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Creep in metals at intermediate temperatures and low stresses: a review
Materials &'ience and Engineering, A 137 ( 1991 ) 1632172
163
Creep in metals at intermediate temperatures and low stresses: a review J. Fiala, L. KIoc and J. (~adek Institute of Physical Metallurgy, Czechoslovak Academy of Sciences, 616 62 Brno (Czechoslovakia)
Abstract In the present paper, some recent results on viscous creep in a-Fe and a-Zr are reviewed and some results of investigations of creep in a Cu-14A1 solid solution alloy at intermediate temperatures and low stresses are presented and discussed. The "low stresses" are specified as those causing steady state creep rates lower than 10- ~0s- 1 at "intermediate temperatures" ranging from 0.4 Tmto 0.6 Tm(Tm is the melting temperature). It is shown that the creep in a-Fe and a-Zr results from stress-directed diffusional transport of matter via the grain boundaries (Coble creep) up to an intercept grain size of about 120/~m, while at the intercept grain sizes above this value Harper-Dorn creep controlled by dislocation core diffusion operates. In creep of a Cu-14AI solid solution alloy, two flow processes act in parallel: (i) the viscous flow process identified with Coble creep and (ii) the non-viscous flow process characterized by a strain rate proportional to applied stress squared, inversely proportional to the intercept grain size cubed, and by the activation energy close to the activation enthalpy of grain boundary diffusion. The latter flow process was identified with the creep due to grain boundary sliding accommodated by slip in thin zones adjoining the sliding boundaries. In the solid solution alloy under consideration, Harper-Dorn creep does not take place up to an intercept grain size of 270 ~m at least.
1. Introduction The low stress high temperature creep mechanisms have been treated in the literature for four decades. Only about ten years ago was it recognized that similar mechanisms may operate at intermediate temperatures (around half the melting temperature). However, the experimental evidence for this recognition is rather limited since the obtention of necessary data is difficult and usually extremely time consuming. Undoubtedly, an understanding of flow mechanisms operating under such conditions of stress and temperature may be of considerable importance for engineering practice, i.e. in design and in the development of performance codes which are used in this practice. A most important common characteristic of all the low stress creep mechanisms consists in a weak dependence of steady state creep rate on applied stress. In fact, the steady state creep rate 0921-5093/91/$3.50
is characterized by a value of the applied stress exponent lower than 3; frequently the creep rate varies with the applied stress linearly, i.e. viscous creep takes place. The mechanisms known at present are listed in Table 1. In this table, T means the absolute temperature, o the applied stress, o 0 the threshold stress and d the mean grain diameter. The creep due to diffusional transport of matter via the lattice (Nabarro-Herring creep) or via the grain boundaries (Coble creep) is one of the few instances in materials science where theoretical predictions have preceded experimental verification. Another deformation mode, known as H a r p e r - D o r n creep and first observed in 1957, has been thought until recently to take place at homologous temperatures higher than about 0.95, stresses lower than about 5 × 1 0 - 6 G ( G is the shear modulus) and mean grain diameters © Elsevier Sequoia/Printed in The Netherlands
164 TABLE 1 Basic mechanisms of creep at low stresses Deformation process
Creep
Rate-controlling process
Accommodating process
Creep rate dependence on the following c7
d
References Theory
Experiment
Nabarro-Herring
Diffusional transport of matter via lattice
Lattice diffusion
Grain boundary sliding
ua
l/d*
[L 21
[3-51
Coble
Diffusional transport of matter via grain boundaries
Grain boundary diffusion
Grain boundary sliding
ua
l/d’
bl
]5>7381
Diffusional interface controlled: solid solution alloys
Diffusional transport of matter via lattice and/or grain boundaries
Emission and/or absorption of vacancies by grain boundaries
Grain boundary sliding
d
Lid
[91
[lo-121
Diffusional interface controlled: particles containing alloys
Diffusional transport of matter via lattice and/or grain boundaries
Emission and/or absorption of vacancies by grain boundaries
Grain boundary sliding
iu- %I2
l/d
[91
[13-151
Diffusional enhanced by grain boundary migration (solid solution alloys)
Diffusional transport of matter enhanced by grain boundary migration
Grain boundary diffusionh
Gram boundary sliding
U’
l/d’
[I61
Grain boundary sliding“
Grain boundary sliding
Grain boundary diffusion
Dislocation slip in narrow zones adjoining grain boundaries
U2
l/d2’
[19-211
[18,22,23]
Motion of lattice dislocations
Lattice diffusion
U
None
[24,25]
[24,26,27]
Motion of lattice dislocations
Dislocation core diffusion
(Ja
None
[281
[29-331
Harper-Darn, temperatures Harper-Darn, intermediate temperatures
high
“Small but non-zero threshold stress is frequently observed experimentally. hThe presence of a component with a significantly higher diffusivity in the system is required. ‘Threshold behaviour can be expected. %reep occurring by a mechanism of superplasticity. r& K 1/d 3 proposed by Sherby and Wadsworth [ 171 and experimentally observed by Kloc et al. [ 181.
larger than about 500 pm. However, in the last decade it has been shown that Harper-Dorn creep can take place as well not only at much lower homologous temperatures but also at much smaller grain diameters. The superplastic deformation mode was first observed more than 70 years ago and it remained a scientific curiosity until about 25 years ago. In all the flow mechanisms listed in Table 1 except the mechanism of Harper-Dorn creep, the grain boundaries play a most important role. This is manifested by variation in creep rate with mean grain diameter. On the contrary, the HarperDorn creep rate is, in a similar way to the power law creep rate, independent of grain diameter. This strongly suggests that Harper-Dorn creep is due to the motion of lattice dislocations.
The creep due to stress-directed diffusional transport of matter involves two consecutive processes: emission and/or absorption of vacancies by grain boundaries and the diffusion of emitted vacancies either via the lattice or via the grain boundaries. The creep is then diffusion controlled only if the grain boundaries act as perfect sources and sinks for vacancies. NabarroHerring or Coble creep then occur. This is, however, almost certainly not the case in metals and alloys containing dispersed particles. In fact, since some of these particles are located at grain boundaries the latter cannot serve as perfect sources and sinks for vacancies, and the creep, although still resulting from diffusional transport of matter, is controlled by the processes at grain boundaries (interface-controlled diffusional
165
creep). The interface control is associated with a significant threshold stress (Bingham behaviour). Also, in some solid solution alloys the diffusional creep can be expected to be interface controlled. However, experimental evidence for interfacecontrolled creep in solid solution alloys is not yet sufficiently convincing. Independently of whether it is interface controlled or not the diffusional creep in solid solution alloys may be enhanced by grain boundary migration. The grain boundary migration enhanced diffusional (diffusioncontrolled) creep in multicomponent systems has been analysed by Chen [16]; however, experimental evidence for its existence in metallic alloys is still missing. On the contrary, there seems to be ample experimental evidence for creep due to a mechanism of superplasticity. In the present paper, some results of recent investigations of viscous creep in a-Fe and a-Zr at temperatures close to 0.5T m are reviewed. Also, some results of investigations of creep in a Cu-14A1 solid solution alloy are presented and discussed. It is shown that while in the above pure metals Coble and Harper-Dorn creep take place, in the solid solution alloy a mechanism of superplasticity operates in parallel with the Coble mechanism, while the Harper-Dorn mechanism essentially does not contribute to the creep rate even at large grain sizes. 2. On the techniques for investigating creep at intermediate temperatures and low stresses
In investigations of viscous (diffusional and Harper-Dorn) creep at temperatures close to or even higher than 0.95Tin the conventional isothermal tensile creep test technique is usually applied. Since the viscous creep strains are generally small and the creep rates are very low, creep specimen gauge lengths several times greater than those typical for conventional power law creep tests are frequently used to increase sensitivity of creep strain measurement. In principle, a similar technique can be applied as well to study viscous creep at much lower temperatures, namely at temperatures close to or even lower than 0.5T m. However, to measure creep rates as low as 10 -]2 S-1 or even lower, extremely long-term creep tests are generally required. This serious disadvantage can be removed using the helicoid spring specimen technique [5]. In fact, the sensitivity of creep strain measurement is almost two orders of magnitude
higher as compared with the tensile creep test, which enables the duration of creep tests to be reduced to quite an acceptable level, especially taking into account the fact that several creep curves (up to 20) for various stresses can be obtained in a single experiment (using a single specimen) at a given temperature. Of course, the stress is distributed nonuniformly across the cross-section of the helicoid spring specimen wire. This has been known [34, 35] to be a serious disadvantage of this technique when the creep behaviour is non-viscous. Recently, however, a new numerical procedure enabling a correction for redistribution of stress across the specimen wire cross-section during non-viscous creep has been developed [35]. The application of this procedure completely eliminates the above disadvantage of this test technique. All the results presented in this paper were obtained using the technique just mentioned. The technique as well as the corresponding apparatus have been described elsewhere [36]. 3. Results
3.1. a-Fe The low stress creep in a-Fe of 99.96% purity was investigated at temperatures of 723 K-918 K (i.e. 0.40 Tin-0.51 Tin) and at stresses ranging from 0.12 to 3.50 MPa. The intercept grain size /~ ranged from 82 # m to 478 pro. (The mean grain diameter d is related to the intercept grain size by d = 1.57L.) Figure 1 shows a creep mechanism map in the coordinates (normalized stress o/G, normalized intercept grain size L/b) (G is the shear modulus, b is the modulus of the Burgers vector). The map was obtained by employing more than
3 - 1 o 6 ~
T
i.I ..a
6
\ "~
10F I
I : I ; Oo HI
o ~
I I
I ~
~
T'
HARPER
-
"ri
-7 5.10
.
