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Influence of stacking fault energy and short-range ordering on dynamic recovery and work hardening behavior of copper alloys
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Scripta Materialia 62 (2010) 693–696 www.elsevier.com/locate/scriptamat
Influence of stacking fault energy and short-range ordering on dynamic recovery and work hardening behavior of copper alloys Farzad Hamdi and Sirous Asgari* Department of Materials Science and Engineering, Sharif University of Technology, Azadi Avenue, Tehran, Iran Received 27 August 2009; revised 15 January 2010; accepted 15 January 2010 Available online 21 January 2010
True stress vs. true strain responses of Cu–6 wt.% Al and Cu–12 wt.% Mn alloys are presented. While Cu–6 wt.% Al alloy shows the typical mechanical response of low stacking fault energy alloys, the Cu–12 wt.% Mn alloy behaved similarly to medium to high stacking fault energy alloys. These findings clearly show that while short-range ordering triggers slip planarity, it has a minor effect on total dynamic recovery of these copper alloys. Ó 2010 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Copper alloys; Short-range ordering; Stacking fault energy; Plastic deformation
The work hardening response of low stacking fault face-centered cubic (low SFE fcc) alloys comprises four distinct stages during simple compression testing, as depicted in Figure 1 [1–7]. These stages of work hardening are summarized as follows. Stage A hardening starts instantly after yielding of the material and is characterized by a gradual decrease in hardening rate. The nature of this stage is not unambiguous. Many researchers believe that this behavior is simply an elasto-plastic transition and development of slip systems in the elastically deformed crystals is considered to be the main cause of the occurrence of this stage of hardening [8]. Although recent investigations by Feaugas [9] and Feaugas and Haddou [10] supported this idea, observation of this stage up to rather large strains is questionable [1]. Stage B hardening is reported to emerge after some plastic straining (mainly before a strain level of 0.1 in simple compression) [1–7]. During this stage of hardening the gradual decreasing in hardening rate remains at a relatively constant value (h = 0.025–0.030l, in which h is the work hardening rate and l is the shear modulus of the material) and a plateau forms in the work hardening rate vs. strain diagram (i.e. linear hardening). A classical description of this behavior was given by Kocks and Mecking based on Taylor type hardening [8]. This model is simplified and some parameters such as deformation twinning are not included. A recent description of * Corresponding author. Tel.: +98(21)6616 5238; fax: +98(21)6600 5717; e-mail: [email protected]
this stage of hardening included the effect of mechanical twinning [1,3–7,11–14]. It was reported that deformation twinning developed concurrent with Stage B hardening in a number of low SFE fcc alloys, such as MP35N and a-brass [1] and different TWIP steels [11– 14]. Formation of deformation twinning leads to a reduction in slip length and Stage B hardening emerges as a result of a phenomenon known as the dynamic Hall–Petch effect [4,11–14]. Stage C hardening is characterized by breakdown of the constant hardening rate observed in Stage B hardening. This phenomenon can be explained by activation of cross-slip processes as a dynamic recovery mechanism [8,15]. The stress and strain level corresponding to the transition from Stage B to Stage C was found to be inversely dependent on temperature and stacking fault energy [15]. It is noteworthy that the hardening response of higher SFE alloys, e.g. aluminum and copper, does not include Stage B hardening, as depicted in Figure 1. The stress range of Stage B decreases with increasing SFE value and may even vanish for high SFE fcc alloys. Stage D hardening is explained based on saturation of the hardening rate to a finite value [1]. This value is different from saturation for higher SFE fcc alloys, in which the work hardening rate approaches 0 [1–7]. It has been suggested that activation of a secondary twining system further reduces slip lengths and causes additional hardening [1]. The occurrence of linear hardening behavior has been considered to be a sign of deformation twinning in a number of investigations [3–7]. In a recent study by
1359-6462/$ - see front matter Ó 2010 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.scriptamat.2010.01.031
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F. Hamdi, S. Asgari / Scripta Materialia 62 (2010) 693–696
Figure 1. Schematic representation of four stage strain hardening response of low SFE alloys vs. work hardening response of higher SFE alloys.
