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The grain size dependence of flow stress in a Cu–1%Cd alloy
Materials Science and Engineering A 466 (2007) 68–70
The grain size dependence of flow stress in a Cu–1%Cd alloy S. Nagarjuna a,∗ , J.T. Evans b a
b
Defence Metallurgical Research Laboratory, Kanchanbagh PO, Hyderabad 500058, India School of Mechanical and Systems Engineering, University of Newcastle, Newcastle upon Tyne NE1 7RU, UK Received 18 November 2006; received in revised form 7 February 2007; accepted 5 April 2007
Abstract The dependence of flow stress on grain size was studied in a Cu–1%Cd alloy. The Hall–Petch constants are related to true strain in such a way that σ 0 (ε) is proportional to ε and k(ε) having negative variation with ε1/2 . A new equation for flow stress as a function of true strain (ε) and grain size (d) has been evolved for this alloy. © 2007 Elsevier B.V. All rights reserved. Keywords: Cu–1Cd alloy; Annealing; Grain size; Tensile testing; Flow stress equation
1. Introduction Copper–1% cadmium alloy enjoys specialised commercial use because of the combination of good mechanical properties and high electrical conductivity. Addition of cadmium to copper greatly improves the mechanical properties with little effect on conductivity [1]. Its major use is in the production of high conductivity rods and wires where good tensile strength and fatigue resistance are required. Behnood et al. [2] deduced that these beneficial effects arise from the dislocation pinning by cadmium due to its large atomic misfit in the copper lattice and that the lower resistivity of Cu–Cd alloy arises because the addition of Cd to Cu does not raise the electron to atom ratio markedly. Grain size strengthening is an important strengthening mechanism in a number of metals and alloys. Armstrong et al. [3] proposed that the flow stress σ(ε) is related to the mean grain diameter d, according to the equation: σ(ε) = σ0 (ε) + k(ε)d −1/2
(1)
where ε is the plastic strain; σ 0 (ε) the grain size independent component of the flow stress and k(ε) is the Hall–Petch parameter that reflects the intensity of the grain size strengthening. It was reported that the yield and flow stresses of Cu–1%Cd alloy obey the Hall–Petch relation [2].
Phillips and Armstrong [4] and Meakin and Petch [5] proposed the following equation for flow stress for ␣-brass (Cu–30Zn alloy): σ(ε) = σ0 + Aε + Bε1/2 d −1/2 + ky d −1/2
(2)
Nagarjuna and co-workers derived similar equations for flow stress in Cu–Ti alloys containing Ti < 3.0 wt% [6] and Cu–26Ni–17Zn alloy [7]. However, little data has been reported on such an equation for flow stress in Cu–1%Cd alloy. In the present paper, an equation for flow stress as a function of true strain and grain size (d) has been developed for Cu–1%Cd alloy. 2. Experimental The material used for this investigation was Cu–1%Cd alloy with the following chemical composition (by wt%): Cd, 0.98; Sn, 0.005; Zn, 0.002; balance, Cu. The alloy was hot rolled initially to 12.5 mm diameter rods and cold drawn subsequently to rods, 6 mm diameter. Cylindrical tensile specimens (gauge length 25.4 mm and gauge diameter 3.20 mm) were machined from cold drawn bar. They were annealed in an atmosphere of argon in the temperature range 500–950 ◦ C to obtain grain sizes ranging from 8.5 to 50 m. Specimens were tested at a nominal strain rate of 6.6 × 10−4 s−1 at room temperature. 3. Results
∗
Corresponding author. Tel.: +91 40 24342655; fax: +91 40 24340559. E-mail address: [email protected] (S. Nagarjuna).
