Evolution of stress and strain in 2219 aluminum alloy ring during roll-bending process

The residual stress of large aluminum alloy ring after heat treatment will cause serious deformation in subsequent processing. The conventional methods for reduction of residual stress (such as stepwise cold pressing and bulging) have little effect in the residual stress reduction for large-scale ring component and will induce inhomogeneous stress distribution. In this paper, roll bending process is adopted to reduce the quenched residual stress of 2219 aluminum alloy super-large ring. The numerical model of roll bending process was established, and the evolution and distribution of stress and strain after roll bending were studied. The influence of roll winding number on the uniformity of stress and strain was analyzed. The results show that the arch-shaped quenched residual stress of the ring changes to N-shaped distribution from inside to outside after roll bending process. The value of the residual stress reduces from ± 180 MPa in quenched state to the value within ± 50 MPa in roll bended state. With the increase of roll winding number, the stress uniformity is improved, but the stress reduction amplitude is basically the same. By analyzing the elastic–plastic strain distribution characteristics and strain springback law of the ring after roll bending, the formation mechanism of N-shaped residual stress distribution after roll bending is revealed.


Introduction
2219 aluminum alloy is widely used in rocket fuel tank, supersonic aircraft skin, and other fields due to its excellent comprehensive performance [1][2][3][4]. In order to improve the comprehensive properties of aluminum alloy materials, aluminum alloy materials are usually solution strengthened [5][6][7][8]. However, in the process of quenching, due to the rapid cooling rate of the outer layer and the slow cooling rate of the inner layer, residual stress is inevitably introduced in the material, accompanied by deformation. The material warping and subsequent machining deformation caused by quenching residual stress is an urgent problem to be solved in the aerospace field. It is of great significance to control the quenching residual stress of aluminum alloy effectively.
According to the principle of residual stress relief, the current residual stress relief process can be divided into two categories, thermal stress reduction method, and mechanical stress reduction method [9]. Wu et al. [10] studied the effect of thermal stress relief treatment on reducing residual stress and homogenization of rolling process. Gelfi et al. [11] developed a new residual stress assessment method based on single Debye ring analysis. This method is especially suitable for stress analysis of coatings and samples with complex geometry. LV et al. [12] studied a long-term natural aging treatment method, explored, and verified the effect of natural aging and artificial aging on homogenizing and reducing in ferrule. Ma et al. [13] used indentation strain gauge method to study the residual stress of 2219 aluminum alloy super large rolled ring. Dolan et al. [14] uses quenching factor analysis to slowly cool the alloy from the solution heat treatment temperature to the intermediate temperature above the critical temperature region of the C curve. In the quenching process, the numerical value of residual stress generated is reduced. Yang et al. [15] used finite element method and finite element simulation method to study the reduction of residual stress in the cold stretching process.
Due to the large size and special application field of largesize ring parts, the research on controlling the residual stress of large-size ring parts is less. In recent years, radial bulging method [16] and roll bending method [17] used in controlling the residual stress of large-size ring parts are reported. However, it is difficult to provide scientific guidance for the internal stress reduction process of the ring from the mechanism due to the lack of research on the evolution of the internal stress of the ring by these stress reduction processes.
In this work, the roll bending process of large size 2219 aluminum alloy ring was analyzed by using finite element method, and the evolution of stress and strain was obtained. Then the effect of the number of rolling turns on the uniformity of stress and strain in the ring was studied. It is of great significance to understand the evolution of stress-strain field in the bending process of large-scale ring for effectively formulating the stress reduction process.

Overview of roll bending
This article explores the influence of roll bending process on the quenching residual stress of ring parts. The specific steps of roll bending experiment are as follows: Firstly, the quenching ring is installed in the proper position of the symmetrical three-axis roll bending machine. Make the feed roller and drive roller contact with the inner and outer surfaces of the ring, and ensure that the axis direction of the ring and the axis direction of the three rollers are parallel, as shown in the Fig. 1 to ensure the stability of installation.
Then, a radial downward displacement is applied to the feed roller to make it have a certain amount of pressure down. The pressure down amount used in this experiment is 10 mm, and the ring is marked along the radial direction at the bottom of the ring.
Then two driving rollers were set to rotate at the same angular speed (0.4 rad/s), so that the ring was driven by the friction force of the driving roller to rotate, and then, the feed roller was driven to rotate passively, and the ring was also driven by the friction force.
After the rotation of the ring ends, load is applied to the feed roller moving up in the radial direction to make the feed roller and drive roller leave the ring.
The process parameters selected in this experiment are based on engineering practice and simulation, and the specific experimental parameters come from the research of Gong et al. [18]. The principle of reducing residual stress by roll bending is that the feeding roll presses down the ring under the action of external load, and the ring part in the middle of the three rollers will be affected by bending moment. During the roll bending process, the ring will rotate with the rotation of the driving roll, and the ring will appear continuously in the middle of the three rollers, thus continuously affected by bending moment. When the bending moment reaches a certain value, the ring will have plastic deformation, which will release and reduce the quenching residual stress.

