Prediction of Velocity and Deformation Fields During Multipass Plate Hot Rolling by Novel Mixed Analytical-Numerical Method

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JOURNAL OF IRON AND STEEL RESEARCH, INTERNATIONAL. 2011, 18(7): 2@27

Prediction of Velocity and Deformation Fields During Multipass Plate Hot Rolling by Novel Mixed Analytical-Numerical Method ZHANG Jin-ling,

CUI Zhen-shan

(National Die and Mold CAD Engineering Research Center, Shanghai Jiao Tong University, Shanghai 200030, China)

Abstract: An integrated mathematical model is proposed to predict the velocity field and strain distribution during multi-pass plate hot rolling. This model is a part of the mixed analytical-numerical method (ANM) aiming at prediction of deformation variables, temperature and microstructure evolution for plate hot rolling. First a velocity field with undetermined coefficients is developed according to the principle of volume constancy and characteristics of metal flow during rolling, and then it is solved by minimizing the total energy consumption rate. Meanwhile a thermal model coupling with the plastic deformation is exploited through series function solution to determine temperature distribution and calculate the flow stress. After that, strain rate field is calculated through geometric equations and strain field is derived by means of difference method. This model is employed in simulation of an industrial sevenpass plate hot rolling process. The velocity field result and strain field result are in good agreement with that from

FEM simulation. Furthermore, the rolling force and temperature agree we11 with the measured ones. The comparisons verify the validity of the presented.method. The calculation of temperature, strain and strain rate are helpful in predicting microstructure. Above all, the greatest advantage of the presented method is the high efficiency, it only takes 12 s to simulate a seven-pass schedule, so it is more efficient than other numerical methods such as FEM. Key words: multi-pass plate rolling; analytical-numerical method ( A N M ) ; velocity field; strain field; strain rate

Plate product has been widely used in various engineering fields, such as shipbuilding industry, civil construction, bridge structure, high pressure vessel, and military-industry. Multi-pass rolling schedule is adopted in plate industrial produce. Researches show that the mechanical and processing properties of the plate are determined by the corresponding microstructure of the plate. However, for hot-rolled products with certain chemical composition, the microstructure mainly depends on the temperature, strain, and strain rate during rolling. Online control of heating temperature as well as adjustment of rolling schedule and cooling system are efficient methods to improve products' properties, so it is important to accurately predict procedure parameters such as temperature, strain, strain rate and thermo-mechanical parameters"'. In the earlier stage of rolling research, conventional analytical methods, such as engineering method, slip-line filed method and upper bound method,

were widely used to calculate mechanical and thermal parameters"'. But in order to solve the plastic mechanics equations lots of simplifications were made, leading to lower precision of calculation results, especially for the complicated problems. With development of computer, numerical methods such as FEM and FDM have become more and more popular. FEM has been extensively used in rolling field, which is good at solving complex governing equations and hence variation of temperat u re L 2 p 41 and d e f ~ r m a t i o n [ ~ -as ~ ' well as final microcan be simulated accurately. Nevertheless, it will take too much time t o simulate rolling process through FEM, so it is not appropriate for on-line control, especially for multi-pass rollingCgl. For the purpose of improving computational efficiency, some researchers combined FEM with analytical methods. For example, Hassan SheikhC1'' integrated FEM and upper bound method ( U B M ) t o study temperature field during hot strip rolling, S Serjza-

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Foundation Item: Item Sponsored by Specialized Research Fund for Doctoral Program of Higher Education of China (20050248007) Biography: ZHANG Jin-ling(l980--), Female, Doctor Candidate; E-mail: zjl-hjg@sohu. com; Received Date: July 2, 2010

Prediction of Velocity and Deformation Fields During Multipass Plate Hot Rolling

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deh and Y Mahmoodkhani‘”’ combined UPD and FEM to predict velocity and temperature fields during hot rolling process. The model in S Serajzadeh’s work requires relatively lower computational effort in comparison with standard FEM, but it was only suitable for single pass solution and it still took 6 s for one pass calculation, which can not meet the requirement of on-line control for single pass rolling, let alone multi-pass process. Consequently, it is still necessary to develop a new method, which is more accurate than analytical methods and requires far less calculation time than FEM. In present work, a generalized mathematical model is developed to predict the process parameters and product microstructure, which consists of temperature model, deformation model and rolling-force as well as microstructure models. It only takes 1 2 s to simulate a seven-pass industrial hot rolling process and the results have a good precision comparing with the measured. The temperature and rolling force parts have been depicted in Ref. [ l Z ] and Ref. [13]. This paper will describe deformation field model in detail.

