- Email: [email protected]
or thermal loads
IIECHAlilCS OF MATEIUALS ELSEVIER
Mechanics of Materials 19 (1995) 281-292
Phenomenological analysis on subloops and cyclic behavior in shape memory alloys under mechanical and/or thermal loads K. Tanaka
a,1,
F. Nishimura
a,
T. Hayashi a, H. Tobushi b, C. Lexcellent c
a Department of Aerospace Engineering, Tokyo Metropolitan Institute of Technology, Hino / Tokyo, Japan b Department of Mechanical Engineering, Aichi Institute of Technology, Toyata, Japan c Universitd de Franche-Comt~ UFR des Sciences et des Techniques, Laboratoire de Mdcanique appliqude Associd au CNRS, Besan~on, France Received 16 March 1993; revised version received 28 February 1994
Abstract
A theoretical framework is presented, from the phenomenological point of view, for the cyclic uniaxial deformation in shape memory alloys subjected to the thermal and/or mechanical loads by introducing three internal variables; the local residual stress and strain and the volume fraction of the martensic phase accumulated during cyclic forward and reverse martensitic transformations. The cyclic effect on the stress-strain and strain-temperature hysteresis loops is discussed. The subloops due to incomplete transformations are also analyzed by assuming the transformation starting stress or temperature which depends on the preloading. Numerical results explain qualitatively well the observations on the thermomechanical behaviors of shape memory alloys.
1. Introduction
The thermomechanical response of shape memory alloys is directly associated with the progress of elementary metallurgical processes occurring under thermal a n d / o r mechanical applied loads such as the martensitic transformation or its reverse transformation and the reversible movement of the twin boundaries (Perkins, 1975; Funakubo, 1984; Duerig et al., 1990). Based on the intensive metallurgical studies on the micro-
i Corresponding author: Professor Dr. Kikuaki Tanaka, Department of Aerospace Engineering, Tokyo Metropolitan Institute of Technology, Asahigaoka 6-6, J-191 Hino/Tokyo, Japan.
scopic mechanism of the "memory", macroscopic theories have been proposed, starting from different levels of microstructure in alloys, to predict the thermomechanical behaviors of shape memory alloys (Patoor et al., 1987; Falk, 1989; Miiller, 1989; Tanaka, 1990; Fischer and Tanaka, 1992). The stress-strain-temperature relation, the recovery stress induced in the heating process and the response under thermomechanical constraints are some of the subjects to be discussed from a theoretical point of view. The full stress-strain and strain-temperature hysteresis loops, together with their subloops realized during incomplete transformations for a smaller stress, strain or temperature range, are also being investigated in the course of constructing a unified theory of the strong interaction of
0167-6636/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSD! 0167-6636(94)00038-1
282
K. Tanaka et aL / Mechanics of Materials 19 (1995) 281-292
thermomechanics and transformation, based fully on the continuum thermodynamics (Miiller and Xu, 1991; Raniecki et al., 1992). The experimental observations for the subloops during thermal or mechanical loading contradict each other with respect to the transformation start stress or temperature after an incomplete transformation (Paskal and Monasevich, 1981; Wei and Yang, 1988; Mfiller and Xu, 1991; Amengual et al., 1994). The deformation/transformation response of shape memory alloys exhibits very often a strong cycle dependence, which has always a negative effect on the practical use of the shape memory devices because they must generally show a stable output (Duerig et al., 1990); the stroke, force or the transformation temperatures themselves. In order to acquire a stable response of the alloy during cyclic loading some metallurgical treatments are proposed, and a so-called thermomechanical training is often performed before producing an actual device. The cycle dependence of stress-strain hysteresis loops have been investigated in different alloys (Miyazaki and Otsuka, 1989). The strain-temperature hysteresis loops also depend on the cyclic loading under a constant applied stress (Li and Ansell, 1983; Sade et al., 1988; Contardo and Gu6nin, 1990; Xu and Tan, 1991; Tobushi et al., 1991). Both loops shift gradually with the cyclic progress, and finally reach stable limit loop after scores of cycles. Metallurgical studies have clearly revealed that dislocations are piled up around the defects scattered in the alloy during the cyclic forward and reverse martensitic transformations and that the martensite is locally induced by the local stress accumulated there (Miyazaki et al., 1986). In this paper a macroscopic theoretical framework is presented to explain the hysteresis behavior of shape memory alloys during the thermal a n d / o r mechanical cyclic loadings. The cyclic effect on the stress-strain and strain-temperature hysteresis loops and the subloops due to incomplete transformations arc the main topics to be discussed. The macroscopic theory on the behavior of shape memory alloys presented by the authors (Tanaka, 1990, 1991) is generalized by introducing three internal variables; the local
stress and strain and the residual martensic phase accumulated irreversibly due to the cyclic forward/reverse martensitic transformation. Numerical simulations are carried out to explain qualitatively the experimental results for the thermomechanical cyclic loadings.
