Simulation Study on Jet Formability and Damage Characteristics of a Low-Density Material Liner

materials

Article Simulation Study on Jet Formability and Damage Characteristics of a Low-Density Material Liner

Liangliang Ding ID , Wenhui Tang * and Xianwen Ran

College of Science, National University of Defense Technology, Changsha 410073, China; [email protected] (L.D.); [email protected] (X.R.) * Correspondence: [email protected]; Tel.: +86-181-7512-1477

Received: 12 December 2017; Accepted: 3 January 2018; Published: 4 January 2018

Abstract: The shaped charge tandem warhead is an effective weapon against the ERA (explosive reactive armor). Whether the pre-warhead can reliably initiate the ERA directly determines the entire performance of the tandem warhead. The existing shaped charge pre-warhead mostly adopts a metal shaped jet, which effectively initiates the ERA, but interferes the main shaped jet. This article, on the other hand, explores the possibility of producing a pre-warhead using a low-density material as the liner. The nonlinear dynamic analysis software Autodyn-2D is used to simulate and compare three kinds of low-density shaped jets, including ﬂoatglass, Lucite, and Plexiglas, to the copper shaped jet in the effectiveness of impacting ERA. Based on the integrative criteria (including u-d initiation criterion, explosive reactive degree, explosive pressure, and particle velocity of the panels), it can be determined whether the low-density shaped jet can reliably initiate the sandwich charge. The results show that the three kinds of low-density shaped jets can not only initiate the reaction armor, but are also superior to the existing copper shaped jet in ductility, jet tip velocity, jet tip diameter, and the mass; namely, it is feasible to use the low-density material shaped jet to destroy the ERA.

Keywords: explosive reactive armor; low-density material; shaped jet; liner; tandem warhead

1. Introduction Local wars in recent years have shown that the modern armored targets, such as tanks and armored vehicles, are still the main combat systems on the ground battlefield, which integrates firepower, protection, and mobility. The shaped charge warhead is one of the most important anti-armor ammunition, and a great deal of manpower, material, and financial resources have been spent on improving its power and performance. In 1969, Held found that the sandwich structure (metal panel/explosive/metal panel) can significantly reduce the penetration ability of a shaped jet and kinetic rod, then applied for a patent in 1970, which then became the prototype of ERA (explosive reactive armor). The ERA is a revolution in the field of armor protection for its wide use in main battle tanks and armored vehicles, to name a few: Russia0s T-90 main battle tank, Israel0s MK4 main battle tank, the United States M1A1/M1A2 main battle tank, China0s 99 series main battle tank and Germany0s Leopard II main battle tank. The wide application of ERA has posed a great threat to the traditional anti-armor ammunition. When equipped with ERA, the armored target0s protection ability, battlefield survival ability, and the sustained combat ability are greatly enhanced. The experimental results show that the protective efficiency of the single-layer ERA structure to the conventional shaped charge warhead is 70–90% [1,2]. Therefore, in order to effectively deal with ERA and eliminate the interference of the ERA action field on the shaped jet, the concept of a shaped charge tandem warhead was put forward. At present, the shaped charge tandem warhead that is widely used mainly includes two kinds of structures; it is either the armor piercing-shaped charge type or the shaped charge-shaped charge type. The armor

Materials 2018, 11, 72; doi:10.3390/ma11010072 www.mdpi.com/journal/materials Materials 2018, 11, 72 2 of 17 piercing-shapedMaterials 2018, 11 charge, 72 type tandem warhead is mainly composed of a pre-warhead (armor2 of piercing 17 or EFP)either and the post-warhead armor piercing-shaped (shaped charge charge), type as or shownthe shaped in Figurecharge-shaped1a. Its workingcharge type. principle The armor is that the pre-projectilepiercing-shaped penetrates charge type and tandem passes warhead through is ma theinly ERA composed at a certainof a pre-warhead speed without (armor piercing causing the sandwichedor EFP) and charge post-warhead to explode, (shaped and then charge), the mainas shown charge in Figure is initiated 1a. Its working after a certain principle delay is that and the forms the shapedpre-projectile jet to impactpenetrates the ERAand passes through through the hole the formed ERA at by a thecertain pre-impact; speed without thus, the causing ERA losesthe the abilitysandwiched to interfere charge with to the explode, shaped and jet. then The the shaped main ch charge-shapedarge is initiated after charge a certain type delay tandem and warhead forms is mainlythe composed shaped jet to of impact a pre-warhead the ERA through (shaped the charge) hole formed andpost-warhead by the pre-impact; (shaped thus, the charge), ERA loses as shown the in Figureability1b. Its to working interfere principlewith the shaped is that thejet. The pre-charge shaped ischarge-shaped initiated ﬁrst charge and forms type tandem a shaped warhead jet to initiate is mainly composed of a pre-warhead (shaped charge) and post-warhead (shaped charge), as shown in the sandwich charge placed in the middle of ERA, and after a certain delay the main shaped jet formed Figure 1b. Its working principle is that the pre-charge is initiated first and forms a shaped jet to initiate by the post-charge can penetrate the ERA without interference [3]. the sandwich charge placed in the middle of ERA, and after a certain delay the main shaped jet Heldformed [4 by–11 the] studied post-charge the interactioncan penetrate mechanism the ERA without between interference the armor [3]. and the shaped jet through a large numberHeld [4–11] of experiments, studied the interaction and obtained mechanism the between interference the armor law ofand sandwich the shaped charge jet through thickness, a detonationlarge number velocity, of metalexperiments, plate thickness, and obtained and jetthe angleinterference on the law shaped of sandwich jet. Mayseless charge [12thickness,] conducted experimentsdetonation to velocity, study the metal interaction plate thickness, between and the jet angle shaped on jetthe andshaped the jet. metal Mayseless panel, [12] and conducted proposed the pebbleexperiments interference to study model, the ininteraction which the between metal the panel shaped generates jet and the periodic metal panel, interference and proposed to the the shaped jet. Inpebble the subsequent interference model, study, in Mayseless which the [metal13] also panel found generates that periodic the jet diameterinterference decreases to the shaped after jet. the jet passesIn throughthe subsequent the ERA, study, and Mayseless then established [13] also found a continuousthat the jet diameter interference decreases model after (Grazingthe jet passes model). through the ERA, and then established a continuous interference model (Grazing model). Koch [14] Koch [14] studied the change of the initiation criterion values of ERA with the variation of the shaped studied the change of the initiation criterion values of ERA with the variation of the shaped jet jet penetrationpenetration normalnormal throughthrough experi experiments.ments. Helte Helte [15] [15 carried] carried out outa series a series of experiments of experiments on the on the penetrationpenetration of ERA of ERA by by the the shaped shaped jet jet of of the the pre-warhead,pre-warhead, and and found found that that the the shaped shaped jet formed jet formed by by the aluminathe alumina powder, powder, aluminum aluminum powder, powder, andand glassglass material material liner liner can can perforate perforate the theERA ERA without without causingcausing explosion. explosion. ThroughThrough the abovethe above literature literature research, research, it it cancan be found that that the the reaction reaction mechanism mechanism and andinitiation initiation criterioncriterion of the of shapedthe shaped jet to jet the to ERAthe ERA have have been been relatively relatively well well studied, studied, and and metal metal shaped shaped jetjet waswas used to damageused to ERA damage in most ERA ofin themost previous of the previous work. In wo therk. future In the developmentfuture development of weapons, of weapons, the warhead the mass,warhead economic mass, cost, economic initiation cost, safety, initiation and safety, other performanceand other performance indicators indicators will become will become more importantmore important factors. Therefore, this article takes the shaped charge-shaped charge type tandem factors. Therefore, this article takes the shaped charge-shaped charge type tandem warhead as the warhead as the research prototype and hopes to explore the application of low-density materials in researchthe prototypeshaped charge and warhead hopes to based explore on the the application damage effectiveness of low-density of a materialslow-density in theshaped shaped jet on charge warheadERA basedtargets. on the damage effectiveness of a low-density shaped jet on ERA targets.