1

Mathematical Model

Kinematically admissible velocity field Derivation of a kinematically admissible velocity field is critical for prediction of deformation. T h e rolling process is assumed as a steady-state so strain and strain rate as well as rolling force remain constant at any spatial point during rolling. Therefore, it is reasonable to solve this problem in Euler coordinate system. Plate width is much bigger than the contact arc length and thickness, therefore the lateral spread is negligible. Schematic illustration of velocity is shown in Fig. 1, where, P and y are the bite angle and neutral angle respectively; uo denotes the entering velocity of the plate; u, is the rolling speed of the 1. 1

Fig. 1

Schematic illustration of velocity field

-

21

work roll; v, is the velocity of the plate in the rolling direction; v, is the velocity of the plate in the reduction direction; v is resultant velocity of v, and v,; ho and hl represent the thickness of the plate at entrance point and exit point; All and A12 are respectively the length of rigid zones before and after rolling; and 1 is the contact arc length. Origin of the coordinates is set at the exit point on the symmetry line of thickness. Due t o the friction at the interface between work roll and plate, deformation of the plate is inhomogeneous along thickness. Therefore, feature of the plate velocity is just shown in Fig. 1. First, in the whole roll-gap, v, decreases gradually along the thickness from the surface until equals t o zero on the symmetry line. Second, v, is homogeneous at the neutral point and in the rigid zones. However, to the left of the point the roll moves faster than the plate; to the right of this point the plate moves faster than the roll. Hence, in backward slip zone, plate is driven by friction force and v, on the surface is a little bigger, so the v, curve shows concave at the central thickness. Contrarily, in the forward slip zone it appears convex in center. Moreover, thanks to the variation of v, and v, , during the whole rolling process v is tangent to the roll surface at the interface and in the backward slip zone v decreases gradually with the decline of v, and v,. In the forward slip zone, v increases with the increment of v, until equals to u ‘ , at the symmetry line. Distribution of the velocity filed behaves the metal flow characteristics during rolling. According t o the principle of second-flow constancy of continuous rolling, at a given time the volume passing through any section of the deformation zone should be constant, that is as Eqn. (1).

s

+hZ:/Z -hz/Z

v,dy=uo ho =u,h,

= u i hi =@

(1)

where, v, and v, are the velocity of the plate at an arbitrary point in the deformation zone and at the neutral point respectively; h, is the thickness of the plate at an arbitrary position; and @ represents the rate of volume passing any position of the deformation zone. Utilizing incompressible volume principle and plane strain hypothesis, at any point of the deformation zone the following equation should be satisfied. €,+Ey=0 (2) where E, and &, are the strain rate in rolling direction and along thickness. In terms of geometric equation and volume con-

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Journal of Iron and Steel Research, International

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stancy principle as well as boundary conditions expressed in Eqn. ( 3 1 , the velocity component along thickness direction can be calculated consequently.

3 1 y = fF = tanp,

V ,

;

vuyI Y = o

(3)

Then velocity model in backward slip zone can be written as follows:

(4)

In the forward slip zone velocity in the rolling direction shows convex along thickness. The corresponding velocity model is expressed as Eqn. (5).

2voho:

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c1 ( x ) appears at different positions of contact arc with change of the undetermined coefficients. So, the velocity field can be adjusted by changing k , and then the minimum value of rolling energy can be searched. T h e steel in rolling is treated as rigid-plastic material, and the flow stress is a function of temperature, strain and strain rate. An interpolation meth-

od is applied to calculate the flow stress according t o the test data measured by thermal-mechanical simulator. According to the Markov variational principle, among all the kinematically admissible velocity fields, the actual one minimizes the total consumed power during processing, which can be expressed as a sum of power required to deform the material plastically W, and the power required to overcome friction Wr as well as the power required to overcome shear forces resulting from velocity discontinuities at entry and exit W,. In the form of mathematical equation:

