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On the applicability of a phenomenological relationship to creep at low stresses and intermediate temperatures
Materials Science and Engineering, 61 (1983) 1-5
1
On the Applicability of a Phenomenological Relationship to Creep at Low Stresses and Intei-J~mdiate Temperatures T. SRITHARAN* and H. JONES
Department of Me tallurgy, University o f Sheffield, St. George's Square, Sheffield $1 3JD (Gt. Britain) (Received January 21, 1983)
SUMMARY In a recent paper, R u a n o and Sherby used a phenomenological relationship o f the form c: 02 derived from superplasticity data to reinterpret some data for creep at low stresses. These data had been represented previously in the fo r m ~ zc 0 -- Oo with 00 as the threshold stress for the operation o f vacancy diffusion creep. In the present paper it is shown that o: 02 in fact gives a poorer representation o f these creep data than does ~ cc 0 -- 00. This conclusion is based on (i) the marked curvature o f p lo ts o f ~1]2 versus 0 consistent with e cc a -- 00, (ii) the unit slope o f log ~ versus log(o -- ao) p lo ts and (iii) the creep rates observed which are two to three orders o f magnitude larger than those predicted by the phenomenological relationship when used with the empirical diffusion coefficient D with which it was originally established. Use o f this original D also virtually eliminates from their deformation map the field o f dominance over Coble creep o f the m ode o f deformation represented by ~ cc 0 2 Analysis finally shows that, when interpreted as ~ cc 0 -- 00, the phenomenological relation should give an apparent Oo which is independent o f grain size, contrary to what is observed.
The effect of operational and material variables on the secondary creep rate ~ can be generalized in the form [ 1-3]
~a -A
Db
kT
(1)
where o is the applied stress, o o is a threshold stress, L is the grain diameter, b is Burgers' *Present address: Department of Civil Engineering, University of Peradeniya, Peradeniya, Sri Lanka. 0025-5416/83/$3.00
vector, Ft is the atomic volume (approximately equal to b3), G is the shear modulus (or alternatively Young's modulus E), D is the operative diffusion coefficient, k is Boltzmann's constant and A, n and p depend on the operative creep mechanism. Characteristically, p tends to zero and n is typically a b o u t 4 f o r the strongly stress-dependent slip c r e e p (power law creep) observed at high stresses while p is between 2 and 3 and n is unity for the strongly grain-size-dependent vacancy diffusion creep observed at low stresses for normal grain sizes. Theoretical analysis for vacancy creep predicts that p is 2 and A is a b o u t 13.3 when the lattice diffusion of vacancies is the controlling mechanism [ 4] (Nabarro-Herring creep) and that p is 3 and A is a b o u t 47.7 when the grain boundary diffusion of vacancies is the controlling mechanism [ 5] (Coble creep). Under such conditions, eqn. (1) thus takes the special form = A
~-~ (a -- a0)
(2)
with d~ da
A[b~P Db kT
(Y
(3)
The evidence for the applicability of eqns. (2) and (3) to creep at low stresses (less than or approximately equal to 10-4G or 10-4E) includes experimental observations of (i) the predicted parametric dependences on stress, grain size and temperature, (ii) values of d~/do usually within a factor of 10 of predictions based on radiotracer measurements o f D and on the measured L values and (iii) thicknesses of precipitate-denuded zones, adjacent to grain boundaries subjected to tensile stress, that correspond to the predicted and observed creep strains [3, 6 - 1 6 ] . The © Elsevier Sequoia/Printed in The Netherlands
observed threshold stresses o0 are typically about 10-5G or 10-SE for pure metals at about 0.5T~ [9] where Tm is the temperature of the melting point and are increased by additions of appropriate second phases [16-18]. In a recent publication, Ruano and Sherby [19] have reinterpreted some data for creep at low stresses in terms of a phenomenological alternative [20-22] to eqn. (2) which can be written in the form 4~ _ A
