Materials Science and Engineering, 84 (1986) 157162
157
Necking Development and Strain to Fracture under Uniaxial Tension JIANSHE LIAN and BERNARD BAUDELET
Gdnie Physique et Mdcanique des Mat$riaux, Unit~ ~,ssoci~e au CNRS 793, GRECO "Grandes Ddformations et Endommagement", Institut National Polytechnique de Grenoble, B.P. 46, 38402 St. Martin d 'H~res (France)
(Received December 4, 1985; in revised form April 21, 1986)
ABSTRACT
An analysis o f necking development under uniaxial tensile deformation is made to derive the analytical expression for the correlation between the elongation e~ to failure, the strain rate sensitivity m and the strainhardening effect n for geometric defects as follows: e~ = {1  (1
__f)llm)m
exp(n)   1
This expression is in agreement with the me~ relationship given by Ghosh and Ayres where n = O, or with the classical analysis o f Considere where m = O. The equation for the strain e~ at fracture using the parameters o f the material is for deformation defects, ef= n   m l n { 1   e x p (   e o e f =  n ) l elm / ) The results calculated using the above equations agree well with the experimental results. Also a modification analysis o f the linearized approximation is made to derive a simple m  n  e l relationship which is the same as the result given by Semiatin and Jonas.
1. INTRODUCTION Superplastic metals are characterized by a very high tensile elongation, a low flow stress and a high strain rate sensitivity of stress u n d er d e f o r m a t i o n . For superplastic alloys with a stable structure (e.g. the materials are insensitive to cavitation and have a stable or quasistable grain size) under tensile deformation, the ductile fracture is mainly controlled by the strain rate sensitivity coefficient m and also by the strain hardening o f material. In the past 15 or m or e years, m a n y papers have been published on studies of the plastic instability and the d e v e l o p m e n t of macro00255416/86/$3.50
scopic necking of superplastic metals as well as on the relationship between the fracture strain and the parameters of the material. All these studies present a good understanding of this subject. Backofen et al. [1] give a phenomenological explanation of the role of strain rate sensitivity coefficient m in the resistance to rapid neck propagation. Hart's [2] work established the basis of the theoretical analysis of plastic instability of a material exhibiting strain hardening and strain rate hardening. Ghosh and Ayres [3] suggested a relationship between the fracture strain ef and the strain rate sensitivity coefficient m. Subsequently, several researchers have used this analysis and included both the strain rate sensitivity and the strain hardening. Using the same m e t h o d as Ghosh as Ayres, which is called the "longwavelength a p p r o x i m a t i o n " , Hutchinson and Neale [4] gave an integral expression including both the strain rate sensitivity and the strain hardening. On the basis of Hart's analysis which is a firstorder approximation, Nichols [5] and Lin et al. [6] also gave two approxi m at e mef relationships including strain hardening % Nichols's result has been modified recently by Semiatin and Jonas [7]. In view of the various analyses, further developments are possible. So the objective of this paper is to present an analysis to derive a simple expression between the ductility and the parameters of the material.
2. ANALYSIS For the phenomenological description of behaviour under uniaxial tensile d e f o r m a t i o n , Hart [8] gave a differential constitutive equation 8(ln (1) = ~, 8e + m 8(ln ~)
(1)
© Elsevier Sequoia/Printed in The Netherlands
158 This expression will be used in the present analysis; this means that, in the derivation, both m and ~, are considered to be constant except for the final equations where V = n/ef is used for a p o w e r law hardening material. Since the initial imperfection in the specimen can be either a geometric defect or a deformation defect, the following analysis will deal with each of these separately.
