Rigid elliptical cross-section ogive-nose projectiles penetration into concrete targets

Xiang-hui Dai,Ke-hui Wang,Ming-rui Li,Jian Duan,Bing-wen Qian,Gang Zhou

Laboratory of Intense Dynamic Loading and Effect,Northwest Institute of Nuclear Technology,Xi’an,Shaanxi,710024,PR China

Keywords:Elliptical cross-section Projectile Penetration Concrete target Dynamic cavity-expansion

ABSTRACT The elliptical cross-section ogive-nose projectile(ECOP)has recently attracted attention because it is well suited to the flattened shape of earth-penetrating weapons.However,the penetration performance of ECOPs has not been completely understood.The objective of this study was to investigate the penetration performance of ECOPs into concrete targets using a theoretical method.A general geometric model of ECOPs was introduced,and closed-form penetration equations were derived according to the dynamic cavity-expansion theory.The model was validated by comparing the predicted penetration depths with test data,and the maximum deviation was 15.8%.The increment in the penetration depth of the ECOP was evaluated using the proposed model,and the effect of the major-minor axis ratio on the increment was examined.Additionally,the mechanism of the penetration-depth increment was investigated with respect to the caliber radius head,axial stress,and resistance.

1.Introduction

Earth-penetrating weapons(EPWs)are effective for attacking underground buildings and are designed with flattened shapes to improve their stealth performance.If the projectile mounted on an EPW has an elliptical cross section instead of a circular cross section,the installation space can be more effectively utilized.It is possible to increase the mass of the projectile by increasing the dimensions of its outline and to eventually increase the initial kinetic energy for penetration into the concrete target.However,the penetration performance of elliptical cross-section ogive-nose projectiles(ECOPs)into concrete targets has still not been completely understood,because of their unique shape.

The use of analytical methods for penetration mechanics began with the study of Bishop et al.[1].They developed equations for the quasi-static expansion of cylindrical and spherical cavities,and these equations were used to estimate the force acting on the conical nose when it punches slowly into a metal target.Hill[2]and Hopkins[3]developed dynamic spherically symmetric cavityexpansion equations for an incompressible target material.Goodier[4]applied these equations and developed a model to predict the penetration depth of rigid spheres into metal targets.Furthermore,Forrestal et al.[5-8]and Warren et al.[9]presented a series of formulas to predict the penetration depth of rigid circular cross-section ogive-nose projectiles(CCOPs)into soil[5],ductile metal[6-8],and rock[9]targets.

In the recent past,the cavity-expansion theory has also been widely applied to study the deep penetration of semi-infinite concrete targets.First,by idealizing concrete constitutive descriptions,Forrestal et al.[10]developed a spherical cavityexpansion penetration model for concrete targets.By assuming that the expansion stresses of the spherical or cylindrical cavity were equal to the normal stresses acting on the projectile nose,Forrestal et al.[11]proposed penetration formulae to predict the penetration depth of rigid CCOPs into concrete targets.By introducing a dimensionless empirical parameterSthat depends only on the unconfined compressive strength of the concrete target,Forrestal et al.[11,12]and Frew et al.[13]simplified the formulae of the cavity-expansion stress and applied them on the projectile during the penetration process.Moreover,by considering the diameter scale effect,Forrestal et al.[14]and Frew et al.[15]proposed another dimensionless parameterRto describe the penetration resistance.Second,by introducing two dimensionless numbers,the geometry functionNand impact factorI,and considering the projectile-target interfacial frictions,Li and Chen[16-19]extended the Forrestal formula to a dimensionless form that can be applied for projectiles with arbitrary nose profiles penetrating into diversified targets.Third,by assuming that the force acting on the projectile remains unchanged during the penetration process,Peng et al.[20]proposed a new mean resistance approach based on the dynamic cavity-expansion theory,and a simple unified model was further presented to predict the penetration depth.By introducing a hyperbolic yield criterion and the Murnaghan equation of state to describe the plastic behavior of concrete under projectile penetration,Kong et al.[21]proposed an extended dynamic cavityexpansion theory and formulated a unified one-dimensional resistance of the concrete target to rigid and eroding projectile penetrations.Based on the cavity-expansion theory,Liu et al.[22]proposed a circular cross-section double-ogive-nose projectile scheme with a smaller penetration resistance.

In general,according to the cavity-expansion theory,the mechanism of rigid CCOP penetration into concrete targets is fully understood.Relevant theories have been established,and excellent agreement has been obtained between high-precision experimental data and predictions employing the cavity-expansion theory.This lays a theoretical foundation for the study of the penetration performance of ECOPs.