(4)
Db
where ~ is the atomic volume (approximately equal to b a) and A is a b o u t 6 X 10 s when grain boundary diffusion is the controlling mechanis m (p = 3) and is an order of magnitude larger when lattice diffusion controls (p = 2). For the particular case of grain boundary diffusion control, eqn. (4) thus reduces to = A ~-~
(5)
Equation (5) was used to correlate superplasticity data mainly for Zn-A1 eutectoid or Sn-Pb eutectic alloys at (0.5-0.9)T~ for the stress range ( 1 0 - L 1 0 - a ) E or (10-L10-3)G using an empirical relationship D
e x p [ - llTm~
2
T jcm s-
1
[20] for grain boundary diffusivity and estimates for E and b. The correlation fits the observed ~ data typically within a factor of 10 over the range covered, as does eqn. (3) in relation to the appropriate creep data. Ruano and Sherby [ 19] suggest that eqn. (5) indicates control of deformation by grain-boundarydiffusion-controlled grain boundary sliding a c c o m m o d a t e d by slip, on the basis that theoretical predictions [23-25] from that standpoint yield 4 cc o 2 (n = 2 in eqn. (1)), although they differ from eqn. (5) in that (i) cc L -2 (p = 2 in eqn. (1)) is predicted rather than the e cc L-3 dependence of eqn. (5), (ii) the ~xtra variable factor G ~ / k T of eqn. (1) whigh is absent in eqns. (4) and (5)* is included and (iii) 4 values several orders of magnitude lower than that in eqn. (5) are pre*Exclusion of the factor G $ ~ / k T was based o n t h e
fact that it gave a worse fit with the superplasticity data ~vhen included.
dicted. In spite of these disparities with available theory, Ruano and Sherby go on to apply eqn. (5) in particular to some of our results for fi-Co [12], claiming that our results exhibit a parabolic relation with stress and the magnitudes of strain rate given by eqn. (5). They base this conclusion on four approaches: (i) that a Plot of 4~2 versus o for our test SC2-3 at 1073 K can be extrapolated linearly through the origin, (ii) that plots of log versus log(o/E) for our test SC2-3 at 1073 K and test SC7-1 at 773 K show slopes with a value nearer to 2 than unity, (iii) that plots of log(~L3/Db) versus l o g ( o / E ) for these t w o tests show good agreement with the predictions of eqn, (5) using the radiotracer data o f Brik et al. [26] for D b and (iv) that the conditions in tests on fi-Co, ~-Fe, AISI type 304 stainless steel, magnesium and copper, previously interpreted in terms of Coble creep with a threshold stress, fall within the range of eqn. (5) rather than within the region of Coble creep when they are plotted on a deformation map of l o g ( L / b ) versus l o g ( o / E ) at 0.55Tin. We shall deal with each of these points in turn. Concerning item (i), the same plot of ~12 versus o for test SC7-1 at 773 K (Fig. 1, line b), n o t included by Ruano and Sherby, can be extrapolated linearly through the origin only if the points for the four lowest stresses are excluded; these points exhibit the downward curvature consistent with zero strain rate below a threshold stress of 2.7 MPa indicated by the corresponding plot of versus o (Fig. 1, line a). Further examples from other tests on ~-Co are shown in Figs. 2 and 3. Concerning items (ii) and (iii), this
,
•
,
•
,
,
,
.
,
Dj ~. 9 p "6
6
f
(~-%-~) 4 2
e
•
I
~ - - -
-
7J ~
I
4
I
i
6
i
i
a
o'(Mh)
|
10
Fig. 1. Creep data for ~-Co plotted as ~ (o, line a) and 51/2 (e, line b) vs. a (test SC7-1: temperature T = 773 K; mean linear intercept grain size L = 21 #m). T h e data were taken from ref. 12.
~6 S" ° /
6
(s")
(xm'%-l) 4
(x~-s.-~) 2 0
0
0
•i f~ldSjJ ~ ~ aO'{M~}
|0
wO
Fig. 2. tks for Fig. 1 b u t for test SC6-1 ( T = 803 K; /7, = 26 pm). The data were taken f r o m ref. 12.
