Scripta METALLURGICA
Vol. 5, pp. 137-142, 1971 Printed in the United States
Pergamon Press, Inc
ON THE DETERMINATION OF ACTIVATION AREA IN HIGH T~PERATURE CREEP
M. Pahutov~, J. ~adek and P. Ry~ Czechoslovak Academy of Sciences, Institute of Physical Metallurgy, Brno, Czechoslovakia (Received December 22, 1970) Assuming a single rate controlling process the creep rate is described by the equation
where
~
rate at
is the free activation energy, ~o is the maximum attainable creep ~$ = 0
and kT
has its usual meaning of Boltzmann constant and tem-
perature in °K. The factor ~o is expressed as
~o =¢.~,.b.~ where ~ is a geometric factor, ~
(2)
is the moving dislocation density, % is the
maximum attainable dislocation velocity at ~G - 0 and ~ is Burgers vector. From Equation (1) and (2) it follows that
IT --~--\-T~--~/T A% =
= -~-
A~ b e i n g the a c t i v a t i o n ctive will
stress
area,
~ bel~
(3)
component o f the e f f e -
( ~" i s the c o r r e s p o n d i n g component o f the a p p l i e d s t r e s s which
be taken t o be equal to ~/~
stress). Should
~
i n what f o l l o w s ,
~ being the normal a p p l i e d
not instantly respond the stress change there would be A"
p~'ovlded that
the r e s p e c t i v e
,
kt l@~,m~ ~ ~
"-~-~&
kt [~n,~ I
--~ ~-~-JT
(4)
the change of the effective stressA~ is equal to the change
of the applied stress ~ * stress change is ~ a l l
. This is supposed to be the case if the applied
and if it is performed quickly enough for the internal
stress to remain unchanged. Equation (4) has been many times applied to mea137
138
ACTIVATION AREA IN HIGH TEMPERATURE CREEP
Vol. 5, No.2
sure the activation area for low temperature deformation, where it is a common practice to assume that the dislocation structure remains essentially constant during a stress change. Recently, Li [1] and Balasubramanian and Li [2] have applied Equation (4) to determine the activation area for high temperature creep using literature data on applied stress dependence of steady state creep rate ~ k"- kT
. In this case
--
(5)
where
and if ~ ÷~(~') , A" decreases with the increasing stress according to a byperbolic law. The assumption of stress independence of moving dislocation density has been discussed recently by Hirth and Nix [33 who have concluded that it is almost certainly not valid. The density of dislocations generated in creep and unbound in sub-boundaries depends on about the second power of the applied stress [4,5]. It is always only a fraction of these dislocations that moves at any instant in time. Frequently it is supposed (e.g. [6]) that the ratio of the moving dislocation density ~
to the density
~t of dislocations unbound
in sub-boundaries is stress independent. From this it follows that the moving dislocation density should depend also on about the second power of the applied stress. This assumption has not received any experimental support as yet; however, it can be hardly expected that ~
is stress independent despite
of
?t quickly increasing with the increasing stress. Moreover, recently, Solomon and Nix [7] have shown that the dislocation structure changes even during relaxation and strain rate sensitivity tests at high temperatures. However, the method of activation area determination based on Equation (4) can easily be modified to avoid the stress dependence of the moving dislocation density. If the stress change performed to measure strain rate change for determination by means of Equation (4) approaches zero that of moving dislocation density approaches zero as well. Therefore, if
a~
6
is obtained for
Vol. 5, No. 2
ACTIVATION AREA IN HIGH TEMPERATURE CREEP
several different positive and negative ~ decrements, the ratio ( ~ $ ~ / ~ ) to the curve
~C
139
, i.e. stress increments and stress
can be determined as the slope of the tangent
vs. a ~ i n the point ~ - 0
,a~
-0 ; the branches of
the curve corresponding to the positive and negative values of ~
must be
extrapolated to this point in such a way as to have a common tangent in that point. The method was used to determine the stress dependence of activation area in creep of iron at 873 OK. Creep specimens of Armco iron of 50.0 mm in gauge length and 9.0 x 3.2 mm 2 in cross section were mashined from a sheet 4 mm in thickness. Before grind. ing specimens were decarburized by annealing in wet hydrogen. The carbon content in the annealed specimens was 0.008 pct. +), the mean grain diameter was 0.50 ram. Constant stress creep tests were performed in purified dried hydrogen. The temperature 873 OK was stabilized within + 0.5 OK. Creep strain was measured by means of a variable linear differential transformer with a sensitivity of about 10"5. The activation ares was determined for three values of the normal applied stress: 4.0; 5.5 and 8.0 kgmm "2. It was found that the ratio of strain rates measured immediately before and ~-,,ediately after the stress change, ~L and g~, respectively, did not depend on the creep strain at which the stress change was performed, at least for creep strains from 0.15 to 0.35. This can be seen from Fig. I in which the ratio 6L / ~ strain rates
is plotted against creep strain ~ . The
~L and ~ c o r r e s p o n d to ~L and
" ~ (~L ÷ ~ ) changed from ~
. The value to ~
All the values of ~ / ~
of
~/~
~
, respectively;
~L ) ~
,
depends on whether the stress is
, i.e, decreased, or from 6~ to
~
, i.e. increased.
