Theoretical and numerical simulation study on jet formation and penetration of different liner structures driven by electromagnetic pressure

Jian-hao Dou,Xin Jia,Zheng-xiang Huang,Xiao-hui Gu,Ying-min Zheng,Bin Ma,Qiang-qiang Xiao

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

Keywords:Electromagnetic field Shaped charge jet formation Theoretical calculation Numerical simulation

ABSTRACT The use of a shaped liner driven by electromagnetic force is a new means of forming jets.To study the mechanism of jet formation driven by electromagnetic force,we considered the current skin effect and the characteristics of electromagnetic loading and established a coupling model of“Electric-Magnetic-Force”and the theoretical model of jet formation under electromagnetic force.The jet formation and penetration of conical and trumpet liners have been calculated.Then,a numerical simulation of liner collapse under electromagnetic force,jet generation,and the stretching motion were performed using an ANSYS multiphysics processor.The calculated jet velocity,jet shape,and depth of penetration were consistent with the experimental results,with a relative error of less than 10%.In addition,we calculated the jet formation of different curvature trumpet liners driven by the same loading condition and obtained the influence rule of the curvature of the liner on jet formation.Results show that the theoretical model and the ANSYS multiphysics numerical method can effectively calculate the jet formation of liners driven by electromagnetic force,and in a certain range,the greater the curvature of the liner is,the greater the jet velocity is.

1.Introduction

Shaped charges(SCs)are cylindrical explosive filled charges with a cavity at one end and a detonating device at the other end[1].They have been extensively used in military and industry for their great penetration capacity,such as high explosive anti-tank and oil well perforators[2,3].According to the forming mechanism,the projectile formed by SC is mainly divided into explosive formed projectile(EFP)[4]and shaped charge jet(SCJ)[5].The velocity of SCJ can reach 2000 m/s to 10,000 m/s much higher than that of EFP,and is often used to penetrate high strength target such as RHA and ultra-high strength concrete[6-8].

Since the 19th century,scholars in various countries have conducted research on SCJ and SCJ penetration.For example,Davison et al.

[9]replaced the traditional conical liner with bell-shaped liner with variable thickness,which produced a jet with 10% more kinetic energy than before,and the penetration into concrete targets was increased by 28% from that of a reference range.Huerta[10]et al.designed a 0.7 m conical SC,the field test by firing it into tuff rocks showed that the hole-depth extended to 5.9 m,his research indicated that the power of SCJ can be increased by increase the diameter of SC.However,when the diameter of the liner is constant,the power of SCJ cannot be increased by increasing the charge amount after a certain range.Rassokha[11]studied shaped charges with shear-formed liners,and found that show that the penetration depth increases 9%-15% compared to a liner without the selfspinning effect when the self-spinning jet rotates oppositely to the shaped charge.

It is true that the power of SCJ can be improved by designing the liner structure,but the traditional SCJ is obtained by driving the liner with explosive detonation,so the power of SCJ is limited by the power of explosive.With continuous research,the power of explosives is gradually increasing.Velocity of detonation(VOD)is an important parameter to measure the power of explosives,conventional explosives,such as TNT,RDX,HMX,have a maximum detonation velocity of 8980 m/s[12].Jafari et al.

[12]studied a new tetrazole base high energy density materials,whose VOD could reach 9040 m/s.Zou et al.

[13]Prepared aε-CL-20-based polymerbonded,whose VOD is larger than 9100 m/s.It can be seen that the explosive power is growing slowly,which makes it difficult to increase the power of SCJ significantly.With new armor appears constantly,and the stopping power of armor has increased[14-17],so the penetration power of jet needs to be further improved.

With the development of technology,electromagnetic field is gradually applied to SC to improve the power of SCJ.Littlefield[18]theoretically analyzed the stability of rapidly stretching and perfectly plastic jets when they were subjected to axial magnetic fields,and found the axial magnetic field imposed on the SCJ could effectively inhibit the growth rates of perturbation.Fedorov et al.[19,20]showed that jet stretching with a diffused magnetic field is accompanied by magnetic field compression inside the jet,thereby generating radial stretching electromagnetic forces.Ma et al.[21,22]found that applying external electromagnetic field on the formed jet can accelerate the jet,delay the jet fracture,and inhibit jet turnover,thus increasing the jet power.Degnan[23]studied the implosion of a spherical liner using 4.8 MJ electromagnetic energy.The implosion speed of the inner surface can reach 25-30 km/s,which is much high than the collapse velocity of liner under explosive load.Due to PER theory,velocity of SCJ is related to collapse velocity,so high jet velocity and great penetration power can be obtained easily by electromagnetic loading theoretically.

