quantitative framework for HVI-induced size effects governing damage assessment. Parameter λ was defined to distinguish two forms of destruction, Class I (stripping-dominated) and Class II (cratering-dominated). The size effect on the failure phenomenon of rock subjected to shock waves is worthy of attention. This research also provides a valuable reference for understanding the evolution and formation mechanisms of binary asteroids.