Experimental Investigation of the Crater Caused...

Research ArticleExperimental Investigation of the Crater Caused byHypervelocityRod Projectile Impacting on Rocks

Yanyu Qiu12 Songlin Yue 2 Mingyang Wang 12 Gan Li12 Yihao Cheng2

Zhangyong Zhao2 and Zhongwei Zhang2

1School of Mechanical Engineering Nanjing University of Science and Technology Nanjing 210094 China2State Key Laboratory for Disaster Prevention amp Mitigation of Explosion amp Impact College of Defense EngineeringArmy Engineering University of PLA Nanjing 210007 China

Correspondence should be addressed to Songlin Yue yslseuhotmailcom and Mingyang Wang wmyrf163com

Received 12 December 2019 Revised 5 April 2020 Accepted 29 June 2020 Published 24 July 2020

Academic Editor M I Herreros

Copyright copy 2020 Yanyu Qiu et al is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

To investigate the cratering effects of hypervelocity rod projectile impacting on rocks a two-stage light gas gun was used to carryout 10 groups of small-scale experiments whose velocity ranges from 15 kms to 41 kms After each experiment themorphologyand size of the hypervelocity impacting crater were accurately obtained by using a device for image scanning According to themorphology of the final crater the impact crater can be divided into crushing area spallation area and radial crack area Based onthe experimental results of steel projectile vertical impacting on granite targets the relationship between the depth and thediameter of the crater is analyzed ie hDasymp01sim02 it shows that the depth of the crater is much smaller than the diameter of thecrater and the crater seems to be a shallow dish e relation between the kinetic energy of the projectile and the size of the craterwas discussed With the increase of the projectile kinetic energy it is uncertain whether the depth of the crater increases but thevolume of the crater will increase Lastly dimensionless analysis of the impact crater was carried out Specifically the limitations ofpoint source solutions to hypervelocity rod projectile impact cratering have been proved and there is no essential difference tocalculate the final crater by using the energy scale or the momentum scale

1 Introduction

eproblemof hypervelocity impact on rockymedia originatedfrom the study ofGeology and Planetary Science [1 2]With therise of hypervelocity kinetic weapons the study of suchproblems began to attract more and more attention [3ndash5] ecratering phenomenon is one of the important effects of hy-pervelocity impact and the crater size can be used as a bridge tocalculate the ground shock effects by establishing equivalentmodels Moreover the impact crater size including depth anddiameter is restricted by the hydrodynamic limit of materialserefore it is very important to study the mechanism andpropagation process of cratering caused by hypervelocity im-pact which is also vital for understanding the damage effect ofthe hypervelocity kinetic energy weapon

Generally hypervelocity impact is a high-temperaturehigh-pressure and high strain-rate process caused by the rapidrelease of energy It is convenient to divide cratering processes

into two stages ie a relatively short initial high-pressure stageand a longer cratering flow stage [6] In the initial high-pressurestage a strong shock produced by the hypervelocity impactpropagates outward from the source creating a crushing regionWhen the high pressure reflects on the surface it becomes therarefaction wave which leads to spallatione crater continuesto form after shock crushing and spallation Melosh [7] clarifiedthe relation between the detached shock and the excavationflow e initial intensity and the propagation law of the shockwave can be easily and successfully measured by a variety oftechniques However less success had been achieved in pre-dicting the final crater size e author found that the exca-vation flow velocity was a small fraction of the particle velocitywhich was sensitive to the constitutive equation of materials Itwas difficult to calculate cratering by simplistic constitutiveequations erefore experimental investigation based on thephysical model and dimensional analysis is necessary to studythe problem of hypervelocity impact cratering

HindawiShock and VibrationVolume 2020 Article ID 9768745 11 pageshttpsdoiorg10115520209768745

is cratering problem can be conveniently divided intothree regimes In the ldquoearly stagerdquo the impact velocity andthe impedances of the two materials determine the shockpropagation speed In the ldquointermediate stagerdquo the pressuresare still large compared to either material strength viscosityor gravity-induced stresses so that the response is still muchlike that of an inviscid compressible fluid In the ldquolate stagerdquothe pressures decay down to the kilo bar level or less wheredepending upon the problem material strength viscosity orgravity forces act to retard and ultimately stop the cratergrowth at its final configuration An interesting phenome-non was discovered that both small-scale impact craters inthe laboratory and bowl-shaped craters which are less than5 km in diameter on the Earth are controlled by strength (ofrocks) [8] It is reasonable that the processes which lead tothe final crater are strongly influenced by low-stress materialproperties and for large craters by gravity In almost allcases any variable in early and intermediate time regimescan be assumed to be independent of viscosity and gravitye experiments [9ndash12] showed that small-scale craters werestrength-dominated whereas large-scale craters weregravity-dominated us point source assumption is de-veloped and proved to define a single scalar ldquocoupling pa-rameterrdquo measure let the dimensions of C be given by(length)l (velocity)μ (mass density)φ where μ and φ areconstants for given materials Note that with μ 23 andφ l3 C has the units of (energy)l3 With μ minus13 andφ 13 it has the units of (momentum)13 However a pointsource solution is to be a limit of solutions as the impactorradius r0⟶0erefore some researchers considered pointsource limits for specific materials e most common is forperfect gases and fluids where as already mentioned thepoint source solution for one-dimensional spherical ex-plosion as given by Taylor (1950) and Sedov (1946) is nowclassical However to date there are a large number ofdifferent phenomena and results that can be successfullycorrelated and explained by the concept [6] Generally itsfinal utility of applicability must be judged for cases

At present there are lots of investigations on themechanism and propagation process of hypervelocity im-pact cratering including results of small-scale experimentsin the laboratory and large-scale meteorite impact events inthe nature However the physical model and dimensionalanalysis were always established on the spherical projectileand a lot of studies did not consider cratering caused by therod projectile Actually hypervelocity kinetic energyweapons are always rod projectiles Besides it is nearlyimpossible to build a theoretical model to calculate hyper-velocity rod projectile impact cratering with simplisticconstitutive equations erefore it is important to inves-tigate the cratering effects caused by the hypervelocity rodprojectile which should be based on small-scale experimentsand dimensional analysis

2 Experimental Procedure of HypervelocityProjectile Impacting on Rocks

A two-stage light gas gun (Figure 1) was used which canaccelerate a projectile whose weight in grams to 3 to 5 kms

e projectile velocity is obtained through the method ofusing the laser velocimeter whose effective measuring rangeis 01 kmssim10 kms e target is placed horizontally in anenclosed target chamber After each experiment the residualprojectile and the target are recovered

e projectile has an ogive nose shape whose caliberradius head (CRH) equals to 3 and the diameter and thelength are equal to 72mm and 36mm respectively (ie thelength-diameter ratio is 5) e material is 30CrMnSiNiA themass of the projectile is 967 g and the density is 7850 kgm3e projectile is mounted in a polycarbonate sabot with adiameter of 18mm e sabot is formed as a three-lobedstructure and a mechanical shelling device (Figure 2) eprojectile penetrates the target after the sabot is decorticatedby the shelling device

e target material of all experiments is granite eparameters of granite measured before the experiment aredensity elastic P-wave velocity uniaxial compressivestrength ultimate shear strength shear modulus andPoissonrsquos ratio As shown in Figure 3 granite blocks areenclosed by C30 concrete and placed in a steel barrel As thewave impedance of C30 concrete is close to that of granite itis equivalent to increasing the size of the granite target

e target is made of granite with a density (ρt) of2670 kgm3 a longitudinal wave velocity (ct) of 4900ms auniaxial compressive strength (fc) of 150MPa a ultimateshear strength (τs) of 10GPa a shear modulus (G) of270 GPa a Poissonrsquos ratio (]) of 02 and Hugoniot elasticlimit (H) of 236sim263GPae cross section of the target is asquare with a side length of 600mm and a total thickness of800mme target is surrounded by a steel cylinder with aninner diameter of 900mm and a wall thickness of 10mm(Figure 3)

ere were 10 groups of hypervelocity impact experimentsconducted whose velocity range was 15 kmssim41 kms At theend of each experiment the recovery of the projectile and themeasurement of the target were carried out In all the ex-periments no residual projectile was found therefore erosiondamage of projectiles occurredere were no tensile cracks onthe edge of the rock target which indicated that the target waslarge enough to be considered as a semi-infinite target withoutbeing affected by the boundary reflection wave

3 Results and Discussion of the Impact Crater

31 5e Morphology of the Crater e concepts of thetransient crater and final crater are very important for brittlemedia such as rocks Because the tensile strength of thebrittle medium is far lower than the compressive strengththe reflected tensile wave on the surface of the target willcause irregular fragmentation of the target which is calledldquospallingrdquo If the zone of the crater caused by spalling isexcluded a bowl-shaped crater will be obtained is part ofthe crater is formed entirely by the flow of the mediumdriven by the shock compression wave which can be calledas the ldquotransient craterrdquo Because of the existence of spallingthe diameter and volume of the crater will be greatly in-creased and the crater as a whole will present a very roughand irregular shallow dish shape on the inner surface e

2 Shock and Vibration

final morphology of the crater affected by shock compres-sion and spalling is called as the ldquofinal craterrdquo

By observation of the final crater it can be found that theimpact crater can be divided into three typical zones after thehypervelocity penetration Each zone owns its obviouscharacteristics which are as follows

(1) e first zone is the crushing area As it is shown inFigure 4 the center of the crater is concave down-ward and there are lots of fine powders at the centerof the crater is results in the fact that the materialis intensely compressed and the original graniteparticles become fine powders