-6 10
1
0
CREEP
i
,,,, i* I .~,I
~
-T i o, -=1
I
3
' ',,
DORN
% II~ 1 ~, ~,I I! ,-'
I
5
-6 5,10
I
-5 10
~
15.0
5/6
Fig. 1. A map of creep mechanisms in a-Fe (99.96 mass%) for a temperature of 843 K in the coordinates (normalized intercept grain size/2/b, normalized stress a/G).
166
250 experimental creep curves relating to a temperature of 843 K, i.e. 0.47 Tm. The boundaries in the map delineate regions of conditions of the stresses and the intercept grain sizes under which particular creep mechanisms are dominating. In the map, also the curves representing various steady state creep rates ranging from 5 x 10 -12 s -~ to 1 x 10 -1° S - 1 a r e shown. As expected, Coble creep dominates both Harper-Dorn creep and power law creep at small values of Lib and/ or o/G. The Coble creep rate is proportional to stress and inversely proportional to the third power of the mean grain diameter. For its activation energy a value of 150+ 15 kJ mo1-1 was found, which is in very good agreement with the values of activation enthalpy of grain boundary diffusion reported by various researchers [37-41 ] and ranging from 140.0 kJ mo1-1 to 173.8 kJ mol- 1. The results obtained for Coble creep differ from the predictions of the theory [6] in two respects: (i) a threshold stress o 0 is observed below which Coble creep does not proceed, and (ii) the creep rates measured are about two orders of magnitude higher than those calculated on the basis of Coble theory. The threshold stress for diffusional creep in metals has been reported by a number of researchers [9, 42-45]. However, none of its proposed interpretations is free of objections. The threshold stress for Coble creep in a-Fe seems to be somehow related to the role of impurities. This is illustrated in Fig. 2, in which the threshold stress o 0 is plotted against the reciprocal 1/T of temperature for irons of various purities. The threshold stress apparently increases with decreasing purity at any given temperature. Also, the above-mentioned difference between measured creep rates and the creep rates calculated from Coble theory is not surprising, since a 10 0
i
i
I
I
I
I
10.1
16o.9
I
1.0
1.1 1.2 103/T [K-I]
I
1.3
1.4
Fig. 2. The threshold stress a 0 for diffusionai creep plotted against the reciprocal of temperature for irons of different purities: e, 99.8 mass% [45]; o, 99.96 mass% [33].
similar difference was reported by other workers for various metals. Again, the attempts to account for this difference have not yet been successful. Probably, the main reason consists in the difference between the model structure of uniform spherical grains assumed in the theory and the real grain structure. However, a relation expressing an effective value of the shape constant in the Coble equation in terms of measurable parameters of real structure characterized by a distribution of dimensions of grains varying in shapes is not yet available. For Harper-Dorn creep operating at intercept grain sizes L larger than 128 p m a value of the activation energy Qc = 153.3 + 9.3 kJ mol- 1 was found. This value is much lower than that of the activation enthalpy of lattice diffusion. Although it is close to the value of the activation enthalpy for grain boundary diffusion the creep under consideration is certainly not controlled by grain boundary diffusion since its rate does not depend on grain size. Thus it seems quite natural to identify the activation energy of creep with the activation enthalpy A H c of dislocation core diffusion. The value of A H c for a-Fe is not known from experiment. However, it can be estimated using the relation [46] A H c = 11RT mwhere R is the gas constant. Accordingly, AH~ should be close to 165 kJ mol-1, which is only slightly higher than the above value of Qc. Thus, it can be concluded that Harper-Dorn creep in a-Fe at temperatures close to 0.5 Tm is probably dislocation core diffusion controlled. It is interesting to note that lattice-diffusioncontrolled Nabarro-Herring creep does not operate in the region of conditions of temperatures, stresses and intercept grain sizes considered here. At stresses only slightly higher than the threshold stress, Coble creep operates at normalized intercept grain sizes [,/b as large as 3 x 106. At normalized stresses in the interval 2 x 10-6-8 × 10 -6, a direct transition from Coble to Harper-Dorn creep occurs at a value of Lib as l o w as 5 × 10 -6 ( Z = 123 ~m). At L/bhigher than the latter value a transition from Harper-Dorn creep to power law creep with o / G increasing is observed; at lower values of L/b, Coble creep acts in parallel with power law creep. The transition from viscous creep to power law creep is illustrated in Fig. 3 (it should be mentioned that the power law creep rate does not depend on grain size). Despite a gap between the measurements obtained by means of the helicoid
167 I
i05
POWER-LAW CREEP ks = 5Ee
o/ 10 6
6
O
1
9 9
oo
//
157 6aI ? ,.-, u
COBLE/CREEP
//
(-9
P /o
,~~,e/
/
.S-"
/
1 |
I
1./"
lu j--"
J~%" 1
-
//
/-"I
-'~//" /
//
\
•~ 10~
J
//
..:4 " "iU-
I
1~ e
/ j
j-/jJT.
10 p
101( 5• 5o
i011 ,,..
IC]'
/I
' 2.0
-
NO CREEP
212
~ 2.4
TraIT
1012 n ''~" I 1(] 1 10 0
K 101
I 10 2
10 3
[MPo] Fig. 3. Stress dependence of steady state creep rate in a-Fe. Two sets of data were obtained by different creep test techniques. Power law creep, the tensile stress creep tests technique [47]: o, T= 900 K,/~ = 30/~. Transition from viscous to power law cre_ep, the helicoid spring specimen technique. 0, T=918 K, L = l 1 8 /~m; ", T=843 K, L = 1 2 2 /~m; ~z, T = 823 K,/~ = 82 #m.
spring specimen technique (transition from viscous to power law creep) and those obtained by means of the constant tensile stress isothermal test technique (power law creep), the agreement between both sets of measurements seems to be very good. 3.2. ct-Zr The creep in zirconium was investigated at temperatures ranging from 748 K to 1023 K (0.40Tm to 0.54Tin), stresses in the interval 0.08 MPa-2.19 MPa and intercept grain sizes ranging from 48 ~m to 342/~m. Some of the results are presented in Fig. 4 in the form of a creep mechanism map in the coordinates (normalized stress (r/G, reciprocal Tm/T of homologous temperature). The map relates to the mean intercept grain size of 100 #m. Using the above coordinates, only two of the three creep mechanisms observed (Coble, Harper-Dorn and power law creep mechanisms) can be represented by a single map for any intercept grain size. Harper-Dorn creep operates only at intercept grain sizes larger than 127 ~tm, while Coble creep operates at intercept grain sizes smaller than this value. To determine
Fig. 4. A m_apof creep mechanismsin a-Zr for an intercept grain size L = 100/am in the coordinates(normalizedstress a/G, reciprocal Tm/T of homologoustemperature). precisely enough the field of power law creep, the data published by Pahutovfi and Cadek [48] (obtained by a constant tensile stress isothermal creep test technique) were used. The value of the activation energy obtained for Coble creep was found to be 123.9+3.2 kJ mol-1, i.e. close to the expected value of the activation enthalpy of grain boundary diffusion [49]. For Harper-Dorn creep a value of the activation energy much lower than the activation enthalpy of lattice diffusion was found, namely 121.8+11.2 kJ mo1-1 Again, this value was identified with that of the activation enthalpy of dislocation core diffusion, although an estimate using the above relation [46] AH c = 11RT m provides a value as high as 171.8 kJ mol-1. However, Sargent and Ashby [50] deduced from creep data a significantly lower value of 124 kJ mol -~, to which the obtained value of the activation energy of Harper-Dorn creep is very close. In the same way as in a-Fe, Nabarro-Herring creep in a-Zr was not observed under the above experimental conditions. However, at high enough temperatures, Nabarro-Herring creep can be expected to take place at the expense of Coble, and especially Harper-Dorn, creep. 3.3. Cu-14Al solid solution alloy The emission and/or absorption of vacancies by grain boundaries is thought to be associated with motion of grain boundary dislocations. Consequently, grain boundary dislocation solute drag
168 100 / l
,
"
"
-
I
I
POWER-LAWCREEP ~ 65
10
"--....
