Simulation of ring rolling using a rigid-plastic finite element model

Int. J. Mech. Sci. Vol. 33, No. 5, pp. 393-401, 1991

0020-7403/91 $3.00 + .00 © 1991 Pergamon Press plc

Printed in Great Britain.

SIMULATION

OF RING ROLLING USING A RIGID-PLASTIC FINITE ELEMENT MODEL S. G. Xu, J. C. LIAN and J. B. HAWKYARD t

Department of Mechanical Engineering, Yanshan University, Qinhuangdao, China; and tDepartment of Mechanical Engineering, University of Manchester Institute of Science and Technology, P.O. Box 88, Sackville Street, Manchester M60 1QD, U.K.

(Received 30 May 1990; and in revised form 22 November 1990) Abstract This paper describes details of a finite element model for analysing three-dimensional flow in the roll gap during radial ring rolling between plain cylindrical rolls. The model considers such factors as curvature of the workpiece, unequal roll diameters and the fact that the inner roll or mandrel is unpowered. The results of the analysis compare favourably with experimental observations. Fishtailing is predicted satisfactorily, also the form of the pressure distribution in the roll gap, which shows twin peaks under some rolling conditions. The method has the merits of easy handling and short calculation time. It should be a useful aid to the design of ring rolling and other asymmetric rolling processes. NOTATION b ¢r'ij a,. g k'~j ~v V St Sf T~ Vi zf j Avr, Avrj r ho h Rj hj Bo, B R~, Ro 090, o91, ~2 (u, v, w) (uj, vj, wj) ~j urj # k p 6b ,5h

representative stress deviatoric stress hydrostatic stress small positive constant equivalent strain rate deviatoric strain rate volumetric strain rate functional volume of body surface on which traction is prescribed interface of roll and ring traction velocity on S t frictional stress subscript indicating driven roll or mandrel side (j = 1, 2) relative velocity at roll surface radial thickness reduction initial radial thickness radial thickness after rolling radius of rolls reduction of rolls original and final axial width inner radius and outer radius of ring angular velocity of ring, driven roll and mandrel respectively velocity components velocity components at roll surfaces angular position of node tangential velocity of rolls coefficient of friction shear yield stress normal pressure on roll surface vector of all nodal velocities relative width spread relative radial position of points INTRODUCTION

Although ring rolling has been established in industry for more than 100 years and continues to be a major means of ring production [1-1 little attention has been paid to the mechanics of the process, as compared with fiat rolling, for example. 393

394

S.G. Xu

et al.

Johnson et al. [2-4] presented an elementary analysis and experimental results for rings rolled in a small experimental mill. Mamalis et al. [5-7] described a pressure pin measuring technique and experimental results of normal pressure distribution during plain and profiled ring rolling. With the assumption that the deformation field in the roll gap is similar to that for plane strain indentation by opposed fiat indentors, expressions for roll force and torque based on slip line field solutions were derived in Ref. [8]. Yang et al. [9] and Ryoo et al. [10] used the Upper Bound Method to determine roll torque. Because of the complex deformation pattern in ring rolling, a three-dimensional analysis is relatively difficult. In this respect, Hayama [11] carried out an axial spread analysis of plain rings in cylindrical coordinates using an Energy Method, and Lugora et al. [12] used Hill's General Method to predict average axial spread. For the study of edge profiles of rolled rings, a dual-stream function is used in Ref. [13] to derive a quasi-threedimensional velocity field. In Ref. [14] the Upper Bound Element Technique (UBET) is employed to model roll force and torque, the shape of the plastic region in the roll gap and other parameters. Recently various workers have made studies of ring rolling, including Boucly [15], Hirai [16, 17] and Yun [18]. Yang and Kim [19] describe a rigid-plastic finite element analysis for plane strain ring rolling from which predictions of strain rate distribution in the roll gap, roll separating force, driven roll torque and normal pressure distribution are computed. In the present paper, the ring rolling process is modelled using a rigid-plastic finite element method, which should also be suitable for other asymmetric rolling processes.

