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Stress–strain–temperature hysteresis and martensite start line in an Fe-based shape memory alloy
Materials Science and Engineering A238 (1997) 367 – 376
Stress – strain–temperature hysteresis and martensite start line in an Fe-based shape memory alloy Fumihito Nishimura, Noriko Watanabe, Kikuaki Tanaka * Department of Aerospace Engineering, Tokyo Metropolitan Institute of Technology, Asahigaoka 6 -6, J-191 Hino/Tokyo, Japan Received 14 April 1997
Abstract The uniaxial stress–strain–temperature hysteretic behavior in an Fe – 9%Cr – 5%Ni – 14%Mn – 6%Si polycrystalline shape memory alloy is investigated under successive tensile – compressive mechanical loading and thermal loading, especially in relation to the transformation lines; the martensite start line and the austenite start/finish lines on the stress – temperature plane. The form of the thermomechanical hysteresis loops, the strain recovery during heating after forward and reverse transformations induced by the mechanical loading and the recovery stress under strain-constrained heating are the points to be studied. The dependence of the martensite start stress on the extent of prior martensitic transformations is determined from the direct strain-recovery measurement during heating, which agrees well with the estimation by means of the reverse transformation lines coupled with the assumption of the ‘isotropic’ hardening in transformation. © 1997 Elsevier Science S.A. Keywords: Fe-based shape memory alloy; Hysteresis loop; Recovery stress; Thermomechanical loading; Transformation lines; Uniaxial tension and compression
1. Introduction The martensitic transformation condition, which predicts the start of the transformation in the materials under thermomechanical loading, has been studied for years in metallurgy. The importance of the transformation driving force, which depends on both the temperature and the applied stress state, has been clarified [1,2], also from the continuum mechanical point of view [3 –7]. In recent years, this condition has attracted attention when investigating the thermomechanical behavior of such advanced materials as shape memory alloys, zirconia ceramics and TRIP steels [3 –9]. The trend has been strongly motivated by rapid technological development which has realized the manufacturing processes of materials with specified microscopic structures, not only in composite materials but also in conventional metallic alloys [10]. The thermomechanical theories of transforming materials are currently * Corresponding author: Tel.: +81 425835111; 425835119; e-mail: [email protected]
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0921-5093/97/$17.00 © 1997 Elsevier Science S.A. All rights reserved. PII S 0 9 2 1 - 5 0 9 3 ( 9 7 ) 0 0 4 6 1 - 9
being developed in order to describe the material behavior by taking the microstructures of the material into account and to utilize it in the material design with some requirements of the strength, the toughness, and so on. The idea of the transformation condition is always considered in relation to the yield condition in plasticity, which is employed as a potential function in the theory to derive the evolution equations of the internal variables introduced. In fact, Tanaka et al. [8,9] have proposed a unified theory of TRIP, which describes the plastic deformation of the materials in the process of martensitic transformation, by employing the transformation condition and the yield condition simultaneously. Taking such a theoretical and technological background into account, the present authors have been investigating the thermomechanical hysteretic behavior in an Fe–9%Cr–5%Ni–14%Mn–6%Si polycrystalline shape memory alloy, always in relation to the transformation condition [11–17]. The phase diagram of the present alloy under uniaxial loading is represented by three straight lines on the stress–temperature plane; the
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martensite start line and the austenite start and finish lines with the same slope. The reverse transformation zone, bounded by the austenite start/finish lines, increases its width depending on the extent of prior martensitic transformation, whereas their slope stays unchanged. The transformation lines after compressive loading constitute a mirror image of the lines after tensile loading with respect to the stress-free axis, but it is worth noting that the two sets of lines are not symmetric. This fact is especially important when discussing the thermomechanical hysteresis during tensile– compressive loading, monotonic and cyclic [16,17]. During mechanical loading, the transformation start state is determined as a point on the stress – temperature plane at which the thermomechanical load point and the transformation start line meet. The subsequent thermomechanical behavior is also strongly influenced by the shift of the transformation lines, depending on the prior transformation history. The progress of the transformations between the phases, the parent phase and the martensite phases induced in tension and/or compression, can be well explained both from the hysteresis loops and the transformation diagram. In this paper, following the preceding study [16], the stress–strain–temperature hysteretic behavior in the same alloy material is investigated in detail under the uniaxial tensile–compressive and thermal loading, in special relation to the transformation lines. The form of the hysteresis loop and the characteristics of the strain recovery during heating are explained by the shift of the reverse transformation lines during loading. The evolution of the recovery stress is also investigated in the process of strain-constrained heating. The shift of the martensite start line is estimated by both the data of the austenite start temperature and the direct measurement of the martensite start stress from the dilatation curves during heating.
3. Transformation lines When the alloy specimen is loaded mechanically at a test temperature Th up to a maximum stress s + max in tension or down to a maximum stress s − max in compres− sion, a martensite start stress s + Ms or s Ms is determined on the tensile or the compressive branch, respectively, as illustrated in Fig. 1. Here, and henceforth, the superscripts ‘+ ’ and ‘− ’ represent that the quantity corresponds to the tensile loading and the compressive loading, respectively. It should be noted that the martensite M + means here the variants formed when the specimen is loaded in tension, and the martensite M − the variants induced when the specimen is loaded in compression. The point to be emphasized in the present study is that, during reverse transformation, in other words, when the specimen is heated, the martensite M + gives rise to a macroscopic compressive deformation in the specimen, and the martensite M − a macroscopic tensile deformation. − Both stresses s + Ms and s Ms strongly depend on the prior history of transformations that the specimen has experienced so far [11–16]. If the martensite phase is induced in the specimen by mechanical loading, a reverse transformation zone can be determined by measuring an austenite start temperature and an austenite finish temperature along a heating path up to a maximum temperature Tmax under a constant applied stress sh. Fig. 1 plots schematically these temperatures as T + As and T + Af, respectively. The stresses and the temperatures thus determined constitute the transformation lines: the martensite start line (Ms + -line), the austenite start line (As + -line) and the austenite finish line (Af + -line). The As + - and Af + -lines bound the reverse transformation zone. The same is true for the transformation lines corresponding to compressive loading, as illustrated in Fig. 1.
2. Alloy specimen The test specimen, 6 mm in diameter with 20 mm gauge length, of an Fe – 9%Cr – 5%Ni – 14%Mn–6%Si polycrystalline shape memory alloy is the same as that used in a previous study of the same authors [16]. The mechanical properties of the alloy are reported there together with the heat treatment and the experimental procedures. The specimens are, prior to the tests, subjected to a thermomechanical training, details of which is explained in Refs. [16,17], to establish a stable thermomechanical response. When the specimen is heated up above the austenite finish temperature, the alloy returns to the original state metallurgically and mechanically even after some mechanical loading in tension and compression.
Fig. 1. Thermomechanical loading path and transformation lines.
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Fig. 2. Thermomechanical hysteresis loops under tension – compression at RT and heating.
