A new analytical model for the low-velocity perforation of thin steel plates by hemispherical-nosed projectiles

Chng-hi Chen,Xi Zhu,Hi-ling Hou,Xue-ing Tin,Xio-le Shen

aDepartment of Naval Architecture Engineering,Naval University of Engineering,Wuhan,Hubei,PR China

bLvshun Proving Ground,Naval Unit No.91439,Dalian 116041,PR China

A new analytical model for the low-velocity perforation of thin steel plates by hemispherical-nosed projectiles

Chang-hai Chena,*,Xi Zhua,Hai-liang Houa,Xue-bing Tianb,Xiao-le Shenb

aDepartment of Naval Architecture Engineering,Naval University of Engineering,Wuhan,Hubei,PR China

bLvshun Proving Ground,Naval Unit No.91439,Dalian 116041,PR China

1.Introduction

The penetration and perforation of thin metallic plates by projectiles have long been an interesting research topic and have been the subject of numerous studies over the past several decades.Comprehensive reviews on this subject are available in many published works[1-4].This problem is complex for it involves many factors,such as the impact velocities and nose shapes of projectiles,non-linearities of target materials and geometries,and strain rate sensitivity during the impact process.Therefore,earlier studies mostly focus on plates that are within a particular range of thickness and are impacted by projectiles with a special nose shape in a certain velocity range.However,extensive research indicates that the deformation and failure modes of metallic plates vary with the impact velocity and nose shape of projectiles,as well as the ratio of projectile diameter to plate thickness.In the case of theperforation of thin metallic plates in a sub-ordnance range,bluntnosed projectiles tend to cause failure by plugging[3,5],whereas conical projectiles are likely to cause failure through the petalling of the target material[3,6].

Given the complexity of the perforation of thin plates,previous analytical models of the deformation and perforation of plates usually consider only one type of perforation mechanism,and some of these models neglect the local plate deformation in the contact area.In Refs.[7-9],all models were produced with a bending-only method,whereas in Refs.[10,11,13],the interaction of the projectile and the plate following a membrane-only assumption was considered.In the theoretical analyses conducted in Refs.[14-16],both bending and membrane stretching,as well as the local deformation of the target plate,were considered[14].Accordingly,a good agreement was obtained between the theoretical predictions and experimental results.Another aspect of the complexity of the perforation problem mainly involves the influence of the structural response of the surrounding plate on the deformation and failure modes of the plate.Thus,in numerous actual impact situations,the deformation and failure modes of plates mostly involve various types and combinations of perforation mechanisms[17,18].As plate thickness and impact velocity of a projectile increase,a change in the failure modes from tensile failure to shear failure occurs in thin plates[11,12]and intermediate plates[20]impacted by blunt-nosed projectiles.Thus,the selection of failure criteria in analytical models exerts a significant influence on the applicability of models and the accuracy of predictions.A criterion of tensile failure was proposed in Refs.[11,13]for membrane or thin plate perforation,whereas a shear failure criterion was employed in Refs.[21,22].Thus,the applications of these analytical models were all restricted to particular ranges of plate thickness.In the analytical models presented in Refs.[15,16],an equivalent strain failure criterion was used.This criterion incorporated both the effect of membrane forces and the local shear effect.Although extensive studies have been conducted,they mostly focus on the problem of penetration or perforation of thin plates by blunt-nosed[8,13,16,21,22]or pointed-nosed[7,10,14,17-19]projectiles,and studies on the penetration or perforation of thin plates by hemispherical-nosed projectiles remain relatively scanty.Levy and Goldsmith[23,24]analytically and experimentally investigated the normal impact and perforation of thin metallic plates by hemispherical-nosed projectiles.On the basis of a lumped parameter system,an expression was derived for the force-time history involved in thin plates that are normally impacted by hemispherical-nosed projectiles;this expression is totally predictive below the ballistic limit.However,slightly large deviations above the ballistic limit were obtained for mild steel plates.Moreover,the strain rate effects and the influence of various failure mechanisms were neglected in their analytical model.In recent years,a large number of experimental[25-27]and numerical[26-29]studies have been conducted on the perforation of thin plates by hemispherical-nosed projectiles,whereas theoretical analyses of this subject are rare.In Ref.[30],the dynamic plastic response of thin steel plates impacted by hemispherical-nosed projectiles at low velocities was theoretically analyzed,and the effects of shear,bending,and membrane stretching were considered.The perforation process was divided into three stages,namely,bulging deformation,dishing deformation,and perforation.