ANALYSIS

Basic theory In many metal forming processes, particularly hot forming, the plastic strains in the deformation zone greatly exceed the elastic strains and it is possible to neglect the elastic strain components and consider the deforming body to be rigid plastic, obeying Von Mises' yield criterion and the associated flow rule. In applying a finite element formulation to such a material using the extremum principle, the condition of incompressibility can be satisfied by introducing a Lagrangian multiplier or penalty function into the functional. In the present work a method proposed by Mori et al. [20-22] is used, whereby the material is assumed to be slightly compressible during plastic deformation. The relationships are summarized as follows: The yield criterion is if2 = ~O'ijO'ij 3 , , -~- go.2

(1)

,

where # is representative stress, cr'ij is deviatoric stress, a m is hydrostatic stress and g is chosen to be a small, positive constant. The equivalent strain rate is ~ij~ij + ,~ ~,

=

(2)

and stress alj is related to strain rate by O'ij = ~

3 iij "~ •ij

--

9 iv

(3)

where ~u is strain rate, ~'ij is deviatoric strain rate, ~v is volumetric strain rate and 6ij is the Kronecker delta. The functional to be minimized is given by

,=

fv#~dV+

fs rfAv, d S - fs ~

(4)

where V is the volume of the body, T~ is the traction prescribed on the surface where the velocity is vi and Avr is the relative velocity at the interface where the frictional stress is Tf.

Simulation of ring rolling

395

Boundary conditions The analysis is concerned with the radial rolling of rings from blanks of rectangular section, so that there is symmetry about the central plane and only half the workpiece needs to be considered. A typical initial mesh arrangement is shown in Fig. 1. The origin of the Cartesian coordinate system is the centre of the ring, the z-axis is along the line joining the roll centres and the x-axis is in the rolling direction. On the roll surface the velocities u and w in the x and z directions satisfy the following relationship wj = uj tan ~j

(5)

where subscript j is 1 for the driven roll, 2 for the mandrel, and ~j is the angle of the nodal point on the roll surface with respect to the line joining the roll centres, measured clockwise as shown. The relative velocity Avr~ between roll and ring is given by Avrj = [(uj - urj cos ~)e + v~ + (us tan ~j - u~j sin ~j)211/2

(6)

where urj is the tangential velocity of the rolls. The frictional shear stress zf at the roll surface is assumed to obey the Coulomb friction law: "of =

{fp#p>.k

(7)

where # is the coefficient of friction, k is the yield shear stress and p is the normal roll pressure determined in the previous stage of iteration. The value of p used initially over the whole contact area of the rolls is a suitable value of yield stress. In ring rolling the ring rotates about its centre and the entry and exit boundary conditions are determined as follows:

0 FIG. 1. Finite element mesh,

×

396

S.G.

Xu

et al.

In the section a'b'c'd' at entry Z ,7 o

U=R

f

v= 0 X

W z

In the section abcd at exit

(8)

--

UO

. ZO

7 U~N

1 2" 1

v= 0

19) X

W~U

1 Zl

In the plane of symmetry v = 0.

~10}

In Eqns (8) and (9) u o and u 1 are the velocities of the arbitrarily chosen reference points in sections a'b'c'd' and abcd and 2"0, zl are their z coordinates. Tangential velocity of mandrel and distribution of reduction between main roll and mandrel

In ring rolling the driven main roll velocity is given and the mandrel tangential velocity is unknown. From Eqn (6) it is evident that the relative velocity at the mandrel surface Avr2 varies as the mandrel velocity ur2 changes. Hence the value of ur2 affects the characteristics of the functional ~. After the finite element discretization, the functional ~ is a function of not only the nodal velocities but also the mandrel tangential velocity, i.e. O = q b ( f ' , u r 2)

(11)

where ,~, is the vector of nodal velocities. This technique was used by Yang and Kim [19] in their analysis of plane strain ring rolling. Applying the stationary condition to the functional gives 8i', = 0 (12t c9~ = 0. (JUr2

Equation 12 is a non-linear system of equations with n + 1 variables and n + 1 equations. which is solved by the Newton-Raphson iteration method.

