strip layer method for analysis of the three-dimensional

CHINESE JOURNAL OF MECHANICAL ENGINEERING Vol. 28,aNo. 3,a2015

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DOI: 10.3901/CJME.2014.1210.177, available online at www.springerlink.com; www.cjmenet.com; www.cjme.com.cn

Strip Layer Method for Analysis of the Three-Dimensional Stresses and Spread of Large Cylindrical Shell Rolling

LIU Hongmin, CHEN Suwen*, PENG Yan, and SUN Jianliang

1 National Engineering Research Center for Equipment and Technology of Cold Strip Rolling, Yanshan University, Qinhuangdao 066004, China

2 State Key Laboratory of Metastable Materials Science & Technology, Yanshan University, Qinhuangdao 066004, China

Received June 30, 2014; revised December 4, 2014; accepted December 10, 2014

Abstract: As the traditional forging process has many problems such as low efficiency, high consumption of material and energy, large

cylindrical shell rolling is introduced. Large cylindrical shell rolling is a typical rotary forming technology, and the upper and lower rolls

have different radii and speeds. To quickly predict the three-dimensional stresses and eliminate fishtail defect, an improved strip layer

method is developed, in which the asymmetry of the upper and lower rolls, non-uniform deformation and stress, as well as the

asymmetrical spread on the end surface are considered. The deformation zone is divided into a certain number of layers and strips along

the thickness and width, respectively. The transverse displacement model is constructed by polynomial function, in order to increase the

computation speed greatly. From the metal plastic mechanics principle, the three-dimensional stress models are established. The genetic

algorithm is used for optimization calculation in an industrial experiment example. The results show that the rolling pressure, the normal

stresses, the upper and lower friction stress distributions are not similar with those of a general plate rolling. There are two relative

maximum values in rolling pressure distribution. The upper and lower longitudinal friction stresses change direction nearby the upper

and lower neutral points, respectively. The fishtail profile of spread on the end surface is predicted satisfactorily. The reduction could be

helpful to eliminate fishtail defect. The large cylindrical shell rolling example illustrates the calculation results acquired rapidly are good

agreements with the finite element simulation and experimental values of previous study. A highly effective and reliable

three-dimensional simulation method is proposed for large cylindrical shell rolling and other asymmetrical rolling.

Keywords: large cylindrical shell rolling, strip layer method, three-dimensional stresses, rolling pressure, friction stress, spread

1 Introduction

Large cylindrical shell is a crucial part of heavy pressure vessel, which is widely used in nuclear power, coal chemical and petrochemical industries etc. The cylindrical shell measures up to 8000 mm in diameter and 3500 mm in axial width, and the thickwall is greater than 300 mm, as shown in Fig. 1. As environment pollution becomes increasingly serious, energy structure has begun a transition toward clean energy, therefore the development of nuclear power is rapid, and the demand for large cylindrical shell in nuclear power becomes more and more in our country.

For now, large cylindrical shell has been usually produced by hammer forging[1], which includes a few steps: upsetting, punching and hole expansion, heating the

* Corresponding author. E-mail: [email protected] Supported by National Science and Technology Major Project of

China(Grant No. 2011ZX04002-101), National Science and Technology Support Plan of China(Grant No. 2011BAF15B02), and National Natural Science Foundation of China(Grant No. 51305388)

© Chinese Mechanical Engineering Society and Springer-Verlag Berlin Heidelberg 2015

material 5 times. These technology processes lead to the production efficiency low, the consumption of energy and material high. As an advanced forming technique, large cylindrical shell rolling is introduced. Large cylindrical shell is continuously drawn into the roll gap, and the incremental deformation of wall-thickness decrease and diameter expansion is generated. This rolling technology can improve the procuction efficiency and reduce the loss of material and energy.

