Investigate the effects of magnetic fields on the penetration ability of a shaped charge jet at different standoffs

Bin M ,Zheng-xing Hung ,* ,Zhong-wei Gun ,Xin Ji ,Xu-dong Zu ,Qing-qing Xio

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

b School of Engineering,University of Liverpool,Liverpool,L69 3GQ,UK

Keywords: Shaped charge jet Stability Penetration Magnetic field Coupling mechanism

ABSTRACT The influence of a magnetic field on the stability of a shaped charge jet is experimentally investigated at standoffs of 490,650 and 800 mm.The experimental results without and with the magnetic field are compared in terms of the shaped charge jet form,stability and penetration ability.A theoretical model based on one-dimension fluid dynamics is then developed to assess the depth of penetration of the shaped charge at those three standoffs and magnetic conditions.The results show that the penetration capability can be enhanced in more than 70% by the magnetic field.The theoretical calculations are compared with the experimental results with reasonably good correlation.In addition,the parameters introduced in the theory are discussed together with the experiments at three standoffs studied.

1.Introduction

A shaped charge(SC)plays a crucial role in military and civilian sectors due to their distinguished penetration ability.A shaped charge jet (SCJ) is to be slenderized gradually because of the existence of axial velocity gradient until it breaks up into a column of particles with similar size [1].However,the SCJ particles are not moving along its axis but start rotation and drift after breakup[2].The ability of SCJ penetration is positively related to its effective length [3],but experimental results show that the effective length of a SCJ can be reduced due to its breakup,rotation and drift.As a result,the instability and breakup of a SCJ are important factors influencing its penetration capability.

In order to improve a SC performance as much as possible,the processes of formation,elongation and instability of a SCJ have received a great attention in recent years.However,because of the limitations of processing precision and the special nature of the SC structure,the traditional methods cannot be used to improve the penetration ability effectively.Recently attention has been focused on the technology of the magnetic field coupling with SCJ for stabilization.Electromagnetic force will be appeared because of electromagnetic induction between a SCJ and magnetic field,whose direction is opposite to the one of SCJ necking force.In addition,SCJ particles after breakup inside the magnetic field will be remain the same axis due to inhibition effect of electromagnetic torque.As a result,the effect of magnetic field can improve the stability of a SCJ,and enhance the SCJ penetration ability.By coupling action of the magnetic field,it is hoped that the breakup time can be delayed and collimation of SCJ particles after breakup can be improved.Consequently,the depth of penetration of SCJ can be increased through the coupling effect of the magnetic field.Littlefield [3]employed an axial magnetic field imposed on a SCJ with small disturbances,and found that the magnetic field could weaken the growth of the SCJ instability.He applied the magnetohydrodynamic method to explain the stabilizing mechanism of a magnetic field on SCJs.Fedorov et al.[4] raised a possibility of a magnetic field to stabilize a SCJ based on previous exploratory work,and carried out the relevant theoretical analyses to describe this phenomenon.Fedorov et al.[5]investigated the influence of a varied longitudinal magnetic field on a SCJ stretching.Their study showed that the diffusion of the magnetic field was imposed during the SCJ stretching process.Then the consequence was that the radially electromagnetic forces opposite to the necking forces appeared,which would stabilize the SCJ and result in increasing its effective length and penetration ability.In addition,their experimental results indicated that the SCJ penetration ability was increased by 10%when experiencing the magnetic induction from 1 to 10 Tesla.Xiang et al.[6,7] analyzed the influence of the timing sequence between a SCJ and a pulsed magnetic field on the performance of a metal SCJ through experimental testing and numerical modelling.They draw a conclusion that the effect of a pulsed magnetic field could extend the effective length of a metal SCJ.Based on their research,they pointed out that the optimized delay time was the coincidence moment between the peak point of the discharge current and the center sections of a SCJ.Fedorov [8] investigated the increase of the SCJ effective length under different parameters of magnetic fields by numerical modeling.His calculations showed that the effective length of a SCJ could be more than doubled when a magnetic field was tens of Tesla.Ma et al.[9-11] explored the stabilizing mechanisms of the external magnetic field on SCJs.Their results showed that the coupling effect of the magnetic field could delay the SCJ breakup,increase its axial velocity and inhibit the rotation of particles after breakup.Furthermore,experiments were conducted to verify the stabilizing effect of a magnetic field on a SCJ.

Up to date,there is limited research work carried out to investigate the effect of magnetic fields on penetration capability of a shaped charge jet at different standoffs.Therefore,this paper aims to fill this gap.Here,the experiments are carried out at different standoffs for SCJs without and with the magnetic field,which are used to verify the stabilizing effect of the magnetic field on SCJs.A theoretical model for calculating the depth of penetration of a SCJ is then developed,in which parameters of S and α are introduced to describe the influence of standoffs and the SCJ particles flip on the penetration capability of the SCJ.

