Edge stresses in wide strip hot rolling

Pergamon

PII : S0020-7403(96)00042-2

Int. J. Mech. Sci Vol. 39, No. 4, pp. 397 408, 1997 Copyr igh t i 1996 Elsevier Science Lid

Pr inted in G r e a t Britain. All rights reserved 0020 7403/97 $17.00 + 0.00

E D G E S T R E S S E S I N W I D E S T R I P H O T R O L L I N G

LAILA S. BAYOUMI Department of Mechanical Design and Production, Faculty of Engineering, Cairo University, Giza, Egypt.

(Received 30 October 1995; in revisedJorm 28 March 1996)

Abs t rac t - - ln the rolling of wide strip, longitudinal tensile stresses develop along the edges which enhance edge cracking. Theoretical assessment of these stresses and the factors affecting their magni tude and spread are not emphasized in existing solutions, since most of the solutions are based on the assumption of plane strain along the entire width. In the present work a flowline field solution is developed to analyze the flow and stresses in steady state wide strip hot rolling based on the concept of a plane-strain middle zone and two lateral flow edge zones from which edge stresses and edge zones width have been determined. The results obtained show that the maximum value of the longitudinal edge stresses is 1/,,,,"3 the flow stress and occurs at the neutral point where the width of edge zone is maximum. The influence of the coefficient of friction at roll strip interface, roll diameter and thickness reduction ratio on the magnitude and spread of edge stresses is not appreciable. Copyright @ 1996 Elsevier Science Ltd.

Keywords: metal working, hot forming, strip rolling, plastic flow, edge stresses.

2b 2c

C 0 , C l ~C2

2h L m

P q P t"

R T

U

ll x, /?y, Z' z

/;O ~ Ue

['r

W

Fo, Fo

/1

(7 m

(7

"Cx, "gz

Subscripts b C

e

0

N O T A T I O N

strip width width of plane strain zone at any cross-section; 2c = 2b - 2w flowline filed coefficients strip thickness at any cross-section length of contact friction shear factor interfacial pressure flow factor rolling load thickness reduction ratio; r = (he - h,)/ho roll radius rolling torque velocity vector velocity components in the x, y, z directions, respectively strip entry and exist velocities, respectively roll surface velocity width of edge zone at any cross-section deformation zone boundaries at entry and exit effective strain effective strain rate coefficient of friction mean stress effective stress friction shear traction in the x and z directions, respectively

strip edge end of plane strain zone strip exit cross-section strip entry cross-section

1. I N T R O D U C T I O N

In the rolling of wide strip there is no noticeable increase in width, however the edges develop a bulge profile indicating the occurrence of lateral flow. Experimental investigations [1] have shown that such lateral flow is limited to a small portion of the width at the edges; the rest of the strip width which represents the major portion is laterally constrained. Although such lateral edge flow is very small compared to the flow in the longitudinal or thickness directions it has a significant effect on the stresses. Along the laterally constrained portion of the width, lateral and longitudinal stresses are

397

398 L.S. Bayoumi

compressive and uniformly distributed along the width. At the edges, however, lateral stresses drop to zero, roll pressure decreases and longitudinal compressive stresses become tensile [-1]. With the presence of initial fracture centers in the cast slab structure these longitudinal tensile stresses will enhance edge cracking during hot rolling [2]. In this case the width of the trimmed material at the edges has to be increased to more than crack length which means a drop in mill output. The width of edge zones is not constant along the bite length; studies of the flow pattern indicate that the width is maximum at the neutral line of a value about half the roll-bite length decreasing towards the entry and exit ends [3]. Assessment of edge zones width and stresses are not emphasized in existing theoretical solutions since most of the solutions are based on the assumption of plane strain along the entire width [4-8] and in three-dimensional solutions [-9, 10] lateral flow is assumed to be distributed along the entire strip width. In these solutions, particularly finite element simulations, a hypothetical expression of frictional stresses is a p r i o r i implemented along the roll interface which may not be compatible with prevailing kinematical conditions.

