Int. J . Mech. Sei. Pergamon Press. 1968. Vol. 10, pp. 15-28. Printed in Great Britain
T H E I N C O R P O R A T I O N OF W O R K H A R D E N I N G A N D
R E D U N D A N T W O R K IN R O D - D R A W I N G A N A L Y S E S
A. G. ATKINS and R. M. CADDELL
D e p a r t m e n t of Mechanical Engineering, The U n i v e r s i t y of Michigan,
Ann Arbor, Michigan, 48104, U.S.A.
(Received 20 Jm~e 1967)
S u m m a r y - - W i t h few exceptions, m e t a l deformation analyses e m p l o y a c o n s t a n t yield
stress (rigid, perfectly plastic metal) which ignores strain hardening, or a " m e a n " yield
stress which a t t e m p t s to a c c o m m o d a t e strain hardening in a simplified manner. Since
strain h a r d e n i n g is of interest here, little reference will be m a d e to a rigid, plastic t y p e of
behavior.
The first p a r t of this paper d e m o n s t r a t e s t h a t the use of a m e a n yield stress underestimates the working loads (or stresses) needed to draw metal t h r o u g h conical dies as
c o m p a r e d to those loads predicted by more " e x a c t " analyses. I n this c o n t e x t " e x a c t "
refers to those solutions obtained by incorporating the strain hardening relationship in the
governing "force b a l a n c e " differential equations prior to tile integration of the said
equations. I t is shown, however, t h a t the error introduced by the use of a m e a n yield
stress is no more t h a n some 8 per cent for conditions t h a t t y p i f y actual practice. Since
analyses of other m e t a l - w o r k i n g processes, such as rolling and extrusion, e m p l o y the same
sort of differential equation, it is felt t h a t these results are applicable there also.
The second part of this paper shows t h a t r e d u n d a n t work in rod drawing m a y be
a p p r o a c h e d either from considerations of the mechanical properties t h a t result~ after the
metal is d r a w n or from considerations of the stress necessary to draw down the rod.
C o n t r a r y to what is implied in the literature, it is shown t h a t these two approaches lead to
different interpretations of t he " r e d u n d a n t work factor". Relationships are given between
the two for metals that are assume(l to strain harden in certain simple ways.
NOTATION
o~
D
d
semi-angle of (tie cone
d i a m e t e r of rod beh)re drawing
d i a m e t e r of rod after drawing
decimal reduction of area = 1 -- (-~.)
2._-
B
gross, average coefficient of friction between rod and die
drawing stress with no back tension
m e a n uniaxial yield stress (dependent upon strain hardening and r e d u n d a n t
work)
/z cot
/i:h
homogeneous true strain = 2 ln(D) = ln(~--~_r)
(Yd
~0
a,b
0"0, ~/~
f,g,h
true strain induced prior to drawing
constants associated with linear strain h a r d e n i n g (i.e. Y -- a + be)
constants associated with power-law strain h a r d e n i n g (i.e. Y = a0 e ~)
true stress
constants associated with exponential strain hardoning [i.e.
Y = f - g exp ( - he)]
15
16
A.G. ATKL~S and R. M. CADDELL
¢ redundant work factor based upon superI)osition of stress-strain curves,*
e* equivalent, strain = (~eh
C1, C2 empirical coefficients associated with ¢
(D+d~
A geometric configuration factor = \~--~--~]sin a
(1) redundant work factor based upon drawing stresses*
KI, K 2 empirical coefficients associated with ¢
P A R T I.
DRAWING ANALYSES
1. Introduction
M v c ~ of t h e previous w o r k d e v o t e d to drawing m e t a l t h r o u g h conical dies has
b e e n s u m m a r i z e d b y MacLellan 1 and, later, Wistreich2; t o g e t h e r these p a p e r s
contain an e x t e n s i v e list of references relating to this area of s t u d y . :Previous
publications which h a v e some bearing on this p r e s e n t p a p e r will be referred to,
otherwise the r e a d e r can check 1, 2 for a m o r e c o m p l e t e source of i n f o r m a t i o n .
A l t h o u g h Shield a has p r o v i d e d a solution to a x i s y m m e t r i c d r a w i n g t h r o u g h
conical dies of semi-infinite length, Wistreich 2 points o u t t h a t t h e results h a v e
limited p r a c t i c a l v a l u e because of restrictions regarding end effects. I n fact, no
rigorous a n d t r u l y a c c e p t a b l e solution to the plasticity p r o b l e m posed b y
a x i s y m m c t r i c d r a w i n g has y e t been found. T h e literature does contain a
n u m b e r of analyses, however, t h a t p r o v i d e a definite insight to this p r o b l e m ;
some e m p l o y the use of t w o - d i m e n s i o n a l p l a n e - s t r a i n drawing as a guideline
b u t these results do not, of course, p r o v i d e a one-to-one correspondence with the
a x i s y m m e t r i c case. T h e work of Hill a n d T u p p e r ~, W h i t t o n 5 a n d Green G
d e m o n s t r a t e (though not exclusively) some of t h e i m p o r t a n t previous studies.
F o r the d e v e l o p m e n t of the objectives of this p r e s e n t p a p e r , the t h e o r y
originally p r c s e n t e d b y Sachs 7 will be of principal concern. Modifications of
this t h e o r y (e.g. K S r b e r a n d Eictfinger s) will be ignored e x c e p t for reference to
t h e w o r k of D a v i s a n d D o k o s 9. Additionally, use will be m a d e of t h e t h e o r y
d e v e l o p e d b y Hill a n d T u p p e r 4 (see B a r o n a n d T h o m p s o n 1°) which was b a s e d
u p o n a slip-line field analysis. I t should be noted t h a t this s a m e result h a d earlier
been p r o p o s e d b y Sachs and Van ] lorn n, who used a force b a l a n c e analysis.
2. General consideration.¢
Throughout this paper it is assumed that there is no back tension applied to the metal
as it is drawn through a conical (lie. In terms of the parameters of semi-angle of the die (c0,
reduction of area (r) and an average value of coefficient of friction at the dio--workpiec( •
interface (ix), expressions for the drawing stress (as) have been developed a,~ follows :
c~a = ':,,(1 + B)ln(~_r ) = I:,,(1 + B)E~
(ra = /~,,,(-~--)[1--(1--r)'] = ~,
[l--exp (--Be',,)]
(1) ~
(2) 7
where r is the decimal equivalent to the percent reduction of area, en is the homogeneous,
logarithmic strain based upon the reduction of area, Y,n is some constant yield stress, and
t One referee ho~ suggested the adoption of the following terminology:
¢~--redundant defort~ultion factor (based upon sui)erposition of stress-strain eurv~.s);
(I)-redundant work factor (based upon drawing stresses).
