Experimental and numerical study on the influence of shaped charge liner cavity filing on jet penetration characteristics in steel targets

Paweł ochowski,Radosław Warchoł

Military Institute of Armament Technology, 7 Wyszynskiego Str., Zielonka, 05-220, Poland

Keywords: Shaped charge jet PG-7 grenade Armor steel target Finite element modeling Penetration process

ABSTRACT Penetration characteristic(size and shape of penetration craters made in high hardness ARMSTAL 30PM steel)of shaped charge jets formed after detonations of modified PG-7VM warheads was analyzed in the article.Modifications consisted in removing the frontal part of the grenade (fuse,ballistic cap and conductive cone) and introducing of the liner cavity filling made of polyacetal copolymer POM-C.The filings in the form of solid cones with three different heights (33%,66% and 100% of H -the height of original PG-7VM liner)were placed inside of the hollow cone shaped charge liner.As opposed to the vast majority of previously published works (in which warhead optimization studies were focused on increasing of the depth of penetration in rolled homogeneous armor steel)the main aim of the presented modifications was to maximize the damage ratio (diameters of craters,inlet and outlet holes) of target perforated by shaped charge jet at the cost of the loss of part of the jet penetration capability.According to the best knowledge of the authors such approach to the use of the old PG-7VM warheads has not been analyzed so far.Taking into consideration high stock levels of PG-7VM warheads,and the fact that they are continuously being replaced by more efficient and more sophisticated high-explosive anti-tank warheads,it seems reasonable to look for alternate applications of the warheads withdrawn from the service.Thanks to the introduction of proposed modifications the warheads could be used by special forces or other assault units as directional mines or statically detonated cutting shaped charges as well as by combat engineers as universal charges used in various types of engineering or sapper works.The research included experimental penetration tests and their numerical reproduction in the LS-Dyna software with the simulation methodology defined and validated in previous works of the authors.Small differences (average error=10-20%) were identified between the experimental and numerical results(dimensions of craters made in steel targets were compared)what confirmed the reliability of the modelling methodology and enabled its use for further optimization of the shapes of fillings.Within the analyzed variants of warheads modifications maximum diameters of penetration craters were obtained for the filling of the height of h=2/3H.The diameters of holes in individual steel plates were increased by 164%,70%,65% (for the first,second and third plate,respectively) in relation to the variant without filling.The results of the study indicated that with the use of different materials of fillings and their various heights it is possible to control the shape of penetration craters pierced in the steel targets.

1.Introduction

Shaped charges (SC) have been successfully used in various branches of engineering and different applications including perforation,drilling or cutting devices for more than 70 years[1-6].Principles of SCs based on high explosive containing cavity have already been well understood and described[7,8].Various shapes of liners can be used in such devices [9]but the most common are hollow metal cones.Thanks to the cavity the energy of detonation products can be focused on a significantly smaller area.Additionally,when the SC include metal liner,the liner is deformed and stretched by the interacting detonation products and the shaped charge jet(SCJ)is generated.High penetration capability of the SCJ makes that one of the main application of SCs is military industry.SCs are used in various types of warheads,projectiles [7-13]in which different numbers of charges [14,15],liner shapes [16-18],liner materials[19-21],and types of explosives used[21,22]can be used.Review of designs of SCs currently used in military applications indicates the main factors affecting quality and effectiveness of every SC,which are as follows:

· Shape,amount,grain size and type of explosive[7,21-23];

· Geometry,density,material and microstructure of liner[16-21,24-26];

· Type and material of SC casing [27];

· Standoff distance (between target surface and the base of the liner) [28];

· Qquality and precision of assembly of the warhead [23];

· The way of explosive initiation MW [29,30];

· Type and geometry of wave-shaper [31];

· Rotary motion(in case of shaped charges of artillery shells)[32].

The effectiveness of military SCs is assessed on the basis of the depth of penetration value -depth of a crater made in thick steel target with a hardness of 280-500 HB.This value is usually related to the diameter of SC and is highly affected by the standoff distance[33].For more than 70 years military SCs have been optimized in order to achieve the highest possible depth of penetration of rolled homogeneous armor steel target (RHA).The most modern HEAT warheads thanks to their technologically advanced and optimized designs obtain a penetration capability of up to 1500 mm of RHA[11].In the light of those numbers the effectiveness of the old and simple PG-7VM grenade that can penetrate about 300 mm of armor steel does not look impressive.Despite the fact that PG-7VM warheads are continuously being replaced by more efficient and more sophisticated high-explosive anti-tank warheads,there is still a very large number of RPG-7 being in use and constituting a real threat[34,35].

