Cold ring rolling of a polymer

Int. J. Mach. Tool Des. Res. Vol. 20, pp. 97 109. Pergamon Press Ltd. 1980. Printed in Great Britain.

COLD

RING

ROLLING

OF

A POLYMER

K. VIHARI* and S. KUMARt (Received 20 September 1979; in final form 19 December 1979) Abstract - The feasibility of the cold processing of certain polymers has been demonstrated. Considering the process

of the rolling of a polymer ring, the problems of pressure distribution at the roll-workpiece interface, roll force and roll torque have been analysed. The assumptions are that the process is similar to rolling with one of the unequal rolls driven under the constraints imposed by the compression, caused by the progressive decrease in radial thickness as the roll gap is reduced. The correctness of the yield criterion and flow rules used during the analysis have been verified experimentally. The results so obtained are discussed critically to illustrate the interaction of the different parameters involved and are presented graphically for a particular case of polymer ring rolling.

NOMENCLATURE A Ao A, ri ro

h h~ b r2

I X

fIN P a 61,62~63

Gyc ayt

#

~o n

a p p a r e n t area of contact dimensionless A,/ A real area of c o n t a c t current inner ring radius current outer ring radius radial thickness of the ring at any instant on the roll work piece interface current radial thickness of the ring width of the ring radius of main roll radius of undriven roll projected c o n t a c t length between roll and ring distance roll closure rate per revolution roll pressure shear stress n o r m a l stress principal stresses uniaxial yield stress in compression for the polymer uniaxial yield stress in tension for the polymer coefficient of sliding friction angle specific cohesion of contact surface power index, whose value is m u c h greater than one and depends u p o n the processing condition flow stress for the polymer INTRODUCTION

DURING the last few decades polymers have assumed an i m p o r t a n t position in industry, as they are being used increasingly to replace metallic parts in a wide range of applications. P o l y m e r processing is an engineering speciality concerned with the operations carried out on polymeric materials or systems to increase their utility. The rotary forming processes have aroused appreciable commercial interest in recent years, as a means of producing c o m p o n e n t s m o r e accurately and effectively than hitherto, as each process forms the c o m p o n e n t progressively over a n u m b e r of rotations or oscillations by rotating the * Department of Mechanical Engineering, Birla Institute of Technology, Mesra, Ranchi-835215, India. t Department of Production Engineering, Birla Institute of Technology, Mesra, Ranchi-835215, India. 97

98

K. VIHARIand S. KUMAR

plastically-deformed zone relatively through the work piece. The mechanics of the deformations are complex and generally not well understood. Ring rolling is a special forging process performed on a vertical two-high mill with one of the rolls driven. For a metallic ring, the operation can be carried out in the hot or cold state, the hot process usually involving a large expansion of diameter from the thickwalled forged blank and in the cold process a small expansion accompanying precision profile forming. The mechanics of deformation [1] in the plastic zone in ring rolling are more complex than other rotary forming processes as the process differs in relation to work piece shape, unequal roll diameters with one roll undriven, and in that the reduction per pass is relatively small since the required size is achieved by a number of ring rotations. Some useful information in connection with the rolling of metallic rings is provided [2]. The majority of investigations carried out in various polymer processing are based on the assumption that rheological behaviour of polymers will be approximated by fluids [3, 4]. However, there are certain polymers which are ductile in nature and can be cold formed. Recently the formation of a flow zone at the work-die interface and its analysis during polymer processing has been analysed [5]. Unfortunately, the details of cold-ring rolling of polymers are practically unknown. The principal aim of this work is to analyse the problems of pressure distribution at roll workpiece interface, roll force and roll torque. The assumptions are that the process is similar to rolling with one of the unequal rolls driven under the constraints imposed by the compression, caused by the progressive decrease in radial thickness as the roll gap is reduced. During analysis the modified Von Mises criterion and composite friction law have been used. The appropriateness of the proposed modified Von Mises criterion and composite friction law was verified experimentally. The results so obtained are analysed and discussed critically for a particular case of polymer ring rolling. This work will be of a great practical importance in designing required tooling for ring rolling mills for polymers. COLD FORMABILITY OF A POLYMER One important object of testing the material is to aid in predicting or assuring the desired formability of the materials. With the above view in mind standard specimens of Garfion (Nylon group polymer) were prepared from the original and the deformed material and tested in a universal testing machine for both tension and compression. Load extension diagrams so obtained are presented in Fig. 1.