2.Experiments

2.1.Experimental equipment preparation

The φ56 mm SC is extensively applied to basic researches on interaction mechanism between a SCJ and different targets because of its significant stable performance [12-14].During this experimental study,the φ56 mm SC is employed to investigate the effects of a magnetic field on penetration ability of a shaped charge jet at different standoffs.The structure and the product of the φ56 mm SC are shown in Fig.1.

The X-ray experimental results obtained by Ma et al.[9]showed that the SCJ tip and tail velocity were 6.3 and 1.2 mm/μs respectively,and the SCJ tip and tail diameters were captured at 30 μs after detonation are 2 and 14 mm.In addition,the SCJ lengths were also obtained as 113 and 219 mm at 30 and 50 μs,respectively.

In order to obtain the influencing characteristics of a magnetic field on SCJ penetration ability at different standoffs,two structures of solenoids were designed to produce the magnetic field used for coupling with the SCJ.Fig.2 exhibits the configurations of solenoids involved in this study.For solenoids involved in this work,they are constituted by copper wire,and two parallel winding wires.Lengths of the coils part for two type solenoid are 150 mm,and the central aperture diameter is 40 mm for I-type solenoid and 50 mm for II-type solenoid.Fig.5 shows the structure of a high field magnet and its physical characteristics in detail.The electrical parameters are shown in Table 1.

During the experiments,the I-type solenoid was employed in the experiments with the standoff of 490 mm,and the II-type solenoid was used at standoffs of 650 and 800 mm.

2.2.Experiments on penetration depth under the natural conditions

To obtain the penetration ability of the φ56 mm SC under the natural conditions(i.e.without the magnetic field)at the standoffs involved in this work,the experiments on depth of penetration(DOP) were conducted at standoffs of 490,650 and 800 mm respectively.Fig.3 provides the experimental arrangements for the above three standoffs under the natural conditions.

In the experiments,the targets used to measure the penetration ability of the φ56 mm SC were made of 45#steel.The parameters of the crater were measured to evaluate the SCJ stability and penetration ability.

2.3.Magnetic field - SCJ coupling experiments

2.3.1.Experimental arrangements

The coupling experiments between the magnetic field and the SCJ were designed to investigate the effects of a magnetic field on penetration ability of a shaped charge jet at different standoffs.Fig.4 exhibits the experimental setups with the existence of the magnetic field at different standoffs involved in this work.

The coupling circuit includes the capacitance as an initial energy source,the solenoid for producing the magnetic field and the explosion switch for controlling the action timing.During the experiments,the flame-proof sandbag was placed to prevent the mutual interference between the explosion switch and the shaped charge.

Fig.1.Structure and product of the ?56 mm shaped charge.

Fig.2.Configurations and products of solenoids.

2.3.2.Analysis of coupling circuit

For the coupling between the magnetic field and the SCJ,the whole circuit can be equivalent to aRLCone.Fig.5 displays the coupling circuit of the interaction between the magnetic field and the SCJ.

The circuit parameters measured are presented in Table 1.In the table,L,RandCrepresent circuit resistance,induction and capacitance,respectively.

During the experiments,an ElectroBoost is employed to provide a high voltage at both ends of the capacitor.The explosion switch(E-S)is used to close the coupling circuit at the right time to benefit the coupling between the magnetic field and the SCJ.The oscillating current through the solenoid produces a magnetic field when the coupling circuit is closed.

The sequential control is a critical factor influencing the coupling performance,which needs a reasonable timing sequence to be designed for different standoff-conditions.The kinetic parameters of the SCJ and the electrical parameters of the electrocircuit are essential to control the timing sequences of the interaction process between the SCJ and the magnetic field.For the standoffs studied,the optimized timing sequences are provided during the experiments,which are related to those essential parameters.To thoroughly understand the coupling process,the durations of SCJ elements in the solenoid for different standoffs are shown in Table 2.Here,Vis the velocity of the SCJ elements,andT1andT2are the time of the SCJ element with different velocities entering and leaving the magnetic field,respectively.ΔTis the duration time of the element in the field.

Table 2 Time history of SCJ elements with different velocities passing the solenoid.

