Opening behavior of PELE perforation on reinforced concrete target

Li-zhi Xu,Jiang-bo Wang,Xiao-dong Wang,Wen-ke Ren,Tian-xing Zhang,Zhong-hua Du,Guang-fa Gao

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

Keywords:PELE Penetration Reinforced concrete Dimensional analysis

ABSTRACT Impact velocity(v0),target strength(fc)and target thickness(hc)are important factors affecting opening damageof PELE penetration into RC target.In this paper,based on the three influence factors of v0,fc and hc,experimental and numerical simulation studies on PELE penetration into RC target were carried out.The study results show that:(1)Since interaction force(or penetration resistance)between projectile and target is positively correlated with v0 and fc,with the increase of v0 and fc,deformation mode of jacket is changed from small bending deformation to large bending deformation and then to curling deformation.Therefore,the variation of jacket deformation mode causes opening diameter of RC target to increase first and then to decrease.It is found that the two factors approximately satisfy a quadratic function relationship,respectively.(2)For PELE projectile penetrating RC targets with thickness of 80-400 mm,the opening diameter of six sets of RC targets grows from 240 to 500 mm,and hc withD approximately satisfy a linear relationship.(3)Based on the above study results,the relationship between two dimensionless parametersand dimensionless opening diameterwas determined.Combined with the results of previous research,a dimensionless opening diameter modelwas established.By tests verified,the test results are all within±10% error of the theoretical model,which verifies the accuracy of the model.

1.Introduction

A penetrator with enhanced lateral effect(PELE)is a new type of penetrator consisting of a low-density material in its core(or called a filling)surrounded by an outer jacket with high-density material.Once perforation of a metallic target,the jacket forced radially by the filling is broken into a large number of fragments[1].However,when a PELE projectile penetrates a finite thickness reinforced concrete(RC)target,the projectile can cause large opening damage to the target[2].The damage effect is significantly different from the PELE projectile penetrating a metal target.The principle of PELE penetration into concrete target is shown in Fig.1.When the PELE projectile impacts on the target instantly,huge pressure is generated between the projectile and the target,and front crater damage is also produced on the target(Fig.1(a)).The filling material subjected to the impact pressure is radially expanded due to the Poisson effect,and the jacket also undergoes radial expansion under its action,which results in progressively larger opening damage to the target(Fig.1(b)).With the further interaction between the PELE and the target,the filling continues to expand and deform,and the jacket undergoes bending deformation under the double action of the filling expansion force and target resistance.At the moment,the target begins to exhibit shear damage,so that the target resistance gradually decreases(Fig.1(c)).Due to the occurrence of the shear failure,an impact plug of the target is gradually formed,and the projectile pushes the plug to move together.Finally,the target is perforated to form a failure mode including front crater,opening and back crater(Fig.1(d)).

Fig.1.Process of PELE penetrating reinforced concrete target.

At present,the research on the mechanism of PELE is mainly focused on its penetration into metal target[3-7],but there are few reports on its penetration into RC target.Two materials(nylon and rubber)were chosen as the filling of PELE,and the influence of the two materials on penetration of PELE into concrete target was studied by Fan et al.[8].It was found that the damage area of targets and the number of fragments were larger for the filling of PELE with nylon material than that with rubber material.Ye et al.[9]simulated the damage of RC targets subjected to PELE projectiles at different velocities.The simulation results showed that the opening size of RC targets first increased and then decreased,with the increase of the impact velocity.Our research group first optimized the structure of the PELE projectile[10,11],and set prefabricated grooves(pre-grooves)with different depths on the jacket.The study results show that the PELE projectile with pre-grooves causes greater opening damage to the RC target than that of the original structure PELE(without pre-grooves),and the projectile with through pre-grooves produces the best opening damage to the target.Therefore,based on the through pre-grooves structure of the PELE projectile,the main influence factors of the projectile penetrating RC target on the opening size(D)were determined in previous work[12],including:mass(m),impact velocity(v0),outer diameter(d1),inner diameter(d2),effective length(or filling length)(l),number of through pre-groove(N),jacket compressive strength(Y1)and filling compressive strength(Y2)of the projectile;target compressive strength(fc)and thickness(hc)of the target.Thus,the opening size(D)of PELE penetrating RC target was expressed as follows:

Based on the method of dimensional analysis,Eq.(1)was transformed into the following dimensionless form:

