Numerical Simulation of Jet Penetration for Typical Targets

In view of a typical target, the numerical simulation of the jet penetration produced by a working part is carried out. Based on the fluid-solid coupling method, the process of the jet forming and penetration into a target by a front stage working section in series are calculated and simulated by using the MSC.DYTRAN numerical simulation software. On the basis of the calculation and simulation results, the penetration effects of the front stage working section in series on the targets with different thicknesses in the air-medium and water-medium environments are comparatively analyzed.


Introduction
In order to improve the action effect, the energy-gathering working part is usually designed in the form of a front stage section and a rear stage section connected in series.Its working principle is firstly to ignite the front stage working section, and then to ignite the rear stage working section in a delay time through a fuze mechanism.The front stage working section mainly penetrates the medium to open a channel and performs the first action on the target, while the rear stage working section performs the second action on the target after passing through the non-energy dissipation channel, thereby improving the action effect of the whole energy-gathering working part [1][2][3][4].
In this study, based on the fluid-solid coupling method, the process of the jet forming and penetration into a target by a front stage working section in series are calculated and simulated by using the MSC.DYTRAN numerical simulation software.Based on the simulation calculation results, the penetration effects of the front stage working section in series on the targets with different thicknesses in the air-medium and water-medium environments are comparatively analyzed.

Mathematical Model
Jet forming and target penetration are transient dynamics problems.The MSC.DYTRAN software uses the explicit time integration to solve the transient dynamic problems, which needs not only to discretize them in the spatial domain of distribution, but also to convert the continuous differential equations into finite order algebraic equations, and then discretize them in the time domain.The basic principles of the explicit finite element algorithm are as follows.
If the known time of the current system state is t n , the approximate values of the physical parameters should satisfy the following differential equation of motion: Rewrite the above equation to get: Where, M  is the inverse matrix of M. The acceleration can be found by inverting the mass matrix and multiplying it by the residual force vector.
The central difference method is adopted for the time advancement.

  
That is, it is assumed that the acceleration is constant within a time step.
For the explicit time integration, in order to maintain the calculation stability, the integration time step must be smaller than the time of the minimum element length e L of stress wave across the grid, that is: The mass and momentum conservation equation is as follow.
  The internal energy conservation equation is: where, 0 p , V , 0 e and 0 q are the initial pressure, initial volume, initial specific internal energy and chemical specific internal energy of the shock wave front, respectively; p , V , and e are the pressure, volume and specific internal energy of the product, respectively; D is the velocity, and the pressure produced is described by the JWL equation (Jones-Wilkins-Lee equation): where p is the combustion pressure, e is the internal energy per unit mass, 0  is the reference density,  is the overall material density, In the MSC.DYTRAN software, the states of the air, the water body and the steel structure target are described as follows.
The state of air can be described by the ideal equation of state (gamma-law equation of state).
  Where, e is the specific internal energy of air,  is the current density of air, and γ is the specific heat ratio.The state of water body is described by the polynomial constitutive equation of state.In the polynomial equation of state, the water body pressure is a polynomial function relative to the volume and specific internal energy.
The compressed state ( 0 The tensile state ( 0 Where,  is overall material density, 0  is reference density, e is internal energy per unit mass, As in the case of water body, the state of the steel structure material is also described by the polynomial equation of state, and the Johnson-Cook yield model is also used to describe the constitutive relation of the metal material under the conditions of large deformation, high strain rate and high temperature.
Johnson-Cook yield mode:  is the equivalent plastic strain,  is the equivalent strain rate,  is the reference strain rate, T is the temperature, r T is the room temperature, and m T is the melting temperature.

Physical Model
The specific structure of the front stage working section is as shown in Figure 1, and the internal main structure is composed of a fuze, a transmission assembly, an energy-containing material, a shell of charge and a shaped charge liner, etc.Two kinds of media, air and water, 20mm and 40mm thick steel plates are selected as targets to simulate the process of jet penetration into the typical targets.According to the simulation results, the effect laws of different media and different target thicknesses on the jet penetration results are comparatively analyzed.In order to fully simulate the jet penetration process and at the same time, effectively reduce the calculation workload, the axisymmetric modeling was adopted.Finite element models were established for the above-mentioned four working conditions, as shown in (a), (b), (c) and (d) of Figure 2, respectively.The size of the whole fluid domain was 300mm × 70mm, and the target sizes were 20 mm × 70 mm and 40 mm × 70 mm, respectively.In order to test the velocity of jet forming projectile under different conditions, the corresponding monitoring points were defined in the model.The monitoring point 0 was at the flush of the shaped charge liner, the monitoring point 1 was at the junction of air and water surface, with a distance of 50mm from the monitoring point 0. The monitoring point 2 was at the target surface, with a distance of 50mm from the monitoring point 1, and the monitoring point 3 was behind the target, with a distance of 50mm from the monitoring point 2. By measuring the velocities at the three typical positions, the influence of water and target thickness in the jetting process were judged.Location for four monitoring points is shown in Figure 3.

