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Influence of dynamic strain aging on the transition of creep characteristics of a solid solution alloy at various temperatures
Mawrials Scienee and Engineering, A 110 (1989) 125- 130
125
Influence of Dynamic Strain Aging on the Transition of Creep Characteristics of a Solid Solution Alloy at Various Temperatures S. 1. HONG
Department of Materials Seienee and Engineering, University of Pennsylvania, l'hiladelphia, PA 19104 (U.S.A.) (Received March 3.1988; in revised form August 2, 1988)
Abstract
In this paper the temperature dependence of creep rates of A l - l M g alloy is examined in order to test the concept of dynamic strain aging with respect to temperature. The change in the stress exponent is attributed to dynamic strain aging. The equations for the creep rates at various temperatures are in good agreement with the published data available. A reasonably good fit to a straight line with a slope of 4. 76 is obtained if the temperature-compensated creep rate is plotted against the normalized effective stress. The dynamic strainaging model of this study is in good agreement with the computer simulation result of Takeuchi and Argon. The validity of the viscous glide model as the dynamic strain-aging mechanism for the high temperature creep of AI-Mg alloys is also discussed.
1. Introduction
Recently, Hong has attempted to explain the creep behavior of solid solution alloys in terms of dynamic strain aging [1-3]. In ref. 1 he showed that changes in the stress exponent for creep in A1-Mg alloys is attributed to dynamic strain aging. The predicted dislocation structure from this model [1] generally agrees with the electron micrographs of Yavari and Langdon [4]. It was also shown that the stress exponent for creep will always be 4.76 if the appropriate value of the stress ocrf(= O a p p - - O'D) is used. In a related study [3] it was found that this model predicts the creep ductility minimum as well as the creep ductility maximum, both of which were observed experimentally by Mullendore and Grant [5]. More recently, Chaudhury and Mohamed also observed the creep ductility maximum and minimum for an A1-Cu solid solution alloy [6]. 0921-5107/89/$3,50
One controversial prediction [2] of this model is that the magnitude of the measured activation energies is dependent on the temperature intervals used for their measurement. If temperature intervals larger than 10 K are used, the model predicts normal activation energies (i.e. equal to those for self-diffusion) in the stress region of socalled class I creep and very high activation energies at higher stresses. However, if temperature intervals smaller than 10K are used, the predicted activation energies contrast sharply with the established activation energies. (In this case the model predicts unusually large activation energies in the region of class I creep and unusually small activation energies at higher stresses.) Because of the possible errors involved in the measurement of temperatures, investigators have generally used temperature intervals of 20-100 K. To the knowledge of the author, only Botch et al. [7] have measured the creep activation energy for an AI-Mg alloy using temperature intervals less than 10 K. They observed unusually large activation energies, which were explained by Hong [8] in terms of dynamic strain aging. Recently, Chaudhury and Mohamed observed a high value of activation energy (250 kJ mol-~) measured in the high stress region using temperature intervals of 40 K. They claimed that the creep characteristics in this region are not entirely consistent with any of the existing high stress creep mechanisms. However, the model of Hong [2] predicts higher activation energies than those of diffusion in the high stress region if temperature intervals larger than 10 K are used. It seems that this model predicts activation energies which a r e in good agreement with the reported activation energies of aluminum-base solid solution alloys. To verify the dependence of creep activation energies on the temperature intervals of the © Elsevier Sequoia/Printed in The Netherlands
126 measurements, more systematic works are desirable. In this study, published creep data of A1-Mg alloys [9, 10] at three different temperatures are used to test the concept of dynamic strain aging with respect to temperature. The temperature dependence of the creep rate of an A1-Mg alloy is examined over a wide range of stresses. The validity of the viscous glide model as the dynamic strain-aging mechanism for high temperature deformation of A1-Mg alloys is also discussed in the light of the criticisms [11-14] of this model. 2. Analysis
Hong suggested the following creep equation for A1-Mg solid solution alloys [1]:
i--
1"09x10211 ( Q~ TG3.76 ((Yapp - O'D)476C-°96exp --