773K
"tn .it)
2
0
I
6 - 60
i 50
NO CREEP
i 100
I 200
I# //o
/
~,o,
~---
COBLE CREEP
0.1
/
///"
@737f'~
1 .............................
l
tO t..J
;%
NON-VISCOUSCREEP
LJ
I
~:r
,
1
I
2
I
3
0 [MPa] 500
Fig. 6. Stress dependence of steady state creep rate in a
Fig. 5. A map of creep mechanisms in Cu-14AI solid solution alloy for temperatures of 773 K and 873 K in the coordinates (applied stress o, mean intercept grain size/~ ).
Cu-14Al solid solution alloy at a temperature of 848 K for a mean intercept grain size of 145 ~tm: o, measured values of strain rate ~R at various stresses ORE on the specimen wire surface; , the relation g(a) calculated from the relation eR(aRE) using the procedure described in ref. 35.
occasionally results in interface control of diffusional creep [3] (Table 1 ). Another effect of a solute may consist in enhancement of diffusional creep by grain boundary migration [16] if this solute exhibits a significantly higher diffusivity than the solvent (in the case of binary solid solution alloy)(Table 1 ). Thus, the knowledge of diffusional creep in pure metals is in general not directly transferable to the solid solution alloys. Of course, this may also apply to Harper-Dorn creep. This is why an extensive investigation of low stress creep in solid solution alloys at intermediate temperatures has been undertaken by the present authors. In this section, some results for a Cu-14AI (6.5 mass.% AI) solid solution alloy are presented. Using again the helicoid spring specimen technique, the creep was investigated at temperatures ranging from 723 K to 873 K (0.55 Tmto 0.66 Tm), stresses ranging from 0.2 MPa to 5.0 MPa and intercept grain sizes/~ in the interval 71 p m - 2 7 2 pm. Some of the results are presented in Fig. 5 in the form of a map of creep mechanisms in the coordinates (intercept grain size L, applied stress a), for temperatures of 773 K (full lines) and 873 K (broken lines). It can be seen that, at 773 K, Coble creep takes place at stresses lower than about 1 MPa while a non-viscous creep occurring presumably by a mechanism of superplasticity is observed to take place at higher stresses, at least up to an intercept grain size of about 300 pm. This type of non-viscous creep will be discussed later in this section. At still higher stresses, power law creep sets in.
Thus, diffusional creep in the Cu-14A1 alloy under consideration is not affected by either interface control or grain boundary migration. This creep of Coble type is associated with a threshold stress in a way similar to that in a-Fe and a-Zr. However, this threshold stress does not depend either on temperature or on intercept grain size. One remarkable feature of the creep behaviour of the Cu-14AI alloy investigated is a complete absence of Harper-Dorn creep. In fact, the viscous Harper-Dorn creep is replaced by a nonviscous creep occurring by a mechanism different from that of power law creep. In what follows, this non-viscous creep will be briefly described and discussed. In Fig. 6, a typical relation between the creep rate and stress is illustrated for a temperature of 848 K and an intercept grain size of 145/~m. It can be seen that a viscous creep predominates at stresses lower than about 1 MPa. At stresses higher than this value, a deviation from the viscous behaviour (i.e. from linearity of the ~ vs. o relation) is obvious. In the figure, the experimental points represent the measured values of ~Rand aRE, i.e. the values of the strain rate on the specimen wire surface and the initial elastic stress resulting from a constant loading moment respectively. A broken curve is drawn through the experimental points. The full curve represents the relation g(a) between steady state creep rate and applied stress calculated from the ga(OaE) data by a numerical procedure already mentioned and described in detail in a previous paper [35].
[prn3
169
From Fig. 6 it clearly follows that the creep under the conditions given occurs by a viscous mechanism up to a stress of about 1 MPa (the relation between i s and o is finear); this mechanism exhibits Bingham behaviour (note a threshold stress). At stresses higher than about 1 MPa is increases with o non-linearly (the stress exponent of i s is greater than 1), which suggests that a nonviscous flow mechanism is acting in addition to the viscous mechanism. If it is assumed that the viscous and the non-viscous mechanisms act in parallel the steady state creep rate is
102°
105
1(521
ld 16 r"l
2
r'-I
% 11322
1(5'7
t..d
\,, 0
1(~2~-
i tiC,
2.8~°x\ I \
ld,~ "-> .W
)>o \
is = iv + iNv
where iv and iNv represent the contributions of viscous and non-viscous mechanisms respectively to the steady state creep rate. On describing the steady state creep rate by the phenomenological equation [28] i s = A ~-TD (b)g ( G - G ) "
(2)
where D is an "effective" diffusion coefficient, A is a dimensionless constant, g and n are grain size exponent and stress exponent respectively, and k is the Boltzmann constant, the relations
,v Zv OV(¢ o-a0
G'-lkT
and
o"
G"-lkT
\
(1)
(4)
can be easily obtained. In eqns. (3) and (4) A v and Ayv are dimensionless constants, D v and D ~v are the coefficients of diffusion controlling viscous and non-viscous creep mechanisms respectively (D in eqn. (2) is represented by a combination of D v and D Nv ) and gv and gNv are the exponents characterizing the grain size sensitivity of viscous and non-viscous mechanisms respectively. In Fig. 7, the viscous and the non-viscous stress compensated creep rates iv and iNV respectively are plotted against intercept grain size L (the mean grain diameter d is proportional to /]). For the viscous mechanism, a value of gV equal to 2.95 _+0.5 and a value of the activation energy Qv of 136+_10 kJ tool -1 were obtained. Since the value of the grain size exponent is close
a62
260
5001019
[ E/am] Fig. 7. Cu-14A1 solid solution alloy. Relations between stress-compensated creep rates gv/( O-oo) ( zx ) and gNv/ O2 (o) and the mean interceptgrain size L for a temperatureof 773 K[51].
to 3 and that of activation energy is close to 135 kJ mol-1, i.e. to the value reported for the activation enthalpy of grain boundary diffusion of copper [52], this viscous mechanism was quite naturally identified with the Coble mechanism associated with a threshold stress (Bingham behaviour) (Fig. 6). For a temperature of 833 K, the value of the grain boundary diffusion coefficient of copper in the Cu-14A1 solid solution alloy investigated was determined by Kloc et al. [18]. This value of D~ was used to calculate the creep rate is by means of the original Coble equation. This creep rate was found to be higher than the measured rate by a factor of 2.5 only. This agreement between calculated and measured creep rate is excellent compared with that for a-Fe (cf. Section 3.1 ). For the exponents n and gNV values of 2.06 + 0.45 and 2.8 + 1.1 respectively were found using the procedure [35] mentioned above. The grain size dependence of the creep rate component due to the non-viscous creep mechanism is thus very strong. This clearly points to an important role of grain boundaries in this non-viscous creep mechanism. A relation similar to that found in the present work, is oc o2/d 3, was proposed by Sherby and Wadsworth [17] for a mechanism of superplastic deformation. However, the proposed relation is purely phenomenological. Other researchers [ 19-21] suggested as a mechanism of superplastic deformation grain boundary sliding accommo-
170
llJ 25 t-I
v
1(~
4. Discussion and concluding remarks (1) The diffusional Nabarro-Herring creep was not observed to take place in a-Fe up to a homologous temperature of 0.51 and in a-Zr up to a homologous temperature of 0.48. Coble creep and Nabarro-Herring creep act in parallel and thus, because of different temperature dependences of Coble and Nabarro-Herring creep rates, Nabarro-Herring creep can be expected to dominate Coble creep at temperatures
0
Z
1027 % ~j
1028 > z
1(j2g
1.1
~
1.2
1.3
1.4
103/T [K 4] Fig. 8. Cu-14AI solid solution alloy. A relation between the stress, temperature and mean intercept grain size compensated creep rate gNv[,=T/a2 and the reciprocal of temperature: o, experimental creep data; o, data calculated by means of eqn. (5) [51].
