- Email: [email protected]
Analysis of the Stress-Strain Curves of a Fe-Mn-Si Shape Memory Alloy
Analysis of the Stress-Strain Curves of a Fe-Mn-Si
Shape Memory Alloy
Jihua Zhang, Shuchuan Chen, Lei Li, and T.Y. Hsu (Xu Zuyao) Department of Materials Science, Shanghai Jiao Tong University, Shanghai 200030, People’s Republic of China The stress-induced martensitic transformation of an Fe-30(wt.%)Mn-6(wt.%)Si shape memory alloy has been studied through the tensile deformation and the recovery processes. The results show that the deformation process of Fe-Mn-Si shape memory alloys can be divided into three stages with different mechanisms. The first stage is the elastic stage, the second one is related to the g → e phase transformation, and the third one can be described as plastic deformation. An appropriate thermomechanical training procedure can markedly increase the recovery rate to close to 100%. © Elsevier Science Inc., 1998
INTRODUCTION
Generally, the SME of an Fe-Mn-Si alloy is evaluated through measurement of the percentage of recoverable strain or the recovery rate, h, of a specimen subjected to various treatments. This method is direct but tedious. In consideration of the mechanical behavior of SMA, several important problems still await attention by the research community. These problems include assessment of the recovery rate, exploration of the relation between deformation and strain recovery, estimation of the amount of the stress-induced e-martensite, and correlation of the critical stress required for initiation of martensite and the stacking fault energy of the alloy. The present paper attempts to reveal the deformation process through analysis of the stress-strain curves of the Fe-Mn-Si alloy under various conditions and to characterize the shape memory properties from a mathematical approach to the tensile deformation curves.
It is well known that the deformation process of common metals includes an elastic stage, a plastic stage associated with the multiplication of dislocations, and fracture. However, the deformation behavior of shape memory alloys (SMAs) shows somewhat different aspects from those just stated. In alloys with a thermoelastic martensitic transformation, such as Ni-Ti and the Cubased alloys, as deformation takes place in the martensitic state, reorientation of the martensite crystals occurs, while deformation at temperatures below Md stress-induced martensite (SIM) is formed. The shape memory effect (SME) of the Fe-Mn-Si–based alloys results from the martensitic transformation fcc (g) ↔ hcp (e) induced by the reverse motion of the Shockley partial dislocations [1–4]. However, the amount of thermally induced martensite in Fe-Mn-Si– based alloys is very limited. A dynamic mechanical study in Fe-Mn-Si [5] indicated that the amount of SIM should be ten times that of thermally induced martensite, although appropriate thermomechanical training will markedly increase the amount of SIM that can be reversely transformed, significantly improving the SME.
EXPERIMENTAL PROCEDURE An Fe-30(wt.%)Mn-6(wt.%)Si alloy was used for this study. Tensile testing was conducted in a Shimadzu AG-1OTA test ma37
MATERIALS CHARACTERIZATION 40:37–41 (1998) © Elsevier Science Inc., 1998 655 Avenue of the Americas, New York, NY 10010
1044-5803/98/$19.00 PII S1044-5803(97)00104-6
J. Zhang et al.
38
chine by using (5.0mm 3 4.98mm 3 0.62 mm) specimens and a strain rate of 2 3 1025/s. The strain was measured by an extensometer with a rear sight distance of 25mm. The strain was fixed at 2.5% for each cycle. The prestrain gauge length, Ln21, the length after tension, Ln, and the length after recovery at a high temperature, Ln11, were measured by a tool microscope. Let n be the number of cycles. The recovery rate, h, representing the shape memory effect, is obtained through the following equation: η = ( Ln – Ln + 1 ) ⁄ ( Ln – Ln – 1 ) .
(1)
Each training cycle of a specimen, after being annealed at 9008C for 360 s and then air cooled, includes 2.5% tensile strain at room temperature followed by heating to 6008C, holding for 60 s, and then cooling to room temperature.
