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A statistical theory of steady-state creep and its application to type 316 steel, zinc, magnox AL80 and nickel
A Mech.
Phys.Solids,1965, Vol. 13,pp.69to 75. Peqgnmon Pnss Ltd. Printedin GreatBritain.
A STATISTICAL THEORY OF STEADY-STATE CREEP AND ITS APPLICATION TO TYPE 316 STEEL, ZINC, MAGNOX ALso AND NICKEL By J. H. G~TTUS U.K.A.E.A., Springtfelds R.F.L., Salwick, Preston (Remhed 44h Novmaber,
1904)
SUMMARY
AN EXPRESSION is derived for the creep of a specimen containing obstacles which offer variable resistance to the passage of dislocations. The expression accurately represents the available data for the steady-state creep of type 818 steel, Zinc, Magnox AL80 and Nickel for temperaturecompensated strain-rates ranging over up to ten orders of magnitude. The materials behave as though they comprise a statistical assembly of obstacles having norm~y-debuted potential energies which the moving disloeations either circumvent at a tern~~t~-de~ndent rate determined by the Maxwell-Bolt~nlann di~ribution or penetrate with a probability determined by the applied stress.
TItE steady-state creep-rate IFof metals and alloys is strongly dependent on the applied stress ucl, equations of the following type being commonly found to apply over limited ranges of i : i cc aan (1)
i cc exp 0,.
(9)
W~~RTMAN (1957) has proposed a theory that explains (1) in terms of the effects of the applied stress on the height to which a dislocation in a piled-up group must climb, if it is to surmount the barrier presented by the stress field of the group, and on the rate of climb. For comparatively high values of a, (2) approximates the EYRING (1936) rate-process viscosity equation, applied to metals by KAUZMANN (1941). These theories do not take account of the essentially statistical nature of the creep process. Thus, although it is widely accepted that the temperature dependence of creep-rate is attributable to the statistical distribution of thermal energy, little attention has been paid to the possibility of analogous explanations for the stress-dependence of steady-state creep, although the application of the concept of Statisti~alIy-distributed activation-stresses to Andrade creep and logarithmic creep has had some success (KENNIDY 1962). Accordingly an attempt has been made to evolve and test a model for steady state creep which is based on the concept that metals and alloys contain obstacles to deformation of statistically variable efficacy.
69
J. H. GITKS
70 2.
T~IF, STATISTICAL
MODEL
The consensus resulting from metallographic: observations is that metals and alloys contain obstacles that oppose creep by preventing the free passage of dislocations. It is also established that in a given specimen these obstacles are not all equally effective. In developing it will be convenient to consider stress
a creep mode1 based on these premises and temperature separately, as though
their effects were consecutive. Consider a random line of unit length drawn
through the volume of a creepspecimen. Let the total number of obstacles to deformation which it intersects be N, of which a number noa,a are too weak to oppose the movement of dislocations when a stress of ocris applied to the specimen. Then
and the average distance between the lrrn,ocrobstacles of strength lying between ua and co will be
If the random
line is one along which segments
a stress oa is applied to the specimen
then the strain
of dislocations de produced
move
when
will be propor-
tional to IIO&O1earn = n,,,W
-
n%,o ) = ~~~,O~(l -
P,,o)
/N_ the probability that a given obstacle will be too weak to prohibit passage of dislocations under the applied stress. At this juncture, in the absence of thermal activation, deformation would cease, all of the dislocations being confronted by obstacles which prohibit their further movement. However, during a subsequent period of time dt, a fraction of the dislocations FD proportional to dt exp (- Q/M’) will climb away from the obstacles into positions where they will be confronted by a distribution of obstacles identical to that which prevailed at the outset. Accordingly, under the influence of the applied stress they will again move, producing a strain proportional to
wheref’,,, = n,,,
Once more further deformation is prohibited, a fraction (l -PO) of the dislocations being trapped by the original obstacles and a fraction FD by the new obstacles. Since there is no difference between the original obstacles and the new ones, the condition of the specimen is now the same as it was before thermally-activated climb occurred and the processes of climb and slip will repeat identically, the same increment of strain being produced in each period dt. That is, (3) 2 exp (Q/W = Go P,,d(l - Po,,J where P,, is the value of i exp (Q~RT) when P,,, creep-law given by the present model.