In the present work a flowline field solution is developed to analyze the flow and stresses in steady state wide strip hot rolling with the aim of assessing the magnitude and zone width of edge stresses. The flowline field formulation is based on earlier experimental work by the author E11] in which measurements were carried out to analyze deformed grids laid originally parallel to the direction of rolling on the two planes of symmetry and also on the outer surfaces of lead-tin alloy rectangular billets of different aspect ratios. The billets were rolled at different height-reduction ratios on an experimental mill equipped with smooth rolls and the deformed grids were measured using a high-resolution scanner. The measurements have established certain characteristics of the flow pattern in the deformation zone; namely that the flowtines are almost identical in planes normal to the roll axis in spite of lateral spread and represent a set of second order polynomials. Based on these characteristics the flowline field has been formulated to satisfy geometrical and kinematical conditions.

The solution obtained in the present analysis is based on the concept of plane-strain middle zone and two lateral flow edge zones. The flowline field parameters are determined by satisfying static and friction boundary conditions. The stress components, as welt as friction shear traction, are obtained by substitution of the strain rate components in Levy-Mises flow rule and integration of the equations of equilibrium along the flowlines. Edge stresses and width of edge zones are determined through a direct equilibrium approach. The solution is applied to a case of wide strip hot rolling given in the literature [10] in which the results are obtained by using finite element simulation. The magnitude and zone width of edge stresses are investigated in relation to friction and roll-bite geometry.

2. P R O B L E M F O R M U L A T I O N

Isothermal steady state flow is considered with no front and back tensions. Elastic roll deforma- tions, due to transverse deflection and flattening are neglected. The strip material is assumed to be homogeneous, isotropic, rigid-plastic and incompressible. The flow stress 6- is a function of the effective strain ~,, effective strain-rate { and temperature. Referring to Fig. 1, coordinate axes are chosen with the origin at the midpoint of the entry plane. The axes x, y and z are the directions of length, height and width, respectively. The strip is symmetrical about the x - y and x - z planes. The entry plane, x = 0, and the exit plane, x = L, are not the rigid-plastic boundaries; the deformation zone boundaries Fo, Fe, are determined from Eqn (3). The axial velocity is uniform across the strip width but is nonuniform through the strip thickness and the flowline field in the deformation zone in planes parallel to the plane of symmetry x y is independent of z and is described by a set of second order polynomials in x [11] as:

[ L ' 1 yj =f j 1 + c o q j -- ~ o ( 1 + clqj)x q- 2~h~o(1 + c 2 q j ) x 2 (1)

where 2R = roll diameter, 2h0 = initial strip thickness, and L = contact length, qj is a flow factor expressed as

Edge stresses in wide strip hot rolling 399

h o

I

Y

_2 !

I

h

L O X ~

he

1 X

Y

b IWb

C = =

Fig. 1. Roll-bite geometry.

lk ." l'o r

Fig. 2. Flowlines and rigid-plastic boundaries 2h o = 4.2 ram, R = 375 mm, r = 0.2857, # = 0.25.

wherefj is the specific flow = material volume flow below a flowlinej for a unit strip width per unit entry velocity. A flowline is a constant f-line for which f = Yo so that ho >~fj ~> 0. Co, ca, c2 are positive flow coefficients determined from static and friction boundary conditions. The flowlines are virtually circular arcs of radii Rj having their centers at x = xCj obtained from Eqn (1) as

1 fj L(1 4- Clq j )

Rj - RTo (1 + c2qJ)' Xcj - (1 4- c2qj)

The x - z plane of symmetry, f = 0, and the roll interface, f = ho are boundary flow surfaces. Substituting in Eqns (2) and (1) gives:

At f = 0 , q = 1, y = 0 , 1,/R j = 0 .

At f = h o , q = O , y = h o - ( L / R ) x + ( 1 / ' 2 R ) x 2 = h , 1 / R j = I / R , x c j = L

which satisfies geometrical boundary conditions. The flowline field at the entry and exit rigid zones are sets of parallel lines given by the expressions:

Yoj =f j at entry, 3,'ej = (hdho)f j at exit.