(a) (b)

Figure 1. (a) Schematic diagram of the armor piercing-shaped charge type tandem warhead; and Figure 1. (a) Schematic diagram of the armor piercing-shaped charge type tandem warhead; (b) schematic diagram of the shaped charge-shaped charge type tandem warhead. and (b) schematic diagram of the shaped charge-shaped charge type tandem warhead. 2. Initiation Criteria of ERA 2. Initiation Criteria of ERA To initiate the ERA by shaped jet is actually to initiate the charge in between the sandwich Tostructure initiate (front the panel/sandwich ERA by shaped charge/rear jet is actually panel). Whether to initiate the the sandwiched charge in charge between can be the initiated sandwich structureby a (frontshaped panel/sandwich jet is affected by charge/rearthe following panel). factors: Whether the jet velocity, the sandwiched jet diameter, charge thickness can beof initiatedthe by asandwich shaped jet charge, is affected thickness by of the panels, following and the factors: material theof the jet shaped velocity, jet [16]. jet diameter, thickness of the sandwichIn charge, the interaction thickness between of panels, shaped and jet the and material sandwich of the charge, shaped the jet jet [ 16velocity]. determines the Inmagnitude the interaction of the incident between impact shaped force. The jet andhigher sandwich the jet velocity, charge, the higher the jet the velocity pressure determines generated the by the impact and the stronger the excitation reaction. Therefore, jet velocity is the main control magnitude of the incident impact force. The higher the jet velocity, the higher the pressure generated by the impact and the stronger the excitation reaction. Therefore, jet velocity is the main control parameter in the process of initiating ERA, which has been reflected in the critical criteria for the Materials 2018, 11, 72 3 of 17 detonation of explosives. The jet diameter mainly influences the jet precursor wave; since the jet precursor wave is generally curved, the greater the jet diameter, the smaller the curvature of the precursor wave, and less easily is the precursor wave zone affected by lateral rarefaction waves. At the same time, the jet penetrates into the ERA with high temperature, high pressure, and high strain rate, and the influence of the jet density on the penetration process is also very important, which mainly affects the penetration depth. In addition, with the increase of thickness and density of the sandwich charge, it is more difficult to initiate. According to the above analysis, many scholars have studied the mechanism and process of the anti-ERA, the initiation criteria and threshold under many specific conditions were obtained, and many mature empirical formulas and critical conditions were established. Several widely used initiation criteria listed below.

2.1. p-τ Initiation Criterion Walker and Wasley [17] proposed a one-dimensional short pulse initiation criterion in 1969, which was expressed as the critical energy on unit area. The expression of critical energy Ecr is as follows:

2 Ecr = p τ/ρ0U (1) where p is the initial shock wave pressure, τ is the duration of the pressure p in the explosive, U is the shock wave velocity, and ρ0 is the explosive density. According to the critical energy theory of the impact initiation of heterogeneous explosives, when p2τ exceeds the critical explosive initiation threshold, it can arouse the detonation of explosives. Thus, 2 the relationship between the critical initiation energy Ec and p τ is as follows:

2 Ec = p τ = const (2)

2.2. v-d Initiation Criterion In general, the loading area is very small under the impact of a projectile, fragments, or shaped jet. According to Held0s study [18], the inﬂuence of the pressure pulse width τ can be ignored when the aspect ratio of projectile is greater than 1/5. Held proposed an initiation criterion for high energy explosives by the shaped jet in two-dimensional loading case:

K = v2d (3) where K is the critical initiation threshold of explosives, v is the tip velocity of shaped jet, and d is the diameter of shaped jet. In engineering applications, if the cross-section of the penetrator is not circular, so the criterion is revised as follows: √ K = v2 A (4) where A is the cross-sectional area of the penetrating body.

2.3. u-d Initiation Criterion In the follow-up research, it was found that the jet density has a certain influence on the threshold velocity. Based on the numerical calculation, Mader [19,20] used a linear function to fit and revise the v-d initiation criterion: 2 K = ρpv d (5) At the same time, Chick and Hatt [21] fit and revised the v-d initiation criterion based on experimental results of steel and aluminum jet:

p 2 K = ρjv d (6) Materials 2018, 11, 72 4 of 17 Materials 2018, 11, 72 4 of 17

It isIt is clear clear that that the the two two revised revised criteria criteria aboveabove presentpresent significance signiﬁcance in in the the initiation initiation thresholds thresholds for for thethe same same explosives. explosives. Through Through comparative comparative analysis,analysis, Held Held [22,23] [22,23 ]concluded concluded the the general general rule rule that that the the stagnationstagnation point point pressure pressure determined determined the the difﬁculty difficulty of of initiating initiating the the high high explosive, explosive, and and he he revised revised the v-dtheinitiation v-d initiation criterion criterion as follows: as follows: K = u2d (7) Kud 2 (7) The above equation is the u-d initiation criterion, where u is the penetration velocity of the jet. The above equation is the u-d initiation criterion, where u is the penetration velocity of the jet. In the hydrodynamic process, the penetration velocity u can be given by the Bernoulli equation: In the hydrodynamic process, the penetration velocity u can be given by the Bernoulli equation: q u =uvv/ 11+ ρ /ρ (8) HEHE j j (8) wherewherev isv is the the tip tip velocity velocityof of shapedshaped jet, dd isis the the diameter diameter of of shaped shaped jet, jet, HEρHE isis the the density density of ofhigh high explosive charge, and ρ is the density of the shaped jet. explosive charge, and j j is the density of the shaped jet.

3. Simulation 3. Simulation TheThe numerical numerical simulations simulations were were carriedcarried out usin usingg Autodyn-2D Autodyn-2D (Century (Century Dynamics, Dynamics, Fort Fort Worth, Worth, TX,TX, USA), USA), which which is is an an interactive interactive non-linearnon-linear explicitexplicit dynamics analysis analysis software, software, widely widely used used in in simulatingsimulating detonation, detonation, impact, impact, armor-piercing, armor-piercing, ballistics, and and other other problems. problems. The The software software includes includes itsits own own material material library, library, and and provides provides differentdifferent state equations, strength strength models models and and failure failure and and erosionerosion models. models.

3.1.3.1. FE FE Model Model TheThe entire entire model model mainly mainly consists consists ofof seven parts, namely, namely, air, air, main main charge, charge, shell, shell, liner, liner, sandwiched sandwiched charge,charge, front front panel, panel, and and rear rear panel. panel. AsAs thethe entireentire finitefinite element element model model has has the the characteristics characteristics of ofaxial axial symmetry,symmetry, the the two-dimensional two-dimensional axisymmetricaxisymmetric model is is used used in in the the simulation simulation for for computational computational efficiency.efficiency. The The finite finite element element model model is is shown shown inin FigureFigure2 2..

FigureFigure 2.2. Finite element model. model.

The explosion of the shaped charge and the collapse of the liner involve great deformation, so The explosion of the shaped charge and the collapse of the liner involve great deformation, so the the fluid-solid coupling method was used in this article. The entire model was divided into two parts, fluid-solid coupling method was used in this article. The entire model was divided into two parts, he Lagrange part and Euler part. The main charge, air, liner, and sandwich charge were meshed as he Lagrange part and Euler part. The main charge, air, liner, and sandwich charge were meshed as the the Euler part to deal with great deformation, while the shell, front panel, and rear panel were meshed Euleras the part Lagrange to deal with part great for fracture deformation, and fragmentation. while the shell, To front ensure panel, the and accuracy rear panel of the were numerical meshed as thesimulation, Lagrange partthe air for domain fracture adopts and fragmentation.the center region To encrypted ensure the gradient accuracy mesh. of theThe numerical flow-out boundary simulation, thewas air set domain for the adopts Euler boundary the center to regioneliminate encrypted the influence gradient of the mesh. boundary The effect. flow-out The boundarycontact between was set forthe the Lagrange Euler boundary mesh and to the eliminate Euler mesh the was influence defined of as theautomatic, boundary and effect. the initiation The contact mode was between center the Lagrangepoint initiation. mesh and In theaddition, Euler meshthe unit was sy definedstem of FE as automatic,model was andchosen the as initiation cm-g-μs. mode was center point initiation.The In geometrical addition, theparameters unit system of the of entire FE model model was are chosen shown asin cm-g-Tableµ 1.s. A series of Gauss points wereThe set geometrical in the model parameters to facilitate of the the analysis entire model of simulation are shown results, in Table and 1the. A specific series of numbers Gauss pointsand werecoordinates set in the of model these Gauss to facilitate points theare shown analysis in ofTable simulation 2. results, and the specific numbers and coordinates of these Gauss points are shown in Table2.

