Issue 
Metall. Res. Technol.
Volume 117, Number 3, 2020



Article Number  301  
Number of page(s)  12  
DOI  https://doi.org/10.1051/metal/2020019  
Published online  30 April 2020 
Regular Article
The mechanical parameters modeling of heavy steel plate snake/gradient temperature rolling with the same roll diameters
^{1}
School of Mechanical Engineering, Taiyuan University of Science and Technology,
Taiyuan
030024, PR China
^{2}
State Key Laboratory of Rolling and Automation, Northeastern University,
Shenyang
110819, PR China
^{*} email: neu2015@163.com
Received:
19
November
2019
Accepted:
16
March
2020
The grains in the center of the heavy steel plate can be refined by the snake/gradient temperature rolling, and the deformation penetration, the microstructure, and the properties of the steel plate will be improved. The existing rolling mechanical models are not suitable for the snake/gradient temperature rolling, so it is necessary to establish the mechanical parameters model of the snake/gradient temperature rolling to instruct production. The yield criterion of rolled material was modified based on the idea of equivalent flow stress. The element stress analyses were carried out based on the uniform normal stress and nonuniform shear stress in the vertical sides of each slab. Then the equilibrium equation of the unit pressure based on the slab method was established on this basis. The deformation region was divided into three layers (the top layer, the bottom layer, and the central layer) and maximum four zones (back slip zone, front slip zone, cross shear zone, and reverse deflection zone) according to the temperature distribution and position of the neutral point, and then the 12 zones were formed during the snake/gradient temperature rolling. The boundary conditions of the existence of the back slip zone, the front slip zone, and the cross shear zone were established according to the relationship between the threading angle and the neutral angle. The accurate mechanical parameters model of the rolling force and rolling torque of the snake/gradient temperature rolling with the same roll diameters was set up on this basis. The ANSYS software has been used in the rolling process simulation by many scholars, and the calculating precision has been verified. So the rolling processes were simulated by the ANSYS software to validate the model precision. The results show that the maximum relative deviation of the rolling force analytic model is less than 7% compared with the numerical method, and the maximum relative deviation of the rolling torque analytic model is less than 11% compared with the measured results. The mechanical parameters model can accurately predict the rolling force and rolling torque during the snake/gradient temperature rolling with the same roll diameters, so as to provide a theoretical basis for the design of rolling mill and the setup of the process parameters.
Key words: heavy steel plate / snake/gradient temperature rolling / slab method / rolling force / rolling torque
© EDP Sciences, 2020
1 Introduction
Steel is known as an industrial food as industrial food, and it is the most used metal material by humans. Various fields of social production and life have a wide range of applications. There is a great demand for heavy steel plates with fine microstructure and properties in all walks of life, such as construction machinery, mining machinery, pressure vessel, bridge, warship, armor, and so on [1]. The maximum thickness of continuous casting slab is 350 mm. So, in the case of thick finished products, the required rolling reduction ratio cannot be easily satisfied. The deformation in the inner region is insufficient, the microstructure and properties between the surface and inner region are nonuniform. Then, the heavy steel plate with a fine performance in the inner region cannot be easily to be obtained because of the small total rolling reduction ratio and the nonuniform deformation. The insufficient deformation in the inner region becomes the key technology bottleneck to produce the heavy plates with fine property. It is of vital importance to solve the nonuniform deformation problem to improve the microstructure and properties in the inner region of the heavy steel plate [2].
The grain of the asymmetrically rolled steel strip was refined compared to the symmetric rolling which was proved by many experimental tests on microstructure and properties [3]. The plate bending phenomenon will be appearing because of the different rolling linear velocity between the top and bottom work rolls [4,5]. The bending phenomenon can be avoided by the strip tension during steel strip production, but it is difficult to avoid this phenomenon during heavy steel plate production by this tension, as it can affect the plate turning and the threading process. In order to solve the heavy plate bending problem, the slow work roll moves to a distance along the rolling direction. A new rolling method which is called snake rolling was formed. The reverse deflection zone during snake rolling was formed to prevent bending deformation based on the back slip zone, the front slip zone, and the cross shear zone.