To analyze the influence of the jacket strength(Y1),jacket wall thickness(B),number of through pre-groove(N)and filling strength(Y2),related research was conducted under condition of specific impact velocity and target parameters.Therefore,Eq.(2)was converted intowhereis regarded as constant[12].Aiming at the influence ofY1,B,NandY2on the opening damage of the PELE penetrating RC target,it was found that bending strength of jacket was positively correlated with the three factors ofY1,BandN.As the bending strength of jacket declined,the bending deformation of jacket gradually enlarged,and then curling deformation occurred on the jacket,as shown in Fig.2a-c.The change in deformation mode of jacket caused the opening diameter of RC target to increase first and then to decrease.Based on the experimental and simulation results,two dimensionless infulence parameterswere determined,and a dimensionless opening diameter model was also established:

However,the application scope ofis very small due toregarded as constant.Therefore,the present paper aims to continue to study the influence factors of impact velocity(v0),target strength(fc)and target thickness(hc).The effect of dimensionless impact factorIand dimensionless thicknesson the opening diameter were analyzed by experiments and numerical simulations.Combined with the results of previous studies,a dimensionless opening diameter model considering the influence factors ofQ、G、IandHwere established,and its accuracy was verified by other test results.

2.Experimental details

2.1.Projectile structures and materials

In the present study,the PELE projectile is mainly composed of three parts:a projectile ogive,a filling and a jacket,as shown in Fig.3.Six through pre-grooves are evenly distributed on the jacket.The part of the projectile ogive is designed to ensure the flight stability of the projectile.The detail dimensions of the projectile are also displayed in Fig.3,which includes the outer diameter(d1=105 mm),inner diameter(d2=80 mm),total length of the jacket(L=400 mm),length of the filling(l=300 mm),length of the pre-groove(l1=240 mm)and number of the pre-grooves(N=6)(or angle of the single-piece jacketθ=60°).

The jacket was machined from 50SiMnVB steel,whose mechanical properties were determined by quasi-static compression test,and the stress-strain cure of the material is shown in Fig.4(a).The material possesses a yield strengthY1=1018 MPa and exhibits significant plastic hardening properties.Its plastic modulus is about 0.28 GPa.The filling was made of nylon 66 material,whose mechanical properties were studied under different strain rates.The stress-strain curves are plotted in Fig.4(b),and the material has a yield strengthY2=90 MPa under quasi-static conditions.

Fig.2.Deformation mode of jacket.

Fig.3.Projectile structures and dimensions.

Fig.4.Mechanical properties of materials:(a)uniaxial compression properties of 50SiMnVB;(b)uniaxial compression properties of nylon 66.

2.2.Experimental arrangement

Fig.5 illustrates the schematic diagram of the present penetration test,where the projectile was fired by a 105 mm tank gun and impacted normally on the RC target.A speed measurement system was placed between the tank gun and the RC target,and it wasS1=144 mm away from the RC target.Thus,the impacting velocity of the projectile(v35)was measured by the speed measurement system.According to the previous work[12],the impact velocity of the projectile(v0)was calculated by v0=v35-S1×0.217.In this study,14 shots were carried out to study the influence of these projectile factors.

Fig.5.Schematic diagram of the test arrangement.

Three sets of RC targets with different thicknesses were designed in this paper,whose structure diagrams are shown in Fig.6.Their length and width were all 2.0 m,and their thicknesses were 160,240 and 360 mm,respectively.The RC targets were reinforced by double-layer steel mesh,and each layer of the steel mesh was 20 mm away from the target surface.The mesh size was 200×200 mm,and the diameter of the reinforcing bar was 12 mm.The compressive strength(fc)of the target was determined to be about 42 MPa by uniaxial compressive experiment[12].Moreover,three sets of test schemes with different impact velocities were designed.Reinforced concrete targets with thickness of 240 mm were used in this group of tests.All the specific test schemes are listed in Table 1.

3.Numerical simulations

3.1.Description of the simulation model

In order to gain further insight onto the damage mechanism of the PELE on the RC target,the experiments and a wider range of the PELE penetrating the RC target were simulated by using a commercial LS-DYNA software.As the model of the PELE penetrating the RC target satisfied the axisymmetric structure,a 1/2 model was established to save calculation time,as shown in Fig.7.The simulation model is the same as the one which has been built and used in the previous work[12],so the same modeling methods and element sizes was employed again in this study.

