Study on jet formation and penetration performance of shaped charge with untypical double-cone titanium liner

Based on the platform of AUTODYN-2D code, the shaped charge with untypical double-cone titanium liner penetrating concrete targets are investigated. The influence of the titanium liner thickness and the height ratio of top cone on the jet tip-velocity and jet characteristics are analyzed. The titanium liner thickness, the height ratio of top cone, and the standoff affecting on the penetration performance of the concrete targets are also conducted. The numerical simulations indicated that, with increasing the titanium liner thickness, the jet tip-velocity gradually decreased, and the penetration depth of the concrete targets first increased and then decreased. When the height ratio of top cone was 0.5, the shaped charge with untypical double-cone titanium liner can combine the advantage of the small cone-angle liner and the large cone-angle liner, which will ensure the sufficient penetration depth and hole diameter. Furthermore, the shaped charge with untypical double-cone titanium liner can penetrate the concrete with a thickness of 1000mm at the standoff within a region between 2.0 CD and 2.5 CD.


Introduction
The shaped charge warhead is the core damage component of the anti-concrete/reinforced concrete and other solid targets.Modern military targets' robustness puts higher requirements for their comprehensive damage power.Researchers at home and abroad have put forward a variety of methods to improve the power of the warhead, mainly by changing the structure and charge structure of the shaped charge [1][2][3][4] to improve the utilization rate of the shaped charge, so that it can produce higher jet tip-velocity, better velocity gradient, and continuous jet penetrator, thereby further improving the comprehensive damage performance of the jet to the concrete medium.
In the shaped charge warhead, the commonly used shape of the liners are small angle conical liner, large angle conical liner, horn-shaped liner, and hemispherical liner, while the large angle liner and hemispherical liner are mainly used in the shaped charge of the explosively formed projectile, and the small angle liner and horn-shaped liner are mainly used to form the high-velocity jet penetrator [5][6][7].The processing technology of the small cone angle liner is relatively simple, and it is widely used in the active-shaped charge armor-piercing projectile, but the utilization rate of the small cone angle liner is too low.Although the jet head formed by the trumpet-shaped liner has a higher velocity, its processing technology is relatively complex, and the requirements for symmetry of the cartridge-shaped liner are very high, so it is rare in practical applications.Later, it was found that by changing one cone angle, the trumpet-shaped liner could be changed into a double-cone-shaped liner with a small cone angle upper and a large cone angle lower [8,9], which not only retained the advantages of the trumpet-shaped liner to form a high-velocity jet but also simplified its processing technology.Although the penetration performance of the shaped charge with the double-cone shaped charge liner to steel targets has been improved a lot, the jet is easy to break and is only suitable for the penetration of reactive armor targets with the small standoff.Therefore, it is suggested that the upper cone angle of the double-cone liner should be appropriately greater than the lower cone angle by changing the cone angle, which can reduce the overall velocity gradient of the jet and delay the jet breaking time based on ensuring that the jet tipvelocity meets the requirements, thus enhancing the matching performance of the penetration depth and the crater diameter to the concrete target [10].
In this paper, numerical simulation is used to study the penetration performance of shaped charge with untypical double-cone titanium liner into concrete medium, mainly analyzing the influence of the liner thickness and the height ratio of the top cone on the jet formation and the penetration performance of concrete, as well as the influence of the standoff on the penetration performance.

Material models and equation of states
The material of the liner used in this paper is titanium.To study the influence of the upper and lower cone angles of the shaped charge liner on the jet formation and penetration performance of concrete, the case material used is steel, the selected explosive is 8701, and the target is 35MPA concrete medium.AUTODYN software has a very rich material library.The materials selected in this paper are commonly used in the AUTODYN material library, and the material parameters are also included in the material library, which can ensure the accuracy of simulation to the greatest extent.Table 1 lists all materials to be used in this simulation and their corresponding equations of state and strength models.Tables 2 and  3 lists the main material parameters of concrete.