/

Extension

Compression

FIG. 1. (a) Tensiletest. (b) Compressive test of a polymer.

Cold Ring Roiling of a Polymer

99

/ /

f

--1T

U

~

Normal pressure

Ap

]

p P

T1-r

----

-

I

pp

i

I

rr

1mr

Ar<
Ar
Ar=A Plastic flow in bulk

~

p

A = Apparent area of contact Ar: Real area of contact

FIG. 2. Three regimes of surface contact.

The characteristic properties of Garflon are given in Appendix A. The particular polymer is subjected to about 55% elongation before fracture and the yield stress after processing of the polymer is more or less the same as that of the original one. Hence it can be concluded that the particular polymer can be processed under cold conditions both efficiently and economically.

INTER-FACIAL

FRICTION

One of the very important problems which requires special consideration during ring rolling of a polymer is that of inter-facial friction, as it will determine the success or failure of the operation. But the frictional phenomenon is not completely understood and for actual cases an exact formulation of frictional relationship can not be made in the analysis. Relative velocities between the workpiece material and the roll surface combined with high interface pressure and/or deformation modes will create the condition essential for adhesion, and probably the most important problem is that of adhesion between work piece and the roll. The basic affinity of the rolls and the workpiece material will determine the friction value. For such a mechanism of composite friction which usually occurs in ring rolling [6] the shear equation becomes = ~(p + .4oq~o),

(1)

where the first term on the right gives the sliding friction and the second the friction due to adhesion. It is found that the real area of contact grows and approaches the apparent one as force increases (Fig. 2). This approach will probably be an asymptotic one [-7]. Therefore, shear equation (1) becomes

{

[

" r = # p+Ao~b o 1 - \ l / /

JJ'

where n >> 1, the value depending upon the processing conditions.

(2)

100

K. VIHARI and S. KUMAR YIELD CRITERIA

The yield criterion used would have to contain the hydrostatic stress as the yielding of the group of polymers under consideration is not completely insensitive to the hydrostatic stress imposed [8-1. Hence Von Mises criterion is modified as 20.yc0.yt ~--" E(0.1 -- 0"2) 2 -I- (0.2 -- 0.3) 2 -'I- (0.3 -- 0.1) 2] "{- 2(0.yc- 0.yt)(0.1 ~- 0.2 q- 0.3).

(3)

If the yield stress is the same in tension and compression the usual Von Mises condition is recovered. For plain strain, the principal stress will be 0.x = 0.~, a2 = - p ; a3 = (0.1 + 0.2)/2 and (3) reduces to 3(0.1

- - 0 . 2 ) 2 -}-

6@(0.1

-- 0.2) +

12@tr2 -- 40.rc0.yt = 0,

(4)

where ~b = 0.y~ - ay, = a constant. ANALYSIS FOR PRESSURE DISTRIBUTION, FORCE AND ROLL TORQUE

ROLL

In addition to the usual assumptions during rolling [9], the following assumptions are made in the analysis: (1) friction due to adhesion is not constant but is a function of real area of contact ; (2) the yield criterion is not insensitive to hydrostatic stress components imposed during processing. In the case of ring rolling, the external roll is driven by motor, whilst the other roll rotates only as a result of friction arising between the workpiece being rolled and the free running roll. The roll force is applied hydraulically with valves providing control of the roll closure rate and roll force, causing a progressive decrease in radial thickness and increase in the diameter of the ring as the roll gap is reduced. Apart from the obvious differences of workpiece geometry, unequal rolls etc. as compared to conventional rolling, the main difference is that of the draft, which is small in ring rolling, since the required size is achieved by a larger number of rotations of the ring. As compression followed by rolling takes place in each pass and the amount of reduction per pass is very small, the process may be considered

Polymer

'~uide roll

Idling

/Work

i

FIG. 3. Ring rolling process.

