The Construction of a Small-Caliber Barrel Wear Model and a Study of the Barrel Wear Rule

Abstract

The wear of small-caliber barrels is one of the key factors affecting barrel life. Based on the Archard wear model, a high-temperature pin plate wear experiment was carried out, and wear models of chrome-plated layers and gun barrel materials were established. In addition, a finite element model of the interaction between the bullet and the barrel was established. The movement of the projectile along the barrel was simulated and analyzed, and the force distribution of the spatial geometry structure of the rifling was mastered through simulation. The wear law of the gun barrel along the axial direction was obtained based on the wear model of the chrome-plated layer and gun barrel material. A position 100 mm away from the barrel breech wears very fast; this position is where the cone of the bullet is engraved in the barrel. At the position 150–350 mm away from the barrel breech, the barrel bore wears even faster. The barrel chrome layer is mainly affected by the gunpowder impact and projectile engraving, which is consistent with the actual failure of the coating. When the distance to the barrel breech is 350 m, the wear becomes stable. Through an analysis of the diameter of the barrel, it was found that, when the diameter of the barrel exceeded 12.85 mm, the barrel reached the end of its life.

1. Introduction

The decrease in gun barrel life is mainly caused by the combination of heat chemical factors and mechanical factors [1]. Mechanical wear, including impact wear and friction wear, is the most important mechanism for the change in the size of the barrel bore. When the projectile engraves in the rifling, it has a certain muzzle velocity, which will have an impact on the rifling origin and result in impact wear. When the projectile moves along the barrel bore, the driving side of the rifling is subjected to large positive pressure and relative slip speed, which intensifies the friction and wear. Surface damage to the barrel will directly change its structural parameters, resulting in an increase in the radial size of the barrel bore, which will affect the performance of the internal ballistic and limit the service life of the gun barrel. The test showed the following: the inside bore of the barrel was seriously damaged, the breech was mainly eroded, and the mouth experienced a lot of wear [2,3,4]. When the projectile quantity was 1200 rounds, the dimensional wear of the muzzle accelerated with the destruction of the breech of the barrel. Damage to the breech of the barrel leads to a shorter barrel life.

Using chrome coating in the barrel bore is an important method for improving the wear resistance of the barrel. The wear of the barrel can be divided into two stages: the wear stage of the chrome layer, and the wear stage of the gun barrel material. Regarding aspects of friction and the wear model, research into gun barrel wear has mainly focused on the prediction of gun barrel life. Dong-Yoon Chun et al. [5] were the first to carry out research on bore wear, and they proposed and developed a precision wear measurement system to measure the wear rate of a high-friction and high-pressurized gun barrel. Liqun Wang et al. [6] proposed a novel mechanical friction wear model affected by temperature changes. P. Sequard-Base et al. [7] presented a friction model that showed a correlation between bullet velocity and friction force in a barrel. Wei et al. [8] employed an experimental approach to investigate abrasive wear and adhesive wear mechanisms, and they primarily observed these mechanisms in 30CrNi2MoVA gun steel in an air environment during initial engraving. Bin Wu et al. [9] found that the strain rate and temperature have significant effects on the deformation behavior of rotating bands during the engraving process by performing quasi-static and dynamic push tests. Ji-sheng Ma [10] created an empirical formula for the number of rounds and the amount of barrel wear based on test data from the firing of large-caliber barrels. Shuli Li et al. [11] proposed a general surface damage computational model of a gun barrel that could predict the surface damage of mechanical structures caused by the coupling of thermochemical erosion and mechanical wear. Less research has been conducted on the wear of chrome on gun barrels. But in other mechanical fields, variations in the friction coefficient, mass loss, and surface morphologies of the tested samples were systemically investigated and analyzed [12,13,14]. Abdullah et al. [15] investigated the mechanical damage of a hard chromium coating as a function of the thickness of 416 stainless steel with plated chromium and investigated the mechanical damage of a hard chromium coating as a function of thickness. Miroslav et al. [16] conducted wear tests on a UMT TriboLab universal tribometer and analyzed and compared the wear conditions of 30CrNiMo8 and 42CrMo4 steel in contact with G40 ball bearings with a diameter of 4.76 mm.

In the shooting process of a small-caliber barrel, due to the serious wear of the gun barrel, a large amount of the propellant gas leaks, leading to a drop in the muzzle velocity and the accuracy of the projectile. Metal patch bullets further increase the wear on the barrel bore. Therefore, research on the friction and wear of the barrel with the projectile at high temperatures, high speeds, and high pressure, and the establishment of the barrel wear model, have theoretical guiding significance for the development of a wear-resistant barrel suitable for high-pressure firing conditions.