dated by dislocation slip in narrow zones adjoining grain boundaries. For this mechanism the relation gs oc a2/d 2 was obtained. Because of the error in the determination of the grain size exponent gNv such a relation and thus this deformation mechanism cannot be excluded. For this mechanism, the equation for steady state creep rate derived by Kajbyshev et al. [21], if slightly modified [18], can be written as
gS-l.25a2kT
-
In eqn. (5), dB is the effective grain boundary width, b B is the modulus of the Burgers vector of grain boundary dislocations and a B is a stress required for grain boundary dislocation sources to operate. This equation was applied to calculate the steady state creep rate using a value of the grain boundary diffusional conductance DB6 B as determined for the solid solution alloy under consideration by Kloc et al. [18] and setting a~--0. From Fig. 8 it follows that agreement between calculated and measured steady state creep rates is very good. Some assumptions accepted by Kajbyshev et al. [21] are not free of objections (cfi refs. 51 and 53). Nevertheless, the model of grain boundary sliding accommodated by dislocation slip in narrow zones adjoining grain boundaries enables the present results to be accounted for quite acceptably, although microstructural evidence supporting this model has not yet been obtained.
T>
AHL-AHB R ln(BcDoBdB/BNu~dDoL)
(6)
In eqn. (6), BNH = 14 and B c = 148 are the constant in the Nabarro-Herring and Coble creep equations respectively, and D0L and Don a r e the pre-exponential factors in relations describing temperature dependences of lattice and grain boundary diffusion coefficients and AH L is the activation enthalpy of lattice diffusion. From eqn. (6) it follows that for the value of intercept grain size above which Harper-Dorn creep dominates Coble creep in a-Fe, i.e. L - - 1 3 0 pm, the temperature would have to be higher than 1080 K for Nabarro-Herring creep to dominate Coble creep, i.e. much higher than 918 K, the highest temperature at which creep in a-Fe was investigated (see Section 3.1). (2) There can be little doubt that Harper-Dorn creep is controlled by lattice diffusion at very high temperatures. However, the results presented in Sections 3.1 and 3.2 clearly demonstrate that this is not the case at intermediate temperatures, at least in a-Fe and a-Zr. Instead, Harper-Dorn creep in a-Fe and a-Zr is probably controlled by dislocation core diffusion at these temperatures. The temperature of transition from lattice diffusion control to the dislocation core diffusion control can be easily calculated [54]. The Harper-Dorn steady state creep rate can be described by the phenomenological equation es =AHD
k-----~
-
where A HD is a dimensionless constant. The lattice diffusion coefficient D E can be replaced by an effective diffusion coefficient Deff=fLDL + fcDc
(8)
where fL and fc are the fractions of atoms taking
171 i/
part in lattice diffusion and dislocation core diffusion respectively. The fraction fL is very close to unity and the fraction fc can be expressed as
/
C u - 14 A[ T =773K
fc=acp
(9)
In eqn. (9), a c is the cross-sectional area of the dislocation core through which diffusion occurs and p is the dislocation density. Thus, the creep rate is
,.:AHoGb,OL+ac Oc,(
(10)
For dislocation core diffusion to dominate lattice diffusion, the following relation must be fulfilled:
acpD c > DI.
( 11 )
Hence, the condition for dislocation core diffusion control can be expressed as T<
AHL-AHc R ln(DoLpac/Doc)
10e
/
108
1610
16,2
1 _
01
I
•
1
1
10
100
Fig. 9. Cu-14AI solid solution alloy. Stress dependence of steady state creep rate according to the following: data obtained by the helicoid spring specimen technique for T = 773 K and/~ = 71 ~m (o) and for T= 773 K and L = 267 #m (u); curves A and B, calculated using the procedure described in ref. 35 (see also Fig. 6); data obtained by the tensile stress creep test technique [55] for T= 773 K and /]= 240 #m (z~).
(12)
where D0c is the pre-exponential factor in the relation expressing the temperature dependence of the dislocation core diffusion coefficient D o From this condition a "transition" temperature 966 K follows for the observed dislocation density of about 1012 m -2. This temperature is higher than 943 K, i.e. the highest at which creep in a-Fe was investigated (Section 3.1). A similar result was obtained for a-Zr. (3) In Section 3.3, the non-viscous creep in a Cu-14A1 solid solution alloy was interpreted in terms of grain boundary sliding accommodated by dislocation slip in narrow zones adjoining grain boundaries, i.e. in terms of a mechanism of superplastic deformation. However, two objections might be raised against such an interpretation. First, the deviation from viscous behaviour (Fig. 6) may be caused by gradual setting in power law creep. That this is not the case is, apart from the strong grain size dependence of steady state creep rate, clearly demonstrated in Fig. 9. In this figure, relations between gs and o for two grain sizes are shown together with a relation for the power law creep (gs oc os) obtained for the same alloy. The experimental curves distinctly differ from those obtained under the assumption of a superposition of viscous (diffusional) and power law creep rates. Second, the value of a stress exponent
obtained may be an apparent value if the diffusional creep is associated with a sufficiently high threshold stress. In fact, the true stress exponent may then be equal to unity and thus the creep may be viscous. Since the diffusional creep in the solid solution alloy may be enhanced by grain boundary migration and thus associated with a threshold stress for grain boundary migration [16], this type of creep should be considered. However, no grain boundary migration was observed experimentally during creep of the alloy under consideration. Hence, the diffusional creep enhanced by grain boundary migration can be excluded as operating in the Cu-14A1 solid solution alloy. (4) The question quite naturally arises of why the Harper-Dorn creep takes place in a-Fe and a-Zr but not in a Cu-14A1 solid solution alloy under similar conditions of temperatures, stresses and grain sizes. An answer should be perhaps sought in the low stacking fault energy in the solid solution alloy under consideration (cf. ref. 56). The diffusion along dissociated dislocations can be expected to be much slower than that along non-dissociated dislocations [57]. Since latticediffusion-controlled Harper-Dorn creep cannot take place to a noticeable extent at intermediate temperatures and also the dislocation core diffusion controlled Harper-Dorn creep can proceed with a very low rate, another mechanism controlled by grain boundary diffusion starts to
172
dominate Coble creep at stresses higher than about 1 MPa.
References 1 E R. N. Nabarro, Rep. of Conf. on Strength of Solids, Physical Society, London, 1948, p. 75. 2 C. Herring, J. Appl. Phys., 21 (1950) 437. 3 R. Borch and J. R. Hauber, Trans. Metall. Soc. AIME, 242(1968) 1933. 4 J. E. Harris, R. B. Jones, G. W. Greenwood and M. J. Ward, J. Aust. Inst. Met., 14 (1969) 154. 5 B. Burton and G. W. Greenwood, Met. Sci. J., 4 (1970) 215. 6 R.L. Coble, J.Appl. Phys., 34(1963) 1679. 7 R. B. Jones, Nature (London), 207(1965) 70. 8 I. M. Bernstein, Trans. Metall. Soc. AIME, 239 (1967) 1518. 9 E. Arzt, M. E Ashby and R. A. Verrall, Acta Metall., 31 (1983) 1977. 10 I. G. Crossland, in J. E. Harris and E. C. Sykes (eds.), Physical Metallurgy of Reactor Fuel Elements, Metals Society, London, 1975, pp. 66, 109. 11 B. Burton, I. G. Crossland and G. W. Greenwood, Met. Sci., 14(1980) 134. 12 I. M. Bernstein, Trans. Metall. Soc. A1ME, 239 (1967) 1518. 13 B. Burton, Met. Sci. J., 5 (1971 ) 11. 14 R. G. Stang and C. R. Barret, Scr. Metall., 7(1973) 233. 15 W.J. Clegg and J. W. Martin, Met. Sci., 16 (1982) 65. 16 I.-W.Chen, Acta Metall., 30 (1982) 1317. 17 O.D. Sherby and J. Wadsworth, in G. Krauss (ed.), Deformation Processing and Structure, American Society for Metals, Metals Park, OH, 1984, p. 355. 18 L. Kloc, J. Fiala and J. t~ermak, Met. Mater., 28 (1990) 478. 19 A. Ball and M. M. Hutchinson, Met. Sci. J., 3(1969) 1. 20 R.C. Gifkins, Metall. Trans. A, 7(1976) 1225. 21 O. A. Kajbyshev, R. Z. Valiev and A. K. Emaletdinov, Phys. Status Solidi A, 90 (1985) 197. 22 R. H. Johnson, C. M. Packer, U J. Anderson and O. D. Sherby, Philos. Mag., 18(1968) 1309. 23 K. Matsuki, H. Morita, M. Yamada and Y. Murakami, Met. Sci., 11 (1977) 156. 24 J.G. Harper and J. E. Dorn, Acta Metall., 5 (1957) 654. 25 P. Yavari, D. A. Miller and T. G. Langdon, Acta Metall., 30(1982) 871. 26 J. G. Harper, L. A. Shepard and J. E. Dorn, Acta Metall., 6(1958) 509. 27 K.L. Murty, Mater. Sci. Eng., 14 (1974) 169.