The authors fcel that this suggestion has definite merit as it might lead to far greater
clarity in future disc.u.~ions of "redundant work factors".
Incorporation of work hardening and redundant work in rod-drawing analyses
17
B = / z cot c~. If one notes that
1-exp (-Be) = Be-U+.
2o
°"
(3)
it becomes apparent that as Be becomes small, equation (2) eventually reduces to
equation (1).
An even more pertinent point to note is that equation (2) was derived from a stress
theory based upon the following differential equation, which can allow for variation of the
flow stress with strain:
do" d
de b Baa = Y(e) (l + B)
(4)
If Y(e) is takcn as some mean (or constant) flow stress, the solution of (4) leads to (2). If,
however, the ftmctional dependence of Y on e is known, that is, if strain hardening can be
handled rationally, the use of a mean yield s t r e ~ would be unnecessary. Equation (4)
could be solved [using the integrating factor exp (Be)] to give
ad exp
(Boa) = (1 + B) f : a Y(e) exp (Be) de
(5)
For simplicity, it is assumed throughout this paper (unless specifically stated otherwise)
that the material before drawing contains no prior work hardening, hence, the lower limit
of zero in equation (5). Inserting the function Y(e) inside the integral sign would then
enable one to obtain a solution that accounts for strain hardening. The obvious problem,
of course, is to choose a proper function for Y(e). Davis and Dokos' have provided one
such solution.
Pertinent assumptions made in the stress formulation that lead to equation (2) are well
documented elsewherO ° and need not be repeated here.
Hill and T u p p e r ' used a slip-line analysis for plane-strain drawing as a model for the
axisymmetric ease. They proposed a method for handling strain hardening that, in
essence, provides a correction to equation (1). A very thorough coverage of plane-strain
drawing, and suggested extensions of past theories, has been published by Green'.
3. The effect of strain hardening t
I n the approach of Hill and Tupper', where no differential equation exists after the
formulation of the problem, the effect of strain hardening can be handled only by employing
a type of mean yield stress for the range of equivalent strain induced by the drawing
operation. With the Sachs treatment, this restriction is not confronted; in fact, Davis and
Dokos provided a solution by assuming the metal was a rigid, linear, work-hardening solid
(i.e. Y = a + be, where a and b are constants). Using this relationship in equation (5) they
found the following:
aa = ( I ~--B-B) { [ 1 - e x p ( - Ben)] (a - b ) +ben}
(6)
The present authors feel that the oft-quoted but less often used power law form of strain
hardening very well typifies the strain hardening behavior of m a n y common metals.
Here, Y = a0e "~, where a0 and m are considered as constant properties of the metal.++
t The influence of strain-rate and temperature are not considered in this paper.
I n the form Y = a0 em it is implicit that this equation predicts the behavior of a
metal containing no initial cold work. If the metal had been subjected to prior
work hardening equal to a strain e0, then its subsequent behavior is depicted by
Y = a0(e0+ e) ~. I n this manner, a0 and m are considered to be material constants
which do not change with the condition of initial strain hardening.
If, however, for an initially strained metal, a typical tensile test is conducted
and l o g s is plotted against logo {i.e. e0 is ignored), the points always lie on a
shallow concave-up curve with no straight portion; this is due to the nonlinear
log scales. To quote average values of a0 and m for various states of strain
hardening, from plots which are definitely not straight lines, seems very
questionable practice.
2
18
A . G . ATKI~S and R. M. CADDELL
Voce12. z3 proposed the stress-strain relationship, Y = f - g exp ( - h e ) , where f, g and h are
a r b i t r a r y constants. A l t h o u g h this provides a flexible a r r a n g e m e n t t h a t p e r m i t s close
fitting to m a n y curves, the simpler power law form describes the strain hardening b e h a v i o r
of metals sufficiently well. To illustrate the principal point u n d e r consideration in this p a r t
of t h e paper, it is unnecessary to e m p l o y a v a r i e t y of possible forms depicting strainhardening characteristics. Therefore, further discussion will be restricted solely to the
two forms t h a t follow
Y = a+be
(linear work hardening)
(77
Y = aa e"
(power-law work hardening)
(8)
I f e q u a t i o n (8) were substituted in (5), one would get
an = ( I + B ) [exp (--Beh)]
a0e
cxp (Be) de
(9)
Since m is always fractional (0 < m < 0-6 for materials whose e x p e r i m e n t a l values of m are
q u o t e d in the literature), e q u a t i o n (9) cannot be solved in closed form and so is best
expressed as
B
~
B" e ''+"+z
a~ = a o ( l + B ) [ e x p ( e^)l~_0n ! ( m + n + l }
(10)
F o r t t m a t e l y , practical values of (Be) are usually less t h a n u n i t y and rapid convergence
of the series occurs after e x p a n d i n g a b o u t four terms.
W h e n an analytical form for the stress-strain curve is not known, an analog c o m p u t e r
can be usefully e m p l o y e d to integrate e q u a t i o n (4), as recently shown b y R o w e 14,
Chapter 12 of his book gives e x a m p l e s of the technique.
4. The mean yield stress method
Much of the previous work related to rod or wire drawing entails the use of a m e a n
yield stress of the d r a w n m e t a l in order to solve for t h e drawing stress. Such a procedure,
in essence, a t t e m p t s to consider the effects of strain h a r d e n i n g b y correcting for it after the
fact. One m i g h t certainly ask just how a c c u r a t e is this a p p r o a c h c o m p a r e d to one t h a t
considers strain h a r d e n i n g in the basic governing equation. (Note t h a t in order to present
this clearly, one m u s t r e v e r t to an e q u a t i o n such as (4) which is based on a stress analysis.)
F o r the linear w o r k - h a r d e n i n g material, the m e a n yield stress from equation (77
would be
= --1 Ioh (a+be) de = a + - )beh
(ll)
fm
Eh L/V
and inserting (11) in (5) would giw ~,
[a+y} rl - e x p
( - Beh)]
(127
F o r the power-law form of strain hardening, the m e a n yield stress is found from
1 C"
ao e~
Y,, = - - | aoemdz =
ehdo
m+l
(137
t h a t is, the m e a n yield stress is simply the flow stress at a strain ca, divided by the factor
( m + 1). V~rhen the results of (137 axe combined with (5) one gets
aa
: (~-)
( a ° e ' ~ [ , - - e x p (-- Be.)]
