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Strain hardening parameters of some low carbon manganese steels
Scripta METALLURGICA
Vol. II, pp. 405-409, P r i n t e d in the U n i t e d
1977 States
Pergamon
Press,
Inc
STRAIN HARDENING PARAMETERS OF SOME LOU CARBON MANGANESE STEELS
R. Vetter, A.R. ~Jachters and P. Jongenburger
Laboratory of Metallurgy, Delft University of Technology Rotterdamseweg 137, Delft, The Netherlands ~Received March
8, 1977)
Introduction The shape o f the s t r e s s - s t r a i n curve is o f p r a c t i c a l importance f o r deep-drawing t e c h n i q u e s . I t would t h e r e f o r e be u s e f u l i f i t were p o s s i b l e to c o n t r o l t h i s shape r a t h e r than having to accept i t as an e m p l r l c a l f a c t . I t is a l r e a d y known t h a t the g r a i n s i z e i n f l u e n c e s m a l n l y the f r i c t i o n s t r e s s , w h i l e the s t r a i n hardening is governed by the d i s l o c a t l o n d e n s i t y i n c r e a s e . A model has r e c e n t l y been proposed [1] which v e r y w e l l describes the dependence o f d i s l o c a t l o n d e n s i t y on s t r a i n . In the p r e s e n t l e t t e r an a t t e m p t is made to c o r r e l a t e the s t r a i n hardening parameters o f t h i s model w i t h the v a r i a t l o n o f manganese c o n t e n t o f low carbon s t e e l s . Curve fitting is used in o r d e r to see whether i t is p o s s i b l e to r e f i n e the assumption made in the rhode1 t h a t the d l s l o c a t i o n mean f r e e path is c o n s t a n t . I t is i n v e s t i g a t e d whether curve f i t t i n g is a d m i s s i b l e w i t h o u t a c t u a l d l s l o c a t i o n d e n s i t y measurements. Models The Ludwik-Hollomon r e l a t i o n
between t r u e s t r e s s a and t r u e s t r a i n
a = alo. + K n ,
c, (I)
makes i t p o s s i b l e to c h a r a c t e r i z e a s t r e s s - s t r a i n curve o f bcc p o l y c r y s t a l s by t h r e e e m p i r i c a l constants a i o , K and n, but gives no i n s i g h t i n t o the p h y s i c a l mechanisms u n d e r l y i n g the s t r e s s s t r a i n c u r v e s . A semi-phenomenological d e s c r i p t i o n o f these mechanisms, in terms o f the b e h a v i o u r o f the mean d i s l o c a t i o n d e n s i t y , was proposed by BergstrSm and c o l l e a g u e s in a s e r i e s o f papers s t a r t i n g w i t h r e f . [ 1 ] . T h e i r s t r e s s - s t r a i n r e l a t i o n reads
a(~) = aio + ~Gb {~(I - e -~) + Po e-~E} ½,
(2)
where ~ is a constant, G the glide modulus, b the Burgers vector, U a parameter describing dislocation production rates, ~ the probability of remobilization of immobilized dislocations, Po the initial density of dislocations. The relation (2) is based on the general equation o = a.I O + ~Gb/ p.
(3)
Eq. (2) can be m o d i f i e d by t a k i n g the d i s l o c a t i o n mean f r e e path i n v e r s e l y p r o p o r t i o n a l d e n s i t y , i n s t e a d o f a c o n s t a n t p a t h . This leads ( V e t t e r and van den Beukel [ 2 ] ) to A 0 = a l o + ~Gb { ~ (I - e ' ½ ~ ) +
po ½ e - ½ ~ } ,
405
(4)
t o the
406
STRAIN HARDENING
OF LOW CARBON STEELS
Vol.
II, No.
5
where a new constant~ A, replaces the old U. The d i s l o c a t i o n balance model was used by BergstrSm and Roberts [3] to deduce a r e l a t i o n between the parameters Olo, U and ;; on the one hand and the elongation to necking parameter nL of the Ludwik expression nL o = KL c
(5)
on the other hand. Equation (5) has no physical meaning; for deep-drawing purposes i t is important to obtain high values of nL, According to ref. [3] a reduction, of ~ with o.= .and U kept constant would be the most promising. I t is not known how to manipulate the remo~=lization p r o b a b i l i t y ~. The present authors therefore t r i e d to study the influence of the manganese content o~ the hardening parameters.
