Effects of projectile parameters on the momentum transfer and projectile melting during hypervelocity impact

Wenjin Liu , Qingming Zhng ,*, Renrong Long ,**, Zizheng Gong , Ren Jinkng ,Xin Hu , Siyun Ren , Qing Wu , Gungming Song

a State Key Laboratory of Explosion Science and Technology, Beijing Institute of Technology, Beijing,100081, China

b Beijing Institute of Spacecraft Environment Engineering, China Academy of Space Technology, Beijing,100094, China

Keywords:Hypervelocity impact Energy partitioning Impact melting Momentum transfer

ABSTRACT The effects of projectile/target impedance matching and projectile shape on energy,momentum transfer and projectile melting during collisions are investigated by numerical simulation.By comparing the computation results with the experimental results, the correctness of the calculation and the statistical method of momentum transfer coefficient is verified.Different shapes of aluminum, copper and heavy tungsten alloy projectiles striking aluminum, basalt, and pumice target for impacts up to 10 km/s are simulated.The influence mechanism of the shape of the projectile and projectile/target density on the momentum transfer was obtained.With an increase in projectile density and length-diameter ratio, the energy transfer time between the projectile and targets is prolonged.The projectile decelerates slowly,resulting in a larger cratering depth.The energy consumed by the projectile in the excavation stage increased, resulting in lower mass-velocity of ejecta and momentum transfer coefficient.The numerical simulation results demonstrated that for different projectile/target combinations, the higher the wave impedance of the projectile, the higher the initial phase transition velocity and the smaller the mass of phase transition.The results can provide theoretical guidance for kinetic impactor design and material selection.

1.Introduction

The Double Asteroid Redirection Test (DART) spacecraft successfully conducted the first full-scale planetary defense test on September 26,2022,effectively verification of kinetic energy impact as a means of asteroid deflection [1].As the impact progresses, the ejecta leaves the surface of the asteroids and is thrown into space,forming a crater on the surface of the asteroids [2].Due to the generation of the ejecta,the momentum transferred to the asteroid by the impactor is greater than the impactor initial momentum[3].The asteroid gets a velocity change Δv and deviates from its original orbit [4].The ratio of the asteroid momentum change to the impactor momentum is commonly called the momentum transfer factor,which is expressed by a symbol β[5].The momentum transfer factor β is an essential parameter in evaluating the velocity change and kinetic impact effect of asteroids.

The kinetic impactor usually consists of a cube body with various loads and pores inside and solar panels, such as DART spacecraft[6].Experimental[7,8]and numerical simulation[9-11]results showed that the geometry and density of the impactor influence the momentum transfer coefficient and mass-velocity distribution of ejecta.Under the same impact velocity,the momentum transfer coefficient generated by the spherical impactor is the highest,and the momentum transfer coefficient is the largest when the impactor is closest to the target density[11].Although there are many experiments and numerical simulation studies on the influence of projectile shape and density on momentum transfer coefficient, there are few analyses on the energy transfer process between impactor and asteroid, and the influence mechanism of impactor shape and density on momentum and energy transfer is still unclear.The law of energy transfer between the impactor and asteroid is very important for designing kinetic impactors and selecting materials.The projectile kinetic energy is transferred to the target in the form of target kinetic energy,internal energy,and elastic compressive energy [12].The compression energy is dissipated by plastic work [13].Gault and Heitowit [14] proposed the first semi-analytical energy balance model, assuming that exactly half of the initial kinetic energy of the projectile is transferred to the target.Walker [15] analyzed that the proportion of energy dissipation changes with the impact velocity through the analytical model.Signetti and Heine [16] studied the impact kinetic energy distribution mechanism of different projectile/target combinations of metals through theoretical analysis.The conversion ratio is difficult to calculate theoretically and the influence of projectile density and shape on energy and momentum distribution is difficult to quantify, but this is done automatically in numerical simulation.According to the orbital parameters of asteroids,the kinetic impactor deflects the asteroids at a speed between 5 km/s and 20 km/s [17].In this velocity range, the energy involved in the impact was enough to melt, vaporize and ionize some of the impactor and asteroid material and the resulting plasma [18,19].The melting characteristics of projectile are beneficial to the selection materials of kinetic impactors.Due to the limitation of experimental conditions,it was difficult to quantitatively study the phase change mass of the impactor in the impact process.Numerical simulation usually adopts two methods [20], the peak pressure method [21,22] and the critical entropy method [23], to obtain the materials phase change information and quantitatively study the phase change mass and distribution in the impact process.Raducan et al.[9]used the peak pressure method to calculate the melting mass of Al6061 impactors with different shapes at the impact velocity of 6.5 km/s.The results showed that the projectile geometry had a particular influence on the melting mass of the projectile and the melting distribution of the projectile is not clear.