The shear stress increases due to the existence of the cross shear zone, which is beneficial to improve the throughthickness homogeneity of the shear strain [6,7]. Li et al. [8] and Yang et al. [9] studied the snake rolling by the finite element method and experiment, and the plate curvature became well by adjusting the work roll offset and the roll speed ratio. Jiang et al. [10] studied the plate curvature affecting the law during the snake rolling process by the finite element method. The results showed that the snake rolling achieved asymmetrical rolling and plate curvature control, and the plate curvature can satisfy the requirements of the rolling process. Snake rolling will play an important role in the heavy steel plate production.
There is another way to improve the nonuniform deformation which is called gradient temperature rolling (GTR) [11]. The great temperature gradient along the thickness of the heavy steel plate can be formed during the ultrafast cooling, and then the deformation resistance gradient along the thickness will be formed. The deformation resistance of the top and bottom surface layers is higher than the central part. So, the deformation becomes difficult to be appeared in the surface layer compared with the central part, where the deformation in the central part of the steel plate can be improved.
Xie et al. [12] studied the highstrength ultraheavy plate processed with GTR, intercritical quenching and tempering, and the results showed that the GTR process could effectively refine prior austenite grain size in half and quarter thickness of the asrolled plate, and improved the fraction of intercritical ferrite and highangle boundaries in the asquenched specimens. Yu et al. [13], Li et al. [14,15], Xu et al. [16], and Ding et al. [17] studied the microstructure and properties of the GTR. The microstructure of the GTR process was analyzed by the finite element method and the experiments. The results showed that the austenite grains in the center part of the heavy plate can be refined by the GTR, and the microstructure and properties of the thick plate core were improved. Li et al. [18] conducted an experimental study on the appropriate cooling of the GTR. It was found that the plate prepared by GTR and air cooling had the best integrated property, and the mechanical properties of the rolled plate cooled in water were insufficient.
The heavy steel plate snake/gradient temperature rolling which synthesized the snake rolling and GTR was formed in order to improve the inner deformation and the plate curvature. It is necessary to establish a mechanical parameters calculation model of the snake/gradient temperature rolling to predict the rolling force and rolling torque during the rolling process. Presently, the slab method is the most commonly used analytical method to calculate the rolling force. Wang et al. [19] established a model to calculate the unit pressure during sandwich synchronous rolling with two different thicknesses of the outer layers by the slab method. Afrouz and Parvizi [20,21] set up a new rolling force analytical solution based on the modified slab method for asymmetrical clad sheet rolling. Qwamizadeh et al. [22] established a theoretical model based on the slab method to predicted the rolling force and the curvature at the exit for asymmetrical rolling of bonded twolayer sheets. Cao et al. [23] proposed a new logarithmic velocity field in the study of the hot tandem rolling, and obtained the rolling force and rolling torque model. Zhang et al. [24] introduced a deformation penetration coefficient in the process of studying the synchronous rolling of ultraheavy plate, and an analytical model of the rolling force was obtained by the variational method. Huang et al. [25] studied the rolling force of aluminum alloys during hot rough rolling, and a prediction procedure of the rolling force was developed by means of the MATLAB software.
The traditional synchronous rolling region is divided into two zones (the back slip zone and the front slip zone) and the asymmetrical rolling deformation region is divided into three zones (the back slip zone, the front slip zone, and the cross shear zone). But the snake/gradient temperature rolling deformation region will be divided into four zones (the back slip zone, the front slip zone, the cross shear zone, and the reverse deflection zone) and three layers (the top layer, the bottom layer, and the central layer). The deformation region becomes much more difficult compared with the traditional synchronous rolling and the asymmetrical rolling because of the offset distance of work rolls and the deformation resistance gradient. So the rolling mechanical model described above is not suitable for the snake/gradient temperature rolling. The model used to calculate the snake/gradient temperature rolling mechanical parameters that take into account the composition forms of the deformation region and deformation resistance gradient should be setup in the present paper.
2 Mechanical parameters modeling
The following assumptions were made to establish the mechanical parameters model of the snake/gradient temperature rolling:

The width of the heavy steel plate is much larger than the thickness, and the widthtothickness ratio of the heavy steel plate is great enough to simplify the heavy steel plate as a plane strain problem and the width spread can be neglected.