obtained during the creep test were taken to deter-
mine the mean value of this ratio. In Pig. 2 the values of
+)
~/~
are plotted against stress change for the
The contents in pct. of other impurities: 0.27 Nn; 0.01 Si; 0.017 P; 0.019 S; 0.07 Cu; 0.05 Ni; 0.01 Cr; 0.0059 N2; 0.0043 02 .
140
ACTIVATION AREA IN HIGH TEMPERATURE CREEP
Vol. 5, No.2
,o' ~0 z
I
1
I
T.8?3"K
~/ -
e . s s ~ ,,;,"
~.~45
I
I
3
I
1
,o
T= 873 "K Ae .0.S kgmr~* o A
.,,,T 0
0
0---
-fO
1
e~-eu
//
o ~-el 0
i
0
I
0.4
.40 z -2
I
0.2
0.3
0.4
CREEP S T R A I N .C
Fig. 1
I 0
I ¢
Fig. 2
&
Relation between 6~/~., ratio and stress change ~~ for +
the applied stress
~
2.
STRESS CHANGE A 6 = f f l - 6 u
Relation between£L/~ratio and creep strain
i -~
= 5.5 kgmm "2. The nonlinearity of
a ~
=
5.5
6 vs. a ~
-2
rela-
tion is believed to manifest changes in moving dislocation density during stress changes.
The value of the activation area was determined from the slope
of the tangent to
a~
6
vs. ~
curve in the point
~-
O
, A~
The relation between activation area and the mean effective stress measured by the stress dip test technique
- 0 F~
• as
[8] is shown in Fig. 3. The activa-
tion area decreases with the increasing effective stress according to a hyperbolic law. In Pig. with
4 the quantity
kT ~ , Equation (5), is plotted +) together b ¢ A* determined in this work against applied stress ~ . The difference
between these two relations is most probably caused by the stress dependence of structure,
including the moving dislocation density not taken into account
in the activation area determination based on Equation (5).
+)
The exponent ~ determined for the iron investigated increases with the applied stress [9].
260
141
ACTIVATION AREA IN HIGH TEMPERATURE CREEP
Vol. 5, No. 2 I
\
350
I
I
I
250\ .Q
3OO
,Ik
#
\
~11"1
2,~0-
A
\
_
\
@
2k.T n_ b ~r
_
\ \
D-
250
230-
t~ ,q
\
220 -
2¢0 0.4
I
I
0.8
1.2
\
2OO
46
~50
MEAN EFFECTIVE STRESS
......
3
I
I
6
9
APPLIED STRESS
Fig. 3
Mean effective stress dependence of activation area A~
Fig. 4
\
I
'12 -Z • ckgmm
Comparison of applied stress dependence of activation area ~determined in this work and quantity (kT/b)(~/~), Eq. (7)
R e f e r e n c e s
[I]
J.C.M.Li, Dislocation Dynamics, Ed. A.R. Rosenfield et al., Mc Graw-Hill Book Comp., (1969), 87
[2]
N.Balasubramanian, J.C.M.Li, J. Mat. Sci. ~, (1970), 434
[3]
J.P°Hirth, W.D.Nix, phys. stat. sol. 35, (1969), 177
[4]
D.McLean, K,F.Hale, Structural Processes in Creep, Ir.St.Inst.Spec.Rep.
To 70, (1961), 19 [5]
C.R.Barrett, Trans. Met. Soc. AIME, 239, (1967), 1927
[6]
C.N.Ahlqulst, R.Gasca-Neri, W.D.Nix, Acta Met. 18, (1970), 663
[7]
A.A.Solomon, W.D.Nix, Acta Met. 18, (1970), 863
[8]
C.N.Ahlquist, W.D.Nix, Scripta Met. ~, (1969), 679
[9]
M.Pahutov~, J. ~adek, P. Ry§, to be published.
¢5