In this paper, the transfer and partitioning of energy of projectile/target were studied by numerical simulation,and the influence of projectile density and shape on energy distribution and momentum transfer coefficient were analyzed.The minimum impact pressure of three metal projectile materials required for each phase transformation state was solved.The impact pressure of phase transition was written into AUTODYN-2D subroutine as the criterion,and the phase distribution of the projectile at a different time and the change of phase transition mass of projectile with impact conditions were obtained.

2.Numerical model

The AUTODYN was used to build 2D axisymmetric models to numerically simulate the impact of projectiles on targets with different materials.The AUTODYN-2D has been benchmarked against other codes for crater size, peak shock pressure, and momentum enhancement factor[10,24].To save computing resources and computing time, the using finite element method (FEM) and smooth particle hydrodynamics (SPH) coupled numerical simulation method is adopted in this paper to quantitatively study the effects of the projectile and target materials on the phase change of projectile, energy transfer, and momentum enhancement.

As shown in Fig.1,SPH particles were used for modeling within the projectile range and the crater (30 times the radius of the sphere projectile).In contrast,conventional FEM mesh was used to model the small deformation area far away from the crater area.FEM mesh size is twice that of SPH particle size [25].The consolidation coupling is set between SPH particles and FEM elements through the Join function button.The coupling diagram of the SPH and FEM contact surface is shown in Fig.1.The bonded particles transfer the motion equation and other information from other particles to the finite element mesh [26].

Fig.1.Calculation model.

2.1.Projectile and target properties

2.1.1.Projectile geometry and material model

As shown in Fig.1 and Table 1,the projectile is modeled in three shapes:sphere,cylinder,and plate,which are made of Al2024.The diameter of the spherical projectile is 6 mm,and the diameter and height of the cylindrical projectile are half and 2.5 times the diameter of the spherical projectile, respectively.The diameter of the plate projectile is 12 mm,and its height is 1.05 mm.To keep the mass of the three metal projectiles with different densities the same,the diameter of the WHA and the copper projectile is 3.3 mm and 4.2 mm,respectively.The shape parameters of projectiles with different densities and shapes are shown in Table 1.Three different metal materials,Al2024,Copper,and heavy tungsten alloy(WHA),were used for the sphere projectile and modeled with the Mie-Grüneisen equation of state (EOS).The Steinberg-Guinan strength model is adopted for Al2024 and copper.The WHA adopts the Johnson-Cook material model.The material parameters are summarized in Tables 2-4.The maximum tensile stress failure criterion was adopted to describe the three metal materials.The Al2024,Copper, and WHA failure tensile stress were 1.5 GPa, 2 GPa, and 3.5 GPa, respectively[27,28].

2.1.2.Target materials model

The three target materials were Al2024, basalt, and pumice.Basalt and pumice were used to simulate dense and porous rocky asteroids.The target Al2024 material model and equation of state parameters are the same as the projectile.The Mie-Grüneisen EOS and Drucker-Prager strength model is used for basalt and pumice.The slope of the Drucker-Prager strength model for basalt and pumice is 0.6 [32].The yield strength of basalt and pumice is 145.5 MPa and 1 MPa, respectively.The basalt and pumice failure tensile stress is 50 MPa and 0.5 MPa, respectively [11,33].Pumice with 50% porosity and the P-alpha model represents the porosity[33].The expansion parameter α is introduced in the P-alpha model to separate the volume change into two parts: the volumechange caused by compression and pore collapse.The P-alpha model parameters are given in Table 5.

Table 1 Impactor parameters.

Table 2 Impactor and target parameters of different materials [29,30].

Table 4 The Johnson-Cook model material model parameters for WHA[27].

Table 5 P-alpha model for Pumice [33].

2.2.Hypervelocity impact benchmarking experiment

2.2.1.Experimental setup and method

As shown in Fig.2(a), sphere 2024 aluminum alloy projectiles with a diameter of 6 mm were used.The sphere projectile was launched with the polycarbonate sabot.The velocities of the projectiles were measured by an electromagnetic velocimeter,then the projectile and sabot were separated by a forced separation device.To calculate the speed of the sabot without the projectile,the sabot is divided into upper and lower parts(Fig.2(b)).A flute is cut in the center of the sabot, and an aluminum plate is placed to trigger magnetic velocity measurement and measure the velocity of the sabot.After obtaining the speed of the sabot,the sabot is stopped by the forced separation device.

The size of the basalt cylindrical targets is φ123.3×123.3 mm.The bulk density of the basalt was 2.89±0.01 g/cm3.The average uniaxial compressive strength and tensile strength of basalt are 144.5 and 9.69 MPa,respectively,according to uniaxial compressive and Brazilian tests.To prevent the material from falling and disruption under hypervelocity impact,several circles of fiber-glass reinforced plastic tape were wound around the side of basalt targets(Fig.3)The target is hung on two nylon ropes of 40 cm to form a ballistic pendulum,which is suspended in the center of the target chamber (Fig.3).

