A theory of tube sinking

MECH. RES. COMM.

A THEORY

OF

Vol.2,

TUBE

233-238,

1975.

Pergamon Press.

Printed in USA.

SINKING

N. Cristescu University of Bucharest, Romania (Received 5 May 1975; accepted as ready for print 28 July 1975)

1.

Introduction

The theory considered here assumes that the procass of tube sinking (decrease of its outer diameter) through a conical die will be accompanied by a reduction of the wall thickness. A land of die as well as a mandrel will also be considered (see Fig.1 where the notations are selfexplanatory). The geometry of the process is similar to the one assumed by Avitzur [1] Ch.12 : in the region II the material is deforming plastically,and the velocity field in spherical coordinates is •

v-

-vfr

2 cos0 ~

,

v 0 = v~ = o .

(1.1)

In the regions I and III the material is assumed to be rigid and moving with the velocity -v o and -vf respectively. Other obvious geometrical relations are tf

Rof

Vo

R2 if

(1.2)

2 Rof

1

It is also assumed that at the spherical surface ~ the velocity has a tangential discontinuity v t = -vfsin@ . Sim~'larly at ro the tangential discontinuity of the velocity is v t = VoSin6 If the length of the land of die is L, the mandrel will be in contact with the inner surface of the tube along a cylindrical surface of the length L +Z~L where

V2 I L o %f

cos (~" ~ i) sin2 ~

Rif~2

(.I - ~ j

•

(1.3)

. . . . . . . . . . .

A Bingham type constitutive equation ly as it was done for the drawing of constitutive equation with (1.1) one stress along ~I and ~2 which in turn Scientific Communication

233

will be used here similarbars [2]. Combining this can obtain the shearing can be used to estimate

234

N. C R I S T E S C U

Vol.2, N o . 5 / 6

r4

the power consumed at the surf ac~s L I and I 2. The drawing stress will be obtained from the theorem of power expended apolied to the volume of the tube comprised between the limits xf and x o. 2. The dErawing stress Using a similar procedure as in [2] the rate-of-work dissipated in domain II due to viscoplastic deformation is obtained as = 2 ~ v f R 2 f ( l n R°f~o)f (o(,o~i)+ 2?Tvf~Rof

g ( e ~ i)

(2.1)

where f (°~'c(i)= 2 ~

1_^ [cos,~c0s2~+ I 1 ln(cosc(+~cos2o(+ ~ ) sin"(XL "~ + "~

- c°sC(i~ c°s2~i + 1-W

_

1-~

g(~,oq) : y s~n~[-~ is the mean yield stress and ~ the viscosity coefficient. The friction law which is used along ~3 is a generalization for the visco-plastic model of the so-called "coostant friction factor" which is taken in the form ~ = m I ~ s where II s is the second invariant of the stress deviator and m a constant friction coefficient. It has been shown for the drawing of bars [2] that this law and the Coulomb law are leading to practically the same results. The dissipated rate-of-work due to friction along ~3 is then W~3=-2SmvfR2fcoto(

In ROfRo +

~+1

,(2.3)

In a similar way the dissipated rate-of-work due to friction along ~4 is

w~4 =

~n

~ vf2~RofT

(2.4)

where n is the constant friction coefficient (n=m or n#m), and along ~6

w•Q

~ -n ~~ vf2~ Rif (L + A L )

•

(2.5)

Vol.2, No.5/6

235

THEORY OF TUBE SINKING

Along the surfaces ~5' ~ and F'8 we have wr5, wc7 = Wrs. o. The power consumed at ~I and ~2 can be computed following a procedure given in [5]: --v~of 2{ I P(~'
~vf q(~'~i ) } + off

T'f(%,]~

" 2~ VfRof 2 II p(~,o(i) WF2=

(2.6)

}

where

1 [cos~ff 2. 1~ - ~ ~sin~ [ 2 ~ T

cos~. "'

1

2223in(2V11(11cos~+1)

_1__ 2_~ ln(2~11(qlcos~i+l)+22cos~(i) ) J

q(~,0(i) =

coso( + ~

cos~i-

cos~ i

(2.7)

The power of the applied stress is

(oZT~/

1 -

] ~T~

(2.8)

and the power of the back tension

2[

(2.9)

All these formulae introduced in the theorem of power expended yields

+Nq(~'qi)

+" 0

Rif L + ~n I~of +~-F-(

Rof

~

g(~,O(i ) + 2P(~,C(f) +

~ cot0( In

[I_

(2.1o)

where ~vf

(2.11)

236

N. CRISTESCU

is the dimensionless

Vol.2, No. 5/6

"speed effectiveness

parameter"

(see [2]).

