A lower upper-bound solution of cold strip-rolling

A lower upper-bound solution of cold strip-rolling

Citation for published version (APA):

Mot, E. (1968). A lower upper-bound solution of cold strip-rolling. (TH Eindhoven. Afd. Werktuigbouwkunde, Laboratorium voor mechanische technologie en werkplaatstechniek : WT rapporten; Vol. WT0194). Technische Hogeschool Eindhoven.

Document status and date: Published: 01/01/1968

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-A lower upper-bound solution of cold strip-rolling. E. Mot •

Summary

In the plastic zone of a strip, a kinematically admissible velocity field is formulated as a 4th-degree polynominal v (x, y) and v (x, y). All coefficients except one are found _x

y

from the boundary conditions and the continuity requirement. Then the last coefficient is determined by a generalised version of the lower upper-bound theorem as formulated by Prager and Hodge [1J.

This could be accomplished by introducing the formulation of the external load as calculated by Bland and Ford

(2].

Throughout the analysis, the material was considered to be

- ":'1D

strain-hardening according to a

=

c6 •

It was found that in this way the extremum could be determined' very accurately. Next, the velocity field was integrated along streamlines in such a way that the deformation of

an

initially straight line could be determined. The form of this line was compared with experimental results found by Tarnovskii,

Pozdereyev and Lyashkov

[3]

t and the results matched reasonably

well •

•

Department of Production Engineering, Prof.dr. P.C. Veenstra Technological University Eindhoven.

(delta) (epsilon) (mu) (sigma) (tau) (phi) (omega) •• List of symbols. a b c coefficients for coefficients for effective stress Vx v y -if 6 • 1 h local thickness of strip

k maximum.shear stress m strain hardening exponent p normal pressure on the roll t time

v velocity

x,y,z cartesian coordinates (also suffixes)

R

radius of roll

logarithmic strains

6

effective logarithmic strain quadratic strain rate tensor coefficient of Coulomb friction normal stresses

a

effective stress shear 'stress pola~ coordinate

surface velocity of rolls suffixes.

0 denotes lien try"

1 denotes "exit"

n denotes "neutral point" s denotes "streamline"

denotes "between 0 and

+ denotes "between n and

•

denotes "kinematically ntl

1"

Basic assumptions.

(1) The broadening of the strip is negligible with respect to the other deformations.

(2) The rolls are considered to be rigid bodies (For the cases presented, the error due to roll deformation was calculated and appeared to be of the order of magnitude of 0.1

%).

(3) The area

ABeD

in Fig.

1

becomes fully plastic, all other material remains rigidly elastic.

(4) The friction between rolls and strip can sufficiently accurately be represented by a constant coefficient of friction l.L.

(5)

Inertia forces can be neglected

(6) 'Po ~ sin 'PO Ii'4 tan 'Po

(7) The flow condition can be described by Von Mises'law -2 2 2 2

a = a + a + a - a a - a a - a a (1)

x y Z x 7 Y z z x

if x,7 and z are principal directions.

(8) Strain hardening is given by

(9) (10) (11)

a

==

c"'i?'

where 6 _x = -In.!L· _{h t} o h 6

=

+In

h

y 0 ; 0 z == 0

Bland and Ford [2] make use of three additional assump-tions which we will not maintain but only mention since the results of their calculations are used as boundary conditions:

'txy == 0

a is· independen t of the X -coord:\.nate.

y

Plane sections remain plane during the entire process. Finally, we assume that

(12) the loads round the plastic zone as calculated by Bland and Ford are sufficiently accurate to be used as boundary conditions for a two-d~mensional analysis of the process, in which assumptions

9

to 11 are no longer maintained.

Fig. 1. The velocity field

1

=

1(x,y)

in the strip-rolling process.

The velocity field.

We represent the velocities by

. 2 2 'J§ 2 2

Vx = _{aOO+a10x+a01y+a2Qx +a11X1+a02Y +a30X'y+a21 x y+a12xy +}

.3 ,. . 'J§ 22 3 4 (

+a03Jr+a40x +~1X'y+a22x _{y +a13xy-+aQ4Y} 4)

From symmetry.

v is an even function of y_ _x Therefore,

Vy is an odd function of y.

Therefore.

Further. v does not depend on _x y if x

=

O. Hence, while Or, v _y = 0 b01

=

b₀₃

=

0 if x

=

o.

With (6), (7), (8) and (9) we find for (4) and (5):

. 2 'J§ 2 ,. 2 2

v x

=

_{aOO+a10x+a20x +a30X' +a12X1 +a,.Ox +a22x y}

Vy •

b11X1+b21X2Y+b31r'y+b13~

(6)

(8)

(10)

5

-Since the material is incompressible, we must satisfy the continuity equation Viti. 0 for all x and y.

