The theoretical model of the effective strain establishment for the plate during the snake rolling

The snake rolling for the heavy plate rolling has been veri�ed that it can re�ne the grain and solve the problem of insu�cient deformation in the center of the plate to some extent. The effective strain along the thickness is mainly obtained by the �nite element method and the experiments which cost a long time and great material resources. A theoretical model of the effective strain is required to solve the problem. The metal �ow model including the shear deformation is established according to the actual asymmetric distribution of stress and strain in the deformation zone. The parabolic roll gap inlet boundary equation is proposed according to the kinematic characteristics of plate rolling. The plastic deformation power, shear power and friction power models are established according to the �rst variational principle of the rigid plasticity and the principle of virtual velocity. The analytic model to calculate the effective strain of the snake rolling has been established by the principle of minimum energy and the theory of continuum mechanics. The model accuracy has also been veri�ed by the �nite element method and the experiments. The minimum and the maximum relative error is 1.33% and 13.44% which is acceptable for industrial applications. This research will provide an important guidance for the process and process parameters setup.


Introduction
The heavy plate (the thicker plate) has been widely used in the aerospace, shipping, transportation and so on. There is strong demand for the heavy plate with high-performance. The insu cient deformation appeared in the center of the plate will greatly affect the strength of the plate. The thickness of continuous casting slab limits the maximum rolling reduction that the mill can achieve. So the little total rolling reduction will cause the central deformation to be insu cient and the performance in the center to be lower than the surface. The asynchronous rolling can produce great shear stress which is better for the improvement of the central deformation [1,2] . However, the plate will be bent after asynchronous rolling [3] . The snake rolling combines the asynchronous rolling and the plate straightening together. The snake rolling can not only improve the central deformation of the plate, but also keep the plate straightness. [4] . As a new and effective rolling process, the deformation calculation and analysis of the snake rolling are less [5] , so a great deal research should be conducted to instruct the production.
On the basis of asynchronous rolling, the slow work roll is moved a certain distance along the rolling direction to form the snake rolling. With the development of nite element method, many scholars have used the numerical simulation methods to simulate the snake rolling process to study of the center deformation and the plate curvature. Zhang et al. [6] used the nite element method to analyze the stress and strain distribution of 7050 aluminum alloy plate during the snake rolling and symmetric rolling. The results show that the center shear strain of the plate during snake rolling is bigger than the symmetric rolling. Yang et al. [7] studied the effects of roll offset, reduction and work roll velocity ratio on the strain distribution of AA7050 aluminum plate by the nite element method and experiments. Tamimi et al. [8] studied the effect of asymmetric rolling with different parameter on the shear strain along the thickness. Hao et al. [9] simulated the snake rolling process with different rolling parameters and established a formula to calculate the exit thickness after snake rolling. Ling et al. [10] studied the effects of velocity ratio, reduction and roll offset on the curvature of the aluminum alloy plate by nite element method. The above research ndings have high accuracy and provide a direct method for the analysis of rolling process. The study for snake rolling process cannot only rely on simulation because of its complex, so the theoretical analysis method as an effective and feasible idea is also required by many scholars and technicians.
Some scholars have used the energy method to establish the plastic forming mechanism models. The energy method consists of the ow function [11] , the upper and down boundary method [12] , the rigid plastic nite element method [13] and so on. Fu et al. [14] established a nite element model based on the plane strain assumption. The effects of roll offset, speed ratio, depression amount and roll radius on the curvature of the plate are studied by using the nite element model during the snake rolling process. Jiang et al. [15] studied the modeling method of the equivalent strain of the synchronous rolling based on the upper bound principle and ow function method. Sun et al. [16] used the ow function method to analyze the rheological properties of the cast-rolled deformation zone, and established the velocity eld of the cast-rolled zone. Mao et al. [17] established a stress model for high-temperature alloys using the ow function method, and analyzed the velocity ow line of the metal during the stable extrusion. Wang et al. [18] established a model to predict the free boundary pro le of metal forming based on the ow function method. Wang et al. [19] proposed a method to solve the plane strain rolling of ideal rigid-plastic material by ow function method. The method was used to calculate the velocity eld, stress eld, front and rear boundaries of the plastic zone, and the location of the neutral point on the contact arc and the unit pressure. Yang et al. [20] combined the ow function and nite element method to predict the volume free boundary into a forming pro le. The above research provides theoretical support for the strain calculation of the snake rolling, and also provides new ideas for the study of the snake rolling mechanism.
In summary, the strain calculation during the snake rolling process has not been analyzed in depth. The existing research is mainly focus on the nite element software and experiments. However, the previous research on the strain calculation is no longer applicable because of the great difference in the deformation feature for the snake rolling. It is necessary to establish an effective strain calculation model according to the deformation characteristics of the snake rolling. In this paper, the effective strain calculation model of the plate deformation area is established by the ow function and the upper bound method. An additional ow function is introduced into the ow function to improve the model accuracy. The rigid-plastic boundary is assumed to be a plane for the previous research. In this paper, the rigid-plastic boundary function is calculated by the relation between the ow function of rigid zone and plastic zone. And this rigid plastic interface can improve the model accuracy. The complex integration problem in the modeling process is solved by using multi-node Gauss-Legendre quadrature method. This method simpli es the integration process and shortens the solution time. This method can calculate the results online. The model can be used to calculate the effective strain at any point in the rolling deformation zone. It will play an important role in the optimization and the establishment of the rolling process.