the present authors, however, it was shown that in some commercial alloys, such as IN625, four-stage hardening behavior can be seen in the absence of deformation twinning [2]. It has been suggested that parameters which effectively inhibit cross-slip processes may have a similar effect on the work hardening behavior of fcc alloys. These parameters are short-range ordering (SRO), low temperature, high friction stresses, a high shear modulus and a low SFE value of the material [16,17]. The variety of these parameters casts substantial doubt on the classical calculation of the stress level corresponding to the transition from Stage B to Stage C hardening based only on SFE value and temperature [15]. To the knowledge of the authors there has been no effort to evaluate the effect of each mentioned parameter in isolation on the Stage B–Stage C transition stress. In the present report we have experimentally evaluated two parameters, i.e. SRO and SFE value, separately. For this purpose two specifically designed copper alloys were prepared and their deformation responses were investigated. It is possible to alter the SFE value and SRO of copper alloys independently [4,18]. Figure 2 shows the effect of manganese and aluminum addition on the SFE value of copper alloys. As
Figure 2. Variation in SFE values of copper alloys as a function of alloying element concentration [18].
is clearly evident from these data, manganese addition has a minor effect on the SFE value of copper alloys. On the other hand, it has been reported that the addition of 10% manganese causes SRO and slip planarity in copper alloys [17]. Cu–12 wt.% Mn and Cu–6 wt.% Al alloys were prepared from virgin high purity metals including oxygen free high conductivity (OFHC) copper, electrolytic manganese and refined pure aluminum by induction melting under an inert gas atmosphere and cryolite flux in a graphite crucible. Molten alloys were cast in a round graphite mold and rapidly cooled after a holding time to allow possible inclusions float off and dissolve in the cryolite flux. Cast round bars were solution treated in 800 °C under a protective deoxygenized argon atmosphere in a tube furnace for 24 h. Bars were plastically deformed via swaging and uniaxial pressing. Samples were recrystallized in a furnace at 500 °C for 90 min. X-ray diffraction and transmission electron microscopy (TEM) investigations confirmed single phase and annealed microstructure in the samples. Round cylindrical samples were cut from the prepared bars with an aspect ratio of 1.2 by means of electrical discharge machining (EDM). Room temperature uniaxial compression tests were carried out on the cylinders using an INSTRON 4208 unit at an initial strain rate of 103 s–1. Sample surfaces were polished and lubricated using Teflon high pressure grease before compression testing. Almost no barreling was observed up to a strain of 0.5. Force displacement data were recorded and true stress vs. true strain (r–e) curves and consequently the work hardening rate vs. true strain (h–e) response of the materials were calculated numerically. For compressive strains >0.5 plots of r–e and h–e might have been influenced by frictional effects due to the observed barreling, although these effects were not expected to change the observed trends in these plots. Deformed samples at a strain of 0.2 were prepared for TEM studies. Slices 500 lm thick were cut from deformed samples normal to the loading direction using EDM and were then chemically thinned to a thickness of 70–120 lm. Discs of 3 mm diameter were punched from the slices and were perforated using a Tenupol-5 jet-polisher with Struers D2 electrolyte at a voltage of 6 V and temperature of 5 °C. TEM studies were performed with a Philips CM200 at an operating voltage of 200 keV. Figure 3a shows r–e curves for the Cu–12 wt.% Mn and Cu–6 wt.% Al alloys. The Cu–6 wt.% Al alloy showed the typical hardening response of low SFE fcc metals. This behavior is characterized by a linear segment in the r–e curve. This linearity is more evident in the h–e graph of the alloys, as shown in Figure 3b. This trend was previously reported by Rohatgi et al. [4] and was expected due to the low SFE value of the alloy (see Fig. 2). Figure 4a shows the main features of the microstructure of the deformed Cu–6 wt.% Al alloy at a strain level corresponding to Stage B hardening (e = 0.2). The microstructure of the deformed Cu– 6 wt.% Al alloy at this strain level included planar arrays of dislocations and deformation twinning in a considerable number of grains. Both deformation twinning and slip planarity are expected to cause a linear hardening
F. Hamdi, S. Asgari / Scripta Materialia 62 (2010) 693–696
Figure 3. (a) True stress vs. true strain (r–e) and (b) work hardening vs. true strain (r–e) curves for the two alloys examined.
Figure 4. (a) Deformation twinning in Cu–6 wt.% Al alloy at a strain of about 0.2. (b) Planar dislocation structure in Cu–12 wt.% Mn at a strain of about 0.2.