Several studies have been conducted recently to evaluate the penetration performance of ECOPs into concrete targets.According to the dynamic spherical cavity-expansion theory,Wang et al.[23]presented an analytical model to calculate the penetration depth of an ECOP.They performed comparative penetration experiments to investigate the penetration performance of these projectiles.However,simplifications were made in the solution process,and the analytical model has room for improvement.Additionally,although a qualitative analysis of the advantages associated with the penetration of ECOPs was performed on the basis of test data,the penetration mechanism was not discussed.Dong et al.[24]and Liu et al.[25]conducted a series of penetration experiments involving two elliptical cross-section projectiles with major-minor axis ratios of 1.5 and 2.0 and one circular cross-section projectile.They studied the stress characteristics of an ECOP during the penetration process via numerical simulations.According to the assumption of the relationship between the normal velocity and normal stress of the projectile nose,a penetration-depth formula was established.The test data indicated that the elliptical crosssection projectiles had excellent penetration performance and trajectory stability within an initial velocity range of 700-1000 m/s.However,the penetration-depth formula and relevant conclusions based on the numerical-simulation results require verification.Guo et al.[26]investigated the penetration performance of ECOPs via experimental and numerical methods.The experimental results revealed that the penetration of the ECOP with a major-minor axis ratio of 2.8 was 17.5% deeper than that of the equivalent CCOPs when the initial velocity was approximately 800 m/s.However,no further research on the mechanism of the penetration-depth increment was conducted in the study.In general,although various analytical models for ECOP penetration into concrete target have been established[23-25]on the basis of simplifications and assumptions,an accurate and reliable analytical model that describes the penetration process is yet to be developed.Additionally,although scholars have determined the advantage of the penetration performance of ECOPs by comparing the penetration depths[24-26],the mechanism of the penetration-depth increment is not completely understood.

In the present study,a general geometry function was introduced to define the geometric characteristics of ECOPs.According to the dynamic cavity-expansion theory,a theoretical model for rigid ECOP penetration into concrete targets was developed.The model was validated using test data.Then,the effect of the major-minor axis ratio on the penetration performance of ECOPs was examined.Finally,the mechanism of the penetration-depth increment in ECOPs was described.

2.Theoretical model

Fig.1 shows the geometry of a rigid ECOP.The ogival curve is the arc of a circle and is tangent to the shank of the projectile;aandbrepresent the semi-major and semi-minor axes of the initial crosssection ellipse,respectively;lrepresents the length of the projectile nose;srepresents the radius of curvature of the arbitrary ogival curve;θrepresents the azimuth between the plane of the arbitrary ogival curve and the XOZ plane;φrepresents the angle between the outward normal vector of the ogive-nose surface and the projectile axis;randr1represent the radii of the initial cross-section ellipse and the arbitrary cross-section ellipse,respectively;andSArepresents the surface area of the ogive nose.The arrow indicates the movement direction of the projectile.

For a rigid ECOP that impacts a uniform concrete target at normal incidence with an initial striking velocity ofV0and proceeds to penetrate at an instantaneous velocity ofV,the penetration process parameters can be calculated when the resistance of the projectile is known.First,we establish the ogive-nose surface geometry function.Then,we model the target resistance.Finally,we calculate the penetration process parameters.

2.1.Geometry of ECOP

2.1.1.Ogive-nose surface geometry function

According to the equation of the initial cross-section ellipse in Fig.1,

where

Fig.1.ECOP geometry.

From the geometry shown in Fig.1,

wherer1=(x2+y2)1/2.

Substituting Eq.(3)into Eq.(4a)yields

The ogive-nose surface geometry function is given by

2.1.2.Caliber radius head(CRH)

The CRH of a CCOP was defined by Forrestal et al.[11].In the present study,the CRH of an ECOP was defined as follows:

Substituting Eqs.(1)and(4a)into Eq.(8)yields

We assume thatrcis the radius of a circle that has the same area as an ellipse,i.e.,rc=(ab)1/2,which is called the equivalent radius of the ellipse.λis the major-minor axis ratio,i.e.,λ=a/b;thus,a=λ1/2rcandb=λ-1/2rc.When the major-minor axis ratio is 1,a=b=rc;the cross-section ellipse degenerates into a circle,and the corresponding projectile is called the equivalent CCOP.The CRH of the equivalent CCOP is given by

2.2.Resistance on ECOP

The normal expansion velocityVnof the projectile-target interface caused by the projectile penetrating at rigid-body velocityVis

where

Here,fx(x,y,z),fy(x,y,z),andfz(x,y,z)are the derivatives off(x,y,z)with respect tox,y,andz,respectively.