The deformation of the work rolls is too small compared with the steel plate, so the deformation of the work rolls is neglected, and the work rolls can be set as a rigid body. The diameters of the top and bottom work rolls are the same, and the linear velocity of the top work roll is less than the bottom work roll.

The temperature of the top and bottom layers decreased greatly, but the central temperature is kept nearly the same after the ultrafast cooling. The temperature in the center of the steel plate is unchanged basically, the temperature of the surface layers keep constant after cooling until the rolling process finished. Therefore, the steel plate can be divided into three layers along the thickness direction (the top layer, the bottom layer, and the central layer).

The top and bottom layers and the central layer of the steel plate obey the VonMises yield criterion.

The friction between the heavy steel plate and the work rolls is sticking friction, and the friction can be calculated by the Tresca friction theorem.
2.1 Gradient temperature analysis
The temperature of the top and bottom layers is lower than the central layer part after ultrafast cooling. The deformation resistance in the top and bottom surface layers is higher than the central layer because of the existence of temperature gradient. Thus, the transient analysis of the ultrafast cooling process of the steel plate was performed with the ANSYS software. The length of the steel plate was 2000 mm as the basic parameter. The temperature distributions along the thickness of the steel plate with different parameters are shown in Figure 1. The temperature distribution of the vertical section of the steel plate is shown in Figure 2.
Figures 1 and 2 show that the temperature along the thickness of the steel plate appears as an obvious stratification phenomenon. The temperature drop mainly appears in the surface of the steel plate, and the stratification phenomenon cannot be changed with the adjustment of the parameters with the working conditions required for the snake/gradient temperature rolling, such as the cooling time, the thickness, the initial temperature, and the heat transfer coefficient. Thus, the steel plate can be divided into top and bottom layers where the temperature drop occurs, and the central layer whose temperature remains substantially unchanged according to the numerical results, and the thickness of the top or bottom layers is about 1/10 of the total thickness of the steel plate. Thereby, the mechanical parameters model of the snake/gradient temperature rolling with the same roll diameters can be simplified.
Fig. 1 Temperature distribution along the thickness with different (a) cooling times; (b) thickness; (c) initial temperature; and (d) heat transfer coefficients. 
Fig. 2 Temperature distribution of the steel plate vertical profile (°C). 
2.2 Deformation region analysis
There is a roll offset between the top and bottom work rolls during the snake/gradient temperature rolling. The linear velocity of the bottom work roll is higher than the top work roll, so there are two neutral points (x_{n}_{1} and x_{n}_{2}, in Fig. 3) in the steel plate. The plastic deformation can be divided into four zones according to the position of these two neutral points. The schematic of the plastic deformation region of the snake/gradient temperature rolling with the same roll diameters is shown in Figure 3 according to the assumptions. The positive direction of x is to the left.
In zone I, the direction of the friction force of the top and bottom surfaces of the steel plate is the same as the rolling direction. The steel plate is threaded into the roll gap synchronously, which is called the back slip zone. In zone II, the velocity of the steel plate is higher than the top work roll but it is less than the bottom roll. The direction of the friction force of the top surface of the steel plate is opposite to the rolling direction, but the direction of the friction force of the bottom surface is the same as the rolling direction. The steel plate receives a strong shear action, which is called the cross shear zone. In zone III, the direction of the friction force of the top and bottom surfaces of the steel plate is opposite to the rolling direction, which is called the front slip zone. In zone IV, the velocity of the steel plate is higher than the work rolls, and the steel plate is only subjected to the pressure of the top roll, which is called the reverse deflection zone. Every zone can be divided into three layers because of the temperature gradient along the thickness of the steel plate.