Materials 2018, 11, 72 5 of 17

Table 1. Geometrical parameters of the entire model.

Cone Front Sandwich Rear Shell Shell Shell Liner Bottom Panel Charge Panel Length Thickness Diameter Thickness Diameter Thickness Thickness Thickness 135 mm 3 mm 60 mm 54 mm 3 mm 2 mm 4 mm 2 mm

Table 2. The speciﬁc numbers and coordinates of Gauss points.

Gauss Point Gauss Point Coordinate INTERVAL Gauss Point Part Number x/mm y/mm ∆x/mm ∆y/mm Type Front panel 1–11 29.8 0–70 - 7 Moving Rear panel 12–22 30.4 0–70 - 7 Moving Sandwich 23–33 30.1 0–68 - 6.8 Fixed charge Air 34–153 93–450 0 3 - Fixed

3.2. Modeling Material Here, the main charge material was chosen as compB explosive, the sandwich charge as compBJJ1 explosive, the liner as copper, ﬂoatglass, Lucite, and Plexiglas, and the shell and panel as steel_1006. Each part of the corresponding material parameters and material model were taken from the Autodyn database, the material models and related parameters used in each section will be described below.

3.2.1. Material Properties of Air The equation of state of air was chosen as Ideal Gas. The model needs to be given an initial temperature of 15 ◦C and an initial internal energy of 2.068 × 105 kJ·kg−1.

P = ρ(γ − 1)E (9)

γ = Cp/Cv (10)

E = CvT (11) where P, ρ, and γ are the pressure, density, and polytropic index of the gas, respectively, Cp and Cv are the specific heat at constant pressure and specific heat at constant volume, T is the temperature, and E is the internal energy of gas. The specific material parameters are shown in Table3.

Table 3. Material properties of air.

3 −1 Material ρ (kg/m ) γ Cp (kJ/kg·K) Cv (kJ/kg·K) T (K) E0 (kJ·kg ) air 1.225 1.4 1.005 0.718 288.2 2.068 × 105

3.2.2. Material Properties of the Main Charge The Jones-Wilkins-Lee equation of state (EOS_JWL) was chosen to describe the material properties of the compB explosive. The EOS_JWL can accurately describe the volume, pressure, and energy characteristics of gas products in the process of detonation, which is expressed as follows:

ω ω ωE P = A 1 − e−R1V + B 1 − e−R2V + 0 (12) R1V R2V V where A, B, R1, R2, ω are material constants, V is the initial relative volume, E0 is the initial speciﬁc internal energy, and P is the detonation pressure. The speciﬁc parameters are listed in Table4. Materials 2018, 11, 72 6 of 17

Table 4. The JWL parameters and C-J parameters of the compB explosive.

A B C ρ P D Material R R ω 0 CJ (Mbar) (Mbar) (Mbar) 1 2 (g/cm3) (Gpa) (m/s) CompB 5.242 0.07678 0.01082 4.20 1.10 0.34 1.717 29.5 7980

3.2.3. Material Properties of the Sandwich Charge The shock initiation process of heterogeneous explosives is divided into three stages, namely, ignition, growth, and completion. At present, the numerical model that can describe the behavior and characteristics of the impact initiation process is the Lee-Tarver ignition growth model. In the Lee-Tarver model, the state of explosive may be completely reactive, unreacted, or both coexist. Therefore, the Lee-Tarver model was selected to characterize the impact initiation process of compBJJ1 explosives. The reaction rate equation of the model is as follows:

x dλ b ρ c d y e g z = I(1 − λ) − 1 − a + G1(1 − λ) λ p + G2(1 − λ) λ p (13) dt ρ0 where I, G1, G2, a, b, c, d, e, g, x, y, z are adjustable parameters. a is the critical degree of compression; when the explosive is compressed to a certain value, it begins to ignite. In general, the fuel consumption power in ignition and combustion terms is b = c. The speciﬁc parameters of the compBJJ1 explosive are listed in Table5.

Table 5. Parameters of Lee-Tarver ignition growth model for the compBJJ1 explosive.

A B C G σ JWL R R v s ω (Mbar) (Mbar) 1 2 (Mbar/K) (Mbar) (Mbar) Unreacted 778.1 −0.0503 11.3 1.13 2.487 × 10−5 0.0354 0.002 0.894 Product 5.242 0.0768 4.2 1.1 1.0 × 10−5 - - 0.34 −1 a b c d e g I (µs ) FG1max 0.01 0.222 0.222 0.667 0.0 0.0 44 1.0 Reaction G G Rates FG Fig x y z 1 2 2min max (Mbar−2·µs−1) (Mbar−2·µs−1) 1.0 0.3 4 2 0.0 414 0.0

3.2.4. Material Properties of the Shell, Panels, and Liner In this article, both the shell and front and rear panel materials were chosen as steel_1006, the material exhibits a certain strain rate effect. The Johnson-Cook model shows very good advantages in describing the strain rate effect of metal materials. Therefore, the equation of state was chosen as the shock equation of state incorporating the Johnson Cook strength model as follows:

n . ∗ ∗m σy = A + Bεp 1 + C ln ε (1 + T ) (14) where σy is the yield stress, A, B, C, n, m are the material parameters, εp is the equivalent plastic . ∗ strain, ε is the equivalent strain rate, and T∗m is the reduced temperature. The liner includes four kinds of materials, copper, ﬂoatglass, Lucite, and Plexiglas. The equation of state of copper, Lucite, and Plexiglas is the shock equation of state, while the equation of state of ﬂoatglass is the polynomial equation of state and the strength model, the failure model of which is Johnson-Holmquist.

4. Analysis of the Numerical Simulation Results

4.1. Comparison of Shaped Jet Formability When the detonation wave reaches the top of liner, the liner begins to be crushed under the action of detonation pressure and then closes. Subsequently, collision and squeezing occur on the axis, and the Materials 2018, 11, 72 7 of 17 shaped jet with high temperature and pressure is formed. The forming states of four different types of Materials 2018, 11, 72 7 of 17 shapedMaterials jets 2018 at t, =11, 35 72 µ s are shown in Table6. In addition, by extracting the historical information7 of 17 of the Gauss points, the jet tip velocity corresponding to the four different liner material types is shown information of the Gauss points, the jet tip velocity corresponding to the four different liner material information of the Gauss points, the jet tip velocity corresponding to the four different liner material in Figuretypes3 ,is and shown the in velocity Figure history3, and the of velocity four kinds histor ofy jetsof four corresponding kinds of jets tocorresponding different stand-offs to different (1 CD, types is shown in Figure 3, and the velocity history of four kinds of jets corresponding to different 2 CD,stand-offs 3 CD, 4 CD,(1 CD, where 2 CD, CD 3 isCD, the 4 abbreviationCD, where CD for is thethe chargeabbreviation diameter) for the were charge obtained, diameter) as were shown in stand-offs (1 CD, 2 CD, 3 CD, 4 CD, where CD is the abbreviation for the charge diameter) were Figureobtained,4. as shown in Figure 4. obtained, as shown in Figure 4.

Figure 3. Jet tip velocity corresponding to the four different liner materials. FigureFigure 3. Jet 3. Jet tip tip velocity velocity corresponding corresponding to to the the four four different different liner liner materials. materials.

(a) (b) (a) (b)

(c) (d) (c) (d) Figure 4. The velocity history corresponding to different stand-offs and different liner materials: Figure 4. The velocity history corresponding to different stand-offs and different liner materials: Figure(a) 4.copper;The velocity(b) floatglass; history (c) Lucite; corresponding and (d) Plexiglas. to different stand-offs and different liner materials: (a) copper; (b) floatglass; (c) Lucite; and (d) Plexiglas. (a) copper; (b) ﬂoatglass; (c) Lucite; and (d) Plexiglas.

