Strain concentration caused by the closed end contributes to cartridge case failure at the bottom

Song Cai, Jie Feng , Hui Xu, Kun Liu, Zhong-xin Li, Zhi-lin Wu

School of Mechanical Engineering, Nanjing University of Science and Technology, Nanjing, 210094, Jiangsu, China

Keywords:Closed end Cartridge case failures Strain concentration Bending deformation

A B S T R A C T Ruptures at the bottom of cartridges are a common cause of failure of ammunitions, which directly threatens the safety of weapons and shooters.Based on plastic tube theory,this study analyses the radial and axial deformation of a cartridge,considering the radial constraint of the closed end at the bottom of the cartridge. Owing to the influence of the closed end, the bottom of a cartridge does not establish complete contact with the chamber. Owing to strain concentration in the non-contact area, this area is more amenable to the occurrence of cartridge rupture. This theory predicts the location of the fracture more accurately than the traditional theory. The maximum axial deformation of a cartridge comprises bending and friction deformation.The maximum strain at the bottom of the cartridge increased by 135%owing to the introduction of bending strain caused by the closed end. The strain distribution of a cartridge was measured using digital image correlation technology,and the measured result was consistent with the predicted results of the bending deformation theory and rupture case. The effects of wall thickness, radial clearance, friction coefficient, and axial clearance on the axial deformation of the cylinder were studied. Increasing the wall thickness and reducing radial clearance were found to reduce bending deformation; furthermore, lubrication and reduction in axial clearance reduce frictional deformation, which in turn reduce cartridge rupture.

1. Introduction

A typical cartridge case of firearm ammunition packaging comprises a projectile,propellant substance,and primer.The metal or plastic cartridge case is precisely made to fit within the barrel chamber of a breech loading gun for the practical purpose of convenient transportation and handling during shooting. Existing research regarding lightweight cartridges demonstrates that cartridge case head separation is a common universal phenomenon.Irrespective of the material (aluminium alloy [1], brass [2,3], steel[4],etc.)used,head separation results in damage to the firearm and injury to the shooter.

The study of cartridge deformation is the basis for the optimization and lightweight design of a cartridge structure.A reasonable structure can reduce the weight of the cartridge and improve the carrying capacity of individual ammunition,thereby improving the survivability of soldiers on the battlefield under the premise of avoiding serious failure of cartridges.The deformation process of a cartridge in the chamber is very complicated. Over the last few decades,a significant number of studies have investigated cartridge case deformation under blast loading. Carlucci et al. [5] used the combined thin-walled cylinder theory to analyse the radial deformation of cartridges and calculated the minimum radial clearance of a cartridge.Wang[6]calculated the contact pressure and friction between a cartridge and the chamber, and verified the shell extraction resistance of the cartridge through experiments. The contact pressure cannot accurately predict the thicker region at the bottom of the cartridge because a thin-walled cylinder is assumed.Zhao [7] obtained the plastic strain of the cartridge by measuring the strain on the external surface of the cartridge chamber and simulating it.However,owing to the indirect measurement method used,the obtained result had some deviation.The priming process of a cartridge under the action of gunpowder gas can be simplified to the plastic deformation process of the combined tube with gap.A typical application of this process is to the development of a tubesheet, which is a hydraulically expanded tube. Bouzid [8-10],Huang[11]and Laghzale[12,13]calculated the contact pressure and residual stress between the elastoplastic tube sheets by thickwalled cylinder theory. The calculated results were verified by experiments [14] and simulation results [15]. Tube-to-tubesheet theory can accurately calculate the contact deformation of thickwalled tube; however, it is generally used in open-closed models and,thus,ignores axial deformation.Many scholars believe that the failure of a cartridge is related to the axial deformation. The relationship between the length of the cartridge and the maximum axial stress is calculated using the thin-walled cylinder theory in the literature [16]; it is considered that the axial stress of the cartridge is greater than the yield strength of the material when the cartridge is broken. Because this theory adopts an elastic-plastic material model and ignores the initial axial gap between cartridge and chamber it is often used to estimate the limit length of the cartridge.Ou[17]calculated the axial stress on each section of the cartridge,a model which considered the axial clearance and the plastic deformation of cartridge. The rupture of the cartridge is determined when the axial deformation of the cartridge is greater than the axial clearance. The fracture position of the cartridge predicted by Ou’s theory, is moderately different from the actual position. Additionally, to theoretical analysis, many experiments have been proposed to solve the fracture issue.Minisi[2]believed that friction was the main reason for the rupture of the cartridge,and reasonable lubrication could reduce the rupture of the cartridge. Tasson [18] solved the rupture problem of the aluminium alloy cartridge on M16A3 by increasing the thickness of the bottom.This method could not solve the rupture problem of the cartridge on M249, and it sacrificed the initial velocity of ammunition and increased the mass of the cartridge. Sharma [3] concluded that stress corrosion cracking of brass caused the rupture of the cartridge;however,he could not explain why the crack of the cartridge was concentrated at the bottom of the cartridge.Cai[1,19]and Luo[20] believed that the axial clearance was the main reason for the failure of the cartridge and experimentally proved that the failure could be significantly reduced by reducing the axial clearance.Although the existing study analyses the effects of various factors on the failure of the cartridge and present improvement measures,there is no unified theory to explain the concentration failure of the cartridge at the bottom.