Based on the dynamic cavity-expansion theory,the normal stressσnacting on the projectile nose and the cavity-expansion velocityVnsatisfy the following[11]:

whereρtrepresents the density of the concrete target,andRrepresents the resistance parameter of the concrete target.According to Frew et al.[13],R=Sfc,whereSis an empirical constant for the concrete target,andfcrepresents the unconfined compressive strength of the concrete target.S=82.6fc-0.544andS=72.0fc-0.5were reported by Frew et al.[13]and Li and Chen[17],respectively.We selectedS=82.6fc-0.544in the present study.

Because the sliding frictional resistance on the projectile-target interface was considered by Frew et al.[13]when fittingS,it was neglected in the present study.Substituting Eq.(11)into Eq.(14)and integrating yields the axial resistance on the projectile nose,which is given by

whereΩrepresents the entire ogive-nose surface,andσzrepresents the axial stress acting on the projectile nose.

When the surface integral is converted into a double integral,Eq.(15)becomes

wherezx(x,y)andzy(x,y)are the derivatives ofz(x,y)with respect toxandy,respectively.Differentiating both sides of Eq.(6)with respect toxandyyields

respectively.To solve Eq.(16)more conveniently,the generalized polar-coordinate transformation is performed as follows:

where 0≤ρ≤1 and 0≤α≤2π.

Substituting Eq.(18)into Eqs.(12)and(13)yields

wherel1is a length variable,which is defined as follows to simplify the equations in the present study:

Substituting Eqs.(18)and(20)into Eq.(17)yields

According to the polar-coordinate transformation rule of double integration,Eq.(16)becomes

Eq.(22)takes the following form:

whereArepresents the static resistance,andBis the dynamic resistance coefficient.

Substituting Eqs.(19)and(21)into Eq.(22)and combining Eq.(23)yields

2.3.Penetration process

Consider a projectile impacting a concrete target normally with the initial velocityV0.If the thickness of the concrete target is sufficient,the projectile stops and is embedded in the target.The two-stage(cratering+tunneling)penetration model of the CCOP is always utilized for thick concrete targets[11,17].A conical crater with depthkdand a tunnel with diameterd,wherek=2.0 andk=0.707+l/dwere suggested by Forrestal et al.[11]and Li et al.[17],respectively.In the present study,a two-stage penetration model was utilized for the ECOP,as shown in Fig.2,whereHrepresents the height of the concrete target,H*represents the final penetration depth,andd=2(ab)1/2represents the diameter of the equivalent CCOP.k=2.0 was adopted in the present study.

Fig.2.Schematic for normal penetration of an ECOP into a thick concrete target.

The axial resistance on the projectile nose during the cratering and tunneling processes are given as follows[11]:

wherecis a constant,andz1represents the instantaneous penetration depth.

The deceleration vs.time is given as follows:

wheremrepresents the projectile mass.

The rigid-body velocity vs.time is given as follows:

The displacement vs.time is given as follows:

The final penetration depth is

In these equations,

The total penetration time is

3.Validation

In this section,the ogive-nose surface geometry function,penetration-depth predictions for the CCOP,and experimental analyses for the ECOP are employed to validate the present model.

3.1.Ogive-nose surface geometry function

According to Eq.(7),the nose shape of an ECOP witha=34 mm,b=17 mm,andl=79.7 mm was drawn accurately,as shown in Fig.3.It indicates that the ogive-nose surface geometry function and the geometric modeling process are correct.

3.2.Penetration-depth predictions for CCOPs

When the semi-major axis is equal in length to the semi-minor axis,the ECOP degenerates into a CCOP.The penetration depths of the CCOPs can be calculated using the Forrestal formula[11];then,they are employed to validate the present model.

The calculation parameters for the projectiles and concrete targets are as follows:a=b=24 mm;m=1.8 kg;andl=63.6,79.7,93.1,and 104.8 mm,corresponding toψc=2,3,4,and 5,respectively;ρt=2450 kg/m3;andfc=40 MPa.

The penetration depth of CCOPs can be calculated using Eq.(29)and the Forrestal formula[11].Fig.4 shows a comparison of predictions based on the present model and the Forrestal formula.The predictions of the present model are consistent with those of the Forrestal formula,confirming that the present model incorporates the Forrestal formula.

Fig.3.Nose shape of an ECOP.

Fig.4.Penetration depth vs.the striking velocity for CCOPs.