EXPERIMENTAL RESULTS True stress–true strain curves of the annealed specimen (0) and specimens that had undergone from 1 to 5 cycles of training are shown in Fig. 1. These curves indicate that the alloy softened as the number of cycles increased. Figure 2 shows the true stress– true strain curve of an annealed Fe-Mn-Si
FIG. 1. Stress-strain curve of Fe-Mn-Si alloy after annealing (0) and from 1 to 5 cycles of thermomechanical training.
FIG. 2. Stress-stain curve (A) and regression curve (B) of an Fe-Mn-Si alloy after annealing.
alloy under tension (curve A) as well as the stress-strain curve obtained through regression analysis with the Hollomon equation s 5 ken [6] beyond the elastic stage (curve B). From Fig. 2, it can be seen that the regression-fitted curve deviates from the experimental data curve. Figure 3 is a replotting of curve A in Fig. 2 as log stress versus log strain. This reveals that, beyond the elastic stage (o→a), the curve may be divided into two stages: (a→b) and (b→c). Because the Fe-Mn-Si alloy possesses low stacking fault energy, the nucleation of the martensite may occur by means of the stacking fault mechanism [7] and SIM will appear through the g → e transformation under deformation below the Md temperature. Obviously, the deformation behavior during the (a→b) stage in Fig. 3 corresponds to the stress-induced martensitic transformation. The corresponding points a, b, and c in the true stress-strain curve shown in Fig. 2 can be determined from points a, b, and c in Fig. 3. After points a, b, and c are fixed in curve A in Fig. 2, the three stages of the stress-strain curve can be clearly observed. Projecting points a, b, and c to the strain axis in Fig. 2 produces strain values of a9, b9, and c9, from which can be calculated the value of (a9b9)/(a9c9), where (a9b9) 5 b9 2 a9 and (a9c9) 5 c9 2 a9. The
Shape Memory Alloy Stress-Strain Curves
FIG. 3. Log stress–log strain curve plotted from the stress-strain curve (A) in Fig. 2.
value thus obtained, 0.58, is close to the value h 5 0.574 determined from Eq. (1) after substituting the measured data Ln, Ln11, and Ln21. Figure 4 shows the true stress–true strain curve (A) of the specimen that had undergone three cycles of training and the corresponding curve B obtained through regression by using the equation s 5 ken. Obviously, these curves fit each other very well. Replotting curve A in Fig. 4 as log stress ver-
39
sus log strain, it is found that the length of the stage a→b is greatly extended. This means that the formation stage of SIM is prolonged, or the amount of SIM increases, as alluded to in the experimental results from the dynamic mechanical analysis in ref. 5. Further, with the use of the values of a9, b9, and c9 obtained from the log stress– log strain plot of curve A in Fig. 4, the calculated value of the ratio (a9b9)/(a9c9) turns out to be approximately equal to the value of h, 0.964, calculated from Eq. (1). This shows that the amount of (g → e) stress-induced martensite is increased, as is the recovery rate, h. Table 1 shows various mechanical parameters of the annealed specimen and specimens that had undergone from 1 to 5 cycles of training. In Table 1, n2 and n3 represent the hardening exponent of the second and third stages, respectively, K2 and K3 refer to the coefficient K for the second and third stages, respectively, and n* and K*, the hardening exponent n and the coefficient K, respectively, for the combined second and third stages. The critical stress, sc, is the stress required to induce 0.0008 strain (ec), in most cases the point at which the stress begins to deviate from the elastic stage. The sc first drops and then rises with the increase in the number of cycles. The n2 maintains a nearly stable value of about 0.43, but both n3 and n* increase as the number of cycles increases.
DISCUSSION CHARACTERISTICS OF THE STRESS-STRAIN CURVE
FIG. 4. Stress-strain curve (A) and regression curve (B) of an Fe-Mn-Si alloy after three cycles of thermomechanical training.