The rigid-plastic boundaries Fo at entry and F~ at exit, Fig. 2, are the intersections of the parallel flowlines at entry and exit respectively with curves given by Eqn (1) which yields:

L 1 coqj - ~oo(1 + c~qj)x + 2 ~ o ( 1 + c2qj)x z = 0 for Fo

he) L 1 1 - ho + coqj - ~ o ( 1 + clqj )x + 2 ~ 0 ( 1 + c2q j ) -x:2 = 0 for Fe.

(3)

Along the strip width, there are two deformation zones; a large plane-strain zone at the middle, 0 ~< z ~< c, and a small lateral flow zone at the edges, c ~< z ~< b. Each zone is treated separately.

2.1. Plane-strain zone 0 <~ z <<. c In this zone there is no lateral displacement, the flow is two-dimensional occurring in planes

parallel to the x-y plane.

400 L.S. Bayoumi

The velocity components and the strain rate components are derived in the Appendix as

UO F o Y l t,~ = - - , r ~ - v~ = 0 ( 4 )

Y2 Y2

~;x - - UY2 i v = - - g x ' gz = 0

Y2 ~ "

where for

Vxz )'1 ,; ' - - 0 5'xy = - - @ Uyl - - Uy2 - - , , 'zx = ) 'yz - -

Y2 Y2

(5)

/" ,2 2 //1 -2 ~;2 o = v ' 3 ( a ~ +~xy), [ = . , '~(4e,, + ,,,y)

The normal stresses are expressed as a summation of the stress deviations, Eqn (6), and the mean stress Or, obtained from solving the stress equation of equilibrium in the x-direction,

~,Ox & x y ~Zz~

~7~ + t ~ - + 77-. = °

by transformation from (x, y, z) to (x,f, z) and substituting the static boundary condition that there is no front and back tensions as

f2o J° o oxdzdy = at x = ;b

0 0. (7)

The interracial pressure p and friction shear traction r~ are obtained from the stress components along the roll-interface, f = ho, as:

1 1 a ~ 1 - - 3:2 2 ) ' 1

v,,' . 2 + TxY

"rz=O

where rz is the friction shear traction along the width. The resultant friction shear stress r is expressed by the relation

/ ' ~ 2 = ,~,, r; + rz (9)

and friction boundary conditions along the roll-interface are stated in terms of the friction shear factor m as

~L L

j I ldx= " i odx (10) o , / 5 :o

rather than z = m ~ / v / 3 since the distribution of z is obtained by substituting a', ~'y and ~-xy from Eqn (6) in Eqn (8).

(8)

~v ~3: v = y ( x , f ) : .vl = - ~ , )'2 -

cx ~f

C'U x CUy C'Uy

It can be easily shown from Eqns (3) and (4) that the velocity components normal to each of Fo and Fo satisfy material continuity condition.

The stress deviations a'x, a'y, cr'z and shear stress rxy are obtained from Levy-Mises flow-rule as:

2# #

(6) where


In many finite element appl icat ions friction bounda ry condit ions are expressed in terms of a coefficient of sliding friction # which varies along the roll strip interface [12] with a t ransi t ion from C o u l o m b friction to constant friction shear stress at a certain value of t [-8]. In present work I~ is calculated as the average value of r /p along a small length of contac t f rom entry, taken arbi trar i ly as 0.2L, expressed by the relation:

1 f~'2L~dx. (11) l, = o75-/ r

2.2. Edge-zone c <~ z <~ b Lateral flow occurs; the edges become bulged without appreciable increase in width. Hence ~v~/'Ox

and ~vz,,'Oz can be neglected with respect to C~rz/'?y. For constant v~, Vy along z, (£v~/?z) = (~Vy/'?z) = 0 and hence

- - I z x ~ - - - . ( 1 2 )

' ~ Y - ~v

It may be noted that this is not a state of plane strain. Consider the stress equat ion of equil ibrium in the z-direction