(1) where g is the creep rate, G is the shear modulus ( G ( M P a ) = 3 . 0 2 2 x 1 0 4 - 1 6 T ) , (Yapp is the applied stress, oD is the contribution of dynamic strain aging to the total stress, C is the mole fraction of magnesium atoms (0.0109), Q* is the activation energy ( 143.4 kJ mol- 1) for lattice selfdiffusion of pure aluminum [15, 16], and R and T have their usual meanings. The strengthening t e r m oD due to dynamic strain aging can be considered to be a frictional stress for the movement of dislocations [1, 17]. Cottrell and Jaswon [18] showed that the stress increase due to dragging solute atoms increases almost proportionally to the dislocation velocity when the velocity is sufficiently low relative to the diffusivity of solute atoms. However, in the temperature and strain rate ranges where the diffusion rate of solute atoms is comparable with the dislocation velocity, the atmosphere which can follow the dislocation motion will be reduced in size and concentration with increasing velocity of dislocations. Thus the dragging stress will decrease as the velocity of dislocations increases in this range [19-21]. Takeuchi and Argon [21] simulated the steady viscous motion of a straight edge dislocation surrounded by a Cottrell atmosphere and calculated the drag force using a computer. (This model will be called the modified Cottrell model.) They found that the dislocation velocity vc at which the Cottrell atmosphere produces a maximum drag
force is approximately D k T / A , where A is the interaction parameter between the dislocation and solute atoms due to a size misfit. According to the results of Takeuchi and Argon [21], the maximum frictional stress o ° can be calculated from the following equation:
F= o°b, l+v - a - -
:r(1 - v)
GbeC
(2)
where v is Poisson's ratio, G is the shear modulus, b is the Burgers' vector and e is the size misfit parameter. Apparently OD decreases from o ° as the strain rate deviates from the critical velocity v c. The critical dislocation velocity v c at which the atmosphere produces a maximum drag force is given by the following equation [2, 21]: :~(1- v)kT/)_/3 e x p ( - ~ - i ) Vc-(1 + v ) G b Q e
(3)
where k is the Boltzmann constant,/5 is the diffusivity of magnesium in aluminum [22], Q is the atomic volume, Q is the activation energy (130 kJ mole- 1) for diffusion of magnesium in aluminum, and fl = er(1 - v)kTDo/(1 + v)Gbfae(= 1.824 x 105 m s-l at 600 K). From eqn. (3) we can calculate the critical strain rate at which the maximum drag force o ° occurs as follows:
i~ = pbv~ = p b f l e x p ( - ~Q-i.)= B*exp
(4)
where p is the dislocation density. B* was determined to be 2.25 x 108 by fitting the calculation of Hong [1] to the data of Oikawa et al. [23]. From eqn. (4) and the data of Oikawa et al. [23] the dislocation density is calculated to be 4.31 x 1012 m-2, which is in good agreement with the experimental result of Horiuchi and Otsuku [24] (1012-1013 m-2). fl is a function of temperature whereas B* is considered to be independent of temperature. The constancy of B* has been confirmed experimentally by some investigators [25-27]. The discrepancy could be due to the temperature dependence of the dislocation density. The increase in/3 with temperature may be compensated for by the decrease in dislocation density.
127
The frictional stress aD due to dynamic strain aging has its maximum potency at some intermediate temperatures and can be represented by a statistical distribution function. The reason why the frictional stress OD due to dynamic strain aging has the form of the statistical distribution has been given elsewhere [1-3, 8, 17, 26-29]. The temperature dependence of the frictional stress for dynamic strain aging can be determined experimentally by the subtraction method [17, 26-29]. The flow stress vs. temperature plots for aluminum alloys [7, 30] show that the frictional stress oo due to dynamic strain aging is almost symmetrical about the temperature /" where the maximum of o D occurs. Therefore, irrespective of the detailed mechanisms of dynamic strain aging, the frictional stress aD for aluminum alloys can be approximately represented by the following equation:
where B measures the width of the distribution about T. From eqns. (2) and (5) the following equation can be obtained: l+v ((T-~')2) o D = et er( 1 - v--~)G b e C exp B
(6)
To make cr° = 32 MPa for the best fit between the prediction and the data for AI-3%Mg alloy [1], the correlation factor a was chosen to be 0.62, which is in good agreement with the results of Takeuchi and Argon (a=0.1-1). For A1-Mg alloys, v=0.34 [31], b = 2 . 8 6 3 × 1 0 -~° m and e=0.12 [32]. In this study, B = 6 × 103 was used for a good fit between the prediction and the data of Oikawa et al. By incorporating eqns. (4) and (6), the a o vs. g or o D vs. v relations can be obtained. Takeuchi and Argon [21] showed that the oD vs. v relation is approximately symmetrical about the critical velocity v c where the maximum of OD occurs. The predictions [2] from eqns. (4) and (6) are in good agreement with the results of Takeuchi and Argon. Therefore their model can be conveniently modelled by a statistical distribution function to simplify the calculations [2]. a ° and B obtained by the subtraction method from the flow stress vs. temperature plot (Fig. 4 of ref. 7) were 36 MPa and 1.8 × 1 0 4 K 2 respectively [2] for AI-3.23mol.%Mg alloy, which is in reasonably good agreement with the values