MaterialsMaterials2018 2018, 11, 11,, 72 72 8 of 17 8 of 17 MaterialsMaterials 2018 2018, ,11 11, ,72 72 88 of of 17 17 Materials 2018, 11, 72 8 of 17 Table 6. Configuration and performance parameters of shaped jets. Table 6. Configuration and performance parameters of shaped jets. TableTable 6.6. ConfigurationConfiguration andand performanceperformance parametersparameters ofof shapedshaped jets.jets. Table 6. Conﬁguration and performanceJet Tip parametersJet Tail ofJet shaped Pestle jets. JJetet Tip Tip JJetet Tail Tail JetJet PestlePestle Material Shaped Jet Configuration VelocityJet Tip VelocityJet Tail LengthJet LengthPestle MaterialMaterial Shaped Shaped Jet Jet Configuration Configuration VelocityVelocity VelocityVelocity LengthLength LengthLength Material Shaped Jet Configuration Velocity(m/s)Jet Tip Velocity(m/s) Jet TailLength(mm) JetLength(mm) Length Pestle Length Material Shaped Jet Conﬁguration (m/s) (m/s) (mm) (mm) Velocity(m/s)(m/s) (m/s)(m/s)(m/s)Velocity (mm)(mm) (m/s) (mm)(mm) (mm) Copper 6088 138 84 56 Copper 6088 138 84 56 CopperCopperCopper 608860886088 138138 1388484 56 8456 56

Floatglass 7421 953 98 58 Floatglass 7421 953 98 58 FloatglassFloatglassFloatglass 742174217421 953953 9539898 58 9858 58

Lucite 9715 667 137 77 LuciteLucite 97159715 667 667137 77 137 77 LuciteLucite 97159715 667667 137137 7777

PlexiglasPlexiglas 98719871 632 632141 14178 78 Plexiglas 9871 632 141 78 PlexiglasPlexiglas 98719871 632632 141141 7878

