- Email: [email protected]
On the cavity-growth-model-based prediction of the stress dependence of time to creep fracture
Materials Science and Engineering, A117 (1989) L5-L9
L5
Letter
On the cavity-growth-model-based prediction of the stress dependence of time to creep fracture J. CADEK Institute of Physical Metallurgy, Czechoslovak Academy of Sciences, 616 62 Brno (Czechoslovakia)
(Received April 7, 1989)
Abstract A model describing intergranular cavity growth and cavity shape evolution during its growth is applied to the numerical study of the growth of cavities by coupled diffusion and power-law creep, and to predict the stress dependence of time to creep fracture. It is shown that at stresses at which the power-law creep represents an important component of the growth the predicted time to fracture is inversely proportional to the third power of stress independent of the values of the ratio of surface diffusion to grain boundary diffusion conductance, while the minimum creep rate varies with approximately the fifth power of stress. Even if the cavity nucleation process is taken into account, assuming that the nucleation rate is proportional to the power-law creep rate, the predicted time to fracture is not inversely proportional to the creep rate, i.e. to the fifth power of stress. This contradicts the commonly observed proportionality of the product of time to fracture and minimum creep rate to the creep strain to fracture (modified MonkmanGrant (M-G) relationship). The analysis made suggests that the experimentally observed validity of the modified M - G relationship is always associated with the constrained cavity growth which cannot proceed unless accommodated by powerlaw creep.
boundaries. Until recently, the time to intergranular creep fracture (creep life) has been widely believed to depend primarily on the rate of diffusional cavity growth processes. This explains why so much attention has been paid to these processes in the last two decades, which has led to remarkable success in their understanding [1-3]. Recently, a model of unconstrained intergranular cavity growth embracing all the important features of the current models was developed [4]. The model describes the cavity growth in a broad region of conditions and makes it possible to study cavity shape evolution during the growth. In the present paper, the model is applied to study cavity growth by coupled diffusion and power-law creep with special reference to the model-based prediction of the stress dependence of time to creep fracture, taking the process of cavity nucleation into account under the assumption that the cavity nucleation rate is proportional to the creep rate. The cavity growth model is described in ref. 4. All the required information on the cavity growth can be obtained by solving numerically several equations of the model. The numerical method [5, 6] employed is based on the finite difference technique and is similar to that used by Martinez and Nix [7].
2. Conditions of cavity growth modelling For the cavity growth modelling, the data available for a 21Cr-37Ni stainless steel were used. These data, given elsewhere [8], are the grain boundary free energy VB, the surface free energy Ys, the grain boundary diffusion conductance DB6 B at 973 K and the constants go, ao and n in the power-law creep equation
1. Introduction Fracture of polycrystalline metallic materials undergoing high temperature creep under typical service conditions is most frequently intergranular and occurs by the processes of nucleation, growth and coalescence of cavities on grain 0921-5093/89/$3.50
go
( °\ o0/ ¢
Ill
in which a~o is the remotely applied stress and gco is the remote creep rate. The surface diffusion conductance D s 6 s is not known experimentally. © Elsevier Sequoia/Printed in The Netherlands
L6
Therefore, it was varied in a very broad interval so that the ratio
a
Ds 6s -
DB 6 B
(2)
ranged from 10-5 to 103 though only the values of A in the interval from 10-1 to 10 should be considered to be realistic. In the model [4], the concept of a characteristic diffusion path A is applied. This path is defined as [91 ( D B + s a O + ) 1/3
[ limbs - 4.6E-23 Im3s-;I [ Dsbs = 4 6 E - 2 2 [m3s-~] /6 = 135.00 [MPa] 4! A'+ = 1.34 ram] ]1TIME TO FRACTURE t+ =
(3)
. A 2 ~-Z_ZL . _ ~ _ - . . . . . . .
3. Results
The results of cavity growth modelling for aav=135 MPa and A = 1 0 are illustrated in Fig. 1. In Fig. 1, the first quadrant of the axisymmetric cavity axial section is shown revealing cavity profiles in various stages of cavity growth (Curves A). These stages are characterized by times of growth t shown in Fig. 1. The profiles of the local normal stress field corresponding to the respective cavity growth stages are also shown in Fig. 1 (curves B). Using equality a = b as the frac-
.
,+ -
, ~
Z
'
/
,o
,I
~
_1
[1111~ I [k$]
2
0
4
6 8 10 CAVITY RAI)IUS a [am]
Fig. 1. Profiles of the axial section of an axially symmetric cavity in various stages of its growth (curves A) and corresponding profiles of the local normal stress field (curves B). The diffusional conductance ratio A = Dsrs/DB6 B = 10; the conditions of modelling are given in the figure. The time to fracture tf was computed using the fracture criterion a = b.