It has been reported [8, 9] that there exists a “double-n” phenomenon in the plastic stage in some common metals (e.g., Fe, Cu, and Al), which was identified as resulting from different processes, such as single slip and cross slip. As mentioned earlier, the experiments show that there were three stages in the stress-strain curves in a Fe-Mn-Si alloy that had undergone various treatments. These three stages of deformation ought to
J. Zhang et al.
40
Table 1 The Mechanical Parameters of the Three Stages on the Stress-Strain Curve Number of cycle (n)
0 1 2 3 4 5
Hardening exponent
Strength coefficient
Critical stress s 0.0008 (MPa)
n2
n3
n*
K2
K3
K*
Recovery rate h(%)
142.873 122.462 119.322 101.424 103.154 109.648
0.45 0.46 0.41 0.43 0.43 0.45
0.19 0.36 0.39 0.42 0.43 0.46
0.30 0.40 0.42 0.43 0.44 0.45
487 344 242 257 252 263
118 208 225 241 254 257
188 208 243 256 252 262
57.4 84.2 96.4 96.4 94.6 95.7
be induced by different processes. Prior to the plastic deformation, stress-induced martensitic transformation will obviously appear in the Fe-Mn-Si alloy. Consequently, the three stages in the stress-strain curve of the Fe-Mn-Si alloy may be classified: the first stage is the elastic deformation stage; the second stage is the phase transformation g → e; and the third stage is plastic deformation. With the use of the stress-strain curves, it is possible to obtain values for the projected points a9, b9, and c9, as shown in Figs. 2 and 4, and thus calculate the ratio (a9b9)/ (a9c9). Such values are close to the measured recovery rate h of the alloy and thus it is easy to estimate the SME of an alloy that has undergone various treatments by analyzing the stress-strain curve. The second stage of the stress-strain curve expands with the increase in the number of cycles, indicating that there is an increase of the amount of SIM. Thus the stress-strain curve may give information on SIM. The longer this second stage is, the greater the amount of SIM formed. Consequently, it is possible to estimate the recovery rate, h, of a specimen from some tensile true stress–true strain curves of the Fe-Mn-Si SMA or SME after training. The hardening effect induced by the phase transformation is related to the transformation strain, which keeps nearly constant as the number of cycles increases for a certain material. That is, the value of n2 in Table 1 varies only slightly as the value of n increases. The hardening exponent of the third stage is determined by the
behavior of plastic deformation. In alloys with low stacking fault energy, cross slip is difficult to develop; thus an increase in the number of cycles may lower the stacking fault energy. This in turn may increase the formation of e-martensite and the hardening exponent of the third stage; that is, the plastic deformation stage. THE CRITICAL STRESS REQUIRED FOR INDUCING e-MARTENSITE RELATED TO THE STACKING FAULT ENERGY Because the second stage in the stressstrain curve represents the g → e transformation, sc may be recognized as the critical stress required for inducing the formation of e-martensite. From Table 1, it can be seen that the critical stress for SIM is lowered as the number of cycles increases. The equation for the approximate estimation of the stacking fault energy, g, is given by [10]: 2
γ = Ga dρ ⁄ 24πα ,
(2)
in which G is the shear modulus, a the lattice parameter, d the crystal plane space, r the dislocation density, and a the probability of stacking faults. a may be directly proportional to the nucleation rate, N, of the e-martensite, and it can be assumed that N 5 Aa, where A is a constant. As indicated in ref. 7, the critical driving force for the e-martensite transformation may be expressed as ∆G = Bγ + C ,
(3)
where B is a constant, C may refer to the strain energy, and g is the stacking fault en-
Shape Memory Alloy Stress-Strain Curves
41
ergy as indicated in Eq (2). In the formation of stress-induced martensite, DG 5 scec/ 2 5 0.0004sc and N 5 Aa. Substituting these equations and Eq. (2) in Eq. (3), we have σ c = Dρ ⁄ N + C ,
(4)
where D is a constant. Because the recovery temperature is fixed at 6008C and the duration at 60 s, it may be assumed that the dislocation density in the various treatments in Table 1 is approximately constant. Thus Eq. (4) indicates that the critical stress intensity is inversely proportional to the nucleation rate of e-martensite. That is, the higher the e-martensite nucleation rate, the lower will by sc and the larger the amount of e-martensite formed (or the longer the second stage in the true stress–true strain curve will appear). This is consistent with the results given by Reyhani and McCormick [11], who revealed that the critical stress for the formation of SIM decreases and the stacking fault density increases as the number of the thermomechanical training cycles increases. Table 1 shows that, after three cycles of training, the sc increases as the number of cycles, n, increases. This may be due to the fact that the first three cycles of the thermomechanical training enhance the recovery rate, raising it to a saturation value (nearly 100%) and subsequent cycles may lead to incremental increases of residual permanent strain.
