Fragmentation behavior of large-caliber PELE impacting RHA plate at low velocity

Mo-ang Lei,Hai-fu Wang,Qing-bo Yu,Yuan-feng Zheng

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

ABSTRACT Impact experiments of large-caliber PELE with various inner-outer diameter ratio perforating RHA plate at low velocity were performed.Experimental results showed the size of perforated holes on plate,average diameter of damage area on witness plate,and number of behind-armor fragments will increase as d/D increasing from 0.72 to 0.84.Expansion and fragmentation of large-caliber PELE in this condition were also numerically studied with ANSYS Autodyn.Then,an analytical model accounting for an additional radial shock wave was presented to predict radial expansion velocity and fragmentation of jacket,as well as an empirical approach to estimate diameter of damage area.Calculation results by these approaches were in good agreement with experiments and numerical simulations.Further discussion revealed that Shock/rarefaction wave interactions behavior varying with inner-outer diameter ratio is an important mechanism resulting in different lateral effect by PELE projectiles with various configurations.

Keywords:PELE Enhanced lateral effect Penetration mechanics Behind-armor fragments

1. Introduction

Penetrator with Enhanced Lateral Effect(PELE)is a kind of ammunition which consists of a low-density material as the filling and a high-density material as the jacket[1,2].When PELE impact target,the jacket is radially accelerated,expanded and fractured because of distinctive lateral effect.Behind-armor fragments by PELE will scatter with higher radial velocity than that by other ammunition such as kinetic energy metal penetrators.This higher radial velocity of fragments will result in enhanced behind-armor damage effect.

Paulus and Schirm[3]conducted PELE experiments by smallcaliber projectiles with polyethylene(PE)/aluminum(AG-3)filling and tungsten alloy(D180k)jacket against aluminum(A-U4G)/steel(XC48)target with impact velocity ranging from 900 to 3000 m⋅s-1.Expansion and fragmentation of these projectiles were monitored by X-ray photographs,as well as axial residual velocity and radial velocity of behind-armor fragments. Based on these detailed experimental data,Paulus numerically investigated lateral expansion behavior and presented an analytical approach of radial expansion velocity under assumptions of weak shock and acoustic approximation.

According to Paulus'data,Verreault[4-6]conducted further numerical and theoretical investigation on lateral expansion and fragmentation of PELE jacket.Considering propagation and interaction of shock/rarefaction waves in PELE without assumptions of weak shock or acoustic approximation, Verreault managed to describe impact-induced pressure evolution in PELE filling more accurately,and developed an improved model to predict maximum radial velocity and fragmentation of PELE jacket. Verreault's analytical results were in relatively better agreement with Paulus'experimental data.This indicated model built by Verreault can effectively describe lateral effect of PELE,especially for smallcaliber PELE projectile against thin target at high velocity.Besides,Zijian Fan and Xiwen Ran[7]developed another analytical model,which also predicted maximum radial velocity more accurately than Paulus'estimation,by assuming kinetic energy accumulated as compression energy before being perforated will be completely released as radial velocity of PELE fragments.

More experiments of PELE with various jacket and filling materials with impact velocity ranging from 400 to 900 m⋅s-1were carried out by Zhu[8].Witness plates were setup very near to main target plates,in this case axial velocity differences between behindarmor fragments can be ignored,to indirectly acquire maximum radial velocity of PELE fragments according to distribution of perforated holes on witness plates. Zhu also developed an analytical model for maximum axial and radial velocities of behindarmor fragments under assumption of weak shock wave and ignorance of energy loss.Furthermore,fragmentation behavior of PELE with rotation against aluminum target at various impact angle was numerically investigated by Jiang and Zhang with stochastic model[9].

Fig.1.Diagram of experimental arrangement.

Fig.2.PELE projectiles.

