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Effect of strain rate on Lüders band propagating velocity and Lüders strain for annealed mild steel under uniaxial tension
Materials Letters 57 (2003) 4535 – 4539 www.elsevier.com/locate/matlet
Effect of strain rate on Lu¨ders band propagating velocity and Lu¨ders strain for annealed mild steel under uniaxial tension H.B. Sun a,*, F. Yoshida a, M. Ohmori b, X. Ma a a b
Department of Mechanical System Engineering, Hiroshima University, 1-4-1, Kagamiyama, Higashi-Hiroshima 739-8527, Japan Department of Mechanical Engineering, Hiroshima Kokusai Gakuin University, 6-20-1, Nakano Aki, Hiroshima 739-0321, Japan Received 30 January 2003; accepted 18 April 2003
Abstract For annealed mild steel that exhibits yield-point phenomena under uniaxial tension, physical equations expressing the strain-rate dependencies of the strain and the Lu¨ders-band velocity have been proposed in this work. In these expressions, both the Lu¨ders strain and the Lu¨ders-band velocity increase with gauge-length strain rate in the form of exponent functions. The proposed equations have been verified by our experimental data of uniaxial tension on mild steels, as well as by those reported by some other researchers. The proposed equations may provide a simple method to determine the stress-rate sensitivity exponent, in the equation of stress-dependent dislocation velocity, which is usually difficult to be measured by experiments. D 2003 Elsevier Science B.V. All rights reserved. Keywords: Yield point phenomena; Annealed mild steel; Lu¨ders-band propagation; Lu¨ders strain; Strain rate
1. Introduction In stress – strain curves under uniaxial tension of some metallic materials, typically annealed steels, yield-point phenomena characterized by a sharp yield point and the subsequent yield plateau are observed, where a localized plastic deformation appears as a result of Lu¨ders-band propagation under a constant lower yield stress [1]. To the authors’ knowledge, Sylwestrowicz and Hall [2] reported the first research work on the velocity of Lu¨ders-band front. By measuring the propagation of a single Lu¨ders band in a
specimen of iron, they proposed the following relationship between the constant crosshead velocity V of the testing machine, the resulting Lu¨ders strain eL and the Lu¨ders-band velocity s˙L: s˙ L ¼
ð1Þ
Subsequently, by replacing the crosshead velocity V with gauge-length strain rate e˙, and further considering the case of occurring several bands, Eq. (1) was modified to: s˙ L ¼
* Corresponding author. Tel.: +81-824-24-7541; fax: +81-82422-7193. E-mail address: [email protected] (H.B. Sun).
V eL
V e˙ l0 ¼ N eL N eL
ð2Þ
where l0 is the initial gauge length and N is the number of bands [3,4]. Here, it should be noted that
0167-577X/03/$ - see front matter D 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0167-577X(03)00358-6
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H.B. Sun et al. / Materials Letters 57 (2003) 4535–4539
s˙L and eL are strongly strain-rate dependent, i.e., the higher rate e˙ the larger eL [5 –8], and as a result, the Lu¨ders-band velocity s˙ L is also rate-dependent. Although some empirical equations for rate dependency of the Lu¨ders strain had been reported (e.g., Yoshida [7]), physical-based equations for the strainrate dependencies of eL and s˙L have never been proposed. In the present work, upon a sound physical foundation, an explicit relationship between the strain rate and the Lu¨ders strain has been established. Moreover, a similar relationship for the Lu¨ders-band velocity is also derived. The proposed equations have been verified by our experimental data of uniaxial tension on annealed steels, as well as by those reported by some other researchers [3 –5].
ence of dislocation velocity (Gilman and Johnson [11]: m ¼ Drn
where n is the stress-rate sensitivity exponent, we can obtain s˙ L ¼ kDC n e˙ mn
The lower yield stress rly is often expressed by a power law of the gauge-length strain rate e˙ as follows (e.g., [10]): rly ¼ C e˙ m
ð6Þ
More generally, if numbers of band appear (N z 1), Eq. (6) could be rewritten as: s˙ L ¼
kDC n mn e˙ N
ð7Þ
By combining Eqs. (2) and (7), the rate dependency of the Lu¨ders strain is described by the equation: eL ¼
2. Theoretical considerations
ð5Þ
l0 e˙ 1mn kDC n
ð8Þ
Eqs. (7) and (8) indicate that both s˙L and eL increase with e˙ in the form of an exponent function, where the corresponding exponents are mn and 1 mn, respectively.