As shown in Fig.4,the experiment was carried out on the twostage light-gas gun with a 14.5 mm launch tube inner diameter.The ambient air pressure in the target chamber is kept at about 40-60 Pa.A high-speed camera was used to record the trajectory pendulum motion.The frame rate of the high-speed camera was 10,000 fps.An LED light is set opposite the high-speed camera.A circular baffle with a hole in the center is placed at the outlet of the magnetic velocimeter to allow the projectile to pass through,which can reduce the influence of the gas of the pump tube on the target and reduce the black smoke entering the target chamber and affect the shooting of high-speed camera.To quantitatively estimate the effect of pump-tube gas on momentum measurement,no projectile was used in the blow-down tests 3 and 4.The chamber pressure,hydrogen pump-tube pressure,and target were the same as in tests 1 and 2.The horizontal displacement of the target caused by the gas in the pump is obtained at different velocities.The maximum horizontal displacement was recorded, and the momentum enhancement β was calculated according to Eq.(1)

where M is the mass of the pendulum and target after impact,mpis the mass of the projectile, U is the impact speed, g =980 cm/s2is the acceleration of gravity, l=40 cm is the length of the ballistic swing rope,xmaxis the maximum horizontal displacement,and xbdis the blow-down test maximum horizontal displacement.The horizontal displacement caused by air flow is subtracted from the swing,as shown in Eq.,and the magnitude of the actual momentum transfer coefficient is calculated.

2.2.2.Experimental results

The crater diameter,depth and momentum transfer coefficients at different impact velocities are shown in Table 6.The impact velocity ranges from 3.47 to 3.9 km/s,β increasing from 2.39 to 2.51.As illustrated in Fig.5,the cavity formed on basalt targets consists of a small central pit of a bowl shape (within a thin white dashed circle in Fig.5)and a large shallow exfoliation area(within a white and red dashed circle in Fig.5),which are consistent with previous hypervelocity impacts on dense land rocks[34].These two parts are produced by the shock excavation flow, which crushes the target material, and the interaction of the shock wave with the free surface, that is, the spallation, respectively[35].

2.3.Phase transition criterion and momentum transfer calculation and set up

2.3.1.Phase transition criterion calculation

The spherical projectile impacting a semi-infinite target can be approximated as a one-dimensional plane shock at the contact interface [36].There is a conservation of mass, momentum, and energy in the shock wave front.

where u0and upis the particle velocity of the compressed material before and behind the shock, respectively.usis the shock wave velocity,v is the specific volume,ρ represents the material density,p denotes the shock pressure, and e is the specific internal energy.The subscript 0 indicates the initial condition in the unshocked material.The initial particle velocity u0, internal energy e0, and pressure p0are assumed to be zero.

Table 3 Parameters of Steinberg-Guinan strength model for Al 2024 and Copper [31].

Fig.2.Photograph of projectile and sabot: (a) Projectile and sabot; (b) Sabot and aluminum plate.

Fig.3.Picture of basalt pendulum in the target chamber.

Fig.5.Impact craters formed in basalt: (a) Impact velocity is 3.47 km/s; (b) Impact velocity is 3.90 km/s.

Fig.4.Schematic figures of the experimental apparatuses of a two-stage light-gas gun.

Table 6 The experimental parameters in hypervelocity impact conditions and results.

The shock wave and particle velocity satisfy the Hugoniot relationship over a wide range of pressures [37].

where c and s is the speed of sound and material constant,respectively.

The Hugoniot curve in Fig.6 can be obtained according to Eqs.(2)-(5).When the projectile impacts the target at hypervelocity,the initial impact process is adiabatic and non-isentropic.Therefore, the entropy of the material increases after the shock loading and isentropic unloading process,and the residual specific internal energy (eR) will heat the material, leading to the temperature rise of the material and then melting or vaporization.The increase in specific internal energy(eH)due to impact loading is the triangular ABC area in Fig.6.The decrease in internal energy(eIesn)due to isentropic unloading is the area under the isentropic unloading curve BD, which can be calculated by finite difference method.The residual specific internal energy is equal to the triangle ABC area minus the area under the curve BD,eR= eH- eIsen,which is the shaded area in Fig.6.

The material reaches the initial melting state when the residual internal energy in the material meets the following conditions:

where T0=298 K, Tmis the melting temperature of the material and Cs(T)is the specific heat capacity of solid material varying with temperature.

When the initial melting temperature of the material is reached,more energy is required for the material to melt completely, and the temperature of the material remains unchanged.Complete melting eventually occurs when the following condition is satisfied:

where Tvis the material vaporization temperature, and Clis the specific heat capacity of liquid.

