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Upper yield stress increase during strain ageing of α-iron
ScriptaMETALLURGICA
Vol. 3, pp. 865-870, 1969 Printed in the United States
Pergamon
Press,
UPPER YIELD STRESS INCREASE DURING STRAIN AGEING OF G-IRON Y. BergstrGm and 0o Vingsbo Institute of Physics, University of Uppsala, Uppsala, Sweden
(Received
September
29,
1969)
Introduction The resolved shear stress ~ in a dislocation glide plane can be separated into components according to T = T~ + T i + Td
(1)
The effective resolved shear stress T~ is the temperature dependent stress, required to force the dislocation through stress fields of short range. At a constant temperature it depends on the shear strain rate 9 and the density L of mobile dislocations according to I •~
(2)
= c(~) ~*-
where m* is the dislocation velocity exponent and C is a material constant ~. The formation of Cot%tell and/or Snoek atmospheres during strain ageing causes a decrease in the mobile dislocation density, and a corresponding increase in T~ according to eq. (2). Ti is the stress representing long range stress fields of strain independent distribution, and is not supposed to be affected by strain ageing. T d ~ / p f is the stress necessary to overcome dislocation-dislocation long range interaction. It is strain dependent through the dislocation density pf in tangled regions, and represents the work hardening, pf is not mobile, and Td is strain ageing sensitive only after so long ageing times that a redistribution of the dislocation structure can take place.
I ,
m @
is obtained by substituting the Johnston-Gilman (I) relation
~'= (T_~)m* Tu
into the strain rate expression ~ = bL~, where b is the length of the Burgers vector, ~ is the average dislocation velocity and ~u is the stress giving unit velocity.
865
Inc
866
STRAIN A G E I N G OF a - I R O N
Vol.
3, No.
The purpose of the present work has been to derive the time dependence of the increase in upper yield stress AT during strain ageing of ~-iron, in terms of dislocation locking parameters, i.e. under such conditions that A~Ct)
=
AT~(t)
Deriva_tion_of__the_U~?er Yield Stress IncreaSe as a Function of Ageing Time At the beginning of a tensile test of a strain aged iron specimen (or a very perfect crystal)~ the effective stress T~-=O corresponding to the upper yield point, is very high because of the low density of mobile dislocations. A rapid dislocation multiplication then induces the yield drcp~ during which T z- falls to a strain independent value T~, corresponding to a constant mobile dislocation length Lo. T~ has been measured in ~-iron by Michalak (2), and was found to depend on strain rate according to eq. (2), but to be constant in strain. The way in which L approaches Lo has been discussed by BergstrSm and Vingsbo (5). Because of the inhomogeneous deformation during the LGders strain, howeverf the present work will deal with prestrains ¢ > CLGd. ~ so that at the beginning of the ageing pause of a strain ageing test L = Lo in all grains~ and I 7-=
~* --
c(~) m*
(5)
Assume that the fraction g of the total dislocation density p that has been
locked by a t m o s p h e r e s , i s a rune%ion of a g e i n g
time
t . The i n c r e a s e i n y i e l d s%~ess
AT~~ then is a function of t according to m*
,,~*(t) = ~(t)
.t ] - 7. = c {V.oE1-~t)']
- T.* = ~* {[~-¢(t)]
m4~
- ~]
(4)
The main difficulty in finding an expression for the function g(t) is to decide, when a certain length of initially glissile dislocation shall be considered as locked. Let us make the simplifying assumption that an average of Z interstitial atoms per atomic plane cut by a dislocation, are needed to prevent the dislocation from escaping from its atmosphere under an applied stress. The number n of locking interstitials in atmospheres after an ageing time t then is
n(t)
z.~t).o
(5)
d
where d is the mean distance between two atomic planes, threaded by a dislocation. n(t) can be estimated from the equation suggested by Harper (4) to describe the diffusion of interstitial atoms in the long range stress fields of dislocations
In[1 -
-~n(t)]=
ADt 2/5
- e'p(~-)
(6)
ii
Vol.