CONCLUSIONS 1. The deformation process of Fe-Mn-Si SMAs can be divided into three stages, the first stage of which is the elastic stage of the parent phase. The second stage expresses the formation of stress-induced e-martensite, and the third one defines the plastic deformation of the parent phase and the martensite phase. 2. The stress at which there is a deviation from the elastic deformation of the first stage, sc, is the critical stress for inducing the formation of e-martensite. This stress increases as the dislocation density of the parent phase increases and de-
creases with increases in the nucleation rate or the stacking fault density of e-martensite. 3. The recovery rate of the Fe-Mn-Si alloy studied can be estimated directly by the ratio of (a9b9)/(a9c9), by using values of a9, b9, and c9 corresponding to projected strains from the true stress–true strain curves. The authors would like to express their appreciation to the National Natural Science Foundation of China for its support of this study.
References 1. K. Enami, A. Nagusara, and S. Nenno: Reversible shape memory effect in Fe-base alloys. Scr. Metall. 9:941–948 (1975). 2. A. Sato, E. Chishima, K. Soma, and T. Mori: Shape memory effect in g ↔ e transformation in Fe30Mn-1Si alloy single crystals. Acta Metall. 30: 1177–1183 (1982). 3. A. Sato, K. Soma, E. Chishima, and T. Mori: Shape memory effect and mechanical behavior of an Fe30Mn-1Si alloy single crystal. J. Phys. C4 43:797– 802 (1982). 4. A. Sato, E. Chishima, Y. Yamaji, and T. Mori: Orientation and composition dependencies of shape memory effect in Fe-Mn-Si alloys. Acta Metall. 32: 539–547 (1984). 5. C. Y. Chung, S. Chen, and T. Y. Hsu (Xu Zuyao): Thermomechanical training behavior and its dynamic analysis in an Fe-Mn-Si shape memory alloy. Mater. Char. 37:227–236 (1996). 6. J. H. Hollomon: Tensile deformation. Trans. AIME 162:268–290 (1945). 7. T. Y. Hsu (Xu Zuyao): Thermodynamics of the martensitic transformation fcc (b or g) → hcp (e). Acta Metall. Sinica 16:430–434 (1980) (in Chinese). 8. W. B. Morrison: The effect of grain size on the stress-strain relationship in low-carbon steel. Trans. ASM 59:824–846 (1966). 9. H. Mingzhi, L. Jingxi, and H. Baoping: Stage on strain hardening curve and ultimate uniform strain. Acta Metall. Sinica 9:A291–A298 (1983) (in Chinese). 10. N. I. Noskova and V. A. Pavlov: Stacking faults in nickel solid solutions. Phys. Metals Metall. 14:87–89 (1962). 11. M. M. Reyhani and P. G. McCormick: Mechanical and shape memory behaviour in an Fe-Mn-Si-CrNi alloy. Mater. Sci. Eng. A160:57–61 (1993). Received June 1997; accepted August 1997.