(7)

W= Wp + Wf + W,

1

where W, where c1 (x)and c2 (x) are shape functions of velocity, which takes x as argument and includes undetermined coefficients. According to the characteristics of metal flow, the functions should meet the homogeneous requirement at the entry and exit points as well as the neutral point. So they are expressed in the following style. c1 ( x ) = k , [x+ (l+Al, ( x n - x >+k, [x+

>IZ

(l+ALi>](.~,-x)~ c2 ( x ) =kg ( x - x - c , > (AL, z -x)+kq

(6)

(x-xn)

(AL2 - x l z where, x n donates the coordinate of neutral point; and k , , k z , k , , k , are undeterm'ined coefficients. Shape functions will differ according to the undetermined coefficients. Variation of c1 ( 2 ) along backward slip zone is shown in Fig. 2. As shown in illustration, the maximum value of

W,=

1'

=

s,

&dV,

rf 1 A s

Wr=

I

dS1, and

S1

rkAvkdS2 5 is the effective stress, which

s2

is determined by strain, strain rate and temperature. In present program, O = D ( E , E, T ) =A€%" e x p ( Q / R T ) , where A , m , n are linear regression coefficients from experimental data; tive strain rate,

= .@-/,. 3

is the effecAnd the strain

rate can be derived through geometric equations. In backward slip zone:

In forward slip zone: 0.20 A

v " 0.10

0

Fig. 2

0.03

0.05 0.07 Are lengrk'm

0.09

Effect of undetermined coefficients on shape functions

(9) where, rf and Tk are respectively the friction shear stress on the roll-plate interface and yield shear stress on

Issue 7

Prediction of Velocity and Deformation Fields During Multipass Plate Hot Rolling

the velocity discontinuities section; shear friction law is adopted so rf=mrk ; m is the shear friction coefficient ; Av, is the relative slip velocity between the work roll and the plate, Avf = v, - d m ; and Avk is the velocity discontinuity at entry and exit. Therefore, according to extreme value theorem the following variational equation should be satisfied.

6W=

i,88 E d V f S

S1

rial Avf I dS,

js2 rk8AvkdSz = o

+

(10)

Thanks to the undetermined coefficients in velocity model, the above equation can be converted into a system of equations of k, , k, , k, , k,. For plain strain problem, the above integration will be reduced to two-dimension through deformation zone and one-dimension along contact arc as well as along thickness expressed as following.

23

field by means of geometric equations. However, the mentioned models of velocity and strain rate are set up in Euler coordinate system, for a mass point, strain rate is the material time derivative of the strain:

d x ,,t>=

D&,( X I ,t ) ( X I ,t ) - a&, Dt

at

+a€,

(XI

,t )

.

axk

vk ( X I 9 t ) (12) For a fixed point, during steady rolling strain a&,(x,,t ) value keeps constant so = O . Then strain at

components and effective strain model can be derived according to strain rate expression via Eqn. ( 1 0 1 , but the derivation is still difficult. Therefore, difference method is employed instead of direct derivation. To set up the difference expression, the deformation zone is mapped into a rectangular as shown in Fig. 3 ( b ) , where i and j are the node number along rolling direction and thickness.

(11)

where, Ah and A, respectively represents the area in backward slip zone and forward slip zone; L h 1 , l b 2 donate the boundary line at the interface between roll and plate and along thickness in backward slip zone, so does I f , , Lfz in the forward slip zone. The deformation zone is meshed as shown in Fig. 3 ( a ) and numerical evaluation is adopted to solve the integration items. Moreover, N-R iteration method is employed to solve the undetermined coefficients corresponding to the minimization of power. Consequently, the actual velocity field is obtained.

Deformation field Then the strain rate component and effective strain rate can be calculated according t o velocity

1. 2

(a) Original deformation zone shape; ( b ) Rectangle after mapping.

Fig. 3

Mapping sketch of the deformation zone

T h e corresponding strain rate components in the rectangular are yielded:

(13)

For any point in the rectangular zone, strain rate can be calculated through forward difference method. And then the recurrence formula of strain

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Journal of Iron and Steel Research, International ~

field can be derived. Take the strain in the rolling direction for example.