Taking into consideration high stock levels of PG-7VM warheads,and the fact that due to their relatively low penetration capability (in relation to modern warheads) they are continuously being replaced by more efficient and more sophisticated HEAT warheads,the main objective of the presented work was to modify the PG-7VM warheads in a way that will allow their unconventional and completely different use.As opposed to the vast majority of previously published works (in which warhead optimization studies were focused on obtaining of the deepest penetration crater in RHA) the main aim of the proposed modifications was to maximize the damage ratio (diameter of crater,inlet and outlet holes) of target perforated by shaped charge jet at the cost of the loss of part of its depth of penetration capability.Modification consisted in removing the frontal part of the grenade(fuse,ballistic cap and conductive cone)and introducing of the cavity filling made of polyacetal copolymer POM-C.The filing in the form of solid cones with three different heights(33%,66%and 100%of the original PG-7VM liner height) were placed inside of the hollow cone shaped charge liner.According to the best knowledge of the authors such approach to the use of the old PG-7VM warheads has not been analyzed so far.This is understandable because from the warhead depth of penetration capability point of view(that is the main aim in the most of studies)decreasing of standoff as well as introducing of foreign objects in the liner cavity area significantly reduces depth of penetration of RHA that can be obtained by SCJ.However,the penetration crater made by SCJ formed from conventional warheads is usually very deep but small in transverse direction(diameter of about 10 mm).That is why,one of the aim of current work was to check if it is possible to introduce such modifications of the warhead (different materials of fillings and their various heights) that will allow its unconventional use in order to control the shape of the penetration crater pierced in the target in case when the growth of the radius of damage is more important than depth of penetration of target.Such modified warheads could be used by special forces or other assault units as statically detonated cutting shaped charges(Fig.1(a))or directional mines(Fig.1(b))as well as by combat engineers as universal charges used in various types of engineering or sapper works (Fig.1(c)).

Fig.1.Potential applications of the analyzed modifications of warheads as (a) directional mines;(b) Cutting shaped charges;(c) Universal charges used in various types of engineering or sapper works.

The idea of increasing of the penetration crater diameter by using of shaped charge liner cone filing is entirely novel and has not been analyzed so far.The only one reference found in which similar concept was analyzed was the work [36]published in 1977.However,the purpose of using of liner cone fillings was completely different.In the work the influence of liner cone filling on the character and parameters of jet tip was assessed.The SC analyzed in the study consist of copper liner (base diameter 81.3 mm,height 91.19 mm and apex angle 42°) elaborated with Composition B explosive.Impulsive radiography was used for observation and measurement of jet tip particles velocity for six different variants of SC in which the apex end of the liner was filled to different heights(12.7 mm,25.4 mm,38.1 mm,50.8 mm,63.5 mm) with Wood's metal (a metal alloy used in soldering with density of 9.7 g/cm3).Even though the aim of that study was different (the authors wanted to prove that approximately 40% of the liner height at the apex end of the cone does not provide an efficient jet as far as penetration depth is concerned)it was observed that the velocity of jet tip decreased together with the increase in the liner cone filling.For the abovementioned heights of filling the velocity of jet tip equaled 7,81 km/s;7,52 km/s;7,53 km/s;6,53 km/s;5,44 km/s;3,25 km/s respectively.On the basis of the theoretical calculations the authors claimed that the kinetic energy of the jet tip formed from the liner cone filled in 30% of its height would be smaller by 16%in relation to unfilled liner.Unfortunately conclusions from the work were drawn on the basis of theoretical considerations only since no experimental depth of penetration tests were performed in the study.

Different approach to modification of SC but based on similar assumptions of introducing additional elements in the liner area was presented in work[37].The SC analyzed by the authors beside the conventional copper liner comprised the second outer liner made of PTFE (polytetrafluoroethylene)/Al reactive material.Experimental results showed that such modified SC produced a deeper penetration depth compared with the single liner shaped charge jet.But opposite to the current study,achieving of the deepest possible depth of penetration was the goal of the referenced work.Taking into consideration lack of information about similar solutions based on SC with liner cone fillings,the approach of SC modification presented and analyzed in the article were patented in the Patent Office of the Republic of Poland [38].

