Formation behaviors of rod-like reactive shaped charge penetrator and their effects on damage capability

To Sun , Hifu Wng , Shipeng Wng , Cho Ge , Die Hu , Pengwn Chen ,Yunfeng Zheng ,*

a Beijing Institute of Technology, State Key Laboratory of Explosion Science and Technology, Beijing,100081, China

b Xi'an Institute of Electromechanical Information Technology, Xi'an, 710065, China

Keywords:Reactive materials Al-PTFE composites Reactive liner X-ray Penetration

ABSTRACT Formation behaviors of rod-like reactive shaped charge penetrator (RRSCP) and their effects on damage capability are investigated by experiments and numerical simulations.The pulsed X-ray technology and a spaced aluminum/steel plate with the thicknesses of 5 mm/100 mm are used.Three types of sphericalsegment aluminum-polytetrafluoroethylene-copper (Al-PTFE-Cu) reactive liners with Cu contents of 0%,46.6%,and 66%are fabricated and tested.The experimental results show that the reactive liners can form excellent rod-shaped penetrators with tail skirts under the shaped charge effect, but the tail skirts disappear over time.Moreover, rupturing damage to the aluminum plate and penetration to the steel plate are caused by the RRSCP impact.From simulation analysis, the RRSCP is formed by a mechanically and chemically coupled response with the reactive liner activated by shock in its outer walls and bottom and then backward overturning, forming a leading reactive penetrator and a following chemical energy cluster.The unique formation structure determines the damage modes of the aluminum plate and the steel plate.Further analysis indicates that the formation behaviors and damage capability of Al-PTFE-Cu RRSCP strongly depend on Cu content.With increasing Cu content, the velocity, activation extent, and reaction extent of Al-PTFE-Cu RRSCP decrease,which contribute to elongation and alleviate the negative effects of chemical reactions on elongation, significantly increasing the length-diameter ratio and thus enhancing the capability of steel plate penetration.However, the lower activation extent and energetic density will weaken the RRSCP's capability of causing rupturing damage to the aluminum plate.

1.Introduction

Shaped charges have been widely used to attach armors and concrete fortifications due to their high energy utilization and excellent penetration capability [1].Traditional shaped charges with metal liners can form three classical penetrators: shaped charge jet (SCJ), jetting projectile charge (JPC), and explosively formed projectile (EFP).Much research has been conducted on their formation characteristics and penetration performance[2-4].However, due to the limitations of the single kinetic energy penetration mechanism and mechanical perforation damage mode,it is difficult to significantly improve the perforation hole diameter and behind-plate effect of traditional metal shaped charge penetrators[5,6].

Reactive materials and their applications in shaped charges are current research hotspots.Reactive shaped charge penetrators(RSCPs, such as reactive jets [7] and EFPs [8]) formed by reactive material liner not only have metal-like penetration capability, but will also spontaneously induce explosive-like deflagration reactions during penetration or after perforation,resulting in greater structural damage to the targets.The combined damage capability of kinetic and chemical energy of RSCPs strongly depend on their formation characteristics, such as the morphology and velocity.Similar to penetration,the formation process of RSCPs is also highly dynamic and involves high temperatures,high pressures,and high strain rates.The reactive liner undergoes significant plastic deformation during formation, as well as being activated by shock and going through chemical reactions [9].It is more difficult to understand the formation behaviors of RSCPs due to their unique formation mechanism with mechanically and chemically coupled responses.

