Effect of projectile parameters on opening behavior of PELE penetrating RC target

Li-zhi Xu,Cheng-xin Du,Chun Cheng,Xiao-dong Wang,Jiang-bo Wang,Zhong-hua Du,Guang-fa Gao

School of Mechanical Engineering,Nanjing University of Science and Technology,Nanjing,210094,China

Keywords:Penetrator with enhanced lateral effect Opening behaviour Reinforced concrete target Dimensional analysis

ABSTRACT When a penetrator with enhanced lateral effect(PELE)impacts on a reinforced concrete(RC)target,the target is damaged with a large opening.An understanding of how PELE projectile parameters affect the opening dimension,is essential for effective design of the PELE projectile.In this study,under the condition that the impact velocity and target parameters(strength and thickness)were fixed values,the important influence factors of the PELE(jacket wall thickness B,jacket material strength Y1,filling material strength Y2 and angle of monolithic jacketθ)were determined by a dimensional analysis.Tests and simulations of the PELE penetrating the RC target were conducted to analyze the influence of these factors on opening diameter(an equivalent diameter under relative kinetic energy).Based on the test and simulation results,it is found that the influence of these factors B,Y1 andθon the deformation mode of the jacket shows a similar trend:as values of the three factors decrease,the jacket deforms from small bending deformation to large one,and then to curling deformation.This causes the opening diameter to first increase with the decrease of these three factors,and then decreases.It is well known that the bending resistance of the jacket is related to these factors B,Y1 andθ.Therefore,a plastic limit bending moment(M0)of the jacket was quoted to characterize the influence of these factors on the bending deformation of the jacket and the opening diameter of the target.The influence factor Y2 causesto first increase with the increase of Y2,and then decreases.A formula was developed to predict the opening diameter,whose influence parameters were considered in a dimensionless way.It has been shown that the dimensionless opening diameter D/d1 is dependent on two dimensionless parametersandwhere d1 and fc are the outer diameter of the projectile and the compressive strength of the target,respectively.

1.Introduction

A penetrator with enhanced lateral effect(PELE)is a new type of penetrator consisting of a low-density material in its core(the filling)surrounded by an outer jacket of a high-density material[1].Once perforation of a metallic target,the jacket forced radially by the filling is broken into a large number of fragments,which have a radial velocity component in addition to a residual velocity component.Based on the mechanism of the PELE,it has been applied in a minor-caliber ammunition to attack gunships and cruise missiles in the field of anti-aircraft[2].Under the background of the urban war,the PELE is also used in a heavy caliber ammunition to cause large opening damage to buildings.The size of the large opening is approximate 3-4 times diameter of the projectile[2].Therefore,the damage mode of the PELE penetrating a reinforced concrete(RC)target is different from that of the PELE penetrating the metallic target,which has also aroused extensive attention.

For the research of the PELE penetrating the metallic target,a series of tests including filling materials,target thicknesses,target materials and impact velocities,were conducted by Paulus.et al.[3]to study how these factors affect formation and distribution of fragments,and then a theoretical model to predict the maximum radial velocity of the fragment was established based on the elastic wave theory.Zhu.et al.[4]derived an improved model by considering a shear resistance of the metallic target to predict the maximum radial velocity and residual velocity of the fragment.In Ref.[5],we proposed a model,which considered the influence of the radial motion of the jacket on the maximum radial velocity of the fragment.Verreault[6]found that above analytical models all used the approximate elastic wave theory,and the multiple wave interaction at the filling/target interface were neglected.Therefore,he provided an improved analytical model by using the Rankine-Hugoniot relation of the shock wave to determine the material state.However,this improved model did not consider the effect of the axial compression force of the jacket on the fragment radial velocity.To evaluate the effect of the axial compression force,Fan.et al.[7]developed a theoretical model using the shockwave and energy conservation theories to estimate the maximum radial velocity of the fragment.

For the research of the PELE penetrating the RC target,we attempted to optimize the structure of the PELE by engraving prefabricated grooves on the jacket[8].Based on test results,it was found that the prefabricated grooves added on the jacket is beneficial to the opening behavior of the PELE penetrating the RC.However,the influence of some projectile parameters(such as jacket material strength and filling material)on opening behavior of the PELE penetrating the RC target,until now,has not been studied.Therefore,the goal of this investigation is to analyze the influence of the projectile parameters on the opening behavior,and to provide a model to predict the opening diameter.

