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Influence of dynamic strain aging on the apparent activation energy for creep
Materials Science and Engineering, 64 (1984) L19-L21
L19
Letter
Influence o f dynamic strain aging on the apparent activation energy for creep
SUN IG HONG Korea Advanced Energy Research Institute, P.O. Box 7, Daeduk Danji, Chung-Nam (South Korea) (Received February 27, 1984) SUMMARY The influence o f dynamic strain aging on the apparent activation energy for the creep o f Zircaloy-2 is examined on the basis o f the published data available. The expression for the apparent activation energy in the temperature range where dynamic strain aging occurs is given in terms o f the solute strengthening term a D due to dynamic strain aging. The equation produces realistic simulations o f the negative apparent activation energy and the peak in apparent activation energy observed in Zircaloy-2. The negative apparent activation energy and the peak result from the strong temperature dependence o f the solute strengthening term due to dynamic strain aging. 1. INTRODUCTION Unusually high activation energies for creep have been observed for an A1-3%Mg alloy [1] and Zircaloy-2 [2-4] in the temperature range in which dynamic strain aging occurs. Furthermore, negative activation energies have been reported for Zr-2.5%Nb alloy [5]. So the physical validity of the activation energy determined under these conditions has been called into question. The activation energy peaks in Zircaloy-2 have been considered meaningless because the creep rate drops to very low values during dynamic strain aging [6]. In this letter a simple explanation of these high creep activation energies and negative 0025-5416/84/$3.00
activation energies is proposed. It is suggested that the peak and the negative values axe caused by the solute strengthening term due to dynamic strain aging. 2. ANALYSES Since creep involves thermally activated processes, the creep rate for pure metals can be expressed by
(1) where ~ is the steady state creep rate, K is a constant, o is the applied stress, n is the stress exponent (equal to about 5), Q is the activation energy for self-diffusion, and R and T have their usual significance. The applied stress for materials containing impurities and alloying elements is usually comprised of two separable parts [7-10] in a region of dynamic strain aging: O'app =
0
(2)
-~- 0 D
(]rap p is the applied stress, o is the basic stress expected in the absence of a solute strengthening term due to dynamic strain aging and OD is the contribution of dynamic strain aging. Equation (2) is substituted into eqn. (1) to give where
K(a + OD) n exp(-- Qa~PPl (3) \ RT! where Qapp is the apparent activation energy for creep. The apparent activation energy can be calculated by differentiating eqn. (3) with respect to 1/T:
_ I~(Ine)1 Qapp
----
p{In(o + aD)}]
R ( b ( 1 / T ) j a + nRL
= Q*
~I]T)
Jo
nRT2 t ~(o + oD) t ~a 0+0 D I n R T 2 do D O'app
dT
© Elsevier Sequoia/Printed
(4) in The Netherlands
L20
where Q* is the true activation energy expected in the absence of dynamic strain aging, which is identical with the activation energy for self-diffusion. The solute strengthening term OD due to dynamic strain aging has the form of the statistical normal distribution function [7-11] since dynamic strain aging dies out at high and low temperatures and has its m a x i m u m potency at some intermediate temperature: OD (Pa) = OD° exp
B
(5)
where OD° is the m a x i m u m value of OD, T is the temperature at which this m a x i m u m occurs and B measures the width of the distribution about T. It is generally accepted that the solutes responsible for dynamic strain aging in Zircaloy are oxygen atoms [8, 9, 1214]. The derivative of OD with respect to temperature is substituted into eqn. (4) to give
Oapp = Q* + +
2nROD°T2(T - T) exp{--(T-- T)2/B} BOapp