1~8 B = 4.6E-23 ImPs-l] I ~&s = 4.6E'23[m3s-~] 4"A6 " = 1 3 5 ' 0 0 [MPa]
(4)
where O,v is in megapascals. In this expression, the average normal stress G~ acting on the grain boundary facet was set for the remotely applied stress 000. The stress aav follows from the model [4]. The inititial cavity radius a t was chosen as 0.099/~m and the half cavity spacing b was fixed to 10.0 pro. From eqn. (4) it follows that the stress aav = 34.6 MPa corresponds to A = b (sintering stress G ° = 32.9 MPa). At this stress, the cavities grow by diffusion alone. At stresses aa~ higher than 34.6 MPa the characteristic diffusion path A < b and therefore cavities grow by coupled diffusion and power-law creep. In the present paper, the average normal stresses ranging from 34.6 to 135 MPa are considered. At O~v= 135 MPa the characteristic diffusion path A = 0.134b.
B ,
1400~
100 ~
where ~ is the atomic volume and k T has its usual meaning. Using eqn. (1) and the appropriate data mentioned, the following expression for A (in micrometres) is obtained A = 1790/aav 3/2
, 7~
e~ = 0.10 [pm] b = 100 Cram] A - CAVITY PROFILE B - LOCAL STRESS ~7178Iks]
[TII
=
1.34
ai b A B
Ilam]
= = -
0.10 [~m] 10.0 Cue] CAVITY PROFILE LOCAL STRESS
T I !
1400'~ ~ a.
200~
!100.a ::c e
,0 0
2
4
6 8 CAVITY RADIUS a Ilam]
Fig. 2. As in Fig. 1 but for A = 1.
ture criterion, the time to fracture tf given in Fig. 1 was computed. Since A < b, the cavity growth occurs by coupled diffusion and powerlaw creep. When the diffusion conductances ratio A ~ 1, the growing cavity becomes crack-like rather quickly. This is illustrated in Figs. 2 and 3 respectively for A = I and A=0.1 and Oav = 135 MPa. Figure 4 shows the time dependences of the current cavity radius a, the jacking or grain boundary contribution z, the radial cavity growth rate ti and the jacking or grain boundary opening velocity g for A = I and aav=135MPa (see Fig. 2). From Fig. 4 it can be seen that no steady state radial cavity growth takes place and therefore only the minimum radial cavity growth rate ami n c a n be defined. It can be seen that at t/tf > 0.8 the current radial cavity growth rate is m a n y times higher than ami n.
L7
a, = 0.18 [pea] b = 10.0 [~ea]
! l)ea e = 4.6E-;~3[ea3S-']
]
: Bs~ s = 4 . 6 E - 2 4 i d~ = 135.018
[eaz~t]
[MPa] ~4~/t = 1.34 [wm] i TIME TO FRACTURE t~ = tJ i
"1:
/ 1
A - CAUITY PROFILE B -
STRESS
LOCAL
138.54
Iks]
,,
/
i
7
~
;
,'
I'
I
i
7481D'~ ]
i
% .
~ "o J381~
i
--
A =1790 6 ~
i
]
1BB
%--. e
2
"%--a\ o \ ' 1 ~ -
6
4
6 10 CAVITY RAglUS a [pm] Q =
IJ
b =
100.urn
Fig. 3. A s in Fig. 1 but for A = 0.1.
1
If in
ks
De6e=/,6xiO m S A - D s 6slO be
Boy in MPo,
[ De&a = 4.6E-23 [m~-1] 0[ DsSs= 4.6E-23 [m3s-11 i
~A6~ F,'at 6~b [!t, [i
--
1 // //"
/ ,,'.'
__:___ :/......"
i ',,
0
•
z2
Fig. 5. Relationship between time to fracture t~ and the average normal stress acting on the grain boundary facet o~.
x .N
The characteristic diffusion path A was calculated using eqn. (4). C u r v e a, A = 1 ; curve b, the same value of A, A = b (cavities are assumed to grow by diffusion alone); curve c,
41~'" x .t0
J
t/t~
2o log 6ov
6
/./,,;.."
3
I
-I
~[22~" 1
~
to
E
== 135.00 1.34 tMPa] tum] = 0.10 [gm] = 10.00 [.m] = 286.34 tk$] --a-ai /
~ "~,.
