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Stress saturation in a nickel-base superalloy, under different aging treatments
Materials Science and Engineering A319– 321 (2001) 643– 646 www.elsevier.com/locate/msea
Stress saturation in a nickel-base superalloy, under different aging treatments J.A. del Valle a,b, R. Romero a,c, A.C. Picasso a,c,* a
Instituto de Fı´sica de Materiales Tandil (IFIMAT), Uni6ersidad Nacional del Centro de la Pro6incia de Buenos Aires, Pinto 399, Tandil (7000), Pro6incia de Buenos Aires, Argentina b Consejo Nacional de In6estigaciones Cientı´ficas y Te´cnicas (CONICET), Argentina c Comisio´n de In6estigaciones Cientı´ficas de la Pro6incia de Buenos Aires (CICPBA), Buenos Aires, Argentina
Abstract The effect of g%-precipitates on the saturation stress, extrapolated from stage III of work hardening (WH), has been studied in the superalloy Inconel X-750 on two situations, the solution heat treated, and peakaged-overaged samples. The WH was analysed by means of diagrams q= d|/dm versus |, and [d =d|d/dm versus |d, where | is the flow stress and |d is the dislocation related stress component of |, obtained by the application of a superposition rule. The results indicate that, for the overaged alloy, the storage of dislocation loops leads to a high initial WH rate; but during stage III a strong dynamic recovery reduces the effect of the precipitates on the saturation stress. The influence of the superposition rules, used in the data analysis for the interpretation of the saturation stress in terms of the dislocation density, was also discussed. © 2001 Elsevier Science B.V. All rights reserved. Keywords: Work hardening; Precipitation strengthening; Nickel-base superalloy
1. Introduction The effect of the g%-precipitates on stage II of work hardening (WH) of the Inconel X-750 alloy has been recently investigated [1]. It has been shown that, in the overaged samples, a steep increase in the WH rate is due to the storage of Orowan loops around the unshearable g%-precipitates. In the present work we have analysed the effect of the g%-precipitates, in peakaged and overaged samples, on stage III of WH, focusing on the saturation stress |s extrapolated at zero WH rate. The results are also compared with the solution heat treated Inconel X-750 alloy (SHT samples) and with the Inconel 600 alloy, whose composition is near the matrix phase composition of the aged Inconel X-750 alloy. In pure metals, the flow stress | is given by the well established relation [2]: |= |o + MhGbz 1/2
(1)
where |o is a friction stress, M the Taylor factor, b the Burgers vector, G the shear modulus, h a constant, and * Corresponding author. Fax: + 54-02293-444190. E-mail address: [email protected] (A.C. Picasso).
z is the dislocation density. Therefore, the study of the rate q d|/dm 8 (1/z 1/2)dz/dm by means of plots q−| and q| − |, enables us to investigate the WH stages and z-evolution. It is well known [2] that, in pure metals, stage III of WH can be described by the linear relation:
q=qo 1−
| |s
(2)
where qo is the athermal WH rate limit for stage III and |s is the saturation stress, which contains the temperature dependence of the WH. The Eq. (2) was rationalised by Kocks, Mecking and Estrin [3–5] as the result of the competition of the dislocation storage and dynamic recovery processes. According to the recent review of Gil Sevillano [2], the micromechanism involved with dislocation storage is the formation of Orowan loops around impenetrable volumes and the recovery is related to the annihilation of these loops with other dislocations. Therefore, the saturation stress would be physically interpreted as the situation when a reciprocal cancellation of these processes occurs.
0921-5093/01/$ - see front matter © 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 5 0 9 3 ( 0 1 ) 0 1 0 3 2 - 2
J.A. del Valle et al. / Materials Science and Engineering A319–321 (2001) 643–646
644
Before the saturation is reached, a transition to stage IV has been observed [2]. However, the extrapolated saturation stress, |s, can be used as a reference value for scaling the plastic properties of different materials [6] or, as in the case we show here, to study an alloy under different aging treatments.