(2) e second zone is the spallation area As shown inFigure 5 spallation is obvious around the concaveregion at the center of the impact crater As thetensile strength of granite is much lower than thecompressive strength spallation occurs on the freesurface under the action of shock wave propagationand reflection which also lead to an irregular shapeof the impact crater

(3) e third zone is the radial crack area When theimpact speed is relatively high radial cracks aredistributed radially on the target As shown inFigure 6 on the edge of the impact crater obviousradial cracks can be seen before the rock blocks were

High-pressure chamber Piston First-stage barrel High-pressure cone

Projectile Second-stage barrel

Laservelocimeter Sheller

Projectile

Target

Target chamberSync lineOscilloscope

Sync line

Laser source

High-speed camera

Figure 1 Schematic diagram of the experimental system for a two-stage light gas gun

(a) (b)

Figure 2 Photographs of the projectile with sabots

Figure 3 Photograph of targets

Powders

Figure 4 Crushing area at the bottom of the crater

Spallation

Spallation

Figure 5 Spallation area around the center of the crater

Shock and Vibration 3

stripped off As shown in Figure 7 the impact crateris of a regular circular shape initially While theblocks of the radial crack area were stripped off theshape of the impact crater is no longer regular andthe diameter increases apparently

32 5e Size of the Crater A device for image scanning ofsurfacemorphology was used tomeasure the crater size and thescanned image and profile of the crater are shown in Figure 8

As brittle material can be destroyed and peeled on thefree surface by the tension wave the crater shape is veryirregular erefore each target is scanned by three inde-pendent cross sections to obtain the maximum depth of thecrater and the average diameter of the crater and the cratervolume can be calculated by numerical calculation estatistics in Table 1 are the experiment results As shown inTable 1 when the velocity of the projectile is over 2231 kmsthe depth of the crater increases with the increase of velocitye depth of the crater at the velocity of 1555 kms is greaterthan that at the velocity of 1829 kms and 2231 kms eexperimental results are in good agreement with the resultsof depth inversion in the range of 16 kms to 18 kms eaverage diameter of the crater increases with the increase ofprojectile velocity e shape of the crater is irregular due tothe spallation around the crater In Table 1D is the diameterof the crater h is the depth of the crater and V is the volumeof the crater and all of these parameters are the measurableapparent crater size At this time crater is also called the finalcrater

According to the experimental results the relationshipbetween the depth and the diameter of the crater is analyzedie hD asymp 01 sim 02 It shows that the depth of the crater ismuch smaller than the diameter of the crater and the craterseems to be a shallow dish e relationship between nor-malized diameter and depth is also shown in Figure 9Besides the experimental results of hypervelocity projectilesimpacting on gabbro by Polanskey and Ahrens [13] are alsogiven where the projectiles are spherical and the targets areSan Marcos gabbro and the impact velocity ranges from17 kms to 65 kms

Some interesting conclusions can be drawn fromFigure 9 (1) the depth of the crater which is caused by therod projectile (in this article) is 1-2 times of the length ofthe projectile while the depth of the crater caused by the

spherical projectile is 3ndash10 times of the length of theprojectile (2) From the shape of the impact crater thediameter of the crater caused by the spherical projectile isclose to the depth of the crater while the diameter of thecrater caused by the rod projectile (in this paper) is muchlarger than the depth of the crater (3) No matter whetherthe projectile is rod or spherical and no matter whether thetarget medium is granite or gabbro the diameter of thecrater is much larger than the diameter of the projectilewhich is more than 10 times (the prerequisite is thatimpact velocity is more than 5 Mach) (4) From the resultsof the crater the shape of the crater caused by the sphericalprojectile is more regular while the shape of the cratercaused by the rod projectile is more discrete

Due to the limitation of observation and measurementlittle is known about the actual situation of transient craterformation In recent years MEMIN a German researchteam had carried out a series of hypervelocity impact ex-periments of tuff sandstone and quartzite which measuredthe process and morphology of transient cratering Kenk-mann et al [14] and Dufresne et al [15] used a three-di-mensional laser to measure the inner wall of the crater andconsidered that the contour of the transient crater can befitted by a quadratic paraboloid ie the parabola can bedetermined by the central depression of the crater and theangle of the splash For the experimental results parabolawas used to fit the central concave part of the crater to get thetransient crater e typical section is shown in Figure 10which gives the fitting result of the crater at a projectilevelocity of 2231 kms

e parabolic equation for fitting the transient crater inFigure 10 is as follows

y 00155x2+ 002554x minus 4523118 (1)

e volume of the transient crater (Vtc) is as follows

Vtc π 4ac minus b2( 1113857

2

32a3 (2)

where a b and c are coefficients of the parabolic equationwhich can be obtained from equation (1) e results of thefinal and transient crater under different impact velocitieshave been listed in Table 2

33 Relation between the Kinetic Energy of the Projectile andthe Size of the Crater Combining Tables 1 and 2 the kineticenergy of the projectile can be known easily en therelation between the kinetic energy of the projectile and thesize of the crater is shown in Figures 11ndash13

Figure 11 and Table 2 show that there is little differencein volume of the transient crater and final crater under thesame kinetic energy of the projectile when it is less than40 kJ However when the kinetic energy increases thevolume deviates greatly and the discreteness is large eresults indicate that when the kinetic energy is small theshape of the impact crater at the bottom of the target wassimilar to a rotating paraboloid However with the increaseof the kinetic energy the shape of the impact crater is nolonger regular us the method of rotating paraboloid

Radial cracks

Figure 6 Radial cracks on the free surface


fitting the transient crater is not suitable for the impact cratercaused by the projectile with a large kinetic energy

In Figure 12 the relation between the kinetic energy ofthe projectile and the volume of the final crater in loga-rithmic coordinates is given As shown both the experi-mental results of this article and the experimental results byLange et al [16] are in the same fitting curve It indicates thatthe volume of the final crater exponentially increased withthe increase of the kinetic energy of the projectile and thepower function of the fitting curve has an exponent of 12(mathematical correlation coefficient R2 096) It is obviousthat the volume of the impact crater is always increasing withthe increase of the impact kinetic energy For hypervelocityimpacting on rocks it is uncertain whether the depth of thecrater increases by improving the kinetic energy of the

projectile but the volume of the crater will increase is isbecause the increase in the kinetic energy leads to muchmore increase in the diameter of the crater As shown inFigure 13 the kinetic energy of the projectile and the di-ameter of the final crater (the diameter data are shown inTable 2) satisfy a linear relationship (linear correlation co-efficient R2 0976)

4 Dimensionless Analysis of the Impact Crater

41 5e Limitations of Point Source Solutions Dimensionalanalysis is a method to determine the similarity criteria byusing dimension theory It is a common method to deter-mine the conditions of cratering in studies [17] For aspherical projectile Holsapple and Schmidt [6] believe that

Befo

re st

rippi

ng

(a)

Afte

r str

ippi

ng

(b)

Figure 7 e influence of stripping on the shape of the crater

(a) (b)

ndash300 ndash200 ndash100 0 100 200 300ndash300

ndash200

ndash100

0

100

200

300

The profile of the crater

Vert

ical

coor

dina

te (m

m)

Horizontal coordinate (mm)

(c)

Figure 8 Cratering test on granite under hypervelocity projectile impact (a) Photograph of a crater (b) 3D scanned image of the crater(c) Crater profile

Table 1 Experimental results of the final crater

NoParameter of the projectile Size of the final crater

ρp (gcm3) dp (cm) mp (g) υi (kms) D (cm) h (cm) V (cm3) Dh1 785 072 967 15550 167 4770 372 3502 785 072 967 18294 185 4468 503 4143 785 072 967 22310 275 4606 652 5974 785 072 967 28069 285 5137 715 5555 785 072 967 28782 335 6260 1575 5356 785 072 967 31478 405 6068 1391 6677 785 072 967 31996 387 5979 1716 6478 785 072 967 35421 470 6222 2863 7559 785 072 967 35584 461 6584 2761 70010 785 072 967 41356 597 6565 6299 909


the shape and size of the final crater can be determined by acoupling parameter which can characterize the total kineticenergy or momentum of the projectile e similar condi-tions of cratering in hypervelocity impact events can bededuced according to dimensional analysis which areshown as follows

V f r0 υi ρp1113966 1113967 ρt Y1113864 1113865 g1113960 1113961 (3)

where V denotes the crater volume r0 denotes the radius ofthe projectile υi denotes the velocity of the projectile ρpdenotes the density of the projectile g denotes gravity Ydenotes the strength of the target and ρt denotes the densityof the target

erefore for any given materials note parameter C as acoupling parameterus let the dimensions of C be given by(length)l (velocity)μ (mass density)φ where μ and φ areconstants for given materials Note that with μ 23 andφ l3 C has the units of (energy)l3 With μ 13 and φ l3it has the units of (momentum)13

C r0υiμρp

ψ (4)

According to the point source theory and power lawequation (3) can be written as follows

V f r0υμi ρ

ψp ρt Y g1113960 1113961 (5)

0 10 20 30 40 50 60 70 80 90 1000

2

4

6

8

10

12

14

Long rod projectile (this article)Spherical projectile (Polanskey and Ahrens)Fitting curve

hL

Dd

Slope = 015R2 = 096

Slope = 001R2 = 076

Figure 9 Relation between the dimensionless diameter and the dimensionless depth

Dep

th (m

m)