Result Analysis of Jet Penetration Simulation
The action of the energy-gathering working part on the target is the coupling of various action effects, and the characterization of its action power is a very complex problem.The jet shape will directly affect the perforation radius formed by the jet penetration into the target by the working part, while the jet velocity will directly affect the penetration depth of the working part.In this paper, the jet velocities and jet diameters under the four conditions are compared.

Overall Process of Jet Penetration
For the four working conditions, the processes of jet forming and target penetration are basically similar, and the difference lies in the penetration effect, that is, in the size of expanded hole.Taking the process   Table 2 shows the maximum velocity at each monitoring point under the four working conditions.It can be seen from Table 2 that due to the influence of the medium and target resistance, the maximum jet velocities under the four working conditions gradually decrease from the monitoring point 1 to the monitoring point 2 and then to the monitoring point 3. From the comparison of the results under the working conditions 1 and 2, it can be seen that the maximum velocity at the monitoring point 3 under the working condition 2 is smaller than that at the monitoring point 3 under the working condition 1 because the resistance of water to the jet is greater than that of air to the jet.From the comparison of the results under the working conditions 1 and 3, it can be seen that the maximum velocity at the monitoring point 3 under the working condition 3 is smaller than that at the monitoring point 3 under the working condition 1 because the thickness of the target under the working condition 3 is larger than that under the working condition 1.Among the four working conditions, the working condition 4 has the largest resistance to the jet during the whole penetration process because the medium of the environment is water and the target thickness is the largest, and therefore, the maximum velocity at the monitoring point 3 under the working condition 4 is the smallest.It can be seen from Table 3 that due to the influence of the densities of the medium and the target, the maximum jet perforation radiuses are different under the four working conditions.From the comparison of the results under the working conditions 1 and 2, it can be seen that the maximum perforation radius under the working condition 2 is greater than that under the working condition 1 because the resistance of water to the jet is greater than that of air to the jet.From the comparison of the results under the working conditions 1 and 3, it can be seen that the maximum perforation radius under the working condition 3 is smaller than that under the working condition 1 because the thickness of the target under the working condition 3 is larger than that under the working condition 1.Among the four working conditions, the working condition 3 has an air medium environment and the largest target thickness, so the smallest maximum perforation radius in the whole penetration process is under the working condition 3.

Conclusion
Based on the fluid-solid coupling method, the process of jet forming and penetration into the target by a front stage working section in series are calculated and simulated by using the MSC.DYTRAN numerical simulation software.The simulation results show that the final velocity of jet penetration is related to the density of the medium in which the jet travels and the thickness of the target, and the greater the density of the medium and the greater the thickness of the target is, the smaller the final velocity of jet penetration will be; the perforation radius produced by jet penetration is related to the density of the medium where the jet is and the thickness of the target, and the smaller the density of the medium and the greater the thickness of the target is, the smaller the perforation radius of jet penetration will be.
and ignition point of the working part are defined, and the ignition time of the working unit is determined according to the distance and velocity from the centroid of such unit to the ignition point.The conservations of the mass, momentum and internal energy of the matter flowing through the action region are expressed by the Rankine-Hugoniot equations.

Figure 1 .
Figure 1.The specific structure of the front stage working section.

Figure 2 .
Figure 2. Finite element models section for four working conditions.

Figure 3 .
Figure 3. Location for four monitoring points.

FMIA- 2023
Journal of Physics: Conference Series 2599 (2023) 012029 of a 20mm air target being penetrated by the jet as an example, a cloud image showing the whole process of jet forming and penetration into the target is given in Figure4.The whole process of jet forming and penetration into the target can be clearly seen from Figure4.

Figure 4 .
Figure 4.The whole process of jet forming and penetration into the target.
Where, M is the structural mass matrix, C is the structural damping matrix, K is the structural stiffness matrix, 2

Table 1 .
Different working conditions.

Table 2 .
Maximum speed of monitoring point under different working conditions.

Table 3 .
Table3shows the maximum perforation radius after target penetration under the four working conditions.Maximum perforation radius for different conditions.