695
effect (i.e. the constant hardening rate observed during Stage B hardening) due to a reduction in dislocation glide length and dynamic recovery rate [2]. The Cu–12 wt.% Mn alloy, however, showed a different mechanical response and microstructural evolution. The microstructure of the deformed Cu–12 wt.% Mn is depicted in Figure 4b. Planar arrays of dislocations appeared in the microstructure, but no deformation twinning was observed. Planarity of slip in this alloy showed the difficulty of cross-slip processes. The SFE value of the alloy was high and was approximately equal to that of pure copper. The dislocation arrangement of the alloy, however, differed from that of deformed copper at the same strain level [8]. Deformed copper samples typically show a tendency to form cellular dislocation substructures which are more localized compared with the planar arrangement of dislocations in the Cu–12 wt.% Mn alloy. The latter resembled dislocation substructures usually seen in low SFE alloys [2]. The origin of planarity of dislocation structures in the Cu–12 wt.% Mn alloy is a phenomenon termed glide plane softening [17]. In alloys containing SRO glide of the first dislocation in a virgin atomic plane disturbs the local ordering of solute atoms and leaves behind a diffuse antiphase boundary. This effect causes an additional energy barrier to the passage of dislocations on every fresh slip plane. As a result, the next active dislocations prefer to chase the first to avoid this additional barrier energy to slip. In ordinary alloys, where no ordering occurs, cross-slipping may take place in every dislocation glide plane. In alloys having SRO, however, the occurrence of cross-slipping is restricted to previously swept planes. Despite a planar dislocation structure of the deformed Cu–12 wt.% Mn alloy, no signs of problems with the dynamic recovery process of the alloy were evident in r–e and h–e plots (Fig. 3a and b). These observations may be explained using the Kocks–Meacking model (KM model) of work hardening. In the KM model stress is related to the forest dislocation density and work hardening is derived in terms of dislocation density evolution [8]. This model has been widely used to model work hardening of a wide range of alloys, including both low SFE and medium to high fcc single crystals and polycrystals [8,19]. Dislocation density evolution is described in the KM model as a result of two dislocation generation and dislocation annihilation rates as: þ dq dq dq þ ð1Þ ¼ dc dc dc where q and c are dislocation density and shear strain, respectively. The dislocation annihilation rate (the minus term) corresponds to dynamic recovery processes. It is noteworthy that in the KM model dynamic recovery is not considered a stage but is a process operative throughout plastic deformation [8]. The strain at which dynamic recovery overcomes dislocation storage mechanisms is considered as the end point of Stage B [15]. The main mechanism of dislocation annihilation at low homologous temperatures is dislocation cross-slip [8]. Therefore, the dislocation annihilation rate is proportional to the probability of opposition of dislocation dipoles in two
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F. Hamdi, S. Asgari / Scripta Materialia 62 (2010) 693–696
adjacent slip planes. This is linearly proportional to the dislocation density in polycrystals [15,19–22]. dq ¼ k d N cs q ð2Þ dc where kd is a constant and Ncs represents the average dislocation number per unit strain interval which is successfully annihilated at each potential recovery site [23]. In a constant strain rate regime Ncs may be expressed as: N cs ¼ P cs N d
ð3Þ
Pcs represents the probability of successful attempts at cross-slipping and Nd is the number of total attempts at cross-slipping within a unit strain interval. According to Gerold and Karnthaler [17] and Hong and Laird [16] different mechanisms may hinder crossslipping. Each mechanism may impose a specific probability factor to Eq. (3) due to general probability theory. Hence: P cs ¼ P SRO P elastic P SFE P friction :::: ¼ P SRO P |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
ð4Þ
P
