The effects of compressibility and strength on penetration of long rod and jet

Weng-jie Song,Xio-wei Chen,Pu Chen

aDepartment of Mechanics and Engineering Science,College of Engineering,Peking University,Beijing 100871,China

bCentre for Applied Physics and Technology(CAPT),Peking University,Beijing 100871,China

cState Key Lab of Explosion Science and Technology,Beijing Institute of Technology,Beijing 100081,China

1.Introduction

Penetration byprojectile with velocityof a few km/s intovarious targets is an important problem.The representative projectiles are long rod penetrator and high explosive anti-tank warhead which uses the jet formed by shaped charge to penetrate target.

The process of long rod or jet penetrating semi-in finite target is usually analyzed by incompressible hydrodynamic theory[1-5].The long rod is usually made of the material with high strength,high density and high bulk modulus.So in practical ordnance velocity,the volumetric strain of long rod is small and the relative theories about penetration by long rod treat it as incompressible material.Meanwhile the strength has a significant effect.However the research and development of kinetic-energy weapon never cease,such as the electromagnetic rail gun and new ultra highenergetic materials.The future long rod can reach higher velocity and the penetration is hypervelocity,in which the pressure at the rod/target interface is extremely high and so are the volumetric strains of the rod and target.Anderson and Orphal[6]conducted numerical simulations to examine the effect of compressibility at 1.5 km/s to 6 km/s and both the pressure and density at the rod/target interface deviate more from incompressible hydrodynamic theoryat higher impactvelocity.Thus,long rodcannot be treated as incompressible and the effect of compressibility on penetration by hypervelocity long rod has to be considered.On the other hand,the velocity of tip of the jet formed by ordinary military shaped charge is up to 8 km/s and even can be larger than 10 km/s for special design.In such hypervelocity penetration,the compressibility of the projectile and target cannot be really ignored.

WHA(tungsten heavy alloy)is a common material for long rod and copper is the most frequently-used material for the liner of shaped charge,so it is necessary to study the effects of compressibility and strength on the hypervelocity penetration by WHA long rod and copper jet.In the present work,we use the approximate compressible model to study the effects of compressibility and strength on hypervelocity penetration by WHA long rod in detail and clarify howcompressibilityaffects the penetration efficiency by changing the stagnation pressures of the rod and target.For the hypervelocity copper jet,the effects of compressibility and penetration resistance of the target on penetration efficiency are also studied.In which,the virtual origin model is adopted,and the compressibility and strength are considered by the linear relation between the penetration velocity and impact velocity.

2.Basic theory

Birkhoff et al.[1]and Hill et al.[2]suggested a hydrodynamic theory of penetration(HTP)during WWII,respectively,invoking the incompressible Bernoulli equation

where ρpand ρtstand for the projectile and target densities,respectively,and are assumed to be constants.V and U are the impact and penetration velocity,respectively.

The penetration efficiency is de fined as the increment ratio of the penetration depth to the erosion length of rod

where P is the penetration depth and l is the length of rod.For the hydrodynamic limit,according to Eq.(1)we can get

For penetration by metallic jets,Eichelberger[3]considered the strengths of jet and target as constant initial pressures in the incompressible Bernoulli equation

where Ypis the dynamic yield strength of the jet,Rtis the penetration resistance of the target and picis the pressure at the rod/target interface,i.e.,the stagnation pressure.The subscript ic represents the incompressible model with strength.According to the above equation,we can get complete compressible model firstly by adopting the compressible Bernoulli equation and treating the shockwave as the stationary wave.Then Flis and Chou[8]studied the effect of different EOS(equation of state).

Osipenko and Simonov[9]applied the linear Hugoniot relation between the shockwave velocity and the particle velocity into both the treatment of shockwave and the Mie-Grüneisen EOS,and thus the model is completely self-consistent.Flis[10,11]extended the compressible model to a quadratic dependence of shockwave velocity on particle velocity and considered the strengths of projectile and target.Flis[10,11]also performed CTH code(the Eulerian shock physics analysis package developed by Sandia National Laboratory)simulations to compare with the compressible model.The results predicted by the compressible model were in good agreement with the CTH simulations,showing the validity of the model.Federov and Bayanova[12]adopted the Murnaghan EOS to simplify the compressible model.Flis[13]simplify the compressible model by ignoring the shockwave and using the Hugoniot curve to approximate the EOS.The compressible models described above contain many complex equations and have to be solved by numerical method.