2. Archard Wear Model

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).

$$V=k{\displaystyle \frac{pl}{H}}$$

(1)

In the formula, $V$ is the wear volume (g); $k$ is the dimensionless wear coefficient; $p$ is the normal load (N); $l$ is the slip distance (mm); and $H$ 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):

$$m={\displaystyle \frac{\rho k}{H}}pl$$

(2)

The above formula can be further amended to produce Formula (3).

$$\dot{w}=k\cdot v_{\mathrm{s}}^{\alpha }\cdot p^{\beta }\cdot H^{\chi }$$

(3)

In the formula, $\dot{w}$ is the wear rate of the barrel material (g/min); $v_{\mathrm{s}}$ is the relative sliding speed; and $k$, $\alpha$, $\beta$, and $\chi$ 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. Hardness of hexavalent chromium at different temperatures.

Temperature/°C	25	100	200	300	400	500	600	700
Hardness/Hv	840	1040	1015	1012	889	761	703	443

.

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

$$H=3.424-06T^3-0.00626T^2+2.221T+813.6$$

(4)

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. High-temperature wear experiment parameters.

Number	Temperature	Load	Rotational Speed	Linear Velocity
1	550 °C	400 N	191 r/min	200 mm/s
2	400 °C	400 N	191 r/min	200 mm/s
3	250 °C	400 N	191 r/min	200 mm/s
4	550 °C	100 N	191 r/min	200 mm/s
5	550 °C	200 N	191 r/min	200 mm/s
6	550 °C	400 N	382 r/min	400 mm/s

.

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. Experimental wear amount.

Number	Temperature	Load	Linear Velocity	Lost Mass/g
1	550 °C	400 N	200 mm/s	−0.0022
2	400 °C	400 N	200 mm/s	−0.0020
3	250 °C	400 N	200 mm/s	−0.0011
4	550 °C	100 N	200 mm/s	−0.0050
5	250 °C	100 N	200 mm/s	−0.0032
6	400 °C	100 N	200 mm/s	−0.0041
7	550 °C	200 N	200 mm/s	−0.0199
8	250 °C	200 N	400 mm/s	−0.0182
9	400 °C	200 N	400 mm/s	−0.0221
10	550 °C	400 N	400 mm/s	−0.0130

. 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.

$$h={\displaystyle \frac{m}{\rho S}}$$

(5)

In the formula, $h$ is the wear depth, $m$ is the coating change quality, $\rho$ is the coating density, and $S$ is the wear area.

According to Formula (3), the coefficient $k$, $\alpha$, $\beta$, $\chi$ 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 $k=4.3437\times {10}^{-14}$, $a=3.9458$, $b=0.1459$, and $\chi =-0.0019$.

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: $k=2.7184\times {10}^{-13}$, $a=2.9483$, $\beta =1.5344$, $\chi =-0.0019$.

Therefore, the Archard wear model of the chrome coating of the gun barrel is obtained following Equation (6):

$${\dot{w}}_m=4.3437\times {10}^{-14}\cdot v_{\mathrm{s}}^{3.9458}\cdot p^{0.1459}\cdot H{(t)}^{-0.0019}$$

(6)

The Archard wear model of the gun barrel matrix material is shown in Equation (7).

$${\dot{w}}_m=2.7184\times {10}^{-13}\cdot v_{\mathrm{s}}^{2.9483}\cdot p^{1.5344}\cdot H{(t)}^{-0.0019}$$

(7)

In the above formula, ${\dot{w}}_m$ is the wear rate (g/min), $v_{\mathrm{s}}$ is the relative sliding speed, and $H(t)$ 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. Amount of wear calculated by the fitting formula.

Number	Hardness/Hv	Pressure/Pa	Linear Velocity/(m/s)	Lost Mass/(m/s)	Wear Rate Calculated by Fitting Formula/(m/s)	Error × 100%
(a)	730	1.0742 × 106	0.2	7.4444 × 10−6	6.7778 × 10−6	−9.0814
(b)	889	1.0742 × 106	0.2	7.1111 × 10−6	6.7778 × 10−6	−3.7693
(c)	1015	1.0742 × 106	0.2	6.2222 × 10−6	6.7778 × 10−6	9.0339
(d)	730	2.6857 × 106	0.2	2.6111 × 10−5	2.7667 × 10−5	5.9733
(e)	730	5.3714 × 106	0.2	8.0556 × 10−5	8.0222 × 10−5	−0.3115
(f)	730	1.0742 × 106	0.2	5.2667 × 10−5	5.2444 × 10−5	−0.5391

. From Equations (6) and (7), according to the values of $\alpha$, $\beta$, and $\chi$, 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.