Shape Memory Alloy
Jihua Zhang, Shuchuan Chen, Lei Li, and T.Y. Hsu (Xu Zuyao) Department of Materials Science, Shanghai Jiao Tong University, Shanghai 200030, People’s Republic of China The stress-induced martensitic transformation of an Fe-30(wt.%)Mn-6(wt.%)Si shape memory alloy has been studied through the tensile deformation and the recovery processes. The results show that the deformation process of Fe-Mn-Si shape memory alloys can be divided into three stages with different mechanisms. The first stage is the elastic stage, the second one is related to the g → e phase transformation, and the third one can be described as plastic deformation. An appropriate thermomechanical training procedure can markedly increase the recovery rate to close to 100%. © Elsevier Science Inc., 1998
INTRODUCTION
Generally, the SME of an Fe-Mn-Si alloy is evaluated through measurement of the percentage of recoverable strain or the recovery rate, h, of a specimen subjected to various treatments. This method is direct but tedious. In consideration of the mechanical behavior of SMA, several important problems still await attention by the research community. These problems include assessment of the recovery rate, exploration of the relation between deformation and strain recovery, estimation of the amount of the stress-induced e-martensite, and correlation of the critical stress required for initiation of martensite and the stacking fault energy of the alloy. The present paper attempts to reveal the deformation process through analysis of the stress-strain curves of the Fe-Mn-Si alloy under various conditions and to characterize the shape memory properties from a mathematical approach to the tensile deformation curves.
It is well known that the deformation process of common metals includes an elastic stage, a plastic stage associated with the multiplication of dislocations, and fracture. However, the deformation behavior of shape memory alloys (SMAs) shows somewhat different aspects from those just stated. In alloys with a thermoelastic martensitic transformation, such as Ni-Ti and the Cubased alloys, as deformation takes place in the martensitic state, reorientation of the martensite crystals occurs, while deformation at temperatures below Md stress-induced martensite (SIM) is formed. The shape memory effect (SME) of the Fe-Mn-Si–based alloys results from the martensitic transformation fcc (g) ↔ hcp (e) induced by the reverse motion of the Shockley partial dislocations [1–4]. However, the amount of thermally induced martensite in Fe-Mn-Si– based alloys is very limited. A dynamic mechanical study in Fe-Mn-Si [5] indicated that the amount of SIM should be ten times that of thermally induced martensite, although appropriate thermomechanical training will markedly increase the amount of SIM that can be reversely transformed, significantly improving the SME.
EXPERIMENTAL PROCEDURE An Fe-30(wt.%)Mn-6(wt.%)Si alloy was used for this study. Tensile testing was conducted in a Shimadzu AG-1OTA test ma37
MATERIALS CHARACTERIZATION 40:37–41 (1998) © Elsevier Science Inc., 1998 655 Avenue of the Americas, New York, NY 10010
1044-5803/98/$19.00 PII S1044-5803(97)00104-6
J. Zhang et al.
38
chine by using (5.0mm 3 4.98mm 3 0.62 mm) specimens and a strain rate of 2 3 1025/s. The strain was measured by an extensometer with a rear sight distance of 25mm. The strain was fixed at 2.5% for each cycle. The prestrain gauge length, Ln21, the length after tension, Ln, and the length after recovery at a high temperature, Ln11, were measured by a tool microscope. Let n be the number of cycles. The recovery rate, h, representing the shape memory effect, is obtained through the following equation: η = ( Ln – Ln + 1 ) ⁄ ( Ln – Ln – 1 ) .
(1)
Each training cycle of a specimen, after being annealed at 9008C for 360 s and then air cooled, includes 2.5% tensile strain at room temperature followed by heating to 6008C, holding for 60 s, and then cooling to room temperature.