3.Theoretical approach

3.1.Evolution of the current and magnetic induction intensity

The current in the coupling circuit is a precondition to calculate the magnetic field.According to Fig.5,the coupling circuit can be equivalent to aRLCoscillation circuit.Based on the theory forRLCcircuit,the expression of the circuit discharge current can be expressed as follows:

whereI(t),U0,R,LandCrepresent the real-time circuital current,initial discharge voltage of capacitors,working resistance,induction and capacitance,respectively.is an attenuation coefficient,andis oscillation frequency.

A reference point is required to calculate the magnetic induction because of the influence of the temporal relationship between the SCJ movement and the generation of the magnetic field.Based on the virtual origin theory,the virtual origin is set as the reference point,which is selected on the vertex of the liner.Then the magnetic induction intensity along the solenoid axial can be derived as follows:

Fig.4.Experimental setups at the different standoffs with the action of the magnetic field.

Fig.5.Coupling circuit of the interaction between the magnetic field and the SCJ.

Here,the circuit closing moment is set to time zero,z1=z-（ΔL+z0）is the co-ordinate along with the solenoid axis,zis the coordinate from the vertex of the liner,ΔLis the length from the solenoid entrance to the virtual origin,andz0is the displacement of the virtual origin.In addition,bis the solenoid length,andR1andR2are inner and outer radius of the solenoid,respectively.

3.2.The penetration model with considering the standoff and stability characteristics

Theoretical prediction is an important means to evaluate the SC performance.The 1D hydrodynamic penetration theory proposed by Birkhoff et al.[15]and modified by Pack et al.[16]has been used extensively.According to the 1D hydrodynamic penetration theory,the maximum penetration will present when the SCJ is completely broken up.However,the penetration depth firstly increases to a maximum and then decreases with the increase of standoff in practice.Normally,the SCJ penetration ability decreases significantly for a long standoff.As mentioned in Ref.[2],the SCJ particles are not along with the same axis but tumbling after breakup,which reduces the SCJ acting length and leads to the deceleration of the SCJ penetration.Based on the 1D hydrodynamic penetration theory,parameters of α andSare introduced to evaluate the influence of the coaxiality of the SCJ particles as well as the standoff on the SCJ penetration [17],i.e.

Here,ρjis the density of the SCJ,ρtis the density of the target andleffis the effective length of the SCJ.λ is the constant,and its value is one for continuous SCJ to penetrate a target and two for the broken up SCJ.If the SCJ is between perfectly continuous and completely broken,the value range of λ is between 1 and 2;Srepresents the standoff and α is a constant obtained by the experiments to characterize the coaxiality of the SCJ particles.

3.3.Effective length of a SCJ with the influence of a magnetic field

The effective length of a SCJ is one of critical factors affecting the SCJ penetration ability.During the deformation of the SCJ inside a magnetic field,the corresponding magnetic flux flowing through the SCJ cross section will change due to the decrease of the SCJ cross-section area during the coupling process.Therefore,the electromagnetic induction caused by the variation of the magnetic flux will lead to emerging of the electromagnetic force inside the SCJ.In this work,the axial component of the magnetic induction is considered,so that the direction of the electromagnetic force produced is along the SCJ radial.Through the Ampere’s law and the right-hand screw rule,it can be judged that the direction of the electromagnetic force is opposite to that of the force which causes the SCJ necking and furthermore,the breakup of the SCJ can be delayed,which is shown in Fig.6.

The particles after SCJ breakup will be flipping but not maintaining along the same axis due to the influence of the irregular breakup section and air resistance.However,the torque induced by the electromagnetic force can inhibit effectively the flipping of the particles after the SCJ breakup to enhance the SCJ penetration capability.Fig.7 shows the mechanism of a magnetic field inhibiting the flipping of SCJ particles.

Based on the kinetic characteristics of the SCJ material under the natural conditions,the kinematic equation of the SCJ material inside the magnetic field is derived as [17]:

By integration,the radial velocity of the SCJ material can be obtained:

Here ρ and η are the density and the resistivity of the SCJ material,respectively;V0is the initial radial velocity on the SCJ surface;is the initial strain rate;R0is the initial radius;B0is the magnetic induction intensity,andtis the time.

Based on Eq.(5),when the SCJ radius at the neck goes to zero,the cumulative breakup timetbcan be calculated as

Fig.6.Magnetic field which delays the SCJ breakup.

The cumulative breakup time is the most adopted parameter used to evaluate a SCJ stretching property[18].The effective length can be calculated by multiplying the cumulative breakup timetbby the velocity increment from the SCJ tip (Vj) to the tail (Vs),i.e.

Based on the theoretical analysis,the penetration capability of a SCJ influenced by a magnetic field can be evaluated with the related parameters.

