Analysis of the velocity relationship and deceleration of long-rod penetration

W.J.Jiao·X.W.Chen

Abstract The relationship between the average penetration velocity,ˉU, and the initial impact velocity,V0,in long-rod penetration has been studied recently.Experimental and simulation results all show a linear relationship between and V0 over a wide range of V0 for different combinations of rod and target materials.However,the physical essence has not been fully revealed.In this paper,the ?V0 relationship is comprehensively analyzed using the hydrodynamic model and the Alekseevskii–Tatemodel.In particular,the explicitˉU?V0 relationships are derived from approximate solutions of the Alekseevskii–Tate model.In addition,the deceleration in long-rod penetration is discussed.The deceleration degree is quantified by a deceleration index,α?which is mainly related to the impact velocity,rod strength, and rod/target densities.Thus,the state of the penetration process can be identified and designed in experiments.

Keywords Long-rod penetration ·Average penetration velocity ·Initial impact velocity ·Alekseevskii–Tate model·Deceleration

1 Introduction

As a kinetic-energy weapon,long-rod projectiles are made with high-density metals and have a large length-to-diameter ratio.Because of the large cross section kinetic energy,long rods exhibit excellent performance in penetration and perforation at hypervelocity around 1.5–3.0 km/s.When a target is impacted by a long rod at hypervelocity,both the rod and target deform in a semi-fluid manner because the pressure acting on the surface is higher than the dynamic strength of the rod/target materials.

Long-rod penetration was first introduced in the 1960s[1–4], and many hypervelocity ballistic experiments have been conducted since the 1970s[5,6].In the 1990s,combining reverse ballistic experiments with flash X-ray,information was obtained regarding the deformation and movement of rods and targets during the penetration process[7–9].Meanwhile,numerical simulation has become an important method for studying long-rod penetration[10–12].Based on the analysis of experimental phenomena and simulation results,theoretical models are established by abstraction and approximation.They can reflect typical physical characteristics and have predictive ability.Theoretical models of long-rod penetration are developed from a hydrodynamic model to the Alekseevskii–Tate model and the n to more complex models[1–4,12–15].Among the m,the Alekseevskii–Tate model is one of the most successful models and is often used even now.Although it cannot explain the two-dimensional(2D)effect of long-rod penetration as a one-dimensional(1D)model,the Alekseevskii–Tate model is proven to be valid in describing the primary phase in the long-rod penetration process by numerous experiments and simulations.However,the Alekseevskii–Tatemodel cannot be solved explicitly due to the nonlinear nature of the equations, and this problem has been solved by explicit approximate solutions proposed by Jiao and Chen[16].

Recent experiments show that the average penetration velocity is a linear function of the initial impact velocity.This phenomenon was first reported by Subramanian and Bless[7]in an experiment of tungsten long rods penetrating semi-infinite AD995 alumina targets.Subsequently,Orphal et al.[17–19]conducted a series of experiments of tungsten long rods penetrating different ceramic targets, and the same conclusion was drawn.Experiments with other combinations of rods and targets have led to the same phenomenon[8,20–22].The linear relationship between the average penetration velocity and the initial impact velocity was also observed in the numerical simulations conducted by Orphal and Anderson[23].

The linear relationship was briefly analyzed using a hydrodynamic model by Orphal and Anderson[23],but the physical essence has not been fully revealed.The validity of using the hydrodynamic model in long-rod penetration is limited because the material strength is ignored, and thus the Alekseevskii–Tate model is more reasonable for analyzing this problem.However,due to the nonlinear nature of the Alekseevskii–Tate equations,the re is no explicit expression of instant penetration velocity, and thus the explicit?V0relationship cannot be derived directly.Therefore,in this paper,approximate solutions proposed by Jiao and Chen[16]are used to obtain the explicit?V0relationship, and the physical essence is revealed.

Furthermore,the deviation of the long-rod penetration process from steady state is observed[23,24] and briefly discussed[25].As the main cause of the deviation,the deceleration is specifically and comprehensively analyzed in this paper.The deceleration degree is quantified by a deceleration index α, and thus the state of the penetration process can be identified.It is of great significance for guiding experimental design,because the state of the penetration process can be designed by choosing the appropriate corresponding parameters for the experiments.