A separated modeling method was applied to separately build concrete and reinforcing bars,which were interacted by defining“tied surface to surface”contact mode.Besides,an eroding surfaceto-surface contact was employed to define the contact behavior between the projectile and the RC target.Based on the result of the mesh convergence study[12],a local mesh refinement method was employed on the concrete target,and the specific element sizes were introduced:the minimum element size of the target was 5 mm near the impact location,which was enlarged to 10 mm after a distance beyond 400 mm of the target center.Moreover,the reinforcing bars were meshed to be 2.65 mm element size,and the projectile was modeled using size of 5 mm elements.3D solid element(Type-164 in LS-DYNA)was used for this simulation model.

Fig.6.Schematic diagram of reinforced concrete target.

Table 1Tests data of different target thicknesses and impact velocities.

Fig.7.Schematic diagram of simulation model.

Table 2Johnson-Cook model parameters of 50SiMnVB.

3.2.Material models

In the present study,the materials of the projectile and the target are the same as the previous work[12],so the identical material models and parameters of the projectile and the target were used again in this investigation,and a brief introduction aboutthose models and their related parameters were done as follows.The jacket(50SiMnVB)of the PELE projectile was characterized by a Johnson-Cook constitutive model(Mat_015 in LS-DYNA)[13],and its related parameters were listed in Table 2.A constitutive model was built to characterize the filling material[12,14],and it was implemented into a LS-DYNA software through UMAT.Its related parameters were listed in Table 3.

Table 3Umat41 model parameters of nylon 66.

A Karagozian & Case(K&C)concrete model(Mat_072R3 in LSDYNA)was used to present the behavior of the concrete material[15].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 and the density.Moreover,there were no available element erosion criteria in the K&C model,so the maximum and minimum principal strain criterion was adopted by using*MAT_ADD_EROSION in LS-DYNA.Therefore,some basic mechanical parameters and two principal strain values of the concrete are listed in Table 4.

The grade of the reinforcing steel bars(HRB 400)used in this study were the same as the literature[16],and a Plastic-Kinematic model(Mat_003 in LS-DYNA)was employed to calculate the yielding and plastic flow of the bars.According to the literature[16],the parameters of the model are determined and listed in Table 5.

4.Results and analysis

Previous research results suggest that the typical opening mode of the PELE penetrating the RC target was approximately an elliptical in shape,so the maximum size(D1)and minimum size(D2)of the opening were measured after the test.An equivalent diameterwas introduced to characterize the opening damage.Figs.8-14 illustrate the front and back damages of each group ofthe RC targets,and the equivalent diameters of the openings in each group of tests.

Table 4Mechanical parameters of concrete.

Table 5Plastic-Kinematic model parameters of HRB 400.

In order to exclude the influence of the impact velocity(or initial kinetic energy),the equivalent diameter of the opening was converted to an equivalent diameter under relative kinetic energy,which was defnied asThe impact velocities of these tests fluctuate around 670 m/s,so the impact velocityvr=670 m/s was taken as a reference.

4.1.Influence of the impact velocities

Fig.8 shows the test results of the PELE perforating the RC target,it is found that the increase of the impact velocity(v0)from 510 to 785 m/s,causes the equivalent opening diameterto increase from 300 to 452 mm.Meanwhile,combined with the fragments recovered from tests as shown in Fig.9,it is found that the deformation degree of the jacket also increases with the increase of the impact velocity,which indicates that the deformation degree of the jacket determines the opening diameter of the RC target plate.In this range of the impact velocity(500-800 m/s),increasing the impact velocity of the PELE projectile is conducive to improving the damage degree.

In order to study the effect of the impact velocity on the damage degree in a larger impact velocity range,six groups of numerical simulation were carried out in the range of 400-1000 m/s.Fig.9 illustrates the relationship between the impact velocity and the opening diameter,and the comparison between the simulation and test results.Firstly,the deformation degree of the jackets was obtained from the numerical simulation under the condition ofv0=510,665 and 785 m/s,respectively.It is found that a higher impact velocity makes the jacket deform larger,and the simulation results of the jacket deformation mode and quantity basically coincide with that of the corresponding test results.Secondly,under the condition of the three kinds of impact velocities,the simulation results of the opening diameter are 327,415 and 437 mm,whose relative differences with the corresponding test results are 10.7%,1.7% and 5.0%,respectively.It demonstrates that the simulation results are in good agreement with those of the tests,and the simulation model is able to simulate the PELE penetration into the RC target.Finally,when the impact velocity improves from 400 to 785 m/s,the opening diameter enlarges from 277 to 437 mm.However,as the impact velocity continues to increase to 1000 m/s,the opening diameter of the target decreases to 384 mm.