[
MPa Soo
400
<~
t
~
.
.
/
/
.
3ooi/'-~ ~
o
.j
4
FIG. 1. True s t r e s s - s t r a i n curves of some low carbon manganese steels. The manganese increases in the order in which the curves are numbered ( c i r c l e s indicate the stresses ca]culated from d i s l o c a t i o n densities and regression parameters, see t e x t ) .
3
i/
o,1
o.~ Experimental d e t a i l s
Four steels with the compositions given in Table I were prepared s t a r t i n g from Armco iron. The important v a r i a b l e is the manganese content, which is approximately 0, 0.5, 1 and 2~ by weight. TABLE I. Composition of four low carbon steels in percent by weight. Haterial
C
S[
1 2 3 4
0.036 0.042 0.044 0.065
0.010 0.025 0.024 0.037
Hn 0.038 0.39 1.02 2.06
P
S
0.013 0.009 0.010 0.008
0.011 0.013 0.009 0.009
grain size 23 pm 25 22 10
After homogenizing f o r 24 hours at 1050°C the ingots were h o t - r o l l e d to slabs 8 mm thick. From these specimens with dimensions 5 x 8 x 50 mm were m i l l e d , with enlarged ends, for better gripping. A f t e r this treatment the specimens were normalized f o r h a l f an hour at 950°C, a f t e r which the microstructure was checked. The t e n s i l e tests were performed on an Instron testing machine, with a cross head v e l o c i t y of = 3 x 10-~ s =1. Some tests were stopped at a chosen s t r a i n and the specimens were then reduced by spark erosion and chemical etching to small discs for observation by electron microscopy. Employing an advanced algorithm ( r e f . [ 4 ] ) , using on average 30 points of each stress = s t r a i n curve, least=squares f i t s of the expressions (1), (2) and (4) were found by varying for each expression 3 appropriate hardening parametersj which were respectively (Oio , K, n), (o.I , a G b / U, ~) and (Oio , aGbA, ~) In ~qs. (2) and (4) Po was set equal to zero, since for well-annealed materia]s Po is generally of the order of 1012 m-2 ( B e r g s t r ~ and Aronsson [ 5 ] ) . Therefore i t was thought that its contribution could safely be neglected fo r strains beyond 2~.
Vol, 11, No, 5
STRAIN HARDENING OF LOW CARBON STEELS
407
Results In Fig. I four representative stress-strain curves are p]otted corresponding to the four Mn contents given in Table I. From the intersection of the stress-strain curve and the plot of its derivative the elongation to necking En was obtained. The dependenceof this parameter on the manganese content is plotted in Fig. 2, where the hardening parameters of Eq. (2) are also shown. Fig. 3 gives the dislocation density, obtained by electron microcopy, of the steels wlth the highest and lowest manganese contents (materials I and 4). x8o. NPa
o
~
8
/
~
6 g4l
leo
0
~'
To"
o
)~
,
k
LZL)~
o
i
7oe /
i
3
OIM,,I i
•
,
i
FIG. 2. S t r a i n hardening parameters as a f u n c t i o n of manganese c o n t e n t . a. b c d.
E l o n g a t i o n to necking En. (Dotted c i r c l e : c o r r e c t e d f o r g r a i n s i z e ) . F r i c t i o n s t r e s s o~^. (Dotted c i r c l e : c o r r e c t e d f o r g r a i n s i z e ) . Steepness p a r a m e t ~ ~Gb/ U. R e m o b i l i z a t i o n parameter G.
T.
lOC,~~.
FIG. 3. .
•
o.o5
o~
~
,
. ~15
D i s l o c a t i o n d e n s i t y as a f u n c t i o n o f t r u e s t r a i n f o r the h i g h e s t and lowest manganese contents (upper and tower curve r e s p e c t i v e l y ) .