2 Velocity relationship in long-rod penetration

2.1 Experimental and simulation results

In the reverse ballistic experiments,flash X-rays are used to provide the position,p, of the rod inside the target or the depth of penetration as a function of time,t,during the penetration process.It is found for all the experiments in Refs.[7–9,17–22]that the penetration depth is a linear function of time.The average penetration velocity,ˉU,is obtained by the slope of the p?t curve(d p/d t)for a given initial impact velocity,V0,in these experiments.Therefore,the ˉU?V0relationship is obtained.

The penetration performance of tungsten long rods against AD995 alumina targets was measured through reverse ballistic testing with flash X-rays by Subramanian and Bless[7], and the average penetration velocity was observed to be a linear function of initial impact velocity from 1.5 km/s to 3.5 km/s.A similar linear function was observed in experiments of tungsten long rods penetrating aluminum targets[8].Likewise,Orphal et al.[17–19]conducted a series of penetration experiments of tungsten long rods against A lN,SiC, and B4C targets, and Behner et al.[20]tested the penetration performance of gold rods(much weaker than tungsten rods)against SiC targets over a larger range of impact velocities.These experiments all show a linear relationship between the average penetration velocity and the initial impact velocity.This phenomenon was also recently reported in penetration experiments of borosilicate glass by Behner et al.[21] and Orphal et al.[22].

To sum up,the linear relationship between the average penetration velocity,ˉU, and the initial impact velocity,V0,can be expressed as

where the coefficients a and b are related to the material properties of the rods and targets.

The linear relationships betweenˉU and V0in these experiments are summarized in Table1.Different combinations of rod/target materials(including metal/ceramic,metal/metal, and metal/glass) and a large range of initial impact velocities are included.However,given the limited capability of flash X-rays,high-density rods are chosen to penetrate relatively low-density targets in all these experiments.It should be mentioned that gold and copper rods are approximately assumed to have no strength because the y are relatively weak.Two different rod lengths are used in the experiments by Orphal et al.[19] and Behner et al.[21],while a single rod length is used in other experiments.

Additional combinations of rod/target materials are studied through numerical simulation by Orphal and Anderson[23].The linear relationships are observed in the simulation results listed in Table2.In contrast to the experimental results listed in Table1,there is a great difference among the quantities of coefficients a and b in the simulation results.

It should be noted that the instant tail and nose(penetration)velocity changes very slowly during the primary phase of the long-rod penetration process.Thus,the penetration process can be approximately represented by the average penetration velocity.

The primary phase of long-rod penetration is quasi-steady state,defined by an approximately constant pressure level and penetration velocity[12].In particular,the penetration process is steady state when the re is no strength for the penetrator.The deviation of the general long-rod penetration process from steady state will be identified in Sect.5, and its effect on the ˉU?V0relationship will be discussed.

In addition,as the instant penetration velocity changes significantly in the transient and secondary phases,it is mean-ingless to use the final penetration depth to analyze the ˉU?V0relationship.This implies that the analysis of the ˉU?V0relationship should be limited to the primary phase.To weaken the influence of the transient and secondary phases,a large length-to-diameter ratio rod is used in the experiments to ensure that the penetration process mainly occurs in the primary phase.

Table 1 Compilation of ˉU B a + bV0 equations from ballistic experiments

2.2 Velocity relationship in the hydrodynamic model

The hydrodynamic model was used to analyze the ˉU?V0relationship by Orphal and Anderson[23].According to the assumption of ignoring the strength effect,the instant tail(impact) and nose(penetration)velocities stay constant during the penetration process,i.e.the penetration process is steady state.Therefore,the ˉU?V0relationship can be obtained from the Bernoulli equation

Substituting Eq.(2)into Eq.(1),the coefficients are obtained as a?0 and b?(1+μ)?1≡ bh(bhis the slope of the hydrodynamic curve,ˉU?bhV0).This implies that the hydrodynamic model can be regarded as a special form of linear?V0relationship.

Comparing the linear relationship in the experimental and simulation results(listed in Tables 1,2,respectively)with the hydrodynamic model,all of the cases show that a≤0 and b≥bh>0,except for the last two cases in Table 2.Actually,as is easily drawn from Eq.(1),V0??a?b≥0 when?0.This implies that the rod must have a relatively high velocity to penetrate the target.Therefore,a≤0 and b>0 are related to the physical essence.In addition,b≥bhmeans that the measured?V0curve in these experiments and simulations approaches the hydrodynamic curve from below.Contrarily,two exceptional cases(gold penetrating water and aluminum penetrating gold)show that a>0 and bh>b>0,which means that the?V0curve approaches the hydrodynamic curve from above.For>0 when V0?0,it violates the physical essence.It was noted by Orphal and Anderson[23]that the rod has strength and the target has no(or little)strength in these two cases,but otherwise the physical reasons for this different behavior are not yet understood.