The Effective Strain Modeling
The following assumptions were made for the effective strain modeling during the snake rolling with the same work roll diameter according to the actual model.
(1) The width spreading of the plate is too small to be ignored for the large width-to-thickness ratio, so the threedimensional rolling model is simpli ed to a two-dimensional plane strain model.
(2) The elastic recovery after hot rolling is much smaller than the plastic deformation, so the plate is considered as an ideal rigid-plastic material.
(3) Compared to the plate, the deformation of the work roll can be ignored, so the work roll is considered a rigid material.

Partition of deformation zone
The deformation zone is divided into back slip zone I, cross shear zone II, front slip zone III and reverse de ection zone IV according to the stress distribution within the deformation zone. The deformation zone can be divided into the following four cases according to the location of the neutral point x n1 (slow roll side) and x n2 (fast roll side): the rst case: x n1 and x n2 locate inside the deformation zone which consists of back slip zone I, cross shear zone II, front slip zone III and reverse de ection zone IV; the second case: x n1 locates outside the outlet of the roll gap and x n2 locates inside the entrance of the roll gap, and the deformation zone consists of back slip zone I, cross shear zone II and reverse de ection zone IV; the third case: x n1 and x n2 locate outside the deformation zone, and the deformation zone consists of cross shear zone II and reverse de ection zone IV only. In this paper, we focus on the rst and the second case because the third case almost does not occur in the normal process parameters. Figure 1 shows the two-dimensional schematic diagram of the deformation zone with the same work roll diameter. The plate is symmetrically dragged into the gap without de ection. The point of intersection between the vertical centerline of the bottom work roll and the horizontal centerline of the plate after rolling is taken as the origin of the coordinate system. The y-axis pointing to the top work roll and the direction of the x-axis is opposite to the rolling direction. Γ 1 and Γ 2 represent the rigid-plastic boundary line for the top and bottom part at the entrance respectively. The angle between the tangent at any point on Γ 1 and Γ 2 and the horizontal line is α and β. The deformation zones along the rolling direction are the entrance rigid zone RZ -plastic zone PZ -exit rigid zone EZ. In the zone RZ and zone EZ, material with velocity v 0 and v 1 respectively to do the rigid motion parallel to the axis. In the zone PZ, the material along the ow line obeys the permissible velocity eld. where ϕ RZ−top , ϕ PZ−top , ϕ EZ−top , ϕ RZ−bot , ϕ PZ−bot and ϕ EZ−bot represent the metal ow of the top and bottom parts of the inlet rigid zone, plastic zone and outlet rigid zone in the cross section, respectively. φ indicates the ow ratio on the cross section; In the plastic zone, φ(ax 2 + b)y[y-y 1 (x)] represents the additional ow function in the top half part of the plate;

Two-dimensional ow function model
(ax 2 + b) represents the shape function of the top half; y[y-y 1 (x)] is the control function of the contact surface in the top part of the plate; φ(cx 2 + d)y[y-y 2 (x)] represents the additional ow function in the bottom part of the plate; (cx 2 + d) represents the shape function of the bottom half; y[y-y 2 (x)] is the boundary control function of the contact surface in the bottom part of the plate.

Boundary model
The contact surface between the plate and the top and bottom rolls are treated as parabolic, and the geometric Eq. (3) and Eq. (4) are shown as follows.
For the top work roll: For the bottom work roll: The geometric equations of velocity discontinuous lines y 3 (x) and y 4 (x) (Γ 1 The strain rate component along the thickness direction of the top and bottom halves of the plate is Eq. (10). is the contact boundary equation between the bottom roll and the plate. The shear power is shown in Eq. (13).