In this relationship Pelastic, PSFE, Pfriction and PSRO are the probabilities of overcoming elastic, stacking fault energy, friction stress and SRO energy barriers, respectively. A careful deliberation on each mechanism reveals a substantial difference between the contribution of SRO and the others, i.e. PSRO only applies to the first dislocation while the other parameters apply to all glide dislocations. Consider a dislocation pile-up in a short-range ordered matrix. Cross-slip is plausible for each dislocation, but the first dislocation is more likely to cross-slip due to very high local stresses [24]. Therefore, it is reasonable to assume that the cross-slip process in a pileup is practically confined to the pile-up front. The other assumption is that the strain rate is low enough to allow a considerable number of dislocations (n) to pass a cross-slip position. As a result, by averaging cross-slip probabilities over a strain interval one obtains the following equation for cross-slip probability: 1 1 1 P cs ¼ P P SRO þ P þ þ P n n n |fflfflfflfflfflffl{zfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflffl ffl{zfflfflfflfflfflfflfflfflfflfflffl} Pioneering
ð5Þ
n1
which for large n values is simplified to: n 1 P SRO P þ P cs ¼ P n n
ð6Þ
The significance of Eq. (6) is that the contribution of SRO in hindering the total dynamic recovery processes is negligible, while it still triggers slip planarity. In other words, after successful cross-slipping of the first dislocation, cross-slipping of further dislocations is similar to that in alloys without SRO. Another important conclusion from the present observations is that the classical relationship between
SFE value and transition stress between Stage B and Stage C [15] should be still valid for the copper alloys used in this study. The reason is that the total dislocation annihilation rate is not affected by SRO. Other mechanisms, such as high misfit solute elements, which are believed to hinder cross-slip processes in all dislocations, may, however, influence the transition stress. As a final remark, it is noteworthy that the mechanism of glide plane softening, suggested by Gerold and Karnthaler [17] implies a long-range energy barrier to cross-slipping and dynamic recovery in the presence of SRO. Passage of dislocations through this energy barrier, however, should involve a series of short-range phenomena. The magnitude of PSRO, therefore, should be highly dependent on the local stress. As a result, a precise evaluation of the contribution of SRO requires a relation between the local stress, temperature and strain rate. The authors wish to thank Sharif University of Technology for providing financial support for this project. Mrs P. Amini of Sharif University of Technology is gratefully acknowledged for her assistance in preparing TEM samples. [1] S. Asgari, E. El-Danaf, S.R. Kalidindi, R.D. Doherty, Met. Mat. Trans. A 28A (1997) 1781. [2] F. Hamdi, S. Asgari, Met. Mat. Trans. A 39A (2008) 294. [3] E. El-Danaf, S.R. Kalidindi, R.D. Doherty, Met. Mat. Trans. A 30A (1999) 1223. [4] A. Rohatgi, K.S. Vecchio, G.T. Gray, Met. Mat. Trans. A 32A (2001) 135. [5] E. El-Danaf, S.R. Kalidindi, R.D. Doherty, Int. J. Plast. 17 (2001) 1245. [6] S.R. Kalidindi, Int. J. Plast. 14 (1998) 1265. [7] S.R. Kalidindi, Int J Plast. 17 (2001) 837. [8] U.F. Kocks, H. Mecking, Prog. Mater. Sci. 48 (2003) 171. [9] X. Feaugas, Acta Mater. 47 (1999) 3617. [10] X. Feaugas, H. Haddou, Met. Mat. Trans. A 34A (2003) 2329. [11] J.G. Sevillano, Scripta Mater. 60 (2009) 336. [12] O. Bouaziz, N. Guelton, Mater. Sci. Eng. A 246 (2001) 319. [13] S. Allain, J.P. Chateau, O. Bouaziz, Mater. Sci. Eng. A 143 (2004) 387. [14] O. Bouaziz, S. Allain, C. Scott, Scripta Mater. 58 (2008) 484. [15] E. Nes, Prog. Mater. Sci. 41 (1998) 129. [16] S.I. Hong, C. Laird, Acta Metall. 38 (1990) 1581. [17] V. Gerold, H.P. Karnthaler, Acta Metall. 37 (1989) 2177. [18] O. Engler, Acta Mater. 48 (2000) 4827. [19] H. Mecking, U.F. Kocks, Acta Metall. 29 (1981) 1865. [20] E. Nes, T. Pettersen, K. Marthinsen, Scripta Mater. 43 (2000) 55. [21] E. Nes, K. Marthinsen, Mater. Sci. Eng. A 322 (2002) 176. [22] E. Nes, K. Marthinsen, Y. Brechet, Scripta Mater. 47 (2002) 607. [23] U.F. Kocks, J. Eng. Mater. Tech. 98 (1976) 76. [24] J.P. Hirth, J. Loth, Theory of Dislocations, McGraw-Hill, New York, 1968, p. 694.