3~-
. . . . . . . .
,/
-- ~-~0"~:~-' 4
~"""~ e '°'l;
t
(a)
.~a
/
-.
60.(MPa~
|0
12
Fig. 3./ks for Fig. 1 b u t for test SC3-1 ( T = 9 6 3 K ; /~= 3 4 p m ) . The data were taken f r o m ref. 12.
,[ ./. ??. ~'~ (b)
. ..
~
~-4 -~
log(~L3/Db) versus log{(o -- ao)/E} in contrast
Fig. 4. L o g a r i t h m (a) of the strain rate ~ and (b) of ~L3/Db vs. log(((~ -- Oo)/E} for test SC2-3 ( T = 1073 K ; / ~ -- 125 pro) and test SC7-1 ( T = 773 K; £ = 21pro) for ~-Co. The test data were taken f r o m ref. 12, with the mean grain diameter L taken as 1.776/~ and Db = 2.5 X 10 -14 e x p ( - - l l T m / T ) m 3 s-1 [20] w i t h T m = 1765 K for cobalt. The stress O0 is t h a t for zero ~ on plots o f ~ vs. o and the lines shown are for a slope n equal to u n i t y consistent w i t h the operation of vacancy diffusion creep.
are linear with a slope n of u n i t y as shown in Fig. 4, fully consistent with the operation of diffusion creep. In this connection, a further inconsistency concerning the application of eqn. (5) by Ruano and Sherby to predict log(~L3/Db) versus log(alE) is their use of radiotracer data [26] for D rather than the empirical e x p ( - - l l T m / T ) cm 2 s-z [20] on which eqn. (5) was based. Use of the empirical D instead of radiotracer data in their Fig. 4 places the experimental values of ~L3/Db two to three orders of magnitude above those predicted by eqn. (5), indicating that creep is occurring at t w o to three orders of magnitude faster than predicted by eqn. (5) as originally conceived. In respect of item (iv), the value chosen for D in eqn. (5) will correspondingly affect the relative sizes of the fields of dominance of each mechanism in the deformation
map for T/Tm -- 0.55 plotted by Ruano and Sherby as their Fig. 5. The empirical D is 2 X 10 -14 m 2 s-1 at T/Tm = 0.55 compared with 2 X 1 0 -12 m 2 s-1 as used by Ruano and Sherby in their Fig. 5. It is evident that use of this empirical D in eqn. (5) shifts the boundary between the field of dominance of Cable creep and that of eqn. (5) identified with grain boundary sliding on their Fig. 5 from a o/E value of 7 X 10 -5 to a a/E value of 7 X 10 -3, virtually eliminating the field of eqn. (5) from the map and placing the conditions in the tests on ~-Co, ~-Fe, AISI type 304 stainless steel, magnesium and copper all well within the field of dominance of Cable creep. Furthermore, eqn. (5) sheds no light on the
downward curvature of points at low stresses in such tests is evident for test SC7-1 at 773 K even in plots of log ~ and log(~L3/Db) versus log(o/E) (ref. 19, Figs. 3 a n d 4), thereby in-
validating these plots. Such plots of log ~ and
large apparent enhancement of creep rate, by between 40 and 140 times the Coble rate, found for ~-Fe and attributed earlier to transient operation of dislocations as sources, sinks and short-circuit paths for vacancies [9]. Equation (5) predicts creep rates a similar amount lower than observed using radiotracer data for D, the discrepancy being even larger if the empirical value of D is employed. Finally, it is possible to evaluate apparent values of a0 for each test on H-Co on the a s s u m p t i o n that eqn. (5) is applicable and that we fitted in ref. 12 a straight line to results falling on the parabola given by eqn. (5) in a plot of ~ versus o. Thus, for Coble creep with a threshold stress,
A1Db~2(o -- Oo) =aa+b where AI ~ 50, a = A I D b ~ / L 3 k T Equation (5) gives
and b =
- - aOo.
