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

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.

式中: Q(t)=[vEINS/GPS vNINS/GPS pEINS/GPS pNINS/GPS];

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

音乐是一门听觉的艺术，声响在时间中缓缓流动，所以这门学科和多媒体技术相结合是最恰当的，也是最好的，音乐教学中利用音响辅助文化理解也是非常重要的。教师可以通过网络收集音乐课堂需要的一些音乐，来帮助学生理解和拓展，例如在《哆来咪》一课，教师可以收集《音乐之声》中学生比较熟悉的歌曲《雪绒花》《孤独的牧羊人》《晚安歌》等音乐，通过聆听，学生对电影作品以及歌曲《哆来咪》会有更深层次的理解。

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

whereand the subscript h represents the HTP.

四极透镜是各向异性聚焦系统中的核心部件，是控制束流横向运动最有力、也最灵活的元件.在各向异性聚焦条纹变像管中，静电四极透镜可实现电子束空间方向(沿x方向)的聚焦.另外，四极透镜也决定空间方向交叉点位置，因此可通过调节其压控制系统放大倍率.静电四极透镜在垂直于z轴的x-y二维平面上的场分布满足双曲分布，当电极间的距离为2a，电极电压为Va时，忽略边缘场效应，电四极透镜内部的电位分布为

我国已经进入人口老龄化社会，按照目前的老龄化现状及发展趋势推算，只有不到20年的时间，我国将全面进入超老龄社会，老龄化人口中失能老人、高龄老人、空巢老人将会占据更大比重。目前为止，“三多”问题已经较为普遍，大多数老人生活自理能力差，日常生活通常需要他人照料。综上所述，当前我国老年人生活照料服务不稳定的问题表现在：其一，老年人生活照料服务来源渠道单一；其二，老年人生活照料服务提供者的现行压力大；其三，老年人生活照料服务需求大。

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,ρand 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

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.

3.High-velocity WHA long rod penetration

3.1.Penetration efficiency

We adopt the approximate compressible model to study the

部分航运船公司存在重生产、轻安全的思想，公司安全管理不到位，对船舶防台投入不够；船员安全意识淡薄，设备维护、管理跟不上，防台值班纪律松懈。

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).

通常来说，发展成熟的多元化企业，其经营战略可以分为三个层次：即企业总体经营战略层次、事业部发展战略层次以及职能战略层次。其中，第一层次是企业战略体系的主体和基础，具体又分为稳定型、增长型和收缩型三种战略；第二层次具体规定了各项经营事业的目标和战略；第三层次又进一步将企业战略按照专门职能进行落实和具体化。本文在系统地分析了企业总体经营管理战略的基础上，结合企业生命周期的四阶段理论，得出了企业处于生命周期的不同发展阶段所应采取的不同经营战略。

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.

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.

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.

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

近两年，中国石化黑龙江石油把全面可持续发展，迈向高质量发展作为第一要务，以打造“一流企业”为愿景，以建设“大一企业”为目标，坚决落实集团公司工作部署，围绕“改革、管理、创新、发展”工作方针，坚持稳中求进工作总基调，统筹拓市场、促改革、抓运行，持续推进新业务转型突破，主营业务扎实开展。1—8月，公司实现经营总量72.5万吨，同比增幅17%。零售总量同比增加9.2万吨，同比增幅24.3%。非油品交易额1.28亿元，同比增幅13%。零售总量增幅排名系统第一，市场份额达到17%，荣获销售企业月度红旗四面。

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.

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.

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 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.

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

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.

2.1 常规方法收集处女蝇的特点 常规方法收集处女蝇是通过清除亲本，控制培养时间，每隔8～10h分别收集新羽化出来的果蝇，麻醉后在解剖镜下观察，将果蝇雌雄分开，这时得到的雌果蝇应该全部都是处女蝇。

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

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%.

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.

运价是影响路径经济性的主要因素，也是货主选择路径首先考虑的因素，以东莞集装箱通过水上“巴士”经枢纽港中转出口美国为例，进行具体分析：

理工科高校通识教育存在的问题与改进对策 ………………………………………… 纪光欣，刘兴波（5．108）

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.

浙江省永康市地处钱塘江和瓯江的分水岭，素有“鲤鱼背”之称，水资源相对短缺，人均水资源占有量不到全国平均水平的一半。同时，永康市也是全国知名的“百工之乡”“五金之都”，位列全国经济实力百强县。面对水资源禀赋不足和经济的快速发展，永康市积极推进最严格水资源管理制度试点建设，建立组织领导机构，完善基础支撑体系；实施用水总量控制，提高水资源优化配置能力；通过节水减污控制，提高用水效率；通过纳污总量控制提升水环境质量，为永康市社会经济可持续发展提供可靠的水资源保障。

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.

偶尔迷茫的王老师（以下称王老师）：工作还算顺利，就是有一个问题让我很困惑。现在英语教学强调培养学生的语感和交际能力，提出了“淡化语法”的主张。根据课程标准和教材内容，课堂教学的大量时间都让学生练习会话，没有专门进行语法知识教学，导致学生在课堂发言时经常出现语法毛病。我不明白为什么要“淡化语法”，语法明明很重要啊！

按照《关于节能新能源车船享受车船税优惠政策的通知》(财税〔2018〕74号)相关要求，工信部官方网站对《享受车船税减免优惠的节约能源使用新能源汽车车型目录》(第一批)名单进行了公示。本次将排量小于1.6L的车型纳入车船税减半政策适用范围，在满足排量要求的同时，车辆还需符合综合工况油耗的相应标准，所以并不是所有1.6L排量以下的车型都可以享受此政策。此外，对于符合新能源产品技术标准以及相关质量要求的纯电动商用车、插电式(含增程式)混合动力汽车、燃料电池商用车，免征车船税，目前市场中绝大部分插电式混合动力车型均在免征范围内，主要包括中国生产的自主、合资车，而进口插混车型不在此范围内。

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.

