During the rolling of doubly-curved sheet metal surfaces, the flow of metal is very complicated and will affect by the dimension of rolling deformation zone. The dimension of rolling deformation zone is mainly affected by the rolling reduction and the central cross-section diameter of the rollers. In this paper, the influence of central cross-section diameter of the rollers on the longitudinal bending deformation of formed spherical surfaces and the stability of the rolling process is studied combined with the rolling reduction. The current research results show that the larger the central cross-section diameter of the rollers, the more stable the rolling process, and the better the shape accuracy of the formed spherical surface. When the central cross-section diameter of the rollers is relatively small, the longitudinal bending deformation of the formed spherical surface increases with the increase of the central cross-section diameter of the rollers. However, an excessively large roller diameter will restrain the longitudinal bending deformation of the sheet. When the central cross-section diameter of the rollers is relatively large, the longitudinal bending deformation of the formed spherical surface will decrease with the increase of the central cross-section diameter of the rollers. The results in this study can be used as a reference for the design of roller to ensure that the formed doubly-curved surface can obtain greater longitudinal bending deformation and better shape accuracy under the same rolling reduction.
Research Article
The effects of central cross-section diameter of the rollers on the doubly-curved surface rolling
https://doi.org/10.21203/rs.3.rs-1446242/v1
This work is licensed under a CC BY 4.0 License
Journal Publication
published 28 Mar, 2023
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Doubly-curved surface
Rolling
Surface forming
Flexible forming
Numerical Simulation
Doubly-curved surfaces are widely used in aviation, shipbuilding, high-speed trains, construction and other industries. Traditional machining methods can realize mass precision machining of doubly-curved surfaces. Due to the long development cycle and high production costs of fixed molds, traditional surface forming methods are not suitable for the manufacture of small-batches surfaces with multiple specifications. Therefore, in recent years, a variety of flexible forming technology of doubly-curved surfaces has been widely used in different industrial fields [1–4].
The doubly-curved surface rolling process based on the line forming method, due to its high production efficiency, low equipment cost, and good surface quality of the formed surfaces [5], has attracted the attention of many scholars in recent years. In 1988, Yamashita and Yamakawa [6] first proposed the use of roll forming to manufacture doubly-curved surfaces. In 2003, Yoon et al. [7] developed an incremental roll forming process for doubly-curved surfaces. In 2007, Li et al. [8] developed a flexible roll forming technology using three flexible wire soft shafts as forming rollers. In 2009, Shim et al. [9] developed an array roller forming technology for manufacturing doubly-curved surfaces. In 2012, Li et al. [10, 11] proposed a flexible rolling technology for doubly-curved surfaces, which only use two bendable rollers as forming tools. The method could roll doubly-curved surfaces through unevenly distributed roll gaps formed by bendable work rollers [12]. During the rolling process, the sheet metal passes through two flexible rollers that can be bent transversely (perpendicular to the rolling direction), and elongated unevenly in the longitudinal direction (rolling direction) while deformed in the transverse direction, forming the required doubly-curved surfaces. The sheet metal passes through an unevenly distributed roll gap formed by two rollers that can be bent transversely (perpendicular to the rolling direction). When the sheet metal is bent transversely, it will also produce uneven elongation in the longitudinal direction (rolling direction), and finally is formed into a doubly-curved surface. The flexible rolling technology of doubly-curved surfaces has high flexibility and could process different types and specifications of doubly-curved surfaces [13–18].
In 2019, Li and his colleagues [19, 20] developed a doubly-curved surfaces rolling technology based on variable section arc rollers on the basis of flexible rolling technology. The forming method has lower equipment cost and better rigidity, and could quickly process spherical surfaces and saddle-shaped surfaces with different specifications. The forming process uses a pair of variable section arc rollers (usually a convex roller and a concave roller) as forming tools to form an unevenly distributed roll gap, thereby realizing the rolling of doubly-curved surfaces. When the rollers form a roll gap, which is small in the middle and big in the edges, the formed surface is a spherical surface (see Fig. 1a); when the rollers form a roll gap, which is big in the middle and small in the edges, the formed surface is a saddle-shaped surface (see Fig. 1b). The longitudinal curvature of the formed doubly-curved surfaces could be changed by adjusting the rolling reduction [21].