"%-23 3
A = 103; curve d, A = 103, A = b; C u r v e e, A = 10 5; C u r v e f, A = 10-5, A = b.
0 1
---
....
i
i
i
m=3
t
i
~
Fig. 4. A n example of the variation of the current cavity
radius a, the jacking or grain boundary opening contribution z, the current radial cavity growth rate d and the jacking or grain boundary opening velocity ~ with time of growth t normalized to the time to fracture tf computed using the fracture criterion a = b. The conditions of modelling are the same as in Fig. 2.
\
~o3/2°~.~..O~o.~ i
I0 G
tfoc l/Oav m
(5)
In Fig. 5, the computed times to fracture are plotted against stress o~v for various values of the diffusion conductance ratio A (eqn. (2)). Let us first consider the log tf vs. IOgOav plot for A = 1 (curve a). The plot is not linear, the exponent m increases from a value close to ~ at 0 , v = 3 4 . 6 MPa to a value close to three at Oav = 135 MPa. In Fig. 5, the linear relationship between logtf and logoav obtained assuming that A = b, i.e. that cavities grow by diffusion alone, is shown for comparison (curve b). In this case, m -~ 3 is in agreement with the results [10] shown in Fig. 6. At A = 103 the exponent m decreases from a value close to unity at a,~ = 34.6 MPa to a value
100~arn
A = 100/Jn)
E
..... I
When predicted on the basis of a cavity growth model, the time to fracture tf is inversely proportional to the mth power of stress
b = ~NNN "
I
l
104
10-2
o~o..
m =1
;
I
100
102
I
I¢
& Ds6s IDa 6e =
Fig. 6. Variation of the exponent m in eqn. (5) with the diffusion conductance ratio A. The cavities are assumed to grow by diffusion alone (A = b) [10].
close to three at o,v = 135 MPa (curve c). Again, the relationship between logt, and 1OgOav obtained assuming A = b is shown for comparison (curve d). In this case, the exponent m is equal to 1.06 which is in agreement with the results [10] shown in Fig. 6. When the surface diffusion conductance D s 6 s is much less than the grain boundary diffusion conductance DBOB,e.g. when A = 10 -5, the relationship between log r e and log O,v is linear (curve c); thus, the exponent m does not vary with O,v and its value is close to three. The relationship is practically identical with that obtained assuming A to be equal to b (curve f, cf Fig. 6).
L8
Thus, from the results presented in Fig. 5 it follows that at the stresses aav at which the diffusion paths A is much shorter than the half cavity spacing a, and at which the power-law creep thus represents an important component of the growth, the value of the exponent m in eqn. (5) is close to three independent of the value of A. At very low values of A, the power-law creep does not affect the predicted time to fracture and its stress dependence at all.
4. Discussion
One of the most remarkable features of the intergranular fracture in high temperature creep of metallic materials is the rather general validity of the well-known Monkman-Grant (M-G) relationship [11] which, as modified by Dobe~ and Mili~ka [12], is expressed as tf = C o c f / ~ m i n m°
(6)
where ~minis the minimum creep rate, ef the creep strain to fracture and CO and m 0 are constants. There is ample experimental evidence (see, for example, refs. 13 and 14) that m 0 is close to unity. Thus the time to fracture should be inversely proportional to the power-law creep rate and, hence, to the nth power of applied stress o~o provided the stress dependence of ef is insignificant compared with that of ~min, which is usually the case [13, 14]. The above predictions of stress dependence of time to fracture based on the cavity growth model are not in accord with the rather general validity of the modified M - G relationship even at stresses at which power-law creep represents a significant component of cavity growth. Hence, it may seem that it is the process of cavity nucleation that is primarily responsible for the validity of the M - G relationship. It is well known that the cavity nucleation proceeds continuously during creep and that the rate of cavity nucleation is related to the power-law creep rate [2]. In fact, experimental evidence is available that the cavity nucleation rate varies with the creep rate linearly (see, for example, ref. 15). However, from the recent theoretical analysis of Yu and Chung [16] (see also ref. 3) it follows that when the cavity nucleation rate is proportional to the power-law creep rate and the time to fracture predicted on the basis of a cavity growth model is inversely proportional to the third power of stress, the expo-
nent m' in the relationship tf oc 1/ 0 ~ m'
(7)
is , 6+5n m 9
(8)
(cf. ref. 3). Now, if n = 5.4, which is the case of the 21Cr-37Ni stainless steel [8], m'=3.8. Hence again, the M - G relationship does not hold. This strongly suggests that the experimentally observed validity of the modified M - G relationship with m 0 ~-1 is associated with the constrained cavity growth which cannot proceed unless accommodated by power-law creep [17, 18, 2].