2. Constitutive relation
The authors (Tanaka, 1990, 1991; Tanaka et al., 1992) have shown from a phenomenological point of view that the uniaxial thermomechanical behavior of shape memory alloys can be well described by the constitutive relation consisting of the constitutive equation in rate form ~ = D ~ + O~/"+ D~,
(1)
and the transformation kinetics for the martensitic transformation
1-f
= bMCMT
-- b M d - >__0,
(2)
or, for the reverse transformation, -- -- = b A C A T -- bAd" >_ O,
(3)
where or, • and T stand for the stress, strain and temperature, respectively, while D denotes the Young's modulus and -O/D and -DID are the coefficient of linear expansion and the strain due to transformation which is often called the eigenstrain in micromechanics, respectively. The symbol ~ represents the volume fraction of the martensite phase. The volume fraction of the parent phase, the austenite phase actually, is, therefore, 1 - s~. The material parameters in the kinetics, b M, CM, bA and CA, can be determined from experiments. Their physical meaning will become clear later. Eq. (1) governs the thermomechanical behavior of the alloys, while Eq. (2) or (3) describes the progress of the martensitic transformation or the reverse transformation under a given thermomechanical process. The inequalities in Eqs. (2) and (3) determine which transformation, the martensitic or the reverse one, or none of them, pro-
K~ Tanaka et al. /Mechanics of Materials 19 (1995) 281-292
gresses at each instant of the thermomechanical loading. All material parameters in the equations can be determined by a series of simple experiments. Let us now consider minutely a reverse transformation process starting from the initial values ((r0, To, ~M0), where ~M0 stands for the fraction of martensite at the start of the reverse transformation. If the transformation kinetics (3) is integrated from the initial values to the current values (cr, T, ~) for constant material parameters bA and c A, one finally obtains an explicit form of the transformation kinetics;
283
Af A,
M, I
0
I
0.5
Mf
I
1
Volume fraction of residual martensite ~ 0 Fig. 1. Transformation start temperatures (schematic).
sc = ~M0 exp[bACA(To -- T ) + bA(O"-- o'0) ] .
(4)
The trace (o-0, T0) in the stress-temperature plane represents a line, the reverse transformation start line or the As-line, at which the reverse transformation starts. Now if we assume that the A~-line does not depend on the amount of residual martensite ~M0 at the start of the reverse transformation, and can be expressed by ~o =
CA( To - As),
(5)
where A s stands for the austenite start temperature under free stress, the final transformation kinetics can be written as ~: = ~:M0 e x p [ b A c A ( A s -- T ) + bAO"] .
(6)
It is, however, much more rational to assume that the A~-line depends on the amount of residual martensite through the form; cro = cA(To -- a(~M0)),
(7)
where a(~M o) represents the austenite start temperature that depends on the amount of residual martensite. The slope of the As-line, CA, is assumed to be constant. In this case, instead of Eq. (6), the following final transformation kinetics can be obtained by integrating Eq. (3): s¢ = ~:M0 exp[bACA(a(~MO) -- T ) + bA~r] .
(8)
The reverse transformation finish line, the Afline, can now be read as 1 (r = CAT-- 7---In(100SCM0) - CAOt(SeM0),
OA
(9)
where ~ = 0.01 is regarded as the completion of the reverse transformation following the usual understanding in metallurgy (Ashby and Jones, 1986). When the Af-line is assumed, for simplicity, to be independent of ~M0, the material parameter ba is determined by ln(100~:m0 ) bA = CA(A f _ a ( ~ M 0 ) ) ,
(10)
where Af stands for the austenite finish temperature under stress free conditions. The austenite start temperatures in both cases are schematically illustrated in Fig. 1. The previous analysis by the present authors was based on Eqs. (5) and (6), where the As-line is a fixed line in the stress-temperature plane, being independent of the amount of residual martensite. The broken line in Fig. 1 shows this case. Paskal and Monasevich (1981) have actually observed such behavior in a TiNi shape memory alloy. Amengual et al. (1994) have, on the other hand, very clearly shown in C u - Z n - A l and C u - A I - M n shape memory alloys that the austenite start temperature, and the martensite start temperature as well, is strongly dependent on the amount of preloading, strictly speaking, on ~M0 in our context, during the isostatic thermal loading. The present authors (Tanaka et al., 1993) have also observed that the same is true in an Fe-based polycrystalline shape memory alloy. This case corresponds to the solid curve in Fig. 1. The dependence of the A s- and Af-lines on the preloading
K. Tanaka et aL /Mechanics of Materials 19 (1995) 281-292
284
has also been reported in an F e - M n - S i alloy (Robinson and McCormick, 1990) and an F e M n - S i - C r - N i - C o alloy (Tan and Yang, 1992). It should be noted that the thin solid line in the figure represents the shift in the austenite start temperature predicted by the thermodynamic theory of shape memory alloys (see, e.g., MOiler, 1989; MOiler and Xu, 1991; Raniecki et al. 1992). If a similar assumption, i.e. ~ o = C M ( TO --
~'(~A0)),
(11)
is imposed for the martensitic transformation with a constant material parameter C M , where /*(~:A0) stands for the martensite start temperature being dependent on the amount of residual austenite CA0 at the start of the martensitic transformation, the final transformation kinetics can be written as = 1 -~A0 exp[bMCM(/Z(~A0) -- T) + bMO"] . (12) When the martensite finish line or the Mf-line, determined from the condition £ = 0.99 (Ashby and Jones, 1986), is assumed to be immovable, its expression in the tr-T plane and the material parameter b M are given by o" = C M T --
1 ~M ln(100~:a0) -- CM/~(£A0) ,
ln(100sCn0) b M = CM(Mf _/Z(£A0)),
(13)
(14)
where Mf (and Ms, which appears later) stand for the martensite finish (and start) temperatures under free stress. The dependence of the martensite start temperature on the amount of residual austenite, £A0 = 1 - £M0, is schematically given in Fig. 1, where the solid line represents the theory to be developed here while the broken line and thin solid line correspond to the theory with a constant value of the martensite start temperature and the thermodynamic theory, respectively. The present paper is devoted to the consequences of Eqs. (7)-(10) and Eqs. (11)-(14). The behavior of shape memory alloys during thermal a n d / o r mechanical cyclic loadings is the topic to be investigated.