Fig.8.Equivalent diameters and damages of the RC targets in test schemes:IV-1～I(xiàn)V-3.

Fig.9.Relationship between impact velocity and opening diameter.

During the PELE penetrating the RC target,the deformation of the jacket at position of the maximum acceleration directly determines the opening size of the RC target[12].Therefore,in order to analyze the influence mechanism of the impact velocity on the PELE penetrating the RC target,the deformation of jackets at the position of the maximum acceleration were gotten at different impact velocities,and displayed in Fig.10.To quantify the deformation degree of the jacket,two bending angles(αandβ)and an opening mouth(nd1)were defined to measure the bending deformation of jacket in this paper,as shown in Fig.11.(I)When the impact velocity increases from 400 to 1000 m/s,the bending deformation angleβaugments from 3°to 20°.In the impact velocity range of 400-670 m/s,the bending deformation angleα increases from 14°to 80°,and when the impact velocity is 785 m/s,the front end of the jacket presents a larger bending deformation(α is about 95°).(II)Because the interaction force between the projectile and the target is positively correlated with the impact velocity,the increase of penetration resistance makesαandβgrowth in the range of 400-785 m/s,which brings the degree of opening mouth to widen from 1.56(d1/2)to 3.14(d1/2).(III)As the impact velocity continues to rise(≥785 m/s),the penetration resistance also continues to increase,which results in the transformation of the jacket deformation mode from bending deformation to curling deformation,and even erosion destruction of the jacket.Therefore,in the range of 785-1000 m/s,the degree of opening mouth reduces from 3.14(d1/2)to 2.62(d1/2),due to the curl deformation and erosion damage of the jacket.Based on the above analysis,it is found that the variation of the jacket opening mouth is in good agreement with that of the opening diameter of RC target.Therefore,the impact velocity affects the deformation of jacket by changing the interaction force(or penetration resistance)between projectile and target,and then the deformation of jacket determines the opening diameter of RC target.

4.2.Influence of the target strength

Based on the influence analysis of the impact velocity on PELE penetrating RC target,the impact velocity is an important factor on affecting the force(or penetration resistance)between target and projectile.Since target strength(fc)is another influence factor on the penetration resistance,a numerical simulation method was used to study the influence of the target strength on PELE penetrating RC target.Five series of simulation studies on the target strength were carried out in this paper,which arefc=20,42,60,80 and 100 MPa.The material model parameters of concrete target(K&C model)are also automatically output byfcvalue through LSDYNA.The value of the maximum and minimum principal strain criterion is the same as those offc=42 MPa(Section 3.2).In the simulation model,a PELE projectile with mass of 15.6 kg penetrates a 240 mm thickness RC target at impact velocity of 670 m/s.

Fig.12 illustrates the relationship between opening diameter of RC target and target strength.When the strength of target plate increases from 20 to 60 MPa,the opening diameter of RC target increases from 357 to 422 mm.However,as the strength of target plate continues to increase to 100 MPa,the opening diameter of target plate decreases to 380 mm.

Fig.10.Deformation of jackets at different impact velocities.