Researches aforementioned provide effective numerical/theoretical descriptions of expansion behavior of PELE when smallcaliber projectiles penetrating thin target at high impact velocity.Here we presented a similar study on large-caliber PELE projectile(130 mm)with various thickness steel jacket against 30 mm RHA plates at low impact velocity(415 m⋅s-1).By considering an additional radial shock wave originating from transmission at jacket/filling interface,an analytical model developed by Verreault was presented to accurately describe pressure evolution, lateral expansion and jacket fragmentation in this condition.To predict damage area on behind-armor witness target that was far away(2 m)from target plate,empirical investigation were conducted,focusing on relation between maximum emission angle of PELE behind-armor fragments and maximum axial/radial velocity.Further comparisons between analytical and experimental/numerical data verified above models and then revealed influence ofd/Don lateral expansion and jacket fragmentation of large-caliber PELE.

Fig.3.Target and witness plate.

2. Experiments

2.1. Experimental setup

Experimental setup of large-caliber PELE projectile against Rolled homogeneous armour(RHA)plate is presented in Fig.1.By 130 mm caliber launching platform,PELE projectiles were launched at an initial velocity of 430±10 m⋅s-1,then impact target 170 m away at 415±10 m⋅s-1.Initial velocity and impact velocity were measured and recorded by radar speedometer.Penetration of PELE against RHA plate was recorded by high-speed photography.Witness plate were set behind target 2 m away.

As showed in Fig.2,experimental 130 mm caliber PELE projectile consisted of filling,jacket and wind cap,which were made of polyethylene, steel and aluminum alloy, respectively. Lengthdiameter ratio of projectile(without wind cap)was 3.5.Target plate was 1600 mm×1200 mm and made of RHA with 30 mm thickness.Witness plate was 6000 mm×2000 mm and placed as an arc behind target by 2 mm thick steel plates.Target and witness plate were demonstrated in Fig.3.

2.2. Experimental results

Fig.4.Typical high-speed photography frames of PELE impact target at low velocity.

Typical high-speed photography frames of PELE with 0.84d/D（inner-outer diameter ratio）impact target at 413.1 m⋅s-1are presented as Fig.4.It can be observed that PELE penetrated target normally as showed in Fig.4(a).After target were perforated,great quantities of behind-armor fragments were generated as showed in Fig.4(b).Due to insufficient height of target plate,some fragments with downward emission angle hit ground,while most fragments penetrated witness plate,as showed in Fig.4(c).Typical damage effect on target and witness plate were present in Fig.5.On target plate,a perforated hole with a diameter of 200 mm and apparent deformation and tears with a diameter of 400 mm can be observed,in Fig.5(a).According to plug of target observed in high-speed photography frames and damage effect on target, it's can be concluded that PELE penetration resulted in plugging-failure.The plug,residual penetrator and scattering fragments impacted and perforated witness plate in large area of 2000 mm×2300 mm,as showed in Fig.5(b).Above observation indicated that this largecaliber PELE projectile against target at 415 m⋅s-1impact velocity can induce remarkable lateral effect and generate lots of scattering fragments with high radial velocity to enhance behind-armor damage effect.

Fig.5.Typical damage effect on target and witness plate by PELE with 0.84 d/D.

To study influence ofd/Don lateral effect,further experiments were carried out by PELE projectiles withd/Dranging from 0.72 to 0.84.Typical experimental phenomenon is presented in Fig.6 and Fig.7.More perforated holes that distributed in larger areas on witness plates can be observed in Fig.6(a)than that in Fig.6(b)and(c).Above observation indicate that thed/Dof PELE projectile is positively related to scattering of behind-armor fragments.In Fig.7,18/12/10 jacket fragments were recycled behind witness target whend/Dwere 0.84/0.80/0.76,respectively.Besides,average size of fragments tended to become larger asd/Ddecreasing.According to above observations,it's can be concluded that lateral effect caused by penetration of PELE will make thinner jacket(corresponding to greaterd/Dprojectile)suffer more significant fragmentation and expansion,resulting in relatively bigger perforated hole on target,more scattering fragments with higher radial velocity and larger damage area on witness plate.This conclusion is corresponding to detailed experimental results listed in Table 1,wherevinitis initial velocity of PELE projectile，v0is impact velocity，wtis diameter ofperforated hole on target，Dfris average diameter of damage area on witness plate.Considering an ellipse enclosing all perforated holes on witness plates,Dfris the average of minor axis and major axis of this ellipse.Leaving a few data affected by experimental random error out of account,a decreasing trend ofwtandDfrasd/Ddecreasing can be observed.