Cold Ring Rolling of a Polymer

101

ol

\\

T (0)

°; O-X+ d o ' x~~

"fiX

7

h

,

h+dh

P2

(c) FlG. 4. (a) Ring and rolls just at the instant of initial feeding phase. (b) Enlarged view of the material in the roll gap. (c) The differential element.

to be equivalent to rolling with one of the unequal rolls driven with the constraint imposed by compression during each pass. A general arrangement of the ring rolling is shown in Fig. 3.

Initial feeding phase The roll, workpiece, differential element and co-ordinate system are shown in Fig. 4. The deformation in the roll gap at the instant of initial feeding will be symmetrical about the line joining the centres of the rolls. Considering the equilibrium of the forces in the x direction which is assumed to be the direction of principal stress ; (ax + dax)(h + dh) - a~ dh + dx(p I tan cqx + P2 tan Ct2x)- (z I + z2)dx -- 0, or

hd0-~-F0"xdh+dx(px tan~qx+p2tan~2~)-I~dx

pl + Ao~o 1 + p. + Ao4)o

\ll)

1

[ ?

1 - V./

AJ = O,

(5)

where dh = dx (tan ~1~ + tan ~zx). Since closure rate of roll gap, i.e. feed per pass is very small, ~lx and 0~2xare small, therefore, Pl ~ P2 = P"

Hence (5) reduces to hdax + 0-xdh + p d h -

# dh tan alx + tan a2x

2p+Aoq5 o 2

\11/

-\12)

JJ =0.

(6)

From (4); 0-1 -- 0"2 = --I// ~___~ 1 (

, 3 2I/] -- 12ff21~ + 40"yc0"yt) = 21(say).

(7)

As the yielding of the material at a point during the process will take place at a particular maximum value of 0-2, hence 21 will be a constant for the particular material and for given processing conditions (see Appendix B).

102

K. VIHARIand S. KUMAR

Therefore 0"1

a2 = 21 ; Here al = ax and 0"2

-

-=

- p , therefore,

0"x = 21 - P

(8)

d a . = -- dp

(9)

and Using (8) and (9) in (6);

- hdp + dh 2 1 - t a n o t l ~ + t a n a 2 ~

{2P +

Ao~bo[2 -

(x/ll) 1/" -- (x/12)l/"]} 1 = O.

(10)

Since the feed is very small, the arc of contact may be equated to a parabola to get sufficiently accurate results for practical purposes, hence the thickness h at any instant may be expressed as

h=h,+\

(Aht Ah2"~ 2 1(1 1 12 +--~-2 ) x = h i + ~ \ ~ 0 + - + rl

1 --

1)

r2

rif

h,(1 + tan 2

x2=

14;),

(11)

where tan cqx + tan 0~2x tan w - x/[2(1/r ° + l/rx _ l/ri + 1/r2)hJ tan ctlx =

(1+1)

x;

t a n 0~2x ~--

(11)

x

tan w

w

11

tan wl

wl

x

tan w

w

12

tan w2

w2

ll

[ r l ( r i _ r 2 ~ 2r ° ]1/3

-

(12) X

(13)

/2 = ~ 2 \ r o ~ , ] ~ ]

"

As the ring rolling operation is carried out at constant main roll speed and constant closure speed of the roll gap, f, the thickness reduction (draft) will increase as the ring diameter increases. F o r a m o m e n t neglecting the slip between main driving roll and the ring [2], Ah-

f r° N r1

Differentiating (11) dh dw

-

2h~ tan w (1 + tan 2 w) = 2h tan w.

(14)

(jw

(15)

Hence (10) becomes

where 4 #

k=~/I2(lll~0

+ - 1 - -r,

(16)

l')h']'r2/ d

Integrating (15) on simplifications neglecting higher order terms gives P

Ce-kw -- 2(1 =

--kw)--

/wlJn w,Jn

2. . . . \Wl, /

\W2//

+

(17)

Cold Ring Rolling of a Polymer

103

where C = a constant of integration. Using the boundary condition at entrance on the upper roll, Pl 21

--=

1

at w = w

1.