The emergence of electromagnetic loading technology provides a new means of jet forming,that is to use electromagnetic force to replace explosive detonation to drive the liner forming jets.Grace et al.[24]used the direct loading approach and allowed megaampere current to flow through the liner,thereby generating a magnetic field;the current then acts with the magnetic field to form an electromagnetic force.The force acts on the liner and collapses the liner to form the jet.Grace applied current on liners of different sizes and shapes and obtained an effective jet,proving that a liner can form a jet in such manner.Grace also developed a model of how electromagnetic force acts on a liner to form jets,but the motion of the liner during the loading process was not considered,and theoretical calculation was not performed.However,no relevant experimental research has been conducted after that of Grace.Then,Quan[25]introduced a numerical simulation method of liner load by electromagnetic energy using an ANSYS multiphysics solution.The method can calculate the jet generation and velocity,but the method that loads the liner with static electromagnetic force has some error.And Rabczuk[26,27]described a method for treating fluid-structure interaction of fracturing structures under impulsive loads and a new approach for modelling discrete cracks in meshfree particle methods in three dimensions,which were formulated for the problems with high-pressure,large deformations and arbitrary nonlinear and rate-dependent materials.Dou et al.[28]is interested in this kind of jet formation method.He combined experimental results of Grace to establish a simple theoretical model of jet formation through a strong electromagnetic loading liner;however,the skin effect under the high frequency current and the deformation of the liner in the collapse process were not considered.

The electromagnetic loading technology can easily control the loading energy,expand the jet forming technology,and pave the way for new jet research.In addition,it can address the difficulty of improving traditional jet power,which is crucial in dealing with the advancement of armor technology and preparing for future wars.In addition,the safety of ammunition has gradually attracted people’s attention,insensitive munitions have become an important research[29,30].The energy of electromagnetic loading is controlled by the switch,and it gets rid of explosives,which greatly improves the safety of future ammunition.

In this work,On the basis of the characteristics of electromagnetic loading and PER jet forming theory,the current skin effect and the liner deformation in the loading process were considered,and the jet forming theoretical model of the liner under electromagnetic force loading was established.Furthermore,a numerical simulation of the liner driven by electromagnetic energy was performed through ANSYS Maxwell and ANSYS Autodyn.ANSYS Maxwell was used to calculate the static and dynamic electromagnetic fields of two and three dimensions.ANSYS Autodyn could be used to simulate jet formation[31].In this study,a relatively simple numerical simulation method was proposed to calculate the jet formation of the liner under the action of electromagnetic force by combining two software products.The common conical and trumpet liners were selected for theoretical calculation and numerical simulation,and the results were compared with the experimental results in Ref.24.The jet formation process and the jet shape and velocity were analyzed.

2.Theoretical model

A schematic of a shaped charge liner driven by electromagnetic energy is presented in Fig.1.A strong current flows through the liner along the axial direction and generates a circumferential magnetic field.The ampere force is generated by the interaction between the current and the magnetic field.According to the righthand rule,the ampere force is perpendicular to the surface of the liner.The liner collapses to the axis due to the force and collides at the axis to form jets and slugs.

2.1.Electric-Magnetic-Force coupling model

The loading device for the electromagnetically driven shaped charge liner is equivalent to an resistance--inductance-capacitance(RLC)circuit model.According to the characteristics of the current,the discharge current can be expressed as

Fig.1.Diagram of a liner driven by electromagnetic force.I is the current following through the liner device,B is the magnetic flux density generated by the current,and F is the magnetic force.Arrows at corresponding position represent the direction of I,B and F,respectively.

wheretp=π/ωis the current maintaining time(approximately half cycle),ζ=R/2Lis the attenuation coefficient,andω=is the oscillation frequency.