408
STRAIN HARDENING OF LOW CARBON STEELS
Vol, ii, No, S
Discussion Although there is c o n s i d e r a b l e s c a t t e r , a l l 4 parameters o f Fig. 2 show manganese dependence. In o r d e r t o apply a c o r r e c t i o n to c n f o r the d e v i a t i n g g r a i n s i z e o f the 2~ Mn m a t e r i a l , i t was assumed t h a t the corresponding curve, i n c l u d i n g i t s Oio , had u n i f o r m l y risen by 50 MPa in accordance w i t h a Hall-Perch f l o w constant o f about kf = 15 MPa mm½. The c o r r e c t e d Oio is i n d i c a t e d by an arrow in Fig. 2b. From the s t r e s s - s t r a i n curve and i t s d e r i v a t i v e , using the assumed s h i f t o f o, a c o r r e c t e d value o f c n o f about 0.0175 can be found. This is i n d i c a t e d by a d o t t e d c i r c l e in Fig. 2a. Thus the net e f f e c t o f Mn on cn is A~n/~[Mn ] = -0.028 per wt~ Mn. This is much less than the value o f - 0.044 per wt~ Mn t h a t can be c a l c u l a t e d from the data o f Masui e t a l . using Eq. 6 o f r e f . [ 6 ] . T h e i r data were, however, based on a s m a l l e r i n t e r v a l o f 0.5 wt~ Mn. In Fig. 2 only the hardening parameters o f Eq. (2) are shown. The parameters o f Eqs. (1) and (4) show s i m i l a r behaviour. In the case o f copper the assumption o f constant mean f r e e path f o r d i s l o c a t i o n s is known to be inadequate, but t a k i n g in to be i n v e r s e l y p r o p o r t i o n a l to the d e n s i t y , Eq. (4) was found by V e t t e r and van den Beukel [2] to y i e l d a good f i t and they have shown t h a t the same mlght apply to i r o n . This is confirmed by the present study, the standard d e v i a t i o n being only s l i g h t l y l a r g e r than w i t h the f i t t i n g o f the unmodified d i s l o c a t i o n balance model o f Bergstr~rn (Eq. ( 2 ) ) . The d i f f e r e n c e is not large enough to consider i t as an argument in favour o f one o f the models. More i n f o r m a t i o n about d i s l o c a t i o n d e n s i t i e s is needed in o r d e r to s e t t l e t h i s q u e s t i o n , but at present i t is u n l i k e l y t h a t the counting o f d i s l o c a t i o n s by e l e c t r o n microscopy w i l l y i e l d the r e q u i r e d accuracy. Nevertheless, d e n s i t y countings were c a r r i e d out since i t was f e l t t h a t one should not apply Eq. ( 2 ) , as sometimes is done, in complete ignorance o f the actual d e n s i t i e s - even when i t was found to f i t to the curves w i t h a standard d e v i a t i o n o f t e n as small as 0.4 MPa. (This f i g u r e is to be compared w i t h the + 3 MPa claimed by Bergstr~m et a l . , e . g . r e f . [ 7 ] ) . From a p l o t o f / p ( ~ ) vs o(~) one can o b t a i n a(Eq. ( 3 ) ) . The s t r e s s ~ could not be measured e i t h e r at ¢ = O, i . e . at the p o i n t Po, o r a t ~-values less than the Luders s t r a i n . No p o i n t s could t h e r e f o r e be found in Fig. 4 a t /p s m a l l e r than 10 x 106 m-1. E x t r a p o l a t i o n to zero t h e r e f o r e had to be c a r r i e d out somewhat s p e c u l a t i v e l y . In s p i t e o f t h i s the r e s u l t s compare reasonably w e l l w i t h the l i t e r a t u r e (see r e f . [ 8 ] ) on pure iron (Fig. 5) where o i o v a r i e s between -20 and +60 MPa. From Fig. 4 the values o f a are 1.3 and 1.4 f o r 0 and 2~ Mn r e s p e c t i v e l y , corresponding to ~ i o - v a l u e s o f about 0 and 80 MPa. For pure iron ~ ranges from 0.76 to 1.4 ( r e f s . [9] and [ 1 0 ] . The present values o f ~ t h e r e f o r e seem reasonable.
