2.1. Plane Shock Wave Assumption of PELE Projectile Penetrating Against Target Plate
Since the PELE projectile has a symmetrical structure, the collision between the projectile and the target plate can be simplified to a plane collision problem. When the PELE projectile impacts the target plate, two shock waves are generated at the contact interface between the projectile and target plate. The shock waves propagate to the rear of the projectile and the target plate respectively, as shown in
Figure 3a. Since the target thickness is much smaller than the projectile length, the shock waves in the target plate first reach the free interface of the target plate and reflect rarefaction waves to the contact interface, as shown in
Figure 3b. In order to facilitate the division of the penetration process, the following definitions are made:
t0 represents the moment when the projectile has just touched the target plate;
t1 represents the moment when the rarefaction waves from the back side of the target plate reach the contact interface;
t2 represents the moment when the PELE projectile just perforates the target plate. The time range
t0–
t1 is defined as the impact stage, and the time range
t1–
t2 is defined as the plug shear stage. Thus, the process of the PELE projectile penetrating the target plate can be divided into two stages: projectile impact stage and plug shear stage.
On the moment of impact between the PELE projectile and target plate, one shock wave is produced in the target plate and propagates to the back of the target plate, and another shock wave, opposite to the projectile’s flight direction, is generated in the projectile. Due to the inner core material being different from the outer casing, the intensity and shock wave velocity of the inner core and outer casing are different. After the plane shock wave, the rarefaction wave propagates radially from the interface to the projectile interior, which causes the projectile to expand radially. The plane shock wave relationship is obtained under the one-dimensional stress condition. To analyze the process of a PELE penetrating a target plate using the plane shock wave theory, the following assumptions are required:
(1) Neglecting the influence of shearing force on the impact wave intensity. The shearing force is caused by the difference of particle velocity between the projectile and target plate, which makes the impact velocity decrease and the impact pressure increase. For a thin target plate, the shearing force is smaller than the impact pressure, so it can be neglected.
(2) The radial expansion of projectile is neglected before the projectile perforates the target plate. After the shock wave, the projectile casing will expand radially under the action of the rarefaction wave. However, within the time range of investigation, the radial expansion value of the projectile is limited, and it is considered that the radial deformation of the outer casing is limited by the target plate. Therefore, the radial expansion of the projectile is neglected in the perforation process.
Since both the PELE projectile and the target plate have symmetrical structures, when analyzing the stress state of the projectile and the target plate, it can be simplified into a 2D (two-dimensional) plane problem and the 2D plane is axisymmetric. In this paper, the axial residual velocity of the projectile is mainly studied, not the radial dispersion of fragments after the projectile perforates the target plate. Therefore, combined with the above two basic assumptions, the study of the axial residual velocity of projectile can be further simplified to a 1D (one-dimensional) collision problem. The penetration process of the PELE projectile against a thin metal target plate is similar to that of the flat head projectile, and the penetration process can be analyzed using the shear resistance model of the flat head projectile. The shear resistance model divides the energy loss during the penetration process into two parts: the impact energy dissipation between the projectile and the target plate, and the shear energy dissipation of the plug in the perforation process.
According to shock wave theory, the impact energy dissipation can be divided into the following parts: impact compression potential energy of the plug, the kinetic energy of particles after the shock wave, and the impact compression potential energy of the outer casing and inner core etc. Based on the characteristics of the PELE projectile, the shear energy dissipation of the plug is divided separately into the inner edge and outer edge of projectile casing shear of the target plate.
2.2. Internal Energy and Kinetic Energy Increment of the Target Plate Plug after Shock Waves
Before the rarefaction wave reflects from the back side of the target plate, the impact between the projectile and the target plate can be approximately considered a coaxial impact problem, as shown in
Figure 4. The projectile comprises two parts, outer casing and inner core, and their internal shock wave intensities are different because the materials are different. Correspondingly, the shock wave intensities in different impact regions of the target plate are different.