(6) The self-diffusion activation energy for zirconium as reported by Kidson [15] is 190 kJ mo1-1 and the applied stress aapp given by Fidleris [4] is 138 MPa. The value of OD° is that given by Hong et al. [8] as 91.4 MPa for Zircaloy-4. B and T are expected to be lower in this study than the values predicted by Hong et al. [8] because the strain rate range of the experiment performed by Fidleris [4] is lower by four or five orders of magnitude than that carried out by Hong et al. [8]. In general, the lower the strain rate, the lower the temperature T at which dynamic strain aging has its maximum potency [7-11], and for any given material the width of the distribution of o D due to dynamic strain aging becomes smaller with decreasing temperature [11]. For a good fit of the value of the peak in apparent activation energy predicted by eqn. (6) to the data of Fidleris [4], B and were calculated to be 9.14 × 102 and 600 K respectively. Equation (6) indicates that the second term of the right-hand side is negative at temperatures below T and positive at temperatures
above T. At a temperature T where the maxim u m of the solute strengthening term occurs, doD/dT in eqn. (4) equals zero and the second term of the right-hand side in eqn. (6) also equals zero. In Fig. 1 the apparent activation energy versus temperature curve predicted by eqn. (6) is plotted together with the data of Fidleris [4]. The predicted line is in good agreement with the experimental data above 500 K. Below 500 K the observed activation energies deviate from the predicted line, which might be explained by the increased importance of dislocation pipe diffusion with the decrease in temperature. In Fig. 2 a sudden decrease in creep rate with temperature is observed in the vicinity of 530 K, implying a negative activation energy, and a rapid increase in creep rate with temperature is observed in the vicinity of 620 K, implying a high activation energy. Equation (6) predicts both the peak in activation energy and the negative apparent activation energy as shown in Fig. 1. The solute strengthening term OD is gradually increasing with increasing temperature at temperatures below and reaches the m a x i m u m value at temperature T. Therefore, the decrease in creep rate
50(
400
--
300
0
"~ 200 O I00
--I00
.,~o
5 O'O
~o
7OO '
I 800
I 900
TEMPERATURE (K)
Fig. 1. Variation in activation energy for the creep of Zircaloy-2 with temperature at 138 MPa [4 ]: - - , predicted; o, transverse, data of Fidleris; A, longitudinal, data of Fidleris.
L21 10-5
and the negative activation energy. The results of this investigation illustrate convincingly that dynamic strain aging is an important factor in the creep behaviour of Zircaloy-2. The variation in the solute strengthening term OD due to dynamic strain aging with temperature leads to the negative activation energy and the peak in apparent activation energy. This model appears to be applicable to simulating the peaks in the apparent activation energy for the creep of other alloys such as A1-3%Mg.
IO-S
'~ 10-7 UJ In., 10- ! W 0
IO-Ii
I0-'(
~
300
I
,oo
I
5oo
I
60o
REFERENCES
TEMPERATURE (K)
Fig. 2. Variation in creep rate of Zircaloy-2 with temperature at 138 MPa [4]: , predicted; ©, transverse, data of Fidleris; ~, Longitudinal, data of Fidleris.
and the minimum in creep rate in Fig. 2 are attributed to the strengthening effect due to dynamic strain aging. The temperature range where the minimum in creep rate is observed coincides with the temperature range when the maximum in OD occurs. In conclusion, the equation described in this letter appears to be useful not only in simulating the peak and the negative activation energy but also in possibly aiding our understanding of their physical causes.
3. CONCLUDING REMARKS
The equation derived in this letter predicts both the peak in apparent activation energy
1 N. R. Botch, L. A. Shepard and J. E. Dorn, Trans. Am. Soc. Met., 52 (1960) 494. 2 J. J. Holmes, J. Nucl. Mater., 13 (1964) 137. 3 V. Fidleris, J. Nucl. Mater., 26 (1968) 51. 4 V. Fidleris, A S T M Spec. Tech. Publ. 458, 1969, p. 1. 5 M. Pahutova, V. Cenry and J. Cadek, J. Nucl. Mater., 61 (1976)285. 6 D. L. Douglass, The Metallurgy o f Zirconium, International Atomic Energy Agency, Vienna, 1971, p. 270. 7 K. W Qian and R. E. Reed-Hill, Acta Metall., 31 (1983) 87. 8 S.I. Hong, W. S. Ryu and C. S. Rim, J. Nucl. Mater., 116 (1983) 314. 9 S.I. Hong, W. S. Ryu and C. S. Rim, J. Nucl. Mater., to be published. 10 S. I. Hong, submitted to Scr. Metall. 11 A. K. Miller and O. D. Sherby, Acta Metall., 26 (1978) 289. 12 W. R. Thorpe and I. O. Smiths, J. Nucl. Mater., 78 (1978) 49. 13 O. D. Sherby and A. K. Miller, Development of the materials code, MATMOD, EPRI Rep. NP-567, 1977 (Electric Power Research Institute). 14 J. L. Derep, S. Ibrahim, R. Rouby and G. Fantozzi, Acta Metall., 28 (1980) 607. 15 G. V. Kidson, Electrochem. Technol., 4 (1966) 193.