Investigating the ballistic penetration and perforation of thin steel plates by low-velocity hemispherical-nosed projectiles is interesting and remains relevant.Many theoretical studies on this subject are available,as reviewed above.However,the experimental results in Refs.[26,27]and those in the present study indicate that early analytical models are incomplete and perhaps oversimplified in terms of the following important points:

(1)First,no analytical attention is given to the fact that the radius of the bulging region changes along with the impact velocity of the projectile.In early models,the change relation between the radius of the bulging region and the impact velocity of the projectile is obtained experimentally or empirically.

(2)Most early models neglect the strain rate effects and energy

dissipation during the impact process resulting from the further deformation of the dishing region after plate dishing.

(3)In most early models,the plastic deformation energy of the dishing region and the energy dissipated by a thin plate during ductile hole enlargement,including the energy required for the propagation of radial cracks,are incomprehensively or unreasonably considered.

To overcome the limitations of early models,the present study conducts ballistic impact tests on mild steel plates measuring 1.36,1.90,and 2.86 mm thick and impacted normally by hemisphericalnosed projectiles at velocities ranging from200 m/s to400 m/s.The deformation and failure modes of thin steel plates are analyzed.On the basis of the experimental results,a new method is presented to calculate the radius of the bulging region of thin steel plates that are normally impacted by hemispherical-nosed projectiles.In establishing this new method,a dynamic method and plastic wave propagation concept based on rigid plastic assumption are adopted.According to the experimental results and the analysis of the perforation process,a new model is developed to predict the residual velocities of thin steel plates that are normally perforated by hemispherical-nosed projectiles on the basis of the energy conservation principle.Theoretical predictions are compared with experimental results in terms of both the radius of the bulging region and the residual velocity of the projectile.Finally,the range of the applicability of the present model is discussed.

2.Experimental procedure

A smoothbore 15 mm caliber powder gun of a fixed barrel was fired at the projectiles at velocities ranging from 200 m/s to 400 m/s,which is a range that generally exceeds the ballistic limit of the target.The initial and residual velocities of the projectiles were measured through the oscilloscopically recorded(Hitachi VC7104 with the highest frequency of 100 MHz)voltage changes produced in two sets of 6μm-thick aluminum foil screens only before and behind the target plate.The schematic of the experimental setup is shown in Fig.1.

The projectiles used in the present tests showed a hemispherical nose,a diameter of 14.9 mm,and a length of 21.4 mm.The nominal massof the projectiles was 25.8 g.The schematic and photograph of the projectile are presented in Fig.2.The material of the projectiles consisted of quenched 45 steel,which is a type of hardened tool steel with a yield strength of 355 MPa and ultimate tensile strength ranging from 450 MPa to 685 MPa;it was used in as-received condition.

The target plates were fully clamped by steel strips along their four edges,and all of these plates had square dimensions of 350 mmX350 mm.The plates employed were cut out of commercial Q235 mild steel sheets with thicknesses of 1.36,1.90,and 2.86 mm.The square targets were made from these plates and were used in as-received condition.The strength and other mechanical parameters of the target materials obtained from the quasi-static uniaxial tests are listed in Table 1.

Fig.1.Schematic of the experimental arrangement.

Fig.2.(a)Schematic and(b)photograph of the projectile.

Table 1 Mechanical properties of the target plates.

3.Experimental results

The experimental results of the response of the target plates with three thicknesses are presented in Table 2.In this table,h0is the thickness of the target plate,mpis the projectile mass,and viand vrare the initial impact and residual velocities of the projectile,respectively.The masses of the projectiles in each run were measured,and the results are also given in Table 2.

The representative appearances of the deformation and failure of the target plates are shown in Fig.3.In the case of the slightlythick plate impacted by a hemispherical-nosed projectile at a relatively low velocity,necking occurs at the periphery of the contact region,and large deformation is produced in the remaining part of the target plate(as shown in Fig.3(a)).The deformation and failure mode of the target plate in test no.14 is a bulging-dishing deformation[28].As the initial impact velocity of the projectile increases and the plate thickness decreases,the failure modes of the thin steel plates change from bulging-dishing-perforation[28](Fig.3(b))to bulging-perforation accompanied by shearing to some extent(Fig.3(c)).Thinning occurs around the perforation in the case of these two failure modes,thus indicating the necking of the target materials.This phenomenon is due to the tensile stretching at the periphery of the bulging region that occurs before the failure of the target plates.Furthermore,the lower impact velocity of the projectile,the higher degree of necking of the target plate.As shown in Fig.3,the overall structural deformation in the dishing region of the target plate decreases with an increase in the initial impact velocity of the projectile.In addition,the response of the target plate impacted bya projectile at a relatively high velocity tends to be a local perforation response.