0

ndash10

ndash20

ndash30

ndash40

ndash50

Final craterTransient crater

Spallation

Radial dimension (mm)ndash250 ndash200 ndash150 ndash100 ndash50 0 50 100 150 200 250

Scanned data of final craterFitting curve of transient crater

Figure 10 Parabolic fitting of transient craters

Table 2 Experimental results of the transient crater

No υ0 (ms) V (cm3) Vtc (cm3) Dtc (cm) VtcV DtcD1 15550 372 105 76 028 0462 18294 503 182 104 036 0563 22310 652 207 109 032 0404 28069 715 288 126 040 0445 28782 1575 248 101 016 0306 31478 1391 376 127 027 0317 31996 1716 320 125 019 0328 35421 2863 1775 283 062 0609 35584 2761 997 200 036 04410 41356 6299 mdash mdash mdash mdash


where μ and ψ are constants determined by the experimentis expression involves five quantities and three inde-pendent dimensional units and can therefore be written interms of two dimensionless variables in various butequivalent ways One such form is

Vρ3φt Yρt( 11138573μ2

r0υμi ρ

φp1113872 1113873

3 fυ2i g( 1113857 Yρt( 1113857

μ2ρφtr0υ

μi ρ

φp

1113890 1113891 (6)

Equation (6) can be written as

V

r30

Y

ρtυ2i1113888 1113889

3μ2 ρt

ρp

1113888 1113889

3ψ

fυ2i

gr0

Y

ρtυ2i1113888 1113889

μ2 ρt

ρp

1113888 1113889

ψ⎡⎣ ⎤⎦ (7)

en equation (7) can be transformed into

ρtV

mp

Y

ρtυ2i1113888 1113889

3μ2 ρt

ρp

1113888 1113889

3ψminus1

Fgr0

υ2i

Y

ρtυ2i1113888 1113889

minusμ2 ρt

ρp

1113888 1113889

minusψ⎡⎣ ⎤⎦

(8)

where mp αρpr30 which denotes the mass of the projectileF(X) αminus1f(Xminus1) and α is the coefficient of correctionwhich is the mass ratio of the irregular projectile to thespherical projectile

e experiments [9ndash12] showed that small-scale craterswere strength-dominated whereas sufficiently large craterswere gravity-dominated erefore these are strength-dominated craters and it is reasonable to assume no gravitydependence us F(0) at the right part of formula (8) is aconstant and then

ρtV

mp

propY

ρtυ2i1113888 1113889

minus3μ2 ρt

ρp

1113888 1113889

1minus3ψ

(9)

where πV (ρtVmp) which is called crater efficiency andit is the mass ratio of the crater to the projectileπ3 (Yρtυ2i ) is called dimensionless strength and it is theratio of target material strength to initial dynamic pressureπ4 (ρtρp) is the density ratio of the target to the projectileand it is constant for given materials of the projectile and thetarget erefore equation (9) can be written as

πVprop πminus3μ23 (10)

ere are several models to determine material strengthHousen and Holsapple [18] believed that it is dominated bythe friction angle in the MohrndashCoulomb strength modelHowever Huirong [19] believed that it can be indicated asuniaxial compressive strength According to Tobias Hoerthaet al [20] the strength of the target (Y) was characterized bythe uniaxial compressive strength (UCS) for calculation of

0 10 20 30 40 50 60 70 80ndash1000

0

1000

2000

3000

4000


Volu

me (

cm3 )

Kinetic energy (kJ)

Figure 11 e volumes of the final and transient crater changewith the kinetic energy of the projectile

1 10 10010

100

1000

10000

100000

Experimental results of this articleLange and Ahrens (1983)Power function fitting curve

Slope = 12R2 = 096

Volu

me (

cm3 )

Kinetic energy (kJ)

Figure 12 Relation between the kinetic energy of the projectile andthe volume of the final crater

0 10 20 30 40 50 60 70 80 9010

20

30

40

50

60

70

Experimental results of this articleLinear fitting curve

Dia

met

er (c

m)

Slope = 0603R2 = 0976

Kinetic energy (kJ)

Figure 13 Relation between the kinetic energy of the projectile andthe diameter of the final crater


the strength-term π3 e measured μ-values are located inthe theorectical range ie 13lt μlt 23 erefore it isreasonable to determine material strength by the uniaxialcompressive strength For the almost nonporous granite orgabbro the product of crater volume and target density usedfor calculation of the cratering efficiency (πV) agrees wellwith the ejected mass erefore the relationship of πV andπ3 is shown in Figure 14

In Figure 14 the relation between normalized cratervolume (ie crater efficiency) and dimensionless strength isgivene higher the dimensionless strength is the lower thecrater efficiency is e normalized crater volume rangesfrom 101 to 103

Fitting curves of the impact crater caused by thespherical projectile (ie results of Polanskey and Ahrens)and rod projectile (ie results of this article) are both in thepower-law form as equations (11) and (12) respectively

πV 028πminus1103 (11)

πV 018πminus1333 (12)

According to the slope of equations (11) and (12) thecoupling parameter μ equals to 071 and 089 respectivelywhich are not located in the theoretical range (13lt μlt 23)Both the measured μ-values are greater than 23 ereforethe point source solutions cannot be applicable for the rodprojectile

As stated the only point source limit that exists is forr0⟶0 at fixed source energy For the simple physics of aperfect gas in a one-dimensional problem one can provethat a point source solution exists by analytical means Formore general materials and for two-dimensional problemssuch as the cratering problem there is no actual proof of theexistence of such solutions Generally the point sourcesolution cannot be used to analyze hypervelocity rod pro-jectile impact cratering

42 5e Energy Scale and Momentum Scale of HypervelocityImpact Cratering In the problem of cratering caused by thehypervelocity impact of the rod projectile the ldquofunda-mentalrdquo independent variables which will affect the di-mensional analysis process and the scaling law should becarefully considered When considering a projectile of agiven material (density ρp) and shape (length L and diameterr0) with massm and velocity υi impacts at a given target of agiven material (density ρt and strength Y) with an overlyingatmosphere (pressure Pa) e target is assumed to be ini-tially in lithostatic equilibrium under the influence ofgravitational field strength (g) In contrast the choice of thedependent variables is arbitrary such as the final cratervolume (V) or diameter (R)

A simple case occurs for a final crater geometry measuresuch as the volume V erefore it is given by a form such as

V ϕ L r0 υi ρp1113966 1113967 ρt Y1113864 1113865 Pa g1113960 1113961 (13)

In a length-force-time system the independent and de-pendent variables were developed according to dimensional

analysis [21] is expression involves nine quantities andthree independent dimensional units and can therefore bewritten in terms of six dimensionless variables in various butequivalent ways One such form is

V

Lr20 ϕ

ρpυ2iY

1113888 1113889ρp

ρt

1113888 1113889L

r01113888 1113889

gL

υ2i1113888 1113889

ρtgL

Pa

1113888 11138891113890 1113891 (14)

421 5e Energy Scale Equation (14) can be transformed asfollows

ρpV

mp

ρpυ2i

Y1113888 1113889

β

middotΦρp

ρt

1113888 1113889L

r01113888 1113889

gL

υ2i1113888 1113889

ρtgL

Pa

1113888 11138891113890 1113891 (15)

where mp (π4)ρpLr20 ΠV (ρpVmp) which is thecrater efficiency and it is the volume ratio of the crater to theprojectile and ΠY (ρpυ2i Y) which is the dimensionlessstrength and it is the ratio of initial dynamic pressure tomaterial strength it is obvious that ΠV and ΠY are powerexponential relations with an exponent of β

Small-scale craters are strength-dominated whereassufficiently large craters are gravity-dominated ereforethese are strength-dominated craters and it is reasonable toassume no gravity or atmosphere dependence In additionthe dimensionless terms (ρpρt) and (Lr0) are constant forgivenmaterials of the projectile and the targetusΦ [X] atthe right part of equation (15) is a constant and then

ρpV

mp

αρpυ2i

Y1113888 1113889

β

(16)

According to the qualitative relationship between thecrater efficiency and the dimensionless strength the lowerthe strength of the material is the greater the final cratervolume is erefore there must be such conditions 0lt βen it is possible to discuss different forms of expression(16) under different β-value conditions

10ndash3 10ndash2 10ndash1101

102

103

Slope = ndash110 (μ = 071)R2 = 096

π3 = Yρtvi2

π V =

ρtV

mp

Slope = ndash133 (μ = 089)R2 = 086

Results of this articleResults of Polanskey and Ahrens (1990)Fitting curves

Figure 14 e relationship of dimensionless πV and π3


(1) For 1le β equation (16) can be rewritten as follows

1 mpυ2iY middot V

middot αρpυ2i

Y1113888 1113889

βminus 1

(17)

Referring to the studies done by Wang et al and Liand Chen [22 23] ΠE (mpυ2i Y middot V) 43 which iswidely known as the dimensionless impact factor ordimensionless energy factor It denotes that the finalcrater volume increases with the increase of thekinetic energy of the projectile and the decrease ofmaterial strength [24 25]

(2) For 0lt βlt 1 equation (16) can be rewritten asfollows

1 mpυ2iY middot V

middot αY

ρpυ2i1113888 1113889

1minus β

(18)

From equation (18) the final crater volume can also beexpressed by the kinetic energy of the projectile and thematerial strength [26 27]

erefore no matter what the β-value is the final cratercan always be described by the dimensionless impact factoror dimensionless energy factor If the scaling law expressedas (17) or (18) is named as the energy scale then what formis the momentum scale