~Z C~ZX ~rz~r - - + ~ + - ~ = 0 . (?z cx cy

From Eqns (6) and (12) t ~ = 0 which gives

~0" z ~r~y

8z Py (13)

and hence from Eqn (13)

~'zy Z'z

8 3 , h

~20" z T z - - - ( 1 4 ) 8z h

In tegra t ion of Eqn (14) along z not ing that a~ = azo at z = c gives

1 e .7

a z = a z ~ - ~ | rzdz (15) d c

where 2c = width of plane strain zone which varies along x and y. Substi tut ing in Eqn (15) crz = 0 at z = b yields

o-z~ = ~ rzdz. (16)

Consider ing the lateral friction shear tract ion rz to p redomina te at the edge zones and neglecting the var ia t ion of 6 along z gives

f b - r z d z = m ~ - ( b - c ) = m ~w hence Eqn (16) yields

,¢ /3 ,¢/3 (17)

O'zc N///'3 W - - 11

m

where w = width of edge zone which varies a long x and y.

~zy = 0 at y = 0 and equals rz at ), = h. In strip rolling analysis the distr ibution of shear stresses is considered to be linear through the thickness [14, 15], and hence rzy can be expressed as

402 L.S. Bayoumi

At the neutral line, a~/6 is maximum at the roll-interface which by substitution in Eqn (17) gives Wm,~ at h = h, where 2h, = strip thickness at the neutral line.

Again az = a'~ + O'm and from Eqns (6) and (12) o-'z = 0 hence

O'z = O'm, O'zc = o-mc, where o-rnc ~ o-m at z = c .

Substituting in Eqn (15) gives

1 (*z

am = am¢ - ~ j~ rzdz.

With no variation of a', and a; along z

a, = G~ - ~ %dz

( 1 8 )

% = aye - rzdz.

At the edges z = b, the edge stresses o-,b and ayu are obtained by substituting Eqn (16) in Eqn (18) so that

0"xb = O'xc - - O'zc = 0 x - - o-'z ~- o-x

o-yu = o-yo - o-zo = 4 - 4 = 4 .

(19)

From the expression of 6 in Eqn (6) a'x/6" is maximum at the neutral point, where rx = 0 and hence the maximum value of the edge stresses is determined as:

(O'xb t = __(ayb) __ 17 O" / m a x \ ~" / m a x 5 3 "

(2o)

3. METHOD OF SOLUTION

A case of wide strip hot rolling is defined by 2b, 2ho, r, R, v~, /~ together with the material constitutive law at the rolling temperature. In order to obtain a solution, the flowline field coefficients Co, Cl, c2 are to be determined. This is carried out numerically by iteration to satisfy the following three boundary conditions relating the distribution of interfacial pressure p and friction shear stress rx along bite length L:

(1) f2(PYa + rx)dx = 0

(2) (p - rxyl) (L - x)dx = R rxx//1 + y2dx

f~ m f~' 1 (.o.2L rx (3) f r ddx = ~ 6dx, ~ = ~ f£ Jo pdX.

The first equation states that the resultant of the components of interfacial pressure and shear traction along the x direction integrated over the contact length is zero since there is no front and back tensions. The second equation is the moment equation for the equilibrium of the roll. The third equation is an assessment of friction boundary conditions at the roll interface.

The rolling load P and torque T are determined from the relations

j.L P = 2 b ( p - r x y t ) d x , T = 4 b R rxx/1 +yZdx

0

where P is the resultant of the components of interracial pressure and shear traction along y direction integrated over the roll contact surface and T is the moment of the shear traction about the roll axis integrated over the two roll contact surfaces.