ð3Þ 3. Experimental observations
where C is a material constant and m is the strain-rate sensitivity exponent. From the pioneering works of Cottrell and Bilby [8] and Johnson and Gilman [9], the yield-point phenomena are explained as a consequence of the unlocking and multiplication of dislocations. The initiation of Lu¨ders band, namely, the sharp yield drop from the upper yield stress to the lower yield stress, could be attributed to the rapid dislocation multiplication [9]. The mechanism of Lu¨ders-band propagation is understood as a process of the dislocation generation within the band and the injection of dislocations to adjacent regions a small distance ahead of the band front. Consequently, it would be reasonable to postulate that the band velocity s˙L is directly related to the dislocation velocity m. For this, Hahn [10] proposed a linear relationship: s˙ L ¼ km under r ¼ rly
3.1. Experimental methods Two types of low carbon steels, one is a 0.07% C steel sheet and the other is a 0.036% C steel wire, which exhibit typical yield-point phenomena, were
ð4Þ
where k is a material constant. Combining Eqs. (3) and (4) with the following equation of stress depend-
Fig. 1. Schematic of specimens: (a) 0.07% C steel sheet and (b) 0.036% C steel round wire.
H.B. Sun et al. / Materials Letters 57 (2003) 4535–4539
4537
captured by a digital camera [12]. The band velocity was calculated from the recorded band length and its propagating time. The tensile loads and crosshead displacements were measured and recorded by the test machine automatically. The displacement in the gauge length was calculated assuming that it was proportional to the crosshead displacement. 3.2. Experimental results Typical stress –strain curves of the sheet and wire specimens under uniaxial tension were illustrated in Fig. 2a and b, respectively, where the crosshead speeds were 0.05 mm/s for the sheet and 1 mm/min
Fig. 2. Typical experimental results of stress vs. strain curves: (a) 0.07% C steel sheet at crosshead speed of 0.05 mm/s and (b) 0.036% C steel wire at crosshead speed of 1 mm/min.
used in this work. For the sheet specimen, as shown in Fig. 1a, it was annealed at 973 K for an hour in a vacuum furnace and then furnace-cooled to room temperature. As a result, an average grain size of 11.5 Am was obtained. A servo-controlled testing machine was employed for the uniaxial tension test with different constant crosshead speeds of 0.0005, 0.001, 0.005, 0.01, 0.025, 0.05 and 0.1 mm/s. For the wire specimen, of which shape is shown in Fig. 1b, it was annealed at 1023 K for an hour and then furnacecooled. Uniaxial tension tests were performed at crosshead speeds of 0.1, 1, 10 and 100 mm/min. For both cases, all the annealed specimens had been polished before they were tested. All the tension tests were conducted at room temperature. The pictures of formation and propagation of Lu¨ders bands were
Fig. 3. Experimental data obtained from 0.07% C steel sheet to demonstrate the validity of Eqs. (7) and (8): (a) gauge-length strain rate vs. Lu¨ders strain and (b) gauge-length strain rate vs. Lu¨dersband velocity.
