Modeling and dynamic simulation on engraving process of rotating band into rifled barrel using three different numerical methods

2.1. Geometry of gun barrel and projectile

The schematic diagram of the rifled barrel is shown in Fig. 1(a). $D_3$ is the chamber diameter and $L_{Fc}$ is forcing cone length. The cross-sectional view of the rifling is shown in A-A of Fig. 1(a). The lands are considered as trapezoid and the grooves are considered as rectangle. The number of lands in the studies is 36, the main parameters include the width of the land $a_L$, the width of the groove $b_G$, the depth of the groove ${\mathrm{t}}_{\mathrm{D}}$, the land diameter $D_2$ and the groove diameter $D_1$. The structure of the rotating band is shown in Fig. 1(b). The main parameters are the diameter of the rotating band $D_r$ and the length of the rotating band $L_r$.

2.2. Engraving process

The engraving process is shown in Fig. 2. At the beginning, the projectile is loaded to its designated position, in Fig. 2(a). When the firing starts, as propellant gas pressure in the chamber rises, the projectile starts to move. As propellant gas pressure increases, the rotating band is subjected to plastic deformation and is engraved partly into the rifling, in Fig. 2(b). It indicates the end of the engraving process when the rotating band is completely engraved into the rifling, in Fig. 2(c).

Fig. 1a) The structures of rifled barrel and b) rotating band

a)

b)

Fig. 2Rotating band engraving process

a)

b)

c)

2.3. Basic assumptions

According to the main features of the engraving process, some assumptions are made to simplify the numerical calculation [12]:

1) There do not have initial stress and deformation on the rotating band when it is loaded into its designated position.

2) The aerodynamic force in front of the projectile, the dynamic unbalance and the gravity of the projectile are neglected. The recoil of the gun tube is neglected.

3) The influences of the thermal field are ignored.

3.1. Plastic model of rotating band

The Johnson-Cook plastic model [13] is used to describe the plastic behavior of the rotating band, its expression is Eq. (1):

where ${\epsilon }_p$ and ${\dot{\epsilon }}_p$ are the effective plastic strain and the effective plastic strain rate respectively, ${\dot{\epsilon }}_0$ is the reference strain rate, $T$, $T_r$, $T_m$ are respectively the actual temperature, room temperature and the melting temperature in the absolute scale, and $A$, $B$, $C$, $m$, $n$ are material constants.

3.2. Failure criteria of rotating band

The Johnson-Cook failure model [14] is used to perform the material failure of the rotating band. The effective failure strain is used to determine the damage parameters. The expression is shown by Eq. (2):

where $d_1\text{,…,}{d}_5$ are the constants of material failure, ${\sigma }^{\mathrm{*}}$ is the stress triaxiality defined by ${\sigma }^{\mathrm{*}}={\sigma }_H/{\sigma }_{eq}$ in which ${\sigma }_H$ is the hydrostatic stress and ${\sigma }_{eq}$ is the equivalent von Mises stress.

The damage accumulation is expressed by Eq. (3):

where $D$ is the damage parameter to material elements. Failure happens once $D=\text{1}$, and in the FEM simulations element erosion is used to remove elements that have reached the critical damage. The Johnson-Cook plastic and failure constitutive model parameters for CuZn10 brass are shown in Table 1.

4.1. Element mesh

Eight node hexahedral elements are used to build finite element model which has been shown in Fig. 3 is meshed by the HyperMesh software. Numerical simulations are performed by ABAQUS. The rotating band and the front of the rifling are composed of deformable elements with a finer mesh. The width of the band is 22 mm. There is a total of 582,830 elements, and the rotating band has 479,412 elements. This model will be used during the studies of the engraving process except CEL analysis. In CEL analysis, as a Eulerian section, the rotating band must be built in ABAQUS [15].

Fig. 3Mesh of finite element model

4.2. Contact between the rotating band and the tube

The contact between the rotating band and the inner wall of the tube is modeled using a *GENERIAL CONTACT algorithm available in ABAQUS. The implementation of Eulerian-Lagrangian contact is an extension of general contact in Abaqus/Explicit. The finite element models are solved by the explicit integration algorithm. For purpose of promoting the increase stability and efficiency of the calculation, the single point integration and the stiffness hourglass control methods were introduced.

4.3. Loads and boundary conditions

As is known to all, the projectile is pushed by pressure of combustion gas during live firing. This pressure is the main driving power during internal phase. The projectile base pressure is given by Fig. 4. It was obtained by live firing. Projectile base pressure is loaded as *PRESSURE on the base of the projectile in ABAQUS.

According to the assumptions presented in Section 2.3, the front end face of the barrel is fixed.