= 0.5.
Equation (3) is the
A statisticaltheory of steady-state 3.
71
creep and its application
OF THE STATISTICALMODEL TO MAGNOX ALSO, ETC.
APPLICATION
TO apply (3) to a specific alloy we need to know i,, and also the stress-dependence of p,, 0’ Data for Type 316 stainless-steel (GARAFALO and RICHMOND 1963), Zinc (GILMAN 1956), Magnox AL80 (HARRIS and JONES 1963; CIIURCHMAN et al. 196%) and Nickel (WEERTMAB and SIIAIIINIAN 1956) have been used to test the feasibility of the model. By repeated approximation it was established that the best fit to each set of data is given by a normal distribution of log a,, with standard deviation u* and lgo set at the values listed in Table 1. The excellent fit so produced is shown in Fig. 1 where mean values of Pom,o have been plotted against log a, on probability
TABLE 1.
Values
Q MateriO 316 steel
cal mole-1 75 x 103
of
Q, a8 and i,, for various makriakr urn
08 log10 a
650
log10 OJII
0.21
6 x 10’0
34M log10psi
vi
---
4300
Zinc
38.1
x 10s
0.30
4 x 108
6.9 log19 dyne cm-9
115
Magnox AL80
32.1 x 109
0.26
6 x 109
2.47 log19 psi
296
Nickel
65 x 10’
0,23
6 x 109
7.92 log19 dyne cm-s
1210
paper : if the model is correct, points in this diagram should (and in fact do) lie on straight lines. None of the materials conformed to equation (1) or equation (Q), a feature which is exemplified for Magnox AL80 in Fig. 2. Accordingly it is concluded that, over the range of values covered by the data presented in Fig. I, the creep-rate, d per hour, is given within the reproducibility of replicate tests by the expression
with
where Q is an activation energy (Table 1), R is the gas constant (- Q), T is the absolute temperature, iso = P exp [Q/m] when P,,,= 0.50 (Table 1), a8 is the standard deviation of log,, a, (Table 1), ucris the externally applied stress, Us is the stress that produces a creep-rate i, (Table 1).
72
J. H. WITUS a999999 a99999
a999
3990
Id 39.00
35
BO
50
2:
5
I
5-I P
FIQ. 1.
,0x10*
A statistical theory of steady-state creep and ita application 4.
78
DISCUSSION
The finding, that the four materials of Table 1 all behave in steady state creep as though they contain obstacles, whose deformation-resistance conforms to a log-normal distribution, is somewhat analogous to the situation prevailing in the interpretation of the creep of polymers. Thus, for such materials, successful models have been proposed which involve the concept of component elements having a log-normal distribution of relaxation-times (KENNEDY 1962; FELTHAM 1955). 10mo-
L-
’OOh
3.0 ‘00
U.T. S,
lb/in2
FIQ. 8. In the present case the implication is that (for example) there are as many obstacles of twice the average strength as there are obstacles with half the average strength, while the normal form of the distribution indicates that many random factors operate simultaneously to determine the strength of an obstacle. As would be expected, CQ,,correlates with several time-independent measures of deformation-resistance. For example, Fig. 6 shows how it varies with the room-temperature ultimate tensile-strengths of the four materials studied in this work. At strain rates lower than those covered by the present data, a deviation from (3) can be expected to be produced by the increasing dominance of a creepmechanism involving stress-directed diffusion such as has been proposed by NABARRO (1948) and later elaborated by HERRING (1950) and MCLEAN (1958).