A comment on the transformation start lines: as Raniecki et al. (1992) have clearly discussed, the transformation start lines, both martensitic and reverse, can be derived from the driving force of the transformation (Kaufman and Hillert, 1992) as a condition of constrained equilibrium in continuum thermodynamics (Kestin and Rice, 1970; Rice, 1971). Their thermodynamic theory of transformation could, therefore, be extended rationally to include the effect of preloading by introducing additional terms into the driving force. Derivation of such a new driving force and its physical interpretation are a problem to be studied in thermomechanics, and in metallurgy as well (Segul et al., 1992).
3. Analysis of cyclic deformation/transformation During thermal a n d / o r mechanical cyclic loading the austenite/martensite phase interfaces travel forward and backward in the alloy. The dislocations are inevitably piled up cycle by cycle around the defects scattered in the alloy. From direct observations of the dislocations metallurgists have concluded that the microscopic residual stresses are induced there and are accumulated cycle by cycle (Miyazaki et al., 1986). They have also shown that the local residual stresses trigger off the formation and accumulation of the martensite in the vicinity of the defects, and that this local martensite does not take part in the subsequent martensitic/reverse transformations (Li and Ansell, 1983; Perkins and Muesing, 1983). In order to take into account these metallurgical phenomena observed during cyclic loading we introduce into the theory formulated in Section 2 the microscopic residual stress trir and the corresponding microscopic residual strain eir, both accumulated in the alloy due to cycling, as the internal variables. We also define as another internal variable the macroscopic volume fraction of the martensite phase, £ir, which is obtained by averaging the local martensite phase explained above over the whole specimen. This martensite is understood not to take part in the subsequent transformations.
K. Tanaka et al. / Mechanics of Materials 19 (1995) 281-292
The alloy is then assumed to have a proper condition to start the transformations, which depends on the stress and the temperature actually induced at the local material point considered and never changes even under cyclic loading. The alloy behavior is, therefore, still governed by Eq. (1) and Eqs. (7)-(10) or Eqs. (11)-(14), but the stress cr and the strain E must now be understood as the local values. The global stress ,~ and the global strain E measured in the macroscopic tests are proposed to be determined from the following compatibility formula:
285
"d
Number of cycles N Fig. 2. Accumulation of residual stress during cycling (schematic).
O" = ~¢ -[- Orir , e = E -
Eir.
(15)
The evolution of the residual stress and strain are assumed to be governed by the evolution equations O;'ir = ( S - O'ir)///-' , • = ( n / A ) ( c r i r / A ) n - 1 . Orir -[- Eooir , Eir
(16)
where S, g, A and n are the material parameters to be determined from the experiments. The time derivative (denoted by the dot) is carried out with respect to the "intrinsic" time z which flows only when the transformations take place. Since the cycle dependence, not the time dependence, of the thermomechanical behavior is usually of interest in the shape memory alloys, the derivative should be carried out with respect to the "intrinsic" number of cycles. Eq. (16) 2 is generalized from the former equation presented by the same authors (Tanaka et al., 1992) so as to be consistent with any temporal transformation. The term ~ooir represents the case observed in an Fe-based polycrystalline shape memory alloy (Tanaka et al., 1993), in which the local residual strain still increases even after the accumulation of the local stress has almost stopped. If all the material parameters are assumed to be constant for simplicity, Eq. (16) 1 can be solved to be Orir= S[1 - exp( - T / u ) ] ,
(17)
which clearly shows the accumulation of the residual stress gi~ during cycling (cf. Fig. 2). The
material parameter S corresponds to the final limit value of the local residual stress, while u governs the "speed" of the accumulation. Evolution of the residual martensite phase due to local residual stress is, for simplicity, assumed to be given by ~ir = ~*O'ir
(18)
with a constant material parameter ~. Micromechanical verification of the assumptions (16) and (18) is the next theme to be investigated.
4. Numerical
illustrations
In order to show how the theory developed works, at least qualitatively, uniaxial deformation analyses are carried out under various thermomechanical loadings. Unless otherwise stated, the material data tabulated in Table 1 are used in the simulation for alloy A and alloy B. The "initial" transformation lines of the alloys are shown in Figs. 3 and 4. Although the data for the alloy A are partly taken from experiments in a Cu-based polycrystalline alloy (Eisenwasser and Brown, 1972) and for alloy B from experiments in an Fe-based polycrystalline alloy (Tanaka et al., 1993), the results given here should be understood to be a qualitative description of the behavior in shape memory alloys in general. Quantitative discussion for each practical alloy is another theme in the future.