Fig. 4Projectile base pressure

5.1. FEM-SPH adaptive coupling method

The SPH method was first proposed by L. B. Lucy [16]. Since then, the computational cost has still been a significant problem. The FEM-SPH adaptive coupling method [17] can take advantage of the intrinsic strengths of both Lagrangian finite element and SPH methods when modeling a body. We can define the model with Lagrangian finite elements and convert them to SPH particles either at the beginning of an analysis or after the deformation becomes significant. The conversion functionality is intended to be used when the deformations in the original finite element mesh are significant and elements may distort. Traditionally, in such cases deletion of the soon-to-be distorted Lagrangian elements would be the only choice to allow analyses to continue. Converting to SPH particles provides an improvement over the element deletion option because the generated particles are able to provide resistance to deformation beyond finite element distortion levels. Therefore, it can address the elements distortion on one hand, and improve the computational efficiency on the other hand [15].

In order to automatically transform the element to the particle, an effective criterion is needed to determine the degree of distortion of the elements. In the study of high-speed impact, the distortion degree can be judged based on the physical criterion or the geometric criterion. Some scholars have tried to use the equivalent plastic strain [17], maximum principal stress [18] or other physical quantities to determine the elements which have to be converted to particles.

In ABAQUS, when using the conversion technique, particles are generated internally at the beginning of the preprocessing phase of an analysis, and they are placed in an inactive or dormant state. The particles are attached to the parent elements in a similar fashion as the nodes of embedded elements are attached, as shown in Fig. 5, and they follow the motion of the parent element nodes in an average sense. Upon conversion the state information (such as stress or equivalent plastic strain) associated with the element being converted is transferred to the generated particles [15].

In this paper, the rotating band will be set as the convertible region. Other parts of the engraving system will still utilize Lagrangian finite elements. The law of conversion is Mises Stress.

Fig. 5Internally generated particles per parent element

5.2. CEL method

In the traditional Lagrangian analyses, nodes are fixed within the material, and elements deformation equal to material deformation. Lagrangian element is always 100 % full of material, so the material boundary and elements boundary are the same. In the Eulerian analysis is just the opposite. In this case, nodes are fixed in space. Material can flow through elements and do not deform [19, 20]. The Eulerian time incrementation algorithm which is known as ‘Lagrange-plus-remap’ is based on an operator split of the governing equations, leading to a traditional Lagrangian phase followed by a Eulerian phase. As we have mentioned above, during the Lagrangian phase of the time increments nodes are assumed to be fixed within the material temporarily. But during the Eulerian phase of the time increments, deformation being suspended, elements with extreme deformation are automatically remeshed, and the corresponding material flows between the elements are computed. The different features of Lagrangian description and Eulerian description have been shown by Fig. 6 [21].

The CEL method was first proposed by Noh in 1964 [22]. It picks up the strengths of both Lagrangian and Eulerian method. In ABAQUS, material is determined by calculating its Eulerian Volume Fraction (EVF) in every element. By definition, the EVF is 1 if an element is completely filled, and vice versa. As it has shown in Fig. 7, the dark grey region on the left is filled with material, and EVFs in this region are all larger than zero. In the case of this paper, in order to get a relatively accurate boundary at the beginning of the analysis, the Eulerian model will be cut deliberately, more details are given by Fig. 8. The top right part of Fig. 8 which will be filled with material is the true shape of the rotating band.

Fig. 6Feature of Lagrangian algorithm and Eulerian algorithm

a) Lagrangian description

b) Eulerian description

Fig. 7Eulerian volume fraction

Fig. 8The rotating band of Eulerian analysis

Eulerian-Lagrangian formulation is based on the enhanced contact immersed boundary method. In this method, the Lagrangian mesh occupies void area of the Eularian gird. In ABAQUS, the contact algorithm automatically calculates and tracks the interfaces among Eulerian materials and Lagrangian structure [23].

The CEL method applies explicit dynamic method which is based on the central difference criterion to solve the nonlinear differential equations:

where $u^n$, ${\dot{u}}^n$, ${\ddot{u}}^n$indicate displacement, velocity, and acceleration of $n$th freedom respectively. $∆t$ is the time increment. $i$ is the moment of $i$th increment.

6.1. Deformation process

6.1.1. FEM-based results

Given that the FEM has been widely utilized in engraving simulation. In this study, the FEM-based simulation is used to verify the results of the proposed methods. Fig. 9 is the evolution of von Mises stress of rotation band using FE method.

Because the Lagrangian finite element mesh may cause simulation termination in the case of large deformation, the plastic strain accumulation criterion presented in Section 3.2 was introduced to delete the failed elements. Elements forced to be deleted are shown in Fig. 10. Totally 1232 elements, 0.257 % of the total elements of the rotating band, have been forced to be deleted due to the extreme distortion. These deleted elements’ interior attributes are also deleted. Therefore, it inevitably caused energy loss. Although the energy loss is low, we cannot conclude that the energy will not affect the kinematics of the projectile. From Fig. 10, it can be seen that the failure of the rotating band element occurs more often near the rear of the band. It shows the fact that the material of the rotating band is forced to flow to the rear at first, then, the element failure occurs as the rotating band keeps moving forward.