Song et al.[14,15]analyzed the effects of different factors in the compressible model,developed a simplified approximate model with an algebraic solution and analyzed the precision and applicable range of the approximate model.Unlike the complex models of other researchers,the simple solution of the approximate model can be easily used in the engineering problems without any numerical method.

In the simplified approximate compressible model of Song et al.[15],the Murnaghan EOS is used

where p0is the initial pressure,ρa(bǔ)nd ρ0are the density of compressed material and initial density,respectively.n=4λ-1,A=λis the slope of the linear relation between the shockwave velocity and the particle velocity and C0is the initial sound speed.The initial pressure p0is taken as Ypfor rod and Rtfor target,respectively.The equation of pressure-equilibrium across the projectile/target interface is

where pcis the stagnation pressure in the approximate compressible model.The subscript c represents the approximate compress-

However,in the case of hypervelocity penetration,the pressure at the rod/target interface is extremely high and the volumetric strain of rod or target is especially significant,so the incompressible assumption doesn't apply.Haugstad and Dullum[7]developed a ible model hereinafter.The solution of the above equation is assumed to be

whereUcisthe penetration velocity in the approximate compressible model.This solution is essentially a modification to the incompressible model.Substitute Eq.(9)into the first order Taylor series expansion about U for Eq.(8)and rearrange the result to get

3.High-velocity WHA long rod penetration

3.1.Penetration efficiency

We adopt the approximate compressible model to study the

The solution of the approximate compressible model is based on the solution of the incompressible model and Taylor series expansion,so it is very convenient to evaluate the error of penetration efficiency due to the compressibility for the incompressible model.Combine Eq.(2)and Eq.(9)to get the error of penetration efficiency for the incompressible model

Compared to the complete compressible model[11],the approximate compressible model of Song et al.[15]doesn't need numerical iteration and can be easily used in the engineering problems.With the velocitychange of 1.6times of theinitial sound speed for the projectile or target,the error of stagnation pressure for the approximate compressible modelis about 1%andthecorresponding errorofpenetrationefficiencyisabout0.5%.Forthecommonmetallic rod-target combinations,the approximate compressible model is applicable even at the impact velocity of 12 km/s.To the authors'knowledge,there is no experimental study on long-rod penetration upto 10 km/s.However Flis[10,11]performed CTH code simulations to compare with the compressible model up to 12 km/s.The results predicted by the compressible model were in good agreement with the CTH simulations,showing the validity of the model.In Ref.[15],results of the approximate compressible model were in great agreement with that of the complete compressible model[11].

In the present work,we use the approximate compressible model[15]to study the effects of compressibility and strength on hypervelocity penetrations by WHA long rod and copper jet in detail and clarify how compressibility affects the penetration efficiency by changing the stagnation pressure of the projectile and target.For the hypervelocity penetration bycopper jet,according to the existing experiments and simulations[16-22],the linear relation between the penetration velocity and impact velocity is adopted in the virtual origin model,and the effects of compressibility and strength on the hypervelocity penetration by copper jet are also studied in detail.penetrations by WHA[23]long rod(Yp=2.0 GPa)into semiin finite 4340 Steel[23](Rt=4.5×0.792 GPa),6061-T6 Al[24](Rt=4.5×0.3 GPa)and PMMA[8](polymethyl methacrylate,Rt=4.5×0.07 GPa)at the impact velocity of 2 km/s＜V＜5 km/s.The material properties are shown in Table 1.

The ratios ofpenetration efficiencies predicted by the compressible model and the incompressible model to the hydrodynamic limit are shown in Fig.1.The compressible model represents the approximate model of Song et al.[15]hereinafter.WHA is less compressible than the other three materials,so the penetration efficiencies predicted by the compressible model are lower than that predicted by the incompressible model.At V=5 km/s,we have(PEc-PEic)/PEh=0.964-0.979=-0.014 for 4340 Steel,(PEc-PEic)/PEh=0.984-1.019=-0.035 for 6061-T6 Al and(PEc-PEic)/PEh=0.944-1.099=-0.155 for PMMA.The deviation of red dashdot lines from the unit is attributed to the effect of strength.The strength has a measurable effect in the range of 2 km/s＜V＜5 km/s and the effect becomes weaker with increasing the impact velocity,just the same conclusion as the numerical simulations by Anderson et al.[25].With increasing the impact velocity,the results of the incompressible model approach the hydrodynamic limit but can not intersect.However the results of the compressible model intersect with the hydrodynamic limit with increasing the impact velocity in Fig.1(b)and(c).