The strength effect should be considered in an actual penetration process,especially at low speed.Experimental results are compared with their hydrodynamic model predictions[Eq.(2)]in Fig.1,where the solid point represents experimental data and the dash-dotted line represents model prediction.Discrepancies between the experimental results and model predictions for different combinations of rod and target materials can be observed:the discrepancy is relatively large for tungsten rods penetrating AD995 alumina targets and is relatively small for tungsten rods penetrating aluminum targets.In addition,the re is an obvious discrepancy between the experimental?V0curves in the tungsten rods penetrating SiC-B targets and the gold rods penetrating SiC-N targets.The discrepancy is related to their differences in material strength,as these two cases have the same density ratios for rod and target materials,which leads to the same hydrodynamic curve.

Table2 Compilation ofˉU?a+bV0 equations from numerical simulations[23]

Fig.1 Comparison between hydrodynamic model prediction and experimental results

Therefore,coefficients a and b must be related to the strengths of the rod and target materials.The hydrodynamic model ignores the strength effect,so it can only be used to briefly analyze the?V0relationship.Another theoretical model considering the strength effect is needed to accurately explain and predict the?V0relationship.

3 Velocity relationship in the Alekseevskii–Tate model

Because of the limitations of the hydrodynamic model,the Alekseevskii–Tate model is used to further analyze the?V0relationship.

The Alekseevskii–Tate model,formulated by Alekseevskii[2] and Tate[3,4]independently and almost simultaneously,is one of the most successful theoretical models in long-rod penetration.Based on the hydrodynamic model,the Alekseevskii–Tate model uses rod and target strengths to modify the Bernoulli equation.In this semifluid model,the rod is assumed to be rigid except for an infinitesimally thin region near the rod–target interface,where erosion is occurring in a fluid manner.The model also assumes balanced stress and continuous velocities at the interface,which means that the penetration velocity is equal to the nose velocity of the rod, and the deceleration of the rear end of rod is governed by the dynamic flow stress of the rod material.The governing equations are given as follows

where Ypis the dynamic flow stress of the rod material and Rtis the penetration resistance of target.p and l?st and for the instant depth of penetration and residual rod length,respectively,which are all functions of the time t.

The relationship between u and v can be derived from Eq.(3)

where Vc?represents the critical penetration velocity.Long-rod penetration stops when V0≤Vc.

The validity of the Alekseevskii–Tate model for describing the primary phase in the long-rod penetration process has been proven by numerous experiments and simulations.Therefore,the u?v relationship is accurately given by Eq.(7a).

It should be noted that Eq.(7a)shows the relationship between the instant velocities.However,the?V0relationship reported by experiments and simulations is that between the average penetration velocity and the initial impact velocity.Replacing the instant velocities u and v with initial velocities U0and V0,respectively,the relationship between initial velocities can be expressed as

However,due to the existence of deceleration of instant velocities during the penetration process in the primary phase,the?V0relationship is different from the U0?V0relationship given by Eq.(7b).

The average penetration velocity can be obtained by integrating instant penetration velocity and the n dividing it by the total duration of the primary phase,T,i.e.,

Unfortunately,the instant penetration velocity cannot be explicitly expressed as a function of time because of the nonlinear nature of the governing equations[Eqs.(3)–(6)].Therefore,the explicit expression ofˉU cannot be derived directly from the Alekseevskii–Tatemodel.The ˉU?V0relationship can only be obtained numerically using software such as Matlab.

To verify the predictions of different theoretical models,the experimental results listed in Table 1 are used as a case study.It should be noted that the quantity of target resistance,Rt,varies with the variation in impact velocity,V0.To simply the discussion,the variation of Rtis ignored in this paper, and the average quantity of Rtis chosen for all these cases,which is termedˉRtin Table 1.However,the case of tungsten long rods penetrating aluminum targets is not used here,because the suggested quantities of Ypand Rtare unknown.