Scripta Materialia 62 (2010) 693–696 www.elsevier.com/locate/scriptamat
Influence of stacking fault energy and short-range ordering on dynamic recovery and work hardening behavior of copper alloys Farzad Hamdi and Sirous Asgari* Department of Materials Science and Engineering, Sharif University of Technology, Azadi Avenue, Tehran, Iran Received 27 August 2009; revised 15 January 2010; accepted 15 January 2010 Available online 21 January 2010
True stress vs. true strain responses of Cu–6 wt.% Al and Cu–12 wt.% Mn alloys are presented. While Cu–6 wt.% Al alloy shows the typical mechanical response of low stacking fault energy alloys, the Cu–12 wt.% Mn alloy behaved similarly to medium to high stacking fault energy alloys. These findings clearly show that while short-range ordering triggers slip planarity, it has a minor effect on total dynamic recovery of these copper alloys. Ó 2010 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Copper alloys; Short-range ordering; Stacking fault energy; Plastic deformation
The work hardening response of low stacking fault face-centered cubic (low SFE fcc) alloys comprises four distinct stages during simple compression testing, as depicted in Figure 1 [1–7]. These stages of work hardening are summarized as follows. Stage A hardening starts instantly after yielding of the material and is characterized by a gradual decrease in hardening rate. The nature of this stage is not unambiguous. Many researchers believe that this behavior is simply an elasto-plastic transition and development of slip systems in the elastically deformed crystals is considered to be the main cause of the occurrence of this stage of hardening [8]. Although recent investigations by Feaugas [9] and Feaugas and Haddou [10] supported this idea, observation of this stage up to rather large strains is questionable [1]. Stage B hardening is reported to emerge after some plastic straining (mainly before a strain level of 0.1 in simple compression) [1–7]. During this stage of hardening the gradual decreasing in hardening rate remains at a relatively constant value (h = 0.025–0.030l, in which h is the work hardening rate and l is the shear modulus of the material) and a plateau forms in the work hardening rate vs. strain diagram (i.e. linear hardening). A classical description of this behavior was given by Kocks and Mecking based on Taylor type hardening [8]. This model is simplified and some parameters such as deformation twinning are not included. A recent description of * Corresponding author. Tel.: +98(21)6616 5238; fax: +98(21)6600 5717; e-mail: [email protected]
this stage of hardening included the effect of mechanical twinning [1,3–7,11–14]. It was reported that deformation twinning developed concurrent with Stage B hardening in a number of low SFE fcc alloys, such as MP35N and a-brass [1] and different TWIP steels [11– 14]. Formation of deformation twinning leads to a reduction in slip length and Stage B hardening emerges as a result of a phenomenon known as the dynamic Hall–Petch effect [4,11–14]. Stage C hardening is characterized by breakdown of the constant hardening rate observed in Stage B hardening. This phenomenon can be explained by activation of cross-slip processes as a dynamic recovery mechanism [8,15]. The stress and strain level corresponding to the transition from Stage B to Stage C was found to be inversely dependent on temperature and stacking fault energy [15]. It is noteworthy that the hardening response of higher SFE alloys, e.g. aluminum and copper, does not include Stage B hardening, as depicted in Figure 1. The stress range of Stage B decreases with increasing SFE value and may even vanish for high SFE fcc alloys. Stage D hardening is explained based on saturation of the hardening rate to a finite value [1]. This value is different from saturation for higher SFE fcc alloys, in which the work hardening rate approaches 0 [1–7]. It has been suggested that activation of a secondary twining system further reduces slip lengths and causes additional hardening [1]. The occurrence of linear hardening behavior has been considered to be a sign of deformation twinning in a number of investigations [3–7]. In a recent study by
1359-6462/$ - see front matter Ó 2010 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.scriptamat.2010.01.031
694
F. Hamdi, S. Asgari / Scripta Materialia 62 (2010) 693–696
Figure 1. Schematic representation of four stage strain hardening response of low SFE alloys vs. work hardening response of higher SFE alloys.