From the shaped jet configuration shown in Table 6, it can be seen that the jet length FromFrom thethe shapedshaped jetjet configurationconfiguration shownshown inin TaTableble 6,6, itit cancan bebe seenseen thatthat thethe jetjet lengthlength correspondingFromFrom the the shaped toshaped the four jet jet configurationdifferent configuration liner materials shownshown atin in the Ta Table sameble 66, time, itit can canis L Plexiglas bebe seenseen ≈ LLucite thatthat > L thethefloatglass jetjet > length lengthLcopper, corresponding correspondingcorresponding to to the the four four different different liner liner materials materials at at the the same same time time is is L LPlexiglasPlexiglas ≈ ≈ L LLuciteLucite > > L Lfloatglassfloatglass > > L Lcoppercopper, , to thewhichcorresponding four indicates different to that the liner fourthe ductilitydifferent materials ofliner Plexiglas at materials the sameand at Lucitethe time same is is timecloser,LPlexiglas is LwhilePlexiglas≈ floatglass≈ LLucite > L >isfloatglass Lworse,floatglass > L copperand,> Lcopper, which whichwhich indicatesindicates thatthat thethe ductilityductility ofof PlexiglasPlexiglas andand LuciteLucite isis closer,closer, whilewhile floatglassfloatglass isis worse,worse, andand indicatescopperwhich thatindicatesis the theworst. that ductility There the ductilityexists of a Plexiglas temperature of Plexiglas and rise and Lucite during Lucite the isis closer,processcloser, whileof while shaped floatglass floatglass jet formation, is worse, is worse,which and and copper is coppercopper is is the the worst. worst. There There exists exists a a temperature temperature rise rise during during the the process process of of shaped shaped jet jet formation, formation, which which mainlycopper iscomes the worst. from Theretwo parts: exists the a temperature shockwave riseand during plastic the strain. process Under of shaped normal jet circumstances, formation, which the the worst.mainlymainly comescomes There fromfrom exists twotwo a parts:parts: temperature thethe shockwaveshockwave rise and duringand plasticplastic the strain.strain. process UnderUnder of normalnormal shaped circumstances,circumstances, jet formation, thethe which mainly shapedmainly comesjet temperature from two distribution parts: the shockwave characteristic and is plasticas follows: strain. the Under surface normal temperature circumstances, is generally the shapedshaped jet jet temperature temperature distribution distribution characteristic characteristic is is as as follows: follows: the the surface surface temperature temperature is is generally generally comesaroundshaped from 400–500jet twotemperature parts:°C, much distribution the lower shockwave than characteristic the center and temperature plasticis as follows: strain. of abtheout surface Under 900–1000 temperature normal °C. Since circumstances, isthe generally melting the shaped around 400–500 °C, much lower than the center temperature of about 900–1000 °C. Since the melting jet temperaturepointaroundaround of 400–500400–500copper distribution is °C,°C, 1083 muchmuch °C, lowerlowerit does characteristic thth notanan thereachthe centercenter the melting istemperaturetemperature as follows: point of ofunder abab theoutout the surface900–1000900–1000 action of °C. temperature°C. the SinceSince shaped thethe meltingcharge.melting is generally around pointpoint of of copper copper is is 1083 1083 ° °C,C, it it does does not not reach reach the the melting melting point point under under the the action action of of the the shaped shaped charge. charge. 400–500Therefore,point ◦ofC, copper muchthe copper is lower1083 shaped °C, than it does jet the isnot not center reach in a the temperaturemelted melting state, point but of under in about a plastic the 900–1000action state of of the high-speed◦ C.shaped Since charge. flow, the melting point of Therefore,Therefore, thethe coppercopper shapedshaped jetjet isis notnot inin aa meltedmelted state,state, butbut inin aa plasticplastic statestate ofof high-speedhigh-speed flow,flow, whichTherefore, is different the◦ copper from shaped the general jet is fluid.not in The a melted Vicar softeningstate, but temperaturein a plastic stateof floatglass, of high-speed Lucite, flow, and copperwhichwhich is is 1083is different differentC, itfrom from does the the not general general reach fluid. fluid. the The The melting Vicar Vicar softening softening point under temperature temperature the action of of floatglass, floatglass, of the Lucite, shapedLucite, and and charge. Therefore, Plexiglaswhich is differentare about from 720–730 the general°C, ~113 fluid. °C, and The ~108 Vicar °C, softening respectively. temperature When the of amorphous floatglass, Lucite, material and is PlexiglasPlexiglas are are about about 720–730 720–730 °C, °C, ~113 ~113 °C, °C, and and ~108 ~108 °C, °C, respectively. respectively. When When the the amorphous amorphous material material is is the copperheatedPlexiglas to shaped arethe aboutVicar jet 720–730softening is not °C, in temperature, a~113 melted °C, and state,the ~108 liner but°C, material respectively. in a plastic will soften, When state theand of amorphous high-speedthen continue material flow,to heat is which is different heatedheated toto thethe VicarVicar softeningsoftening temperature,temperature, thethe linerliner material material willwill soften,soften, and and thenthen continuecontinue to to heatheat fromup,heated the which general to thewill Vicar gradually fluid. softening The be able Vicartemperature, to flow, softening and the the liner hi temperaturegher material the temperature, will of soften, floatglass, the and better then Lucite, the continue fluidity. and to From heat Plexiglas are about up,up, which which will will gradually gradually be be able able to to flow, flow, and and the the hi highergher the the temperature, temperature, the the better better the the fluidity. fluidity. From From theup, abovewhich◦ analysiswill gradually◦ based onbe ablemelting to◦ flow, point and and the softening higher the point, temperature, it is obvious the that better the the ductility fluidity. of Fromthree 720–730the aboveC, analysis ~113 C,based and on~108 meltingC, point respectively. and softening When point, it the is obvious amorphous that the materialductility of is three heated to the Vicar low-densitythethe aboveabove analysisanalysis amorphous basedbased materials onon meltingmelting under pointpoint the andand action softeningsoftening of the point,point,shaped itit ischargeis obviousobvious is superior thatthat thethe to ductilityductility that of ofcopper.of threethree softeninglow-densitylow-density temperature, amorphous amorphous the materials materials liner materialunder under the the action willaction soften, of of th thee shaped shaped and thencharge charge continue is is superior superior to to to heat that that of up,of copper. copper. which will gradually low-densityIn addition, amorphous it can materialsbe seen from under Figures the action 3 and of th 4,e shapedand Table charge 6 that is superior the jet totip that velocity of copper. was InIn addition,addition, itit cancan bebe seenseen fromfrom FiguresFigures 33 andand 4,4, andand TableTable 66 thatthat thethe jetjet tiptip velocityvelocity waswas be ableVPlexiglas toIn flow,≈addition, VLucite and > Vitfloatglass thecan higher be> V seencopper ,the fromandtemperature, theFigures corresponding 3 and the4, andjet better mass Table tran the 6 sferthat fluidity. ratethe ofjet From thetip fourvelocity the different above was analysis based VVPlexiglasPlexiglas ≈≈ VVLuciteLucite >> VVfloatglassfloatglass >> VVcoppercopper, , andand thethe correspondingcorresponding jetjet massmass trantransfersfer raterate ofof thethe fourfour differentdifferent on meltinglinerVPlexiglas materials ≈ pointVLucite also>and Vfloatglass showed softening > V ηcopperPlexiglas, andpoint, ≈ η Lucitethe it>corresponding η isfloatglass obvious > ηcopper jet that. The mass theeffective tran ductilitysfer charge rate of at the threethe fourtop low-densityof different liner is amorphous linerliner materials materials also also showed showed η ηPlexiglasPlexiglas ≈ ≈ η ηLuciteLucite > > η ηfloatglassfloatglass > > η ηcoppercopper. .The The effective effective charge charge at at the the top top of of liner liner is is relativelyliner materials greater, also while showed the η Plexiglaseffective ≈ η Lucitecharge > η floatglassat the > bottomηcopper. The of theeffective liner chargeis relatively at the less,top ofthus liner the is materialsrelativelyrelatively under greater,greater, the whilewhile action thethe of effectiveeffective the shaped chargecharge chargeatat ththee bottombottom is superior ofof thethe linerliner to that isis relativelyrelatively of copper. less,less, thusthus thethe crushingrelatively velocity greater, andwhile jet thevelocity effective is lower charge than at thethe former. bottom It of can the be liner seen is that relatively the jet tipless, velocity thus the is crushingcrushingIn addition, velocity velocity itand and can jet jet velocity bevelocity seen is is lower fromlower than than Figures the the former.former.3 and It It4 can ,can and be be seen seen Table that that6 the thethat jet jet tip thetip velocity velocity jet tip is is velocity was highercrushing than velocity the tail and velocity, jet velocity namely, is lowerthere isthan a velocity the former. gradient, It can so be that seen the that entire the jet jet is tipcontinuously velocity is higher than the tail velocity, namely, there is a velocity gradient, so that the entire jet is continuously VPlexiglaselongatedhigherhigher≈ thanthan Vuntil Lucite thethe tailthetail > velocity,jetvelocity,V isfloatglass pulled namely,namely, off.> Moreover, V theretherecopper isis aa, velocityvelocityth ande lower the gradient,gradient, the corresponding density soso thatthat of thethe entireentireliner jet material, massjetjet isis continuouslycontinuously transfer the more rate of the four elongatedelongated until until the the jet jet is is pulled pulled off. off. Moreover, Moreover, th thee lower lower the the density density of of the the liner liner material, material, the the more more differenteasilyelongated linerit moves until materials theforward jet is pulled alsounder showed off.the Moreover, detonationηPlexiglas th ewave, lower≈ ηwhich theLucite density means> ηfloatglass of that the linerthe> velocityηmaterial,copper .and the The moremass effective charge at easilyeasily itit movesmoves forwardforward underunder thethe detonationdetonation wave,wave, whichwhich meansmeans thatthat thethe velocityvelocity andand massmass conversioneasily it moves rate of forward the low-density under the shaped detonation jet is higher. wave, which means that the velocity and mass the topconversionconversion of liner rate rate is of of relatively the the low-density low-density greater, shaped shaped while jet jet is is higher. thehigher. effective charge at the bottom of the liner is relatively conversionFor most rate liner of the materials, low-density when shaped the top jet of is thehigher. liner is subjected to detonation waves, the liner less, thusForFor the mostmost crushing linerliner materials,materials, velocity whenwhen and thethe top jettop velocityofof thethe lilinerner is isis lower subjectedsubjected than toto detodeto thenationnation former. waves,waves, It can thethe linerliner be seen that the jet beginsFor to most be crushed liner materials, and deformed, when whichthe top results of the in li nera rise is insubjected the density to deto of thenation liner waves, material the at liner this beginsbegins to to be be crushed crushed and and deformed, deformed, which which results results in in a a rise rise in in the the density density of of the the liner liner material material at at this this tip velocitypoint.begins However,to isbe highercrushed due thanandto the deformed, the presence tail which velocity, of the results veloci namely, inty a gradient, rise there in the the isdensity material a velocity of the is graduallyliner gradient, material stretched. so at thatthis the entire jet is point.point. However,However, duedue toto thethe presencepresence ofof thethe velocivelocityty gradient,gradient, thethe materialmaterial isis graduallygradually stretched.stretched. continuouslyTherepoint. willHowever, be elongated a coexistence due to untilthe of presence compression the jet of is the pulled and veloci tensil off.tye gradient,states, Moreover, which the causesmaterial the lower a sharpis gradually the fluctuation density stretched. in of the the liner material, There will be a coexistence of compression and tensile states, which causes a sharp fluctuation in the jetThereThere density, willwill bebe but aa coexistence coexistenceas the shaped ofof compressioncompression jet is stretched, andand tensiltensil the eedensity states,states, whichwhichdecreases causescauses rapidly aa sharpsharp in fluctuation fluctuationthe fluctuation. inin thethe the morejetjet density,density, easily butbut it as movesas thethe shapedshaped forward jetjet is underis stretched,stretched, the detonationthethe densitydensity decreasesdecreases wave, which rapidlyrapidly means inin thethe thatfluctuation.fluctuation. the velocity and mass Eventually,jet density, whenbut as the the shaped shaped jet breaks,jet is stretched, the material the density density remains decreases essentially rapidly constant. in the Thefluctuation. shaped conversionEventually,Eventually, rate when when of the thethe shaped shaped low-density jet jet breaks, breaks, shaped the the material material jet isdensity density higher. remains remains essentially essentially constant. constant. The The shaped shaped jetEventually, density state when at the the shaped distance jet ofbreaks, 3 CD the is shown material in density Figure remains5, corresponding essentially to constant. four different The shaped types jetjet density density state state at at the the distance distance of of 3 3 CD CD is is shown shown in in Figure Figure 5, 5, corresponding corresponding to to four four different different types types ofjetFor liner.density most state liner at the materials, distance of when 3 CD is the shown top in of Figure the liner5, corresponding is subjected to four to detonationdifferent types waves, the liner of liner. beginsofof liner.liner. to be crushed and deformed, which results in a rise in the density of the liner material at this

point. However, due to the presence of the velocity gradient, the material is gradually stretched. There will be a coexistence of compression and tensile states, which causes a sharp ﬂuctuation in the jet density, but as the shaped jet is stretched, the density decreases rapidly in the ﬂuctuation. Eventually, when the shaped jet breaks, the material density remains essentially constant. The shaped jet density state at the distance of 3 CD is shown in Figure5, corresponding to four different types of liner. Materials 2018, 11, 72 9 of 17 MaterialsMaterials 20182018,, 1111,, 7272 99 ofof 1717

((aa)) ( (bb)) ( (cc)) ( (dd)) Figure 5. The jet density state at the distance of 3 CD for different liner materials: (a) copper; FigureFigure 5. 5. TheThe jet jet density density state state at at the the distance distance of of 3 3 CD CD for for different different liner liner materials: materials: (a (a) )copper; copper; (b) floatglass; (c) Lucite; and (d) Plexiglas. ((bb)) floatglass; ﬂoatglass; ( (cc)) Lucite; Lucite; and and ( (dd)) Plexiglas. Plexiglas.