the same ~, and m values. So we integrate eqn. (6) along the tensile axis from the homogeneous section A to the necking section AN and define the integral constant by putting .4 = 4N while A = AN. This integration gives
2.1. Geometric defect
dN ~ A ~l~)lrnl Ao.~/m
In a specimen under tension the initial cross section and length are A0 and L 0 respectively and the corresponding instantaneous values are A and L. On the assumptions that there is an initial geometric defect f and that only the plastic deformation is considered, from the constantvolume condition for a geometrical defect the relation can be expressed as follows: 6e = 6(In A0)  5(In A)
(2)
5(In ~) = 6(In ~i)  6(In A)
(3)
The substitution of eqns. (2) and (3) into eqn. (1) gives
Defining the initial geometric defect as
f= _
5A0
Ao (Ao
 ANO)
Ao ANO
(9)
 1
Ao
Substituting eqn. (9) into eqn. (8) and remembering that A = A0 exp (e), we obtain
__f)l/m
exp{'eN~T)}
deN
(4)
Under uniaxial tension, the condition of load equilibrium along the specimen can be expressed as
6(ln A) + 6(ln o) = 0
(7)
A0 J
From A =  dA and AN =  ~NAN, eqn. (7) can be rewritten as
(1
6(ln a ) = m 6(In ~ i )   ( m + ~,) 6 ( l n A ) + ~, 6(In A0)
=
(5)
The combination of eqns. (4) and (5) gives
= e x p {  e ( 1 ~)}de m
(10)
For the description of necking development, assuming also that ~t is constant during the deformation (see later for comments) and integrating both sides of eqn. (10) beginning at e = 0 and eN 0 respectively gives ~
m 6(ln A) = (m + 7  1) 6(ln A) ~' 6(ln A0)
11
(6) This must indicate that 6 expresses the differential along the tensile axis. According to Hart, eqn. (6) is the relationship between the differential of A and the differential of .~ under the load equilibrium condition at each step of the deformation. At the stable or quasistable stage of deformation, as the difference between the t w o sections (the homogeneous section and the neck section) is n o t great, the difference in 7 between these t w o sections (if ~, =n/e is accepted) is also n o t great; therefore, it can be considered that both the homogeneous section and the necking section (where there is an initial geometric defect and a neck will develop) have
te(1  ~ )
= exp,.
~n
}1
(11)
Equation (11) shows that, when there is a geometric defect f, the necking will develop b u t will be controlled by m and V. When failure occurs at eN + ¢¢ [3, 5, 6, 9], the strain in the homogeneous section to failure is obtained: m e~   In{1  (1 __f)11m} ~I
(12)
This is an m'yef relationship. For a power. law hardening material, the final 7 ( n/ef)
159
value is used in eqn. (12) as an "average" 3, value [5]. We obtain e~ = n   m ln{1   ( 1 _ f ) l l . , }
1f
CN = C 0 :
(13)
Because f is very small, we take only the first term of the Taylor series: (1  f)X/m
eqn. (20) is integrated beginning at e = 0 and
f e x p (   e N l m ' ) ' ) deN Go ef
(14)
,~ _ _ m
f
e x p (   e l   7d )e m
0
Corresponding to this approximation, we have a simpler expression
This gives
ef~n+
ef
m In(f)
(15)
If the contribution of the neck section to elongation is neglected, the total elongation (the engineering strain) for both eqn. (13) and eqn. (15) is ef = ( 1  (1  f)llm}m exp(n)  1
(16)
or
ef =
exp(n)  1
(17)
Since the error in the simplification of eqn. (14) is very small, it can be seen that the two equations, eqns. (16) and {17), are very similar.
2.2. Deformation defect For a deformation defect, instead of eqn. (2) we have [9, 10] tie =  8(ln A)
(18)
This equation was first used by Hart [2] and, according to Jonas et al. [10] and Ghosh [9], it is valid for a deformation defect (e.g. a hammer blow). All the derivations carried o u t above from eqn. (1) to eqn. (8) for a geometric defect are repeated here using eqn. (18) for a deformation defect. This gives dN =
(I

7)/rn
A_
(19)
Since we are considering a specimen without an initial geometric defect, we have exp(eN~)de
N = exp(e l~/)de
(20)
A deformation defect can be considered as a predeformation in a local region of the specimen, which is the initial place for necking. So
(21)

In 1   e x p
eo

(22)
or, with a powerlaw material, ef=nmln
1exp
eo
reef / )
Comparing eqn. (22) with eqn. (12), we find that the difference between the equation for a geometric defect and that for a deformation defect is that the former has a (1  f)l/m term and the latter an exp{e0(1   7 ) / m ) term. These are the relationships between the strain at fracture, the strain rate sensitivity coefficient m and the strainhardening exponent n for either a geometric defect or a deformation defect. Of course, the present analysis provides only the limiting strain to fracture outside the necked region which is not exactly the total elongation. However, if the ratio of the length to the diameter (or the width) of the specimen is large, the error between the limiting strain outside the neck and the total elongation can be ignored. The calculation of Lee and Zaverl [11] indicated that, although the necking strain is significant, it does not make a major change to the type of curve plotted by Woodford [12] (i.e. a loglog form).