8~,2

8g,Su~

(320

~_.2~

l{} {} Ur2 {n)

=

Ur2 (n 1)

A~'

= _

AUr2 (,,,

{}

+ O~n

AUr~

(n)

?'~' t3~

,,3,

c~, is the optimal step length in the Newton direction, in order to make the functional @ smallest in the current iteration, thus saving calculation time. Equation (13) is the final form of the rigid-plastic finite element iterative formula incorporating the unknown mandrel velocity.

S i m u l a t i o n of r i n g rolling

397

Assuming that no slip occurs at the roll surfaces, the following formula was derived in Refs [11, 13] to calculate the initial value of the mandrel tangential velocity Ur2 = ( K 2 e / K I e ) O l

~o1 is the angular velocity of the driven roll and K,e, R o c o s f l , - B xsin Eft, + gle

=

KEe =

( R E - g 2 s i n E fll)l/2

g i cos fiE -- BE sin E fl2 (R22 -- Ri2 sinEflz) x/E

KEe are

given by

B2 sine fll cos fll (R21 - B E s i n a f l , ) l / a ( R 2 _ R E s i n a f l l ) U 2

BE sin2 fiE COS fl2 - - ( R ~ -- B 2 s i n E f l z ) l / E ( g 2 -- g 2

fll = c o s - ' [(R 2

+ B2 -

fiE = c o s - ' [(R 2

+ B 2 -- RE)/(2RiBE)]

B1

Ahl

= Ro + R1 -

(14)

RE

sinEflE) '/E

(15)

RE)/(ERoB1)]

B E = R i - R 2 - Ah 2 Since the diameters of the driven roll and the mandrel are not generally equal, the amount of reduction due to each roll is not known a p r i o r i , although the total reduction is given. In the calculation, the apportioning of reduction between the rolls is decided arbitrarily at first and when the iteration has converged the forces on the two rolls are compared. If the difference is not sufficiently small the reductions are adjusted and the process repeated until the forces are sufficiently closely matched. RESULTS

AND

DISCUSSION

To examine the results of the analysis, calculated values are compared with those obtained from various experiments, the conditions of which are given in Table 1. Results 1 and 2 were obtained from the authors' experiments, results 3, 4 and 5 are due to H a y a m a [11] and 6 is due to Polukin [23]. The stress-strain relationship for the test material is not given in Ref. [22] and the relationship given in Ref. [11] is used in the calculations. Comparisons between experimental and theoretical results for roll force are shown in Figs 2 and 3, while roll torque results are given in Fig. 4. G o o d correlation is evident, better than 3% for force and 5% for torque. In Fig. 5 are shown patterns of lateral deformation, i.e. fishtailing. Relative width spread b b and relative radial position 6h are given by ~b --

B - Bo

X 100%,

(~h

Ro - R -

-

B

Ro

-

-

R i

where Bo is the original width, B is the final width, R o is the outer radius, R i the inner radius and R is the radius of the element, width B. Calculated results compare well with experimental measurements and it is seen that with increase of the ratio of inner ring radius

TABLE 1. DETAILS OF TEST RESULTS D i m e n s i o n s in m m Test no. 1 2 3 4 5 6

MS

33:5-F

RI 65 65 75 75 75 111 - 230

RE 40 40 22.5 15 22.5 61 - 180

Ro 89 91.5 39 39 39 112.2 - 590

RI

B0

70.4 70.6 34 34 34 62.2 - 540

27.2 30.2 12 12 l0 40

6 (MPa)

Material

24.6e T M

Lead

118e °1T

Aluminium

398

S.G. Xu et al.

750 [

o I~ 250

,~/~No. / ~f,'~

Experiment Theory ' 0

I

4

8

12

16

Y FIG. 2. Experimental and theoretical roll force results for leadl

t

f

./ , J

I~ 200

Experiment Theory • o --.~

5 4

i

0

J

OI tn

0.2

(ho/h )

FIG. 3. Experimental and theoretical roll force results for aluminium.

,o]

~'

"f O

___~/"

I

No. Exper2, nt Theory

:

OI t.n (ho/h)

I

02

Fla. 4. Experimental and theoretical roll torque results.

to mandrel radius R i / R 2 , the width spread increases near the inner surface and decreases near the outer surface. In ring rolling, roll friction affects the lateral deformation significantly and since friction is likely to vary over the roll surface it is not possible to specify it exactly in the calculations.