Table 1Experimental data.

Fig.6.Damage effects on witness plates by PELE with typical d/D against 30 mm RHA plate.

Fig.7.Recycled jacket fragments and residual filling material by PELE with typical d/D.

3. Numerical simulations

3.1. Finite element model

To further study fragmentation behavior of PELE,finite-element numerical simulations were performed by Autodyn.Numerical model consisted with three main parts:filling,jacket and target,which were all modeled with Lagrangian elements.Element size for all parts was 2 mm,and two symmetry planes were used to simulate one quarter of the full domain.Geometry Sizes of numerical projectiles were same to experimental ones.A typical finite element model grids of jacket and filling were illustrated in Fig.8(a).Three main parts and Gauges points for recording pressure histories in varies filling locations were arranged as Fig. 8(b)showed.Filling,jacket and target were made of polyethylene(PE),steel 4340,RHA,respectively.Material properties and material models for each parts were listed in Table 2 and Table 3,respectively.Failure model of target and jacket and corresponding Principal Tensile Failure Stress were determined by similar simulation[10].Rest materials models and parameters were obtained from Autodyn material library[11-13].In addition,stochastic failure was added to numerical model with following equation to describe probability of jacket elements failure:

Where ε is strain,λ is an empirical parameter.Cmust be set to ensurePfailure(1)=0.5.In this paper,λ=10，C=3.14×10-4.

3.2. Numerical results

Typical pressure contours of PELE after impact are displayed in Fig.9(Detailed numerical pressure evolution will be presented in section 4.2).It can be observed that filling material was compressed by both axial and radial shock wave.After swept by shock wave,radial expansion of filling and jacket can be observed.Meanwhile,rarefaction waves(axial one reflected from target free surface and radial one from jacket outer surface)were propagating into filling and unloading pressure.The radial rarefaction wave,entered filling earlier and weakened radial shock wave immediately.At later time,radial shock/rarefaction waves will disappear,thus only axial ones will affect filling material at rear part of PELE.

Typical process of PELE with 0.78d/Dperforated target plate was illustrated in Fig.10.Distinct expansion and fragmentation of jacket can be observed,especially at the front part of PELE,where the filling was affected by stronger shock wave.According to the above observation,it can be infer that impact-induced wave interactions are strongly related to lateral effect.

Fig.8.Numerical model and arrangement of Gauges points.

Table 2Material properties for analytical and numerical model.

Table 3Material model for numerical simulation.

Fig.9.Pressure contours of PELE with 0.76 d/D at typical moments.

Fig.10.Typical process of PELE with 0.78 d/D perforated target plate.

Comparison of expansion and fragmentation of jacket with typicald/Dat 250/500/750 μs after impact are presented in Fig.11.Significant radial expansion can be observed for both PELE projectiles,especially at the front end.More cracks and severer fragmentation caused by lateral expansion can be observed in jacket of PELE with 0.84d/D.Furthermore,this PELE projectile generated higher maximum radial velocity (203.45 m⋅s-1) than that(119.30 m⋅s-1)of 0.74d/D.More numerical axial/radial velocity data for variousd/Dwill be presented in section 4.2.Corresponding to experimental data,above observation indicated thatd/Dhas influence on radial expansion and jacket fragmentation.