2.3. A firstorder approximation analysis In the firstorder approximation analysis of Lin et al. there is a constant term in their equation which is chosen to be 67 in order to fit the experiments. Furthermore the constant has no exact physical meaning. Also, for the types of analysis e m p l o y e d by Nichols [5], Lin et al. [6] and Semiatin and Jonas [7], only the limiting strains to failure were given and they did n o t describe the process of necking development. So here we also try to derive
160
the type of analysis which not only gives the limiting strain to failure but also presents a description of necking development. From the equation given by Duncombe [13, 14],
~A ~Ao (Ao~(1~,/m A  A0 ~A]
(24)
We define 6Ao/Ao as an initial inhomogeneity (eqn. (9)) and 5A/A as an instantaneous inhomogeneity after tensile deformation of the homogeneous section from A0 to A. So we rewrite eqn. (22) as
@)(1


7)Ira
f(e) =
f
(25)
where f is defined by eqn. (9). f(e) can be expressed as 5A f(e) =  A
A  A N A = 1  (1  f) exp(e  eN)
(26)
So eqn. (25) becomes
1(1f)
exp(e eN) = e x p { e ( l m: 7 ) I f , (27)
This is an equation describing the process of necking development based on the firstorder approximation. The limiting strain at fracture can be obtained from eN + ~o: m In(;) ef  1 ')' and
(28)
;)m/(1  7) 1
ef =
Using 7  n/e~ we have
e~=n+ e~ =
m ln(;) exp(n)  1
(29) (30)
Equations (29) and (30) are the same as the modified Nichols result derived by Semiatin and Jonas [7]. However, in the above derivation the necking process can be described.
3. COMPARISON WITH OTHER THEORIES AND EXPERIMENTS
Table 1 shows the various relationships of theoretical analyses given by others and the present analysis. From the viewpoint of theoretical analysis, those of Hutchinson and Neale [4] or Ghosh and Ayres [3] are accurate integral analyses of the "longwavelength approximation" of one dimension. However, unfortunately, the resultant equation of Hutchinson and Neale is an integral one which cannot be solved in an analytical form. The analyses given by Nichols [5], Lin et al. [6], Semiatin and Jonas [7] or in the present work (Section 2.3) based on Hart's approach are a linearized or firstorder perturbation in which the necking p h e n o m e n o n is described by a gradient approach. The present analysis (Sections 2.1 and 2.2) is based on Hart's constitutive equation of differential form. The relation for uniaxial load equilibrium is used with two successive integrations, which makes the analysis more accurate than those of Nichols and Lin et al. Of course, some simplification was introduced in order to achieve an analytical expression. Figure 1 is plotted using eqn. (17) and eqn. (23) with n  0 and n = 0.2 labelled LB; Woodford's [12] experimental data (labelled W) and Hutchinson and Neale's theoretical results (labelled HN) are also plotted for comparison. From Fig. 1. we can define that, for m > 0.1, the strainhardening exponent n has a small influence on the elongation, and both of the theoretical lines are in agreem e n t with the experimental data of Woodford. For m < 0.1, the strainhardening effect has an increasing influence on total elongation and the lines for n  0 and n = 0.2 included most of the experimental data. Figure 1 shows that the present analysis gives the same result as that of Hutchinson and Neale when n = 0 (which is also the same as that given by Ghosh and Ayres), and for n  0.2 our analysis gives a good approximation to Hutchinson and Neale's integral analysis. When m is small (e.g. between 0.01 and 0.1), eqns. (17) and (23) do contain some errors in contrast with Hutchinson and Neale's results. The errors may arise from the assumption in the integrations of the above derivation that 7 is constant, especially in the integration of eqn. (10), which is in fact n o t true.