5. Conclusions
(1) The above cavity-growth-model-based prediction of time to fracture (Fig. 5) starts with the assumption that at the time t = 0 all the cavities (having an equilibrium shape and the radius chosen) are already present. Thus, the process of cavity nucleation is avoided in the prediction similarly as in creep fracture experiments on metals and alloys with cavities implanted prior to creep. Such experiments, performed for instance on silver [19] and copper [20] at stresses low enough to avoid additional cavity nucleation, show that the time to fracture is reduced considerably and is inversely proportional to the reciprocal third power of stress, while the creep rate depends on the fifth power of stress. Hence, for metals with cavities implanted prior to creep the M - G relationship does not hold. (2) The relationship between time to fracture and the reciprocal of the third power of stress found for the above metals was interpreted [19, 20] in terms of surface diffusion controlled growth by diffusion alone. However, as can be seen from Fig. 6, such an interpretation requires A(=Ds6s/DBdB) to be at least as low as 10 -4 while it is well known [21] that the surface diffusion conductance is higher than the grain boundary diffusion conductance in pure metals. The results presented above show that under the condition of avoided nucleation the time to fracture may be inversely proportional to the third power of stress even if Dsd s > DBdiB, e.g. A = 10. This suggests another interpretation of the above-mentioned stress dependence of time
L9
to creep fracture of silver and copper with cavities implanted prior to creep, i.e. the interpretation in terms of the cavity growth by coupled diffusion and power-law creep. Obviously, to support this suggestion, a detailed analysis of the creep and the cavity growth conditions in the experiments of Nix et al. [19, 20] is necessary. (3) Contemporary models of constrained cavity growth (see, for example, ref. 22) make it possible to predict the time to cavity coalescence, but not the time to fracture. However, even such predictions must consider the process of cavity nucleation [3]. Acknowledgment The author thanks Dr. J. Svoboda for performing the computations on which the present paper is based.
References 1 A. C. F. Cocks and M. F. Ashby, Prog. Mater. Sci., 27 (1982) 189. 2 A. S. Argon, in B. Wilshire and D. R. J. Owen (eds.), Recent Advances in Creep and Fracture of Engineering Materials and Structures, Pineridge Press, Swansea, 1982, p. 1.
3 H. Riedel, Fracture at High Temperature, Springer, Berlin, 1987. 4 J. Svoboda and J. Cadek, Mater. Sci. Eng., 93 (1987) 135. 5 J. Svoboda and J. Cadek, Mater. Sci. Eng., 96 (1987) 65. 6 J. Svoboda and J. ~adek, Acta Tech. (Prague), 32 (1987) 637. 7 L. Martinez and W. D. Nix, Metall. Trans. A, 13 (1982) 427. 8 J. Svoboda and J. Cadek in, E O. Kettunen, T. K. Lepist6 and M. E. Lehtonen (eds.), Proc. 8th Int. Conf. on the Strength of Metals and Alloys, Pergamon, Oxford, 1988, Vol. 2, p. 911. 9 A. Needleman and J. R. Rice, Acta Metall., 28 (1980) 1315. 10 J. Svoboda and J. Cadek, Res. Rep. VZ 747/879, 1988 (Institute of Physical Metallurgy, Czechoslovak Academy of Sciences, Brno, Czechoslovakia). 11 E C, Monkman and N. J. Grant, Proc. ASTM, 56 (1956) 593. 12 E Dobe~ and K. Mili6ka, Met. Sci. J., 10(1976) 382. 13 K. Mili6ka and E Dobe~, Met. Mater., 15 (1977) 186. 14 M. Pahutov~, K. Kucha~ov~iand J. ~adek, J. Nucl. Mater., 131 (1985) 20. 15 N. C. Needham and T. Gladman, Met. Sci. J., 14 (1980) 64. 16 Jin Yu and J. O. Chung, Scr. Metall., 22 (1988) 27. 17 B.F. Dyson, Met. Sci. J., 10(1976) 349. 18 J. R. Rice, Acta Metall., 29(1981)675. 19 S.H. Goods and W. D. Nix, Acta Metall., 26 (1978) 753. 20 T. G. Nieh and W. D. Nix, Acta Metall., 28 (1980) 557. 21 T.-J. Chung, K. J. Kagawa, J. R. Rice and L. B. Sills, Acta Metall., 27(1979) 265. 22 I.-W.Chen, Scr. Metall., 17 ( 1983) 17.