Material parameters
The chemical composition of 2219 aluminum alloy is shown in Table 1, and the physical properties of 2219 aluminum alloy are shown in Table 2.

Finite element model
Ring roll bending is an unsteady three-dimensional deformation process with geometric nonlinearity, physical nonlinearity, and unstable boundary conditions; the roll bending model of the ring is established, as shown in the Fig. 2. The outer two driving rollers rotate counterclockwise to provide rotating power for the clockwise rotation of the ring. The feeding roller completes the continuous rolling extrusion of the ring through radial downward and rotary motion to produce plastic deformation. The simulation parameters are shown in Table 3. The research method in this paper is also suitable for roll bending simulation of other large diameter rings. The initial state of the roll bending is shown in Fig. 3. A fixed virtual ring surface with the same diameter of the aluminum alloy ring is defined to analyze the roll bending process of the ring. The virtual ring surface is divided into eight equal parts along the circumferential direction, and the equinoxes are marked as A1 to A8 successively. Along the clockwise direction, A1 to A8 correspond to 0°, 45°, 90°, 135°, 180°, 225°, 270°, and 315°, respectively. A1 to A8 are used as the reference points for the analysis of the roll bending of the large ring, and the positions of A1 to A8 remain the same. A section S1 on the ring is selected to track and analyze the stress and strain evolution during the roll bending process, as shown in Fig. 4. The initial position of section S1 locates at A1 (0°), while the position of the working roll is fixed at A3 (90°). During the roll bending process, the large ring rotates clockwise, and the section S1 passes through point A1(0°) to A8(315°) in turn and go on; that is, the angle positions of section-1 are corresponding to 0°, 45°, 90°, 135°, 180°, 225°, 270°, and 315°, respectively, and go on. When section S1 locates at the position of 90°, it is just in the bending roll working area. In order to further study the law of stress and strain, the residual stress was studied by taking the axial center line of the radial section of the ring along the radial path L1 from the inside of the ring to the outside of the ring.

Experiments
Surface residual stress of the large ring was measured using X-ray diffractometer (XRD, STRESSTECH OY, Jyväskylä, Finland), and the measurement result was compared with FEM result. The measurement area is shown in Fig. 5. The measurement was carried out in steps of 10-mm interval in the radial and axial directions. The diffractometer parameters used in the measurement are shown in Table 4 (Fig. 6).

Residual stress
The simulated results of quenching residual stress are shown in Fig. 7. The circumferential stress and axial stress of the ring are composed of internal tensile stress and compressive stress of the surface. The peak compressive stress is 140 MPa, and the peak tensile stress is 130 MPa. The axial stress is relatively small on the upper and lower edges, and the axial tensile stress is less than the circumferential tensile stress. The distribution range of the Mises stress is 40 ~ 140 MPa. As there is significant difference between the radial size and axial size of the ring, the distribution of the temperature and the stress gradient is quite different along the radial direction; as a result, the radial stress shows quite different distribution compared to that in the axial and circumferential of the ring. In this study, the circumferential stress and axial stress of the ring are mainly analyzed as the value of residual stress in the radial direction is much smaller.
The radial and circumferential stresses of the simulated results and the measurement results were compared as shown in Fig. 8. It can be seen from the figure that the measurement results are in good agreement with the simulated results, which verifies the feasibility and reliability of the numerical model.