The methodology of the presented work included experimental penetration tests and finite element modeling of the phenomenon in the LS-Dyna software.Such combined approach is widely used for similar studies because it allows to collect more data from the analyzed phenomenon.In the literature,various numerical methods and elements formulations were used for reproduction of SCJs formation and penetration processes: Lagrangian [39],Eulerian two-dimensional [40-42]and three-dimensional [43],meshless based on smoothed particle hydrodynamics (SPH) method[44].All of the mentioned methods have their specific pros and cons but without a doubt they constitute a strong support of experimental tests.Numerical model of the phenomena of SCJ formation and penetration into high-hardness armor steel that was used in the current study had already been defined and verified by the team of the authors in their previous work[45].

2.Materials and experimental methods

2.1. Experimental tests methodology

Experimental tests (static detonations of modified warheads)were carried out in order to determine the penetration characteristics (size and shape of penetration crater) of SCJ formed after detonation of modified PG-7VM warheads.The results of experiments were used also during validation and verification of the numerical model of the phenomenon.Modification of warheads consisted in removing the frontal part of the grenade(fuse,ballistic cap and conductive cone -Fig.1) and introducing of the cavity filling made of polyacetal copolymer POM-C.The filing in the form of solid cones with three different heights (33%,66% and 100% of the original PG-7VM liner height)were placed inside of the hollow cone shaped charge liner.In order to check the influence of the fillings on the penetration characteristics analogical tests were carried out for the shortened PG-7VM warhead without fillings.During the tests,the modified PG-7VM warheads were placed at the frontal surface of the first target plate (Fig.2).The target consisted of three ARMSTAL 30PM[46]high-hardness(500 HB)armor steel plates(thickness of 5 mm,10 mm,10 mm respectively)and a 4 mm-thick witness plate placed 100 mm behind the target plates.All of the plates were of 100 × 100 mm2in size.Three static detonations were carried out for each of warhead modification variant.The penetration characteristics of shaped charge jet formed after detonation of modified PG-7VM warheads was determined on the basis of the size and shape of penetration crater measured in the target.The arrangement of the experimental test components was shown in Fig.2.

Fig.2.Scheme of the experimental tests: (a) PG-7VM warhead;(b) Modified PG-7VM warhead;(c) Components of tests ready for detonation;(d) Cross section of the testing assembly:1-Electric fuse,2-Pilot sleeve,3-Warhead body,4-Indirect detonator(pressed pentryt),5-Wave shaper,6-explosive A-IX-1,7-Liner inert filling(h),8-Liner,9-Isolator ring,10 -Target plates: 5 mm,10 mm,10 mm (Armstal 30PM),11 -PCV distance cylinder 100 mm,12 -Witness plate 3 mm (Armstal 30PM).

2.2. PG-7VM warhead

The proposed modifications were incorporated into the PG-7VM grenade.It is an improved,and more effective version of the standard PG-7V type grenade.The assembly of the original PG-7VM grenade (without modifications) was presented in Fig.3.The principles of the PG-7VM grenade functioning can be found in the previous work of the authors [45].

Fig.3.The assembly of the original grenade (without modifications).

Fig.4.The PG-7VM liner cone fillings analyzed in the study: (a) The way of various fillings arrangement;(b) Variants of fillings.

The PG-7VM warhead is filled with A-IX-1 explosive composition consisting of 94-96% RDX (hexogen) and 4-6% wax [34]and density of 1.63 g/cm3.According to the relation between the detonation velocity and density of the phlegmatized RDX [23,47]the detonation velocity of the mentioned composition was assumed as 8100-8300 m/s.

2.3. Material of liner filings

The PG-7VM liner cavity filling analyzed in the study was made of polyacetal copolymer POM-C (TECAFORM AH) inert material.Effectiveness of different filling materials was analyzed (Al 2024-T3,S355JR steel,M1E copper) but POM-C provided the highest damage ratio boost.POM-C is a thermoplastic produced by the polymerization of formaldehyde.POM-C is an engineering plastic for a wide range of applications in different branches of industry.Structural formula and basic properties of POM-C were shown in Table 1.