The literature published so far has mainly focused on the fabrication, mechanical properties, and shock response of reactive materials, as well as the formation behaviors and penetration performance of reactive jets.Jossi [10] patented the preparation process of Al-PTFE, a typical reactive material, which involves powder mixing, molding, and sintering hardening.Using this preparation approach, Li [11] and Guo [12] fabricated reactive material liners with higher density by introducing high-density inert metal particles (such as copper and plumbum) into Al-PTFE.Raftenberg [13] investigated experimentally the mechanical properties of Al-PTFE(with a tested quasi-static mechanical strength of only 8 MPa),and then determined the parameters for the Johnson-Cook constitutive model to supporting numerical simulation.Sun[14] found that the strength of reactive materials is almost independent of the proportion of high-density inert metals.Osborne[15] described the thermal reaction behaviors of Al-PTFE.Zhang[16] and Tang [17] investigated the shock response of reactive materials using theoretical models and mesoscale simulations,and revealed the shock-induced reaction mechanism.Guo [18,19]observed through X-ray experiments that the Al-PTFE reactive material liner can form a reactive jet with excellent performance under shaped charge effect, and found that a relative larger hole diameter but lower depth accompanying with fragmentation effects of penetrating steel plates are produced by reactive jets compared with traditional metal jets.She assumed that there is a time delay in the chemical reaction of the reactive jet during formation.Before this delay, the reactive jet remains inert, but when the reaction delay time is reached, the reactive jet immediately undergoes a deflagration reaction.Based on the assumption, she developed a penetration model and discussed the effect of standoff and reaction delay time on the penetration performance.In addition,she also investigated the effects of wave shaper on reactive jet formation and its penetration performance through experiments and simulations [20].

However, there have been relatively few studies on the formation behaviors of rod-like reactive shaped charge penetrators(RRSCPs).The influence mechanism of chemical reactions on the formation of RRSCPs is especially unclear.Further research should be also required on the formation behaviors of RRSCPs with higher density and their role in affecting the damage capability.

This study begins with X-ray and penetration experiments to investigate the formation characteristics and damage capability of Al-PTFE-Cu RRSCP with three different Cu contents.Then, simulations are conducted and analyzed to reveal the mechanism of how the RRSCP is formed by a mechanically and chemically coupled response.The effect of Cu content on the formation characteristics such as velocity distribution, structure, and length-diameter ratio are also discussed in detail.Lastly, the rupturing damage made to aluminum plates by the RRSCP and the enhanced penetration capability it has for steel plates are discussed based on the formation characteristics.

2.Experimental

2.1.Shaped charge with a liner made of reactive materials

The discussed shaped charge consists of explosive,a baffle ring,a reactive material liner, and a detonator, as shown in Fig.1.A powerful explosive, 8701, is molded at a pressure of 200 MPa and initiated by a detonator.The density, diameter, and height of the explosive are 1.71 g/cm3,50 mm,and 50 mm.The reactive material liner is a spherical segment with an even wall thickness of 4 mm and curvature radius of 45 mm.The 45#steel baffle ring is used to fix the reactive liner by gluing,with an inner and outer diameter of 45 mm and 50 mm, respectively.

Fig.1.Structure of reactive liner shaped charge: (a) Schematic and (b) typical physical photograph.

The fabrication method proposed in Ref.[10]is used for the Al-PTFE-Cu reactive material liner.The preparation process mainly includes powder mixing,molding,and sintering.First,Al,PTFE,and Cu powders are added to a ball mill at a certain mass ratio and mixed for 3 h.The average particle diameters of the Al,PTFE,and Cu powders are 44 μm,100 nm, and 24 μm, respectively.It should be noted that the relative mass of Al and PTFE always follows the stoichiometric ratio of the chemical reaction under zero oxygen balance(i.e.,26.5 wt%:73.5 wt%).Then,the mixture is placed in a vacuum drying oven at 82°C and dried for roughly 24 h.Third,the uniform mixture is placed in a self-designed steel mold and pressed at a pressure of 200 MPa for 30 s.Fourth,the molded liner samples are sintered in a nitrogen-filled furnace at a maximum temperature of 380°C.In this study, three types of Al-PTFE-Cu reactive material liners with Cu contents of 0%, 46.6%, and 66%are fabricated.Table 1 displays the material compositions, densities, and photographs of the fabricated reactive liners.