Fig. 1. Large cylindrical shell

CHINESE JOURNAL OF MECHANICAL ENGINEERING

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Large cylindrical shell rolling is characterized by the asymmetry of upper and lower rolls, the huge blank size, and the three-dimensional uneven deformations and stresses. Therefore, the fast and accurate simulation of the large cylindrical shell rolling is of major technological interest as it provides a basic understanding of the rolling mechanics and an insight into how fishtail defect is generated and eliminated.

Large cylindrical shell rolling is similar with ring rolling. By now, some research has been conducted on ring part rolling based on experiment[2–4], analytical method[5–7] and numerical method[8–10]. Although the experimental investigations could provide some basic insights into rolling process, a few limitations to one kind of rolling technology and material exist. Based on the slab method, PARVIZI, et al[11], predicted ring rolling pressure distribution, in which the constant shear friction was assumed. PARVIZI and ABRINIA[12] proposed the upper bound method to predict the rolling force and torque with experiment and FEM verifications.

Apart from the above methods, there is growing interest in numerical method to better predict and optimize the ring rolling process. YEA, et al[13], predicted the rolling pressure, force and spread of the plain and T-shaped rings using rigid-plastic finite element method. ANJAMI and BASTI[14] investigated the effects of roll size on strain, stress and temperature distributions and side spread by three- dimensional FE simulation. CERETTI, et al[15], established a 3D FE model to simulate a hot ring rolling industrial process, and the FE model was compared with the industrial experiment results from the geometrical and physical aspects.

In order to further shorten the computing time, the explicit finite element method is used. ZHOU, et al[16], developed a 3D elastic-plastic explicit finite element model of radial-axial ring rolling and investigated the roll size effects on the ring rolling process. WANG, et al[17], analyzed the effect of the relative velocity on the ring rolling process using the coupled thermo-mechanical explicit three-dimensional finite element simulation. Although the FEM may be accurate in simulation of rolling process, it involves excessive runtime and convergence problems, especially for large cylindrical shell.

The strip element method can predict rapidly the metal three-dimensional deformations and stresses. The rectangular strip element method[18] was employed to analysis of the cold strip rolling, together with streamline strip element method[19]. There were also stream surface strip element method[20]for simulating hot plate rolling, together with strip layer method[21]. In these methods, the spline interpolation function is used to construct transverse displacement model, which includes the more unknown parameters affecting the computation speed. Moreover, they are symmetrical rolling of lower and upper rolls in previous studies, but there is few strip element analysis of asymmetrical rolling and large cylindrical shell rolling.

Due to the unequal speed and radius of the lower and upper rolls in cylindrical shell rolling, the deformation zone is asymmetrically rolled in the thickness direction. The author proposed a two-dimensional model of cylindrical shell rolling[7] to predict the rolling force. In order to improve the accuracy and research the metal transverse flow, the three -dimensional deformation analysis is necessary. Although strip layer method has been tentatively used for cylindrical shell rolling by the author[22], inner and outer surface asymmetrical spread and three-dimensional stresses is not provided.

In this paper, for large cylindrical shell rolling, an improved strip layer method is developed to study the three-dimensional stresses and spread, taken into account the asymmetry of lower and upper rolls, the non-uniform deformation and stress, as well as the asymmetrical spread of inner and outer surfaces. The three-dimensional stresses and spread can be obtained rapidly. The reliability of this method is verified by the experiment and FE simulation.

2 Strip Layer Method

Fig. 2 illustrates the schematic diagram of large cylindrical

shell rolling, where the shadow area is just the deformation

Fig. 2. Schematic diagram of large cylindrical shell rolling

LIU Hongmin, et al: Strip Layer Method for Analysis of the Three-Dimensional Stresses and Spread of Large Cylindrical Shell Rolling

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zone. The entry and exit cross sections are assumed to be vertical. The steady rolling process is supposed. The lower and upper rolls are rigid, and the cylindrical shell in the deformation zone is rigid-plastic, and it is elastic deformation outside the deformation zone. The deformation zone is symmetrical about the axial width center y=0 (Fig. 2).