28 J. (~adek, Creep in Metallic Materials, Elsevier, Amsterdam, 1988. 29 G. Malakondaiah and P. Rama Rao, Scr. Metall., 13 (1979) 1187. 30 G. Malakondaiah and P. Rama Rao, Acta Metall., 29 (1981)1263. 31 G. Malakondaiah and P. Rama Rao, Mater. Sci. Eng., 52 (1982) 207. 32 J. Novotn~, J. Fiala and J. ~adek, Acta Metall., 33 (1985) 905. 33 J. Fiala, J. Novotn2~ and J. (~adek, Mater. Sci. Eng., 60 (1983) 195. 34 T. Sritharan and H. Jones, Res. Mech., 2 ( 1981 ) 295. 35 L. Kloc, J. Fiala and J. (~adek, Mater. Sci. Eng., A130 (1990)61. 36 J. Fiala and J. (2adek, Met. Mater., 20 (1982) 277. 37 V. T. Borisov, V. I. Golikov and G. V. Schebedinski, Fiz. Met. Metalloved., 17(1964) 881. 38 J. Fridberg, Z. E. Thorndahl and T. Hellert, Jernkontorets Ann., 153(1969) 263. 39 T.-J. Chuang, K. J. Kagawa, J. R. Rice and L. B. Sills, Acta Metall., 27(1979) 265. 40 J. I. Federer and T. S. Lundy, Trans. Metall. Soc. A1ME, 227(1969) 592. 41 D. W. James and G. M. Leak, Philos. Mag., 12 (1965) 491. 42 R.G. Peters, Scr. Metall., 17(1983) 587. 43 B. Burton, in R. E. Smallman and J. E. Harris (eds.), Vacancies 76, Metals Society, London, 1977, p. 156. 44 B. Burton, Mater. Sci. Eng., 10(1972) 9. 45 D.J. Towle and H. Jones, Acta Metall., 24 (1976) 399. 46 O. D. Sherby and J. Weertmann, Acta Metall., 27(1979) 387. 47 J. (~adek and K. Mili6ka, Czech. J. Phys. B, 18 (1968) 1156. 48 M. Pahutov~ and J. (~adek, Mater. Sci. Eng., 11 (1973) 151. 49 J. C. Hwang and R. W. Balluffi, Scr. Metall., 12 (1978) 709. 50 P. M. Sargent and M. E Ashby, Scr. Metall., 16 (1982) 1415. 51 L. Kloc, J. Fiala and J. (2adek, Mater. Sci. Eng., A130 (1990) 165. 52 J. Qenn~k, Phys. Status Solidi A, 121 (1990) K31. 53 R.A. Varin, K. J. Kurzydiowski and K. Tangri, Mater. Sci. Eng., 85(1987) 115. 54 A.H. Chokshi, Scr. Metall., 19(1985) 529. 55 K. Kuchafovfi, Institute of Physical Metallurgy, Czechoslovak Academy of Sciences, Brno, personal communication, 1990. 56 A. Howie and P. R. Swann, Philos. Mag., 6 ( 1961 ) 1215. 57 J. Weertman, Trans. ASM, 61 ( 1968) 681.
163
Creep in metals at intermediate temperatures and low stresses: a review J. Fiala, L. KIoc and J. (~adek Institute of Physical Metallurgy, Czechoslovak Academy of Sciences, 616 62 Brno (Czechoslovakia)
Abstract In the present paper, some recent results on viscous creep in a-Fe and a-Zr are reviewed and some results of investigations of creep in a Cu-14A1 solid solution alloy at intermediate temperatures and low stresses are presented and discussed. The "low stresses" are specified as those causing steady state creep rates lower than 10- ~0s- 1 at "intermediate temperatures" ranging from 0.4 Tmto 0.6 Tm(Tm is the melting temperature). It is shown that the creep in a-Fe and a-Zr results from stress-directed diffusional transport of matter via the grain boundaries (Coble creep) up to an intercept grain size of about 120/~m, while at the intercept grain sizes above this value Harper-Dorn creep controlled by dislocation core diffusion operates. In creep of a Cu-14AI solid solution alloy, two flow processes act in parallel: (i) the viscous flow process identified with Coble creep and (ii) the non-viscous flow process characterized by a strain rate proportional to applied stress squared, inversely proportional to the intercept grain size cubed, and by the activation energy close to the activation enthalpy of grain boundary diffusion. The latter flow process was identified with the creep due to grain boundary sliding accommodated by slip in thin zones adjoining the sliding boundaries. In the solid solution alloy under consideration, Harper-Dorn creep does not take place up to an intercept grain size of 270 ~m at least.
1. Introduction The low stress high temperature creep mechanisms have been treated in the literature for four decades. Only about ten years ago was it recognized that similar mechanisms may operate at intermediate temperatures (around half the melting temperature). However, the experimental evidence for this recognition is rather limited since the obtention of necessary data is difficult and usually extremely time consuming. Undoubtedly, an understanding of flow mechanisms operating under such conditions of stress and temperature may be of considerable importance for engineering practice, i.e. in design and in the development of performance codes which are used in this practice. A most important common characteristic of all the low stress creep mechanisms consists in a weak dependence of steady state creep rate on applied stress. In fact, the steady state creep rate 0921-5093/91/$3.50
is characterized by a value of the applied stress exponent lower than 3; frequently the creep rate varies with the applied stress linearly, i.e. viscous creep takes place. The mechanisms known at present are listed in Table 1. In this table, T means the absolute temperature, o the applied stress, o 0 the threshold stress and d the mean grain diameter. The creep due to diffusional transport of matter via the lattice (Nabarro-Herring creep) or via the grain boundaries (Coble creep) is one of the few instances in materials science where theoretical predictions have preceded experimental verification. Another deformation mode, known as H a r p e r - D o r n creep and first observed in 1957, has been thought until recently to take place at homologous temperatures higher than about 0.95, stresses lower than about 5 × 1 0 - 6 G ( G is the shear modulus) and mean grain diameters © Elsevier Sequoia/Printed in The Netherlands
164 TABLE 1 Basic mechanisms of creep at low stresses Deformation process
Creep
Rate-controlling process
Accommodating process
Creep rate dependence on the following c7
d
References Theory
Experiment
Nabarro-Herring
Diffusional transport of matter via lattice
Lattice diffusion
Grain boundary sliding
ua
l/d*
[L 21
[3-51
Coble
Diffusional transport of matter via grain boundaries
Grain boundary diffusion
Grain boundary sliding
ua
l/d’
bl
]5>7381
Diffusional interface controlled: solid solution alloys
Diffusional transport of matter via lattice and/or grain boundaries
Emission and/or absorption of vacancies by grain boundaries
Grain boundary sliding
d
Lid
[91
[lo-121
Diffusional interface controlled: particles containing alloys
Diffusional transport of matter via lattice and/or grain boundaries
Emission and/or absorption of vacancies by grain boundaries
Grain boundary sliding
iu- %I2
l/d
[91
[13-151
Diffusional enhanced by grain boundary migration (solid solution alloys)
Diffusional transport of matter enhanced by grain boundary migration
Grain boundary diffusionh
Gram boundary sliding
U’
l/d’
[I61
Grain boundary sliding“
Grain boundary sliding
Grain boundary diffusion
Dislocation slip in narrow zones adjoining grain boundaries
U2
l/d2’
[19-211
[18,22,23]
Motion of lattice dislocations
Lattice diffusion
U
None
[24,25]
[24,26,27]
Motion of lattice dislocations
Dislocation core diffusion
(Ja
None
[281
[29-331
Harper-Darn, temperatures Harper-Darn, intermediate temperatures
high
“Small but non-zero threshold stress is frequently observed experimentally. hThe presence of a component with a significantly higher diffusivity in the system is required. ‘Threshold behaviour can be expected. %reep occurring by a mechanism of superplasticity. r& K 1/d 3 proposed by Sherby and Wadsworth [ 171 and experimentally observed by Kloc et al. [ 181.
larger than about 500 pm. However, in the last decade it has been shown that Harper-Dorn creep can take place as well not only at much lower homologous temperatures but also at much smaller grain diameters. The superplastic deformation mode was first observed more than 70 years ago and it remained a scientific curiosity until about 25 years ago. In all the flow mechanisms listed in Table 1 except the mechanism of Harper-Dorn creep, the grain boundaries play a most important role. This is manifested by variation in creep rate with mean grain diameter. On the contrary, the HarperDorn creep rate is, in a similar way to the power law creep rate, independent of grain diameter. This strongly suggests that Harper-Dorn creep is due to the motion of lattice dislocations.