2.2.Setup of X-ray and penetration experiments

The experimental setup is illustrated in Fig.2.The reactive liner shaped charge is placed on a hollow standoff cylinder made of nylon.The height of the standoff cylinder is 500 mm,which is 10 times the charge diameter (CD).The pulsed X-ray technology is used to observe the formation of each RRSCP.After the charge is initiated, a RRSCP is formed and moves within the standoff cylinder.As two X-ray tubes with different delay settings emit X-rays successively,the morphology and positioning of the RRSCP at two distinct moments are recorded.The characteristic parameters of the RRSCP at a specific time, including its length, diameter, and average velocity, can be obtained based on the geometric similarity principle.A spaced 2024 aluminum/45#steel plate is placed below the standoff cylinder to verify the damage behaviors of the RRSCP.The thickness of the aluminum and steel plates is 5 mm and 100 mm, respectively.The aluminum and steel plates are separated by a foam with a distance of 100 mm.

2.3.Experimental results of formation and damage effects

The RRSCP formation images captured by X-rays are presented in Fig.3.At the time of t1, the spherical-segment reactive liner forms a rod-shaped penetrator with a tail skirt.The RRSCP has a clear and symmetric outline, yet the tail skirt is severely broken,dispersing into a conical cloud of debris.The clarity of the X-ray image improves as the Cu content in the RRSCP increases.This is determined by the material density of the reactive liner.At the time of t2, the RRSCP becomes longer.More interestingly, the broken tail skirt almost disappears, and the bright rod tail turns blurry.This may be caused by the chemical reaction of reactive materials.In comparison,the outlines of Al-PTFE-Cu RRSCPs with Cu content of 46.6% and 66% are still clearly visible, while that of the Al-PTFE RRSCP is severely blurred.

Define vhas the average head velocity of the RRSCP, L as the length of the RRSCP, and the width d at half of the length as the average diameter of the RRSCP.The characteristic parameters of the RRSCPs at two exposure times are listed in Table 2.In terms of the formed velocity,the head velocity of the RRSCP decreases from 3667 m/s to 3344 m/s and 3224 m/s, respectively, as the Cu content increases from 0% to 46.6% and 66%.This is consistent with the velocity-changing trend of metal shaped charge penetrators with various densities [21,22].Comparing the morphology characteristics of RRSCP at two different times, the length increases but the diameter decreases, resulting in an increased lengthdiameter ratio.This is also consistent with the morphological evolution law of metal shaped charge penetrators during elongation [21,22].

The above experimental results demonstrate that Al-PTFE-Cu reactive material liners with Cu content from 0% to 66% are able to form RRSCPs with excellent performance under the shaped charge effect.However,their formation behaviors are much more complex than those of traditional metal shaped charge penetrators, especially regarding the morphology evolution, which may be accompanied by the chemical reactions of reactivematerials.Moreover,the velocity and morphology characteristics of Al-PTFE-Cu RRSCP are significantly related to the Cu content.

Table 1 The fabricated Al-PTFE-Cu reactive material liners with different Cu contents.

Fig.2.Experimental setup: (a) Schematic; (b) Physical photograph.

Fig.3.X-ray images of RRSCP with different materials.

Table 2 Experimental data of RRSCP characteristic parameters.

Fig.4.Damage effects of aluminum and steel plates impacted by RRSCPs:(a)Al-PTFE RRSCP;(b)Al-PTFE-Cu RRSCP with Cu content of 46.6%;(c)Al-PTFE-Cu RRSCP with Cu content of 66%.

The aluminum and steel plates impacted by RRSCPs are shown in Fig.4.All three Al-PTFE-Cu RRSCPs perforate the front aluminum plate and penetrate the rear steel plate to a certain depth, indicating that RRSCPs still have an effective penetration capability at a large standoff of 10 CD.In particular, the aluminum plate perforation shows a typical irregular large rupturing hole with a maximum diameter of more than 2 CD.The rupturing areas on the aluminum plates are obtained by the method proposed by Lu [23].It can be found that as the Cu content increases from 0 to 66%,the rupturing area on the aluminum plate decreases from 99.6 cm2to 44.3 cm2,a sharp decline of 55.5%.This indicates that the increasing Cu content weakens the rupturing damage capability of Al-PTFE-Cu RRSCPs.However, the penetration depth on the steel plate increases from 7 mm to 38 mm as the Cu content increases from 0% to 66%, a growth of 4.4 times.This indicates that the increasing Cu content significantly enhances the penetration capability of Al-PTFE-Cu RRSCPs.