74
J. H. Grrrus
The model may also be expected to break down at stresses approaching the yield-point. The model does not predict the form of the P,,, v emus ucrdistribution, which therefore may be different for materials other than those studied here. However, as yet there are few other materials whose creep properties have been determined over a range of temperature-compensated strain-rates wide enough to permit determination of their P,,, distribution functions with precision. The fact that it is the logarithm of a,, and not aa itself, which exhibits a normal distribution suggests the existence of some other normally-distributed parameter, related to log aaS Now cr@is the maximum force needed to drive a dislocation through an obstacle. In the simple case where the obstacle is another dislocation then when the distance of separation is x, the force c acting between them will be given by a = K/x where K is a constant, and oa = K/X%, where x= is the distance of closest approach. Hence the work done is =a
w=
K
I
~=10g%=10g0, -logOa X Xl
where 0% is the stress experienced by the dislocation at a distance x1 from the opposing dislocation. Therefore W is normally distributed. This is a reasonable conclusion since it is plausible that the potential energies of the obstacles conform to conditions which GAUSS (1821)showed to be the premises from which the normal distribution function derives, namely : (a) {b)
ZAW = 0, where AW is the deviation of the potential energy of an obstacle from the mean value, The mean value of the potential energy has maximum probability.
Theoretical arguments leading to the normal distribution function are renowned for their circular nature and recourse has often been had to experimental data as a means for its ultimate verification. For example, JEFFREYS (1939) describes some experiments planned to reproduce the conditions of certain observations in astronomy and analyses the results with the conclusion that the deviations from the mean follow the normal law closely over the range - 2% to + Z&J. Viewed in this light, the data of Fig. 1 approximately verify the normal law over the range - it.Eia to -+ 2-90 (i.e. 994 per cent of the total probability).
5.
CONCLUSIONS
A model has been proposed which predicts that a metal or alloy containing obstacles to dislocation-movement of statistically-variable efficacy will creep at a steady state rate, 2, given by
where Pas, o is the probability that a given obstacle will be unable to prevent dislocations from moving when the specimen is subjected to a stress ua and i, is a material constant. This equation accurately represents the steady-state creep of type 816 steel,
A statistical theory of steady-state creep and its application
75
Zinc, Magnox AL80 and Nickel, for which materials the logarithm of aa proves to be normally distributed and is shown to be a measure of the potential energy of an obstacle. Accordingly a metal or alloy may now be visualized as a statistical assembly of obstacles whose potential energy is normally distributed. During creep, dislocations either circumvent these obstacles at a temperature-dependent rate determined by the Maxwell-Boltzmann distribution or penetrate them with a probability determined by the applied stress. ACKNOWIEDGMENT I am grateful to Mr. R. V. Moore, Managing Director of the U.K.A.E.A. permission to publish this paper.
Reactor Group, for
REFERENCES CHURCEMAN, A. T., GREENWOOD,G. W., B~DDE~Y, J. H., HESKETH,R. V., EDMONDSON,B. and WmAMSON, G. K.
EYRINQ, H. FELTIXAM,P. GAXLOFALO, F., RIcnMoND, c., Dams, W. F., VON GEMMINGEN,F.
1964
1986 1955
1988
1821 1956
Fundamental studies of materiuls problems in gas-cooled reactors. A/CONF.28/A/156. 8rd U.N. Conf. on the Peaceful Uses of Atomic Energy. J. Chem. Phys. 4, 288. BriL J. Appl. Phys. 6, 26.
Strain-lime. rate-stress and rats-temperature relations during large defomuziions in creep. Joint International Conf. on Creep. Paper 80, Am. Soe. Mech. Eng. Theoria Combinationis Observationum (Gottingen). J. Metals 8, 1826.
GAUSS, C. F. GILMAN, J. J. HNUUS, J. E. and JONES, R. B. HERRING,C. JEFPREYS,H. KAUZXANN, W. KENNEDY, A.J.
1963 1950 1989 1941 1962
MCLEAN, D.
1958
NABARRO, F.R.N. WEERTMAN,J. WEERTMAN,J. and SHAHINIAN P.
1945 1957
C.E.G.B. Report RDIBIR.144. J. Appl. Phys. 21, 487. Phil. Trans. Boy. Sot. A 237, 231. Trans. Amer. Inst. Min. Metal Erg. 143, 57. Processes of creep and fatigue in metals. (Oliver & Boyd, London). Inst. Met. Symposium, Vacancies and other point defects in Metals and Allays, 159. Proc. Bristol Conf. Strength of Solids, 75. J. Appl. Phys. 28, 1185.