3, No.
ii
S T R A I N A G E I N G OF a - I R O N
867
where no is the initial number of solute interstitials per unit volume, ~ is a numerical constant, A is a constant determined from the lattice strain due to one solute atom, D is the diffusion coefficient for the interstitial in m~trix, k is Boltzmann's constant and T is the ageing temperature. In spite of the fact that this equation does not take into account the development of concentration gradients during the diffusion process, it is found to be valid as an acceptable approximation, which is in good agreement with experimental observations. Substitution of (6) and (5) into (4) then gives the time dependence of the increase in yield stress. I nod - m-q.j &T(t) = ~ - ~{1 -~.p. [1 - exp(- Kpt2/3)]] - I~ (7)
AD 2/3
where K = o~(--=-~
For short ageing times eq. (7) can be expanded in series, and higher order terms neglected, so that
T~o dK T
=
-- z ' - ' m ~ - -
t2/3
(8)
•
The t 2/3 proportionality is well illustrated by the diagrams of the present work, and has been experimentally verified by Nakada and Keh (5) and by Wilson and Russel (6).
Another interesting feature of eq. (8) is that AT is directly proportional to the solute content no, as experimentally observed by e.g. Wilson and Russel (6), but independent cf P, i.e. independent of prestrain, which is in agreement with the experimental results of Josefsson and BackstrGm (7). Eq. (8) also orovides a possibility of determining Z from the slope of the experimental A~ vs t2/3 line. With the data of Nakada and Keh (5) this calculation gives Z = 0.12 (the applied numerical values of the remaining parameters are given in the Appendix) for room temperature and nitrogen interstitials. This is in agreement with estimations by Wilson and Russel (8), who found an average of about I nitrogen atom per atomic ~lane after the completion of atmosphere formation, which gives an upper limit of the quantity Z introduced by us.
There is no reason to believe that the Z parameter should be time dependent. Therefore, the value obtained from eq. (8) can be used for numerical investigations of AT after any ageing time. In figs. I - 4 AT has been plotted vs t 2/3 according
868
to
STRAIN AGEING OF a - I R O N
Vol.
3, No.
eq. (7) for Z=0.1 and nitrogen interstitials, with T ~ 330 K and t ~ 2 h. in
order to make the neglection of A7 d acceptable. (For numerical values, see the Appendix.) In fig. la 0 has been varied between 2.1010 and 1011 cm -2 for a constant nitrogen content of 2.1017 cm -3 at room temperature. It is seen that for high p values AT asymptotically approaches a constant, typical for the no- 0 combination, while for p ~ 4.1010 AT rapidly increases towards very high values. The same two types of ageing behaviour can be obtained by varying no for constant p, as shown iu fig. lb.
: / _
|i
B lO
"'
AGEING TIME IN 52/3 AGEING TIME IN S 2/3
FIG. Ib Constant total dislocation density p=4.10 I0 cm -2. The nitrogen.content n° is given in units of 10If cm-3 at each curve. Room temperature.
FIG. la cm_3" Constant nitrogen content n.=2.1017 The total dislocation density 0 is given in units of 1010 cm-2 at each curve. Room temperature.
This can be understood by studying eq. (7) for so long ageing times to that the exponential term is neglectable, and no d
to)
.
no
=
= constant
(9)
If g(to) < I, i.e. no < Zo/d , there is an interstitial undersaturation with respect to the nlmber of atmosphere sites corresponding to complete dislocation locking. For t > to a constant fraction of L is still glissile, and the AT(t) curve has an asymptot parallel to the time axis. This case will be referred to as under.saturated ageing. If, however, no > Zp/d, i.e. if g(t) can reach I, the whole dislocation length L can be locked. The curves then have no inflexion point, and AT(t) approaches a value, related to the stress necessary to unpinn the dislocations from their atmospheres or to activate new dislocation sources. This case corresponds to a saturation or oversaturation of no with respect to Z0/d, and will be referred to as
saturated or e ~ m ~ a ~
ageing respectively.
ii
Vol.