Table 1 Structural formula and basic properties of the POM-C material used for liner cone fillings provided by the manufacturer[48].

The fillings had the form of solid cones with three different heightsh=1/3H,h=2/3Handh=H(33%,66% and 100% ofH-original PG-7VM liner height) and were placed inside the hollow cone shaped charge liner (Figs.2 and 4).

In order to determine the stress-strain curves of POM-C material required to define numerical model of the material,it was necessary to carry out various material characterization tests.Quasistatic tension and compression tests were carried out.Samples with a geometry presented in Fig.5 were tested.Tests were registered with a camera in order to record the values of actual crosssections of samples that were changing together with the movement of jaws due to the barreling/necking effects caused by friction at specimen/jaws interfaces.

Fig.5.Mechanical testing stand: (a) Zwick Z100 testing machine;(b) Machine jaws with a sample between them;(c) Samples geometry.

Mechanical tests were carried out in order to collect data(forcedisplacement curves) allowing subsequent definition and validation of numerical models of the POM-C material.The tests of the mechanical properties of target material were carried out on a Zwick Z100 universal testing machine with a hydraulic drive(Fig.5).The results of the tests were shown in Table 2 and in Fig.6.

Fig.6.True stress-strain curves of the POM-C material calculated on the basis of mechanical test results (strain rate 10-3 s-1): (a) Tension;(b) Compression.

As it can be seen in Fig.6 the POM-C material exhibits significantly different behavior depending on the load variant(tension or compression).For example,yield strength during compression was significantly higher (by 50%) than in tensile test.The samples during tensile tests were damaged at relatively low strains,just above the elastic range.Opposite,during compression tests samples remained undamaged even at very high plastic strains.Additionally,after failure in tensile test,sample made of POM-C material exhibited high elastic recovery ratio.The maximal length of the measuring part of the sample during tension test was 138 mm(15%growth in relation to initial lengthL0=120 mm).After failure the length of the sample decreased toL=123 mm (2.5% growth in relation to initial length).Such a high elastic recovery was not observed during compression tests during which plastic strains dominated and sample was permanently deformed.

2.4. ARMSTAL 30PM armor steel target

High-hardness ARMSTAL 30PM armor steel plates were used as a target during the experiments.According to the data provided by the manufacturer[46],after optimized heat treatment the material exhibits uniform microstructure and a right balance of strength(Rm=1600 MPa)and ductility (Re02=1300 MPa,A5=min 8%).

Mechanical properties of the material(helpful in definition and validation of the numerical equivalent of the Armstal 30PM target)were investigated during material characterization experiments.The testing procedure included: hardness measurement(H=498-512 HB);quasi-static tension (Re02=1363-1401 MPa andRm=1623-1645 MPa) and compression tests(Re02=1501-1523 MPa andRm=1765-1782 MPa);impact strength tests.The detailed description of the testing procedure can be found in the work[45]and was not repeated here.The values of mechanical parameters obtained during the tests were higher than those provided by the manufacturer and slightly lower than in the works [49-52].There were relatively small differences in the results obtained in case of the samples with different thickness and extracted from different directions(parallel and perpendicular to the rolling direction) what testified about high uniformity and low anisotropy of the material.

3.Methodology of numerical simulations

3.1. Discretization, contacts, initial and boundary conditions

The phenomenon of a SCJ formation (as a results of detonation of modified PG-7VM warheads)and its subsequent penetration of a relatively thin high-hardness armor steel target was numerically analyzed in the study.Simulations were carried out in order to collect more information about the course of the penetration process.Due to the extremely high dynamic of the analyzed phenomenon the amount of data that could have been obtained during experiments was significantly reduced and was based on observations of the final deformations of target plates mainly.In the analyses the deformation ratios of the target plates were measured.The quantitative parameters assessed included diameters of the inlet and outlet holes in the plates.From the qualitative point of view,the way of target plates failure,cracks generation and propagations were analyzed.

