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Internal stress superplasticity in Al–Be eutectic alloy during triangular temperature profile
PII:
Acta mater. Vol. 46, No. 1, pp. 207±213, 1998 # 1997 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved Printed in Great Britain 1359-6454/98 $19.00 + 0.00 S1359-6454(97)00216-4
INTERNAL STRESS SUPERPLASTICITY IN Al±Be EUTECTIC ALLOY DURING TRIANGULAR TEMPERATURE PROFILE K. KITAZONO and E. SATO The Institute of Space and Astronautical Science, 3-1-1 Yoshinodai, Sagamihara, Kanagawa 229, Japan (Received 16 April 1997; accepted 23 June 1997) AbstractÐMaterials containing second phases show superplastic behavior during thermal cycling (internal stress superplasticity). In the present study, this behavior has been investigated by thermal cycling creep tests with a triangular wave having ®xed heating and cooling rates, using a Be-particle±dispersed Al matrix alloy. The thermal cycling creep rates were much higher than the isothermal creep rates and proportional to the applied stress at intermediate stresses. It was interpreted by the previous theoretical model of internal stress superplasticity, and the values of the thermal cycling creep rate were about 60% of the theoretical one. The discrepancy was explained by the transition after each temperature reversal until achieving the quasi-steady state stress distribution during thermal cycling. # 1997 Acta Metallurgica Inc.
1. INTRODUCTION
Two dierent types of superplasticity [1] have been observed in crystalline materials: ®ne structure superplasticity and internal stress superplasticity. The former refers to extensive elongation in materials with ®ne-grained crystal structures, and the latter in materials having high internal stresses. If an external stress is applied on some kinds of materials during thermal cycling, they possess high internal stresses and deform superplastically with an average strain rate which is proportional to the applied stress and much faster than the isothermal creep rate. Internal stress superplasticity is classi®ed according to materials as follows. (i) Materials having phase transformations in a thermal cycling regime such as steel (transformation superplasticity) [2, 3] (ii) Materials with anisotropic thermal expansion coecients such as a-uranium and pure zinc [4, 5] (iii) Materials consisting of phases with dierent thermal expansion coecients such as metal matrix composites [6±11] The third type is investigated in the present paper. Metal matrix composites (MMCs) are engineering materials in which hard components are dispersed in a ductile metal matrix to produce desirable strength and stiness, though, in general, they exhibit very low tensile ductility. Although many attempts have been made to apply the concept of ®ne structure superplasticity to MMCs [12±15], fewer attempts have been made to apply that of internal stress superplasticity. One example is SiC207
whisker±reinforced Al matrix composite, the strain of which was extended, under tension, to about 160% without fracture using thermal cycling [6]. Several theoretical models have been proposed for internal stress superplasticity. One model proposed by Hong et al. [8] predicts a linear relationship between the average strain rate and the applied stress, assuming that half of the moving dislocations are in¯uenced by the internal stress that aids their motion, and the remaining half are in¯uenced by the internal stress that opposes their motion. However, it cannot predict the realistic distribution and magnitude of the internal stresses. The old models for transformation superplasticity [2, 3] can predict an enhancement of the yield region adjacent to the transformed phase by a transformation strain, but cannot predict the stress±strain rate relation. Simulations of the ®nite-element method on two dimensions [16] have been reported for the internal stress superplasticity. The authors [17] have recently proposed a new theoretical model for internal stress superplasticity based on continuum micromechanics. They analyzed quasi-steady state stress ®elds in a material consisting of a matrix with spherical inclusions during a temperature change, and then calculated the average strain rate under a given applied stress and a given heating or cooling rate. A signi®cant eect of heating and cooling rates on the average strain rate is predicted. However, in the previous experimental investigations [6±11], the temperature pro®les were those in which heating and cooling rates were not constant, but were large at just after each temperature reversal and decreased gradually. For quantitative discussion, the temperature pro®le
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of triangular-wave with ®xed heating and cooling rates is desirable. The purpose of this paper is to experimentally analyze internal stress superplasticity in an ideal case of the theoretical model. Constant heating and cooling rates, i.e. triangular temperature pro®le, were achieved using induction heating and cooling gas. Al±Be eutectic alloy was adopted as the model material of a simple metal matrix composite. The alloy consists of Be particles and Al matrix, which are mutually insoluble, whose interface is incoherent but hard to break, and the former of which has a volume fraction as low as 0.012 [18]. In addition, Be particles have a size around 1 mm, with which the strain mismatch by thermal expansion is hardly relaxed by volume diusion in the matrix but the strain mismatch by matrix plastic ¯ow is easily relaxed by interface diusion [17]. (For the particular case of this study, it is reexamined in Appendix A.) Experimental results of isothermal and thermal cycling creep were, thus, quantitatively compared with the theoretical model. Preliminary results had been reported [19] and the present paper analyzed the eect of the temperature pro®le and discussed the origin of the discrepancy from the theoretical prediction. On the other hand, Wakashima et al. [20] analyzed stress-enhanced thermal ratcheting in directionally solidi®ed Al±Al3Ni eutectic alloy. This phenomenon is, however, strictly distinguished from internal stress superplasticity. Thermal ratcheting occurs even under zero applied stress and only in materials having an isotropic second phase. On the contrary, internal stress superplasticity does not occur under zero applied stress, but occurs even in materials having an isotropic second phase. 2. EXPERIMENTAL
2.1. Material Ingots of Al±0.8 mass% Be eutectic alloy were cast in ambient atmosphere using Al±2.5 mass% Be mother alloy and 99.99 mass% pure Al ingots. They were hot extruded with an extrusion ratio of 18, and annealed at 873 K for 250 ks. A scanning electron micrograph of the inside of a grain of this material is shown in Fig. 1(a). Be particles precipitate both at the matrix grain boundaries and inside the matrix grains; the average particle size at the grain boundaries is 2.4 mm, that inside the grains is 0.5 mm, and the ratio of the volume fraction between them is 1:2. Because the internal stress generated by Be particles at the grain boundaries is easily accommodated by fast grain boundary diusion, the volume fraction of Be particles, ¦, is regarded as 0.008, two third of the total volume fraction of 0.012, in the further discussion. Figure 1(b) shows a scanning electron micrograph of electron channeling contrast of the Al matrix observed
Fig. 1. SEM micrographs of creep test specimens. (a) Be particles dispersed in the Al matrix grains and (b) electron channeling contrast of Al matrix grains.