ThereThere isis aa certaincertain gradientgradient inin thethe overalloverall densitydensity ofof thethe shapedshaped jetjet atat thethe distancedistance ofof 33 CDCD asas There is a certain gradient in the overall density of the shaped jet at the distance of 3 CD as shown shownshown inin FigureFigure 5.5. However,However, thethe floatglassfloatglass′′ss overalloverall jetjet densitydensity isis consistent.consistent. TheThe densitydensity variationvariation ofof in Figure5. However, the ﬂoatglass 0s overall jet density is consistent. The density variation of four fourfour differentdifferent jetjet typestypes isis shownshown inin TableTable 7.7. SinceSince thethe jetjet densitydensity hashas changedchanged duringduring thethe formationformation ofof different jet types is shown in Table7. Since the jet density has changed during the formation of the the shaped jet, j should denote the density at the moment when the shaped jet impacts the target the shaped jet, j should denote the density at the moment when the shaped jet impacts the target shaped jet, ρj should denote the density at the moment when the shaped jet impacts the target2 plate 2 2 plate instead of the initial density in the u-d initiation criterion Kudv2h 1q i2 d. plate instead of the initial density in the u-d initiation criterion 2Kudv1  HE j d. instead of the initial density in the u-d initiation criterion K = u d = v/ 1 +ρHEHE/ρj j d.

TableTableTable 77 7... TheTheThe densitydensity density variationvariation variation ofof of fourfour differentdifferent shapedshaped jetjet types.types. Initial Density Current Density Variable Quantity of Density Material Initial Density Current Density Variable Quantity of Density Material Initial Density3 Current Density3 Variable Quantity of Density Material ρinitial (g/cm3 ) 3 ρcurrent (g/cm33 ) Δρ (%) ρinitialρinitial (g/cm(g/cm) ) ρρcurrentcurrent (g/cm(g/cm ) ∆ρΔ(%)ρ (%) Copper 8.93 7.02 −21.4% CopperCopper 8.93 8.93 7.02 −21.4%−21.4% FloatglassFloatglassFloatglass 2.532.53 2.53 2.53 2.53 0% 0% 0% LuciteLuciteLucite 1.1811.181 1.181 0.763 0.763 −35.4%−−35.4%35.4% PlexiglasPlexiglasPlexiglas 1.1861.186 1.186 0.776 0.776 −34.6%−−34.6%34.6%

4.2.4.2.4.2. ResponseResponse Response AnalysisAnalysis Analysis ofof of thethe the ERAERA ERA InInIn thethe the numericalnumerical numerical simulation,simulation, simulation, itit it isis is possiblepossible possible toto to juju judgedgedge whetherwhether whether thethe the sandwichsandwich sandwich chargecharge charge reactsreacts reacts accordingaccording according tototo thethe the particleparticle particle velocityvelocity velocity ofof of panel,panel, panel, thethe the pressurepressure pressure andand and reactionreaction reaction degreedegree degree ααα ofofof sandwichsandwich sandwich charge.charge. charge. Specifically,Specifically, Speciﬁcally, ααα representsrepresentsrepresents the the ratio ratio of of the the reacted reacted partpart tototo theththee wholewholewhole part partpart in inin the thethe explosive, explosive,explosive, and andand the thethe range rangerange of ofofα ααis isis from fromfrom 0 00to toto 1. 1.1. If IfIfα α=α = 0,= 0,0, it it indicatesit indicatesindicates that thatthat the thethe explosive explosiveexplosive does doesdoes not notnot react; react;react; 0 < 0α0 <<1,< αα it<1,<1, indicates itit indicatesindicates that thatthat the explosive thethe explosiveexplosive does doesdoesnot react notnot completely;reactreact completely;completely; and if andandα = 1,ifif itαα indicates == 1,1, itit indicatesindicates that the explosivethatthat thethe exex hasplosiveplosive completely hashas completelycompletely reacted. Therefore, reacted.reacted. Therefore,Therefore,a series of a Gaussa seriesseries points ofof GaussGauss were pointspoints set inside werewere the setset ERA insideinside to the observethe ERAERA the toto observeobserve particle the velocitythe particleparticle of panel, velocityvelocity the ofof pressure panel,panel, thetheand pressurepressure reaction andand degree reactionreactionα ofsandwich degreedegree αα of charge,of sandwichsandwich as shown charge,charge, in Figureasas shownshown6. inin FigureFigure 6.6.

Figure 6. The distribution of Gauss points in ERA. FigureFigure 6.6. TheThe distributiondistribution ofof GaussGauss pointspoints inin ERA.ERA.

4.2.1.4.2.1. ReactionReaction DegreeDegree αα ofof thethe SandwichedSandwiched ChargeCharge InIn orderorder toto studystudy thethe reactionreaction processprocess ofof sandwichedsandwiched chargecharge underunder thethe impactimpact ofof shapedshaped jet,jet, thethe reactionreaction degreedegree ofof sandwichsandwich chargecharge correspondingcorresponding toto fourfour differentdifferent typestypes ofof jetsjets atat differentdifferent timestimes

Materials 2018, 11, 72 10 of 17

4.2.1. Reaction Degree α of the Sandwiched Charge In order to study the reaction process of sandwiched charge under the impact of shaped jet, Materials 2018, 11, 72 10 of 17 the reaction degree of sandwich charge corresponding to four different types of jets at different times was studied. There are six different moments for each set of working conditions, and the entire was studied. There are six different moments for each set of working conditions, and the entire time timerange range covers covers the theformation formation of hot of hot spots spots to tothe the complete complete detonation detonation of ofthe the sandwiched sandwiched charge. charge. TheThe reaction reaction degree degree contour contour of theof the sandwiched sandwiched charge charge corresponding corresponding to different to different types types of jet of is shownjet is inshown Figure in7. Figure 7.

t = 49.0 μs t = 51.0 μs t = 53.0 μs t = 55.0 μs t = 57.0 μs t = 59.0 μs (a)

t = 42.0 μs t = 44.0 μs t = 46.0 μs t = 48.0 μs t = 50.0 μs t = 52.0 μs (b)

t = 34.0 μs t = 35.7 μs t = 37.4 μs t = 39.1 μs t = 40.8 μs t = 42.5 μs (c)

Figure 7. Cont.

Materials 2018, 11, 72 11 of 17 Materials 2018, 11, 72 11 of 17

t = 33.4 μs t = 35.2 μs t = 37.0 μs t = 38.8 μs t = 40.6 μs t = 42.4 μs (d)

FigureFigure 7. 7.The The reaction reaction degree degreecontour contour ofof thethe sandwichedsandwiched charge corresponding corresponding to to different different jet jet types: types: (a()a copper;) copper; (b ()b ﬂoatglass;) floatglass; ( c()c) Lucite; Lucite; and and ( d(d)) Plexiglas.Plexiglas.