1956
J. Metals
8, 1223.
Phys.Solids,1965, Vol. 13,pp.69to 75. Peqgnmon Pnss Ltd. Printedin GreatBritain.
A STATISTICAL THEORY OF STEADY-STATE CREEP AND ITS APPLICATION TO TYPE 316 STEEL, ZINC, MAGNOX ALso AND NICKEL By J. H. G~TTUS U.K.A.E.A., Springtfelds R.F.L., Salwick, Preston (Remhed 44h Novmaber,
1904)
SUMMARY
AN EXPRESSION is derived for the creep of a specimen containing obstacles which offer variable resistance to the passage of dislocations. The expression accurately represents the available data for the steady-state creep of type 818 steel, Zinc, Magnox AL80 and Nickel for temperaturecompensated strain-rates ranging over up to ten orders of magnitude. The materials behave as though they comprise a statistical assembly of obstacles having norm~y-debuted potential energies which the moving disloeations either circumvent at a tern~~t~-de~ndent rate determined by the Maxwell-Bolt~nlann di~ribution or penetrate with a probability determined by the applied stress.
TItE steady-state creep-rate IFof metals and alloys is strongly dependent on the applied stress ucl, equations of the following type being commonly found to apply over limited ranges of i : i cc aan (1)
i cc exp 0,.
(9)
W~~RTMAN (1957) has proposed a theory that explains (1) in terms of the effects of the applied stress on the height to which a dislocation in a piled-up group must climb, if it is to surmount the barrier presented by the stress field of the group, and on the rate of climb. For comparatively high values of a, (2) approximates the EYRING (1936) rate-process viscosity equation, applied to metals by KAUZMANN (1941). These theories do not take account of the essentially statistical nature of the creep process. Thus, although it is widely accepted that the temperature dependence of creep-rate is attributable to the statistical distribution of thermal energy, little attention has been paid to the possibility of analogous explanations for the stress-dependence of steady-state creep, although the application of the concept of Statisti~alIy-distributed activation-stresses to Andrade creep and logarithmic creep has had some success (KENNIDY 1962). Accordingly an attempt has been made to evolve and test a model for steady state creep which is based on the concept that metals and alloys contain obstacles to deformation of statistically variable efficacy.
69
J. H. GITKS
70 2.
T~IF, STATISTICAL
MODEL
The consensus resulting from metallographic: observations is that metals and alloys contain obstacles that oppose creep by preventing the free passage of dislocations. It is also established that in a given specimen these obstacles are not all equally effective. In developing it will be convenient to consider stress
a creep mode1 based on these premises and temperature separately, as though
their effects were consecutive. Consider a random line of unit length drawn
through the volume of a creepspecimen. Let the total number of obstacles to deformation which it intersects be N, of which a number noa,a are too weak to oppose the movement of dislocations when a stress of ocris applied to the specimen. Then
and the average distance between the lrrn,ocrobstacles of strength lying between ua and co will be
If the random
line is one along which segments
a stress oa is applied to the specimen
then the strain
of dislocations de produced
move
when
will be propor-
tional to IIO&O1earn = n,,,W
-
n%,o ) = ~~~,O~(l -
P,,o)
/N_ the probability that a given obstacle will be too weak to prohibit passage of dislocations under the applied stress. At this juncture, in the absence of thermal activation, deformation would cease, all of the dislocations being confronted by obstacles which prohibit their further movement. However, during a subsequent period of time dt, a fraction of the dislocations FD proportional to dt exp (- Q/M’) will climb away from the obstacles into positions where they will be confronted by a distribution of obstacles identical to that which prevailed at the outset. Accordingly, under the influence of the applied stress they will again move, producing a strain proportional to
wheref’,,, = n,,,
Once more further deformation is prohibited, a fraction (l -PO) of the dislocations being trapped by the original obstacles and a fraction FD by the new obstacles. Since there is no difference between the original obstacles and the new ones, the condition of the specimen is now the same as it was before thermally-activated climb occurred and the processes of climb and slip will repeat identically, the same increment of strain being produced in each period dt. That is, (3) 2 exp (Q/W = Go P,,d(l - Po,,J where P,, is the value of i exp (Q~RT) when P,,, creep-law given by the present model.