2. Experimental The nominal composition of the Inconel X-750 superalloy (INCO trademark), which was supplied in sheet form, is as follows (wt.%), 0.05 C, 1 max Co, 14–17 Cr, 5–9 Fe, 2.25 – 2.75 Ti, 0.4– 1 Al, 0.7–1.2 Nb plus Ta, 1 max Mn, 0.5 max Si, 0.01 max S, and balance Ni. The flat samples used for tensile tests were spark-machined in order to get a gauge length of 24.5
mm and cross section of 4× 1.8 mm2. The samples were solution heat treated at 1363 K for 2 h, and then water-quenched. To get different g%-precipitate sizes, the samples were aged at 1010 K for different times, the mean precipitate radius was calculated from the coarsening law detailed in [7], these were r=17 (near the peakaged condition), 36, 55, and 80 nm, the volume fraction of the g%-phase was estimated to be fv =0.15. The average grain size of the solution treated alloy was d: 100 mm, and did not suffer noticeable modifications during the aging treatment. The tensile tests were performed in a Shimadzu machine at room temperature. The initial strain rate was m= 3.3× 10 − 4 s − 1. Strain measurements were made with an extensometer. The load–elongation curve was measured using a data acquisition board. Moreover, tests on SHT and quenched samples of the Inconel X-750 alloy, and on homogenised Inconel 600 alloy (whose composition is in wt.%, 14–17 Cr, 5–9 Fe, and balance Ni) were also performed.
3. Experimental results and discussion
Fig. 1. True stress vs. true plastic strain curves.
Fig. 2. WH behaviour of Inconel 600 ( ) and Inconel X-750, SHT ( ) and aged to r =17 nm ( ), r= 36 nm ( ), r= 55 nm (— ), r=80 nm ( ).
Fig. 1 shows the true stress versus true plastic strain curves for aged samples, a SHT sample, and an Inconel 600 sample. The hardening rate, d|/dm, and the 0.2% flow stress, |0.2, have been obtained from these curves. Firstly, to analyse the WH, we assume the flow stress Eq. (1) and use the experimental value of the 0.2% flow stress to estimate the friction stress |o. In this way, the difference |− |0.2 can be correlated to z 1/2, and the extrapolation to zero WH rate, |s − |0.2, can be correlated to a ‘saturation’ dislocation structure. In Fig. 2, the d|/dm versus |− |0.2 diagrams are shown. The SHT and the Inconel 600 curves have a similar trend during stage II, showing that the dislocation athermal storage is not sensitive to the addition of Al and Ti to the matrix. Conversely, during stage III the WH decreases in a slighter way in the Inconel X-750 curve, leading to a higher saturation stress. This observation would be associated with a decrease in the stacking-fault energy (SFE) due to the addition of Al and Ti in the SHT alloy. The SFE of Inconel 600 is about 0.12 Jm − 2 and the SFE of SHT samples of Inconel X-750 near 0.09 Jm − 2 [8]. Therefore, a higher SFE would lead to a faster dynamic recovery by favouring the cross-slip processes. From Fig. 2, an increase in stage II plateau is produced with the size of the unshearable particles due to the Orowan loops accumulation around the particles [1]. The critical stress (|−|0.2) to trigger stage III is lower in aged samples than in SHT and Inconel 600 samples. As can be seen, stage III is not strictly linear in the aged samples, the dynamic recovery produces a
J.A. del Valle et al. / Materials Science and Engineering A319–321 (2001) 643–646
645
one must take into account that glide dislocations interact with several obstacles at a time, and in some cases the superposition law of the different contributions could not be linear. The superposition problem, in most studies [2,9,10], has been reduced to estimate the individual strengthening contributions and the superposition law describing their combined effect. A previous analysis [1] suggested that the following constitutive equation might be used in the case of aged samples: |= |f + |ss + |b + (| 2g% + | 2d)1/2
Fig. 3. The back stress obtained by means of measurements of the permanent softening in Bauschinger tests [11].
Fig. 4. [d vs. |d diagram. The symbols correspond to Fig. 2. In the insert, a |d[d vs. |d diagram.