The creep due to stress-directed diffusional transport of matter involves two consecutive processes: emission and/or absorption of vacancies by grain boundaries and the diffusion of emitted vacancies either via the lattice or via the grain boundaries. The creep is then diffusion controlled only if the grain boundaries act as perfect sources and sinks for vacancies. NabarroHerring or Coble creep then occur. This is, however, almost certainly not the case in metals and alloys containing dispersed particles. In fact, since some of these particles are located at grain boundaries the latter cannot serve as perfect sources and sinks for vacancies, and the creep, although still resulting from diffusional transport of matter, is controlled by the processes at grain boundaries (interface-controlled diffusional
165
creep). The interface control is associated with a significant threshold stress (Bingham behaviour). Also, in some solid solution alloys the diffusional creep can be expected to be interface controlled. However, experimental evidence for interfacecontrolled creep in solid solution alloys is not yet sufficiently convincing. Independently of whether it is interface controlled or not the diffusional creep in solid solution alloys may be enhanced by grain boundary migration. The grain boundary migration enhanced diffusional (diffusioncontrolled) creep in multicomponent systems has been analysed by Chen [16]; however, experimental evidence for its existence in metallic alloys is still missing. On the contrary, there seems to be ample experimental evidence for creep due to a mechanism of superplasticity. In the present paper, some results of recent investigations of viscous creep in a-Fe and a-Zr at temperatures close to 0.5T m are reviewed. Also, some results of investigations of creep in a Cu-14A1 solid solution alloy are presented and discussed. It is shown that while in the above pure metals Coble and Harper-Dorn creep take place, in the solid solution alloy a mechanism of superplasticity operates in parallel with the Coble mechanism, while the Harper-Dorn mechanism essentially does not contribute to the creep rate even at large grain sizes. 2. On the techniques for investigating creep at intermediate temperatures and low stresses
In investigations of viscous (diffusional and Harper-Dorn) creep at temperatures close to or even higher than 0.95Tin the conventional isothermal tensile creep test technique is usually applied. Since the viscous creep strains are generally small and the creep rates are very low, creep specimen gauge lengths several times greater than those typical for conventional power law creep tests are frequently used to increase sensitivity of creep strain measurement. In principle, a similar technique can be applied as well to study viscous creep at much lower temperatures, namely at temperatures close to or even lower than 0.5T m. However, to measure creep rates as low as 10 -]2 S-1 or even lower, extremely long-term creep tests are generally required. This serious disadvantage can be removed using the helicoid spring specimen technique [5]. In fact, the sensitivity of creep strain measurement is almost two orders of magnitude
higher as compared with the tensile creep test, which enables the duration of creep tests to be reduced to quite an acceptable level, especially taking into account the fact that several creep curves (up to 20) for various stresses can be obtained in a single experiment (using a single specimen) at a given temperature. Of course, the stress is distributed nonuniformly across the cross-section of the helicoid spring specimen wire. This has been known [34, 35] to be a serious disadvantage of this technique when the creep behaviour is non-viscous. Recently, however, a new numerical procedure enabling a correction for redistribution of stress across the specimen wire cross-section during non-viscous creep has been developed [35]. The application of this procedure completely eliminates the above disadvantage of this test technique. All the results presented in this paper were obtained using the technique just mentioned. The technique as well as the corresponding apparatus have been described elsewhere [36]. 3. Results
3.1. a-Fe The low stress creep in a-Fe of 99.96% purity was investigated at temperatures of 723 K-918 K (i.e. 0.40 Tin-0.51 Tin) and at stresses ranging from 0.12 to 3.50 MPa. The intercept grain size /~ ranged from 82 # m to 478 pro. (The mean grain diameter d is related to the intercept grain size by d = 1.57L.) Figure 1 shows a creep mechanism map in the coordinates (normalized stress o/G, normalized intercept grain size L/b) (G is the shear modulus, b is the modulus of the Burgers vector). The map was obtained by employing more than
3 - 1 o 6 ~
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Fig. 1. A map of creep mechanisms in a-Fe (99.96 mass%) for a temperature of 843 K in the coordinates (normalized intercept grain size/2/b, normalized stress a/G).
166
250 experimental creep curves relating to a temperature of 843 K, i.e. 0.47 Tm. The boundaries in the map delineate regions of conditions of the stresses and the intercept grain sizes under which particular creep mechanisms are dominating. In the map, also the curves representing various steady state creep rates ranging from 5 x 10 -12 s -~ to 1 x 10 -1° S - 1 a r e shown. As expected, Coble creep dominates both Harper-Dorn creep and power law creep at small values of Lib and/ or o/G. The Coble creep rate is proportional to stress and inversely proportional to the third power of the mean grain diameter. For its activation energy a value of 150+ 15 kJ mo1-1 was found, which is in very good agreement with the values of activation enthalpy of grain boundary diffusion reported by various researchers [37-41 ] and ranging from 140.0 kJ mo1-1 to 173.8 kJ mol- 1. The results obtained for Coble creep differ from the predictions of the theory [6] in two respects: (i) a threshold stress o 0 is observed below which Coble creep does not proceed, and (ii) the creep rates measured are about two orders of magnitude higher than those calculated on the basis of Coble theory. The threshold stress for diffusional creep in metals has been reported by a number of researchers [9, 42-45]. However, none of its proposed interpretations is free of objections. The threshold stress for Coble creep in a-Fe seems to be somehow related to the role of impurities. This is illustrated in Fig. 2, in which the threshold stress o 0 is plotted against the reciprocal 1/T of temperature for irons of various purities. The threshold stress apparently increases with decreasing purity at any given temperature. Also, the above-mentioned difference between measured creep rates and the creep rates calculated from Coble theory is not surprising, since a 10 0
i
i
I
I
I
I
10.1
16o.9
I
1.0
1.1 1.2 103/T [K-I]
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1.3
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Fig. 2. The threshold stress a 0 for diffusionai creep plotted against the reciprocal of temperature for irons of different purities: e, 99.8 mass% [45]; o, 99.96 mass% [33].
similar difference was reported by other workers for various metals. Again, the attempts to account for this difference have not yet been successful. Probably, the main reason consists in the difference between the model structure of uniform spherical grains assumed in the theory and the real grain structure. However, a relation expressing an effective value of the shape constant in the Coble equation in terms of measurable parameters of real structure characterized by a distribution of dimensions of grains varying in shapes is not yet available. For Harper-Dorn creep operating at intercept grain sizes L larger than 128 p m a value of the activation energy Qc = 153.3 + 9.3 kJ mol- 1 was found. This value is much lower than that of the activation enthalpy of lattice diffusion. Although it is close to the value of the activation enthalpy for grain boundary diffusion the creep under consideration is certainly not controlled by grain boundary diffusion since its rate does not depend on grain size. Thus it seems quite natural to identify the activation energy of creep with the activation enthalpy A H c of dislocation core diffusion. The value of A H c for a-Fe is not known from experiment. However, it can be estimated using the relation [46] A H c = 11RT mwhere R is the gas constant. Accordingly, AH~ should be close to 165 kJ mol-1, which is only slightly higher than the above value of Qc. Thus, it can be concluded that Harper-Dorn creep in a-Fe at temperatures close to 0.5 Tm is probably dislocation core diffusion controlled. It is interesting to note that lattice-diffusioncontrolled Nabarro-Herring creep does not operate in the region of conditions of temperatures, stresses and intercept grain sizes considered here. At stresses only slightly higher than the threshold stress, Coble creep operates at normalized intercept grain sizes [,/b as large as 3 x 106. At normalized stresses in the interval 2 x 10-6-8 × 10 -6, a direct transition from Coble to Harper-Dorn creep occurs at a value of Lib as l o w as 5 × 10 -6 ( Z = 123 ~m). At L/bhigher than the latter value a transition from Harper-Dorn creep to power law creep with o / G increasing is observed; at lower values of L/b, Coble creep acts in parallel with power law creep. The transition from viscous creep to power law creep is illustrated in Fig. 3 (it should be mentioned that the power law creep rate does not depend on grain size). Despite a gap between the measurements obtained by means of the helicoid
167 I
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POWER-LAW CREEP ks = 5Ee
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-
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[MPo] Fig. 3. Stress dependence of steady state creep rate in a-Fe. Two sets of data were obtained by different creep test techniques. Power law creep, the tensile stress creep tests technique [47]: o, T= 900 K,/~ = 30/~. Transition from viscous to power law cre_ep, the helicoid spring specimen technique. 0, T=918 K, L = l 1 8 /~m; ", T=843 K, L = 1 2 2 /~m; ~z, T = 823 K,/~ = 82 #m.