The shock wave and thermodynamic parameters of three metal projectile materials are shown in Tables 2 and 7.According to Eqs.(2)-(5), the residual internal energy and peak pressure of three metal projectiles impacting the Al2024 target at different velocities were obtained(Fig.7).According to Eqs.(6)-(8),the peak pressure required for the initial melting and complete melting and initial vaporization of the material was solved,as shown in Table 8,which could be used as the criterion for the unloading phase transition of the material.

Fig.6.The relationship between the pressure and specific volume [36].

2.3.2.Phase transition criterion and momentum transfer set up

Based on the secondary development function of AUTODYN software, the maximum impact pressure criterion of material transformation was embedded into the AUTODYN subroutine as the phase transformation criterion.The subroutine will record the maximum impact pressure experienced by SPH particles and compare it with the phase change criterion to determine the states of all SPH particle materials.The spatial positions, masses, velocities,and phase states of all SPH particles at different times can also be derived by the subroutine, and the phase distribution cloud diagrams of projectiles at different times can be obtained.According to all ejecta mass and velocity information,the momentum components of all ejecta particles produced in the impact direction are added.The momentum transfer factor β-1 is equal to the total momentum of the ejecta in the direction of impact divided by the initial momentum of the projectile [40].

2.4.Resolution

Previous simulations show that the optimal resolution of the spherical projectile is 20 particles per projectile radius (ppr) using SPH calculation of cratering size and momentum enhancement[10].Projectile phase change mass is sensitive to resolution.Fig.8 shows the projectile phase transition results of the simulations with resolutions ranging from 10 to 50 ppr.Mass of each state of the projectile changes with increasing resolution until converging at resolutions above approximately 40 ppr.We adopted the resolution of 20 ppr to calculate crater and ejecta and 40 ppr to calculate the phase change of the projectile.

2.5.Numerical simulation verification

To verify the accuracy of the numerical simulation and the statistical method of momentum transfer coefficient, the final crater size and β was compared between the experiment and simulations.As shown in Table 9, simulations using AUTODYN predicted crater diameter and depth between 7%-15%and 3%-7%different that the experiment for aluminum spheres striking basalts,respectively.The maximum error between the momentum transfer coefficient obtained by numerical simulation and the experimental is 4.2% at different velocities.The modeling and statistical methods for momentum transfer coefficients were effectively validated by comparison with different velocities of experiment results.We compare the experimental and simulated results of 6.3 km/s aluminum spheres striking basalts.As shown in Fig.9,tensile cracks were formed in the target,resulting in the formation of a large mass of fragmentations.The formation of spalling fragments increases the value of momentum enhancement.As shown in Fig.10,at the impact velocity of 6.3 km/s, the momentum transfer coefficients of calculated and uncalculated spalling momenta are 3.01 and 4.37, respectively,which is larger than the Gault et al.[41] experimental data 3.5.Tensile fracture stress was the lone failure criterion in the simulation.The momentum enhancement is very sensitive to the tensile strength of the target,the tensile fracture stress determines the mass of the fragmentation [42].At the impact velocity of 6.3 km/s, the maximum tensile stress of basalt increases from 50 to 100 MPa,the momentum transfer coefficients of calculated and uncalculated spalling momenta are 2.76 and 3.32, respectively.The momentum transfer coefficient decreases from 4.37 to 3.32,which is close to the Gault et al.[41] experimental result.Hence, at a higher calculation speed,the failure stress of basalt is set at 100 MPa.

Fig.10 shows the mass-velocity distribution of the ejecta at different velocities.The mass velocity distribution of the ejecta formed by the shear excavation mechanism at different velocities is the same.The velocity of the spalling fragment is significantlylower than that of the ejecta formed by shear ejection,but the mass is significantly greater than the mass of the ejecta formed by shear ejection.Above the horizontal line are the mass and momentum of the spalling fragment.Spalling controls the results of the small-size crater,the spalling fragmentation greatly improves the mass of the ejecta, and the momentum transferred to the target (Fig.10(b)).When the crater reaches a few meters in diameter, a large area of damage forms on the crater surface, where the ejecta cannot gain enough speed to leave the asteroid surface [32,43].Therefore, the extrapolation of small-size experiments and simulations to the asteroid scale requires modification of the experimental and simulation results to remove the contribution of the ejecta generated by the spalling mechanism to the momentum transfer.Because of the low velocity of the spalling fragmentation, it takes longer for the fragment to completely leave the surface of the target.The duration of the spalling process is longer with the increase of the impact velocity.Therefore, to save calculation time and resources, the higher velocity calculation counts only the momentum contribution of the ejecta formed by the shear ejection.

Table 7 Thermodynamic parameter of three metal materials [16,38,39].

π Group scaling relationships were used to explain our simulation results, Holsapple and Housen [5] gives the scaling relationships of momentum transfer in the strength regime.

where μ and ν are constants scaling exponent from the pointsource measure [44], n = 3μ.A fixed value of v = 0.4 for different materials suggested by previous experimental studies[44].U and δ are impactor velocity and density, respectively.Y and ρ are target strength and density, respectively.Ksand Kvsare empirically determined constants, obtained by experiment or numerical simulation.

where m,Me,v*are the impactor mass,the cumulative mass of the ejecta and the minimum velocity of the ejecta, respectively.