In this paper, the process parameter of the central cross-section diameter of the variable section arc rollers are investigated. A series of simulations and experiments were carried out with different central cross-section diameter of the rollers. Taking spherical surfaces as an example, the longitudinal bending deformation of surfaces formed by different central cross-section diameters of the rollers was compared, and the influence of different central cross-section diameters of the rollers on the stability of the rolling process was analyzed. And through the numerical simulation results, the difference of the forming results with different central cross-section diameters of the rollers is analyzed.
When rolling spherical surfaces, the roll gap is large in the middle and small in the edges, the rolling reduction in the center of the sheet is the largest, and the central cross-section diameter of the rollers has the greatest influence on the dimension of deformation zone. Figure 2 shows the rolling deformation zone on the center symmetry plane. The length of the deformation zone along the rolling direction increases with the increase of the central cross-section diameter of the rollers (L’>L). The variation in the dimension of the deformation zone will affect the flow of the metal inside the sheet during the rolling process, which will affect the longitudinal bending deformation of the formed doubly-curved surface.
It can be seen from Fig. 2 that the longitudinal curvature radius of the formed spherical surface must be smaller than the central-section radius of the upper roller. An excessively large central cross-section diameter of the rollers will cause a certain degree of mechanical interference to the longitudinal bending of the sheet metal during rolling, thereby limiting the longitudinal bending deformation of the formed surface.
The doubly-curved surfaces rolling equipment based on variable section arc rollers is shown in Fig. 3. The rolling equipment uses a convex roller and a concave roller as forming tool. The generatrixes of the convex roller and the concave roller are arcs with radii of 206 and 212mm, respectively. The central cross-section diameters of the rollers are both 60mm. The equipment could accurately adjust the displacement of the convex roller along the vertical direction. Each time the crank handle rotates 360°, the convex roller moves up or down 0.01mm. The material of the metal plate used in the experiment is 2024 pure aluminum, and its mechanical properties at room temperature are shown in Table 1.
|
Material |
Density (kg/m3) |
Elastic modulus (Mpa) |
Poisson's ratio |
Yield limit (Mpa) |
|---|---|---|---|---|
|
2024AL-O |
2780 |
72400 |
0.33 |
77 |
4.1 Finite element modeling
A series of elastoplastic finite element models of doubly-curved surface rolling with different central cross-section diameters of the rollers are established by ABAQUS/Explicit. The models consist of a sheet metal, a convex roller and a concave roller. The sheet metal with size of 250×80×2.7 mm is a 3D deformable solid and is meshed into four layers in the thickness direction. The mesh sizes in the length and width directions of the sheet are 0.5 and 1.5 mm, respectively. And element type of the sheet is set as the C3D8R which is an eight-node linear brick, reduced integral, and hourglass control element.
The generatrix radii of the convex and concave rollers are 206 and 212 mm, respectively. The convex and concave rollers are 3D discrete rigid shells and are meshed by R3D4 (four-node, 3D, bilinear quadrilateral) rigid body elements, which size is 0.6×0.6 mm. The forming speed is set to 300 mm/s according to the quasi-static conditions recommended by Abaqus Analysis User's Guide.
The penalty friction formula and the Coulomb friction model are used, and set the friction coefficient to 0.12. Due to the symmetry of the sheet and rollers, their boundaries and load conditions, a 3D axisymmetric model was chosen to improve calculation efficiency.
4.2 Experimental verification of finite element model
In order to verify the reliability of the finite element model, experiments with different rolling reductions were carried out. Three different maximum rolling reductions of 0.06, 0.08 and 0.10 mm were selected to form spherical surfaces. The spherical surfaces formed by numerical simulation and experiment are shown in Fig. 5 (h is the maximum rolling reduction). The longitudinal curvature of the formed spherical surface increases as the rolling reduction increases. The plastic strain of the formed spherical surfaces is slender and uniformly distributed along the rolling direction (see Fig. 6), indicating that the quality of the formed surface is good. Figure 7 shows the longitudinal curvature of the formed spherical surface along the rolling direction. It can be seen from Fig. 7 that the finite element simulation results are in good agreement with the experimental results, which proves that the finite element model is reliable.