Input of parameters; Initial pass number N=l Initial iteration number i=O

Setting initial value of the undetermined coefficients KO

Initial condition should be given in order to complete the recurrence formula.

I'

(15)

Initial velocity field and corresponding strain rate and strain are solved

+

It is worth to note that in order to derive the strain model at thickness center the following coefficients expressions have been adopted:

i=i+l N=N+1

IN Because there is no deformation in the rigid zone, there is no strain and strain rate here either. Therefore, strain model corresponding to i = 1 can be expressed as Eqn. (14).

analysis

t

-52

pass number

(17)

Until now the recurrence formula of E, is completed. Moreover, other strain components such as E , and y,, a s well as T can also be derived in the same way. Besides the deformation distribution, temperature prediction is also particularly important for accurate calculation of mechanical and energy parameters because of strong dependence of yield stress and plate microstructure on it. In present research, series function method is adopted and compatible relationships of constants in the solutions are successfully established between passes and operations, so the continuous computation of temperatures in the whole process is achieved. More details can be found in Ref. [lZ]. The program flow chart of continuous simulation of multi-pass hot rolling by presented ANM is given in Fig. 4. Practice has proved that a proper initial value of k will decrease iteration number observably. Take the convergence of pass 3 for example as described in Fig. 5. In Fig. 5 , the step rate means the change rate of k after each iteration, that is [ k ( i + l ) - k ( i ) ] / k ( i ) . As the step rate decreases the convergence becomes better. From Fig. 5 , the initial value of k in Case 1 induced about 9 iterations before convergence, however, Case 2 caused about 13 iterations. So it is nec-

velocity field

interstand section

I End the program I

Fig. 4

-!i

Illustration of the program employed in the present model

0.002 0

.

O.Ool

-

0.6 Case 1: k, step rate Case 1:k, step rate Case 2 kl step rate Case 2 k2 step rate 0.4 E:

8.-

0

.ti 0.001 2 -

-Y

--Y

%

u

3 0.0008 -

- 0.2 3

E

2 0.0004 -

a 8 cn

cn

0

2

4 6 8 1 2 1 4 1 6 Number of iteration

k1(0)=313914. 0, kz(0)=1054777. 0 , k3(0)=17858499.0, k 4 ( 0 ) = 1 6 6 6 5 4 0 9 . 0 ; Case 2: k 1 ( 0 ) = 5 . 0 , Kz(0)=5.0, k ~ ( 0 ) = 1 0 . 0 , k g ( O ) = l O . O . Case 1:

Fig. 5

Effect of the initial value of k on convergence

essary for us t o estimate a proper initial value of k according t o minimum energy principle rather than evaluate it arbitrarily.

2

Application and Discussion

To verify the present model, a seven-pass hot rolling process is simulated. The rolling schedule is

Issue 7

Prediction of Velocity and Deformation Fields During Multipass Plate Hot Rolling

25

-

given in Table 1 ,which is simulated simultaneously by the present method and by FEM software MSC. MARC respectively under the very same boundary conditions. Comparison of deformation field resulted from different methods is given below. And the validity of FEM simulation has been verified in Ref. [ S ] . Fig. 6 shows the velocity field of some passes calculated respectively by the present ANM and by

2DFEM. 0.93 0.89 0.85 0.81 0.77 0.73 0.G9

0.65 0.61

Table 1

*

Rolling schedule adopted

Pass

Plate thickness/m

Reduction/ m

Diameter of roll/m

Rolling speed/

1

0.048 55

0.019 70

0.817 83

2. 288 98

2

0.028 84

0.01086

0. 821 53

3. 678 50

3

0.017 98

0. 005 70

0. 827 30

5.400 70

4

0.012 28

0.002 95

0. 817 07

7.284 56

5

0.009 33

0.002 14

0.640 29

12.069 53

6

0.007 19

0.001 42

0.644 01

15.033 93

7

0.005 77

0.000 68

0.626 42

17.805 95

(r

*

s-l)

1.01 0.97 0.92 0.88 0.84 0.80 0.76 0.71 0.67 0.&? 0.54

3.96 3.84 3.73 3.62 3.51 3.39 3.28 3.16 3.06 (a) v, of

pass 1 resulted from present program;

( c ) vr of pass 5 resulted from present program;

Fig. 6

( b ) uz of pass 1 resulted from 2DFEM; (d) u, of pass 5 resulted from BDFEM.