Fig. 3 Schematic of the deformation region. 
2.3 Von Mises yield criterion
The flow stress of the top surface layer is σ_{1}, the thickness is h_{1}, the ratio of thickness to total thickness is β_{1} = h_{1}/H, and the shear yield stress is . The flow stress of the central layer is σ_{2}, the thickness is h_{2}, and the shear yield stress is . The flow stress of the bottom surface layer is σ_{3}, the thickness is h_{3}, the ratio of thickness to total thickness is β_{2} = h_{3}/H, and the shear yield stress is . Then, the equivalent flow stress of the steel plate is (1)
Any point in the deformation region according with the VonMises yield criterion is listed in (2)
Under the plane strain conditions, the shear stress in the contact surface is τ_{yz} = τ_{zx} = 0. According to the flow criterion σ_{z} = (σ_{x} + σ_{y})/2, the two equations are substituted into equation (2) and obtained the following equation: (3) where k_{e} is the equivalent shear yield stress, . The average shear stress of the top and bottom halves of the element is expressed as (4)where c_{1} and c_{2} are coefficients introduced by Salimi and Sassani [26], it is considered to be c_{1} = c_{2} = 1 in the cross shear zone and c_{1} = c_{2} = 0.5 in other zones.
Substituting equation (4) into equation (3) and the yield criterion for the top and bottom portions of the rolled plate is obtained, as is listed in (5)
Considering , , the yield criterion of the rolled piece can be obtained as is shown in (6)
The heavy plate in zone IV is only in contact with the top work roll and it is only subjected to the shear stress and normal stress from the top work roll. The shear stress of the bottom work roll is zero, and the average shear stress in zone IV is . Then, the yield criterion (7) of the heavy plate in zone IV is obtained by introducing it into equation (3) (7)
2.4 Unit pressure modeling
The deformation region is divided into 12 zones by the four zones and the three layers, as shown in Figure 3. Twelve elements are taken out for unit pressure modeling from the 12 zones one by one, as shown in Figure 4. The element in the top layer of the rolled piece in zone I is taken out for indepth analysis, as shown in Figure 5.
In Figure 5, the horizontal force acting on the BD plane is σ_{x}_{1}h_{x}_{1}, the horizontal force acting on the AC plane is (σ_{x} + dx)(h_{x}_{1} + dh_{x}_{1}), the horizontal force acting on the AB plane is , and the horizontal force acting on the CD plane is .
The sum of forces acting along the horizontal direction of the element in Figure 5 is (8)
The force equilibrium equation along the horizontal direction of the element is obtained by sorting equation (8) (9)
Similarly, the force equilibrium equation of other elements shown in Figure 4 can be obtained by this method as indicated above.
The threading angle of the snake/gradient temperature rolling is generally less than 30°, so the contact arc can be replaced by a parabola rather than a line segment. Then, the rolling reduction of the top and bottom work rolls is expressed as (10) (11)
The total thickness of the heavy plate in the deformation region as shown in Figure 3, is obtained as (12) according to equations (10) and (11) (12)
Fig. 4 Twelve elements taken out from the deformation region. 
Fig. 5 Force analysis of the top element in zone I. 
2.4.1 Unit pressure in zone I
The horizontal direction force equilibrium equations of the three elements in zone I with three layers are shown in (13) where , .
As a result of h_{x}σ_{x} = h_{x1}σ_{x1} + h_{x3}σ_{x2} + h_{x3}σ_{x3}_{, p1 = p − τ1tanθ1}, p_{2} = p − τ_{2}tanθ_{2}, and through the integration of x, the unit pressure of zone I is obtained as (14) where, , , ,, and .
2.4.2 Unit pressure in zone II
The horizontal direction force equilibrium equations of the three elements in zone II with three layers are shown in (15) where tan θ_{1} = x/R, .
As a result of h_{x}σ_{x} = h_{x1}σ_{x1} + h_{x3}σ_{x2} + h_{x3}σ_{x3}_{, p1 = p + τ1tanθ1}, p_{2} = p − τ_{2}tanθ_{2}, and through the integration of x, the unit pressure of zone II is obtained as (16) where, , and .