In this study, the plastic tube theory is used to analyse the deformation of the cartridge, and the axial strain distribution on the surface of the cylinder is calculated under the constraint of the closed end; furthermore, the phenomenon of strain concentration at the bottom of the cartridge is explained. The theoretical results were verified by finite element analysis and digital image correlation (DIC) residual strain measurement in the cartridge. The new theory is used to discuss the traditional methods of increasing wall thickness,lubrication,reducing radial clearance,and reducing axial clearance to reduce the rupture.

2. Theoretical calculation

2.1. Fundamental assumptions

The deformation and movement of a cartridge in a gun chamber under the influence of propellant gas is complex.Under the action of pressure, the cartridge undergoes radial displacement, which eliminates the radial clearance and ensures that the cartridge fits within the cartridge chamber. Owing to the radial displacement constraint at the closed end at the bottom of the cartridge, the bottom of the cartridge cannot produce radial displacement to establish complete contact with the chamber. According to the continuity principles, the cartridge near the bottom is radially deformed by the effect of bending in the bending expansion,gradually eliminating the initial radial gap.The non-contact zone is called the bending zone, as shown in Fig. 1. In addition to the bending stress,the bending zone is subjected to bending and axial tensile stress in a very short range, and is likely to result in strain concentration, which causes cartridge failure. For ease of application and discussion of the equations, the following assumptions have been made in this study:

Fig.1. Simplified model of cartridge and chamber.

1) The cartridge and chamber are cylinders with uniform wall thickness; the cartridge is an axisymmetric, thick-walled cylinder, while the chamber is an inner conical tube; the shoulder and neck of the cylinders are neglected.

2) The cartridge and chamber materials are isotropic;the cartridge is a bilinear material, while the chamber is a theoretical elastoplastic material.

3) Only the process of increasing pressure is considered, because the maximum plastic strain occurs at maximum pressure.

4) Only the axial deformation of the cartridge is considered and the axial movement of the cartridge ignored.

2.2. Calculation model

According to the basic assumption, the deformation of the cartridge in the chamber can be simplified as follows: The inner and outer radius of the cartridge is riand ro,respectively.Furthermore,its full length is l,bottom thickness is l4,elastic modulus is Et,yield stress is σs, and tangent modulus is Ett. The Poisson’s ratio of the elastic and plastic stages of the cartridge is vtand vp, respectively.The inner and outer radius of the chamber is Riand Ro,respectively,and its Poisson’s ratio is vs.The initial radial clearance between the cartridge case and the chamber is Δr,and the axial clearance at the bottom of the cartridge is Δ.The coordinate system is established as follows:The bottom coordinate of the cartridge is 0,and the Y-axis points to the mouth of the cartridge. The model used for the calculation is shown in Fig.1.

2.2.1. Axial strain before closing of the initial radius clearance

Uniform expansion pressure is applied to the inner surface of the cartridge, which causes the cartridge to expand and deform.Because the influence area of the closed end is limited [15], the elastic-plastic calculation formula[21]can be used to calculate the pressure of the cartridge entering the complete plastic stage, Pat

In the elastic range, the cartridge outer radius displacement is given by Lame’s equations:

where Peis the pressure in the cartridge.