The electromagnetic field is assumed to be magnetic quasi-static[28].In accordance with the Ampere circuital theorem,the magnetic flux density is expressed as

whereμ0is the vacuum permeability,iis the current contained in a ring with radiusr,andris the radius of the liner element.The current and the magnetic field interact and generate the ampere force,F=BIL.A random element is taken from the liner,and the view from the axis direction is shown in Fig.2.In accordance with Eq.(2),the magnetic field is distributed in the space where the liner is located,the magnetic flux density is related to the radius and the current within the radius,and the current is not evenly distributed in the section.Therefore,the magnetic induction intensity is different in the radial direction.The integral method must be used to calculate the electromagnetic force on the liner.

The skin effect should be considered at high frequency current,and most of the current is concentrated on the surface layer with depthδ.The calculation shows that current densityjdecays exponentially with the increase in depth[32].

wherej2is the current density at the outer diameter,r2is the outer radius,andδis the skin depth.

Fig.2.Diagram of electromagnetic force.The gray color represents the current flow area.The current is distributed on the outer surface of the liner due to skin effect.Red represents the element layer,the black dots represent the direction of current flow,and the black arrows represent the direction of magnetic flux density.

whereωis the angular frequency of the current,σis the conductivity of the liner material,andμis the permeability.The integration of Eq.(3)in the direction of the cross-section radius of the liner element represents the current flowing through the element.Then,the current density from the outside diameter can be obtained as

wherer1is the inner radius of the liner.

When the thickness of the conductord≥5δ,the current can be equivalent to evenly distributing current densityj0on the surface layer with a circumference of 2πr2and a thickness ofδ,as shown in Fig.2.In this manner,the error is less than 1%[33].

A layer of the element in the radial direction is taken,as shown in the red part of Fig.2.The electromagnetic force of the layer of unit length is

where diis the current flow though the layer element

By taking Eq.(6)and Eq.(8)into Eq.(7)and integrating dFin the radial direction,we can obtain the electromagnetic force of the element of unit length as follows:

When r?2/3δ,Eq.(9)can be simplified as

Fig.3.Velocity relation at collision point.V1 is the velocity of collision point in the earth coordinate system,that is,the velocity of the dynamic coordinate system.V2 is the collapse velocity in the dynamic coordinate system,andβis the collapse angle.

2.2.Jet formation theory

According to momentum theorem,collapse velocity is expressed as[28].

whereβ*is the angle between the tangent line of the liner surface and the axis,andβ*for the conical liner is the half cone angle.

Eq.(11)indicates that during the collapse to the axis of the liner,a gradient in the radial direction of the collapse velocity of the liner elements exists due to deformation.We define a centroid layer of the element whose radius isr0,which satisfiesr22-r20=r20-r21.The velocity of the centroid layer is taken as the average collapse velocity(V0)of the element and expressed as

wheret0is the time when the element collapses to the axis.The coordinate system is established on collision point B,and the geometric relationship of the velocity is shown in Fig.3.AJ is expressed as the collapse velocity(V0)at the earth coordinate system[34].

V1andV2are expressed as follows:

given the extremely large pressure of the liner at the collision point,we assume that the liner is a constant and incompressible ideal fluid,and the Bernoulli equation can be used to obtain the left and right flow velocities at the collision point.The formation of jet and slug at the collision point B is shown in Fig.4[34].

If current exists when the element collapses to the axis,Bernoulli’s equation is expressed as

Fig.4.Formation of jet and slug.V2a is the velocity of fluid flowing out to the left from the collision point in the dynamic coordinate system,and that to the right is V2b.B is the collision point.

whereP2,P2a,andP2bare the presses of the fluid whose velocities areV2,V2a,andV2b,respectively.The presses are expressed as

Combining Eq.(14)and Eq.(15)obtains

By integrating the traditional calculation method,the jet and slug velocities in the earth coordinate system are

The masses of the jet and the slug are expressed as

2.3.Penetration theory

Virtual origin theory[35]is often used to calculate the penetration of the jet into the target.Assuming that the virtual origin point position of the jet is(xa,ta)and the stand-off isH,and the A-T model[36]is used to calculate the penetration of the jet into the

Fig.5.Current waveform in the experiment.Peak current of the conical and trumpet liner are 7.23 MA and 8.01 MA,respectively,and half cycle of the current of conical and trumpet liner are 18μs and 14μs

Fig.6.Structure and size of the liners(unit:mm):(a)conical liner;(b)trumpet liner.

Table 1Circuit parameters.