It can be seen from Figure 7, when the impact intensity reaches a certain value under the impact of shapedIt can be jet, seen the from local Figure high temperature7, when the impactarea is intensityformed, which reaches is a called certain a valuehot spot. under With the further impact ofpenetration shaped jet, of the the local shaped high jet, temperature the pressure, area densi is formed,ty, and whichtemperature is called at the a hot wave spot. front With increase further penetrationsharply, which of the breaks shaped the jet, thermal the pressure, balance density, around and the temperaturehot spot regions at the inside wave the front charge. increase When sharply, the whichhot spot breaks temperature the thermal is higher balance than around the thermal the hot spotdecomposition regions inside temperature the charge. of the When explosive, the hot spotthe temperatureexplosive decomposes is higher than and thegenerates thermal energy, decomposition with heat generating temperature and of expanding the explosive, at the thecenter explosive of the decomposeshot spot to andthe surrounding generates energy, explosive, with heatwhich generating will eventually and expanding lead to the at hot the spot center growth. of the When hot spot the to theexplosive surrounding breaks explosive, down to a whichcertain willextent, eventually the hot sp leadots will to the begin hot to spot connect, growth. and When the strong the explosivereaction breaksformed down by the to hot a certain spots extent,will lead the the hot explosive spots will into begin low-speed to connect, detonation, and the and strong then reactiongrow into formed a stable by thedetonation. hot spots willAs leada result, the explosivethe reaction into degree low-speed α reaches detonation, 1 rapidly, and then it spreads grow into to athe stable surrounding detonation. Asdiffusion a result, at the a certain reaction rate, degree and αthenreaches extends 1 rapidly, to the itentire spreads area toof thethe surroundingsandwiched diffusioncharge. In ataddition, a certain rate,by observing and then extendsthe reaction to the degree entire α of area Gauss of the points sandwiched in the middle charge. of the In sandwich addition, charge, by observing it can be the reactionseen from degree Figureα of 8 that Gauss the points sandwich in the charge middle has of achieved the sandwich the complete charge, detonation it can be seen process. from Figure8 that theAccording sandwich to charge the kinetic has achieved energy thetheorem, complete the detonationkinetic energy process. of the jet is proportional to the squareAccording of the velocity, to the kinetic so the energy larger theorem, the velocity the kineticis, the greater energy ofthe the kinetic jet is energy proportional is. The to density the square of ofthree the velocity,amorphous so thematerials larger is the lower, velocity the jet is, velocity the greater obtained the kineticis higher, energy so its is.energy The is density higher, of and three it amorphousis more likely materials to initiate is lower, the sandwich the jet velocity charge obtained for the isthree higher, amorphous so its energy materi isals. higher, The andtime it of is four more likelyshaped to initiatejets from the impacting sandwich the charge sandwich for thecharge three to fully amorphous detonating materials. the sandwich The time charge of four is tfloatglass shaped = 9.5 μs, tLucite = 9.0 μs, tPlexiglas = 8.9 μs, tcopper = 10.7 μs, respectively. jets from impacting the sandwich charge to fully detonating the sandwich charge is tﬂoatglass = 9.5 µs, Based on the simulation results of the tip velocity, tip diameter and density of shaped jet, it is tLucite = 9.0 µs, tPlexiglas = 8.9 µs, tcopper = 10.7 µs, respectively. possibleBased to on judge the simulationwhether the results jet can ofinitiate the tip the velocity, sandwich tip charge diameter from and the density theoretical of shaped point of jet, view it is 2 possible to judge whether the jet can initiate the2 sandwich charge from the theoretical point of view according to the u-d initiation criterion Kudv1 HE j d. For the compBJJ1 explosive, 2 2 h q i accordingits critical to initiation the u-d initiation threshold criterion is K = 23K mm= u3/μds=2, andv/ the1 + calculationρHE/ρj resultsd. For are the shown compBJJ1 in Table explosive, 8. its critical initiation threshold is K = 23 mm3/µs2, and the calculation results are shown in Table8.

Table 8. The calculation results of the u-d initiation criterion.

Initiation Jet Tip Velocity Jet Tip Diameter Density Detonation Material Criterion v (m/s) d (mm) ρ (g/cm3) Situation t t u2d (mm3/µs2) Copper 6042 2.6 7.02 42.49 Yes Floatglass 7464 5.2 2.53 87.09 Yes Lucite 9785 6.0 0.763 91.91 Yes Plexiglas 9849 6.0 0.776 94.06 Yes

Materials 2018, 11, 72 12 of 17 Materials 2018, 11, 72 12 of 17

(a) (b)

Figure 8. The reaction degree α of Gauss points in the middle of the sandwich charge for different Figure 8. The reaction degree α of Gauss points in the middle of the sandwich charge for different liner liner materials: (a) copper; (b) floatglass; (c) Lucite; and (d) Plexiglas. materials: (a) copper; (b) ﬂoatglass; (c) Lucite; and (d) Plexiglas.

Table 8. The calculation results of the u-d initiation criterion. It can be seen from Table8 that the values of u2d corresponding to the four different material Jet Tip liners are greaterJet than Tip 23Velocity mm3 /µs2, which shows thatDensity the sandwich Initiation charge Criterion can be reliablyDetonation initiated Material Diameter from the theoreticalv pointt (m/s) of view. In addition, the numericalρ (g/cm3) simulationu2d (mm results3/μs2) in FigureSituation8 show that dt (mm) the four kinds of jets all initiate the sandwich charge, which indicates that the numerical simulation Copper 6042 2.6 7.02 42.49 Yes methodFloatglass in this article7464 is reliable. 5.2 2.53 87.09 Yes Lucite 9785 6.0 0.763 91.91 Yes 4.2.2.Plexiglas Detonation Pressure9849 of the Sandwiched 6.0 Charge 0.776 94.06 Yes In addition, the detonation pressure can also be used to determine whether the sandwiched charge is initiated.It can be The seen pressure from Table contours 8 that at thethe samevalues time of u as2d thosecorresponding in Figure7 to were the extracted,four different as shown material in linersFigure are9. greater than 23 mm3/μs2, which shows that the sandwich charge can be reliably initiated from Thethe theoretical formation andpoint growth of view. into In stable addition, detonation the numerical waves of simulation hot spots inresults the sandwiched in Figure 8 show are clearly that theshown four in kinds Figure of 9jets. Since all initiate the tip the of thesandwich ﬂoatglass charge, shaped which jet isindicates hollow, that it is the equivalent numerical to twosimulation shock methodwaves propagated in this article into is the reliable. sandwich charge in the two-dimensional plane. The two shock waves must be superimposed on the axis position, and the pressure state corresponding to t = 44.0 µs in Figure9b 4.2.2.can be Detonation well illustrated Pressure at this of the point. Sandwiched In order toCharge analyze the change of the detonation pressure in the sandwichIn addition, charge the more detonation intuitively, pressure the pressure can also history be used curve to showndetermine in Figure whether 10 wasthe sandwiched obtained by chargeextracting is initiated. the pressure The history pressure of thecontours Gauss pointsat the insame the sandwichtime as those charge. in Figure 7 were extracted, as shown in Figure 9. The formation and growth into stable detonation waves of hot spots in the sandwiched are clearly shown in Figure 9. Since the tip of the floatglass shaped jet is hollow, it is equivalent to two

Materials 2018, 11, 72 13 of 17

shock waves propagated into the sandwich charge in the two-dimensional plane. The two shock waves must be superimposed on the axis position, and the pressure state corresponding to t = 44.0 μs in Figure 9b can be well illustrated at this point. In order to analyze the change of the detonation

Materialspressure2018 in ,the11, 72sandwich charge more intuitively, the pressure history curve shown in Figure 1013 ofwas 17 obtained by extracting the pressure history of the Gauss points in the sandwich charge. It can be seen from Figures 9 and 10 that the sandwiched charge is initiated by the shaped jet whenIt the can impact be seen pressure from Figures reaches9 and the initiation10 that the pres sandwichedsure threshold. charge In isaddition, initiated the by pressure the shaped within jet whenthe charge the impact increases pressure instantaneously reaches the initiationand is close pressure to the detonation threshold. Inpressure. addition, With the pressurefurther penetration within the chargeof the shaped increases jet, instantaneously the stable detonation and is closewave to formed the detonation in charge pressure. is transmitted With further to both penetration ends at a steady of the shapedspeed. jet, the stable detonation wave formed in charge is transmitted to both ends at a steady speed.

t = 49.0 μs t = 51.0 μs t = 53.0 μs t = 55.0 μs t = 57.0 μs t = 59.0 μs (a)

t = 42.0 μs t = 44.0 μs t =46.0 μs t = 48.0. μs t = 50.0 μs t = 52.0 μs (b)

t = 34.0 μs t = 35.7 μs t = 37.4 μs t = 39.1 μs t = 40.8 μs t = 42.5 μs (c)

Figure 9. Cont.

Materials 2018, 11, 72 14 of 17 Materials 2018, 11, 72 14 of 17 Materials 2018, 11, 72 14 of 17

t = 33.4 μs t = 35.2 μs t = 37.0 μs t = 38.8 μs t = 40.6 μs t = 42.4 μs t = 33.4 μs t = 35.2 μs t = 37.0 μs t = 38.8 μs t = 40.6 μs t = 42.4 μs (d) (d) Figure 9. The pressure contour of the sandwich charge corresponding to different jet types: (a) copper; FigureFigure 9. 9.The The pressurepressure contourcontour of the sandwich char chargege corresponding corresponding to to different different jet jet types: types: (a (a) )copper; copper; (b) floatglass; (c) Lucite; and (d) Plexiglas. (b(b)) ﬂoatglass; floatglass; ( c(c)) Lucite; Lucite; and and ( (dd)) Plexiglas.Plexiglas.