These transformation lines shift on the stress–temperature plane depending on the prior transformation history, which was the theme of the study by Nishimura et al. [16]. The points to be noted for the present study are summarized as follows: the austenite start line shifts to the lower temperature side when the amount of − pre-stressing, measured here by s + max or s max, becomes larger, but no change in slope of the line is observed. Similarly, the austenite finish line shifts to the higher temperature side without change in slope when the amount of pre-stressing becomes larger. Both start line and finish line have the same slope. The As + - and Af + -lines are extended to the negative stress side without changing their slope. Similarly, the As − - and Af − lines are extended to the positive stress side without changing their slope, and no change is observed in slope on both the tensile and compressive stress sides. The drift of the start line is much more sensitive to s + max or s − max than is the finish line. The reverse transforma− tion zone, therefore, becomes wider with s + max or s max, meaning that at Th =RT (room temperature, 303 K) + the austenite finish stress s + Af decreases with s max, and − finally becomes lower than the P M transformation + start stress s − Ms when s max is large enough. Similar behavior was observed for the transformation lines corresponding to compressive loading: at Th = RT the − austenite finish stress s − Af increases with s max, and + finally becomes larger than the PM transformation − start stress s + Ms when s max is small enough.
4. Thermomechanical hysteresis Fig. 2 plots the thermomechanical hysteresis loops during successive tensile loading/unloading, s + max = 150, 200 and 250 MPa at Th =RT, and subsequent compressive loading/unloading, s − max = −300 MPa at Th = RT,
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followed by heating/cooling, Tmax = 873 K under sh =0 MPa. The first isothermal mechanical loop is often called the ferroelastic loop. It should be noted that, after being loaded up to s + max in tension which produces the stress-induced M + martensite in the specimen, the alloy has, as explained in the previous section, the wider reverse M + P transformation zone depending on s + max, at the start of the subsequent compressive loading. In the case of s + max = 150 MPa, the austenite finish stress s + Af is larger than the martensite start stress + in compression s − P Ms, meaning that the reverse M − transformation zone and the Ms -line are not overlapped at RT. The elastic compressive response is, therefore, expected during loading from s + Af down to s− Ms, though the data reveals it in a very small part. This alloy response diminishes as s + max becomes large + since the s + Af stress shifts down at RT with s max, and + finally in the case of s max = 250 MPa, the P M − transformation starts nearly at the same moment when the reverse M + P transformation finishes. The anomalous performance of the alloy indicated by an arrow in the figure is attributed to this metallurgical process. Since the reverse M + P transformation has already finished at s − max = − 300 MPa for all three loading cases, the strain amplitude in the compressive branch is almost the same for all cases. So is the response during the subsequent thermal run. It is usually observed in the Fe-based alloys that the reorientation of the martensite variants, M + M − and M − M + processes in the present notation, is not observed [18]. The case is assumed to be true in this alloy also. At higher test temperature, as shown in Fig. 6 in Nishimura et al. [16], the reverse transformation zone for the M + martensite is fully apart from the PM − transformation zone in compression. Therefore, the reverse transformation finishes far before the start of the P M − transformation in compression. The elastic unloading of the specimen composed only of the P phase, is expected during a part of the compressive − loading, from s + Af down to s Ms. Fig. 3 reveals that the response is actually observed at Th = 343 K at the part arrowed in the ferroelastic hysteresis loops. The subsequent performance of the alloy is the same, being independent of s + max. Fig. 4 illustrates the alloy response when the specimen is firstly loaded in compression to produce the M − martensite and then reloaded in tension. Judging from the transformation diagram determined in the previous study, Fig. 7 in Nishimura et al. [16], the reverse M − P transformation and the P M + transformation progress simultaneously during the tensile loading. The smooth response of the alloy is due to the fact that, contrary to the cases in Figs. 2 and 3, the reverse transformation zone and the M + transformation zone almost overlap at RT. Since the reverse M − P trans-
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Fig. 3. Thermomechanical hysteresis loops under tension – compression at 343 K.
formation finishes for all s − max cases before the stress reaches the maximum value s + max =300 MPa, almost similar hysteresis loops are obtained in the subsequent unloading and heating/cooling processes. The alloy performs differently in the case of smaller s − max, which will be discussed later in Fig. 9.
5. Strain recovery during heating The alloy response illustrated in Figs. 2 and 4 suggest that during the mechanical loading process the alloy is composed of the M − martensite, the M + martensite and the P phase, changing their fractions with the progress of loading. The situation is clearly proved by examining the strain recovery during the subsequent heating, shown in Fig. 5. The specimen, compressed down to s − max = −280 MPa, is then
Fig. 4. Thermomechanical hysteresis loops under compression – tension at RT and heating.
Fig. 5. Strain recovery during heating after compressive –tensile loading at RT.
loaded in tension up to the different values of s + max = 180, 220, 230 and 300 MPa. After unloaded, the specimen is subjected to the thermal run of heating/cooling. The M − martensite produced during compressive loading is the cause of the tensile deformation of the specimen in the subsequent heating process. As s + max becomes larger, the amount of the M − martensite decreases due to the progress of the stress-induced reverse M − P transformation in tension, resulting in the smaller amount of tensile deformation during heating. − In case s + martensite max is sufficiently large, the M disappears in the specimen while mechanical loading in tension, whereas both the M + martensite and the P phase still remain. The M + martensite increases with s+ max, inducing the larger amount of compressive deformation during heating. The experiment shows that no residual strain is observed after the mechanical run when s + max =220 MPa. The subsequent thermal run clearly shows that the internal structure of the alloy at the start of heating is a mixture of the M + martensite, the M − martensite and most likely the P phase, too. Actually, the dilatation curve exhibits a compensation of the compressive deformation due to the M + martensite and the tensile deformation due to the M − martensite. The M + martensite first transforms back to the P phase during heating. The M − martensite then transforms back to the P phase at a higher temperature range, inducing a tensile deformation. These two deformations in the opposite direction complete during heating and, finally, when all the M + and M − martensites transform back to the P phase, the dilatation curve converges to the thermal expansion linear line. A similar alloy performance is observed when the specimen is first loaded/unloaded in tension, up to s+ max = 250 MPa at Th = RT, and then loaded/un-
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Fig. 6. Strain recovery during heating after tensile–compressive loading at RT.
loaded in compression, down to s − max = −220, −240 and −260 MPa at Th =RT, before heating/cooling (cf. Fig. 6).
6. Recovery stress If the deformation of the specimen is constrained during heating in the tests shown in Figs. 5 and 6, a stress is induced in the specimen due to the transformation strain produced in the process of reverse transformation. The stress, called the recovery stress [19,20], is nothing other than the driving force of the shape memory devices. In order to simplify the situation, following cases are investigated: the specimen is heated with the strain constrained after being loaded mechanically in tension at Th =RT up to s + max. Fig. 7 shows the increase in the recovery stress s rec during heating for each case of s + max. The size of the
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Fig. 8. Maximum recovery stress depending on prior martensitic transformation in tension.
reverse transformation zone depends on the prior history of transformation, on s + max in the present context (cf. Section 2), as indicated in the figure. For the sufficiently small values of s + max, the austenite start temperature As + under no applied stress is higher than RT, meaning that at the early stage of heating a negative stress is induced, due not to the reverse transformation but simply to the constrained thermal expansion. This type of alloy behavior is not observed in the wire specimen [21]. The transformation contraction then follows from As + on, resulting in a sharp increase in the recovery stress. The experiments reveal that this alloy performance diminishes as s + max approaches to 260 MPa, under which the reverse transformation starts just at the moment of start heating. When s + max is higher than this value, in the case of s + max =300 MPa, for example, the reverse transformation starts already at the last stage of the mechanical run. This is the reason why the maximum value of the recovery stress + s rec max saturates as s max becomes larger, which will be plotted later in Fig. 8. Fig. 7 also reveals very clearly that the recovery stress is due to the reverse M + P transformation since its increase finishes at the austenite finish line, the Af + -line. From this point on, the recovery stress decreases due to the constrained thermal expansion of the P phase. The maximum value of the recovery stress s rec max depends on the prior martensitic transformation, on s+ max actually, as illustrated in Fig. 8.