Comparison between the predictions of different theoretical models and the experimental results in different cases are shown in Fig.2,where the experimental data(solid dot)are given by Eq.(1), and the prediction of the hydrodynamic model(black dash-dotted line)is given by Eq.(2).The predictions of the Alekseevskii–Tate model are expressed in two different forms:the explicit U0?V0relationship(blue dashdotted line)is given by Eq.(9b), and the implicitˉU?V0relationship(red dash-dotted line)is numerically evaluated by combining Eqs.(3)–(6),(8).The explicitˉU?V0relationship represented by the green dash-dotted line will be obtained and analyzed in the next section.

As can be seen from Fig.2,for all these cases the re is a clear discrepancy between the hydrodynamic model prediction and the experimental results,while the Alekseevskii–Tatemodel predictions(both U0?V0and ˉU?V0)are relatively close to the experimental results,especially at hypervelocity.For the first four cases,the ˉU?V0curve is below the U0?V0curve in the whole range of initial impact velocities and their discrepancy is smaller when V0is higher;for the last four cases,these two curves are coincident with each other.

It should be emphasized that the relationship between penetration velocity and impact velocity reported by the experiments is actually the ˉU?V0relationship.However,due to the nonlinear nature of the governing equations,the explicit expression cannot be derived directly from the Alekseevskii–Tate model.The explicit U0? V0relationship is taken as an approximation of the ˉU?V0relationship to explain the experimental results by some researchers.The discrepancy between the m is caused by the deceleration of instant penetration velocity.For the last four cases in Fig.2,U0?V0curves are coincident withˉU?V0curves.This is because the re is no deceleration when the rod strengths are zero.The determinants and effects of deceleration will be further discussed,so the validity of using the U0?V0relationship[Eq.(9)]to analyze and predict the experimental results is obtained.

Fig.2 Comparison between the predictions of different theoretical models and experimental results in different cases

It should be noted that,as quantities of Rtare determined from the experimentally obtained?V0relationship and the Alekseevskii–Tate model,good agreement between the predicted and experimental?V0relationship may need further independent assessment.The use of the average quantities of Rti.e.in the model weakens the dependence of the prediction on Rt.The sensitivities of the predicted?V0relationship to Rtare shown in Fig.3.In all cases,the predicted?V0relationship at Rt?and Rt?is close to the experimental results at the highest and lowest impact velocity,respectively.Therefore,Rt?is an acceptable approximation for analysis in the present paper.In addition,although the effects of the Rtvalue on the discrepancy between predicted and experimental?V0relationships are different in different cases,the Alekseevskii–Tatemodel predictions are much closer to experimental results than the hydrodynamic model predictions in all cases,especially when the impact velocity is relatively low.

Fig.2 continued

4 Velocity relationship in approximate solutions of the Alekseevskii–Tate model

Because of the nonlinear nature of the governing equations of the Alekseevskii–Tatemodel,the tail velocity,penetration velocity,rod length, and penetration depth were obtained implicitly as a function of time and solved numerically.Therefore,variation laws and influence factors for these variables cannot be directly obtained.By employing a linear approximation to the logarithmic relative residual rod length,ln?l?/L?(L is the initial rod length),two sets of explicit approximate algebraic solutions were obtained by Jiao and Chen[16].

The explicit expressions of instant penetration velocity as a function of time are given by Eqs.(25) and (36)in Ref.[23].Substituting Eq.(25)in Ref.[16]into Eq.(8),the explicitˉU?V0relationship is derived from approximate solution 1 of the Alekseevskii–Tate model

According to Eq.(36)in Ref.[16],the average penetration velocity given by approximate solution 2 can be rewritten as

Therefore,two explicit expression of the ˉU?V0relationship are correspondingly derived from two sets of approximate solutions.

In fact,due to the definition of K(Eq.(38)in Ref.[16]),approximate solutions 1 and 2 have the same penetration depth at the terminal time t?T for the same initial impact velocity.According to Eqs.(6) and (8),the average penetration velocities given by approximate solutions 1 and 2 are the same,so Eqs.(9) and (10)are equivalent.The validity of this conclusion can be proven by substituting Eq.(38)in Ref.[16]into Eqs.(9) and (10).

