A novel method for determination of lethal radius for high-explosive artillery projectiles

Alan Catovic ,Elvedin Kljuno

b Mechanical Engineering Faculty,University of Sarajevo,Sarajevo,Bosnia and Herzegovina

Keywords: High explosive projectiles Lethal radius Fragments Modeling

ABSTRACT A new model has been defined that enables the estimation of the lethal radius(radius of ef ficiency)of HE(High Explosive)artillery projectiles against human targets.The model is made of several modules:CAD(Computer Aided Design)modeling,fragment mass distribution estimation,fragment initial velocity prediction,fragment trajectory calculation,effective fragment density estimation,and high explosive projectile lethal radius estimation.The results were compared with the experimental results obtained based on tests in the arena used in our country,and the agreement of the results was good.This model can be used in any terminal-ballistics scenario for high explosive projectiles since it is general,parametric,fast and relatively easy to implement.

1.Introduction

High explosive projectile lethal radius is an important terminalballistics parameter.When the projectile has a larger lethal radius,fewer artillery projectiles are necessary to eliminate the targets.From a tactical point of view,this can be signi ficant since it reduces logistical burden.There are many factors in fluencing projectile efficiency and because of the complexity of these parameters,there is currently no comprehensive model for estimation of its terminalballistics parameters.

In the literature,one can find different models for prediction of high explosive projectile ef ficiency,and one of the notable is american model[1],which de fines a lethal area as a measure of the fragment casualty producing potential of an exploding projectile when employed against soldiers.Lethal area in this model is the expected number of incapacitations,after the detonation of a single projectile,using the certain level of density of enemy soldiers.Incapacitation refers to a debilitating injury(ie,hit into a vital body area after which the soldier is inacapable of fighting).This american model is a probability model,where a probability that the person will be incapacitated is calculated using the probability of incapacitation due to shockwave and probability due to fragments.Zecevic et al.[2,3]were also developing a terminal-ballistics model for high explosive projectiles,where a lethal zone of naturally fragmenting high explosive warheads was determined using the coupled numerical-CAD(Computer Aided Design)technique.These models and several other models mentioned in the literature[4-6]depend primarily on experimental data from the arena fragmentation test.These tests are very expensive,time consuming and labor intensive.

Our approach was to develop a new analytical model for prediction of high explosive projectiles lethal radius,where there is no need for experimental tests in inital phase of design.This way an ammunition designer is provided with preliminary projectile ef ficiency data before conducting large scale experimental fragmentation tests(pit and arena),which saves both money and time.

Some of the major contributions in the paper are related to the calculation of the movement of fragments:1)using aerodynamic forces and moments rather than the force coef ficient(s),and 2)using complete 6DOF(Six Degrees of Freedom)equations for fragment trajectory estimation.

The model,presented in this work,is in some ways similar to fragment hazard method(DDESB;Department of Defense Explosives Safety Board)adopted by US DoD(United States Department of Defense)and other similar organizations years ago which is detailed in many technical reports(ie reference[8]).The major difference between the method described herein and the widely accepted method is the trajectory module in our model,whereby the kinetic energy of fragment is calculated accurately during the ballistic travel of fragment,by solving all necessary dynamic equations(fully coupled 6DOF/Six Degrees of Freedom equations),whereas the DDESB(Department of Defense Explosives Safety Board)methods employ an analytic expression of kinetic energy as a function of distance and some other approximations which can lead to errors.

In that regard,we observed following possible de ficiencies in the currently accepted DDESB/Department of Defense Explosives Safety Board fragment hazard model:

-It considers only the drag force and the force of gravity.Since this is a 2-dimensional model it cannot resolve the spatial motion of fragments(ie it cannot model side drift of the body).

-The aerodynamic moment is not calculated,so the equation of change of angular momentum is not taken into account.This means that the model does not address body rotation;and even if it is to take the moment into account,with the 2D modelspatial rotation cannot be solved,but only rotation about an axis perpendicular to the plane of motion.

-The drag force coef ficient(CD)is taken as a constant value(ie a rather small value of 0.8)which can introduce large errors in the estimation of aerodynamic force,especially in the calculation of entire trajectories(where theCD(Mach)curve varies considerably,especially in a transonic zone).It is known that theCDchanges due to a change in the shape of the exposed surface of the body in the current direction of movement due to its rotation and due to a change in the speed of the body.

-DDESB/Department of Defense Explosives Safety Board model uses fragment form factorkwith values from 2609 kg/m3(forged projectiles)to 4740 kg/m3(steel spheres).This means that for the lower boundary of the shape factor it takes into account the type and method of processing the body material,and for the upper boundary it takes the shape of the fragment,which means that it is not a unique parameter taken for comparison.This factor is also taken as a constant value,and these approximations can lead to further calculation errors.

-DDESB(Department of Defense Explosives Safety Board)model is applicable only for the high-order detonation of a cylindrical casing with evenly distributed explosives.The more an item deviates from this ideal,the less reliable this methodology becomes[8].

-This model does not consider the fragment hazard presented by base plates,nose plugs,boattails,or other fragments from the non-cylindrical portions of the case.These unique fragments can travel to signi ficantly greater distances(distances greater than 3050 m).These special fragments should be considered on a case-by-case,site-speci fic analysis[8].

-To determine the maximum fragment range,the range of the maximum mass fragment for each region of the model can be calculated using the initial velocity for that region and the largest of these ranges is defined as the maximum fragment range of the munition.Since upper bound estimates are used,the resulting range that is computed represents a conservative estimate.In all cases where the data allow the comparisons to be made,the methodology in DDESB/Department of Defense Explosives Safety Board produces conservative results,i.e.ranges greater than the measured range[8].

-The program(code)is not available for public use,and the literature does not cite comparisons of results obtained from this model with numerical simulations for fragments.

-Crull&Swisdak[8]report validation of DDESB(Department of Defense Explosives Safety Board)model by comparing the results to the distances listed in several literature references for several munitions.Comparisons have been made for both the Hazardous Fragment Distance and the Maximum fragment Range.In all cases where the data allow the comparisons to be made,the DDESB(Department of Defense Explosives Safety Board)model produces conservative results,i.e.,ranges greater than the measured range.For example,for 105 mm M1 artillery projectile relative difference for maximum fragment range between DDESB model and literature data is around 90%,and for 81 mm M56 mortar projectile this difference is 110%[8].

2.Physical model

The lethal radius of a high explosive projectile against human targets in the model is defined as a distance from the explosion at which there is an ef ficient fragment(with a kinetic energy of at least 80 J)density of 1 frag/m2.It is assumed that the human target exposed area(to the projectile)is 1 m2.This area can be reduced(ie when it is assumed the soldier is in prone or some other position on the ground).Incapacitation in the model is related to a hit into a human target with a fragment of 80 J kinetic energy(adopted NATO inacapacitation criterium).The model does not take into account the condition and age of the human target,its protection,and the location of the impact of the fragment into the human body.

It is assumed that the position of a projectile upon detonation is vertical to the ground.That means lethal radius represents a radius of a circular lethal zone-a maximal possible lethal zone of projectile on the ground(real artillery projectiles have smaller impact angles).This parameter(lethal radius)can be used in an initial phase of design to compare the ef ficiency of different high explosive projectiles.