7. Martensite start line
Fig. 7. Recovery stress and transformation zone.
The martensite start lines, the Ms + -line and the Ms − -line in Fig. 1, are determined with the specimens which have no prior history of transformations [12,13,16]. The lines correspond to the initial yield surface in plasticity. As in plasticity, the martensite
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start lines shift during subsequent loading, exhibiting the subsequent martensite start line. Fig. 19 in the later discussion shows the increase in the subsequent martensite start stress in the process of uniaxial loading in tension; in other words, the alloy exhibits the ‘hardening’ during transformation. In order to establish a rational concept of the transformation surface, which is a generalized figure of the transformation line in the multiaxial stress– temperature space, one has firstly to know in the uniaxial case the effect of the prior martensitic transformation in the opposite direction of loading on the martensite start stress. This is the theme to be investigated in this section. Fig. 9 shows the ferroelastic hysteresis loops under the successive compressive loading/unloading, the tensile loading/unloading and the heating/cooling. The mechanical run is carried out at Th =RT, and the thermal run under sh =0 MPa. The maximum stress in tension is always kept constant to s + max =300 MPa, whereas the maximum stress in compression s − max changes loop by loop. On loading in tension, both the reverse M − P transformation and the P M + transformation progress simultaneously, and the M − martensite fully transforms back to the P phase by the end of the tensile loading. Since the reverse M + P transformation completes during heating, no residual strain is observed after the whole thermomechanical run. The maximum strain in tension o + max, defined in Fig. 10, is measured in Fig. 9, and is plotted in Fig. 10. It should be noted that the strain o + max is a good measure of the extent of the PM + transformation since the M − martensites fully transform back to the P phase during the tensile loading. This is actually the case, as proved in Fig. 5 for sufficiently large s + max. One could conclude for now that the M + martensite start stress − s+ transforMs increases when the extent of the P M mation becomes large, and equivalently when s − max
Fig. 9. Thermomechanical hysteresis loops under compression – tension at RT and heating.
Fig. 10. Maximum strain in tension and maximum stress in compression.
decreases. In deriving this estimation, one should take into account the fact that the decrease in o + max, in other words, the decrease in the extent of the martensitic + + transformation s + max –s Ms results in the increase in s Ms when s + max is kept constant. The 0.05% proof stresses during tensile loading s0.05%, indicated by the circles in Fig. 9, exhibit in Fig. 11 a clear linear change versus the transformation strain in compression o − T defined in the figure, even when the test temperature is different. However, no definite reason is yet given as to whether this stress is the reverse M − P transformation start stress s − As or the P M + transformation start stress s + Ms in question, although in the later discussion it will be proved to be the former stress. The austenite start temperature As + of the M + martensite is measured, as shown in Fig. 12 by the circles, after successive compressive loading down to + s− max = − 280 MPa and tensile loading up to s max. The + temperature As must depend on the progress of both
Fig. 11. ‘Yield stress’ in tension after compressive loading at RT.
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Fig. 12. Austenite start temperature after compressive–tensile loading at RT.
the PM − transformation in compression and the P M + transformation in tension, directly speaking, + on both s − max and s max. Actually, the result summarized in Fig. 13 reveals that As + depends on s − max. This eventually means that the M + martensite start stress s+ Ms also depends on the extent of loading in compression, measured in the present context by s − max. In order to prove this statement directly, Fig. 13 is replotted, in − the case of s + max =200 MPa, versus s max in Fig. 14. One gets an estimation for the slope of the lines in the figure, DAs + = − 0.11 K MPa − 1 Ds − max
(1)
On the other hand, the data on the reverse transformation temperatures by Nishimura et al. [16], displayed again in Fig. 15, reveal the following estimation:
Fig. 13. Austenite start temperature depending on prior transformations.
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Fig. 14. Austenite start temperature depending on prior martensitic transformation in compression.
DAs DAs + = = − 0.55 K MPa − 1 + DST D(s max − s + ) Ms
(2)
where the transformation stress ST represents the extent of the martensitic transformation and is defined in tension by + ST+ = s + max − s Ms
(3)
and in compression by 1 − ST− = (s − Ms − s max). k
(4)
The material constant k in the equation, which is determined by the slopes of the martensitic transformation lines, denotes the difference in size of the martensitic transformation zones in tension and compression. Now Eq. (2) is combined with Eq. (1) between the As + and s − max, and one reaches the final formula stating that the P M + transformation start stress depends on the extent of the P M − transformation:
Fig. 15. Reverse transformation zone after martensitic transformation in tension.
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Fig. 18. Transformation diagram after compressive loading at RT. Fig. 16. Strain recovery during heating after compressive – tensile loading at 288 K.
Ds + Ms =0.20 MPa MPa − 1 Ds − max
(5)
The estimation in Eq. (5) can directly be approved in Figs. 16 and 17. The thermomechanical ferroelastic loops in the figure correspond to the case in Fig. 5, in which the specimen is composed, at the start of heating, of the M + martensite, the M − martensite and the P − phase. When changing the value of s + max, while s max = − 280 MPa is kept constant, the dilatation curve exhibits complicated characteristics (cf. also Figs. 5 and 6), depending on the amount of the phases the specimen has at the start of heating. One can determine a s + max value above which the dilatation curve falls at the early stage of heating on the shorter strain side of the thermal expansion line, indicating that a transformation contraction takes place. This s + max value is nothing other than the M + martensite start stress s + Ms, at which the M + martensite starts being produced. Fig. 17, summarizing the measured s + Ms values, presents an estimation
Fig. 17. Martensite start stress of M + martensite depending on prior martensitic transformation in compression.
Ds + Ms = − 0.27 MPa MPa − 1 Ds − max
(6)
which has a good agreement with the value determined above in Eq. (5). The results in Figs. 11, 15 and 17 are plotted together in Fig. 18. The shadow part, which is a replot of Fig. 15 − and is bounded by the s − Af- and s As-lines, represents the region in which the stress-induced reverse M − P transformation progresses along a constant s − max path. The solid circles, determined in Figs. 9 and 11 as the ‘yield’ stress s0.05%, now turn out to represent the reverse M − P transformation start stress s − As. The fact that the stresses s0.05% almost coincide with the s − As-line in the figure, determined from the reverse M − P transformation start temperature, is worth emphasizing in the sense that almost similar transformation lines are obtained in both the isothermal mechanical loading tests and the thermal loading tests under constant stress. In the test of s − max = − 250 MPa, for example, while loading mechanically, the reverse M − P transformation starts at about 65 MPa (s − As), and before finishing the reverse transformation, the PM + transformation starts at about 170 MPa (s + Ms). From this stress on, both the reverse M − P transformation and the PM + transformation progress simultaneously until the reverse M − P transformation finishes at about 203 MPa + (s − transformation Af). From then on, only the P M takes place. It should be noticed that the actual austenite finish stress may change from the value in the figure because the s − Af stresses in the figure are determined with the alloy specimens which contain only the M − martensite, whereas in the present case the M + martensite is formed and increases during loading from s + Ms = 170 MPa on. As the data (hollow circles) in Fig. 18 show, and as has already been explained in Eqs. (5) and (6), s + Ms clearly depends on s − max, in other words, on the prior history of the PM − transformation in compression.