used in this study ( a ° = 3 2 MPa for AI-3%Mg alloy, B = 6 x 10 3 K2). The differences could be due to the difference in grain size [2] and other unknown factors. Before comparing the prediction of this study with the data of Oikawa et al. [9, 10], the validity of the modified Cottrell model as a dynamic strain-aging mechanism for the high temperature creep of A1-Mg alloys must be discussed. In the 1970s many investigators [11-14] criticized the Cottrell mechanism and, by modifying the ideas of Sleeswyk [11], successfully predicted the condition for the Protevin-Le Chatelier effect. (These models will be called the modified Sleeswyk models hereafter.) Many researchers [11-14] claimed that the modified Sleeswyk models are more accurate in describing the deformation behavior of alloys and that the Cottrell mechanism is only valid under specific conditions. In this study it was shown that the dynamic strain-aging models of the previous study [1] and of this study are generally in good agreement with the computer simulation result of Takeuchi and Argon (the modified Cottrell model). Recently, Hong [1] showed, using the symmetrical distribution function for OD, that dislocations move smoothly at the average dislocation velocity in the stress region of stress exponent 3. Many researchers [4, 24, 32-35] observed a homogeneous distribution of dislocations in this region. As stress increases (in the latter half of region H in Fig. 1 of ref. 1), dislocation motion is predicted to be rather jerky and dislocations tend to segregate [1]. Therefore dislocation motion at high stresses is different from the basic assumption of the modified Cottrell mechanism (steady viscous glide motion of dislocations)[21]. However, the frictional stress o D predicted by the modified Cottrell model could provide a good approximation of the average frictional stress of many mobile dislocations. For creep deformation in the high stress region (in the latter half of region H in Fig. 1 of ref. 1), dynamic strain aging increases the tendency for the dislocation velocity to increase in the low internal stress field and to decrease in the high internal stress field [1]. Therefore in this region, at a certain moment, some dislocations move at a very high speed and some dislocations move at a very low speed, depending on where they are located. To calculate the average frictional stress due to dynamic strain aging, fast-moving dislocations as well as slow-moving dislocations must be con-
128
sidered. The average frictional stress due to dynamic strain aging might be predicted by assuming that dislocations move steadily at the average dislocation velocity as in the work by Takeuchi and Argon [21]. One interesting point of this model [1] is that it. predicts the variation of the dislocation velocity in the high stress region although it is based on the steady viscous motion of dislocations. Therefore the frictional stress at) predicted from the assumption that dislocations move smoothly at the average velocity could represent the average frictional stress on the mobile dislocations in the stress and temperature range of this study. The good fit between the pre-. diction of this study and the data of Oikawa et al. [9, 10] supports this assumption (see Fig. 1). At low stresses where: the stress exponent is approximately 3, as stated, dislocations tend to move at the average dislocation velocity irrespective of the internal stress fields [1]. The dislocation motion in this region is very similar to the basic assumption (steady viscous glide motion) of the Cottrell mechanism. Many researchers [2, 4, 9, 23, 36-39] believe that the most likely controlling mechanism for the dislocation motion in this region is the Cottrell mechanism. Since the experiments of Oikawa et al. [9, 10] were performed at relatively high temperatures (0.63-0.7 Tin), the modified Cottrell model could be a reasonable mechanism for dynamic strain aging. The modified Sleeswyk models [12-14], on the other hand, are based on experiments performed on A1-Mg alloys at around 0.3 Tm, where discontinuous deformation (i.e. yield drop or serration) is manifested. For the high temperature creep deformation of A1-Mg alloys, no such kind of discontinuous deformation (i.e. strain burst) has been reported [1-6, 9, 10, 23, 24]. At very high stresses where the stress exponent approaches infinity (i.e. the strain rate sensitivity approaches zero), the instability of deformation occurs and the modified Sleeswyk model might be important for the high temperature creep deformation. At intermediate temperatures (0.4Tm), Kirk et al. [40] observed strain burst during creep deformation of AI-4.6%Mg alloy, which may be explained by the modified Sleeswyk model. The stress dependence of the creep rate at various temperatures can be obtained from eqns. (1)-(4) and (6). The creep rates predicted from eqns. (1), (4) and (6) are plotted in Fig. 1 together with the data of Oikawa et al. [9, 10] obtained at
/x 10 -n
10 -4
/://
,lO
.tO
lO-S
lO-e
2//
2
. . . . .5 . .
101
,
!
, = ,,
,J,~,150
0", MPa
Fig. 1. Steady state creep rate for A I - I M g alloy at various temperatures: A, 570 K; e, 600 K; XT,640 K; - - , predicted from eqns. (1), (4)and (6).