From the geometrical relationship, the length L of the deformation zone can be obtained[23]:

1 2

1 1 1 12 ,L H

R R R r

æ ö÷ç ÷= + + -ç ÷ç ÷çè ø (1)

where R is outside radius, r is the inside radius, R1 is the upper roll radius, R2 is the lower roll radius, and ΔH is the feeding reduction per pass.

2.1 Transverse displacement model

As the cylindrical shell is of such great thickness and width, the non-uniform deformation and stress occur at the roll gap. Therefore, as shown in Fig. 3, the deformation zone is divided into m layer elements in the thickness direction, and each layer is divided into n strip elements in the axial width direction. The node co-ordinates at the exit are shown yi and zj (i=0, 1, , n; j=0, 1, , m). n and m are even numbers.

Fig. 3. Strip layer element division of the deformation zone

Taken into account the non-uniform deformation along

the thickness direction at the roll gap, the transverse displacement function is assumed to be

( ) ( ), , ( , ).W x y z f x U y z= (2)

From Ref. [12], f(x) can be got:

( ) ( ) ( )3 41 4 3 .f x x L x L= - + (3)

The exit transverse displacement function U(y, z) is

separated:

( , ) ( ) ( ),U y z u y g z= (4)

where u(y)and g(z) are the components of exit transverse

displacement in the width direction y (transverse) and thickness direction z, respectively, and can be expressed as

3 5

1 3 52 2 2

( ) ,y y y

u y a a aB B B

æ ö æ ö æ ö÷ ÷ ÷ç ç ç= + +÷ ÷ ÷ç ç ç÷ ÷ ÷ç ç çè ø è ø è ø (5)

( )3

0 1

30 2

( 0),

( 0),

c c z zg z

c c z z

≥ìï +ïï=íï + <ïïî

c0>0, c1>0, c2>0, (6)

where B is the blank width, a1, a3, a5, c0, c1 and c2 are 6 unknown optimization parameters.

The transverse displacement function W(x, y, z) satisfies the boundary conditions:

( )

( )

( , , ), , 0, 0, ,

(0, , )0, , ( , ), 0, 0.

W L y zW L y z x L

xW y z

W y z U y z xx

ì ¶ïï = = =ïï ¶ïíï ¶ï = = =ïï ¶ïî

(7)

According to the constant volume principle and the

relationships among the flow speed, strain rate and the transverse displacement, the flow speeds vx, vy, vz, the strain rates ,x ,y ,z

and the shearing strain rates xy , yz , zx

can be derived[22].

2.2 Three-dimensional stresses From the yield condition of Von-Mises, the plastic flow

equation of Levy-Mises and the assumption of z p =- , the three-dimensional stresses are expressed as

2 2

(2 ), ( 2 ),

, , ,

s sx x y y x y

s s sxy xy yz yz zx zx

k kp p

k k k

=- + + =- + +

= = =

(8)

where σx, σy, σz are the normal stresses in three directions, τxy, τyz, τzx are shear stresses in three planes, respectively, p

is the rolling pressure on the contact surface, ks is the yield strength of shear stress, and Г is the shear strain rate strength.

2.3 Rolling pressure and friction stresses

There are two neutral points within the roll gap at different vertical planes in asymmetrical rolling[24]. Thus, beside the backward-slip and forward-slip zones, the cross shear zone is constructed between the two neutral planes, where the frictions on the lower and upper interfaces act in opposite directions. Fig. 4 shows the element stress states at the roll gap. It is supposed that the rolling pressures on lower and upper contact surfaces are equal.

From Fig. 4, the longitudinal differential equilibrium in the deformation zone is derived:

1 2( ) ( ) 0,x xy x x

HH H p

x y x ¶ ¶ ¶

+ + + + =¶ ¶ ¶

(9)


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where x is the average normal stress in the longitudinal direction x, and xy is the average shearing stress, τx1 and τx2 are the upper and lower longitudinal friction stresses, in the backward-slip zone, 1 1x x p =- , 2 2x x p =- , in the cross shear zone, 1 1x x p = , 2 2x x p =- , in the forward-slip zone, 1 1x x p = , 2 2x x p = .