Thus far,based on two sets of approximate solutions,the same explicit?V0relationship has been obtained.To analyze its validity,the experimental results in Table1 are used.The explicit?V0relationship given by Eq.(9)or(10)is represented by the green dash-dotted line in Fig.2.In all these cases,the explicit?V0relationship is very close to the implicit?V0relationship(red dash-dotted line),except for the discrepancy at very low initial impact velocity in the last two cases.The limitations and application scope of approximate solutions have been discussed in Ref.[16].The lowest permitted impact velocities of approximate solutions are higher than that of the theoretical solution,i.e.,the critical impact velocity,Vc.

Fig.3 Sensitivities of predictedˉU?V0 relationship to Rt in different cases

Therefore,the explicitˉU?V0relationship given by Eq.(9)or(10)is a reasonable approximation of the implicitˉU?V0relationship, and it can be used to analyze the experimental results.

Fig.3 continued

Using the approximate expression of K(Eq.(48)in Ref.[16])at hypervelocity to simplify Eq.(9),a simpler explicitˉU?V0relationship can be obtained

where the average penetration velocity is expressed as a sum of the positive and negative first power of initial impact velocity.The coefficient of the positive first power,(1+μ)?1,is equal to the linear coefficient in the hydrodynamic model,bh.Thus,the first power of V0is the hydrodynamic term.Correspondingly,the negative first power of V0is the strength term,because the target strength Rtis included.Therefore,the average penetration velocity is expressed as the sum of the hydrodynamic term and the strength term.

With an increase in initial impact velocity,the hydrodynamic term increases and the strength term decreases.Therefore,the strength term can be ignored at hypervelocity, and Eq.(11)degenerates into the?V0relationship given by the hydrodynamic model,Eq.(2).From Eq.(11),we can conclude that the Alekseevskii–Tate model is applicable even at hypervelocity,where the strength effect still plays a minor role.This is related to the observation of Fig.2,where the predictions of the Alekseevskii–Tatemodel approach the hydrodynamic model at hypervelocity, and their discrepancy still exists at velocities up to 6 km/s.

It can be observed in Fig.2 that the explicit?V0curves in all cases are approximately linear and of very good linearity in hypervelocity.This is related to the observation that Eqs.(9)–(11)all have the same form,?AV0?, and the positive first power of V0is larger than the negative first power,especially in the hypervelocity regime.For example,the ratio of the positive and negative first power in Eq.(11)is computed as λ?Substituting the corresponding parameters in the case of tungsten rods penetrating SiC targets in Ref.[18],the quantity of λ drops from 0.556 to 0.059 when the initial impact velocity increases from 1.5 km/s to 4.6 km/s.

It can be clearly concluded from Fig.2 that the Alekseevskii–Tate model yields better predictions for the experimental results than the hydrodynamic model.For example,for two cases with the same density ratio of rod/target materials and different strengths(W/SiC-B and Au/SiC-N),the prediction of Eq.(11)shows the difference between these two cases,while the hydrodynamic model prediction[Eq.(2)]does not.Therefore,the explicit?V0relationship given by Eq.(11)reveals the physical essence of the linear relationship reported by the experiments and simulations and can be used for engineering prediction.

5 Discussion o f the deceleration in long-rod penetration

5.1 Deceleration degree of the long-rod penetration process

After the instant velocities are explicitly expressed as a function of time in approximate solutions,the deceleration rate of the instant velocities during the long-rod penetration process is analyzed in Ref.[16].Based on Eqs.(40) and (41)in Ref.[16],the initial deceleration rate of the instant tail and penetration velocity,i.e.,and d u??are independent of the initial impact velocity,V0.However,according to the observation of experimental data,the degree of deceleration of the penetration process is actually related to the initial impact velocity.Meanwhile,d v??andhave the same dimension as the acceleration,so it is difficult tome a sure the effect of deceleration on long-rod penetration.Therefore,a dimensionless parameter needs to be introduced as the characteristic index of the deceleration degree.

Fig.4 Schematic of the deceleration of tail velocity during long-rod penetration process

The velocity–time curve of approximate solution 1 in Fig.A1(see Appendix)is carefully analyzed.A tangent line of the curve from point A(0,V0)intersects with lineat point C in Fig.4,in which AB??T and BD?V0.Combining the physical meaning of the initial deceleration rate of instant tail velocity and the geometric relationship,we get BD?V0and

According to Eq.(40)in Ref.[16],d v??is independent of the initial impact velocity,V0.However,for a given initial rod length,L,the duration of the penetration process,?T,decreases with the increase in V0.Correspondingly,in Fig.4,BD increases while AB and BC decrease with the increase of V0.

