2.1. Fundamental Equation
The high-temperature and high-speed friction and wear behavior between the barrel bore and the projectile can become extremely complicated [
17]. The Archard wear model has a wide range of applications in engineering fields [
18,
19,
20,
21]. The wear of the barrel bore usually occurs under high-speed dry contact conditions between the muzzle and the bullet, which is more suitable for describing the wear phenomenon. The dry condition is more consistent with the actual wear of the barrel bore. Its general form is shown in Equation (1).
In the formula, is the wear volume (g); is the dimensionless wear coefficient; is the normal load (N); is the slip distance (mm); and is the material hardness (Hv).
It can be seen from the above formula that the wear amount is directly proportional to the normal load and the slip distance and inversely proportional to the material hardness. The left and right sides of Formula (1) are multiplied by the same density to obtain Formula (2):
The above formula can be further amended to produce Formula (3).
In the formula, is the wear rate of the barrel material (g/min); is the relative sliding speed; and , , , and and are undetermined coefficients.
2.2. Chromium Layer Hardness Test and Calculation
The hardness of the coating was determined with an HVS-1000 digital microhardness tester. In order to avoid the influence of the substrate, the hardness of the coating was measured along the cross section. Before measurement, the coating sample was cut along the cross section, sealed with epoxy resin, ground and polished, and then measured. Each sample measured 8 to 10 points, and the average value was taken as the final data. The chrome-plated samples were heated at 25 °C, 200 °C, 300 °C, 400 °C, 500 °C, 600 °C, and 700 °C, and then air-cooled to room temperature.
The coating material was hexavalent chromium, and the hardness of hexavalent chromium at different temperatures was tested with a Vickers hardness tester, as shown in
Table 1.
Data fitting was performed on the chromium hardness data, and the curve of hardness variation with the temperature of the coating material in the temperature range of 25 to 700 °C was obtained, as shown in
Figure 1. The formula of hardness variation with temperature after fitting is
In the formula, T is the material temperature.
According to the one-dimensional heat conduction equation of the barrel [
22], the temperature variation curve of the coating interface at 200–463 mm distance from the barrel breech under the condition of 150 consecutive shots (including 10 short shots and 20 and 30 long shots) was calculated by taking the maximum temperature of the gunpowder gas as the heat source and considering the effect of the heat transfer coefficient of the gas–solid interface. After 150 rounds of firing, the barrel surface temperature reached 700 °C, and the temperature at the interface between the coating and the substrate material was about 420 °C, as shown in
Figure 2. It can be seen from the temperature curve that the temperature change in the barrel will affect the hardness of the coating.
In order to obtain the relationship between the coating temperature and hardness changes at different positions of the gun barrel axis, the average temperature and hardness of the coating material were obtained from the surface and interface temperatures of the coating according to the calculation results of the gun barrel temperature field. The temperature change curve of the gun barrel coating from 200 mm to 450 mm is shown in
Figure 3a. The average coating temperature is about 119 °C. Combined with Formula (4), the coating hardness values of different coaxial positions of the barrel (200 mm, 250 mm, 300 mm, 350 mm, 400 mm, and 450 mm) can be calculated, as shown in
Figure 3b.
As can be seen from
Figure 3b, the hardness of the coating at the gun barrel mouth varied from 991 Hv to 995 Hv, with a small variation range.
2.3. Archard Wear Model Coefficient Correction
During the firing of the barrel, under the action of high-speed friction and wear of the bullets, the surface temperature of the barrel inner bore rises sharply, and the initial temperature reaches more than 200 °C [
17]. Therefore, only wear test studies at high temperatures were carried out.
In order to determine the coefficient of the Archard wear model of the coating and the gun steel base material, and to verify the applicability of the Archard wear model, a high-temperature pin plate wear test was carried out. High-temperature pin rotation wear was carried out on the electroplated sample of gun steel with a pin plate wear tester, and the wear of the plate (gun steel) and pin (bullet) was analyzed. The test coating material was hexavalent chromium and the friction material was 08Al steel; the friction pin is shown in
Figure 4. The diameter of the wear mark was 20 mm, and the wear time was 15 min. The experimental parameters of wear are shown in
Table 2.