3.3.Experimental analyses for ECOPs

In this section,existing penetration-test data of ECOPs are employed to validate the present model.A collection of test data from Refs.[23,24,26]are compared with the penetration-depth predictions of the present model in Table 1,whereLrepresents the length of the projectile,andεrepresents the deviation of the prediction from the test data.The predictions agree well with the test data,and the maximum deviation is 15.8%.Generally,the deviations are within the acceptable range.

Fig.5 shows a comparison between the test data and predictions based on Eq.(29),with an average value ofm=448 g for type nos.T1 and T2,m=222.57 g for type no.E1,andm=222.28 g for type no.E2(from Table 1).Although there are significant differences in the projectile parameters and impact velocities,the predictions agree well with the test data,particularly for T1 and T2(which had larger dimensions).Thus,the efficacy of the present model was confirmed by its agreement with test data.

Fig.5.Test data and predictions.

The ogive-nose surface geometry function is validated by drawing the nose shape accurately.Additionally,according to the Forrestal formula and test data,the present model is further validated via a comparison of the penetration depth.

4.Penetration performance

In this section,according to the present model,the penetration performance of the ECOP with different major-minor axis ratios is examined.

Fig.6 shows a comparison between the penetration depth and the striking velocity for different major-minor axis ratios(from Eq.(29)).The calculation parameters for the projectiles and concrete targets are identical to those presented in Subsection 3.2,except forrc=24 mm.The penetration depth increases exponentially with the increasing striking velocity,and it increases with the increasing major-minor axis ratio,but the increasing tendency becomes insignificant with the increasing nose length.

To quantitatively evaluate the penetration-depth increment of ECOPs,the penetration-depth increment vs.the major-minor axis ratio for different striking velocities is shown in Fig.7.ηrepresents the penetration-depth increment of an ECOP relative to theequivalent CCOP,which is calculated asη=(H*n-H*1)/H*1×100%,whereH*n andH*1 represent the penetration depths corresponding toλ=n andλ=1,respectively,and the subscript“n”is a real number greater than or equal to 1.

Table 1Comparison of the penetration depth.

Fig.6.Penetration depth vs.the striking velocity for different major-minor axis ratios.

The results in Fig.7 indicate that the penetration-depth increment increases with the major-minor axis ratio.The increasing tendency becomes more significant with the increasing striking velocity,but it decreases with the increasing nose length.For instance,when the nose length is 63.6 mm,the striking velocity is 1000 m/s,and the major-minor axis ratio is 5,the penetrationdepth increment reaches approximately 10%.However,when the nose length is 104.8 mm,the striking velocity is 400 m/s,and the major-minor axis ratio is 2,the penetration-depth increment is only approximately 0.1%.

Figs.6 and 7 indicate that the ECOP has better penetration performance than the equivalent CCOP,particularly for a higher striking velocity and larger major-minor axis ratio.However,if the nose length is sufficient,the penetration performance of the equivalent CCOP is adequate;thus,the penetration-depth increment becomes insignificant even if the major-minor axis ratio reaches 5.These conclusions can be used to guide the structural design of the ECOP.

5.Mechanism of penetration-depth increment

The mechanism of the penetration-depth increment is investigated with respect to the CRH,axial stress,and resistance of the ECOP.

5.1.CRH

The penetration performance of the CCOP significantly depends on the CRH.As discussed in Refs.[11-16,27],a CCOP with a larger CRH has better penetration performance.

According to Eq.(9),the CRH of an ECOP is not a constant,and it is closely related to the azimuth.Fig.8 shows a comparison between the CRH and the azimuth for different major-minor axis ratios.The CRH changes periodically(with a period of 180°),and it gradually increases from the semi-major axis(θ=0°)to the semiminor axis(θ=90°)along the circumference.The CRH of an ECOP is larger than that of the equivalent CCOP within a broad range of azimuths,which accounts for 60%-73% of the cycle.Furthermore,the peak CRH increases significantly with an increase in the major-minor axis ratio.For instance,the peak CRH of an ECOP with λ=5 is close to five times that of the equivalent CCOP.Moreover,the CRH increases with the nose length,particularly for ECOPs with large major-minor axis ratios.When the major-minor axis ratio is 5,the peak CRH of an ECOP withl=104.8 mm is nearly 15 times that of an ECOP withl=63.6 mm.

5.2.Axial stress

Using Eq.(15),we can calculate the axial stress along the circumference at thez=l/2 cross section in the projectile nose.The calculation parameters for the projectiles and concrete targets are identical to those presented in Section 4.