K. Tanaka et al. /Mechanics o f Materials 19 (1995) 281-292
286
4.1. Full loops during isothermal cyclic loading
Table 1 Material parameters Alloy A D [MPa] /2 [MPa] 19 [ M P a / K ] M t [K] M s [K] A s [K] A f [K] cA [MPa/K] c M [MPa/K] S [MPa] v A [MPa] n -~ [ M P a - 1] E~ir
7.0× -7.0× -7.0× 233 243 253 263 1.5 1.5 40 1.5 2.0× 1.5 1.5 × 0
Alloy B 103 102 10 - 2
When a specimen is subjected to cyclic loading under isothermal condition, stress-strain hysteresis loops (i.e. pseudoelastic loops) are observed. According to the experimental observation in a TiNi alloy by Tobushi et al. (1991) the loop shifts to the larger strain side and the lower stress side with the number of cycles. Fig. 5 illustrates the stress-strain hystereses in alloy A at a test temperature T h = 303 K under the cyclic loading between 0 and 120 MPa, which fully covers the martensitic/reverse transformation zones. In the figure the elastic response above the martensite finish stress is omitted for simplicity. The hysteresis loop shifts with the number of cycles and tends gradually to a limit stationary loop. The characteristics, the monotonic decrease of the transformation start and finish stresses with the number of cycles, may be so rephrased phenomenologically that the transformation lines drawn in the global stress ,~ - temperature T plane shifts to the higher temperature side. The applied stress ,~ necessary to start the transformation decreases with the number of cyclessince the microscopic residual stresses accumulated during cycling elevates the stress state, as shown in Fig. 2.
1.7× 105 - 1.7x 104 -3.4 3 188 293 458 2.5 2.5 10 15 1.0× 103 1.0 1.5 × 10-3 a = 3.5 × 10-9 b = 30
103 10-3
Alloy A
60 M e M, A~
2[,
e~
30
0
Af
20
270
22(
4.2. Full loops during isostatic cyclic loading
Temperature T, K Fig. 3. Transformation lines (alloy A).
Alloy B 400 Mf
M s
/ 200 r r,o
0 100
300 Temperature T, K
Fig. 4. Transformation lines (alloy B).
7
5OO
The strain-temperature hysteresis loops are obtained when a specimen is subjected to a temperature cycling between the minimum temperature Train and maximum temperature Tm~x under a constant hold stress "~n. If the temperature range AT = Tmax - Tmin is wide enough to cover the whole martensitic/reverse transformation zones during the whole process of cycling, the full loops are obtained as given in Fig. 6 for the alloy A, where the results are shown for the two types of loading; Xh=19"5MPa,
Tmi. = 2 4 8 K ,
Xh = 49.4 MPa,
Tmin =
268 K,
Tm~=313K, Tm~x = 333 K.
The loops shift to the higher strain and higher temperature sides, and finally converge to a limit
287
K. Tanaka et al. / Mechanics of Materials 19 (1995) 281-292
Alloy A 1 2 0 [ Th = 303 K
M
60
0
0.015
0.03
prior to the tests to have a limit stationary loop are always used for the experimental study of the subloops. The coupling of the cyclic effect and the incomplete transformations will be discussed in Section 4.4. Employing the assumption for the transformation start temperatures in Eqs. (7) and (11) for alloy A;
=
1[
{ms+
(As-ms)(1 --*n0)}
Strain E Fig. 5. Stress-strain hysteresis during cyclic isothermal loading (alloy A).
stationary loop. The position of the loops in the strain-temperature plane depends on the hold stress.
4.3. Subloops
Alloy A 0.02
=
_ . ~ ~ 9 . 4
M~a
0.01
0 240
(
Ms +
ln,,0, /1
'
1[
a(SCM0) = ~ {As + (Ms - A s ) ( 1 - ~:M0)}
+ A s + (A s - A f ) ln(lO0)
If the range of the cyclic thermomechanical loading does not fully cover the transformation zones, the martensitic a n d / o r reverse transformations are not complete in each cycle, resulting in subloops. To simplify the discussion the cyclic effect discussed in Section 3 is not taken into account for the moment. In other words, the alloy behavior with a limit stationary loop is considered here. Actually speaking, the specimens trained
m
+
I
Xh = 19.5 MPa I I 290
340
Temperature T, K Fig. 6. Strain-temperature hysteresis during cyclic isostatic loading (alloy A).
which corresponds to the thick solid curves in Fig. 2, the strain-temperature subloops in Figs. 7 and 8 are obtained for the hold stress 2h = 22.5 MPa. Fig. 7 represents the results for the isostatic loading with the different values of Tmin between Me and M s and a constant Tm~, = 278 K which is equal to the reverse transformation finish temperature under "~h- The martensitic transformation is incomplete in the process of cooling while the reverse transformation completes in each cycle. Fig. 8, on the other hand, shows the cases in which the reverse transformation stops on the way while the martensitic transformation always completes. The martensite and austenite start temperatures determined from the figures, which actually corresponds to Eq. (19), are shown by the broken lines in Fig. 9. The figure qualitatively explains the data of Amengual et al. (1994). For the same alloy the stress-strain subloops given in Figs. 10 and 11 are calculated at Th = 268 K when the martensitic and reverse transformation is incomplete, respectively, in the process of the isothermal loading-unloading. The transformation start stress in Figs. 10 and 11 is plotted with the broken lines in Fig. 12.