Simulation of ring rolling (o)

399

(b)

6 ~

Theory

"~ - - c4~

,

~~Experiment

,

6 ~

~x~

Theory

co~4~ ~ X ~

0

Experiment

0 I

0

t

02

I

04

I

0.6

0,8

]

I

1.0

I

0.2

c)

L

I

0.4

8h

6

--A--

I

0.6

0.8

I

1.0

8h

Id)

Theory

Experiment ~ = ~

Theory

6

Experiment --.o.--

,

0

. I

I

0

0.2

.

.

.

I

I

0.4

0.6

.

0 l

0.8

J

1.0

I

0

I

0.2

I

0.4

Bh

I

0.6

I

0.8

1.0

8h

FIG. 5. Width spread (fishtailing)Test No. 6.

(o)

,~ 3o E

3o (-b)

Rolling direction r =8.6%

r=5%

E

Rolling

-~ I0

direction

r=8.6%

r=5%

~ I0 n,,"

o

I

Z l(mm)

3

4 (Exit)

o

i

2

3

4

5

~(mm)

6 (Exit)

F=O. 6. N o r m a l pressure distribution along arc o f contact: (a) on driven roll; and (b) on mandrel.

The value used was/t = 0.28 and the differences between the calculated and experimental results in Fig. 5 might be partly accountable to discrepancies in friction. Figure 6 shows the theoretical curves of normal pressure distribution along the arc of contact, for the driven roll and the mandrel. The pressure increases rapidly to a maximum after initial contact, falls a little and then increases slightly near the exit point. Normal pressure distributions in the lateral direction, parallel to the ring axis, are shown in Fig. 7. The pressure at any position is the average value along the arc of contact. The pressure is maximum at mid length, falling slightly towards the ends of the ring. The calculated pressure distributions in both directions are seen to be similar to the experimental results reported by Mamalis [6-1. Figure 8 shows the distribution of equivalent strain rate on the mid-plane. Near the initial contact points with the roll there are regions of high strain rate indicating severe deformation. The strain rate is generally higher at the surface of the main roll than at the mandrel surface.

400

S.G. Xu et al.

Driven roLL E

15 E-

r=86%

~,° g.

5

I ©

',

I

15

5

J 45

P 6

Y (oxioL direction)

FIG. 7. Normal pressure distribution across ring width.

d

FIG. 8. Equivalent strain rate distribution in roll gap.

CONCLUSIONS

In the analysis presented here, both the nodal velocities and the mandrel velocity are considered as unknown variables to be solved in the iteration process. Thus with only one variable added to the vector of nodal velocities, the mandrel velocity is easily calculated. For determining the amount of reduction due to each roll, the distribution is chosen arbitrarily at first and the roll forces are compared. With iteration the forces converge and finally match. These methods for dealing with the unknown roll velocity and the reduction under each roll are applicable also to other asymmetric rolling processes. Comparison between calculated and experimental results indicate that the finite element analysis accurately models the mechanics of the process. In general the calculation time is relatively long and to save the CPU time it is suggested that different meshes are used according to the purpose of the analysis. For example if only the roll force and torque are required a small number of elements might yield sufficiently accurate results, while for studying flow patterns relatively fine meshes are required. The program developed here can select different mesh sizes automatically. The analysis so far has been concerned with single pass rolling of plain rings and further development will be concerned with continuous multipass profile rolling. Acknowledgement The authors wish to thank the Computer Centre of Yanshan University for the use of computer facilities•

REFERENCES 1, W. JOHNSON and A. G. MAMALIS, Rolling of rings. Int. Met. Rev. 24, 4, 137 (1979). 2, W. JOHNSON and G. NEEDHAM, Experiments on ring rolling. Int. J. Mech. Sci. 10, 95 (1968).