The evolution of inner stress during the roll bending process
As showed in Fig. 4, the radial path L1 is taken on the section S1 of the ring to analyze the change rule of stress at each node on the path during the process of 1-turn roll bending. As can be observed in Figs. 9 and 10, as the roll bending progresses, the distribution of circumferential stress along the path L1 gradually evolves from a parabola to an N-shape, and the change rule of axial stress is consistent with that of circumferential stress. When the cross section S1 is at 0 and the feed roll is pressed down 20 mm, the distribution trend of the anisotropic stress remains unchanged, but the numerical value decreases slightly. When the cross section S1 reaches the position of A1(45°), although the cross section is not in the roll bending zone, the stress has begun to decrease, the maximum tensile stress and the maximum compressive stress have decreased by about 10%, and the stress distribution is still parabolic. When the section S1 reaches the position of A3(90° roll bending zone), the stress distribution presents the compressive stress on the inner surface of  the ring and the tensile stress on the outer surface, due to the large pressure generated by the feed roller. When the section S1 arrived at A4(135°), at this time as a result of roll bending process in section S1 surrounding area is introduced into the plastic deformation, stress reduction amplitude is larger; the stress distribution by parabola into N shape, but because of the close section distance driven roller, work stress has not released, is still in a state of large, large ring inside for a larger compressive stress, the lateral tensile stress. When the section S1 is located at A6(225°), the section has left the roll bending zone and is not affected by excess working stress. Meanwhile, the stress value decreases but the overall trend does not change, and the stress distribution is basically stable. When the section S1 is located at A8(315°), it is far away from the drive roller, the stress value and law tend to be stable, and the stress size is basically the same. Meanwhile, the stress is quite small and the reduction effect is good, presenting an obvious N-shaped distribution.

Evolution of internal strain
When the roll bending occurs, the ring produces elastic-plastic strain. Although the tensile stress and compressive stress exceed the yield stress, in fact, when the tensile stress inside the ring transits to the external compressive stress, a large plastic deformation occurs outside the ring, corresponding to its plastic strain. After the feed roll is lifted, the plastic strain is retained, but the neutral layer does not produce plastic deformation or small plastic deformation, Fig. 7 Maps of residual stress distribution after quenching and there is no plastic strain or small plastic strain. It is almost all elastic deformation. Corresponding to the elastic strain, the ring reaches the stable region. When the bending load is unloaded, the neutral layer material will inevitably rebound, and the stress is unchanged. Therefore, it is necessary to study the elastic-plastic strain law. Figure 11 is the contour map of equivalent strain after S1 section. Path L1 is taken at the S1 section of the ring, and the results are shown in Fig. 12. Figure 11 shows the plastic strain nephogram of section S1 after roll bending. The results demonstrate that the maximum plastic strain value is at the outer contour of the ring, and the strain value is 0.0324. The minimum strain is at the core of the ring and near the inner part of the ring, and the smallest plastic strain is 0.00098. This is because insufficient pressure of the feed roller in the roll bending process does not affect the core of the ring enough to make it produce sufficient plastic deformation.
In order to discover the distribution rule, the equivalent plastic strain and the anisotropic plastic strain were studied by taking the radial path 1. Figure 12 shows the equivalent plastic strain distribution along the selected path in the longitudinal section of the ring after roll bending. The results show that the plastic deformation generated by radial path 1 present an asymmetric "U" shape along the radial direction; that is, the middle and low ends are higher, and the difference is obvious. This is explained by the large radial strain difference between the inner and outer rings, so the region cannot reach the center, and plastic deformation occurs only near the inner and outer surfaces of the rings.   In the roll bending process, the ring not only produces plastic strain, because the reduction of the feed roller does not reach enough to lead to the change of its shape; therefore, there is also the corresponding elastic strain. In order to study the elastic strain law of the ring, the elastic deformation cloud image in the roll bending process is taken as follow: Figure 13 shows the elastic strain nephogram of S1 section. The results show that the maximum elastic strain was located in the neutral layer of the ring, the elastic strain is 0.001, the minimum elastic strain is located in the inner surface of the ring, and the minimum elastic strain is 0.00073. In order to study the rule of path 1, Fig. 14 shows the elastic strain distribution along the selected path on the longitudinal section of the ring after roll bending. The results show that the elastic deformation of the middle part is large, and the elastic deformation of the inner and outer contour is smaller than that of the middle part, but the overall difference is not large.