Applying Eq.(11)into these three cases,the error of penetration efficiency for the incompressible model is shown in Fig.2.The targets are all more compressible than WHA long rod.And the greater the difference between the compressibility of the rod and target is,the more the penetration efficiency of the compressible model decreases.PMMA is much more compressible than WHA,so the compressibility has a great effect on the penetration by WHA rod into PMMA.For WHA rod penetrating PMMA,with increasing the impact velocity,the error of penetration efficiency for the incompressible model decreases first and then increases,and the error is always higher than 10%.

Table 1Material properties.

3.2.Lower-limit critical velocity

When the target has a greater effective strength than that of the rod,i.e.,Rt＞Yp,there is a critical velocity Vminbelow which the rod cannot penetrate the target and the target stays rigid,i.e.,U=0.And rod's stagnation pressure equals to the penetration resistance of the target.Vminis the lower-limit critical velocity at which the target starts to deform and flow.For the incompressible model,Forthecompressiblemodel[15],combine U=0 and Eq.(8)to get

where the approximately equal sign is suitable for （Rt-Yp）/Ap?1 due to the Taylor series expansion.

On the other hand,when the rod has a greater effective strength than that of the target,i.e.,Rt＜Yp,there is a critical velocity Vrigid,below which the rod penetrate the target rigidly(U=V).And target's stagnation pressure equals to the rod's strength.Vrigidis the lower-limit critical velocity at which the rod starts to deform.For the incompressible model,For the compressible model[15],combine U=V and Eq.(8)to get

where the approximately equal sign is suitable for （Yp-Rt）/At?1 due to the Taylor series expansion.

The region distributions of rod and target for WHA rod of strength Yp=2.0 GPa penetrating different targets of various strengths are shownin Fig.3,respectively.The strengthof WHA rod is fixed as Yp=2.0 GPa but the ratio Rt/Ypchanges.The regionbelow the left curve in the plots represents that the rod penetrates the target rigidly but the target flows.The region below the right curve represents that the target behaves rigidly but the rod flows.The region above both pairs of curves represents that both the rod and target flow hydrodynamically.

When Rt/Yp＞1,according to Eq.(12)the in fluence of compressibility on Vminis about （Rt-Yp）/（4npAp）.And the initial bulk modulus of WHA is quite high(npAp=ρ0pC20p=302.1 GPa),so the curves of Vminpredicted by the incompressible model and the compressible model almost overlap each other.That means,the in fluence of compressibility on the lower-limit critical velocity,at which the target starts to flow,is negligible.

When Rt/Yp＜1,according to Eq.(13)the in fluence of compressibility on Vrigidis about （Yp-Rt）/（4ntAt）.And the bulk moduli of both 4340 Steel and 6061-T6 Al are much higher than the strength of WHA,so the curves of Vrigidpredicted by the incompressible model and the compressible model in Fig.3(a)and(b)almost overlap each other,i.e.,similarly the in fluence of compressibility on the lower-limit critical velocity for rod's flowing is negligible.However,the bulk modulus of PMMA is just 8.9 GPa,which is inthesameorderofmagnitudecomparedtothestrengthof WHA,so the curves of Vrigidpredicted by the incompressible model and the compressible model in Fig.3(c)have a slight difference.

Actually there isa transition zone between rigid and hydrodynamic penetration modes.The corresponding models and details can be found in Refs.[26-28].

The effect of compressibility on the two lower-limit critical velocities Vminand Vrigidis in the order of ratio of strength difference to the bulk modulus of the rod andtarget,respectively.The strength difference between the rod and target is usually quite smaller than the bulk modulus.Therefore the effect of compressibility on the two lower-limit critical velocities is negligible.For WHA rod penetrating 4340 Steel and 6061-T6 Al at 2 km/s＜V＜5 km/s,the effect of compressibilityonpenetration efficiency is small.For WHA rod penetrating PMMA,the effect of compressibility on penetration efficiency is great due to the great difference between compressibility of the rod and target.