3, No.
ii
STRAIN AGEING OF ~ - I R O N
869
Fig. 2 shows the ingluence of the ageing temperature. As expected, the asymptote of the undersaturated case (fi<~. 2a) is approached faste~ at high tenpozature. The aslmptotic value i~self is also ~ffected by the temperature dependence of ~
as well as m': (see App.). Si'~i!arly, disloeatioi~ locking takes place more
rapidly at higher temperatures ~Iso "~ the oversaturated case (fig. 2b).
/ ),,o/ jr
31o
/ /
/
/
/
9
~zso
z~oo
375o
~ooo
sz~
~oo
~7~o
1250
2500
3750
5000
6250
'7500
8750
AGEING TIME IN S 2/3 AGEING T I M E IN S 213
FIG. 2a ConsLant nitrogen content n =2.1017cm -5 and dislocation deusihy 0=4~.4.10I0 cm "2 The temperature is given in Kelvin at each curve. Undersaturated case.
FIG. 2b Constant nitrogen content no=2.1.1017c~ 5 and dislocation density 0=4.1010 cm "2. The temperature is given in Kelvin at each o-~rv.-. Ove z~tur::~ted c~se.
A~(t) is also sensitive to the Z parameter. Z in its turn depends e.g. on the ageing temperature (which has not been considered in fig. 2), or on the type of in.erstzt_al" atoms that constitute the atmospheres. To demonstrate this A~(t 213) has Seen plotted for different Z values between 0.05 a~d 0.5 in fig, 5, with all other parameters constant, and the ao/0 ratio so chosen that the limit between ~Ider- and oversaturation is passed. ,,o,
o o5 oo6 oo'~ oos
oo~
/
jY-
z
j
t
ol
z
c~ O2
N 0
7250
25G0
3750
5000
AGEING TIME tN$
6250 2/]
7500
8750
FIG. 5 Constant nitrogen content n =2 IO17cm-3 and dislocation de:IsTtj ~=4~2.;0 ~O c:~-2. The Z value is given at each curve. Room temperature.
870
STRAIN AGEING
OF c~-IRON
Vol.
3, No.
The authors are greatly indebted to Dr Bertil Aronsson for introducing us into the field of strain ageing and for helpful discussions. We want to thank ~{r Lennart Vilander for skilful help in working out the computer proo~rams. One of us (Y.B.) has received partial financial support from the Swedish Board for Teclhuical Development. We also want to express our gTatitude to Professor 01of Beclauan for continuous interest and encouragement.
Temperature dependent parameters (derived from the curves of reference 2): T
250
260
270
280
290
500
310
320
330
T~
4.2
3.6
3.1
2.6
2.1
1.8
1.4
1.2
1.0
m*
7.?
?.3
6.8
6.4
6.0
5.5
5.0
4.6
4.2
Temperature independent parameters: A = 3.10 -20 dyne.cm 2
D = 3.10-3.exp(-1S200/RT)
cm 2.s -I
~=5 d = 2.10 -8 cm References I.
W.C. Jolmston and J.J. Gilman, J. Appl. Phys.
2.
J.T. Nichalak, Ac~a ~eb.
30, 129 (1959)
3.
Y. Bergstr~m and 0. Vingsbo, to be published
4.
S. Harper, Phys. Rev. 83, 709 (1951)
5.
Y. Nakada and A.S. Keh, Edgar C. Hain Laboratory, ~#onroeville, Pennsylvania,
15, 213 (1965)
Preprint 373(1967) 6,
D.V. Wilson and B. Russel, Acta Met. Z, 628 (1959)
?.
~. Josefsson and B. BackstrGm, 1#edd. Jernkentorets Tekniska R&d 27~, 597 (1965)
8.
