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Constitutive description of high temperature creep in mechanically alloyed AlCO alloys
Materials Science and Engineering, A 174 (1994) 37-43
37
Constitutive description of high temperature creep in mechanically alloyed A1-C-O alloys F. D o b e L K. Kucha~ovfi, A Orlovfi, K. M i l i 6 k a a n d J. t 2 a d e k Institute of PhysicalMetallurgy, Academy of Sciences of the Czech Republic, 616 62 Brno (Czech Republic) (Received May 26, 1993)
Abstract Creep curves of four mechanically alloyed aluminum alloys were studied within the range of stresses from 20 to 180 MPa at temperatures of 623 and 723 K. Creep curves of pure aluminum at temperatures of 473-623 K were taken as reference material data. The curves were described by the McVetty equation. The primary and steady-state stages of creep can be interpreted as a result of changes in the internal stress following from competition between work hardening and recovery processes. The analysis makes it possible to divide the internal stress into two components, one being due to stress fields of dislocations and the other to the existence of dispersed particles. The latter component equals the threshold stress in steady-state creep.
1. Introduction In many areas of modern technology, the demands 1o raise working temperatures have initiated a development of new materials with enhanced creep resistance. As a rule, these materials owe their improved properties to the existence of a dispersion of fine particles in the matrix. An extensive review, focused specifically on creep properties of dispersion strengthened materials, was recently given by Arzt [1]. This author reviewed apparently all the available papers devoted to creep of mechanically alloyed aluminum alloys (henceforth referred as MA alloys) and concluded that "the creep behaviour of polycrystalline dispersion strengthened materials is insufficiently understood." In addition, the present authors were not able to find any paper dealing more deeply with the primary stage of creep, or, generally speaking, with creep curves in these alloys. That is why this study was undertaken. Its purpose is to give the results of the time dependence of creep strain in dispersion strengthened aluminum alloys with varying contents of A1203 and A14C 3.
2. Experimental material and testing technique The subject of testing was four A1-C-O mechanically alloyed materials (marked AIC0, AIC1, AIC2 and AIC202), supplied by Erbsl6h, FRG. They were produced from aluminum powder containing about 0.4 wt.% 02 by milling with graphite in an attritor under 0921-5093/94/$7.00
argon atmosphere, cold isostatic pressing to a bar, annealing for 6-8 h at 823 K and final extrusion at 773 K. The microstructure of the resulting materials consisted of fine (sub)grains of micrometer or submicrometer size. It contained several types of dispersoid particles in quantities related to the content of carbon and oxygen additions. The resultant carbon and oxygen content and the volume fractions of A14C 3 and AI203 which can be formed by a complete transformation are given in Table 1. The creep tests were performed on cylindrical specimens of gage length 50 mm and diameter 5 mm which were machined from the extruded bars. The technique of isothermal tensile creep tests under constant applied stress was used, the engineering strain was measured with a sensitivity of 10-5. Details of the creep testing equipment are given in ref. 2. For the sake of comparison, the investigation of A I - C - O alloys was complemented by a set of creep tests of aluminum of 99.99% purity in the temperature range 473-623 K. The specimens were machined from hot-rolled slabs so that the
TABLE 1. Chemical compositions and nominal volume fractions of dispersed phases in mechanically alloyed alloys Alloy
O (wt.%)
C (wt.%)
fv
A1C0 AIC1 A1C2 A1C202
0.8 0.8 0.8 2.0
0.0 1.0 2.0 2.0
0.0116 0.0484 0.0855 0.1039
© 1994 - Elsevier Sequoia. All rights reserved
38
F. Dobe# et al.
/
High temperature creep in A I - C - O alloy
tensile stress axis was parallel to the rolling direction. The grain size of creep specimens with rectangular cross-section 8 x 3.2 mm 2 was 80 #m. For a description of the primary stage of creep curves the equation proposed by McVetty [3] completed by a steady-state term was chosen, e = e 0+ e l [ 1 - e x p ( - t/r1)]+ gs t
(1)
where t is the time passed from the start of the creep test, el is the limiting primary strain, rl is the relaxation time of the primary stage and gs is the steady-state creep rate. The values of e 0, el, r~ and gs were found by the Newton-Marquard least squares method.
3. Experimental results The alloys were creep tested at temperatures of 623 and 7 2 3 K within the interval of applied stresses to which the minimum creep rates ranging from 10- a~ to 10- 2 s- a correspond. Examples of creep curves of the alloys are given in Fig. 1. As a general conclusion, it can be said that, unless the fracture strain exceeds the value of approximately 1%, the creep curves at both the testing temperatures are characterized by a more significant primary creep stage and less important secondary and tertiary stages. However, in the region of creep conditions corresponding to a fracture strain which increases steeply above the value of 1%, creep
curves with a suppressed primary stage and an extensive tertiary stage are observed. The values of calculated primary strain are given in Fig. 2. Whereas in pure aluminum the primary strain el increases with increasing applied stress, the stress dependence of e~ in M A alloys can hardly be defined. The values of ,61 in the alloys are approximately two orders of magnitude smaller than those in pure aluminum. The relaxation time q decreases with increasing both applied stress and temperature (see Fig. 3). It can be described by the equation r I =Z~a-"" exp(Q~ / \Kll
(2)
where a is the applied stress, R is the universal gas constant and T is the absolute temperature. The values of the stress exponent n~ and activation energy Q~ found by multiple linear regression are given in Table 2. The
,
, , ,,i
,
,
i
,
, , ,,i
-1 10 ++ +/,'1~ I x / x~ / +//
,S Z .< rv" I--0,'3 >IM .<
10- 2
AI
v 47,3 • 52,3 + 573 x 62,3
K K K K
x
"...4,,W t
cl
623 723K$ o • AICO •
?&&A&&~
l
AICI
0.01
,
,
T =
,
625
u
K ,~====::~
AICO AIC 1 ~ AIC2
IN -4_
,
• • I
AIC2 AIC202
i i ii
i
i
i
i
i
i i ii
10
|
100
STRESS (~ [MPo] 0.00 0.12
'
'
'
.