4.Hypervelocity WHA long rod penetration

For WHA rod penetrating 4340 Steel and 6061-T6 Al at 2 km/s＜V＜5 km/s,the effect of compressibility on penetration efficiency is small and ignored.In higher range of impact velocity 2 km/s＜V＜10 km/s,the ratios of penetration efficiencies predicted by the compressible model and the incompressible model to the hydrodynamic limit are shown in Fig.4.The solid lines with consideration of strength in the range of 2 km/s＜V＜5 km/s are just the same as Fig.1(a)and(b),respectively.The dash-dot line and dash line represent the results without consideration of strength,i.e.,Yp=Rt=0 GPa.The difference between the black solid line and black dash-dot line is attributed to the difference of strength in the compressible model.At lower impact velocity,the effectof strength is quitestrong.Howeverat higher impact velocity,the effect of strength becomes weaker.The trend is same for the incompressible model,i.e.,the effect of strength is strong at lower impact velocity and is weak at higher impact velocity.The difference between the black solid line and red solid line is attributed to the compressibility.The effect of compressibility is weak at lower impact velocity and strong at higher impact velocity,which is opposite to the trend of strength.For WHA rod penetrating 4340 Steel and 6061-T6 Al with consideration of strength at V=10 km/s,we have(PEc-PEic)/PEh=0.964-0.979=-0.036 and(PEc-PEic)/PEh=-0.076,respectively.

In general,at lower impact velocity,the effect of strength is strong and the effect of compressibility is negligible.At higher impact velocity,the effect of strength is weak and the effect of compressibility becomes stronger.

For the problem of penetration,the pressure equilibrium across the rod/target interface is an important condition.The relation between the stagnationpressure and change of velocity W is shown in Fig.5.For the rod and target,the changes of velocity are W=V-U and W=U,respectively.The abscissa is the ratio of change of velocity to the initial sound speed.The ordinates in Fig.5(a)and(b)are the stagnation pressure and ratio of stagnation pressure predicted by the compressible model to the incompressible model,respectively.The stagnationpressure predictedbythe compressible model is obviously higher than that predicted by the incompressible model.And at higher impact velocity,the pressure increment attributed to the compressibility is higher.As shown in the dimensionless plot of Fig.5(b),the proportional increments of stagnation pressure for different materials are similar.

The compressibility has effect on both the stagnation pressure and penetration efficiency.Analogous to the graphic solution in planar impact,Fig.6 shows how compressibility affects the penetration efficiency by changing the stagnation pressures of the rod and target for 5-km/s WHA rod penetrating PMMA.For the incompressible model,the stagnation pressure curves of the WHA and PMMA intersect at Uic=4.07 km/s and Pic=10.12 GPa.With consideration of compressibility,the stagnation pressure curve of WHA changes little but the stagnation pressure curve of PMMA changes a lot.So the intersection point for the stagnation pressure curves moves towards the left to Uc=3.95 km/s and Pc=12.46 GPa.The stagnation pressure increases and penetration velocity decreases.According to Eq.(2),the penetration efficiency decreases.

For PMMA target and WHA,we have W/C0=1.44 and W/C0=0.26,respectively.According to Fig.5(b),the proportional increment of stagnation pressure for PMMA is much higher than that for WHA,so the penetration efficiency obviously decreases,i.e.,(PEc-PEic)/PEh=-0.155.For 10-km/s WHA rod penetrating 4340 Steel and 6061-T6 Al,the pressure states are shown in Fig.7(a)and(b),respectively.

For WHA rod penetrating 4340 Steel,we have Uc=5.97 km/s,W/C0=1.00 for the rod and W/C0=1.30 for the target,respectively.The dimensionless changes of velocity for the rod and target are comparative,so the reduction of penetration efficiency is small,i.e.,(PEc-PEic)/PEh=-0.036.

For WHA rod penetrating 6061-T6 Al,we have Uc=7.10 km/s,W/C0=0.72 for the rod and W/C0=1.35 for the target.The difference between the dimensionless changes of velocity for the rod and target is intermediate between the above two cases and the reduction of penetration efficiency is also intermediate,i.e.,(PEc-PEic)/PEh=-0.076.

5.Copper jet penetration

For the jet formed by shaped charge,the velocity changes with position,so the jet cannot be treated as long rod.Allison and Bryan[29] firstly introduced the concept of virtual origin.Then Allison and Vitali[30]and Schwartz[31]developed the model with consideration of the velocity gradient and the stand-off distance between the virtual origin and the target surface for the penetration bycontinuous and particulated jets.In the virtual origin model,all of the jet elements are assumed to originate at a virtual origin,located a distance Z0in front of the target surface,and then move at their own velocities.The model doesn't take consideration of the strength and compressibility of the rod and target,and the penetration velocity is predicted by the HTP.