0921-5093/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2007.04.019
The quantity [σ 0 (ε) − σ Y ] is found to be proportional to and increased linearly with true strain ε, as shown in Fig. 1. The
S. Nagarjuna, J.T. Evans / Materials Science and Engineering A 466 (2007) 68–70
69
here that the value of B for Cu–1%Cd alloy is negative, as k(ε) decreased with increasing strain at 300 K. 4. Discussion The present investigation reveals that the addition of 1% Cd to Cu has resulted in an interesting relationship between the Hall–Petch constant ky and the true strain ε. The relationship between (σ 0 (ε) − σ Y ) and strain ε in Cu–1%Cd alloy is approximately linear (Fig. 1) and is similar to the behaviour reported by Nagarjuna et al. for Cu–Ti [6] and Cu–Ni–Zn alloys [7]. This represents work hardening behaviour in Cu–1%Cd alloy. Further, σ 0 (ε) increases with strain as a result of strain hardening with in the grain volumes. The variation of k(ε) − ky with strain (ε1/2 ) shows a negative increase in Cu–Cd alloy and the observation is different from that observed in Cu–30Zn [4,5], Cu–Ti [6] and Cu–Ni–Zn [7] alloys wherein k(ε) − ky was reported to increase with strain (ε1/2 ) linearly. This difference in the behaviour between Cu–1%Cd alloy and other copper alloys under reference is attributed to the decrease in k(ε) with strain as reported by Behnood et al. [2]. Substitution of Eqs. (3) and (4) in Eq. (1) resulted in Eq. (5). Eq. (5) for flow stress of Cu–1%Cd alloy is similar to that reported by Phillips and Armstrong [4] and Meakin and Petch [5] for brasses and Nagarjuna et al for Cu–Ti alloys containing Ti < 3.0 wt% [6] and Cu–26Ni–17Zn alloy [7]. On substitution of numerical values for σ Y , ky , A and B in Eq. (5) resulted in Eq. (6), as given below:
Fig. 1. Variation of (σ 0 (ε) − σ Y ) with true strain, ε in Cu–1%Cd alloy.
Fig. 2. Variation of k(ε) − ky with ε1/2 in Cu–1%Cd alloy.
σ(ε) = 26 + 903ε + (−0.3)ε1/2 d −1/2 + 0.25d −1/2
linear portion of the variation can be represented by σ0 (ε) = σY + Aε
(3)
where A is a constant. Fig. 2 shows the variation of k(ε) − kY with ε1/2 . The linear portion of the variation can be represented by k(ε) = kY + Bε1/2
(4)
σ(ε) = σY + Aε + Bε
1/2 −1/2
d
+ ky d
−1/2
Eq. (6) is similar to Eq. (5) except that the value for B is negative (−0.3). The negative value for B for Cu–1%Cd alloy is a result of decrease of k(ε) with strain as reported by Behanood et al. [2] and is associated with strong dislocation locking, a sharp yield point and a Luders extension on initial yield. An interesting observation noticed here is that at a strain (ε) value of 1.0, Eq. (6) changes to σ(ε) = 929 + (−0.05)d −1/2
where B is a constant. Substituting Eqs. (3) and (4) in Eq. (1) will give (5)
where σ(ε) is the flow stress, ε the true strain, d the grain diameter, σ Y and ky are Hall–Petch constants for 0.5% yielding and A and B are constants. Numerical values for σ Y , ky , A and B of Cu–1%Cd alloy are compared with other copper base alloys in Table 1. It is observed
(6)
(7)
σ (ε) turns out to be almost a constant at 929 MPa, as the addition of two terms, i.e. (−0.3)d−1/2 and 0.25d−1/2 reduces to (−0.05)d−1/2 at ε = 1.0 and the grain size has very little influence on σ(ε). Table 1 gives a comparison of the numerical values for σ Y , ky , A and B. While σ Y and A for Cu–Cd alloy are comparable, B has a negative value and does not match with those for Cu–30Zn or Cu–26Ni–17Zn alloys. The magnitude of ky for Cu–Cd is
Table 1 Comparison of numerical values of σ Y , ky , A and B in Eq. (5) of Cu–1%Cd alloy with other copper base alloys Alloy
σ Y (MPa)
A (MPa)
B (MN m−3/2 )
ky (MN m−3/2 )
Reference
Cu–30Zn Cu–30Zn Cu–26Ni–17Zn Cu–1%Cd
33 44 140 26
746 1403 1354 903
0.5 0.47 1 −0.3
0.22 0.28 0.31 0.25
Phillips and Armstrong [4] Meakin and Petch [5] Nagarjuna et al. [7] Present work
70
S. Nagarjuna, J.T. Evans / Materials Science and Engineering A 466 (2007) 68–70
5. Conclusions
Fig. 3. Variation of k(ε) with ε1/2 in Cu–1%Cd alloy.