Fig.7.Diagram of the simulation model:(a)the coppery part is the liner device and the blue part is the insulator made of polyethylene;(b)the arrows represent the direction of the current.

Fig.8.Set of external circuit.It is a RLC circuit,Wingding1 represents the liner device,equivalent to an inductance.

steel target.

whereρjis the density of the jet,vjis the velocity,ρsis the density of the steel target,uis the penetrating velocity of the jet,andRtis the penetration resistance of the steel target.Thus,the penetrating velocity can be obtained as follows:

Fig.9.Equivalent electromagnetic pressure(1μs).Abscissa represents the number of the element.

Fig.10.Model of the conical liner.The yellow indicates the pressure acting on the surface of the liner,the wathet and yellow parts are meshed with Lagrange grids and filled with CU-OFHC,the green and dark blue parts are meshed with Euler grids and filled with CU-OFHC and AIR,respectively.

Assuming that the moment the jet reacts to the target ist0,the velocity of the jet approaching the target at momenttiisvji,and the depth of penetration(DOP)isPi.Thus,

whereΔPiis the penetration depth,andΔtiis the penetration time of the ith jet infinitesimal element.

If the jet is continuous,then using the virtual origin method,the velocity and time of the(i+1)th jet infinitesimal element are

If the jet has broken up attf,then the velocity and time of the(i+1)th jet infinitesimal element are

due to the invariant length of the jet.

The breakup time of the jet is calculated by empirical formula performed by Ref.[37].

wheredj0is initial diameter of jet,Dis a constant.

Table 3Parameters in the ideal-gas of AIR.

Fig.11.Diagram of the current calculated by theory model and simlation.Peak current of the conical liner under theoretical calculation and simulation are 7.2 MA and 7.0 MA,respectively.That of the trumpet liner are 7.9 MA and 7.8 MA,respectively.

3.Theoretical calculation and numerical simulation

3.1.Theoretical calculation

On the basis of the loading current of the conical liner with variable wall thickness and the trumpet liner in Ref.[24],the jet formation of these two kinds of liner driven by strong current is calculated using the previous model.The current curve in the experiment is shown in Fig.5[24],and the structural parameters and the coordinate system settings of the liners are shown in Fig.6.

The theoretical calculation of the two kinds of liner is performed using MATLAB software.The liner is divided into 47 elements in the axial direction.Assuming each element does not affect each other in the collapse process,we calculate the jet formation of each element.

3.2.Numerical simulation

Given that ANSYS Maxwell can only simulate the electromagnetic field of the static structure,the software is used to simulate the current and the electromagnetic field generated by the current.Then,the nonlinear dynamics of the jet formation is simulated by ANSYS Autodyn.Eq.(1)is used for backstepping the circuit parameters,and the static circuit parameters are shown in Table 1.Fig.7 shows the electromagnetic field simulation model of the conical liner.The current flows in from the top of the liner,and then flows through the liner in the positive direction of theYaxis and outfrom the outer end of the top.

Table 2Parameters in the Johnson-cook of CU-OFHC.

Fig.12.Diagram of magnetic field of simulation at 8μs:(a)elevation view;(b)Cross-sectional view.

Fig.13.Change of magnetic flux density along the axis at the outer surface of the liner(8μs).

Fig.14.Change of static magnetic flux density at the outer surface of the top of the liner(point A in Fig.12).

In the simulation,the external circuit program is set in accordance with the electrical parameters,and the circuit conditions are consistent with the actual conditions.The external circuit is mainly composed of four parts:capacitor bank,transmission line inductance,resistance,and load.The main electrical parameters are the resistance,inductance,capacitance,and voltage at both ends of the capacitance.The external circuit loading setting of the conical liner is shown in Fig.8.

In accordance with the current calculated in ANSYS Maxwell,the equivalent electromagnetic pressure of each element is calculated by the external MATLAB program,as shown in Fig.9.Then,the dynamic mechanical behavior of the liner is simulated in Autodyn.Given that the liner is an axisymmetric model,we use twodimensional modeling in Autodyn and divide the thin-walled part of the liner into 47 grids in the axial direction using the Lagrange algorithm.Electromagnetic pressure is applied on each grid,and the electromagnetic pressure on each grid of the liner cylinder is equal.The remaining part of the liner and the air area are filled with Eluer.The simulation model of the conical liner is shown in Fig.10.A gradual grid is adopted for the grid.From the top position of the liner to the upper right corner of the air area,the grid gradually changes from small to large.The minimum size is 0.25 mm×0.25 mm,and the maximum grid size is 1 mm×1 mm.CU-OFHC and AIR is selected from the Autodyn material library as the material of the liner,the Johnson Cook strength model is used as the strength model.And the material parameters of CU-OFHC and AIR are listed in Table 2 and Table 3.