Many experiments[16-21]showed that there is a linear relation between the penetration velocity U and impact velocity V for long-rod penetrations into brittle ceramic targets in a great velocity range.Clayton[32]discussed the results of these experiments.Besides,Orphal and Anderson[22]conducted numerical simulations of long-rod penetrations into ductile targets at 2 km/s＜V＜8 km/s and the results showed the clear linear relation between the penetration velocity U and impact velocity V.However the hypervelocity long-rod penetration and jet penetration share the same penetration mechanism,i.e.,both the projectile and target behave like fluid.So the experimental conclusions and the corresponding analytical models apply to both the hypervelocity longrod penetration and jet penetration no matter whether the target is ductile or brittle.The linear U-V relation is adopted in the jet penetration.For the hypervelocity copper jet,the virtual origin model is also adopted here,and the compressibility and strength are considered implictly by the linear relation between the penetration velocity and impact velocity.Thus we can study the effects of compressibility and penetration resistance of the target on penetration efficiency.

5.1.The virtual origin model with considering the strength and compressibility

The schematic diagram of the virtual origin model is shown in Fig.8,where Z0is the distance from the virtual origin point to the target surface,t is the penetration time,P(t)is the penetration depth at time t and V(t)is the impact velocity.

Table 2Coefficients of linear fit for Cu rod penetrating different targets.

According to the geometrical relationship,we have

From the de finition of the penetration efficiency,i.e.,Eq.(2),we get

The change of velocity for jet with length of dl is dV,so

Substitute Eqs.(14)and(16)into Eq.(15)to get

Experiments[16-21]and numerical simulation[22]showed the linear relation between the penetration velocity U and impact velocity V for long-rod penetration.However the long-rod penetration and jet penetration share the same penetration mechanism.So the linear U-V relation is also adopted in the jet penetration.

where a and b are constants.This linear relation is the result of real strength and compressibility.Former researchers did not simultaneously consider both strength and compressibility in the virtual origin model.We substitute this linear relation into Eq.(17),i.e.,the effects of strength and compressibility are implicitly considered.Integrate the result to get the penetration depth

where Vtipand Vtailare the velocities of tip and tail of the jet,respectively.The constants are evaluated by the compressible model with strength and compressibility.

For the hydrodynamic limit,i.e.,the strength and compressibility are not considered,the penetration velocity is U=kV/(k+1)and substitute it into Eq.(17)to get the penetration depth

Substitute the hydrodynamic limit of U=kV/(k+1),i.e.,ah=0 and bh=k/(k+1),into Eq.(19)and the result can degenerate to the hydrodynamic result Eq.(20).

Combining Eq.(19)and Eq.(20),we can get the ratio of penetration depth with consideration of strength and compressibility to the hydrodynamic penetration depth

5.2.Cases

We adopt the above virtual origin model with considering strength and compressibility to study the penetration by copper jet[8](Yp=0 GPa)into 4340 Steel[23](Rt=4.5×0.792 GPa),6061-T6Al[24](Rt=4.5×0.3 GPa)and PMMA[8](Rt=4.5×0.07 GPa).The material properties are shown in Table 1.

In order to use the virtual origin model,we must get the U-V relation,i.e.,a and b,for copper jet-target intersections.The penetration velocity U for copper jet with impact velocity V is assumed to be equal to the penetration velocity for the copper rod with same impact velocity.With consideration of strength and compressibility,the penetration velocitiespredicted by the compressible model for copper jet penetrating different targets are shown in Fig.9.The effect of strength on the penetration velocity is measurable at lower impact velocity and becomes weak at higher impact velocity.We conduct the linear fitting between the penetration velocity and impact velocity and the corresponding coefficients are listed in Table 2.The last column is the hydrodynamic limit bh=k/(k+1).For strengthless 4340 Steel and 6061-T6 Al,the slopes of the linear relation are quite similar to the hydrodynamic limit and the intercepts are also very small.

We fix the tail velocity of copper jet at Vtail=2 km/s and change the tip velocity in the range of 3 km/s＜ Vtip＜8 km/s.For example Vtip=5 km/s presents the copper jet with tail velocity of 2 km/s and tip velocity of 5 km/s and linear velocity distribution among the jet.

Substitute the coefficients in Table 2 into Eq.(21)to get the ratio of penetration depth Pcwith consideration of strength and compressibility to the hydrodynamic penetration depth Phand the results are shown in Fig.10.The solid line and dash-dot line with same color represent the results with and without strength,respectively.The comparison between the solid line,dash-dot line and unit 1.0 decouples the effects of strength and compressibility.Three couples of solid line and dash-dot line all keep distinct gap in the range of 3 km/s＜ Vtip＜8 km/s.