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Let us trace in the figure an experiment of isothermal compression at Th =RT. The s − max value decreases, indicating the progress of compression, and when the compression stops at a s − max value and the unloading and subsequent tensile loading follow, the generic point in the figure moves up along the constant s − max line as demonstrated above. Now till the s − value reaches max s− = − 170 MPa during compression, the martensite Ms start stress s¯s¯do3(s + Ms) stays constant at an initial martensite start stress s + Ms-ini =153 MPa since no transformation yet progresses in compression. From − 170 MPa down in compression the alloy experiences the P M − transformation, which, due to the ‘hardening effect’ explained below, raises the s + Ms value. The experimental result shows that the dependence of s + Ms on the prior transformation in compression, on s − max, is almost linear. It is worth noting that the martensite start stress s + Ms measured here is different from the martensite start + stress SMs shown in the figure, which the alloy has just in the process of compressive PM − transformation, since the alloy state cannot come from the compressed + state at the SMs stress on the positive stress side without experiencing the reverse M − P transformation. This + reverse M − P transformation changes the SMs value + to the s Ms value actually measured in Fig. 18. The + martensite start stress SMs depends on the prior transformation history in compression, on s − max shortly. Let us study in detail whether this dependence can be understood by the concept of ‘isotropic hardening’ in plasticity. Fig. 19 shows an isothermal stress – strain curve obtained during a repeated loading/unloading in tension at Th =RT with increasing maximum stress, representing the ‘hardening’ of the martensite start stress. Assuming the ‘isotropic hardening’ of the martensite start + stress, SMs increases in the process of transformation in compression and is given by
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Table 1 Materials parameters a (K MPa−1)
b (K MPa−1)
k
−1 c− ) A (MPa K
0.55
0.35
1.3
−2.4
1 − + − + SMs = s+ (s Ms − s − Ms-ini + ST = s Ms-ini + max) k
(7)
where s + Ms-ini stands for the initial martensite start stress (see Fig. 19), corresponding to the initial yield stress in plasticity. + reWhen the reverse transformation finishes, SMs . It is, therefore, reasonturns to the initial value s + Ms-ini able to assume that in case the reverse transformation + decreases depending linearly on stops on the way, SMs the extent of the reverse transformation: − + s− SMs − s+ Af − s As Ms-ini = + + − s Ms − s As SMs − s + Ms
(8)
which leads to an expression for s + Ms; s+ Ms =
+ − − + + SMs (s − Af − s As)+ s As(SMs − s Ms-ini) − + + (s − Af − s As)+ (SMs − s Ms-ini)
(9)
As explained in Section 2, the reverse transformation zone becomes broader with the progress of martensitic transformation, which can be formulated as [16]: − − s− As = c A (TAi − aST )
and
− − s− Af = c A (TAi +bST ) (10)
Here the material parameter c − A denotes the slope of the Af − - or As − -line, and a and b are the constant material parameters. Eqs. (9) and (10) combine to yield an estimation − bc − Ds + Ms A = − − Ds max k(ac A + bc − A − 1)
(11)
Substitution of the values of all material constants, shown in Table 1, into Eq. (11) gives Ds + Ms = − 0.20 MPa MPa − 1 Ds − max
(12)
which, representing the linear line in Fig. 18, agrees well with the experimental result denoted by the hollow circles. The estimation also coincides surprisingly well with Eq. (5) which is derived from the experimental data of the austenite start temperature. The discussion reveals that the postulate of ‘isotropic hardening’ of the martensite start stress is acceptable.
8. Concluding remarks
Fig. 19. Stress – strain curve under repeated loading–unloading at RT with increasing maximum stress.
The uniaxial stress–strain–temperature hysteresis is investigated in an Fe–9%Cr–5%Ni–14%Mn–6%Si polycrystalline shape memory alloy under tensile–com-
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pressive and thermal loading. The form of the hysteresis loops is shown to be well characterized by means of the transformation lines, the martensite start line and the austenite start/finish lines, which represent the progress of the transformations between the phases; the parent phase, the martensite phases induced by tensile loading or compressive loading. The lines shift on the stress – temperature plane, depending on the extent of prior transformations. The martensite start stress depends on the extent of prior martensitic transformation during mechanical loading in the opposite direction. The dependence is determined both from the direct strain-recovery measurement during heating and from the data on the reverse transformation lines coupled with the assumption of the ‘isotropic’ hardening in transformation. Both results agree well, suggesting that the concept of ‘isotropic’ hardening is acceptable. When the Fe-based shape memory alloys are mechanically loaded, the thin o martensite plates grow from the grain boundaries in the parent phase [22–24]. The local internal stresses, the back stresses, induced at the tip of the plates play an essential role in the strain recovery process during subsequent heating [25,26]. The extent and direction of this back stress, at least its averaged value over the active variants, can be evaluated macroscopically if the strain recovery is measured in the process of heating under the hold stress sh. This should be a theme to be investigated in the next paper.
Acknowledgements The authors express their thanks to Professor H. Inagaki (Shonan Institute of Technology) for his constructive discussion. Part of this work was financially supported by the Special Research Fund of the Tokyo Metropolitan Government as well as by a Grant-inAid for Scientific Research (No. 08650117) from the Ministry of Education, Science and Culture, Japan. The supply of the alloy specimens by the Steel Research Center of the NKK Corporation is gratefully acknowledged.
.