161 TABLE 1 Various relationships of theoretical analyses Relationship
Reference
Year
ef = {1  f)llm}rn _ 1
Ghosh and Ayres [3 ]
1976
Hutchinson and Neate [4]
1977
Nichols [5 ]
1980
Lin et al. [6]
1981
Semiatin and Jonas [ 7 ]; present work
1984
~ (sei)k
exp(seD
k=O
1 s=, m
 (1  (1 /,)s)
k!
n p=m
el=t7 ]
~
ef = (1 +
eo)~7rnl(1~)  1
77= 67 ef =
exp(n)  1
e f = {1  (1
_f)llrn}m
ei = ( f ) m "  exp(n) n
;
i
Present work Present work
1

rn
1
ein)t
]~Jl~ll
i
i
JJJll~
0.1
001
(n) 1
exp
I
Present work
i
,
z JJlltl
r
Ill,',
I
I
I I,,HI 1000
I
I
ef
%
o " ~t'N
Z
gg~
0.001
l
i
t IlIHI
I 10
i
iilllll 100
ill 10000
Fig. 1. The theoretical results o f the m  e l relationship calculated from eqns. (17) and (23) (Lian
Baudelet (LB)) and from Hutchinson and Neale's [4] analysis (HN) in comparison with the experimental data collected by Woodford (W) [12] (f = 0.005 for n = 0.2 (eqn. (23)); Co= 0.005). H o w e v e r , f o r a m a t e r i a l having a large strain r a t e sensitivity, t h e strain at f r a c t u r e is larger t h a n t h e s t r a i n  h a r d e n i n g e x p o n e n t n; f o r t h e d e f o r m a t i o n p r o c e s s w h e r e e > n, t h e variat i o n in 3' w i t h e b e c o m e s s m o o t h a n d in this
case t h e a s s u m p t i o n t h a t 7 is a c o n s t a n t is q u i t e r e a s o n a b l e . This is w h y , f o r large m , t h e results o f o u r analysis give q u i t e a g o o d a p p r o x i m a t i o n in c o m p a r i s o n w i t h H u t c h i n son and N e a l e ' s n u m e r i c a l results. T h e r e f o r e , w h e n m is small, t h e results o f p r e s e n t analysis c o n t a i n s o m e errors in c o m p a r i s o n w i t h t h e results o f H u t c h i n s o n a n d Neale b e c a u s e o f t h e a s s u m p t i o n . T h e a s s u m p t i o n t h a t 7 is c o n s t a n t has b e e n a c c e p t e d b y o t h e r researchers in t h e s e t y p e s o f analysis [ 5  7 , 1 5 ] . We a c c e p t this a p p r o x i m a t i o n in t h e a b o v e derivat i o n in o r d e r t o arrive at a simple a n a l y t i c a l e x p r e s s i o n w h i c h can, in general, give a g o o d d e s c r i p t i o n o f t h e e x p e r i m e n t a l t r e n d s prov i d e d b y W o o d f o r d . Since t h e r e are also o t h e r f a c t o r s (e.g. stress t r i a x i a l i t y , c a v i t y g r o w t h and structure change) which influence the n e c k i n g p r o c e s s , t h e small errors d u e to t h e a s s u m p t i o n in t h e p r e s e n t d e r i v a t i o n m a y n o t be serious. I t can also be seen t h a t o u r analysis is in a g r e e m e n t w i t h t h e classical analysis given b y C o n s i d e r e [16] f o r rn = 0. T h e results o f eqn. (30) are s h o w n in Fig. 2 w i t h t h e s a m e p a r a m e t e r c h o i c e as in Fig. 1.