~]quation (2.10) gives the drawing stress as a function drawing speed, reduction, tion coefficient influence

wall thickness,

semicone

angle, fric-

and length of the land of die. By N the speed

is introduced in (2.11). Thus this

influence depends

not on the working speed alone, but essentially here the measure

of :

of ths rate dependence

to a lesser degree, N depends

on ~

which is

of the material considered;

on the mean yield stress and Rof too.

3. ,i~amples In order to show the influence meters

of the previously mentioned para-

on the drawing stress some numerical examples were computed

for a certain sort of aluminium.

The constants used are: ~

= 16

kg r~ -2 , 6-xb= 0, m = n = 0.Ol~, L = 2 ram, Rof = 15 mm. To obtain various wall thicknesses was varied.

in (2.10), Rof was kept constant

It was thought that the results are more suE;gestive

if instead of the total drawing force stress,

and Rif

a dimensionless

is used. Very low speeds N = 0.149 and N = 0.3

or the relative

drawing force defined (i.e. N ~ O ) angle

(Ri/Ro = 0.2 and 0.8)

10'~ a~d 4 ~

where -=

by F = ~ / ( 2 ~ R 2 o f ~ )

of tl~e dimensionless

o~ is shown in Fig.2

thicknesses

1oo

and two reductions

1

drawing

have been compared with

• The variation

force with the semicone

Full lines correspond

~

-

i

--J

drawing

for two wall in area r =

•

to N=O, broken lines to N=0.149,

and dotted

lines to N=0.3. The minimum points on these curves correspond to the optimum semicone

an~:le O~opt; this angle is obviously depen-

dent on N and the thickness. cant for ~igher reductions dimensionless same values

The speed influence

is more signifi-

and thicker tubes. The variation of the

drawing force with the wall thickness for o( =6 ° and

of r and N as above, is shown on Fig.3. Thus the

"speed influence" is more si~:nificant for thicker tubes and bigger reductions. values

of O~

The maximum possible and wall thickness

reduction

in area for various

is shown on Fi~j.4, for N=O and

N=0.149 and for several wall thicknesses

(Ri/Ro = 0.1; 0.5 ; 0.7;

Vol.2, No.5/6

237

THEORY OF TUBE SINKING

0.9). Points wh.ic~ are under the curves shown are possible reduc tions. The curves have been obtained from the condition O-xf= (see [2]). The computatioos have been done starting from a certain value of o( on. For a given ol the 1~ximum reduction is for thin tubes much smaller than for thick tubes, mainly due to the significant increase of the internal dissipation. There is, however, a certain increase due to the velocity discontinuity and to friction too. The speed influence is for small oL negligible but increases significantly with o~ . References 1. B.Avitzur, ~;etal Forming Processes and Analysis, ~icGraw Hill Book Comp., New ~ork (1968) 2~ N. Cristescu, Int.J.~ecb.Sci., 17, #25 (1975) 9. ~. Cristescu,

(to be published)

--~I ~ L ~ - -

, J7

a \

_ Vo

I & • R~¢ &¢

t

I" zp

-

-.o--41

0

Fig.1

zo

Geometry of tube sinking

238

N. CRISTESCU

Vol.2, No.516

F U.8

--.°oo. ~ "'"°...o..

p~o

Q6

~q

~<~,.~, "~ak., °(=6° ~ ~ .

02

~[ ~

I0 ................""22"22-,TZ j

i

I

I

I

O.Z

O,~

~6

0.8

l.O

RiCo Fig.2 Variation of dimensionless

Fig.5 Variation of dimension-

drawing force with cC for various

less drawing force with thick-

drawing speeds ,tube thicknesses

ness of tube wall for r=I0%o and

and reductions in area

r=40% and three drawing speeds

f

o

Fig.4 Maximum possible reduction as function of o~ for various wall thicknesses and two drawing speeds