with an accelerated voltage of 3 kV. It is found that the average grain size of the Al matrix is 50 mm, which is too large for ®ne structure superplasticity [1]. 2.2. Creep tests To compare isothermal creep with thermal cycling creep, the equivalent temperature, Teq, is de®ned as Z Tmax Q 1 Q dT; exp ÿ exp ÿ RTeq Tmax ÿ Tmin Tmin RT 1 where Tmin and Tmax are the minimum and maximum temperatures during thermal cycling, respectively, R is the gas constant, T is the absolute temperature and Q is the apparent activation energy of isothermal creep, 146 kJ/mol [21]. The equivalent temperature for thermal cycling creep tested at 573± 673 K is calculated as 638 K. Compression creep apparatus was designed specially for this study. The uniaxial load was applied to the specimen through a quartz cylinder. The specimen was heated using a high-frequency induction coil and cooled by cooling gas. A thermocouple was spot-welded on a hollow cylinder specimen of 10 mm in length, 6 mm in outside
KITAZONO and SATO et al.: INTERNAL STRESS SUPERPLASTICITY
Fig. 2. Typical experimental result of thermal cycling creep under a constant applied stress of 1.2 10ÿ5 E. The heating and cooling rates and temperature range for thermal cycling are 5 K/s and 573±673 K, respectively. Broken line shows an average strain rate of thermal cycling creep.
diameter and 4 mm in inside diameter. The typical temperature pro®le and the corresponding specimen length change are shown in Fig. 2. The temperature pro®le for thermal cycling was triangular, and the maximum controlling error, which occurred at points where heating was reversed to cooling, was less than 22 K. The thermal cycling strain rate was measured as an average strain rate shown schematically by the broken line in Fig. 2; the measurement was performed after a period sucient for achieving the quasi-steady state. The tensile creep test was performed to con®rm the large tensile elongation during thermal cycling, using a servo hydraulic testing machine equipped with an induction heating facility. The specimen had a gage length of 20 mm and a diameter of 9 mm. 3. RESULTS
Isothermal and thermal cycling compression creep rates, eÇ , are shown in Fig. 3 as functions of applied stress, sA/E, which is compensated by Young's modulus of E = 5.26 1010 Pa for pure Al at 638 K [22]. Three heating and cooling rates, vTÇ v = 10, 5 and 2 K/s, were selected in the thermal cycling creep tests performed within the temperature range of 573±673 K. The isothermal creep test was performed at the corresponding equivalent temperature of 638 K. Under isothermal conditions, an area with a stress exponent of seven is observed at high stress, and with a decrease in the applied stress, the stress exponent decreases to one. At high applied stresses,
209
thermal cycling creep rates are close to isothermal creep rates for all heating and cooling rates. For vTÇ v = 10 and 5 K/s, with a decrease in the applied stress, the thermal cycling creep rate becomes much higher than the isothermal creep rate, where thermal cycling creep shows a stress exponent of one. These areas can be considered to exhibit internal stress superplasticity, as discussed in detail in Section 4.2. With a further decrease in the applied stress, thermal cycling creep rates again drop sharply to the same values as those of isothermal creep rates. On the other hand, for vTÇ v = 2 K/s, no region of a stress exponent of one, but a region of low stress exponents (1.5±2.0) is observed at around sA/E = 5 10ÿ5. The specimen after the thermal cycling tensile creep test is shown in Fig. 4. A ®xed load of 108 N, which corresponds to an initial stress 4.4 10ÿ5E, was applied to the specimen. The temperature range and heating and cooling rates were programmed as 673±773 K and 15 K/s, respectively, but they were not achieved perfectly because of the large size of the specimen. The specimen fractured after 120 h with about 200% elongation when the controlling thermocouple moved beyond the heating zone. Neither necking nor sharp crack was observed in the specimen. SEM observation showed that the dispersion state of Be particles was identical to that before the test.
4. DISCUSSION
4.1. Isothermal creep Under the isothermal condition, a high value of stress exponent, n = 7, is observed at high stresses and a stress exponent of one is observed at low stresses (Fig. 3). At high stresses, an experimental constitutive equation is expressed by e_ iso B0 snA
n 7;
2
where B0 is a constant depending on the temperature. The high value of stress exponent at high stresses is understood in terms of the particle dispersion strengthening mechanism of dislocation creep. At low applied stress, diusional creep equation [23, 24] given by e_ iso 47Dgb d 13Dv d
O 1 sA kT d 3
3
becomes predominant compared to dislocation creep, where Dgb and Dv are grain boundary and volume diusion coecients, respectively, d is the matrix grain size, d is the width of the matrix grain boundary, k is the Boltzmann constant and O is the atomic volume. The calculated results, shown by the broken line in Fig. 3, are close to the experimental results within one order. The values used for the calculation are listed in Appendix A.
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Fig. 3. Thermal cycling and isothermal creep rates as a function of the modulus compensated applied stress in Al±Be alloys. The equivalent temperature of thermal cycling creep tests was calculated as 638 K, and the temperature range, DT, was ®xed at 100 K. Heating and cooling rates were ®xed at (a) 10 K/s, (b) 5 K/s and (c) 2 K/s. Broken lines show theoretical values calculated using equation (5).
4.2. Comparison between thermal cycling creep data and the theoretical model The authors [17] proposed a new theoretical model for internal stress superplasticity based on continuum micromechanics. Unlike the previous models, this model can be used to calculate quasisteady state stress and strain rate distributions, which are nonuniform in space but stationary in time, during heating or cooling at a ®xed rate under a uniaxial applied stress. These, in turn, lead to the volume-averaged strain rate of the material, Çeth. Here it is assumed that the material consists of a
matrix containing spherical inclusions with dierence in thermal expansion coecient, Da. The matrix is assumed to deform according to a multiaxial power-law equation with a stress exponent, n, and a constant depending on the temperature, B; B = (1 ÿ ¦)nB0. In addition, the volume component of the strain mismatch is assumed not to be relaxed by volume diusion, but the shear component of that is assumed to be completely relaxed by interface diusion. The last assumption of fast interface diusion and null volume diusion is con®rmed in Appendix
KITAZONO and SATO et al.: INTERNAL STRESS SUPERPLASTICITY
211
Fig. 4. Photograph of thermal cycling tensile creep specimens deformed to fracture. Initial applied stress was 4.4 10ÿ5 E, temperature range was 673±773 K, heating and cooling rates were 15 K/s, and total elongation to fracture was about 200%.