5oQ
-
FIG. 4. Square root o f d i s l o c a t i o n d e n s i t y of Fig. 3 ( c i r c l e s ) p l o t t e d a g a i n s t true s t r e s s . (Crosses i n d i c a t e s t r e s s a t zero s t r a i n , see t e x t ) ,
zN
/
/
/
~p _~
e
o
.o~
/
~
le
1'$ x,e~ ;'
Vol.
II,
No.
5
STRAIN
HARDENING
OF LOW
CARBON
STEELS
409
When the measured dislocation densities are substituted in Eq. (3) with the values of the parameters o|m and ~Gb as found from the parameter fitting (Fig. 2), U being taken as 2 x 1015 m -L (ref. [I~) calculated stresses can be obtained (small circles in Fig. I). In the case of 2% Mn steel in particular the calculated points lay off the curve seriously in excess of the error limits. In Fig. 3, however, a non-negligible grown-in dislocation density was present, which is generally not expected in well-annealed materials (compare ref. [5]) and this could account quite well for the discrepancy. ~/hen in the case of 2% Mn an experimentally found constant term Po = 29 x 1012 m is introduced in Eq. (2) but its exponential dependence left out following Roberts and Bergstr6m [11], regression yields o. = 80 MPa. This fits well with Fig. 4. From this follows a stress at zero strain of o = 220 ~ a , indicated by a cross in Fig. 4. The standard deviation of the fitting procedure is now not so low as in the case where Po would be zero, although it is still quite acceptable. This means that Po could not have been found simply by curve fitting without microscopic density measurements. Hhen o(~) is now calculated from / p with the new values of Oio and ~Gb, the points coincide very well with the stress-strain curve (Fig. I, dotted circles). In a similar way the coincidence for the points on the 0% Mn curve could be improved. As a consequence of this the uncorrected values of Oio (Fig. 2) are considered to be too high by about 90 MPa. The influence on ~ is less dramatic: it rises by 1.5 in the case of 2% Mn, which is about equal to the order of accuracy obtainable in this three-parameter fitting. The influence on U cannot be calculated, as parameter fitting yields only the product ~Gb/ U, but the absolute value of U can indeed be obtained once ~ is known: U = 2.5 x 1015 m "2 in the case of 2% Mn. This compares well wi~h Reid et el. (ref. [12]) where data collected for iron range from U I x up to 3 x 1015 m-- per unit strain. Acknowle.d~eme.nt The authors are indebted to Dr. H. Nieswaag for preparing the specimen materials and to Mr. C.D. de Haan for performing the dislocation density measurements.
300 MPa
Ta//~ lOO / o
FIG.
2oc
5.
The linear relation between stress and square root of dislocation density of pure iron found by different authors according to ref~ [8]. a. Ba~lon e t a ] . b. Dingley et al. c. Keh et el. d. Tanaka et el.
/ ,
\P
D
References I. 2. 3. 4. 56. 78. 9.
Y. BergstrSm, Mater. Sci. Eng. ~ (1969/1970) 193. R. Vetter and Ao van den Beukel, Scripta Met. l] (]977) 143. Y. Bergstr6m and W. Roberts, Scripta Met. 5 (I~-I) 459. R. Fletcher and M.J.D. Powell, Computer Jo'urnal 6 (1963) 163. Y. BergstrSm and B. Aronsson, Metallurg. T. 3 (1972) 1951. H. Masui, M. Kawaharada and H. Takechi, TranSactions ISIJ 15 (1975) 553. W. Roberts and Y. Bergs tr6m, Z. Metallkde 62 (1971) 752. K. Tanaka and T. Watanabe, Jap. J. Appl. P'~s. 11 (1972) 1429. A.S. Keh and S. Weissmann, in G. Thomas and J. Wa'--shburn (eds.), Electron microscopy and strength of crystals, Interscience 1963, p. 231. 10. D.J. Dingley and D. McLean, Acta Met. 15 (1967) 885. 11. W. Roberts, S. Karlsson and Y. Bergstr~, Mater. Sci. Eng. 11 (1973) 247. 12. C.N. Reid, A. Gilbert and A.R. Rosenfield, Phil. Mag. 12 (19-~'5) 409.