\ m + 1/
(14)
Comparison of (12) with (6) and (147 with (10) shows t h a t the use of a m e a n yield stress
does n o t produce a solution e q u i v a l e n t to the more e x a c t analysis.
R a t h e r t h a n c o m p u t i n g differences at this point, let us rearrange (6) as follows:
I n c o r p o r a t i o n of w o r k hardening and r e d u n d a n t work in rod-drawing analyses
19
Making use of t h e a p p r o x i m a t i o n (see Green 6)
1
1/
Be\
2
1-cxp (-B~) =~( I +K~ +T)
which is a c c u r a t e to 1 per c e n t for Be = 2 and Tt6 per c e n t for Be = 1, e q u a t i o n (15) m a y b e
reduced to
As Be becomes smMler, e q u a t i o n (16) derived from the e x a c t analysis reduces to t h e
" m e a n " yield stress expression as given by e q u a t i o n (12).
CO
c
~r
(D
1.0
fllssO
lit ~ 0.1
m • 0.5
I11~ 0 . 5
E
o
a.
U.
0,9,
0.8'
0.7.
.2
o
o
oq
o:z
o'.3 6.4
0:5
BE " ~ . ( C o t
o'.s o.Y
~ )
c¢5
0"9
~o
Fxo. 1. Comparison of predicted drawing stress using a " m e a n " yield
stress and a more " e x a c t " analysis for various combinations of (Be)
and strain-hardening rates.
This same general finding can be shown by c o m p a r i n g (14) with (10) although t h e
m a t h e m a t i c a l m a n i p u l a t i o n becomes l e n g t h y and cumbersome. I t is more c o n v e n i e n t to
w o r k with t h e ratio of the " e x a c t " to " a p p r o x i m a t e " drawing stress [i.e. ratio of e q u a t i o n
(10) to (14)].
o, from (10)
oa from (14)
(1 + B ) [exp-- (Beh)]
=
[1+/~
aoer
VT-)
~0
'K"/
r
Bnrlm+n+l}
oh.
"1
/
°°~0t,~ Ira+,*+ 1)J
(17)
[1
Simplifying gives
O,d f r o m
(10)
aa from (14)
(m-~)~0
I n ! (B~h)n+l
(m~n~
l)]
(18)
exp (B~h)-- 1
I f one takes the reciprocal form of (18), in order to judge how closely the ratio of
a p p r o x i m a t e stress to e x a c t stress approaches unity, t h e results as a function of (Be) are
shown in Fig. 1. Several different work-hardening exponents h a v e been chosen and it m a y
be observed t h a t t h e greatest d e v i a t i o n occurs at large values of (Be) w i t h large values of
m. I t was found sufficient to use a m a x i m u m of five t e r m s of the series in (18) since r a p i d
convergence occurs; u n d e r t h e least favourable c o m b i n a t i o n of conditions (Be = 1 a n d
m = 0.5), t h e use of a m e a n yield stress provides predicted drawing stresses w i t h i n
10 per c e n t as c o m p a r e d to the m o r e e x a c t solution.
F r o m t h e a b o v e one can infer t h a t regardless of the shape of the s t r e s s - s t r a i n c u r v e ,
t h e use of a m e a n yield stress will always u n d e r e s t i m a t e the m a g n i t u d e of t h e r e q u i r e d
d r a w i n g stress. I n this light, t h e results presented in Fig. 12.5 of t h e t e x t b y R o w e 14 a r e
20
A. G. ATKrNS and R. M. CADDEI.L
of interest and lend confirmation to these conclusions. There, comparison is m a d e between
a c o m p u t e r solution of e q u a t i o n (5) and a m a n u a l solution of e q u a t i o n (14) ; with reasonable
practical values of Be (up to ~0.5) excellent a g r e e m e n t was found. Fig. 1 of this paper
would predict a drawing stress from e q u a t i o n (14) to be a b o u t 98 per c e n t of t h a t predicted
from e q u a t i o n (10), since R o w e ' s stress-strain d a t a for mild steel (Fig. 12.2a of his text)
fit the e q u a t i o n a = 107,000e °'23.
E m p h a s i s m u s t be placed on the o b s e r v a t i o n t h a t Fig. 1 was derived for a material
containing no initial strain hardening: if this condition is n o t m e t , it is obvious t h a t the
lower limit on the various integrals is finite and n o t zero. A l t h o u g h no calculations arc
given here, it may, however, be shown t h a t the differences between predictions based upon
" m e a n " yield stress versus " e x a c t " analysis diminishes with increasing initial cold work.
This comes a b o u t because the stress-strain curves, depicting strain hardening characteristics, tend to flatten because of the influence of e0, and the b e h a v i o r approaches t h a t of a
rigid, plastic material (that is, loosely speaking, m approaches zero).
:In closing this section of the paper, it seems proper to n o t e one i m p o r t a n t observation.
Due to t h e perhaps fortuitous fact t h a t practical m a g n i t u d e s of m and (Be) fall within the
range of those shown in Fig. 1, the use of a m e a n yield stress will generally provide sensible
predictions of drawing stress. (This, of course, implies t h a t values of/z, as used in these
calculations, are realistic.) Thus, the e x t r a m a t h e m a t i c a l m a n i p u l a t i o n t h a t enters with
more e x a c t analyses is, in general, unjustified. U p to this point, reference to " r e d u n d a n t
w o r k " has been a v o i d e d ; this will now be discussed.
PART
II.
REDUNDANT
WORK
IN ROD
DRAWING
1. General considerations
W h e n a rod of d i a m e t e r D is reduced to a d i a m e t e r d b y drawing t h r o u g h a die, it
suffers a strain which is a c t u a l l y greater t h a n would be e x p e c t e d m e r e l y from considerations
of homogeneous plastic deformation. The origin of this effect m a y be found in the
" r e d u n d a n t w o r k " p e r f o r m e d in the die, where the e x t e r n a l constraint causes appreciable
internal distortion of the workpiece b e y o n d t h a t strictly necessary for shape change alone.