Fig. 4. Element stress states of the deformation zone

1 1 2 21. ; 2. ; 3. ; 4. ; 5. ; 6. ; 7. d ;

8. d ; 9. d ; 10. ; 11. ; 12. ;

13. d ; 14. d ; 15. d ;

16. ; 17. ; 18. .

xx y y x x

xy xzxy xz x xy xz

y xy yzy xy yz

y xy yz

p p xx

x xx x

y y yy y y

¶+

¶¶ ¶

+ +¶ ¶¶ ¶ ¶

+ + +¶ ¶ ¶

The upper and lower longitudinal friction coefficients μx1,

μx2 are expressed:

1 21 2

1 2

min , , min , ,sx sxs sx x

s s

v vk k

v p v p

ì ü ì üï ï ï ïï ï ï ï= =í ý í ýï ï ï ïï ï ï ïî þ î þ (10)

where μ is the frictional coefficient, vsx1, vsx2 are the longitudinal tangent sliding speeds of cylindrical shell on the upper and lower contact surfaces relative to rolls, and vs1, vs2 are the resultant sliding speeds.

On the upper and lower surfaces, the transverse friction stresses are expressed as

1 1 1 2 2 2sgn( ), sgn( ),y y sy y y syp v p v =- =- (11)

where vsy1, vsy2, are the transverse tangent sliding speed, sign(vsy1)means a sign (positive or negative) of vsy1, sign(vsy2)means a sign (positive or negative) of vsy2, the upper and lower transverse friction coefficients μy1, μy2 are expressed:

1 21 2

1 2

min , , min , .sy sys s

y ys s

v vk k

v p v p

ì ü ì üï ï ï ïï ï ï ï= =í ý í ýï ï ï ïï ï ï ïî þ î þ (12)

The resultant interface friction stresses are

2 21 1 1 ,x y = + 2 2

2 2 2 .x y = + (13)

At the exit and entrance, the boundary conditions are

1

0

2 , 0,

2 , ,

s

s

k xp

k x L

ìï- + =ï=íï- + =ïî (14)

where 2

2

21d

Hx y

H zH

-

æ ö+ ÷ç ÷= ç ÷ç ÷çè øò ,

the mean back tension stress 0 and mean front tension stress 1 are

0 01 1

1 1,

n m

iji jn m

= =

æ ö÷ç ÷ç= ÷ç ÷ç ÷çè øå å 1 1

1 1

1 1,

n m

iji jn m

= =

æ ö÷ç ÷ç= ÷ç ÷ç ÷çè øå å (15)

where the back tension stress σ0ij is at some point of the entrance cross section, the front tension stress σ1ij is at some point of the exit cross section.

From Eq. (9) and the boundary conditions Eq. (14), the rolling pressure p can be computed by the difference method. In the differential calculation process, the deformation zone is discretized along x direction, and its length is divided into a number of differential elements. The differential calculation is made from the entrance and exit of the deformation zone respectively to the internal zone, and then the rolling pressure distribution can be obtained. By integrating the rolling pressure p over the contact area, the rolling force can be acquired.

2.4 Front and back tension stresses

The back tension stress is influenced by the non-uniform entry flow speed and the residual stress of the incoming material. Thus, assuming no residual stress, the back tension stress is derived, at the jth layer,

( ) 0 00 2

0

( )

1• ,j j

jj

v y vEy

v

-=

- j=1, 2, , m, (16)

where 0 0,j jv v are the longitudinal flow speed at the entrance and its average value at the jth layer, respectively. is the Poisson coefficient and E is the elastic modulus of workpiece, respectively.

The front tension stress is influenced by the non-uniform exit flow speed. Thus, the front tension stress is derived, at the jth layer,

( ) 1 11 2

1

(

1•

),j j

jj

v v yEy

v

-=

- j=1, 2, , m, (17)

where 1 1,j jv v are the longitudinal flow speed at the exit and its average value at the jth layer.

