Effects of material relaxation on the mechanics of strip winding

MECHANICS RESEARCH COMMUNICATIONS Vol. 17 (4), 223-230, 1990. Printed in the USA. 0093-6413/90 $3.00 + .00 Copyright (c) 1990 Pergamon Press plc

EFFECTS OF MATERIAL RELAXATION ON THE MECHANICS OF STRIP WINDING

Terenziano RAPAPRIJ.I Department of Mechanics, Torino, Italy

Politecnico di Torino,

(Received 16January 1990; accep~d ~r print 19 January 1990)

Introduction Many areas of industry (textiles, paper, printing machinery, plastic films, etc.) produce coils or reglets by winding materials in strip form on to rigid cores. Typical problems encountered with these products arise essentially as a result of core loss with consequent unwinding, and of the high radial stress reached near the core, which may damage the strip coating. It is thus necessary to determine the radial and tangential stresses in the coil. As the case of a perfect elastic material has been analyzed (see reference [8]), this paper will address the problem of material which shows relaxation over time.

Elastic field stress distribution Let

it be the longitudinal modulus of elasticity,

v Poisson's modulus, o r and

a t the normal radial and tangential stresses, Cr and v t the associated percentage elongation , r the generic radius, Wr the radius variation at r, r i and r e the inside and autside radii of the coil.

The following formulas apply

to a coil in static conditions: r ct

(dor/dr)

+ or

= 1/E (o t

~r = 1/E

- ot

- vo r)

(o r

= 0

(1)

ffi A r / r

(2)

= dAr/dr

(3)

vo t)

whence: E

dAr

Ar

dr

r

(4)

or = l-v 2

223

224

T. RAPARELLI

E

Ar

at = l-v 2 The

stress

+ v -dr

r

state

dAr (5)

of stress of a strip element may be considered as the sum

of

the

due to strip pull and the stress pruduced on that element by the strip

layers wound on top of it.

We can thus accept the equation o t = o 0 + o[ where

o 0 is the tangential stress due to strip pull and

o~ is the tangential stress

due to the overlying strip layers. Considering

the coil core to be indeformable (&r = 0 for r = ri)

and

radial

stress on the outside edge to be nil (o r = 0 for r = re), the solution for (i) (given in [8]) yields the following expressions: 1 + p2/N =

1/2

1

+

in

--

p2

°t --

[ =

i

-

I/2

N I

-

Oo

(6) 1 + p=/N

]

--

I + P2/N

p2

where

(7)

in 1 + p2/N

P = r/ri , Pe = re/ri , N = (l-v)/(l+v)

Material relaxation 5

To obtain data relating to the problem at hand, we considered a coil (inside radius and

autside radius r e = 40 mm)

plastic

film

r i = i0 ~

consisting

of

iO0~/min 4~

(width 8 mm and thickness 15 ~m)

wound on a rigid steel core. Tensile and creep tests were carried out on the plastic film,

while the core was subjected

to

radial compression tests. Figure

I shows tensile force versus elongation

for 200 mm long specimens and pull rates of mm

per minute and

300 mm per minute.

be noted that elastic modulus pull rate.

20

2-

It will

increases

with

Figure 2 shows creep test curves. Constant loads of 0.50, 1.00 and 2.00 N were applied to three

550

mm

long

tape

specimens

and

deformation was recorded over a period of time. While elongation along the specimen axis was considerable and easily measured (elongation increases over time to reach a certain limit

50

100

FIG. i Tensile force /

150

mm

200

elongation,

MATERIAL RELAXATION IN STRIP WINDING

value),

tape

constant.

width

remained

Consequently,

contraction

can

zero.

testing

Film

material

practically

tape

transverse

reasonably be

regarded

indicated

150

as

viscoplastic

behaviour and a value of

Imm}

'°°!/

Poisson's

modulus close to zero. Radial the

225

compression tests were carried out on coil

in

order

to

obtain

characterization data (i.e. stress

and

deformation)

material

relation between

in relation to

50

the

geometrical form of the finished product. The

test fixture consisted of

metal to

two

bands wrapped around the coil and free

slide

relative to each

other.The

~0.SN

f

flexible 0

bands

1

2

3

4

5"

6

7

t(hours)

were connected by two moveable crossmembers. During the test, the two bands were put under tension.