Although, as H u n d y and Singer is h a v e shown, the induced strain can be quite inhomogeneons under certain conditions, being greater on t h e outside of the workpiece t h a n at the
center, it is nevertheless c o n v e n i e n t to write t h e strain a c t u a l l y imposed as e* = ~ea,
where ¢ is called the " r e d u n d a n t w o r k f a c t o r " .
The existence of r e d u n d a n t w o r k will, of course, necessarily increase drawing stresses,
a b o v e the sort of values t h a t are predicted by the equations given in t h e preceding sections.
This provides one a p p r o a c h towards defining the r e d u n d a n t w o r k factor and will bc
discussed subsequently. This second p a r t of the p a p e r really concerns itself with the
c o m p a t i b i l i t y of two different ways of defining t h e r e d u n d a n t w o r k factor.
2. Redundant work from mechanical properties
Because of the " e n h a n c e d s t r a i n " in the drawn rod, it has, tor example, a greater yield
strength t h a n homogeneous d e f o r m a t i o n would predict, and this suggests one w a y of
d e t e r m i n i n g ¢ e x p e r i m e n t a l l y for different combinations of materials, reductions and die
angles. I f the stress-strain curve of the drawn m e t a l be superposed on the stress-strain
curve of the u n d r a w n metal, ¢ m a y be e v a l u a t e d directly from the ratio of " e q u i v a l e n t
s t r a i n " (e*) to the homogeneous strain (ca), where E* is the abscissa value by which the
curve of the d r a w n m a t e r i a l m u s t be shifted to line up w i t h the curve of the u n d r a w n
material. A l t h o u g h such curve-fitting can be quite capricious, the technique has been
used quite often.
A r a t h e r nice p o i n t presents itself in the m e a s u r e m e n t of the mechanical properties of
the d r a w n rod because t h e question arises w h e t h e r one should m a k e s t a n d a r d tensile test
pieces with c u t - a w a y gauge sections, which r e m o v e s t h e m o r e highly w o r k e d o u t e r layers,
or whether one should test the rod " w h o l e " . Certainly, s t a n d a r d tensile pieces give lower
values for the yield s t r e n g t h of the d r a w n m e t a l w h e n r e d u n d a n t work has been large.
Consequently, it seems b e t t e r to tensile test the whole d r a w n rod, hoping t h a t necking
does n o t occur in the grips. I f it does, one has to test anew, a l t h o u g h a simple w a y to
Incorporation of work hardening and redundant work in rod-drawing analyses
21
minimize fracture in the grips is to make steps on the rod before drawing; this gives a test
piece with more highly worked ends. (See Caddell and Atkins16.)
F r o m yield stress measurements, two main parameters seem to influence the magnitude
o f ¢ , namely, the angle of the die, and the percent reduction taken. I t has been shown that
the relationship between these quantities m a y be expressed as follows:
'
= Cl+C, I A
(19)
where A = \(D
D -+dd~
/ sin a. This may be interpreted physically as the ratio of the mean
cross-section diameter of the conical die to the frustum slant length of contact between the
die and workpiece. Both Ct and C~ v a r y with the work-hardening characteristics of the
metal being drax~-a; although Ct does not vary appreciably from a value of about 0"9,
C~ can vary considerably. I n a recent study of redundant work using commercially pure
aluminum, Armco iron, an austenitic stainless steel, and an age-hardening aluminum
alloy, 16 none of which contained initial cold work, empirical relationships for these
coefficients were found as follows :
C l = 3.70m 0'28 a -0"10
(20)
C2 =
0 . 4 8 m 0"Ts 0 -0.054
(21 )
where o0 and m have been defined previously.
Other relations have been reported for ~ (e.g. Linicus and Sachst:t ), but such expressions
may be shown to be equivalent ix) equation (19) as indicated in another work. le Moreover,
the A form as in (19) appears to be preferable because of the analogy with the more exact
form for two-dimensional strip drawing. (See, for example, Greene.)
As to the friction at the die-work interface, ff it does influence 4, it m a y well be a
second-order effect. Wistreich is showed t h a t the yield stxongths of wires, drawn down by
the same amounts, were slmi!m" whether drawing was done dry or woU lubricated (of
course, the drawing stresses and clio pressures were quite different). However, later in that
paper, and in his review paper, s Wistreich indicates t h a t redundant work should be
influenced by friction. Different opinions on this point m a y be found in the literature, but
let it suffice to observe t h a t there is to date no complete agreement. I n the study 16 t h a t led
to equations (20) and (21), the possible effects of friction were not investigated since only
one condition of lubrication was employed.
3. Redundant work from drawing s~resses
Much of the published work on wire-drawing has concerned itself more with predicting
the stress necessary to draw rod through dies, rather than considering the subsequent
mechanical properties of the drawn wire. The empirical theories of KSrber and Eichinger e
and Siebel l°, which are discussed by Wistreich t, are the only analyses which formally take
redundant deformation into account. The expressions for drawing stress are somewhat like
equation (2) with an additional term for redundant work added on. However, both
Wistreich 2 and Whitton ~ point out the potential shortcomings of both of these empirical
theories.
In P a r t I of this paper, some typical analyses relating drawing stress with die angle,
percent reduction taken and die-workpiece frictional effects were discussed. Those
analyses would necessarily give underestimates for drawing stresses because the effort in
t Linicus and Sachs, by measuring mechanical properties of drawn wire, give
A
¢~1+8
for initially annealed brass.
gives
5a
8
Again, Wistreieh 2, by measuring drawing stresses,
(I) = 0 . 8 7 - ~
A
....
4
2
for initially worked copper. I t m a y be noted that this latter expression is developed
from a drawing-stress equation [equation (27) of Wistreieh TM] using the
approximation A ~ 4a/e.
A. G. ATKINS anti R. M. CADDELL
22
performing r e d u n d a n t work had n o t been t a k e n into account. (This discrepancy, of course,
has nothing to do with the fact t h a t the theories m a y give different predictions for
particular combinations of t* and ~, and also slightly different answers depending on the
w a y work hardening is treated.)
Despite these last problems, to m a k e up for the fact t h a t ad would be u n d e r e s t i m a t e d
by one's inability to cope with r e d u n d a n t work in a force-balance analysis, the drawingstr(,~c~s (~(tuatiolm are often multiplied by a factor we shall ('.all (1). For reasons t h a t will I)(,
clear later, this p a r a m e t e r is distinguished from ~, although the literature does not (h) this.