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H.B. Sun et al. / Materials Letters 57 (2003) 4535–4539
for the wire. In both cases, the yield point phenomena were clearly observed. It should be noted that the number of Lu¨ders bands appearing in experiments was not always the same, even the test condition was fixed. This is because the appearance of Lu¨ders band is quite sensitive to the level of difference in stress concentration existing in the specimen. The present authors [12] discussed this point by means of finite element simulation, as well as from experimental observations, and found that, if there is a certain level of such difference, only a single band will be formed and propagate. Most of the experiments using the sheet specimen are just in this case because it is difficult to keep the same fixing forces of the chucks acting on the two ends of the
Fig. 5. Experimental data of e˙ vs. eL from Sylwestrowicz and Hall [2] and Fujita and Miyazaki [5].
specimen. On the other hand, if the difference of stress concentration is small, double Lu¨ders bands would initiate from both ends of the specimen and then propagate until they met with each other. This is the case of the wire specimen. For the sheet specimen, experimental data of Lu¨ders strain eL and the band velocity s˙L as a function of the gauge-length strain rate e˙ are shown in Fig. 3a and b, respectively. In these figures, all the experimental plots are the case of single Lu¨ders band. Fig. 4a and b shows the results of the wire specimen. In these figures, regression lines based upon Eqs. (7) and (8) are drawn. The good agreements between the experimental and regression results demonstrate the validity of the proposed equations.
Fig. 4. Experimental data obtained from 0.036% C steel wire to demonstrate the validity of Eqs. (7) and (8): (a) gauge-length strain rate vs. Lu¨ders strain and (b) gauge-length strain rate vs. Lu¨dersband velocity.
Fig. 6. Lower yield stress vs. gauge-length strain rate for 0.036% C steel wire.
H.B. Sun et al. / Materials Letters 57 (2003) 4535–4539
4. Discussions To further evaluate the validity of the proposed equations, experimental data obtained by some other researchers [2,5] are also examined here. As shown in Fig. 5, the reported data of gauge-length strain rate vs. Lu¨ders strain are well analyzed by Eq. (8). Now let us consider some applications of the proposed equations. As mentioned above, dislocation movement is a dominant factor controlling the yieldpoint phenomena. Dislocation velocity is strongly dependent on the stress level, as shown in Eq. (5), and the stress-rate sensitivity exponent n is an important parameter for the corresponding viscoplastic constitutive model (e.g., [7]). However, it is not so easy to directly determine the value of n due to the difficulty in experimentally measuring the dislocation velocity for engineering materials. For this problem, the present work may provide a simple solution. From the recorded load and crosshead displacement during a uniaxial tension test, it is easy to calculate the corresponding gauge-length strain rate e˙, stress r and Lu¨ders strain eL. According to Eqs. (3) and (8), with the available data at different crosshead speeds V, the values of m and 1 mn could be obtained, then the value of n could be indirectly determined. An example, for the wire specimen, as shown in Fig. 6,
4539
where the experimental data of the lower yield stress vs. gauge-length strain rate (rly vs. e˙) are plotted, from which we can obtain m = 0.046. From the data of eL vs. e˙, as shown in Fig. 4a, we have 1 mn = 0.197, then n = 17.5 is obtained for this material, which is a reasonable value for the stress-rate sensitivity exponent of mild steel [10].
References [1] E.O. Hall, Yield Point Phenomena in Metals and Alloys, Plenum, New York, 1970. [2] W. Sylwestrowicz, E.O. Hall, Proc. Phys. Soc. (London) 64B (1951) 495. [3] J.C. Fisher, H.G. Rogers, Acta Metall. 4 (1956) 180. [4] H. Conrad, G. Stone, J. Mech. Phys. Solids 12 (1964) 139. [5] H. Fujita, S. Miyazaki, Acta Metall. 26 (1978) 1273. [6] F. Yoshida, H. Murakami, in: S. Tanimura, A.S. Khan (Eds.), Proc. 5th Int. Symp. On Plasticity and Its Current Applications, Gordon and Breach Publishers, 1995, p. 696. [7] F. Yoshida, Int. J. Plast. 16 (2000) 359. [8] A.H. Cottrell, B.A. Bilby, Phys. Soc. (London) 62A (1949) 49. [9] W.G. Johnson, J.J. Gilman, J. Appl. Phys. 30 (1959) 129. [10] G.T. Hahn, Acta Metall. 10 (1962) 727. [11] J.J. Gilman, W.G. Johnson, J. Appl. Phys. 31 (1960) 687. [12] H.B. Sun, F. Yoshida, X. Ma, T. Kamei, M. Ohmori, Mater. Lett. 57 (2003) 3206.