Vol. II, pp. 405-409, P r i n t e d in the U n i t e d
1977 States
Pergamon
Press,
Inc
STRAIN HARDENING PARAMETERS OF SOME LOU CARBON MANGANESE STEELS
R. Vetter, A.R. ~Jachters and P. Jongenburger
Laboratory of Metallurgy, Delft University of Technology Rotterdamseweg 137, Delft, The Netherlands ~Received March
8, 1977)
Introduction The shape o f the s t r e s s - s t r a i n curve is o f p r a c t i c a l importance f o r deep-drawing t e c h n i q u e s . I t would t h e r e f o r e be u s e f u l i f i t were p o s s i b l e to c o n t r o l t h i s shape r a t h e r than having to accept i t as an e m p l r l c a l f a c t . I t is a l r e a d y known t h a t the g r a i n s i z e i n f l u e n c e s m a l n l y the f r i c t i o n s t r e s s , w h i l e the s t r a i n hardening is governed by the d i s l o c a t l o n d e n s i t y i n c r e a s e . A model has r e c e n t l y been proposed [1] which v e r y w e l l describes the dependence o f d i s l o c a t l o n d e n s i t y on s t r a i n . In the p r e s e n t l e t t e r an a t t e m p t is made to c o r r e l a t e the s t r a i n hardening parameters o f t h i s model w i t h the v a r i a t l o n o f manganese c o n t e n t o f low carbon s t e e l s . Curve fitting is used in o r d e r to see whether i t is p o s s i b l e to r e f i n e the assumption made in the rhode1 t h a t the d l s l o c a t i o n mean f r e e path is c o n s t a n t . I t is i n v e s t i g a t e d whether curve f i t t i n g is a d m i s s i b l e w i t h o u t a c t u a l d l s l o c a t i o n d e n s i t y measurements. Models The Ludwik-Hollomon r e l a t i o n
between t r u e s t r e s s a and t r u e s t r a i n
a = alo. + K n ,
c, (I)
makes i t p o s s i b l e to c h a r a c t e r i z e a s t r e s s - s t r a i n curve o f bcc p o l y c r y s t a l s by t h r e e e m p i r i c a l constants a i o , K and n, but gives no i n s i g h t i n t o the p h y s i c a l mechanisms u n d e r l y i n g the s t r e s s s t r a i n c u r v e s . A semi-phenomenological d e s c r i p t i o n o f these mechanisms, in terms o f the b e h a v i o u r o f the mean d i s l o c a t i o n d e n s i t y , was proposed by BergstrSm and c o l l e a g u e s in a s e r i e s o f papers s t a r t i n g w i t h r e f . [ 1 ] . T h e i r s t r e s s - s t r a i n r e l a t i o n reads
a(~) = aio + ~Gb {~(I - e -~) + Po e-~E} ½,
(2)
where ~ is a constant, G the glide modulus, b the Burgers vector, U a parameter describing dislocation production rates, ~ the probability of remobilization of immobilized dislocations, Po the initial density of dislocations. The relation (2) is based on the general equation o = a.I O + ~Gb/ p.
(3)
Eq. (2) can be m o d i f i e d by t a k i n g the d i s l o c a t i o n mean f r e e path i n v e r s e l y p r o p o r t i o n a l d e n s i t y , i n s t e a d o f a c o n s t a n t p a t h . This leads ( V e t t e r and van den Beukel [ 2 ] ) to A 0 = a l o + ~Gb { ~ (I - e ' ½ ~ ) +
po ½ e - ½ ~ } ,
405
(4)
t o the
406
STRAIN HARDENING
OF LOW CARBON STEELS
Vol.
II, No.
5
where a new constant~ A, replaces the old U. The d i s l o c a t i o n balance model was used by BergstrSm and Roberts [3] to deduce a r e l a t i o n between the parameters Olo, U and ;; on the one hand and the elongation to necking parameter nL of the Ludwik expression nL o = KL c
(5)
on the other hand. Equation (5) has no physical meaning; for deep-drawing purposes i t is important to obtain high values of nL, According to ref. [3] a reduction, of ~ with o.= .and U kept constant would be the most promising. I t is not known how to manipulate the remo~=lization p r o b a b i l i t y ~. The present authors therefore t r i e d to study the influence of the manganese content o~ the hardening parameters.