2.4.3 Unit pressure in zone III
The horizontal direction force equilibrium equations of the three elements in zone III with three layers are shown in (17) where tanθ_{1} = x/R, tanθ_{2} = (x − d)/R.
As a result of h_{x}σ_{x} = h_{x1}σ_{x1} + h_{x3}σ_{x2} + h_{x3}σ_{x3}_{, p1 = p + τ1tanθ1}, p_{2} = p + τ_{2}tanθ_{2}, and through the integration of x, the unit pressure of zone III is obtained as (18) where .
2.4.4 Unit pressure in zone IV
The horizontal direction force equilibrium equations of the three elements in zone IV with three layers are shown in (19) where .
As a result of h_{x}σ_{x} = h_{x1}σ_{x1} + h_{x3}σ_{x2} + h_{x3}σ_{x3}, p_{1} = p + τ_{1}tanθ_{1}, p_{2} = 0, , and through the integration of x, the unit pressure of zone IV is obtained as (20) where .
2.5 Rolling force and rolling torque modeling
It is assumed that the rolling deformation region is composed of the back slip zone, the cross shear zone, the front slip zone, and the reverse deflection zone. Equation (21) is the model to calculate the length of the rolling deformation region [27,28] (21)
The boundary conditions are x = 0 and q = 0 at the exit of the top work roll. The unit pressure of the reverse deflection zone is , and C_{IV} can be obtained by substituting it into equation (20).
The boundary conditions at the entrance of the rolling deformation region are x = l and q = 0. The unit pressure of the back slip zone is , and C_{I} can be obtained by substituting it into equation (14). Since p_{III}(x = d) = p_{IV}(x = d) at x = d, C_{III} can be obtained.
In the position of x = x_{n}_{1}, p_{I}(x = x_{n}_{1}) = p_{II}(x = x_{n}_{1}), and then C_{II}(x = x_{n}_{1}) is obtained. In the position of x = x_{n}_{2}, p_{II}(x = x_{n2}) = p_{III}(x = x_{n}_{2}), and then C_{II} (x = x_{n}_{2}) is obtained. The unit pressure is continuous at the neutral point of the top and bottom work rolls, so (22)
Equation (23) can be obtained according to the principle that the volume of the heavy plate remains unchanged during the rolling process (23) and can be obtained according to the geometric relationship, and then equation (24) can be obtained by substituting equation (12) into equation (23) (24)x_{n}_{1}, x_{n}_{2}, and C_{II} can be obtained by combining equations (22) and (24).
The four zones of the rolling deformation region may not exist at the same time due to the movement of the neutral point. The composition of the deformation region needs to be determined, and the solving flowchart is shown in Figure 6.
Case 1: The rolling deformation region is composed of the back slip zone, the cross shear zone, the front slip zone, and the reverse deflection zone when d < x_{n}_{1} < l, d < x_{n}_{2} < x_{n}_{1}. The unit pressure can be integrated along the contact arc, and then equation (25) used to calculate the snake/gradient temperature rolling force will that is obtained as (25)
The rolling torque can be calculated by integrating the friction force along the contact arc of the top and bottom work rolls. Equations (26) and (27) used to calculate the rolling torque of the top and bottom work rolls is obtained, respectively, (26) (27)
Case 2: The rolling deformation region is composed of the back slip zone, the cross shear zone, and the reverse deflection zone when d < x_{n}_{1} < l, x_{n}_{2} ≤ d. The boundary condition in the entrance of the rolling deformation region is x = l and q = 0. The unit pressure of the back slip zone can be known,, and the C_{I} can be obtained by substituting it into equation (14). p_{II}(x = d) = p_{IV}(x = d) at x = d, and C_{II} is obtained. p_{I}(x = x_{n}_{1}) = p_{II}(x = x_{n}_{1}) at x = x_{n}_{1}, and then x_{n}_{1} can be obtained by calculation.
The rolling force is obtained by integrating the unit pressure along the contact arc (28)
The rolling torque can be calculated by integrating the friction force along the contact arc of the top and bottom work rolls. Equations (29) and (30) are used to calculate the rolling torque of the top and bottom work rolls is obtained, respectively, (29) (30)