.
T = Z
0.10
I'-(/3
0.08
'
.
.
Fig. 2. Stress dependence of primary strain.
'
/ / A I C O
/
72,3 K
0.06 /
'
.
"
108
,,,,I
623 723K
o O u
~-' 1° 6 AIC 1
F
V,5
102
0.02
,cU 0"00.0
0.2
0.4
0.6
O.
1.0
.2
[~
v'(
~.
104 oo,
\
100
AI:
t/b
Fig. 1. Examples of creep curves of the alloys (applied stress, time to fracture tf at 623 and 723 K respectively): AIC0 (40 MPa, 5.06 x 105 and 246 s), AIC1 (60 MPa, 5.32 x 104 and 7.5 s), A1C2 (75 MPa, 2.23 x 104 and 235.4 s) and AIC202 (120 MPa, 4.1 x 103 and 33.4 s).
10- 2
AIC0 AlCl AIC2 AIC202
i
i
~
,,,,i
IIIIII
@
473 523 573 623
K K
K
I I llllll
10
100
STRESS
I
v • + .
100
[MPa]
Fig. 3. Stress dependence of the relaxation time of primary creep.
F. Dobe~ et al.
/
39
High temperature creep in A I - C - O alloy
TABLE 2. Stress sensitivity parameters and activation energies of the steady state creep rate, relaxation time of primary creep and recovery rate Alloy
nc
Qc (kJ mol- 1)
nT
Q, (kJ mol- 1)
nr
Qr (kJ mol-t)
Al A1C0 A1C1 A1C2 A1C202
4.9 23.6 30.7 25.7 17.6
161 311 362 327 229
7.5 13.3 34.3 24.3 20.1
159 249 397 297 255
7.1 14.2 40.2 29.9 24.4
161 214 462 348 293
loI
T=623 K
oo c
lO
0
T=723
-3
E
.CO
LJ_I
I..<::: Or"
lo - 5
EL LLI 10- 7
~
lo-9
o2 (D
L/j;y /
L.tJ
_e
.
z 2~ -1 10 30
50
100
200 20
30
50
100
STRESS o [MPo] Fig. 4. Applied stress dependence of the minimum creep rate in MA alloys.
values for MA alloys are significantly greater than those for pure aluminum, but otherwise the dependence on chemical composition is non-monotonic. However, the values of stress exponent n T, activation energy QT and pre-exponential factor A t correlate very strongly. Figure 4 presents the applied stress dependence of the minimum creep rate Eminof the alloys obtained at the respective temperatures. The diagrams illustrate (i) a growing influence of the dispersoid strengthening, which corresponds to the growth of the dispersoid content, (ii) a tendency to threshold behavior at the "actual" dispersoid strengthened alloys A1C1, AIC2 and A 1 C 2 0 2 in comparison with the behavior of the alloy AIC0 containing only the technologically unavoidable content of A1203, and (iii) in spite of a scatter of data, "jumps" in the applied stress dependence of gm~nmay be detected, indicating a change in
the applied stress sensitivity of ~min" The last fact is taken into account by using a broken line to approximate roughly the course of emi, vs. o curves. Phenomenologically, the applied stress and temperature dependence of gm~, can be characterized by the parameter of applied stress sensitivity nc and apparent activation energy Qc corresponding to the equation gmin= Ac One e x p [ - R ~ ]
(3)
where A c is a constant. In the range of higher applied stresses, where the effect of the threshold behavior does not intervene in the course of the stress and temperature dependence, this equation can approximate the curves with constant parameters no, Qc and A ~. The values of nc in Table 2 indicate a high applied stress sensitivity apparently dependent on the chemical
40
17. Dobeg et al.
/
High temperature creep in A I - C - O alloy
composition of the alloy. The values of Qc are also influenced by the alloy composition. The greatest values of n c and Qc are found in the ahoy AIC 1. In this respect, the behavior of n c and Qc characterizing the minimum creep rate is similar to that of equivalent characteristics of the relaxation time of primary creep and of the recovery rate. The apparent activation energies exceed the value of activation enthalpy of lattice diffusion A H D = 142 kJ m o l - 1, i.e. 1.6 AHD ~< Qc ~<2.5AHD. The alloys investigated are known for their superplastic behavior at high applied stresses [4]. The same tendency was observed in our experiments, especially for A1C0 and A1C1 with lower dispersoid content. Whereas at low stresses the samples broke abruptly at low strains, at high stress levels the tertiary creep commenced relatively early, probably before steadystate creep was reached. The measured minimum creep rates are then presumably greater than the relevant steady-state creep rates. This could be a cause of the steep slopes in log Ernin VS. log o plots observed by Otsuka et al. [5] at intermediate stresses and also of the "jumps" mentioned as item (iii) above. To minimize the effect of early onset of fracture mechanisms we tried to calculate the steady-state creep rate by fitting experimental creep curves to eqn. (1). Unfortunately, the differences between the measured minimum and calculated steady-state creep rate were not too significant.