Numerical analyses were carried out in the LS-Dyna software[53].The methodology of simulations that was used in the current study had already been verified and proved its efficiency during previous works of the team of the authors.Therefore,only basic parameters of the model were explained here and more details can be found in the work [45].The model was defined as threedimensional (3D) with two symmetry planes.Contacts between components described with multi-material arbitrary Lagrangian Eulerian(MMALE)(air,explosive,liner)and Lagrangian(remaining components) element formulation was modeled with*Constrained_Lagrange_In_Solid fluid-structure interaction algorithm.Discretization of the simulation components,performed using ALTAIR HyperMesh software,was shown in Fig.7.In the discretization the distance between the neighboring nodes of components of the warhead and the target plates equaled Δx=Δy=Δz=0.5 mm.Finally,all the components of simulation were discretized with 1,195,432 elements (635,432 elements for the warhead/domain system and 560,000 elements for the target).

Fig.7.Discretization of the simulation components.

3.2. Constitutive equations

The components of the analyzed phenomenon were made of metals (steel,copper and aluminum alloys),plastics,high explosives and air.Common constitutive equations were used to define numerical equivalents of those materials including: Johnson-Cook(J-C),High-explosive burn (HEB),Jones-Wilkins-Lee (JWL),Gruneisen equations of state (EOS) [[46,54-69]].The values of parameters used in the mentioned equations can be found in the work [45]where they were determined on the basis of the literature data [56-69]and own experimental tests.Only the liner fillings were made of new material (POM-C) that was not previously used in the referenced study [45].Since the POM-C material exhibited significantly different behavior during tension and compression (Fig.6) the *MAT_155_PLASTICITY_COMPRESSION_TENSION_EOS material model together with a polynomial equations of state (EOS) was used to reproduce its behavior.It is an isotropic elastic-plastic material where unique yield stress as a function of plastic strain curves can be defined for compression and tension.The main input to the model constituted stress-strain curves determined during experimental material characterization tests (Fig.6).The values for the remaining parameters were listed in Table 3.

Table 3 Parameters of the POM-C material model used in simulations.

4.Results and discussion

4.1. Results of depth of penetration tests and simulations

The results of penetration tests and simulations were shown in Fig.8-Fig.12.The following data was included in the figures:

Fig.8.Results of experiments and simulations for the PG-7VM warhead without filling:(a)Deformations of components and maps of velocities obtained in simulations for chosen moment after detonation;(b) Shapes of the inlet and outlet holes in the target plates;(c) Profiles of penetration craters.

Fig.9.Results of analyses for the PG-7VM warhead with the POM-C filling of the height h=1/3H:(a)Deformations of components and maps of velocities obtained in simulations for chosen moment after detonation;(b) Shapes of the inlet and outlet holes in the target plates;(c) Profiles of penetration craters.

Fig.11.Results of analyses for the POM-C filling of the height h= H: (a) Deformations of components and maps of velocities obtained in simulations for chosen moment after detonation;(b) Shapes of the inlet and outlet holes in the target plates;(c) Profiles of penetration craters.

Fig.12.Diameters of holes in target plates obtained in experiments and simulations.

· Deformations of simulation components at different times after detonation of warhead;

· Shapes of the scjs at different times after the detonation (just before the SCJ interaction with the first plate and after perforation or stopping of SCJ on the witness plate;

· Axial and radial velocity distribution in the SCJ at the abovementioned moments of time;

· Shapes of the inlet and outlet holes in the individual target plates and their buckling;

· Profiles of the penetration craters in the target made of ARMSTAL 30PM steel plates (experiments -black colour;simulations -green,red and blue colour).