A 2Dbo 2
LaE2
e2 -
= CO 2
with A2 = 108 and c = A2bD]L3E 2. When the line el = aa + b is fitted by the least-squares m e t h o d between the limits 01 and a2, Q = Z(d2--dl) 2 a2
=;
(e2--el) 2d° al a2
= f
(Ca 2 - a a - b )
2dO
al
To find b = -- aoo, we put dQ/db = O. N o w
dQ -- f~ ~d (CO 2 - - a o - - b ) 2 d o
(o2- ol) 3
{ 2 c ( a 2 2 4- 0"10 2 -t-
-- 3a(o2 + ol) -- 6b} =0
b .0 0
=
- -
_ a
= 1 (o2 + ol)
-
1 f(022_1_ 0102-1- 012)
Since c/a is independent of grain size, the resulting a0 should be independent of grain size at a fixed temperature T and in the stress range o2 :> o > a l . On the contrary we find (ref. 12, Fig. 5b) that o0 is n o t only strongly dependent on grain size b u t inversely proportional to it as predicted by the grain boundary dislocation climb source model.
REFERENCES
LakT
~1 :
Thus
0"12 )
-
-
1 T. G. I.~ngdon, in J. C. M. Li and A. K. Mukherjee (eds.), Rate Processes in Plastic Deformation o f Materials, American Society for Metals, Metals Park, OH, 1975, p. 410;Met. Forum, 1 (2) (1978) 59. 2 J. C. Gibeling and W. D. Nix, Mater. Sci. Eng., 45 (1980) 123. 3 T. Sritharan and-H.-Jones, in Mechanical Behaviour and Nuclear Applications o f Stainless Steel at Elevated Temperatures, Metals Society, London, 1982, p. 99. 4 C. Herring, J. Appl. Phys., 21 (1950) 437. 5 R. Coble, J. Appl. Phys., 34 (1963) 1679. 6 H. Jones, Mater. Sci. Eng., 4 (1969) 106. 7 R. B. Jones, J. Sheffield Univ. Metall. Soc., 12 (1973) 34. 8 I. G. Crossland, in J. E. Harris and E. C. Sykes (eds.), Physical Metallurgy o f Reactor Fuel Elements, Metals Society, London, 1975, p. 66. 9 D. J. Towle and H. Jones, Acta Metall., 24 (1976) 399. 10 B. Burton, Diffusional Creep o f Polycrystalline Materials, Trans Tech Publications, Aedermannsdorf, 1977. 11 I. G. Crossland and R. B. Jones, Met. Sci., 11 (1977) 504. 12 T. Sritharan and H. Jones, Acta Metall., 27 (1979) 1293. 13 B. Burton, I. G. Crossland and G. W. Greenwood, Met. Sci., 7 (1980) 134. 14 T. Sritharan and H. Jones, Acta MetaU., 28 (1980) 1633. 15 T. Sritharan and H. Jones, Met. Sci., 15 (1981) 365. 16 B. Burton and W. Beer~, Met. Sci., 12 (1978) 71. 17 W.J. Clegg and J. W. Martin, Met. Sci., 16 (1982) 65. 18 E. Arzt and M. F. Ashby, Scr. Metall., 16 (1982) 1285. 19 O. A. Ruano and O. D. Sherby, Mater. Sci. Eng., 56 (1982) 167.
20 R0 A. White, Ph.D. Thesis, Department of Materials Science, Stanford University, 1978, 21 H. Liithy, R. A. White and O. D. Sherby, Mater. Sc£ Eng., 39 (1979) 211. 22 O. A. Ruano, A. K. Miller and O. D. Sherby, Mater. Sci. Eng., 51 (1981) 9.
23 A. Ball and M. M. Hutchison, Met. Sei. J., 3 (1969) 1, 24 A. K. Mukherjee, Mater. Sci. Eng., 8 (1971) 83. 25 R. C. Gifkins, Metall. Trans. A, 7 (1976) 1225. 26 V. B. Brik, L. N. Larikov and V. Falchenko, Ukr. Fiz. Zh., 20 (1975) 397.