(18)

To evaluate the constant C and putting the value of C in (17) gives the distribution of pressure on upper roll as 2--~=ek0~'-w)

1+

1--\WE/

(1-kwl)+~

1-~

2

+

1

1 +

1-n

Similarly the pressure distribution on the lower roll, 2 - ~ : e ktw~-w, 1 +

k2 (1 -

1-(w2~1/"(1\

(1-kw2)+

kw)

\Wll

22:

1

+

1

~2/n

\w2/t

The initial feeding force is co-linear to the line joining the centres of roll and is given by P=2b

p dx c o s ~ + z \

COS ~

sin~t cos

(21) which on integration gives

[k-~ ( 1 - e - k W ' ) - - k 5 - ( 2 - k w l ) 221 { [ (\w2/ lw,y/"l(I kw, l~-n n )} ro+r i Ix{B[1 ( kw,)] 2wl{l k~,) +# rorl wi -~x -~--e-kw' .1+ 1

P=2bl,S1 -

-

- A°q~°22~w,{[1

A°q~°

k2

+

1 \w2/ d (l+n)kw~

-

2+

1 + -

~ - \2

n] - [i

l+2n

(w,y/"l~ll '

2n l+-2nkWs] j j j j

,22)

where

B = e k W ' { l + ~ - - i ( 1 - k W l ) + ~ I 1-(wl~l/"(1--\w2/ k

1

,23)

Rolling phase Immediately after initial feeding, the re-distribution of pressure on the contact surface takes place due to rolling, causing the resultants of the pressure of the workpiece on the rolls, though equal to each other, to depart from the vertical. On the free running roll the pressure resultant is directed radially to the roll, whilst on the driven roll this resultant is directed along the line drawn through the centre of the free running roll and the point of application of the resultant of the workpiece pressure on the rolls. In this case, the rolling process becomes unsymmetric relative to both rolls. During rolling under these conditions, the rolled material slips along the driven roll, the slip taking place over the entire arc of contact in the same direction. Consequently, there is no zone of forward slip on the contact surface of the driven roll and the zone of the backward slip extends over the entire arc of the contact. Similarly the entire contact surface of the free running roll will be the zone of the forward slip [-10]. The rolls, workpiece, differential element and coordinate system during rolling are shown M.T.D.R. 20/2--B

104

K. VIHARI a n d S. KUMAR

/

,-./

IDrivenroll

~

alx~",..~t_

\

P

Pt,-

'

Idle roll \

h+dh

2
~

(o)

(c) FIG. 5. (a) Ring a n d rolls d u r i n g rolling. (b) Differential element. (c) O r i e n t a t i o n of resultant pressure m a t e r i a l on roll.

in Fig. 5. H o w e v e r just at the instant of initial feeding, the compressed material is symmetrical to the line joining the centre line of the rolls, but the initiation of rolling makes the portion on one side of the centre line effective. N o w , considering the equilibrium of the forces in the x direction (try, + dtrx)(h + dh) - a':,h + dx(p'x tan ~lx + P~ tan 0{2x)

--

(Z";

z~)dx = 0,

--

(24)

or

h dtr'x + tz'~dh + dx(p'x tan cq~ + p~ tan ~2x) --

+

--

-P~-

-\12J

JJ=0"

(25)

Since ~lx and 0¢2x are small; p'~ ~ p~ = p. F r o m (4) O"1

- o-2 = - ¢

_ ~33 4(31/t2 - 12o'2@ + 4trrcar, ) =/12 (say).

(26)

Following the earlier a r g u m e n t 22 will be a constant. Neglecting the variation of tr 2 due to redistribution of stresses, 22 = 21 = 2. Therefore, O"i - - 0"2 = 2 ;

0"2 = --p.

O"1 = O'x,

(27)

Since arc of contact and closure rate of roll g a p is very small, (11), (12) and (13) will also hold g o o d for the case. Hence putting the values from (11), (12), (13) and (27) into (25) gives

dp-

. 2w + ~ Ao(aOk\w2/

.

.

\wl/ .

lJ_]

o.

(28)

Cold Ring Rolling of a Polymer

105

On integration it gives, P

w2 +

A°cp° kn~(wl

=

;

2II+

1 "]w'l+")/" + C, /"

(29,

w~/"/

where C' is a constant of integration. Since the ring is being rolled under the constraints imposed by initial feeding, during each pass the boundary condition at exit is a~=2-pl

at w = w e = 0 .