References [1] J. Patel, M. Cohen, Acta Metall. 1 (1953) 531. [2] L. Kaufman, M. Hillert, in: G.B. Olson, W.S. Owen (Eds.), Martensite, ASM International, 1992, p. 41. [3] B. Raniecki, C. Lexcellent, K. Tanaka, Arch. Mech. 44 (1992) 261. [4] Q.P. Sun, K.C. Hwang, J. Mech. Phys. Solids 41 (1993) 1. [5] E. Patoor, A. Eberhardt, M. Berveiller, in: L.C. Brinson, B. Moran (Eds.), Mechanics of Phase Transformations and Shape Memory Alloys, AMD-vol. 189/PVP-vol. 292, ASME, 1994, p. 23. [6] S. Leclercq, C. Lexcellent, J. Mech. Phys. Solids 44 (1996) 953. [7] V.I. Levitas, Int. J. Eng. Sci. 33 (1995) 921, 947. [8] K. Tanaka, E.R. Oberaigner, F.D. Fischer, in: L.C. Brinson, B. Moran (Eds.), Mechanics of Phase Transformations and Shape Memory Alloys, AMD-vol. 189/PVP-vol. 292, ASME, 1994, p. 151. [9] F.D. Fischer, K. Tanaka, in: T. Abe, T. Tsuta (Eds.), Advances in Engineering Plasticity and its Applications, Pergamon Press, Amsterdam, 1996, p. 185. [10] B.H. Rabin, I. Shiota (Eds.), Functionally Gradient Materials, Mater. Res. Soc. Bull. XX (1995). [11] K. Tanaka, T. Hayashi, Y. Aida, H. Tobushi, J. Intell. Mater. Syst. Struct. 4 (1993) 567. [12] K. Tanaka, T. Hayashi, F. Nishimura, H. Tobushi, J. Mater. Eng. Performance 3 (1994) 135. [13] K. Tanaka, T. Hayashi, F.D. Fischer, B. Buchmayr, Z. Metallkd. 85 (1994) 122. [14] K. Tanaka, F. Nishimura, H. Tobushi, Z. Metallkd. 86 (1995) 211. [15] K. Tanaka, F. Nishimura, H. Tobushi, E.R. Oberaigner, F.D. Fischer, in: Proc. ICOMAT 95, J. Phys. IV (Paris), Coll. 5–8 (1995) 463. [16] F. Nishimura, N. Watanabe, K. Tanaka, Mater. Sci. Eng. A221 (1996) 134. [17] F. Nishimura, N. Watanabe, K. Tanaka, Mater. Sci. Res. Int. 3 (1997) 23. [18] L. Federzoni, G. Gue´nin, Scr. Metall. Mater. 31 (1994) 25. [19] L. Delaey, R.V. Krishnan, H. Tas, H. Warlimont, J. Mater. Sci. 9 (1974) 1521. [20] K. Tanaka, J. Pressure Vessel Tech. 112 (1990) 158. [21] H. Tobushi, S. Yamada, T. Hachisuka, A. Ikai, K. Tanaka, Smart Mater. Struct. 5 (1996) 788. [22] N. Bergeon, G. Gue´nin, in: Proc. ICOMAT 95, J. Phys. IV (Paris), Coll. 5 – 8 (1995) 439. [23] H. Inagaki, Z. Metallkd. 83 (1992) 2. [24] T. Kikuchi, S. Kajiwara, Y. Tomota, in: Proc. ICOMAT 95, J. Phys. IV (Paris), Coll. 5 – 8 (1995) 445. [25] K. Tsuzaki, Y. Natsume, T. Maki, in: Proc. ICOMAT 95, J. Phys. IV (Paris), Coll. 5 – 8 (1995) 409. ˚ gren, in: Proc. ICOMAT 95, J. Phys. IV [26] M. Andersson, J. A (Paris), Coll. 5 – 8 (1995) 457.
Stress – strain–temperature hysteresis and martensite start line in an Fe-based shape memory alloy Fumihito Nishimura, Noriko Watanabe, Kikuaki Tanaka * Department of Aerospace Engineering, Tokyo Metropolitan Institute of Technology, Asahigaoka 6 -6, J-191 Hino/Tokyo, Japan Received 14 April 1997
Abstract The uniaxial stress–strain–temperature hysteretic behavior in an Fe – 9%Cr – 5%Ni – 14%Mn – 6%Si polycrystalline shape memory alloy is investigated under successive tensile – compressive mechanical loading and thermal loading, especially in relation to the transformation lines; the martensite start line and the austenite start/finish lines on the stress – temperature plane. The form of the thermomechanical hysteresis loops, the strain recovery during heating after forward and reverse transformations induced by the mechanical loading and the recovery stress under strain-constrained heating are the points to be studied. The dependence of the martensite start stress on the extent of prior martensitic transformations is determined from the direct strain-recovery measurement during heating, which agrees well with the estimation by means of the reverse transformation lines coupled with the assumption of the ‘isotropic’ hardening in transformation. © 1997 Elsevier Science S.A. Keywords: Fe-based shape memory alloy; Hysteresis loop; Recovery stress; Thermomechanical loading; Transformation lines; Uniaxial tension and compression
1. Introduction The martensitic transformation condition, which predicts the start of the transformation in the materials under thermomechanical loading, has been studied for years in metallurgy. The importance of the transformation driving force, which depends on both the temperature and the applied stress state, has been clarified [1,2], also from the continuum mechanical point of view [3 –7]. In recent years, this condition has attracted attention when investigating the thermomechanical behavior of such advanced materials as shape memory alloys, zirconia ceramics and TRIP steels [3 –9]. The trend has been strongly motivated by rapid technological development which has realized the manufacturing processes of materials with specified microscopic structures, not only in composite materials but also in conventional metallic alloys [10]. The thermomechanical theories of transforming materials are currently * Corresponding author: Tel.: +81 425835111; 425835119; e-mail: [email protected]
fax:
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0921-5093/97/$17.00 © 1997 Elsevier Science S.A. All rights reserved. PII S 0 9 2 1 - 5 0 9 3 ( 9 7 ) 0 0 4 6 1 - 9
being developed in order to describe the material behavior by taking the microstructures of the material into account and to utilize it in the material design with some requirements of the strength, the toughness, and so on. The idea of the transformation condition is always considered in relation to the yield condition in plasticity, which is employed as a potential function in the theory to derive the evolution equations of the internal variables introduced. In fact, Tanaka et al. [8,9] have proposed a unified theory of TRIP, which describes the plastic deformation of the materials in the process of martensitic transformation, by employing the transformation condition and the yield condition simultaneously. Taking such a theoretical and technological background into account, the present authors have been investigating the thermomechanical hysteretic behavior in an Fe–9%Cr–5%Ni–14%Mn–6%Si polycrystalline shape memory alloy, always in relation to the transformation condition [11–17]. The phase diagram of the present alloy under uniaxial loading is represented by three straight lines on the stress–temperature plane; the
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martensite start line and the austenite start and finish lines with the same slope. The reverse transformation zone, bounded by the austenite start/finish lines, increases its width depending on the extent of prior martensitic transformation, whereas their slope stays unchanged. The transformation lines after compressive loading constitute a mirror image of the lines after tensile loading with respect to the stress-free axis, but it is worth noting that the two sets of lines are not symmetric. This fact is especially important when discussing the thermomechanical hysteresis during tensile– compressive loading, monotonic and cyclic [16,17]. During mechanical loading, the transformation start state is determined as a point on the stress – temperature plane at which the thermomechanical load point and the transformation start line meet. The subsequent thermomechanical behavior is also strongly influenced by the shift of the transformation lines, depending on the prior transformation history. The progress of the transformations between the phases, the parent phase and the martensite phases induced in tension and/or compression, can be well explained both from the hysteresis loops and the transformation diagram. In this paper, following the preceding study [16], the stress–strain–temperature hysteretic behavior in the same alloy material is investigated in detail under the uniaxial tensile–compressive and thermal loading, in special relation to the transformation lines. The form of the hysteresis loop and the characteristics of the strain recovery during heating are explained by the shift of the reverse transformation lines during loading. The evolution of the recovery stress is also investigated in the process of strain-constrained heating. The shift of the martensite start line is estimated by both the data of the austenite start temperature and the direct measurement of the martensite start stress from the dilatation curves during heating.
3. Transformation lines When the alloy specimen is loaded mechanically at a test temperature Th up to a maximum stress s + max in tension or down to a maximum stress s − max in compres− sion, a martensite start stress s + Ms or s Ms is determined on the tensile or the compressive branch, respectively, as illustrated in Fig. 1. Here, and henceforth, the superscripts ‘+ ’ and ‘− ’ represent that the quantity corresponds to the tensile loading and the compressive loading, respectively. It should be noted that the martensite M + means here the variants formed when the specimen is loaded in tension, and the martensite M − the variants induced when the specimen is loaded in compression. The point to be emphasized in the present study is that, during reverse transformation, in other words, when the specimen is heated, the martensite M + gives rise to a macroscopic compressive deformation in the specimen, and the martensite M − a macroscopic tensile deformation. − Both stresses s + Ms and s Ms strongly depend on the prior history of transformations that the specimen has experienced so far [11–16]. If the martensite phase is induced in the specimen by mechanical loading, a reverse transformation zone can be determined by measuring an austenite start temperature and an austenite finish temperature along a heating path up to a maximum temperature Tmax under a constant applied stress sh. Fig. 1 plots schematically these temperatures as T + As and T + Af, respectively. The stresses and the temperatures thus determined constitute the transformation lines: the martensite start line (Ms + -line), the austenite start line (As + -line) and the austenite finish line (Af + -line). The As + - and Af + -lines bound the reverse transformation zone. The same is true for the transformation lines corresponding to compressive loading, as illustrated in Fig. 1.