Our model for evaluating the fragmentation ef ficiency is based on analytical,numerical and CAD/Computer Aided Design methods,and consists of several modules:

1.CAD(Computer Aided Design)modeling module(high explosive projectile).

2.A module for estimation of mass distribution of fragments.

3.A module for predicting the initial velocity of fragments.

4.Fragment trajectory calculation module.

5.Module for estimating the density of ef ficient fragments,and

6.Module for estimating the lethal radius of high explosive projectiles.

2.1.Module for CAD(Computer Aided Design)modeling

Within this module,in some of the CAD(Computer Aided Design)software,one determines the mass(body masses and explosives and mass ratioC/M)and the geometric ratiot/d(average body thicknes to average explosive diameter ratio).The CAD(Computer Aided Design)method is also used to split the 3D model of the projectile into cylindrical segments(Fig.1).We have segmented projectile bodies into a large number of segments because of their variable geometry.In this way,an attempt was made to appreciate the geometry of the body as close as possible to reality(especially when the geometry is rapidly changing,ie at the end of ogive part).

Fig.1.Drawing,122 mm OF-462 projectile model(3-Dimensional)and central body segments(up to the rotating band,16-31 segments)used in the model to evaluate its ef ficiency.

The geometric parametert/d,using the body and explosive charge volume data obtained from CAD(Computer Aided Design)software,is analytically determined by first calculating the equivalent diameter of the explosive d(the explosive charge is converted to a cylinder):

HereLexpis explosive charge length andVexpis explosive charge volume,determined based on the CAD(Computer Aided Design)model of the projectile.

The average(so-called equivalent)thicknesstof the projectile body is given by:

HereDis the average(equivalent)diameter of the projectile,determined by the expression:

HereVtis the volume of high explosive projectile body containing the explosive(determined by the Computer Aided Design method).The volume of the bodyVtcan also be determined analytically by dividing the projectile into segments and summing the volumes of the segments.CAD(Computer Aided Design)models of projectiles(such as one in Fig.1)are also used for calculation of the initial velocity of projectile fragments(splinters),as well as to de fine initial conditions(the initial coordinates of fragments,depending on their position on the body of the projectile)in a generalized mechanics model(6DOF/Six Degrees of Freedom model)for fragment trajectory.CAD(Computer Aided Design)models are also used to evaluate,in a real(1:1 scale)fragmentation test(arena)model,depending on the calculated range of fragments,whether fragments of individual masses can reach arena panels.Here,the Computer Aided Design model is used to calculate the polar angles that appear in expressions for estimation of the density of ef ficient fragments.CAD(Computer Aided Design)software can also be used to model the fragments themselves(stochastic shape),either by drawing a fragment,based on its projections[7],or indirectly after 3D scanning.

2.2.A module for estimation of mass distribution of fragments

To estimate the numberN(m)of fragments from a high explosive projectile,for each quasi-cylindrical segment of the body,the Mott formula is used[9,16]:

The massMof the projectile body is known,based on CAD/Computer Aided Design model of projectile(this information is given in the test protocol,for example,in the fragmentation pit,but also in the factory prospects of ammunition).Based on expression(4),for the condition whenm→0,a formula is obtained for the total number of fragmentsN0resulting from the fragmentation of the body of a high explosive projectile:

Expression(5)is also used to estimate the total number of fragments from certain cylindrical segments of the projectile wherein then the massMcorresponds to the mass of the given segment.Using Mott’s formula(4),whenN=1,the expression for calculating the mass of the largest fragment from the body of a given projectile can be obtained:

This mass can later be also used to estimate the maximum possible range of fragments resulting from fragmentation of a given projectile.The characteristic Mott parameterμfrom the above expressions is determined by the formula:

HereBMis the Mott constant that depends on the material of the projectile body and the explosive charge,whiletis the equivalent thickness of the projectile body anddis the equivalent diameter of the explosives determined by the previously described method(Computer Aided Design modeling of the projectile).When fragmentation data is available for a projectile(testing in the pit),optimization of the Mott constant(for a given projectile)can be performed,using the expression:

In this way(by optimizing the Mott constant)the agreement between the experimental data obtained in the pit test and the Mott model is better than using the value of theBMconstant reported in the available literature.Data on the values of the optimizedBM_optconstant for individual high explosive projectiles used in our country can be found in Refs.[7].

When analyzing the parameters of a new projectile(eg in the preliminary design process for new ammunition),it is suggested that if the experimental data obtained by fragmentation a given projectile in the pit are not available,the averaged values of theBMsrconstant be adopted for the given projectile caliber and appropriate explosive charge for projectiles of the same(or closest)caliber of the same type of ammunition and of the same explosive charge and body material.

Since the Mott constant is not the same for any one projectile(even for the same type of ammunition,in different tests),it is possible,for example,to use the Monte Carlo method to estimate the value of a given constant in the appropriateBMrange obtained experimentally using a random number method.

When all the necessary parameters are known,the Mott mass distribution parameterμfor a given projectile is first determined.Therefore,based on the model,it is possible to calculate the average mass of fragments(2μ)obtained by fragmentation(this data can be compared with the data from the tests,if available).

The next step in the model is to apply the Mott method to selected segments of high explosive projectiles that presumably participate in penetrations into the target(Fig.1,for example,segments 16-31 on the projectile body to the rotating band).In this way,the cumulative distribution of the number of fragments for the given segments is obtained,using the Mott formula(expression 4).These data are used further in the model when estimating the density of ef ficient fragments.

The model assumes that only fragments from the central part of the body to the rotating band(Fig.1)are involved in penetrations,ie it is assumed that fragments from the front and rear spray do not signi ficantly contribute to the ef ficiency of the high explosive projectile.

The model assumes that the projectile is in a vertical position(90°impact angle simulation).In this way,the lethal radius of the projectile is ultimately obtained,and the results can be compared,for example,with the results from the quarter-circle arena used in our country.

2.3.A module for predicting the initial velocity of fragments

When,based on the Mott method,the number of fragments from individual projectile segments is known,the initial velocity of these fragments,required to calculate their trajectory through the atmosphere to the target,is determined.

The initial velocities of fragments are estimated by the Gurney model[17]:

Here,MandCare the masses of the projectile body and explosive,respectively,andis Gurney’s constant.Detonation velocityDof explosive charge can be determined by software(in EXPLO5[12],for a given charge density-determined from the table data for charge mass and charge volume determined in the CAD(Computer Aided Design)system-based on projectile drawings).The Gurney constant can be determined analytically,e.g.based on detonation velocity of explosives, using the expression=0.338D[13],where Cooper averaged test data for a twentyfive conventional explosives.In our country mainly TNT and Comp.B were used in the projectiles mentioned in the paper so we limited the model for these explosives(as per Cooper suggestion).This relationship is capable of accurately predictingof sensitive and moderately sensitive near ideal explosives;however,this equation is somewhat unsuitable for predictingof insensitive and nonideal explosives(ie.NTO,PBX-9502,AFX-902),because it overestimatesof some insensitive explosives.As a result,the model may overestimate the lethal radius of projectiles containing insensitive explosives such as the new M795 155 mm artillery projectile filled with IMX-101(DNAN/NTO/NQ)or even the lethal radius of the 122 mm OF-462 filled with amatol(nonideal explosive).