422 5e Momentum Scale Certainly equation (14) canalso be transformed as follows

ρtV

mp

ρtυ2iY

1113888 1113889

β

middotΦρp

ρt

1113888 1113889L

r01113888 1113889

gL

υ2i1113888 11138891113890 1113891 (19)

where πV (ρtVmp) and πY (ρtυ2i Y) Similarly it is easyto simplify equation (19) as follows

ρtV

mp

αρtυ2iY

1113888 1113889

β

(20)

e same as the above there must be such conditions0lt β en it is possible to discuss different forms of ex-pression (20) under different β-value conditions [28 29]


1 mpυi

VρtY

1113968 middot αρtυ2iY

1113888 1113889

βminus 05

(21)

where ΠM (mpυiVρtY

1113968) which is named as the

dimensionless momentum factor It denotes that thefinal crater volume increases with the increase ofmomentum of the projectile and the decrease ofmaterial strength and material density


1 mpυi

VρtY

1113968 middot αY

ρtυ2i1113888 1113889

05minus β

(22)

erefore no matter what the β-value is the final cratercan always be described by the dimensionless momentumfactor

423 5e Equivalent Effect Generally equation (16) can beregarded as the energy-scale expression which is describedby dimensionless ΠV and ΠY And equation (20) can beregarded as the momentum-scale expression which is de-scribed by dimensionless πV and πY

e relation between energy scale and momentum scalecan be built up as follows

ΠM ΠE middot Πm

1113968

mpυ2iYV

mp

ρtV

1113971

mpυi

VρtY

1113968 (23)

where Πm (mpρtV) ΠM mpυiVρtY

1113968 and

ΠE mpυ2i YV us it is obvious that the dimensionlessenergy factor and dimensionless momentum factor are notmutually independent in contrast they can be converted toeach other erefore there is an equivalent effect by usingthe energy scale or the momentum scale

According to equation (16) it is easy to establish therelation between dimensionless ΠV and ΠY based on theenergy scale which is shown in Figure 15 In the same wayaccording to equation (20) it is easy to establish the relation

000 002 004 006 008000

002

004

006

008

010

012

014

016

Experimental resultsof this articleExperimental resultsof Polanskey and Ahrens(1990)

Fitting curve ofthis article

=ρpυi2prodY Y

=ρpVprodV mp

Slope = 101R2 = 0996

Slope = 099R2 = 0910

Fitting curve of Polanskeyand Ahrens (1990)

Figure 15 e relation between dimensionless ΠV and ΠY basedon the energy scale


between dimensionless πV and πY based on the momentumscale which is shown in Figure 16

From Figure 15 the fitting curves of the relation betweendimensionless ΠV and ΠY are shown as follows

ΠV 125Π099Y R2 0910( 1113857 this article

ΠV 220Π101Y R2 0996( 1113857 Polanskey andAhrens

⎧⎨

⎩

(24)

From Figure 16 the fitting curves of the relation betweendimensionless πV and πY are shown as follows

πV 125π099Y R2 0910( 1113857 this article

πV 220π101Y R2 0996( 1113857 Polanskey andAhrens

⎧⎨

⎩

(25)

Combining Figures 15 and 16 and equations (24) and(25) it can be concluded that the relation between di-mensionless ΠV and ΠY based on the energy scale are thesame as the relation between dimensionless πV and πY basedon the momentum scale In dimensional analysis of hy-pervelocity impact cratering it is equivalent to calculate thefinal crater by using the energy scale or the momentum scale

5 Conclusion

(1) According to the morphology of the crater theimpact crater can be divided into crushing areaspallation area and radial crack area

(2) When a hypervelocity rod projectile impacts on agranite target spallation occurs on the free surfaceunder the action of shock wave propagation andreflection which also lead to an irregular shape of the

impact crater It shows that the depth of the crater ismuch smaller than the diameter of the crater and thecrater seems to be a shallow dish According to theexperimental results the relationship between thedepth and the diameter of the crater is analyzed iehD asymp 01 sim 02

(3) With the increase of the projectile kinetic energy it isuncertain whether the depth of the crater increasesbut the volume of the crater will increase is isbecause the increase of the kinetic energy leads tomuch more increase of the diameter of the crater

(4) e volume of the transient and final crater increaseswith the increase of the projectile kinetic energy andthe contribution of spallation to the volume isgrowing more rapidly

(5) When calculating the relationship between dimen-sionless crater efficiency and dimensionless strengthby the dimensional analysis method the point sourcesolution cannot be used to analyze the problem ofcratering caused by the hypervelocity rod projectileDimensional analysis was redesigned and the sim-ilarity law was re-established Another interestingand reasonable conclusion had been proved by ex-perimental investigation in which it is equivalent tocalculate the final crater by using the energy scale orthe momentum scale

Data Availability

e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

e authors acknowledge the financial support receivedfrom the Natural Science Foundation of China (Grant nos51808552 51808553 and 11602303) the China PostdoctoralScience Foundation (Grant nos 2017M621752 and2018M643853) and the Natural Science Foundation ofJiangsu Province (Grant no BK20190570)

References

[1] W Herrmann and J S Wilbeck ldquoReview of hypervelocitypenetration theoriesrdquo International Journal of Impact Engi-neering vol 5 no 1 pp 307ndash322 1986

[2] H J Melosh Impact Cratering Oxford University Press NewYork NY USA 1989

[3] D E Gault and E D Heitowit ldquoe partition of energy forhypervelocity impact craters formed in rockrdquo Sixth Hyper-velocity Impact Symposium Cleveland Ohio vol 2 pp 219ndash546 1963

[4] T H Antoun L A Glenn O R Walton P GoldsteinI N Lomov and B Liu ldquoSimulation of hypervelocity

0000 0005 0010 0015 0020 0025000000050010001500200025003000350040004500500055

Slope = 101R2 = 0996

Slope = 099R2 = 0910

=ρtVπV mp

=ρtυi2πY Y


Fitting curve ofthis articleFitting curve of Polanskeyand Ahrens (1990)

Figure 16 e relation between dimensionless πV and πY based onthe momentum scale


penetration in limestonerdquo International Journal of ImpactEngineering vol 33 no 1ndash12 pp 45ndash52 2006

[5] L I Zheng Y Liu M Hu et al ldquoDamage effect evaluation ofGod stick space-based kinetic energy weaponsrdquo Journal ofVibration and Shock vol 35 no 18 pp 159ndash180 2016

[6] K A Holsapple and R M Schmidt ldquoPoint source solutionsand coupling parameters in cratering mechanicsrdquo Journal ofGeophysical Research vol 92 no B7 pp 6350ndash6376 1987

[7] H J Melosh ldquoImpact cratering mechanics relationship be-tween the shock wave and excavation flowrdquo Icarus vol 62no 2 pp 339ndash343 1985

[8] T J Ahrens K Xia and D Cokert ldquoDepth of cracking be-neath impact craters new constraint for impact velocityrdquo inAIP Conference Proceedings Zacatecas Mexico July 2002

[9] K A Holsapple and R M Schmidt ldquoOn the scaling of craterdimensions 1 Explosive processesrdquo Journal of GeophysicalResearch vol 85 no B12 pp 7247ndash7256 1980

[10] K A Holsapple and R M Schmidt ldquoOn the scaling of craterdimensions 2 Impact processesrdquo Journal of GeophysicalResearch vol 87 no B3 pp 1849ndash1870 1982

[11] K A Holsapple ldquoe scaling of impact processes in planetarysciencesrdquo Annual Review of Earth and Planetary Sciencesvol 21 no 1 pp 333ndash373 1993

[12] K R Housen R M Schmidt and K A Holsapple ldquoCraterejecta scaling laws-Fundamental forms based on dimensionalanalysisrdquo Journal of Geophysical Research vol 88 no 17pp 2485ndash2499 1983

[13] C A Polanskey and T J Ahrens ldquoImpact spallation ex-periments fracture patterns and spall velocitiesrdquo Icarusvol 87 no 1 pp 140ndash155 1990

[14] T Kenkmann K Wunnemann A Deutsch M H PoelchauF Schafer and K oma ldquoImpact cratering in sandstone theMEMIN pilot study on the effect of pore waterrdquoMeteoritics ampPlanetary Science vol 46 no 6 pp 890ndash902 2011

[15] A Dufresne M H Poelchau T Kenkmann et al ldquoCratermorphology in sandstone targets the MEMIN impact pa-rameter studyrdquoMeteoritics amp Planetary Science vol 48 no 1pp 50ndash70 2013

[16] M A Lange T J Ahrens and M B Boslough ldquoImpactcratering and spall failure of gabbrordquo Icarus vol 58 no 3pp 383ndash395 1984

[17] J K Dienes and J M Walsh ldquoeory of impact some generalprinciples and the method of Eulerian codesrdquo High-velocityImpact Phenomenon pp 50ndash61 Academic Press New YorkNY USA 1970

[18] K R Housen and K A Holsapple ldquoEjecta from impactcratersrdquo Icarus vol 211 no 1 pp 856ndash875 2011

[19] A Huirong Shock-Induced Damage in Rocks Application toImpact Cratering pp 5ndash33 California Institute of Technol-ogy Pasadena CA USA 2006

[20] T F Hoerth J Hupfer O Millon and M Wickert ldquoMo-mentum transfer in hypervelocity impact experiments on rocktargetsrdquo Procedia Engineering vol 103 pp 197ndash204 2015