D.V. Wilson and B. Russel, Acta Net. ~, 468 (1960)
ii
Vol. 3, pp. 865-870, 1969 Printed in the United States
Pergamon
Press,
UPPER YIELD STRESS INCREASE DURING STRAIN AGEING OF G-IRON Y. BergstrGm and 0o Vingsbo Institute of Physics, University of Uppsala, Uppsala, Sweden
(Received
September
29,
1969)
Introduction The resolved shear stress ~ in a dislocation glide plane can be separated into components according to T = T~ + T i + Td
(1)
The effective resolved shear stress T~ is the temperature dependent stress, required to force the dislocation through stress fields of short range. At a constant temperature it depends on the shear strain rate 9 and the density L of mobile dislocations according to I •~
(2)
= c(~) ~*-
where m* is the dislocation velocity exponent and C is a material constant ~. The formation of Cot%tell and/or Snoek atmospheres during strain ageing causes a decrease in the mobile dislocation density, and a corresponding increase in T~ according to eq. (2). Ti is the stress representing long range stress fields of strain independent distribution, and is not supposed to be affected by strain ageing. T d ~ / p f is the stress necessary to overcome dislocation-dislocation long range interaction. It is strain dependent through the dislocation density pf in tangled regions, and represents the work hardening, pf is not mobile, and Td is strain ageing sensitive only after so long ageing times that a redistribution of the dislocation structure can take place.
I ,
m @
is obtained by substituting the Johnston-Gilman (I) relation
~'= (T_~)m* Tu
into the strain rate expression ~ = bL~, where b is the length of the Burgers vector, ~ is the average dislocation velocity and ~u is the stress giving unit velocity.
865
Inc
866
STRAIN A G E I N G OF a - I R O N
Vol.
3, No.
The purpose of the present work has been to derive the time dependence of the increase in upper yield stress AT during strain ageing of ~-iron, in terms of dislocation locking parameters, i.e. under such conditions that A~Ct)
=
AT~(t)
Deriva_tion_of__the_U~?er Yield Stress IncreaSe as a Function of Ageing Time At the beginning of a tensile test of a strain aged iron specimen (or a very perfect crystal)~ the effective stress T~-=O corresponding to the upper yield point, is very high because of the low density of mobile dislocations. A rapid dislocation multiplication then induces the yield drcp~ during which T z- falls to a strain independent value T~, corresponding to a constant mobile dislocation length Lo. T~ has been measured in ~-iron by Michalak (2), and was found to depend on strain rate according to eq. (2), but to be constant in strain. The way in which L approaches Lo has been discussed by BergstrSm and Vingsbo (5). Because of the inhomogeneous deformation during the LGders strain, howeverf the present work will deal with prestrains ¢ > CLGd. ~ so that at the beginning of the ageing pause of a strain ageing test L = Lo in all grains~ and I 7-=
~* --
c(~) m*
(5)
Assume that the fraction g of the total dislocation density p that has been
locked by a t m o s p h e r e s , i s a rune%ion of a g e i n g
time
t . The i n c r e a s e i n y i e l d s%~ess
AT~~ then is a function of t according to m*
,,~*(t) = ~(t)
.t ] - 7. = c {V.oE1-~t)']
- T.* = ~* {[~-¢(t)]
m4~
- ~]
(4)
The main difficulty in finding an expression for the function g(t) is to decide, when a certain length of initially glissile dislocation shall be considered as locked. Let us make the simplifying assumption that an average of Z interstitial atoms per atomic plane cut by a dislocation, are needed to prevent the dislocation from escaping from its atmosphere under an applied stress. The number n of locking interstitials in atmospheres after an ageing time t then is
n(t)
z.~t).o
(5)
d
where d is the mean distance between two atomic planes, threaded by a dislocation. n(t) can be estimated from the equation suggested by Harper (4) to describe the diffusion of interstitial atoms in the long range stress fields of dislocations
In[1 -
-~n(t)]=
ADt 2/5
- e'p(~-)
(6)
ii
Vol.