In the model presented in the paper,a simultaneous overall detonation of explosive is assumed,and initial velocity varies along the axial coordinate of projectile only due to the varying shell thickness(ratioC/M).Regarding accuracy of the Gurney model,Walters and Zukas[14]state: “The Gurney approximation is based on the conservation of momentum and energy,and the results represent excellent engineering approximation,within 10%of the experimental results or detailed numerical result.

Basically,none of the Gurney-like(modi fications)models provide signi ficant improvement over the basic Gurney expressions,but they serve to extend the Gurney method to geometries not covered by the Gurney assumptions,for example,implosive geometries,or extension of theM/Crange of validity.Karpp and Predebon showed that fragment velocity predictions based on Gurney formula is adequate for cases when flow is onedimensional(radial)and for practical ranges of charge to metal ratios(0.1

Fig.2.Initial velocities of fragments for projectile 122 mm OF-462.

Fig.2 shows the curve of the fragment’s initial velocity for the 122 mm high explosive OF-462 projectile as a function of position along the projectile(relative centers of segments on the projectile body).With this projectile,the wall in the central part of the body is relatively thick(relative to,for example,a 122 mm M76 projectile),so in these body segments,the initial velocities of the fragments are slightly less than in some segments on the frontal part.

The front part of the 122 mm OF-462 projectile(Fig.2)has slightly higher initial velocities,but for the assumed position of the projectile(90°impact angle),most of this forward spray of fragments,as previously assumed,ends in the ground and does not reach the target(e.g.,quarter-arena panels).The model assumes that there is no ricochetting of fragments,which is a reasonable assumption for terrestrial terrain,but in urban conditions,ricochetting from the ground can occur,especially if the soil is very hard(eg granite slabs).On the other hand,Fig.2 shows that the end spray of fragments has much less initial velocity.It is assumed that this part of the beam of fragments does not affect the parameters of the ef ficiency of the projectile,although fragments from the bottom of the projectile may have a relatively long range(some authors report the range of fragments fromthe bottom of the projectile over 3 km).

The initial velocity of fragments,determined by the Gurney method,for quasi-cylindrical projectile elements(ratio of body mass to mass of explosive chargeM/Cgreater than 0,5-usually the case for high explosive projectiles)represents the upper limit(maximum)of the fragment velocity because there are energy losses due to the fracture of the projectile body and the early leakage of gaseous detonation products.On the other hand,for relatively thin projectile walls,whereM/C<0.5(which is rarely the case in practice),the real initial velocities of the fragments are slightly higher than obtained using the Gurney method.

Similar to the research performed by Crull and Swisdak for US DDESB(Department of Defense Explosives Safety Board)[8],projectile segments are also approximated by cylindrical elements in this model,with the initial velocity fragment vector perpendicular to the projectile axis.The expansion of the projectile body is not taken into account.The obtained data on the initial velocities of fragments are further used in the model for the calculation of the trajectory of fragments,for realistic initial conditions.

Fig.3 provides a different way of displaying the initial velocities of high explosive projectile fragments(comparing to Fig.2),represented via a polar diagram(Grapher?software was used)in which one axis represents the initial velocities of the fragment(units m/s)and the angular axis of the diagram represents the angles measured from the projectile axis to the centers of the individual projectile segments.The point of origin(lower part of Fig.3)is represented by the center of mass of the projectile.The directions in the lower part of Fig.3(on the projectile)should not be confused with the initial velocity vectors;they serve here only to estimate the angles measured from the projectile axes to the centers of the individual projectile segments(in order to construct the diagram in Fig.3)Such diagrams can serve to better visualize the parameters of the terminal ballistics of a high explosive projectile.

Fig.3.Variation of initial fragment velocity for 122 mm OF-462 projectile as a function of segment position,presented in the polar diagram.

2.4.Fragment trajectory calculation module

Trajectories are calculated for representative fragments by geometrically arranged segments per projectile body,with a discrete variation of the mass of fragments by mass groups,so that the computation of trajectories for all fragments is reduced to an acceptable number of trajectories,much smaller than the actual number of fragments.The projectile body is divided into a number of segments,and then a representative fragment is determined at the center position of each body segment.Projectile segments are broken down into numerous fragments,which are divided into a number of mass groups.The calculation of the trajectories of all fragments for one segment of the projectile body is reduced to the calculation of the trajectory of one representative fragment,with the variation of mass by a certain number of mass groups.

There are two variations in the number of fragment trajectories:

a)position of fragments representative for each projectile segment,

b)a representative mass of a particular mass group of fragments.

For example,if the projectile body is divided into 6 segments and the mass of fragments into 5 mass groups,this means that a total of 30 representative trajectories must be calculated.Since the program calculates a single trajectory in a relatively short time(less than a minute),then the automated process of calculating all representative trajectories is performed within the acceptable operating time.

The intensity of the initial velocity of a representative fragment is determined algorithmically according to formula(9)and Fig.2,and the direction of the initial velocity is algorithmically determined according to the relative position of the fragment on the projectile body,according to Fig.3.Based on these initial kinematic conditions,with the variation of mass by mass groups,all representative trajectories are determined.

Thus,fragment trajectories were obtained for each segment of the projectile’s body,for several characteristic mass groups,so that the total number of trajectories to be determined is reduced to a signi ficantly reduced number relative to the actual total number of fragments.

The calculation of the trajectory and changes in the kinetic energy of fragments as they move through the atmosphere is determined on the basis of a parametric 6DOF(Six Degrees of Freedom)model of fragment flight mechanics.

Generally,determining the position of the body during the motion through atmosphere includes determining the position of the moving coordinate system(fixed to the rigid body)in relation to the fixed(inertial)reference(coordinate)system.The orientation of the body in relation to the reference coordinate system is determined by three Euler angles(ψ,φ,θ)and three coordinates(x,y,z)that de fine the position of the body relative to the given system.

Fig.4 shows the coordinate system tied to the Earth(xyz)and the moving system tied to the fragment(ξηζ).The figure also shows the vectors of the total aerodynamic forceFadand momentMad(determined using our model described in earlier paper[11]),gravity forcemg,and angular velocity vectorsωin the initial and arbitrary times,and Euler’s angles:ψ(angle of precession),θ(angle of nutation)andφ(angle of intrinsic rotation).These angles determine the spherical movement around the center of the body mass,i.e.,de fines the position of the body relative to the reference coordinate system tied to the Earth.

Fig.4.Visualization of coordinate systems,dynamic and kinematic parameters during the movement of fragment through the atmosphere.

Differential equations of the translational motion component in a coordinate system rigidly connected to the body,are set using the law of the body center of mass movement:

where:Fiis external forces,mis body(fragment)mass,aandVare acceleration and velocity of the body center of mass.

External forces acting on a fragment are an aerodynamic force and a force of gravity,so the expression(10)can be expressed as:

By projecting equation(11)on axes of the inertial(fixed)coordinate systemoxyz(Fig.4),the following expressions are obtained:

The model uses a novel approach,in the first place in de fining the aerodynamic force that was used for trajectory calculations.We defined our model for aerodynamic force and moment estimation based on several geometrical body parameters,as well as several kinematic and parameters of the media where the body motion occurs.This model has two major parts:(a.)geometrical modeling,where the surface of the irregularly shaped body is approximated,via continuous surface,by a triaxial ellipsoidal surface(b.)aerodynamic force modeling based on momentum change of fluid particles colliding with the irregularly shaped body at high velocity.