[21] S L Schafer and Y Y Qiu ldquoModelling experiment methodsfor cratering effects of explosions in rocks and comparativeanalysisrdquo Chinese Journal of Rock Mechanics and Engineeringvol 33 no 9 2014

[22] M Wang Y Qiu and S Yue ldquoSimilitude laws and modelingexperiments of explosion cratering in multi-layered geo-technical mediardquo International Journal of Impact Engineeringvol 117 pp 32ndash47 2018

[23] Q M Li and X W Chen ldquoDimensionless formulae forpenetration depth of concrete target impacted by a non-

deformable projectilerdquo International Journal of Impact En-gineering vol 28 no 1 pp 93ndash116 2003

[24] C E Anderson Jr D L Littlefield and J D Walker ldquoLong-rod penetration target resistance and hypervelocity impactrdquoInternational Journal of Impact Engineering vol 14 no 1ndash4pp 1ndash12 1993

[25] H Wu L-L Chen and Q Fang ldquoStability analyses of themass abrasive projectile high-speed penetrating into concretetarget Part I engineering model for the mass loss and nose-blunting of ogive-nosed projectilesrdquo Acta Mechanica Sinicavol 30 no 6 pp 933ndash942 2014

[26] H Wu Q Chen and L-L He ldquoStability analyses of the massabrasive projectile high-speed penetrating into concrete tar-get Part II structural stability analysesrdquo Acta MechanicaSinica vol 30 no 6 pp 943ndash955 2014

[27] G Ben-Dor T A Dubinsky and T Elperin ldquoHigh-speedpenetration modeling and shape optimization of the projectilepenetrating into concrete shieldsrdquo Mechanics Based Designof Structures and Machines vol 37 no 4 pp 538ndash549 2009

[28] Y Peng Q Fang H Wu et al ldquoeoretical analyses forterminal ballistic of the projectiles with different nose ge-ometries penetrating into concrete targetsrdquo Binggong XuebaoActa Armamentarii vol 35 pp 128ndash134 2014

[29] C S Meyer ldquoModeling experiments of hypervelocity pene-tration of adobe by spheres and rodsrdquo Procedia Engineeringvol 58 pp 138ndash146 2013


is cratering problem can be conveniently divided intothree regimes In the ldquoearly stagerdquo the impact velocity andthe impedances of the two materials determine the shockpropagation speed In the ldquointermediate stagerdquo the pressuresare still large compared to either material strength viscosityor gravity-induced stresses so that the response is still muchlike that of an inviscid compressible fluid In the ldquolate stagerdquothe pressures decay down to the kilo bar level or less wheredepending upon the problem material strength viscosity orgravity forces act to retard and ultimately stop the cratergrowth at its final configuration An interesting phenome-non was discovered that both small-scale impact craters inthe laboratory and bowl-shaped craters which are less than5 km in diameter on the Earth are controlled by strength (ofrocks) [8] It is reasonable that the processes which lead tothe final crater are strongly influenced by low-stress materialproperties and for large craters by gravity In almost allcases any variable in early and intermediate time regimescan be assumed to be independent of viscosity and gravitye experiments [9ndash12] showed that small-scale craters werestrength-dominated whereas large-scale craters weregravity-dominated us point source assumption is de-veloped and proved to define a single scalar ldquocoupling pa-rameterrdquo measure let the dimensions of C be given by(length)l (velocity)μ (mass density)φ where μ and φ areconstants for given materials Note that with μ 23 andφ l3 C has the units of (energy)l3 With μ minus13 andφ 13 it has the units of (momentum)13 However a pointsource solution is to be a limit of solutions as the impactorradius r0⟶0erefore some researchers considered pointsource limits for specific materials e most common is forperfect gases and fluids where as already mentioned thepoint source solution for one-dimensional spherical ex-plosion as given by Taylor (1950) and Sedov (1946) is nowclassical However to date there are a large number ofdifferent phenomena and results that can be successfullycorrelated and explained by the concept [6] Generally itsfinal utility of applicability must be judged for cases

At present there are lots of investigations on themechanism and propagation process of hypervelocity im-pact cratering including results of small-scale experimentsin the laboratory and large-scale meteorite impact events inthe nature However the physical model and dimensionalanalysis were always established on the spherical projectileand a lot of studies did not consider cratering caused by therod projectile Actually hypervelocity kinetic energyweapons are always rod projectiles Besides it is nearlyimpossible to build a theoretical model to calculate hyper-velocity rod projectile impact cratering with simplisticconstitutive equations erefore it is important to inves-tigate the cratering effects caused by the hypervelocity rodprojectile which should be based on small-scale experimentsand dimensional analysis

2 Experimental Procedure of HypervelocityProjectile Impacting on Rocks

A two-stage light gas gun (Figure 1) was used which canaccelerate a projectile whose weight in grams to 3 to 5 kms

e projectile velocity is obtained through the method ofusing the laser velocimeter whose effective measuring rangeis 01 kmssim10 kms e target is placed horizontally in anenclosed target chamber After each experiment the residualprojectile and the target are recovered

e projectile has an ogive nose shape whose caliberradius head (CRH) equals to 3 and the diameter and thelength are equal to 72mm and 36mm respectively (ie thelength-diameter ratio is 5) e material is 30CrMnSiNiA themass of the projectile is 967 g and the density is 7850 kgm3e projectile is mounted in a polycarbonate sabot with adiameter of 18mm e sabot is formed as a three-lobedstructure and a mechanical shelling device (Figure 2) eprojectile penetrates the target after the sabot is decorticatedby the shelling device

e target material of all experiments is granite eparameters of granite measured before the experiment aredensity elastic P-wave velocity uniaxial compressivestrength ultimate shear strength shear modulus andPoissonrsquos ratio As shown in Figure 3 granite blocks areenclosed by C30 concrete and placed in a steel barrel As thewave impedance of C30 concrete is close to that of granite itis equivalent to increasing the size of the granite target

e target is made of granite with a density (ρt) of2670 kgm3 a longitudinal wave velocity (ct) of 4900ms auniaxial compressive strength (fc) of 150MPa a ultimateshear strength (τs) of 10GPa a shear modulus (G) of270 GPa a Poissonrsquos ratio (]) of 02 and Hugoniot elasticlimit (H) of 236sim263GPae cross section of the target is asquare with a side length of 600mm and a total thickness of800mme target is surrounded by a steel cylinder with aninner diameter of 900mm and a wall thickness of 10mm(Figure 3)

ere were 10 groups of hypervelocity impact experimentsconducted whose velocity range was 15 kmssim41 kms At theend of each experiment the recovery of the projectile and themeasurement of the target were carried out In all the ex-periments no residual projectile was found therefore erosiondamage of projectiles occurredere were no tensile cracks onthe edge of the rock target which indicated that the target waslarge enough to be considered as a semi-infinite target withoutbeing affected by the boundary reflection wave

3 Results and Discussion of the Impact Crater

31 5e Morphology of the Crater e concepts of thetransient crater and final crater are very important for brittlemedia such as rocks Because the tensile strength of thebrittle medium is far lower than the compressive strengththe reflected tensile wave on the surface of the target willcause irregular fragmentation of the target which is calledldquospallingrdquo If the zone of the crater caused by spalling isexcluded a bowl-shaped crater will be obtained is part ofthe crater is formed entirely by the flow of the mediumdriven by the shock compression wave which can be calledas the ldquotransient craterrdquo Because of the existence of spallingthe diameter and volume of the crater will be greatly in-creased and the crater as a whole will present a very roughand irregular shallow dish shape on the inner surface e










Projectile

Target


Sync line

Laser source

High-speed camera


(a) (b)



Powders


Spallation

Spallation











y 00155x2+ 002554x minus 4523118 (1)



2

32a3 (2)




Radial cracks








Befo

re st

rippi

ng

(a)

Afte

r str

ippi

ng

(b)


(a) (b)


ndash200

ndash100

0

100

200

300


Vert

ical

coor

dina

te (m

m)


(c)







V f r0 υi ρp1113966 1113967 ρt Y1113864 1113865 g1113960 1113961 (3)



C r0υiμρp

ψ (4)


V f r0υμi ρ

ψp ρt Y g1113960 1113961 (5)

0 10 20 30 40 50 60 70 80 90 1000

2

4

6

8

10

12

14


hL

Dd

Slope = 015R2 = 096

Slope = 001R2 = 076


Dep

th (m

m)

0

ndash10

ndash20

ndash30

ndash40

ndash50


Spallation








Vρ3φt Yρt( 11138573μ2

r0υμi ρ

φp1113872 1113873

3 fυ2i g( 1113857 Yρt( 1113857

μ2ρφtr0υ

μi ρ

φp

1113890 1113891 (6)


V

r30

Y

ρtυ2i1113888 1113889

3μ2 ρt

ρp

1113888 1113889

3ψ

fυ2i

gr0

Y

ρtυ2i1113888 1113889

μ2 ρt

ρp

1113888 1113889

ψ⎡⎣ ⎤⎦ (7)


ρtV

mp

Y

ρtυ2i1113888 1113889

3μ2 ρt

ρp

1113888 1113889

3ψminus1

Fgr0

υ2i

Y

ρtυ2i1113888 1113889

minusμ2 ρt

ρp

1113888 1113889


(8)



ρtV

mp

propY

ρtυ2i1113888 1113889

minus3μ2 ρt

ρp

1113888 1113889

1minus3ψ

(9)




0 10 20 30 40 50 60 70 80ndash1000

0

1000

2000

3000

4000


Volu

me (

cm3 )