spring specimen technique (transition from viscous to power law creep) and those obtained by means of the constant tensile stress isothermal test technique (power law creep), the agreement between both sets of measurements seems to be very good. 3.2. ct-Zr The creep in zirconium was investigated at temperatures ranging from 748 K to 1023 K (0.40Tm to 0.54Tin), stresses in the interval 0.08 MPa-2.19 MPa and intercept grain sizes ranging from 48 ~m to 342/~m. Some of the results are presented in Fig. 4 in the form of a creep mechanism map in the coordinates (normalized stress (r/G, reciprocal Tm/T of homologous temperature). The map relates to the mean intercept grain size of 100 #m. Using the above coordinates, only two of the three creep mechanisms observed (Coble, Harper-Dorn and power law creep mechanisms) can be represented by a single map for any intercept grain size. Harper-Dorn creep operates only at intercept grain sizes larger than 127 ~tm, while Coble creep operates at intercept grain sizes smaller than this value. To determine
Fig. 4. A m_apof creep mechanismsin a-Zr for an intercept grain size L = 100/am in the coordinates(normalizedstress a/G, reciprocal Tm/T of homologoustemperature). precisely enough the field of power law creep, the data published by Pahutovfi and Cadek [48] (obtained by a constant tensile stress isothermal creep test technique) were used. The value of the activation energy obtained for Coble creep was found to be 123.9+3.2 kJ mol-1, i.e. close to the expected value of the activation enthalpy of grain boundary diffusion [49]. For Harper-Dorn creep a value of the activation energy much lower than the activation enthalpy of lattice diffusion was found, namely 121.8+11.2 kJ mo1-1 Again, this value was identified with that of the activation enthalpy of dislocation core diffusion, although an estimate using the above relation [46] AH c = 11RT m provides a value as high as 171.8 kJ mol-1. However, Sargent and Ashby [50] deduced from creep data a significantly lower value of 124 kJ mol -~, to which the obtained value of the activation energy of Harper-Dorn creep is very close. In the same way as in a-Fe, Nabarro-Herring creep in a-Zr was not observed under the above experimental conditions. However, at high enough temperatures, Nabarro-Herring creep can be expected to take place at the expense of Coble, and especially Harper-Dorn, creep. 3.3. Cu-14Al solid solution alloy The emission and/or absorption of vacancies by grain boundaries is thought to be associated with motion of grain boundary dislocations. Consequently, grain boundary dislocation solute drag
168 100 / l
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Fig. 6. Stress dependence of steady state creep rate in a
Fig. 5. A map of creep mechanisms in Cu-14AI solid solution alloy for temperatures of 773 K and 873 K in the coordinates (applied stress o, mean intercept grain size/~ ).
Cu-14Al solid solution alloy at a temperature of 848 K for a mean intercept grain size of 145 ~tm: o, measured values of strain rate ~R at various stresses ORE on the specimen wire surface; , the relation g(a) calculated from the relation eR(aRE) using the procedure described in ref. 35.
occasionally results in interface control of diffusional creep [3] (Table 1 ). Another effect of a solute may consist in enhancement of diffusional creep by grain boundary migration [16] if this solute exhibits a significantly higher diffusivity than the solvent (in the case of binary solid solution alloy)(Table 1 ). Thus, the knowledge of diffusional creep in pure metals is in general not directly transferable to the solid solution alloys. Of course, this may also apply to Harper-Dorn creep. This is why an extensive investigation of low stress creep in solid solution alloys at intermediate temperatures has been undertaken by the present authors. In this section, some results for a Cu-14AI (6.5 mass.% AI) solid solution alloy are presented. Using again the helicoid spring specimen technique, the creep was investigated at temperatures ranging from 723 K to 873 K (0.55 Tmto 0.66 Tm), stresses ranging from 0.2 MPa to 5.0 MPa and intercept grain sizes/~ in the interval 71 p m - 2 7 2 pm. Some of the results are presented in Fig. 5 in the form of a map of creep mechanisms in the coordinates (intercept grain size L, applied stress a), for temperatures of 773 K (full lines) and 873 K (broken lines). It can be seen that, at 773 K, Coble creep takes place at stresses lower than about 1 MPa while a non-viscous creep occurring presumably by a mechanism of superplasticity is observed to take place at higher stresses, at least up to an intercept grain size of about 300 pm. This type of non-viscous creep will be discussed later in this section. At still higher stresses, power law creep sets in.
Thus, diffusional creep in the Cu-14A1 alloy under consideration is not affected by either interface control or grain boundary migration. This creep of Coble type is associated with a threshold stress in a way similar to that in a-Fe and a-Zr. However, this threshold stress does not depend either on temperature or on intercept grain size. One remarkable feature of the creep behaviour of the Cu-14AI alloy investigated is a complete absence of Harper-Dorn creep. In fact, the viscous Harper-Dorn creep is replaced by a nonviscous creep occurring by a mechanism different from that of power law creep. In what follows, this non-viscous creep will be briefly described and discussed. In Fig. 6, a typical relation between the creep rate and stress is illustrated for a temperature of 848 K and an intercept grain size of 145/~m. It can be seen that a viscous creep predominates at stresses lower than about 1 MPa. At stresses higher than this value, a deviation from the viscous behaviour (i.e. from linearity of the ~ vs. o relation) is obvious. In the figure, the experimental points represent the measured values of ~Rand aRE, i.e. the values of the strain rate on the specimen wire surface and the initial elastic stress resulting from a constant loading moment respectively. A broken curve is drawn through the experimental points. The full curve represents the relation g(a) between steady state creep rate and applied stress calculated from the ga(OaE) data by a numerical procedure already mentioned and described in detail in a previous paper [35].
[prn3
169
From Fig. 6 it clearly follows that the creep under the conditions given occurs by a viscous mechanism up to a stress of about 1 MPa (the relation between i s and o is finear); this mechanism exhibits Bingham behaviour (note a threshold stress). At stresses higher than about 1 MPa is increases with o non-linearly (the stress exponent of i s is greater than 1), which suggests that a nonviscous flow mechanism is acting in addition to the viscous mechanism. If it is assumed that the viscous and the non-viscous mechanisms act in parallel the steady state creep rate is
102°
105
1(521
ld 16 r"l
2
r'-I
% 11322
1(5'7
t..d
\,, 0
1(~2~-
i tiC,
2.8~°x\ I \
ld,~ "-> .W
)>o \
is = iv + iNv
where iv and iNv represent the contributions of viscous and non-viscous mechanisms respectively to the steady state creep rate. On describing the steady state creep rate by the phenomenological equation [28] i s = A ~-TD (b)g ( G - G ) "
(2)
where D is an "effective" diffusion coefficient, A is a dimensionless constant, g and n are grain size exponent and stress exponent respectively, and k is the Boltzmann constant, the relations
,v Zv OV(¢ o-a0
G'-lkT
and
o"
G"-lkT
\
(1)
(4)
can be easily obtained. In eqns. (3) and (4) A v and Ayv are dimensionless constants, D v and D ~v are the coefficients of diffusion controlling viscous and non-viscous creep mechanisms respectively (D in eqn. (2) is represented by a combination of D v and D Nv ) and gv and gNv are the exponents characterizing the grain size sensitivity of viscous and non-viscous mechanisms respectively. In Fig. 7, the viscous and the non-viscous stress compensated creep rates iv and iNV respectively are plotted against intercept grain size L (the mean grain diameter d is proportional to /]). For the viscous mechanism, a value of gV equal to 2.95 _+0.5 and a value of the activation energy Qv of 136+_10 kJ tool -1 were obtained. Since the value of the grain size exponent is close
a62
260
5001019
[ E/am] Fig. 7. Cu-14A1 solid solution alloy. Relations between stress-compensated creep rates gv/( O-oo) ( zx ) and gNv/ O2 (o) and the mean interceptgrain size L for a temperatureof 773 K[51].