Fig.11 shows the total ejecta mass uncalculated spalling mass with velocity greater than the velocity v of basalt impacted by the aluminum sphere.The slope of the line - 3μ = - 1.65.μ = 0.55 consistent with the previous experimental results[44].The knee inthe limit envelope for basalt impacted by the aluminum sphere has Ks=0.23 and Kvs= 0.51.Put scaled variables and the materials parameters of the projectile and basalt into Eq., the momentum transfer curve of the impact on basalt is obtained (U in m/s):

Table 8 Relevant pressure and energy of phase transition of three metal projectiles.

Fig.8.Effect of the resolution on the phase transition mass of the projectile, 6 mm Al2024 projectile impact on basalt target at 10 km/s.

In the range of 10 km/s impact velocity, the maximum error between the momentum transfer coefficient simulation results uncalculated spalling momenta and the law of similarity results is between 10% (see Fig.12).Which indicated that the calculation results are in good agreement with the scaling results.

Additional simulations for impacts into aluminum targets were also performed and compared with experimental results[45].Both the projectile and the target are made of 2024 aluminum.Denardop et al.[45] performed 6.35 mm aluminum spheres hypervelocityimpact experiments on 2024 aluminum targets at speeds ranging from 2 to 5 km/s.As shown in Fig.13,the maximum error between the momentum transfer coefficient obtained by numerical simulation under consistent conditions and the experimental is within 6.5% at different velocities.

Table 9 Comparison between numerical simulation and experimental of spheres striking basalt results.

Fig.9.Crater results of basalts: (a) Impact velocity is 3.9 km/s; (b) Impact velocity is 6.3 km/s.

Fig.10.The mass-velocity distribution of the ejecta of aluminum spheres striking basalt targets:(a)Cumulative ejected mass M(v),normalized by the mass of the projectile,m,as a function of normalized ejection velocity v/U; (b) Total ejected momentum β-1 , as a function of the normalized velocity, v/U.

To verify the accuracy of pumice parameters, 2D axisymmetric simulations were performed with the 2.54 cm and 0.32 cm aluminum spheres impacting pumice.As shown in Table 10, the simulated value of the momentum transfer coefficient of 2.54 cm aluminum sphere impacting pumice at 2.1 km/s is 1.22, which is close to the Walker et al.[46] experimental average data 1.3.The crater depth and diameter predicted by the code for the 2.54 cm sphere impacting pumice at 2.1 km/s were 30 and 18 cm, respectively.Simulations using AUTODYN predicted crater width and depth 20%and 30%different that the experiment,respectively.The pumice was hit with 0.32 cm aluminum projectile at 4.0 km/s,and the experimental results of mean momentum transfer coefficient were 2.15 [47] and the simulation results are 1.94, which is lower than the experimental result.The reason for this phenomenon may be that the porosity of the pumice target in the experiment ranges from 77 to 82%[47],while the porosity of the pumice target in the simulation is higher, which is 87%.

3.Results and discussions

3.1.Energy transfer and momentum enhancement

A detailed simulation of projectile shape and projectile/target density on energy transfer and momentum enhancement is explored in this section.The process and distribution of energy transfer between targets and projectiles, the mass and velocity distribution of ejecta are displayed and analyzed.

3.1.1.Effects of projectile shape

Fig.11.The mass with ejection velocity >v plotted in the strength-scaled form using logarithmic axes.

Fig.12.The momentum enhancement factor β vs impact velocity.

Fig.13.Momentum transfer coefficients of spheres striking 2024 aluminum.

Fig.14 shows the function of the kinetic energy of the projectile and target with time.When the upward propagating shock wave reaches the free surface of the projectile,it is reflected back and side to the projectile in the form of release wave.The release waves from the sides may reach the projectile target contact before the back surface release wave in a long but narrow projectile [48].The release wave reflected from the back and side of the projectile reaches the interface of the projectile and target,the projectile itself is completely unloaded, and does not act on the target.As the projectile height increases,projectile unloading time increases,the interaction time between the projectile and the target increases,and more projectile kinetic energy is converted into the target kinetic energy(Fig.14).At different speeds,with the increase of the long-radius ratio of the projectile, the projectile shape changes from flat plate to sphere and cylinder, and the target obtaining maximum kinetic energy increases (Fig.15).Compared with spherical projectiles, the crater formed by the flat projectile impacting the target is wide and shallow,and the mass and velocity of the ejecta produced by the flat projectile are smaller than those of the spherical projectile(Fig.16(a)).Compared with the spherical projectile, the cylindrical projectile transmits more kinetic energy to the target,and the crater is deeper,which is not conducive to the generation of ejecta, and the mass-velocity of the ejecta is smaller than that of the spherical projectile (Fig.16(a)).As shown in Fig.16(b), flat plate projectile produces the least momentum enhancement and the spherical projectile produces largest momentum enhancement [11].