Simulation model
The penetration characteristics of shaped charge of untypical double-cone titanium liner into concrete are studied.By changing the structure of the liner, the laws of jet formation and their influence on target penetration are simulated.Figure 1 shows the geometry of the shaped charge of untypical double-cone titanium liner.The charge diameter D is 120 mm, the case thickness is t, the liner thickness is δ, the upper cone angle is α 1 , the lower cone angle is α 2, the upper cone height is h 1 , the total height of the shaped charge is h 2 , and the charge length is L. The AUTODYN-2D software platform is used for simulation.The simulation principle and model of untypical double-cone titanium shaped charge penetrating concrete are shown in Figure 2. The case, titanium liner, and explosive all use the Euler algorithm.The lagrangian algorithm is used for concrete, and the whole model is a fluid-structure coupling contact algorithm.The boundary condition of the set air part is "Flow out (ALL equal)".The concrete used in this paper has a diameter of 1000 mm and a thickness of 1000 mm.To ensure the simulation accuracy and speed, the mesh size of the concrete target is 2.5 mm.The variable density grid division method is used in the Euler part.In order to ensure that the grid density at the top of the liner is not less than 5, the area where the middle jet acts are relatively dense, and the grid size on both sides is relatively large.

Influence of the Liner Thickness
To study the influence of δ on the jet formation, the thickness is selected as 3 mm, 4 mm, 5 mm, and 6 mm in turn.The structural parameters of other shaped charge are as follows: α 1 is 80 °, α 2 is 50 °, the ratio of h 1 to h 2 is 0.5, the charge is 8701, the charge length-diameter ratio is 1.5 CD, the case material is 45 # steel, and the case thickness is 3 mm.The velocity distribution of the jets with four thickness liners at 1 and 2 CD standoff is shown in Figure 3, and the relationship between the jet tip-velocity and time is shown in Figure 4.It can be seen from Figure 3 and Figure 4 that the thickness of the untypical double-cone titanium liner has a significant influence on the jet formation, and the jet tip-velocity decreases significantly with the increase of δ.When δ is 3 mm, the formed jet is slender and uniform, and the jet velocity is high.However, because the jet quality is too small, the penetration performance will be extremely unstable, which is not conducive to jet penetration; When the thickness is 4 mm and 5 mm, the jet has a good shape, sufficient velocity and quality, and the penetration performance is relatively stable; When the thickness is 6 mm, the part of the slug is thick, which affects the jet stretch, the jet shape is poor, and the velocity is low, which may be unfavorable for the penetration of concrete.To sum up, the thickness of untypical doublecone titanium liners is generally 4 mm~5 mm.

Influence of the Height Ratio of Top Cone
To study the influence of the ratio of h 1 to h 2 on the jet formation, based on the previous study of δ, δ is 5 mm, and the ratio of h 1 to h 2 is selected as 0.2, 0.4, 0.5, 0.6, 0.8 in turn, and other parameters of the shaped charge structure are unchanged.The velocity distribution nephogram of the jet formed by the shaped charge of untypical double-cone titanium with five proportions at 1 and 2 CD the standoff is shown in Figure 6 and the relationship between the jet tip-velocity and time is shown in Figure 7.It can be seen from analyzing Figure 7 that the tip-velocity of the jet decreases with the increase of the proportion of h1.The variation from 0.2 to 0.4 is very obvious, and the variation from 0.4 to 0.6 is relatively weak.Figure 6 shows that when the ratio is 0.2, the proportion of the jet slug is larger and the effective jet mass is less at the height of 2 CD standoff, but the jet tip-velocity is much higher than in other conditions, which may be due to the small proportion of h1, and the jet characteristics formed by the untypical double-cone liner tend to the jet characteristics of the single-cone liner with small cone angle; However, when the ratio is 0.8, the proportion of the slug of the formed jet penetrator decreases significantly, the proportion of the effective jet mass increases, and the jet diameter is thicker, but the jet tip-velocity decreases significantly, which may affect its penetration performance into concrete; When the ratio of h1 to h2 is 0.4~0.6, the velocity difference of the formed jet penetrator is not very large, and the effective jet mass and the ratio of the slug are relatively appropriate.