L5
Letter
On the cavity-growth-model-based prediction of the stress dependence of time to creep fracture J. CADEK Institute of Physical Metallurgy, Czechoslovak Academy of Sciences, 616 62 Brno (Czechoslovakia)
(Received April 7, 1989)
Abstract A model describing intergranular cavity growth and cavity shape evolution during its growth is applied to the numerical study of the growth of cavities by coupled diffusion and power-law creep, and to predict the stress dependence of time to creep fracture. It is shown that at stresses at which the power-law creep represents an important component of the growth the predicted time to fracture is inversely proportional to the third power of stress independent of the values of the ratio of surface diffusion to grain boundary diffusion conductance, while the minimum creep rate varies with approximately the fifth power of stress. Even if the cavity nucleation process is taken into account, assuming that the nucleation rate is proportional to the power-law creep rate, the predicted time to fracture is not inversely proportional to the creep rate, i.e. to the fifth power of stress. This contradicts the commonly observed proportionality of the product of time to fracture and minimum creep rate to the creep strain to fracture (modified MonkmanGrant (M-G) relationship). The analysis made suggests that the experimentally observed validity of the modified M - G relationship is always associated with the constrained cavity growth which cannot proceed unless accommodated by powerlaw creep.
boundaries. Until recently, the time to intergranular creep fracture (creep life) has been widely believed to depend primarily on the rate of diffusional cavity growth processes. This explains why so much attention has been paid to these processes in the last two decades, which has led to remarkable success in their understanding [1-3]. Recently, a model of unconstrained intergranular cavity growth embracing all the important features of the current models was developed [4]. The model describes the cavity growth in a broad region of conditions and makes it possible to study cavity shape evolution during the growth. In the present paper, the model is applied to study cavity growth by coupled diffusion and power-law creep with special reference to the model-based prediction of the stress dependence of time to creep fracture, taking the process of cavity nucleation into account under the assumption that the cavity nucleation rate is proportional to the creep rate. The cavity growth model is described in ref. 4. All the required information on the cavity growth can be obtained by solving numerically several equations of the model. The numerical method [5, 6] employed is based on the finite difference technique and is similar to that used by Martinez and Nix [7].
2. Conditions of cavity growth modelling For the cavity growth modelling, the data available for a 21Cr-37Ni stainless steel were used. These data, given elsewhere [8], are the grain boundary free energy VB, the surface free energy Ys, the grain boundary diffusion conductance DB6 B at 973 K and the constants go, ao and n in the power-law creep equation
1. Introduction Fracture of polycrystalline metallic materials undergoing high temperature creep under typical service conditions is most frequently intergranular and occurs by the processes of nucleation, growth and coalescence of cavities on grain 0921-5093/89/$3.50
go
( °\ o0/ ¢
Ill
in which a~o is the remotely applied stress and gco is the remote creep rate. The surface diffusion conductance D s 6 s is not known experimentally. © Elsevier Sequoia/Printed in The Netherlands
L6
Therefore, it was varied in a very broad interval so that the ratio
a
Ds 6s -
DB 6 B
(2)
ranged from 10-5 to 103 though only the values of A in the interval from 10-1 to 10 should be considered to be realistic. In the model [4], the concept of a characteristic diffusion path A is applied. This path is defined as [91 ( D B + s a O + ) 1/3
[ limbs - 4.6E-23 Im3s-;I [ Dsbs = 4 6 E - 2 2 [m3s-~] /6 = 135.00 [MPa] 4! A'+ = 1.34 ram] ]1TIME TO FRACTURE t+ =
(3)
. A 2 ~-Z_ZL . _ ~ _ - . . . . . . .
3. Results
The results of cavity growth modelling for aav=135 MPa and A = 1 0 are illustrated in Fig. 1. In Fig. 1, the first quadrant of the axisymmetric cavity axial section is shown revealing cavity profiles in various stages of cavity growth (Curves A). These stages are characterized by times of growth t shown in Fig. 1. The profiles of the local normal stress field corresponding to the respective cavity growth stages are also shown in Fig. 1 (curves B). Using equality a = b as the frac-
.
,+ -
, ~
Z
'
/
,o
,I
~
_1
[1111~ I [k$]
2
0
4
6 8 10 CAVITY RAI)IUS a [am]
Fig. 1. Profiles of the axial section of an axially symmetric cavity in various stages of its growth (curves A) and corresponding profiles of the local normal stress field (curves B). The diffusional conductance ratio A = Dsrs/DB6 B = 10; the conditions of modelling are given in the figure. The time to fracture tf was computed using the fracture criterion a = b.