A. The calculated relaxation time for interface diusion, ti, is so small that the creep rate is not dominated by the relaxation kinetics and thus independent of the particle size. The power-law equation of n = 7 for the matrix is justi®ed because the equivalent stresses in the matrix during thermal cycling is so high that they are in the high stress region of n = 7, as con®rmed in Appendix B. Using the above analysis, the volume-averaged uniaxial strain rate of the material is given by
Under an applied stress much higher than the internal stress, the thermal cycling creep rate [equation (4)] coincides with the isothermal creep
rate [equation (2)], and the relation of B = (1 ÿ ¦)nB0 is obtained. At low applied stresses, it is proportional to the applied stress, and depends on the heating and cooling rates [equation (5)]; this region is regarded as the superplastic region. It is worth mentioning the obtained thermal cycling creep equations [equations (4) and (5)] contain no unknown parameters if the isothermal creep equation [equation (2)] has been obtained experimentally. The theoretical prediction for the superplastic region is plotted in Fig. 3 by a broken line. The values of ¦, Da and B are summarized in Appendix B. In the case of vTÇ v = 10 and 5 K/s, experimental results for a stress exponent of one at applied stress sA/E = 7 10ÿ6±3 10ÿ5 agree well with theoretical prediction calculated using equation (5). For vTÇ v = 2 K/s, the low stress exponent region at around sA/E = 5 10ÿ5 locates near the line calculated using equation (5).
Fig. 5. Relation between thermal cycling creep rates and absolute value of heating and cooling rates subjected to sA=10ÿ5 E.
Fig. 6. Relation between thermal cycling creep rates and temperature range subjected to sA=3.6 10ÿ5 E. Heating and cooling rates were ®xed at 5 K/s. Experimental results were normalized by a theoretical value. Solid line is the curve ®tted using equation (7).
e_ th B0 snA ;
for sA >> jDaT_ j=B0 1=n ; 4 1=n
2n n 4 1 ÿ f f sA e_ th jDaT_ j1ÿ1=n B1=n 0 21=n 5 1 ÿ f f 1=n for sA << jDaT_ j=B0 1=n ; 5
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The relation between strain rate and heating and cooling rates is depicted in Fig. 5 with a ®xed applied stress of sA/E = 10ÿ5. The broken line is the theoretical prediction using equation (5). The measured creep rate increases with an increase in heating and cooling rates, and is about 60% of the corresponding theoretical value. It is clear that high heating and cooling rates are required for high strain rates in internal stress superplasticity. 4.3. Eect of reversing the direction of temperature change during thermal cycling Experimental results of thermal cycling strain rates measured in the superplastic region are about 60% of the theoretical prediction calculated using equation (5). This discrepancy is caused by reversing the direction of temperature change during thermal cycling; while the theoretical analysis treats only the quasi-steady state during heating or cooling, a stationary internal stress distribution is achieved after a certain period after each reversal of temperature change. Figure 6 shows the relation between the strain rate and the temperature range of thermal cycling with a ®xed equivalent temperature of 638 K, ®xed heating and cooling rates of 5 K/s and ®xed applied stress of 3.6 10ÿ5 E. The thermal cycling strain rate at D T = 0 K corresponds to the isothermal strain rate. Here the authors assume that the instantaneous strain rate, Çe*, increases exponentially with time, t, after each temperature reversal, from the corresponding isothermal creep rate, Çeiso, given by equation (2) towards the theoretical stationary creep rate, Çeth, given by equation (5) with a constant C. e_ e_ iso e_ th ÿ e_ iso 1 ÿ exp ÿCjT_ jt
6
The average strain rate, Çe, which is one and only measurable value during thermal cycling in the temperature range, D T, is given by Z B0 DT e_ e_ 1 d jT_ jt DT 0 B0 e_ th ÿ
1 e_ th ÿ e_ iso 1 ÿ exp ÿCDT CDT
7
The solid line in Fig. 6 is the ®tted curve of equation (7) with 1/C = 80 K. It indicates that after each temperature reversal, the quasi-steady state stress and strain rate distributions are achieved in a transient time, (CvTÇ v)ÿ1, of 16 s. Daehn and Gonzalez-Doncel [9] investigated the eect of the temperature range during thermal cycling in A1±SiC alloy. However, because they did not consider the heating and cooling rates, the heating and cooling rates changed with a change in the temperature range in their experiments, and then, they could not reach the present discussion.
4.4. Thermal cycling under low applied stress The superplastic region of a stress exponent of one in thermal cycling creep appears in the region of intermediate applied stresses, and with a further decrease in the applied stress, the thermal cycling creep rate drops sharply to close to the isothermal creep rate (Fig. 3). At present the authors cannot explain the reason of this drop, and are planning to study the behavior at dierent equivalent temperature or with dierent matrix grain size to understand it. 5. SUMMARY
Internal stress superplasticity in Al±Be eutectic alloy was investigated. The applied temperature pro®le was a triangular wave with ®xed heating and cooling rates. The thermal cycling creep rates were much higher than the isothermal creep rates and proportional to the applied stress at intermediate stresses; this region was considered to be the internal stress superplastic region. At an applied stress much higher than the internal stress, the thermal cycling creep rate became close to the isothermal one. The internal stress region was interpreted by the previous theoretical model of internal stress superplasticity, and the value of the thermal cycling creep rate was about 60% of the theoretical one. The discrepancy was explained by the transition after each temperature reversal until achieving the quasi-steady state stress distribution during thermal cycling. Internal stress superplasticity of high strain rate needs high heating and cooling rates and a wide temperature range under appropriate applied stresses. AcknowledgementsÐThis work was supported by Grant-in Aid for Scienti®c Research on Priority Area ``Innovation in Superplasticity'', the Ministry of Education, Science, Sports and Culture, Japan. REFERENCES 1. Sherby, O. D. and Wadsworth, J., Progress in Materials Science, 1989, 33, 169 (As a review). 2. de Jong, M. and Rathenau, G. W., Acta metall., 1961, 9, 714. 3. Greenwood, G. W. and Johnson, R. H., Proc. Roy. Soc. London, 1965, 283A, 403. 4. Lobb, R. C., Sykes, E. C. and Johnson, R. H., Metal Sci. J., 1972, 6, 33. 5. Wu, M. Y., Wadsworth, J. and Sherby, O. D., Metall. Trans., 1987, 18A, 451. 6. Wu, M. Y. and Sherby, O. D., Scripta metall., 1984, 18, 773. 7. Le Flour, J. C. and Locicero, R., Scripta metall., 1987, 21, 1071. 8. Hong, S. H., Sherby, O. D., Divecha, A. P., Karmarkar, S. D. and MacDonald, B. A., J. Comp. Mater., 1988, 22, 102. 9. Daehn, G. S. and Gonzalez-Doncel, G., Metall. Trans., 1989, 20A, 2355. 10. Pickard, S. M. and Derby, B., Acta metall. mater., 1990, 38, 2537.