Jet penetration characteristics
The standoff is the vertical distance between the bottom diameter end surface of the shaped charge liner and the concrete target plate.A large number of experiments show that the standoff has a great influence on the length and diameter of the jet penetrator, thus affecting the penetration depth and crater diameter of the jet penetrator to the concrete target.The standoff of shaped charge is generally 1~3 times the charge diameter, which is conducive to the formation of the jet from each micro-element of the liner, and is conducive to the jet stretching, so as to ensure that the formed jet penetrator is not dispersed and not broken when hitting the target, so as to maximize the penetration effect on the target.In this section, by selecting different standoffs, the influence of standoff on the penetration performance of shaped charge with untypical double-cone titanium liner into concrete is studied, so as to determine the range of advantageous standoff.Based on the previous research on the structure and charge of the shaped charge, α 1 is 80 °, α 2 is 50 °, the ratio of h 1 to h 2 is 0.5, the charge is 8701, the charge length is 1.5 CD, and with center point detonates.The relationship between the penetration depth of the shaped charge with untypical double-cone titanium liner into concrete and the standoff is shown in Figure 9 and the penetration damage of the concrete is shown in Figure 10   As can be seen from Figure 9, with the increase of the standoff, the penetration depth of the untypical double-cone titanium shaped charge into the concrete target increases first and then decreases.The influence of the standoff on the penetration ability of the untypical double-cone liner jet can be analyzed from two aspects: ① with the increase of the standoff, the jet is effectively stretched, thus improving the penetration depth; ② With the increase of the standoff, the extended jet will produce radial dispersion and swing and will produce necking or even fracture after extending to a certain extent, which will reduce the stability of the jet and reduce the penetration depth.
According to the damage in Figure 10, the crater diameter of the jet to the concrete gradually decreases with the increase of the standoff.This is because after the increase of the standoff, the jet movement distance is lengthened, resulting in the gradual extension of the jet penetrator, leading to the reduction of the jet diameter and the tip-velocity of the jet, and the crater diameter is proportional to the jet diameter and velocity.Therefore, it is better to take 2.0 ~ 2.5 CD as the advantageous standoff by comprehensively considering the penetration depth and crater diameter of the shaped charge with untypical double-cone titanium liner into concrete.

Conclusion
(1) The thickness of the untypical double-cone titanium liner should not be too thin, because thin thickness results in the low mass of the jet penetrator.Generally, the optimal thickness of the untypical double-cone titanium liner is 5 mm~6 mm.
(2) The ratio of the upper cone height to the liner height of the untypical double-cone liner is generally 0.5.This type of liner can combine the advantages of the small cone angle liner and the large cone angle liner, which is conducive to matching the penetration depth and crater diameter of the concrete.
(3) When the standoff is 2.0 ~ 2.5 CD, the penetration depth and the crater diameter of the shaped charge with untypical double-cone titanium liner into the concrete match well.

Figure 1 .
Figure 1.Geometry and simulated model of the shaped charge with untypical double-cone titanium liner against concrete: (a) Geometry, (b) Simulated model.

Figure 2 .
Figure 2. The shapes of jet formation under different δ.

Figure 3 .
Figure 3. Jet tip velocity to time under different δ.

Figure 4 .
Figure 4.The shapes of jet formation under different height ratio of top cones.

Figure 5 .
Figure 5.The curve of jet tip velocity and time under different height ratio of top cones.

Figure 7 .
Figure 7. Damage pictures of concretes impacted by shaped charge under different standoff.

Table 1 .
Material strength models and EOSs of shaped charge each part

Table 2 .
Parameters of EOS for concrete material.

Table 3 .
Parameters of strength and erosion model for concrete material.