In the present paper,the important influence factors of the PELE projectile were determined based on the dimensional analysis.Three groups of tests were performed to analyze the influence law of factors(jacket material strength,jacket wall thickness and filling material strength)on the opening diameter.Combing with the method of numerical simulation,the influence law of the prefabricated groove number was also studied,and the influence mechanism of these factors on the opening diameter was further analyzed in detail.Based on the test and simulation results,a formula was established to predict the opening diameter of the PELE projectile impacting the RC target,whose influence parameters were considered in a dimensionless way.

2.Dimensional analysis

Some analysis and assumptions about the PELE projectile penetrating the RC target were made as follows:(i)the PELE projectile normally penetrated the RC target,without considering an impact angle of the projectile;(ii)Both the projectile and target produced large deformation and damage during the whole penetration process,so the effect of elastic parameters of the projectile and target,such as Young’s modulus and Poisson’s ratio,was not considered;(iii)The reinforcement ratio of the target plate is about 3%,so the reinforcing bars were only considered to ensure the integrity of the concrete target,and the influence of the reinforcing bars on the local response of the target was ignored.Therefore,the main influence factors of the projectile and target were listed as follows.

The influence factors of the projectile:mass(m),impact velocity(v0),outer diameter(d1),inner diameter(d2),effective length of the jacket(or filling length)(l),prefabricated groove number(N),jacket material strength(Y1),filling material strength(Y2).The projectile structure is drawn in Fig.1.

The influence factors of the target:strength(fc)and thickness(hc).The target structure is shown in Fig.2.For the PELE projectile penetrating the RC target,the opening diameter(D)of the RC target was generally determined by:

Based on the dimensional analysis theory,the mass(M),length(L)and time(T)were selected as the basic dimensions to analyze the influence factors of the projectile and target,as listed in Table 1.Moreover,the parameters ofm,d1andfcwere selected as three basic references,so Eq.(1)was transformed into the following dimensionless form:

The effect of the geometric and material parameters of the projectile on the opening behavior are the focus of this study.From Eq.(1),the four dimensionless parameters:mainly include the geometric and material parameters of the projectile:d1,d2,N,Y1andY2.Therefore,the parameters ofm,v0,d1,l,fcandhcrelated towere set to fixed values in this study,and Eq.(2)could be decomposed into:

Fig.2.Schematic diagrams of RC target.

Table 1Dimensions of projectile and target influence factors.

whereKwas constant.Now,the emphasis was to analyze the influence of the projectile factors on the opening behavior,and then to confirm the functional expression of Eq.(3).Thus,tests and simulations of the PELE projectile penetrating the RC target were carried out in Section 3 and Section 4.

3.Penetration test

3.1.Projectile structure

As shown in Fig.1(a),a PELE projectile was mainly composed of three parts:a projectile ogive,filling and jacket engraved with prefabricated grooves.The projectile ogive was designed to ensure the flight stability.Fig.1(b)illustrates the dimensions of the projectile,including the outer diameter(d1),inner diameter(d2),total length of the jacket(L),effective length of the jacket(or filling length)(l),prefabricated groove length(l1)and prefabricated groove number(N).It should be pointed out that the parameterNwas able to be represented by an angle of monolithic jacket(θ),because the prefabricated grooves were uniformly distributed on the jacket.In all tests and simulations of this study,the values of the outer diameter,effective length and prefabricated groove length and prefabricated groove number of the projectile were fixed,which wered1=105 mm,l=300 mm,l1=240 mm,andN=6,respectively.The materials and other parameters of the projectile were introduced in detail in Section 3.3.

3.2.Target

The dimensions of the RC target are illustrated in Fig.2,where both length and width of the target were 2.0 m,and the thickness was 240 mm.The RC target was reinforced by a double-layer steel mesh with a spacing of 200 mm,whose mesh size was 200 mm×200 mm,and the diameter of the reinforcing bar was 12 mm.The unconfined cylinder compressive strength(fc)of the concrete was approximately 42 MPa.

3.3.Setup

The schematic diagram of the PELE projectile penetration test is depicted in Fig.3.The projectile was fired by a tank gun with 105 mm caliber and impacted normally on the RC target,which wasS1=180 m away from the muzzle.The velocity(v35)of the projectile was measured by a speed measurement at distance ofS2=34 m from the muzzle.Due to the influence of air resistance on the projectile velocity in flight[9],an attenuation coefficient of the projectile velocity was determined to bew=0.217[(m/s)/m].Therefore,the impact velocity(v0)of projectile was calculated by Eq.(4)based on the value ofv35.In this study,14 shots were carried out to study the influence of these projectile factors.

