## 5 Numerical Results and Discussion
### Investigation of the Ratcheting Behavior of the Pipe Considering the Assembly Contact Effect  
Figure 7 shows the hoop ratcheting strain distribution between Position 1 and Position 4 after 45 cycles. In this figure, it is clear that the hoop ratcheting strain increases with an increase in internal pressure. In addition, it should be pointed out that the valley value of the hoop ratcheting strain is almost on Position 2 under an internal pressure of 10 MPa, while it is on the left of Position 2 under internal pressures of 21 and 28 MPa. This suggests that, as the internal pressure increases, the valley value moves further from Position 2. This is due to axial deformation of the pipe, caused by the increase in internal pressure. As a result, Position 2 moves far away from the sleeve end as the internal pressure increases. However, the valley value of the hoop ratcheting strain is still on the node of the pipe which is closest to the sleeve end. Thus, axial deformation of the pipe with assembly parts had no effect on the above conclusion that assembly contact between pipe and sleeve suppresses cumulative plastic deformation. In addition, the position of the peak value of the hoop ratcheting strain hardly changed during the 45 cycles under different internal pressures. Figure 8 presents the change in outside diameter of the pipe with assembly parts between Position 1 and Position 4 under different internal pressure levels, for which the trend has high consistency with the hoop ratcheting strain distribution in Figure 7. This indicates that the hoop ratcheting strain's increase with increasing internal pressure led to the outside diameter of the pipe increasing [33], and also gives a reasonable explanation for the experimental results presented in Table 2. It is important to state that the negative change in the outer diameter of the pipe under the 10 MPa internal pressure in Figure 8 was caused by ovalization of the pipe's cross-section [36], which reached a maximum at Position 2 due to the large bending deformation. However, an increase of the internal pressure and hoop ratcheting strain resists negative deformation, making the outer diameter increase. Figure 6: The effect of the sleeve on the hoop ratcheting strain after 45 cycles. Figure 7: The ratcheting strain distribution between Position 1 and Position 4 under different internal pressures after 45 cycles. As mentioned above, an increase of the bending moment leads to the hoop ratcheting strain increasing, and assembly contact between the sleeve and pipe can suppress the hoop ratcheting response, thereby changing the monotonic trend of the ratcheting strain. There are peak and valley values of the hoop ratcheting strain and the outside diameter change in Figures 7 and 8. Overall, it can be deduced that the ratcheting response between the valley value and the peak value is controlled and suppressed by assembly contact between the sleeve and pipe. The ratcheting response in other areas was dominated by the bending moment, and was found to increase as the bending moment increased. Thus, the valley value and peak value are the transition points between the two control modes. Figure 9 exhibits the ratcheting response of the pipe with assembly parts at Position 1, Position 2, and Position 3 under the internal pressures of 10 MPa, 21 MPa and 28 MPa. It can be seen that the axial ratcheting strain is much smaller than the hoop ratcheting strain, indicating that the hoop ratcheting strain plays a dominant role in cumulative plastic deformation. This also conforms with the view presented in [PERSON] and colleagues' papers [13; 35]. Figure 8: The outside diameter change distribution between Position 1 and Position 3 under different internal pressures after 45 cycles. Figure 9: _Cont._  
As displayed in Figure 9, after 45 cycles and under the internal pressures of 10 MPa, 21 MPa, and 28 MPa, the bending moment at Position 3 is smaller than that at Position 2, although the hoop ratcheting strain at Position 3 is larger than that at Position 2.