Consequently, for a rigid-plastic, non-work-hardening solid, the " r e a l " drawing stress
is said to be either e q u a t i o n (1) or (2) multiplied by some empirical (P. W h e n this is
e x t e n d e d to a work-hardening material, Hill's t r e a t m e n t employs a mean yield stress.
again applied to e q u a t i o n (1), b u t with the Sachs a p p r o a c h one has a choice as to whether
equat ions (6) or (12) or equations (10) or (14) might be used (of course, other variations arc
possible depending upon the form of the strain-hardening b e h a v i o r (Icemed most
appropriate).
I f one mea.nures/z (e.g. MacLellan's 2° sl)lit-die technique as employed by Wistreich TM
and Yang~l), t h e n (l) m a y I)e deduee.d by comparison of measured and predicted w~lues (,f
drawing stre.~. In this way, tl) can be related to a and r, much like the expre~'.,~ion fiw
in equation (19), i.e.
(I) = K I + K 2 5
(22)
I t is at once a p p a r e n t , however, t h a t the m a g n i t u d e of ~) may depend markedly on the
p a r t i c u l a r analysis used to predict the drawing strew. Perhaps of e v e n greater concern is
t h a t w i t h o u t measured values of friction, one really has two unknowns, (P and/z. For this
reason, there has been the t e n d e n c y to e m p l o y the w~rious drawing-stress equations to
solve for bL under those conditions where (I) is a p p r o x i m a t e l y u n i t y (i.e. combinations of
large reductions and small (tie angles).
4. Compatibility of ~ and d)
The literature would seem to imply t h a t ~ and (I) are the .same p a r a m e t e r , but in a
recent, study, 1~ where various ways of e s t i m a t i n g r e d u n d a n t work were investigated, it
became a p p a r e n t t h a t these were not identical. This c a m e about as follows. Consider
t h a t ~b is our basic definition in terms of the actual strain induced by drawing being greater
t h a n the homogeneous strain. F u r t h e r , asmuno t h a t (~ is known f(~r various eombinatious
of die angle and reduction. Then, instead of using a factor like (I) to modify the drawingstress equations to compensate for r e d u n d a n t work, let us a t t e m p t to incorporate the
effects of r e d u n d a n t work in the deriwttion of the drawing-stress e q u a t i o n .
The yield stress of a metal which has b(..en subjected to the homogeneous strain, ~'h, will
actually be the flow stress at (~h because of r e d u n d a n t work. E m p l o y i n g the definition of
m e a n yieht stress used previously, this means
t;. = (~gT~hl20f0~ Y(e) d,:
rather than
1;,, = e,,l J0["~Y(g) tie
(23)
]{eturning to the form of Saehs's equation (5) and considering the effect of r e d u n d a n t work
on the yield stress, there results
g,, exp
(B~gh) =
(i + B) IO"^Y(g) exp (Be) (te
(24)
,,0
~'()w in P a r t I of this paper it, was shown t h a t even under imfavorable conditions (large
reductions with metals t h a t strain harden s(~verely), the error introduced by using a mean
yield str(~s was usually well under 10 per cent as compared to a more e x a c t approach. It
is felt t h a t these same c o m m e n t s are appropriate here, so instead of using expressions |'~)r
Y(e), such as equations (7) or (8), in e q u a t i o n (24) the more simplified approach using a
mean yield stress will be followed. Certainly, the comparison of (~ with (I) does not a p p c a r
to be invalidated by using this simpler concept, since it is felt t h a t differences in drawingstress predictions do not affect the (1)-(~ relationship.
Incorporation of work hardening and redundant work in rod-drawing analyses
23
For the strain-hardening behavior given by (11) and (13), and considering equivalent
strain rather than homogeneous strain, we may write
y,~ = a + b ~ - ~
(25)
Y,n = ~°¢~e~*
(26)
m+l
These may be substituted directly in equation (2) to give
o', = (1 ~B)(a_i_bC2eh ) [1--exp (-B¢g~)]
,27)
and
0e:
Cl-exp
(28)
If, however, one used the factor • to correct for redundant work, the corresponding
expressions [i.e. equations (12) and (14)] would be
a~ ----(I)
(a+ ,5-) [1--cxp (-Bea)]
(29)
and
a~ = ~ [ ~ )
[1-exp(-BeD]
(30)
Now if~ and • are identical then equivalence must exist between (27) and (29) and between
(28) and (30). To demonstrate that this is not the c~e, it will be sufficient to consider
only (28) and (30).
Combining these expressions, one gets
[~-
~xp ( - ¢ m ~ ) ]
~
(I) = [ i ~ - i : : ~ j ~
(31)
As (Be) becomes small, as in equation (3), equation (31) approaches
= ~,
(32)
Clearly, ¢ = ~ only when m = 0 (rigid, plastic material); in fact, when (Be) is not small,
~ even if m = 0. The same conclusion (i.e. (I)~ 4) will result if a comparison is made
between equations (27) and (29); the details are left to the reader.
To provide a quantitative comparison, the relationship in equation (31) could be v.ued,
but it seems more appropriate to introduce the parameter A since expressions like (19)
and (22) are generally found in the literature. In order to t)ursue this, (Be) in (31) must be
replaced by an equivalent expression containing A. This can be approximated as follows :
(Be) ~/~cot ct(4:~_ ~) _ 4/~cosct
~4/zA
~~-
(33)
I t will be shown subsequently that the above approximation introduces no serious error
even up to values of ct that are much larger than those used in practice. Introducing (33)
into (31) gives
I n using equation (34), values of A from 2 to 20 were first substituted into equations
patterned after (19). The empirical expressions used for 303 stainless steel and
commercially pure aluminum were reported in another study; 16 the pertinent values for
m (0.52 for the stainless and 0.23 for the aluminum) were also included in that same study.
24
A . G . A~,_I~s a n d R . M. CADDELL
T h e s e expressions for ¢ were as follows:
¢ = 0.87 + 0" 15A
(stainless)
(35)
¢ = 0.89+0.0925
(aluminum)
(36)
To c h e c k t h e influence of/z on t h e ( I ) - ¢ c o m p a r i s o n , values of 0, 0"05, 0.10, 0.20 a n d 0-50
were e m p l o y e d . T h i s r a n g e would seem to e n c o m p a s s p r a c t i c a l l y all v a l u e s of ~ for
d r a w i n g t h a t are r e p o r t e d in t h e l i t e r a t u r e . T h e c o m p l e t e sot of values d e t e r m i n e d for t h e
r a n g e of A a n d /z m e n t i o n e d a b o v e are listed in T a b l e 1. To j u s t i f y t h e a p p r o x i m a t i o n
TABLE l.