3.3.1.Tests of jacket material strength

Six shots with different material strength of the jackets were conducted to study the effect of the material strength on the opening behavior.In this group of tests,Nylon 1010(compressive strength was about 52 MPa)was selected as the filling material,and its inner diameter wasd2=80 mm.A relationship between hardness and strength of metallic material has been established.Therefore,the material strength of the jacket was determined by different heat treatment methods to achieve different hardness values.The jackets of the test schemes from ST-1 to ST-3 chose 50SiMnVB as the jacket material,whose Rockwell hardness values(Hr)obtained by heat treatment were 43HRC,41HRC,36HRC,36.4HRC,27.3HRC and 27HRC,respectively.According to Ref.[11],these Rockwell hardness values were translated to Vickers hardness values(Hv),as shown in Eq.(5),and then the strength values(Y1)were calculated by a widely used relationship that strength is triple relation of Vickers hardness[12,13],listed in Table 2.It should be noted that the plastic hardening properties of 50SiMnVB material were basically consistent under different hardness conditions[10],so the yield strength of 50SiMnVB was considered to be main variable.

3.3.2.Tests of jacket wall thickness

Four shots with different wall thickness of the jacketswere performed to analyze the effect of wall thickness on the opening behavior.Detailed test conditions and parameters of this set of test schemes(TH-1 and TH-2)are listed in Table 2,whereBof the two sets of test schemes were 10.5 mm(d2=84 mm)and 11.5 mm(d2=82 mm),respectively.It should be noted that the consistency of the projectile mass was ensured in about 15.5±0.2 kg by adjusting the total length(L)of the jacket.The jacket and filling material adopted 50SiMnVB and Nylon 1010,respectively.The same heat treatment method as test scheme ST-2-1 was used to reach 1018 MPa of the jacket material.

Fig.3.Schematic diagram of the projectile penetration test.

3.3.3.Tests offilling material strength

Based on above two sets of tests,two other kinds of materials(polyethylene and Nylon 66)were selected as filling materials to analyze the effect of filling material strength,because Nylon1010[14],polyethylene and Nylon 66[15]possessed the similar acoustic wave velocity(1000-1300 m/s)and Poisson’s ratio(0.42-0.45)parameters,but they varied in the material strength(listed in Table 2).50SiMnVB with material strength 1018 MPa was also applied to be the jacket material.Structure sizes of the projectile were exactly the same as these in Section 3.3.1.The results of this group of tests would be compared with schemes ST-2(selecting Nylon 1010 as filling material)to study the effect of filling material properties on opening behavior of PELE penetrating RC target.

4.Numerical modeling and simulation

4.1.Modeling of PELE penetrating RC target

Based on the test conditions,finite element modeling of the PELE penetrating the RC target was carried out using the LS-DYNA software.The model was axisymmetric,and a 1/2 model was established to save calculation time,as shown in Fig.4.In the model,the projectile ogive was designed to ensure the flight stability,and this part had little effect on the test results,so it was not modeled.A separate modeling method was used to separately build the concrete and reinforcing bars,which interacted by defining a“tied surface-to-surface”contact mode.It should be noted that the section of the reinforcing bars was a square with a side length of 10.6 mm(determined by the section area)in the simulation model,which facilitated the meshing of the concrete and reinforcing bars.To save computational time and ensure precision simultaneously,the local mesh refinement was employed for the concrete target.The minimum element size for the target was selected to beenear the impact location,which was enlarged to doubleeafter a distance beyond 400 mm from the target center.The reinforcing bars were meshed to be half ofe,and the projectile was modeled using an element size ofe.A mesh convergence study was performed(taking ST-2 as an example),and Fig.5 shows the influence of element size(e)on the simulation results.As the element size decreases,the opening diameter increases,and the simulation results are stable ate=5 mm.It means that the mesh starts to converge ate=5 mm,which was used in this study.Three-dimensional elements(Type-164 in LS-DYNA)were used for this model,and an eroding surfaceto-surface contact model was employed to define the contact behavior between the projectile and the RC target.According to the test results,the element sizes,element types and contact modes were determined by trial and error,and the simulation accuracy were described in Section 5.

Fig.4.Schematic diagram of simulation model.

Fig.5.Verification of the mesh convergence.