Before closing the initial clearance, the pressure is increased beyond Pat, the cartridge is in the plastic range, the cartridge deformation are the same as the elastic range,with Young modulus Etreplaced by the tangent modulus Ett,the Poisson’s ratio vtby the Poisson ratio of plastic vp= 0.5, the pressure Peby Pe- Pat, and displacement u by u-uatin Eq. (2), the displacement is thus expressed by Ref. [12]:

Replacing Peby Pl,and u by Δr,the pressure when closing of the initial radius clearance Plis given by:

where Δris the initial radius clearance.

The chamber is only affected by the internal pressure in the axial direction before closing the initial clearance; thus, the axial strain caused by the axial pressure is

According to Lame’s equations, the axial strain caused by circumferential deformation and radial deformation is

Combining Eqs.(5)and(6),the axial strain before closing of the initial radius clearance ε1is

The axial deformation before closing of the initial radius clearance is

where l1is the depth of cartridge,or the difference between the full length and the bottom thickness, l1= l- l4.

2.2.2. Axial strain caused by the friction

According to the elastic-plastic calculation formula [12], the contact pressure between the chamber and the cartridge(Pc) is

When there is no constraint at the bottom of the cartridge, the cartridge is only affected by the mouth constraint, internal pressure, and friction. Using the pressure from the initial clearance to replace the mouth constraint pressure, according to the force balance equationthen the length of the sliding friction of the cartridge can be obtained as

The maximum axial strain caused by sliding friction is

The coefficient of the sliding friction strain replaced by coefficient of the static friction strain in Eq. (11), the axial strain caused by static friction is

where a′is the coefficient of the static friction strain.

2.2.3. Axial strain cause by the bending

Owing to the introduction of the end face, the bottom of the cartridge cannot produce radial displacement and produces bending deformation. The micro-element is obtained from the bottom of the cartridge and the angle of the micro-element is dθ.The cross and longitudinal sections of the micro-element are shown in Fig. 2. On the micro-element, the deformation at the bottom of the cartridge can be simplified as the bending deformation of the cantilever beam.

The uniformly distributed load on the micro-element is dq =Peridθ, which can be integrated to obtain a uniform load on the bending section as

According to bending beam formula [22], the bending displacement of the cylinder can be obtained as

Because the cartridge is in a completely plastic state, the bending displacement of the cylinder can be obtained as

Knowing that the maximum strain position is near the bottom of the cartridge,and the displacement on the external surface of the cartridge is the same as the initial clearance, ut= Δrcan be substituted into Eq.(16)and the bending length of the cartridge is

Because the chamber can be elastically deformed, there is a transition section on the cartridge, and the radial displacement of cartridge gradually transitions from Δrto Δr+ us. Because the stress of the transition section is complex and difficult to calculate,the length is assumed to be lwand contact pressure of the segment is uniform at 0.5Pc; thus, the bending moment of the maximum bending point isand the axial strain[22]caused by bending is

2.2.4. Axial strain of the cartridge

The bending section of the cartridge is still subject to axial tension; thus, the maximum axial strain on the cartridge is,

The axial deformation of the bending section is,

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The axial deformation of the friction section is,

The axial deformation of the cartridge satisfies the deformation coordination equation

Combining Eqs. (12), (20)-(22),

We can obtain the axial strain at any position on the outer surface of the cartridge using,

Fig. 2. Bending deformation of the micro-element near the bottom of the cartridge (a - longitudinal section; b - cross section).

3. Validation using finite element modelling

Based on the basic assumptions, the 12.7 mm cartridge is simplified.The simplified structure and material parameters of the cartridge and chamber are listed in Table 1.

The axisymmetric plane element PLANE183 with eight nodes were used to model the cartridge and the chamber, and the axial displacement of the chamber was constrained, as shown in Fig. 3.The contact between the cartridge and the chamber was calculated using the penalty function method, and the maximum pressure was 300 MPa; furthermore, a transient analysis was conducted using a non-load method, and the calculation time was 0.8 ms.