3.Analysis and discussion

3.1.Numerical method of formation

Numerical simulations are carried out to further understand the formation behaviors of Al-PTFE-Cu RRSCPs using AUTODYN-3D software.It should be noted that the Al-PTFE-Cu reactive material liner is a powder liner in nature.For the numerical calculation of such liners, previous investigations indicated that the smoothed particle hydrodynamics (SPH) algorithm delivers more accurate results than the Euler algorithm[24].Therefore,the SPH algorithm is used in this study.Fig.5 shows the numerical model of the reactive liner shaped charge.Comprehensively considering the less calculation time,better formation effect and acceptable calculation difference caused by particle size difference,the particle size of the explosive and the baffle ring are set to be 0.5 mm, and that of the reactive material liner is set to be 0.25 mm.Taking structural symmetry into account,1/2 of the geometry is used to simplify the analysis and improve the computational efficiency.

Fig.5.Numerical model of reactive liner shaped charge.

Table 3 Material parameters of 8701 explosive [25].

The Jones-Wilkins-Lee (JWL) equation of state (EOS) is used to describe the 8701 explosive as follows

where Pais the hydrostatic pressure,V is the specific volume, Eais the specific internal energy, and A1, B1, R1, R2and ω are material constants.The material parameters of the 8701 explosive are listed in Table 3,where ρc,D and PCJare the explosive density,detonation velocity and detonation pressure, respectively.

The Johnson-Cook strength model and the shock EOS are used to characterize the dynamic mechanical properties of reactive liner and baffle ring materials under high strains, strain rates and temperatures.

The Johnson-Cook strength model can be expressed as

where A, B, C, n and m are material constants, which are generally determined by experiments.σ is the equivalent plastic stress, εpis the equivalent plastic strain, ˙ε is the current strain rate, ˙ε0= 1.0,s-1is the reference strain rate,T is the current temperature,and Tmand Trare the melting temperature and room temperature,respectively.It should be noted that the strength of the reactive liner is much lower than that of the metal liner.Although the addition of high-density inert metal particles improves the strength of the reactive liner, the improvement is minor [14].Therefore,the strength effect caused by Cu content is ignored in the simulations, and identical strength model parameters as those of Al-PTFE liners are used for Al-PTFE-Cu reactive liners with different Cu contents.The strength model parameters of each material are listed in Table 4.

Table 4 Parameters of strength models for reactive liner and baffle ring materials[13,25].

The shock EOS can be expressed as

where Pbis the shock pressure, ρ0is the initial density, C0is the sound speed of the material, s is the linear Hugoniot slope coefficient, η = (ρ/ ρ0-1) where ρ is the current density, γ0is the Grüneisen coefficient,and Ebis the internal energy.It should be noted that the formation process of RRSCP is approximately regarded as an adiabatic process and its shock temperature rise is closely related to parameters of EOS and heat capacity.For Al-PTFE-Cu reactive mixtures with different Cu contents, the sound velocity C0, Hugoniot coefficient s and heat capacity cvare approximately obtained by the classic Voigt-Reuss mixing law [26], which has been successfully applied to the numerical studies of dynamic mechanical properties and shock responses of multi-component mixtures including reactive materials [27].The Voigt-Reuss mixing equation is expressed as

where Mris the mixture modulus,f is the component mass ratio,M is the component modulus,and N is the component number.After obtaining the Hugoniot coefficient s of the reactive mixture, the Grüneisen coefficient γ0of the reactive mixture can be estimated as γ0=2s-1.The EOS parameters and heat capacity of each material are listed in Table 5.It should be noted that the shock EOS used for reactive materials in the present study cannot describe the chemical reaction behavior.However, the simulation results are still meaningful for the formation study of RRSCP.