Effect of roll bending cycle on stress
It can be seen from the evolution of the stress and strain in the ring that the stress and strain in the roll bending process are uniform. In order to examine the influence of roll bending on the uniformity of stress distribution of ring parts, four rings with the same quenching process were subjected to continuous roll bending for 1 to 4 times, respectively. Eight points was uniformly distributed on the end faces of ring parts as shown in Fig. 3, and the surface stress distribution at each point of the four rings after roll bending was compared.
According to the characteristics of residual stress distribution of 2219 aluminum alloy rings, the circumferential stress and axial stress of 2219 aluminum alloy rings at different points were compared and analyzed by drawing a radar chart. The closer the position of eight points to the uniform octagon, the more homogeneous the residual stress of the ring is, and the more consistent the anisotropic structure is. It can be seen from Figs. 15 and 16 that the residual stress at each analysis point of the ring after different number of roll bending turns is far less than the original quenching residual stress. After the first roll bending, the change of residual stress of the ring is the largest, but the distribution of residual stress is uneven. With the increase of the number of roll bends, the residual stress changes little, but the residual stress tends to be uniform.
In Figs. 15 and 16, the distribution of stress in all directions appears a "sharp angle" phenomenon; that is, residual stress at 1 ~ 2 test points is significantly higher or lower than other test points. After the second roll bending, the stress reduction value decreases, the axial stress is on the brink of the normal octagon, but the circumferential stress still has  In order to quantify the influence of the number of turns in the roll bending process on the stress inhomogeneity, stress standard deviation S is used to evaluate and describe the distribution uniformity of residual stress on the above points. The formula for standard deviation is as follows: where S is the standard deviation of each point, n is the study point, x i is the stress value of the study point, and x is the mean stress value. The result is shown in Figs. 17 and 18: It can be seen from the standard deviation analysis shown in Figs. 17 and 18 that there are great differences in the distribution uniformity of ring residual stress after different number of roll bending. After one roll bending, although the stress reduction is very large, the distribution of stress at each part of the ring is also quite different, so the standard deviation of stress at each part is much larger than that in the quenched state, with the standard deviation of stress distribution of 17.35 MPa, the inhomogeneity increased by 71.6%. After 2 turns of roll bending, the standard deviation of circumferential stress decreases and reaches 12.2 MPa. Meanwhile, the sharp angle phenomenon is partially improved, but the fluctuation is still large, and the unevenness increases by 59.5%. After 3 turns of roll bending, the circumferential stress homogeneity decreases and reaches 2.21 MPa, and the homogeneity is enhanced by 55.1%, showing good uniformity. After 4 rolls, due to the local deformation of the roll bending process, excessive number of turns will not result in the decrease of stress uniformity. As can be observed in Fig. 18, the distribution uniformity of residual stress in the third roll bending circle has the most significant difference, reaching 1.27 MPa, indicating good stress uniformity. The standard deviation of axial stress has decreased by 75.7%. Therefore, it can be assumed that the uniformity of the roll bending to the third turn is better.

Effect of roll bending cycle on strain
According to the strain distribution law, the maximum deformation occurred at the outer contour of the ring, while the minimum plastic deformation occurred at the core of the ring. In order to analyze the influence of different number of roll bending turns on the strain uniformity, the ratio e of the minimum strain at the core of the large ring section (aforementioned S1) and the maximum strain at the surface was used to represent the strain uniformity The strain nonuniformity coefficient e was used to evaluate the distribution uniformity of plastic deformation in the ring. After adopting different roll bending processes, e ratio is shown in Table 5. In the process of roll bending with different roll bending turns, e gradually decreases. The e value of the first circle is very large, reaching 2.9%, and the strain is not uniform, but the stress reduction effect is very good. In the second lap, e gradually decreases to 1.7%, the uniformity increases, but the stress reduction effect decreases. From the third turn, e reaches 1.3%, and the strain coefficient tends to be stable after that, so it can be considered that the uniformity is good at the third cycles.

Evolution of N-shape stress
The ring roll bending process is similar to the three-roll bending process for beam parts as described by Abvabi et al. [19]. In this section, a simplified three-point roll bending model of the ring is established as shown in Fig. 21, and the evolution of stress distribution from parabola to N-shape is analyzed. The roll bending part is divided into four areas: entry area, contact area, departure area, and stable area. When the ring material passes through the four areas, the stress distribution along path L1 is shown in Figs. 19 and 20.
In the process of roll bending, when the ring material is located at the entering point, the inner side and the outer side of the ring are subjected to working tensile stress and (2) e = min max working compressive stress, respectively, under the support of the driving roll.
When the ring material reaches the contact area, the ring will bulge from the inner side to the outer side under the pressure of the feeding roller, and the inner and outer sides of the ring are subjected to working compressive stress and the working tensile stress respectively, which is opposite to the direction of the working stress produced by the driving roller at the feeding point.
When the ring material reaches the departure area, the direction of the working stress generated by the driving roller in the ring is similar to that at the entering point (Fig. 21).
When the ring material reaches the stable area, the tensile elastic stress caused by the tensile working load of the inner ring material, and the compressive elastic stress caused by the compressive working load are released, so that the approximate Z-type stress distribution of the previous step (at the departure area) evolves into N-type stress distribution, which is consistent with the research results of Peng et al. [20].
After adding quenching stress, the roll bending results are shown in Fig. 20. At the entering point, due to the existence of quenching residual stress, the overall stress is greater than that in the non-stress state. However, after entering the contact point, due to the pressure of the feed roller, the ring stress changes uniformly, presenting the stress distribution of internal pressure and external tension. At this point, the stress of the ring without prestressed roll bending and the ring with quenching stress roll bending presents the same trend under the same feed roller, and the difference of stress values is small. When entering the departure point, due to the quality of the ring itself and the pressure of the feed roller, the internal stress of the ring presents the stress distribution of internal tension and external pressure. Meanwhile, the stress value of the two states is basically the same. Until the ring reaches the stable area after the end of the roll bending and the elastic deformation is released, the stress distribution law of the two states presents the same, and the difference of stress value is small.