By integrating Eq.(24),an axial displacement at any position on the surface of the cartridge can be obtained,as shown in Fig.4.The axial deformation of the cartridge is concentrated in the area of bending and friction. According to Eqs. (10) and (17), the total length of the bending and friction area is 21.28 mm, length of the axial deformation caused by bending and friction is 0.353 mm,and 88.25% axial deformation occurred in 1/5 of the cartridge length.Therefore, in conclusion, the friction and bending deformation together eliminate the axial clearance;the axial deformation of the cartridge is concentrated in the area of bending and friction deformation.

Table 1 Geometry and material properties.

Fig. 3. Cartridge case and Chamber FE models.

Fig. 4. Distribution of axial displacement on the outer surface of cartridge with Y coordinate.

Under the action of pressure,the cartridge is deformed radially and fully contact the cartridge chamber, as shown in Fig. 5, where the Y-coordinate is greater than 12.6 mm. Considering the radial constraint of the closed end, there is an evident non-contact zone(Contact status = non-contact) near the bottom of the cartridge,and the maximum strain of the cartridge is located at the intersection of the non-contact segment and the contact segment,which is consistent with the results of theoretical analysis, as shown in Fig. 5.

Fig. 5. Distribution of axial strain and contact status on the outer surface of cartridge with Y-coordinate.

The axial strain of the cartridge is concentrated near the bottom of the cartridge,as shown in Fig.5.According to the FEM results,the cartridge in the Y-coordinate section,15-32 mm,was subjected to friction stress, and the friction strain on the surface presented an approximate linear rule while maximum axial strain in this section was 2.78%. Theoretically, the length of the friction section is 17.7 mm, while the maximum axial strain is 2.53%.

In the FEM results, the Y-coordinate 11-15 mm axial strain increased sharply; the maximum strain was 6.20%, and the strain increased by 3.42%. According to theoretical analysis, the bending strain of the cartridge in the non-contact section is 3.43% and maximum strain is 5.96%. The simulation results are consistent with the theoretical results,as shown in Fig. 5.

The conventional theory believes that the axial strain of the cartridge is caused by friction [16,17]. Compared with the traditional theory, the maximum axial strain is 2.53%. However, owing to the introduction of the bending strain,the maximum axial strain of the cartridge increased from 2.53%to 5.96%,an increase of 135%,which was closer to the simulation calculation of 6.2%. Bending deformation is an important part of the maximum strain. The bending deformation near the bottom of the cartridge intensifies the concentration of strain near the bottom, resulting in the transverse fracture of the cartridge.

4. Experimental verification

The deformation of the cartridge in the chamber cannot be measured directly.In this study,the DIC(Digital Image Correlation)principle is used to measure the axial residual deformation of the cartridge. By comparing the variation in the speckle before and after the shooting, the axial residual strain of the cartridge can be calculated and tested on a 12.7 mm ballistic gun with the 12.7 mm × 108 mm steel cartridge. To ensure that the relative position of the cartridge before and after deformation does not change significantly,a special cartridge fixture was designed to fix the position of the cartridge, as shown in Fig. 6 below.

As shown in Fig. 7(a), as measured by DIC test, the maximum axial residual strain of the cartridge is near the bottom of the cartridge, and there is an evident strain concentration area in which the strain increases sharply.

The concentrated area of strain is consistent with the area of maximum axial strain(shown in Fig.7(b))in finite element analysis and is consistent with the bottom rupture position of the cartridge(Fig. 7(c)). The rupture position of the cartridge and the strain concentration point are a certain distance from the inner plane of the cartridge, which is moderately different from the traditional theory.Owing to the introduction of the bending deformation,the maximum axial strain position of the cartridge has moved further away from the bottom, while the bending deformation increased the concentration of axial strain of the cartridge, resulting in transverse fracture of the cartridge.

5. Results and discussion

An existing analysis solved the problem of the rupture of the cartridge to some extent, by increasing the thickness of the cartridge wall [18] and reducing the radial clearance [16], friction [2],as well as axial clearance [1]. This study investigates the internal relationship between these factors and the rupture of the cartridge through theoretical analysis and simulation analysis.