EXPERIMENTAL RESULTS True stress–true strain curves of the annealed specimen (0) and specimens that had undergone from 1 to 5 cycles of training are shown in Fig. 1. These curves indicate that the alloy softened as the number of cycles increased. Figure 2 shows the true stress– true strain curve of an annealed Fe-Mn-Si
FIG. 1. Stress-strain curve of Fe-Mn-Si alloy after annealing (0) and from 1 to 5 cycles of thermomechanical training.
FIG. 2. Stress-stain curve (A) and regression curve (B) of an Fe-Mn-Si alloy after annealing.
alloy under tension (curve A) as well as the stress-strain curve obtained through regression analysis with the Hollomon equation s 5 ken [6] beyond the elastic stage (curve B). From Fig. 2, it can be seen that the regression-fitted curve deviates from the experimental data curve. Figure 3 is a replotting of curve A in Fig. 2 as log stress versus log strain. This reveals that, beyond the elastic stage (o→a), the curve may be divided into two stages: (a→b) and (b→c). Because the Fe-Mn-Si alloy possesses low stacking fault energy, the nucleation of the martensite may occur by means of the stacking fault mechanism [7] and SIM will appear through the g → e transformation under deformation below the Md temperature. Obviously, the deformation behavior during the (a→b) stage in Fig. 3 corresponds to the stress-induced martensitic transformation. The corresponding points a, b, and c in the true stress-strain curve shown in Fig. 2 can be determined from points a, b, and c in Fig. 3. After points a, b, and c are fixed in curve A in Fig. 2, the three stages of the stress-strain curve can be clearly observed. Projecting points a, b, and c to the strain axis in Fig. 2 produces strain values of a9, b9, and c9, from which can be calculated the value of (a9b9)/(a9c9), where (a9b9) 5 b9 2 a9 and (a9c9) 5 c9 2 a9. The
Shape Memory Alloy Stress-Strain Curves
FIG. 3. Log stress–log strain curve plotted from the stress-strain curve (A) in Fig. 2.
value thus obtained, 0.58, is close to the value h 5 0.574 determined from Eq. (1) after substituting the measured data Ln, Ln11, and Ln21. Figure 4 shows the true stress–true strain curve (A) of the specimen that had undergone three cycles of training and the corresponding curve B obtained through regression by using the equation s 5 ken. Obviously, these curves fit each other very well. Replotting curve A in Fig. 4 as log stress ver-
39
sus log strain, it is found that the length of the stage a→b is greatly extended. This means that the formation stage of SIM is prolonged, or the amount of SIM increases, as alluded to in the experimental results from the dynamic mechanical analysis in ref. 5. Further, with the use of the values of a9, b9, and c9 obtained from the log stress– log strain plot of curve A in Fig. 4, the calculated value of the ratio (a9b9)/(a9c9) turns out to be approximately equal to the value of h, 0.964, calculated from Eq. (1). This shows that the amount of (g → e) stress-induced martensite is increased, as is the recovery rate, h. Table 1 shows various mechanical parameters of the annealed specimen and specimens that had undergone from 1 to 5 cycles of training. In Table 1, n2 and n3 represent the hardening exponent of the second and third stages, respectively, K2 and K3 refer to the coefficient K for the second and third stages, respectively, and n* and K*, the hardening exponent n and the coefficient K, respectively, for the combined second and third stages. The critical stress, sc, is the stress required to induce 0.0008 strain (ec), in most cases the point at which the stress begins to deviate from the elastic stage. The sc first drops and then rises with the increase in the number of cycles. The n2 maintains a nearly stable value of about 0.43, but both n3 and n* increase as the number of cycles increases.
DISCUSSION CHARACTERISTICS OF THE STRESS-STRAIN CURVE
FIG. 4. Stress-strain curve (A) and regression curve (B) of an Fe-Mn-Si alloy after three cycles of thermomechanical training.