References

[1]Birkhoff G,MacDougall DP,Pugh EM,Taylor G.Explosives with lined cavities.J Appl Phys 1948;19:563-82.

[2]Hill R,Mott NF,Pack DC.Theoretical research report No.2/44,Jan.1944 and 13/44.UK Armament Research Department;Mar.1944.

[3]Eichelberger RJ.Experimental test of the theory of penetration by metallic jets.J Appl Phys 1956;27(1):63-8.

[4]Alekseevskii VP.Penetration of a rod into target at high velocity.Combust Explo Shock+1956;2(2):63-6.

[5]Tate A.A theory for the deceleration of long rods after impact.J Mech Phys Solid 1967;15(6):387-99.

[6]Anderson CE,Orphal DL.An examination of deviations from hydrodynamic penetration theory.Int J Impact Eng 2008;35:1386-92.

[7]Haugstad BS,Dullum OS.Finite compressibility in shaped charge jet and long rod penetration-the effect of shocks.J Appl Phys 1981;52(8):5066-71.

[8]Flis WJ,Chou PC.Penetration of compressible materials by shaped-charge jets.In:7th int symp ballistics;1983.The Hague,The Netherlands.

[9]Osipenko KY,Simonov IV.On the jet collision:general model and reduction to the Mie-Grüneisen state equation.Mech Solid 2009;44(4):639-48.

[10]Flis WJ.A model of compressible jet penetration.In:Proc 26th int symp ballistics;2011.Miami,FL.

[11]Flis WJ.A jet penetration model incorporating effects of compressibility and target strength.Process Eng 2013;58:204-13.

[12]Fedorov SV,Bayanova YM.Hydrodynamic model for penetration of extended projectiles with consideration of material compressibility.In:Proc 25th int symp ballistics;2010.Beijing,China.

[13]Flis WJ.A simplified approximate model of compressible jet penetration.In:Proc 27th int symp ballistics;2013.Freiburg,Germany.

[14]Song WJ,Chen XW,Chen P.Effect of compressibility on the hypervelocity penetration.Acta Mech Sin 2017.https://doi.org/10.1007/s10409-017-0688-1.

[15]Song WJ,Chen XW,Chen P.A simplified approximate model of compressible hypervelocity penetration.Acta Mech Sin 2017(under review).

[16]Subramanian R,Bless S.Penetration of semi-in finite AD995 alumina targets by tungsten long rod penetrators from 1.5 to 3.5 km/s.Int J Impact Eng 1995;17(4-6):807-16.

[17]Subramanian R,Bless S,Czamias J,et al.Reverse impact experiments against tungsten rods and results for aluminum penetration between 1.5 and 4.2 km/s.Int J Impact Eng 1995;17(4):817-24.

[18]Orphal D,Franzen R,Piekutowski A,et al.Penetration of con fined aluminum nitride targets by tungsten long rods at 1.5-4.5 km/s.Int J Impact Eng 1996;18(4):355-68.

[19]Orphal D,Franzen R.Penetration of con fined silicon carbide targets by tungsten long rods at impact velocities from 1.5 to 4.6 km/s.Int J Impact Eng 1997;19(1):1-13.

[20]Orphal D,Franzen R,Charters A,et al.Penetration of con fined boron carbide targets by tungsten long rods at impact velocities from 1.5 to 5.0 km/s.Int J Impact Eng 1997;19(1):15-29.

[21]Behner T,Orphal D,Hohler V,et al.Hypervelocity penetration of gold rods into SiC-N for impact velocities from 2.0 to 6.2 km/s.Int J Impact Eng 2006;33(1):68-79.

[22]Orphal D,Anderson C.The dependence of penetration velocity on impact velocity.Int J Impact Eng 2006;33(1):546-54.

[23]Steinberg DJ.Equation of state and strength properties of selected materials.Lawrence Livermore National Laboratory;1996.UCRL-MA-106439.

[24]Corbett BM.Numerical simulations of target hole diameters for hypervelocity impacts into elevated and room temperature bumpers.Int J Impact Eng 2006;33(1):431-40.

[25]Anderson CE,Orphal DL,Franzen RR,Walker JD.On the hydrodynamic approximation for long-rod penetration.Int J Impact Eng 1999;22(1):23-43.

[26]Chen XW,Li QM.Transition from non-deformable projectile penetration to semi-hydrodynamic penetration.J Eng Mech 2004;130(1):123-7.

[27]Segletes SB.The erosion transition of tungsten-alloy long rods into aluminum targets.Int J Solid Struct 2007;44:2168-91.

[28]Lou JF,Zhang YG,Wang Z,Hong T,Zhang XL,Zhang SD.Long-rod penetration:the transition zone between rigid and hydrodynamic penetration modes.Defence Technol 2014;10:239-44.

[29]Allison FE,Bryan GM.Cratering by a train of hypervelocity fragments.Proc 2nd Hypervelocity Impact Effects Symp 1957;1:81.

[30]Allison FE,Vitali R.A new method of computing penetration variables for shaped charge jets.Ballistic Research Laboratories Internal Report.BRL;1963.

[31]Schwartz W.Modified SDM model for the calculation of shaped charge hole pro files.Propell Explos Pyrot 1994;19(4):192-201.

[32]Clayton JD.Dimensional analysis and extended hydrodynamic theory applied to long-rod penetration of ceramics.Defence Technol 2016;12:334-42.