Fig. 6. Measurement of axial strain of cartridge by DIC. (a-relative position of cartridge and camera; b-state of cartridge on fixture).

Fig. 7. Distribution of axial strain and rupture of cartridge. (a-distribution of axial strain of cartridge measured by DIC test; b-distribution of axial strain of FEM; c-rupture of cartridge).

According to the traditional theory [16], the initial radial clearance of the cartridge is 0.1-0.2 mm, and with an increase of the radial clearance,the maximum strain of the cartridge increases,as shown in Fig. 8(a). When the radial clearance increased from 0.05 mm to 0.25 mm, the maximum axial strain of the cartridge increased from 5.64% to 6.52%. With the increase of radial clearance, the bending deformation near the bottom of the cartridge also increased, the bending strain increased from 2.50% to 4.91%,and the proportion of bending strain in the maximum strain increased from 46.58%to 72.00%,as shown in Fig.8(b).An increase of radial clearance reveals strain concentration near the bottom of the cartridge.The radial deformation of the cartridge increases with radial clearance; thus, the probability of transverse fracture increases with radial clearance.

Fig. 9. Variation in axial strain with thickness. (a-Variation in maximum strain,bending strain and friction strain with thickness; b - variation in bending strain/maximum strain with thickness).

When the thickness of cartridge increases from 1.5 mm to 2.1 mm,the maximum axial strain on the cartridge decreases from 6.27% to 5.62%. When the thickness of the cartridge increases, the bending strain of the cartridge decreases from 3.87%to 2.90%, and the proportion of bending strain in the maximum strain decreases from 60.74% to 53.10% and the friction strain of the cartridge remains constant, as shown in Fig. 9.

The method of increasing wall thickness reduces the bending deformation near the bottom of the cartridge, reducing the maximum strain near the bottom of the cartridge and,thus,solving the fracture phenomenon of the aluminium alloy cartridge on M16A2 [16].

According to Eq. (19), the bending strain of the cartridge is independent of the friction coefficient and axial clearance.When the axial clearance increases from 0.25 mm to 0.45 mm,the maximum axial strain of the cartridge increases from 4.54%to 6.43%,bending strain remains constant,and friction strain increases accordingly,as shown in Fig.10. As shown in Fig.11, when the friction coefficient increases from 0.1 to 0.5,the maximum axial strain of the cartridge increases from 4.14% to 7.86%, while the bending strain remains constant and the friction strain increases accordingly. The method for reducing axial clearance and friction significantly reduces the friction strain on the surface of the cartridge, reducing the maximum strain in the strain-concentrated area,and thus,avoiding the rupture of the cartridge.

6. Conclusions

Fig. 10. Variation in maximum strain, bending strain and friction strain with axial clearance.

In this study, the plastic tube theory is used to analyse and calculate the radial deformation and axial deformation of the cartridge considering the influence of the closed end constraint on the deformation of the cartridge. Owing to the radial constraint of the end face,there is a non-contact zone at the bottom of the cartridge.The bending deformation of the non-contact zone causes strain concentration at the bottom. Under the combined action of frictional deformation and bending deformation, the maximum axial strain in the non-contact zone causes the rupture of the cartridge.FEM,DIC residual strain test,and rupture cases all verify the strain concentration phenomenon in the theoretical analysis.Compared with traditional theory,this bending deformation theory more accurately predicts the rupture position. By comparing the theoretical and simulation studies on radial clearance, axial clearance,wall thickness,friction,and other factors,this theory suggests that reducing the axial friction strain by reducing friction and axial clearance can solve the problem of cartridge rupture.The bending strain can be lowered by reducing radial clearance and increasing thickness, while the concentration of strain can be decreased by reducing the axial bending strain and friction strain. The scope of application of existing working calculation is limited and further efforts are being invested to expand its scope of application.

Fig.11. Variation in maximum strain, bending strain and friction strain with friction coefficient.

Acknowledgements

The authors are grateful for the reviewers’ instructive suggestions and careful proofreading. This work was supported by the Equipment Development Department of the Central Military Commission of China (grant nos. 301090702), and the Foundation of National Laboratory (grant nos. 61426060102162606005 and JCKYS2019209C001).