3.2.Velocity distribution

Velocity distribution of a typical RRSCP during formation is presented in Fig.6.The formation process can be divided into two stages.The first is the detonation driving stage (within about 5 μs～30 μs).Under the shaped charge effect,the centroid of the liner develops into the head of RRSCP at a faster speed,while the bottom of the liner moves relatively slower and develops into the tail of RRSCP.At the same time,the elements on both sides of the axis are crushed,further increasing the velocity gap between the head and the tail.Overall, the reactive liner exhibits a mixing deformation mode including backward overturning,closing,and stretching.The second is the free elongation stage, in which the RRSCP freely elongates in accordance with its own velocity gradient.The velocity at the head slightly decreases while it increases at the tail.The velocity of the RRSCP tends to stabilize after 50 μs.

The axial velocity distribution of Al-PTFE-Cu RRSCP with different Cu contents at 50 μs is illustrated in Fig.7.The velocity of RRSCP increases approximately linearly from the tail to head.The tail velocities of the Al-PTFE-Cu RRSCP with three different Cucontents are extremely similar, at about 870 m/s.However, the velocities of their compacted sections differ greatly.The compacted section of the Al-PTFE RRSCP has the highest head and tail velocities, which are 3627 m/s and 2064 m/s, respectively.Its head and tail velocity gap is also the largest, reaching 1563 m/s.As the Cu content increases to 46.6%, the head and tail velocities of the compacted sections decrease to 3126 m/s and 1713 m/s, respectively,with a smaller velocity gap of 1413 m/s.When the Cu content reaches 66%, the head and tail velocities further decrease to 2948 m/s and 1565 m/s, and the corresponding velocity gap decreases to 1383 m/s.The results above indicate that with the increase of Cu content, the velocity of the RRSCP significantly decreases while its velocity gradient of the compacted section does not change much.

Table 5 EOS parameters and heat capacity of materials [17,25,27].

Fig.7.Velocity distribution of RRSCPs at 50 μs: (a) Al-PTFE RRSCP; (b) Al-PTFE-Cu RRSCP with Cu content of 46.6%; (c) Al-PTFE-Cu RRSCP with Cu content of 66%.

The velocity characteristics of the RRSCP can be analyzed from the following three aspects.First, the RRSCP tail is formed by the bottom section of liner.This portion of the material will be separated from the liner in a form of collapse or fragmentation,due to the small effective charge action and the rapid unloading of the radial rarefaction wave.The bottom velocity of the liner can be basically considered as independent from that of the liner material.So, the Al-PTFE-Cu RRSCPs with different Cu contents have nearly identical tail velocities.Second, the velocity of the RRSCP compacted section depends mainly on the liner density for a shaped charge with a definite shape and size.The density of the reactive liner increases with the Cu content, resulting in an increased liner mass driven by detonation.According to energy conservation, the corresponding RRSCP velocity will continuously decrease.Finally,the strength of Al-PTFE-Cu reactive material is not sensitive to Cu content.This means that no more reactive materials will be crushed and move to the axis with the increase of the Cu content.Therefore,the velocity gradients of the compacted sections of Al-PTFE-Cu RRSCPs with different Cu contents are not significantly different.It is also found from Fig.7 that the numerical predicted head velocities are close to the experimental ones with a maximum deviation of 8.5%, indicating a good agreement between the simulated and experimental results.

According to the fluid dynamics theory on penetration[28],the penetration performance depends on the penetration velocity and penetration time of each element on the RRSCP.

where H is the penetration depth, uiand τiare the penetration velocity and the penetration time of element i, which can be expressed as

where v and ρjare the formation velocity and the density of Al-PTFE-Cu RRSCP.ρtand Rtare the density and the resistance of target.