This elemental effect(momentum change)is integrated over the exposed surface of the moving body and obtain the total aerodynamic force and the total aerodynamic moment for the center of mass.In this way,the model does not require force coef ficients but directly obtains the drag force.The kinetic energy of the fragment(parameter important for lethal radius estimation)at different distances was determined using this model,as the fragment mass and its velocity during a flight are known.This model has been validated with experimental data and numerical simulations using Ansys Fluent.More details on this model can be found in our previous paper[11].

Differential equations of rotation around the body center of the mass are obtained using the law of change of angular momentum:

HereMis vector of the aerodynamic moment,aLis angular momentum vector in moving coordinate system tied to the body.

Aerodynamic momentMadacts on a fragment,so expression(13)can be written as:

After the neccesary tranformations and kinematic expressions for(13),ψis angle of precession,θisangle of nutation andφis angle of intrinsic rotation are obtained.

In order to obtain the value of the aerodynamic forceFad,projected on the fixed coordinate systemoxyz,the transformation matrixRis used:

So the aerodynamic force is obtained by:

individual mass groups)may correspond to the mean masses of

Similarly,the following can be written for the aerodynamic moment:

fragments obtained after fragmentation of this projectile in the pit or taken arbitrarily for a given mass group.Fragments of all

The model assumes that the fragments are approximated using the triaxial ellipsoid.An ellipsoid has three pairwise perpendicular axes of symmetry which intersect at a center of symmetry,called the center of the ellipsoid.The line segments that are delimited on the axes of symmetry by the ellipsoid are called the principal axes.If the three axes have different lengths,the ellipsoid is said to be triaxial,and the axes are uniquely defined.Semi-axes of the ellipsoid,a,b,care half the length of the principal axes.They correspond to the semi-major axis and semi-minor axis of an ellipse.Dimensionais the largest,andcthe smallest.More details on fragment shape and determination of fragment reference area(projected exposed area)can be found in our earlier paper[15].

This module for determination of fragments trajectory is implemented in Matlab program,written by authors.The basic parameters of the fragments(the dimensions of the half-axis of the ellipsoid by which the fragment is approximated)can be arbitrarily changed in the analysis.Based on their values and values of initial conditions(direction of the initial velocity vector,components of initial translational and angular velocity,orientation and position of fragments with respect to coordinate origin),the trajectory of fragments and changes of its kinetic energy during flight through the atmosphere are determined.The calculation of path elements is obtained by software,eg in Matlab.

From the point of view of evaluating the effectiveness of high explosive projectiles against human targets,it is important to evaluate the trajectory elements of fragments at ranges up to 40-50 m.In this regard,the calculation of trajectory elements for a particular projectile is performed for fragments that start from the segment centers on a given projectile.The calculation should try to get as close as possible to the real conditions that may occur after the detonation of fragments.For example,the following initial conditions can be set in the calculation:

-The shape of fragments is defined according to the dimensions of the real fragments that are available(eg.by measuring fragments-by measuring the maximum dimensions of fragments in three mutually perpendicular directions)generated by detonation of a particular high explosive projectile,the mean values of thea/banda/cdimension ratios for real fragments are obtained.In this regard,in the calculation of the trajectories,for each mass group,the shapes(dimensions)of the fragments corresponding to the real relationsa/banda/care defined.Fragment masses(in mass groups up to 100 g are considered in the analysis.It is assumed that fragments of mass above 100 g do not signi ficantly affect the ef ficiency of the high explosive projectile.

If one,for example,looks at the fragmentation data of the 122 mm OF-462 projectile in the pit[7],it can be concluded that this projectile has the highest number of fragments of very low mass(<0.5 g),so that the assumption of using fragments of mass less than 0,5 g for the purpose of calculating the trajectories of the fragments is justi fied.In addition,these fragments,although of very low mass,have relatively high kinetic energy and can be effective against unprotected human targets(for cases whereEkin>80 J).

-The initial height of the fragments should correspond to the position of the center of each segment separately(eg.segments 16-31,Fig.1)on the cylindrical part of the projectile,which is positioned vertically(top down)with respect to the ground(with the fuze).A stationary coordinate system is placed at the top of the fuze,on the ground.For each segment of the projectile,calculations of the trajectories of fragments of all mass groups are performed(for given dimensions of the fragments)and the level of their kinetic energy is estimated at certain distances from the center of the explosion.

-The initial elevation angle of the fragments is assumed to be zero in the trajectory simulations-because the projectile is positioned vertically with respect to the ground,and it is assumed that the initial velocity vectors of the fragments(which are perpendicular to the axis of the projectile)of the cylindrical part of the projectile are parallel to the ground(planexyof a fixed coordinate system,Fig.5).

-The initial translational velocity vector is set in the positive direction of thex-axis(Fig.4).The intensities of the initial velocity are given exactly for each segment of the projectile,according to the values obtained by calculation using the Gurney method(Fig.2).

-The initial angular velocity of fragments can have any value(it is presumed that the order of magnitude of the maximum intensity value of the initial angular velocity is 50 rev/s).In the videos,recording the process of fragmentation of the high explosive projectile,one can observe the general trend of movement of individual fragments in the first phase after the detonation.The footage shows that the fragments rotate at a certain angular velocity(mostly around all three axes).Angular velocity values cannot be determined precisely because fragments are moving at high velocities and the ability to estimate angular velocities based on videos depends primarily on the frame rate per second the camera enables.For example,it may be possible to rotate a fragment several times while changing one frame on the camera and to prevent proper estimation of the angular velocity of the body.If professional ultra-fast cameras are available,this information can be more accurately determined.In the model,for example,it can be assumed that all components of the initial angular velocity are the same and have a value of 50 rev/s(314.16 rad/s).In this case,the total intensity of the initial angular velocity is 544.14 rad/s.

Fig.5.Initial orientation of fragments(vector of fragment initial translational velocity is in the direction of x-axis).

-The initial orientation of the fragments(Fig.5)is also a stochastic parameter and,in general,the initial roll angleα0,pitch angleβ0,and the yaw angleγ0can take on any value.The fragment is unlikely to start from an idealized orientation when,for example,its initial projected surface(surface in a plane perpendicular to the velocity direction) is minimal(α0=β0=γ0=0)or when its initial exposed surface is maximum(α0=γ0=0°,β0=90°)-these orientations(and angles)refer to the given initial conditions when the initial velocity vector is directed in the positive direction of thex-axis(Fig.5).It can be assumed in the analysis(within the model)that the fragments begin their movement from an arbitrarily chosen orientation(eg,these can be anglesα0=γ0=β0=45°;Fig.5,right).It should be noted thatα0,γ0,andβ0do not represent the angles of the longitudinal axis of the fragment according to the coordinate axes,i.e.,cosα0+cosγ0+cosβ0≠1.At this orientation,the projected surface of the fragment is neither maximal nor minimal but represents the stochastic(random)value of the exposed surface.