KITAZONO and SATO et al.: INTERNAL STRESS SUPERPLASTICITY 11. Dunand, D. C. and Bedell, C. M., Acta metall. mater., 1996, 44, 1063. 12. Nieh, T. G., Henshall, C. A. and Wadsworth, J., Scr. Metall, 1984, 18, 1405. 13. Mahoney, M. W. and Ghosh, A. K., Metall. Trans., 1987, 18A, 653. 14. Imai, T., Mabuchi, M., Tozawa, Y. and Yamada, M., J. Mater. Sci. Lett., 1990, 9, 255. 15. Higashi K, and Mabuchi M. Advanced composites '93. TMS, Pennsylvania. 1993. p. 35 (As a review). 16. Zhang, H., Daehn, G. S. and Wagoner, R. H., Scripta Metall. Mater., 1990, 24, 2151. 17. Sato, E. and Kuribayashi, K., Acta metall. mater., 1993, 41, 1759. 18. Massalski T. B., Binary Alloy Phase Diagrams. Vol. 1. ASM, Ohio, 1986, p. 92. 19. Kitazono K, Sato E and Kuribayashi K. J. Japan Inst. Metals, 1996, 60, 441 (in Japanese). 20. Wakashima, K., Choi, B. H. and Lee, S. H., Composites '86: Recent Advances in Japan and the United States Japan Soc. Comp. Mater, Tokyo 1986, p. 579. 21. Otsuka M, Nozue A, and Horiuchi R. J. Japan Inst. Metals. 1982, 46, 353. (in Japanese). 22. Abe K, Tanji Y, Yoshinaga H, and Morozumi S. J. Japan Inst. Light Metals, 1977, 27, 279. (in Japanese). 23. Herring, C., J. Appl. Phys., 1950, 21, 437. 24. Coble, R. L., J. Appl. Phys., 1963, 34, 1679. 25. Mori, T., Okabe, M. and Mura, T., Acta metall., 1980, 28, 319. 26. Onaka, S., Miura, S. and Kato, M., Mechanics of Materials, 1990, 8, 285. 27. Luthy, H., Miller, A. K. and Sherby, O. D., Acta metall., 1980, 28, 169. 28. Mohamed, F. A. and Langdon, T. G., Metall. Trans., 1974, 5, 2339. 29. Onaka, S., Okada, T. and Kato, M., Acta metall. mater., 1991, 39, 971.
APPENDIX A The relaxation times of the two kinds of diusional accommodation which relax the mis®t strain left around the inclusions are calculated as 1 8 ÿ 10 4a3 kT ti G G 7 ÿ 5 3p2 Di dO interface diffusion 25 tv
A1
1 kTa2 G 1 v 2G 1 ÿ 2v 4 Dv O GG 1 v volume diffusion 26
A2
where Di, Dv, n, n*, G, G* and a are the interface diusion constant, the volume diusion constant, Poisson ratio of
213
the matrix, Poisson ratio of the inclusion, the shear modulus of the matrix, the shear modulus of the inclusion and the radius of the inclusion, respectively. In the case of this study, i.e. one unit Al±Be eutectic alloy consisting of Al matrix and Be particles of radius 0.25 mm at 638 K, ti and tv are calculated as 0.0059 s and 1.3 s, respectively. Here Di is assumed to equal Dgb, the grain boundary diusion constant, and the following values are adopted; Dv=1.7 10ÿ4exp(±Qv/RT) m2/s, Qv=142 kJ/mol [27], Dgb=1.86 10ÿ4exp(Qgb/RT) m2/s, Qgb=86 kJ/mol [28], O = 1.66 10ÿ29 m3, d = 2b, b = 2.86 10ÿ10 m, n = 0.34, n* = 0.041, G = 1.96 1010Pa (at 638 K) [22], and G* = 1.47 1011 Pa (at room temperature). When creep of the material is limited by relaxation kinetics caused of interface diusion around spherical inclusions under uniaxial applied stress, the steady state creep rate is given as [29] 5 1 ÿ v sA : A3 e_ 2ti G 7 ÿ 5v In the case of this study as shown in Fig. 3, the relaxationlimited creep rate is calculated as 1.4 10ÿ2 sÿ1 at sA/ E = 10ÿ4 which is much faster than the experimental creep rates of 2.0 10ÿ5 sÿ1 (thermal cycling) and 7.0 10ÿ6 sÿ1 (isothermal) (Fig. 3). APPENDIX B During heating or cooling at vT_ v under an applied stress, sAij, the quasi-steady state deviatric stress distribution in the matrix in the superplastic region is given by [17] 1=n 1 1 DaT_ jDaT_ j 2a3 sAij s0ij n 1ÿf 3 jDaT_ j 1 ÿ f B0 x3 3xi xj 1=n _ dij ÿ 2 ; for sA << jDaTj=B B1 0 x where xi is a point vector from the center of the inclusion, x is its magnitude, dij is the Kronecker delta and the summation convection for the repeated indices is employed. The equivalent stress se in the limit of null sAij is, then, given by 1=n 3 0 0 1=2 jDaT_ j 2a3 : B2 skl skl se 2 1 ÿ f n B0 x3 This se shows its maximum at x = a (at the interface) and its minimum at x = ¦ÿ1/3a (at the pseudo-outside of the matrix possessed by an inclusion). In the case of Al±Be alloy at vTÇ v = 5 K/s, the maximum and the minimum are calculated as 1.5 10ÿ4E and 7.7 10ÿ5E, respectively. and Here, ¦ = 0.008, Da = 1.34 10ÿ5 Kÿ1 B0=6.24 10ÿ53 sÿ1 Paÿ7 are used. Because of this high internal stress, dislocations in the matrix can move even under a very low applied stress during thermal cycling obeying the power-law equation of n = 7.
