All bodies filled with explosives or some other substance capable of causing a certain effect on the target are called by a common name – ammunition. Today, a large number of different types of ammunition are in use and are distinguished from each other by purpose, shape, structural parameters, launch method, effect on the target, etc.

This type of projectile is used in cannons from 20mm caliber up to the heaviest ones, that is, in all cannons where it is possible to give the projectile such an initial speed that the trajectory is flat, and the impact speed is such that the kinetic energy of the projectile is sufficient to overcome the resistance of a hard obstacle – body armor.

The armor-piercing projectile penetrates the armor thanks to the kinetic energy it has at the moment of collision with the armor and the great endurance of its body.

Penetration, i.e., breaking through, is the process of movement of the penetrator through an obstacle. Every movable body designed for penetration is called a penetrator, and the target body exposed to the influence of a moving penetrator is called an obstacle. The study of the penetration process is of great importance both in the field of civilian application and in the field of military technology.

Terminal ballistics is one of the basic disciplines that deals with the definition of penetration mechanisms, which significantly contributes to the optimization of the design of projectiles with a penetrating and destructive effect, as well as to the design of armor protection.

On the other hand, there are also numerous civilian applications of the penetration process, such as the protection of buildings, etc. Applications for military purposes are certainly a priority and the most important driver of research in the field of penetration (Elek, 2018).

The basic types of penetration processes characteristic of armor- piercing ammunition are shown in Figure 4.

Figure 4

Рис. 4 – Основные типы процессов проникновения

Слика 4 – Основни типови пенетрационих процеса

Johnson and Cook proposed a semi-experimental constitutive model for metals characterized by high stresses, high strain rates and high temperatures. Each of the phenomena (strain hardening, strain hardening rate and thermal softening) is represented by an independent factor. Taking all factors into account, yield stress is obtained as a function of effective plastic strain, rate of plastic strain and temperature (Wang & Shi, 2013; Liu et al, 2012).

Johnson and Cook represent yield stress by equation 1:

(1)

where A is the initial yield stress, B is the reinforcement coefficient, n is the amplification exponent, c is the deformation rate constant and m is the thermal softening exponent.

is the effective plastic deformation and is the effective plastic strain rate for given by equation 2:

(2)

Temperature is given by equation 3:

(3)

where 𝑇_{𝑟𝑜𝑜𝑚} is the room temperature and 𝑇_{𝑚𝑒𝑙𝑡} is the material melting temperature.

The constants for materials are determined by various types of tests, such as tensile test, Hopkinson rod test, etc.

The deformation of the material during damage is given by equation 4:

(4)

where 𝐷_𝑖 , 𝑖 = 1, . . . ,5 are the parameters that define the material damage criteria and 𝜎^∗ is the ratio of the pressure divided by the effective stress.

Material failure occurs when the parameter (equation 5):

(5)

reaches the value of 1.

The Mie–Grüneisen equation of state represents the relationship between pressure and volume of a solid at a given temperature. It is used to determine the pressure in solids exposed to high pressure for a short period of time. There are several different relations that define a given dependency. Grüneisen's model can be presented in the form given by equation 6 (Heuzé, 2012; Wilkins, 1999):

(6)

where 𝛤₀ is the Grüneisen parameter which represents the thermal pressure arising from a set of vibrating atoms, 𝑉 is the volume, 𝑝 is the pressure and 𝐸 is the internal energy.

If it is assumed that 𝛤₀ does not depend on pressure and internal energy, it can be written (equation 7):

(7)

where 𝑝₀ is the reference pressure at a temperature 𝑇 = 0𝐾 and 𝐸₀ is the reference internal energy at a temperature 𝑇 = 0𝐾.

In that case, 𝑝₀ and 𝐸₀ are also independent of temperature, so the values of these parameters can be estimated from Hugoniot's equation - equation 8, 9 and 10 (Heuzé, 2012; Wilkins, 1999).