2. Alloy specimen The test specimen, 6 mm in diameter with 20 mm gauge length, of an Fe – 9%Cr – 5%Ni – 14%Mn–6%Si polycrystalline shape memory alloy is the same as that used in a previous study of the same authors [16]. The mechanical properties of the alloy are reported there together with the heat treatment and the experimental procedures. The specimens are, prior to the tests, subjected to a thermomechanical training, details of which is explained in Refs. [16,17], to establish a stable thermomechanical response. When the specimen is heated up above the austenite finish temperature, the alloy returns to the original state metallurgically and mechanically even after some mechanical loading in tension and compression.
Fig. 1. Thermomechanical loading path and transformation lines.
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Fig. 2. Thermomechanical hysteresis loops under tension – compression at RT and heating.
These transformation lines shift on the stress–temperature plane depending on the prior transformation history, which was the theme of the study by Nishimura et al. [16]. The points to be noted for the present study are summarized as follows: the austenite start line shifts to the lower temperature side when the amount of − pre-stressing, measured here by s + max or s max, becomes larger, but no change in slope of the line is observed. Similarly, the austenite finish line shifts to the higher temperature side without change in slope when the amount of pre-stressing becomes larger. Both start line and finish line have the same slope. The As + - and Af + -lines are extended to the negative stress side without changing their slope. Similarly, the As − - and Af − lines are extended to the positive stress side without changing their slope, and no change is observed in slope on both the tensile and compressive stress sides. The drift of the start line is much more sensitive to s + max or s − max than is the finish line. The reverse transforma− tion zone, therefore, becomes wider with s + max or s max, meaning that at Th =RT (room temperature, 303 K) + the austenite finish stress s + Af decreases with s max, and − finally becomes lower than the P M transformation + start stress s − Ms when s max is large enough. Similar behavior was observed for the transformation lines corresponding to compressive loading: at Th = RT the − austenite finish stress s − Af increases with s max, and + finally becomes larger than the PM transformation − start stress s + Ms when s max is small enough.
4. Thermomechanical hysteresis Fig. 2 plots the thermomechanical hysteresis loops during successive tensile loading/unloading, s + max = 150, 200 and 250 MPa at Th =RT, and subsequent compressive loading/unloading, s − max = −300 MPa at Th = RT,
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followed by heating/cooling, Tmax = 873 K under sh =0 MPa. The first isothermal mechanical loop is often called the ferroelastic loop. It should be noted that, after being loaded up to s + max in tension which produces the stress-induced M + martensite in the specimen, the alloy has, as explained in the previous section, the wider reverse M + P transformation zone depending on s + max, at the start of the subsequent compressive loading. In the case of s + max = 150 MPa, the austenite finish stress s + Af is larger than the martensite start stress + in compression s − P Ms, meaning that the reverse M − transformation zone and the Ms -line are not overlapped at RT. The elastic compressive response is, therefore, expected during loading from s + Af down to s− Ms, though the data reveals it in a very small part. This alloy response diminishes as s + max becomes large + since the s + Af stress shifts down at RT with s max, and + finally in the case of s max = 250 MPa, the P M − transformation starts nearly at the same moment when the reverse M + P transformation finishes. The anomalous performance of the alloy indicated by an arrow in the figure is attributed to this metallurgical process. Since the reverse M + P transformation has already finished at s − max = − 300 MPa for all three loading cases, the strain amplitude in the compressive branch is almost the same for all cases. So is the response during the subsequent thermal run. It is usually observed in the Fe-based alloys that the reorientation of the martensite variants, M + M − and M − M + processes in the present notation, is not observed [18]. The case is assumed to be true in this alloy also. At higher test temperature, as shown in Fig. 6 in Nishimura et al. [16], the reverse transformation zone for the M + martensite is fully apart from the PM − transformation zone in compression. Therefore, the reverse transformation finishes far before the start of the P M − transformation in compression. The elastic unloading of the specimen composed only of the P phase, is expected during a part of the compressive − loading, from s + Af down to s Ms. Fig. 3 reveals that the response is actually observed at Th = 343 K at the part arrowed in the ferroelastic hysteresis loops. The subsequent performance of the alloy is the same, being independent of s + max. Fig. 4 illustrates the alloy response when the specimen is firstly loaded in compression to produce the M − martensite and then reloaded in tension. Judging from the transformation diagram determined in the previous study, Fig. 7 in Nishimura et al. [16], the reverse M − P transformation and the P M + transformation progress simultaneously during the tensile loading. The smooth response of the alloy is due to the fact that, contrary to the cases in Figs. 2 and 3, the reverse transformation zone and the M + transformation zone almost overlap at RT. Since the reverse M − P trans-
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Fig. 3. Thermomechanical hysteresis loops under tension – compression at 343 K.
formation finishes for all s − max cases before the stress reaches the maximum value s + max =300 MPa, almost similar hysteresis loops are obtained in the subsequent unloading and heating/cooling processes. The alloy performs differently in the case of smaller s − max, which will be discussed later in Fig. 9.
5. Strain recovery during heating The alloy response illustrated in Figs. 2 and 4 suggest that during the mechanical loading process the alloy is composed of the M − martensite, the M + martensite and the P phase, changing their fractions with the progress of loading. The situation is clearly proved by examining the strain recovery during the subsequent heating, shown in Fig. 5. The specimen, compressed down to s − max = −280 MPa, is then
Fig. 4. Thermomechanical hysteresis loops under compression – tension at RT and heating.
Fig. 5. Strain recovery during heating after compressive –tensile loading at RT.
loaded in tension up to the different values of s + max = 180, 220, 230 and 300 MPa. After unloaded, the specimen is subjected to the thermal run of heating/cooling. The M − martensite produced during compressive loading is the cause of the tensile deformation of the specimen in the subsequent heating process. As s + max becomes larger, the amount of the M − martensite decreases due to the progress of the stress-induced reverse M − P transformation in tension, resulting in the smaller amount of tensile deformation during heating. − In case s + martensite max is sufficiently large, the M disappears in the specimen while mechanical loading in tension, whereas both the M + martensite and the P phase still remain. The M + martensite increases with s+ max, inducing the larger amount of compressive deformation during heating. The experiment shows that no residual strain is observed after the mechanical run when s + max =220 MPa. The subsequent thermal run clearly shows that the internal structure of the alloy at the start of heating is a mixture of the M + martensite, the M − martensite and most likely the P phase, too. Actually, the dilatation curve exhibits a compensation of the compressive deformation due to the M + martensite and the tensile deformation due to the M − martensite. The M + martensite first transforms back to the P phase during heating. The M − martensite then transforms back to the P phase at a higher temperature range, inducing a tensile deformation. These two deformations in the opposite direction complete during heating and, finally, when all the M + and M − martensites transform back to the P phase, the dilatation curve converges to the thermal expansion linear line. A similar alloy performance is observed when the specimen is first loaded/unloaded in tension, up to s+ max = 250 MPa at Th = RT, and then loaded/un-
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Fig. 6. Strain recovery during heating after tensile–compressive loading at RT.
loaded in compression, down to s − max = −220, −240 and −260 MPa at Th =RT, before heating/cooling (cf. Fig. 6).