4.Results and discussion

4.1.Calculation of current and magnetic field

Fig.11 shows the comparison of the theoretical current with that calculated in ANSYS Maxwell.The difference of the current is insignificant for conical and trumpet liners.The spatial distribution of the electromagnetic field at 8μs is shown in Fig.12.As shown in Fig.12(a),the current is perpendicular to the paper surface along they-axis from the outer part to the inner part,and the magnetic flux density is clockwise,which conforms to the right-hand rule.The results show that the magnetic field is distributed in the space between the liner and the current output end,and the closer the space position is to the axis of the liner,the greater the magnetic flux density is.In the horizontal direction,the magnetic flux density is the same,and the direction of magnetic flux density is the circumference direction of the liner.

Fig.15.Collapse process of liners from time 0-20μs:(a)conical liner;(b)trumpet liner.The velocity represented by each color is listed beside to the jet at the moment.

Table 4Jet parameters.

Fig.13 shows the theoretical and simulation results of the change in magnetic flux density from the top of the outer surface to the base of the liner(the yellow line as shown in Fig.12)at 8μs Except for the relative difference in magnetic flux density at the base of the conical liner,the rest of the positions are consistent.The large difference at the base may be analyzed as follows:1.Free mesh is adopted in Maxwell,and the small size of the insulator at the base position results in few grids and inaccurate calculation.2.The calculation at the boundary of the calculation domain may be inaccurate.The change trend of the magnetic flux density calculated by the two methods is the same.Thus,we can infer that at any time,the magnetic flux density at the surface of the shaped charge liner decreases from the top to the base.The electromagnetic force also show the same trend under the same current.Therefore,the top of the liner first collapses to the axis,and then the latter elements collapse to the axis successively.

Fig.16.Result of the simulation and experiment:(a)t=13μs;(b)t=17μs The blue dotted line ensures that the same position of the liner is aligned in the vertical direction,and the red circle indicates that the size of same position of the liner is the same.

Fig.17.Diagram of jet velocity of each conical and trumpet liner element.

Fig.14 shows the theoretical and simulation results of the static magnetic flux density changes at the top of the outer surface(point A as shown in Fig.12).The change trend of magnetic induction intensity is the same as that of the current,and the theoretical calculation is consistent with the simulation.

4.2.Formation of SCJs

Fig.15 shows the forming state of the conical and trumpet liners at different times under numerical simulation,respectively.The figures show that although the liner starts to collapse under the action of electromagnetic force,the liner at the top collapses to the axis to form jets earlier than the liner at the base.This phenomenon occurs because the electromagnetic force at the top is greater than that at the base.At 20μs,the two kinds of liner form well-shaped jets.The conical liner almost completely collapses to form jets under the action of electromagnetic force,whereas compare with the shape of uncollapsed trumpet liner at 15μs and 20μs,we found they were almost the same,it indicated that the base of the trumpet liner was unable to collapse to the axis to form jets.From Eq.(10),the electromagnetic force is related to the radius of liner element,the radius at the base of the liner is large,which results in relatively small electromagnetic force at base.Due to the internal stress and deformation,energy consumed in the collapsing process,the electromagnetic force is not big enough to collapse the element to axis.As shown in Fig.15(b),the jet tip is bifurcated,which may be due to nonagglomeration caused by the large collapse velocity.Another interesting phenomenon we must pay attention,a cavity appears in the cylindrical parts of conical and trumpet liner devices in Fig.15.We speculated that the plastic flow occurs due to the compression of the cylindrical part under electromagnetic force,and the liner collided at the axis formed slug with opposite velocity,the slug penetrated the cylinder made the cavity.

The parameters of the jet shape of the theoretical and simulated calculation of conical liner at 26μs and the trumpet liner at 28μs are presented in Table 4.The theoretical calculation of the jet tip diameter is quite different from the simulation diameter,the tail diameter is close relatively,and the relative error of the length is about 10%.Because the jet tip diameter is small,and the size of the grid and the element also has a great influence on the calculation.Overall theoretical calculation is in good agreement with the simulation calculation in jet shape.