Kinetic energy (kJ)


1 10 10010

100

1000

10000

100000


Slope = 12R2 = 096

Volu

me (

cm3 )

Kinetic energy (kJ)


0 10 20 30 40 50 60 70 80 9010

20

30

40

50

60

70


Dia

met

er (c

m)

Slope = 0603R2 = 0976

Kinetic energy (kJ)






πV 028πminus1103 (11)

πV 018πminus1333 (12)





V ϕ L r0 υi ρp1113966 1113967 ρt Y1113864 1113865 Pa g1113960 1113961 (13)



V

Lr20 ϕ

ρpυ2iY

1113888 1113889ρp

ρt

1113888 1113889L

r01113888 1113889

gL

υ2i1113888 1113889

ρtgL

Pa

1113888 11138891113890 1113891 (14)


ρpV

mp

ρpυ2i

Y1113888 1113889

β

middotΦρp

ρt

1113888 1113889L

r01113888 1113889

gL

υ2i1113888 1113889

ρtgL

Pa

1113888 11138891113890 1113891 (15)



ρpV

mp

αρpυ2i

Y1113888 1113889

β

(16)



102

103

Slope = ndash110 (μ = 071)R2 = 096

π3 = Yρtvi2

π V =

ρtV

mp

Slope = ndash133 (μ = 089)R2 = 086





1 mpυ2iY middot V

middot αρpυ2i

Y1113888 1113889

βminus 1

(17)



1 mpυ2iY middot V

middot αY

ρpυ2i1113888 1113889

1minus β

(18)




ρtV

mp

ρtυ2iY

1113888 1113889

β

middotΦρp

ρt

1113888 1113889L

r01113888 1113889

gL

υ2i1113888 11138891113890 1113891 (19)


ρtV

mp

αρtυ2iY

1113888 1113889

β

(20)



1 mpυi

VρtY


1113888 1113889

βminus 05

(21)





1 mpυi

VρtY

1113968 middot αY

ρtυ2i1113888 1113889

05minus β

(22)




ΠM ΠE middot Πm

1113968

mpυ2iYV

mp

ρtV

1113971

mpυi

VρtY

1113968 (23)


1113968 and



000 002 004 006 008000

002

004

006

008

010

012

014

016



=ρpυi2prodY Y

=ρpVprodV mp

Slope = 101R2 = 0996

Slope = 099R2 = 0910








⎧⎨

⎩

(24)




⎧⎨

⎩

(25)


5 Conclusion







Data Availability




Acknowledgments


References





0000 0005 0010 0015 0020 0025000000050010001500200025003000350040004500500055

Slope = 101R2 = 0996

Slope = 099R2 = 0910

=ρtVπV mp

=ρtυi2πY Y









































Projectile

Target


Sync line

Laser source

High-speed camera


(a) (b)



Powders


Spallation

Spallation











y 00155x2+ 002554x minus 4523118 (1)



2

32a3 (2)




Radial cracks








Befo

re st

rippi

ng

(a)

Afte

r str

ippi

ng

(b)


(a) (b)


ndash200

ndash100

0

100

200

300


Vert

ical

coor

dina

te (m

m)


(c)







V f r0 υi ρp1113966 1113967 ρt Y1113864 1113865 g1113960 1113961 (3)



C r0υiμρp

ψ (4)


V f r0υμi ρ

ψp ρt Y g1113960 1113961 (5)

0 10 20 30 40 50 60 70 80 90 1000

2

4

6

8

10

12

14


hL

Dd

Slope = 015R2 = 096

Slope = 001R2 = 076


Dep

th (m

m)

0

ndash10

ndash20

ndash30

ndash40

ndash50


Spallation








Vρ3φt Yρt( 11138573μ2

r0υμi ρ

φp1113872 1113873

3 fυ2i g( 1113857 Yρt( 1113857

μ2ρφtr0υ

μi ρ

φp

1113890 1113891 (6)


V

r30

Y

ρtυ2i1113888 1113889

3μ2 ρt

ρp

1113888 1113889

3ψ

fυ2i

gr0

Y

ρtυ2i1113888 1113889

μ2 ρt

ρp

1113888 1113889

ψ⎡⎣ ⎤⎦ (7)


ρtV

mp

Y

ρtυ2i1113888 1113889

3μ2 ρt

ρp

1113888 1113889

3ψminus1

Fgr0

υ2i

Y

ρtυ2i1113888 1113889

minusμ2 ρt

ρp

1113888 1113889


(8)



ρtV

mp

propY

ρtυ2i1113888 1113889

minus3μ2 ρt

ρp

1113888 1113889

1minus3ψ

(9)




0 10 20 30 40 50 60 70 80ndash1000

0

1000

2000

3000

4000


Volu

me (

cm3 )

Kinetic energy (kJ)


1 10 10010

100

1000

10000

100000


Slope = 12R2 = 096

Volu

me (

cm3 )

Kinetic energy (kJ)


0 10 20 30 40 50 60 70 80 9010

20

30

40

50

60

70


Dia

met

er (c

m)

Slope = 0603R2 = 0976

Kinetic energy (kJ)






πV 028πminus1103 (11)

πV 018πminus1333 (12)





V ϕ L r0 υi ρp1113966 1113967 ρt Y1113864 1113865 Pa g1113960 1113961 (13)



V

Lr20 ϕ

ρpυ2iY

1113888 1113889ρp

ρt

1113888 1113889L

r01113888 1113889

gL

υ2i1113888 1113889

ρtgL

Pa

1113888 11138891113890 1113891 (14)


ρpV

mp

ρpυ2i

Y1113888 1113889

β

middotΦρp

ρt

1113888 1113889L

r01113888 1113889

gL

υ2i1113888 1113889

ρtgL

Pa

1113888 11138891113890 1113891 (15)



ρpV

mp

αρpυ2i

Y1113888 1113889

β

(16)



102

103

Slope = ndash110 (μ = 071)R2 = 096

π3 = Yρtvi2

π V =

ρtV

mp

Slope = ndash133 (μ = 089)R2 = 086





1 mpυ2iY middot V

middot αρpυ2i

Y1113888 1113889

βminus 1

(17)



1 mpυ2iY middot V

middot αY

ρpυ2i1113888 1113889

1minus β

(18)




ρtV

mp

ρtυ2iY

1113888 1113889

β

middotΦρp

ρt

1113888 1113889L

r01113888 1113889

gL

υ2i1113888 11138891113890 1113891 (19)


ρtV

mp

αρtυ2iY

1113888 1113889

β

(20)



1 mpυi

VρtY


1113888 1113889

βminus 05

(21)





1 mpυi

VρtY

1113968 middot αY

ρtυ2i1113888 1113889

05minus β

(22)




ΠM ΠE middot Πm

1113968

mpυ2iYV

mp

ρtV

1113971

mpυi

VρtY

1113968 (23)


1113968 and



000 002 004 006 008000

002

004

006

008

010

012

014

016



=ρpυi2prodY Y

=ρpVprodV mp

Slope = 101R2 = 0996

Slope = 099R2 = 0910








⎧⎨

⎩

(24)




⎧⎨

⎩

(25)


5 Conclusion







Data Availability




Acknowledgments


References





0000 0005 0010 0015 0020 0025000000050010001500200025003000350040004500500055

Slope = 101R2 = 0996

Slope = 099R2 = 0910

=ρtVπV mp

=ρtυi2πY Y









































y 00155x2+ 002554x minus 4523118 (1)



2

32a3 (2)




Radial cracks








Befo

re st

rippi

ng

(a)

Afte

r str

ippi

ng

(b)


(a) (b)


ndash200

ndash100

0

100

200

300


Vert

ical

coor

dina

te (m

m)


(c)







V f r0 υi ρp1113966 1113967 ρt Y1113864 1113865 g1113960 1113961 (3)



C r0υiμρp

ψ (4)


V f r0υμi ρ

ψp ρt Y g1113960 1113961 (5)

0 10 20 30 40 50 60 70 80 90 1000

2

4

6

8

10

12

14


hL

Dd

Slope = 015R2 = 096

Slope = 001R2 = 076


Dep

th (m

m)

0

ndash10

ndash20

ndash30

ndash40

ndash50


Spallation








Vρ3φt Yρt( 11138573μ2

r0υμi ρ

φp1113872 1113873

3 fυ2i g( 1113857 Yρt( 1113857

μ2ρφtr0υ

μi ρ

φp

1113890 1113891 (6)


V

r30

Y

ρtυ2i1113888 1113889

3μ2 ρt

ρp

1113888 1113889

3ψ

fυ2i

gr0

Y

ρtυ2i1113888 1113889

μ2 ρt

ρp

1113888 1113889

ψ⎡⎣ ⎤⎦ (7)


ρtV

mp

Y

ρtυ2i1113888 1113889

3μ2 ρt

ρp

1113888 1113889

3ψminus1

Fgr0

υ2i

Y

ρtυ2i1113888 1113889

minusμ2 ρt

ρp

1113888 1113889


(8)



ρtV

mp

propY

ρtυ2i1113888 1113889

minus3μ2 ρt

ρp

1113888 1113889

1minus3ψ

(9)




0 10 20 30 40 50 60 70 80ndash1000

0

1000

2000

3000

4000


Volu

me (

cm3 )

Kinetic energy (kJ)


1 10 10010

100

1000

10000

100000


Slope = 12R2 = 096

Volu

me (

cm3 )

Kinetic energy (kJ)


0 10 20 30 40 50 60 70 80 9010

20

30

40

50

60

70


Dia

met

er (c

m)