After de fining the initial geometric-inertial and kinematic conditions,it is possible to calculate the trajectories and changes in the kinetic energy of fragments of high explosive projectiles.Thus,for example,Fig.6 shows the trajectories of fragments of different masses and shapes,which begin the movement from a particular segment on the cylindrical part of the body of the 122 mm OF-462 projectile.The diagram in Fig.6 is shown(without sacri ficing details)to see the trajectories as closely as possible,but it should be borne in mind that in this case the maximum value at the ordinate is 0.25 m and the abscissa is 35 m.Based on the results of the program for calculation of the trajectories of fragments of individual masses and shapes,and for the given initial conditions,it is possible to estimate which fragments can reach certain distances,and on this basis,also the data on the number of fragments from individual segments of the projectile(Mott method),to estimate the density of ef ficient fragments in space.

Fig.6.Trajectories of fragments of different masses and shapes,starting from a segment on the cylindrical part of the 122 mm OF-462 projectile body.

Based on the data obtained,the change in kinetic energy of the fragments during the flight can also be estimated.Thus,for example,Fig.7 shows the drop in kinetic energy of fragments of different mass and shape,which begin their movement from segment no.16(Fig.1)on the cylindrical part of the 122 mm OF-462 projectile body.The diagram shows that fragments(especially fragments of larger masses)have high kinetic energy and do not lose it signi ficantly up to a distance of 30 m through the atmosphere.

In order to better see the kinetic energy drop curves for smaller mass fragments(up to 2 g),the ordinates and abscissas of Fig.7 can be scaled and presented in another way.Thus,in Fig.8(Fig.8 is just “zoomed in” Fig.7,where we tried to better represent(visualize)the trajectories of smaller fragments)it can be seen that even a fragment of mass 0.28 g has kinetic energy sufficient to incapacitate human targets(Ekin>80 J).However,smaller mass fragments also have a smaller range,which is particularly important,for example,for the 122 mm OF-462 projectile,which after fragmentation has a very large number of fragments(about 40%of all fragments)with a mass below 0.5 g,which can signi ficantly affect the effectiveness of projectiles because fragments of a given mass group generally have a short range.

Fig.7.Drop in kinetic energy of fragments of different masses and shapes,starting from the segment on the cylindrical part of the 122 mm OF-462 projectile body.

Fig.8.The drop in kinetic energy of fragments of different masses and shapes,which start their movement from segment no.16 on the cylindrical part of the 122 mm OF-462 projectile body.

Based on the results of the kinetic energy levels of fragments of given shapes,for given initial conditions,it is possible,using the Mott method,to estimate the density of ef ficient fragments at certain distances from the detonation center and subsequently the lethal radius of this projectile.For the 122 mm OF-462 projectile,for example,in which 16 segments on the body are analyzed(16-31,Fig.1),and for the given mass groups(12 of them),a total of 192 trajectory calculations were performed.

The calculation of the trajectories for the given fragments,under such initial conditions defined,takes very short time-about 1 s(the initial coordinatesz0have low values and the flight time of the fragments is short),with trajectories for each projectile segment(12 trajectories because there are 12 mass groups)are written to a single document(MS Excel)for easier analysis.

We will give here a comparison between our model for determining aerodynamic force(necessary for trajectory calculations)and model where the force is determined via drag coef ficient,since determining the aerodynamic force is the single most important task when estimating the trajectory of a body and hence model accuracy.Our model also includes compressibility effects which are important in supersonic flow of high explosive projectile fragments.

First,the results of our model for estimation of aerodynamic force(where fragment is approximated with triaxial ellipsoid,Fig.9)will be compared with the results obtained by CFD(eng.Computer Fluid Dynamics)numerical simulations(Ansys Fluent)for aerodynamic forces acting on an irregularly shaped fragment with jagged,iregular surface(Fig.10).The flow was directed towards positive and negativeyandzaxes(shown in Fig.9).

Fig.9.Computer Aided Design model of projectile fragment,approximated with an ellipsoid.

Fig.10.Re finement of mesh around irregularly shaped fragment(Ansys Fluent).

Table 1 shows comparisons of results for the aerodynamic force acting on the fragment(Fig.9)obtained by numeric simulations(for real fragment)with the results for force obtained using the our aerodynamic force model(correction due to compressibility of air taken into account).

The results in Table 1 show there are no large deviations(relative difference of 5.3%-23.6%)for flow in the direction ofzaxis for this fragment(flow towards largest exposed area of fragment).In case of flow towards the smaller exposed surface of the fragment(ie in the direction of the positive and negativeyaxis;Table 1,Fig.9),the relative differences are somewhat larger(23.3%-59.6%).Also for different directions of fluid flow,even in the same direction(same axis),the results may be signi ficantly different(Table 1).It should be noted that every possible fragment(even with the same general dimensions)will have somewhat different values of these forces because of the stochastic nature of the natural fragmentation process.

Table 1 Comparison of results for aerodynamic force acting on fragment,obtained by numerical simulations and developed model.

Next,the results from numerical simulations for real fragments were then compared with the results for the aerodynamic force acting on fragment-determined using the “classical” model(F=0.5ρACDv2)where force is determined via drag force coef ficient(to establish the order of errors).In this analysis,the fragment was approximated with parallelepiped(Fig.11),as this shape approximation is suggested by some authors.The largest projection of the exposed surface of the fragment was represented withAmax=ab(Fig.11),the average projection of fragment exposed surface is assumedAaver=ac;smallest projection of the exposed surface of the fragment,in this case,isAmin=bc(Fig.11).These projected surfaces were considered so the results can also be compared with results obtained with our model(shown in Table 1).Dimensions of the fragment were the same as in Fig.9.Air density was taken as value at sea level,ρ=1.225 kg/m3.

Fig.11.Fragment approximated with a parallelepiped.

Experimental[18]values(CDmax,CDaver)were taken as an approximateCDvalues for this fragment.Fig.12 presents these data.It is assumed that values ofCDmaxcorrespond for the case when the fragment was exposed to flow with the largest surface projection(Amax),andCDaverfor the case when the fragment was exposed to flow with an average surface projection(Aaver).

Fig.12.Experimental data for drag force coef ficient CD for fragments[18].

Table 2 presents the comparison of results for the aerodynamic force acting on the fragment(from Fig.11),obtained by numerical simulations(Fig.10),and results using the “classical” model(force determined via drag force coef ficient),for two directions of flow(towards positive/negativeyandzaxis,Fig.11).Relative differences between the results for aerodynamic force obtained by the “classical” approach and numerical simulation(for a fragment)for flows towards theyaxis were from 192.8%to 386.3%and for flow towards thezaxis from 128.5%to 206.5%(Table 2).These errors can be larger or smaller depending on adoptedCDvalues as well as adopted values of the reference area(ie.projected area of a body).WhenCDvalues are adopted as constant(ie such as in DDESB/Department of Defense Explosives Safety Board model),the errors can grow signi ficantly.

As can be seen from Tables 1 and 2,using the “classical” approach for the determination of aerodynamic force acting on a fragment can give signi ficantly larger errors(192%-386%)than using our physical model(where rel.differences were<60%,Table 1),which can lead to erroneous(less accurate)results for fragment trajectory.