where K is a constant. The effective stress dependence of the creep rate is usually described as in ref. 7
(5)
g = A ' ( o - oi) n
Integrating the linearized equations (4) and (5) with the initial conditions t = 0, e = e0, cr~= oi0 one obtains eqn. ( 1 ) in which gs = A ' ( o - O'iS) r1=
(6a)
1/K
(6b)
e i = A'( ais - o i 0 ) / K = g s r~ Ois - oi~ O-
(6c)
OiS
In dispersion strengthened systems, the internal stress is often supposed to be a superposition of two components [8]: O"t = O'ip "+- O'iD
(7)
The first component is due to the presence of dispersed particles impeding the dislocation motion, and the second component is induced by long-range stress fields of neighboring dislocations. Whereas for thermodynamically stable particles the former component is independent of strain, the latter changes from an initial to a steady-state value. For the steady-state value of dislocation internal stress, from the above relations we obtain El((y -- O.ip) --[- g S r l O'iD0 aiD S
(8) Et + ~sZ'l
4. Discussion In the last two decades, creep data have been frequently analyzed on the basis of the internal stress concept [6]. The concept starts from the postulate that the creep is driven by a difference between the externally applied stress and an internal stress, resulting from resistance of the structure to dislocation motion. The difference is called the effective stress. During the creep test, the initial dislocation structure gradually develops towards its steady-state configuration. Simultaneously the internal stress also changes towards the steady-state value. Depending on its initial value, the internal stress can either increase or decrease. The phenomenon manifests itself ostensibly as a normal primary creep with a decelerating strain rate in the former case and as an inverse primary creep with an accelerating rate in the latter case. Let us assume that the time derivative of the internal stress is proportional to the difference between the steady-state value and the instantaneous value d~7 i
dt
= K(ois - oi)
(4)
where oio0 is the initial value of dislocation induced internal stress. For single-phase alloys this equation reduces to the relation introduced by Ion et al. [9] provided that, as assumed in their paper, the initial dislocation internal stress can be neglected. Equation (8), unlike Ion's equation, does not enable us to calculate the internal stresses Oios or oie directly from parameters of the creep curves. We suggest the following procedure to estimate the values of Oios and oie. The dependence of E1/(E 1 q- ES 2"1) VS. applied stress can be extrapolated to the level e l / ( e l + g s r l ) = 1. A t t h i s level, the product gsr~ must be equal to zero and from eqn. (8)it follows that GriDS = Gr -- O'ip
(9)
Thus, the effective stress is equal to zero at this level. It is generally accepted that there is no threshold stress for dislocation creep in single-phase materials. In other words, for zero effective stress the internal stress OiDs must be equal to zero. This conclusion is supported by the results of direct measurement of the internal stress in steady-state creep of pure metals and class II solid solution alloys by the dip test technique. The ratio of
F. Dobeg et al.
/
41
High temperature creep in A I - C - O alloy
1.2
1.2
1.0
|llO0
,l,
1.0
flip
,l,
O0
o
723 K •
623 K
0.8
0.8 723 K AICO
co
0.6
o
AIC1
~'O
4'o
5'O
6o 0.6.36
7'o
+ 1.2
1.0
1.2
P
$
GIp
$
1.0
0.8
0.8
0.6
0.6
$
O'ip
$
72.3 K
•\ AIC202 0.440
6`0
8'O
,oo
STRESS
0.450
1
100
150
200
(7 [ M P o ]
Fig. 5. Determination of the particle internal stress from creep curve parameters.
effective to internal stress is either a constant independent of the applied stress [10] or it approaches zero at zero applied stress only [11]. Therefore, the applied stress found by the above extrapolation is equal to the internal stress induced by the dispersed particles Oip. The procedure is illustrated in Fig. 5. With the exception of the alloy A1C0, linear extrapolation of the experimental data is very plausible. The values obtained of the particle internal stress O~p are close to the threshold stress determined from the applied stress dependence of the steady-state creep rate. For the known value of Oip, the internal stress O~DS can be calculated from the parameters of the creep curves by means of eqn. (8) provided the term gs q OiD0/(el + iS q ) can be neglected. The assumption of negligible initial dislocation internal stress O~D0was introduced tacitly by Ion et aL [9]. The factor gSrl/ (e t + g s q ) is usually substantially less than unity. The above mentioned neglection seems, therefore, to be quite plausible. The results of calculation for aiDS are given in Fig. 6 as a function of the difference a - Oie, which represents a "net" applied stress available for dislocations to move. The dislocation internal stress is
8O
r~ EL 60
623 0 a z,
723 0 • •
K AIC1 AIC2 AIC202
u?
#'u''L)l"d-
40 ~
z
~ E]
.< Z
•
•
2O
Ld E--
z
0
20
40
i 60
80
DIFFERENCE o'-o'ip [MPo]
Fig. 6. The dependence of the dislocation internal stress in the steady state on the difference between the applied stress and particle internal stress.
obviously not dependent on either the temperature or the composition of the alloys. In this respect it corresponds to observed dislocation densities.