Case 3: There are the cross shear zone and the reverse deflection zone when x_{n}_{1} ≥ l and x_{n}_{2} ≤ d. The boundary conditions at the entrance of the rolling deformation region are x = l and q = 0. The unit pressure of the rolling zone can be obtained, and C_{II} can be obtained by substituting it into equation (16). p_{II}(x = d) = p_{IV}(x = d) at x = d, and C_{II} can also be obtained. The value of C_{II} was calculated by the two methods may be different because of the solving precision. So the average value of C_{II} obtained by the two boundary conditions was taken to reduce the error. Equation (31) used to calculate the snake/gradient temperature rolling force will be obtained by integrating the unit pressure along the contact arc (31)
The rolling torque can be calculated by integrating the friction force along the contact arc of the top and bottom work rolls. Equations (32) and (33) used to calculate the rolling torque of the top and bottom work rolls is obtained, respectively, (32) (33)
Fig. 6 Solving the flowchart. 
3 Results and discussion
Comparisons of analytical results from other researchers’ models with those from the experimental studies and the numerical method have shown good agreements [20,21]. As a new rolling technology, the heavy steel plate snake/gradient temperature rolling experiment is difficult to be conducted in the laboratory because of the great equipment casts. The numerical method is reliable and it can be used to simulate the snake/gradient temperature rolling process to verify the model precision.
ANSYS LSDYNA is a finite element method for dynamic analysis and it can be used for metal plastic forming simulations. The snake/gradient temperature rolling with different process parameters was simulated by the ANSYS LSDYNA software. In the simulation, the work roll material is defined as rigid, but the elastic modulus also needs to be input into the ANSYS software interface. The dynamic friction coefficient and static friction coefficient need to be input synchronously. The data required for the simulation is shown in Table 1. The finite element method model used for simulations is shown in Figure 7. The rolling force can be obtained and the rolling forces with different process parameters were obtained as shown in Figures 8–13.
The snake/gradient temperature rolling processes were simulated when the linear velocity of the bottom work roll was 1.5, 1.51, 1.52, 1.53 and 1.65 m/s respectively. The rolling force was obtained from these simulations. Figure 8 shows the curve of the rolling force with different roll speed ratios. It can be seen that the rolling force gradually decreases as the roll speed ratio increases. The rolling force will be decreased with the increase of the roll speed ratio during the asymmetrical rolling process, which is also applicable to the snake/gradient temperature rolling. The two neutral points (x_{n}_{1}, x_{n}_{2}) move to the inlet and outlet of the deformation region, respectively, with the increase of the roll speed ratio and the cross shear zone also increases correspondingly. The shearing action is enhanced and the rolling force is reduced correspondingly. The maximum relative deviation between the analytic calculation results and the numerical calculation results is 5.41%.
The snake/gradient temperature rolling processes were simulated when the roll offset was 10, 15, 18, 20 and 22 mm respectively. Figure 9 shows the rolling force with different roll offsets, where the rolling force is increasing with the increase of the roll offset. The length of the reverse deflection zone will be increased with the increase of the roll offset, but the other deformation region length keeps nearly the same, so the rolling force will be increased slightly. The maximum relative deviation between the analytic calculation results and the numerical calculation results is 2.11%.
When the dynamic friction coefficient was 0.2, 0.25, 0.3, 0.35 and 0.4 respectively, the rolling force during the snake/gradient temperature rolling process is shown in Figure 10. The rolling force decreases gradually with the increase of friction coefficient. This is because that the neutral points move to the inlet and outlet of the roll gap and the length of the cross shear zone become longer. Then, the shearing effect is enhanced that the rolling force will be decreased correspondingly. The maximum relative deviation between the analytic calculation results and the numerical calculation results is 6.54%.