[
MPa Soo
400
<~
t
~
.
.
/
/
.
3ooi/'-~ ~
o
.j
4
FIG. 1. True s t r e s s - s t r a i n curves of some low carbon manganese steels. The manganese increases in the order in which the curves are numbered ( c i r c l e s indicate the stresses ca]culated from d i s l o c a t i o n densities and regression parameters, see t e x t ) .
3
i/
o,1
o.~ Experimental d e t a i l s
Four steels with the compositions given in Table I were prepared s t a r t i n g from Armco iron. The important v a r i a b l e is the manganese content, which is approximately 0, 0.5, 1 and 2~ by weight. TABLE I. Composition of four low carbon steels in percent by weight. Haterial
C
S[
1 2 3 4
0.036 0.042 0.044 0.065
0.010 0.025 0.024 0.037
Hn 0.038 0.39 1.02 2.06
P
S
0.013 0.009 0.010 0.008
0.011 0.013 0.009 0.009
grain size 23 pm 25 22 10
After homogenizing f o r 24 hours at 1050°C the ingots were h o t - r o l l e d to slabs 8 mm thick. From these specimens with dimensions 5 x 8 x 50 mm were m i l l e d , with enlarged ends, for better gripping. A f t e r this treatment the specimens were normalized f o r h a l f an hour at 950°C, a f t e r which the microstructure was checked. The t e n s i l e tests were performed on an Instron testing machine, with a cross head v e l o c i t y of = 3 x 10-~ s =1. Some tests were stopped at a chosen s t r a i n and the specimens were then reduced by spark erosion and chemical etching to small discs for observation by electron microscopy. Employing an advanced algorithm ( r e f . [ 4 ] ) , using on average 30 points of each stress = s t r a i n curve, least=squares f i t s of the expressions (1), (2) and (4) were found by varying for each expression 3 appropriate hardening parametersj which were respectively (Oio , K, n), (o.I , a G b / U, ~) and (Oio , aGbA, ~) In ~qs. (2) and (4) Po was set equal to zero, since for well-annealed materia]s Po is generally of the order of 1012 m-2 ( B e r g s t r ~ and Aronsson [ 5 ] ) . Therefore i t was thought that its contribution could safely be neglected fo r strains beyond 2~.
Vol, 11, No, 5
STRAIN HARDENING OF LOW CARBON STEELS
407
Results In Fig. I four representative stress-strain curves are p]otted corresponding to the four Mn contents given in Table I. From the intersection of the stress-strain curve and the plot of its derivative the elongation to necking En was obtained. The dependenceof this parameter on the manganese content is plotted in Fig. 2, where the hardening parameters of Eq. (2) are also shown. Fig. 3 gives the dislocation density, obtained by electron microcopy, of the steels wlth the highest and lowest manganese contents (materials I and 4). x8o. NPa
o
~
8
/
~
6 g4l
leo
0
~'
To"
o
)~
,
k
LZL)~
o
i
7oe /
i
3
OIM,,I i
•
,
i
FIG. 2. S t r a i n hardening parameters as a f u n c t i o n of manganese c o n t e n t . a. b c d.
E l o n g a t i o n to necking En. (Dotted c i r c l e : c o r r e c t e d f o r g r a i n s i z e ) . F r i c t i o n s t r e s s o~^. (Dotted c i r c l e : c o r r e c t e d f o r g r a i n s i z e ) . Steepness p a r a m e t ~ ~Gb/ U. R e m o b i l i z a t i o n parameter G.
T.
lOC,~~.
FIG. 3. .
•
o.o5
o~
~
,
. ~15
D i s l o c a t i o n d e n s i t y as a f u n c t i o n o f t r u e s t r a i n f o r the h i g h e s t and lowest manganese contents (upper and tower curve r e s p e c t i v e l y ) .