3 Determination of the Exit Transverse

Displacement For large cylindrical shell rolling, the energy function N

of the includes five parts: the plastic deformation power Np, the interface frictional power Nf between the cylindrical


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shell and rolls, the power Nσ0 of back tension stress, the power Nσ1 of front tension stress, and the shear power Ns caused by the discontinuous flow speed at the entrance:

0 1

1

1

1 1

1

01 0

1

1 1 2 20

0

1 1

0 0 0 1 1 11 0 1 0

d d

( )d d

d d

i

i

i

i

i i

i i

p f s

m n y L

sy

j i

n y L

s sy

i

m n m ny y

j j j jy y

j i j i

N N N N N N

k h x y

v v x y

v h y v h y

+

+

+ +

-

= =

-

=

- -

= = = =

= + + - + =

æ ö÷ç +÷ç ÷çè ø

+ +

- +

ååò ò

åò ò

åå ååò ò1

-1

1

0=1 0

d d ,i j

i j

m n y z

s zy z

j i

k v z y+

-

=

æ ö÷ç ÷ç ÷ç ÷çè øååò ò (18)

where vz0 is the vertical flow speed at the entry.

According to the minimum principle of energy variation, the unknown coefficients should satisfy Eq. (19):

( )1 3 5 0 1 2min , , , , , .N N a a a c c c= (19)

The solution of this optimum problem is obtained by a

genetic algorithm method. A genetic algorithm method is a search technique for solving the global optimization problem, which gets to work form an initial population representing possible solutions of the problem. In this paper, based on an optimization strategy, a genetic algorithm method is presented by reserving the best solution in the evolutionary process.

The fitness function is defined as Eq. (19), which is the objective function of the problem.

Taking 6 unknown coefficients as the design variables and setting the search domain, various solutions are displayed by a suitable code format named the genotype. For every one of the design variables, a binary code is applied by various numbers of bits.

The genetic algorithm method is based on three operators: selection, crossover and mutation processed by an optimization strategy, which reserves the best individuals of the population. Generally, the crossover rate is about from 0.4 to 0.9, and mutation rate is about from 0.005 to 0.1.

This optimum problem has converged after 150 generations. The best solution is finally got as a1=6.345 7, a3=2.801 6, a5=1.798 5, c0=0.410 2, c1=0.034 2, c2= 0.098 9.

Compared with the traditional n unknown parameters, there are only 6 optimized parameters, so the computation speed is accelerated greatly.

4 Results and Discussions

4.1 Stresses distribution Table 1 represents the rolling parameters. The

constitutive model of the rolled material named 2.25Cr1Mo0.25V was selected from Ref. [22].

Table 1. Rolling parameters of the example

Parameter Value

Upper roll radius R1/mm 900 Lower roll radius R2/mm 1000 Outer radius of cylindrical shell blank R/mm 2450 Inner radius of cylindrical shell blank r/mm 1950 Upper roll speed vr1/(mm·s–1) 148 Lower roll speed vr2/(mm·s–1) 150 Initial width B/mm 2680 Temperature T/℃ 1120

Fig. 5 presents the distribution of the rolling pressure p

on the contact surface. There are two relative maximum values of rolling pressure in the rolling direction, which is unlike common plate rolling. As the reverse direction of the frictional stresses on the lower and upper surfaces in the cross shear zone, friction-peak of rolling pressure is cut down greatly. Moreover, shear action will accelerate the metal deformation, resulting in lower rolling pressure. The computed rolling pressure distribution is similar with the finite element simulation result[13]. Besides, rolling pressure gradually increases from the edge to the center along the axial width.

Fig. 5. Rolling pressure distribution on the contact surface Fig. 6 shows the longitudinal normal stress σx near the

contact interface. In the rolling direction, the longitudinal normal stress σx appears two different valley values nearby the upper and lower neutral points, which is caused by the friction stresses on the contact interface. The longitudinal normal stress σx decreases from the edge to the center along the axial width.