570, 600 and 640 K. Figure 1 shows a good fit between the predicted line and the experimental data over a wide stress region for three different temperatures. This model apparently simulates the transition of the stress exponent of A I - I M g alloy at three different temperatures. The attempt to explain the creep behavior of solid solution alloys in terms of dynamic strain aging seems valid in regard to the effect of temperature. Since the predicted creep rates at various temperatures are in good agreement with the data of Oikawa et al. [9, 10], the creep activation energies predicted from this model seem reasonable. Since quite a few papers [2, 6, 7-10, 41-48] have reported anomalies of creep activation energies for solid solution alloys, more systematic experimental works are desirable to investigate the dependence
129
of creep activation energies on the temperature intervals of measurement. In eqn. (1) it is shown that the stress exponent will always be 4.76 if the appropriate value of stress, namely the effective s t r e s s Oeff( = Oap p --O'D) , is used. Equation (1) can be modified as follows:
where D is the seif-diffusivity in pure aluminum (=lxl0 4 e x p ( _ 1 4 3 . 4 / R T ) m 2s ~).InFig. 2 the temperature-compensated creep rates are plotted against the normalized effective stress. A reasonably good fit to a straight line with a slope of 4.76 is obtained. The results shown in Fig. 2
10 -s
support the presumption that the change in the stress exponent occurs because of the contribution of the dynamic strain aging to the stress exponent. According to this model, as oi~ changes nonlinearly with temperature, the dependence of apparent activation energies on the temperature interval used for the measurement is expected. Actually the internal stress field is modified by the frictional stress due to dynamic strain aging [1, 17] and the modification of the internal stress field is greatly temperature dependent as well as strain rate dependent. Hence activation parameters determined under these conditions should be Considered with caution because the temperature dependence of o D greatly alters the activation energy of the basic process. Without accurate knowledge of solute effects on the short-range order internal stress field, it is impossible to associate the activation parameters with a single process unambiguously. If the effective stress defined in this paper is used for the measurement of activation energies, the activation energies for the creep in the absence of dynamic strain aging can be obtained (eqn. (7i and Fig. 2). 3. Conclusions
10-s
Attempts were made to explain the change in the stress exponent at three different temperatures in terms of dynamic strain aging. The predicted creep rate is in good agreement with the data of Oikawa et al. [9, 10]. A reasonably good fit to a straight line with a slope of 4.76 is obtained if the temperature-compensated creep rate is plotted against the normalized effective stress. Since the frictional stress due to dynamic strain aging modifies the internal stress field, activation parameters determined for alloys should be considered with caution.
el
oCI
a
10-x ,.x
Iff1 I
References
10 -+2
1 2 3 4
! I
i
f , l ~
!
I
i
|
10-4
J ,l.I
i
i
i
i
lO-a
O'e.IG Fig. 2. Plot of temperature-compensated creep rate vs. normalized effective stress: A , 570 K; e, 600 K; ~, 640 K; - - , predicted from eqn, (7).
S. 1. Hong, Mater. Sci. Eng., 82(1986) 175. S.I. Hong, Mater. Sci. Eng.. 86 (1987) 211. S. I. Hong, Mater. Sci. Eng., 91 (1987) 137. E Yavari and T. G. Langdon, Acta Metall., 30 (1982) 2181. 5 A.W. Mullendore and N. J. Grant, in A. M. Freudehthal, B. A. Boley and H. Liebowitz (eds.), Proc. 3rd 3~vmp. on Naval Structural Mechanics, Pergamon, Oxford, 1964, p. 169. 6 P. K. Chaudhury and F. A. Mohamed, Metall. Trans. A, 18(1987) 2105.
130 7 N. R. Borch, L. A. Shepard and J, E. Dorn, Trans. ASM, 52 (1960) 494. 8 S. I. Hong, Scr. Metall., 18(1984) 1351. 9 H. Oikawa, H. Sato and K. Maruyama, Mater. Sci. Eng., 75(1985)21. 10 H. Sato and H. Oikawa, Scr. Metall., 22 ( 1988) 87. 11 A.W. Sleeswyk, Acta Metall., 6 ( 1958) 598. 12 P.G. McCormick, Acta Metall., 20 (1972) 351. 13 A. van den Beukel, Phys. Status Solidi A, 30 ( 1975) 197. 14 R. A. Mulford and U. E Kocks, Acta Metall., 27(1979) 1125. 15 T. S. Lundy and J. E Murdock, J. Appl. Phys., 33 (1962) 1671. 16 M. Beyeler and Y. Adda, J. Phys. (Paris), 29(1968) 345. 17 S. I. Hong, Mater. Sci. Eng., 79(1986) t. 18 A. H. Cottrell and M. A. Jaswon, Proc. R. Soc. London, Ser. A, 109(1949) 104. 19 A. H. Cottrell, Dislocation and Plastic Flow in Crystals, Oxford University Press, Oxford, 1953. 20 J.P. Hirth and J, Lothe, Theory of Dislocations, McGrawHill, New York, 1968. 21 S. Takeuchi and A. S. Argon, Philos. Mag., 40(1979) 65. 22 S. J. Rothman, N. L. Peterson, L, J. Nowicki and L. C. Robinson, Phys. Status Solidi (b), 63 (1974) K29. 23 H. Oikawa, K. Honda and S. Ito, Mater. Sci. Eng., 64 (1984) 237. 24 R. Horiuchi and M. Otsuku, Trans. Jpn. Inst. Met., 13 (1972)284. 25 M. Doner and H. Conrad, Metall. Trans., 4(1973) 2809. 26 S.I. Hong, Mater. Sci. Eng., 76(1985) 77. 27 S. I. Hong, W. S. Ryu and C. S. Rim, J. Nucl. Mater., 116 (1983)314.