288
K. Tanaka et al. /Mechanics of Materials 19 (1995) 281-292
Alloy A 0.016 ~ m .~
0.008
Alloy A
]
~
2
60
~ = 22.5 MPa 7
8
K
-
Ix,
M
0 240
1
I 260
30
L
I
I
280
I
0.01
Temperature T, K
0.02
Strain E
Fig. 7. Strain-temperature subloops under an incomplete martensitic transformation (alloy A).
Fig. 10. Stress-strain subloops under incomplete martensitic transformation (alloy A).
Alloy A
Alloy A 0.016 _
=
60
] ~
~ = 22.5 MPa
t~
~4
0.008 -
30
re]
0 240
I
I 260
S I
I
T h = 268 K I
280
Fig. 8. Strain-temperature subloops under an incomplete reverse transformation (alloy A).
Fig. 11. Stress-strain subloops under incomplete reverse transformation (alloy A).
Alloy A
~
0.008 r/'J
0 240
f
Alloy A 60 5 dPa
M
30
?=
I
260
0.02
Strain E
Temperature T, K
0.016
I
0.01
I
280
Temperature T, K Fig. 9. Transformation start temperature (alloy A).
0
I
0.01
I
0.02
Strain E Fig. 12. Transformation start stress (alloy A).
K Tanaka et aL/Mechanics of Materials 19 (1995) 281-292
To show the stress-strain-temperature response predicted by the thermodynamic theory (Miiller and Xu, 1991; Raniecki et al., 1992; Brandon and Roger, 1992), which corresponds to the thin straight line in Fig. 2, is a simple task.
289
Alloy A 0.016
[-O e~
0.008
4. 4. Coupling of cyclic effect and incomplete transformations Eh = 19.5 MPa The alloy response due to incomplete transformations during cyclic loading has of course to be analyzed as a problem of subloops, but by taking account of the cyclic effect formulated in Section 3. If we regard the M s, Mf, A s and Af temperatures in Eq. (19) be the transformation start temperatures determined for the fully hysteresis loop in each cycle, the whole discussion on the subloops in Section 2 can cooperate without any problem with the theory on the cyclic deformation developed in Section 3. Some illustrative examples are given below for alloy A. When the test temperatures Th is low enough during isothermal loading, the reverse transformation lines cross over the test temperature on the way of cyclic loading. Fig. 13 shows the alloy response in such a case mechanically cycled at T h = 268 K (note Af = 263 K). As the cycle progresses the stress-strain hysteretic behavior gradually transfers from the full pseudoelasticity explained in Section 4.1 to the partial pseudoelasticity in which the strain recovers only partly
Alloy A 60 T. = 268 K
M
30
r~ 0
0.01
0.02
Strain E Fig. 13. Stress-strain hysteresis under a test temperature just above Af (alloy A).
0 240
I
Tm~ = 283 K I
I
270
300
Temperature T, K Fig. 14. Strain-temperature hysteresis for an upper limit temperature just above Af (alloy A).
during unloading due to the incomplete reverse transformation. Subloops are then formed under the condition of decreasing ~M0 cycle by cycle. It should be noted that the martensitic transformation is always complete during loading since the upper limit stress of cycling is high enough. The strain range of the hysteresis loop quickly decreases with cycling, due partly to the cyclic effect but mainly to the decrease of the range in reverse transformation. The transition stops just when the A s reaches the test temperature. The alloy behavior exhibits the shape memory effect because it does not undergo the reverse transformation in the process of unloading. The elastic loading-unloading of the martensite phase is finally observed for the subsequent cycling. This phenomenon has been reported in a Cu-based alloy (Vacher and Lexcellent, 1990). Let us next consider the behavior of the alloy subjected to isostatic loading in which Tma~ is just above the austenite finish temperature while Tmin is low enough than the martensite finish temperature. The strain-temperature hysteretic behavior for alloy A is given in Fig. 14 for the isostatic loading path in Fig. 3 under -~n = 19.5 MPa with Tmin = 248 K and Tmax = 283 K. As explained in Section 4.2, the full loop shifts to the higher strain and higher temperature sides at the first stage of cycling. After the austenite finish temperature reaches a n d / o r passes Tm~x, the reverse transformation is always incomplete during heat-
K. Tanaka et al. /Mechanics of Materials 19 (1995) 281-292
290
ing. The strain recovery during heating becomes, therefore, more imperfect as the cycle progresses. This is the reason why the lower branch of the hysteresis loops goes up quickly after the 4th cycle while the upper branch still continues to rise slowly due to the cyclic effect. The hysteresis finally converges to a linear thermal c o n t r a c t i o n / expansion line when the austenite start temperature reaches Tmax. The phenomenon have actually been observed in a TiNi alloy by Tobushi et al. (1991).