4. Discussion

4.1. Analytical model for PELE behavior

4.1.1. Wave propagation and interaction

Despite of complexity of behaviors of penetration and fragmentation,radial expansion of PELE is mainly determined by wave interactions during penetration,according to simulation results in section and pervious works by Paulus[3]and Verreault[4].According to pressure contours presented in previous section,a qualitative description can be presented under simplified conditions of ignoring mass loss and energy loss during penetration.Schematic of interactions between shock waves and rarefaction waves is presented in Fig.12.When PELE projectile impact target,axial shock waves are generated in filling(labeled as ASf)and in jacket,respectively,as showed in Fig.12(a).The shock wave in jacket,which is much stronger than that in filling due to different material properties,transmits into filling as radial shock wave(labeled as RSf).Above shock waves including ASfand RSfwill compressed filling material,result in lateral expansion which will be calculated by radial acceleration in section 4.1.2.

Fig.11.Comparison of jacket radial expansion between PELE with 0.74 d/D and 0.84 d/D.

Fig.12.Schematic of interactions between shock waves and rarefaction waves.

Impact also generate shock wave in target material(labeled as ASt1),which will reflect at free surface of target plate as an axial rarefaction wave (labeled as ARt1). When ARt1propagates to interface between target and projectile,a reflected shock wave in target(labeled as ASt2)will be generated,as well as a transmitted axial rarefaction wave in filling(labeled as ARf1).ASt2will reflect at free surface and transmit at interface to generate another axial rarefaction wave.Above axial reflection/transmission progress will continuously repeat until filling is completely unloaded.Meanwhile,radial rarefaction wave(labeled as RR)from free surface of jacket caused by radial expansion starts to propagate inward immediately after impact,then transmits into filling,as Fig.12(b)showed.Due to higher propagating velocity,wavefront of radial rarefaction wave(labeled as RRfront)will catch up and weaken RSf.At a later time,radial waves,including RR and RSf,will reach axis of filling atx=0 and then axially propagates,as showed in Fig.12(c).RSfwill soon be released and has no further effect on the rest of filling,as Fig.12(d)showed,ASfwill be weakened by RR and ARftill all shock wave unloaded.

Affected by shock waves including ASfand RSf,filling material will be compressed and results in radial acceleration because of Poisson's effect.Once swept by rarefaction waves,compressed filling starts to be unloaded and radial acceleration starts decreasing to null,radial expansion velocity in this location will tend to become a constant.Thus,to acquire radial expansion of PELE,it's necessary to obtain sates of shock/rarefaction wave and pressure evolution in filling.

Initial pressure and propagating velocity of shock waves ASfis determined by shock states of materials,which can be calculated based on conservation of mass and momentum as well as EOS of material[14],which are expressed as:

Where ρ is density,Pis pressure,Uis shock wave velocity anduis particle velocity,candsare constant parameters to certain material.The subscripts 0 and 1 refer to unshocked and shocked sates in materials,respectively.In this case,relations between particle velocity and pressure in filling and target can be expressed as:

Where subscriptsfand t refer to filling and target material,Pf1andUf1is initial pressure and velocity of ASf.After impact,particle velocity and pressure of each materials between interface must be equal(v0-uf1=ut1andPf1=Pt1).Taking these boundary conditions into consideration,above system of equations are closed and all shock states can be obtained.According to this procedure,impactinduced shock wave in jacketPj1can also be acquired,as well as the pressure and propagating velocity of RSf,which are referred asPftranandUftran,respectively.

Pftran/Pf1(ratio of initial pressure of shock wave RSfand ASf)is calculated at impact velocity ranging from 300 to 3000 m⋅s-1with various filling/jacket/target materials which were mentioned in this paper and Paulus'experiments,as listed in Table 4.Calculation results showed thatPftran/Pf1only slightly varying as impact velocity increasing.Thus,average value ofPftran/Pf1by various materials configurations were listed in Table 4.It can be observed thatPftran/Pf1of steel jacket PELE(Case#1),which is discussed in this paper,is higher than that of tungsten jacket PELE(Case#2 ～#5),which were used for Paulus’experiments.Due to relatively strongerPftranin steel jacket PELE than that in tungsten jacket one,effect of RSfshould be taken into consideration for steel-jacket projectiles.