It has been reported [8, 9] that there exists a “double-n” phenomenon in the plastic stage in some common metals (e.g., Fe, Cu, and Al), which was identified as resulting from different processes, such as single slip and cross slip. As mentioned earlier, the experiments show that there were three stages in the stress-strain curves in a Fe-Mn-Si alloy that had undergone various treatments. These three stages of deformation ought to
J. Zhang et al.
40
Table 1 The Mechanical Parameters of the Three Stages on the Stress-Strain Curve Number of cycle (n)
0 1 2 3 4 5
Hardening exponent
Strength coefficient
Critical stress s 0.0008 (MPa)
n2
n3
n*
K2
K3
K*
Recovery rate h(%)
142.873 122.462 119.322 101.424 103.154 109.648
0.45 0.46 0.41 0.43 0.43 0.45
0.19 0.36 0.39 0.42 0.43 0.46
0.30 0.40 0.42 0.43 0.44 0.45
487 344 242 257 252 263
118 208 225 241 254 257
188 208 243 256 252 262
57.4 84.2 96.4 96.4 94.6 95.7
be induced by different processes. Prior to the plastic deformation, stress-induced martensitic transformation will obviously appear in the Fe-Mn-Si alloy. Consequently, the three stages in the stress-strain curve of the Fe-Mn-Si alloy may be classified: the first stage is the elastic deformation stage; the second stage is the phase transformation g → e; and the third stage is plastic deformation. With the use of the stress-strain curves, it is possible to obtain values for the projected points a9, b9, and c9, as shown in Figs. 2 and 4, and thus calculate the ratio (a9b9)/ (a9c9). Such values are close to the measured recovery rate h of the alloy and thus it is easy to estimate the SME of an alloy that has undergone various treatments by analyzing the stress-strain curve. The second stage of the stress-strain curve expands with the increase in the number of cycles, indicating that there is an increase of the amount of SIM. Thus the stress-strain curve may give information on SIM. The longer this second stage is, the greater the amount of SIM formed. Consequently, it is possible to estimate the recovery rate, h, of a specimen from some tensile true stress–true strain curves of the Fe-Mn-Si SMA or SME after training. The hardening effect induced by the phase transformation is related to the transformation strain, which keeps nearly constant as the number of cycles increases for a certain material. That is, the value of n2 in Table 1 varies only slightly as the value of n increases. The hardening exponent of the third stage is determined by the
behavior of plastic deformation. In alloys with low stacking fault energy, cross slip is difficult to develop; thus an increase in the number of cycles may lower the stacking fault energy. This in turn may increase the formation of e-martensite and the hardening exponent of the third stage; that is, the plastic deformation stage. THE CRITICAL STRESS REQUIRED FOR INDUCING e-MARTENSITE RELATED TO THE STACKING FAULT ENERGY Because the second stage in the stressstrain curve represents the g → e transformation, sc may be recognized as the critical stress required for inducing the formation of e-martensite. From Table 1, it can be seen that the critical stress for SIM is lowered as the number of cycles increases. The equation for the approximate estimation of the stacking fault energy, g, is given by [10]: 2
γ = Ga dρ ⁄ 24πα ,
(2)
in which G is the shear modulus, a the lattice parameter, d the crystal plane space, r the dislocation density, and a the probability of stacking faults. a may be directly proportional to the nucleation rate, N, of the e-martensite, and it can be assumed that N 5 Aa, where A is a constant. As indicated in ref. 7, the critical driving force for the e-martensite transformation may be expressed as ∆G = Bγ + C ,
(3)
where B is a constant, C may refer to the strain energy, and g is the stacking fault en-
Shape Memory Alloy Stress-Strain Curves
41
ergy as indicated in Eq (2). In the formation of stress-induced martensite, DG 5 scec/ 2 5 0.0004sc and N 5 Aa. Substituting these equations and Eq. (2) in Eq. (3), we have σ c = Dρ ⁄ N + C ,
(4)
where D is a constant. Because the recovery temperature is fixed at 6008C and the duration at 60 s, it may be assumed that the dislocation density in the various treatments in Table 1 is approximately constant. Thus Eq. (4) indicates that the critical stress intensity is inversely proportional to the nucleation rate of e-martensite. That is, the higher the e-martensite nucleation rate, the lower will by sc and the larger the amount of e-martensite formed (or the longer the second stage in the true stress–true strain curve will appear). This is consistent with the results given by Reyhani and McCormick [11], who revealed that the critical stress for the formation of SIM decreases and the stacking fault density increases as the number of the thermomechanical training cycles increases. Table 1 shows that, after three cycles of training, the sc increases as the number of cycles, n, increases. This may be due to the fact that the first three cycles of the thermomechanical training enhance the recovery rate, raising it to a saturation value (nearly 100%) and subsequent cycles may lead to incremental increases of residual permanent strain.