1
On the Applicability of a Phenomenological Relationship to Creep at Low Stresses and Intei-J~mdiate Temperatures T. SRITHARAN* and H. JONES
Department of Me tallurgy, University o f Sheffield, St. George's Square, Sheffield $1 3JD (Gt. Britain) (Received January 21, 1983)
SUMMARY In a recent paper, R u a n o and Sherby used a phenomenological relationship o f the form c: 02 derived from superplasticity data to reinterpret some data for creep at low stresses. These data had been represented previously in the fo r m ~ zc 0 -- Oo with 00 as the threshold stress for the operation o f vacancy diffusion creep. In the present paper it is shown that o: 02 in fact gives a poorer representation o f these creep data than does ~ cc 0 -- 00. This conclusion is based on (i) the marked curvature o f p lo ts o f ~1]2 versus 0 consistent with e cc a -- 00, (ii) the unit slope o f log ~ versus log(o -- ao) p lo ts and (iii) the creep rates observed which are two to three orders o f magnitude larger than those predicted by the phenomenological relationship when used with the empirical diffusion coefficient D with which it was originally established. Use o f this original D also virtually eliminates from their deformation map the field o f dominance over Coble creep o f the m ode o f deformation represented by ~ cc 0 2 Analysis finally shows that, when interpreted as ~ cc 0 -- 00, the phenomenological relation should give an apparent Oo which is independent o f grain size, contrary to what is observed.
The effect of operational and material variables on the secondary creep rate ~ can be generalized in the form [ 1-3]
~a -A
Db
kT
(1)
where o is the applied stress, o o is a threshold stress, L is the grain diameter, b is Burgers' *Present address: Department of Civil Engineering, University of Peradeniya, Peradeniya, Sri Lanka. 0025-5416/83/$3.00
vector, Ft is the atomic volume (approximately equal to b3), G is the shear modulus (or alternatively Young's modulus E), D is the operative diffusion coefficient, k is Boltzmann's constant and A, n and p depend on the operative creep mechanism. Characteristically, p tends to zero and n is typically a b o u t 4 f o r the strongly stress-dependent slip c r e e p (power law creep) observed at high stresses while p is between 2 and 3 and n is unity for the strongly grain-size-dependent vacancy diffusion creep observed at low stresses for normal grain sizes. Theoretical analysis for vacancy creep predicts that p is 2 and A is a b o u t 13.3 when the lattice diffusion of vacancies is the controlling mechanism [ 4] (Nabarro-Herring creep) and that p is 3 and A is a b o u t 47.7 when the grain boundary diffusion of vacancies is the controlling mechanism [ 5] (Coble creep). Under such conditions, eqn. (1) thus takes the special form = A
~-~ (a -- a0)
(2)
with d~ da
A[b~P Db kT
(Y
(3)
The evidence for the applicability of eqns. (2) and (3) to creep at low stresses (less than or approximately equal to 10-4G or 10-4E) includes experimental observations of (i) the predicted parametric dependences on stress, grain size and temperature, (ii) values of d~/do usually within a factor of 10 of predictions based on radiotracer measurements o f D and on the measured L values and (iii) thicknesses of precipitate-denuded zones, adjacent to grain boundaries subjected to tensile stress, that correspond to the predicted and observed creep strains [3, 6 - 1 6 ] . The © Elsevier Sequoia/Printed in The Netherlands
observed threshold stresses o0 are typically about 10-5G or 10-SE for pure metals at about 0.5T~ [9] where Tm is the temperature of the melting point and are increased by additions of appropriate second phases [16-18]. In a recent publication, Ruano and Sherby [19] have reinterpreted some data for creep at low stresses in terms of a phenomenological alternative [20-22] to eqn. (2) which can be written in the form 4~ _ A