Alloy B 450 I Ema~=350MPa
M
Th =288K
250
50 288
4.5. Hysteresis under thermomechanical loading 388 The Fe-based shape memory alloys usually have wide transformation zones, as shown in Fig. 4, compared to the other TiNi or Cu-based shape memory alloys; the value of Ms - My and Af - A ~ are of the order of 102 K in the usual case. And the martensitic transformation zone and the austenite transformation zone are apart to each other compared with these alloys. In the Fe-based alloys, therefore, isothermal pseudoelasticity is difficult to observe. The thermomechanical loading illustrated in Fig. 15 is considered to realize the cyclic martensitic/reverse transformations, each cycle of which is composed of a mechanical loading up t o "~max at the test temperature T h followed by a thermal loading up to Tmax under the hold stress Xh. Fig. 16 shows the stress-strain-temperature hysteresis loops in alloy B during thermomechan-
;0 MPa : 473 K
E
¢.
488 0
0.0l
0.02
0.03
0.04
--] 0.05
Strain E Fig. 16. Stress-strain-temperature hysteresis during thermomechanical loading (alloy B).
ical cyclic loading characterized by T h -- 288 K,
Xmax =
350 MPa,
X h =
50 MPa,
Tma~ = 473 K. The transformation kinetics in alloy B is heuristically, as determined from the experimental data (Tanaka et al., 1993), so that Eq. (12) is replaced by = 1 - 1.05~:A0/{1 + exp[bMCM(/X(~A0 ) -- T ) +bM~r + In 0.05]}
(20)
for the martensitic transformation, and Eq. (8) by Zma
= 1.05~M0/{1 + exp[bACA(a(,~MO ) -- T) +bA~r + In 0.051}
,.a
Zh
//-Th
(21)
for the reverse transformation. The strain Eo~ir in the evolutional Eq. (16) 2 is expressed as T~
Temperature T Fig. 15. Thermomechanicalcyclicloading (schematic).
e=ir = a ( N h + b)
(22)
in order to take into account the effect of the hold stress Nh. The value of the material parameters a and b is tabulated in Table 1.
1(2 Tanaka et al. / Mechanics of Materials 19 (1995) 281-292
Alloy B 0.004 - -
_
0.002
0
20
291
isothermal stress-strain hysteresis and the isostatic strain-temperature hysteresis during cycling, and their convergence to a limit loop. The formation of subloops due to the incomplete transformation was also well demonstrated, both in the isothermal and isostatic cases. The theory was successfully generalized to describe the hysteretic behavior in an Fe-based shape memory alloy during thermomechanical loading.
40
Number of cycles N Fig. 17. Change in residual strain during thermomechanical loading (alloy B).
It should be noted that the martensitic transformation is always incomplete in each cycle while the reverse transformation is complete in the thermal loading process. A limit stationary loop is not obtained, but the final loop continues shifting without changing its shape to the higher strain side by a certain amount cycle by cycle. The hold stress Zh is an important factor for predicting the progress of strain, as shown in Fig. 17 for the residual strain ER defined in the figure. These phenomena have actually been found in an Febased polycrystalline shape memory alloy (Tanaka et al., 1993).
5. Conclusions A phenomenological theory has been developed to describe the uniaxial deformation in shape memory alloys during cyclic thermal a n d / o r mechanical loading. The cyclic effect on the hysteretic behavior and the effect of the prior incomplete transformation on the subloops were the topics to be investigated. In order to characterize the microscopic processes progressing in the alloys during cyclic loading the local residual stress, local residual strain and the volume fraction of the martensite phase which takes no part in the subsequent transformations, were introduced as the internal variables. The simulation by means of the theory developed has clearly explained the shift of the
Acknowledgements Part of this work was financially supported by the Amada Foundation for Metal Work Technology, Japan as well as by the Grant-in-Aid for Scientific Research (No. 04550093) through the Ministry of Education, Science and Culture, Japan. One of the authors (K. Tanaka) is grateful for the travel grant from the Japan Society for Promotion of Sciences, which enabled the authors to complete this work.
References Ashby, M.F. and D.R.H. Jones (1986), Engineering Materials 2: An Introduction to Microstructures, Processing and Design, Ch. 8, Pergamon Press, New York. Amengual, A., E. Cesari and C. Segul (1993), Subloop behaviour in the thermoelastic martensitic transformations, in: C.M. Wayrnan and J. Perkins, eds., Proc. ICOMAT 92, Monterey Institute for Advanced Studies, Monterey, p. 377. Brandon, D. and R.C. Rogers (1992), Constitutive laws for pseudo-elastic materials, J. Intel. Mater. Syst. Struct. 3, 255. Contardo, L. and G. Gu~nin (1990), Training and two way memory effect in C u - Z n - A I alloy, Acta Metall. Mater. 38, 1267. Duerig, T.W., K.N. Melton, D. St6ckel and C.W. Wayman, eds. (1990), Engineering Aspects of Shape Memory Alloys, Butterworth-Heinemann, London. Eisenwasser, J.D. and L.C. Brown (1972), Pseudoelasticity and the strain-memory effect in C u - Z n - S n alloy, Metall. Trans. 3, 1359. Falk, F. (1989), Pseudoelastic stress-strain curves of polycrystalline shape memory alloys calculated from single crystal data, Int. J. Eng. Sci. 27, 277. Fischer, F.D. and K. Tanaka (1992), A micromechanical model for the kinetics of martensitic transformation, Int. Z Solids Struct. 29, 1723.