Acta mater. Vol. 46, No. 1, pp. 207±213, 1998 # 1997 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved Printed in Great Britain 1359-6454/98 $19.00 + 0.00 S1359-6454(97)00216-4
INTERNAL STRESS SUPERPLASTICITY IN Al±Be EUTECTIC ALLOY DURING TRIANGULAR TEMPERATURE PROFILE K. KITAZONO and E. SATO The Institute of Space and Astronautical Science, 3-1-1 Yoshinodai, Sagamihara, Kanagawa 229, Japan (Received 16 April 1997; accepted 23 June 1997) AbstractÐMaterials containing second phases show superplastic behavior during thermal cycling (internal stress superplasticity). In the present study, this behavior has been investigated by thermal cycling creep tests with a triangular wave having ®xed heating and cooling rates, using a Be-particle±dispersed Al matrix alloy. The thermal cycling creep rates were much higher than the isothermal creep rates and proportional to the applied stress at intermediate stresses. It was interpreted by the previous theoretical model of internal stress superplasticity, and the values of the thermal cycling creep rate were about 60% of the theoretical one. The discrepancy was explained by the transition after each temperature reversal until achieving the quasi-steady state stress distribution during thermal cycling. # 1997 Acta Metallurgica Inc.
1. INTRODUCTION
Two dierent types of superplasticity [1] have been observed in crystalline materials: ®ne structure superplasticity and internal stress superplasticity. The former refers to extensive elongation in materials with ®ne-grained crystal structures, and the latter in materials having high internal stresses. If an external stress is applied on some kinds of materials during thermal cycling, they possess high internal stresses and deform superplastically with an average strain rate which is proportional to the applied stress and much faster than the isothermal creep rate. Internal stress superplasticity is classi®ed according to materials as follows. (i) Materials having phase transformations in a thermal cycling regime such as steel (transformation superplasticity) [2, 3] (ii) Materials with anisotropic thermal expansion coecients such as a-uranium and pure zinc [4, 5] (iii) Materials consisting of phases with dierent thermal expansion coecients such as metal matrix composites [6±11] The third type is investigated in the present paper. Metal matrix composites (MMCs) are engineering materials in which hard components are dispersed in a ductile metal matrix to produce desirable strength and stiness, though, in general, they exhibit very low tensile ductility. Although many attempts have been made to apply the concept of ®ne structure superplasticity to MMCs [12±15], fewer attempts have been made to apply that of internal stress superplasticity. One example is SiC207
whisker±reinforced Al matrix composite, the strain of which was extended, under tension, to about 160% without fracture using thermal cycling [6]. Several theoretical models have been proposed for internal stress superplasticity. One model proposed by Hong et al. [8] predicts a linear relationship between the average strain rate and the applied stress, assuming that half of the moving dislocations are in¯uenced by the internal stress that aids their motion, and the remaining half are in¯uenced by the internal stress that opposes their motion. However, it cannot predict the realistic distribution and magnitude of the internal stresses. The old models for transformation superplasticity [2, 3] can predict an enhancement of the yield region adjacent to the transformed phase by a transformation strain, but cannot predict the stress±strain rate relation. Simulations of the ®nite-element method on two dimensions [16] have been reported for the internal stress superplasticity. The authors [17] have recently proposed a new theoretical model for internal stress superplasticity based on continuum micromechanics. They analyzed quasi-steady state stress ®elds in a material consisting of a matrix with spherical inclusions during a temperature change, and then calculated the average strain rate under a given applied stress and a given heating or cooling rate. A signi®cant eect of heating and cooling rates on the average strain rate is predicted. However, in the previous experimental investigations [6±11], the temperature pro®les were those in which heating and cooling rates were not constant, but were large at just after each temperature reversal and decreased gradually. For quantitative discussion, the temperature pro®le
208
KITAZONO and SATO et al.: INTERNAL STRESS SUPERPLASTICITY
of triangular-wave with ®xed heating and cooling rates is desirable. The purpose of this paper is to experimentally analyze internal stress superplasticity in an ideal case of the theoretical model. Constant heating and cooling rates, i.e. triangular temperature pro®le, were achieved using induction heating and cooling gas. Al±Be eutectic alloy was adopted as the model material of a simple metal matrix composite. The alloy consists of Be particles and Al matrix, which are mutually insoluble, whose interface is incoherent but hard to break, and the former of which has a volume fraction as low as 0.012 [18]. In addition, Be particles have a size around 1 mm, with which the strain mismatch by thermal expansion is hardly relaxed by volume diusion in the matrix but the strain mismatch by matrix plastic ¯ow is easily relaxed by interface diusion [17]. (For the particular case of this study, it is reexamined in Appendix A.) Experimental results of isothermal and thermal cycling creep were, thus, quantitatively compared with the theoretical model. Preliminary results had been reported [19] and the present paper analyzed the eect of the temperature pro®le and discussed the origin of the discrepancy from the theoretical prediction. On the other hand, Wakashima et al. [20] analyzed stress-enhanced thermal ratcheting in directionally solidi®ed Al±Al3Ni eutectic alloy. This phenomenon is, however, strictly distinguished from internal stress superplasticity. Thermal ratcheting occurs even under zero applied stress and only in materials having an isotropic second phase. On the contrary, internal stress superplasticity does not occur under zero applied stress, but occurs even in materials having an isotropic second phase. 2. EXPERIMENTAL
2.1. Material Ingots of Al±0.8 mass% Be eutectic alloy were cast in ambient atmosphere using Al±2.5 mass% Be mother alloy and 99.99 mass% pure Al ingots. They were hot extruded with an extrusion ratio of 18, and annealed at 873 K for 250 ks. A scanning electron micrograph of the inside of a grain of this material is shown in Fig. 1(a). Be particles precipitate both at the matrix grain boundaries and inside the matrix grains; the average particle size at the grain boundaries is 2.4 mm, that inside the grains is 0.5 mm, and the ratio of the volume fraction between them is 1:2. Because the internal stress generated by Be particles at the grain boundaries is easily accommodated by fast grain boundary diusion, the volume fraction of Be particles, ¦, is regarded as 0.008, two third of the total volume fraction of 0.012, in the further discussion. Figure 1(b) shows a scanning electron micrograph of electron channeling contrast of the Al matrix observed
Fig. 1. SEM micrographs of creep test specimens. (a) Be particles dispersed in the Al matrix grains and (b) electron channeling contrast of Al matrix grains.