Table 2 Experimental results and correlative parameters.

Fig.4.Several cap-like plugs collected after the tests.

Plugs of cap-like shapes with reduced thicknesses and diameters are produced after the perforation of the target plates in the contact region.This phenomenon is obviously due to thinning,followed by the tensile tearing of the target materials at the periphery of the contact region.The plugs detached from the target plates are collected after the tests.These plugs are shown in Fig.4.The ejected plug and the projectile are fused together in two cases.As depicted in Fig.4,the sizes of the plugs increase as the impact velocities of the projectiles increase.

4.Formulation

Fig.3.Deformation and damage view of target plates.

Fig.5.Schematic of the penetration model and four stages.

Experimental observations show that the impact responses of the target plates above the ballistic limit are local perforation responses and that the influence of the boundary conditions on the penetration process can be neglected.Therefore,a circular plate of thickness h0and radius R(R→∞)that is normally impacted by a hemispherical-nosed projectile of mass mpand radius rp(2rp>>h0)at an initial velocity of viis considered,as shown in Fig.5(a).According to the present experimental results,along with correlative studies[27-30],the failure modes of the thin metallic plates normally impacted by hemispherical-nosed projectiles are related to initial projectile velocities.In Ref.[28],numerical simulations were conducted on thin steel plates impacted by hemisphericalnosed projectiles,and the perforation process was considered to be divided into bulging deformation,dishing deformation,and projectile perforation.Subsequently,the dynamic plastic responses of the thin target plates were analyzed on the basis of the threestage model in Ref.[30].However,in the same reference,the energy absorbed as a result of the further deformation of the dishing region during the third stage was neglected.In the present theoretical analysis,a four-stage model is developed to perforate thin steel plates by hemispherical-nosed projectiles at low velocities.The analysis relies on the following perforation phenomenon.When a thin steel plate is normally impacted by a hemisphericalnosed projectile at low velocity,bulging deformation first occurs in the contact region.As the projectile moves forward,dishing deformation occurs in the remaining part of the plate apart from the contact region.After the occurrence of failure at the periphery of the bulging region,an initial hole is created and enlarged in a ductile manner by the projectile nose until the diameter of the enlarged hole is equal to the projectile diameter.Finally,the projectile entirely perforates the target plate,followed by projectile exit.Accordingly,the four stages in the present model are bulging deformation,dishing deformation,ductile hole enlargement,and projectile exit(see Fig.5(b-f)).

The calculations of the energy absorbed during the perforation process for each stage are conducted with the following assumptions:

(1)The amount of energy absorbed by the projectiles is ignored;that is,the projectiles are considered to be rigid and nondeformable.

(2)The projectile and plug move with the same velocity during the entire perforation process after the initiation of contact.

(3)Only plastic deformation is considered in the target plate,elastic deformation is ignored,and the plate material is considered to be rigid plastic with a quasi-static flow stress of σ0.

(4)The energy dissipated by thermal effects for the target and projectile is neglected;that is,the thermomechanical coupling between the target and the projectile in the case of low velocity penetration is ignored.

4.1.Stage 1:bulging deformation

After the initiation of contact between the projectile nose and the target plate,the target materials in the contact region move concurrently with the projectile.As the projectile progresses,bulging deformation occurs in the contact region of the target plate(Fig.5(b)),and compressive forces are produced by moving the projectile.Simultaneously,plastic waves propagate from the initial contact point in a radial direction.When the radius of the bulging region equals the propagation radius of the plastic waves,the remaining part of the target plate starts to dish.At this time,the first stage ends,and the second stage commences.Hence,the element of the mass of the target material in the bulging region is assumed to be set into motion at the same velocity of the projectile,whereas the remainder of the target material is assumed to remain at rest.

4.1.1.Determination of the radius of the bulging region

The radius of the bulging region changes along with the initial projectile velocity,and a high initial projectile velocity equates to a large radius of the bulging region.When the initial projectile velocity is substantially high,the radius of the bulging region is close to the projectile radius.In the present study,the radius of the bulging region of the target plate perforated by hemisphericalnosed projectiles at various velocities is determined using the dynamic method combined with the plastic wave propagation concept.