408
STRAIN HARDENING OF LOW CARBON STEELS
Vol, ii, No, S
Discussion Although there is c o n s i d e r a b l e s c a t t e r , a l l 4 parameters o f Fig. 2 show manganese dependence. In o r d e r t o apply a c o r r e c t i o n to c n f o r the d e v i a t i n g g r a i n s i z e o f the 2~ Mn m a t e r i a l , i t was assumed t h a t the corresponding curve, i n c l u d i n g i t s Oio , had u n i f o r m l y risen by 50 MPa in accordance w i t h a Hall-Perch f l o w constant o f about kf = 15 MPa mm½. The c o r r e c t e d Oio is i n d i c a t e d by an arrow in Fig. 2b. From the s t r e s s - s t r a i n curve and i t s d e r i v a t i v e , using the assumed s h i f t o f o, a c o r r e c t e d value o f c n o f about 0.0175 can be found. This is i n d i c a t e d by a d o t t e d c i r c l e in Fig. 2a. Thus the net e f f e c t o f Mn on cn is A~n/~[Mn ] = -0.028 per wt~ Mn. This is much less than the value o f - 0.044 per wt~ Mn t h a t can be c a l c u l a t e d from the data o f Masui e t a l . using Eq. 6 o f r e f . [ 6 ] . T h e i r data were, however, based on a s m a l l e r i n t e r v a l o f 0.5 wt~ Mn. In Fig. 2 only the hardening parameters o f Eq. (2) are shown. The parameters o f Eqs. (1) and (4) show s i m i l a r behaviour. In the case o f copper the assumption o f constant mean f r e e path f o r d i s l o c a t i o n s is known to be inadequate, but t a k i n g in to be i n v e r s e l y p r o p o r t i o n a l to the d e n s i t y , Eq. (4) was found by V e t t e r and van den Beukel [2] to y i e l d a good f i t and they have shown t h a t the same mlght apply to i r o n . This is confirmed by the present study, the standard d e v i a t i o n being only s l i g h t l y l a r g e r than w i t h the f i t t i n g o f the unmodified d i s l o c a t i o n balance model o f Bergstr~rn (Eq. ( 2 ) ) . The d i f f e r e n c e is not large enough to consider i t as an argument in favour o f one o f the models. More i n f o r m a t i o n about d i s l o c a t i o n d e n s i t i e s is needed in o r d e r to s e t t l e t h i s q u e s t i o n , but at present i t is u n l i k e l y t h a t the counting o f d i s l o c a t i o n s by e l e c t r o n microscopy w i l l y i e l d the r e q u i r e d accuracy. Nevertheless, d e n s i t y countings were c a r r i e d out since i t was f e l t t h a t one should not apply Eq. ( 2 ) , as sometimes is done, in complete ignorance o f the actual d e n s i t i e s - even when i t was found to f i t to the curves w i t h a standard d e v i a t i o n o f t e n as small as 0.4 MPa. (This f i g u r e is to be compared w i t h the + 3 MPa claimed by Bergstr~m et a l . , e . g . r e f . [ 7 ] ) . From a p l o t o f / p ( ~ ) vs o(~) one can o b t a i n a(Eq. ( 3 ) ) . The s t r e s s ~ could not be measured e i t h e r at ¢ = O, i . e . at the p o i n t Po, o r a t ~-values less than the Luders s t r a i n . No p o i n t s could t h e r e f o r e be found in Fig. 4 a t /p s m a l l e r than 10 x 106 m-1. E x t r a p o l a t i o n to zero t h e r e f o r e had to be c a r r i e d out somewhat s p e c u l a t i v e l y . In s p i t e o f t h i s the r e s u l t s compare reasonably w e l l w i t h the l i t e r a t u r e (see r e f . [ 8 ] ) on pure iron (Fig. 5) where o i o v a r i e s between -20 and +60 MPa. From Fig. 4 the values o f a are 1.3 and 1.4 f o r 0 and 2~ Mn r e s p e c t i v e l y , corresponding to ~ i o - v a l u e s o f about 0 and 80 MPa. For pure iron ~ ranges from 0.76 to 1.4 ( r e f s . [9] and [ 1 0 ] . The present values o f ~ t h e r e f o r e seem reasonable.