3. Results of Calculations

3.1. Friction Velocity Calculation

During the interior trajectory period, the friction velocity of each part of the barrel is different from that of the projectile. The friction velocity can be calculated from the axial velocity of the projectile along the barrel and the twist angle of the rifling.

$$v_f=v/\cos \alpha$$

(8)

In the formula, $v_f$ is the friction velocity, $v$ is the projectile speed along the axis direction, and $\alpha$ is the rifling angle.

According to the calculation of the internal trajectory, the projectile muzzle velocity (that is, the velocity change curve along the axis direction) can be obtained and the friction velocity change curve can be calculated according to the change in rifling angle, as shown in Figure 11.

3.2. Contact Pressure in Barrel Rifling

The force law at rifling was obtained according to the finite element calculation results, as shown in Figure 12.

At the same cross section position, the contact pressure of the rifling driving side was the largest, because the rifling driving side bears the torsional action of the projectile directly; the maximum contact pressure of the rifling driving side was about 248 MPa.

In order to more intuitively understand the force law of rifling, we selected the path (gun barrel muzzle) shown in Figure 13a, where the rifling along the ring normalized distance (where the normalized distance refers to the circumference of the rifling) is composed of six rifling lines. We drew the contact pressure distribution curve on the path at which the projectile passes through this cross section, as shown in Figure 13b.

When the projectile moves along the barrel, the contact pressure distribution of the rifling changes in an oscillating manner. The high-speed contact collision between the projectile and the barrel results in an uneven force distribution for rifling, but the contact pressure curve of each rifling groove has the same trend (there were six rifling grooves in the barrel). After further processing, the contact pressure distribution curve of a single rifling groove was obtained, as shown in Figure 14.

As can be seen from Figure 14, the contact pressure on the rifling driving side is the largest, and the rifling contact pressure increases from the non-driving side to the driving side. After the projectile experiences a groove caused by the armor material, the excess material is squeezed into the rifling groove, forming an extrusion of the rifling groove, resulting in greater contact pressure in the middle of the rifling groove than on either side of it. The analysis of force distribution in rifling shows that rifling geometry is the key factor affecting rifling force, and rifling force can be optimized by improving the rifling structure. Figure 15 shows the wear morphology of a mid-life gun barrel. From the view of cross section morphology, the chromium layer on the rifling land and groove is thinned, the positive line is thicker than the Cr layer, the thickness of the Cr layer is about 40 μm, and the coating on the side of the positive line is worn to be flush with the negative line, exposing the matrix.

Since the rifling changes in spatial geometry, the rifling forces are further processed along the axis and radial direction of the barrel to obtain the spatial distribution of rifling forces in the barrel, as shown in Figure 16.

As can be seen from the figure, along the axial direction of the barrel, the closer to the muzzle, the greater the force on the rifling and the more intense the wear. In the process of projectile launching, as the chamber pressure increases, the projectile accelerates. At the muzzle position, the projectile velocity is the largest, and the slip velocity between the projectile and the barrel is also larger. The gun barrel is similar to the cantilever beam structure when shooting, and the muzzle vibration attitude is large, which intensifies the rifling force.

3.3. The Pattern of Barrel Wear

According to the wear morphology at the end of the barrel, as shown in Figure 17, and according to the force analysis of rifling, the friction velocity changes along the barrel axis and the friction velocity is the same everywhere along the ring of the same section. The material hardness varies with the temperature and the temperature distribution of each section of the barrel is different, but the temperature of the same section of the barrel is roughly the same. In addition, the contact pressure changes along the barrel axial direction, and the contact pressure of different sections has the same distribution trend along the inner wall circumferential direction: the contact pressure of the driving side is the largest, the pressure in the middle of the rifling groove is larger, and the contact pressure of the rifling land decreases from the driving side to the non-driving side. Therefore, the annular wear variation in each section is only related to the distribution of contact pressure, and the contact pressure distribution trend of each section is the same, so the wear morphology evolution of each section is also the same. The wear morphology of the gun barrel at different life stages can be represented according to the wear morphology of the gun barrel muzzle. The calculated rules are in agreement with the actual phenomena, which shows the accuracy of the research method in this paper.

Through the analysis of the barrel wear morphology at the middle and end of life, the calculated results are in good agreement with the actual phenomena, which indicates the accuracy of the research method in this paper.

Based on the Archard wear model of the chrome layer and the finite element model of the interaction between the projectile and the barrel, the variation trend of the contact force along the axis was analyzed, and the wear rate from the middle of the barrel to the barrel muzzle was calculated. The calculation results are shown in

Table 5. The average wear rate of the coating from the middle to the barrel muzzle sections.