6. Recovery stress If the deformation of the specimen is constrained during heating in the tests shown in Figs. 5 and 6, a stress is induced in the specimen due to the transformation strain produced in the process of reverse transformation. The stress, called the recovery stress [19,20], is nothing other than the driving force of the shape memory devices. In order to simplify the situation, following cases are investigated: the specimen is heated with the strain constrained after being loaded mechanically in tension at Th =RT up to s + max. Fig. 7 shows the increase in the recovery stress s rec during heating for each case of s + max. The size of the
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Fig. 8. Maximum recovery stress depending on prior martensitic transformation in tension.
reverse transformation zone depends on the prior history of transformation, on s + max in the present context (cf. Section 2), as indicated in the figure. For the sufficiently small values of s + max, the austenite start temperature As + under no applied stress is higher than RT, meaning that at the early stage of heating a negative stress is induced, due not to the reverse transformation but simply to the constrained thermal expansion. This type of alloy behavior is not observed in the wire specimen [21]. The transformation contraction then follows from As + on, resulting in a sharp increase in the recovery stress. The experiments reveal that this alloy performance diminishes as s + max approaches to 260 MPa, under which the reverse transformation starts just at the moment of start heating. When s + max is higher than this value, in the case of s + max =300 MPa, for example, the reverse transformation starts already at the last stage of the mechanical run. This is the reason why the maximum value of the recovery stress + s rec max saturates as s max becomes larger, which will be plotted later in Fig. 8. Fig. 7 also reveals very clearly that the recovery stress is due to the reverse M + P transformation since its increase finishes at the austenite finish line, the Af + -line. From this point on, the recovery stress decreases due to the constrained thermal expansion of the P phase. The maximum value of the recovery stress s rec max depends on the prior martensitic transformation, on s+ max actually, as illustrated in Fig. 8.
7. Martensite start line
Fig. 7. Recovery stress and transformation zone.
The martensite start lines, the Ms + -line and the Ms − -line in Fig. 1, are determined with the specimens which have no prior history of transformations [12,13,16]. The lines correspond to the initial yield surface in plasticity. As in plasticity, the martensite
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start lines shift during subsequent loading, exhibiting the subsequent martensite start line. Fig. 19 in the later discussion shows the increase in the subsequent martensite start stress in the process of uniaxial loading in tension; in other words, the alloy exhibits the ‘hardening’ during transformation. In order to establish a rational concept of the transformation surface, which is a generalized figure of the transformation line in the multiaxial stress– temperature space, one has firstly to know in the uniaxial case the effect of the prior martensitic transformation in the opposite direction of loading on the martensite start stress. This is the theme to be investigated in this section. Fig. 9 shows the ferroelastic hysteresis loops under the successive compressive loading/unloading, the tensile loading/unloading and the heating/cooling. The mechanical run is carried out at Th =RT, and the thermal run under sh =0 MPa. The maximum stress in tension is always kept constant to s + max =300 MPa, whereas the maximum stress in compression s − max changes loop by loop. On loading in tension, both the reverse M − P transformation and the P M + transformation progress simultaneously, and the M − martensite fully transforms back to the P phase by the end of the tensile loading. Since the reverse M + P transformation completes during heating, no residual strain is observed after the whole thermomechanical run. The maximum strain in tension o + max, defined in Fig. 10, is measured in Fig. 9, and is plotted in Fig. 10. It should be noted that the strain o + max is a good measure of the extent of the PM + transformation since the M − martensites fully transform back to the P phase during the tensile loading. This is actually the case, as proved in Fig. 5 for sufficiently large s + max. One could conclude for now that the M + martensite start stress − s+ transforMs increases when the extent of the P M mation becomes large, and equivalently when s − max
Fig. 9. Thermomechanical hysteresis loops under compression – tension at RT and heating.
Fig. 10. Maximum strain in tension and maximum stress in compression.
decreases. In deriving this estimation, one should take into account the fact that the decrease in o + max, in other words, the decrease in the extent of the martensitic + + transformation s + max –s Ms results in the increase in s Ms when s + max is kept constant. The 0.05% proof stresses during tensile loading s0.05%, indicated by the circles in Fig. 9, exhibit in Fig. 11 a clear linear change versus the transformation strain in compression o − T defined in the figure, even when the test temperature is different. However, no definite reason is yet given as to whether this stress is the reverse M − P transformation start stress s − As or the P M + transformation start stress s + Ms in question, although in the later discussion it will be proved to be the former stress. The austenite start temperature As + of the M + martensite is measured, as shown in Fig. 12 by the circles, after successive compressive loading down to + s− max = − 280 MPa and tensile loading up to s max. The + temperature As must depend on the progress of both
Fig. 11. ‘Yield stress’ in tension after compressive loading at RT.
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Fig. 12. Austenite start temperature after compressive–tensile loading at RT.
the PM − transformation in compression and the P M + transformation in tension, directly speaking, + on both s − max and s max. Actually, the result summarized in Fig. 13 reveals that As + depends on s − max. This eventually means that the M + martensite start stress s+ Ms also depends on the extent of loading in compression, measured in the present context by s − max. In order to prove this statement directly, Fig. 13 is replotted, in − the case of s + max =200 MPa, versus s max in Fig. 14. One gets an estimation for the slope of the lines in the figure, DAs + = − 0.11 K MPa − 1 Ds − max
(1)
On the other hand, the data on the reverse transformation temperatures by Nishimura et al. [16], displayed again in Fig. 15, reveal the following estimation:
Fig. 13. Austenite start temperature depending on prior transformations.
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Fig. 14. Austenite start temperature depending on prior martensitic transformation in compression.
DAs DAs + = = − 0.55 K MPa − 1 + DST D(s max − s + ) Ms
(2)
where the transformation stress ST represents the extent of the martensitic transformation and is defined in tension by + ST+ = s + max − s Ms
(3)
and in compression by 1 − ST− = (s − Ms − s max). k
(4)
The material constant k in the equation, which is determined by the slopes of the martensitic transformation lines, denotes the difference in size of the martensitic transformation zones in tension and compression. Now Eq. (2) is combined with Eq. (1) between the As + and s − max, and one reaches the final formula stating that the P M + transformation start stress depends on the extent of the P M − transformation:
Fig. 15. Reverse transformation zone after martensitic transformation in tension.
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Fig. 18. Transformation diagram after compressive loading at RT. Fig. 16. Strain recovery during heating after compressive – tensile loading at 288 K.
Ds + Ms =0.20 MPa MPa − 1 Ds − max
(5)
The estimation in Eq. (5) can directly be approved in Figs. 16 and 17. The thermomechanical ferroelastic loops in the figure correspond to the case in Fig. 5, in which the specimen is composed, at the start of heating, of the M + martensite, the M − martensite and the P − phase. When changing the value of s + max, while s max = − 280 MPa is kept constant, the dilatation curve exhibits complicated characteristics (cf. also Figs. 5 and 6), depending on the amount of the phases the specimen has at the start of heating. One can determine a s + max value above which the dilatation curve falls at the early stage of heating on the shorter strain side of the thermal expansion line, indicating that a transformation contraction takes place. This s + max value is nothing other than the M + martensite start stress s + Ms, at which the M + martensite starts being produced. Fig. 17, summarizing the measured s + Ms values, presents an estimation
Fig. 17. Martensite start stress of M + martensite depending on prior martensitic transformation in compression.