3.1.2.Effects of projectile density

Consistent with the previous simulation results,the momentum transfer coefficient is the largest when the projectile density is closest to the target density at different impact velocities [11].As shown in Fig.17,with the increase of projectile/target density ratio,the internal energy of the projectile and the energy converted into the target decrease, resulting in a longer energy transfer time between the projectile and the target (Fig.17(a)), and the projectile decelerates slowly,leading to greater cratering depth;more energy is expended in the cratering phase and transferred to the target kinetic energy decreases(Fig.17(b)),resulting in lower velocity and less mass of the ejecta (Fig.18(a)).Therefore, at the same impactvelocity, the higher the projectile density, the lower the momentum transfer coefficient (Fig.18(b)).

Table 10 Comparison between numerical simulation and experimental of spheres striking pumice results.

3.1.3.Effects of target density

Fig.19 shows the kinetic energy, internal energy and plastic work of the Al2024 projectile and target with different densities as a function of time for impacts at 6.5 km/s.The energies have been normalized by the initial kinetic energy of the projectile.As shown in Fig.19,the energy transfer between the projectile and the target plate is very fast, and the time scale of energy deposition depends on the time required for the shock wave to reach the free surface of the projectile.When the release wave reaches the interface between the projectile and the target material, the initial energy transfer between the projectile and the target is completed.With the decrease of the wave impedance of the target material, the shock wave velocity in the projectile decreases,and the time for the target to obtain the maximum kinetic energy is prolonged.Compared with basalt and aluminum, pumice material has low wave impedance, and the time for the target to obtain maximum kinetic energy is doubled.After the initial energy transfer between the projectile and the target,the projectile and the target particles continue to crater under the action of the remaining kinetic energy,and the kinetic energy of the projectile and the target is converted into internal energy and plastic work of the target.The cratering flow stops under the action of the target strength, and the energy conversion time is prolonged with the decrease of the target strength.The aluminum projectile hits the aluminum, basalt, and pumice target at 6.5 km/s,and the energy stabilization time is 15 μs,20 μs,and 30 μs,respectively.At this moment,the energy transfer between the projectile and target has been completed, and the energy distribution can represent the final energy distribution.

Fig.20 shows the kinetic energy and internal energy of the target,normalized by the initial kinetic energy of the projectile,at the time of the target obtaining maximum kinetic energy.For the impact of Al2024 on Al2024, more than half of the kinetic energy of the projectile is converted to the target kinetic energy at different impact speeds,and the proportion of the kinetic energy of the projectile to the target kinetic energy is greater than that of the target internal energy.The increase in impact velocity translates into an increase in energy ratio within the target.For Al2024 impact basalt, with the increase of impact velocity, the proportion of projectile to target kinetic energy is decreased,and the proportion of projectile to target internal energy is increased.For impacting aluminum pumice, the ratio of the kinetic energy of the projectile to target kinetic energy and internal energy is lower than that of aluminum impacting basalt.For pumice, a porous material with high porosity, part of the projectile energy is consumed by pore collapse and deformation.On the contrary, under different impact velocities, the ratio of the kinetic energy of the projectile to the target internal energy is greater than that of the target kinetic energy.

Fig.14.Kinetic energy in the projectile and target as a function of time,normalized by the initial kinetic energy of the projectile,for different shape projectile impacting into a basalt target at 6.5 km/s: (a) Projectile kinetic energy; (b) Target kinetic energy.

Fig.15.The target obtained maximum kinetic energy and the crater depth for different shape projectiles impacting basalt targets at different velocities: (a) Target obtained maximum kinetic energy; (b) Crater depth.

Fig.16.Ejecta distribution from impacting into basalt target at 6.5 km/s: (a) Cumulative ejected mass, M(>v), normalized by the mass of the projectile, m, as a function of normalized ejection velocity, v/U; (b) Total ejected momentum β- 1, as a function of the normalized velocity v/U.

Fig.17.Kinetic energy in the projectile and target,normalized by the initial kinetic energy of the projectile, as a function of time, for 6.5 km/s different density spheres projectile striking basalt: (a) Projectile kinetic energy; (b) Target kinetic energy.

Fig.18.Ejecta distribution from different density spheres projectile impacting into basalt target:(a)Cumulative ejected mass,M(>v),normalized by the mass of the projectile,m,as a function of normalized ejection velocity, v/U; (b) Total ejected momentum, β- 1, as a function of the normalized ejection velocity, v/U.

Fig.19.Kinetic energy, internal energy and plastic work in the projectile and target, normalized by the initial kinetic energy of the projectile, as a function of time, for Al2024 spheres striking three different materials at 6.5 km/s: (a) Target is Al2024; (b) Target is basalt; (c) Target is pumice.