14
where k is the effective shear yield stress, k = σ s / , and σ s is the yield stress. dΓ denotes the unit length of the inlet boundary Γ, as shown in Eq. (14).

15
The velocity components vertical to the velocity discontinuity plane must be equal to ensure material continuity according to the principle of the constant volume. Therefore, the Eq.
The friction power is generated by the contact between the plate and the work rolls. The friction power between the plate and the top work roll is W f1 . The friction power between the plate and the bottom work roll is W f2 . W f1 and W f2 can be expressed as Eq. (19). 19 The , and the dy 1 (x) and dy 2 (x) are shown in Eq. (20).

22
(2) The deformation zone is composed of the back slip zone I, cross shear zone II and reverse de ection zone IV. The friction power of top and bottom work rolls is expressed as Eq. (23). .

23
The total power can be obtained by adding the plastic deformation power, the shear power and the friction power together as shown in Eq. (11). And the unknown parameters a, b, c and d were obtained by the principle of minimum energy on the basis of Eq. (11) according to the initial and boundary conditions. The velocity and strain rate components in every direction can be calculated by taking a, b, c and d into the velocity and strain rate model.

Post-rolling strain model
The effective strain is calculated as shown in Eq. (24) for the plane strain model.

24
where Λ denotes the generalized shear strain and ε e denotes the effective strain.
Combining the strain rate models derived from Eq. (7)

Results And Model Accuracy Veri cation
The snake rolling process is simulated by Ansys software. The effective plastic strain along the thickness after rolling can be calculated by Ansys software and the analytic model established in this paper. The analytic model accuracy of the strain can be veri ed by the numerical results and the experiments. The snake rolling process with different rolling parameters, including the rolling reduction, roll offset and the velocity ratio, were simulated in this paper to verify the model accuracy. The Aluminum alloy (1060 raw materials) plate with thickness of 10 mm, the yield strength of 39MPa, the reduction ratio of 20%, 30%, 40%, and 50%, the roll offset of 5 mm, 10   material. The element type is 2D Solid162. The element size is 1×1mm. The work rolls can only rotate around its axis, and other degrees of freedom are constrained. A horizontal initial velocity is applied to the plate to enter the roll-gap.

Model accuracy veri cation of the rolling reduction ratio
The effective plastic strain of the snake rolling with different reduction ratio was calculated by the analytic model established in this paper. And the effective plastic strain was also obtained by the Ansys software. The initial thickness of the plate is 10 mm, the roll offset is 10 mm, the velocity ratio is 1.2, the diameter of work roll is 450 mm, and the reduction ratio are 20%, 25%, 30%, 35%, 40%, 45% and 50%. The curves of the reduction ratio and the effective plastic strain obtained from the simulated results and the analytic results were shown in Fig. 3.
The top and bottom neutral points x n1 and x n2 are all in the deformation zone with the least rolling reduction ratio 20% according to the analytic calculation, so the neutral points x n1 and x n2 are all still in the deformation zone with the increase of the rolling reduction ratio when the other parameters kept constant. And the deformation zone is composed of the back-slip zone I, cross shear zone II, front-slip zone III and reverse de ection zone IV. The effective plastic strain calculated by the nite element method and the analytic model established in this paper and their relative errors are shown in Table 1. Figure3 shows that the effective plastic strain and the rolling reduction ratio present the linear relationship. It can be found that the analytic results are consistent with the analytic results. And the effective plastic strain of the 1/4, 1/2, and 3/4 thickness increase with the reduction increases. The analytic results are the closest to the analytic results when the reduction is 20%. The effective plastic strain of the 1/4, 1/2, and 3/4 thickness were selected for the analytic model accuracy veri cation. It can be known from Fig. 3 and Table 1 that the minimum, maximum and the average relative errors are 4.93%, 11.92% and 9.42% respectively.