4. R E S U L T S AND D I S C U S S I O N

In o r d e r to inves t iga te edge stresses, the a b o v e ana lys i s is app l i ed to a case o f wide s t r ip ho t ro l l ing

r e p o r t e d in the l i t e ra tu re [10], in wh ich the resul ts were o b t a i n e d by us ing f ini te e l e m e n t s i m u l a t i o n

for the fo l l owing ro l l ing data :

2b = 1000 lnm, 2ho = 4.2 ram, r = 0.2857, vr = 4 m / s e c

W o r k roll d i a m e t e r s = 400, 575, 750 m m

B a c k - u p rol l d i a m e t e r = 1150 m m

Coef f ic ien t of f r ic t ion = 0.25

M a t e r i a l cons t i t u t i ve law:

6 = 44.13 (1 + 2g) °'21 ~o.31 M P a .

Th is r e l a t ion c o r r e s p o n d s to s ta inless steel SS 316 at 1 0 0 f f C [13].

Tab le 1 gives the ca lcu la ted results of the f lowline field so lu t ion for the a b o v e case which include

coefficients Co, c l , Ca, ro l l ing load, ro l l ing torque , f r ic t ion shear fac tor m and the m a x i m u m long i tud ina l

Table 1. Flowline field results (2h 0 = 4.2 mm, r = 0.2857, p = 0.25)

Work roll diameter (2R) mm 400 575 750 Roll-bite length (L) mm 15.490 18.574 21.213 Flowline field coefficient Co 0.01300 0.01230 0.01172 Flowline field coefficient c, 0.05669 0.05442 0.05229 Flowline field coefficient c2 0.05569 0.05557 0.05452 Rolling load (P) MN: Flowlines 5.420 6.746 7.961

FEM [10] 5.510 7.030 8.700* Rolling torque (T) kN.m: Flowlines 86.65 127.07 170.83

FEM [10] 288.13* Torque arm ratio (a/L}: Flowlines 0.457 0.507 0.506

FEM [10] 0.7810" Friction shear factor m: Flowlines 0.449 0.465 0.476

FEM [lO] (axb)~,x MPa: Flowlines 116.4 109 104.2

FEM [10] -- 105 ** Wm~x mm: Flowlines 11.09 12.38 13.51

FEM [10] ~ 40 ~. 40 ~ 40

*Calculated by the author **Occurs at exit

MPa Ro l l i ng d i r e c t i o n 6 0 0 -

200 ~

T

R=37 - 6 0 0 ~

x/L

Fig. 3. Variation of pressure, shear traction and normal stresses along roll-interface at the middle width of the strip {flowline field solution).

404 L.S. Bayoumi

M P a 800]

6004

4 0 0 -

R o l l i n g d i r e c t i o n

200

0

-200

-400-

-600

-801:

J

T

o.'2 o.4 o.-7-_-_-_-.~--~,-f-7~

R = 3 7 5 m ~ n , r = 0 . 2 8 5 7 , # = 0 . 2 5

x/L

Fig. 4. Variation of pressure, shear traction and normal stresses along roll-interface at the middle width of the strip (FEM Ref. [10]).

a=b ( M P a ) R o l l i n g d i r e c t i o n 200[

so ] . . . . . .

1 O0 - - - - -- ' : v - , , . ,

EM

-5o- ~ - - /

- 1 O0

- 1 5 0 -

-200

- - R = 3 7 5 m m - - R = 2 8 7 . 5 m r r t

R = 2 0 0 m m

r = 0 . 2 8 5 7 , # = 0 . 2 5

Fig. 5. Variation of longitudinal edge stresses along roll-interface.

tensile edge stresses and maximum widths of edge zones compared to results available in Ref. [10] obtained from finite element simulation.

From Table 1, the rolling loads given by the flowline solution are in fairly good agreement with the FEM solution. The rolling torque and torque arm ratio obtained from the flowline field solution agree with the values gives by practical hot rolling formulas [1, 3, 14] while those obtained from the FEM solution are 70% higher. The friction shear factor m is almost equal to 0.5 which is the value quoted in the literature for such rolling conditions [-4, 5]. The maximum width of the edge zone given by the flowline field solution is about three times the strip thickness, which agrees with the experimental results given in Ref. [-3] compared to about 10 times the strip thickness in the FEM solution in which lateral flow is assumed to be along the entire strip width.