quick decrease in the hardening rate. On the other hand, the extrapolated saturation stress values of the aged alloy, fall between the saturation stresses of the SHT and Inconel 600 samples. Then, the effect of the precipitates on the saturation stress would produce a change in an amount equal to the yield stress increase. It is interesting to note that the curves of the aged alloy converge on a common behaviour through stage III in spite of starting from different athermal hardening values in stage II. These results suggest that on the saturation stage the same dislocation density is presented in both types of samples (i.e. with and without precipitates), in spite of the loops storage around the particles. However, in alloys with a complex microstructure, the previous interpretation of |− |0.2 in terms of z 1/2 is not as straightforward as in the pure metal case. In this case
(3)
where we have assumed that it is possible to add the contributions of the lattice friction, |f, the dislocation– solute atom interaction, |ss, and the back stress, |b, which arises from the elastic stresses supported by the precipitates. The g%-precipitate contribution, |g%, is given by the Orowan looping stress in the overaged samples. The dislocation component is given by |d = MhGbz 1/2. In Eq. (3), the |g% and |d components were combined by means of a quadratic superposition rule suitable for two sets of obstacles of similar strength. To study the WH in terms of the z-evolution, it is necessary to obtain the dislocation contribution, |d, from the total flow stress. It was performed by means of the superposition rule of Eq. (3) using the experimental values of the various contributions. In the case of Inconel X-750 aged samples, the value of |f + |ss (220 MPa) was estimated from the 0.2% flow stress of the Inconel 600 alloy, which has a quite similar composition compared with the matrix of the Inconel X-750 alloy after the precipitation treatment. Furthermore, the precipitate stress component has been estimated using |g%(r)= |0.2(r)− 220 MPa, where |0.2(r) is the 0.2% flow stress of the aged samples. We have estimated the back stress component, |b, from our experimental results on the permanent softening measured in Bauschinger test, Fig. 3 [11]. It can be noted that |b produces only about 15% of the total WH, both in SHT and aged samples. Minor differences in the results can only arise if this contribution is neglected in the present analysis. In the case of the solution hardened alloys, SHT and Inconel 600, Eq. (3) reduces to |= |f + |ss +|b +|d. We derive |f + |ss from the 0.2% flow stresses (|0.2 = 283 MPa for SHT). In the SHT sample, we also estimated |b from our experimental results given in Fig. 3. In Fig. 4, the diagrams [d = d|d/dm against |d are shown, it can be noted that in the precipitate hardened alloy, stage II is not clearly defined. However, in the |d[d versus |d diagram (insert of Fig. 4) stage II can be distinguished as a linear zone with a positive intercept with the ordinate axis [1]. On the other hand, Fig. 4 shows that stage III of the aged alloy converge on a common saturation stress, |d,s, higher than that one of the alloys without precipitates. Of course, the analysis
646
J.A. del Valle et al. / Materials Science and Engineering A319–321 (2001) 643–646
developed here depends on the validity of the flow stress Eq. (3) throughout stage III up to the saturation stage.
4. Conclusions The results indicate that, for the overaged alloy at low strains in stage II of WH, the storage of dislocation loops leads to a high WH rate. However, a strong dynamic recovery during stage III reduces rapidly the hardening rate, leading to an almost independent behaviour with the precipitate size. This study shows that, in precipitate-hardened alloys, the analysis of the saturation stresses is sensitive to the superposition laws used to obtain the dislocation related stress component. If a linear superposition is applied, the results could indicate that the precipitates have no influence on the accumulated dislocations in the saturation. However, an analysis based on quadratic rule leads to a higher density of accumulated dislocations in the saturation, compared with the alloys without precipitates.
Acknowledgements The authors acknowledge the financial support of,
Consejo Nacional de Investigaciones Cientı´ficas y Te´ cnicas (PIP/BID No. 4318/97); Agencia Nacional de Promocio´ n Cientı´fica y Tecnolo´ gica (PICT No. 0192/98); CICPBA and SeCyT-UNCentro Argentina. The authors are grateful to Dr A. Cuniberti for careful reviews of the manuscript and helpful comments.
References [1] J.A. del Valle, A.C. Picasso, R. Romero, Acta Mater. 46 (1998) 1981. [2] J. Gil Sevillano, in: R.W. Cahn, P. Haasen, E.J. Kramer (Eds.), Materials Science and Technology, vol. 6, VCH, Weinhein, 1993. [3] U.F. Kocks, J. Eng. Mater. Techol. 98 (1976) 76. [4] H. Mecking, U.F. Kocks, Acta Metall. 29 (1981) 1865. [5] Y. Estrin, H. Mecking, Acta Metall. 32 (1984) 57. [6] H. Mecking, B. Nicklas, N. Zarubova, U.F. Kocks, Acta Metall. 34 (1986) 527. [7] J.A. del Valle, A.C. Picasso, I. Alvarez, R. Romero, Scr. Mater. 41 (1999) 237. [8] R.A. Mulford, Acta Metall. 27 (1978) 1115. [9] U.F. Kocks, Proceedings of the Fifth International Conference on the Strength of Metals and Alloys, Pergamon Press, 1979, p. 1661. [10] L.M. Brown, R.K. Ham, in: A. Kelly, R.B. Nicholson (Eds.), Strengthening Methods in Crystals, Applied Science Publishers, London, 1971. [11] J.A. del Valle, R. Romero, A.C. Picasso, Mater. Sci. Eng. A 311 (2001) 100.