(8)

(9)

(10)

where 𝐶₀ is the speed of sound through the material, 𝜌₀ is the initial density of the material, 𝜌 is the current density of the material, 𝛤₀ is the Grüneisen parameter, 𝑠 is the linear slope coefficient of Hugoniot's line, 𝑈_𝑠 is the shock wave speed, 𝑈_𝑝 is the particle velocity and 𝐸 is the internal energy per unit of reference volume.

The Johnson-Cook's parameters for different types of materials used in the numerical simulation of the penetration of a 30mm armor-piercing projectile are defined in Table 2 and the temperature parameters are given in Table 3. In Table 4, the damage parameters for the same materials are defined. These parameters are defining the Johnson–Cook material model used in numerical simulations (Murthy & Santhanakrishnanan, 2020; Bataev et al, 2019; Champagneet al, 2010; Rezasefatet al, 2018).

Table 2

Таблица 2 – Параметры Джонсона–Кука для различных материалов

Taбела 2 – Џонсон–Кукови параметри за различите материјале

Table 3

Таблица 3 – Тепловые характеристики различных материалов

Taбела 3 – Термичке карактеристике за различите материјале

Table 4

Таблица 4 – Параметры повреждения различных материалов

Taбела 4 – Параметри оштећења за различите материјале

Table 5 defines the parameters of the equation of state for different materials used in the numerical simulation.

Table 5

Таблица 5 – Параметры уравнения состояния различных материалов

Taбела 5 – Параметри једначине стања за различите материјале

The creation of the model using the finite element method is performed on the basis of the existing projectile 3D model. Since the analysis system has two symmetry planes (along the projectile axis), a quarter model is created to obtain faster calculation results. In accordance with the shape and construction of the tested projectiles and the plate, in order to properly define the network, a 3D eight-nodes element type hexa is used. The projectile and plate models are shown in Figures 5, 6 and 7. The finite element model of the projectile is modeled using 141000 nodes and 127000 elements, while the 10mm thick plate is modeled using 1275000 nodes and 1200000 elements. The average size of the elements is 0.5mm.

Figure 5

Рис. 5 – МКЭ-модель снаряда и пластины, изометрия

Слика 5 – МКЕ модел пројектила и плоче, изометрија

Figures 8-13 show the Von Misses equivalent stress and penetration effect for armor plate Weldox 460 with a thickness of 30mm.

Figure 8

Рис. 8 – Эквивалентное напряжение Фон Мизиса, шаг 1 – случай 1

Слика 8 – Вон Мисесов еквивалентни напон, корак 1 – случај 1

Figures 16-21 show the Von Misses equivalent stress and penetration effect for armor plate Weldox 460 with a thickness of 40mm.

Figure 16

Рис. 16 – Эквивалентное напряжение Фон Мизиса, шаг 1 – случай 2

Слика 16 – Вон Мисесов еквивалентни напон, корак 1 – случај 2

After numerical simulation and result analysis, the conclusion is that a 30mm armor-piercing projectile at an impact speed of 750m/s achieves the full penetration effect on 30mm thick plates while in the case of a 40mm thickness, it jams into the plate.

For different purposes, it is of great importance to determine the limit (maximum) plate thickness for which the projectile with defined ballistic and material characteristics is able to achieve the full penetration effect.

For the additional simulation cases, the same input parameters are used for the projectile, and the only difference is the armor plate thickness.

Additional numerical simulations are carried out in accordance with the previously defined models using the same initial and boundary conditions. It is found that the full penetration effect is achieved on plates with a thickness of up to 33mm, and after increasing the thickness to higher values, a limited penetration effect then occurs.

Figures 24-29 show the Von Misses equivalent stress and penetration effect for armor plate Weldox 460 with a thickness of 33mm.

Figure 24

Рис. 24 – Эквивалентное напряжение Фон Мизиса, шаг 1 – случай 3

Слика 24 – Вон Мисесов еквивалентни напон, корак 1 – случај 3

Figures 32-37 show the Von Misses equivalent stress and penetration effect for armor plate Weldox 460 with a thickness of 34mm.

Figure 32

Рис. 32 – Эквивалентное напряжение Фон Мизиса, шаг 1 – дело 4

Слика 32 – Вон Мисесов еквивалентни напон, корак 1 – случај 4