292
IC Tanaka et al. / Mechanics of Materials 19 (1995) 281-292
Funakubo, H., ed. (1984), Shape Memory Alloys, Gordon and Breach, New York. Kaufman, L. and M. HiUert (1992), Thermodynamics of martensitic transformations, in: G.B. Olson and W.S. Owen, eds., Martensite, ASM International, Ch. 4. Kestin, J. and J.R. Rice (1970), Paradoxes in the application of thermodynamics to strained solids, in: E.B. Stuart, B. Gal-Or and A.J. Brainard, eds., A Critical Review of Thermodynamics, Mono Book Corp., Baltimore, p. 275. Li, J.C. and G.S. Ansell (1983), The effect of thermal cycling on the thermoelastic martensitic transformation in a C u Z n - A I alloy, Metall. Trans. A 14A, 1293. Miyazaki, S. and K. Otsuka (1989), Development of shape memory alloys, ISIJ Int. 29, 353. Miyazaki, S., T. Imai, Y. Igo and K. Otsuka (1986), Effect of cyclic deformation in the pseudoelasticity characteristics of TiNi alloys, Metall. Trans. A 17A, 115. Miiller, I. (1989), On the size of the hysteresis in pseudoelasticity, Contin. Mech. Thermodyn. 1, 125. Miiller, I. and H.B. Xu (1991), On the pseudo-elastic hysteresis, Acta Metall. Mater. 36, 263. Paskal, Yu.I. and L.A. Monasevich (1981), Hysteresis features of the martensitic transformation of Titanium Nicklide, Phys. Met. Metall. 53, 95. Patoor, E., A. Eberhardt and M. Berveiller (1987), Potentiel pseudoelastique et plasticite de transformation martensitique dans les mono- et polycristaux metalliques, Acta Metall. 35, 2779. Perkins, J., ed. (1975), Shape Memory Effects inAlloys, Plenum Press, New York. Perkins, J. and W.E. Muesing (1983), Martensitic transformation cycling effects in Cu-Zn-AI shape memory alloys, Metall. Trans. A 14A, 33. Raniecki, B., C. LexceUent and K. Tanaka (1992), Thermodynamic model of pseudoelastic behaviour of shape memory alloys, Arch. Mech. 44, 261. Rice, J.R. (1971), Inelastic constitute relations for solids: an internal variable theory and its application to metal plasticity, J. Mech. Phys. Solids 19, 433.
Robinson, J.S. and P.G. McCormick (1990), Shape memory in an F e - M n - S i alloy, Mater. Sci. Forum 56-58, 649. Sade, M., K. Halter and E. Hornbogen (1988), The effect of thermal cycling on the transformation behaviour of F e Mn-Si shape memory alloys, Z. MetaUkde. 79, 487. Segul, C., E. Cesari and J. Pons (1992), Phenomenological modelling of the hysteresis loop in thermoelastic martensitic transformations, Mater. Trans. JIM 33, 650. Tan, Shiming and Shiwei Yang (1992), Effect of pre-strain on shape memory behaviour of an Fe-Mn-Si-Cr-Ni-Co alloy, Scripta Metall. Mater. 27, 229. Tanaka, K. (1990), A phenomenological description on thermomechanical behavior of shape memory alloys, J. Pressure Vessel Technol. 112, 158. Tanaka, K. (1991), Analysis of recovery stress and cyclic deformation in shape memory alloys, in: O. Briiller, V. Mannl and J. Najar, eds., Advances in Continuum Mechanics, Springer, Berlin-Heidelberg-New York, p. 441. Tanaka, K., T. Hayashi, Y. Itoh and H. Tobushi (1992), Analysis of thermomechanical behavior of shape memory alloys, Mech. Mater. 13, 207. Tanaka, K., T. Hayashi, Y. Aida and H. Toshushi (1993), Thermomechanical behavior of an F e - C r - N i - M n - S i polycrystalline shape memory alloy, J. Intel. Mater. Syst. Struct. 4, 567. Tobushi, H., H. Iwanaga, K. Tanaka, T. Hori and T. Sawada (1991), Deformation behavior of TiNi shape memory alloy subjected to variable stress and temperature, Contin. Mech. Thermodyn. 3, 79. Vacher, P. and C. Lexcellent (1990), Comportent pseudo61astique d'un alliage polycrystallin Cu Zn A1, ~ m6moire de forme, Rev. Phys. AppL 25, 783. Wei, Zhongguo and Dazhi Yang (1988), On the hysteresis loops and characteristic temperatures of thermoelastic martensitic transformations, Scr. Metall. 22, 1245. Xu, Huibin and Shusong Tan (1991), Relation between temperature hysteresis and recovery strain in a CuZnAI alloy, Scr. Metall. Mater 25, 1507.
