Simulation and control of high-order flatness in rolling wide titanium strip with 20-high mill

: In wide titanium strip cold rolling process, the high-order flatness defect is one of the most difficult problem to be solved. Based on finite element method, considering the anisotropic mechanical characteristics of titanium, an implicit integration calculation model of rolls-strip for 20-high mill was developed, which can simulate the dynamic rolling process. The model was used to analyze the adjustment characteristics of high-order flatness on the 20-high mill. The simulation revealed as the increasing of the 1# & 7# AS-U or 2# & 6# AS-U press adjustment, the high-order flatness defect was more aggravated; And as the increasing of 3# & 5# AS-U or 4# AS-U press adjustment, the high-order flatness defect was alleviated to some extent. In addition, the high-order flatness cannot be effectively adjusted by roll shifting. Finally, the industrial test showed that increasing 4# AS-U press adjustment can effectively relieve the high-order flatness defect.


Introduction
Commercially pure titanium strip is a widely used titanium metal material, and its mechanical properties anisotropy and microstructure evolution law have attracted the attention of numerous scholars. Liu et al. [1] studied the anisotropy and forming characteristics of commercially pure titanium sheet under different rolling paths and found that cross rolling showed a smaller planar anisotropy. Liu et al. [2] studied the effects of cold rolling and annealing on the microstructure evolution of commercially pure titanium sheet and found that twinning was the dominant deformation mechanism when the cold rolling reduction is less then 40%, that 20% cold rolled sheet recrystallized almost completely when the annealing time reached 60min.
Roth et.al. [3] studied the anisotropy of mechanical behavior of commercially pure α-Ti samples cut along the rolling and transverse direction under tensile tests and suggested a mechanism based on the consideration of the dislocation glide anisotropy.
Compared to the research on the microstructure of commercially pure titanium, the research on the shape control of titanium strips in cold rolling process is not flattening model was established. In addition, the 20-high mill simulation model based on the finite element method has high calculation accuracy because of few assumptions, but it is suitable for offline calculation due to the lengthy calculation time. Zhang et.al [10] established a threedimensional 20-high mill finite element model with ANSYS and studied the contact pressure between rolls and the shape of the load roll gap. Li et.al [11] used the finite element method to establish a 20-high mill roll system and strip integrated elastic-plastic deformation statics model.
Besides the research on the simulation model of a 20high mill, some scholars have also studied the quarter waves control of the 20-high mill. Hara et.al. [12] found   The complex high-order flatness defect mentioned above proposes a new research subject, and it is necessary to study the high-order flatness defect control characteristics of 20-high mill. The roll system of the 20high rolling mill has a complex structure, and the small roll diameter work roll is more prone to high-order deflection under the action of the rolling force. Coupled with the characteristics of anisotropic material in the elastoplastic stage of commercially pure titanium, the deformation behavior of the roll system and strip becomes quite complicated. Therefore, the study of high-order flatness defect requires a higher-precision calculation model. (b) compressive stress area high-order flatness defect (a) high-order flatness defect However, the rolls-strip coupling model established by many scholars for 20-high mill with FEM belongs to the static model, which cannot realize the dynamic rolling between work rolls and strip. As a result, the contact arc length under the static model is twice than the actual situation, as shown in Figure 3. Thus, this research uses ABAQUS finite element software to establish an implicit integration calculation model of rolls-strip for 20-high mill that can realize dynamic rolling between work rolls and strip and analyzes the adjustment characteristics of shape control means. and C backup rolls. The AS-U adjustment mechanism had seven segments eccentric rings that could realize independent adjustment, as shown in Figure 5. Relative to the zero position of eccentric ring, the seven segments can realize eccentricity in the range of -0.28 mm to +0.28 mm, and the percentage can be adjusted from -100% to +100%.

The model geometric parameters
The geometric dimensions of the rolls of 20-high mill are given in Table 1. The backup roll is comprised of mandrel, eccentric ring, backing bearing, and saddle.
Ignoring the influence of the gap between these components, the backup roll is simplified into an integrated stepped structure, as shown in Figure 6

The model material parameters
In order to calculate the deflection and flattening state of the roll more accurately, the roll is defined as an elastic material in this model, as shown in Table 2.  [15] studied the relationship between engineering constants and elastic matrix with the hexagonal crystal system structure. Wang et.al [16] proposed that the elastic matrix of the hexagonal crystal system has 5 independent components. The elastic matrix of commercially pure titanium hexagonal crystal system expressed by engineering constants can be given: 13 where 1 is the rolling direction, 2 is the lateral direction, and 3 is the thickness direction, 1, 2, 3 are the orthotropic principal axes, the rolling direction is the direction of the symmetrical principal axis of the hexagonal crystal system.
Ei is the elastic modulus in the i direction, Gij is the shear modulus between i and j direction, μij is the Poisson's ratio of the shrinkage (extension) in the j direction caused by the tensile (compressive) stress in the i direction.
There are 5 independent elastic coefficients of commercially pure titanium TA2, which are E1, E2, μ12, μ23, G12, the elastic stage parameters of commercially pure titanium TA2 can be obtained through the room temperature tensile test, as shown in Table 3. The deformation resistance of commercially pure titanium TA2 in the rolling direction is obtained through the compression test at room temperature, as shown in Table 4. In the research on plastic deformation of commercially pure titanium with significant anisotropy, Toussaint et.al [17] used the Hill'48 anisotropic yield criterion to simulate the stamping process of pure titanium scoliosis. Xue et.al [18] considered the material anisotropy when studying the deep drawing of commercially pure titanium and adopted the Hill'48 anisotropic yield criterion in the finite element simulation. In this study, the Hill'48 anisotropic yield criterion is used to describe the plastic deformation process of titanium strip. The Hill'48 anisotropic yield criterion is expressed by the following equation: where F, G, H, L, M, N is the material parameters characterizing the anisotropy.
The material parameters characterizing the anisotropy can be calculated: where σ0 is the reference yield stress, the yield stress in the rolling direction is used as the reference yield stress.
The anisotropic yield stress ratio can be calculated: where σX, σY, σZ is the yield stress in anisotropic principal axis direction, τT, τR, τS is the shear yield stress relative to the anisotropic principal axis. Because the metal sheet is thin, τR and τS cannot be obtained, this study assumes τR=τS=τT.
Wang et.al [19] studied the anisotropic yield stress of commercially pure titanium TA2. This study determined the anisotropic yield stress ratio based on its research results, as shown in Table 5.