the present authors, however, it was shown that in some commercial alloys, such as IN625, four-stage hardening behavior can be seen in the absence of deformation twinning [2]. It has been suggested that parameters which effectively inhibit cross-slip processes may have a similar effect on the work hardening behavior of fcc alloys. These parameters are short-range ordering (SRO), low temperature, high friction stresses, a high shear modulus and a low SFE value of the material [16,17]. The variety of these parameters casts substantial doubt on the classical calculation of the stress level corresponding to the transition from Stage B to Stage C hardening based only on SFE value and temperature [15]. To the knowledge of the authors there has been no effort to evaluate the effect of each mentioned parameter in isolation on the Stage B–Stage C transition stress. In the present report we have experimentally evaluated two parameters, i.e. SRO and SFE value, separately. For this purpose two specifically designed copper alloys were prepared and their deformation responses were investigated. It is possible to alter the SFE value and SRO of copper alloys independently [4,18]. Figure 2 shows the effect of manganese and aluminum addition on the SFE value of copper alloys. As
Figure 2. Variation in SFE values of copper alloys as a function of alloying element concentration [18].
is clearly evident from these data, manganese addition has a minor effect on the SFE value of copper alloys. On the other hand, it has been reported that the addition of 10% manganese causes SRO and slip planarity in copper alloys [17]. Cu–12 wt.% Mn and Cu–6 wt.% Al alloys were prepared from virgin high purity metals including oxygen free high conductivity (OFHC) copper, electrolytic manganese and refined pure aluminum by induction melting under an inert gas atmosphere and cryolite flux in a graphite crucible. Molten alloys were cast in a round graphite mold and rapidly cooled after a holding time to allow possible inclusions float off and dissolve in the cryolite flux. Cast round bars were solution treated in 800 °C under a protective deoxygenized argon atmosphere in a tube furnace for 24 h. Bars were plastically deformed via swaging and uniaxial pressing. Samples were recrystallized in a furnace at 500 °C for 90 min. X-ray diffraction and transmission electron microscopy (TEM) investigations confirmed single phase and annealed microstructure in the samples. Round cylindrical samples were cut from the prepared bars with an aspect ratio of 1.2 by means of electrical discharge machining (EDM). Room temperature uniaxial compression tests were carried out on the cylinders using an INSTRON 4208 unit at an initial strain rate of 103 s–1. Sample surfaces were polished and lubricated using Teflon high pressure grease before compression testing. Almost no barreling was observed up to a strain of 0.5. Force displacement data were recorded and true stress vs. true strain (r–e) curves and consequently the work hardening rate vs. true strain (h–e) response of the materials were calculated numerically. For compressive strains >0.5 plots of r–e and h–e might have been influenced by frictional effects due to the observed barreling, although these effects were not expected to change the observed trends in these plots. Deformed samples at a strain of 0.2 were prepared for TEM studies. Slices 500 lm thick were cut from deformed samples normal to the loading direction using EDM and were then chemically thinned to a thickness of 70–120 lm. Discs of 3 mm diameter were punched from the slices and were perforated using a Tenupol-5 jet-polisher with Struers D2 electrolyte at a voltage of 6 V and temperature of 5 °C. TEM studies were performed with a Philips CM200 at an operating voltage of 200 keV. Figure 3a shows r–e curves for the Cu–12 wt.% Mn and Cu–6 wt.% Al alloys. The Cu–6 wt.% Al alloy showed the typical hardening response of low SFE fcc metals. This behavior is characterized by a linear segment in the r–e curve. This linearity is more evident in the h–e graph of the alloys, as shown in Figure 3b. This trend was previously reported by Rohatgi et al. [4] and was expected due to the low SFE value of the alloy (see Fig. 2). Figure 4a shows the main features of the microstructure of the deformed Cu–6 wt.% Al alloy at a strain level corresponding to Stage B hardening (e = 0.2). The microstructure of the deformed Cu– 6 wt.% Al alloy at this strain level included planar arrays of dislocations and deformation twinning in a considerable number of grains. Both deformation twinning and slip planarity are expected to cause a linear hardening
F. Hamdi, S. Asgari / Scripta Materialia 62 (2010) 693–696
Figure 3. (a) True stress vs. true strain (r–e) and (b) work hardening vs. true strain (r–e) curves for the two alloys examined.
Figure 4. (a) Deformation twinning in Cu–6 wt.% Al alloy at a strain of about 0.2. (b) Planar dislocation structure in Cu–12 wt.% Mn at a strain of about 0.2.