Table 2 Comparison of results for force acting on fragment,obtained by “classical” model(force determined via drag force coef ficient)and numerical simulations.

2.5.Module for estimating the density of ef ficient fragments

In order to be able to estimate the ef ficiency zone of a high explosive projectile,with the knowledge of the above parameters,it is necessary to first determine the density of effective fragments at different distances from the detonation point.

The estimation of the ef ficiency of a high explosive projectile is based,first of all,on the knowledge of the density function of ef ficient fragments from a high explosive projectile depending on the distance from the explosion center.The model assumes that the distribution of fragments in space is symmetrical with respect to the axis of symmetry of the projectile(Fig.13).

Fig.13.Polar zones in determining the number of fragments in space.

Previous models of researchers in the world,which defined the density of ef ficient fragments in space,are based largely on the results of experimental studies in fragmentation arenas,which,although irreplaceable,are quite expensive and time-consuming,and in addition require the work of larger number of people(counting penetrations in wooden panels).

The total number of fragmentsNukin space can be estimated by summing the number of fragmentsNiin projectile segments:Nuk=whereNi=niSΩi/Sm,whereniis the number of fragments hitting the targetSm,andSΩiis the surface area of a sphere cut off by a spatial angle in a given polar zone.These methods for estimating the density of fragments are generally based on knowing(experimentally-via tests in the fragm.arena)the parameterni-number of fragments that hit the arena panels.In our new model,on the other hand,it is necessary to determine exactly the number of fragmentsniin the i-th polar zone that hit the target,for the known total number of fragmentsNiin thei-th polar zone,using the expression:

The total number of fragmentsNifrom expression(18)is determined by the Mott method,in interaction with the module for predicting the trajectory of fragments,taking into account only fragments of a certain mass-fragments that,based on the results of the model for estimating the trajectory of fragments,reach the distance for which the density of fragments is determined.

The target surfaceSmfrom expression(18)is taken arbitrarily,and this area may be the area of the panel in the semicircular arena,the area of the sector in the quarter-circular arena,the exposed surface of the standing man,or any other value.The ratioSΩi/Smfrom expression(18)is the ratio of the surface area of a part of a sphere cut off by a spatial angle in a given polar zone to the surface of the target(eg.panel or arena sector).

The general expression for de fining the area of a part of a sphere cut off by two adjacent space angles for the i-th polar zone(Fig.13)is given by the expressionSΩi=2πRk2（cosθi-cosθi+1）,where the anglesθde fine the angular width of the target(eg.panel,sector),looking from the projectile segments[7].

Fig.13 shows the geometric parameters(polar angles-θ,areas of part of a sphere cut off by the spatial angleΩin a given polar zone-SΩ,target surface-Sm,distance to the target-Rk)required to determine the total number of fragments that occur after fragmentation of high explosive projectile.

The formula for determining the number of fragmentsniini-th polar zone hitting the target with areaSmat the distanceRkcan be represented in the form:

When the value of the number of fragmentsniin i-th polar zone is known,estimation of the density of fragments at a certain distance from the detonation center can be determined.Namely,the density of fragmentsqiat a certain distance from the center of detonation can be expressed via the formulaqi=ni/Sm,so the density of(all)fragmentsqiat a certain distance from the center of detonation,and for the i-th polar zone,can be written in the form:

In our model it is necessary to determine the density of ef ficient(energies greater than 80 J)fragmentsqef-i,so expression(20)can be represented in the following form:

HereNef-iis the total number of effective fragments in a given polar zone around the projectile.In the model,polar anglesθiare measured from the projectile axis to the directions connecting the centers of the selected segments with the top and bottom of the arena(Fig.14).These angles are precisely determined in CAD(Computer Aided Design)software,based on the projectile and arena drawings(1:1 scale).Fig.14 shows,for example,that the anglesθ1=81.2°andθ2=91.4°for the first sector of the arena(distance from the projectile is 10 m),for the case of the 122 mm OF-462 projectile.The angles for the other two sectors of the quarter-circular arena(used in our country)are similarly determined.These angles,together with information for the radius of fragm.arenas,are used in an expression(21).

Fig.14.Example of determination of polar angles-angles measured from the axis of a 122 mm OF-462 projectile to the top and bottom of the 1st sector of the quarter-circular arena(10 m).

The total number of effective fragmentsNef-ifrom expression(21)is determined using the mass distribution estimation module(Mott’s method)and the fragment trajectory prediction module.Speci fically,the Mott method estimates the number of fragments of a given mass group,as well as the total number of fragmentsNifor a given projectile segment(the projectile is divided into a larger number of segments in CAD(Computer Aided Design)software,Fig.1).

When the total number of fragments(and the number of fragments by speci fic mass groups,ie the number of fragments of mass greater than massm)generated from a given segment of the HE projectile is known,the trajectory simulations for different mass groups(and fragments shapes)determine the trajectory and a decrease in the velocity and kinetic energy of the fragments for each segment of the projectile body.The kinetic energy of a fragment at a certain distance can be determined,using the model results for estimating the fragment trajectory,since the mass of the fragment and its translational velocity during flight at any given time are known.The rotational kinetic energy of a fragment is negligible relative to its translational kinetic energy[7].

The next step in the model is to determine,based on the results obtained from the calculation in the fragment trajectory program,which fragments have kinetic energy greater than 80 J(from fragments whose range is greater than the distance for which the density of ef ficient fragments is considered)for incapacitation of human targets(eg.soldiers on the battle field).

After the values of the number of effective fragments for certain segments of the projectile at a given distance are known,the density of effective fragments at that distance(expression 21),for a given polar zone,is determined.The density function of ef ficient fragments has a declining character,and in the regression analysis it is usually an exponential or power function(depending on the correlation coef ficient).

Regarding the criteria of the kinetic energy of the fragments(80 J)used in the model,this is the energy level required for the fragment to be considered effective(incapacitating human targets)and has been accepted in most NATO countries.Also,in the arena tests in the former Yugoslavia,it was considered that only fragments with energy greater than 80 J could penetrate 2.5 cm thick wooden panels.

A number of parameters affect the ef ficiency of fragments,the most important being the kinetic energy,the shape of the fragment,the current orientation of the fragment at the time of impact,and the location of the impact of the fragment.Since it is almost impossible to predict precisely all these parameters,the procedure is simpli fied by determining the reference level of kinetic energy.All fragments with kinetic energy above the reference are considered effective in the stated sense of penetration ef ficiency of wooden panels of a certain thickness.

2.6.Module for estimating the lethal radius of HE projectiles

Based on the known value of effective fragments density,it is possible to determine the lethal radius(and area of the high explosive projectile ef ficiency zone)against human targets.The area of ef ficiency of a high explosive projectile is a circular surface(zone)within which fragments with greater than 80 J are present and within which the density of effective fragments is greater than 1 fragment per m2of surface area.The boundary curve(circle)of the ef ficiency zone is obtained provided that the density of ef ficient fragments is equal to 1 fragment/m2-it is assumed that a standing soldier,with an exposed area of 1 m2towards the center of the explosion,receives at least one effective fragment.Of course,analysis can also be carried out according to the needs of the research with the assumption that the density of ef ficient fragments is greater or less than 1 frag/m2.