4.2.Material models

4.2.1.Material model of the jacket

Johnson-Cook model[16]was employed to model the flow stress behavior of 50SiMnVB.Since the impact tests were conducted at room temperature,the Johnson-Cook model describes the flow stress as:

4.2.2.Material model of thefilling

Referred to previous research[15],Nylon 66 material was studied under different strain rates,and a one-dimensional constitutive model has been established to characterize the material.The model contains viscoelastic and viscoplastic components and it was given by:

where,εeandεpare the elastic and plastic strain,respectively;E1(˙ε)is defined asE1(˙ε)=p·[exp(lg(˙ε/˙ε0))q-1],τs(˙ε)is defined as lgτs(˙ε)=lgα-βlg(˙ε/˙ε0),and others are material parameters.

Table 3Parameters for constitutive model of Nylon 66.

According to the mechanical properties of Nylon 66 under different strain rates,these parameters were determined and listed in Table 3.

To investigate the effect of hydrostatic pressure on yield responses of polymers,a Drucker-Prager model was used to describe the semi-crystalline polymers’yield behavior by:

whereα0andα1are material parameters,J2andI1are second deviatoric stress invariant and the first invariant of stress,respectively.Since the basic mechanical properties(density,Young’s modulus and yield strength,etc.)of the Nylon 66 material used in the tests were similar to these of the literature[17],the test results of Nylon 66 under complex stress state were used to confirm the material parameters ofα0andα1.There the model was implemented employed into LS-DYNA to characterize the filling material,and the parameter of yield strength was only adjusted according to the filling materials.

4.2.3.Material model of the target

K&C model was employed to present the behavior of concrete material[18].Three independent strength surfaces of compressive meridians,namely,the initial yield strength surfaceΔσy,the maximum strength surfaceΔσmand the residual strength surface Δσrwere defined by

Fig.6.Quasi-static compressive response of concrete.

Table 4Parameters of K&C model for 42 MPa concrete.

whereai,aiyandaif(i=0,1,2)are constants determined from a suitable set of triaxial compression test data.fyc=0.45fc,fcandTare the initial yield strength,unconfined uniaxial compressive strength and tensile strength,respectively.pis the hydrostatic pressure,and ψdenotes the tensile-to-compressive meridian ratio.

The current failure surface of the model was determined as follows:

whereJ2is the second deviatoric stress invariant,r’is the ratio of the current meridian to the compressive meridian,ηis the yield scale factor related to the damage functionλ,which was determined by:

wherekdis the internal scalar multiplier,εVis the volumetric strain,andεV，yieldis the volumetric strain at yield.The damage constantsb1,b2,andb3usually have different values for compression and tension with considering different damage evolutions of concrete.According to the Keyword Manual of LS-DYNA,all parameters of the K&C model can be generated automatically by LS-DYNA software using the unconfined compressive strength(also including density 2300 kg/m3).Therefore,a series of uniaxial compressive experiments of the concrete specimens directly taken from the test target were conducted,and the stress-strain curves are shown in Fig.6.The compressive strength(fc)of the target was approximately 42 MPa,so the parameters of K&C model for 42 MPa are listed in Table 4.Since there were no available element erosion criteria in the K&C model,the maximum and minimum principal strain criterion was adopted by using*MAT_ADD_EROSION in LS-DYNA.In this manuscript,the maximum and minimum principal strain values are MXEPS=0.35 and MNEPS=-0.70,respectively.

Table 5Parameters for constitutive model of reinforcing bar[16].

Fig.7.Schematic of typical opening model.

4.2.4.Material model of the reinforcing bar

For the reinforcing steel bar,it is a standard material,which can be divided into three grades(HRB 335,HRB 400 and HRB 500)according to its yield strength.In our tests,the reinforcing steel bars with grade of HRB 400(i.e.yield strength 400 MPa)were used in the targets.In the literature[19],they used the same grade of reinforcing steel bars as ours,and the Plastic-Kinematic model(Mat_003 in LS-DYNA)was employed to calculate the yielding and plastic flow of the bars.The model assumes the equivalent flow stressin the following form:

5.Results and analysis

The opening diameter of the RC target was recorded and considered as an important index to analyze the opening behavior of PELE penetrating RC target.A typical opening mode of PELE penetrating RC target is shown in Fig.7.The opening shape is approximately an ellipse,so the maximum(D1)and the minimum(D2)sizes of the opening were measured after tests.In this study,an equivalent diameter(D)was introduced and calculated by Eq.(13).It means that the ellipse shape of the opening was equivalent to a circle shape.