Effect of springback on bending
During the roll bending process of the ring, the outer material is subjected to tensile stress, and the inner material is subjected to compressive stress. The surface with zero stress of the stress neutral layer in the thickness direction of the ring is divided into tensile deformation zone and compressive deformation zone. Under the action of pure bending moment, the values of tensile, compressive stress. and strain at the upper and lower parts of the neutral layer are equal, and the elastic part under the two stress states should be restored, which will produce an opposite torque and cause springback. Since the springback of each part of the profile is different, the springback of the middle part with large curvature is small, and the springback of the two ends with small curvature is large, the maximum distance between the two ends of the profile before and after unloading is defined as the springback, as shown in Fig. 22. When elastic-plastic bending occurs in the roll bending process of the material, the elastic strain affects the inner wall 1, the middle part 2, and the outer part 3 of the ring material, respectively. The plastic deformation of the parts 1 and 3 is retained after unloading at the end of the bending deformation, while the elastic deformation of the parts 2 is completely disappeared. At the same time, the elastic deformation components in the deformation area and the outer area produce the elastic stress opposite to the direction of the plastic deformation stress. After the ring is removed from the roll bending zone, the corresponding deformation (springback) is generated under the action of elastic bending moment to reach a new equilibrium state. Corresponding to the departure point, the inner compressive stress and the outer tensile stress of each part of the ring bending.
It can be seen from the Fig. 22 that the main springback areas are located at positions 1, 2, and 3, and the position of elastic stress generated by the springback amount of 1 and 3 is basically consistent with the position of the reverse stress. Since 2 was located in the middle part, the plastic deformation is small and the elastic deformation is large, so the stress distribution in the two parts is approximately linear, forming an N-shaped distribution. The law of strain reflects the change of stress.

Conclusions
The residual stress of 2219 aluminum alloy ring after quenching was reduced by roll bending process. The evolution of stress and strain during roll bending and the distribution of residual stress after roll bending were studied. The results are summarized as follows: (1) When the ring is rotated from 0 to 360° (the roll position is 90°), the arch-shaped quenched residual stress of the ring changes to N-shaped distribution from inside to outside after roll bending process. The value of the residual stress reduces from ± 180 MPa in quenched state to the value within ± 50 MPa in roll bended state.
(2) Based on elastoplastic strain analysis during roll bending, the inner mechanism of N-type stress distribution during roll bending was revealed. Elastoplastic deformation exists in the roll bending process. Small feed of the feed roll results in large elastic deformation in the inner part of the ring, while it is small in the outside part. The distribution of the plastic deformation is opposite to that of the elastic deformation. When the pressure of ring is released after the roll bending process, an N-shaped distribution of the residual stress is generated.
(3) With the increase of the number of roll bends, the stress uniformity is improved, but the stress decrease is basically unchanged. After one turn of roll bending, the stress decreased to within ± 50 MPa, but the stress uniformity increased, the standard deviation of principal stress reached 17.35, and the strain uniformity coefficient was 2.9%. After the second roll bending, the stress uniformity is improved to some extent, but the stress reduction is close to that of the first roll bending. The standard deviation of the principal stress is 12.2, and the strain non-uniformity coefficient is 1.7%, but it is still higher than the standard deviation of the principal stress when quenching is 4.93. After three cycles, the standard deviation of the principal stress and strain inhomogeneity coefficient is 1.27 and 1.3%, respectively. The stress uniformity has reached a stable state, and increasing the number of roll bends has little influence on the stress and strain. (4) The roll bending method proposed in this paper can effectively reduce the quenching residual stress of large size 2219 aluminum alloy ring, and the stress reduction rate can reach 50%. In addition, this method can also be used as a reference method for eliminating residual stress of other annular components. It is of great significance to understand the evolution law of stress-strain field during the bending process of large ring parts for formulating effective stress reduction process of ring parts.