= 0.5.
Equation (3) is the
A statisticaltheory of steady-state 3.
71
creep and its application
OF THE STATISTICALMODEL TO MAGNOX ALSO, ETC.
APPLICATION
TO apply (3) to a specific alloy we need to know i,, and also the stress-dependence of p,, 0’ Data for Type 316 stainless-steel (GARAFALO and RICHMOND 1963), Zinc (GILMAN 1956), Magnox AL80 (HARRIS and JONES 1963; CIIURCHMAN et al. 196%) and Nickel (WEERTMAB and SIIAIIINIAN 1956) have been used to test the feasibility of the model. By repeated approximation it was established that the best fit to each set of data is given by a normal distribution of log a,, with standard deviation u* and lgo set at the values listed in Table 1. The excellent fit so produced is shown in Fig. 1 where mean values of Pom,o have been plotted against log a, on probability
TABLE 1.
Values
Q MateriO 316 steel
cal mole-1 75 x 103
of
Q, a8 and i,, for various makriakr urn
08 log10 a
650
log10 OJII
0.21
6 x 10’0
34M log10psi
vi
---
4300
Zinc
38.1
x 10s
0.30
4 x 108
6.9 log19 dyne cm-9
115
Magnox AL80
32.1 x 109
0.26
6 x 109
2.47 log19 psi
296
Nickel
65 x 10’
0,23
6 x 109
7.92 log19 dyne cm-s
1210
paper : if the model is correct, points in this diagram should (and in fact do) lie on straight lines. None of the materials conformed to equation (1) or equation (Q), a feature which is exemplified for Magnox AL80 in Fig. 2. Accordingly it is concluded that, over the range of values covered by the data presented in Fig. I, the creep-rate, d per hour, is given within the reproducibility of replicate tests by the expression
with
where Q is an activation energy (Table 1), R is the gas constant (- Q), T is the absolute temperature, iso = P exp [Q/m] when P,,,= 0.50 (Table 1), a8 is the standard deviation of log,, a, (Table 1), ucris the externally applied stress, Us is the stress that produces a creep-rate i, (Table 1).
72
J. H. WITUS a999999 a99999
a999
3990
Id 39.00
35
BO
50
2:
5
I
5-I P
FIQ. 1.
,0x10*
A statistical theory of steady-state creep and ita application 4.
78
DISCUSSION
The finding, that the four materials of Table 1 all behave in steady state creep as though they contain obstacles, whose deformation-resistance conforms to a log-normal distribution, is somewhat analogous to the situation prevailing in the interpretation of the creep of polymers. Thus, for such materials, successful models have been proposed which involve the concept of component elements having a log-normal distribution of relaxation-times (KENNEDY 1962; FELTHAM 1955). 10mo-
L-
’OOh
3.0 ‘00
U.T. S,
lb/in2
FIQ. 8. In the present case the implication is that (for example) there are as many obstacles of twice the average strength as there are obstacles with half the average strength, while the normal form of the distribution indicates that many random factors operate simultaneously to determine the strength of an obstacle. As would be expected, CQ,,correlates with several time-independent measures of deformation-resistance. For example, Fig. 6 shows how it varies with the room-temperature ultimate tensile-strengths of the four materials studied in this work. At strain rates lower than those covered by the present data, a deviation from (3) can be expected to be produced by the increasing dominance of a creepmechanism involving stress-directed diffusion such as has been proposed by NABARRO (1948) and later elaborated by HERRING (1950) and MCLEAN (1958).
