The Central Strain Analytical Modeling and Analysis for the Plate Rolling Process

: The strain after rolling plays an important role in the prediction of the microstructure and properties and plate deformation permeability. So it is necessary to establish a more accurate theoretical strain model for the rolling process. This paper studies the modeling method of the equivalent strain based on the upper bound principle and stream function method. The rolling deformation region is divided into three zones (inlet rigid zone, plastic zone, and outlet rigid zone) according to the kinematics. The boundary conditions of adjacent deformation zones are modified according to the characteristics of each deformation zone. A near-real kinematics admissible velocity field is established by the stream function method on this basis. The geometric boundary conditions of the deformation region are obtained. The deformation power, friction power and velocity discontinuous power are calculated according to the redefined geometric boundary conditions. On this basis, the generalized shear strain rate intensity is calculated according to the minimum energy principle. Finally, the equivalent strain model after rolling is obtained by integrating the generalized shear strain rate in time. The plate rolling experiments of AA1060 and the numerical simulations are carried out with different rolling reductions to verify the analytic model precision of the equivalent strain. The results show that the minimum and maximum relative equivalent strain deviation between the analytic model and the experiment is 0.52% and 9.96%, respectively. The numerical calculation and experimental results show that the model can accurately calculate the strain along the plate thickness. This model can provide an important reference for the rolling process setup and the microstructure and properties prediction.  Generalized shear strain rate

H Initial plate thickness. h Plate thickness after rolling.
L Length of the deformation region.  k Shear yield strength.
m Friction coefficient.
x, y Cartesian coordinates.

a, b
Correction factor of the streamline. α The angle between the tangent of velocity discontinuity (Г 1 ) and horizontal direction. β The angle between the tangent of velocity discontinuity (Г 2 ) and horizontal direction.

W Г1
Power consumed in inlet velocity discontinuities.

W Г2
Power consumed in outlet velocity discontinuities.

Introduction
The metal plate which is the basic industrial material has been widely used in plenty fields, such as aerospace, bridge, shipping and so on [1][2]. A majority metal plate is produced by the rolling method. There is not a reliable method to measure the equivalent strain during rolling online, so the strain is unknown during the rolling production process [3]. So the soft measuring method can be introduced to predict the strain along the thickness, which is the key to predict the microstructure and properties and the reference for the process parameters setup. The soft measuring method needs a reliable analytic strain model, so the key problem about how to establish a reliable analytic strain model will be solved in this paper.
As the most convenient method, the numerical method has been widely used in the analysis of the metal rolling process. Hacquin et al. [4] simulated thermo-elastic rolling process with a steady-state thermo-elastoviscoplastic finite element model, and obtained the stress-strain variation law for the strip rolling process. Su et al. [5] developed a Fortran program that combined the three-dimensional rigid-plastic finite element method and slab method. This program can quickly calculate the strain during the rolling process. Denis pustovoytov et al. [6] used rigid plastic finite element method to simulate the asymmetric rolling process of magnesium-aluminum alloy, and the relationship between strain gradient and speed ratio was obtained. Bian et al. [7] simulated the gradient temperature rolling (GTR) and obtained the strain affecting rule. Matthias Schmidtchen et al. [8] established a fast finite element model of asymmetric compound rolling based on the improved slab theory and the actual situation of non-uniform plane strain deformation. These results can provide a good basis for the analysis of the rolling process, especially for the stain calculation.
Many scholars have also established some analytic models based on slab method, upper bound method and energy method, such as the strain, rolling force and so on, to improve the calculation speed of rolling process parameters. Jiang et al. [9] modified Von-Mises yield criterion and calculated mechanical parameters of the snake rolling process by the slab method.
The accuracy of the model is in good agreement with the numerical results. Zhao et al. [10] established the power consumption model by the upper bound method and the linearized yield criterion, the rolling force was calculated with good precision compared to the data measured in the factory. Zhang et al. [11] obtained the functional relationship between the stress state coefficient and the crack defect by using the upper bound theorem. The shape dependent criterion of the closed rectangular defect in the hot rolling process was established on this basis.
The success of these calculation methods provides an effective and feasible idea for the rolling strain calculation.
The stream function method can be used to analyze the kinematic process of metal plastic forming as an upper bound method. S.I. Oh and Kobayashi [12] taken the extremum principle to analyze the three-dimensional rolling process for the ideal rigid plastic materials. Bayoumi [13] established the metal flow model of the bar during the rolling process by the flowline field method. The velocity strain rate and stress components were solved, and the process parameters affecting the law were analyzed. Wang et al. [14] built a model to predict the free boundary profile for the metal forming based on the stream function method. Maity et al. [15] made an upper bound analysis for the billet extrusion. The kinematically admissible velocity field was calculated by using the dual stream function (DSF), and the deformation power consumption was calculated. Aksakal et al. [16] studied the metal flow in the polygonal block forging process by the dual stream function and the upper bound method. The kinematically admissible velocity field of incompressible material was established.
The successful application of the stream function method in the forging and extrusion field makes many researchers study its application in the rolling process. Hwang et al. [17] analyzed the rolling process of the clad plate by the stream function method. Since then, Hwang et al. [18] established the mathematical model to calculate the velocity field for the plate rolling by using the double stream function method and cylindrical coordinates. The plastic deformation behavior of the sheet in the roll gap was studied. Sezek et al. [19] modified the inlet and outlet boundary conditions based on Hwang [17] and calculated the rolling power and rolling force.