to 3 and that of activation energy is close to 135 kJ mol-1, i.e. to the value reported for the activation enthalpy of grain boundary diffusion of copper [52], this viscous mechanism was quite naturally identified with the Coble mechanism associated with a threshold stress (Bingham behaviour) (Fig. 6). For a temperature of 833 K, the value of the grain boundary diffusion coefficient of copper in the Cu-14A1 solid solution alloy investigated was determined by Kloc et al. [18]. This value of D~ was used to calculate the creep rate is by means of the original Coble equation. This creep rate was found to be higher than the measured rate by a factor of 2.5 only. This agreement between calculated and measured creep rate is excellent compared with that for a-Fe (cf. Section 3.1 ). For the exponents n and gNV values of 2.06 + 0.45 and 2.8 + 1.1 respectively were found using the procedure [35] mentioned above. The grain size dependence of the creep rate component due to the non-viscous creep mechanism is thus very strong. This clearly points to an important role of grain boundaries in this non-viscous creep mechanism. A relation similar to that found in the present work, is oc o2/d 3, was proposed by Sherby and Wadsworth [17] for a mechanism of superplastic deformation. However, the proposed relation is purely phenomenological. Other researchers [ 19-21] suggested as a mechanism of superplastic deformation grain boundary sliding accommo-
170
llJ 25 t-I
v
1(~
4. Discussion and concluding remarks (1) The diffusional Nabarro-Herring creep was not observed to take place in a-Fe up to a homologous temperature of 0.51 and in a-Zr up to a homologous temperature of 0.48. Coble creep and Nabarro-Herring creep act in parallel and thus, because of different temperature dependences of Coble and Nabarro-Herring creep rates, Nabarro-Herring creep can be expected to dominate Coble creep at temperatures
0
Z
1027 % ~j
1028 > z
1(j2g
1.1
~
1.2
1.3
1.4
103/T [K 4] Fig. 8. Cu-14AI solid solution alloy. A relation between the stress, temperature and mean intercept grain size compensated creep rate gNv[,=T/a2 and the reciprocal of temperature: o, experimental creep data; o, data calculated by means of eqn. (5) [51].
dated by dislocation slip in narrow zones adjoining grain boundaries. For this mechanism the relation gs oc a2/d 2 was obtained. Because of the error in the determination of the grain size exponent gNv such a relation and thus this deformation mechanism cannot be excluded. For this mechanism, the equation for steady state creep rate derived by Kajbyshev et al. [21], if slightly modified [18], can be written as
gS-l.25a2kT
-
In eqn. (5), dB is the effective grain boundary width, b B is the modulus of the Burgers vector of grain boundary dislocations and a B is a stress required for grain boundary dislocation sources to operate. This equation was applied to calculate the steady state creep rate using a value of the grain boundary diffusional conductance DB6 B as determined for the solid solution alloy under consideration by Kloc et al. [18] and setting a~--0. From Fig. 8 it follows that agreement between calculated and measured steady state creep rates is very good. Some assumptions accepted by Kajbyshev et al. [21] are not free of objections (cfi refs. 51 and 53). Nevertheless, the model of grain boundary sliding accommodated by dislocation slip in narrow zones adjoining grain boundaries enables the present results to be accounted for quite acceptably, although microstructural evidence supporting this model has not yet been obtained.
T>
AHL-AHB R ln(BcDoBdB/BNu~dDoL)
(6)
In eqn. (6), BNH = 14 and B c = 148 are the constant in the Nabarro-Herring and Coble creep equations respectively, and D0L and Don a r e the pre-exponential factors in relations describing temperature dependences of lattice and grain boundary diffusion coefficients and AH L is the activation enthalpy of lattice diffusion. From eqn. (6) it follows that for the value of intercept grain size above which Harper-Dorn creep dominates Coble creep in a-Fe, i.e. L - - 1 3 0 pm, the temperature would have to be higher than 1080 K for Nabarro-Herring creep to dominate Coble creep, i.e. much higher than 918 K, the highest temperature at which creep in a-Fe was investigated (see Section 3.1). (2) There can be little doubt that Harper-Dorn creep is controlled by lattice diffusion at very high temperatures. However, the results presented in Sections 3.1 and 3.2 clearly demonstrate that this is not the case at intermediate temperatures, at least in a-Fe and a-Zr. Instead, Harper-Dorn creep in a-Fe and a-Zr is probably controlled by dislocation core diffusion at these temperatures. The temperature of transition from lattice diffusion control to the dislocation core diffusion control can be easily calculated [54]. The Harper-Dorn steady state creep rate can be described by the phenomenological equation es =AHD
k-----~
-
where A HD is a dimensionless constant. The lattice diffusion coefficient D E can be replaced by an effective diffusion coefficient Deff=fLDL + fcDc
(8)
where fL and fc are the fractions of atoms taking
171 i/
part in lattice diffusion and dislocation core diffusion respectively. The fraction fL is very close to unity and the fraction fc can be expressed as
/
C u - 14 A[ T =773K
fc=acp
(9)
In eqn. (9), a c is the cross-sectional area of the dislocation core through which diffusion occurs and p is the dislocation density. Thus, the creep rate is
,.:AHoGb,OL+ac Oc,(
(10)
For dislocation core diffusion to dominate lattice diffusion, the following relation must be fulfilled:
acpD c > DI.
( 11 )
Hence, the condition for dislocation core diffusion control can be expressed as T<
AHL-AHc R ln(DoLpac/Doc)
10e
/
108
1610
16,2
1 _
01
I
•
1
1
10
100
Fig. 9. Cu-14AI solid solution alloy. Stress dependence of steady state creep rate according to the following: data obtained by the helicoid spring specimen technique for T = 773 K and/~ = 71 ~m (o) and for T= 773 K and L = 267 #m (u); curves A and B, calculated using the procedure described in ref. 35 (see also Fig. 6); data obtained by the tensile stress creep test technique [55] for T= 773 K and /]= 240 #m (z~).
(12)
where D0c is the pre-exponential factor in the relation expressing the temperature dependence of the dislocation core diffusion coefficient D o From this condition a "transition" temperature 966 K follows for the observed dislocation density of about 1012 m -2. This temperature is higher than 943 K, i.e. the highest at which creep in a-Fe was investigated (Section 3.1). A similar result was obtained for a-Zr. (3) In Section 3.3, the non-viscous creep in a Cu-14A1 solid solution alloy was interpreted in terms of grain boundary sliding accommodated by dislocation slip in narrow zones adjoining grain boundaries, i.e. in terms of a mechanism of superplastic deformation. However, two objections might be raised against such an interpretation. First, the deviation from viscous behaviour (Fig. 6) may be caused by gradual setting in power law creep. That this is not the case is, apart from the strong grain size dependence of steady state creep rate, clearly demonstrated in Fig. 9. In this figure, relations between gs and o for two grain sizes are shown together with a relation for the power law creep (gs oc os) obtained for the same alloy. The experimental curves distinctly differ from those obtained under the assumption of a superposition of viscous (diffusional) and power law creep rates. Second, the value of a stress exponent
obtained may be an apparent value if the diffusional creep is associated with a sufficiently high threshold stress. In fact, the true stress exponent may then be equal to unity and thus the creep may be viscous. Since the diffusional creep in the solid solution alloy may be enhanced by grain boundary migration and thus associated with a threshold stress for grain boundary migration [16], this type of creep should be considered. However, no grain boundary migration was observed experimentally during creep of the alloy under consideration. Hence, the diffusional creep enhanced by grain boundary migration can be excluded as operating in the Cu-14A1 solid solution alloy. (4) The question quite naturally arises of why the Harper-Dorn creep takes place in a-Fe and a-Zr but not in a Cu-14A1 solid solution alloy under similar conditions of temperatures, stresses and grain sizes. An answer should be perhaps sought in the low stacking fault energy in the solid solution alloy under consideration (cf. ref. 56). The diffusion along dissociated dislocations can be expected to be much slower than that along non-dissociated dislocations [57]. Since latticediffusion-controlled Harper-Dorn creep cannot take place to a noticeable extent at intermediate temperatures and also the dislocation core diffusion controlled Harper-Dorn creep can proceed with a very low rate, another mechanism controlled by grain boundary diffusion starts to
172
dominate Coble creep at stresses higher than about 1 MPa.
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