42
F. Dobeg et al. /x
, , ,,,,,i
,
"~
,,,,,i
101
J
/
High temperature creep in A I - C - O alloy
, , , ,,,,,i 623 723K o • AICO @ @ AIC1 o • AIC2 * AIC202
,
gV /
t--m o
10
d,'
6
10 4
L
LO
Ld
1 0- 2
/
of
F ~
(D
10-5 AI:
tad
-8
, i Jl,,ll
100
J%N J I I I I l l v
v
473
•
523 57,3
K K K
623
K
+ × ,
of 10
623 o O
E
E3 rY T
100
10 o 50
STRESS ~ [MPo]
-hg-r
(10)
where h is the work hardening coefficient and r the recovery rate. Coupled integration of eqns. (5) and (10) again reveals eqn. ( 1 ) with t7
h-
-
-
(lla)
e I + gsrt
r=
Ogs -
-
~1 +
(lib)
~STI
The calculated values of r and h are presented in Figs. 7 and 8. The work hardening coefficient in pure aluminum increases slightly with increasing stress, whereas in M A alloys a stress dependence cannot be ascertained. However, the behavior of the recovery rate is more transparent. In both pure aluminum and MA alloys it increases with increasing both stress and temperature. When it is described by the equation
r = A r o "~ e x p ( - k ~ )
o a
• •
AIC2 AIC202
Ah
v
473 525 573
K K K
623
K
4o
200
Fig. 8. Comparison of work hardening coefficients in pure aluminum and in mechanically alloyed materials.
Alternatively, eqn. ( 1 ) can be derived on the basis of the frequently applied concept of mutual competition between the processes of recovery and work hardening [12]. The evolution of the internal stress is then described by [6]
dt
AICO AIC1
STRESS G [MPo]
Fig. 7. Stress dependence of the recovery rate.
do i
~
•
x
<
, , ,,,,,i
10
10 2
723K
(12)
multiple linear regression reveals the values of the stress exponent n r and activation energy Qr given in Table 2. The correlation between these parameters and the pre-exponential factor A r is similar to that observed for the relaxation time rl. The behavior of the work hardening coefficient and recovery rates as a whole corresponds qualitatively
very well to the behavior that is found experimentally by means of small stress changes (for a review see, for example, ref. 13). Comparing the absolute values of the work hardening coefficients and recovery rates, a remarkable effect can be noted for the values of work hardening coefficients in pure aluminum and in MA alloys. The effect is illustrated in Fig. 8. The work hardening coefficient in MA alloys is about two orders of magnitude greater than in pure aluminum. The beneficial effect of the specific microstructure and of carbide and oxide particles on the creep strength of M A alloys is in good correspondence with the enhanced work hardening in these alloys. An appreciation of the influence of the recovery rate on the creep behavior is evidently not so straightforward. The steady-state creep rate in MA alloys at the same normalized stress o / G is suppressed by (at least) 10 orders of magnitude in the alloy A1C0 and the effect increases with increased alloying (see Fig. 4). For the steady state (i.e. d o i / d t = 0 ) the well known Bailey-Orowan equation follows from eqn. (10) =--
gs
r h
(13)
The depression of the recovery rate must therefore be responsible for a greater part of the improvement of creep resistance.
5. Conclusions
( 1 ) Primary creep in mechanically alloyed aluminum alloys studied in the present work can be described by the McVetty equation. (2) The maximum strain occurring in the primary stage is approximately independent of both the applied
F. Dobe~ et al.
/
High temperature creep in A I - C - O alloy
stress and temperature. The relaxation time of primary creep is characterized by a power-law stress dependence with a power ranging from 13.3 to 34.3. (3) The primary and steady-state stages of creep can be interpreted as a consequence of changes in the internal stress. Analysis of the creep curves thus enables the internal stress to be partitioned into components due to stress fields of dislocations and to the presence of dispersed particles. (4) The evolution of the internal stress can be described as a result of a competition between work hardening and recovery processes. The characteristics of these processes were established by an analysis of the primary creep curve.
References I E. Arzt, ResMech., 31 (1991)399.
43
2 T. Hostinsk)~ and J. (~adek, J. Test. EvaL, 4 (1976) 26. 3 P.G. McVetty, Mech. Eng., 56 (1934) 149. 4 T. R. Bieler and A. K. Mukherjee, Mater. Sci. Eng., A128 (199(/) 171. 5 M. Otsuka, Y. Abe and R. Horiuchi, in B. Wilshire and R. W. Evans (eds.), Proc. 3rd Int. Conf on Creep and Fracture of Engineering Materials and Structures, The Institute of Metals, London, 1987, p. 307. 6 C. N. Ahlquist, R. Gasca-Neri and W. D. Nix, Acta Memll., 19 (1970) 663. 7 W. D. Nix and B. IIschner, in R Haasen, V. Gerold and G. Kostorz (eds.), Proc. 5th Int. Conf. on the Strength of Metals andAlloys, Vol. 3, Pergamon, Oxford, 1980, p. 1503. 8 R. Lagneborg and B. Bergman, Proc. 3rd Int. ('onf on the Strength of Metals and Alloys, Vol. 1, Institute of Metals and Iron and Steel Institute, London, 1973, p. 316. 9 J. C. Ion, A. Barbosa, M. F. Ashby, B. F. Dyson and M. McLean, National Physical Laboratot?', Teddington, UK Rep. DMA A l l 5 , 1986. 10 W. Blum and A. Finkel, Acta Metall., 30 {,1982) 17(15. 11 S. Takeuchi and A. S. Argon, J. Mater. Sci., 11 (1976) 1542. 12 S.K. Mitra and D. McLean, Met. Sci. J., 1 (1967) 192. 13 J. ('?adek, Creep in Metallic Materials. Elsevier Academia, Amsterdam, Prague, 1988, p. 76.