[For upper roll to be obtained from (20).] From (20) and (29) we get

{

C ' = e kw' l + ~ - ( 1 - k w 0 +

1--

1-

w2/

,t+ 'I} .0,30, k2

From (29) and (30) the pressure distribution at the contact surface on the upper roll is given : P_a2= ekw' 1 +

1-\w2i-- 1 - ~

(1-kwl)+~-

2 Aoc~oI

kn

+

( 1

1 "~w~+,v,]+w2.

(31)

Similarly, the pressure distribution at the contact surface on the lower roll is given by . P22

\wx/(w-2~/"(\lnkw£1)+nk~w211}

ek'~2{l+k . .2~-(1 .k w 2 ' + ~ - 2 ~ ° [ 1

2 Ao¢o[

k2

;

1

kn ( 1

1 )w,l+,,)/,,] w2"

w~2/"

2(1 + n~ w]/"

+

(32)

(The difference in value of flow-stress 2 at the upper and the lower roll has been neglected i.e. '~pper ~ ~'|ower

= ~')'

ROLL FORCE To calculate the value of roll force, the action of the top roll on the material, expressed in the form of the elementary forces applied to the arc of contact can be represented by a single resultant force P~ applied, say, at the point A (Fig. 5). Neglecting the variation of normal pressure and tangential stress over the width of the ring, the resultant force is given below:

PR

PR = b

Pc-~sa

bll WI

;?

+

z c-~s~/ d

(

(p2 d-

--COS Ct

(p2 + z2)t/2 dw.

Since e is small, cos e --- 1.

PR=--w~ b/1 f:l

b

1S2"~1/2dw----Wl bill:l( p+

p 1+~-]

(33)

T2)

1 / 2 p dw ~g2

(34, Again from (31) Pl = P = 2[B1 +

Cxw(1+"~/"+ w2]

z la).{B,+C,w'+"/"+wz+A°~2°[l (w'~'/"l~ \wl/ J) =

-

,

(35)

106

K. VIHAR1and S. KUMAR

where

Bi=ekW'{l+ff~(1-kw')+~[1-(wlyin(1-~)+n~-~l\ }}\w2) C'-

2

2(1+

wl/"

2 k2

Ao~o J. (36)

w~/" "

Transferring the values from (35) to (34), integrating and simplifying, one arrives at the following:

PR = 2bll ×

[(1+ ~# )[Bx + l + 2n

[(")1 -

1

~

2

nC1

-

C~ll(-1+2n I'I

w( 1 +n)ln +

1

1- ~

+ ~t 2

#'l )W(ll+n,ln}} Wl(l ~ B1\3

n) --o' +

-~

B1

. )}

2(1+n)

l+3n

1 2)(Bl + l ~nn C1w(ll+")/"+w-2"]+l~2A°~°(1-~n 2 \ _,,_bll2[(l+~# n )J

(37)

ROLL TORQUE The torque necessary to rotate the driven roll (upper roll) is equal to the product of resultant force PR by its lever arm a 2 as shown in Fig. 5. Neglecting the friction forces in the journal of the roll Torque

T1

PR(rl + r2 + hi) "sin eR,

= PR ' a2 =

(38)

where eR = angle characterizing the point of application of the resultant force material on the free running roll. sin • = Pn/PR where PH = Horizontal component of force PR

;(

Pn = b

dx sin • - ~

P cos ~

PR of the

sin

cos

=b(ro+r, li ;;t pwdw---w,l, f;' zdw ) . \ ro~2 "w~

(39)

The transfer of the values from (35) to (39) and the consequent integration give:

pH = 2bll ~Blr( r°+rl ~l 1 _ #1 + nClw~l+,)/m F( ll ~(ro+rl ~ I. Lk2ro r, ) L k l + 3 n / \ rorl ] _.

# 1

1+2n

.