28 A. K. Miller and O. D. Sherby, Acta Metall., 26 (1978) 289. 29 S. I. Hong, W. S. Ryu and C. S. Rim, J. Nucl. Mater., 120 (1984) 1. 30 D. Walton, L. Shepard and J. E. Dorn, Trans. AIME, 221 (1961)458. 31 ASM Committee on Aluminum and Aluminum Alloys, Metals Handbook, Vol. 2, Nonferrous Alloys and Pure Metals, ASM, Metals Park, OH, 1979. 32 F.A. Mohamed, Metall. Trans. A, 9(1978) 1013. 33 B.L. Jones and C. M. Sellars, Met. Sci. J., 4 (1970) 96. 34 A. Orlova and J. Cadek, Z. Metallkd., 65 (1974) 200. 35 S. Takeuchi and A. S. Argon, J. Mater. Sci., 11 (1976) 1542. 36 F.A. Mohamed, Mater Sci. Eng., 38(1979) 73. 37 H. Oikawa, J. Matsumo and S. Karashima, Met. Sci. J., 9 (1975)209. 38 S.H. Hong and J. Weertman, Acta Metall., 34 (1986) 742. 39 E A. Mohamed, Mater. Sci. Eng., 61 (1983) 149. 40 J. L. Kirk, D. K. Matlock, G. R. Edwards and W. L. Bradley, Metall. Trans. A, 8(1977) 2030. 41 A. Lawley, J. A. Coll and R. W. Cahn, Trans. AIME, 218 (1960) 166. 42 W. R. Johnson, Metall. Trans., 3(1972) 963. 43 S. 1. Hong, S. W. Nam and I. S. Choi, J. Mater. Sci., 19 (1984) 1672. 44 R. G. Stang, J. Inst. Met., 90(1962) 329. 45 C.M. SeUar, Metall. Trans., 4(1973) 1693. 46 V. Fidleris, ASTMSpec. Tech. Publ. 458, 1969, p. 1. 47 S. I. Hong, Mater. Sci. Eng., 64(1984)L19. 48 B. S. Chin, W. D. Nix and G. M. Pound, Metall. Trans. A, 8(1977) 1523.
125
Influence of Dynamic Strain Aging on the Transition of Creep Characteristics of a Solid Solution Alloy at Various Temperatures S. 1. HONG
Department of Materials Seienee and Engineering, University of Pennsylvania, l'hiladelphia, PA 19104 (U.S.A.) (Received March 3.1988; in revised form August 2, 1988)
Abstract
In this paper the temperature dependence of creep rates of A l - l M g alloy is examined in order to test the concept of dynamic strain aging with respect to temperature. The change in the stress exponent is attributed to dynamic strain aging. The equations for the creep rates at various temperatures are in good agreement with the published data available. A reasonably good fit to a straight line with a slope of 4. 76 is obtained if the temperature-compensated creep rate is plotted against the normalized effective stress. The dynamic strainaging model of this study is in good agreement with the computer simulation result of Takeuchi and Argon. The validity of the viscous glide model as the dynamic strain-aging mechanism for the high temperature creep of AI-Mg alloys is also discussed.
1. Introduction
Recently, Hong has attempted to explain the creep behavior of solid solution alloys in terms of dynamic strain aging [1-3]. In ref. 1 he showed that changes in the stress exponent for creep in A1-Mg alloys is attributed to dynamic strain aging. The predicted dislocation structure from this model [1] generally agrees with the electron micrographs of Yavari and Langdon [4]. It was also shown that the stress exponent for creep will always be 4.76 if the appropriate value of the stress ocrf(= O a p p - - O'D) is used. In a related study [3] it was found that this model predicts the creep ductility minimum as well as the creep ductility maximum, both of which were observed experimentally by Mullendore and Grant [5]. More recently, Chaudhury and Mohamed also observed the creep ductility maximum and minimum for an A1-Cu solid solution alloy [6]. 0921-5107/89/$3,50
One controversial prediction [2] of this model is that the magnitude of the measured activation energies is dependent on the temperature intervals used for their measurement. If temperature intervals larger than 10 K are used, the model predicts normal activation energies (i.e. equal to those for self-diffusion) in the stress region of socalled class I creep and very high activation energies at higher stresses. However, if temperature intervals smaller than 10K are used, the predicted activation energies contrast sharply with the established activation energies. (In this case the model predicts unusually large activation energies in the region of class I creep and unusually small activation energies at higher stresses.) Because of the possible errors involved in the measurement of temperatures, investigators have generally used temperature intervals of 20-100 K. To the knowledge of the author, only Botch et al. [7] have measured the creep activation energy for an AI-Mg alloy using temperature intervals less than 10 K. They observed unusually large activation energies, which were explained by Hong [8] in terms of dynamic strain aging. Recently, Chaudhury and Mohamed observed a high value of activation energy (250 kJ mol-~) measured in the high stress region using temperature intervals of 40 K. They claimed that the creep characteristics in this region are not entirely consistent with any of the existing high stress creep mechanisms. However, the model of Hong [2] predicts higher activation energies than those of diffusion in the high stress region if temperature intervals larger than 10 K are used. It seems that this model predicts activation energies which a r e in good agreement with the reported activation energies of aluminum-base solid solution alloys. To verify the dependence of creep activation energies on the temperature intervals of the © Elsevier Sequoia/Printed in The Netherlands