CONCLUSIONS 1. The deformation process of Fe-Mn-Si SMAs can be divided into three stages, the first stage of which is the elastic stage of the parent phase. The second stage expresses the formation of stress-induced e-martensite, and the third one defines the plastic deformation of the parent phase and the martensite phase. 2. The stress at which there is a deviation from the elastic deformation of the first stage, sc, is the critical stress for inducing the formation of e-martensite. This stress increases as the dislocation density of the parent phase increases and de-
creases with increases in the nucleation rate or the stacking fault density of e-martensite. 3. The recovery rate of the Fe-Mn-Si alloy studied can be estimated directly by the ratio of (a9b9)/(a9c9), by using values of a9, b9, and c9 corresponding to projected strains from the true stress–true strain curves. The authors would like to express their appreciation to the National Natural Science Foundation of China for its support of this study.
References 1. K. Enami, A. Nagusara, and S. Nenno: Reversible shape memory effect in Fe-base alloys. Scr. Metall. 9:941–948 (1975). 2. A. Sato, E. Chishima, K. Soma, and T. Mori: Shape memory effect in g ↔ e transformation in Fe30Mn-1Si alloy single crystals. Acta Metall. 30: 1177–1183 (1982). 3. A. Sato, K. Soma, E. Chishima, and T. Mori: Shape memory effect and mechanical behavior of an Fe30Mn-1Si alloy single crystal. J. Phys. C4 43:797– 802 (1982). 4. A. Sato, E. Chishima, Y. Yamaji, and T. Mori: Orientation and composition dependencies of shape memory effect in Fe-Mn-Si alloys. Acta Metall. 32: 539–547 (1984). 5. C. Y. Chung, S. Chen, and T. Y. Hsu (Xu Zuyao): Thermomechanical training behavior and its dynamic analysis in an Fe-Mn-Si shape memory alloy. Mater. Char. 37:227–236 (1996). 6. J. H. Hollomon: Tensile deformation. Trans. AIME 162:268–290 (1945). 7. T. Y. Hsu (Xu Zuyao): Thermodynamics of the martensitic transformation fcc (b or g) → hcp (e). Acta Metall. Sinica 16:430–434 (1980) (in Chinese). 8. W. B. Morrison: The effect of grain size on the stress-strain relationship in low-carbon steel. Trans. ASM 59:824–846 (1966). 9. H. Mingzhi, L. Jingxi, and H. Baoping: Stage on strain hardening curve and ultimate uniform strain. Acta Metall. Sinica 9:A291–A298 (1983) (in Chinese). 10. N. I. Noskova and V. A. Pavlov: Stacking faults in nickel solid solutions. Phys. Metals Metall. 14:87–89 (1962). 11. M. M. Reyhani and P. G. McCormick: Mechanical and shape memory behaviour in an Fe-Mn-Si-CrNi alloy. Mater. Sci. Eng. A160:57–61 (1993). Received June 1997; accepted August 1997.