Model accuracy veri cation of the roll offset
The effective plastic strain of the snake rolling with different roll offset was calculated by the analytic model established in this paper. And the effective plastic strain was also obtained by Ansys software. The initial thickness of the plate is 10 mm, the roll offset are 5 mm, 10 mm, 15 mm and 20 mm, the velocity ratio is 1.2, the work roll diameter is 450 mm, and the reduction ratio is 30%. The relationship between the roll offset and the effective plastic strain obtained from the simulated results and the analytic results are shown in Fig. 4. Figure 4 shows effective plastic strain of the simulated results and analytic results in the center, quarter, and threequarters thickness with different roll offsets, respectively. It can be seen from the Table 2 that the effective plastic strain decreases with the increase of the roll offset. Table 2 lists the effective plastic strain of the simulated and analytic results in the center, quarter, and three-quarters thickness with different roll offsets. In this case, the top and bottom neutral points are all in the deformation zone, and the deformation zones are composed of the back-slip zone I, the cross shear II, the front-slip zone III and the reverse re ection zone IV. It can be known from Fig. 4 and Table 2 that the minimum, maximum and the average relative errors are 4.4%, 10.73% and 7.57% respectively. In order to verify the accuracy of the model, Aluminum alloy (1060 raw materials) was selected as the experimental material for rolling. The speci c mechanical properties are shown in Table 3  This experiment relies on a two-roll mill with a diameter of 450 mm. The experimental raw material size is 100 mm×60 mm×10 mm. The strain is measured by marking the grid with a laser on the side of the plate, and the speci c parameters and the rolling mill are shown in Fig. 5. The experimental method is single-pass rolling with the roll offset 0 mm, the reduction ratio of 20%, 30%, 40% and 50%, the velocity ratios of 1, 1.05, 1.1, and 1.2, respectively. The simulation results, experimental results and calculation results are shown in Fig. 6. Figure 6 respectively shows the simulated results, experimental results and analytic results of the equivalent strain of the lower core element under different pressure amounts when the ratio of velocity is 1.05, 1.1 and 1.2 In Fig. 6 (a), the maximum relative error between calculation results and experimental results is 13.44%, and the minimum relative error is 1.44%. In Fig. 6 (b), the maximum relative error between theoretical calculation results and experimental results is 10.19% (30% reduction rate), and the minimum relative error is 1.54%. In Fig. 6 (c), the maximum relative error between theoretical calculation results and experimental results is -5.92%, and the minimum relative error is 1.33%. According to the statistics, the strain model established in this paper can accurately predict the equivalent strain after rolling. The model can be used as the principle basis for the formulation of rolling process.

Model accuracy veri cation of the velocity ratio
The effective plastic strain of the snake rolling with different velocity ratio was calculated by the analytic model established in this paper. And the effective plastic strain was also obtained by Ansys software. The initial thickness of the plate is 10 mm, the roll offset is 10 mm, the velocity ratio are1.0, 1.05, 1.1 and 1.2, the work roll diameter is 450 mm, and the reduction ratio is 30%. The relationship between the velocity ratio and the effective plastic strain obtained from the simulated results and the analytic results are shown in Fig. 7.
The component of deformation zones is determined by the location of the neutral points. The top and bottom neutral points are in the deformation zone when the reduction ratio is 30%, the roll offset is 10mm, and the velocity ratio is no more than 1.2. In this case, the deformation zone is composed of the back slip zone I, the cross shear zone II, the front slip zone III and the reverse de ection zone IV. The neutral point on the side of the bottom working roll is far away from the exit when the velocity ratio is more than 1.2 and other parameters kept constant. The deformation zone is composed of the back slip zone I, the cross shear zone II and the reverse de ection zone IV at this time.  Figure 7 shows that the effective plastic strain of the 1/4, 1/2, and 3/4 thickness increase as the velocity ratio increase. Table 4 shows the simulated results and analytic results of plastic effective strain at the center, quarter and threequarters of the plate. The minimum, maximum and the average relative errors are − 1.39%, 10.77% and 4.69%.

Conclusions
This paper used the ow function and the energy method to establish a model for calculating the strain after snake rolling.The main conclusions are as ows.
(1) The parabolic roll gap inlet boundary equation and the boundary condition of the rigid-plastic-rigid zone are proposed according to the kinematic characteristics of plate rolling. The theoretical model of near real kinematic allowable velocity eld which concludes the velocity and the strain rate component is established.
(2) The plastic deformation power, shear power and friction power models are established according to the rst variational principle of rigid plasticity and the principle of virtual velocity. The analytic model to calculate the strain, strain rate and the velocity of the snake rolling are established by the principle of minimum energy and the theory of continuum mechanics.
(3) The effective plastic strain after snake rolling along the thickness are calculated by the analytic model established in this paper and the nite element method. Compared to the experimental results the relative minimum and the maximum relative error is 1.33% and 13.44%. 6955(93)90070-B. Figure 1 Schematic diagram of the snake rolling with same work roll diameters Figure 2 Page 16/17

Figures
The nite element model used for simulation  The equivalent plastic strain of center element with different speed radio and reduction ratio after rolling