Actually,the physical meaning of BC is the deceleration quantity of the tail velocity at the terminal time,when the instant tail velocity is assumed to be decelerated at the initial rate(the variation in the deceleration rate during the penetration process is ignored).Therefore,the ratio between BC and BD can briefly reflect the total deceleration degree of the tail velocity during the primary phase.

For a given initial rod length,L,with the increase of V0,BD increases while BC decreases, and thus BC?BD decreases,which implies that the total deceleration degree of the tail velocity decreases.

For a given V0,with the increase of L,BD does not change,while AB increases.In addition,is proportional to L andis inversely proportional to L,so BC does not change.Therefore,BC BD is unchanged,which implies that the total deceleration degree of the tail velocity is unchanged.

These implications are verified by observing and summarizing the experimental results,so the ratio between BC and BD,reflecting the deceleration degree of the tail velocity,can be defined as a dimensionless deceleration parameter

which is related to the density ratio of the rod and target materials,the Johnson damage number of the rod materials, and the dimensionless linear parameter K.The physical meaning of α can be easily obtained by substituting Eq.(13)into Eq.(24)in Ref.[16]

It should be noted that α can also reflect the deceleration degree of the penetration velocity,which is proportional to the deceleration degree of the instant tail velocity,based on Eq.(41)in Ref.[16].Therefore,α can be defined as a deceleration index of the long-rod penetration process.

When α?0,the instant tail and penetration velocity does not change during the penetration process,i.e.steady-state penetration, and the variation in dominant variables can be described with out deviation by approximate solution 2;when α>0,the penetration process deviates from the steady state, and α actually reflects the deviation degree.Therefore,the deceleration index α can be taken as the application criteria for approximate solution 2.

Corresponding to velocity–time curves such as Fig.A1,α can reflect the shape of the area between the velocity—time curve and the coordinate axes.When α?0,the shape of the area is a rectangle,corresponding to the steady penetration state;when α approaches 0,the shape of this area approaches a rectangle,as the penetration process approaches the steady state.The values of α in cases A(Fig.A1a) and B(Fig.A1b)are2.935% and 10.879%,respectively.Corresponding to Fig.A1,the shape of the area in case A is closer to a rectangle than that in case B,which implies that the penetration process in case A is closer to the steady state,so it can be approximately described by approximate solution 2.

5.2 Deceleration effect on the velocity relationship

For further analysis of the effect of deceleration on the ˉU?V0relationship,the expression of the dimensionless decelera-tion parameter,Eq.(13),is substituted into Eq.(9), and thus the explicit?V0relationship can be further expressed as

Fig.5 Variation of α with the variation of V0,ρp,ρt,Yp, and Rt

which implies that,with the decrease in the deceleration indexα,the slope of the?V0curve increases.In addition,the ratio of the positive and negative first power of V0increases with the decrease in the deceleration index,so the linearity degree of the curve increases.Therefore,both the slope and degree of linearity of the?V0curve are affected by the deceleration.

Substituting corresponding parameters in the experiment in Ref.[18]into Eq.(13),α ?1.55%–8.41%is computed.For all the experiments listed in Table1,the range of deceleration index values α is from 0 to 10.70%.Therefore,we can conclude that the deceleration degrees of all these experiments are relatively small.Observing their parameters listed in Table 1,the high-density and low-strength rod is found to be the common characteristic.This characteristic will lead to a small deceleration degree according to Eq.(13).Combining this with the implication derived from Eq.(15),it will be easier to obtain the linearˉU?V0relationship if high-density and low-strength rods are used in the experiments.

5.3 Determination of deceleration degree:guide for experimental design

From Eq.(13),it is known that the deceleration degree is related to initial impact velocity,density, and strength of the rod and target, and is independent of rod length.The relationships between the deceleration index α and V0,ρp,ρt,Yp, and Rtare studied based on the corresponding parameters of case B in Table A1.The variations in α with all these parameters are shown in Fig.5.It can be observed that the deceleration degree is negatively related to initial impact velocity,density of rod and target,and target strength,while it is positively related to rod strength.Moreover,as can be observed from the first two pictures in Fig.5,the density and strength of the target material play a smaller role in deceleration than those of the rod material,respectively.