The penetration velocity of each element on compacted section of RRSCP is shown in Fig.8.Although the formation velocity of Al-PTFE-Cu RRSCP decreases as the Cu content increases, the penetration velocity does not change due to the increasing density and the close velocity gradient on the compacted section.In fact, according to Eq.(6), lower formation velocities will increase the penetration time and thus enhance the penetration capability.

3.3.Tandem structure feature determined by temperature

Fig.8.Penetration velocity of RRSCP.

Under the shaped charge effect, reactive liner forms a highspeed penetrator, and chemical reactions in the reactive materials are triggered.Thus, the formation behaviors of RRSCP and conventional metal shaped charge penetrators are essentially different.For mechanism considerations, the shock-induced chemical reaction of reactive materials is a complex mechanical-thermalchemical coupling process [16].In the process, the shock first causes the mechanical responses of reactive materials,resulting in a temperature rise, which then causes the chemical reaction.It should be noted that the shock-induced temperature rise is a comprehensive temperature effect caused by many mechanical responses (such as shock wave, plastic deformation, and fracture)and is used to evaluate whether the material is activated to react.Therefore, the temperature rise plays an important role and provides more detailed information concerning the formation mechanism and characteristics of RRSCP.

The temperature diagram of a typical RRSCP during formation is shown in Fig.9.After the charge is initiated for 5 μs～8 μs, the detonation wave sweeps along the top to the bottom of the liner,compressing the liner to deform.Simultaneously, the detonation wave transmits into the liner,forming a shock wave that propagates toward the inner wall of the liner.The temperature of the reactive liner material rises sharply as a result of the shock wave action.Then,the temperature is promptly reduced to a certain level,due to the unloading of the reflected rarefaction wave from the inner wall of the liner.It should be noted that the shock wave pressure is successively released from the inner wall of the liner to the outer wall during the propagation of the reflected rarefaction wave, so the shock time acting on the outer wall is longer than that on the inner wall.As a result, the shock temperature rise decreases gradually from the outer wall of the liner to its inner wall.As the reactive liner overturns backward to deform, the outer wall with higher temperature develops into the inner surface of the RRSCP,and the inner wall with lower temperature into the outer surface.In the process, the temperature of the RRSCP continuously increases due to great plastic deformation.Moreover, the material at the bottom of the liner breaks due to collapse effect, and the corresponding fracture energy is transferred into heat, which further increases the temperature rise of this part of the material.The detonation driving effect disappears at about 30 μs, at which the RRSCP is basically formed.The high temperature zone is mainly concentrated in the tail skirt break zone and the tail of the compacted section.Thereafter, the RRSCP freely elongates by its own velocity gradient.In this process, the tail skirt is evidently broken.

A previous research [15] demonstrated that the chemical reaction of reactive materials is determined by material temperature.When the material temperature reaches the decomposition temperature of PTFE,the PTFE matrix first decomposes to produce the potent oxidizing gas product C2F4,which then undergoes a violent redox reaction with the Al particles, releasing a large number of high-pressure gas products.Therefore, the decomposition temperature (about 800 K [15]) of the PTFE matrix can be approximately considered as the activated temperature threshold of the reactive material.As Fig.10 shows,the high-temperature tail of the RRSCP (i.e.the outer wall and the bottom section of the reactive liner)is activated,while the low-temperature head of the RRSCP is not.The unactivated head remains still inert, while the activated tail will react and release chemical energy.In addition,the chemical reactions of the reactive materials cannot sustain by themselves[29].In other words, the chemical reactions at the tail can hardly trigger the unactivated materials at the head.So overall,the RRSCP exhibits a tandem structure composed of a leading reactive penetrator and a following chemical energy cluster.On the one hand,thanks to the unique tandem structure, the RRSCP is able to penetrate the aluminum plate and the steel plate at a standoff as large as 10 CD,rather than reacting completely.On the other hand,for the thin aluminum plate, the tandem structure allows the RRSCP to first perforate the plate with the leading reactive penetrator like a metal penetrator, and then release chemical energy with the following chemical energy cluster to cause further damage in a form of overpressure.Thus, a rupturing effect is produced on the aluminum plate (see Fig.4).