0.25 MN m−3/2 , which is lower than that of Cu–26Ni–17Zn alloy (0.31) but comparable with that of Cu–30Zn alloy reported by Phillips and Armstrong [4] (0.22) as well as Meakin and Petch [5] (0.28). In order to investigate the applicability of dislocation density model of grain size strengthening proposed by Cottrell [8], Conrad et al. [9] and Ashby [10], variation of k(ε) was studied with ε1/2 here. The following is the Hall–Petch relation based on the dislocation density model C2 1/2 1/2 C1 + σ(ε) = σ0 (ε) + αGbε (8) L d where σ(ε) is the flow stress, σ 0 (ε) the friction stress independent of grain size, α a constant, G the shear modulus, b the burgers vector, L the average slip length, d the grain diameter and C1 and C2 are constants. Fig. 3 shows a plot of k(ε) versus ε1/2 to demonstrate the relationship in accordance with Eq. (8). k(ε) decreased with increasing ε1/2 . The reasoning that was elucidated for the variation of k(ε) − ky with ε1/2 holds here. This leads to the conclusion that dislocation density model of grain size strengthening does not hold good for Cu–1%Cd alloy. This is in contrary to the results that dislocation density model was reported to be valid for Cu–30Zn alloy [4,5] and Cu–26Ni–17Zn alloy [7].
1. The Hall–Petch constants of Cu–1%Cd alloy are related to true strain in such a way that σ 0 (ε) is proportional to ε and k(ε), having negative variation with ε1/2 . 2. A new equation for flow stress as a function of true strain (ε) and grain size (d) has been developed for Cu–1%Cd alloy: viz. σ(ε) = σ Y + Aε + Bε1/2 d−1/2 + ky d−1/2 , where values of the constants are shown in Table 1. 3. At a strain (ε) value of 1.0, Eq. (6) becomes σ(ε) = 929 + (−0.05)d−1/2 which indicates that the grain size dependence of flow stress is negligible. 4. The dislocation density model of grain size strengthening is found invalid for Cu–1%Cd alloy. Acknowledgements This work was done when the first author worked as a Royal Society Visiting Fellow at the University of Newcastle upon Tyne, for which the financial support of the Royal Society, the Indian National Science Academy and the Defence Research and Development Organisation, Government of India is gratefully acknowledged. The first author (SN) is grateful to Director, DMRL for permission to publish this paper. The helpful discussions with Dr. M. Srinivas Scientist of DMRL are acknowledged with thanks. References [1] P. Gregory, A.J. Bangay, T.L. Bird, Metallurgia 71 (1965) 207. [2] N. Behnood, R.M. Douthwaite, J.T. Evans, Acta Metall. 28 (1980) 1133. [3] R.W. Armstrong, I. Codd, R.M. Douthwaite, N.J. Petch, Philos. Mag. 7 (1962) 45. [4] W.L. Phillips, R.W. Armstrong, Metall. Trans. A 3 (1972) 2571. [5] J.D. Meakin, N.J. Petch, Philos. Mag. 29 (1974) 1149. [6] S. Nagarjuna, M. Srinivas, K. Balasubramanian, D.S. Sarma, Acta Mater. 44 (1996) 2285. [7] S. Nagarjuna, M. Srinivas, K.K. Sharma, Acta Mater. 48 (2000) 1807. [8] A.H. Cottrell, The Mechanical Properties of Matter, Wiley, New York, 1964, p. 277. [9] H. Conrad, S. Fuerstein, L. Rice, Mater. Sci. Eng. 2 (1967) 157. [10] M.F. Ashby, Philos. Mag. 21 (1970) 399.