The coil was subjected to

external

FIG.2

pressure corresponding to the applied tensile force,

and

was

left

in

the

Creep test curves.

deformation

conditions so produced. Figure 3 shows applied radial load as a function of time. reacts with a certain internal stress distribution,

The coil initiallly

after with it yields as a

result of relaxation. As will be noted, relaxation produces a drop in load for a

given deformation,

according to an approsimately exponential law where the

exponent is negative over time and tends towards a constant limit value. A

number

of

demonstrated percent

tests that

below

carried out by the

applying

different

external

load level stabilizes at a value which is

the original level.

As there is a proportional

between the radial load on the coil and Young's modulus,

pressures 15

to

relationship

it may be

concluded

that the relaxation effect is due to a reduction of the latter. The

effect of this relaxation

the

distribution

tangential was

of

then

studied

on

radial

stresses in

the

and coil

theoretically above

soo N 600

The effect of Poisson's modulus is

t,0o

using

initial

justified

by

assumptions the

experimental data. considered and

it

to be negligible (v=0) is

assumed

that

the

longitudinal modulus of elasticity passes from an initial value Ez, typical of state i corresponding to

a

freshly wound

coil,

to

a

200 / 0 -200

0

2

4

20

6

$ t {rain)

FIG.3

final value E2, typical of state 2 corresponding to a coil in which

Radial load versus time with

material relaxation is complete.

constant deformation.

226

In

T. RAPARELLI

state

1

conditions,

the radial and tangential

defined by expressions (6) and (7). The corresponding deformations are defined by expression (2) and (3). Whit v=0, STATE i:

distribution

is

radial and tangential

(5) gives:

otl = El Arl/r State

stress

; El

(8)

1 is now considered

The change

in Young's

as the initial

modulus

state for the phenomenon

in question.

from the initial value to the final value causes

radial and tangential deformations to vary with respect to state i. The tangential and radial stress distribution for state 2 can be deduced from these changes: STATE 2 : ot2 = otl + E2/r (Ar2 - Arl) Given

(9)

; E2

Y2 = Ar2/ri Yl = Arl/ri

(9) becomes :

(1o)

°t2 = °tl + E2 [Y2 - Yl ]/p Conversely, Otl --

i :

I

+

--

c0

c0

(1

-

2

Orl --

considering

--

(i

[in(l

(7) and (4) one has:

+ p2)

_

in(l

+

pc2)]

(11)

p2

1 =

I --)

(6),

+

1 --)

E 1 y~

[ i n ( 1 + p2)

_

p2

2

in(l

+

p2)]

(12)

=

cO

dy dAr ! where Yl = -- = - dp dr Yl is explicit

from (12). Thus:

°0 Yl = - JT

1 + --) p2

[ l n ( 1 + p2 ) - ln(1 + pe2)] do =

MATERIAL RELAXATION IN STRIP WINDING

°O =

--

fQ {

ln(1

p2) dp

+

+I Qln(1 + p2)

"I

2El

JI

o0

do

227

IdQp [ Qdp ] -

P2

ln(1

+

O2 )

[

+

.' 1

--

J1

I

= --{(p 2E 1

}

=

p2 1

(13)

- -)in(l + p2) _ 2p + 4arctgp + 2 - ~ - in(l + p2)[p_ _]) p P

Substituting

in (i0) gives:

E2 Y2 at2 = - + f(p) P

(14)

with 4arctgp E__ 2 I i E2 )[(i)in(l+o2)-(l )in(l+p2)] - - - [ - 2 q p E1 ~ ~ 2E 1

1 f(p) = a0{l+-(l2 Substituting

2-~

p

ot2 in (I) with the value given by (14) yields: E2Y2

l

PE2Y~ + E2Y2 -

- -

=

(15)

f(p)

P

or

d

(16)

Y2)

EZP --(Y~ + -dp p

= f(p)

/flp) E2(Y ~ + Y_2) = C'

dp

C'= 2E2C

(C',C = cost.)

P

Y2 F Y2 = - -- + 2C + -P E2

with

The solution of (17) is

Y2=e

[D +/

F

(17)

(d = cost.):

F (2C + --) dp] E9

(18)

228

T. RAPARELLI

or

Y2 = - + Cp + - - ~ P

with

~

(19)

pdp

PE 2

Deriving

(19) for p,

multiplying

by E 2 and dividing

by

o 0,

gives

the

ratio

Or2/O 0. Thus: Or2 E2Y 2, . . . . . .