Fig.9.The temperature profiles of Al-PTFE RRSCP during formation: (a) t = 5 μs; (b) t = 8 μs; (c) t = 16 μs; (d) t = 30 μs; (e) t = 50 μs; (f) t = 85 μs.

Fig.10.The structure feature of RRSCP at 30 μs: (a) Al-PTFE RRSCP; (b) Al-PTFE-Cu RRSCP with Cu content of 46.6%; (c) Al-PTFE-Cu RRSCP with Cu content of 66%.

Fig.10 also shows that the tail skirt and the compacted section tail of Al-PTFE RRSCP are activated.As the Cu content increases to 46.6%, the activated reactive material decreases.When the Cu content increases to 66%,only some broken portion of the tail skirt and the surface of the compacted section tail are activated.The above results indicate that the increase of Cu content leads to a lower activation extent for RRSCPs.According to the laws of thermodynamics[30],the main reason is that the ratio of the Grüneisen coefficient to the specific volume for Al-PTFE-Cu reactive materials increases with the Cu content,which results in a lower temperature rise.Moreover, the potential energetic density of Al-PTFE-Cu decreases with the increase of Cu content.Therefore, as the leading penetrator perforates the aluminum plate, the following chemical energy cluster with a lower activation extent and energetic density will cause a weaker secondary damage to the aluminum plate.This explains the experimental results in Fig.4 well, which shows that the rupturing hole area of the aluminum plate decreases with an increasing Cu content.

3.4.Morphology evolution accompanied by chemical reactions

Under zero oxygen balance, the main reaction of reactive materials is the reaction of Al and PTFE, while Cu almost does not participate in the reaction.Therefore,the influence of Cu content on reaction rate is not considered.According to the Arrhenius law,the reaction rate of the activated reactive materials can be expressed as[31].

where y is the reaction extent, t is the reaction time, Z is the preexponential factor, Ecis the activation energy, R is the molar gas constant, T is the temperature, and k is the coefficient related to boundary conditions.The relevant calculation parameters are displayed in Table 6.

The reaction history of the activated reactive material can be calculated by Eq.(7), as shown in Fig.11.The activation time in Fig.11 is approximated as the termination time(about 8 μs)of the detonation wave action.It is found from Fig.11 that the reaction extents of the activated reactive materials increase over time, but the reaction rates at various temperatures vary greatly.The reaction rate at temperatures below 900 K is extremely low,and the reaction extent is less than 0.1 at 100 μs.The reaction rate increases significantly with the temperature.For example, the reactive materials at 1000 K can reach a reaction extent of 0.58 at 100 μs,and react completely at 80 μs at 1100 K.When the temperature of reactive materials exceeds 1200 K,the reaction rate is so high that the whole reaction can be completed within 30 μs.The resultsabove indicate that the activated reactive materials at the tail will not complete their reaction instantaneously, but will react in batches corresponding to different temperatures.In the reaction process, the activated reactive materials at the tail of the RRSCP gradually evolve from solid form to gaseous form, resulting in a decreased density.This well explains why the tail skirt disappears and the rod tail blurs in the experiments, as shown in Fig.12.

Table 6 Parameters of reaction rate equation [17,32].

Fig.11.Reaction history of activated reactive materials.

As mentioned above, the formation process of the RRSCP is accompanied by chemical reactions.In order to analyze the influence of such reactions on the length-diameter ratio of the RRSCP qualitatively, the simulated results with no chemical reactions taken into account and the experimental results are compared.As Fig.13 shows, due to the large velocity gradient, the lengthdiameter ratio of the RRSCP shows an increasing trend during formation.Moreover,the length-diameter ratio of RRSCP increases with the Cu content at the same standoff.According to Section 3.2,when Cu content increases, the density of Al-PTFE-Cu reactive material increases while its strength does not change much,resulting in a decreased head velocity and close velocity gradient of RRSCP.As a result, the elongation time to reach the same standoff increases and thus the length-diameter ratio increases.However,the simulation results are always higher than the experiments.More specifically, the error between the simulation and experimental data is quite small at time t1.Yet at time t2, the simulation result is evidently higher than the experimental one for Al-PTFE RRSCP.The error decreases as the Cu content increases, and is near zero at a Cu content of 66%.