Effect of strain rate on Lu¨ders band propagating velocity and Lu¨ders strain for annealed mild steel under uniaxial tension H.B. Sun a,*, F. Yoshida a, M. Ohmori b, X. Ma a a b
Department of Mechanical System Engineering, Hiroshima University, 1-4-1, Kagamiyama, Higashi-Hiroshima 739-8527, Japan Department of Mechanical Engineering, Hiroshima Kokusai Gakuin University, 6-20-1, Nakano Aki, Hiroshima 739-0321, Japan Received 30 January 2003; accepted 18 April 2003
Abstract For annealed mild steel that exhibits yield-point phenomena under uniaxial tension, physical equations expressing the strain-rate dependencies of the strain and the Lu¨ders-band velocity have been proposed in this work. In these expressions, both the Lu¨ders strain and the Lu¨ders-band velocity increase with gauge-length strain rate in the form of exponent functions. The proposed equations have been verified by our experimental data of uniaxial tension on mild steels, as well as by those reported by some other researchers. The proposed equations may provide a simple method to determine the stress-rate sensitivity exponent, in the equation of stress-dependent dislocation velocity, which is usually difficult to be measured by experiments. D 2003 Elsevier Science B.V. All rights reserved. Keywords: Yield point phenomena; Annealed mild steel; Lu¨ders-band propagation; Lu¨ders strain; Strain rate
1. Introduction In stress – strain curves under uniaxial tension of some metallic materials, typically annealed steels, yield-point phenomena characterized by a sharp yield point and the subsequent yield plateau are observed, where a localized plastic deformation appears as a result of Lu¨ders-band propagation under a constant lower yield stress [1]. To the authors’ knowledge, Sylwestrowicz and Hall [2] reported the first research work on the velocity of Lu¨ders-band front. By measuring the propagation of a single Lu¨ders band in a
specimen of iron, they proposed the following relationship between the constant crosshead velocity V of the testing machine, the resulting Lu¨ders strain eL and the Lu¨ders-band velocity s˙L: s˙ L ¼
ð1Þ
Subsequently, by replacing the crosshead velocity V with gauge-length strain rate e˙, and further considering the case of occurring several bands, Eq. (1) was modified to: s˙ L ¼
* Corresponding author. Tel.: +81-824-24-7541; fax: +81-82422-7193. E-mail address: [email protected] (H.B. Sun).
V eL
V e˙ l0 ¼ N eL N eL
ð2Þ
where l0 is the initial gauge length and N is the number of bands [3,4]. Here, it should be noted that
0167-577X/03/$ - see front matter D 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0167-577X(03)00358-6
4536
H.B. Sun et al. / Materials Letters 57 (2003) 4535–4539
s˙L and eL are strongly strain-rate dependent, i.e., the higher rate e˙ the larger eL [5 –8], and as a result, the Lu¨ders-band velocity s˙ L is also rate-dependent. Although some empirical equations for rate dependency of the Lu¨ders strain had been reported (e.g., Yoshida [7]), physical-based equations for the strainrate dependencies of eL and s˙L have never been proposed. In the present work, upon a sound physical foundation, an explicit relationship between the strain rate and the Lu¨ders strain has been established. Moreover, a similar relationship for the Lu¨ders-band velocity is also derived. The proposed equations have been verified by our experimental data of uniaxial tension on annealed steels, as well as by those reported by some other researchers [3 –5].
ence of dislocation velocity (Gilman and Johnson [11]: m ¼ Drn
where n is the stress-rate sensitivity exponent, we can obtain s˙ L ¼ kDC n e˙ mn
The lower yield stress rly is often expressed by a power law of the gauge-length strain rate e˙ as follows (e.g., [10]): rly ¼ C e˙ m
ð6Þ
More generally, if numbers of band appear (N z 1), Eq. (6) could be rewritten as: s˙ L ¼
kDC n mn e˙ N
ð7Þ
By combining Eqs. (2) and (7), the rate dependency of the Lu¨ders strain is described by the equation: eL ¼
2. Theoretical considerations
ð5Þ
l0 e˙ 1mn kDC n
ð8Þ
Eqs. (7) and (8) indicate that both s˙L and eL increase with e˙ in the form of an exponent function, where the corresponding exponents are mn and 1 mn, respectively.