(4)
Db
where ~ is the atomic volume (approximately equal to b a) and A is a b o u t 6 X 10 s when grain boundary diffusion is the controlling mechanis m (p = 3) and is an order of magnitude larger when lattice diffusion controls (p = 2). For the particular case of grain boundary diffusion control, eqn. (4) thus reduces to = A ~-~
(5)
Equation (5) was used to correlate superplasticity data mainly for Zn-A1 eutectoid or Sn-Pb eutectic alloys at (0.5-0.9)T~ for the stress range ( 1 0 - L 1 0 - a ) E or (10-L10-3)G using an empirical relationship D
e x p [ - llTm~
2
T jcm s-
1
[20] for grain boundary diffusivity and estimates for E and b. The correlation fits the observed ~ data typically within a factor of 10 over the range covered, as does eqn. (3) in relation to the appropriate creep data. Ruano and Sherby [ 19] suggest that eqn. (5) indicates control of deformation by grain-boundarydiffusion-controlled grain boundary sliding a c c o m m o d a t e d by slip, on the basis that theoretical predictions [23-25] from that standpoint yield 4 cc o 2 (n = 2 in eqn. (1)), although they differ from eqn. (5) in that (i) cc L -2 (p = 2 in eqn. (1)) is predicted rather than the e cc L-3 dependence of eqn. (5), (ii) the ~xtra variable factor G ~ / k T of eqn. (1) whigh is absent in eqns. (4) and (5)* is included and (iii) 4 values several orders of magnitude lower than that in eqn. (5) are pre*Exclusion of the factor G $ ~ / k T was based o n t h e
fact that it gave a worse fit with the superplasticity data ~vhen included.
dicted. In spite of these disparities with available theory, Ruano and Sherby go on to apply eqn. (5) in particular to some of our results for fi-Co [12], claiming that our results exhibit a parabolic relation with stress and the magnitudes of strain rate given by eqn. (5). They base this conclusion on four approaches: (i) that a Plot of 4~2 versus o for our test SC2-3 at 1073 K can be extrapolated linearly through the origin, (ii) that plots of log versus log(o/E) for our test SC2-3 at 1073 K and test SC7-1 at 773 K show slopes with a value nearer to 2 than unity, (iii) that plots of log(~L3/Db) versus l o g ( o / E ) for these t w o tests show good agreement with the predictions of eqn, (5) using the radiotracer data o f Brik et al. [26] for D b and (iv) that the conditions in tests on fi-Co, ~-Fe, AISI type 304 stainless steel, magnesium and copper, previously interpreted in terms of Coble creep with a threshold stress, fall within the range of eqn. (5) rather than within the region of Coble creep when they are plotted on a deformation map of l o g ( L / b ) versus l o g ( o / E ) at 0.55Tin. We shall deal with each of these points in turn. Concerning item (i), the same plot of ~12 versus o for test SC7-1 at 773 K (Fig. 1, line b), n o t included by Ruano and Sherby, can be extrapolated linearly through the origin only if the points for the four lowest stresses are excluded; these points exhibit the downward curvature consistent with zero strain rate below a threshold stress of 2.7 MPa indicated by the corresponding plot of versus o (Fig. 1, line a). Further examples from other tests on ~-Co are shown in Figs. 2 and 3. Concerning items (ii) and (iii), this
,
•
,
•
,
,
,
.
,
Dj ~. 9 p "6
6
f
(~-%-~) 4 2
e
•
I
~ - - -
-
7J ~
I
4
I
i
6
i
i
a
o'(Mh)
|
10
Fig. 1. Creep data for ~-Co plotted as ~ (o, line a) and 51/2 (e, line b) vs. a (test SC7-1: temperature T = 773 K; mean linear intercept grain size L = 21 #m). T h e data were taken from ref. 12.
~6 S" ° /
6
(s")
(xm'%-l) 4
(x~-s.-~) 2 0
0
0
•i f~ldSjJ ~ ~ aO'{M~}
|0
wO
Fig. 2. tks for Fig. 1 b u t for test SC6-1 ( T = 803 K; /7, = 26 pm). The data were taken f r o m ref. 12.
3~-
. . . . . . . .