Fig.16 shows the comparison of the X-ray experimental results[24]of the conical liner at 13 and 17μs,with the results of the simulation and theoretical calculation at the corresponding time.The simulation and theoretical results for the jet diameter and length at 13μs are consistent with the X-ray photograph.At 17μs,the jet tip of the experimental photograph is bifurcated.Furthermore,the jet shape from the simulation and theoretical calculation is near the jet shape in the X-ray in the experiment.Thus,the simulation and theoretical calculation are reliable.

4.3.Jet velocity distribution

Fig.15 also shows the jet velocity distribution at different moments.The velocity of the jets decreased due to the stretch deformation and air resistance,the velocity of conical liner drops rapidly at 10μs-15μs,while that of the trumpet cover is at 5μs-10μs,and then velocity dropping slowed down.At 20μs,the jets almost shaped,the jet tip velocities of conical and trumpet liners are 9560 m/s and 9432 m/s,respectively.

Fig.17 shows the jet velocity distribution of each element of the conical and trumpet liners.No reverse velocity gradient for the jet is formed by both liners.The jet tip velocity is similar;however,the tail velocity is significantly different because the diameter at the base of the trumpet liner is relatively large,and the energy required to collapse to the axis is great.Consequently,the velocity is relatively low at the base.Fig.18 shows that the last element velocity of the trumpet liner is 277 m/s;but no low jet velocity exists in fact.When the liner collides at the axis,only when the collision pressure is more than 10 times of the yield strength of the liner can the deformation of the material be treated as fluid and form the jet[37].However,we assume that all elements can form jets,which do not affect the calculation results.As shown in Fig.15,the base of the conical and trumpet liners cannot collapse to the axis to form a jet.Therefore,for the given loading energy,a reasonable liner structure must be designed to enable the liner to form a jet completely and improve the utilization rate of electromagnetic energy.

Fig.18.Distributions of jet velocity:(a)conical liner at 26μs;(b)trumpet liner at 28μs

Fig.19.3D model of the jet of simulation:(a)conical liner;(b)trumpet liner.

Fig.18 shows the jet shape and distributions of jet velocity under theoretical calculation and simulation.And to observe the jet more clearly,we map the jet to the 3D model in the simulation,as shown in Fig.19.The jet velocity and velocity distribution under theoretical calculation are in good agreement with the simulation.Table 5 shows the jet tip and tail velocities in the theoretical calculation,simulation,and tests,as well as the relative error between the theoretical and simulation calculation and test results.The calculation error of jet tail is slightly higher than that of jet tip,which may be caused by the measured position of jet tail.The maximum error of calculation is 7.2%;thus,the theoretical and simulation results for velocity are accurate.

Table 5Jet parameters.

Table 6Parameters of the liners.

Fig.20.Liner structures and velocity distributions of different jets at 32μs:(a)trumpet 0(conical,K=0);(b)trumpet 1(K=2.5 m-1);(c)trumpet 2(K=3.9 m-1);(d)trumpet 3(K=6.2 m-1).And K is the curvature of the liner outer surface.

Table 7Shape parameters of the jets.

4.4.Influence of the liner shape

To study further the influence of the shape of the liner on the jet formation of the liner under electromagnetic energy,we change the curvature of the liner and perform theoretical calculation and numerical simulation on the jet formation of the liner with different curvatures under the loading condition of the original trumpet liner.For the liner with variable wall thickness,changing the curvature means that the thickness at the same position also changes.To control the unique variable curvature,we change the original trumpet liner to a liner with a uniform wall thickness of 1.71 mm.On this basis,we change the curvature for the calculation.All calculations are performed under ideal conditions.The structural parameters of the different curvatures of the liner are shown in Table 6 and Fig.20,and the calculation results are shown in Fig.20 and Table 7.

Fig.21.DOP of the jet formed by the conical liner penetrate into steel target under theoritical calculation.

Fig.22.DOPs of the jets formed by trumpet liners penetrate into steel target under theoritical calculation.