The results of analyzes for the warhead without filling were shown in Fig.8.Therefore,it was the original PG-7VM warhead in which only the standoff distance was significantly reduced by cutting and shortening the warhead body.As a result,the penetration capability of the warhead was significantly reduced.At the nominal standoff of about 220 mm(resulting from the design of the frontal part of the PG-7VM grenade) the SCJ formed after detonation of PG-7VM warhead penetrates about 280 mm of the ARMSTAL 30PM target [45].Despite the significant reduction of the standoff distance to about 10 mm in the analyzed warhead,the SCJ easily penetrated both the target consisting of three ARMSTAL 30PM plates with a total thickness of 25 mm and the witness plate located 100 mm from the target.The maximum residual velocity of the tip of SCJ after perforation of the witness plate was very high and equaled aboutvres=4600 m/s(which constituted the velocity drop of about 37% in relation to the maximum SCJ velocity).Due to the small diameter of the formed SCJ (DSCJ=4-10 mm) in relation to the variants with fillings,its interaction with the target was more point-like (in a small area).Taking into account the nature of deformation of individual plates(no significant bulges) during the penetration process,it can be presumed that the mechanisms of deformation and failure were dominated by compressive and shear stresses.More accurate observation of the shapes of the holes in the plates revealed brittleness of the steel and its tendency to generate small fragments under impact loads.It was particularly visible in the case of the third plate,where the characteristic tearing out of the rear parts of the plate was observed,and the cracks propagated almost parallel to the plate surface.Disregarding the indentation on the surface of the first plate and the detachment of a fragment from the rear face of the plate,the mean diameter of the crater(observed on the longest section) was aboutD=20-25 mm.On the basis of the comparison of the experimental and numerical results,it should be noted that a high agreement was achieved.Differences in the diameters of craters in individual plates did not exceed 25%(exit hole in the witness plate)and in most cases were lower than 10%.

Fig.9 shows the results of the analyzes for the warhead with the filling ofh=1/3Hheight applied.Also in this case,the formed SCJ easily perforated both the target consisting of three ARMSTAL 30PM plates and the witness plate.However,the velocity of the tip of the SCJ both before interaction with the first plate (vSCJ1/3=6670 m/s) and after perforation of the witness plate (vSCJres1/3=1616 m/s) was much lower than in the case of the SCJ formed from the warhead without filling (vSCJPG=7298 m/s,vSCJresPG=4658 m/s)(Fig.8).Due to the SCJ-target interaction,the velocity of the tip of SCJ decreased by about 75%.The course of the penetration process was significantly different than in the case of the warhead without filling.The presence of POM-C filling with a height ofh=1/3Hand much lower density than that of copper caused that the SCJ generated from POM-C material was the first that reached the front surface of the target.At the same time,the filling limited to some extent the possibility of collapsing of the copper liner to the axis of the warhead,what was evidenced by 23%lower copper liner collapsing velocity (measured along theYaxis)vY1/3=1523 m/s (Fig.9) than for the variant without filling(vYPG=1990 m/s,Fig.8).As a result,both jets interacted with the surface of the target plates in an area with a much larger diameter(D1/3SCJ=12-20 mm) than in the case of unmodified PG-7VM(DPGSCJ=4-10 mm) but at a much lower velocity.In addition,due to the lower density of the POM-C material the shearing of the target material was not as intensive as in case of copper jet,and the jet was significantly deformed and retarded.Accordingly,considerable bulging of the target plates was observed (Fig.9) and its intensity increased with each successive plate.Taking the above into consideration,it can be presumed that the penetration mechanisms in this case relied less on shearing and tensile stresses played a much greater role.No plate cracks were observed in this case.

Thanks to the use of filling of theh=1/3Hheight made of POMC material,the diameters of the inlet and outlet holes in individual plates increased significantly(they ranged from 32 to 45 mm).The average diameter of the crater (observed on the longest section)was aboutD=35-40 mm,which constituted the growth of about 66% in relation to the unmodified variant.When comparing the results obtained experimentally and numerically,it should be noted that a high agreement was achieved.The differences in the diameters of the craters in individual plates in this case did not exceed 25% (the maximum was recorded for the inlet hole in the first plate).

The use of POM-C filling with a height ofh=2/3Hadditionally increased the damage ratio observed in the target plates(Fig.10).It should be noted,however,that in this case the SCJ penetration capability was significantly reduced and the jet did not perforate the witness plate located 100 mm behind the target.Although the velocity of the SCJ tip before the impact into the first plate(vSCJ=5919 m/s)was lower for only about 17% than in the case of SCJ formed from the warhead without filling(vSCJ=7298 m/s),due to the presence of filling made of POM-C the collapse velocity of the copper liner(vY=1200 m/s)was significantly lower(by 40%)than in the case of the unfilled variant (Fig.9).Consequently,the diameter of the combined SCJs (DSCJ2/3=28-30 mm) at the moment of contact with the target was about 300%higher than for the variant without modification (DPGSCJ=4-10 mm).The POM-C SCJ tip,during the penetration process,was additionally flattened by the reaction forces from the armor.The mechanisms of plate failure and deformation that dominated during the penetration process were based on exceeding the limiting tensile stresses of the target material.The plates bulges were large enough to cause the generation and propagation of radial cracks in all of the target plates.The greatest damage was observed in the first and third plate,which were fragmented into several large pieces.This phenomenon was undoubtedly influenced by the small size of the plates (100 × 100 mm2),which insufficiently exceeded the outer diameter of the PG-7VM grenade warhead (D=70.5 mm).Also in this case,the numerical model satisfactorily reproduced the real phenomenon.A high agreement was achieved between the numerical and experimental results both in terms of quantity (diameters of inlet and outlet holes for individual plates) and quantitaty (the way of deformation and failure of plates).The largest difference in the diameters of holes in plates observed in simulations and experiments(about 31%)was recorded for the first plate.In this case,the damage ratio of the plate was the highest and it was not possible to find all the fragments of the plate during the experiments.For the other plates,this error did not exceed 10%.In the simulations,the plates were damaged in a manner similar to that observed during the experiments (Fig.10).