The grain size dependence of flow stress in a Cu–1%Cd alloy S. Nagarjuna a,∗ , J.T. Evans b a
b
Defence Metallurgical Research Laboratory, Kanchanbagh PO, Hyderabad 500058, India School of Mechanical and Systems Engineering, University of Newcastle, Newcastle upon Tyne NE1 7RU, UK Received 18 November 2006; received in revised form 7 February 2007; accepted 5 April 2007
Abstract The dependence of flow stress on grain size was studied in a Cu–1%Cd alloy. The Hall–Petch constants are related to true strain in such a way that σ 0 (ε) is proportional to ε and k(ε) having negative variation with ε1/2 . A new equation for flow stress as a function of true strain (ε) and grain size (d) has been evolved for this alloy. © 2007 Elsevier B.V. All rights reserved. Keywords: Cu–1Cd alloy; Annealing; Grain size; Tensile testing; Flow stress equation
1. Introduction Copper–1% cadmium alloy enjoys specialised commercial use because of the combination of good mechanical properties and high electrical conductivity. Addition of cadmium to copper greatly improves the mechanical properties with little effect on conductivity [1]. Its major use is in the production of high conductivity rods and wires where good tensile strength and fatigue resistance are required. Behnood et al. [2] deduced that these beneficial effects arise from the dislocation pinning by cadmium due to its large atomic misfit in the copper lattice and that the lower resistivity of Cu–Cd alloy arises because the addition of Cd to Cu does not raise the electron to atom ratio markedly. Grain size strengthening is an important strengthening mechanism in a number of metals and alloys. Armstrong et al. [3] proposed that the flow stress σ(ε) is related to the mean grain diameter d, according to the equation: σ(ε) = σ0 (ε) + k(ε)d −1/2
(1)
where ε is the plastic strain; σ 0 (ε) the grain size independent component of the flow stress and k(ε) is the Hall–Petch parameter that reflects the intensity of the grain size strengthening. It was reported that the yield and flow stresses of Cu–1%Cd alloy obey the Hall–Petch relation [2].
Phillips and Armstrong [4] and Meakin and Petch [5] proposed the following equation for flow stress for ␣-brass (Cu–30Zn alloy): σ(ε) = σ0 + Aε + Bε1/2 d −1/2 + ky d −1/2
(2)
Nagarjuna and co-workers derived similar equations for flow stress in Cu–Ti alloys containing Ti < 3.0 wt% [6] and Cu–26Ni–17Zn alloy [7]. However, little data has been reported on such an equation for flow stress in Cu–1%Cd alloy. In the present paper, an equation for flow stress as a function of true strain and grain size (d) has been developed for Cu–1%Cd alloy. 2. Experimental The material used for this investigation was Cu–1%Cd alloy with the following chemical composition (by wt%): Cd, 0.98; Sn, 0.005; Zn, 0.002; balance, Cu. The alloy was hot rolled initially to 12.5 mm diameter rods and cold drawn subsequently to rods, 6 mm diameter. Cylindrical tensile specimens (gauge length 25.4 mm and gauge diameter 3.20 mm) were machined from cold drawn bar. They were annealed in an atmosphere of argon in the temperature range 500–950 ◦ C to obtain grain sizes ranging from 8.5 to 50 m. Specimens were tested at a nominal strain rate of 6.6 × 10−4 s−1 at room temperature. 3. Results
∗
Corresponding author. Tel.: +91 40 24342655; fax: +91 40 24340559. E-mail address: [email protected] (S. Nagarjuna).