To analyze the influence mechanism of target strength on PELE penetration into RC target,Fig.13 also shows the deformation of jacket at the position of the maximum acceleration under the condition of different target strength.(I)Whenfcincreases from 20 to 60 MPa,the bending deformation angleβincreases from 7°to 15°with an increment of 2°/10 MPa.Asfccontinues to increase to 80 MPa and 100 MPa,βfirst increases to 17°(but its increment is 1°/10 MPa),and then it decreases to 13°.(II)According to the principle of PELE projectile penetrating RC target analyzed in Fig.2,the projectile is subjected to both axial resistance(Fa)and radial restraint fore(Fr)of the target plate in the process of PELE penetrating RC target.Facauses radial expansion deformation of the projectile(the jacket and filling),whileFrhinders that behavior.An increase in the target strength causes bothFaandFrto increase.In the range offc=20-60 MPa,because the effect ofFaon the radial expansion deformation of projectile is greater than that ofFr,the bending deformation angleβincreases from 7°to 15°.Asfccontinues to increase to 80 MPa,although the effect ofFais still greater than that ofFr,the relative influence difference betweenFaandFrdiminished,which reduces the increment from 2°/10 MPa to 1°/10 MPa.Withfcincreasing to 100 MPa,the relative influence difference betweenFaandFrdiminished further,soβbegins to decrease relative to the case offc=80 MPa,and it decreases to 13°.(III)In the range offc=20-42 MPa,the bending deformation angleαincreases from 30°to 80°,and the jacket undergoes greater bending deformation(αis about 95°)under the condition offc=60 MPa.Therefore,the bending deformation angleαandβincrease with the increase of the penetration resistance,which results in the degree of opening mouth growing from 2.04(d1/2)to 3.06(d1/2).Since the penetration resistance continues to increase with the increase offc(from 60 to 100 MPa),the deformation mode of jacket transforms from bending deformation to curling deformation,and the jacket accompanies erosion damage.Therefore,the curl deformation of jacket and the decrease ofβlead to the reduction of the opening mouth from 3.06(d1/2)to 2.50(d1/2).Based on the above analysis,it is found that the target strength also affects the deformation of jacket by controlling the penetration resistance,and then the opening diameter of RC target is determined by the opening mouth of jacket.The influence mechanism of the target strength is like that of the impact velocity.

Fig.11.Deformation mode of jacket[12].

Fig.12.Relationship between target strength and opening diameter.

4.3.Influence of the target thicknesses

Fig.14 shows the test results of the PELE perforating the RC target,under the nearly same impact velocity conditions(minimum 662.4 m/s,maximum 671.3 m/s,error 1.32%),when the thickness of target plate increases from 160 to 320 mm,the opening diameter of RC target increases from 325 to 450 mm.Combined with fragments recovered from tests as shown in Fig.15,the bending deformation degree of jackets increases with the increase of the target thickness,which also indicates that the deformation degree of jackets determines the opening diameter.In the range of the target thickness:160-320 mm,a PELE projectile causes greater opening damage to a thicker target plate.

In order to study the opening damage of the PELE projectile to the RC target with other different thicknesses,A series of numerical simulation of PELE projectile penetrating six groups of thickness targets(from 80 to 400 mm)was carried out in this paper.Fig.15 shows the relationship between target thickness and opening diameter,and the comparison between simulation and test results.Firstly,the simulation results of the jacket deformation were obtained in the range of 160-320 mm of target thickness.It indicates that the bending deformation degree of the jackets increases with the increase of target thickness,and the simulation results of the jacket deformation mode and quantity basically coincide with that of corresponding test results.Secondly,when the thicknesses of targets are 160,240 and 320 mm,the maximum errors between simulation and test results for the opening diameter are 1.2%,1.4%and 5.2%,respectively.It proves that the numerical simulation model can better simulate PELE penetration into RC target.Finally,combined with the numerical simulation results of other target thickness conditions(80-400 mm),the opening diameter of RC target increases from 232 to 504 mm,and the relationship between the target thickness and the opening diameter of RC target is approximately linear.

Fig.16 exhibits the deformation of jackets at the position of the maximum acceleration to analyze the damage mechanism of PELE penetrating RC targets with different thicknesses.When PELE projectile penetrates RC targets with thicknesses of 80,120,160 and 240 mm,the bending deformation angleαandβincreases from 8°to 80°and from 1°to 11°,respectively.As a result,the opening mouth degree of the jackets increases from 1.46(d1/2)to 2.96(d1/2).After PELE projectile perforates 320 mm RC target,the jacket produces a larger bending deformation(about 90°)and the bending deformation angleβincreases to 20°,which results in an increase in the opening mouth to 3.27(d1/2).For PELE projectile penetrating 400 mm RC target,although the jacket becomes curled,the jacket is fully opened and the bending deformation angleβ reaches 55°,and the final opening mouth increases to 3.48(d1/2).

Fig.13.Deformation of jackets at different target strength.