Fig.20 shows the velocity distributions of the jet formed by the different curvatures of the trumpet liner at 32μs through theoretical calculation.Although the jet tail velocity is approximately the same,the jet tip velocity varies greatly.The velocity of the jet formed by the trumpet liner is higher than that of the conical liner;the greater the liner surface curvature is,the higher the jet tip velocity and the greater the jet velocity gradient are.Correspondingly,the diameter and length of the jet are also different.The data in Table 7 show that the jet diameter increases negatively,and the jet length increases with the curvature.In general,the greater the surface curvature of the liner is,the higher the jet velocity and the thinner the jet formed are.However,Table 6 indicates that a large curvature means that the cone angle at the top of the liner is small,and the extremely small cone angle of the liner hinders the jet formation.Thus,the liner surface curvature cannot be increased without limitation.A reasonable structural design must be developed in combination with the loading conditions to achieve the optimal liner structure.

4.5.Calculation results of penetrating steel target

To study the influence of the shape of the liner on the penetration power of the jet formed by the liner under electromagnetic loading,we use the penetration experiment data in Ref.[24]to verify the penetration model.Using the jet parameters calculated in Section 3.2,we theoretically calculate the penetration at 394 mm stand-off(8LD(liner diameter)of the conical liner)into a steel target.The calculated DOP is shown in Fig.21.The DOP of the jet formed by the conical liner to the steel target is 205 mm,the experimental result[24]is 193 mm,and the relative error is 6.2%.The theoretical penetration and the bottom hole diameters are 18.7 mm and 8.3 mm,respectively.The experimental results are 24 mm and 10 mm[24],and the calculated results are consistent with the experimental results.

Then,we calculate the penetration to the steel target of four kinds of trumpet liners in Section 4.4.The calculation results are shown in Fig.22,and the detailed data are shown in Table 8.As the liner surface curvature increases,the jet velocity increases,the DOP increases,and the penetration aperture decreases because of the jet diameter decreases.The increase in DOP does not have a simple linear relationship with the change of the curvature.Tables 7 and 8 show that the increase of the curvature results in the increase in jet velocity and penetration depth.However,compared with trumpet 0,the increase in DOP is not as large as jet velocity because the jet diameter is relatively reduced with the increase in jet velocity.Moreover,the jet breaks up early in motion.Eq.(23)to Eq.(26)indicate that breakup time considerably influences DOP.Therefore,increasing the penetration power of the jet by changing the curvature of the liner blindly is impossible.The liner structure must be designed reasonably to increase the utilization of electromagnetic energy and maximize the jet power.Generally,if we want to increase the penetration depth in a certain range,then we use the trumpet liner with a large curvature;if we want to obtain a large opening diameter,then we choose trumpet liner with a smaller curvature or a conical liner.

5.Conclusions

The theoretical model of the liner under the action of electromagnetic force is improved,and the jet formation of conical and trumpet liners under electromagnetic loading in Ref.[24]is calculated theoretically,simulated,and compared with the experimental results.The conclusions are as follows:

(1)A theoretical model of jet formation under electromagnetic energy is established by combining the electromagnetic loading characteristics and the traditional jet forming model.The calculated jet formation results are consistent with the experiment results,and the model can accurately calculate the jet formation of the liner under electromagnetic force.

Table 8Parameters of cavity.

(2)The results of the current and the electromagnetic field are consistent with the experiment results,and the relative error of the jet shape parameters and the velocity is less than 10%.The combination of the simulation method with ANSYS Maxwell and ANSYS Autodyn software can accurately calculate the electromagnetic loading of the liner.

(3)Under the same loading condition,the shape of the liner considerably influences the jet formation under electromagnetic loading.In a certain range,the larger the curvature of the liner surface is,the higher the jet velocity,the thinner the jet formed,and the greater the power of the jet are.However,the curvature cannot be increased without limitation.A reasonable structural design must be developed to achieve the optimal penetration power and utilization ratio of electromagnetic energy.

Declaration of competing interest

No conflict of interest exits in the submission of this manuscript,and manuscript is approved by all authors for publication.I would like to declare on behalf of my co-authors that the work described was original research that has not been published previously,and not under consideration for publication elsewhere,in whole or in part.All the authors listed have approved the manuscript that is enclosed.

Acknowledgments

This research is supported by the Natural Science Funds for Distinguished Young Scholar(Grant No.11602110)and Jiangsu Province Graduate Research and Practice Innovation Program(No.KY CX18_0471).