The rolling force was obtained from these simulations when the rolling reduction was 30, 35, 40, 45 and 50 mm respectively. Figure 11 shows the rolling force with different rolling reduction, where the rolling force gradually increases as the rolling reduction increases. The length of the plastic deformation region increases with an increase of the rolling reduction and then the rolling force increases correspondingly. After comparison, the maximum relative deviation between the analytic calculation results and the numerical results is 5.94%.
Figure 12 shows the rolling force at the plate thickness of 180, 200, 220, 250 and 280 mm respectively. The rolling force increases as the thickness of the plate increases, when the elongation coefficient is constant. The reason is that the length of the deformation region will be improved with the increase of the plate thickness when the elongation coefficient is constant. The thinner the plate and the lower the rolling force during the asymmetrical rolling process when the elongation coefficient and the roll speed ratio are constant. The snake/gradient temperature rolling also conforms to this rule. After comparison, the maximum relative deviation between the analytic calculation results and the numerical results is 3.45%.
The snake/gradient temperature rolling processes were simulated when the deformation resistance in the top and bottom layers was 100, 105, 110, 115 and 120 MPa respectively. Figure 13 shows the rolling force with different deformation resistance. The rolling force increases as the deformation resistance increases. The deformation resistance increases with the decrease of the plate surface temperature because of the existence of gradient temperature. The yield stress will be improved with the increase of the deformation resistance, and then the rolling force will also be improved when the length of the deformation region is kept constant. After comparison, the maximum relative deviation between the analytic calculation results and the numerical results is 2.71%.
Taking the 4300 mm thick plate mill in a factory as an example, the thickness of the finished plate is 120 mm; the work roll diameter is 1070 mm; the size of the continuous casting slab is 320 mm × 2050 mm × 3250 mm; the thickness of the plate after the first shaping rolling is 299 mm. The calculated results of the rolling torque were compared with the measured results. As shown in Table 2, the maximum relative deviation between the calculated results and the measured results is less than 11%.
In summary, the maximum relative deviation of the rolling force analytic model is less than 7% compared with the numerical method, and the maximum relative deviation of the rolling torque analytic model is less than 11% compared with the measured results. So the analytic model, used to calculate the mechanical parameters, is accurate and reliable. It can be used as the theoretical basis for the design of rolling mill and the setup of the process parameters.
Basic data used in simulation.
Fig. 7 Finite element method model used for simulations. 
Fig. 8 Rolling force curves with different roll speed ratio. 
Fig. 9 Rolling force curves with different roll offset. 
Fig. 10 Rolling force curves with different friction coefficient. 
Fig. 11 Rolling force curves with different rolling reduction. 
Fig. 12 Rolling force curves with different plate thickness. 
Fig. 13 Rolling force curves with different deformation resistance. 
Comparison between the calculated rolling torque with the measured results.
4 Conclusion
The temperature drop mainly appears in the surface of the steel plate during ultrafast cooling, and the stratification phenomenon cannot be changed with the adjustment of the parameters under the working conditions required for the snake/gradient temperature rolling. So the deformation region can be divided into three layers to simplify the mechanical parameters modeling process.
The deformation region was divided into three layers and maximum of four zones according to the temperature distribution and the position of the neutral point. The boundary conditions of the existence of the back slip zone, the front slip zone, and the cross shear zone were established according to the relationship between the threading angle and the neutral angle.
The yield criterion of the heavy plate was modified based on the idea of equivalent flow stress. Considering the uniform normal stress and nonuniform shear stress in the vertical sides of each slab, the element stress analyses were carried out. Then, the equilibrium equation of the unit pressure based on the slab method was established on this basis.