Theoretical parametric model 2.1 Geometric model
The following assumptions have been made to set up the equivalent strain calculation model after rolling.
1) The width expansion of the rolled plate with a large width to thickness ratio is so little after rolling. The three-dimensional plate rolling model can be simplified to be a twodimensional planar strain model.
2) The elastic deformation and the elastic recovery after rolling are so little compared to the total deformation, so the rolled plate has been considered to be the ideal rigid-plastic material.
3) The deformation extent of the work rolls is smaller than the rolled piece, so the work rolls are considered as a rigid material.
4) The friction between the rolled piece and the work rolls conforms to Coulomb's friction law.
5) The deformation region of the rolled piece is divided into inlet rigid-plastic-outlet rigid zones according to the kinematics characteristics.
6) The velocity discontinuity is defined as the thin layer region (the thickness limit to zero) where the velocity changes rapidly and continuously at the boundary zone. The region is also the maximum shear stress position if the deformed body is in the plastic yield state. So the velocity discontinuity is used as the boundary between the rigid zone and the plastic zone in this paper.
Inlet rigid zone I Plastic zone II Outlet rigid zone III

Metal flow model
The deformation region is divided into three zones according to the deformation state and material flow characteristics. They are called the inlet rigid zone I, plastic deformation zone II, outlet rigid zone III along the rolling direction as shown in Figure 1. The material flow path in the inlet rigid zone I and outlet rigid zone III are obtained according to the mass flow principle and material streamline in Figure 2.
In inlet rigid zone I In outlet rigid zone III where φ is the mass flow in unit cross section, and φ=V0H=V1h. As shown in Figure 2, the streamline is closed to be parabolic in the plastic zone II. So the additional parabolic flow is the geometric equation of rolling contact arc in plastic deformation zone II which can simplify as follows The deformation zones have the following relationship according to the continuous material The variable y is the only unknown variable in Equation 6 that expresses the coordinates in the plastic zone II.
The velocity discontinuities Γ1 and Γ2 are introduced as the boundary of the plastic zone II The velocity components in plastic zone II can be derived from hydrodynamics theory. The velocity components are determined by the partial derivative function ΦII to x and y ( ) ( ) ( ) On this basis, the strain rate components in plastic zone II can be obtained by the partial derivative of Equation 8. Their expressions are as follows

Power model
The unknown variables a and b need to be confirmed by the optimal total power consumption model and then the mass flow in plastic zone II can be known. The tension is not existed for the plate rolling, so the tension can be ignored during the modeling process. The total power Wall in the half model is It is difficult to calculate the deformation power WV due to the nonlinear Von-Mises yield criterion. The linearized yield criterion proposed by D.W. Zhao et al. [10] can where x2(y) and x3(y) are the inverse functions of geometric boundaries y2(x) and y3(x). It can be obtained from Equation (7). The expanded form of the integral term is where y2(x) and y3(x) are inversely proportional to x 2 in Equation (8).