For the case o fp = 0.25, r = 0.2857 and 2R = 750 mm the distribution along the roll interface for the interracial pressure, shear traction and normal stress components at the strip middle width is given in Fig. 3 for the flowline field solution and in Fig. 4 for FEM solution which show the same


4-

5-

2

1 ."

5-

2-

1

/2ho

0 0,'2 0.'4 0:6 0.'8 -1 l:~/Z 0

- - _ _ - - - - _ r

- - - R = 3 7 5 r a m

- - - R = 2 8 7 . 5 r n r n

. . . . R = 2 0 0 r n r r t

r = 0 . 2 8 5 7 , t x = 0 . 2 5 y / h n

0.'2 0.'4 0.'6 0.'8 1.0

Fig. 6. Variation of edge zone width along roll-interface and through h..

150

100 L-~-L-L-L-~-L-L . . . . . . . . . .

50

}~ .I:

_ _ R = 3 7 5 rnrn _ _ _ R = 2 8 7 . 5 m r n

__ _ R = 2 0 0 r n m r = 0 . 2 8 5 7

0.'3 0.'4 O.

w m ~ / 2 h °

5

-4

-3

-2

-1

0

~z

Fig. 7. Variation of maximum edge stress and maximum width of edge zone with coefficient of friction.

trend. Figure 5 gives the variation of longitudinal edge stresses along the roll contact length for the flowline field solution compared to FEM results which show complete disagreement. This may be explained by the difficulties involved in handling of boundary values by the FEM. Figure 6 gives the variation of edge zone width along roll-interface and through strip thickness indicating that the maximum value occurs at the neutral point at the roll-interface. From Figs 7 and 8 the influence of the coefficient of friction l~, roll radius R and thickness reduction ratio r on the edge stresses and width of edge zones is not appreciable.

5. CONCLUSIONS

A flowline field solution based on the concept of a middle plane-strain zone and two lateral flow edge zones has been developed for wide strip hot rolling from which edge stresses and edge zones width have been determined. The results obtained show that the maximum value of the longitudinal

406 L.S. Bayoumi

~=b),,,,.~ ( M P a )

150

100

50

T_-__T 5 ,T_5 . . . . . . .

_ _ R = 3 7 5 r r r m _ _ _ R = 2 8 7 . 5 mrr~ . . . . R = 2 0 0 rnrr~

~ = 0 . 2 5

T 0.'2 0.'3 0.4

w , ~ / 2 h o 5

-4

3

-2

1

0

Fig. 8. Variation of maximum edge stress and maximum width of edge zone with thickness reduction ratio.

edge s t resses is 1 /x / '3 the f low s t ress a n d o c c u r s a t t he n e u t r a l p o i n t w h e r e the w i d t h of edge z o n e is

m a x i m u m . T h e i n f l u e n c e of the coef f ic ien t of f r i c t i on a t r o l l - s t r i p in te r face , ro l l d i a m e t e r a n d

t h i c k n e s s r e d u c t i o n r a t i o o n t he m a g n i t u d e a n d s p r e a d o f edge s t r e s ses is n o t a p p r e c i a b l e .

6. R E F E R E N C E S

1. A. I. Tselikov, G. S. Nikitin and S. E. Rokotyan, The Theory of Lenothwise Rolling. pp. 128, Mir Publisher Moscow (1981). 2. W. F. Hosford and R. M. Caddell, Metal Forming Mechanics and Metalluroy. pp. 139 Prentice-Hall, Inc. (1983). 3. A. I. Tselikov, Stress and Strain in Metal Rolling. pp. 135, Mir publishers Moscow (1967). 4. S. I. Oh and S. Kobayashi, An approximate method for a three-dimensional analysis of rolling. Int. J. Mech. Sci., pp.

293--305 (1975). 5. Guo-Ji Li and S. Kobayashi, Rigid-plastic finite-element analysis of plane strain rolling. ,l. Eng. Ind. Trans ASME.

Vol. 104, pp. 55-64, February (1982). 6. A. Kumar, I. V. Samarasekere and E, B. Hawbolt, Roll-bite deformation during the hot rolling of steel strip. J. Mat. Proc.