37
Constitutive description of high temperature creep in mechanically alloyed A1-C-O alloys F. D o b e L K. Kucha~ovfi, A Orlovfi, K. M i l i 6 k a a n d J. t 2 a d e k Institute of PhysicalMetallurgy, Academy of Sciences of the Czech Republic, 616 62 Brno (Czech Republic) (Received May 26, 1993)
Abstract Creep curves of four mechanically alloyed aluminum alloys were studied within the range of stresses from 20 to 180 MPa at temperatures of 623 and 723 K. Creep curves of pure aluminum at temperatures of 473-623 K were taken as reference material data. The curves were described by the McVetty equation. The primary and steady-state stages of creep can be interpreted as a result of changes in the internal stress following from competition between work hardening and recovery processes. The analysis makes it possible to divide the internal stress into two components, one being due to stress fields of dislocations and the other to the existence of dispersed particles. The latter component equals the threshold stress in steady-state creep.
1. Introduction In many areas of modern technology, the demands 1o raise working temperatures have initiated a development of new materials with enhanced creep resistance. As a rule, these materials owe their improved properties to the existence of a dispersion of fine particles in the matrix. An extensive review, focused specifically on creep properties of dispersion strengthened materials, was recently given by Arzt [1]. This author reviewed apparently all the available papers devoted to creep of mechanically alloyed aluminum alloys (henceforth referred as MA alloys) and concluded that "the creep behaviour of polycrystalline dispersion strengthened materials is insufficiently understood." In addition, the present authors were not able to find any paper dealing more deeply with the primary stage of creep, or, generally speaking, with creep curves in these alloys. That is why this study was undertaken. Its purpose is to give the results of the time dependence of creep strain in dispersion strengthened aluminum alloys with varying contents of A1203 and A14C 3.
2. Experimental material and testing technique The subject of testing was four A1-C-O mechanically alloyed materials (marked AIC0, AIC1, AIC2 and AIC202), supplied by Erbsl6h, FRG. They were produced from aluminum powder containing about 0.4 wt.% 02 by milling with graphite in an attritor under 0921-5093/94/$7.00
argon atmosphere, cold isostatic pressing to a bar, annealing for 6-8 h at 823 K and final extrusion at 773 K. The microstructure of the resulting materials consisted of fine (sub)grains of micrometer or submicrometer size. It contained several types of dispersoid particles in quantities related to the content of carbon and oxygen additions. The resultant carbon and oxygen content and the volume fractions of A14C 3 and AI203 which can be formed by a complete transformation are given in Table 1. The creep tests were performed on cylindrical specimens of gage length 50 mm and diameter 5 mm which were machined from the extruded bars. The technique of isothermal tensile creep tests under constant applied stress was used, the engineering strain was measured with a sensitivity of 10-5. Details of the creep testing equipment are given in ref. 2. For the sake of comparison, the investigation of A I - C - O alloys was complemented by a set of creep tests of aluminum of 99.99% purity in the temperature range 473-623 K. The specimens were machined from hot-rolled slabs so that the
TABLE 1. Chemical compositions and nominal volume fractions of dispersed phases in mechanically alloyed alloys Alloy
O (wt.%)
C (wt.%)
fv
A1C0 AIC1 A1C2 A1C202
0.8 0.8 0.8 2.0
0.0 1.0 2.0 2.0
0.0116 0.0484 0.0855 0.1039
© 1994 - Elsevier Sequoia. All rights reserved
38
F. Dobe# et al.
/
High temperature creep in A I - C - O alloy
tensile stress axis was parallel to the rolling direction. The grain size of creep specimens with rectangular cross-section 8 x 3.2 mm 2 was 80 #m. For a description of the primary stage of creep curves the equation proposed by McVetty [3] completed by a steady-state term was chosen, e = e 0+ e l [ 1 - e x p ( - t/r1)]+ gs t
(1)
where t is the time passed from the start of the creep test, el is the limiting primary strain, rl is the relaxation time of the primary stage and gs is the steady-state creep rate. The values of e 0, el, r~ and gs were found by the Newton-Marquard least squares method.
3. Experimental results The alloys were creep tested at temperatures of 623 and 7 2 3 K within the interval of applied stresses to which the minimum creep rates ranging from 10- a~ to 10- 2 s- a correspond. Examples of creep curves of the alloys are given in Fig. 1. As a general conclusion, it can be said that, unless the fracture strain exceeds the value of approximately 1%, the creep curves at both the testing temperatures are characterized by a more significant primary creep stage and less important secondary and tertiary stages. However, in the region of creep conditions corresponding to a fracture strain which increases steeply above the value of 1%, creep
curves with a suppressed primary stage and an extensive tertiary stage are observed. The values of calculated primary strain are given in Fig. 2. Whereas in pure aluminum the primary strain el increases with increasing applied stress, the stress dependence of e~ in M A alloys can hardly be defined. The values of ,61 in the alloys are approximately two orders of magnitude smaller than those in pure aluminum. The relaxation time q decreases with increasing both applied stress and temperature (see Fig. 3). It can be described by the equation r I =Z~a-"" exp(Q~ / \Kll
(2)
where a is the applied stress, R is the universal gas constant and T is the absolute temperature. The values of the stress exponent n~ and activation energy Q~ found by multiple linear regression are given in Table 2. The
,
, , ,,i
,
,
i
,
, , ,,i
-1 10 ++ +/,'1~ I x / x~ / +//
,S Z .< rv" I--0,'3 >IM .<
10- 2
AI
v 47,3 • 52,3 + 573 x 62,3
K K K K
x
"...4,,W t
cl
623 723K$ o • AICO •
?&&A&&~
l
AICI
0.01
,
,
T =
,
625
u
K ,~====::~
AICO AIC 1 ~ AIC2
IN -4_
,
• • I
AIC2 AIC202
i i ii
i
i
i
i
i
i i ii
10
|
100
STRESS (~ [MPo] 0.00 0.12
'
'
'
.