4.Results and discussion

4.1.Discharge current and magnetic field intensity

Through the experimental measurements,the circuit discharge voltage is 19.97 kV for the standoff of 490 mm,20.75 kV and 20.03 kV for the standoffs of 650 mm and 800 mm respectively.Combining Eq.(1) and the circuit parameters given in Table 1,the curve of the dynamic current in the coupling circuit can be obtained.Fig.8 exhibits the discharge current results from the calculations and experimental measurements in a good degree of correlation for the three standoffs studied,although there are some discrepancies between.

The theoretical and the experimental results for 800 mm standoff after 300 μs? However,the maximum duration of SCJ elements in the magnetic field is 278 μs(Table 2)and therefore,such the discrepancies will have almost no influence on the evaluation of the coupling process between the magnetic field and the SCJ.

The magnetic induction intensity (B) is a critical parameter to evaluate the influence of the magnetic field on the stability of SCJ,which can be calculated through Eq.(2).The results are shown in Fig.9.

Fig.7.Magnetic field inhibiting the flipping of the SCJ particles.

Fig.8.Curve of the discharge current from the theoretical calculations and experiments.

Fig.9.Changing course of the magnetic induction intensity as the elements with different velocities passing the solenoid.

The calculation results show that the distributions of the magnetic field for standoffs of 650 and 800 mm are more uniform compared with that of 490 mm,which is highly beneficial to the production of the uniform electromagnetic force.The magnetic induction intensity is high,though its distribution is not ideal for the standoff of 490 mm,which is useful to produce larger electromagnetic force to enhance the stability of the SCJ.

As the magnetic field produced by the solenoid withRLCcircuit varies at any moment,the average magnetic induction intensity(B)experienced by the SCJ element needs to be calculated by integral instead of the change,which can simplify the complexity of the issue.Fig.10 displaysBcoupling with the SCJ at different standoffs.The coupling timing has an influence on the evolutionary process of the average magnetic induction intensity exposed to the SCJ.The curves indicate thatBexperienced by the SCJ elements increases to a maximum value firstly and then decreases gradually for all cases.The research results from Held [19] show that the drift velocity decreases with the increase in jet velocity,i.e.the stability of the SCJ elements with high velocity is better than those with low velocity to some extent.Also,the SCJ elements with mid-velocity is very important to influence the SCJ penetration ability.The magnetic field is coupled with the SCJ elements in mid-and low velocity when the magnetic induction intensity is high,which indicates that the distribution of the magnetic field is reasonable to be more beneficial to improve the SCJ performance.

Fig.10.Curves of the elements with different velocity experiencing B.

Fig.11.Targets penetrated at the standoff of 490 mm.

4.2.Analysis of the experimental targets

The craters penetrated by a SCJ on the targets reflect the stability characteristics of the SCJ.For comparing the stability characteristics of the SCJ without and with the magnetic field at different standoffs,the targets after experiments are cut through their central cross-sections in order to obtain the crater shape parameters.Fig.11 shows the sectioned targets penetrated by the SCJ at the standoff of 490 mm.The target marked as #1 is penetrated under natural conditions,and the targets named as #2 and #3 are penetrated by the SCJ coupling with the magnetic field.The average penetration depth of #1 target is 135 mm.For the magnetic field coupling case,the depth of penetration(DOP)is more than 200 mm(i.e.the overall thickness of the target #2) as the target was fully perforated,and the DOP for target #3 is 239 mm (the overall thickness of the target#3 is 300 mm).The actual penetration depth for#2 target cannot be obtained due to the full perforation through the target thickness.

Furthermore,sectioned targets penetrated by the SCJ at the standoff of 650 mm are exhibited in Fig.12.For the targets marked as #4 and #5 subjected to natural conditions,the average penetration depth is 126 mm.The sectioned targets#6 and#7 shown in Fig.12 were penetrated by the SCJ with the coupling action of the magnetic field,with DOPs as 216 and 210 mm respectively and the average as 213 mm.

Fig.12.The sectioned targets penetrated by the SCJ at the standoff of 650 mm.

The stability of a SCJ will be getting worse with the increase of the standoff.In order to study the influence features of the standoff change on the SCJ stability,on top of the experiments with standoffs of 490 and 650 mm,the sectioned targets penetrated by the SCJ at the standoff of 800 mm are given in Fig.13.Here,the sectioned targets marked #8,#9 and #10 are penetrated by the SCJ under natural conditions,with an averaged DOP as 109 mm.The sectioned targets #11 and #12 are penetrated by the SCJ imposed on a magnetic field,with an averaged DOP as 185 mm.