Firstly,the difference between the dash-dot line and unit 1.0 is attributed to the difference between the compressibility of the rod and target.At Vtip=3 km/s,the ratios Pc/Phare very close to unit for copper jets penetrating various strengthless targets.With increasing the tip velocity,the effect of compressibility becomes stronger.At Vtip=8 km/s,the ratios Pc/Phare 1.0065,0.9686 and 0.7470 for copper jet penetrating strengthless 4030 Steel,6061-T6 Al and PMMA,respectively,and the corresponding change(Pc-Ph)/Phare 0.0065,-0.0314 and-0.2530,respectively.However,for copper jet penetrating 4030 Steel and 6061-T6 Al in the velocity range of 3 km/s＜Vtip＜8 km/s,the effect of compressibility is weak.But,for copper jet penetrating PMMA,the effect of compressibility is much stronger due to the huge difference between compressibility of copper and PMMA.

Secondly,the target resistance has a significant effect on penetration by copper jet in the whole range of 3 km/s＜Vtip＜8 km/s.At Vtip=3 km/s,the ratios Pc/Phare 1.001,1.003 and 0.983 for copper jet penetrating strengthless 4030 Steel,6061-T6 Al and PMMA,respectively,and the corresponding ratios Pc/Phare 0.749,0.799 and 0.883 for the corresponding targets with strength,respectively.The difference is attributed to the target resistance and the differencesΔPc/Phare 0.252,0.204 and 0.100,respectively.Accordingly,at Vtip=8 km/s,the differenceΔPc/Phattributed to the target resistance are 0.207,0.198 and 0.088.In summary,the differences ΔPc/Phattributed to the target resistance are about 20%,20%and 10%in the whole range of 3 km/s ＜ Vtip＜8 km/s for copper jet penetrating 4030 Steel,6061-T6 Al and PMMA,respectively.

The velocity in the long rod is distributed uniformly but there is some velocity distribution among the jet.There is a portion of jet with lower velocity even for the hypervelocity jet and strength has a significant effect on penetration with lower impact velocity,so strength always has a significant effect on penetration by copper jet.On the other hand,for hypervelocity copper jet penetrating PMMA,the front portion of jet has hypervelocity and the huge difference between the compressibility of copper and PMMA reduces the penetration velocity.Accordingly the penetration depth reduces.Besides,the reduction of the penetration depth of the front portion of jet will restrain the stretch of jet,which will reduce the penetration depth of the rear portion of jet.So the compressibility has a significant effect on the penetration by the hypervelocity copper jet into PMMA.

6.Conclusions

The simple approximate compressible model is adopted to study the effects of strength and compressibility on the penetration by WHA long rod and copper jet into semi-in finite target in detail.

For WHA rod penetrating PMMA at 2 km/s＜V＜5 km/s,the huge difference between compressibility of WHA and PMMA has a significant effect on the penetration efficiency.Taking the penetration by 5-km/s WHA rod into PMMA as example,we clarify how compressibility affects the penetration efficiency by changing the stagnation pressures of the rod and target.For WHA rod penetrating 4340 Steel and 6061-T6 Al at 2 km/s＜V＜10 km/s,the effect of strength is strong and the effect of compressibility is negligible at lower impact velocity,whilst the effect of strength is weak and the effect of compressibility becomes stronger at higher impact velocity.

The existing researches showed the linear relation between the penetration velocity U and impact velocity V.For the copper jet penetrating 4030 Steel,6061-T6 Al and PMMA,the virtual origin model is adopted,and the compressibility and strength are implicitly considered by the linear relation between the penetrationvelocityand impact velocity.Thus the effects of compressibility and penetration resistance of the target on penetration efficiency are studied.The tail velocity of copper jet is fixed at Vtail=2 km/s and the tip velocity changes in the range of 3 km/s＜ Vtip＜8 km/s.The results show that the target resistance has a significant effect in the whole range of 3 km/s＜Vtip＜8 km/s.For copper jet penetrating 4030 Steel and 6061-T6 Al,the effect of compressibility is weak.However PMMA is much more compressible than copper and the huge difference of compressibility has a significant effect on the penetration by hypervelocity copper jet into PMMA.

Acknowledgements

This work was supported by the National Outstanding Young Scientist Foundation of China(11225213)and the Key Subject“Computational solid mechanics”of China Academy of Engineering Physics.

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