In these 14 shots,the impact velocities fluctuated significantly(the maximum:672.8 m/s and the minimum:640.3 m/s,relative difference:5.1%),which affected the analysis of the test results.In order to exclude the effect of impact velocity(or initial kinetic energy),the equivalent diameter(D)of the opening was transformed into an equivalent diameter under relative kinetic energyas shown in Eq.(14).The impact velocities of these tests fluctuated around 670 m/s,so the impact velocityvr=670 m/s was taken as a reference.

To analyze the damage mechanism of the PELE projectile penetrating the RC target,serial sets of simulations of PELE penetrating RC target were conducted,and a meaningful phenomenon was discovered.When a PELE projectile impacted on an RC target instantly,the acceleration of the projectile reached about 4.0×104g,as shown in the process one of Fig.8(a).For the process two of Fig.8(c),the jacket was radially expanded by the filling due to Poisson effect,which led to an increase on the axial resistance(acceleration)of the projectile.With the penetration going on and the axial resistance growing,the jacket began to occur bending deformation,and presented a double bending deformation mode,as shown in Fig.8(b).For the deformation mode of the jacket,two bending angles(αandβ)and an opening mouth(nd1)were defined to measure the bending deformation of the jacket.At the time of the process three,the acceleration arrived at the peak,and the penetration depth was about 125 mm.After that the acceleration dropped gradually due to appearance of shear failure in the RC target,such as the process four and five.Process six illustrated the damage mode of the target,including front crater,rear crater and opening.The axial position of the opening was approximately 106-136 mm from the front side of the target.It was found that the position of the opening was very close to that of the maximum acceleration,which suggested that the opening diameter was mainly determined by the jacket bending deformation(or the degree of opening mouth)at the position of the maximum acceleration.Therefore,the jacket bending deformation(or the degree of opening mouth)at the position of the maximum acceleration was selected as an index to analyze the damage mechanism of the PELE projectile penetrating the RC target in this study.

5.1.Influence of the jacket material strength

Fig.9 shows the equivalent diameters and damages of the RC targets of the test schemes(ST-1-ST-3).According to Eq.(14),the above equivalent diameters were transformed into(the equivalent diameter of the opening under relative kinetic energy)to exclude the influence of the impact velocity,and Fig.10(a)shows the-Y1relationship.By comparing the test and simulation results,the differences ofwere 6.8%,0.8% and 7.8% to different jackets which were 810 MPa,1018 MPa and 1225 MPa,respectively.Therefore,the simulation results had a good agreement with the tests,and the simulation model was used to investigate the influence ofY1on the opening diameter in a greater range ofY1=600-1400 MPa.It was found thatfirst increased with the increase ofY1,and then decreased.

Fig.9.Equivalent diameters and damages of the RC targets in test schemes:ST-1-ST-3.

From the recovered jackets of the tests,as shown in Fig.10(a),the bending deformation of the jackets got larger with the decrease ofY1,and the jacket which wasY1=810 MPa produced a break-off.The jacket deformation affected the opening diameter.Based on above simulation analysis,it was also verified that the opening diameter was related to the jacket bending deformation at the position of the maximum acceleration,so the jacket deformation was obtained at differentY1to analyze the infulence mechanism ofY1onas exhibited in Fig.10(b).Since the decrease ofY1weakened the anti-bending capability of the jacket,the bending angleβ displayed a dramatic increase withY1dropping.In addition,asY1decreased from 1400 to 1018 MPa,there was also a significant rise of the bending angleα,and the growth of both bending angleαand βled the degree of the opening mouth to expand from 2.21(d1/2)to 2.88(d1/2).However,asY1continued to fall,the decrease of the bending resistance of the jacket madeαexceed 90°,and a greater bending deformation(curling deformation)appeared on the jacket which wasY1=810 MPa,but the curling part were eroded due to large deformation.In the condition ofY1=600 MPa,the curling deformation was not predicted by the simulation model,which was taken placed by more serious erosion.It indicated that the curling deformation brings the degree of opening mouth to descend from 2.88(d1/2)to 2.53(d1/2).The relationship betweenand the opening mouth degree at the position of the maximum acceleration,was a completely positive correction.Therefore,Y1mainly affectedby changing the bending resistance of the jacket.With the decrease ofY1(or the bending resistance),the increase of the bending angleα(less than 90°)andβwas beneficial tobut the growth ofαmore than 90°was bad for