3, No.
ii
S T R A I N A G E I N G OF a - I R O N
867
where no is the initial number of solute interstitials per unit volume, ~ is a numerical constant, A is a constant determined from the lattice strain due to one solute atom, D is the diffusion coefficient for the interstitial in m~trix, k is Boltzmann's constant and T is the ageing temperature. In spite of the fact that this equation does not take into account the development of concentration gradients during the diffusion process, it is found to be valid as an acceptable approximation, which is in good agreement with experimental observations. Substitution of (6) and (5) into (4) then gives the time dependence of the increase in yield stress. I nod - m-q.j &T(t) = ~ - ~{1 -~.p. [1 - exp(- Kpt2/3)]] - I~ (7)
AD 2/3
where K = o~(--=-~
For short ageing times eq. (7) can be expanded in series, and higher order terms neglected, so that
T~o dK T
=
-- z ' - ' m ~ - -
t2/3
(8)
•
The t 2/3 proportionality is well illustrated by the diagrams of the present work, and has been experimentally verified by Nakada and Keh (5) and by Wilson and Russel (6).
Another interesting feature of eq. (8) is that AT is directly proportional to the solute content no, as experimentally observed by e.g. Wilson and Russel (6), but independent cf P, i.e. independent of prestrain, which is in agreement with the experimental results of Josefsson and BackstrGm (7). Eq. (8) also orovides a possibility of determining Z from the slope of the experimental A~ vs t2/3 line. With the data of Nakada and Keh (5) this calculation gives Z = 0.12 (the applied numerical values of the remaining parameters are given in the Appendix) for room temperature and nitrogen interstitials. This is in agreement with estimations by Wilson and Russel (8), who found an average of about I nitrogen atom per atomic ~lane after the completion of atmosphere formation, which gives an upper limit of the quantity Z introduced by us.
There is no reason to believe that the Z parameter should be time dependent. Therefore, the value obtained from eq. (8) can be used for numerical investigations of AT after any ageing time. In figs. I - 4 AT has been plotted vs t 2/3 according
868
to
STRAIN AGEING OF a - I R O N
Vol.
3, No.
eq. (7) for Z=0.1 and nitrogen interstitials, with T ~ 330 K and t ~ 2 h. in
order to make the neglection of A7 d acceptable. (For numerical values, see the Appendix.) In fig. la 0 has been varied between 2.1010 and 1011 cm -2 for a constant nitrogen content of 2.1017 cm -3 at room temperature. It is seen that for high p values AT asymptotically approaches a constant, typical for the no- 0 combination, while for p ~ 4.1010 AT rapidly increases towards very high values. The same two types of ageing behaviour can be obtained by varying no for constant p, as shown iu fig. lb.
: / _
|i
B lO
"'
AGEING TIME IN 52/3 AGEING TIME IN S 2/3
FIG. Ib Constant total dislocation density p=4.10 I0 cm -2. The nitrogen.content n° is given in units of 10If cm-3 at each curve. Room temperature.
FIG. la cm_3" Constant nitrogen content n.=2.1017 The total dislocation density 0 is given in units of 1010 cm-2 at each curve. Room temperature.
This can be understood by studying eq. (7) for so long ageing times to that the exponential term is neglectable, and no d
to)
.
no
=
= constant
(9)
If g(to) < I, i.e. no < Zo/d , there is an interstitial undersaturation with respect to the nlmber of atmosphere sites corresponding to complete dislocation locking. For t > to a constant fraction of L is still glissile, and the AT(t) curve has an asymptot parallel to the time axis. This case will be referred to as under.saturated ageing. If, however, no > Zp/d, i.e. if g(t) can reach I, the whole dislocation length L can be locked. The curves then have no inflexion point, and AT(t) approaches a value, related to the stress necessary to unpinn the dislocations from their atmospheres or to activate new dislocation sources. This case corresponds to a saturation or oversaturation of no with respect to Z0/d, and will be referred to as
saturated or e ~ m ~ a ~
ageing respectively.
ii
Vol.