Herefrom the following relations can be derived: .'0

=

_{0; b11}

=

_{-2.20. b21}

=

_{-3a30, .12}

=

O.

2 b31

= -

4a40; b13

= -

j

_{a 22} v _x

=

v _y

=

(12)

Now we introduce the first boundary condition, viz. that the velocity of the strip material which makes ~ontact

with the rolls, is tangent to the rolls. For reasons of simplicity, we approximate the shape of a roll by a parabola. (This assumption has only little influence on the calculated results, since ~ is small).

o

(16)

Hence

(BC ) (BC: ) .!

v _Y

=

v _x •

R

. We substitute (15) in (13) and (14) and find

3

(Be ) _aZ2h1 3a30h1

Vy

=

-(a_20h1+ 12)x - 2

(Bo)

v •

x

It appears that (17) cannot be entirely satisfied by sub-stituting (18) and (19). Therefore, we make only the first two terms of the same power of (18) equal those of (19). This way of doing will be justified later.

From this we find

(21)

As a second boundary condition we postulate that the elasto-plastic bo~dary at the entry is a parabola which cuts the rolls at right angles.

Such a parabola is 2 x = ~o(-t- +

R),

. 0 formulated by

V

hO-h~

~O = R

This parabola is considered to be a cylindrical surface on which the velocity is discontinuous in such a way that

...

the velocity shock Av is tangent to it. Hence,

v(CD)

=

v(CD)

2y~o

y x · t

o

Using (13), (14), (20), (21) and (23) we find that

Using (24) for the first coefficients of (25) and (26) only, we find (with the help of

~O

=

V

(h

O-h1)/R ..,) aOO+a20R(2hO.-h1 ) a40

=

-R2(3~o-h1)(ho-h1)

If x

=

we have v(BC>( )=

~R

xn ' x x=x n (24) (26) (28)

(it)

...

7

-Summarising, we now haTe 2 4 2 2 Vx

=

_{aOO+a20x +a40x +a22x y} Vy

=

b11X1+b31x'Y+b13xy3

in which all coefficients can be explicitely given as functions of a

20 with the help of (28),

(27),

(21) and (12). The value of a

20 will be determined by means of the lower upper-bound theorem.

Application of the lower upper-bound theorem.

The formulation of this theorem for a strain-hardening material and a discontinuous velocity distribution is as follows

[1],

[4].

2]12... miniI!!s~"!;~ .J!.Jffi!'e.!!'ll-2.n J "

=

\r:; 12

J,

_k

·V·

_clilt .,....,

f"

f· "..

i .c . . dV - T.'Y.dF + 't Ay dF V J ~J F ~ ~ F T D • .;Ie 1

*'

...

.

In wh~ch c ..

=

-2 (v . . +v. i)' V

=

volume, FT ~s the area on

~J ~tJ J,

which the stresses are prescribed, T. are the stresses on F_T,

~ .

..

FD is the area on which a discontinuity of speed AT exists,

't are the shear stresses along

FD-We rewrite (31) as

•

J = I₁-I

2+I₃ (32)

. Since k ₌

'Oro,

we find, using (2) and (3):

in which

h

~

_h1+x2/R (34)

Using (29) and (30) we find

Now, if we replace the parabola CD by a straight line (in

doing so, we make only area is small) we find

VR(h -h 1)'

a small error, since the enclosed

I,

=

f2

J

0 {

o

As for the calculation of 1

2, we will assume 0 _Xo

=

O. For the ultimate speed of the strip we have 'f 1= aOO' Hence,

The initial speed of the strip follows from ~o

=

t1v1/tO. The velocity shock at CD is Av = vosin't'''' vat' 04't~ ~o'

Assuming a shear stress distribution ~

=

~kO/~O along CD,

we find, if 13

=

131+132' that kovO _I to

- · 2

_{~o ~o}

For k we arbitrality choose the maximum shear stress as

o

caused by a plastic deformation of 1

%

m

k

=

£-

(1)2

0.01m

0 ' 1 3 3

Since v c«v t the velocity of the strip along the rolls is

Y x (BC)

approximately equal to v •

x

Hence, for the rolls the shock is IAv'

=

Iv~BC)_Q)R'

For the shear stress, Bland and Ford found ~ = 1J. p+ 0 " ~ , ~ _n ~ = 1J.p_ ~n" ~ ,,~o in which p,H e (40) (41 ) (42) (44)

9

-where

H

=

2

Vi',

arctan

(Vf,

.~)

The neutral point ca~ be found from

( x1 t 1 - _ ) 1 0 2k1 H _n

=

H -₀ -In - - - ' -_{2p. a} t 1(1-

~)

~n ~f:r1

tan{1

Y~T

(46) (48)

Now we can find

tp

I<P. ( ..

R~v ~BC»Rd~+

J

l'P_

(v~BC)

-OIR)Rdq>

o

tpn

With the help of foregoing formulas, the value of a₂₀ for which J- . becomes minimum can be determined.