126 measurements, more systematic works are desirable. In this study, published creep data of A1-Mg alloys [9, 10] at three different temperatures are used to test the concept of dynamic strain aging with respect to temperature. The temperature dependence of the creep rate of an A1-Mg alloy is examined over a wide range of stresses. The validity of the viscous glide model as the dynamic strain-aging mechanism for high temperature deformation of A1-Mg alloys is also discussed in the light of the criticisms [11-14] of this model. 2. Analysis
Hong suggested the following creep equation for A1-Mg solid solution alloys [1]:
i--
1"09x10211 ( Q~ TG3.76 ((Yapp - O'D)476C-°96exp --
(1) where g is the creep rate, G is the shear modulus ( G ( M P a ) = 3 . 0 2 2 x 1 0 4 - 1 6 T ) , (Yapp is the applied stress, oD is the contribution of dynamic strain aging to the total stress, C is the mole fraction of magnesium atoms (0.0109), Q* is the activation energy ( 143.4 kJ mol- 1) for lattice selfdiffusion of pure aluminum [15, 16], and R and T have their usual meanings. The strengthening t e r m oD due to dynamic strain aging can be considered to be a frictional stress for the movement of dislocations [1, 17]. Cottrell and Jaswon [18] showed that the stress increase due to dragging solute atoms increases almost proportionally to the dislocation velocity when the velocity is sufficiently low relative to the diffusivity of solute atoms. However, in the temperature and strain rate ranges where the diffusion rate of solute atoms is comparable with the dislocation velocity, the atmosphere which can follow the dislocation motion will be reduced in size and concentration with increasing velocity of dislocations. Thus the dragging stress will decrease as the velocity of dislocations increases in this range [19-21]. Takeuchi and Argon [21] simulated the steady viscous motion of a straight edge dislocation surrounded by a Cottrell atmosphere and calculated the drag force using a computer. (This model will be called the modified Cottrell model.) They found that the dislocation velocity vc at which the Cottrell atmosphere produces a maximum drag
force is approximately D k T / A , where A is the interaction parameter between the dislocation and solute atoms due to a size misfit. According to the results of Takeuchi and Argon [21], the maximum frictional stress o ° can be calculated from the following equation:
F= o°b, l+v - a - -
:r(1 - v)
GbeC
(2)
where v is Poisson's ratio, G is the shear modulus, b is the Burgers' vector and e is the size misfit parameter. Apparently OD decreases from o ° as the strain rate deviates from the critical velocity v c. The critical dislocation velocity v c at which the atmosphere produces a maximum drag force is given by the following equation [2, 21]: :~(1- v)kT/)_/3 e x p ( - ~ - i ) Vc-(1 + v ) G b Q e
(3)
where k is the Boltzmann constant,/5 is the diffusivity of magnesium in aluminum [22], Q is the atomic volume, Q is the activation energy (130 kJ mole- 1) for diffusion of magnesium in aluminum, and fl = er(1 - v)kTDo/(1 + v)Gbfae(= 1.824 x 105 m s-l at 600 K). From eqn. (3) we can calculate the critical strain rate at which the maximum drag force o ° occurs as follows:
i~ = pbv~ = p b f l e x p ( - ~Q-i.)= B*exp
(4)
where p is the dislocation density. B* was determined to be 2.25 x 108 by fitting the calculation of Hong [1] to the data of Oikawa et al. [23]. From eqn. (4) and the data of Oikawa et al. [23] the dislocation density is calculated to be 4.31 x 1012 m-2, which is in good agreement with the experimental result of Horiuchi and Otsuku [24] (1012-1013 m-2). fl is a function of temperature whereas B* is considered to be independent of temperature. The constancy of B* has been confirmed experimentally by some investigators [25-27]. The discrepancy could be due to the temperature dependence of the dislocation density. The increase in/3 with temperature may be compensated for by the decrease in dislocation density.