74
J. H. Grrrus
The model may also be expected to break down at stresses approaching the yield-point. The model does not predict the form of the P,,, v emus ucrdistribution, which therefore may be different for materials other than those studied here. However, as yet there are few other materials whose creep properties have been determined over a range of temperature-compensated strain-rates wide enough to permit determination of their P,,, distribution functions with precision. The fact that it is the logarithm of a,, and not aa itself, which exhibits a normal distribution suggests the existence of some other normally-distributed parameter, related to log aaS Now cr@is the maximum force needed to drive a dislocation through an obstacle. In the simple case where the obstacle is another dislocation then when the distance of separation is x, the force c acting between them will be given by a = K/x where K is a constant, and oa = K/X%, where x= is the distance of closest approach. Hence the work done is =a
w=
K
I
~=10g%=10g0, -logOa X Xl
where 0% is the stress experienced by the dislocation at a distance x1 from the opposing dislocation. Therefore W is normally distributed. This is a reasonable conclusion since it is plausible that the potential energies of the obstacles conform to conditions which GAUSS (1821)showed to be the premises from which the normal distribution function derives, namely : (a) {b)
ZAW = 0, where AW is the deviation of the potential energy of an obstacle from the mean value, The mean value of the potential energy has maximum probability.
Theoretical arguments leading to the normal distribution function are renowned for their circular nature and recourse has often been had to experimental data as a means for its ultimate verification. For example, JEFFREYS (1939) describes some experiments planned to reproduce the conditions of certain observations in astronomy and analyses the results with the conclusion that the deviations from the mean follow the normal law closely over the range - 2% to + Z&J. Viewed in this light, the data of Fig. 1 approximately verify the normal law over the range - it.Eia to -+ 2-90 (i.e. 994 per cent of the total probability).
5.
CONCLUSIONS
A model has been proposed which predicts that a metal or alloy containing obstacles to dislocation-movement of statistically-variable efficacy will creep at a steady state rate, 2, given by
where Pas, o is the probability that a given obstacle will be unable to prevent dislocations from moving when the specimen is subjected to a stress ua and i, is a material constant. This equation accurately represents the steady-state creep of type 816 steel,
A statistical theory of steady-state creep and its application
75
Zinc, Magnox AL80 and Nickel, for which materials the logarithm of aa proves to be normally distributed and is shown to be a measure of the potential energy of an obstacle. Accordingly a metal or alloy may now be visualized as a statistical assembly of obstacles whose potential energy is normally distributed. During creep, dislocations either circumvent these obstacles at a temperature-dependent rate determined by the Maxwell-Boltzmann distribution or penetrate them with a probability determined by the applied stress. ACKNOWIEDGMENT I am grateful to Mr. R. V. Moore, Managing Director of the U.K.A.E.A. permission to publish this paper.
Reactor Group, for
REFERENCES CHURCEMAN, A. T., GREENWOOD,G. W., B~DDE~Y, J. H., HESKETH,R. V., EDMONDSON,B. and WmAMSON, G. K.
EYRINQ, H. FELTIXAM,P. GAXLOFALO, F., RIcnMoND, c., Dams, W. F., VON GEMMINGEN,F.
1964
1986 1955
1988
1821 1956
Fundamental studies of materiuls problems in gas-cooled reactors. A/CONF.28/A/156. 8rd U.N. Conf. on the Peaceful Uses of Atomic Energy. J. Chem. Phys. 4, 288. BriL J. Appl. Phys. 6, 26.
Strain-lime. rate-stress and rats-temperature relations during large defomuziions in creep. Joint International Conf. on Creep. Paper 80, Am. Soe. Mech. Eng. Theoria Combinationis Observationum (Gottingen). J. Metals 8, 1826.
GAUSS, C. F. GILMAN, J. J. HNUUS, J. E. and JONES, R. B. HERRING,C. JEFPREYS,H. KAUZXANN, W. KENNEDY, A.J.
1963 1950 1989 1941 1962
MCLEAN, D.
1958
NABARRO, F.R.N. WEERTMAN,J. WEERTMAN,J. and SHAHINIAN P.
1945 1957
C.E.G.B. Report RDIBIR.144. J. Appl. Phys. 21, 487. Phil. Trans. Boy. Sot. A 237, 231. Trans. Amer. Inst. Min. Metal Erg. 143, 57. Processes of creep and fatigue in metals. (Oliver & Boyd, London). Inst. Met. Symposium, Vacancies and other point defects in Metals and Allays, 159. Proc. Bristol Conf. Strength of Solids, 75. J. Appl. Phys. 28, 1185.
1956
J. Metals
8, 1223.