with an accelerated voltage of 3 kV. It is found that the average grain size of the Al matrix is 50 mm, which is too large for ®ne structure superplasticity [1]. 2.2. Creep tests To compare isothermal creep with thermal cycling creep, the equivalent temperature, Teq, is de®ned as Z Tmax Q 1 Q dT; exp ÿ exp ÿ RTeq Tmax ÿ Tmin Tmin RT 1 where Tmin and Tmax are the minimum and maximum temperatures during thermal cycling, respectively, R is the gas constant, T is the absolute temperature and Q is the apparent activation energy of isothermal creep, 146 kJ/mol [21]. The equivalent temperature for thermal cycling creep tested at 573± 673 K is calculated as 638 K. Compression creep apparatus was designed specially for this study. The uniaxial load was applied to the specimen through a quartz cylinder. The specimen was heated using a high-frequency induction coil and cooled by cooling gas. A thermocouple was spot-welded on a hollow cylinder specimen of 10 mm in length, 6 mm in outside
KITAZONO and SATO et al.: INTERNAL STRESS SUPERPLASTICITY
Fig. 2. Typical experimental result of thermal cycling creep under a constant applied stress of 1.2 10ÿ5 E. The heating and cooling rates and temperature range for thermal cycling are 5 K/s and 573±673 K, respectively. Broken line shows an average strain rate of thermal cycling creep.
diameter and 4 mm in inside diameter. The typical temperature pro®le and the corresponding specimen length change are shown in Fig. 2. The temperature pro®le for thermal cycling was triangular, and the maximum controlling error, which occurred at points where heating was reversed to cooling, was less than 22 K. The thermal cycling strain rate was measured as an average strain rate shown schematically by the broken line in Fig. 2; the measurement was performed after a period sucient for achieving the quasi-steady state. The tensile creep test was performed to con®rm the large tensile elongation during thermal cycling, using a servo hydraulic testing machine equipped with an induction heating facility. The specimen had a gage length of 20 mm and a diameter of 9 mm. 3. RESULTS
Isothermal and thermal cycling compression creep rates, eÇ , are shown in Fig. 3 as functions of applied stress, sA/E, which is compensated by Young's modulus of E = 5.26 1010 Pa for pure Al at 638 K [22]. Three heating and cooling rates, vTÇ v = 10, 5 and 2 K/s, were selected in the thermal cycling creep tests performed within the temperature range of 573±673 K. The isothermal creep test was performed at the corresponding equivalent temperature of 638 K. Under isothermal conditions, an area with a stress exponent of seven is observed at high stress, and with a decrease in the applied stress, the stress exponent decreases to one. At high applied stresses,
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thermal cycling creep rates are close to isothermal creep rates for all heating and cooling rates. For vTÇ v = 10 and 5 K/s, with a decrease in the applied stress, the thermal cycling creep rate becomes much higher than the isothermal creep rate, where thermal cycling creep shows a stress exponent of one. These areas can be considered to exhibit internal stress superplasticity, as discussed in detail in Section 4.2. With a further decrease in the applied stress, thermal cycling creep rates again drop sharply to the same values as those of isothermal creep rates. On the other hand, for vTÇ v = 2 K/s, no region of a stress exponent of one, but a region of low stress exponents (1.5±2.0) is observed at around sA/E = 5 10ÿ5. The specimen after the thermal cycling tensile creep test is shown in Fig. 4. A ®xed load of 108 N, which corresponds to an initial stress 4.4 10ÿ5E, was applied to the specimen. The temperature range and heating and cooling rates were programmed as 673±773 K and 15 K/s, respectively, but they were not achieved perfectly because of the large size of the specimen. The specimen fractured after 120 h with about 200% elongation when the controlling thermocouple moved beyond the heating zone. Neither necking nor sharp crack was observed in the specimen. SEM observation showed that the dispersion state of Be particles was identical to that before the test.
4. DISCUSSION
4.1. Isothermal creep Under the isothermal condition, a high value of stress exponent, n = 7, is observed at high stresses and a stress exponent of one is observed at low stresses (Fig. 3). At high stresses, an experimental constitutive equation is expressed by e_ iso B0 snA
n 7;
2
where B0 is a constant depending on the temperature. The high value of stress exponent at high stresses is understood in terms of the particle dispersion strengthening mechanism of dislocation creep. At low applied stress, diusional creep equation [23, 24] given by e_ iso 47Dgb d 13Dv d
O 1 sA kT d 3
3
becomes predominant compared to dislocation creep, where Dgb and Dv are grain boundary and volume diusion coecients, respectively, d is the matrix grain size, d is the width of the matrix grain boundary, k is the Boltzmann constant and O is the atomic volume. The calculated results, shown by the broken line in Fig. 3, are close to the experimental results within one order. The values used for the calculation are listed in Appendix A.
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Fig. 3. Thermal cycling and isothermal creep rates as a function of the modulus compensated applied stress in Al±Be alloys. The equivalent temperature of thermal cycling creep tests was calculated as 638 K, and the temperature range, DT, was ®xed at 100 K. Heating and cooling rates were ®xed at (a) 10 K/s, (b) 5 K/s and (c) 2 K/s. Broken lines show theoretical values calculated using equation (5).
4.2. Comparison between thermal cycling creep data and the theoretical model The authors [17] proposed a new theoretical model for internal stress superplasticity based on continuum micromechanics. Unlike the previous models, this model can be used to calculate quasisteady state stress and strain rate distributions, which are nonuniform in space but stationary in time, during heating or cooling at a ®xed rate under a uniaxial applied stress. These, in turn, lead to the volume-averaged strain rate of the material, Çeth. Here it is assumed that the material consists of a
matrix containing spherical inclusions with dierence in thermal expansion coecient, Da. The matrix is assumed to deform according to a multiaxial power-law equation with a stress exponent, n, and a constant depending on the temperature, B; B = (1 ÿ ¦)nB0. In addition, the volume component of the strain mismatch is assumed not to be relaxed by volume diusion, but the shear component of that is assumed to be completely relaxed by interface diusion. The last assumption of fast interface diusion and null volume diusion is con®rmed in Appendix
KITAZONO and SATO et al.: INTERNAL STRESS SUPERPLASTICITY
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Fig. 4. Photograph of thermal cycling tensile creep specimens deformed to fracture. Initial applied stress was 4.4 10ÿ5 E, temperature range was 673±773 K, heating and cooling rates were 15 K/s, and total elongation to fracture was about 200%.