Fig. 6. Longitudinal normal stress near the contact surface


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Fig. 7 presents the transverse normal stress σy near the contact interface. In the transverse direction, the absolute value of transverse normal stress σy increases gradually from the edge to the center of width, which is affected by the metal transverse flow.

Fig. 7. Transverse normal stress near the contact surface

Fig. 8 shows the shearing stress τxy near the contact

interface. In the axial width direction, the absolute value of the shearing stress τxy decreases gradually from the edge to the center of the width, which is also affected by the metal transverse flow. The shearing stress τxy changes the direction from the entry to the exit, which is influenced mainly by the longitudinal distribution of the transverse flow speed vy.

Fig. 8. Shear stress near the contact surface

Fig. 9 shows the longitudinal friction stresses τx1, τx2 on

the upper and lower surfaces. From the entry to the exit, the longitudinal friction stresses on the surfaces change direction nearby the neutral points. As the lower and upper neutral point positions are different, the inflection point positions of the lower and upper longitudinal friction stresses are different. The absolute values of longitudinal friction stresses increase gradually from the edge to the center of the width along the transverse direction.

Fig. 10 shows the upper and lower transverse friction stresses τy1, τy2 on the contact surfaces. At the area of y>0, the transverse friction stresses is negative, which means the metal flows outside. The absolute values of the transverse friction stresses decrease from the edge to the center of the width, and present a parabolic distribution in the rolling

direction.

Fig. 9. Longitudinal friction stresses on the upper and lower surfaces

Fig. 10. Transverse friction stresses on the upper

and lower surfaces


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4.2 Spread Fig. 11 shows the exit transverse displacement

distribution. The transverse displacement varies monotonically in the transverse direction, which means the rolled material flows outside. The end surface appears the fishtail profile of spread. It is found that in the thickness direction, the largest spread occurs in the outer surface, followed by the inside surface and the smallest spread appears in the middle layer. In the practical rolling, due to the friction between workpiece and rolls, contact surface spread is slightly lower than that nearby contact surface. Although the effect of interface friction on spread could not be accurately reflected by the theoretical model, the fishtail profile on the end surface is similar with the experimental results[8]. Based on FEM, XU, et al[25], observed the fishtail phenomenon on the end surface in the ring rolling, and claimed that in some rolling conditions, the outer surface spread is larger than the inner one. The difference may be caused by the different feed amounts on the upper and lower surfaces, which can be described[23]:

2 2

1 1

1 1,

1

1

h R R

h R r

+= =

- (20)

where Δh1 and Δh2 are the feed amounts on the upper and lower surfaces, respectively. When γ is greater than a critical value, the outer surface spread is larger than the inner one.

Fig. 11. Exit transverse displacement distribution

The average spread ΔB reveals the axial flow of the rolled material, and the degree of the fishtail deformation FT shows the unevenness of end surface. They are defined as follows:

max min ,2

B BB

+= (21)

max min ,2

B BFT

-= (22)

where ΔBmax is the maximum axial spread, ΔBmin is the minimum axial spread, as shown in Fig. 12. The smaller the degree of the fishtail deformation FT is, the more even and

higher quality the end surface is.

Fig. 12. Profile of the axial spread

on the end surface

Figs. 13 and 14 present the average spread ΔB and the fishtail FT with the reduction. With the reduction increasing, the average spread ΔB and the fishtail FT increase, which means the bigger metal axial flowing, the more uneven axial deformation. It indicates that increasing the reduction will lead to a poor quality of the end surface. Therefore, making a reasonable technological parameter is very important to eliminate the fishtail defect and improve product quality.

Fig. 13. Variation of the average spread with the reduction

Fig. 14. Variation of the fishtail FT with the reduction


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4.3 Results verification In order to verify the reliability of the developed strip

layer method, a three-dimensional finite element model of large cylindrical shell rolling was established, based on the software DEFORM-3D. The FE simulation conditions were the same as those of the strip layer method. Because the symmetry of the axial width of cylindrical shell, only a half of the axial width was simulated, with the symmetric constrain on the symmetry plane.