127
The frictional stress aD due to dynamic strain aging has its maximum potency at some intermediate temperatures and can be represented by a statistical distribution function. The reason why the frictional stress OD due to dynamic strain aging has the form of the statistical distribution has been given elsewhere [1-3, 8, 17, 26-29]. The temperature dependence of the frictional stress for dynamic strain aging can be determined experimentally by the subtraction method [17, 26-29]. The flow stress vs. temperature plots for aluminum alloys [7, 30] show that the frictional stress oo due to dynamic strain aging is almost symmetrical about the temperature /" where the maximum of o D occurs. Therefore, irrespective of the detailed mechanisms of dynamic strain aging, the frictional stress aD for aluminum alloys can be approximately represented by the following equation:
where B measures the width of the distribution about T. From eqns. (2) and (5) the following equation can be obtained: l+v ((T-~')2) o D = et er( 1 - v--~)G b e C exp B
(6)
To make cr° = 32 MPa for the best fit between the prediction and the data for AI-3%Mg alloy [1], the correlation factor a was chosen to be 0.62, which is in good agreement with the results of Takeuchi and Argon (a=0.1-1). For A1-Mg alloys, v=0.34 [31], b = 2 . 8 6 3 × 1 0 -~° m and e=0.12 [32]. In this study, B = 6 × 103 was used for a good fit between the prediction and the data of Oikawa et al. By incorporating eqns. (4) and (6), the a o vs. g or o D vs. v relations can be obtained. Takeuchi and Argon [21] showed that the oD vs. v relation is approximately symmetrical about the critical velocity v c where the maximum of OD occurs. The predictions [2] from eqns. (4) and (6) are in good agreement with the results of Takeuchi and Argon. Therefore their model can be conveniently modelled by a statistical distribution function to simplify the calculations [2]. a ° and B obtained by the subtraction method from the flow stress vs. temperature plot (Fig. 4 of ref. 7) were 36 MPa and 1.8 × 1 0 4 K 2 respectively [2] for AI-3.23mol.%Mg alloy, which is in reasonably good agreement with the values
used in this study ( a ° = 3 2 MPa for AI-3%Mg alloy, B = 6 x 10 3 K2). The differences could be due to the difference in grain size [2] and other unknown factors. Before comparing the prediction of this study with the data of Oikawa et al. [9, 10], the validity of the modified Cottrell model as a dynamic strain-aging mechanism for the high temperature creep of A1-Mg alloys must be discussed. In the 1970s many investigators [11-14] criticized the Cottrell mechanism and, by modifying the ideas of Sleeswyk [11], successfully predicted the condition for the Protevin-Le Chatelier effect. (These models will be called the modified Sleeswyk models hereafter.) Many researchers [11-14] claimed that the modified Sleeswyk models are more accurate in describing the deformation behavior of alloys and that the Cottrell mechanism is only valid under specific conditions. In this study it was shown that the dynamic strain-aging models of the previous study [1] and of this study are generally in good agreement with the computer simulation result of Takeuchi and Argon (the modified Cottrell model). Recently, Hong [1] showed, using the symmetrical distribution function for OD, that dislocations move smoothly at the average dislocation velocity in the stress region of stress exponent 3. Many researchers [4, 24, 32-35] observed a homogeneous distribution of dislocations in this region. As stress increases (in the latter half of region H in Fig. 1 of ref. 1), dislocation motion is predicted to be rather jerky and dislocations tend to segregate [1]. Therefore dislocation motion at high stresses is different from the basic assumption of the modified Cottrell mechanism (steady viscous glide motion of dislocations)[21]. However, the frictional stress o D predicted by the modified Cottrell model could provide a good approximation of the average frictional stress of many mobile dislocations. For creep deformation in the high stress region (in the latter half of region H in Fig. 1 of ref. 1), dynamic strain aging increases the tendency for the dislocation velocity to increase in the low internal stress field and to decrease in the high internal stress field [1]. Therefore in this region, at a certain moment, some dislocations move at a very high speed and some dislocations move at a very low speed, depending on where they are located. To calculate the average frictional stress due to dynamic strain aging, fast-moving dislocations as well as slow-moving dislocations must be con-
128
sidered. The average frictional stress due to dynamic strain aging might be predicted by assuming that dislocations move steadily at the average dislocation velocity as in the work by Takeuchi and Argon [21]. One interesting point of this model [1] is that it. predicts the variation of the dislocation velocity in the high stress region although it is based on the steady viscous motion of dislocations. Therefore the frictional stress at) predicted from the assumption that dislocations move smoothly at the average velocity could represent the average frictional stress on the mobile dislocations in the stress and temperature range of this study. The good fit between the pre-. diction of this study and the data of Oikawa et al. [9, 10] supports this assumption (see Fig. 1). At low stresses where: the stress exponent is approximately 3, as stated, dislocations tend to move at the average dislocation velocity irrespective of the internal stress fields [1]. The dislocation motion in this region is very similar to the basic assumption (steady viscous glide motion) of the Cottrell mechanism. Many researchers [2, 4, 9, 23, 36-39] believe that the most likely controlling mechanism for the dislocation motion in this region is the Cottrell mechanism. Since the experiments of Oikawa et al. [9, 10] were performed at relatively high temperatures (0.63-0.7 Tin), the modified Cottrell model could be a reasonable mechanism for dynamic strain aging. The modified Sleeswyk models [12-14], on the other hand, are based on experiments performed on A1-Mg alloys at around 0.3 Tm, where discontinuous deformation (i.e. yield drop or serration) is manifested. For the high temperature creep deformation of A1-Mg alloys, no such kind of discontinuous deformation (i.e. strain burst) has been reported [1-6, 9, 10, 23, 24]. At very high stresses where the stress exponent approaches infinity (i.e. the strain rate sensitivity approaches zero), the instability of deformation occurs and the modified Sleeswyk model might be important for the high temperature creep deformation. At intermediate temperatures (0.4Tm), Kirk et al. [40] observed strain burst during creep deformation of AI-4.6%Mg alloy, which may be explained by the modified Sleeswyk model. The stress dependence of the creep rate at various temperatures can be obtained from eqns. (1)-(4) and (6). The creep rates predicted from eqns. (1), (4) and (6) are plotted in Fig. 1 together with the data of Oikawa et al. [9, 10] obtained at
/x 10 -n
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/://
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Fig. 1. Steady state creep rate for A I - I M g alloy at various temperatures: A, 570 K; e, 600 K; XT,640 K; - - , predicted from eqns. (1), (4)and (6).