5.2.Influence of the jacket wall thickness

Fig.11 shows the equivalent diameters and damages of the RC targets of the test schemes(TH-1-TH-2).In order to exclude the influence of the impact velocity,the values ofD(the equivalent diameter)were also transformed into(the equivalent diameter of the opening under relative kinetic energy),and-Brelationship was illustrated in Fig.12(a).There was an obvious fluctuation of(about 18%)in schemes TH-1,so the average values ofof every scheme were calculated as 352 mm,377 mm,and 407.5 mm,respectively.By contrastingof the tests and simulations,the maximum errors between every scheme were 14.9%,3.3%,and 0.8%to jackets which were differentB(10.5 mm,11.5 mm,and 12.5 mm),respectively.However,the error of the simulation result and the mean value of schemes TH-1 was approximate 4.3%.It suggested that the simulation results correspond well with that of the tests,and a wider range of simulations were also conducted to observe the influence ofB(10.5-13.5 mm)onIt was found thatfirst increased with the increase ofY1,and then decreased.

Fig.10.Influence of Y1 on PELE penetrating RC target:(a)-Y1 relationship;(b)deformation of jackets with different Y1.

Fig.11.Equivalent diameters and damages of the RC targets in test schemes:TH-1-TH-3.

Fig.12.Influence of B on PELE penetrating RC target:(a)-B relationship;(b)deformation of jackets with different B.

Fig.13.-θrelationship of PELE penetrating RC target.

From the recovered jackets of the tests,as shown in Fig.12(a),the decrease ofBcaused the jackets to deform from bend to breakoff,which was the same as the influence ofY1on the jacket deformation and the opening diameter.Thus,the jacket bending deformation at the position of the maximum acceleration was also acquired at differentBto analyze the influence mechanism ofBonand Fig.12(b)presents the jacket deformation here for variousB.It is wellknown thattheanti-bending capability of the jacket is relatedtoB,so the bending angleβ(rangingfrom 10°to 22°)got bigger with decrease inB.The bending angleαgrewup from55°to 72°,whenBvaried from 13.5 mm to 12.5 mm.However,asBcontinued to drop(from 11.5 mm to 10.5 mm),the jackets behave curling deformation and more serious erosion during simulation.The variation of both bending angleαandβcaused the degree of opening mouth(or)firstly to rise,and then to fall.The trend ofBaffecting the bending deformation,was similar to that ofY1,becauseBalso had an important influence on the anti-bending ability of jacket.Therefore,Bmainly affectedby changing the bending resistance of the jacket as well.When the value ofB(or the bending resistance)decreases,the increase of the bending angleβ was always beneficial toHowever,the growth ofαabove 90°was bad for

Fig.14.Equivalent diameters and damages of the RC targets in test schemes:FL-1-FL-2.

Fig.15.Influence of Y2 on PELE penetrating RC target:(a)-Y2 relationship;(b)deformation of jackets with different Y2.

Fig.16.Schematic diagram of force analysis of projectile.

Table 6Dimensions of projectile influence parameters.

5.3.Simulation analysis of the influence factorθ

Based on the analysis ofY1andB,both of them affectedby changing the bending resistance of the jacket.It is well known that the angle of monolithic jacket was one of the influence factors on the bending resistance of the jacket,so a group of simulation models(schemes:AN-45 and AN-90)with different angle of monolithic jacket were established,includingθ=45°,60°,90°(corresponding prefabricated groove numberN=8,6,4).Other structure parameters of these simulation models wered1=105 mm,d2=80 mm,l=300 mm,andl1=240 mm,and material strength values of the jacket and the filling were 1018 MPa and 52 MPa,respectively.Mass and initial velocity of the simulation models were 15.6 kg and 670.0 m/s,respectively.

Fig.18.Verification of M0.

Fig.13 shows simulation results ofand the corresponding jacket deformation at the position of the maximum acceleration.It can be seen that the bending angleβandαincreased with the decrease ofθ,while the curling part were eroded due to large deformation.Therefore,the degree of the opening mouth reached a maximum of 2.88×(d1/2)forθ=60°,and the degree of opening mouthreduced to be 2.64×(d1/2)(forθ=45°)and 2.39×(d1/2)(forθ=90°).The variationofwasalsothe same as that ofthe opening mouth degree.It suggested that the influence of the jacket anti-bending ability onwas verified again.

Fig.17.Plastic limit bending moment:(a)relationship between and d1/d2;(b)relationship between andθ.

Table 7Values of dimensionless parameters Q.