Da

oO

p2

+ C a + e'

oo

d i ~' = - - ( - -

with

(20)

--)

dp pE 2 o 0

with C a, D* integration

constants

determined

by putting

Or2 = 0 for p = Pe" Multiplyng (19) by the ratio E2/p and considering

(14),

Y2 = 0 for p = i (and dividing

and

again by

o 0) gives:

ot2

1

Da =

~

o0

+

Ca

+

--

p2

_

Value of constant

~2e

D* =

+

f(p)

(21)

p2 o0 D a is:

I:

1 E2 1 (i + lnp e) + -(I - --)[- ~ ( i i + p2e[2 2 EI 2p2e 2K - 1

_ ~(_i) K-I 2

1 + -(inPe)2(l 4K2(K-l)p~ k 2

I 3 - -(I + --)in(l 4 p~

1

in2 - ~

4K2(K - I)

Value of costant

1

1 1 - --) - _ _ p~ 2

2K + i + p~) + -:(-I)K-14K2p~K+2

:(_l)K-i

2

- 3lnp e) -

1 ! - -(I - --)inPeln(l 2 pe2

I E2 + :(-l)K-l~] - ~[inp 4K 2 2E 1

+ p2e) -

e + i - (-

pg

C e is:

E_2

c* = - D*- -(1 )[ 2 E1

: (-1)K-I

1 ~ 4K2(K-I)

1 in2 + u _ :(-I)K-I~KK2] 2

E2 El

+ 1)ln(1 + p~)]

MATERIAL RELAXATION IN STRIP WINDING

Value of funcion

229

F is:

i E__ 2 [-~ (-1)K-I l 1 1 F = o0{in p + -(i ) + (inp)2+ -(I + ~ ) i n ( l 2 El 2K2p 2K 2 1 -inp-(lnp+ ~ ) i n ( l + p ~ ) ] 2p 2

E2 ~[-21np2E 1

+ p2) _

4 2-7 -arctgp+41np-21n(1+p2) - - - ]} P P

The funcion ~ is:

1 1 1 E2 )K-1 1 1 = o 0 { -p21np- -p2K+-(l)[~(-I - -inp + 2 4 2 E1 4K2(K-l)p 2(K-I) 2 1 1 1 1 1 + _p2(Inp)2_ _p21np+ _p2+ _(l+p2)[in(l+p2)_l]_ ~(_I)K-I 2 2 4 4 4K2p 2K

1 1 1 1 1 1 + -(inp)2- _p21np+ -p2-(-p21np- -p2+ _inp)in(l+p 2)] 2 2 4 2 4 2 E2

i [p21np- -p2-4parctgp+21n(l+p2)-(l+p2)in(l+p2)+(l+p2)-(2-~)p]} 2E 1 2

Results and conclusions Figures 4 and 5 show radial and tangential coil stresses versus ratio r/r i for material with,and without relaxation.On the basis of experimental data, the material's longitudinal modulus of elasticity E 2 (as regards its effect on the relaxation process in thise case ) can be assumed to be around 80% of E l . An examination of the curves will show that the values for the case of material with relaxation are lower than the original values and

3 ~i~ 2 _

without

'/relaxation

i

0

relaxation'

-I

FIG. 4 Radial stress versus ratio r/r i.

230

maintain a qualitatively similar trend. These results, together with the conclusions drawn for the case of perfect elastic material [ 8 ] , indicate the contrasting effectcs of relaxation: on the one hand, the drop in radial stress produces a positive effect, which in the case of rolls of printed matter , for example, can reduce ink transfer; on the other hand, the negative effect of the drop in tangential stress can increase the danger of core loss.

T. RAPARELLI

f o,/~

0.0 I\ ,without relaxation o.5 o.4

0.2

0

! FIG.5 Tangential stress versus ratio r/r i.

References I.Y.N.Robotnov, "On the equations of state for creep", Progress in Applied Mechanics, The Macmillan Company, 1963. 2. F.K.G.Odqvist, J.Erikson, "Influence of redistribution of stress on brittle creep rupture of thick walled tubes under internal pressure", The Macmillan Company,1963. 3. J.Turteltaub, M.A.Bejar, "Balloon collapse in ring spinning", J. of Engineering for Industry, may 1976. 4. J.S.Graham, C.K.Bragg, "Effects of spindle centering on ring spinning tension", J. of Engineering for Industry, february 1987. 5. M.J.Turteltaub, D.W.Lyons, L.S.Cauley, "Calculation of peak yarn tension during ring spinning", J. of Engineering for Industry, may 1978. 6. R.A.Taylor, R.S.Brown, "A design for spinning tension control", J. of Engineering for Industry, may 1985. 7. M.M.Schoppee, "Some problems in winding geometry", J. of Engineering for Industry, november 1981. 8. T.Raparelli,"The mechanics of strip winding " (in italian) IX Congresso Nazionale AIMETA, Bari, Italy, october 1988.