Tech,, 30, 91 114 (1992). 7. S. M. Hwang, M. S. Joun and Y. H. Kang, Finite element analysis of temperatures, metal flow and roll pressure in hot

strip rolling. 3. Eng. Ind. Trans ASME. Vol. 115, pp. 290-299, August (1993). 8. Ann B. Richelsen, Comparison of a numerical analysis of rolling with experimental data. Report No. 493, Technical

University of Denmark. Oct (1994). 9. H. Matsumoto, 2-dimensional lateral-material-flow model reduced from 3-dimensional theory for flat rolling. ISIJ

International, 31(6), 55(~558 (1991). 10. J. Yanagimoto and M. Kiuchi, Three-dimensional coupled simulation of strip and shape rolling processes. Trans.

N A M R I / S M E , pp. 15-22 (1991). 11. kaila S. Bayoumi, Analysis of flow and stresses in flat bar rolling. Ph.D. Thesis, Cairo University, Egypt (1994). 12. P. Montmitonnet and P. Buessler, A review on theoretical analyses of rolling in Europe. 1SIJ International, 31(6),

525 538 (1991}. 13. T. Altan and F. W. Bougler, Flow stress of metals and its application in metal forming analysis. J. Eng. Ind. Trans. A SME

June (1973). 14. C. S. Rudisill and C. F. Zorowski, A three-dimensional theory of hot strip rolling. Proc. Int. Conj. Manufacturing Tech.,

ASTME, pp. 1083 -1096 (1967). 15. L. A, Lalli, An analytical rolling model including through thickness shear stress distributions. J. Eng. Mat. Tech. ASME

Vol. 106, Jan (1984). 16. Z. Wusatowski, Fundamentals of Rollin O. pp. 266 275, Pergamon Press, Katowice (1969).

A P P E N D I X

D E R I V A T I O N OF V E L O C I T Y AND S T R A I N - R A T E C O M P O N E N T S

1. Velocity componenLs For steady state incompressible flow the volume flow before deformation equals the volume flow during deformation.

Consider two successive flowlines f and f + dr(Fig. A l) the change in flow per unit strip width in plane strain is expressed


f=h~

Fig. AI.

by the relation

Substituting Vx = v cos 0 gives

~y since Vy = - v sin 0 = - v x tan 0 = v~ - - hence

?x

rod f = ?y d ~ 0 (A1} v ? f j c o s

~o t~x = ' 2 " "

y

qf

(A2)

let

Hence

For

y = y { x , f )

Py ?y ~=y. ~.=y:

dy = y l d x + y z d f (A6)

?v / ?:v D v ~ U 0 ~ /

• ~x , f f (A3) U z ~ 0

where e0 = strip entry velocity and, t' = velocity along flowline.

2. Strain-rate components

The strain-rate components are obtained from the velocity components derivatives w.r.t, x, y as

(~v X ¢3v v ~x=~--'cx iY=~' iz=0 (A4)

+ 77' ~',z = l

0.

In order to obtain the strain-rate components expressed in terms of x , f it is necessary to perform a transformation of the derivatives.

For a velocity component v = v(x, y)

?v &, dv = --?x dx + ~ dy. (A5)

408 L.S. Bayoumi

By substitution

For

let

Hence

dr = ~-dx + ~-()hdx + y2dj ( X ( ' y

(?v #r) &(~ df

t: = r(x,f)

~ [ , ("p

cx ( t

dr = vldx + z'2d f.

Equating the coefficients of dx and d f i n Eqns (A5)-(A8t of dv gives

?v Yl fr ~ X = 1"1 F2 '

.v2 ?y

Hence

[ '2

Y 2

~" [ 'y2 /~'x = / ' x l - - [:x2 - - - ~ , = - -

.U2 ) ;2

Ux 2 ./' 1 i xv . -

3'2 ./'2

(A7)

(A8)

(A9)

(AI0)

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Edge stresses in wide strip hot rolling