Pressure and propagating velocity of ARf,as well as moments of ARftransmitting into filling, can be obtained by considering continuous reflection/transmission progress at target/filling interface.Due to thickness of jacket is thin,it can be assumed that radial rarefaction wave in jacket RRjis a simple wave propagating at sound velocity of jacket material. Thus, this rarefaction wave transmit into filling att=hj/cj,wherehjis thickness of jacket,cjis sound velocity of jacket.By dividing rarefaction wave into multiple wavelet,propagation and unloading process of rarefaction wave can be mathematically described following a method suggested by Cooper[15].Thus,propagation and interaction of shock/rarefactionwaves can be calculated,as well as pressure evolutions at any position in PELE filling.

Table 4Average Pftran/Pf1 of various PELE materials.

4.1.2. PELE expansion and fragmentation

According to shock states acquired above, radial stress at interface between filling and jacket can be calculated as Verreault suggested[4]:

Where σxfis axial stress in filling,determined by axial component of shock/rarefaction waves affect filling material.μ is Poisson’ratio,Eis Young modulus.Thus,radial velocity of jacketvrcan be expressed as:

WhereAisandAscare jacket inner surface area and crosssectional area,respectively,mjis jacket mass,σscis hoop stress,which is equal to ultimate stress of jacket material until jacket strain reaches its fracture strain.Radial velocityvrkeeps increasing until any of following conditions is met:(1)filling radial strain reaches its maximum value εfmax[4],which can be obtained by contact pressure,Young modulus and Poisson's ratio;(2)pressure in filling is completely unloaded by rarefaction waves.Thus,radial expansion velocity of PELE jacket can be obtained.

Rapidly expansion jacket will result in fragmentation,which can be calculated following a procedure suggested by Verreault[5]:considering a“ring”of jacket,radial velocity of this segment jacket can be obtained by Eq.(4)and Eq.(5);total number of fragments can be statistically determined based on researches of Mott[16]and Grady[17];after estimated average mass of fragments by Gold and Baker[18],total mass and axial length of these fragments can be calculated;repeating above steps to calculate fragmentation of next segment until all jacket are fractured,fragmentation of this PELE projectile can be obtained eventually.

4.1.3. Behind-armor fragments scattering

Once target is perforated,jacket fragments,residual penetrator and target plate plug are scattering behind-armor.Due to interaction with target material,axial velocities of fragmentsvsθ(whose emission angel θ≠0°)are relatively lower than axial velocity of residual penetratorvs,which can be estimated by integrate axial acceleration[8]:

WherePf1andPj1are contact pressure in filling and jacket;Danddare outer and inner diameter of PELE,respectively;Mis total mass of PELE projectile;σYis shearing stress of target material;htis thickness of target;xtis distance from target front surface to PELE/target interface,which is increasing from 0 tohtduring penetration.Whenxt=ht,target plate is perforated andasbecomes null.

At high impact velocity,an expansion ellipsoidal debris clouds,the tip of which is the residual penetrator,will soon be formed.In this case,axial velocity of fragments with θ≠0 can be obtained by Ref.[19]:

Where α is a coefficient can be determined based on experimental results.This equation indicate thatvsθ is positively related tovs.

Fig.13.Behind-armor debris cloud and scattering of fragments.

At low impact velocity,according to numerical phenomenon presented in section 3.2,behind-armor fragments and residual penetrator will take much longer time to form an ellipsoidal debris cloud.To describe fragments scattering in this case,we can assume that the fragments with highest radial velocityscattering at the greatest emission angle θmax, as Fig. 13 showed, where θmax=atan（/vsθmax）·can be calculated by radial expansion,and axial velocity vsθmaxis positively related tovs.Thus we can infer that θmaxis related to/vs,and this relation can be determined by polynomial fitting with experimental data.When the θmaxis obtained,diameter of damage area on witness target can be expressed as:

WhereLwis distance from target plate to witness target.