(40)

TI = PR(ra + r2 + hi)sin eR = Pn(rl + r2 + hi)

(41)

-- (r, + r= + hi)~b,, {B, F?° + r'],, - ~] kk 2rorl /

.j 2fro+r, 3)1• t 4rorx ll

+¢,,w~,+.>i.V ', ~?o+r,~ Lk~/t~) --

p

1

1+ n

+ w~

-

-

1 +2n

--

(42)

EXPERIMENTAL WORK In order to compare the theoretical approach with practical values, experiments were conducted. A universal testing machine with appropriate rolls was used for the test for initial feeding. To carry out the experimental investigations, a number of polymer (Garflon) rings with dimensions, ro = 30 mm, rt -- 25 mm, h = 5 mm and b = 10 mm and two unequal rolls

Cold Ring Rolling of a Polymer

107

of mild steel having radii : r x = 25 m m and r 2 12.5 mm, were prepared. To prevent the ring from wandering a circumferential groove 6 m m deep and 10.10 m m wide was cut in the middle of the driven roll. Polymer (Garflon) rings were compressed (initial feeding) by unequal rolls under different known compressive feeding loads in a universal testing machine. Further, on each compressed ring, the projected length of the deformation zone and the draft were measured after enlarging the deformed profile 25 times on a profile projector. The draft was determined after taking an average of the measurements on three different rings for a particular load. The procedure was repeated for each load. The results of the compression test were presented in Fig. 6, where the experimental loads P are compared with theoretical values. The agreement between theory and experimentation is seen to be very close to each other and the general forms of experimental curve are predicted quite well by the theory as shown in Fig. 6. F r o m Fig. 6, it is further revealed that the increase in feed load results in decrease of adhesion very slightly. Hence the flow rule and approach used for ring rolling is valid for all practical purposes. --

RESULTS AND DISCUSSION The following examples indicate various parameters associated with a typical case of cold ring rolling of a polymer. #=0.11; Closing rate of roll gap/revol =

f/N

A0~bo=0.3p;

= 0.5 mm/rev ; radius of undriven roll = r 2 = 12.5 m m ;

isooI

//

'g/

/

/

o...~i~lij~ ii ,..~,~/"

iI I I •~

/ •~ H

-~

n=3

"/

i X'Y

II I / ~ ~

5OO

/

/ ~"/

Legend ro h b Experirnentol

0.15 3 0 mm 5turn 10 mm : --

Aoq~o=QOO3P Ao~OOO4F A0(~:0-005 P

--A~A-

Ao¢,o:O

/~//" ,

-q

I 0.5

I 1.0

I I I5 2D Draft ( A h ) mm

I 25

30

FIG. 6. Initial feeding force vs draft for verification of flow rules used.

~=4

~-

2 ~

n..

I

0

0~2

04'

x

06

0~.8

f/N

3.Smm~v -I

hi

5 rnm

,.0

L

FIG. 7. Effect of relative roll size

(rl/r2)

o v e r t h e p r e s s u r e d i s t r i b u t i o n o n d r i v e n roll w o r k p i e c e

interface.

108

K. VIHARI and S. KUMAR 50

--

~'~'-"~-'-'-~'

ri -

,~ 2.0

I 4

/

7"~-

Af /[email protected] rev'J o.3P

~

rl

N, r2 hi

2 12.5ram 5turn

I0 L

I

Q2

0

,

04

/

0.6

Q8

I0

X T

FIG. 8. Effect of the ring roll clearance ratio (rl,/r2) over the pressure distribution on the driven roll workpiece interface.

instantaneous thickness of ring = 5 mm; Ring roll clearance ratio rl/r 2 = 1.2, 1.5, 1.8, 2, 2, 5, 3,4,5.

Pressure distribution at roll workpiece interface The effect of relative roll size (rl/Q) for a particular ring roll clearance (rJr2 = 2) at any pass over the relative pressure distribution on the driven roll workpiece interface is shown in Fig. 7. At the same instantaneous thickness of the ring, the effect of ring roll clearance (rJr2) for a particular relative roll size (rl/Q = 2) at a pass over the pressure distribution on the driven roll workpiece interface is shown in Fig. 8. From Fig. 7, it is revealed that with the increase of relative roll size, the relative pressure distribution on the driven roll-workpiece interface increases. Figure 8, reflects a decrease in the relative pressure distribution on the driven roll workpiece interface, as the ring roll clearance increases.

Effective coefficient of friction Figure 9 shows the variation of effective coefficient of friction #' over the roll work piece interface for values of n > 1. It is clear from the figure that effective coefficient of friction increases towards exit side. Further as the value of n (which depends upon rolling condition) increases, effective coefficient of friction decreases.