5.1 Longitudinal bending deformation analysis
Figure 8 shows the longitudinal curvature of spherical surfaces formed by different central-section diameters of the rollers with three maximum reduction conditions of 0.06, 0.08 and 0.10mm. The central cross-section diameters of the rollers are 40, 60, 80, 100 and 120 mm, respectively. When the central cross-section diameter of the rollers is small (less than 80 mm), the longitudinal curvature of the formed spherical surface increases with the increase of the central cross-section diameter; when the central cross-section diameter of the rollers is larger (greater than 80 mm), the longitudinal curvature of the formed spherical surface decreases with the increase of the central cross-section diameter.
It is analyzed that when the central cross-section diameter of the rollers is less than 80 mm, the variation in the dimension of the deformation zone will dominate the longitudinal bending deformation of the formed surface. At this time, the diameter of the rollers is relatively small, which will not inhibit the longitudinal bending of the sheet during rolling, so the longitudinal curvature of the formed surface increases with the increase of the central cross-section diameters. However, when the central cross-section diameter of the rollers is greater than 80 mm, the larger roller diameter will have greater mechanical interference on the longitudinal bending of the sheet metal, thereby inhibiting the longitudinal bending of the sheet (see Fig. 2b), resulting in the longitudinal curvature of the formed curved surface decreases with the increase of the central cross-section diameters.
When the maximum rolling reduction is 0.06, 0.08 and 0.10 mm, the longitudinal curvature range of the spherical surfaces formed with different central-section diameters are 0.124×10− 3, 0.375×10− 3 and 0.676×10− 3 mm− 1, respectively, which shows that the influence of the central cross-section diameter of the rollers on the longitudinal bending deformation of the formed spherical surface increases with the increase of the rolling reduction. That is to say, the greater the rolling reduction (the greater the longitudinal curvature of the target surface), the central cross-section diameter of the rollers has a greater influence on the longitudinal bending deformation of the formed spherical surface (see Fig. 8).
Figure 9 shows the difference between the longitudinal center extension and the longitudinal edge extension of the formed spherical surfaces with different center-section diameters of the rollers. When the central cross-section diameter of the rollers is relatively small (less than 80 mm), the difference in the longitudinal extension of the formed spherical surface increases with the increase of the central cross-section diameter. When the central cross-section diameter of the rollers is relatively large (greater than 80 mm), the difference in the longitudinal extension of the formed spherical surface decreases with the increase of the central cross-section diameter. And the greater the rolling reduction, the greater the influence of the roll diameter on the difference in longitudinal extension of the formed spherical surface (see Fig. 9). Since the longitudinal bending deformation of the formed curved surface is mainly determined by the difference in the longitudinal extension between the center and the edge, the influence of the roller diameter on the longitudinal extension difference of the formed surface is consistent with its influence on the longitudinal curvature of the formed surface.
5.2 Stability analysis of forming process
Figures 10, 11 and 12 are the equivalent plastic strain distributions of formed spherical surfaces with the maximum rolling reduction of 0.06, 0.08 and 0.10 mm, respectively; each figure contains the formed spherical surfaces with five different central cross-section diameters of the arc-shaped rollers (40, 60, 80, 100 and 120mm). When the central cross-section diameter is 40 mm, the equivalent plastic strain fluctuates significantly along the rolling direction. As the central cross-section diameter of the rollers increases, the equivalent plastic strain of the formed spherical surfaces along the rolling direction becomes smoother. This indicates that when the central cross-section diameter of the rollers is small, the stability of the rolling process is insufficient, and the formed surface cannot obtain uniform longitudinal bending deformation.
In order to characterize the stability of the rolling process, select the sheet metal tail node (see Fig. 13), and output its vertical displacement component U2 during the forming process. When using rollers with different central cross-sectional diameters to roll spherical surfaces, the displacement component of the tail node in the vertical direction during the sheet metal forming process is shown in Fig. 14. The first 0.1 second of the forming process is the time for convex roller to press down. After the 0.1 second, the rollers rotate synchronously, and the forming speed is 300 mm/s.