Number	Distance from the End of the Barrel/mm	Projectile Velocity/m/s	Pressure/MPa	Number of Shots When Rifling Is Worn Out/Rounds	Average Wear Rate/(mm/Rounds)	Average Wear Rate/(mm/s)
1	200	659.57	118.55	19,425	8.6104 × 10−10	1.3645 × 10−8
2	250	701.50	105.86	17,833	9.4756 × 10−10	1.2829 × 10−8
3	300	735.11	96.83	14,261	1.2239 × 10−9	1.2242 × 10−8
4	350	762.99	85.19	13,781	1.2736 × 10−9	1.1795 × 10−8
5	400	786.73	76.91	13,137	1.3472 × 10−9	1.1439 × 10−8
6	450	807.39	70.44	12,302	1.4562 × 10−9	1.1147 × 10−8

.

It can be seen from Table 5 that the distribution of the velocity and contact pressure of the projectile through different cross sections of the barrel presents an opposite trend. From the middle of the barrel to the muzzle, the projectile velocity increases, the contact pressure decreases, and the average wear rate increases, indicating that the friction velocity has a strong influence on the wear rate; this is the same as the result reflected in the pin disk wear experiment.

The wear rate from the middle of the barrel to the muzzle was also calculated according to the Archard wear model of the gun steel matrix material, and the calculated results are shown in

Table 6. Average wear rate of the substrate at each cross section from the middle to the muzzle of the barrel.

Number	Distance from the End of the Barrel/mm	Projectile Velocity m/s	Pressure/MPa	Number of Shots When Rifling Is Worn Out/Rounds	Average Wear Rate/(mm/Rounds)	Average Wear Rate/(mm/s)
1	200	659.57	118.55	unworn	—	—
2	250	701.50	105.86	21,133	4.13 × 10−9	5.59 × 10−8
3	300	735.11	96.83	17,766	4.67 × 10−9	4.67 × 10−8
4	350	762.99	85.19	17,766	5.19 × 10−9	4.81 × 10−8
5	400	786.73	76.91	17,766	5.59 × 10−9	4.75 × 10−8
6	450	807.39	70.44	17,766	9.09 × 10−9	6.96 × 10−8

.

Taking the number of projectiles and the axial position of the barrel as sample factors, two-dimensional fitting was carried out on the change data of the barrel size, and an array of changes in the barrel size with the number of projectiles and the axial position of the barrel was obtained, as well as the trend in the size change on one side of the barrel, as shown in Figure 18.

As can be seen from Figure 18, the barrel size changes are most obvious in the 0–50 mm and 200–463 mm segments from the barrel breech, while the barrel size changes are not obvious in the 50–200 mm segment from the barrel breech. The change rule of barrel wear is as follows: the initial wear changes slowly, and when the wear amount reaches 0.02 mm, the expansion of the barrel size is accelerated because the coating and the substrate material are different (the corresponding wear rate is also different). The coating thickness of the small-caliber rifle studied in this paper was 0.02 mm, which can explain the phenomenon whereby, after the coating had worn out, the wear rate of the base material became faster, resulting in accelerated dimensional change.

The dimensions of the barrel bore in the middle and end of life were measured with an actual fire test. The calculated values at the middle and end of barrel life were compared with the tested values, as shown in Figure 19. At the middle and end of life of the barrel, the calculated rifling depth of each section was close to the measured value, and the dimension change curve was in good agreement with the measured curve. According to the actual projectile quantity, the predicted projectile number corresponding to the complete rifling wear in the middle of the barrel and the muzzle was close to the measured projectile number.

4. Conclusions

Based on the Archard wear model, combined with a wear experiment, shooting test, and simulation calculation, the force and rifling wear rule of small-caliber barrel rifling were studied, and the following conclusions were obtained.

• Based on the finite element interaction model of the barrel and the projectile, the force law of rifling was obtained. The contact pressure of rifling was the largest on the driving side and increased from the non-driving side to the driving side. The contact pressure in the middle of rifling was larger than that on both sides of rifling.
• For the rifling friction and wear, the barrel structural size changed most obviously in the 0–50 mm distance from the barrel breech and the 200–463 mm distance from the barrel breech, while the barrel’s structural size changes were not obvious in the 50–200 mm distance from the barrel breech. Due to the different mechanical properties of the coating and the base material, the corresponding wear rate was different. After the coating becomes worn, the base material wears faster, resulting in changes to the barrel rifling size and accelerated wear.
• In order to improve the barrel life, optimizing the geometric structure of the rifling can reduce the forces on the rifling driving side, reduce the rifling contact pressure, and thus reduce the rifling wear.