Slope = 0603R2 = 0976

Kinetic energy (kJ)






πV 028πminus1103 (11)

πV 018πminus1333 (12)





V ϕ L r0 υi ρp1113966 1113967 ρt Y1113864 1113865 Pa g1113960 1113961 (13)



V

Lr20 ϕ

ρpυ2iY

1113888 1113889ρp

ρt

1113888 1113889L

r01113888 1113889

gL

υ2i1113888 1113889

ρtgL

Pa

1113888 11138891113890 1113891 (14)


ρpV

mp

ρpυ2i

Y1113888 1113889

β

middotΦρp

ρt

1113888 1113889L

r01113888 1113889

gL

υ2i1113888 1113889

ρtgL

Pa

1113888 11138891113890 1113891 (15)



ρpV

mp

αρpυ2i

Y1113888 1113889

β

(16)



102

103

Slope = ndash110 (μ = 071)R2 = 096

π3 = Yρtvi2

π V =

ρtV

mp

Slope = ndash133 (μ = 089)R2 = 086





1 mpυ2iY middot V

middot αρpυ2i

Y1113888 1113889

βminus 1

(17)



1 mpυ2iY middot V

middot αY

ρpυ2i1113888 1113889

1minus β

(18)




ρtV

mp

ρtυ2iY

1113888 1113889

β

middotΦρp

ρt

1113888 1113889L

r01113888 1113889

gL

υ2i1113888 11138891113890 1113891 (19)


ρtV

mp

αρtυ2iY

1113888 1113889

β

(20)



1 mpυi

VρtY


1113888 1113889

βminus 05

(21)





1 mpυi

VρtY

1113968 middot αY

ρtυ2i1113888 1113889

05minus β

(22)




ΠM ΠE middot Πm

1113968

mpυ2iYV

mp

ρtV

1113971

mpυi

VρtY

1113968 (23)


1113968 and



000 002 004 006 008000

002

004

006

008

010

012

014

016



=ρpυi2prodY Y

=ρpVprodV mp

Slope = 101R2 = 0996

Slope = 099R2 = 0910








⎧⎨

⎩

(24)




⎧⎨

⎩

(25)


5 Conclusion







Data Availability




Acknowledgments


References





0000 0005 0010 0015 0020 0025000000050010001500200025003000350040004500500055

Slope = 101R2 = 0996

Slope = 099R2 = 0910

=ρtVπV mp

=ρtυi2πY Y






































Befo

re st

rippi

ng

(a)

Afte

r str

ippi

ng

(b)


(a) (b)


ndash200

ndash100

0

100

200

300


Vert

ical

coor

dina

te (m

m)


(c)







V f r0 υi ρp1113966 1113967 ρt Y1113864 1113865 g1113960 1113961 (3)



C r0υiμρp

ψ (4)


V f r0υμi ρ

ψp ρt Y g1113960 1113961 (5)

0 10 20 30 40 50 60 70 80 90 1000

2

4

6

8

10

12

14


hL

Dd

Slope = 015R2 = 096

Slope = 001R2 = 076


Dep

th (m

m)

0

ndash10

ndash20

ndash30

ndash40

ndash50


Spallation








Vρ3φt Yρt( 11138573μ2

r0υμi ρ

φp1113872 1113873

3 fυ2i g( 1113857 Yρt( 1113857

μ2ρφtr0υ

μi ρ

φp

1113890 1113891 (6)


V

r30

Y

ρtυ2i1113888 1113889

3μ2 ρt

ρp

1113888 1113889

3ψ

fυ2i

gr0

Y

ρtυ2i1113888 1113889

μ2 ρt

ρp

1113888 1113889

ψ⎡⎣ ⎤⎦ (7)


ρtV

mp

Y

ρtυ2i1113888 1113889

3μ2 ρt

ρp

1113888 1113889

3ψminus1

Fgr0

υ2i

Y

ρtυ2i1113888 1113889

minusμ2 ρt

ρp

1113888 1113889


(8)



ρtV

mp

propY

ρtυ2i1113888 1113889

minus3μ2 ρt

ρp

1113888 1113889

1minus3ψ

(9)




0 10 20 30 40 50 60 70 80ndash1000

0

1000

2000

3000

4000


Volu

me (

cm3 )

Kinetic energy (kJ)


1 10 10010

100

1000

10000

100000


Slope = 12R2 = 096

Volu

me (

cm3 )

Kinetic energy (kJ)


0 10 20 30 40 50 60 70 80 9010

20

30

40

50

60

70


Dia

met

er (c

m)

Slope = 0603R2 = 0976

Kinetic energy (kJ)






πV 028πminus1103 (11)

πV 018πminus1333 (12)





V ϕ L r0 υi ρp1113966 1113967 ρt Y1113864 1113865 Pa g1113960 1113961 (13)



V

Lr20 ϕ

ρpυ2iY

1113888 1113889ρp

ρt

1113888 1113889L

r01113888 1113889

gL

υ2i1113888 1113889

ρtgL

Pa

1113888 11138891113890 1113891 (14)


ρpV

mp

ρpυ2i

Y1113888 1113889

β

middotΦρp

ρt

1113888 1113889L

r01113888 1113889

gL

υ2i1113888 1113889

ρtgL

Pa

1113888 11138891113890 1113891 (15)



ρpV

mp

αρpυ2i

Y1113888 1113889

β

(16)



102

103

Slope = ndash110 (μ = 071)R2 = 096

π3 = Yρtvi2

π V =

ρtV

mp

Slope = ndash133 (μ = 089)R2 = 086





1 mpυ2iY middot V

middot αρpυ2i

Y1113888 1113889

βminus 1

(17)



1 mpυ2iY middot V

middot αY

ρpυ2i1113888 1113889

1minus β

(18)




ρtV

mp

ρtυ2iY

1113888 1113889

β

middotΦρp

ρt

1113888 1113889L

r01113888 1113889

gL

υ2i1113888 11138891113890 1113891 (19)


ρtV

mp

αρtυ2iY

1113888 1113889

β

(20)



1 mpυi

VρtY


1113888 1113889

βminus 05

(21)





1 mpυi

VρtY

1113968 middot αY

ρtυ2i1113888 1113889

05minus β

(22)




ΠM ΠE middot Πm

1113968

mpυ2iYV

mp

ρtV

1113971

mpυi

VρtY

1113968 (23)


1113968 and



000 002 004 006 008000

002

004

006

008

010

012

014

016



=ρpυi2prodY Y

=ρpVprodV mp

Slope = 101R2 = 0996

Slope = 099R2 = 0910








⎧⎨

⎩

(24)




⎧⎨

⎩

(25)


5 Conclusion







Data Availability




Acknowledgments


References





0000 0005 0010 0015 0020 0025000000050010001500200025003000350040004500500055

Slope = 101R2 = 0996

Slope = 099R2 = 0910

=ρtVπV mp

=ρtυi2πY Y


































V f r0 υi ρp1113966 1113967 ρt Y1113864 1113865 g1113960 1113961 (3)



C r0υiμρp

ψ (4)


V f r0υμi ρ

ψp ρt Y g1113960 1113961 (5)

0 10 20 30 40 50 60 70 80 90 1000

2

4

6

8

10

12

14


hL

Dd

Slope = 015R2 = 096

Slope = 001R2 = 076


Dep

th (m

m)

0

ndash10

ndash20

ndash30

ndash40

ndash50


Spallation








Vρ3φt Yρt( 11138573μ2

r0υμi ρ

φp1113872 1113873

3 fυ2i g( 1113857 Yρt( 1113857

μ2ρφtr0υ

μi ρ

φp

1113890 1113891 (6)


V

r30

Y

ρtυ2i1113888 1113889

3μ2 ρt

ρp

1113888 1113889

3ψ

fυ2i

gr0

Y

ρtυ2i1113888 1113889

μ2 ρt

ρp

1113888 1113889

ψ⎡⎣ ⎤⎦ (7)


ρtV

mp

Y

ρtυ2i1113888 1113889

3μ2 ρt

ρp

1113888 1113889

3ψminus1

Fgr0

υ2i

Y

ρtυ2i1113888 1113889

minusμ2 ρt

ρp

1113888 1113889


(8)



ρtV

mp

propY

ρtυ2i1113888 1113889

minus3μ2 ρt

ρp

1113888 1113889

1minus3ψ

(9)




0 10 20 30 40 50 60 70 80ndash1000

0

1000

2000

3000

4000


Volu

me (

cm3 )

Kinetic energy (kJ)


1 10 10010

100

1000

10000

100000


Slope = 12R2 = 096

Volu

me (

cm3 )

Kinetic energy (kJ)


0 10 20 30 40 50 60 70 80 9010

20

30

40

50

60

70


Dia

met

er (c

m)

Slope = 0603R2 = 0976

Kinetic energy (kJ)






πV 028πminus1103 (11)

πV 018πminus1333 (12)





V ϕ L r0 υi ρp1113966 1113967 ρt Y1113864 1113865 Pa g1113960 1113961 (13)



V

Lr20 ϕ

ρpυ2iY

1113888 1113889ρp

ρt

1113888 1113889L

r01113888 1113889

gL

υ2i1113888 1113889

ρtgL

Pa

1113888 11138891113890 1113891 (14)


ρpV

mp

ρpυ2i

Y1113888 1113889

β

middotΦρp

ρt

1113888 1113889L

r01113888 1113889

gL

υ2i1113888 1113889

ρtgL

Pa

1113888 11138891113890 1113891 (15)