By neglecting the frictional forces between the projectile nose and the target plate,the resultant force acting on the projectile during the bulging deformation stage can be expressed as[31]

where K is a numerical constant depending on the projectile geometry and is set to be 0.5 for hemispherical-nosed projectiles[31],ρis the mass density of the target plate,v1is the instantaneous projectile velocity during the first stage,andσd1is the dynamic yield strength of the target material in the first stage.A1is the instant projected area of the nose of the projectile on the target plate,and it can be determined from the geometrical configuration depicted in Fig.5(b),that is,

where rAis the instant radius of the bulging region,rpis the projectile radius,and w01is the maximum deflection of the bulging region changing with the motion of the projectile.

The equation of motion during the first stage in the direction of motion is

where the effective mass m1includes the original mass of the projectile mpand the target material in the bulging region ms,that is,m1=mp+ms=mp+ρA1h0.Another related assumption is that the effective mass of the projectile increases during the first stage because of the addition of the target material displaced in the direction of motion.The kinetic energy imparted to the added mass by the projectile remains stored in the combined effective mass of the projectile.

The rate of change of the effective mass of the combined projectile and added mass is

Substituting Eq.(4)and the relation

into Eq.(3)yields

The differentiation of Eq.(2)with respect to time t yields

Eliminating A1from Eq.(6)by substituting it with Eqs.(2)and(7)leads to

According to the rigid plastic assumption,the dynamic yield stress of the target plate during the first stage can be determined by the well-known Cowper-Symonds empirical constitutive equation.

where D and q are the empirical constants chosen to describe the strain rate-sensitive behavior of the target material.In the case of mild steel,D and q are set to be 40.4 s-1and 5,respectively.

The mean strain rate in the above equation may be expressed as[14].

Substituting the geometrical relation between rAand w01in Eq.(2)into the above equation yields

Denoting the terms in the square bracket at the right-hand side of Eq.(9)byβ1,

the dynamic yield stress during the first stage can be expressed as

Substituting Eq.(13)into Eq.(8)and rearranging the equation yield

In Eq.(14),the velocity of the combined projectile and added mass v1can be considered a function of the maximum deflection of the bulging region w01,that is,v1=v1(w01).Therefore,the time it takes the combined projectile to penetrate the distance w01can be calculated by the numerical integration of the expression

The extension velocity of the bulging region decreases with the motion of the projectile.When the radius of the bulging region is equal to the propagation radius of the plastic wave,the remaining part of the target plate starts to produce dishing deformation.Moreover,the first stage ends under the following termination condition:

Through the combination of Eqs.(14)and(15)with the termination condition in Eq.(16),the radius and maximum deflection of the bulging region at the end of the first stage can be calculated with the numerical method.If the initial projectile velocity is considerably high such that the propagation radius of the plastic wave is still less than the radius of the bulging region when the radius of the bulging region reaches the projectile radius during the penetration process,that is,cpt1(w01)≤rA(w01),then the termination condition for the first stage is rA=rp.

4.1.2.Energy of bulging deformation

The maximum deflection w01at the end of the first stage can be determined in the preceding section,and the energy absorbed by the target plate during stage 1 can then be calculated.The target plates are thin,and the initial projectile velocities are in the low velocity range;thus,the bending moment and membrane force are the two types of generalized stresses that are predominant in the bulging region[28].As a result,the work conducted due to the shearing stress is neglected,and only the energies of bending and membrane stretching are considered.

The transverse deflection in the bulging region at the end of the first stage can be expressed as

where rpis the projectile radius and r is the arbitrary radial coordinate of the plate.

Accordingly,the energy of the bending deformation U1bduring stage 1 is

The energy of membrane stretching U1mduring the first stage can be calculated by Ref.[32].

whereβ1is obtained with Eq.(12),in which v1and w01are taken to be their respective values at the end of stage 1.

At the end of the first stage,the target material in the bulging region in contact with the projectile nose moves at the same velocity of the projectile,and its mass does not change as the projectile advances.

The total plastic deformation energy E1pabsorbed during the first stage is

4.2.Stage 2:dishing deformation

4.2.1.Energy of dishing deformation

As indicated by the experimental observations,transverse deflections of a certain extent are produced surrounding the perforation hole in the target plate.However,small deflections occur in regions away from the perforation,and they decrease rapidly with an increase in the arbitrary radius of the plate.This phenomenon may be due to the fact that the perforation time is significantly short such that plastic waves and plastic hinges have no sufficient time to propagate in stage 2[28].In the present analysis,the energy of dishing deformation during the second stage is calculated.In this calculation,the effects of the propagation processes of the plastic waves and plastic hinges on the transverse deflections in the remaining part of the target plate are neglected;only the energy of the dishing deformation at the end of the second stage is considered,that is,at the time of failure at the target plate at the periphery of the bulging region(Fig.5(d)).The transverse deflection in the dishing region at the end of stage 2 can be expressed as(Ref.[10])

where w02is the maximum deflection of the dishing region at the end of stage 2,a is a constant that can be evaluated from the profile of the dishing region of the plate,and rAis the radius of the bulging region.