1~8 B = 4.6E-23 ImPs-l] I ~&s = 4.6E'23[m3s-~] 4"A6 " = 1 3 5 ' 0 0 [MPa]
(4)
where O,v is in megapascals. In this expression, the average normal stress G~ acting on the grain boundary facet was set for the remotely applied stress 000. The stress aav follows from the model [4]. The inititial cavity radius a t was chosen as 0.099/~m and the half cavity spacing b was fixed to 10.0 pro. From eqn. (4) it follows that the stress aav = 34.6 MPa corresponds to A = b (sintering stress G ° = 32.9 MPa). At this stress, the cavities grow by diffusion alone. At stresses aa~ higher than 34.6 MPa the characteristic diffusion path A < b and therefore cavities grow by coupled diffusion and power-law creep. In the present paper, the average normal stresses ranging from 34.6 to 135 MPa are considered. At O~v= 135 MPa the characteristic diffusion path A = 0.134b.
B ,
1400~
100 ~
where ~ is the atomic volume and k T has its usual meaning. Using eqn. (1) and the appropriate data mentioned, the following expression for A (in micrometres) is obtained A = 1790/aav 3/2
, 7~
e~ = 0.10 [pm] b = 100 Cram] A - CAVITY PROFILE B - LOCAL STRESS ~7178Iks]
[TII
=
1.34
ai b A B
Ilam]
= = -
0.10 [~m] 10.0 Cue] CAVITY PROFILE LOCAL STRESS
T I !
1400'~ ~ a.
200~
!100.a ::c e
,0 0
2
4
6 8 CAVITY RADIUS a Ilam]
Fig. 2. As in Fig. 1 but for A = 1.
ture criterion, the time to fracture tf given in Fig. 1 was computed. Since A < b, the cavity growth occurs by coupled diffusion and powerlaw creep. When the diffusion conductances ratio A ~ 1, the growing cavity becomes crack-like rather quickly. This is illustrated in Figs. 2 and 3 respectively for A = I and A=0.1 and Oav = 135 MPa. Figure 4 shows the time dependences of the current cavity radius a, the jacking or grain boundary contribution z, the radial cavity growth rate ti and the jacking or grain boundary opening velocity g for A = I and aav=135MPa (see Fig. 2). From Fig. 4 it can be seen that no steady state radial cavity growth takes place and therefore only the minimum radial cavity growth rate ami n c a n be defined. It can be seen that at t/tf > 0.8 the current radial cavity growth rate is m a n y times higher than ami n.
L7
a, = 0.18 [pea] b = 10.0 [~ea]
! l)ea e = 4.6E-;~3[ea3S-']
]
: Bs~ s = 4 . 6 E - 2 4 i d~ = 135.018
[eaz~t]
[MPa] ~4~/t = 1.34 [wm] i TIME TO FRACTURE t~ = tJ i
"1:
/ 1
A - CAUITY PROFILE B -
STRESS
LOCAL
138.54
Iks]
,,
/
i
7
~
;
,'
I'
I
i
7481D'~ ]
i
% .
~ "o J381~
i
--
A =1790 6 ~
i
]
1BB
%--. e
2
"%--a\ o \ ' 1 ~ -
6
4
6 10 CAVITY RAglUS a [pm] Q =
IJ
b =
100.urn
Fig. 3. A s in Fig. 1 but for A = 0.1.
1
If in
ks
De6e=/,6xiO m S A - D s 6slO be
Boy in MPo,
[ De&a = 4.6E-23 [m~-1] 0[ DsSs= 4.6E-23 [m3s-11 i
~A6~ F,'at 6~b [!t, [i
--
1 // //"
/ ,,'.'
__:___ :/......"
i ',,
0
•
z2
Fig. 5. Relationship between time to fracture t~ and the average normal stress acting on the grain boundary facet o~.
x .N
The characteristic diffusion path A was calculated using eqn. (4). C u r v e a, A = 1 ; curve b, the same value of A, A = b (cavities are assumed to grow by diffusion alone); curve c,
41~'" x .t0
J
t/t~
2o log 6ov
6
/./,,;.."
3
I
-I
~[22~" 1
~
to
E
== 135.00 1.34 tMPa] tum] = 0.10 [gm] = 10.00 [.m] = 286.34 tk$] --a-ai /
~ "~,.