Simulation of ring rolling

401

3. W. JOHNSON, I. MACLEOD and G. NEEDHAM, An experimental investigation into the process of ring or metal tyre rolling. Int. J. Mech. Sci. 10, 455 (1968). 4. W. JOHNSON and G. NEEDHAM, Plastic hinges in ring indentation in relation to ring rolling. Int. J. Mech. Sci. 10, 487 (1968). 5. A. G. MAMALIS, J. B. HAWKYARD and W. JOHNSON, On the pressure distribution between stock and rolls in ring rolling. J. Mech. Engng Sci. lg, 4, 184 (1976). 6. A. G. MAMALIS, J. B. HAWKYARD and W. JOHNSON, Pressure distribution, roll force and torque in cold ring rolling. J. Mech. Engng Sci. lg, 4, 197 (1976). 7. A. G. MAMALIS, J. B. HAWKYARD and W. JOHNSON, Spread and flow patterns in ring rolling. Int. J. Mech. Sci. 18, 1, 11 (1976). 8. J. B. HAWKYARD, W. JOHNSON, J. KIRKLAND and E. APPLETON, Analyses for roll force and torque in ring rolling with some supporting experiments. Int. J. Mech. Sci. 15, 873 (1973). 9. D. Y. YANG, D. Y. RYOO, J. C. CHOI and W. JOHNSON, Analysis of roll torque in profile ring rolling of Lsections. Proc. 21st Int. M T D R Conf. pp. 69-74, London (1980). 10. J. S. RYOO, D. Y. YANG and W. JOHNSON, Ring rolling; the inclusion of pressure roll speed for estimating torque by using a velocity superposition method. Proc. 24th Int. M T D R Conf. pp. 19-24, Manchester (1983). 11. M. HAYAMA, Theoretical analysis on ring rolling of plain rings. Bull. Fac. Engng Yokohama Nat. Univ. 31, 131 (1982). 12. C. F. LUGORA and A. N. BRAMLEY, Analysis of spread in ring rolling. Int. J. Mech. Sci. 29, 149 (1987). 13. S.G. Xu, H. W. WANG and G. Z. L1, Analysis of width spread in ring rolling. J. Taiyuan Heavy Mchy Inst. 9, 3, 1 (1988). 14. S. G. Xu, H. W. WANG and Z. P. SHAN,Modelling the process of ring rolling by UBET. J. Taiyuan Heavy Mchy Inst. 10, 1, 1 (1989). 15. P. BOUCLY, J. OUDIN and Y. RAVALARD, Simulation of ring rolling with new wax-based model materials on a flexible experimental machine. J. Mech. Wkg. Tech. 16, 119 (1988). 16. T. HIRAI, Y. MAEKAWA, T. KATAYAMA and N. SAIKI, A study on ring rolling of T shaped rings. Part 1. Deformation patterns in ring rolling of plain rings and T shaped rings. Sci. Eng. Rev. Doshisha Univ. 27, 3, 32, (1986). 17. T. HIRAI, Y. MAEKAWA, T. KATAYAMA and N. SAIKI, A study on ring rolling of T shaped rings. Part 2. Consideration of internal deformation patterns using plasticine rings. Sci. Eng. Rev. Doshisha Univ. 27, 3, 43, (1986). 18. J. S. YUN and H. S. CHO, Optimal control system design for ring rolling processes. Proc. 1st ICTP Conf. Tokyo. 1322 (1984). 19. D. Y. YANG and K. H. KIM, Rigid-plastic finite element analysis of plane strain ring rolling. Int. J. Mech. Sci. 30, 571 (1988). 20. K. MORI, Analysis of three dimensional deformation in rolling by rigid plastic finite element method. J. Jap. Sci. Tech. Plast. 24, 273, 1022 (1983). 21. K. MORI and K. OSAKADA, Approximate analysis of three-dimensional deformation in edge rolling by the rigid plastic finite element method. J. Jap. Sci. Tech. Plast. 23, 260, 897 (1982). 22. K. MOR1, S. SH1MA and K. OSAKADA,Analysis of free forging by rigid-plastic finite element method based on the plasticity equation for porous metals. Bull. J S M E 23, 178, 523 (1980). 23. P. I. POLUKIN, P. K. TETERIN, M. M. SHALYPIN and V. K. VORONTSOV, Investigation of spread when rolling rings. Izvest. Vuz Chem. Met. 9, 69 (1972).