Stress saturation in a nickel-base superalloy, under different aging treatments J.A. del Valle a,b, R. Romero a,c, A.C. Picasso a,c,* a
Instituto de Fı´sica de Materiales Tandil (IFIMAT), Uni6ersidad Nacional del Centro de la Pro6incia de Buenos Aires, Pinto 399, Tandil (7000), Pro6incia de Buenos Aires, Argentina b Consejo Nacional de In6estigaciones Cientı´ficas y Te´cnicas (CONICET), Argentina c Comisio´n de In6estigaciones Cientı´ficas de la Pro6incia de Buenos Aires (CICPBA), Buenos Aires, Argentina
Abstract The effect of g%-precipitates on the saturation stress, extrapolated from stage III of work hardening (WH), has been studied in the superalloy Inconel X-750 on two situations, the solution heat treated, and peakaged-overaged samples. The WH was analysed by means of diagrams q= d|/dm versus |, and [d =d|d/dm versus |d, where | is the flow stress and |d is the dislocation related stress component of |, obtained by the application of a superposition rule. The results indicate that, for the overaged alloy, the storage of dislocation loops leads to a high initial WH rate; but during stage III a strong dynamic recovery reduces the effect of the precipitates on the saturation stress. The influence of the superposition rules, used in the data analysis for the interpretation of the saturation stress in terms of the dislocation density, was also discussed. © 2001 Elsevier Science B.V. All rights reserved. Keywords: Work hardening; Precipitation strengthening; Nickel-base superalloy
1. Introduction The effect of the g%-precipitates on stage II of work hardening (WH) of the Inconel X-750 alloy has been recently investigated [1]. It has been shown that, in the overaged samples, a steep increase in the WH rate is due to the storage of Orowan loops around the unshearable g%-precipitates. In the present work we have analysed the effect of the g%-precipitates, in peakaged and overaged samples, on stage III of WH, focusing on the saturation stress |s extrapolated at zero WH rate. The results are also compared with the solution heat treated Inconel X-750 alloy (SHT samples) and with the Inconel 600 alloy, whose composition is near the matrix phase composition of the aged Inconel X-750 alloy. In pure metals, the flow stress | is given by the well established relation [2]: |= |o + MhGbz 1/2
(1)
where |o is a friction stress, M the Taylor factor, b the Burgers vector, G the shear modulus, h a constant, and * Corresponding author. Fax: + 54-02293-444190. E-mail address: [email protected] (A.C. Picasso).
z is the dislocation density. Therefore, the study of the rate q d|/dm 8 (1/z 1/2)dz/dm by means of plots q−| and q| − |, enables us to investigate the WH stages and z-evolution. It is well known [2] that, in pure metals, stage III of WH can be described by the linear relation:
q=qo 1−
| |s
(2)
where qo is the athermal WH rate limit for stage III and |s is the saturation stress, which contains the temperature dependence of the WH. The Eq. (2) was rationalised by Kocks, Mecking and Estrin [3–5] as the result of the competition of the dislocation storage and dynamic recovery processes. According to the recent review of Gil Sevillano [2], the micromechanism involved with dislocation storage is the formation of Orowan loops around impenetrable volumes and the recovery is related to the annihilation of these loops with other dislocations. Therefore, the saturation stress would be physically interpreted as the situation when a reciprocal cancellation of these processes occurs.
0921-5093/01/$ - see front matter © 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 5 0 9 3 ( 0 1 ) 0 1 0 3 2 - 2
J.A. del Valle et al. / Materials Science and Engineering A319–321 (2001) 643–646
644
Before the saturation is reached, a transition to stage IV has been observed [2]. However, the extrapolated saturation stress, |s, can be used as a reference value for scaling the plastic properties of different materials [6] or, as in the case we show here, to study an alloy under different aging treatments.