,/
-- ~-~0"~:~-' 4
~"""~ e '°'l;
t
(a)
.~a
/
-.
60.(MPa~
|0
12
Fig. 3./ks for Fig. 1 b u t for test SC3-1 ( T = 9 6 3 K ; /~= 3 4 p m ) . The data were taken f r o m ref. 12.
,[ ./. ??. ~'~ (b)
. ..
~
~-4 -~
log(~L3/Db) versus log{(o -- ao)/E} in contrast
Fig. 4. L o g a r i t h m (a) of the strain rate ~ and (b) of ~L3/Db vs. log(((~ -- Oo)/E} for test SC2-3 ( T = 1073 K ; / ~ -- 125 pro) and test SC7-1 ( T = 773 K; £ = 21pro) for ~-Co. The test data were taken f r o m ref. 12, with the mean grain diameter L taken as 1.776/~ and Db = 2.5 X 10 -14 e x p ( - - l l T m / T ) m 3 s-1 [20] w i t h T m = 1765 K for cobalt. The stress O0 is t h a t for zero ~ on plots o f ~ vs. o and the lines shown are for a slope n equal to u n i t y consistent w i t h the operation of vacancy diffusion creep.
are linear with a slope n of u n i t y as shown in Fig. 4, fully consistent with the operation of diffusion creep. In this connection, a further inconsistency concerning the application of eqn. (5) by Ruano and Sherby to predict log(~L3/Db) versus log(alE) is their use of radiotracer data [26] for D rather than the empirical e x p ( - - l l T m / T ) cm 2 s-z [20] on which eqn. (5) was based. Use of the empirical D instead of radiotracer data in their Fig. 4 places the experimental values of ~L3/Db two to three orders of magnitude above those predicted by eqn. (5), indicating that creep is occurring at t w o to three orders of magnitude faster than predicted by eqn. (5) as originally conceived. In respect of item (iv), the value chosen for D in eqn. (5) will correspondingly affect the relative sizes of the fields of dominance of each mechanism in the deformation
map for T/Tm -- 0.55 plotted by Ruano and Sherby as their Fig. 5. The empirical D is 2 X 10 -14 m 2 s-1 at T/Tm = 0.55 compared with 2 X 1 0 -12 m 2 s-1 as used by Ruano and Sherby in their Fig. 5. It is evident that use of this empirical D in eqn. (5) shifts the boundary between the field of dominance of Cable creep and that of eqn. (5) identified with grain boundary sliding on their Fig. 5 from a o/E value of 7 X 10 -5 to a a/E value of 7 X 10 -3, virtually eliminating the field of eqn. (5) from the map and placing the conditions in the tests on ~-Co, ~-Fe, AISI type 304 stainless steel, magnesium and copper all well within the field of dominance of Cable creep. Furthermore, eqn. (5) sheds no light on the
downward curvature of points at low stresses in such tests is evident for test SC7-1 at 773 K even in plots of log ~ and log(~L3/Db) versus log(o/E) (ref. 19, Figs. 3 a n d 4), thereby in-
validating these plots. Such plots of log ~ and
large apparent enhancement of creep rate, by between 40 and 140 times the Coble rate, found for ~-Fe and attributed earlier to transient operation of dislocations as sources, sinks and short-circuit paths for vacancies [9]. Equation (5) predicts creep rates a similar amount lower than observed using radiotracer data for D, the discrepancy being even larger if the empirical value of D is employed. Finally, it is possible to evaluate apparent values of a0 for each test on H-Co on the a s s u m p t i o n that eqn. (5) is applicable and that we fitted in ref. 12 a straight line to results falling on the parabola given by eqn. (5) in a plot of ~ versus o. Thus, for Coble creep with a threshold stress,
A1Db~2(o -- Oo) =aa+b where AI ~ 50, a = A I D b ~ / L 3 k T Equation (5) gives
and b =
- - aOo.