Fig.9 shows the axial stress vs.the azimuth for different major-minor axis ratios,when the instantaneous velocity of the projectiles is 1000 m/s during the tunneling phase.The axial stress changes periodically(with a period of 180°),and it fluctuates around the constant value of the equivalent CCOPs.Furthermore,the axial stress decreases gradually from the semi-major axis to the semi-minor axis along the circumference,in agreement with experimental results[24,25].With a larger major-minor axis ratio,the lower peak axial stress of the ECOP is smaller,and the corresponding azimuth range is broader.The axial stress decreases with an increase in the nose length.Additionally,as the nose length increases,the difference in axial stress between the ECOP and the equivalent CCOP decreases.

5.3.Resistance

Using Eq.(23),we can calculate the resistance on the projectile during the tunneling phase.The calculation parameters for the projectiles and concrete targets are identical to those presented in Section 4.Fig.10 shows a comparison between the resistance and major-minor axis ratio for different nose lengths,when the instantaneous velocity of the projectiles is 1000 m/s.An ECOP with a longer nose has a smaller resistance.Moreover,the resistance decreases with an increase in the major-minor axis ratio,and the decreasing tendency becomes insignificant with the increasing nose length.

The surface area of the projectile nose increases with the major-minor axis ratio,as shown in Fig.10.Therefore,we conclude that the axial-stress-reducing effect on the resistance is greater than the surface-area-increasing effect.

The mechanism of the penetration-depth increment is elucidated as follows:an ECOP with a larger major-minor axis ratio has a larger CRH and suffers smaller stress within a broader range of azimuths;hence,it tends to be subjected to less resistance during the penetration process.Thus,there is a significant increment in the penetration depth.However,as the nose length increases,the difference in resistance between the ECOP and the equivalent CCOP decreases,the penetration-depth increment becomes insignificant for the ECOP.

6.Discussion

In the present model,the erosion and deformation of the projectile are not considered.However,these phenomena have been observed in penetration experiments.Local erosion and melting on the projectile surface were observed in experiments conducted by Forrestal et al.[12]and Frew et al.[13].When the striking velocity reached 1200 m/s,the ogive nose became blunt,and the mass loss was approximately 7%.According to Ref.[28],in penetration tests with 45# steel projectiles and mortar targets with a compressive strength of 50 MPa,apparent erosion and deformation may occur when the impacting velocity exceeds 710 m/s.According to Ref.[29],a 30CrMnSiNi2A high-strength steel alloy projectile can be treated as a rigid body when the striking velocity is＜1402 m/s,and it is eroded but not deformed when the striking velocity reaches 1596 m/s.Because projectiles are often machined from highstrength alloy steel,the erosion and deformation are almost negligible when the striking velocity is＜1000 m/s.Therefore,it is reasonable to consider the projectile as a rigid body in the present study.

Fig.8.CRH vs.the azimuth for different major-minor axis ratios.

Fig.9.Axial stress vs.the azimuth for different major-minor axis ratios.

Fig.10.Resistance and surface area vs.the major-minor axis ratio for different nose lengths.

Additionally,the present model contains a dimensionless empirical parameterSthat was proposed by Frew et al.[13],and it was determined by inferring the penetration data of CCOPs.Shas application limitations,as described by Frew et al.[13],and exceeding these limitations may result in inaccurate predictions.Although the predictions made by the present model agree well with the test data in Refs.[23,24,26],if we wish to obtain more accurate predictions under other conditions(in particular,beyond the application limitations ofS),more test data must be obtained to refitS.

7.Conclusions

A theoretical model of the rigid ECOP penetration into semiinfinite concrete target was presented.On the basis of the present model,the penetration performance and penetration-depth increment mechanism of the ECOP were investigated.The following conclusions are drawn.

(1)The penetration equations were in closed form and depended on the striking velocity,geometry,mass of the projectile,and material properties of the concrete target.

(2)The present model was validated by comparing the predicted penetration depths with test data,and the maximum deviation was 15.8%.

(3)The ECOP had better penetration performance than the equivalent CCOP,and the penetration depth increased with the major-minor axis ratio.

(4)The penetration-depth increment mechanism was elucidated as follows:an ECOP with a larger major-minor axis ratio has a larger CRH and suffers smaller stresses within a broader range of azimuths;hence,it tends to be subjected to less resistance during the penetration process.Accordingly,it has better penetration performance.

To further verify the present model,systematic penetration experiments for a 1.8 kg ECOP with different major-minor axis ratios are urgently needed,and this will be the focus of future research.

Declaration of competing interest

All authors declare no conflict of interest.

Acknowledgments

This work was supported by the National Natural Science Foundation of China(Nos.11772269,11802248,and 11872318).