5.4.Influence of the filling material strength

Fig.14 shows the equivalent diameters and damages of the RC targets of the test schemes(FL-1-FL-2).In order to exclude the influence of initial kinetic energy,the values ofD(the equivalent diameter)were also transformed into(the equivalent diameter of the opening under relativekineticenergy),andthe-Y2relationshipis plotted inFig.15(a),wherethe average valuesofof every scheme were 349 mm,407.5 mm and 417.4 mm in turn.By contrastingof the test and simulation,the maximum errors of every scheme were 9.9%,0.8% and 0.6% to the fillings which were differentY2(10.5 mm,11.5 mm and 12.5 mm),respectively.It proved that the simulation results correspond well with that of the tests,and a series of simulations were performed to observe the influence of a greater range ofY2(200-300 MPa)onThe simulation results are plotted in Fig.15(a).WhenY2continued to increase to 200 MPa and 300 MPa,Ddescends gradually.

Fig.19.Opening diameter model:(a)relationship between and Q;(b)relationship between and G.

Table 8Values of dimensionless parameters G.

To explain the influence mechanism ofY2onthe jacket and the filling deformation at the maximum acceleration are obtained and illustratedinFig.15(b).It wasfound that thebendingangleβ got higherfrom6°to15°witha decreaseinY2.According to the study of the plastic deformation theory in impact[20],the plastic deformation of cylindrical projectile was inversely proportional to the material yield strength(A1/A0=1+ω+whereω=ρv20/Y2),so the decrease ofY2led to a larger expansion of the filling.There should be a larger degree of the opening mouth due to the larger expansion of the filling,but the relationship between the degree of the opening mouth and the bending angleβwas not a positive correlation.Therefore,there must be the other influence factor.From the failure of filling,the residual length of the filling got shorter withY2decreasing.For the filling with 52 MPa and 28 MPa,quick erosion of the filling during penetration reduced the expansion effect of the filling on the jacket,so the jacket also showed a small bending deformation.However,for the filling with 200 MPa and 300 MPa,the filling occurred minor erosion but deformed small,which caused the jacket to bend small.Based on above analysis,Y2mainly affectedby controlling the expansion and erosion of the filling.

6.Opening model

6.1.Determination of plastic limit bending moment

Based on the analysis of the influence ofY1,BandθonD,the three factors all affectedby changing the bending resistance of the jacket.Therefore,a parameter,plastic limit bending moment(M0),was quoted to characterize the bending deformation of the jacket.In this study,the jacket of the PELE projectile was engraved with prefabricated grooves,so the jacket should be assumed to consist of several monolithic jackets which alone penetrate the RC target,as shown in Fig.16.For the structure of the monolithic jacket,plastic limit bending moment is mainly determined by its outer diameter(d1),inner diameter(d2),angle of monolithic jacket(θ)and material strength(Y1).These factors have been proved to have a great influence on the opening behavior of PELE penetrating RC target by tests and simulations.Therefore,it was important to determine the relationship between the plastic limit bending moment and these factors,and to know how the plastic limit bending moment affected the opening behavior.

For the structure of the monolithic jacket,its bending deformation is no longer a pure bending state structure under the action of force,especially at a larger angle(θ)[21],so it is impossible to accurately predict the plastic limit bending moment by the theory of plastic mechanics.In this study,the plastic limit bending moment relationship was established by numerical simulation and dimensional analysis.Referring to dimensional analysis theory,mass(M),length(L)and time(T)were selected as basic dimensions to carry out dimensional analysis of the influence parameters,as listed in Table 6.Taking the plastic limit bending moment as an objective parameter,the general functional relationship betweenM0and above factors was established.

After simplification,it was gotten:

Here,it was assumed that the effects ofandθonM0were decoupled,so Eq.(16)was expressed to

To determine the two unknown formulas in Eq.(17),three-point bending simulation models of the monolithic jacket were established to confirm the plastic limit bending moment values under different conditions.Fig.17(a)and Fig.17(b)show the influence of the ratio of inner diameter to outer diameter and the angle on the plastic limit bending moment,respectively.Based on these two sets of data,Eq.(17)was determined as follows(0.65≤d1/d2≤1.00,0°≤θ≤90°,750 MPa≤Y1≤1750 MPa):

To verify the accuracy of Eq.(18),three-point bending models of the monolithic jacket were simulated under other different conditions,such as a monolithic jacket(d2=75 mm,θ=60°)with different material strength.Fig.18 illustrates these simulation results,which were compared with the plastic limit bending moment values calculated by Eq.(18).It was found that the expression of the plastic limit bending moment was in good agreement with the simulation results.