4.2. Comparison and discussion

To validate analytical model in section 4.1,maximum radial velocities at Paulus’conditions[3]were calculated at impact velocity ranging from 500 m⋅s-1to 3000 m⋅s-1and showed in Fig.14.In general,calculation results are in good agreement with experimental data,especially when target thickness is relatively thin and impact velocity is low.

The pressure evolution of filling at gauges point in typical largecaliber PELE projectile(0.76d/D)by numerical simulation and analytical calculation are showed in Fig.15.In Fig.15(a),it can be observed that calculated total pressure evolutions at various positions of filling were in generally good agreement with numerical ones,especially when pressure increasing.Despite of calculation slightly under-estimate consumed time of unloading,deviation of radial velocity between analytical and experimental should not be significant due to pressure during unloading is relatively small and contributes less to lateral expansion.According to analytical curves,filling material particles which are 5 mm away from the impact interface(referred asx=5 mm) was firstly influenced by ASf.Pressure here jumped to 1.23 GPa,then partly unloaded to 0.93 GPa by the first ARfwavelet.After about 12 μs from ASfreachedx=5 mm,RSfalso propagated to this position and cause pressure increasing to a peak which is 2.44 GPa.By RR following RSf,pressure started decreasing immediately,then unloaded to negligible magnitude at around 21 μs by RR and axial rarefaction wave generated by RR converging at axis of filling.Atx=41 mm,before ARfpropagated to this position,ASfreached at 11.74 μs,RSfreached 5.37 μs later,thus pressure peak increased to 2.48 GPa,higher than that atx=5 mm.By both RR and ARf,pressure was unloaded eventually.Asxincreasing,RSftakes less time to affect filling particle at certain position,and pressure increment caused by its decreasing because RSfis weakened by RR.Atx=113 mm,RSfis reached almost at the same time when ASfis reached while only caused slightly pressure increment.It can be inferred that RSfwas completely unloaded by RR,and won't affect further filling materials.According to above observation,it can be concluded that this analytical model is capable of describing pressure evolution in large-caliber PELE filling.

Fig.14.Comparison of radial velocity between analytical model and experimental results of Paulus[3].

Fig.15.Pressure in filling of PELE with 0.76 d/D by numerical simulation and analytical calculation.

According to eEq.(4)and Eq.(5),radial expansion calculation needs axial stress evolutions,which were recorded and presented in Fig.15(b).Atx=5 mm,direction of RSfand following RR were perpendicular to axis of filling,thus pressure were unloaded from 1.23 GPa only by axial rarefaction wave.RSfand RR affected filling particles atx=41/77/113 mm were obliquely propagating,thus pressure at these positions was increased to relatively higher peaks than that atx=5 mm,and then unloaded both by axial components of these waves.

Jacket fragmentation and corresponding radial velocities of large-caliber PELE with typicald/D(0.76/0.80/0.84)were calculated and presented in Fig.16(a).In this figure,xdistance between two adjacent markers refers to relative axial length of fragments in this segment,numbers beside curves refer to numbers of fragments for certain segment of jacket.Sum of fragments by PELE with 0.84/0.80/0.76d/Dis 55/38/22,ratio of which is 1.72/1.18/1 and shows good agreement with the numbers of recycled fragments which were 18/12/10 as showed in Fig.7.This indicated that fragmentation calculating procedure suggested by Verreault[5]can effectively estimate fragmentation of large-caliber PELE jacket.Estimation of average mass of jacket fragments by PELE with typicald/Dwere calculated and presented in Fig.16(b).For projectile with 0.76/0.80/0.84d/D,average masses are 72/41/21 g at the front and 3052/1712/845 g at the end,respectively.Because more fragments with smaller mass and size but with higher radial velocity can be observed at fore part of PELE,it can be inferred that this part of PELE suffered severer fragmentation and expansion than rear part,PELE with greaterd/Dexpands and fractures more than that with smallerd/D.This conclusion are in good qualitative agreement with numerical phenomenon displayed in Fig.11.