162 1 I
I
+
ll,ltZ
I
I
I
i
l~J+ij
I
I
o
I
Jlllll
o
I
i
I
I 1'.++
I
1
t
'
oe
°° : 0.1
o o%
o
o°
o °
0.01
0.001
I
I
I IJlllJ
10
I
J
ltllill
100
Z
I
I l/Jill
1OO0
'
t
10000
erO/o Fig. 2. C o m p a r i s o n o f the m  e l r e l a t i o n s h i p calculated from eqn. (30) and the experimental data
collected by Woodford (W) [12] ( f  0.005).
Figure 2 shows t hat the firstorder approximation gives slightly larger total elongation than eqn. (17) and eqn. (23) do. F r o m comparison with W o o d f o r d's experimental data it seems th at the f o r m e r analysis fits the experiments a little better. Since there are also o t h e r factors which influence the necking of materials and W o o d f o r d collected data on only a limited n u m b e r o f alloys, we cannot draw any final conclusions a b o u t which is the bet t er equation. It should be indicated t hat the present analysis, like the ot her analyses [ 2 6, 9, 10, 17, 18], is also a longwavelength approximation in which threedimensional effects are neglected. According to several researchers [4, 17, 19], the longwavelength approximation analysis will underestimate the actual total elongation since the effect of stress triaxiality which causes neck propagation towards the ends o f a specimen is n o t considered here. Thus, necking proceeds more rapidly wh en the longwavelength approximation is invoked and the critical state should be attained earlier than in an analysis where threedimensional effects are included.
4. CONCLUSION
On the basis o f Hart's constitutive equation and using the longwavelength uniaxial stress a p p r o x i m a t i o n and load equilibrium relation, an analysis o f the necking d e v e l o p m e n t under
uniaxial tension was given. The relationships between the total elongation and the parameters of material including bot h the strain rate sensitivity rn and the strain hardening 7 or n were derived for both a geometric defect and a d e f o r m a t i o n defect as the initial imperfection The m  n  e ~ relationship for a geometric defect given in this paper is in agreement with the m  e ~ relationship of Ghosh and Ayres for n = 0 and with the classical analysis of Considere for m = 0. It is shown that the present analysis agrees well with the experimental data collected by Woodford. A modification o f the linearized approximation based on D u n c o m b e ' s form ul a was reconsidered; this n o t only presents a s i m p l e m  n  e ~ relationship (the same as t hat of Semiatin and Jonas) but also can describe the process o f necking development.
REFERENCES 1 W. A. Backofen, I. R. Turner and D. H. Avery, Am. Soc. Met. Trans. Q., 57 (1964) 980. 2 E. W. Hart, ActaMetall., 15 (1967) 351. 3 A. K. Ghosh and R. A. Ayres, Metall. Trans. A, 7 (1976) 1589. 4 J. W. Hutchinson and K. W. Neale, Acta Metall., 25 (1977) 839. 5 F.A. Nichols, Acta Metall., 28 (1980) 633. 6 I. H. Lin, J. P. Hirth and E. W. Hart, Acta Metall., 29 (1981) 819. 7 S. L. Semiatin and J. J. Jonas, Formability and Workability o f Metals, American Society for Metals, Metals Park, OH, 1984, p. 155. 8 E. W. Hart, Acta Metall., 18 (1970) 599. 9 A.K. Ghosh, Acta Metall., 25 (1977) 1413. 10 J. J. Jonas, R. A. Holt and C. E. Colemen, Acta Metall., 24 (1976) 911. 11 E. Lee and F. Zaverl, Acta Metall., 28 (1980) 1415. 12 D. A. Woodford, Am. Soc. Met. Trans. Q, 62 (1969) 291. 13 E. Duncombe, Int. J. Mech. Sci., 14 (1972) 325. 14 E. Duncombe, Int. J. Solids Struct., 10 (1974) 1444. 15 G. Ferron and M. MlihaTouati, Scr. Metall., 16 (1982) 911. 16 A. Considere, Ann. Ponts Ch. Ser. 6, 9 (1885) 574. 17 J. J. Jonas and B. Baudelet, Acta Metall., 25 (1977)43. 18 U. F. Kocks, J. J. Jonas and H. Mecking, Acta Metall., 27 (1979) 419. 19 C. G'Sell, N. A. AlyHelal and J. J. Jonas, J. Mater. Sci., 18 (1983) 1731.