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

To further explain the influence mechanism of PELE penetrating RC targets with different thicknesses,the failure mode of targets with different thicknesses are given in Fig.17.(I)For the targets with thicknesses of 80-240 mm,the openings are all at position of 0.5hc(or thickness of conical plug is aboutHc=0.5hc).However,for targets with thicknesses of 320 and 400 mm,the openings are located at the position of 0.4hcand 0.32hc,respectively(or thickness of conical plug is aboutHc=0.6hcandHc=0.68hc,respectively).(II)When a projectile penetrates a concrete target and a cone-like plug is formed between the projectile and the rear surface of the concrete target,the penetration resistance on the projectile at this moment is able to be calculated by Ref.Fa=τf·(dHcπ+H2cπtanψ)[17],which is the maximum shear force on the plugging surface,whereτfis a shear strength of target andψis a conical inclination angle.Therefore,a PELE projectile penetrates a RC target with a thickness of 80 mm.When the penetration depth reaches 40 mm,the penetration resistance on the projectile equals to the maximum shear force on the plugging surface.At this moment,the radial expansion deformation of the projectile is maximum,and the opening diameter of the target reaches the maximum value.As the thickness of the target is increased to 120 mm,the projectile penetrates the target and the penetration depth reaches 40 mm.At this point,the resistance on the projectile is less than the maximum shear force on the plugging surface,due to the increase of the target thickness.Thus,the projectile continues to penetrates the target.The radial expansion deformation,the penetration resistance and the penetration depth continue to increase as well.The projectile produces the maximum opening damage to the target as soon as the penetration resistance equals to the maximum shear force on the plugging surface.Therefore,in the range of 80-240 mm,the jacket deformation,the opening diameter of target and the position of the opening all increase with the increase of the target thickness.(III)The projectile penetrates the targets with thicknesses of 320 and 400 mm,and the penetration depth reaches 120 mm.According to the jacket deformation of the projectile penetrating the target with a thickness of 240 mm,it is found that the larger deformation of the projectile at this moment causes it to be more prone to radial expansion,but the penetration ability of the projectile decreases.Therefore,as the projectile continues to penetrate,the speed of radial expansion deformation of the projectile and the penetration resistance increase sharply.The penetration resistance is also increased sharply,and the force of forming a plug is reached.The projectile produces the maximum opening damage to the target as soon as the penetration resistance equals to the maximum shear force on the plugging surface.In this process,the penetration depth of the target plate increases less,because of the weak penetration ability.Therefore,the position of opening gradually moves up(middle position relative to target thickness),and the opening size continues to increase.

Fig.15.Relationship between target thickness and opening diameter.

5.Opening model

5.1.Analysis of dimensionless parameters I and H

Based on previous studies[9],two dimensionless parametershave been identified and analyzed.According to the dimensional analysis method,these factors of target velocity(v0),target strength(fc),target thickness(hc)and effective length of the jacket(l)also constituted two dimensionless parametersI=Now,according to the analysis ofv0andfcabove,the value of the two parameters and corresponding opening diameter were converted into dimensionless parameters,and the relevant data was listed in Table 6.Fig.18(a)shows the relationship betweenIandAlthough there are differences on concrete target strength and impact velocity,the experimental and simulation data can be better represented by the dimensionless parametersIandand they satisfy a quadratic function relationship.

Fig.16.Deformation of jackets at different target thickness.

Fig.17.Failure mode of targets with different thickness.

Table 6Values of dimensionless parameters I.

Similarly,from the results of the above experimental and simulation studies,the results of the dimensionless target thicknessand the dimensionless opening diameterare listed in Table 7.

Fig.18(b)shows the relationship betweenHandIt is found that the dimensionless target thicknessHand the dimensionless opening diameterapproximately satisfy a linear relationship.

5.2.Establishment of opening model

Under the condition ofKconstant,the functionf(Q，G)has been established[12](hereso the three-dimensional map of the function is draw in Fig.19.WhenQis constant,the relationship between dimensionless opening diameterandGsatis fies a quadratic function,and vice versa.However,WhenKconstant(i.e.IandHchange),thefunction is invalid.Therefore,the application scope ofis very small.In order to established a wider range ofopening diameter model,the following analysis is performed:(1)Since the dimensionless parametersQ,G,andIare affected by the target strengthfc,the effect of the three dimensionless parameters on the dimensionless opening diameteris coupled;(2)From the study of the target thickness,its influence on the opening diameter of RC target is different from that of the other three parameters,so the effect ofHand the other three parameters on the dimensionless opening diameteris decoupled.Based on theabove analysis,Eq.(2)is transformed into:

Table 7Values of dimensionless parameters H.

Fig.18.Effect of dimensionless parameters I and H on dimensionless opening diameter:(a)relationship between and I;(b)relationship between and H.