6.2.Establishment of opening model

Based on Eq.(18),the relationship between the plastic limit bending moment(M0)and the influence factors of the projectile(d1,d2,N,Y1)were built.Therefore,the influence factorsd1,d2,NandY1in Eq.(3)were able to be replaced byM0,and taking as the characteristic parameter,Eq.(3)was simplified to

From Eq.(19),the dimensionless opening diameterwere affected by two dimensionless factorsThe first term in Eq.(4)may be interpreted as the ratio between a resistance force of the targetand a bending force of the jacketM0/d1.The second term was noted as a ration of the filling material strength and target strength,which have been widely used in penetration and perforation mechanics[22].It was assumed that the two dimensionless factors on the opening diameter were decoupled,andand the two dimensionless parameters:satisfied a quadratic functional relationship,so Eq.(19)was expressed as:

To determine the two unknown formulas in Eq.(20),the results of tests and simulations were transformed into dimensionless values.Firstly,M0of the jacket under different jacket wall thickness,jacket material strength and angle of monolithic jacket were calculated,and then dimensionless plastic limit bending momentwere also calculated according tofc=42 MPa andd1=105 mm.Both values of the two parameters and dimensionless opening diameterare listed in Table 7.Based on Eq.(20),the relationship betweenQandis plotted in Fig.19(a).It was found that the test data were very well represented by the proposed two dimensionless parameters,and they approximately satisfied quadratic function relationship.Under the condition ofl=300 mm,d1=105 mm,fc=42 MPa andY2=52 MPa,fwas assumed to be 1,thus the expressions ofQandwere determined by unilinear fitting to

Based on the results of tests,dimensionless filling material strengthand dimensionless opening diameterwere calculated(listed in Table 8),which are plotted in Fig.19(b).Analysis of Fig.19(b)shows that the relationship betweenfc/Y2andapproximately also satisfied a quadratic function,so the unknown formulawas determined by fitting to

whereQ0=3.907.

From Eq.(21)and Eq.(22),the dimensionless opening diameter of Eq.(20)was confirmed.It should be point out that the opening diameter model was obtained within a certain range(3.00≤Q≤6.25 and 0.2≤G≤1.4),so the applicability of this model in other ranges need further verification.

7.Conclusion

Based on the test and simulation study of PELE projectile penetrating RC target,the influence of projectile structure and material parameters on the opening behavior was analyzed.The main conclusions drawn from the test and simulation results are as follows:

(1)A dimensional analysis was conducted on the influence of the PELE projectile and target factors on the opening behavior.In this study,the impact velocity and target parameters(strength and thickness)were fixed values,thus the functionKwas considered to be constant.was determined to be the important influence projectile factors.

(2)From the test and simulation results,including opening diameter and bending deformation of the jacket,the influence factors:the jacket material strength(Y1),jacket wall thickness(B)and angle of monolithic jacket(θ)all affected opening diameterby changing the bending resistance of the jacket.As values of the three factors decrease,the jacket deformed from small bending deformation to large one,and then to curling deformation.This causedto first increase with the decrease of these three factors,and then decreased.

(3)From the test and simulation results,the influence factor:thefillingmaterialstrength(Y2)affected theopeningdiameterbycontrolling theexpansion and erosionofthefilling.For the filling with 200 MPa and 300 MPa,the filling occurred minor erosion but deformed small,which caused the jacket to bend small.For the filling with 52 MPa and 28 MPa,quick erosion of the filling during penetration reduced the expansion effect of the filling on the jacket,so the jacket also showed a small bending deformation.This causedto first increase with the decrease ofY2,and then decreased.

(4)A parameter,plastic limit bending moment(M0)which was related to the jacket structure and material parameters,was quoted to characterize the bending deformation of the jacket.For the structure of the monolithic jacket,the general functional relationshipM0=f(d1，d2，θ，Y2)was established,and the function was confirmed based on three-point bending simulations and dimensional analysis.

(5)A formula was built to predict the opening diameter of the PELE projectile impacting the RC target based on dimensional analysis.It has been shown that the dimensionless opening diameterwas dependent on two dimensionless parameters

Declaration of competing interest

We declare that we have no financial and personal relationships with other people or organizations that can inappropriately influence our work,there is no professional or other personal interest of any nature or kind in any product,service and/or company that could be construed as influencing the position presented in,or the review of,the manuscript entitled,“Effect of projectile parameters on opening behavior of PELE penetrating RC target”.

Acknowledgement

The author(s)disclose the receipt of the following financial support for the research,authorship,and/or publication of this article:The project was supported by the National Natural Science Foundation of China(Grant No:11472008,11772160,11802141),the Opening Project of State Key Laboratory of Explosion Science and Technology(KFJJ18-01M),Beijing Institute of Technology.