Ds + Ms = − 0.27 MPa MPa − 1 Ds − max
(6)
which has a good agreement with the value determined above in Eq. (5). The results in Figs. 11, 15 and 17 are plotted together in Fig. 18. The shadow part, which is a replot of Fig. 15 − and is bounded by the s − Af- and s As-lines, represents the region in which the stress-induced reverse M − P transformation progresses along a constant s − max path. The solid circles, determined in Figs. 9 and 11 as the ‘yield’ stress s0.05%, now turn out to represent the reverse M − P transformation start stress s − As. The fact that the stresses s0.05% almost coincide with the s − As-line in the figure, determined from the reverse M − P transformation start temperature, is worth emphasizing in the sense that almost similar transformation lines are obtained in both the isothermal mechanical loading tests and the thermal loading tests under constant stress. In the test of s − max = − 250 MPa, for example, while loading mechanically, the reverse M − P transformation starts at about 65 MPa (s − As), and before finishing the reverse transformation, the PM + transformation starts at about 170 MPa (s + Ms). From this stress on, both the reverse M − P transformation and the PM + transformation progress simultaneously until the reverse M − P transformation finishes at about 203 MPa + (s − transformation Af). From then on, only the P M takes place. It should be noticed that the actual austenite finish stress may change from the value in the figure because the s − Af stresses in the figure are determined with the alloy specimens which contain only the M − martensite, whereas in the present case the M + martensite is formed and increases during loading from s + Ms = 170 MPa on. As the data (hollow circles) in Fig. 18 show, and as has already been explained in Eqs. (5) and (6), s + Ms clearly depends on s − max, in other words, on the prior history of the PM − transformation in compression.
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Let us trace in the figure an experiment of isothermal compression at Th =RT. The s − max value decreases, indicating the progress of compression, and when the compression stops at a s − max value and the unloading and subsequent tensile loading follow, the generic point in the figure moves up along the constant s − max line as demonstrated above. Now till the s − value reaches max s− = − 170 MPa during compression, the martensite Ms start stress s¯s¯do3(s + Ms) stays constant at an initial martensite start stress s + Ms-ini =153 MPa since no transformation yet progresses in compression. From − 170 MPa down in compression the alloy experiences the P M − transformation, which, due to the ‘hardening effect’ explained below, raises the s + Ms value. The experimental result shows that the dependence of s + Ms on the prior transformation in compression, on s − max, is almost linear. It is worth noting that the martensite start stress s + Ms measured here is different from the martensite start + stress SMs shown in the figure, which the alloy has just in the process of compressive PM − transformation, since the alloy state cannot come from the compressed + state at the SMs stress on the positive stress side without experiencing the reverse M − P transformation. This + reverse M − P transformation changes the SMs value + to the s Ms value actually measured in Fig. 18. The + martensite start stress SMs depends on the prior transformation history in compression, on s − max shortly. Let us study in detail whether this dependence can be understood by the concept of ‘isotropic hardening’ in plasticity. Fig. 19 shows an isothermal stress – strain curve obtained during a repeated loading/unloading in tension at Th =RT with increasing maximum stress, representing the ‘hardening’ of the martensite start stress. Assuming the ‘isotropic hardening’ of the martensite start + stress, SMs increases in the process of transformation in compression and is given by
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Table 1 Materials parameters a (K MPa−1)
b (K MPa−1)
k
−1 c− ) A (MPa K
0.55
0.35
1.3
−2.4
1 − + − + SMs = s+ (s Ms − s − Ms-ini + ST = s Ms-ini + max) k
(7)
where s + Ms-ini stands for the initial martensite start stress (see Fig. 19), corresponding to the initial yield stress in plasticity. + reWhen the reverse transformation finishes, SMs . It is, therefore, reasonturns to the initial value s + Ms-ini able to assume that in case the reverse transformation + decreases depending linearly on stops on the way, SMs the extent of the reverse transformation: − + s− SMs − s+ Af − s As Ms-ini = + + − s Ms − s As SMs − s + Ms
(8)
which leads to an expression for s + Ms; s+ Ms =
+ − − + + SMs (s − Af − s As)+ s As(SMs − s Ms-ini) − + + (s − Af − s As)+ (SMs − s Ms-ini)
(9)
As explained in Section 2, the reverse transformation zone becomes broader with the progress of martensitic transformation, which can be formulated as [16]: − − s− As = c A (TAi − aST )
and
− − s− Af = c A (TAi +bST ) (10)
Here the material parameter c − A denotes the slope of the Af − - or As − -line, and a and b are the constant material parameters. Eqs. (9) and (10) combine to yield an estimation − bc − Ds + Ms A = − − Ds max k(ac A + bc − A − 1)
(11)
Substitution of the values of all material constants, shown in Table 1, into Eq. (11) gives Ds + Ms = − 0.20 MPa MPa − 1 Ds − max
(12)
which, representing the linear line in Fig. 18, agrees well with the experimental result denoted by the hollow circles. The estimation also coincides surprisingly well with Eq. (5) which is derived from the experimental data of the austenite start temperature. The discussion reveals that the postulate of ‘isotropic hardening’ of the martensite start stress is acceptable.
8. Concluding remarks
Fig. 19. Stress – strain curve under repeated loading–unloading at RT with increasing maximum stress.
The uniaxial stress–strain–temperature hysteresis is investigated in an Fe–9%Cr–5%Ni–14%Mn–6%Si polycrystalline shape memory alloy under tensile–com-
F. Nishimura et al. / Materials Science and Engineering A238 (1997) 367–376
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pressive and thermal loading. The form of the hysteresis loops is shown to be well characterized by means of the transformation lines, the martensite start line and the austenite start/finish lines, which represent the progress of the transformations between the phases; the parent phase, the martensite phases induced by tensile loading or compressive loading. The lines shift on the stress – temperature plane, depending on the extent of prior transformations. The martensite start stress depends on the extent of prior martensitic transformation during mechanical loading in the opposite direction. The dependence is determined both from the direct strain-recovery measurement during heating and from the data on the reverse transformation lines coupled with the assumption of the ‘isotropic’ hardening in transformation. Both results agree well, suggesting that the concept of ‘isotropic’ hardening is acceptable. When the Fe-based shape memory alloys are mechanically loaded, the thin o martensite plates grow from the grain boundaries in the parent phase [22–24]. The local internal stresses, the back stresses, induced at the tip of the plates play an essential role in the strain recovery process during subsequent heating [25,26]. The extent and direction of this back stress, at least its averaged value over the active variants, can be evaluated macroscopically if the strain recovery is measured in the process of heating under the hold stress sh. This should be a theme to be investigated in the next paper.
Acknowledgements The authors express their thanks to Professor H. Inagaki (Shonan Institute of Technology) for his constructive discussion. Part of this work was financially supported by the Special Research Fund of the Tokyo Metropolitan Government as well as by a Grant-inAid for Scientific Research (No. 08650117) from the Ministry of Education, Science and Culture, Japan. The supply of the alloy specimens by the Steel Research Center of the NKK Corporation is gratefully acknowledged.
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