Approximate solutions for preform design in rolling

lnt J M a t h T ¢ ~ d l ) c s . Rcs Printed in Great Britain

~,ol.24. No. 3, pp. 215-224.1984.

APPROXIMATE

II 2 -7~'~7~,45 ~ I ~ 1lib P e r g a m o n Pres~ l.td

SOLUTIONS FOR PREFORM ROLLING*

DESIGN

IN

SHIRO KOBAYASHIJ ( Received for publication 5 March 1984)

A ~ t r a c t - - l n ingot rolling the defects are overhang at the front and rear ends of plates. The defect shapes that occur under the plane-strain condition can be eliminated with proper design of end shapes in ingots. This paper presents an approximate solution for design of ingot end shapes. The theory uses a channel flow as a velocity field for non-steady-state deformation of end regions and predicts the overhang contours and end shapes to eliminate overhang at front and back ends. Indications are given to show that the solutions are good for non-workhardening materials.

NOMENCLATURE 0,0 R H1. H, g(0) U 6,0 r(O) Ro ro

n,!o), R2(O) r 1 . r2 a, a'

b,b' eL, B 1

coordinates in polar coordinate systems equivalent semi-die angle roll radius plate half-thicknesses before and after rolling function of 0 velocity inclinations of velocity discontinuities radial coordinates of front or back end contour radial coordinate of discontinuity lines along the line of symmetry (0 = 0) radial coordinate of front or back end along the line of symmetry (0 = 0) radial coordinates of front or back end at entrance and at exit, respectively equal to RI(0) and R2(0), respectively dimensions indicating the discontinuity line with respect to front end position dimensions indicating the discontinuity line with respect to back end position coefficients equal to a'/a and b'/b, respectively time

1. INTRODUCTION

ONE Or the most important aspects in metal forming processes is the design of preforms. It is involved in forging and in shell nosing. Some investigations on preform design in shell nosing have been reported recently [1, 2]. The problem of preform design arises also in other forming processes, such as rolling and sheet metal forming; the present investigation is concerned with the problem in rolling. A large tonnage of ingots is rolled each year in blooming and slabbing mills and significant defects are found to occur in the resulting slabs at the front and rear ends [3]. The actual proportions of the defect shape depend on the rolling schedule. The defect shapes are overhang at the front and along a rolled bar, and overlap and fish tail at the rear end (Fig. 1). These defective end-regions are cropped from rolled slabs and each mav account for up to about 4% of the total rolled weight. The international slab yield is said to be about 90%. Obviously, if the yield can be increased by only a small percentage by reducing the crop loss, then the economic savings where there is a large volume of production would be sizeable. The problem is then to design the shape of ingots with *Part of 3 of "Preform Design in Metal Forming." 7Department of Mechanical Engineering. University of California, Berkeley, California, U.S.A. 215

216

S H m O KOBAYASHI Front End

Rear End

Overhan9

Overlap,

S'o Flo. 1. Defect shapes in slab ingot rolling.

/

/

//

/ //

i

/

R

I I

2H 2

2 iI

i

Flo. 2. Rolling process with approximations.

~

Rdl

-C.-

-,X-~-~--F'°"' ~~ .],

I

.Channel

-

Velocity/v Discontinuity (o)

" , •" ~

;

Roll

.~

Channel Flow

Discontinuity (b)

F=G. 3. Non-steady-state velocity fields for (a) front and (b) back portions under rolling.