The rolling of doubly-curved surfaces belongs to the line forming method, and the formed metal plate is usually a thin plate, and the rolling reduction is usually very small. Therefore, during the rolling process, the length of the deformation zone where the sheet metal is squeezed in the roll gap along the rolling direction is small. Due to the weight of the sheet metal, mechanical vibration and other factors, there is a slight swing at the end of the sheet during rolling. It can be found from Fig. 14 that the larger the central cross-section diameter of the rollers, the smaller the swing amplitude of the sheet tail during rolling, and the more stable the forming process. When the central cross-section diameter of the rollers is 40 mm, the swing amplitude of the sheet tail increases with the increase of the rolling reduction; when the central cross-section diameter of the rollers is 60 mm, the swing amplitude of the sheet tail is not affected by the rolling reduction; when the central cross-section diameter of the rollers is 80, 100 and 120 mm, the swing amplitude of the sheet tail decreases with the increase of the rolling reduction. Obviously, the stability of the rolling process increases with the increase of the central cross-section diameter of the rollers.
In order to characterize the shape accuracy of the formed spherical surface, the longitudinal curvature deviation is defined
(1)In Eq. (1), ELi is the longitudinal curvature deviation of the formed spherical surface; ρLi is the fitted value of the local longitudinal curvature of the formed spherical surface; ρL is the fitted value of the overall longitudinal curvature of the formed spherical surface.
Figure 15 shows the longitudinal curvature deviation of the spherical surfaces formed by rollers with different central cross-sectional diameters. The longitudinal curvature deviation of the formed spherical surface is consistent with the vertical displacement of the sheet tail (compare Fig. 14 and Fig. 15). The greater the vertical displacement of the sheet tail, the greater the longitudinal curvature deviation. When the central cross-section diameter of the rollers is 40 mm, the longitudinal curvature deviation of the formed spherical surface is the largest. The longitudinal curvature deviation of the stable forming area (except the head and tail transition area) of the formed spherical surface decreases with the increase of the central cross-section diameter. And it can be found that when the central cross-section diameter is 80, 100 and 120 mm, their difference in the longitudinal curvature deviation is very small. This shows that when the central cross-section diameter of the rollers increases to a certain extent, the rolling process will reach a relatively stable state. In this state, the longitudinal curvature deviation of the stable forming area is small and the shape accuracy is good for the formed spherical surfaces.
In this paper, the influence of the central cross-section diameter of rollers on the longitudinal bending deformation of the formed spherical surfaces and the stability of the doubly-curved surface rolling process is investigated. The research results are summarized as follows:
1. The appropriate central cross-section diameter of rollers could enable the formed spherical surface to obtain greater longitudinal bending deformation and better shape accuracy under the same rolling reduction.
2. When the central cross-section diameter of rollers is relatively small, the longitudinal bending deformation of the formed spherical surface increases with the increase of the central cross-section diameter; but when the central cross-section diameter increases to a certain extent, the excessively large roller diameter will mechanically interfere with the longitudinal bending deformation of the sheet metal, so that the longitudinal bending deformation of the formed spherical surface decreases with the increase of the central cross-section diameter of the rollers.
3. The larger the central cross-section diameter of rollers, the more stable the forming process. However, when the central cross-section diameter increases to a certain extent, the increasing roller diameter has little effect on the stability of the rolling process.
Ethical Approval The manuscript has not been submitted or published anywhere. It will not be submitted elsewhere as well.
Consent to Participate All authors consent to participate in the author team of this submitted manuscript.
Consent to Publish The manuscript was approved by all authors to publish.
Authors' Contributions Xiang Chang wrote this article, conducted the experiments, established the finite element model, processed the numerical and experimental data, and analyzed the results. Wenzhi Fu provided valuable suggestions on the study, and was responsible for guiding the work. Mingzhe Li proposed the new forming method, theoretically demonstrated the feasibility of this method, provided valuable suggestions on the study, and was responsible for guiding the work. Xintong Wang provided valuable suggestions, and helped to complete the experiments. Weifeng Yang helped to complete the experiments. Yushan Deng provided valuable suggestions on the study, and was responsible for guiding the work.
Funding This work was financially supported by the Project of Jilin Provincial Scientific and Technological Department (20150201005GX).
Competing Interests The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.
Availability of data and materials The authors declare that the data and the materials that support the findings of this study are available within the article and from the corresponding author upon reasonable request.
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