Fig.21 shows the kinetic energy and internal energy in the projectile and target,normalized by the initial kinetic energy of the projectile,for impacts into targets with different densities at 30 μs after impact.At this time, the projectile and the target have completed the energy transfer, and this time energy distribution can represent the final energy distribution.As shown in Fig.21(a),when an aluminum projectiles impact targets with different densities,most of the kinetic energy of the projectile is converted to the internal energy of the target, and the ratio of the residual kinetic energy and kinetic energy of the projectile is less than 0.1.With the increase in impact velocity, the proportion of residual kinetic energy and internal energy to the initial kinetic energy of the projectile decreases.

As shown in Fig.22, the momentum transfer coefficient of impacting porous pumice is significantly lower than that of impacting basalt.As can be seen from Fig.21,the Al2024 projectile impacts basalt, and pumice, and the maximum kinetic energy obtained by porous pumice is lower than that of the basalt.More energy is converted into the internal energy of the pumice, while the kinetic energy obtained by the pumice is less,resulting in lower kinetic energy and smaller momentum of the ejecta (Fig.23).The momentum transfer coefficient of impact pumice is smaller.

3.2.Melting distribution and mass of the projectile

The projectile shape and projectile/target density on phase distribution and mass of the projectile are investigated.The melting distribution and mass of the projectile vary with time are displayed and discussed.

3.2.1.Projectile shape effect on melting mass and distribution

Fig.24 shows phase distribution of the Al2024 projectile varies with time.As shown in Fig.24, the initial impact zone pressure reaches the required impact pressure for initial vaporization and complete melting of projectile material, and the projectile head material completely melts and partially vaporizes.As the shock wave propagates and the pressure peak decays,the material at the edge of the projectile and the upper part of the projectile are in a solid-liquid mixture state.The material at the top of the projectile did not melt and remained solid.

Fig.25 is the phase distribution diagram of projectile particles pushed back to the initial material position after impact 3 μs.After the impact of 3 μs, the projectile has been completely unloaded,and the phase state of the particles inside the projectile will not change.As shown in Fig.25(a) and Fig.25(b), the initial vaporization area of the material is mainly concentrated in the initial contact area of the projectile and target.The initial vaporization area of the projectile is discontinuous, and some particles in the center of the projectile have initial vaporization.Under the joint action of the back surface of the projectile and lateral release wave,a liquid and solid-liquid mixing zone is formed inside the projectiles.The material parameters of basalt and Al2024 are relatively close,and the distribution of the phase transition region inside the projectile impacting the two materials is similar.Compared with basalt and Al2024, pumice has a smaller wave impedance.Therefore, in the impact process, the pressure on the target contact surface is low, the kinetic energy of the projectile is converted to the internal energy of the projectile is low, the phase transformation quality of the projectile is low,and only a small amount of melting occurs at the head of the projectile (Fig.25(c)).

Fig.26 shows the phase states distribution of different shapes Al2024 projectile impacted basalt at 10 km/s.The phase transformation region of the spherical projectile (Fig.25(b)) and the cylinder projectile is mainly concentrated in the head (Fig.26(a)),and the phase transformation region of the flat plate projectile is distributed in layers (Fig.26(b)).The total mass fraction of phase transition is the largest in the flat plate projectile, and phase transition is more likely to occur during impact (Fig.27).Cylinder projectiles have the lowest mass fraction of phase transition and are more likely to survive the impact.

3.2.2.Projectile density effect on melting mass

Fig.20.The kinetic energy and internal energy of the target at the time of the target obtaining maximum kinetic energy: (a) Target is Al2024; (b) Target is basalt; (c) Target is pumice.

Fig.21.Kinetic energy and internal energy in the projectile and target, normalized by the initial kinetic energy of the projectile, for 6 mm Al2024 projectiles impact into targets with different densities at 30 μs after impact: (a) Projectile kinetic energy and internal energy; (b) Target kinetic energy and internal energy.

Fig.22.The β for 6 mm Al2024 spheres striking Al2024,basalt,and pumice targets at different velocities.

Fig.23.Total ejected momentum -1 , as a function of the normalized velocity, v/ U.