An estimation of a

20 is found by

0 0 • • • (50)

Actually, (50) indicates that the polynomal representation of

V

in (4) and (5) is admissible. If (51) held exactly, (21) would yield a

22

=

O. This means that the present theory implies the one-dimensional "plane sections remain plane"-theory, since a

22 is the curvature-parameter. In Fig. 2 a solution for one specific case is presented.

Fig. 2. A lower upper-bound solution for one specific case.

Integration of the velocity field.

For the veiocity field calculated in the former section, we will now calculate streamlines. These are the trajectories of the material elements in the deformed area. It is clear that . each streamline ys(x) must satisfy the equation

Fig.

3.

Streamlines y (x)

s

With the coefficients a

ij and bkl calculated we can solve this differential equation numerically. As a boundary condition, we have

If ~ = 0, the solution is y

=

O. Hence, a material 'element

s

on the x-coordinate moves with a velocity v (x, 0) and remains

x

on it.

If ~

=

1, the solution should be given by (15).

Calculations show an error in the order of magnitude of 5

%.

This is an indication that the inaccuracies in determining the velocity field were reasonably admissible.

An

infinitesimal element of streamline has a length

V

2 I .

dB

=

1+ (dYs/dx) dx and is covered in a time

d . \!1+(dY /dx)2 i dx

dt s

1

s .

=

!!3. {

=1;1

=

Vv~+v;'

Vx

Now we will calculate the shape of the initially straight line R t S' (Fig. 3). The time, needed to cover the distance PQ is

the same as the time needed to cover the distance

HGFi.

Or

to

=

J

<p ·R

o

dx v (x"o) x 11

-(56)

If the point G, with coordinates (xo(t), yo,(t»is the point of intersection between a streamline and the entrance

para-x

o

dx v (x,y ) X is EF

-

-From formulae

(56)

to

(60),

if

can be calculated. If this is done for a range of values of t between 0 and 1, we find the shape of the deformed line R'S' which was initially the unde-formed straight line RS (Fig.

3).

In ref.

[3] ,

the authors show a number of photographs of deformed grit patterns (pp. 68-70) which were applied both on the side surface and along the centre of the width of the specimen.

Since this mod,el assumes plane strain t only the latter gri ts

were considered significant. The material was aluminium, heated at a temperature of

450

oC. For the cases referred to, no friction coefficients were measured. For similar conditions, a separate table (p.

274),

however, mentioned values of ~ be-tween 0.171 and 0.396. Figs.

4, 5

and 6 give a comparison be-tween the average shape of a grit-line as measured from

photo-graphs and the shape as calculated for ~

=

0.3 and.O.4.

The agreement seems to be reasonably good.

Differences may be caused by the following facts.

(a) The theory of Bland and Ford [2J is an approximate one. (b) The coefficient of friction is rather an arbitrary

quan-tity. In fact, near the neutral point, the material will probably stick to the roll and plastic friction with con-stant shear stress will occur.

(c) The assumption of plane strain does not hold exactly. Cd) The elastic recovery has been neglected.

Figs.

4, 5

and

6.

Verification of the strain distri-bution in the rolled strip with the help of photographs of deformed grits.

Conclusion and prospec,ts.

It has been shown that the lower upper-bound theorem can successfully be used for the calculation of strain distri-butions in plastically deformed materials. From the strain distribution, which might also be verified by means of micro-hardness measurements

[5],

it may be possible to calculate residual stresses, which mRy be verified by Rontgen-diffrac-tion measurements.

Generally speaking, however, an "elementary" (e.g. one-dimen-sional) theor~ must be available to produce the boundary conditions for the stresses.

When the velocity-field is found, it can also be used to cal-culate a stress-distribution, satisfying the 'equilibrium

condition, the Levy-Von Mises equations and the flow condition. Thus, improved expressions for roll pressure and torque could be found.

13

-References.

[1] W. Prager und G.P. Hodge,

Theorie ideal plastischer Korper, Springer Verlag, Wient 19540

.(2) D.R. Bland and H. Ford, The calculation of roll force and torque in.cold strip rolling with tensions,

Research on the rolling of strip, a symposium of selected papers 1948-1958; B.I.S.R.A., p. 68-77.

[3]

I~Ya.Tarnovskii, A.A. Pozdeyev and V.B. Lyashkov, Deformation of Metals during Rolling,

Pergamon Press, 1965.

[41

B. Avitzur, An upper-bound approach to cold strip rolling, Jrn. of Eng. for Ind., Trans. ASME-B, Febr. 1964. p.31-46.

[5]

J.A.H. Ramaekers, Het ponsonderzoek; bepaling van de de-formatieverdeling en energiedissipatie in het werkstuk-materiaal, internal report Production Engineering Depart-ment, T.H. Eindhoven, Jan. 1968.

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