L19
Letter
Influence o f dynamic strain aging on the apparent activation energy for creep
SUN IG HONG Korea Advanced Energy Research Institute, P.O. Box 7, Daeduk Danji, Chung-Nam (South Korea) (Received February 27, 1984) SUMMARY The influence o f dynamic strain aging on the apparent activation energy for the creep o f Zircaloy-2 is examined on the basis o f the published data available. The expression for the apparent activation energy in the temperature range where dynamic strain aging occurs is given in terms o f the solute strengthening term a D due to dynamic strain aging. The equation produces realistic simulations o f the negative apparent activation energy and the peak in apparent activation energy observed in Zircaloy-2. The negative apparent activation energy and the peak result from the strong temperature dependence o f the solute strengthening term due to dynamic strain aging. 1. INTRODUCTION Unusually high activation energies for creep have been observed for an A1-3%Mg alloy [1] and Zircaloy-2 [2-4] in the temperature range in which dynamic strain aging occurs. Furthermore, negative activation energies have been reported for Zr-2.5%Nb alloy [5]. So the physical validity of the activation energy determined under these conditions has been called into question. The activation energy peaks in Zircaloy-2 have been considered meaningless because the creep rate drops to very low values during dynamic strain aging [6]. In this letter a simple explanation of these high creep activation energies and negative 0025-5416/84/$3.00
activation energies is proposed. It is suggested that the peak and the negative values axe caused by the solute strengthening term due to dynamic strain aging. 2. ANALYSES Since creep involves thermally activated processes, the creep rate for pure metals can be expressed by
(1) where ~ is the steady state creep rate, K is a constant, o is the applied stress, n is the stress exponent (equal to about 5), Q is the activation energy for self-diffusion, and R and T have their usual significance. The applied stress for materials containing impurities and alloying elements is usually comprised of two separable parts [7-10] in a region of dynamic strain aging: O'app =
0
(2)
-~- 0 D
(]rap p is the applied stress, o is the basic stress expected in the absence of a solute strengthening term due to dynamic strain aging and OD is the contribution of dynamic strain aging. Equation (2) is substituted into eqn. (1) to give where
K(a + OD) n exp(-- Qa~PPl (3) \ RT! where Qapp is the apparent activation energy for creep. The apparent activation energy can be calculated by differentiating eqn. (3) with respect to 1/T:
_ I~(Ine)1 Qapp
----
p{In(o + aD)}]
R ( b ( 1 / T ) j a + nRL
= Q*
~I]T)
Jo
nRT2 t ~(o + oD) t ~a 0+0 D I n R T 2 do D O'app
dT
© Elsevier Sequoia/Printed
(4) in The Netherlands
L20
where Q* is the true activation energy expected in the absence of dynamic strain aging, which is identical with the activation energy for self-diffusion. The solute strengthening term OD due to dynamic strain aging has the form of the statistical normal distribution function [7-11] since dynamic strain aging dies out at high and low temperatures and has its m a x i m u m potency at some intermediate temperature: OD (Pa) = OD° exp
B
(5)
where OD° is the m a x i m u m value of OD, T is the temperature at which this m a x i m u m occurs and B measures the width of the distribution about T. It is generally accepted that the solutes responsible for dynamic strain aging in Zircaloy are oxygen atoms [8, 9, 1214]. The derivative of OD with respect to temperature is substituted into eqn. (4) to give
Oapp = Q* + +
2nROD°T2(T - T) exp{--(T-- T)2/B} BOapp
(6) The self-diffusion activation energy for zirconium as reported by Kidson [15] is 190 kJ mo1-1 and the applied stress aapp given by Fidleris [4] is 138 MPa. The value of OD° is that given by Hong et al. [8] as 91.4 MPa for Zircaloy-4. B and T are expected to be lower in this study than the values predicted by Hong et al. [8] because the strain rate range of the experiment performed by Fidleris [4] is lower by four or five orders of magnitude than that carried out by Hong et al. [8]. In general, the lower the strain rate, the lower the temperature T at which dynamic strain aging has its maximum potency [7-11], and for any given material the width of the distribution of o D due to dynamic strain aging becomes smaller with decreasing temperature [11]. For a good fit of the value of the peak in apparent activation energy predicted by eqn. (6) to the data of Fidleris [4], B and were calculated to be 9.14 × 102 and 600 K respectively. Equation (6) indicates that the second term of the right-hand side is negative at temperatures below T and positive at temperatures