O

r

~ Aogo5-0.3b

S

-tg-

-+

°~I

A¢,=0

"::k

0

I

0.2

I

i

04

0.6

I

0.8

I.O

__x

L

FIG. 9. Variation of effective coefficient of fraction on roll workpiece interface.

Cold Ring Rolling of a Polymer

109

~4 o Ao~So 0.004 Pf~ f IQ4mm rev-I - x - - - x - X--

~-IQ5mrnmv-I - iO.6mmmv-i

cr

_

_

Inifiol dimensions r0 h b 0

iI

I

2

I

3

25 rnm IOmm IOmm q

I

4

5

I

6

7

Ring thickness change ( ~ h ) mm

FIG. 10. Variation of relative roll load and roll torque during ring rolling.

Roll force and roll torque It is of interest to observe the variations of resultant roll force and roll torque throughout an operation of ring rolling and the theoretical curves are shown in Fig. 10 for a ring of initial outer dia 50 mm, wall thickness 10 mm and width 10 mm. The resultant roll force first decreases and then increases as the ring thickness change increases. But the roll torque increases first and then decreases as the ring thickness change increases. From Fig. 10, it is further predicted that both the roll force and roll torque vary with the variation of roll closure rate. Both the roll force and roll torque increase with increase of feed rate. REFERENCES [1] HAROLD V. JOHNSON,Manufacturing Processes, Chas. A. Bennett, Peoria, 111, 61614 (1973). I-2] J. B. HAWKYARD,W. JONSON,et al., Int. J. mech. Sci, 15, 873 (1973). l-3] JAMES M. MCKELVEY, Polymer Processing. John Willey, New York (1962). [4] W. L. WILKINSON, Non-Newtonian Fluids, Vol. I, Pergamon Press, New York (1960). [-5] CHANDRAUMESH,KUMARSURENDERand PRASADS. C., Proc. Int. Conf. Prod. Engng. (1977). 1-6] B.V. DERYAGIN, Izd. Akad. Nauk, U.S.S.R. (1952). [7] T. WONKLIN,Friction at high normal Pressure. Proc. First W C I T Paper No. F-7 (1972). [8] J. G. WILLIAMS,Stress Analysis of Polymers, L o n g m a n G r o u p (1973). [.9] SURENDERKUMAR,Principle of Metal Working. Oxford IBH, New Delhi (1976). [.10] A. TSELIKOV, Stress & Strain in Metal Rolling, Mir, Moscow (1967). APPENDIX

A

The following are the properties ofGarflon (ASTM method 1894): (1) C o m p o s i t i o n - - N y l o n - - 6 ; (2) Tensile yield stress = 500 kg/cm 2 ; (3) Compressive yield stress = 900 kg/cm 2 ; (4) Coefficient of sliding friction against Steel = 0.15, Brass = 0.11, Aluminium = 0.12, Self = 0.24; (5) Percentage elongation = 55 percent. APPENDIX

B

N o doubt, the value o f a 2 ( - p ) varies over the interface from entry to exit. The value of a 2 ( - p ) at the entry when yielding starts is constant for the ring rolling during a pass. Hence the value of 2 has a constant value during a pass, although the value of 2 varies slightly during successive passes of ring rolling due to variation of p in each pass.,tt requires an accurate determination of variation of 2 and/or a2( - p) at the roll-workpiece interface during each pass. But the consideration of the variation will extend the numerical work considerably. As the value of a2, the only variable on the right hand side of (7) for a particular material (i.e. fixed value of arc and ay,) is as a square root, its effect on the value of 2 will be sensibly small. Further, as the roll closure rate is very small, this variation can be neglected, whilst maintaining reasonably accurate results and simplifying the solution. Hence the value of 2 corresponding to the m a x i m u m value of a 2 ( - p ) at entry where yielding starts during first pass is very close to the correct value for the practical case of ring rolling of a polymer. From (8), a~ + p = 2. The m o m e n t yielding starts, practically the value of ~x is negligible in comparison to p. Hence for all practical purposes ax can be neglected. Hence, we get p ~ 2. Putting this value of a 2 ( - p) in (7) we get the value of 2 associated with yielding. Hence (7) on simplification reduces to