R E D U N D A N T WORK FACTORS FOR VARIOUS VALUES OF i
AIN'D ~L
(P
A
¢
2
3
5
7
10
16
20
2
3
5
7
10
16
20
/z~
0
0.05
1.17
1.32
1.62
1.92
2.37
3.27
3.87
1.27
1.52
2.09
2.69
3.70
6.0
7.80
1.27
1.50
2.08
2.71
3.68
6.17
7-67
1.07
1.17
1.35
1.53
1.81
2.36
2'73
1.09
1.21
1.45
1.69
2.08
2.87
3'45
0.10
0'10t
303 Stainless st~¢~[
1.26
1.24
1.51
1.50
2.06
2.03
2.71
2.64
3.70
3-62
5.8
5-62
7.60
7-52
Commercially pure aluminum
1.08
1.10
1.08
1-20
1-21
1-20
1.4S
1.44
1.41
1.66
1.67
1.66
2.07
2.11
2.07
2.90
2.78
2.65
3"38
3"40
3'43
0.20
0.50
0"50t
1.24
1.47
2.01
2.60
3.54
5.73
7.65
1-19
1.36
1.90
2-39
3.24
5.30
6.91
1.19
1.40
1.88
2-43
3-2,~
5.4,~
6.90
1.08
1.19
1.41
1.65
2.04
2.80
3"42
1.06
1-14
1.36
1.58
1.91
2.61
3"20
1.05
1.16
1.37
1.58
1.93
2.70
3'20
"~ ¢ c a l c u l a t e d from e q u a t i o n (31) w h e r e Be = 4tt cos ct/A. All o t h e r c a l c u l a t i o n s for (P
b a s e d u p o n e q u a t i o n (34).
i n d i c a t e d in (33), a d d i t i o n a l c a l c u l a t i o n s were m a d e . B y c h o o s i n g a large v a l u e o f a (22.5~'),
u s i n g 4kt cos a/A r a t h e r t h a n 4/z/A, a n d e m p l o y i n g t w o of t h e values o f / z used p r e v i o u s l y
(0.I0 a n d 0.50), t h e p e r t i n e n t v a l u e s of (I) were f o u n d for t h e .same r a n g e of A values.
T h e s e findings are also i n c l u d e d in T a b l e 1, a n d since all c a l c u l a t i o n s were p e r f o r m e d to
slide-rule a c c u r a c y , t h e a p p r o x i m a t e form g i v e n b y (34) c e r t a i n l y a p p e a r s to i n t r o d u c e n o
significant error. I t s h o u l d be n o t e d t h a t for/z = 0, (P = ¢m~1 regardless o~" t h e m a g n i t u d e
o f 0t.
T h e p l o t s sho~m in Figs. 2 a n d 3 include t h e e x t r e m e lines which b o u n d (I) versus A for
all of t h e v a l u e s listed in T a b l e 1. To plot e v e r y p o i n t WoLlld, possibly, lead to a loss of
clarity. T h e ~b versus A p l o t s on these figures are s i m p l y t h e g r a p h i c a l form of e q u a t i o n s
(35) a n d (36).
N o w s t r i c t l y s p e a k i n g , t h e q ) - A plots, c a l c u l a t e d as a f u n c t i o n of" d). are n o t s t r a i g h t
lines a n d t h e d e p a r t u r e from l i n e a r i t y is m o r e severe w i t h t h e stainless steel (this h a s a
l a r g e r s t r a i n - h a r d e n i n g e x p o n e n t , m). H o w e v e r , if we t a k e t h e l i b e r t y of a s s u m i n g t h a t
t h e s e plots are reasonably linear u p to A values of 10 or so, we c a n w r i t e t h e foll,,wing
expressions :
(P ~ 0 . 8 7 +0"27A (stainless)
(37)
q) ~ 0.89 + 0.12A
(aluminum)
(38)
F r o m Figs. 2 a n d 3, or a c o m p a r i s o n of (37) a n d (38) w i t h (35) a n d (36), it is o b v i o u s t h a t
a n d ~b are n o t t h e s a m e p a r a m e t e r . As p o i n t e d o u t b y G r e e n 6, W i s t r e i c h ' s w~)rk is
I n c o r p o r a t i o n of w o r k h a r d e n i n g a n d r e d u n d a n t w o r k in r o d - d r a w i n g a n a l y s e s
25
i n d i c a t e d t h a t ~) is o v e r e s t i m a t e d w h e n a n n e a l e d m e t a l s are s u b j e c t e d t o low r e d u c t i o n s
{i.e. large A values). T h e r a t h e r d r a s t i c increase of ¢ w i t h i n c r e a s i n g A, as s h o w n i n Figs. 2
a n d 3, m a y b e i n t e r p r e t e d as s u p p o r t of t h a t o b s e r v a t i o n .
8
7
o
6"
)G4
I
nr
o
Iu
5 ¸
]~ FOR p = O
4'
v
o
~
= 0 . 8 9 + o.,2 ~
(TO A = 1
4
)
~
~
~~
~=0.5
2
z
o
z
I
i,I
o
2
~
~
8
rD,
ib
f~
i;,
i~
8
~
FOR ,u = 0 ~ / /
7
-
°0
/ j
~ = 0.87 + 0.27 &
4, (APPROXIMATION UP
o
J
~
iz
az
Q
,~
2"0
C o m p a r i s o n of t h e r e d u n d a n t w o r k factors, (P a n d ~b, for
commercially pure aluminum.
FIG. 2.
,.r
O
f'e
d]
~
~
/j
/ /
-
"~== 0.87. ÷ 0.15. ~
2
I.
0
0
2
~,
6
s
ib
1'2
(4
r6
18
~'0
F.D + d .']
" LD
- - ~ - a - J sin
FIG. 3.
C o m p a r i s o n of t h e r e d u n d a n t w o r k factors, (1) a n d ~b, for
t y p e 303 stainless steel.