ð3Þ 3. Experimental observations
where C is a material constant and m is the strain-rate sensitivity exponent. From the pioneering works of Cottrell and Bilby [8] and Johnson and Gilman [9], the yield-point phenomena are explained as a consequence of the unlocking and multiplication of dislocations. The initiation of Lu¨ders band, namely, the sharp yield drop from the upper yield stress to the lower yield stress, could be attributed to the rapid dislocation multiplication [9]. The mechanism of Lu¨ders-band propagation is understood as a process of the dislocation generation within the band and the injection of dislocations to adjacent regions a small distance ahead of the band front. Consequently, it would be reasonable to postulate that the band velocity s˙L is directly related to the dislocation velocity m. For this, Hahn [10] proposed a linear relationship: s˙ L ¼ km under r ¼ rly
3.1. Experimental methods Two types of low carbon steels, one is a 0.07% C steel sheet and the other is a 0.036% C steel wire, which exhibit typical yield-point phenomena, were
ð4Þ
where k is a material constant. Combining Eqs. (3) and (4) with the following equation of stress depend-
Fig. 1. Schematic of specimens: (a) 0.07% C steel sheet and (b) 0.036% C steel round wire.
H.B. Sun et al. / Materials Letters 57 (2003) 4535–4539
4537
captured by a digital camera [12]. The band velocity was calculated from the recorded band length and its propagating time. The tensile loads and crosshead displacements were measured and recorded by the test machine automatically. The displacement in the gauge length was calculated assuming that it was proportional to the crosshead displacement. 3.2. Experimental results Typical stress –strain curves of the sheet and wire specimens under uniaxial tension were illustrated in Fig. 2a and b, respectively, where the crosshead speeds were 0.05 mm/s for the sheet and 1 mm/min
Fig. 2. Typical experimental results of stress vs. strain curves: (a) 0.07% C steel sheet at crosshead speed of 0.05 mm/s and (b) 0.036% C steel wire at crosshead speed of 1 mm/min.
used in this work. For the sheet specimen, as shown in Fig. 1a, it was annealed at 973 K for an hour in a vacuum furnace and then furnace-cooled to room temperature. As a result, an average grain size of 11.5 Am was obtained. A servo-controlled testing machine was employed for the uniaxial tension test with different constant crosshead speeds of 0.0005, 0.001, 0.005, 0.01, 0.025, 0.05 and 0.1 mm/s. For the wire specimen, of which shape is shown in Fig. 1b, it was annealed at 1023 K for an hour and then furnacecooled. Uniaxial tension tests were performed at crosshead speeds of 0.1, 1, 10 and 100 mm/min. For both cases, all the annealed specimens had been polished before they were tested. All the tension tests were conducted at room temperature. The pictures of formation and propagation of Lu¨ders bands were
Fig. 3. Experimental data obtained from 0.07% C steel sheet to demonstrate the validity of Eqs. (7) and (8): (a) gauge-length strain rate vs. Lu¨ders strain and (b) gauge-length strain rate vs. Lu¨dersband velocity.
4538
H.B. Sun et al. / Materials Letters 57 (2003) 4535–4539
for the wire. In both cases, the yield point phenomena were clearly observed. It should be noted that the number of Lu¨ders bands appearing in experiments was not always the same, even the test condition was fixed. This is because the appearance of Lu¨ders band is quite sensitive to the level of difference in stress concentration existing in the specimen. The present authors [12] discussed this point by means of finite element simulation, as well as from experimental observations, and found that, if there is a certain level of such difference, only a single band will be formed and propagate. Most of the experiments using the sheet specimen are just in this case because it is difficult to keep the same fixing forces of the chucks acting on the two ends of the
Fig. 5. Experimental data of e˙ vs. eL from Sylwestrowicz and Hall [2] and Fujita and Miyazaki [5].