The authors used the effective fragments density criteria of 1 fragm/m2to calculate the radius of ef ficiency since this is the standard used in Bosnia and Herzegovina,ie the former Yugoslavia,similar to the Warsaw Pact countries models(large scale frontal attack).Western countries,on the other hand,take into account a lower density of soldiers per surface area.In addition,this density of 1 fragm/m2was used in the standard arena test in our country to compare the ef ficiency of high explosive projectiles in the development and production phase,and as such is an important parameter in the preliminary analysis of the fragmentation potential of high explosive projectiles.But generally speaking,this parameter is arbitrary and it can be taken into analysis that the density of soldiers in the field is,for example,less than or greater than 1 fragm/m2,depending on the needs of the researcher.

When evaluating the ef ficiency,similar to fragment density,it is assumed that the projectile is positioned vertically to the ground,with the fuze facing the ground.In this way,the maximum ef ficiency zone for a given projectile is obtained,since it is known that by reducing the impact angle,the projectile ef ficiency zone is also reduced.In previous studies,Catovic[10]found that the density of ef ficient fragments is a decreasing function(3D surface)going from the center of detonation and that the highest density of fragments is in the lateral spray.Fig.15 shows the dominant side spray;For clearer visualization,a projectile model(2D section)was added to the diagram,with the tip facing in the appropriate direction.

Fig.15.Effective fragment densities for122 mm OF-462 for different polar zones[10].

The lethal radius is determined by first determining the densities of the effective fragments for different distances from the detonation center,after which regression analysis is used to de fine the curve that best fits the data obtained(usually exponential or power functions;namely,to keep the correlation coef ficient as close as possible to the value 1).

Regression analysis of the effective fragment density data,with the aim of estimating the lethal radius of a HE projectile,is performed by software(for this purpose MatLab is very suitable).After determining the approximation curve by regression analysis,the distance at which the density of effective fragments is equal to 1 frag/m2is determined,thus de fining the radius of the ef ficiency zone of a high explosive projectile against human targets.When the lethal radius is known,the procedure for determining the area of the ef ficiency zone is trivial,using the expression:Aef=Ref2π.

Fig.16 is a schematic representation(in scale)of a 2D model of a 122 mm OF-462 artillery projectile,divided into segments.In red,schematic trajectories of the lateral(dominant)spray of fragments at the initial part of the trajectory are indicated(these trajectories naturally are not straight-line trajectories but represent part of the ballistic trajectory of fragments).This position of the projectile(Fig.16)is suitable so that the results can be compared with the experimental results from the quarter-circular fragmentation arena used in the former Yugoslavia.

Fig.16.122 mm OF-462 projectile division into segments and dominant side spray of fragments with schematically shown trajectories.

As stated earlier,this model assumes that most of the front spray fragments for this high explosive projectile position(the position in which the projectile is placed in the quarter-circular arena,or any similar arena)are pointing toward the ground.Con firmation of this assumption can be seen in Fig.17,which shows the moment after fragmentation of a 122 mm high explosive projectile.The picture shows a video sequence of the impact of fragments into the sectors of the quarter-circular arena after the detonation of a high explosive projectile at the test site of the Pretis Sarajevo factory.The shot was taken with a Canon dSLR camera and a Tamron tele-zoom lens at a distance of about 100 m from the center of the blast.

Fig.17.The sequence of the video showing fragments strikes in the arena sectors after the detonation of the high explosive projectile(Pretis Sarajevo military factory test site).

Fig.17 shows that,after the projectile explosion,many fragments strike the ground(fragments from the front of the projectile body),and detonation products are present that spread rapidly around the explosion site.It is worth mentioning that the detonation pressure for TNT,the charge inside the 122 mm OF-462 projectile,is 17.2 GPa and it is sufficient for the rapid disintegration of the projectile body(the detonation pressure is much greater than the tensile strength of the steel forming the projectile body)and fragmentation into a large number of pieces.The temperature of the detonation products is about 3040 K,and during the detonation,about 580 L of detonation products are developed per 1 kg of an explosive charge(software EXPLO 5).

It is assumed that fragment end spray in the position of the high explosive projectile,as shown in Fig.17,goes above the quartercircular arena sectors.In the real case,this may be considered at least partially correct because the projectile body expands after detonation and the rear spray of fragments for a given impact angle(90°)probably start moving at relatively large elevation angles(relative to the ground).The model,therefore,neglects the front and end spray of fragments from the high explosive projectile when estimating the radius of the ef ficiency zone against human targets.

It should also be noted here that it is sometimes overlooked that the character of the density curve of effective fragments as a function of distance(on the basis of which the radius of the ef ficiency zone against human targets is determined)depends a great deal on data from the closest sector of the arena(first sector in Fig.17).These data can skew results so a special attention should be exercised when counting penetration data.

3.Calculation of the lethal zone radius and analysis of results

Based on the model developed,the lethal radius was evaluated for four artillery projectiles:105 mm M1(TNT),122 mm OF-462(TNT),122 mm M76(Comp.B),130 mm M79(TNT).The results were compared with the experimentally obtained(quarter-circular arena tests used in the former Yugoslavia,Fig.20)lethal radius values for the given projectiles.Fig.18 shows the projectiles used in the analysis,as well as their CAD(Computer Aided Design)models(2D cross-sections).

Fig.18.Artillery projectiles tested[7].

Fig.19 shows the CAD(Computer Aided Design)models of the given artillery projectiles and the central segments of the projectile body considered in the analysis(down to the obturating band).

Fig.19.Four artillery projectiles for which the model has been used to determine the lethal radius(122 mm OF-462,122 mm M76,130 mm M79 and 105 mm M1).

Table 3 shows the characteristics of the projectile components and their ballistic performance.

Table 3 Characteristics of the projectile components and their ballistic performance.

Table 4 gives the mass and geometric parameters(the ratio of the mass of explosives to the mass of bodyC/Mand the ratiot/d)of the projectiles considered.

Table 4 Mass and geometric parameters of the projectiles considered.

This research uses the results of fragmentation tests of high explosive projectiles performed by detonation in a quarter-circle fragmentation arena(Fig.20)at the Pretis factory in Sarajevo.

Fig.20.A quarter-circle fragmentation arena at the Pretis factory in Sarajevo.

The results of these tests are used to evaluate the parameters of the spatial distribution of fragments and the ef ficiency of a high explosive projectile.This fragmentation test is applied in the design phase and serial production of high explosive projectiles.Fragmentation arena test data is used for determination of the number of penetrations of fragments per area of the arena at different distances,the spatial distribution of fragments and lethal radius of the projectile.The warhead or projectile in the test is placed vertically in the center of the arena at ground level,with the fuze facing the ground,and detonated electrically.The number of penetrations per square meter can be determined for each individual projectile,or the average for the group is calculated.The calculation is made for each sector according to the general form:

Hereqiis the number of penetrations per m2,niis the number of penetrations in the sector of radiusRi,Siis an area of the sector(m2)of radiusRi,Giis the number of projectiles fragmented within one group.The number of penetrations per m2is shown graphically by sectors(for different radii).