The above relationships cannot be directly observed from Eq.(13)because of the complex expression of K(Eq.(38)in Ref.[16]).At hypervelocity,K can be replaced with?K given by Eq.(48)in Ref.[16].For example,based on the corresponding parameters of case B in Table A1,the values ofare85.41%,96.22%, and 99.05%at V0?1.5km/s,3.0 km/s, and 6.0 km/s,respectively.

Therefore,replacing K with?K,Eq.(13)can be simplified as

From this equation,it can be easily observed that the deceleration index α is positively related to Ypand negatively related toρp,ρt, and V0.In addition,ρtplays a smaller role in deceleration than ρp, and α is independent of Rtat hypervelocity.

As the degree of deviation of the long-rod penetration process from steady state is reflected by α,the state of the penetration process can be identified, and different penetration states can be designed in experiments by choosing the appropriate corresponding parameters,i.e.V0,ρp,ρt,Yp, and Rt.For example,nine cases with different penetration states are designed.The corresponding parameters of these cases are listed in Table3, and the variations in instant tail velocities in these cases are shown in Fig.6,where V0and L?V0are used to non-dimensionalize v and t,respectively.

Comparing case1 with case2,different rod lengths make no difference in deceleration.As can be seen from cases 1,3, and 4,the penetration process approaches steady state with an increase in impact velocity.In addition,comparing case1 with case5,the degree of deceleration is observed to be negatively related to ρpand ρt.Comparing case 5 withcases7 and 8,a positive relationship between the deceleration degree and rod strength is observed. of note,the penetration process is steady state when Yp?0(case 8).Moreover,comparing case5with case6,the deceleration degree is negatively related to Rt, and the effect of Rtis very weak at hypervelocity,as is observed by comparing case4 with case 9.

Table3 Compilation ofparameters in designed cases

Fig.6 Variation of instant tail velocities in cases with different deceleration degrees

To sum up,the deceleration degree is related to V0,ρp,ρt,Yp, and Rt, and cases with different deceleration degrees can be designed by choosing the appropriate corresponding parameters.Therefore,the deceleration index α is of great significance for guiding the experimental design of different deceleration degrees.

6 Conclusions

In this work,the linearˉU?V0relationships reported by experiments and simulations are summarized and briefly analyzed using a hydrodynamic model.However,without considering the strength effect,the re is a clear discrepancy between the hydrodynamic model prediction and the experimental results.Therefore,the Alekseevskii–Tatemodel with strength effect is used for theoretical analysis, and the model prediction is closer to the experimental results.In addition,it is found that the difference between the initial and average penetration velocity exists because of the deceleration during the penetration process.

Because of the nonlinear nature of the Alekseevskii—Tate equations,the explicitˉU?V0relationship cannot be derived directly.Approximate solutions are used to obtain the explicitˉU?V0relationship, and its physical essence is revealed through simplification.The validity of the expressions is verified by substituting the corresponding experimental parameters.

Further,based on the approximate solutions,the deceleration of long-rod penetration is comprehensively discussed.A deceleration index α is defined to describe the deceleration degree of the penetration process and to identify its degree of deviation from steady state.α?i.e.,the deceleration degree is mostly related to impact velocity,rod strength and rod/target densities.It is found that all the experiments with the ˉU?V0relationship have relatively small deceleration indexes, and the reason is discussed.The state of the penetration process can be designed in experiments by choosing the appropriate corresponding parameters;thus α is of great significance for guiding experimental design.

AcknowledgementsThe work was supported by the National Natural Science Foundation of China(Grant 11872118).The authors want to express deep gratitude to the reviewers for their sound comments and helpful suggestions.

Append ix

In Ref.[16],two typical long-rod penetration cases in Refs.[14,26]are further analyzed.The corresponding parameters are listed in Table A1, and the variations in instant penetration(nose) and tail velocity during the penetration process are shown in Fig.A1.

Comparing the two cases,case A is obviously closer to the steady state,i.e.,the instant penetration and tail velocity decelerate very slowly,so the initial penetration velocity is relatively close to the average penetration velocity.The reason for the difference between their penetration states is further analyzed in Sect.5.1.

Table A1 Compilation of parameters in two long-rod penetration cases

Fig.A1 Penetration(nose) and tail velocity versus time for different cases:comparison of different solutions.a Case A.b Case B