Approximate Solutions for Preform Design in Rolling

217

which the defect shapes are reduced or eliminated on the rolled slabs. In general, end shape changes as they occur will be difficult to predict, since they are of a complicated three-dimensional character. However, the defect shapes that occur under the planestrain condition, such as overhang at the front and rear ends, can be eliminated with proper design of end shapes in ingots. A few investigations have been reported on this aspect of rolling in the past. In particular, a theoretical analysis of overlap formation at the back and front ends by Stahlberg et al. [4] is to be noted. However, these investigators were mainly concerned with prediction of crop losses. In this paper an approximate solution is presented for design of end shapes of ingots. The theory predicts the overhang contours and end shapes to eliminate overhang at front and back ends. 2. PRELIMINARIES In slab ingot rolling, the initial slab thickness of 2H~ is reduced to the thickness of 2H2 through a pair of rolls of diameter 2R. It is assumed that the process of rolling is plane strain and the material is non-workhardening. Several approximations and assumptions for metal flow in rolling are made, First, neglecting the effect of roll curvature, metal flow is approximated by a flow through straight converging dies with semi-die angle b, as shown in Fig. 2. From the geometry in Fig. 2, the angle ~ corresponding to a converging die is expressed by R/H1 cot2~ = 2 1 - H2/H 1

1.

(1)

Next, a velocity field for non-steady-state deformation of the front portion of an ingot is assumed to be a channel flow which is continuous to a rigid-body motion through a straight velocity discontinuity line, as shown in Fig. 3(a). Similarly, a channel flow is assumed for the non-steady-state deformation of the back portion of an ingot (Fig. 3(b)). In channel flow the velocity Uis directed toward the apex O of the channel and is expressed by U-

g(0) P

(2)

in the polar coordinate system (O, 0), where g(0) is a function of 0 only. In the next section the velocity fields shown in Fig. 3 are formulated and equations for design calculation are derived. 3. FORMULATIONS Front end

With reference to Fig. 4(a), a velocity discontinuity line is expressed by R0 sin+ P = sin(+ - 0)~"

(3)

Because the velocity components normal to the discontinuity line on both sides must be continuous, the velocity U along the velocity discontinuity in the region of channel flow is expressed by

U =

g(0) Ro sin6/sin(+ - 0)

sin+ sin(+ - 0)

for the rigid body motion of U = - 1 . Then,

218

SmRO KOBAYASHI

sinZ6 g(0) = - R o sin"(6 - 0)' Along the front end P = r, U = g(O) sin26 r = - sin2(6 - O)

Ro

(4)

r

We focus our attention on the geometrical change of the front end and note that dr

U(r,O) = ~

(5)

for 0 = const.

where t is the time. Since Ro ro

U(ro,O) . . . . .

dro dt

from equations (4) and (5), then ro dro. dt = - R-~-

(6)

Equation (6) replaces the time scale from t to ro. Substituting equation (6) into equation (5),we obtain

U(r,O) = g(O) _ r

Ro dr ro dro

from which sin26 r dr = sine(6 _ 0) ro dro for 0 = const.

(7)

Integration of equation (7) from the entrance of the front end into rolling will result in the front end contour during the non-steady-state deformation, namely,

(o) r dr = f,, sine(6 sinZ6- 0) RT(0)

ro dro.

(8)

rl

The integration limits in equation (8) are illustrated in Fig. 4(b). Noting that angle 6 in equation (8) is a function of ro given by cot6=(Htc°t~-r°) Hi

et

where ot = a'/a (see Fig. 4(a)) and assumed to remain constant during deformation, equation (8) results in Vz(r2 - R]) = (

H,

/2[in(A(0) + ot(r./H,)tan0) \ A(O) + ot(r,IH,) tan0

\ ~si~-/L

/1

1 1 + A(O) t A(O) + o~(rdHt) tan0 - A(0) + ot(rl/Ht) tan0) ,

(9)

Approximate Solutionsfor Preform Design in Rolling

~0 '~

~o-~.I

219

ro

Ro

"I

(o)

,Front end at entrance (time ro=r0

X%'<, ..~_~

/// J~,

/

/"