Assuming the usual thin plate,plane stress,and symmetry conditions,the energy of dishing deformation during stage 2 includes the radial bending energy U2rb,the circumferential bending energy U2θb,and the radial stretching energy U2rmbased on the assumption of εθ=0,using the one-to-one correspondence between the stress and strain characteristic of the deformation theory of plasticity.

The energies of radial and circumferential bending absorbed during stage 2 are as follows:

where M02is the dynamic fully plastic moment per unit length at the end of stage 2,and kr2and kθ2are the radial and circumferential curvatures,respectively.

The energy of radial stretching absorbed during the second stage is

where εrmis the radial stretching strain.For large deflections of thin plates,the radial stretching strain εrmfor small radial displacements can be approximately written as

Substituting Eq.(27)back into Eq.(26)yields

After substituting the expression of w2(r)in Eq.(23)into the above equation,the radial stretching energy U2rmcan be obtained.Thus,the total plastic deformation energy E2pabsorbed during the second stage is

The dynamic yield stressσd2during the second stage can be determined by the well-known Cowper-Symonds empirical constitutive relation

whereβ2is calculated by

Similar to the first stage,the mean strain rate at the end of the second stage can be approximately written as

where w02and v2are the maximum deflection of the dishing region and projectile velocity at the end of the second stage,respectively.rB2represents the radius of the dishing region,which obviously relates to the projectile velocity and plate thickness.If the value of rB2is set to be equal to the plate radius R,then the mean strain rate calculated by Eq.(32)is excessively small.In reality,the assumed deflection profile of the dishing region of the target plate by Eq.(23)is inclusive of the characteristic that the deflections of the target plate in the regions far from the contact region are extremely small,thereby then resulting in a small contribution to the total energy calculations.

To reasonably calculate the mean strain rate for the second stage,the radius of the dishing region is set approximately as the radius of the region in which the deflections are larger than or at least equal to one tenth of the plate thickness.In other words,the transverse deflections at the outer radius of the dishing region are equal to one tenth of the plate thickness,that is,

where h0is the original plate thickness.Hence,the radius of the dishing region at the end of the second stage is

4.2.2.Failure criterion

An effective strain criterion is suggested in Ref.[15]for the perforation of rigid-plastic plate by blunt missiles.This criterion incorporates the influences of both tensile strain and shear strain,that is,

where εe,εrm,γ,and εfare the effective strain,radial tensile strain,shear strain,and failure strain of the target material,respectively.

The present experimental observations indicate that the tensile tearing at the periphery of the bulging region is the predominant failure mode of the thin steel plates that are perforated by hemispherical-nosed projectiles at low velocities.Therefore,in the present analysis,the influence of the shear strain is ignored.Moreover,the rupture of the target plate is considered to occur when the radial tensile strain εrmat the periphery of the bulging region(r=rA)reaches the failure strain of the target material εf,that is,

The above criterion also neglects the part of the strain that is caused by bending because experiments show that for thin plates with large bending deformation,a crack forms near a point of inflection in which the radial curvature vanishes.

By substituting εrmin Eq.(27)related to w02into Eq.(36)and rearranging them,the maximum transverse de flection of the dishing region at the end of the second stage is determined by the following expression:

After substituting w02in the above equation into Eq.(29),the total plastic deformation energy absorbed by the target plate during the second stage can be obtained.

4.3.Stage 3:ductile hole enlargement

At the end of the second stage,the bulging region in contact with projectile nose fails at its periphery.An initial hole,whose diameter is less than the projectile diameter,then forms and leads to a plug with a diameter that is also less than that of the projectile.As the projectile moves,the initial hole is further enlarged in a ductile fashion by the projectile nose(as shown in Fig.5(e)).Simultaneously,the remaining part of the target plate other than the hole,that is,the dishing region,is further deformed.The third stage ends when the diameter of the enlarged hole equals the projectile diameter,during which the projectile nose completely penetrates the target plate.During the third stage,the energy absorbed by the target plate consists of the energy induced by the ductile hole enlargement of initial hole and the energy caused by the further plastic deformation of the dishing region.