570, 600 and 640 K. Figure 1 shows a good fit between the predicted line and the experimental data over a wide stress region for three different temperatures. This model apparently simulates the transition of the stress exponent of A I - I M g alloy at three different temperatures. The attempt to explain the creep behavior of solid solution alloys in terms of dynamic strain aging seems valid in regard to the effect of temperature. Since the predicted creep rates at various temperatures are in good agreement with the data of Oikawa et al. [9, 10], the creep activation energies predicted from this model seem reasonable. Since quite a few papers [2, 6, 7-10, 41-48] have reported anomalies of creep activation energies for solid solution alloys, more systematic experimental works are desirable to investigate the dependence
129
of creep activation energies on the temperature intervals of measurement. In eqn. (1) it is shown that the stress exponent will always be 4.76 if the appropriate value of stress, namely the effective s t r e s s Oeff( = Oap p --O'D) , is used. Equation (1) can be modified as follows:
where D is the seif-diffusivity in pure aluminum (=lxl0 4 e x p ( _ 1 4 3 . 4 / R T ) m 2s ~).InFig. 2 the temperature-compensated creep rates are plotted against the normalized effective stress. A reasonably good fit to a straight line with a slope of 4.76 is obtained. The results shown in Fig. 2
10 -s
support the presumption that the change in the stress exponent occurs because of the contribution of the dynamic strain aging to the stress exponent. According to this model, as oi~ changes nonlinearly with temperature, the dependence of apparent activation energies on the temperature interval used for the measurement is expected. Actually the internal stress field is modified by the frictional stress due to dynamic strain aging [1, 17] and the modification of the internal stress field is greatly temperature dependent as well as strain rate dependent. Hence activation parameters determined under these conditions should be Considered with caution because the temperature dependence of o D greatly alters the activation energy of the basic process. Without accurate knowledge of solute effects on the short-range order internal stress field, it is impossible to associate the activation parameters with a single process unambiguously. If the effective stress defined in this paper is used for the measurement of activation energies, the activation energies for the creep in the absence of dynamic strain aging can be obtained (eqn. (7i and Fig. 2). 3. Conclusions
10-s
Attempts were made to explain the change in the stress exponent at three different temperatures in terms of dynamic strain aging. The predicted creep rate is in good agreement with the data of Oikawa et al. [9, 10]. A reasonably good fit to a straight line with a slope of 4.76 is obtained if the temperature-compensated creep rate is plotted against the normalized effective stress. Since the frictional stress due to dynamic strain aging modifies the internal stress field, activation parameters determined for alloys should be considered with caution.
el
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a
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Iff1 I
References
10 -+2
1 2 3 4
! I
i
f , l ~
!
I
i
|
10-4
J ,l.I
i
i
i
i
lO-a
O'e.IG Fig. 2. Plot of temperature-compensated creep rate vs. normalized effective stress: A , 570 K; e, 600 K; ~, 640 K; - - , predicted from eqn, (7).
S. 1. Hong, Mater. Sci. Eng., 82(1986) 175. S.I. Hong, Mater. Sci. Eng.. 86 (1987) 211. S. I. Hong, Mater. Sci. Eng., 91 (1987) 137. E Yavari and T. G. Langdon, Acta Metall., 30 (1982) 2181. 5 A.W. Mullendore and N. J. Grant, in A. M. Freudehthal, B. A. Boley and H. Liebowitz (eds.), Proc. 3rd 3~vmp. on Naval Structural Mechanics, Pergamon, Oxford, 1964, p. 169. 6 P. K. Chaudhury and F. A. Mohamed, Metall. Trans. A, 18(1987) 2105.
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