((aa)) ( (bb))

((cc)) ( (dd))

FigureFigure 10. 10. PressurePressure Pressure history historyhistory curves curvescurves of of Gauss Gauss pointspoints inin thth thee sandwich sandwich charge charge forfor for differentdifferent different liner liner liner materials: materials: materials: (a()a )copper; copper; ( b(b) ) ﬂoatglass;floatglass; floatglass; ( c(c)) Lucite; Lucite; andand (((dd))) Plexiglas.Plexiglas.Plexiglas.

4.2.3.4.2.3. Particle Particle Velocity Velocity of of the the Front Front andand RearRear PanelsPanels 4.2.3. Particle Velocity of the Front and Rear Panels ForFor the the ERA, ERA, if if the the sandwichedsandwiched chargecharge isis initiated, the particle velocitiesvelocities ofof thethe frontfront andand rear rear For the ERA, if the sandwiched charge is initiated, the particle velocities of the front and rear panelspanels will will increase increase dramatically. dramatically. OnOn thethe contrary,contrary, if there is no explosionexplosion inin thethe sandwichedsandwiched charge, charge, panels will increase dramatically. On the contrary, if there is no explosion in the sandwiched charge, therethere is is almost almost no no change change in in thethe particleparticle velocityvelocity ofof the front and rearrear panels.panels. Therefore,Therefore, in in addition addition toto the the above above criteria criteria which which based based on on thethe reactionreaction degdegree and pressure, thethe particleparticle velocity velocity of of the the front front

Materials 2018, 11, 72 15 of 17

Materialsthere is 2018 almost, 11, 72 no change in the particle velocity of the front and rear panels. Therefore, in addition15 of to17 the above criteria which based on the reaction degree and pressure, the particle velocity of the front and rear panels cancan alsoalso bebe usedused asas a a criterion. criterion. To To this this end, end, the the particle particle velocity velocity of of the the Gauss Gauss points points in inthe the front front and and rear rear panels panels was was extracted, extracted, as shownas shown in Figurein Figure 11 .11.

(a) (b)

Figure 11. The particle velocity of the Gauss points in the front and rear panels for different liners: Figure 11. The particle velocity of the Gauss points in the front and rear panels for different liners: (a) Copper; (b) Floatglass; (c) Lucite; and (d) Plexiglas. (a) Copper; (b) Floatglass; (c) Lucite; and (d) Plexiglas.

The Gauss point distribution of the front and rear panels is shown in Figure 6, where Gauss pointsThe 1–11 Gauss are distributed point distribution in the front of thepanel, front and and Gauss rear points panels 12–22 is shown are distributed in Figure in6, the where rear Gauss panel. However,points 1–11 since are distributed the Gaussian in thepoints front 1 panel,and 12 and are Gausslocated points on the 12–22 axis, are they distributed are rapidly in theeroded rear when panel. theHowever, shaped since jet passes the Gaussian though. points Thus, 1the and particle 12 are velocity located onhistory the axis, of the they Gauss are rapidly points eroded2–11 of whenthe front the panelshaped and jet the passes Gauss though. points Thus, 13–22 the on particle the rear velocity panel were history extracted of the Gauss only pointsin this 2–11article. of the front panel and theIt can Gauss be seen points from 13–22 Figure on the11 that rear the panel maximum were extracted velocity only of the in particle this article. velocity of the front and rear panelsIt can be corresponding seen from Figure to the 11 four that different the maximum working velocity conditions of the particleis almost velocity 1000 m/s. of the The front particle and velocityrear panels of the corresponding front panel is to slightly the four smaller different than working the particle conditions velocity is of almost the rear 1000 panel, m/s. but The it exhibits particle avelocity good ofsymmetry, the front panelwhich is is slightly consis smallertent with than the the actual particle situation. velocity ofSince the rearthe panel,front and but itrear exhibits panel a obtainedgood symmetry, a certain which forward is consistent movement with velocity the actual unde situation.r the impact Since of the shaped front andjet, when rear panel the sandwich obtained chargea certain is initiated, forward movementthe front panel velocity obtains under a larger the impactnegative of velocity shaped and jet, whenthe rear the panel sandwich obtains charge a larger is positiveinitiated, velocity, the front which panel makes obtains the a larger resultant negative velocity velocity of front and panel the rear relatively panel obtains small. aIn larger addition, positive the closervelocity, to whichthe ends makes of the the ERA, resultant the difference velocity of will front become panel relativelysmaller. In small. Figure In addition,11b,d, the the velocity closer of to Gaussthe ends point of the 2 suddenly ERA, the differenceappears unloading, will become becaus smaller.e the In Figuremesh near11b,d, the the point velocity experiences of Gauss a point large 2 distortionsuddenly appearsand the point unloading, is deleted. because the mesh near the point experiences a large distortion and the point is deleted. 5. Conclusions Using a low-density material as the liner of the shaped charge pre-warhead to destroy the ERA is proposed in this article, and its effectiveness is proved by the numerical simulation that it is feasible to perforate and initiate the ERA through the low-density shaped jet. According to the u-d initiation criterion, the u2d of the three low-density shaped jets is greater than the critical value of 23 mm3/μs2.

Materials 2018, 11, 72 16 of 17

5. Conclusions Using a low-density material as the liner of the shaped charge pre-warhead to destroy the ERA is proposed in this article, and its effectiveness is proved by the numerical simulation that it is feasible to perforate and initiate the ERA through the low-density shaped jet. According to the u-d initiation criterion, the u2d of the three low-density shaped jets is greater than the critical value of 23 mm3/µs2. It can be concluded that the low-density shaped jet has the following advantages compared with the conventional metal (copper) shaped jet:

(1) The mass of three kinds of low-density liners are smaller than that of copper liner under the same volume condition. (2) When the shaped jet just contacts the target plate, the tip diameters of copper, floatglass, Lucite, and Plexiglas jets are 2.6 mm, 5.2 mm, 6.0 mm, and 6.0 mm, corresponding to tip velocities of 6042 m/s, 7464 m/s, 9785 m/s, and 9849 m/s, respectively, so the tip diameter and velocity of the three kinds of low-density shaped jets is greater than copper. In addition, the respective shaped jet length of copper, floatglass, Lucite, and Plexiglas is 84 mm, 98 mm, 137 mm, and 141 mm at t = 35 µs, which indicates that the ductility of the three low-density materials is also better than copper in the same structure of the shaped charge. (3) The time of three kinds of low-density shaped jets from impacting the sandwich charge to complete detonation (tfloatglass = 9.5 µs, tLucite = 9.0 µs, tPlexiglas = 8.9 µs) is also shorter than that of copper shaped jet (tcopper = 10.7 µs).

The above advantages make the low-density shaped jet a great application prospect. In the three kinds of low-density materials, the density of Lucite and Plexiglas are close to the minimum. At the same time, the ﬂoatglass shaped jet appears to be hollow in the jet tip during the stretching process; in other words, the ﬂoatglass has no good formablity compared to the other two low-density shaped jets. Therefore, the Lucite and Plexiglas can be considered as the two best low-density materials in this article. In summary, this study provides a certain value on the possible application of low-density liner materials in the shaped charge-shaped charge type tandem warhead, and it would be a plausible and sustainable direction for further research.

Acknowledgments: This work is financially supported by the National Natural Science Foundation of China (grant Nos. 11002162 and 11072262). The financial contributions are gratefully acknowledged. Author Contributions: Liangliang Ding put forward the research ideas of this article, designed the numerical simulation scheme, and wrote the article. Wenhui Tang proposed the modeling methods and identified three analytical criteria. Xianwen Ran provided guidance on the technical problems encountered in the numerical simulation and analyzed the numerical data. Conflicts of Interest: The authors declare no conflict of interest.

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