To investigate mechanisms of influence ofd/Don PELE fragmentation and expansion,axial stress evolution in filling and radial acceleration of jacket of typical PELE projectiles are presented in Fig.17.In Fig.17(a),pressure peak differences can only be observed where RSfcontributes to axial stress,especially atx=41 andx=77 mm.At these position,pressure peak of filling materials decreases asd/Dincreasing.In Fig.17(b),a significant trend that radial acceleration of jacket increases asd/Dincreasing can be observed at any position,obviously due to jacket become thinner and lighter,while at positions affected by RSfsuch asx=41 andx=77 mm,differences between radial accelerations are relatively smaller.Thus,it can be concluded that the pressure evolution varying withd/Dis an important factor of lateral expansion and jacket fragmentation of PELE.

Fig.16.Fragmentation and corresponding radial velocity of PELE with typical d/D.

Fig.17.Axial stress evolution and radial acceleration of PELE with d/D of 0.84/0.80/0.76.

Fig.18.Scattering velocity of fragments and diameter of damage area on witness plate by PELE with various d/D.

By numerical simulations and analytical calculations,maximum radial velocity and residual axial velocity produced by PELE with variousd/Dare presented in Fig.18(a)Analytical and numerical radial velocity(vr)are in generally good agreement,and both increase asd/Dincreasing.The deviation of analytical curves and numerical data may be caused by(a)erosion of jacket fragments in numerical simulation,which is relatively significant when PELE with biggerd/Dand thinner jacket,(b)Analytical model ignores interaction between jacket fragments and target, which may decrease quantities of behind-armor fragments and radial velocity,especially for PELE with thinner jacket against relatively thick target.Asd/Dincreasing,both numerical and analytical maximum residual axial velocity of fragments(vs)are slightly decreasing from 405 m⋅s-1to 400 m⋅s-1,which indicated/Dtakes no significant influence onvs. According to above observations, it can be concluded that analytical models are in generally good agreement with numerical results whend/Dranging from 0.7 to 0.84.

Diameter of damage area on witness plate(Dfr)by scattering fragments generated by PELE with variousd/Dis showed in Fig.18(b).Analytical results and experimental data are generally in good agreement,and both increasing asd/Dincreasing from 0.72 to 0.84,except data ofd/D=0.78 which is about 10%greater than calculated result.This anomalous data may cause by experimental PELE projectile impacted target obliquely due to external ballistic factor,and result in a damaged area with biggerDfron witness target.Above observation indicated more fragments with higher radial velocity was generated by PELE with largerd/D,because of greater radial acceleration and severer fragmentation of thinner jacket.However,thinner jacket will generate small and lighter fragments,which are not likely to penetrate witness target because of small kinetic energy.Thus,it can be infer thatDfrwill meets a limitation eventually asd/Dincreasing from 0.86.This limitation remains unknown for now and needs further investigation.

5. Conclusions

By large-caliber PELE with variousd/Dagainst RHA plate at low velocity of 415 m⋅s-1,impact experiments against 30 mm RHA target were carried out.Numerical simulations were also conducted to monitor progress of PELE expansion and fragmentation in this condition.According to experimental and numerical results,an analytical model that taking an additional radial shock wave into consideration was presented to describe lateral effect,as well as an empirical approach for damage area on witness target.Further comparisons and discussion drew conclusions including:

(1)Impact of large-caliber PELE against RHA plate at low impact velocity of 415 m⋅s-1can generate significant lateral effect and induce enhanced behind-armor damage.Size of perforated holes on target(wt),size of damage area on witness plate(Dfr)and quantities of PELE fragments are all positively related to inner-outer diameter ratios(d/D)of PELE.

(2)Due to relatively higher than tungsten jacket condition,radial shock wave in steel-jacket PELE must be accounted for.Comparisons between calculation and numerical/experimental data indicated that analytical model here can effectively predict expansion and fragmentation of large-caliber PELE,as well as damage area on witness plate.

(3)Shock/rarefaction wave interactions behavior varying withd/Dis an important mechanism that results in differences of lateral effect and behind-armor damage between PELE projectiles with various configurations.