695
effect (i.e. the constant hardening rate observed during Stage B hardening) due to a reduction in dislocation glide length and dynamic recovery rate [2]. The Cu–12 wt.% Mn alloy, however, showed a different mechanical response and microstructural evolution. The microstructure of the deformed Cu–12 wt.% Mn is depicted in Figure 4b. Planar arrays of dislocations appeared in the microstructure, but no deformation twinning was observed. Planarity of slip in this alloy showed the difficulty of cross-slip processes. The SFE value of the alloy was high and was approximately equal to that of pure copper. The dislocation arrangement of the alloy, however, differed from that of deformed copper at the same strain level [8]. Deformed copper samples typically show a tendency to form cellular dislocation substructures which are more localized compared with the planar arrangement of dislocations in the Cu–12 wt.% Mn alloy. The latter resembled dislocation substructures usually seen in low SFE alloys [2]. The origin of planarity of dislocation structures in the Cu–12 wt.% Mn alloy is a phenomenon termed glide plane softening [17]. In alloys containing SRO glide of the first dislocation in a virgin atomic plane disturbs the local ordering of solute atoms and leaves behind a diffuse antiphase boundary. This effect causes an additional energy barrier to the passage of dislocations on every fresh slip plane. As a result, the next active dislocations prefer to chase the first to avoid this additional barrier energy to slip. In ordinary alloys, where no ordering occurs, cross-slipping may take place in every dislocation glide plane. In alloys having SRO, however, the occurrence of cross-slipping is restricted to previously swept planes. Despite a planar dislocation structure of the deformed Cu–12 wt.% Mn alloy, no signs of problems with the dynamic recovery process of the alloy were evident in r–e and h–e plots (Fig. 3a and b). These observations may be explained using the Kocks–Meacking model (KM model) of work hardening. In the KM model stress is related to the forest dislocation density and work hardening is derived in terms of dislocation density evolution [8]. This model has been widely used to model work hardening of a wide range of alloys, including both low SFE and medium to high fcc single crystals and polycrystals [8,19]. Dislocation density evolution is described in the KM model as a result of two dislocation generation and dislocation annihilation rates as: þ dq dq dq þ ð1Þ ¼ dc dc dc where q and c are dislocation density and shear strain, respectively. The dislocation annihilation rate (the minus term) corresponds to dynamic recovery processes. It is noteworthy that in the KM model dynamic recovery is not considered a stage but is a process operative throughout plastic deformation [8]. The strain at which dynamic recovery overcomes dislocation storage mechanisms is considered as the end point of Stage B [15]. The main mechanism of dislocation annihilation at low homologous temperatures is dislocation cross-slip [8]. Therefore, the dislocation annihilation rate is proportional to the probability of opposition of dislocation dipoles in two
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F. Hamdi, S. Asgari / Scripta Materialia 62 (2010) 693–696
adjacent slip planes. This is linearly proportional to the dislocation density in polycrystals [15,19–22]. dq ¼ k d N cs q ð2Þ dc where kd is a constant and Ncs represents the average dislocation number per unit strain interval which is successfully annihilated at each potential recovery site [23]. In a constant strain rate regime Ncs may be expressed as: N cs ¼ P cs N d
ð3Þ
Pcs represents the probability of successful attempts at cross-slipping and Nd is the number of total attempts at cross-slipping within a unit strain interval. According to Gerold and Karnthaler [17] and Hong and Laird [16] different mechanisms may hinder crossslipping. Each mechanism may impose a specific probability factor to Eq. (3) due to general probability theory. Hence: P cs ¼ P SRO P elastic P SFE P friction :::: ¼ P SRO P |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
ð4Þ
P