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FIG. 4. Square root o f d i s l o c a t i o n d e n s i t y of Fig. 3 ( c i r c l e s ) p l o t t e d a g a i n s t true s t r e s s . (Crosses i n d i c a t e s t r e s s a t zero s t r a i n , see t e x t ) ,
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Vol.
II,
No.
5
STRAIN
HARDENING
OF LOW
CARBON
STEELS
409
When the measured dislocation densities are substituted in Eq. (3) with the values of the parameters o|m and ~Gb as found from the parameter fitting (Fig. 2), U being taken as 2 x 1015 m -L (ref. [I~) calculated stresses can be obtained (small circles in Fig. I). In the case of 2% Mn steel in particular the calculated points lay off the curve seriously in excess of the error limits. In Fig. 3, however, a non-negligible grown-in dislocation density was present, which is generally not expected in well-annealed materials (compare ref. [5]) and this could account quite well for the discrepancy. ~/hen in the case of 2% Mn an experimentally found constant term Po = 29 x 1012 m is introduced in Eq. (2) but its exponential dependence left out following Roberts and Bergstr6m [11], regression yields o. = 80 MPa. This fits well with Fig. 4. From this follows a stress at zero strain of o = 220 ~ a , indicated by a cross in Fig. 4. The standard deviation of the fitting procedure is now not so low as in the case where Po would be zero, although it is still quite acceptable. This means that Po could not have been found simply by curve fitting without microscopic density measurements. Hhen o(~) is now calculated from / p with the new values of Oio and ~Gb, the points coincide very well with the stress-strain curve (Fig. I, dotted circles). In a similar way the coincidence for the points on the 0% Mn curve could be improved. As a consequence of this the uncorrected values of Oio (Fig. 2) are considered to be too high by about 90 MPa. The influence on ~ is less dramatic: it rises by 1.5 in the case of 2% Mn, which is about equal to the order of accuracy obtainable in this three-parameter fitting. The influence on U cannot be calculated, as parameter fitting yields only the product ~Gb/ U, but the absolute value of U can indeed be obtained once ~ is known: U = 2.5 x 1015 m "2 in the case of 2% Mn. This compares well wi~h Reid et el. (ref. [12]) where data collected for iron range from U I x up to 3 x 1015 m-- per unit strain. Acknowle.d~eme.nt The authors are indebted to Dr. H. Nieswaag for preparing the specimen materials and to Mr. C.D. de Haan for performing the dislocation density measurements.
300 MPa
Ta//~ lOO / o
FIG.
2oc
5.
The linear relation between stress and square root of dislocation density of pure iron found by different authors according to ref~ [8]. a. Ba~lon e t a ] . b. Dingley et al. c. Keh et el. d. Tanaka et el.
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References I. 2. 3. 4. 56. 78. 9.
Y. BergstrSm, Mater. Sci. Eng. ~ (1969/1970) 193. R. Vetter and Ao van den Beukel, Scripta Met. l] (]977) 143. Y. Bergstr6m and W. Roberts, Scripta Met. 5 (I~-I) 459. R. Fletcher and M.J.D. Powell, Computer Jo'urnal 6 (1963) 163. Y. BergstrSm and B. Aronsson, Metallurg. T. 3 (1972) 1951. H. Masui, M. Kawaharada and H. Takechi, TranSactions ISIJ 15 (1975) 553. W. Roberts and Y. Bergs tr6m, Z. Metallkde 62 (1971) 752. K. Tanaka and T. Watanabe, Jap. J. Appl. P'~s. 11 (1972) 1429. A.S. Keh and S. Weissmann, in G. Thomas and J. Wa'--shburn (eds.), Electron microscopy and strength of crystals, Interscience 1963, p. 231. 10. D.J. Dingley and D. McLean, Acta Met. 15 (1967) 885. 11. W. Roberts, S. Karlsson and Y. Bergstr~, Mater. Sci. Eng. 11 (1973) 247. 12. C.N. Reid, A. Gilbert and A.R. Rosenfield, Phil. Mag. 12 (19-~'5) 409.