The results of the analyzes for the variant in which the entire cavity of the liner was filled with POM-C (h=H) were shown in Fig.11.This type of filling reduced both the intensity of the damage of the target plates and the penetration capability of the SCJ.Due to very low velocity of the combined SCJs(vSCJ3/3=4887 m/s)and the large diameter of the SCJ/target interaction with theDSCJ3/3=28-30 mm (resulting from about 60% decrease in the copper liner collapsing velocity compared to the variant without modification) the stresses generated as a result of interacting SCJ were distributed over a significantly larger area.The penetration capability of the combined SCJ was reduced to such an extent that it could not even perforate the first plate.Only significant bulges of the plates and slight cracks in the first plate were observed,both in experiments and numerical simulations.

Summarizing,on the basis of the analysis results,it was clearly visible that with the increase of the height of the filling cone made of POM-C material,the diameter of the SCJ interaction area on the target surface increased,but at the same time the penetration capability of the SCJ decreased.Therefore,depending on the type of target (material from which it was made and its thickness),the height of the filling of the liner can be properly selected,ensuring the perforation of the target,while maximizing the target damage ratio(diameter of the SCJ/target interaction).In case of the analyzed target made of three ARMSTAL 30PM steel plates,the highest damage ratio was achieved for the variant with POM-C filling of heighth=2/3H.Thanks to the use of such filling,the diameters of the holes in individual steel plates were increased by 164%,70%,65% (Fig.12) (for the first,second and third plate,respectively) in relation to the variant without filling while ensuring perforation of the 25-mm-thick target made of ARMSTAL 30PM steel plate.Unfortunately,on the basis of the collected results,it is not possible to state whether the optimal filling height is higher or lower than 2/3H.Such evaluation requires additional tests.

Bar graph showing the differences in the diameters of the inlet and outlet holes obtained for individual variants of warhead fillings was shown in Fig.12.

Fig.13 shows time histories of the maximum velocity of the SCJ tips during the penetration of analyzed target made of ARMSTAL 30 p.m.steel.It is clearly visible that together with the growth of the height of the filling the maximum velocity achieved by the SCJ decreased(in relation to the unmodified warhead).For this reason as well as due to the growth of the SCJ/target interaction diameter the penetration capability of the combined SCJ decreases together with the growth of the height of the filling.In the extreme case-in variant 3(h=H),this caused that SCJ was stopped on the first target plate.

Fig.13.Course of maximal velocity of the frontal part of the jet in time for analyzed variants of warheads.

4.2. Model sensitivity analysis

Simulations and experiments results indicated that since the brittle failure of target plates based on generation and propagation of cracks occurred during the penetration processes,the values of adopted failure criteria of target material and size of the mesh may strongly affect the results of calculations.In the analyses,the limiting values of strains(different for tension and compression of material)were adopted as a target failure criterion.When the strain in the element exceeded one of the defined limiting values of the MXEPS (maximum principal strain in tension) and MNEPS(maximum principal strain in compression)parameters[53,54],the element was removed from the model.Additionally,in order to reproduce brittle failure of ARMSTAL 30PM steel,a low value of Cockroft-Latham (CL) failure criterion was used [53,54].A set of additional simulations were carried out for the variant with the filling ofh=1/3Hheight in which different values of MXEPS,MNEPS and CL parameters were adopted.The average value of failure strain was determined during the experimental compression tests and equaled εf=0.65,while in the work[52]the authors reported failure strain of ARMSTAL 30PM of εf=0.78.The comparison of the simulation results in which different values of failure criteria were adopted in the target material model were shown in Fig.14(a) and Fig.14(b).