It is found that the Equation 12 is still a nonlinear integral equation after taking Equation 13
into Equation 12. The accuracy of the calculated results can be effectively guaranteed by using Gauss integral method with quintic accuracy.
The power consumed at the boundary of the deformation zone is called the velocity discontinuous power, and their expressed are as follows    The tanα and tanβ presented in Equation 16 can be solved by the characteristics of the velocity discontinuity. Its expression is as follows 0 v n v n  =  , (17) where n represents the normal vector at any node on Γ1, Γ2. So it can be known that and then The friction power generated by the contact surface between the plate and the work rolls is as follows The total consumption power in the half thickness model can be obtained by adding deformation power, friction power and velocity discontinuities power. The a and b are calculated according to the principle of the minimum energy with the optimization theory and MATLAB optimization toolbox.

Equivalent strain model
The specific streamline in the plastic zone II can be calculated with the calculated a and b (in On this basis, it is necessary to obtain the strain from the strain rate model. Λ is the cumulative value of the generalized shear strain rate intensity (  ), and its calculation model is Finally, the equivalent strain model is expressed as follow The flowchart of the model is shown in Figure 3.

Numerical modeling
Ansys/Ls-dyna software has been widely used in the deformation simulation, and the accuracy and reliability have been verified by many scholars according to the experiment. So the Ansys/Ls-dyna software was selected to use for the plate rolling process. The main parameters and speed parameters of the rolling were shown in Table 1 and the two-dimensional finite element model was shown in Figure 4.

Minimum energy principle
Compute velocity field and strain rates.
Establish deformation model; Select the coordinate system.

Compute the deformation parameters (a and b).
Compute the power including W Г1 , W Г2 , W V and W f .

Divide the deformation rigon; Set flow equation
Optimization theory

Linearized yield criterion
Equal flow principle Compute the equivalent strain. Fig. 4 The two-dimensional (2D) FEM model As is shown in Figure 4, the geometric model includes the top work roll, bottom work roll, pinch rolls and the plate. The top and bottom work rolls are defined as rigid materials, and the rolled piece is defined as bilinear isotropic (BISO) material. The element type selected for the simulation is 2D solid 162. The grid of rolled piece and the work rolls are 4 nodes elements (2D). The work rolls can only rotate about their axis. The rolled piece will be threatened into the roll gap by friction. The contact between the work rolls and the rolled piece is defined as 'contact-2D-automatic-single-surface'.

Experiment preparation
The plate rolling experiments were carried out in the laboratory by the rolling mill to ensure the reliability of numerical and analytical calculated results. The diameter of the work roll is 320mm which is the same as the 'value 1' in Table 1 and the other main process parameters were listed in Table 2. The tensile test was conducted for the aluminum alloy1060 plate and the stress-strain curves were obtained as shown in Figure 5.  The deformation characteristics of rolling materials were obtained by marking grids on the side. The grid was marking by laser along the thickness direction of rolled piece as shown in

Calculated results and discussion
In order to verify the accuracy of the model established in this paper, the data in 3.    Table 3 and 4.   Figure 8, and the numerical simulation and results are listed in Table 5. It can be seen that even if the material properties and geometric properties are changed, the accuracy of this model is still guaranteed. The maximum relative error of the strain is 12.07%, and the minimum relative error is 0.98% along the thickness direction.

Conclusion
This paper proposes a successful method to calculate the rolling strain after the plate rolling based on the stream function and the energy method. The main conclusions are as follows.
(1) The boundary conditions of adjacent deformation zones have been calculated and optimized according to the characteristics of each deformation zone, which makes the boundary condition close to the real situation. The kinematics partition method which makes the front slip zone and the back slip zone unified in the plastic zone II is adopted to avoid the complex stress analysis which will provide an important reference for the strain modeling.
(2) The rolling deformation region is divided into three zones (inlet rigid zone I, plastic zone II and outlet rigid zone III) according to the kinematics. The total power, the velocity field, and the shear strain rate were calculated and then the equivalent strain model was established on this basis by the minimum energy principle.
(3) The experiment and numerical simulation of the plate rolling were conducted to verify the strain model precision. The experimental and the numerical results showed that the minimum and maximum relative equivalent strain deviation between the analytic model and the experiment is 0.52% and 9.96% respectively. So the strain model established in this paper is reliable and accurate, which can be used for strain prediction online.