The deviations between simulations and experiments may be attributed to the chemical reaction of the activated reactive material at the compacted section tail.As Fig.12 shows, there are two main influence modes.On the one hand, the chemical reaction of the activated reactive materials at the compacted section tail may lead to a loss of part of the length of the RRSCP during elongation.On the other hand, the activated reactive material will be compressed into the interior of the rod during closure,which may cause radial expansion of the rod.As a result, the increased rate of the length-diameter ratio of RRSCP will be retarded.Therefore, the simulation results are always higher than the experiments.In addition,due to the low temperature and slow reaction rate of the activated reactive materials at the compacted section tail, the chemical reactions at the initial stage have little effect on the length-diameter ratio of the RRSCP.The reactions are intensified over time,resulting in an enhanced influence.Especially for the Al-PTFE RRSCP with a higher shock temperature rise, activation and reaction extents, the effects of chemical reactions on its lengthdiameter ratio is more significant.Therefore, the length-diameter ratio of Al-PTFE RRSCP obtained by simulation is significantly higher than that in the experimental data.As the Cu content increases,the shock temperature rise,activation and reaction extents all decrease, resulting in a lower influence on the length-diameter ratio.When the Cu content reaches 66%,nearly no reactive material is activated, or the reaction extent of the activated reactive materials is extremely low, so the length-diameter ratio is hardly affected.

On the whole,the length-diameter ratios of Al-PTFE-Cu RRSCPs increase continuously with the increase of Cu content at the same standoff, partly due to their formation velocities.The difference even increases due to chemical reactions, and is especially significant at a large standoff.According to the penetration theory[28],a larger length-diameter ratio means a stronger penetration capability.In other words, a higher Cu content will enhance the penetration capability of the Al-PTFE-Cu RRSCP, especially at a large standoff.The results are consistent with the experimental results observed in the impacted steel plates in Fig.4.

Fig.12.RRSCP morphology comparison between X-ray photographs and simulated results.

Fig.13.Length-diameter ratio of RRSCP.

4.Conclusions

Formation behaviors of RRSCPs and their effects on damage capability are investigated by experiments and numerical simulations.The main findings are summarized as follows.

(1) The shaped charge detonation drives the reactive liner to overturn backward, and activates its outer wall and bottom section.The formed RRSCP shows a unique tandem structure composed of a leading reactive penetrator and a following chemical energy cluster.With such a structure, the RRSCP can effectively penetrate the target at a large standoff rather than reacting completely.For thin plates,a rupturing damage is produced by the time-sequenced combined effects of the kinetic energy penetration and chemical energy release of the RRSCP.However, as the Cu content increases, the activation extent and potential energetic density of the Al-PTFECu RRSCP decrease, weakening the rupturing damage capability.

(2) With a higher Cu content, the Al-PTFE-Cu RRSCP shows a lower formation velocity, yet the velocity gradient of its compacted section remains basically unchanged.Instead of changing the penetration velocity, the lower formation velocity will improve the penetration time and thus enhance the penetration capability.

(3) A higher Cu content allows the Al-PTFE-Cu RRSCP to elongate more sufficiently, and greatly alleviates the negative effects of chemical reactions on the elongation,leading to a greater length-diameter ratio and thus significantly enhancing the penetration capability of Al-PTFE-Cu RRSCP.The penetration depth on the steel plate increases by 4.4 times as the Cu content increases from 0 to 66%.

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.

Acknowledgments

This research is supported by the National Natural Science Foundation of China (No.12172052 and No.12132003).