.
T = Z
0.10
I'-(/3
0.08
'
.
.
Fig. 2. Stress dependence of primary strain.
'
/ / A I C O
/
72,3 K
0.06 /
'
.
"
108
,,,,I
623 723K
o O u
~-' 1° 6 AIC 1
F
V,5
102
0.02
,cU 0"00.0
0.2
0.4
0.6
O.
1.0
.2
[~
v'(
~.
104 oo,
\
100
AI:
t/b
Fig. 1. Examples of creep curves of the alloys (applied stress, time to fracture tf at 623 and 723 K respectively): AIC0 (40 MPa, 5.06 x 105 and 246 s), AIC1 (60 MPa, 5.32 x 104 and 7.5 s), A1C2 (75 MPa, 2.23 x 104 and 235.4 s) and AIC202 (120 MPa, 4.1 x 103 and 33.4 s).
10- 2
AIC0 AlCl AIC2 AIC202
i
i
~
,,,,i
IIIIII
@
473 523 573 623
K K
K
I I llllll
10
100
STRESS
I
v • + .
100
[MPa]
Fig. 3. Stress dependence of the relaxation time of primary creep.
F. Dobe~ et al.
/
39
High temperature creep in A I - C - O alloy
TABLE 2. Stress sensitivity parameters and activation energies of the steady state creep rate, relaxation time of primary creep and recovery rate Alloy
nc
Qc (kJ mol- 1)
nT
Q, (kJ mol- 1)
nr
Qr (kJ mol-t)
Al A1C0 A1C1 A1C2 A1C202
4.9 23.6 30.7 25.7 17.6
161 311 362 327 229
7.5 13.3 34.3 24.3 20.1
159 249 397 297 255
7.1 14.2 40.2 29.9 24.4
161 214 462 348 293
loI
T=623 K
oo c
lO
0
T=723
-3
E
.CO
LJ_I
I..<::: Or"
lo - 5
EL LLI 10- 7
~
lo-9
o2 (D
L/j;y /
L.tJ
_e
.
z 2~ -1 10 30
50
100
200 20
30
50
100
STRESS o [MPo] Fig. 4. Applied stress dependence of the minimum creep rate in MA alloys.
values for MA alloys are significantly greater than those for pure aluminum, but otherwise the dependence on chemical composition is non-monotonic. However, the values of stress exponent n T, activation energy QT and pre-exponential factor A t correlate very strongly. Figure 4 presents the applied stress dependence of the minimum creep rate Eminof the alloys obtained at the respective temperatures. The diagrams illustrate (i) a growing influence of the dispersoid strengthening, which corresponds to the growth of the dispersoid content, (ii) a tendency to threshold behavior at the "actual" dispersoid strengthened alloys A1C1, AIC2 and A 1 C 2 0 2 in comparison with the behavior of the alloy AIC0 containing only the technologically unavoidable content of A1203, and (iii) in spite of a scatter of data, "jumps" in the applied stress dependence of gm~nmay be detected, indicating a change in
the applied stress sensitivity of ~min" The last fact is taken into account by using a broken line to approximate roughly the course of emi, vs. o curves. Phenomenologically, the applied stress and temperature dependence of gm~, can be characterized by the parameter of applied stress sensitivity nc and apparent activation energy Qc corresponding to the equation gmin= Ac One e x p [ - R ~ ]
(3)
where A c is a constant. In the range of higher applied stresses, where the effect of the threshold behavior does not intervene in the course of the stress and temperature dependence, this equation can approximate the curves with constant parameters no, Qc and A ~. The values of nc in Table 2 indicate a high applied stress sensitivity apparently dependent on the chemical
40
17. Dobeg et al.
/
High temperature creep in A I - C - O alloy
composition of the alloy. The values of Qc are also influenced by the alloy composition. The greatest values of n c and Qc are found in the ahoy AIC 1. In this respect, the behavior of n c and Qc characterizing the minimum creep rate is similar to that of equivalent characteristics of the relaxation time of primary creep and of the recovery rate. The apparent activation energies exceed the value of activation enthalpy of lattice diffusion A H D = 142 kJ m o l - 1, i.e. 1.6 AHD ~< Qc ~<2.5AHD. The alloys investigated are known for their superplastic behavior at high applied stresses [4]. The same tendency was observed in our experiments, especially for A1C0 and A1C1 with lower dispersoid content. Whereas at low stresses the samples broke abruptly at low strains, at high stress levels the tertiary creep commenced relatively early, probably before steadystate creep was reached. The measured minimum creep rates are then presumably greater than the relevant steady-state creep rates. This could be a cause of the steep slopes in log Ernin VS. log o plots observed by Otsuka et al. [5] at intermediate stresses and also of the "jumps" mentioned as item (iii) above. To minimize the effect of early onset of fracture mechanisms we tried to calculate the steady-state creep rate by fitting experimental creep curves to eqn. (1). Unfortunately, the differences between the measured minimum and calculated steady-state creep rate were not too significant.