The energy W3hcaused by the ductile hole enlargement is mainly the energy W3θmof the circumferential stretching that is caused by the circumferential stress and circumferential strain(i.e.,neglecting the radial strain).If the condition rA(1+εf)≥rpis fulfilled,then W3h=W3θm.However,if the condition rA(1+εf)≥rpis not satisfied,that is,rA(1+εf)

where σd3= β3σ0is the dynamic yield stress for stage 3.The propagation energy of the radial cracks is evaluated as

where N is the number of radial cracks,which can be calculated approximately as[33].

where Gcis the fracture toughness of the target material.

During the third stage,the dishing region other than the contact region is further deformed because of the kinetic energy that it acquires in the second stage and the hole enlargement that is caused by the motion of the projectile.Therefore,the energy that results from further plastic deformation of the dishing region in stage 3 remains.The experimental results indicate that the kinetic energy of the dishing region acquired during stages 2 and 3 is assumed to be completely dissipated by transverse plastic deformation.Furthermore,the diameter of the ultimate perforation is assumed to be equal to the projectile diameter after the projectile nose fully penetrates the target plate.Thus,the deformation of the dishing region after the third stage is the ultimate deformation of the target plate that is perforated by the projectile.As shown in Fig.5(f),the final plate deflection in the dishing region can also be expressed as

where w03can be obtained by the experimental measurement of the maximum deflection of the target plate.

In the third stage,the radial stretching strain is assumed to be equal to zero,and its effect can be neglected.Accordingly,the plastic deformation energy of the dishing region mainly consists of the radial bending energy and circumferential bending energy,which are listed as follows

where M03is the mean dynamic yield stress per unit length during the third stage,and kr3and kθ3are the radial and circumferential curvatures,respectively.Equations(42)and(43)include the partial radial bending energy and circumferential bending energy of the dishing region absorbed during the second stage,respectively.Hence,by subtracting the respective partial bending energy absorbed during the second stage from Eqs.(42)and(43),the radial bending energy U3rband circumferential bending energy U3θb absorbed during stage 3 are

The total plastic deformation energyE3pof the dishing region during the third stage is

The dynamic yield stress in the third stage is still determined by the Cowper-Symonds empirical constitutive relationship

whereβ3is calculated by

Similar to the second stage,the mean strain rate in the third stage can be approximately evaluated as

where w03is the ultimate maximum deflection of the perforated target plate,v2is considered the projectile velocity at the end of stage 2,and rB3is the radius of the dishing region at the end of stage 3,which can be approximately taken as

4.4.Stage 4:projectile exit

The fourth stage starts when the projectile nose completely penetrates the target plate(Fig.5(f)).This stage ends when the projectile entirely perforates the plate,followed by its exit from the target plate.In this stage,no forces act on the projectile nose,and only small frictional force exists between the side face of the projectile shank and the inner surface of the perforation.This finding is confirmed by the hot projectile and the observation that a few of the polished marks are found on the side face of the projectile shank after each test.However,the energy absorbed in friction is generally smaller than that absorbed in plastic deformation and may be completely neglected[34].

4.5.Calculation of residual projectile velocity

Based on the assumption that the projectile is rigid and nondeformable,the lost kinetic energy of the projectile is predominantly absorbed by the target plate in the form of plastic deformation and kinetic energy of the plug.Assuming that the plug detached from the target plate moves at the same velocity of the projectile,using the law of energy conservation yields

Accordingly,the residual velocity of the projectile after the perforation of thin steel plates can be expressed as

Given that the energy calculation for each stage(except the fourth stage)involves the corresponding projectile velocity,the projectile velocities at the end of the first and second stages should be first solved in sequence.Thus,the energy absorbed in each stage can be obtained.After substituting the energy absorbed in each stage into Eq.(52),the residual velocity of the projectile can be determined.Notably,in the calculation process of the projectile velocity at the end of the second stage,the instantaneous kinetic energy in the dishing region must be considered.