Under the condition ofQ=4.537,G=0.083,andI=12,D0=3.91 is calculated by Eq.(5).According to the influence ofHon the dimensionless opening diameter,the data listed in Table 7 is linearly fitted.As shown in Fig.20,the function ofis determined as follows:

where:0.2≤H≤1.5.

From Eqs.(5)and(6),it is possible to determine the specific functional relationship of Eq.(4).The following Eq.(7)is a dimensionless opening diameter model of PELE projectile penetrating RC target:

Fig.20.Linear relationship between dimensionless parameter H and dimensionless opening diameter.

where:3.0≤Q≤6.0;0.2≤G≤1.4;6.0≤I≤20;0.2≤H≤1.5.

5.3.Model validation

To verify the accuracy of dimensionless opening diameter model of PELE projectile penetrating RC target,one group of test data conducted by Ye[18]under different impact velocities(450,693,800 and 840 m/s)was obtained,and the other group of test data(125 mm PELE projectile penetrating 500 mm RC target)performed by Zhu[19]was also gathered.Besides,their related simulations

were conducted.Fig.21(a)displays that the test results conducted by Ye[18]and the related simulation results are within±10%of the theoretical model,and the root mean square(RMS)error for the comparison of the theoretical results with tests and simulations are 0.419 and 0.318,respectively.It indicates that the theoretical model agrees well with the test results.Analysis of Fig.21(b)shows that for two data points under the same condition,one of the tests and the related simulation data points agrees well with the theoretical model,and the RMS error are 0.321 and 0.110,respectively.However,the other data point is significantly higher than the theoretical model and exceeds 10% of the model result.Combined with the target damage corresponding to the two data points,as shown in Fig.22,the first projectile impacted on the center of RC target(Fig.22(a)),so the boundary condition of target has less influence on the test result.However,the upper edge of the second target was destroyed(Fig.22(b))by PELE projectile,and the boundary condition of the target plate affects the test result,which causes it to be significantly higher than the theoretical value.

From the above analysis,it can be seen that in the application range of 3.0≤Q≤6.0;0.2≤G≤1.4;6.0≤I≤20;0.2≤H≤1.5,the dimensionless opening diameter model agrees well with the test results,which verifies the accuracy of the model.

6.Conclusion

Based on the test and simulation study of PELE penetrating RC target,the influence of impact velocity(v0),target strength(fc)and target thickness(hc)on opening damage of PELE penetrating RC target was analyzed.The main conclusions drawn from the test and simulation results are as follows:

(1)From the analysis of test and simulation results,both impact velocity(v0)and target strength(fc)affect opening diameterby changing the interaction force(or penetration resistance)between projectile and target.As values of the two parameters increase(v0=400-1000 m/s andfc=20-100 MPa),the interaction force between target and projectile increases,which causes the jacket to undergo from small bending deformation to large bending deformation,and then to curling deformation.Therefore,the opening diameter of RC target increases from 300 to 430 mm and then decreases to 380 mm with the increase of the two factors,which approximately satisfies a quadratic function relationship.

Fig.21.Comparison of theoretical model and test results:(a)test data conducted by Ye[18];(b)test data conducted by Zhu[19].

Fig.22.Damages of the RC targets:(a)the first test data point;(2)the second test data point.

(2)From the analysis of test and simulation results,opening diameter of PELE projectile penetrating six groups of RC targets with different thickness(hc=80-400 mm)increases from 240 to 500 mm.It is found that the relationship betweenhcandis approximately linear.For the targets with thickness ofhc=80-240 mm,the openings are all at position of 0.5hc(or thickness of conical plug is aboutHc=0.5hc).However,for the targets with thickness of 320 mm and 400 mm,the openings are located at the position of 0.4hcand 0.32hc,respectively(or thickness of conical plug is aboutHc=0.6hcandHc=0.68hc,respectively).

(3)Based on dimensional analysis,two other dimensionless parametersIwere determined.Combined with previous research results[9],an opening diameter model with wider application scopeis established.Comparing with the opening diameter model,the test results are within the error range of(±10%)the theoretical model results,so the opening diameter model is in good agreement with the test results.

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,“Opening behavior of PELE perforation on reinforced concrete 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:11772160,11472008,11802141),and Opening Project of State Key Laboratory of Explosion Science and Technology,Beijing Institute of Technology(KFJJ18-01M).