The accurate calculating model of the rolling force and rolling torque of the snake/gradient temperature rolling with the same roll diameters was set up according to the composition of the rolling deformation region and the boundary condition. The results show that the maximum relative deviation of the rolling force analytic model is less than 7% compared with the numerical method, and the maximum relative deviation of the rolling torque analytic model is less than 11% compared with the measured results. So, the model can accurately predict the rolling force and rolling torque during the snake/gradient temperature rolling with the same roll diameters.
Conflicts of interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Nomenclature
R: Radius of the top and bottom work rolls
n_{1},n_{2}: Rotation rates of the top and bottom work rolls
α_{1},α_{2}: Threading angles of the top and bottom work rolls
γ_{1},γ_{2}: Neutral angles of the top and bottom work rolls
l: Total length of the deformation region
l’: Length of the deformation region of the bottom work roll
x_{n}_{1},x_{n2}: Neutral point of the top and bottom work rolls
h_{0}: Plate thickness after rolling
h_{x}: Steel plate thickness in the x cross section
Δh_{1},Δh_{2}: Reductions of the top and bottom work rolls
p_{1},p_{2}: Top and bottom rolls pressures
p_{3},p_{4}: Pressures in the interface of each layer
θ_{1},θ_{2}: Variable angles of contact of the top and bottom work rolls
θ_{3},θ_{4}: Variable angles of contact of the interface of each layer
τ_{1},τ_{2}: Surface shear stresses at the top and bottom work rolls
τ_{3},τ_{4}: Surface shear stress at the interface of each layer
: Average shear stresses of the top and bottom portions
σ_{1},σ_{2},σ_{3}: Flow stresses of each layer
h_{x1},h_{x2},h_{x3}: Plate thicknesses of each layer
σ_{x1},σ_{x2},σ_{x3}: Horizontal normal stresses in each layer
β_{1},β_{2}: Ratios of the top and bottom layers along the total thickness
k_{1},k_{2},k_{3}: Shear yield stress of each layer
σ_{x},σ_{y},σ_{z}: Normal stresses along the x, y, and zaxis
τ_{xy},τ_{yz},τ_{zx}: Shear stresses in the xy, yz, and zx planes
k_{e}: Equivalent shear yield stress
m_{1},m_{2}: Friction coefficients of the top and bottom work rolls contacting with the surface of the rolled piece
q: Horizontal normal stress in the deformation region
v_{1},v_{2}: Linear velocity of the top and bottom work rolls
T_{1},T_{2}: Rolling torques of the top and bottom work rolls
Acknowledgment
The authors gratefully appreciate the financial support by the National Natural Science Foundation of China under Grant 51804206, the Major Science and Technology Projects in Shanxi under Grant 20181102016, and the Scientific and Technological Innovation Programs of Higher Education Institutions in Shanxi under Grant 2016164.
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Cite this article as: LianYun Jiang, Tao Zhen, Guo Yuan, JinBo Huang, YaoYu Wei, Heng Li, Ping Wang, The mechanical parameters modeling of heavy steel plate snake/gradient temperature rolling with the same roll diameters, Metall. Res. Technol. 117, 301 (2020)
All Tables
All Figures
Fig. 1 Temperature distribution along the thickness with different (a) cooling times; (b) thickness; (c) initial temperature; and (d) heat transfer coefficients. 

In the text 
Fig. 2 Temperature distribution of the steel plate vertical profile (°C). 

In the text 
Fig. 3 Schematic of the deformation region. 

In the text 
Fig. 4 Twelve elements taken out from the deformation region. 

In the text 
Fig. 5 Force analysis of the top element in zone I. 

In the text 
Fig. 6 Solving the flowchart. 

In the text 
Fig. 7 Finite element method model used for simulations. 

In the text 
Fig. 8 Rolling force curves with different roll speed ratio. 

In the text 
Fig. 9 Rolling force curves with different roll offset. 

In the text 
Fig. 10 Rolling force curves with different friction coefficient. 

In the text 
Fig. 11 Rolling force curves with different rolling reduction. 

In the text 
Fig. 12 Rolling force curves with different plate thickness. 

In the text 
Fig. 13 Rolling force curves with different deformation resistance. 

In the text 
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