A t p r e s e n t , t h e r e seems t o be n o o b v i o u s w a y to p r o d u c e a n e x p l i c i t r e l a t i o n s h i p
b e t w e e n U I a n d some r e p r e s e n t a t i v e v a l u e for K , , a l t h o u g h for t h e s e t w o m e t a l s t h e
c o n s t a n t s U1 a n d K 1 a r e p r a c t i c a l l y identical. F u r t h e r e x p e r i m e n t a l w o r k a p p e a r s
n e c e s s a r y before t h i s c o n c e p t c a n b e e x t e n d e d .
26
A. G. ATKXNS a n d R. M. CADDEI.L
One o b s e r v a t i o n is i n c l u d e d for c o n s i d e r a t i o n . T h e a p p a r e n t i n s e n s i t i v i t y of ¢ w i t h
r e g a r d t o / z as seen in T a b l e 1 (certainly u p to values o f / z = 0.20) t e m p t s one to s u g g e s t
t h a t f r i c t i o n ha.~ o n l y a s m a l l influence on r e d u n d a n t work, a l t h o u g h it m u s t be n o t e d t h a t
t h e s e r e s u l t s d e p e n d u p o n one a n a l y s i s used for det(~rmining t h e d r a w i n g stress.
DISCUSSION
"]'he full p r e s e n t a t i o n u n d e r Scction 4 carli~w was b a s e d u p o n m a t e r i a l s c o n t a i n i n g no
s t r a i n h a r d e n i n g p r i o r to d r a w i n g . :In t h e s t u d y le t h a t p r o d u c e d e q u a t i o n s (20), (21), (35)
a n d (36), n o work was d o n e w i t h m e t a l s t h a t wcr(~ s t r a i n e d before b e i n g d r a w n , so t h e
influence of e 0 [from a = a0(e0 ÷ e ) " ] o n C l ~n¢l C 2 in e q u a t i o n (19) is n o t kno~zl. F r o m a
q u a l i t a t i v e v i e w p o i n t , h o w e v e r , t h e effect of p r i o r cold work is to r e d u c e m a n d increase
a0.t I f t h i s follows, t h e n f r o m e q u a t i o n s (20) a n d (21) t h e t e n d e n c y would be for C z to
d i m i n i s h m o r e r a p i d l y t h a n wouht C~. I n e ~ e n c e , t h i s implies t h a t t h e effects o f r c d u m t a n t
work are l~,ss p r o n o u n c e d on p r i o r w o r k e d m e t a l s t h a n on n o n - w o r k e d m(~tals; it wouht
follow, therefore, t h a t t h e p l o t s of (I) a n d ¢ versus A would b e c o m e f l a t t e r a n d t h e difference
b e t w e e n (l) a n d ¢ would lessen, t h e m o r e t h e p r i o r cold work.
To p u r s u e t h i s p o i n t , let us refer to t h e comprchensiw~ set of d a t a r e p o r t e d b y R o w e ~
a n d J o h n s o n 23. T h e i r work e n t a i l e d t h e use of copper, brass, m i l d steel a n d a l u m i n u m , all
of w h i c h h a d b e e n initially s t r a i n h a r d e n e d 25 p e r cent prior to d r a w i n g . A p p a r e n t l y , t h c y
did n o t e m p l o y originally a m m a l e d m a t e r i a l s in t h a t s t u d y . F o r all of t h e s e m e t a l s , t h e y
f o u n d a best-fit r e l a t i o n s h i p as folh)ws:~
(I) = 0 . 8 8 + (0.19-0.22) 5
(39)
Since t h e r e is a t p r e s e n t no e x p e r i m e n t a l d a t a t h a t wouhl p e r m i t one to c o r r e c t e q u a t i o n s
(35) a n d (36) to a c c o u n t for a n y po.,~sible effect of initial s t r a i n h a r d e n i n g , it is n o t fully
possible to m a k e a c o m p l e t e c o m p a r i s o n b e t w e e n ( 1 ) - A r e l a t i o n s h i p s p r e d i c t e d from
e q u a t i o n s {31), (35) a n d (36) a n d R o w e a n d J o h n s o n ' s e x p e r i m e n t a l d ) - 5 form p e r
e q u a t i o n (39). Howew~r, a r o u g h c o m p a r i s o n will n o w be m a d e , b u t this a t t c m p t m u s t n o t
be c o n s t r u e d to be a n y t h i n g m o r e t h a n q u a l i t a t i v e in n a t u r e .
TABLE
Mat(wial
Copper
70-30 Brass
Mild steel
Aluminum
2.
COMPARISON
O F (~ A N D (I) F O R F O U ~ A N N E A L E D
METALS
(~o+
m*
"1~:
c~+
q~§
(l)ll
72,000
105,000
100,000
26,000
0.50
0.52
0'23
0.23
1.00
0.98
0"78
0.88
0.16
0.16
0'09
0.09
I. 95
1.94
1"30
1.31
2.62
2.62
1.34
1.34
Values selected from ] ) a | s k o ~-'~'.
**C a l c u l a t e d from e q u a t i o n s (20) a n d (21).
§ F r o m e q u a t i o n (19), u s i n g A of 6 a n d c 1 a n d c 2 as t a b u l a t e d a b o v e .
]1 F r o m e q u a t i o n (34), u s i n g A of 6, /z of 0.10 a n d ~ as t a b u l a t e d a b o v e .
1 n T a b l e 2, t y p i c a l values fiw ao a n d m are s h o w n for t h e fl)ur m e t a l s used b y J o h n s o n 2 a ;
t h e s e values p e r t a i n to t h e a n n e a l e d m e t a l s s u b j e c t e d to s t a n d a r d tensile tests. A s s u m i n g
t h e reliability of e q u a t i o n s (20) a n d (21) is r e a s o n a b l e , t h e c o r r e s p o n d i n g v a l u e s of C 1 a n d
C a m a y be reaxlily c a l c u l a t e d ; t h e s e are listed in this s a m e t a b l e . Since t h e m a x i m u m
v a l u e of A used b y t h e s e o t h e r a u t h o r s a4'2a would c o r r e s p o n d to a v a l u e of 6 in o u r
definition, t h i s v a l u e was l~se(l to d e t e r m i n e t h e m a g n i t u d e of ¢ as p e r e q u a t i o n (19) w i t h
C 1 a n d C 2 defined in '['able 2. Because t h e r e s u l t s in T a b l e 1 ilhmtra~ed t h e a p p a r e n t
t See f o o t n o t e o n page 17 a n d c o m m e n t s on t h e t o p of page 21.