Comparison of v, of the present program and FEM simulation

From the comparison it can be seen that velocity on the surface in backward slip zone is bigger than that at central, contrarily in the forward slip zone, which follows the same law with FEM simulation result. Furthermore, the value of velocity in each area is close to the FEM result. At neutral point, there is no relative slip between plate and roll so the plate velocity is equal to the rolling speed. Fig. 7 illustrates the distribution of equivalent strain respectively calculated by presented ANM and ZD-FEM. Fig. 7 implies that the ANM results agree well with the FEM ones. Strain field in deformation zone is inharmonious and it increases as reducing and shows a bigger value on the surface than that at central thanks to the shear strain on surface. Moreover, the bigger the reduction is the more inharmonious the strain will be. T h e biggest value appears on the surface of exit region in the deformation zone. Fig. 8 demonstrates the simulated temperature result on plate surface and central layer, and the

surface temperature is compared with the measured data. It is seen that the surface temperature drops dramatically when running into the roll-gap due to the contact to the chilly roll, and rises quickly after running out of the roll-gap due to the heat transfer from central layer. While the central temperature rises a little bit when it is in the gap due t o plastic deformation heat, and drops slowly when it is out of the gap due t o heat transfer. As the plate gets thinner, the temperature difference between surface and central layer gets smaller. T h e temperature result from ANM has an excellent agreement with the measured one. T h e detail of temperature model is described in Ref.1121 and Ref. [13]. Fig. 9 shows the distribution of unit friction pressure and unit rolling pressure of several passes. In backward slip zone, the friction pressure is positive in the same direction with plate moving, and drives the plate. A t the neutral point, there is no relative slip between the plate and roll so friction

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Journal of Iron and Steel Research, International

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0.61

0.68 0.62 0.55 0.49 0.42 0.36 0.29 0.23 0.16 0.10 0.03

0.54 0.49

0.43

0.33

0.35 0.32 0.28 0.25 0.22 0.18 0.15 0.12 0.08 0.05 0.02

0.28 0.23 0.17 0.11 0.05

(b) sof pass 1 calculated by FEM; (d) 5 of pass 4 calculated by FEM.

(a) 5 of pass 1 calculated by ANM; (c) 2 of pass 4 calculated by ANM;

Comparison of equivalent strain of the present program and FEM simulation

Fig. 7

F 1

320

1 080

pressure is equal t o zero. And then it changes into the opposite direction as shown in Fig. 9 and retards the plate. T h e rolling pressure increases with the reduction until the maximum value appears near the neutral point, and then decreases until the exit point, which agrees well with the experiment results of Krasnozavodsk. In Fig. 10, the calculated rolling force is compared with the measured one. T h e maximal difference is seen at the first pass about 15. 6 % . T h a t is because during actual production, there is elastic flattening deformation happening to the roll, especially

Temperature at center Temperature on the surface practical surface temperature

t' 0

6

Fig. 8

10

15 Tim&

20

25

Temperature variation of each pass 60

350 .

40

a

20

300

0

250

-20

200

-40 1201 . -0.05

-0.04

-0.03

-0.02

-0.01

0

150

-60

I

-0.016

-0.008

0

x/m

Fig. 9

(b) Pass 7. (a) Pass 3 ; Unit friction pressure and rolling pressure along rolling direction

in the passes with big reduction. However, in the method, the elastic flattening has not been consid-

ered. T h e detail of rolling force calculation also can be found in Ref. [13].

Prediction of Velocity a n d Deformation Fields During M u l t i p a s s Plate Hot Rolling

Issue 7

2.6 x

2 3

Practical rolling force NCalculated rolling force

107

z.nx i n 7

8 i.5x in7

2

M

G

;?

i.nx in7

-8

p:

5.0 x 106 0

1

Fig. 10

2

3

4 5 Pass number

6

7

*

27

2) Comparisons of the calculated rolling force and temperature results with the measured ones, as well as the comparison of velocity field result and strain field with that from FEM simulation show that the present method has a good accuracy. And it will be meaningful for microstructure prediction. 3 ) Moreover, it only takes 1 2 s t o calculate a seven-pass rolling process by present method, which provides a feasible approach for solving multipass hot rolling. References :

Comparison of rolling force of ANM with practical ones

Besides the high precision of the rolling parameters, the main advantage of the present method is that it requires relatively lower computational effort in comparison with that of standard FEM codes. For the above seven-pass rolling schedule, it only took 1 2 s in total. Table 2 shows calculation cost of each pass respectively by FEM and ANM.