2. Experimental The nominal composition of the Inconel X-750 superalloy (INCO trademark), which was supplied in sheet form, is as follows (wt.%), 0.05 C, 1 max Co, 14–17 Cr, 5–9 Fe, 2.25 – 2.75 Ti, 0.4– 1 Al, 0.7–1.2 Nb plus Ta, 1 max Mn, 0.5 max Si, 0.01 max S, and balance Ni. The flat samples used for tensile tests were spark-machined in order to get a gauge length of 24.5
mm and cross section of 4× 1.8 mm2. The samples were solution heat treated at 1363 K for 2 h, and then water-quenched. To get different g%-precipitate sizes, the samples were aged at 1010 K for different times, the mean precipitate radius was calculated from the coarsening law detailed in [7], these were r=17 (near the peakaged condition), 36, 55, and 80 nm, the volume fraction of the g%-phase was estimated to be fv =0.15. The average grain size of the solution treated alloy was d: 100 mm, and did not suffer noticeable modifications during the aging treatment. The tensile tests were performed in a Shimadzu machine at room temperature. The initial strain rate was m= 3.3× 10 − 4 s − 1. Strain measurements were made with an extensometer. The load–elongation curve was measured using a data acquisition board. Moreover, tests on SHT and quenched samples of the Inconel X-750 alloy, and on homogenised Inconel 600 alloy (whose composition is in wt.%, 14–17 Cr, 5–9 Fe, and balance Ni) were also performed.
3. Experimental results and discussion
Fig. 1. True stress vs. true plastic strain curves.
Fig. 2. WH behaviour of Inconel 600 ( ) and Inconel X-750, SHT ( ) and aged to r =17 nm ( ), r= 36 nm ( ), r= 55 nm (— ), r=80 nm ( ).
Fig. 1 shows the true stress versus true plastic strain curves for aged samples, a SHT sample, and an Inconel 600 sample. The hardening rate, d|/dm, and the 0.2% flow stress, |0.2, have been obtained from these curves. Firstly, to analyse the WH, we assume the flow stress Eq. (1) and use the experimental value of the 0.2% flow stress to estimate the friction stress |o. In this way, the difference |− |0.2 can be correlated to z 1/2, and the extrapolation to zero WH rate, |s − |0.2, can be correlated to a ‘saturation’ dislocation structure. In Fig. 2, the d|/dm versus |− |0.2 diagrams are shown. The SHT and the Inconel 600 curves have a similar trend during stage II, showing that the dislocation athermal storage is not sensitive to the addition of Al and Ti to the matrix. Conversely, during stage III the WH decreases in a slighter way in the Inconel X-750 curve, leading to a higher saturation stress. This observation would be associated with a decrease in the stacking-fault energy (SFE) due to the addition of Al and Ti in the SHT alloy. The SFE of Inconel 600 is about 0.12 Jm − 2 and the SFE of SHT samples of Inconel X-750 near 0.09 Jm − 2 [8]. Therefore, a higher SFE would lead to a faster dynamic recovery by favouring the cross-slip processes. From Fig. 2, an increase in stage II plateau is produced with the size of the unshearable particles due to the Orowan loops accumulation around the particles [1]. The critical stress (|−|0.2) to trigger stage III is lower in aged samples than in SHT and Inconel 600 samples. As can be seen, stage III is not strictly linear in the aged samples, the dynamic recovery produces a
J.A. del Valle et al. / Materials Science and Engineering A319–321 (2001) 643–646
645
one must take into account that glide dislocations interact with several obstacles at a time, and in some cases the superposition law of the different contributions could not be linear. The superposition problem, in most studies [2,9,10], has been reduced to estimate the individual strengthening contributions and the superposition law describing their combined effect. A previous analysis [1] suggested that the following constitutive equation might be used in the case of aged samples: |= |f + |ss + |b + (| 2g% + | 2d)1/2
Fig. 3. The back stress obtained by means of measurements of the permanent softening in Bauschinger tests [11].
Fig. 4. [d vs. |d diagram. The symbols correspond to Fig. 2. In the insert, a |d[d vs. |d diagram.