:]: R o w e xa e m p l o y s a p a r a m e t e r , A, t h a t is o n e - q u a r t e r t h e A v a l u e used in this
p a p e r . T h e p r e s e n t a u t h o r s h a v e c o n v e r t e d a c c o r d i n g l y as R o w e 24 g a v e a coefficient
of 0.76-0.88 in e q u a t i o n (39).
Incorporation of work hardening and redundant work in rod-drawing analyses
27
insensitivity of ¢ with regard to tz, a value of ~ = 0.1 was chosen and ~) was then determined for each of the four annealed materials using equation (34) (at a value of A = 6 also).
By assuming that the q) - A relationship is reasonably linear up to A = 6 (see Figs. 2 and 3),
then the slope of the line can be determined between A = 0 and A = 6. It turns out that
two distinct lines are found ; one fits the copper and brass while the other fits the steel and
aluminum. These were as follows
Copper and brass :
¢I) ~ 1.0 + 0.27A
(40)
St eel and aluminum : ¢ ~ 0"8 + 0.09A
(41 )
The reason why one would expect such differences can be seen in the significant influence
of m and the rather modest effect of ~0 on the coeWmients C 1 and C~. Although beyond
this point it is not presently possible to convert the annealed relations of (40) and (41) to
similar forms for the prior-worked metals, one would qualitatively expect the slopes to
become shallower [i.e. lower coefficients of the A term in (40) and (41)] as the plotted lines
tend to pivot about a rather stationary intercept at A = 0 (i.e. a value of about 0.9 or so).
Thus, it. is easy to visualize that the slope in equation (40) (i.e. 0.27) would become smaller
~m(ter the influence of prior cold work, thereby apt)roaching a value of 0.19-0.22 as per
equation (39); the slope of equation (41) (i.e. 0.09) is already less than that given by (39).
Because of the large influence of m in this analysis, it is difficult to explain this apparent
contradiction.
Sevcral observations are, however, offered in regard to the general findings ext)ressed
by (39). I n that study, ~a the condition of the starting metal (25 per cent coht work) would
introduce a dependency of (1) on m that is much less than that found with metals initially
annealed36 Since the influence of a 0 on redundant work has been shown to be relatively
small, therefore, one should not expect as great a variation in expressions for redtmdant
work with prior-worked metals as compared to annealed metals. Additionally, the range
of A values up to 6 is somewhat restricted; in fact. the present authors noted greater
scatter in their ¢-A plots 16 at small values of A compared to larger values (A ~ 20) which
arise by making small rcductiens with large die angles. According to Fig. 6.8 of his text,
Rowe 1~ indicates that the smallest reduction used by Johnson ~a was 19 per cent. I n
summary, therefore, one might conclude that ext)crimenting with prior-worked metals
over a small range of A could provide a number of test results that might hest bc described
by a single lino regardless of the metal employed. This may be an explanation to ~L~sist in
interpreting the possible reasons that lead to an equation such as (39); it does not, of
course, explain why the discrepancy between the 0.09 value of equation (41) and the
0.19-0.22 value of (39) exists.
CONCLUSIONS
(1) W o r k h a r d e n i n g effects in d r a w i n g t h e o r i e s c a n b e h a n d l e d e i t h e r b y t h e
use o f a p r o p e r m e a n y i e l d stress o r b y i n c o r p o r a t i n g s t r a i n h a r d e n i n g p a r a m e t e r s i n t h e b a s i c force b a l a n c e d i f f e r e n t i a l e q u a t i o n . F o r p r a c t i c a l v a l u e s o f
d r a w i n g p a r a m e t e r s , t h e s e t w o a p p r o a c h e s p r o v i d e p r e d i c t i o n s of d r a w i n g
s t r e s s t h a t a r e v e r y close; t h e use o f a m e a n y i e l d stress a l w a y s e s t a b l i s h e s
lesser m a g n i t u d e s .
(2) T h e first c o n c l u s i o n w o u l d s e e m to b e a p p r o p r i a t e for o t h e r m e t a l w o r k i n g processes w h o s e stress a n a l y s e s are b a s e d u p o n a g o v e r n i n g d i f f e r e n t i a l
e q u a t i o n t h a t is s i m i l a r to t h a t u s e d i n r o d d r a w i n g (e.g. r o l l i n g a n d e x t r u s i o n ) .
(3) C o n s i d e r i n g t h e v a l i d i t y o f t h e " f o r c e b a l a n c e " a n a l y s i s i n p l a s t i c i t y
w o r k , t h e s m a l l differences t h a t arise, u s i n g t h e t w o a p p r o a c h e s m e n t i o n e d i n
t h e first c o n c l u s i o n , do n o t s e n s i b l y j u s t i f y t h e e x t r a m a t h e m a t i c a l c o m p l i c a t i o n s a s s o c i a t e d w i t h t h e m o r e " e x a c t " a p p r o a c h . T h u s , t h e use of a p r o p e r
m e a n y i e l d stress s e e m s q u i t e a d e q u a t e .
28
A. G. ATKINS and R. M. CADDELL
(4) T h e r e d u n d a n t w o r k f a c t o r d e t e r m i n e d b y r e l a t i n g m e a s u r e d a n d
p r e d i c t e d d r a w i n g s t r e s s e s is n o t e q u i v a l e n t t o t h e r e d u n d a n t w o r k f a c t o r
determined by the method of superposition of stress-strain curves.
(5) D i f f e r e n c e s b e t w e e n t h e s e t w o f a c t o r s b e c o m e m o r e p r o n o u n c e d w i t h
metals possessing large strain hardening rates if the geometrical considerations
( p e r c e n t r e d u c t i o n a n d d i e a n g l e ) a r e fixed.
(6) F o r a p a r t i c u l a r m e t a l , d i f f e r e n c e s b e t w e e n t h e s e t w o f a c t o r s b e c o m e
m o r e p r o n o u n c e d i f s m a l l r e d u c t i o n s a r e t a k e n t h r o u g h l a r g e a n g l e d i e s (i.e.
l a r g e g e o m e t r i c a l f a c t o r , A).
Acknowledgements--Since this work was an extended effort of a previous study t h a t was
financially assisted by the Institute of Science and Technology at the University of
Michigan. the authors express their deep appreciation to t h a t organization. Additionally,
the correspondence of Dr. G. W. Rowe of the University of Birmingham, England proved
enlightening and hclpful. His provision of certain unpublished experimental results is
accorded our sincere thanks.
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