A 2Dbo 2
LaE2
e2 -
= CO 2
with A2 = 108 and c = A2bD]L3E 2. When the line el = aa + b is fitted by the least-squares m e t h o d between the limits 01 and a2, Q = Z(d2--dl) 2 a2
=;
(e2--el) 2d° al a2
= f
(Ca 2 - a a - b )
2dO
al
To find b = -- aoo, we put dQ/db = O. N o w
dQ -- f~ ~d (CO 2 - - a o - - b ) 2 d o
(o2- ol) 3
{ 2 c ( a 2 2 4- 0"10 2 -t-
-- 3a(o2 + ol) -- 6b} =0
b .0 0
=
- -
_ a
= 1 (o2 + ol)
-
1 f(022_1_ 0102-1- 012)
Since c/a is independent of grain size, the resulting a0 should be independent of grain size at a fixed temperature T and in the stress range o2 :> o > a l . On the contrary we find (ref. 12, Fig. 5b) that o0 is n o t only strongly dependent on grain size b u t inversely proportional to it as predicted by the grain boundary dislocation climb source model.
REFERENCES
LakT
~1 :
Thus
0"12 )
-
-
1 T. G. I.~ngdon, in J. C. M. Li and A. K. Mukherjee (eds.), Rate Processes in Plastic Deformation o f Materials, American Society for Metals, Metals Park, OH, 1975, p. 410;Met. Forum, 1 (2) (1978) 59. 2 J. C. Gibeling and W. D. Nix, Mater. Sci. Eng., 45 (1980) 123. 3 T. Sritharan and-H.-Jones, in Mechanical Behaviour and Nuclear Applications o f Stainless Steel at Elevated Temperatures, Metals Society, London, 1982, p. 99. 4 C. Herring, J. Appl. Phys., 21 (1950) 437. 5 R. Coble, J. Appl. Phys., 34 (1963) 1679. 6 H. Jones, Mater. Sci. Eng., 4 (1969) 106. 7 R. B. Jones, J. Sheffield Univ. Metall. Soc., 12 (1973) 34. 8 I. G. Crossland, in J. E. Harris and E. C. Sykes (eds.), Physical Metallurgy o f Reactor Fuel Elements, Metals Society, London, 1975, p. 66. 9 D. J. Towle and H. Jones, Acta Metall., 24 (1976) 399. 10 B. Burton, Diffusional Creep o f Polycrystalline Materials, Trans Tech Publications, Aedermannsdorf, 1977. 11 I. G. Crossland and R. B. Jones, Met. Sci., 11 (1977) 504. 12 T. Sritharan and H. Jones, Acta Metall., 27 (1979) 1293. 13 B. Burton, I. G. Crossland and G. W. Greenwood, Met. Sci., 7 (1980) 134. 14 T. Sritharan and H. Jones, Acta MetaU., 28 (1980) 1633. 15 T. Sritharan and H. Jones, Met. Sci., 15 (1981) 365. 16 B. Burton and W. Beer~, Met. Sci., 12 (1978) 71. 17 W.J. Clegg and J. W. Martin, Met. Sci., 16 (1982) 65. 18 E. Arzt and M. F. Ashby, Scr. Metall., 16 (1982) 1285. 19 O. A. Ruano and O. D. Sherby, Mater. Sci. Eng., 56 (1982) 167.
20 R0 A. White, Ph.D. Thesis, Department of Materials Science, Stanford University, 1978, 21 H. Liithy, R. A. White and O. D. Sherby, Mater. Sc£ Eng., 39 (1979) 211. 22 O. A. Ruano, A. K. Miller and O. D. Sherby, Mater. Sci. Eng., 51 (1981) 9.
23 A. Ball and M. M. Hutchison, Met. Sei. J., 3 (1969) 1, 24 A. K. Mukherjee, Mater. Sci. Eng., 8 (1971) 83. 25 R. C. Gifkins, Metall. Trans. A, 7 (1976) 1225. 26 V. B. Brik, L. N. Larikov and V. Falchenko, Ukr. Fiz. Zh., 20 (1975) 397.