ρpV

mp

αρpυ2i

Y1113888 1113889

β

(16)



102

103

Slope = ndash110 (μ = 071)R2 = 096

π3 = Yρtvi2

π V =

ρtV

mp

Slope = ndash133 (μ = 089)R2 = 086





1 mpυ2iY middot V

middot αρpυ2i

Y1113888 1113889

βminus 1

(17)



1 mpυ2iY middot V

middot αY

ρpυ2i1113888 1113889

1minus β

(18)




ρtV

mp

ρtυ2iY

1113888 1113889

β

middotΦρp

ρt

1113888 1113889L

r01113888 1113889

gL

υ2i1113888 11138891113890 1113891 (19)


ρtV

mp

αρtυ2iY

1113888 1113889

β

(20)



1 mpυi

VρtY


1113888 1113889

βminus 05

(21)





1 mpυi

VρtY

1113968 middot αY

ρtυ2i1113888 1113889

05minus β

(22)




ΠM ΠE middot Πm

1113968

mpυ2iYV

mp

ρtV

1113971

mpυi

VρtY

1113968 (23)


1113968 and



000 002 004 006 008000

002

004

006

008

010

012

014

016



=ρpυi2prodY Y

=ρpVprodV mp

Slope = 101R2 = 0996

Slope = 099R2 = 0910








⎧⎨

⎩

(24)




⎧⎨

⎩

(25)


5 Conclusion







Data Availability




Acknowledgments


References





0000 0005 0010 0015 0020 0025000000050010001500200025003000350040004500500055

Slope = 101R2 = 0996

Slope = 099R2 = 0910

=ρtVπV mp

=ρtυi2πY Y


































Vρ3φt Yρt( 11138573μ2

r0υμi ρ

φp1113872 1113873

3 fυ2i g( 1113857 Yρt( 1113857

μ2ρφtr0υ

μi ρ

φp

1113890 1113891 (6)


V

r30

Y

ρtυ2i1113888 1113889

3μ2 ρt

ρp

1113888 1113889

3ψ

fυ2i

gr0

Y

ρtυ2i1113888 1113889

μ2 ρt

ρp

1113888 1113889

ψ⎡⎣ ⎤⎦ (7)


ρtV

mp

Y

ρtυ2i1113888 1113889

3μ2 ρt

ρp

1113888 1113889

3ψminus1

Fgr0

υ2i

Y

ρtυ2i1113888 1113889

minusμ2 ρt

ρp

1113888 1113889


(8)



ρtV

mp

propY

ρtυ2i1113888 1113889

minus3μ2 ρt

ρp

1113888 1113889

1minus3ψ

(9)




0 10 20 30 40 50 60 70 80ndash1000

0

1000

2000

3000

4000


Volu

me (

cm3 )

Kinetic energy (kJ)


1 10 10010

100

1000

10000

100000


Slope = 12R2 = 096

Volu

me (

cm3 )

Kinetic energy (kJ)


0 10 20 30 40 50 60 70 80 9010

20

30

40

50

60

70


Dia

met

er (c

m)

Slope = 0603R2 = 0976

Kinetic energy (kJ)






πV 028πminus1103 (11)

πV 018πminus1333 (12)





V ϕ L r0 υi ρp1113966 1113967 ρt Y1113864 1113865 Pa g1113960 1113961 (13)



V

Lr20 ϕ

ρpυ2iY

1113888 1113889ρp

ρt

1113888 1113889L

r01113888 1113889

gL

υ2i1113888 1113889

ρtgL

Pa

1113888 11138891113890 1113891 (14)


ρpV

mp

ρpυ2i

Y1113888 1113889

β

middotΦρp

ρt

1113888 1113889L

r01113888 1113889

gL

υ2i1113888 1113889

ρtgL

Pa

1113888 11138891113890 1113891 (15)



ρpV

mp

αρpυ2i

Y1113888 1113889

β

(16)



102

103

Slope = ndash110 (μ = 071)R2 = 096

π3 = Yρtvi2

π V =

ρtV

mp

Slope = ndash133 (μ = 089)R2 = 086





1 mpυ2iY middot V

middot αρpυ2i

Y1113888 1113889

βminus 1

(17)



1 mpυ2iY middot V

middot αY

ρpυ2i1113888 1113889

1minus β

(18)




ρtV

mp

ρtυ2iY

1113888 1113889

β

middotΦρp

ρt

1113888 1113889L

r01113888 1113889

gL

υ2i1113888 11138891113890 1113891 (19)


ρtV

mp

αρtυ2iY

1113888 1113889

β

(20)



1 mpυi

VρtY


1113888 1113889

βminus 05

(21)





1 mpυi

VρtY

1113968 middot αY

ρtυ2i1113888 1113889

05minus β

(22)




ΠM ΠE middot Πm

1113968

mpυ2iYV

mp

ρtV

1113971

mpυi

VρtY

1113968 (23)


1113968 and



000 002 004 006 008000

002

004

006

008

010

012

014

016



=ρpυi2prodY Y

=ρpVprodV mp

Slope = 101R2 = 0996

Slope = 099R2 = 0910








⎧⎨

⎩

(24)




⎧⎨

⎩

(25)


5 Conclusion







Data Availability




Acknowledgments


References





0000 0005 0010 0015 0020 0025000000050010001500200025003000350040004500500055

Slope = 101R2 = 0996

Slope = 099R2 = 0910

=ρtVπV mp

=ρtυi2πY Y




































πV 028πminus1103 (11)

πV 018πminus1333 (12)





V ϕ L r0 υi ρp1113966 1113967 ρt Y1113864 1113865 Pa g1113960 1113961 (13)



V

Lr20 ϕ

ρpυ2iY

1113888 1113889ρp

ρt

1113888 1113889L

r01113888 1113889

gL

υ2i1113888 1113889

ρtgL

Pa

1113888 11138891113890 1113891 (14)


ρpV

mp

ρpυ2i

Y1113888 1113889

β

middotΦρp

ρt

1113888 1113889L

r01113888 1113889

gL

υ2i1113888 1113889

ρtgL

Pa

1113888 11138891113890 1113891 (15)



ρpV

mp

αρpυ2i

Y1113888 1113889

β

(16)



102

103

Slope = ndash110 (μ = 071)R2 = 096

π3 = Yρtvi2

π V =

ρtV

mp

Slope = ndash133 (μ = 089)R2 = 086





1 mpυ2iY middot V

middot αρpυ2i

Y1113888 1113889

βminus 1

(17)



1 mpυ2iY middot V

middot αY

ρpυ2i1113888 1113889

1minus β

(18)




ρtV

mp

ρtυ2iY

1113888 1113889

β

middotΦρp

ρt

1113888 1113889L

r01113888 1113889

gL

υ2i1113888 11138891113890 1113891 (19)


ρtV

mp

αρtυ2iY

1113888 1113889

β

(20)



1 mpυi

VρtY


1113888 1113889

βminus 05

(21)





1 mpυi

VρtY

1113968 middot αY

ρtυ2i1113888 1113889

05minus β

(22)




ΠM ΠE middot Πm

1113968

mpυ2iYV

mp

ρtV

1113971

mpυi

VρtY

1113968 (23)


1113968 and



000 002 004 006 008000

002

004

006

008

010

012

014

016



=ρpυi2prodY Y

=ρpVprodV mp

Slope = 101R2 = 0996

Slope = 099R2 = 0910








⎧⎨

⎩

(24)




⎧⎨

⎩

(25)


5 Conclusion







Data Availability




Acknowledgments


References





0000 0005 0010 0015 0020 0025000000050010001500200025003000350040004500500055

Slope = 101R2 = 0996

Slope = 099R2 = 0910

=ρtVπV mp

=ρtυi2πY Y


































1 mpυ2iY middot V

middot αρpυ2i

Y1113888 1113889

βminus 1

(17)



1 mpυ2iY middot V

middot αY

ρpυ2i1113888 1113889

1minus β

(18)




ρtV

mp

ρtυ2iY

1113888 1113889

β

middotΦρp

ρt

1113888 1113889L

r01113888 1113889

gL

υ2i1113888 11138891113890 1113891 (19)


ρtV

mp

αρtυ2iY

1113888 1113889

β

(20)



1 mpυi

VρtY


1113888 1113889

βminus 05

(21)





1 mpυi

VρtY

1113968 middot αY

ρtυ2i1113888 1113889

05minus β

(22)




ΠM ΠE middot Πm

1113968

mpυ2iYV

mp

ρtV

1113971

mpυi

VρtY

1113968 (23)


1113968 and



000 002 004 006 008000

002

004

006

008

010

012

014

016



=ρpυi2prodY Y

=ρpVprodV mp

Slope = 101R2 = 0996

Slope = 099R2 = 0910








⎧⎨

⎩

(24)




⎧⎨

⎩

(25)


5 Conclusion







Data Availability




Acknowledgments


References





0000 0005 0010 0015 0020 0025000000050010001500200025003000350040004500500055

Slope = 101R2 = 0996

Slope = 099R2 = 0910

=ρtVπV mp

=ρtυi2πY Y





































⎧⎨

⎩

(24)




⎧⎨

⎩

(25)


5 Conclusion







Data Availability




Acknowledgments


References





0000 0005 0010 0015 0020 0025000000050010001500200025003000350040004500500055

Slope = 101R2 = 0996

Slope = 099R2 = 0910

=ρtVπV mp

=ρtυi2πY Y

































Experimental Investigation of the Crater Caused...

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