The mass changes in disk and pin samples before and after wear were measured using an analytical balance, and the results are shown in
Table 3. The macroscopic morphology of the wear marks after high-temperature wear is shown in
Figure 5.
The wear morphology of the chrome-plated samples is shown in
Figure 6. It can be seen that the boundary between the abrasion mark and the matrix is clear. The width of the wear mark is 1.123 mm. Combined with the EDS line scanning results, as shown in
Figure 7, it can be seen that when the chrome-plated samples undergo friction, all the samples experience H62 brass adhesion, meaning that the brass adhesion in the middle of the wear marks is relatively concentrated; the brass adhesion width is basically the same as the width of the wear marks, and the chrome layer is covered by brass.
In the friction process, because the copper pin and the chromium plating layer are in contact, the force in the middle of the wear mark is large, the force on both sides is small, and the hardness of the copper pin is less than the hardness of the chromium plating layer. This means that the copper pin will be worn off, but not enough to form a wear mark on the surface of the chromium plating layer. The width of the abrasion mark will be larger than the width of the copper adhesive, indicating that the copper adhesive first appears in the middle part of the heavy force and the adhesion then expands to both sides. At the same time, the thickness of the copper adhesive will also be different under the influence of the flatness of the surface of the chrome layer.
The mass change in chromium coating under different working conditions can be obtained via a pin wear test, as shown in
Table 3. The wear depth of the chromium coating can be obtained with the grinding area and density of the chromium coating.
In the formula, is the wear depth, is the coating change quality, is the coating density, and is the wear area.
According to Formula (3), the coefficient , , , is fitted using the least square method according to the results of the pin wear test. Through MATLAB software using least square method fitting, we can obtain , , , and .
The wear formula of the gun barrel matrix material can be fitted using the same method. Since the influence of rifling shape changes on the contact pressure and velocity is not considered, the influencing factors of wear rate remain unchanged after the coating has worn out, and then the wear rate formula coefficient of the gun steel matrix material is as follows: , , , .
Therefore, the Archard wear model of the chrome coating of the gun barrel is obtained following Equation (6):
The Archard wear model of the gun barrel matrix material is shown in Equation (7).
In the above formula, is the wear rate (g/min), is the relative sliding speed, and is the hardness of the material, which varies with temperature.
The wear models of chrome plating and gun steel obtained by fitting were consistent with the wear test data under corresponding working conditions, and the calculated wear rate errors were less than 10%, as shown in
Table 4. From Equations (6) and (7), according to the values of
,
, and
, it can be concluded that, for the pin disk wear test, the friction speed had the greatest influence on the wear rate, followed by the contact pressure and finally the material hardness.
2.4. Finite Element Model
The coupling finite element model of the gun barrel and projectile was established to calculate the whole process of the projectile moving along the barrel, in order to obtain the contact pressure of the barrel at different axial and circumferential positions.
The finite element grid model of the gun barrel [
23,
24,
25] was established, and C3D8R elements were used to divide the gun barrel and projectile grids. The rifling grids of the gun barrel measured 0.06 mm, the rifling groove grids were 0.1 mm, and the driving sides of the rifling grids were 0.04 mm. Some of the finite element meshes of the gun barrel and the projectile are shown in
Figure 8.
In order to accurately describe the friction and wear law between the projectile and the barrel rifling, it is necessary to accurately obtain the force law of the single rifling structure (driving rifling surface, rifling groove, and rifling land surface), as shown in
Figure 9 and
Figure 10.
In the finite element model, the outer surface of the barrel is completely fixed, and the internal ballistic pressure obtained from the test is applied to the end of the projectile; the temperature load is applied to the inner surface of the barrel.