To compare and demonstrate the effect of plate dishing on the total perforation resistance of the target plate,the empirical model proposed by Ipson and Recht[35](hereafter called I-R model),which is based on the assumption of shear plugging,is used to calculate the residual velocities of the projectiles:

where mpis the projectile mass,is the shear plug mass and vbis the ballistic limit of target plate,which can be determined by the de Marre formula(as seen in Ref.[36])

where d is the projectile diameter,h0is the target thickness,andαis the incident angle,for normal impactα=0.Kdeis a complex coefficient that considers many factors,such as nose shape and target material property.More detailed methods to obtain Kdecan be found in Ref.[36].In the present study,Kdeis taken as(as seen in Ref.[36])

whereσsis the yield stress of the target plate;φ =6160Ce/Cm,Ce=h0/d and Cm=mp/d3(kg/m3)are the relative thicknesses of the target plate and relative mass of the projectile,respectively;and k is the effectiveness coefficient.For blunt-nosed projectiles,k can be expressed as

whereiis the shape coefficient and is determined as

where n1is the ratio of the curvature radius of the projectile nose to the projectile radius and n2is the ratio of the bluntness diameter of the projectile nose to the projectile diameter.

5.Comparison with experimental results and discussion

5.1.Comparison of the radius of the bulging region

To validate the rationality of the calculation method on the radius of bulging region,the radius of the theoretically calculated bulging region is compared with that experimentally measured as demonstrated in Fig.6(a-c)for the target plates of thicknesses 1.36,1.90,and 2.86 mm.The process of measuring the radius of the bulging region is as follows.First,the mass of plug msis measured,and then,based on the relationthe radius of the bulging region can be obtained by calculating the equation rA=(ms/πρh0)0.5.In the theoretical calculation,the following parameters are used:projectile radius rp=7.45 mm and quasi-static flow stressσ0=272 MPa,which is computed based on the mechanical parameters that are given in Table 1.As shown in Fig.6,the theoretical calculations are in reasonably good agreement with the experimental data.In addition,the radius of the bulging region increases as the initial projectile velocity increases.This radius also tends to reach the value of the projectile radius when the initial projectile velocity is substantially high.

5.2.Comparison of the residual projectile velocity

Fig.8 compares the theoretically predicted residual projectile velocities and the experimental results.In the table,h0is the thickness of the target plate,mpis the projectile mass,and viand vrare the initial and residual projectile velocities,respectively.wmaxis the maximum transverse deflection of the target plate,which is experimentally measured.The maximum deflection wmaxof the perforation case is the maximum deflection in the dishing region at the end of the third stage,that is,wmax=w03.In the calculation process,the failure strain εfof the target material is obtained based on Table 1,which is set as εf=ln(1+δs)=0.30.The constant a is evaluated from the experimental results by the least-mean-square fit of the deflection pro file of the plate.A representative best- fit curve for the deflection pro file of the plate in test no.6 is illustrated in Fig.7.Relevant experimental studies[10,24]showed that,for target plates made of the same material and with equal thicknesses perforated by hemispherical-nosed or spherical projectiles with identical diameters,the constant a changes slightly with a change in the initial projectile velocity.This conclusion is also obtained in the present experiments via the analysis of the experimental results.Consequently,in the theoretical calculation,the constant a is set as the same value for the tests in which the target plates are of the same thicknesses.In the present experiments,the values of the constant a for 1.36,1.90,and 2.86 mm-thick mild steel plates are 738,483,and 312 m-1,respectively.

Fig.7.Fitting of the deflections for the target plate in test no.6.

As shown in Fig.8,the theoretical calculations of the residual projectile velocities using the present model,Eq.(52),are in good agreement with the experimental results.For the majority of the tests,the relative errors are less than 10%,except for tests no.1 and 8.The slightly larger discrepancies in tests no.1 and 8 may be attributed to the precision of the measuring equipment for velocity,the change in the dynamic parameters of the target material,and the flying attitude of the projectile.In addition,the residual velocities of the plugs in tests no.1 and 8 may be higher than their corresponding residual projectile velocities.By contrast,the residual velocity of the plug in the present analysis is assumed the same as that of the residual projectile velocity,which in turn leads to a relatively lower kinetic energy of the plug than that experimentally obtained.Thus,the theoretically calculated residual projectile velocities are slightly higher than those experimentally measured for tests no.1 and 8.

The residual projectile velocities that are theoretically predicted by the I-R model,Eq.(53),are also presented in Fig.8.As the table shows,the predictions of the I-R model are all much higher than the experimental results,and large relative errors are gained between them.This result is mainly because the I-R model is based on the assumption of shear plugging and ignores the effect of dishing on the perforation resistance of the plate.Further comparison of the residual velocities that were predicted by the current model and those by the I-R model shows that the current model can predict residual velocities much better than can the I-R model in the case of thin steel plates perforated by hemispherical-nosed projectiles.

Fig.6.Comparison of the experimental and calculated radius of the bulging region.

Fig.8.Comparison of the experimental and calculated residual velocities of projectiles.