Fig. 9Evolution of von Mises stress for rotation band under FE method

a) 5ms

b) 6ms

c) 6.85ms

Fig. 10The deleted elements

Fig. 11The sectional engraved rotating band

6.1.3. CEL-based results

As mentioned in Section 2.2, the Eulerian part must be built in ABAQUS, and the model is given by Fig.8. In this case, there is a total of 695,674 elements in the finite element model, and 592,256 elements among it belong to Eulerian part. But the time cost of this simulation was just 23.5 hours. Fig. 13 shows the evolution of the rotating band during the engraving process simulated by using the CEL method. The engraving process finished in about 6.73 ms.

Fig. 12Evolution of von Mises stress for rotation band under FEM-SPH method

a) 0 ms

b) 5.5 ms

c) 6 ms

d) 6.75 ms

Through observation, the deformation and flow of the material are obtained. The detail of the engraving process is, when the projectile has been loaded, the rotating band kept in contact with the inner wall of the tube closely. Once the projectile base pressure increases to a certain value, the projectile starts to move forward. The diameter of forcing cone is decreasing as the projectile going forward. The front end of the rotating band starts to deform. The elastic and plastic deformation begins to happen, and the original groove of the rotating band starts to be filled as the material is being pushed backward. Then, as the engraving process keeps ongoing, the regions of rotating band where corresponding with rifling lands are forced to be extruded and sheared. Finally, the rotating band is engraved by the rifling lands. The rotating band and the interior wall of the tube are fitted closely. The engraving process ends.

Fig. 13Evolution of von Mises stress for rotation band under CEL method

a) 5 ms

b) 6 ms

c) 6.73 ms

6.1.4. Comparison of groove sizes

Fig. 14 shows a recovered projectile from a live firing. The data of the groove sizes are presented in Table 2. According to the studies of Sun Q. Z. [25], the rotating band is further worn after the engraving process ends. The comparisons in the table also demonstrate this phenomenon: these three results are close or within the range of the experimental data, but they are all close to the lowest value of the experimental data.

Fig. 14Engraved rotating band of a recovered projectile

Table 2Comparison of groove sizes

	FEM (mm)	CEL (mm)	FEM-SPH (mm)	Experimental data (mm)
Depth	1.44	1.44	1.43	1.41-1.53
Width	2.72	2.69	2.71	2.68-2.81

6.2. Projectile motion

The displacement-time curves, velocity-time curves and acceleration-time curves of the projectile are given by Figs. 15-17. We can find that, although the numerical algorithms vary, the trends of the displacement, velocity and acceleration curves of these three simulations are basically the same.

Fig. 15, 16 show the displacement-time curves and the velocity-time curves of the mass center of the projectile respectively. A similarity in trends can be observed from the motion curves. But because of the differences of the numerical methods, the curves are a little bit different, especially the curves obtained from the simulation using FEM. It can be explained by the introduction of the damage accumulation criterion. A research [26] shows that the velocity of the projectile has reached 10 % of the muzzle velocity when the band engraved in rifling completely. The instantaneous velocities of projectile when engraving process finishes versus muzzle velocity are shown in Table 3 respectively.

Fig. 17 shows the acceleration-time curves of the mass center of the projectile. As can be seen, the curves show a similar trend. However, the curves are not as smooth as the displacement and velocity curves. This phenomenon demonstrates that there always have some oscillations during the engraving process.

Fig. 15Displacement-time curves

Fig. 16Velocity-time curves

Fig. 17Acceleration-time curves

Fig. 18Engraving resistance

6.3. Dynamic engraving resistance

The engraving resistance-displacement curves which are shown in Fig. 18 are similar to those obtained by Wu B. [5]. Through these curves, the oscillations during the engraving process can also be reflected. The law of the dynamic engraving resistance is obtained. When the projectile starts to move, the dynamic engraving resistance rises. As the plastic deformation of the rotating band occurs, the dynamic engraving resistance increases dramatically. When the rotating band starts to contact with the lands, slight waves occur. Then, the dynamic engraving resistance continues increasing significantly until it reaches its peak value. After that, the dynamic engraving resistance almost stays the same until the deformation of the rotating band reduces. After the rotating band engraved into the barrel completely, the engraving resistance maintains almost a constant value. Table 3 shows the peak values of the engraving resistance.

Jin Z. M. [26] also presented the conclusion that when the engraving process ends, the engraving pressure has reached 173 MPa which is 65 % of the maximum value. Results of engraving pressure through numerical analysis are also shown in Table 3. These conclusions can somehow demonstrate the validity of simulations in this paper.

Table 3 shows the main comparisons in the study. From the data listed in the table, we can find that the new methods can both satisfy the requirements of the accuracy and improve the efficiency.