As shown in Fig.28(a), the liquid region inside the WHA projectile is not continuous.The phase transition regions of Copper and Al2024 spherical projectiles are similar.WHA material has the lowest mass fraction of phase transition.The phase transition mass fraction of Al2024 material is the highest (Fig.29).With the increase of the wave impedance of the projectile material, the proportion of the kinetic energy converted to the internal energy of the projectile decreases, and the proportion of the phase transformation in the projectile decreases.Under the condition that the target material remains unchanged,and the high-density and highimpedance projectile materials are more difficult to undergo a phase transition.Considering the effect of projectile shape and material on the formation of craters, the craters formed by the projectile with high density and large length-diameter ratio shape are deeper,which is conducive to discovering the internal structure of asteroids and sampling.However,the total mass and velocity of the ejecta formed during cratering formation are low, and the deflection efficiency is low.As shown in Figs.25 and 27, at the impact speed of 10 km/s, the metal projectile has partially melted or even vaporized.The phase transition distribution of projectiles beyond 10 km/s calculated by Mie-Gruneisen EOS may not be accurate.At higher speeds, equations of state, such as Analytic EOS[49],which can describe the treatment of solid,liquid,gas and their mixtures, need to be selected to study the projectile phase distribution.The DART result shows that the deflection velocity of 6.2 km/s is sufficient to melt, vaporize, and ionize some impactor and asteroid material and produce flash [50,51].Hence, the influence of impact melting and vaporization products on momentum enhancement needs further study.

3.2.3.Target density effect on melting mass

Fig.30 shows the relationship between the phase change mass of the projectile impacting different density targets with impact velocity and impact velocity.With the increase of impact velocity,the proportion of phase transformation of the projectile increases obviously when it impinges on aluminum and basalt targets.The initial melting pressure of Al2024 aluminum is reached at the impact velocity of 5 km/s, and more than half of the projectile particles reach the initial melting state at the impact velocity of 8 km/s.However, the proportion of phase transition of the projectile impacting the pumice target does not change significantly with the increase of impact velocity.At the impact velocity of 10 km/s, the mass fraction of projectiles undergoing initial melting and complete melting only accounts for 1.5%of the total mass of projectiles.

3.3.Implications for design and material selection of kinetic impactor

Fig.24.Phase distribution of Al2024 projectile varies with time,projectile diameter is 6 mm,and projectile velocity is 10 km/s:(a)Target is Al2024;(b)Target is basalt;(c)Target is pumice.

Fig.25.The phase distribution of Al2024 projectile; projectile diameter is 6 mm, and projectile velocity is 10 km/s: (a) Target is Al2024; (b) Target is basalt; (c) Target is pumice.

Fig.26.Phase distribution of different shaped projectiles impacted basalt targets at 10 km/s: (a) Projectile shape is a cylinder; (b) Projectile shape is a flat plate.

Fig.27.Different shapes of Al2024 projectile impact basalt phase transition mass at 10 km/s-1.

To get a greater momentum-transfer coefficient, the impactor should be designed to be spherical,choosing materials close to the density of the asteroid.Asteroids are usually covered with regolith and have dense structures at the bottom [52].To achieve impedance matching, the kinetic impactor should be designed in layers,with a low-density material at the front and a high-density material at the back.The results of impedance matching relationships obtained in this work could also be used in creating effective means of protecting satellites from space debris [53].A high-speed impact breaking up the asteroid is also an effective defense method [54].Under the same mass and velocity conditions, the impact crater formed by the high-density cylindrical impactor is deeper and more likely to break the asteroid.When the impactor is used to impact the surface of the asteroid for samples[55],high impedance materials are more difficult to melt and contaminate planetary material samples.The cylindrical impactor can bring the bottom material of the asteroid to the surface of the asteroid, which is conducive to observing the internal structure of the asteroid and obtaining fresh samples at the bottom.

Fig.28.Particle phase distribution, the target is Al2024, and the projectile velocity is 10 km/s: (a) Projectile is WHA; (b) Projectile is Copper.

Fig.29.The mass fraction of phase transition particles in the projectile, impacts the basalt target at 10 km/s.

4.Conclusions

In this paper,the energy transfer process between the projectile and target was studied by numerical simulation,and the influence of the density and shape of the projectile on the phase change mass of the projectile, impact energy distribution, ejecta mass-velocity distribution and the enhancement coefficient of momentum were analyzed.The influence mechanism of the shape of the projectile and projectile/target density ratio on the momentum transfer coefficient was obtained.Compared with the dense material, the impact of porous material will consume more energy in the compression and destruction of the pores,which is transformed into plastic work of the target material, resulting in a lower velocity of the ejecta.With the increase in projectile density and height, the energy transfer time between projectile targets is prolonged.The projectile decelerates slowly, resulting in a larger cratering depth and more energy consumption in the excavation stage, resulting in lower velocity, less mass of the ejecta, and lower momentum transfer coefficient.The efficiency of momentum transfer is highest when the projectile density is close to the target.The numerical simulation results demonstrated that for different projectile/target combinations,the higher the wave impedance of the projectile,the higher the initial phase transition velocity and the smaller the mass of melting.

Fig.30.The phase transition mass of Al2024 spherical projectile impacting different density targets with impact velocity: (a) Target is Al2024; (b) Target is basalt; (c) Target is pumice.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant Nos.62227901,12202068) and the Civil Aerospace Pre-research Project (Grant No.D020304), and the authors would like to thank these foundations for the financial support.