"%-23 3
A = 103; curve d, A = 103, A = b; C u r v e e, A = 10 5; C u r v e f, A = 10-5, A = b.
0 1
---
....
i
i
i
m=3
t
i
~
Fig. 4. A n example of the variation of the current cavity
radius a, the jacking or grain boundary opening contribution z, the current radial cavity growth rate d and the jacking or grain boundary opening velocity ~ with time of growth t normalized to the time to fracture tf computed using the fracture criterion a = b. The conditions of modelling are the same as in Fig. 2.
\
~o3/2°~.~..O~o.~ i
I0 G
tfoc l/Oav m
(5)
In Fig. 5, the computed times to fracture are plotted against stress o~v for various values of the diffusion conductance ratio A (eqn. (2)). Let us first consider the log tf vs. IOgOav plot for A = 1 (curve a). The plot is not linear, the exponent m increases from a value close to ~ at 0 , v = 3 4 . 6 MPa to a value close to three at Oav = 135 MPa. In Fig. 5, the linear relationship between logtf and logoav obtained assuming that A = b, i.e. that cavities grow by diffusion alone, is shown for comparison (curve b). In this case, m -~ 3 is in agreement with the results [10] shown in Fig. 6. At A = 103 the exponent m decreases from a value close to unity at a,~ = 34.6 MPa to a value
100~arn
A = 100/Jn)
E
..... I
When predicted on the basis of a cavity growth model, the time to fracture tf is inversely proportional to the mth power of stress
b = ~NNN "
I
l
104
10-2
o~o..
m =1
;
I
100
102
I
I¢
& Ds6s IDa 6e =
Fig. 6. Variation of the exponent m in eqn. (5) with the diffusion conductance ratio A. The cavities are assumed to grow by diffusion alone (A = b) [10].
close to three at o,v = 135 MPa (curve c). Again, the relationship between logt, and 1OgOav obtained assuming A = b is shown for comparison (curve d). In this case, the exponent m is equal to 1.06 which is in agreement with the results [10] shown in Fig. 6. When the surface diffusion conductance D s 6 s is much less than the grain boundary diffusion conductance DBOB,e.g. when A = 10 -5, the relationship between log r e and log O,v is linear (curve c); thus, the exponent m does not vary with O,v and its value is close to three. The relationship is practically identical with that obtained assuming A to be equal to b (curve f, cf Fig. 6).
L8
Thus, from the results presented in Fig. 5 it follows that at the stresses aav at which the diffusion paths A is much shorter than the half cavity spacing a, and at which the power-law creep thus represents an important component of the growth, the value of the exponent m in eqn. (5) is close to three independent of the value of A. At very low values of A, the power-law creep does not affect the predicted time to fracture and its stress dependence at all.
4. Discussion
One of the most remarkable features of the intergranular fracture in high temperature creep of metallic materials is the rather general validity of the well-known Monkman-Grant (M-G) relationship [11] which, as modified by Dobe~ and Mili~ka [12], is expressed as tf = C o c f / ~ m i n m°
(6)
where ~minis the minimum creep rate, ef the creep strain to fracture and CO and m 0 are constants. There is ample experimental evidence (see, for example, refs. 13 and 14) that m 0 is close to unity. Thus the time to fracture should be inversely proportional to the power-law creep rate and, hence, to the nth power of applied stress o~o provided the stress dependence of ef is insignificant compared with that of ~min, which is usually the case [13, 14]. The above predictions of stress dependence of time to fracture based on the cavity growth model are not in accord with the rather general validity of the modified M - G relationship even at stresses at which power-law creep represents a significant component of cavity growth. Hence, it may seem that it is the process of cavity nucleation that is primarily responsible for the validity of the M - G relationship. It is well known that the cavity nucleation proceeds continuously during creep and that the rate of cavity nucleation is related to the power-law creep rate [2]. In fact, experimental evidence is available that the cavity nucleation rate varies with the creep rate linearly (see, for example, ref. 15). However, from the recent theoretical analysis of Yu and Chung [16] (see also ref. 3) it follows that when the cavity nucleation rate is proportional to the power-law creep rate and the time to fracture predicted on the basis of a cavity growth model is inversely proportional to the third power of stress, the expo-
nent m' in the relationship tf oc 1/ 0 ~ m'
(7)
is , 6+5n m 9
(8)
(cf. ref. 3). Now, if n = 5.4, which is the case of the 21Cr-37Ni stainless steel [8], m'=3.8. Hence again, the M - G relationship does not hold. This strongly suggests that the experimentally observed validity of the modified M - G relationship with m 0 ~-1 is associated with the constrained cavity growth which cannot proceed unless accommodated by power-law creep [17, 18, 2].