In order to accurately determine the lethal radius for high explosive projectiles,regression analysis of the data(e.g.,Matlab,Curve Fitting Toolbox)is performed,and the approximate functions of the ef ficient fragment density are determined.These functions are generally power(y=a·xb)or exponential(y=a·e(b·x)).The functions are chosen so that the correlation coef ficient(between the experimental data and the approximation curve)is as close as possible to 1.The lethal radius of a high explosive projectile is determined experimentally as the distance from the center of detonation at which the density of effective fragments(those that have penetrated the wooden targets of the arena)is equal to 1 fragment/m2.

Using the developed model,described in the paper,with all its modules(Section 2),the radii of ef ficiency for the given artillery projectiles(presented in Fig.18)were calculated and compared with the experimental results[7](fragmentation tests in the Yugoslav arena,Fig.20),and the results are presented in Table 5.

Table 5 Lethal radius of high explosive projectiles against human targets for a given artillery projectile(model comparison with experimental results).

More information on the step by step procedure for calculating the radius of ef ficiency using this model for a given high explosive projectile can be found in Refs.[7],bearing in mind that this reference is written in Bosnian language.

In summary,the results for the high explosive projectile lethal radius,obtained by the described physical model show good agreement(relative differences less than 17%)with the experimental results(it should be noted that,even for the same projectile,signi ficant differences in values of lethal radius during fragmentation tests in the arena are possible[7]),although a relatively large number of assumptions and approximations in the model were adopted.As such,the model developed represents a small step forward in the complex study of the terminal-ballistic parameters of a high explosive projectile.

Future research in this area could focus on:

1.Development of new models:

-Numerical simulation of the projectile body fragmentation,where the initial velocity of fragments,the angle of ejection and deformation of the projectile body can be estimated.For these simulations,it is important to properly select the fracture model of the body material and the equation of state for individual projectile materials and to adequately characterize the material,which requires relatively expensive experimental research.

-Simulation of the impact of fragments of high explosive projectiles into different targets(human targets,wooden and metal targets,concrete structures,etc.).

-Extension of the model for evaluation of ef ficiency to other types of high explosive projectiles(eg.mortar projectiles,rocket warheads).

2.Monte Carlo simulations for statistical determination of fragment trajectories set for different initial conditions,in a certain range of variables,where stochasticity is introduced.

3.Additional experimental tests should be performed to determine what the real kinetic energy level(for the energy of fragments of different mass)is required to penetrate these wooden panels,and how much energy of the fragment is lost after penetration.

4.Supplementing the experimental database:

-Fragmentation tests in the pit and semicircular arena should be performed for other types of high explosive projectiles.

-Development of a technique for experimental measurement of fragment velocity.

-Development of a method for visualizing the trajectory and stability of fragments in flight,using a gas cannon,ultra-fast cameras,and barriers for measuring the speed of fragments during flight.

-Assessment of damage to targets that simulate human targets(ballistic gelatin models)by the effects of fragments of high explosive projectiles.

5.Development of software:

-Development of a program for fast calculation of the lethal radius of a high explosive projectile,based on the developed model.

4.Conclusions

The model is comprehensive,consisting of several modules,where every part has some novelty:CAD(Computer Aided Design)modeling of a real projectile,analytical estimation of fragments number(Mott formula),prediction of fragments initial velocity(Gurney formula),calculation of fragments trajectory(6DOF/Six Degrees of Freedom),estimation of ef ficient fragments density,and estimation of the projectile lethal radius.

The model effectively takes into account the real geometry of projectile via CAD(Computer Aided Design)modeling component.Computer Aided Design models of projectiles are also used in fragment mass distribution(Mott model)determination;the calculation of the initial velocity of fragments;to de fine initial conditions(the initial coordinates of fragments for trajectory,depending on their position on the body of the projectile);to calculate the polar angles that appear in equations for estimation of the ef ficient fragments density.

The model uses(where Pit fragmentation data is available)optimized Mott constant value for a given projectile.In this way,the agreement between the experimental data(Pit tests)and the Mott model is better.The number of fragments is determined analytically using an optimized Mott method for a large number of body segments(the body is sectioned into quasi-cylindrical segments).The model assumes that only fragments from the central part of the body up to the rotating band are involved in ef ficiency calculation(it is assumed that fragments from the front and rear spray do not signi ficantly contribute to the ef ficiency of high-explosive projectiles).

The initial velocities of fragments were determined for a large number of projectile segments,using the Gurney model.In this analysis,the detonation velocity of explosive was determined by software EXPLO5,for a given charge density-determined from the table data for explosive mass and its volume(determined in the Computer Aided Design system-based on projectile drawings).The obtained data on the initial velocities of fragments are used in the model for the calculation of the fragment trajectories,for better realistic initial conditions.

The model uses complete 6DOF(Six Degrees of Freedom)equations for fragment trajectory estimation-using the law of the body center of a mass movement,and the law of change of angular momentum.This model uses a novel approach,in the first place in de fining the aerodynamic force that was used for trajectory calculations.We have performed a thorough literature review and we found no evidence that this problem has been solved appropriately for irregularly shaped bodies.We defined our model for aerodynamic force and moment estimation based on several geometrical body parameters,as well as several kinematic and parameters of the media where the body motion occurs.This model has two major parts:(a.)geometrical modeling,where the surface of the irregularly shaped body is approximated,via continuous surface,e.g.by an ellipsoidal surface(b.)aerodynamic force modeling based on momentum change of fluid particles colliding with the irregularly shaped body at high velocity.This elemental effect(momentum change)is integrated over the exposed surface of the moving body and obtain the total aerodynamic force and the total aerodynamic moment for the center of mass.In this way,the model does not require force coef ficients but directly obtains the drag force.The kinetic energy of the fragment(parameter important for lethal radius estimation)at different distances was determined using this model,as the fragment mass and its velocity during a flight are known.This model also has been successfuly validated with experimental data and numerical simulations using Ansys Fluent.Model for aerodynamic force and moment estimation,which is the most important part of 6DOF trajectory model,showed signi ficantly better accuracy than model that uses drag force coef ficient for calculating the force that acts on a fragment during its motion thrugh the air.

The model for determination of fragments trajectory is implemented in the new MatLab program,written by authors.

Novel estimation of ef ficient fragments density model involves interactively combined:CAD(Computer Aided Design)method,optimized Mott method,6DOF(Six Degrees of Freedom)trajectory model and analytical model for spatial fragment density determination.

The lethal radius of high explosive projectile against human targets is defined as a distance from the explosion at which there is an ef ficient fragment(kinetic energy of at least 80 J)density of 1 frag/m2.It is assumed that the position of a projectile upon detonation is vertical to the ground,so lethal radius represents a radius of a circular lethal zone-a maximal possible lethal zone of a projectile on the ground.This parameter(lethal radius)can be used in an initial phase of design to compare the ef ficiency of different high explosive projectiles.

The model showed overall good agreement with experimental results from arena fragmentation tests(used in our country).

This model can be used in any terminal-ballistics scenario for high explosive projectiles since it is general,parametric,fast and relatively easy to implement.The model also saves money,since it can extrapolate from calculations of some important data and implement them with fewer fragmentation tests(which are generally expensive)when designing new projectiles or modifying the old ones.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to in fluence the work reported in this paper.