quick decrease in the hardening rate. On the other hand, the extrapolated saturation stress values of the aged alloy, fall between the saturation stresses of the SHT and Inconel 600 samples. Then, the effect of the precipitates on the saturation stress would produce a change in an amount equal to the yield stress increase. It is interesting to note that the curves of the aged alloy converge on a common behaviour through stage III in spite of starting from different athermal hardening values in stage II. These results suggest that on the saturation stage the same dislocation density is presented in both types of samples (i.e. with and without precipitates), in spite of the loops storage around the particles. However, in alloys with a complex microstructure, the previous interpretation of |− |0.2 in terms of z 1/2 is not as straightforward as in the pure metal case. In this case
(3)
where we have assumed that it is possible to add the contributions of the lattice friction, |f, the dislocation– solute atom interaction, |ss, and the back stress, |b, which arises from the elastic stresses supported by the precipitates. The g%-precipitate contribution, |g%, is given by the Orowan looping stress in the overaged samples. The dislocation component is given by |d = MhGbz 1/2. In Eq. (3), the |g% and |d components were combined by means of a quadratic superposition rule suitable for two sets of obstacles of similar strength. To study the WH in terms of the z-evolution, it is necessary to obtain the dislocation contribution, |d, from the total flow stress. It was performed by means of the superposition rule of Eq. (3) using the experimental values of the various contributions. In the case of Inconel X-750 aged samples, the value of |f + |ss (220 MPa) was estimated from the 0.2% flow stress of the Inconel 600 alloy, which has a quite similar composition compared with the matrix of the Inconel X-750 alloy after the precipitation treatment. Furthermore, the precipitate stress component has been estimated using |g%(r)= |0.2(r)− 220 MPa, where |0.2(r) is the 0.2% flow stress of the aged samples. We have estimated the back stress component, |b, from our experimental results on the permanent softening measured in Bauschinger test, Fig. 3 [11]. It can be noted that |b produces only about 15% of the total WH, both in SHT and aged samples. Minor differences in the results can only arise if this contribution is neglected in the present analysis. In the case of the solution hardened alloys, SHT and Inconel 600, Eq. (3) reduces to |= |f + |ss +|b +|d. We derive |f + |ss from the 0.2% flow stresses (|0.2 = 283 MPa for SHT). In the SHT sample, we also estimated |b from our experimental results given in Fig. 3. In Fig. 4, the diagrams [d = d|d/dm against |d are shown, it can be noted that in the precipitate hardened alloy, stage II is not clearly defined. However, in the |d[d versus |d diagram (insert of Fig. 4) stage II can be distinguished as a linear zone with a positive intercept with the ordinate axis [1]. On the other hand, Fig. 4 shows that stage III of the aged alloy converge on a common saturation stress, |d,s, higher than that one of the alloys without precipitates. Of course, the analysis
646
J.A. del Valle et al. / Materials Science and Engineering A319–321 (2001) 643–646
developed here depends on the validity of the flow stress Eq. (3) throughout stage III up to the saturation stage.
4. Conclusions The results indicate that, for the overaged alloy at low strains in stage II of WH, the storage of dislocation loops leads to a high WH rate. However, a strong dynamic recovery during stage III reduces rapidly the hardening rate, leading to an almost independent behaviour with the precipitate size. This study shows that, in precipitate-hardened alloys, the analysis of the saturation stresses is sensitive to the superposition laws used to obtain the dislocation related stress component. If a linear superposition is applied, the results could indicate that the precipitates have no influence on the accumulated dislocations in the saturation. However, an analysis based on quadratic rule leads to a higher density of accumulated dislocations in the saturation, compared with the alloys without precipitates.
Acknowledgements The authors acknowledge the financial support of,
Consejo Nacional de Investigaciones Cientı´ficas y Te´ cnicas (PIP/BID No. 4318/97); Agencia Nacional de Promocio´ n Cientı´fica y Tecnolo´ gica (PICT No. 0192/98); CICPBA and SeCyT-UNCentro Argentina. The authors are grateful to Dr A. Cuniberti for careful reviews of the manuscript and helpful comments.
References [1] J.A. del Valle, A.C. Picasso, R. Romero, Acta Mater. 46 (1998) 1981. [2] J. Gil Sevillano, in: R.W. Cahn, P. Haasen, E.J. Kramer (Eds.), Materials Science and Technology, vol. 6, VCH, Weinhein, 1993. [3] U.F. Kocks, J. Eng. Mater. Techol. 98 (1976) 76. [4] H. Mecking, U.F. Kocks, Acta Metall. 29 (1981) 1865. [5] Y. Estrin, H. Mecking, Acta Metall. 32 (1984) 57. [6] H. Mecking, B. Nicklas, N. Zarubova, U.F. Kocks, Acta Metall. 34 (1986) 527. [7] J.A. del Valle, A.C. Picasso, I. Alvarez, R. Romero, Scr. Mater. 41 (1999) 237. [8] R.A. Mulford, Acta Metall. 27 (1978) 1115. [9] U.F. Kocks, Proceedings of the Fifth International Conference on the Strength of Metals and Alloys, Pergamon Press, 1979, p. 1661. [10] L.M. Brown, R.K. Ham, in: A. Kelly, R.B. Nicholson (Eds.), Strengthening Methods in Crystals, Applied Science Publishers, London, 1971. [11] J.A. del Valle, R. Romero, A.C. Picasso, Mater. Sci. Eng. A 311 (2001) 100.