A. The calculated relaxation time for interface diusion, ti, is so small that the creep rate is not dominated by the relaxation kinetics and thus independent of the particle size. The power-law equation of n = 7 for the matrix is justi®ed because the equivalent stresses in the matrix during thermal cycling is so high that they are in the high stress region of n = 7, as con®rmed in Appendix B. Using the above analysis, the volume-averaged uniaxial strain rate of the material is given by
Under an applied stress much higher than the internal stress, the thermal cycling creep rate [equation (4)] coincides with the isothermal creep
rate [equation (2)], and the relation of B = (1 ÿ ¦)nB0 is obtained. At low applied stresses, it is proportional to the applied stress, and depends on the heating and cooling rates [equation (5)]; this region is regarded as the superplastic region. It is worth mentioning the obtained thermal cycling creep equations [equations (4) and (5)] contain no unknown parameters if the isothermal creep equation [equation (2)] has been obtained experimentally. The theoretical prediction for the superplastic region is plotted in Fig. 3 by a broken line. The values of ¦, Da and B are summarized in Appendix B. In the case of vTÇ v = 10 and 5 K/s, experimental results for a stress exponent of one at applied stress sA/E = 7 10ÿ6±3 10ÿ5 agree well with theoretical prediction calculated using equation (5). For vTÇ v = 2 K/s, the low stress exponent region at around sA/E = 5 10ÿ5 locates near the line calculated using equation (5).
Fig. 5. Relation between thermal cycling creep rates and absolute value of heating and cooling rates subjected to sA=10ÿ5 E.
Fig. 6. Relation between thermal cycling creep rates and temperature range subjected to sA=3.6 10ÿ5 E. Heating and cooling rates were ®xed at 5 K/s. Experimental results were normalized by a theoretical value. Solid line is the curve ®tted using equation (7).
e_ th B0 snA ;
for sA >> jDaT_ j=B0 1=n ; 4 1=n
2n n 4 1 ÿ f f sA e_ th jDaT_ j1ÿ1=n B1=n 0 21=n 5 1 ÿ f f 1=n for sA << jDaT_ j=B0 1=n ; 5
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The relation between strain rate and heating and cooling rates is depicted in Fig. 5 with a ®xed applied stress of sA/E = 10ÿ5. The broken line is the theoretical prediction using equation (5). The measured creep rate increases with an increase in heating and cooling rates, and is about 60% of the corresponding theoretical value. It is clear that high heating and cooling rates are required for high strain rates in internal stress superplasticity. 4.3. Eect of reversing the direction of temperature change during thermal cycling Experimental results of thermal cycling strain rates measured in the superplastic region are about 60% of the theoretical prediction calculated using equation (5). This discrepancy is caused by reversing the direction of temperature change during thermal cycling; while the theoretical analysis treats only the quasi-steady state during heating or cooling, a stationary internal stress distribution is achieved after a certain period after each reversal of temperature change. Figure 6 shows the relation between the strain rate and the temperature range of thermal cycling with a ®xed equivalent temperature of 638 K, ®xed heating and cooling rates of 5 K/s and ®xed applied stress of 3.6 10ÿ5 E. The thermal cycling strain rate at D T = 0 K corresponds to the isothermal strain rate. Here the authors assume that the instantaneous strain rate, Çe*, increases exponentially with time, t, after each temperature reversal, from the corresponding isothermal creep rate, Çeiso, given by equation (2) towards the theoretical stationary creep rate, Çeth, given by equation (5) with a constant C. e_ e_ iso e_ th ÿ e_ iso 1 ÿ exp ÿCjT_ jt
6
The average strain rate, Çe, which is one and only measurable value during thermal cycling in the temperature range, D T, is given by Z B0 DT e_ e_ 1 d jT_ jt DT 0 B0 e_ th ÿ
1 e_ th ÿ e_ iso 1 ÿ exp ÿCDT CDT
7
The solid line in Fig. 6 is the ®tted curve of equation (7) with 1/C = 80 K. It indicates that after each temperature reversal, the quasi-steady state stress and strain rate distributions are achieved in a transient time, (CvTÇ v)ÿ1, of 16 s. Daehn and Gonzalez-Doncel [9] investigated the eect of the temperature range during thermal cycling in A1±SiC alloy. However, because they did not consider the heating and cooling rates, the heating and cooling rates changed with a change in the temperature range in their experiments, and then, they could not reach the present discussion.
4.4. Thermal cycling under low applied stress The superplastic region of a stress exponent of one in thermal cycling creep appears in the region of intermediate applied stresses, and with a further decrease in the applied stress, the thermal cycling creep rate drops sharply to close to the isothermal creep rate (Fig. 3). At present the authors cannot explain the reason of this drop, and are planning to study the behavior at dierent equivalent temperature or with dierent matrix grain size to understand it. 5. SUMMARY
Internal stress superplasticity in Al±Be eutectic alloy was investigated. The applied temperature pro®le was a triangular wave with ®xed heating and cooling rates. The thermal cycling creep rates were much higher than the isothermal creep rates and proportional to the applied stress at intermediate stresses; this region was considered to be the internal stress superplastic region. At an applied stress much higher than the internal stress, the thermal cycling creep rate became close to the isothermal one. The internal stress region was interpreted by the previous theoretical model of internal stress superplasticity, and the value of the thermal cycling creep rate was about 60% of the theoretical one. The discrepancy was explained by the transition after each temperature reversal until achieving the quasi-steady state stress distribution during thermal cycling. Internal stress superplasticity of high strain rate needs high heating and cooling rates and a wide temperature range under appropriate applied stresses. AcknowledgementsÐThis work was supported by Grant-in Aid for Scienti®c Research on Priority Area ``Innovation in Superplasticity'', the Ministry of Education, Science, Sports and Culture, Japan. REFERENCES 1. Sherby, O. D. and Wadsworth, J., Progress in Materials Science, 1989, 33, 169 (As a review). 2. de Jong, M. and Rathenau, G. W., Acta metall., 1961, 9, 714. 3. Greenwood, G. W. and Johnson, R. H., Proc. Roy. Soc. London, 1965, 283A, 403. 4. Lobb, R. C., Sykes, E. C. and Johnson, R. H., Metal Sci. J., 1972, 6, 33. 5. Wu, M. Y., Wadsworth, J. and Sherby, O. D., Metall. Trans., 1987, 18A, 451. 6. Wu, M. Y. and Sherby, O. D., Scripta metall., 1984, 18, 773. 7. Le Flour, J. C. and Locicero, R., Scripta metall., 1987, 21, 1071. 8. Hong, S. H., Sherby, O. D., Divecha, A. P., Karmarkar, S. D. and MacDonald, B. A., J. Comp. Mater., 1988, 22, 102. 9. Daehn, G. S. and Gonzalez-Doncel, G., Metall. Trans., 1989, 20A, 2355. 10. Pickard, S. M. and Derby, B., Acta metall. mater., 1990, 38, 2537.