where K is a constant. The effective stress dependence of the creep rate is usually described as in ref. 7
(5)
g = A ' ( o - oi) n
Integrating the linearized equations (4) and (5) with the initial conditions t = 0, e = e0, cr~= oi0 one obtains eqn. ( 1 ) in which gs = A ' ( o - O'iS) r1=
(6a)
1/K
(6b)
e i = A'( ais - o i 0 ) / K = g s r~ Ois - oi~ O-
(6c)
OiS
In dispersion strengthened systems, the internal stress is often supposed to be a superposition of two components [8]: O"t = O'ip "+- O'iD
(7)
The first component is due to the presence of dispersed particles impeding the dislocation motion, and the second component is induced by long-range stress fields of neighboring dislocations. Whereas for thermodynamically stable particles the former component is independent of strain, the latter changes from an initial to a steady-state value. For the steady-state value of dislocation internal stress, from the above relations we obtain El((y -- O.ip) --[- g S r l O'iD0 aiD S
(8) Et + ~sZ'l
4. Discussion In the last two decades, creep data have been frequently analyzed on the basis of the internal stress concept [6]. The concept starts from the postulate that the creep is driven by a difference between the externally applied stress and an internal stress, resulting from resistance of the structure to dislocation motion. The difference is called the effective stress. During the creep test, the initial dislocation structure gradually develops towards its steady-state configuration. Simultaneously the internal stress also changes towards the steady-state value. Depending on its initial value, the internal stress can either increase or decrease. The phenomenon manifests itself ostensibly as a normal primary creep with a decelerating strain rate in the former case and as an inverse primary creep with an accelerating rate in the latter case. Let us assume that the time derivative of the internal stress is proportional to the difference between the steady-state value and the instantaneous value d~7 i
dt
= K(ois - oi)
(4)
where oio0 is the initial value of dislocation induced internal stress. For single-phase alloys this equation reduces to the relation introduced by Ion et al. [9] provided that, as assumed in their paper, the initial dislocation internal stress can be neglected. Equation (8), unlike Ion's equation, does not enable us to calculate the internal stresses Oios or oie directly from parameters of the creep curves. We suggest the following procedure to estimate the values of Oios and oie. The dependence of E1/(E 1 q- ES 2"1) VS. applied stress can be extrapolated to the level e l / ( e l + g s r l ) = 1. A t t h i s level, the product gsr~ must be equal to zero and from eqn. (8)it follows that GriDS = Gr -- O'ip
(9)
Thus, the effective stress is equal to zero at this level. It is generally accepted that there is no threshold stress for dislocation creep in single-phase materials. In other words, for zero effective stress the internal stress OiDs must be equal to zero. This conclusion is supported by the results of direct measurement of the internal stress in steady-state creep of pure metals and class II solid solution alloys by the dip test technique. The ratio of
F. Dobeg et al.
/
41
High temperature creep in A I - C - O alloy
1.2
1.2
1.0
|llO0
,l,
1.0
flip
,l,
O0
o
723 K •
623 K
0.8
0.8 723 K AICO
co
0.6
o
AIC1
~'O
4'o
5'O
6o 0.6.36
7'o
+ 1.2
1.0
1.2
P
$
GIp
$
1.0
0.8
0.8
0.6
0.6
$
O'ip
$
72.3 K
•\ AIC202 0.440
6`0
8'O
,oo
STRESS
0.450
1
100
150
200
(7 [ M P o ]
Fig. 5. Determination of the particle internal stress from creep curve parameters.
effective to internal stress is either a constant independent of the applied stress [10] or it approaches zero at zero applied stress only [11]. Therefore, the applied stress found by the above extrapolation is equal to the internal stress induced by the dispersed particles Oip. The procedure is illustrated in Fig. 5. With the exception of the alloy A1C0, linear extrapolation of the experimental data is very plausible. The values obtained of the particle internal stress O~p are close to the threshold stress determined from the applied stress dependence of the steady-state creep rate. For the known value of Oip, the internal stress O~DS can be calculated from the parameters of the creep curves by means of eqn. (8) provided the term gs q OiD0/(el + iS q ) can be neglected. The assumption of negligible initial dislocation internal stress O~D0was introduced tacitly by Ion et aL [9]. The factor gSrl/ (e t + g s q ) is usually substantially less than unity. The above mentioned neglection seems, therefore, to be quite plausible. The results of calculation for aiDS are given in Fig. 6 as a function of the difference a - Oie, which represents a "net" applied stress available for dislocations to move. The dislocation internal stress is
8O
r~ EL 60
623 0 a z,
723 0 • •
K AIC1 AIC2 AIC202
u?
#'u''L)l"d-
40 ~
z
~ E]
.< Z
•
•
2O
Ld E--
z
0
20
40
i 60
80
DIFFERENCE o'-o'ip [MPo]
Fig. 6. The dependence of the dislocation internal stress in the steady state on the difference between the applied stress and particle internal stress.
obviously not dependent on either the temperature or the composition of the alloys. In this respect it corresponds to observed dislocation densities.