Mesh
In order to ensure the calculation accuracy of the model, the mesh size is refined in the contact area between the rolls and the contact area between the rolls and strip.
Considering that the grid torsion is too large in the process of rolling large deformation, and the linear reduction integral element solves the displacement more accurately when the grid is deformed by torsion. Therefore, the roll and the strip adopt the eight-node linear hexahedral noncoordinating element (C3D8R), as shown in Figure 7.

Boundary conditions and loading settings
The model realizes the contact between rolls, work rolls and the titanium strip by defining contact pairs. There are 26 contact pairs in the model. The model will not consider the friction between the rolls. According to field production experience and the friction coefficient curve of the Stone cold rolling [20], the friction coefficient between the work roll and the titanium strip is set 0.05.
The tie restraint method is adopted in the model to fix

Results and discussion 3.1 Lateral thickness difference after rolling without shape adjustment
In this part, the FEM model established above is used to calculate the rolled lateral thickness difference of the titanium strip without the shape adjustment. The simulation conditions are as shown in Table 7. The lateral thickness difference after rolling is given in Figure 9. There is a local depression near the 100 mm-300 mm from the edge. This study defines the 100 mm-300 mm from the edge as the high-order flatness defect area, which is consistent with the compressive stress of the shape meter.
It shows that the occurrence of high-order flatness defect is related to the local thickness reduction of the strip after rolling. The lowest thickness after rolling in the high-order flatness defect area can be calculated: where hγ is the lowest thickness after rolling in the highorder flatness defect area. haim is the target thickness after rolling in the high-order flatness defect area, this study is 1.211 mm. γ is calculated by formula (5).
The rolled length of the target thickness and the lowest thickness in the high-order flatness defect area can be calculated: where H is the thickness before rolling in the high-order flatness defect area, this study is 1.540 mm. L is the length before rolling in the high-order flatness defect area, this study is 1000 mm. haim and hγ can be calculated by formula (7).

Lateral thickness difference after rolling under AS-U press adjustment
The AS-U press adjustment is one of the essential shape  Table 8. The lateral thickness difference after rolling is shown in  After the symmetrical position AS-U combination control, the value of depression γ and the flatness value caused by the local depression λγ are in Table 9. After 1#&7# AS-U press adjustment, the value γ and λγ increased by 17.6% and after 2#&6# AS-U press adjustment, the value γ and λγ increased by 29.4%, 2#&6# AS-U press adjustment will aggravate the high-order flatness defect more easily. After 3#&5# AS-U and 4# AS-U press adjustment, the value γ and λγ decreased by 41.2%.
In order to better grasp the symmetrical position AS-U press adjustment characteristics on high-order flatness defect area, it is also necessary to master the symmetrical position AS-U press adjustment characteristics on the full width of titanium strip. This study uses the lateral thickness difference between AS-U press adjustment and without the AS-U press adjustment as the variation of thickness profile, as shown in Figure 11. It can be founded that increasing adjustment is 1.6 times of that of 4# AS-U press adjustment.
In the case of large tension in cold rolling, the increase of tensile stress at the edge is easy to cause strip breakage, 4# AS-U press adjustment is more conducive to alleviate the high-order flatness defect.   Table 10. The lateral thickness difference after rolling is shown in Figure 12. After different amount of roll shift adjustment, there is local thickness depression near the 100 mm-300 mm area from the edge, which shows that roll shift adjustment also cannot completely eliminate thickness depression in the high-order flatness defect area.   In order to compare the thickness change across the strip width after roll shift adjustment, this study uses the lateral thickness difference between the roll shift adjustment and without the roll shift adjustment as the variation of thickness profile, as shown in Figure 13.

Industry test of 4# AS-U press adjustment characteristics
This study uses the 20-high mill FEM model to analysis the adjustment characteristics of the high-order flatness defect by two adjustment methods, and finds that the  The stress distribution on the shape meter before and after the industry test is shown in Figure 14. It can be found that the stress distribution has changed after increasing 4# AS-U press adjustment. The variation of stress distribution after 4# AS-U press adjustment is shown in Figure 15.

Availability of data and materials
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