In this relationship Pelastic, PSFE, Pfriction and PSRO are the probabilities of overcoming elastic, stacking fault energy, friction stress and SRO energy barriers, respectively. A careful deliberation on each mechanism reveals a substantial difference between the contribution of SRO and the others, i.e. PSRO only applies to the first dislocation while the other parameters apply to all glide dislocations. Consider a dislocation pile-up in a short-range ordered matrix. Cross-slip is plausible for each dislocation, but the first dislocation is more likely to cross-slip due to very high local stresses [24]. Therefore, it is reasonable to assume that the cross-slip process in a pileup is practically confined to the pile-up front. The other assumption is that the strain rate is low enough to allow a considerable number of dislocations (n) to pass a cross-slip position. As a result, by averaging cross-slip probabilities over a strain interval one obtains the following equation for cross-slip probability: 1 1 1 P cs ¼ P P SRO þ P þ þ P n n n |fflfflfflfflfflffl{zfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflffl ffl{zfflfflfflfflfflfflfflfflfflfflffl} Pioneering
ð5Þ
n1
which for large n values is simplified to: n 1 P SRO P þ P cs ¼ P n n
ð6Þ
The significance of Eq. (6) is that the contribution of SRO in hindering the total dynamic recovery processes is negligible, while it still triggers slip planarity. In other words, after successful cross-slipping of the first dislocation, cross-slipping of further dislocations is similar to that in alloys without SRO. Another important conclusion from the present observations is that the classical relationship between
SFE value and transition stress between Stage B and Stage C [15] should be still valid for the copper alloys used in this study. The reason is that the total dislocation annihilation rate is not affected by SRO. Other mechanisms, such as high misfit solute elements, which are believed to hinder cross-slip processes in all dislocations, may, however, influence the transition stress. As a final remark, it is noteworthy that the mechanism of glide plane softening, suggested by Gerold and Karnthaler [17] implies a long-range energy barrier to cross-slipping and dynamic recovery in the presence of SRO. Passage of dislocations through this energy barrier, however, should involve a series of short-range phenomena. The magnitude of PSRO, therefore, should be highly dependent on the local stress. As a result, a precise evaluation of the contribution of SRO requires a relation between the local stress, temperature and strain rate. The authors wish to thank Sharif University of Technology for providing financial support for this project. Mrs P. Amini of Sharif University of Technology is gratefully acknowledged for her assistance in preparing TEM samples. [1] S. Asgari, E. El-Danaf, S.R. Kalidindi, R.D. Doherty, Met. Mat. Trans. A 28A (1997) 1781. [2] F. Hamdi, S. Asgari, Met. Mat. Trans. A 39A (2008) 294. [3] E. El-Danaf, S.R. Kalidindi, R.D. Doherty, Met. Mat. Trans. A 30A (1999) 1223. [4] A. Rohatgi, K.S. Vecchio, G.T. Gray, Met. Mat. Trans. A 32A (2001) 135. [5] E. El-Danaf, S.R. Kalidindi, R.D. Doherty, Int. J. Plast. 17 (2001) 1245. [6] S.R. Kalidindi, Int. J. Plast. 14 (1998) 1265. [7] S.R. Kalidindi, Int J Plast. 17 (2001) 837. [8] U.F. Kocks, H. Mecking, Prog. Mater. Sci. 48 (2003) 171. [9] X. Feaugas, Acta Mater. 47 (1999) 3617. [10] X. Feaugas, H. Haddou, Met. Mat. Trans. A 34A (2003) 2329. [11] J.G. Sevillano, Scripta Mater. 60 (2009) 336. [12] O. Bouaziz, N. Guelton, Mater. Sci. Eng. A 246 (2001) 319. [13] S. Allain, J.P. Chateau, O. Bouaziz, Mater. Sci. Eng. A 143 (2004) 387. [14] O. Bouaziz, S. Allain, C. Scott, Scripta Mater. 58 (2008) 484. [15] E. Nes, Prog. Mater. Sci. 41 (1998) 129. [16] S.I. Hong, C. Laird, Acta Metall. 38 (1990) 1581. [17] V. Gerold, H.P. Karnthaler, Acta Metall. 37 (1989) 2177. [18] O. Engler, Acta Mater. 48 (2000) 4827. [19] H. Mecking, U.F. Kocks, Acta Metall. 29 (1981) 1865. [20] E. Nes, T. Pettersen, K. Marthinsen, Scripta Mater. 43 (2000) 55. [21] E. Nes, K. Marthinsen, Mater. Sci. Eng. A 322 (2002) 176. [22] E. Nes, K. Marthinsen, Y. Brechet, Scripta Mater. 47 (2002) 607. [23] U.F. Kocks, J. Eng. Mater. Tech. 98 (1976) 76. [24] J.P. Hirth, J. Loth, Theory of Dislocations, McGraw-Hill, New York, 1968, p. 694.