0921-5093/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2007.04.019
The quantity [σ 0 (ε) − σ Y ] is found to be proportional to and increased linearly with true strain ε, as shown in Fig. 1. The
S. Nagarjuna, J.T. Evans / Materials Science and Engineering A 466 (2007) 68–70
69
here that the value of B for Cu–1%Cd alloy is negative, as k(ε) decreased with increasing strain at 300 K. 4. Discussion The present investigation reveals that the addition of 1% Cd to Cu has resulted in an interesting relationship between the Hall–Petch constant ky and the true strain ε. The relationship between (σ 0 (ε) − σ Y ) and strain ε in Cu–1%Cd alloy is approximately linear (Fig. 1) and is similar to the behaviour reported by Nagarjuna et al. for Cu–Ti [6] and Cu–Ni–Zn alloys [7]. This represents work hardening behaviour in Cu–1%Cd alloy. Further, σ 0 (ε) increases with strain as a result of strain hardening with in the grain volumes. The variation of k(ε) − ky with strain (ε1/2 ) shows a negative increase in Cu–Cd alloy and the observation is different from that observed in Cu–30Zn [4,5], Cu–Ti [6] and Cu–Ni–Zn [7] alloys wherein k(ε) − ky was reported to increase with strain (ε1/2 ) linearly. This difference in the behaviour between Cu–1%Cd alloy and other copper alloys under reference is attributed to the decrease in k(ε) with strain as reported by Behnood et al. [2]. Substitution of Eqs. (3) and (4) in Eq. (1) resulted in Eq. (5). Eq. (5) for flow stress of Cu–1%Cd alloy is similar to that reported by Phillips and Armstrong [4] and Meakin and Petch [5] for brasses and Nagarjuna et al for Cu–Ti alloys containing Ti < 3.0 wt% [6] and Cu–26Ni–17Zn alloy [7]. On substitution of numerical values for σ Y , ky , A and B in Eq. (5) resulted in Eq. (6), as given below:
Fig. 1. Variation of (σ 0 (ε) − σ Y ) with true strain, ε in Cu–1%Cd alloy.
Fig. 2. Variation of k(ε) − ky with ε1/2 in Cu–1%Cd alloy.
σ(ε) = 26 + 903ε + (−0.3)ε1/2 d −1/2 + 0.25d −1/2
linear portion of the variation can be represented by σ0 (ε) = σY + Aε
(3)
where A is a constant. Fig. 2 shows the variation of k(ε) − kY with ε1/2 . The linear portion of the variation can be represented by k(ε) = kY + Bε1/2
(4)
σ(ε) = σY + Aε + Bε
1/2 −1/2
d
+ ky d
−1/2
Eq. (6) is similar to Eq. (5) except that the value for B is negative (−0.3). The negative value for B for Cu–1%Cd alloy is a result of decrease of k(ε) with strain as reported by Behanood et al. [2] and is associated with strong dislocation locking, a sharp yield point and a Luders extension on initial yield. An interesting observation noticed here is that at a strain (ε) value of 1.0, Eq. (6) changes to σ(ε) = 929 + (−0.05)d −1/2
where B is a constant. Substituting Eqs. (3) and (4) in Eq. (1) will give (5)
where σ(ε) is the flow stress, ε the true strain, d the grain diameter, σ Y and ky are Hall–Petch constants for 0.5% yielding and A and B are constants. Numerical values for σ Y , ky , A and B of Cu–1%Cd alloy are compared with other copper base alloys in Table 1. It is observed
(6)
(7)
σ (ε) turns out to be almost a constant at 929 MPa, as the addition of two terms, i.e. (−0.3)d−1/2 and 0.25d−1/2 reduces to (−0.05)d−1/2 at ε = 1.0 and the grain size has very little influence on σ(ε). Table 1 gives a comparison of the numerical values for σ Y , ky , A and B. While σ Y and A for Cu–Cd alloy are comparable, B has a negative value and does not match with those for Cu–30Zn or Cu–26Ni–17Zn alloys. The magnitude of ky for Cu–Cd is
Table 1 Comparison of numerical values of σ Y , ky , A and B in Eq. (5) of Cu–1%Cd alloy with other copper base alloys Alloy
σ Y (MPa)
A (MPa)
B (MN m−3/2 )
ky (MN m−3/2 )
Reference
Cu–30Zn Cu–30Zn Cu–26Ni–17Zn Cu–1%Cd
33 44 140 26
746 1403 1354 903
0.5 0.47 1 −0.3
0.22 0.28 0.31 0.25
Phillips and Armstrong [4] Meakin and Petch [5] Nagarjuna et al. [7] Present work
70
S. Nagarjuna, J.T. Evans / Materials Science and Engineering A 466 (2007) 68–70
5. Conclusions
Fig. 3. Variation of k(ε) with ε1/2 in Cu–1%Cd alloy.