/Front encl at time ro=ro / Front end at exit (time roar2) /

(b) FIG. 4. (a) Velocity field and (b) various stages of front end portions during non-steady-state

deformation. where A(O) -- 1 - a cot~ tanO. In deriving equation (9), note that the right-hand side of equation (8) is expressed by fo

rodro ~o rodro COS20(1 - cotStan0) 2 = cos20(1 - et cotB tan0 + ot(ro/Hl) tan0) 2

rl

r1

and that use is made of the integration formula given by xdx 1 (a + bx) 2 = ~

a In la + bxJ + --a

+ bx

x,

x,

In equation (9), if Rt(0) (thus, rl = RI(0)) is specified, then r(0) will provide the front end contour as functions of time (time scale is r0). To calculate the preform contour of the front end, we specify r = R2(0) and corresponding time r0 = r2 = R2(0), then Rl(0) from equation (9) will give the preform contour.

Back end deformation Taking that the rigid body velocity at the exit be - 1 , and requiring the continuity condition along the discontinuity line in Fig. 5(a), the velocity for the channel flow given by equation (2) along the back end becomes sin20 R0 U = sin2(0 + 0) r

(10)

Equations (5) and (6) are also valid for this case and, with equation (10), we have f~, r dr = R~

r,

sin20 ~in2ttlJ+ 0) ro dro. 1

(11)

220

SHine KOBAYASH1

3"-"--

'

•

•

•

j

]

,. . . .

R0

1

r0

(a)

•8ock end at entrance(time to: q) sock ena at t me ro=ro } ~endate~t(timero=r /

/

--'%,

/

~

, r =R1(8)

I

I.

rz r~

J

.I.~

ro

(b}

Fro. 5. (a) Velocity field and (b) various stages of back end portion during non-steady-state deformation. Using the relationship that c o t 0 = ( r o - H E c HE °t~)

13, where 13 = b'/b (see Fig. 5(a))

equation (11) results in l/2(r 2 - R]) = [

H2

~-~] B

/ 2 [ i n / B + 13(ro/H2)tan0

[ \ B + 13(rt/H2) t-~nO] +

B + 13(to/H2) t a n O -

B + 13(rl/H2) tanO

)]

where B= When contours equation equation

1-13cotStan0

.

(12)

we specify RI(0), including rl = R 1 ( 0 ) , then equation (12) gives the overhang of the back end as functions of time re. When we put r = R 2 and re = r2 in (12) and require the back end shape r = 82 to be specific, then R1 from (12) will provide the preform shape. 4. EXAMPLE

For ingot rolling where the overhang formation is critical, the thickness of the plate is relatively large compared to the roll diameter. The ratio of roll diameter to the plate thickness in usual rolling is the order to about 50 or 100, but this ratio is about 1-10 in ingot rolling. In order to examine the approximate solution quantitatively, a numerical example is obtained for the following conditions: 2R Hi - H2 -= 4.00, the reduction = 2H2 Hi

-

0.2

Approximate Solutionsfor PreformDesign in Rolling

221

(refer to Fig. 2). We use equation (9) to calculate front end deformation and equation (12) for back end deformation. In these equations the coefficients ~ and 13 should be given. A reasonable choice for the coefficients (although these coefficients can be considered as parameters) is based on consideration of transient deformation between steady-state and non-steadystate deformation represented in Fig. 6. Considering the transient deformation of the front end from the non-steady-state to the steady-state, the angle ~b in Fig. 6(b) should approach the angle d~ in Fig. 6(a) for the steady state. Similarly, the transient deformation of the back end from the steady to non-steady-states could be characterized by the angle ~ in Fig. 6(c) which would be the same as angle ~J in Fig. 6(a). If we simply assume ~ = ~ under the steady-state deformation in Fig. 6(a), then 0t o

Ha H2 H1 + H2 in equation (9) and [3 = H~ + H2

in equation (12).

Front end overhang When the front end is straight before rolling, the contour of a front end overhang can be calculated by setting rl = H1 cot8 and RI = rl/cos0 in equation (9). The contours of the front end are obtained for variovs values of ro/H1 in Fig. 7. Front end preform In order to design the preform shape of the front end to eliminate the overhang, put r = R2 and r0 = r2 and require that r2 = H= cot8 and R2 = r2/cosO. The preform shape is shown in Fig. 7. With this preform, the front end is expected to be straight after rolling.