It is assumed that then the projectile impacts the target plate with the mass m and velocity u0, the initial velocity and pressure of the target plate and the initial pressure of the projectile are both zero. Here, the following definitions are made. P stands for the pressure per unit area, ρ is the material density, D represents the shock wave velocity, u is the particle velocity, E means the energy, c, and λ represents the Hugoniot constant of the material. In addition, the subscript j is an abbreviation for “jacket,” and it is used to denote the parameters of the projectile outer casing material; the subscript f is an abbreviation for “filling,” and it is used to denote the parameters of the inner core material; the subscript t represents the parameters of the target plate, and it is an abbreviation for “target.” The subscript tj represents the post-wave state of target plate after the collision between the outer casing and target plate, and the subscript tf represents the post-wave state of the target plate after the collision between the inner core and target plate. The subscript 0 indicates the wave-front state of the material, and the subscript 1 indicates the post-wave state of the material. According to the interaction of the shock wave, the following relationships can be obtained.
(1) Take the projectile outer casing as the main body of research. The initial state of the outer casing is
P0j,
ρ0j,
u0j,
c0j, and a left shock wave is generated in the outer casing after the impact, and the post-wave state is
P1j,
ρ1j,
u1j,
c1j. Thus, the following relationships can be obtained through the shock wave discontinuous equation and the linear shock equation.
(2) Take the projectile inner core as the main body of research. The initial state of the inner core is
P0f,
ρ0f,
u0f,
c0f, and a left shock wave is generated in the inner core after the impact, and the post-wave state is
P1f,
ρ1f,
u1f,
c1f. Similarly, the following relationships can be obtained by the shock wave discontinuous equation and the linear shock equation.
(3) Take the target plate as the main body of research. The initial state of the target plate is P0t, ρ0t, u0t, c0t, and a right shock wave is generated in the target plate after the impact. The post-wave state is divided into two regions, as follows.
(a) The interaction region between the outer casing and the target plate. In this case, the corresponding post-wave state is
Ptj,
ρtj,
utj,
ctj. Similarly, the following relationships can be obtained through the shock wave discontinuous equation and the linear shock equation.
(b) The interaction region between the inner core and the target plate. In this case, the corresponding post-wave state is
Ptf,
ρtf,
utf,
ctf. Similarly, the following relationships can be obtained through the shock wave discontinuous equation and the linear shock equation.
According to the mechanical equilibrium conditions of the interaction region, the velocity and pressure on both sides of the discontinuity are equal. Hence, the following relationships can be obtained.
(a) The balance condition between the outer casing and the target plate,
Ptj =
P1j,
utj =
u1j. When Equations (1), (2), (5), and (6) are combined together, the expression of the particle velocity of the target plate after the shock wave is as follows.
By solving Equation (9), the particle velocity after the impact between the outer casing and the target plate can be obtained.
(b) The balance condition between the inner core and the target plate,
Ptf =
P1f,
utf =
u1f. When Equations (3), (4), (7), and (8) are combined together, the particle velocity after the impact between the inner core and the target plate can be obtained as follows.
Thus, the kinetic energy increase of the target plate in the impact region of the projectile and the target plate can be obtained after the shock wave.
where
R is the projectile outer casing radius,
r is the inner core radius, and
h is the target plate thickness.
Under the experimental conditions, the projectile and the target plate system are at normal pressure and normal temperature, and the shock wave pressure formed by the impact process is relatively high,
P >>
P0. The relation between the material pressure and the specific volume under the impact adiabatic condition is shown in
Figure 5. The relationship
P >>
P0 is reflected in
Figure 5 where point
A is very close to point
M, which means that the
AB line can be considered to be the diagonal of the rectangular
MNBC. Under this state, the influence of the initial pressure can be neglected, and it can be assumed that the total work of the shock compression is evenly distributed between the internal energy and kinetic energy.