3, No.
ii
STRAIN AGEING OF ~ - I R O N
869
Fig. 2 shows the ingluence of the ageing temperature. As expected, the asymptote of the undersaturated case (fi<~. 2a) is approached faste~ at high tenpozature. The aslmptotic value i~self is also ~ffected by the temperature dependence of ~
as well as m': (see App.). Si'~i!arly, disloeatioi~ locking takes place more
rapidly at higher temperatures ~Iso "~ the oversaturated case (fig. 2b).
/ ),,o/ jr
31o
/ /
/
/
/
9
~zso
z~oo
375o
~ooo
sz~
~oo
~7~o
1250
2500
3750
5000
6250
'7500
8750
AGEING TIME IN S 2/3 AGEING T I M E IN S 213
FIG. 2a ConsLant nitrogen content n =2.1017cm -5 and dislocation deusihy 0=4~.4.10I0 cm "2 The temperature is given in Kelvin at each curve. Undersaturated case.
FIG. 2b Constant nitrogen content no=2.1.1017c~ 5 and dislocation density 0=4.1010 cm "2. The temperature is given in Kelvin at each o-~rv.-. Ove z~tur::~ted c~se.
A~(t) is also sensitive to the Z parameter. Z in its turn depends e.g. on the ageing temperature (which has not been considered in fig. 2), or on the type of in.erstzt_al" atoms that constitute the atmospheres. To demonstrate this A~(t 213) has Seen plotted for different Z values between 0.05 a~d 0.5 in fig, 5, with all other parameters constant, and the ao/0 ratio so chosen that the limit between ~Ider- and oversaturation is passed. ,,o,
o o5 oo6 oo'~ oos
oo~
/
jY-
z
j
t
ol
z
c~ O2
N 0
7250
25G0
3750
5000
AGEING TIME tN$
6250 2/]
7500
8750
FIG. 5 Constant nitrogen content n =2 IO17cm-3 and dislocation de:IsTtj ~=4~2.;0 ~O c:~-2. The Z value is given at each curve. Room temperature.
870
STRAIN AGEING
OF c~-IRON
Vol.
3, No.
The authors are greatly indebted to Dr Bertil Aronsson for introducing us into the field of strain ageing and for helpful discussions. We want to thank ~{r Lennart Vilander for skilful help in working out the computer proo~rams. One of us (Y.B.) has received partial financial support from the Swedish Board for Teclhuical Development. We also want to express our gTatitude to Professor 01of Beclauan for continuous interest and encouragement.
Temperature dependent parameters (derived from the curves of reference 2): T
250
260
270
280
290
500
310
320
330
T~
4.2
3.6
3.1
2.6
2.1
1.8
1.4
1.2
1.0
m*
7.?
?.3
6.8
6.4
6.0
5.5
5.0
4.6
4.2
Temperature independent parameters: A = 3.10 -20 dyne.cm 2
D = 3.10-3.exp(-1S200/RT)
cm 2.s -I
~=5 d = 2.10 -8 cm References I.
W.C. Jolmston and J.J. Gilman, J. Appl. Phys.
2.
J.T. Nichalak, Ac~a ~eb.
30, 129 (1959)
3.
Y. Bergstr~m and 0. Vingsbo, to be published
4.
S. Harper, Phys. Rev. 83, 709 (1951)
5.
Y. Nakada and A.S. Keh, Edgar C. Hain Laboratory, ~#onroeville, Pennsylvania,
15, 213 (1965)
Preprint 373(1967) 6,
D.V. Wilson and B. Russel, Acta Met. Z, 628 (1959)
?.
~. Josefsson and B. BackstrGm, 1#edd. Jernkentorets Tekniska R&d 27~, 597 (1965)
8.
D.V. Wilson and B. Russel, Acta Net. ~, 468 (1960)
ii