above T. At a temperature T where the maxim u m of the solute strengthening term occurs, doD/dT in eqn. (4) equals zero and the second term of the right-hand side in eqn. (6) also equals zero. In Fig. 1 the apparent activation energy versus temperature curve predicted by eqn. (6) is plotted together with the data of Fidleris [4]. The predicted line is in good agreement with the experimental data above 500 K. Below 500 K the observed activation energies deviate from the predicted line, which might be explained by the increased importance of dislocation pipe diffusion with the decrease in temperature. In Fig. 2 a sudden decrease in creep rate with temperature is observed in the vicinity of 530 K, implying a negative activation energy, and a rapid increase in creep rate with temperature is observed in the vicinity of 620 K, implying a high activation energy. Equation (6) predicts both the peak in activation energy and the negative apparent activation energy as shown in Fig. 1. The solute strengthening term OD is gradually increasing with increasing temperature at temperatures below and reaches the m a x i m u m value at temperature T. Therefore, the decrease in creep rate
50(
400
--
300
0
"~ 200 O I00
--I00
.,~o
5 O'O
~o
7OO '
I 800
I 900
TEMPERATURE (K)
Fig. 1. Variation in activation energy for the creep of Zircaloy-2 with temperature at 138 MPa [4 ]: - - , predicted; o, transverse, data of Fidleris; A, longitudinal, data of Fidleris.
L21 10-5
and the negative activation energy. The results of this investigation illustrate convincingly that dynamic strain aging is an important factor in the creep behaviour of Zircaloy-2. The variation in the solute strengthening term OD due to dynamic strain aging with temperature leads to the negative activation energy and the peak in apparent activation energy. This model appears to be applicable to simulating the peaks in the apparent activation energy for the creep of other alloys such as A1-3%Mg.
IO-S
'~ 10-7 UJ In., 10- ! W 0
IO-Ii
I0-'(
~
300
I
,oo
I
5oo
I
60o
REFERENCES
TEMPERATURE (K)
Fig. 2. Variation in creep rate of Zircaloy-2 with temperature at 138 MPa [4]: , predicted; ©, transverse, data of Fidleris; ~, Longitudinal, data of Fidleris.
and the minimum in creep rate in Fig. 2 are attributed to the strengthening effect due to dynamic strain aging. The temperature range where the minimum in creep rate is observed coincides with the temperature range when the maximum in OD occurs. In conclusion, the equation described in this letter appears to be useful not only in simulating the peak and the negative activation energy but also in possibly aiding our understanding of their physical causes.
3. CONCLUDING REMARKS
The equation derived in this letter predicts both the peak in apparent activation energy
1 N. R. Botch, L. A. Shepard and J. E. Dorn, Trans. Am. Soc. Met., 52 (1960) 494. 2 J. J. Holmes, J. Nucl. Mater., 13 (1964) 137. 3 V. Fidleris, J. Nucl. Mater., 26 (1968) 51. 4 V. Fidleris, A S T M Spec. Tech. Publ. 458, 1969, p. 1. 5 M. Pahutova, V. Cenry and J. Cadek, J. Nucl. Mater., 61 (1976)285. 6 D. L. Douglass, The Metallurgy o f Zirconium, International Atomic Energy Agency, Vienna, 1971, p. 270. 7 K. W Qian and R. E. Reed-Hill, Acta Metall., 31 (1983) 87. 8 S.I. Hong, W. S. Ryu and C. S. Rim, J. Nucl. Mater., 116 (1983) 314. 9 S.I. Hong, W. S. Ryu and C. S. Rim, J. Nucl. Mater., to be published. 10 S. I. Hong, submitted to Scr. Metall. 11 A. K. Miller and O. D. Sherby, Acta Metall., 26 (1978) 289. 12 W. R. Thorpe and I. O. Smiths, J. Nucl. Mater., 78 (1978) 49. 13 O. D. Sherby and A. K. Miller, Development of the materials code, MATMOD, EPRI Rep. NP-567, 1977 (Electric Power Research Institute). 14 J. L. Derep, S. Ibrahim, R. Rouby and G. Fantozzi, Acta Metall., 28 (1980) 607. 15 G. V. Kidson, Electrochem. Technol., 4 (1966) 193.