The experimental results from three standoffs show that with the coupling of the magnetic field,the penetration capability is increased by 77,72 and 75%corresponding to the standoffs of 490,650 and 800 mm,respectively.However,the degree of irregularity of the entrances of craters penetrated by the SCJ is getting worse gradually with the increase of the standoff under the natural conditions(Target#1 in Fig.11,#4 and#5 in Fig.12,#8,#9 and#10 in Fig.13),which indicates that the instability of the SCJ is worse at longer standoff.Furthermore,in the case of the coupling of the magnetic field,the entrances of craters are more inerratic and smaller,the depth of penetration is larger,and the overall size of craters is smaller,in comparison to the natural condition counterparts.Therefore,the morphological change of the craters conforms that the stability of the SCJ is enhanced due to the coupling of the magnetic field.

Fig.13.The sectioned targets under the natural (#8,#9 and #10) and magnetic (#11 and #12) conditions at the standoff of 800 mm.

4.3.Discussion of calculation results

There are a number of factors affecting the stability of a SCJ.Based on one-dimension fluid dynamics,the theory developed in this work considers the influence of the breakup and the particles coaxality of the SCJ on its penetration ability.The parameters considered are related to different standoffs.To obtain the depth of penetration through the theory,the related parameters are given in accordance with the experiments in Table 3,which includes the average magnetic induction intensityB,parameters λ and α during the penetration of the SCJ through the targets at different conditions.

According to parameters shown in Table 3,the value of λ is 1.1 at the standoff of 490 mm and 1.4 for the standoffs of 650 and 800 mm,which indicates that the degree of breakup of the SCJ at standoffs of 650 and 800 mm is even worse than that in the case of 490 mm standoff.Furthermore,the values of α are taken as 6×10-3mm-1in the natural conditions(i.e.without the coupling of the magnetic field) and 4 × 10-3mm-1in the case of the presence of the magnetic field.By comparing the values of α without and with the magnetic field,it shows that the electromagnetic torque produced in the coupling process can suppress effectively the rotation of SCJ particles,thus making the particles move along the SCJ axis.

In addition,the experimental results are used to verify the theoretical calculations,which are shown in Table 4.The results indicate that the discrepancies are only 4.4,1.7 and 1.3% between the experimental and the calculated DOPs in the natural conditionsfor standoffs of 490,650 and 800 mm respectively,and 4.1,3.9 and 2.3% for those standoffs with the effect of the magnetic field.Such the small discrepancies confirm that the theoretical calculations correlate with the experimental results quite well and the theory developed can be used to estimate the DOP of a SCJ through the target.

Table 3 The related parameters obtained based on the experiments.

Table 4 Experimental results and theoretical calculations in different conditions.

5.Conclusions

The effects of magnetic fields on penetration ability of a shaped charge jet at different standoffs have been investigated.The analyses of the parameters introduced have shown that the degree of the SCJ breakup is getting more serious with the increase of the standoff,and the coupling action of a magnetic field can improve the stability of a SCJ and enhance its penetration capability significantly.Based on the study outputs,the following conclusions can be drawn.

(1) Experimental results indicate that the stability of a SCJ is sharply deteriorated with the increase of the standoff,which leads to its penetration capability is decreased significantly.

(2) The effects of magnetic fields on a SCJ have a positive impact improving the stability of the SCJ to enhance its penetration ability significantly at three standoffs studied.Based on experimental results,for the φ56mm shaped charge,its depth of penetration is increased by 77,72 and 75% at the standoffs of 490,650,and 800 mm,respectively.

(3) A theoretical model based on one-dimension fluid dynamics has been developed,in which the parameter λ is introduced to describe the extent of breakup of the SCJ,indicating a higher value of λ corresponding to a more serious breakup.The values of λ are set to 1.1 at the standoff of 490 mm,and 1.4 at the standoffs of 650 and 800 mm,which shows the great extent of the SCJ breakup at the longer standoffs.The values of λ are identical with or without the coupling of the magnetic field at the same standoff,which indicates that the coupling of the magnetic field can only delay rather than prevent the breakup of the SCJ.In addition,the value of α is set to 6 × 10-3mm-1under the natural conditions and 4×10-3mm-1in the existence of the magnetic field.These results show that the coaxiality of the SCJ particles with the magnetic field is better than that in the natural conditions.

Declaration of competing interest

I would like to declare on behalf of my co-authors that no conflict of interest exits in the submission of this manuscript,and manuscript is approved by all authors for publication.

Acknowledgements

The authors would like to acknowledge National Natural Science Foundation of China (Grant No.11972196),Youth fund of Jiangsu Natural Science Foundation(Grant Nos.BK20190433)and National Natural Science Funds for Distinguished Young Scholar of China(Grant No.11702144) to provide fund for conducting experiments.