(o)

(b)

(c) FIG. 6. (a) Steady-state velocity field (assumed to be (b = 0), (b) front end deformation showing transient from non-steady-state to steady state, (c) back end deformation showing transient from steady state to non-steady-state.

~9 2,._

SmROKOBAYASHI 1.0

e el Ri/l-,I I x/H I

5.66 5.63 5.60 5.58 5.56 5.53

6 4 2 0 t

0

¢/HI

0 .027 .052 .077

1.00 .791 589 .390 .100 .194 .J22 0

' ' 'a5

Hi

~

Roll

0.5 ~

5.5

5.0 ro/Hf

4.5

0

FIG. 7. Overhang contours and preform shape of the front end.

O° iO 8 6 4 2 0

x/H I y/H I 0 f.O0 .024 .798 .0491.60(3 .075 .401 .10~.130.201 0

RI/H I 5.66 5.68 5.70 5.73 5.76 5.79

I~,.0-----~

0.5

O. HI

~

"'-~, Roll

~0.5 ~ iliIlj

i, 5.5

,. 5.0

ro/H~

f 4.5

Fro. 8. Overhang contours and preform shape of the back end.

Back end overhang Use the same conditions as those for front end overhang, namely, r t = HI cot6 and R~ = rl/cos0, in equation (12).

Back end preform Use again the condition that r = R2 and ro = r2, where r2 =/-/2 cot5 and R2 = r2/cos0, in equation (12). The contours of back end overhang and the preform shape for the back end are shown in Fig. 8.

5. DISCUSSION Based on a velocity field consisting of a channel flow, the non-steady-state deformations of front and back ends are described in ingot rolling. Because of the channel flow, stream lines are known during the deformation and the displacements of the front and

Approximate Solutions for Preform Design in Rolling

223

FIG. 9. Rolling simulation of plates with preform of front end.

LLII

Ilfllll\-k,_.N...,--~

l

FIG. 10. Rolling simulation of plates with preform of back end.

back ends can be obtained by integrating velocities along stream lines. The overhang contours with initially flat ends were calculated and the preform shapes for eliminating overhangs were designed. The approach to the problem and the results may not be too appealing unless it is shown that the solutions are good. The degree of approximation of the solutions can be examined by comparing with experimental results. It can also be done by the finiteelement simulation. Rolling of plates of preform end shapes obtained in Figs 7 and 8 was simulated by the finite-element method. The results of simulation are shown in Fig. 9 for the front end and Fig. 10 for the back end with a non-workhardening material. The results show that both front and back ends are nearly fiat after roiling and demonstrate that the approximate solutions are good for non-workhardening materials. In concluding, it should be mentioned that the finite-element method not only simulates forming processes but also can be used directly for the problem of preform

224

Smao KOBAYASHI

design. T h e a d v a n c e d m e t h o d for p r e f o r m design in rolling by the use of the finitee l e m e n t m e t h o d is p r e s e n t e d in Part 4 of "'Preform Design in Metal F o r m i n g " . to be published. Acknowledgements--The author wishes to thank the National Science Foundation for its grant MEA-8312062

under which the present investigation was made possible. He also wishes to thank Ikuko Workman for typing the manuscript. REFERENCES [1] J. J. PARK,N. REIIELOand S. KOBAYASHI,Int. J. Mach. Tool Des, Res. 23, 71-79 (1983). [2] S. KOBAYASHi,Int. J. Mach. Tool Des. Res. 23, 111-122 (1983). [3] W. JOHNSON,Mechanics of Solids (edited by H. G. Hopkins and M. J. Sewell), p. 303, Pergamon Press. Oxford (1980). [4] U. STAHLBERG,J.-O. SODENBEROand A. WALLERO,Int. J. Mech. Sci. 23, 243-252 (1981).