5. Conclusions
(1) The above cavity-growth-model-based prediction of time to fracture (Fig. 5) starts with the assumption that at the time t = 0 all the cavities (having an equilibrium shape and the radius chosen) are already present. Thus, the process of cavity nucleation is avoided in the prediction similarly as in creep fracture experiments on metals and alloys with cavities implanted prior to creep. Such experiments, performed for instance on silver [19] and copper [20] at stresses low enough to avoid additional cavity nucleation, show that the time to fracture is reduced considerably and is inversely proportional to the reciprocal third power of stress, while the creep rate depends on the fifth power of stress. Hence, for metals with cavities implanted prior to creep the M - G relationship does not hold. (2) The relationship between time to fracture and the reciprocal of the third power of stress found for the above metals was interpreted [19, 20] in terms of surface diffusion controlled growth by diffusion alone. However, as can be seen from Fig. 6, such an interpretation requires A(=Ds6s/DBdB) to be at least as low as 10 -4 while it is well known [21] that the surface diffusion conductance is higher than the grain boundary diffusion conductance in pure metals. The results presented above show that under the condition of avoided nucleation the time to fracture may be inversely proportional to the third power of stress even if Dsd s > DBdiB, e.g. A = 10. This suggests another interpretation of the above-mentioned stress dependence of time
L9
to creep fracture of silver and copper with cavities implanted prior to creep, i.e. the interpretation in terms of the cavity growth by coupled diffusion and power-law creep. Obviously, to support this suggestion, a detailed analysis of the creep and the cavity growth conditions in the experiments of Nix et al. [19, 20] is necessary. (3) Contemporary models of constrained cavity growth (see, for example, ref. 22) make it possible to predict the time to cavity coalescence, but not the time to fracture. However, even such predictions must consider the process of cavity nucleation [3]. Acknowledgment The author thanks Dr. J. Svoboda for performing the computations on which the present paper is based.
References 1 A. C. F. Cocks and M. F. Ashby, Prog. Mater. Sci., 27 (1982) 189. 2 A. S. Argon, in B. Wilshire and D. R. J. Owen (eds.), Recent Advances in Creep and Fracture of Engineering Materials and Structures, Pineridge Press, Swansea, 1982, p. 1.
3 H. Riedel, Fracture at High Temperature, Springer, Berlin, 1987. 4 J. Svoboda and J. Cadek, Mater. Sci. Eng., 93 (1987) 135. 5 J. Svoboda and J. Cadek, Mater. Sci. Eng., 96 (1987) 65. 6 J. Svoboda and J. ~adek, Acta Tech. (Prague), 32 (1987) 637. 7 L. Martinez and W. D. Nix, Metall. Trans. A, 13 (1982) 427. 8 J. Svoboda and J. Cadek in, E O. Kettunen, T. K. Lepist6 and M. E. Lehtonen (eds.), Proc. 8th Int. Conf. on the Strength of Metals and Alloys, Pergamon, Oxford, 1988, Vol. 2, p. 911. 9 A. Needleman and J. R. Rice, Acta Metall., 28 (1980) 1315. 10 J. Svoboda and J. Cadek, Res. Rep. VZ 747/879, 1988 (Institute of Physical Metallurgy, Czechoslovak Academy of Sciences, Brno, Czechoslovakia). 11 E C, Monkman and N. J. Grant, Proc. ASTM, 56 (1956) 593. 12 E Dobe~ and K. Mili6ka, Met. Sci. J., 10(1976) 382. 13 K. Mili6ka and E Dobe~, Met. Mater., 15 (1977) 186. 14 M. Pahutov~, K. Kucha~ov~iand J. ~adek, J. Nucl. Mater., 131 (1985) 20. 15 N. C. Needham and T. Gladman, Met. Sci. J., 14 (1980) 64. 16 Jin Yu and J. O. Chung, Scr. Metall., 22 (1988) 27. 17 B.F. Dyson, Met. Sci. J., 10(1976) 349. 18 J. R. Rice, Acta Metall., 29(1981)675. 19 S.H. Goods and W. D. Nix, Acta Metall., 26 (1978) 753. 20 T. G. Nieh and W. D. Nix, Acta Metall., 28 (1980) 557. 21 T.-J. Chung, K. J. Kagawa, J. R. Rice and L. B. Sills, Acta Metall., 27(1979) 265. 22 I.-W.Chen, Scr. Metall., 17 ( 1983) 17.