The residual velocities of projectiles calculated by adopting the three-stage model in Ref.[30]were also plotted in Fig.8 for test cases with different plate thicknesses.Fig.8 indicates that the residual velocities of projectiles calculated by the three-stage model were all less than those in the test results.Furthermore,the discrepancies between the calculated and test results exaggerates with the increase of plate thickness.In other words,the energy absorbed by the target plate was underestimated by the three-stage model,and the underestimation extent increased with increasing plate thicknesses.The reason for this result is mainly due to the effects of the two aspects as follows:(1)the strain rates in every stage were not appropriately considered,thereby resulting in under-predicted energy absorption in the earlier penetration stage;(2)the energy dissipated during the stage of ductile hole enlargement in the present four-stage model was ignored in the threestage model,which also rendered the underestimation of the energy absorptions by target plates.Furthermore,in the calculation process by the three-stage model,the dynamic yielding stresses of the target plates were all taken as 2.5 times of quasi-static yielding stresses,that is,σd=2.5σs,and were considered unchanged in every stage.This does not fully simulate a real situation.Moreover,the results that are calculated by the three-stage model depend heavily on the value of the dynamic yielding stress of the target plate,thereby lowering the accuracy of this model.

Considering the dispersion of the ballistic experiments,the analytical model established in the present study can efficiently predict the residual projectile velocities in the case of thin steel plates that are normally perforated by hemispherical-nosed projectiles.However,a large deviation may be found between the theoretical predictions of the current model and the experimental results in the case of the perforation of thick steel plates at relatively high impact velocities.This deviation is mainly attributed to the neglect of the shearing effect in the energy calculation of the target plate and the failure criterion used in the present analytical model.

6.Conclusions

Ballistic experiments are conducted to generate experimental data on the perforation of thin steel plates that are impacted normally by hemispherical-nosed projectiles.Based on the experimentally observed mechanisms of deformation and failure,a fourstage model applicable to these mechanisms is developed to predict the residual velocities of the projectiles.On the basis of the rigid-plastic assumption,a new method is proposed to determine the radius of the bulging region by adopting the dynamic method combined with the plastic wave propagation concept.By dividing the perforation process into the four stages of bulging deformation,dishing deformation,ductile hole enlargement,and projectile exit,a new model is developed to predict the residual projectile velocity using energy balance.A good agreement is obtained between the theoretical predictions and the experimental results in terms of both the radius of the bulging region and the residual projectile velocity.This good agreement proves the reasonability and rationality of the analytical model that has been established in the present study.

Acknowledgements

This work is financially supported by the National Security Major Foundation Research Project(973)of China(6133050102);the National Natural Science Foundation of China(Grant No.51409253),which are gratefully acknowledged.The authors would like to thank Ms.Qiong Yi for her assistance with the test measurements and helpful English revision.

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A R T I C L E I N F O

Article history:

10 April 2017

in revised form

11 June 2017

Accepted 13 June 2017

Available online 15 June 2017

Ballistic impact

Thin steel plate

Perforation

Low velocity

Hemispherical-nosed projectile

Theoretical analysis

Ballistic experiments were conducted on thin steel plates that are normally impacted by hemisphericalnosed projectiles at velocities higher than their ballistic limits.The deformation and failure modes of the thin steel plates were analyzed.A new method was proposed according to the experimental results and the perforation phenomenon of the thin steel plates to determine the radius of the bulging region.In establishing this new method,a dynamic method combined with the plastic wave propagation concept based on the rigid plastic assumption was adopted.The whole perforation process was divided into four consecutive stages,namely,bulging deformation,dishing deformation,ductile hole enlargement,and projectile exit.On the basis of the energy conservation principle,a new model was developed to predict the residual velocities of hemispherical-nosed projectiles that perforate thin steel plates at low velocities.The results obtained from the theoretical calculations by the present model were compared with the experimental results.Theoretical predictions were in good agreement with the experimental results in terms of both the radius of the bulging region and the residual velocity of the projectile when the strain rate effects of the target material during each stage were considered.

?2017 Published by Elsevier Ltd.This is an open access article under the CC BY-NC-ND license(http://creativecommons.org/licenses/by-nc-nd/4.0/).

*Corresponding author.

E-mail addresses:chenchanghai0746@163.com(C.-h.Chen),zhuxi816@126.com(X.Zhu),hou9611106@126.com (H.-l.Hou),xbtian@163.com (X.-b.Tian),Xiaoleshen@126.com(X.-l.Shen).

Peer review under responsibility of China Ordnance Society.