0.25 MN m−3/2 , which is lower than that of Cu–26Ni–17Zn alloy (0.31) but comparable with that of Cu–30Zn alloy reported by Phillips and Armstrong [4] (0.22) as well as Meakin and Petch [5] (0.28). In order to investigate the applicability of dislocation density model of grain size strengthening proposed by Cottrell [8], Conrad et al. [9] and Ashby [10], variation of k(ε) was studied with ε1/2 here. The following is the Hall–Petch relation based on the dislocation density model C2 1/2 1/2 C1 + σ(ε) = σ0 (ε) + αGbε (8) L d where σ(ε) is the flow stress, σ 0 (ε) the friction stress independent of grain size, α a constant, G the shear modulus, b the burgers vector, L the average slip length, d the grain diameter and C1 and C2 are constants. Fig. 3 shows a plot of k(ε) versus ε1/2 to demonstrate the relationship in accordance with Eq. (8). k(ε) decreased with increasing ε1/2 . The reasoning that was elucidated for the variation of k(ε) − ky with ε1/2 holds here. This leads to the conclusion that dislocation density model of grain size strengthening does not hold good for Cu–1%Cd alloy. This is in contrary to the results that dislocation density model was reported to be valid for Cu–30Zn alloy [4,5] and Cu–26Ni–17Zn alloy [7].
1. The Hall–Petch constants of Cu–1%Cd alloy are related to true strain in such a way that σ 0 (ε) is proportional to ε and k(ε), having negative variation with ε1/2 . 2. A new equation for flow stress as a function of true strain (ε) and grain size (d) has been developed for Cu–1%Cd alloy: viz. σ(ε) = σ Y + Aε + Bε1/2 d−1/2 + ky d−1/2 , where values of the constants are shown in Table 1. 3. At a strain (ε) value of 1.0, Eq. (6) becomes σ(ε) = 929 + (−0.05)d−1/2 which indicates that the grain size dependence of flow stress is negligible. 4. The dislocation density model of grain size strengthening is found invalid for Cu–1%Cd alloy. Acknowledgements This work was done when the first author worked as a Royal Society Visiting Fellow at the University of Newcastle upon Tyne, for which the financial support of the Royal Society, the Indian National Science Academy and the Defence Research and Development Organisation, Government of India is gratefully acknowledged. The first author (SN) is grateful to Director, DMRL for permission to publish this paper. The helpful discussions with Dr. M. Srinivas Scientist of DMRL are acknowledged with thanks. References [1] P. Gregory, A.J. Bangay, T.L. Bird, Metallurgia 71 (1965) 207. [2] N. Behnood, R.M. Douthwaite, J.T. Evans, Acta Metall. 28 (1980) 1133. [3] R.W. Armstrong, I. Codd, R.M. Douthwaite, N.J. Petch, Philos. Mag. 7 (1962) 45. [4] W.L. Phillips, R.W. Armstrong, Metall. Trans. A 3 (1972) 2571. [5] J.D. Meakin, N.J. Petch, Philos. Mag. 29 (1974) 1149. [6] S. Nagarjuna, M. Srinivas, K. Balasubramanian, D.S. Sarma, Acta Mater. 44 (1996) 2285. [7] S. Nagarjuna, M. Srinivas, K.K. Sharma, Acta Mater. 48 (2000) 1807. [8] A.H. Cottrell, The Mechanical Properties of Matter, Wiley, New York, 1964, p. 277. [9] H. Conrad, S. Fuerstein, L. Rice, Mater. Sci. Eng. 2 (1967) 157. [10] M.F. Ashby, Philos. Mag. 21 (1970) 399.