KITAZONO and SATO et al.: INTERNAL STRESS SUPERPLASTICITY 11. Dunand, D. C. and Bedell, C. M., Acta metall. mater., 1996, 44, 1063. 12. Nieh, T. G., Henshall, C. A. and Wadsworth, J., Scr. Metall, 1984, 18, 1405. 13. Mahoney, M. W. and Ghosh, A. K., Metall. Trans., 1987, 18A, 653. 14. Imai, T., Mabuchi, M., Tozawa, Y. and Yamada, M., J. Mater. Sci. Lett., 1990, 9, 255. 15. Higashi K, and Mabuchi M. Advanced composites '93. TMS, Pennsylvania. 1993. p. 35 (As a review). 16. Zhang, H., Daehn, G. S. and Wagoner, R. H., Scripta Metall. Mater., 1990, 24, 2151. 17. Sato, E. and Kuribayashi, K., Acta metall. mater., 1993, 41, 1759. 18. Massalski T. B., Binary Alloy Phase Diagrams. Vol. 1. ASM, Ohio, 1986, p. 92. 19. Kitazono K, Sato E and Kuribayashi K. J. Japan Inst. Metals, 1996, 60, 441 (in Japanese). 20. Wakashima, K., Choi, B. H. and Lee, S. H., Composites '86: Recent Advances in Japan and the United States Japan Soc. Comp. Mater, Tokyo 1986, p. 579. 21. Otsuka M, Nozue A, and Horiuchi R. J. Japan Inst. Metals. 1982, 46, 353. (in Japanese). 22. Abe K, Tanji Y, Yoshinaga H, and Morozumi S. J. Japan Inst. Light Metals, 1977, 27, 279. (in Japanese). 23. Herring, C., J. Appl. Phys., 1950, 21, 437. 24. Coble, R. L., J. Appl. Phys., 1963, 34, 1679. 25. Mori, T., Okabe, M. and Mura, T., Acta metall., 1980, 28, 319. 26. Onaka, S., Miura, S. and Kato, M., Mechanics of Materials, 1990, 8, 285. 27. Luthy, H., Miller, A. K. and Sherby, O. D., Acta metall., 1980, 28, 169. 28. Mohamed, F. A. and Langdon, T. G., Metall. Trans., 1974, 5, 2339. 29. Onaka, S., Okada, T. and Kato, M., Acta metall. mater., 1991, 39, 971.
APPENDIX A The relaxation times of the two kinds of diusional accommodation which relax the mis®t strain left around the inclusions are calculated as 1 8 ÿ 10 4a3 kT ti G G 7 ÿ 5 3p2 Di dO interface diffusion 25 tv
A1
1 kTa2 G 1 v 2G 1 ÿ 2v 4 Dv O GG 1 v volume diffusion 26
A2
where Di, Dv, n, n*, G, G* and a are the interface diusion constant, the volume diusion constant, Poisson ratio of
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the matrix, Poisson ratio of the inclusion, the shear modulus of the matrix, the shear modulus of the inclusion and the radius of the inclusion, respectively. In the case of this study, i.e. one unit Al±Be eutectic alloy consisting of Al matrix and Be particles of radius 0.25 mm at 638 K, ti and tv are calculated as 0.0059 s and 1.3 s, respectively. Here Di is assumed to equal Dgb, the grain boundary diusion constant, and the following values are adopted; Dv=1.7 10ÿ4exp(±Qv/RT) m2/s, Qv=142 kJ/mol [27], Dgb=1.86 10ÿ4exp(Qgb/RT) m2/s, Qgb=86 kJ/mol [28], O = 1.66 10ÿ29 m3, d = 2b, b = 2.86 10ÿ10 m, n = 0.34, n* = 0.041, G = 1.96 1010Pa (at 638 K) [22], and G* = 1.47 1011 Pa (at room temperature). When creep of the material is limited by relaxation kinetics caused of interface diusion around spherical inclusions under uniaxial applied stress, the steady state creep rate is given as [29] 5 1 ÿ v sA : A3 e_ 2ti G 7 ÿ 5v In the case of this study as shown in Fig. 3, the relaxationlimited creep rate is calculated as 1.4 10ÿ2 sÿ1 at sA/ E = 10ÿ4 which is much faster than the experimental creep rates of 2.0 10ÿ5 sÿ1 (thermal cycling) and 7.0 10ÿ6 sÿ1 (isothermal) (Fig. 3). APPENDIX B During heating or cooling at vT_ v under an applied stress, sAij, the quasi-steady state deviatric stress distribution in the matrix in the superplastic region is given by [17] 1=n 1 1 DaT_ jDaT_ j 2a3 sAij s0ij n 1ÿf 3 jDaT_ j 1 ÿ f B0 x3 3xi xj 1=n _ dij ÿ 2 ; for sA << jDaTj=B B1 0 x where xi is a point vector from the center of the inclusion, x is its magnitude, dij is the Kronecker delta and the summation convection for the repeated indices is employed. The equivalent stress se in the limit of null sAij is, then, given by 1=n 3 0 0 1=2 jDaT_ j 2a3 : B2 skl skl se 2 1 ÿ f n B0 x3 This se shows its maximum at x = a (at the interface) and its minimum at x = ¦ÿ1/3a (at the pseudo-outside of the matrix possessed by an inclusion). In the case of Al±Be alloy at vTÇ v = 5 K/s, the maximum and the minimum are calculated as 1.5 10ÿ4E and 7.7 10ÿ5E, respectively. and Here, ¦ = 0.008, Da = 1.34 10ÿ5 Kÿ1 B0=6.24 10ÿ53 sÿ1 Paÿ7 are used. Because of this high internal stress, dislocations in the matrix can move even under a very low applied stress during thermal cycling obeying the power-law equation of n = 7.