Fig. 15 shows the comparison of average transverse stress distribution between the FEM and strip layer method.

y is the average transverse stress along the deformation

zone length. It is observed that the trend of the average transverse stress distribution obtained from the strip layer method is in good agreement with FEM results.

Fig. 15. Comparison of average transverse stress distribution

between FEM and strip layer method

Fig. 16 shows the spread along the thickness from FEM

and strip layer method. The comparison of spread between the two methods indicates a good agreement, which verifies the feasibility of strip layer method.

Fig. 16. Spread along the thickness between FEM

and strip layer method

In order to further validate the reliability of the strip

layer method, the comparison between the experimental rolling force of the authors previous study[22]and the predicted result under different reductions is shown in Fig. 17. The outer and inner radii of cylindrical shell were

R=2650 mm, r=2100 mm, and the other rolling condition was the same as Table 1. The comparison result shows good agreement among experiment, strip layer method and FEM, which indicates the developed strip layer method is considered to be reliable.

Fig. 17. Comparison of the rolling force from

the present model, experiment and FEM

5 Conclusions

(1) An improved strip layer method is developed to study the three-dimensional stresses and spread of large cylindrical shell rolling. The main advantage of the proposed method is that three-dimensional stresses and spread can be obtained rapidly and easily, and the accuracy can meet the engineering application.

(2) The simulation results show that the rolling pressure, normal stresses, friction stresses are different from those of a general plate rolling. The friction peak of rolling pressure distribution is greatly cut down. The longitudinal friction stresses on contact surfaces change direction nearby the neutral points, and the upper and lower inflection point positions are different. The friction stress distribution is relevant to the rolling pressure and metal transverse flow. The profile of fishtail spread on the end surface is predicted satisfactorily. Decreasing the reduction can improve the flatness of the end surface.

(3) The proposed method reveals the three-dimensional mechanics simulation of large cylindrical shell rolling and how to eliminate the fishtail defect. This method can also extend to other asymmetrical rolling and ring rolling.

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method for simulation of the three-dimensional deformations of plate and strip rolling[J]. International Journal of Mechanical Sciences, 2003, 45(9): 1541–1561.

[21] LIU Hongmin, JIN Dan, WANG Yingrui. Strip layer method for simulation of the three-dimensional deformations of plate and strip rolling[J]. Communications in Numerical Methods in Engineering, 2004, 20(3):183–191.

[22] CHEN Suwen, LIU Hongmin, PENG Yan, et al. Strip layer method for simulation of the three-dimensional deformations of large cylindrical shell rolling[J]. International Journal of Mechanical Sciences, 2013, 77(12): 113–120.

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[25] XU Siguang, WEINMANN K J, YANG D Y, et al. Simulation of the hot ring rolling process by using a thermo-coupled three-dimensional rigid-viscoplastic finite element method[J]. Journal of Manufacturing Science and Engineering, 1997, 119(4): 542–549.

Biographical notes LIU Hongmin, born in 1959, is currently a professor at Yanshan University, China. He received his PhD degree from Northeast Heavy Machinery Institute, China, in 1988. His research interests include three-dimensional rolling theory and numerical methods, strip shape control, rolling technology and equipment. Tel: +86-335-8057086; E-mail: [email protected] CHEN Suwen, born in 1986, is currently a PhD at Yanshan University, China. Her research interest is large cylindrical shell rolling, three dimensional deformation of metal in the rolling, etc. E-mail: [email protected] PENG Yan, born in 1972, is currently a professor at Yanshan University, China. He received his PhD degree from Yanshan University, China, in 2000. His research interests include three-dimensional rolling theory, dynamic model of rolling mill, etc. Tel: +86-335-8387651; E-mail: [email protected] SUN Jianliang, born in 1981, is an associate professor at Yanshan University, China. He received his PhD degree from Yanshan University, China, in 2010. His research interests include cylindrical shell rolling, dynamic model of rolling mill, strip shape control, etc. E-mail: [email protected]

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