Kim J , Lee J , Hwang S M. An Analytical Model for the P r e diction of Strip Temperatures in Hot Strip Rolling [J]. International Journal of Heat and Mass Transfer, 2009, 52(7/8):

1864. I,IU Cai, CUI Zhen-shan. Thermo Mechanical Coupled FiniteElement Modelling of Slab Hot Rolling [J]. Chinese Journal of Mechanical Engineering, 1998, 34(4): 35 (in Chinese). Vladimir P. Model for Prediction of Strip Temperature in Hot Strip Steel Mill [J]. Applied Thermal Engineering, 2007, 27

(14/15) : 2404. Reza R , Siamak S. ThreeDimensional Model for Hot Rolling of Aluminum Alloys [J]. Materials and Design, 2007, 2 8 ( 8 ) :

2366.

Table 2

Comparison of the computational time of ANM with that of FEM

Pass No.

Computational time/s FEM

1 2 3

221 468 498

4 5 6 7

752 964 571 763 4 237

Total

ANM

The total computational time of ANM is only 1 2 s. However the ZDFEM solution took 4 237 s , which is about 353 times of the ANM. So the present ANM brings forth a great improvement in solution of multi-pass plate hot rolling.

3

Conclusions

1) As a part of the new style analytical-numerical method ( A N M ) , the kinematically admissible velocity field was established in Euler coordinate system with undetermined coefficients according to the metal flow characteristics, which are solved by means of N-R iteration method on the basis of Markov variational principle.

JIANG Zheng-yi, Tieu A K. A Simulation of Three-Dimensional Metal Rolling Processes by Rigid-Plastic Finite Element Method [J]. Journal of Materials Processing Technology,

2001, 112(1): 144. Kazutake K. Simulation of Deformation and Temperature in Multi-Pass ThreeRoll Rolling [J]. Journal of Materials Processing Technology, 1999, 92-93: 450. XIAO Hong, XIE Hong-biao, BI En-fu. Prediction of Rolling Loads, Temperature and Microstructure Variation During Hot Strip Rolling [J]. Iron and Steel, 2003, 38(9): 35 (in Chinese). Jang Y S, KO D C, Kim B M. Application of the Finite Element Method to Predict Microstructure Evolution in the Hot Forging of Steel [J]. Journal of Materials Processing Technology, 2000, lOl(1): 85. ZHANG Jin-ling, CUI Zhen-shan, WANG Ying-jie. Continuous Finite Element Simulation of Plate Multi-Pass Hot Rolling [J]. Journal of Shanghai Jiaotong University, 2009, 43(1): 65 (in Chinese). Sheikh H. Thermal Analysis of Hot Strip Rolling Using Finite Element and Upper Bound Methods [J]. Applied Mathematical Modelling, 2009, 33(5): 2187. Serajzadeh S, Mahmoodkhani Y. A Combined Upper Bound and Finite Element Model For Prediction of Velocity and Temperature Fields During Hot Rolling Process [J]. International Journal of Mechanical Sciences, 2008, 50: 1421. CUI Zhen-shan, ZHANG Jin-ling, LIU Juan. An Mixed Analytical-Numerical Modeling for Plate Under Hot Rolling-Part I: Temperature Model [J/OL]. China’s Scientific and Technical Papers Online [2008-01-10]. http: // www. paper. edu. cn/ index. php/default/releasepaper/content/200901-662. ZHANG Jin-ling, CUI Zhen-shan, H U Hong-xun. Simulation of Multi-Pass Plate Hot Rolling by a Mixed Numerical and Analytic Method [J]. Journal of Shanghai Jiaotong Universit y , 2008, 42(1) : 32 (in Chinese).