Fig.14.Influence of the values of failure criteria on the target material behavior:(a)The way of target failure;(b)SCJ velocity in time;(c)Target erosion ratio expressed in percent of mass;and influence of target mesh size on: (d) The way of target failure;(e) SCJ velocity in time.

The sensitivity study revealed that changes of the failure criteria values of ARMSTAL 30PM target in the range of 0.3-0.6 for the MXEPS,-1.0 to-0.6.For the MNEPS and 300 to 500 for the CL did not significantly affect the results obtained.Decreasing of the failure criteria by 25% increased the SCJ residual velocity by about 8%(from 1600 to 1728 m/s).For the values of failure parameters increased by the 25%,the differences were even lower.SCJ residual velocity decreased by about 6%(from 1600 to 1513 m/s).Obviously when the failure criteria were decreased the target elements eroded faster during the jet penetration process.Therefore,more cracks with greater length were generated in the target plates.Target plates fragmented into smaller fragments.The most similar dimensions of penetration craters in target plates to the experimental ones were obtained for the lowest values of failure criteria.

Based on the mesh sensitivity study(Fig.14(c)and Fig.14(d)),it should be stated that the size of the elements (in the analyzed range)practically does not affect the results regarding SCJ residual velocity.The analyses were carried out for three sizes of the target mesh of elements:Δx=Δy=Δz=0.5 mm,0.375 mm and 0.75 mm.Therefore,the number of elements used in the target models were 590,000;1,418,524 and 175,071 respectively.The differences in the SCJ residual velocity after perforation of target with different size of the elements were smaller than 3%.However,the mesh size significantly affected the way of fragmentation of the target plates.When the element size was decreased the fragmentation of the plate was more intensive.The plate was crushed into larger number of smaller fragments similar like in experiments (large number of tracks after fragments impact was visible on the witness plate -Fig.14(c)).

Small differences in the obtained simulation results(differences in SCJ residual velocity was smaller than 3%)and a large growth of computational time (increase from 19 h for the 0.75 mm mesh to 60 h for the 0.375 mm mesh),indicated that elements with the size of Δx=Δy=Δz=0.375 mm could be recommended only for the cases where a high accuracy of the results is required.

5.Summary and conclusions

On the basis of literature review as well as performed experimental and numerical analyses the following conclusions can be drawn:

· Penetration characteristics of SCJ formed from modified PG-7VM warheads with liner cavity filling made of polyacetal copolymer POM-C was analyzed.

· The results of analyses indicated that with different materials of fillings and their various heights it is possible to control the shape of the penetration crater pierced in the target in case when the growth of the radius of damage is more important than depth of penetration of target.

· With the increase of the height of the filling cone made of POMC material,the diameter of the SCJ interaction area on the target surface increases;

· With the increase of the height of the filling cone made of POMC material the penetration capability of the SCJ decreased.

· Depending on the type of target (material from which it was made and its thickness),the height of the filling of the cumulative liner can be optimized in order to ensure perforation of the target,while maximizing the target damage ratio(diameter of the SCJ/target interaction).

· In the analyzed case of a target built of three ARMSTAL 30PM steel plates (with the thickness of 5 mm,10 mm,10 mm respectively) the highest damage ratio of target was achieved for the variant with POM-C filling with a height ofh=2/3H.

· Thanks to the use POM-C filling with a height ofh=2/3H,the diameters of the holes in individual steel plates were increased by 164%,70%,65%(Fig.12)(for the first,second and third plate,respectively) in relation to the variant without filling while ensuring perforation of the 25-mm-thick target made of ARMSTAL 30PM steel plate.

· Determination of the optimal height of POM-C filling for the analyzed case requires additional tests,because it is not possible to state whether the optimal filling height is higher or lower thanh=2/3H.

· A high agreement was achieved between the numerical and experimental results (average error 10-20%) both in terms of quantity (diameters of inlet and outlet holes for individual plates)and quality(the way of deformation and failure of plates)

Declaration of competing interest

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