42
F. Dobeg et al. /x
, , ,,,,,i
,
"~
,,,,,i
101
J
/
High temperature creep in A I - C - O alloy
, , , ,,,,,i 623 723K o • AICO @ @ AIC1 o • AIC2 * AIC202
,
gV /
t--m o
10
d,'
6
10 4
L
LO
Ld
1 0- 2
/
of
F ~
(D
10-5 AI:
tad
-8
, i Jl,,ll
100
J%N J I I I I l l v
v
473
•
523 57,3
K K K
623
K
+ × ,
of 10
623 o O
E
E3 rY T
100
10 o 50
STRESS ~ [MPo]
-hg-r
(10)
where h is the work hardening coefficient and r the recovery rate. Coupled integration of eqns. (5) and (10) again reveals eqn. ( 1 ) with t7
h-
-
-
(lla)
e I + gsrt
r=
Ogs -
-
~1 +
(lib)
~STI
The calculated values of r and h are presented in Figs. 7 and 8. The work hardening coefficient in pure aluminum increases slightly with increasing stress, whereas in M A alloys a stress dependence cannot be ascertained. However, the behavior of the recovery rate is more transparent. In both pure aluminum and MA alloys it increases with increasing both stress and temperature. When it is described by the equation
r = A r o "~ e x p ( - k ~ )
o a
• •
AIC2 AIC202
Ah
v
473 525 573
K K K
623
K
4o
200
Fig. 8. Comparison of work hardening coefficients in pure aluminum and in mechanically alloyed materials.
Alternatively, eqn. ( 1 ) can be derived on the basis of the frequently applied concept of mutual competition between the processes of recovery and work hardening [12]. The evolution of the internal stress is then described by [6]
dt
AICO AIC1
STRESS G [MPo]
Fig. 7. Stress dependence of the recovery rate.
do i
~
•
x
<
, , ,,,,,i
10
10 2
723K
(12)
multiple linear regression reveals the values of the stress exponent n r and activation energy Qr given in Table 2. The correlation between these parameters and the pre-exponential factor A r is similar to that observed for the relaxation time rl. The behavior of the work hardening coefficient and recovery rates as a whole corresponds qualitatively
very well to the behavior that is found experimentally by means of small stress changes (for a review see, for example, ref. 13). Comparing the absolute values of the work hardening coefficients and recovery rates, a remarkable effect can be noted for the values of work hardening coefficients in pure aluminum and in MA alloys. The effect is illustrated in Fig. 8. The work hardening coefficient in MA alloys is about two orders of magnitude greater than in pure aluminum. The beneficial effect of the specific microstructure and of carbide and oxide particles on the creep strength of M A alloys is in good correspondence with the enhanced work hardening in these alloys. An appreciation of the influence of the recovery rate on the creep behavior is evidently not so straightforward. The steady-state creep rate in MA alloys at the same normalized stress o / G is suppressed by (at least) 10 orders of magnitude in the alloy A1C0 and the effect increases with increased alloying (see Fig. 4). For the steady state (i.e. d o i / d t = 0 ) the well known Bailey-Orowan equation follows from eqn. (10) =--
gs
r h
(13)
The depression of the recovery rate must therefore be responsible for a greater part of the improvement of creep resistance.
5. Conclusions
( 1 ) Primary creep in mechanically alloyed aluminum alloys studied in the present work can be described by the McVetty equation. (2) The maximum strain occurring in the primary stage is approximately independent of both the applied
F. Dobe~ et al.
/
High temperature creep in A I - C - O alloy
stress and temperature. The relaxation time of primary creep is characterized by a power-law stress dependence with a power ranging from 13.3 to 34.3. (3) The primary and steady-state stages of creep can be interpreted as a consequence of changes in the internal stress. Analysis of the creep curves thus enables the internal stress to be partitioned into components due to stress fields of dislocations and to the presence of dispersed particles. (4) The evolution of the internal stress can be described as a result of a competition between work hardening and recovery processes. The characteristics of these processes were established by an analysis of the primary creep curve.
References I E. Arzt, ResMech., 31 (1991)399.
43
2 T. Hostinsk)~ and J. (~adek, J. Test. EvaL, 4 (1976) 26. 3 P.G. McVetty, Mech. Eng., 56 (1934) 149. 4 T. R. Bieler and A. K. Mukherjee, Mater. Sci. Eng., A128 (199(/) 171. 5 M. Otsuka, Y. Abe and R. Horiuchi, in B. Wilshire and R. W. Evans (eds.), Proc. 3rd Int. Conf on Creep and Fracture of Engineering Materials and Structures, The Institute of Metals, London, 1987, p. 307. 6 C. N. Ahlquist, R. Gasca-Neri and W. D. Nix, Acta Memll., 19 (1970) 663. 7 W. D. Nix and B. IIschner, in R Haasen, V. Gerold and G. Kostorz (eds.), Proc. 5th Int. Conf. on the Strength of Metals andAlloys, Vol. 3, Pergamon, Oxford, 1980, p. 1503. 8 R. Lagneborg and B. Bergman, Proc. 3rd Int. ('onf on the Strength of Metals and Alloys, Vol. 1, Institute of Metals and Iron and Steel Institute, London, 1973, p. 316. 9 J. C. Ion, A. Barbosa, M. F. Ashby, B. F. Dyson and M. McLean, National Physical Laboratot?', Teddington, UK Rep. DMA A l l 5 , 1986. 10 W. Blum and A. Finkel, Acta Metall., 30 {,1982) 17(15. 11 S. Takeuchi and A. S. Argon, J. Mater. Sci., 11 (1976) 1542. 12 S.K. Mitra and D. McLean, Met. Sci. J., 1 (1967) 192. 13 J. ('?adek, Creep in Metallic Materials. Elsevier Academia, Amsterdam, Prague, 1988, p. 76.