specimen. On the other hand, if the difference of stress concentration is small, double Lu¨ders bands would initiate from both ends of the specimen and then propagate until they met with each other. This is the case of the wire specimen. For the sheet specimen, experimental data of Lu¨ders strain eL and the band velocity s˙L as a function of the gauge-length strain rate e˙ are shown in Fig. 3a and b, respectively. In these figures, all the experimental plots are the case of single Lu¨ders band. Fig. 4a and b shows the results of the wire specimen. In these figures, regression lines based upon Eqs. (7) and (8) are drawn. The good agreements between the experimental and regression results demonstrate the validity of the proposed equations.
Fig. 4. Experimental data obtained from 0.036% C steel wire to demonstrate the validity of Eqs. (7) and (8): (a) gauge-length strain rate vs. Lu¨ders strain and (b) gauge-length strain rate vs. Lu¨dersband velocity.
Fig. 6. Lower yield stress vs. gauge-length strain rate for 0.036% C steel wire.
H.B. Sun et al. / Materials Letters 57 (2003) 4535–4539
4. Discussions To further evaluate the validity of the proposed equations, experimental data obtained by some other researchers [2,5] are also examined here. As shown in Fig. 5, the reported data of gauge-length strain rate vs. Lu¨ders strain are well analyzed by Eq. (8). Now let us consider some applications of the proposed equations. As mentioned above, dislocation movement is a dominant factor controlling the yieldpoint phenomena. Dislocation velocity is strongly dependent on the stress level, as shown in Eq. (5), and the stress-rate sensitivity exponent n is an important parameter for the corresponding viscoplastic constitutive model (e.g., [7]). However, it is not so easy to directly determine the value of n due to the difficulty in experimentally measuring the dislocation velocity for engineering materials. For this problem, the present work may provide a simple solution. From the recorded load and crosshead displacement during a uniaxial tension test, it is easy to calculate the corresponding gauge-length strain rate e˙, stress r and Lu¨ders strain eL. According to Eqs. (3) and (8), with the available data at different crosshead speeds V, the values of m and 1 mn could be obtained, then the value of n could be indirectly determined. An example, for the wire specimen, as shown in Fig. 6,
4539
where the experimental data of the lower yield stress vs. gauge-length strain rate (rly vs. e˙) are plotted, from which we can obtain m = 0.046. From the data of eL vs. e˙, as shown in Fig. 4a, we have 1 mn = 0.197, then n = 17.5 is obtained for this material, which is a reasonable value for the stress-rate sensitivity exponent of mild steel [10].
References [1] E.O. Hall, Yield Point Phenomena in Metals and Alloys, Plenum, New York, 1970. [2] W. Sylwestrowicz, E.O. Hall, Proc. Phys. Soc. (London) 64B (1951) 495. [3] J.C. Fisher, H.G. Rogers, Acta Metall. 4 (1956) 180. [4] H. Conrad, G. Stone, J. Mech. Phys. Solids 12 (1964) 139. [5] H. Fujita, S. Miyazaki, Acta Metall. 26 (1978) 1273. [6] F. Yoshida, H. Murakami, in: S. Tanimura, A.S. Khan (Eds.), Proc. 5th Int. Symp. On Plasticity and Its Current Applications, Gordon and Breach Publishers, 1995, p. 696. [7] F. Yoshida, Int. J. Plast. 16 (2000) 359. [8] A.H. Cottrell, B.A. Bilby, Phys. Soc. (London) 62A (1949) 49. [9] W.G. Johnson, J.J. Gilman, J. Appl. Phys. 30 (1959) 129. [10] G.T. Hahn, Acta Metall. 10 (1962) 727. [11] J.J. Gilman, W.G. Johnson, J. Appl. Phys. 31 (1960) 687. [12] H.B. Sun, F. Yoshida, X. Ma, T. Kamei, M. Ohmori, Mater. Lett. 57 (2003) 3206.