Mechanics of Materials 19 (1995) 281-292
Phenomenological analysis on subloops and cyclic behavior in shape memory alloys under mechanical and/or thermal loads K. Tanaka
a,1,
F. Nishimura
a,
T. Hayashi a, H. Tobushi b, C. Lexcellent c
a Department of Aerospace Engineering, Tokyo Metropolitan Institute of Technology, Hino / Tokyo, Japan b Department of Mechanical Engineering, Aichi Institute of Technology, Toyata, Japan c Universitd de Franche-Comt~ UFR des Sciences et des Techniques, Laboratoire de Mdcanique appliqude Associd au CNRS, Besan~on, France Received 16 March 1993; revised version received 28 February 1994
Abstract
A theoretical framework is presented, from the phenomenological point of view, for the cyclic uniaxial deformation in shape memory alloys subjected to the thermal and/or mechanical loads by introducing three internal variables; the local residual stress and strain and the volume fraction of the martensic phase accumulated during cyclic forward and reverse martensitic transformations. The cyclic effect on the stress-strain and strain-temperature hysteresis loops is discussed. The subloops due to incomplete transformations are also analyzed by assuming the transformation starting stress or temperature which depends on the preloading. Numerical results explain qualitatively well the observations on the thermomechanical behaviors of shape memory alloys.
1. Introduction
The thermomechanical response of shape memory alloys is directly associated with the progress of elementary metallurgical processes occurring under thermal a n d / o r mechanical applied loads such as the martensitic transformation or its reverse transformation and the reversible movement of the twin boundaries (Perkins, 1975; Funakubo, 1984; Duerig et al., 1990). Based on the intensive metallurgical studies on the micro-
i Corresponding author: Professor Dr. Kikuaki Tanaka, Department of Aerospace Engineering, Tokyo Metropolitan Institute of Technology, Asahigaoka 6-6, J-191 Hino/Tokyo, Japan.
scopic mechanism of the "memory", macroscopic theories have been proposed, starting from different levels of microstructure in alloys, to predict the thermomechanical behaviors of shape memory alloys (Patoor et al., 1987; Falk, 1989; Miiller, 1989; Tanaka, 1990; Fischer and Tanaka, 1992). The stress-strain-temperature relation, the recovery stress induced in the heating process and the response under thermomechanical constraints are some of the subjects to be discussed from a theoretical point of view. The full stress-strain and strain-temperature hysteresis loops, together with their subloops realized during incomplete transformations for a smaller stress, strain or temperature range, are also being investigated in the course of constructing a unified theory of the strong interaction of
0167-6636/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSD! 0167-6636(94)00038-1
282
K. Tanaka et aL / Mechanics of Materials 19 (1995) 281-292
thermomechanics and transformation, based fully on the continuum thermodynamics (Miiller and Xu, 1991; Raniecki et al., 1992). The experimental observations for the subloops during thermal or mechanical loading contradict each other with respect to the transformation start stress or temperature after an incomplete transformation (Paskal and Monasevich, 1981; Wei and Yang, 1988; Mfiller and Xu, 1991; Amengual et al., 1994). The deformation/transformation response of shape memory alloys exhibits very often a strong cycle dependence, which has always a negative effect on the practical use of the shape memory devices because they must generally show a stable output (Duerig et al., 1990); the stroke, force or the transformation temperatures themselves. In order to acquire a stable response of the alloy during cyclic loading some metallurgical treatments are proposed, and a so-called thermomechanical training is often performed before producing an actual device. The cycle dependence of stress-strain hysteresis loops have been investigated in different alloys (Miyazaki and Otsuka, 1989). The strain-temperature hysteresis loops also depend on the cyclic loading under a constant applied stress (Li and Ansell, 1983; Sade et al., 1988; Contardo and Gu6nin, 1990; Xu and Tan, 1991; Tobushi et al., 1991). Both loops shift gradually with the cyclic progress, and finally reach stable limit loop after scores of cycles. Metallurgical studies have clearly revealed that dislocations are piled up around the defects scattered in the alloy during the cyclic forward and reverse martensitic transformations and that the martensite is locally induced by the local stress accumulated there (Miyazaki et al., 1986). In this paper a macroscopic theoretical framework is presented to explain the hysteresis behavior of shape memory alloys during the thermal a n d / o r mechanical cyclic loadings. The cyclic effect on the stress-strain and strain-temperature hysteresis loops and the subloops due to incomplete transformations arc the main topics to be discussed. The macroscopic theory on the behavior of shape memory alloys presented by the authors (Tanaka, 1990, 1991) is generalized by introducing three internal variables; the local
stress and strain and the residual martensic phase accumulated irreversibly due to the cyclic forward/reverse martensitic transformation. Numerical simulations are carried out to explain qualitatively the experimental results for the thermomechanical cyclic loadings.
2. Constitutive relation
The authors (Tanaka, 1990, 1991; Tanaka et al., 1992) have shown from a phenomenological point of view that the uniaxial thermomechanical behavior of shape memory alloys can be well described by the constitutive relation consisting of the constitutive equation in rate form ~ = D ~ + O~/"+ D~,
(1)
and the transformation kinetics for the martensitic transformation
1-f
= bMCMT
-- b M d - >__0,
(2)
or, for the reverse transformation, -- -- = b A C A T -- bAd" >_ O,
(3)
where or, • and T stand for the stress, strain and temperature, respectively, while D denotes the Young's modulus and -O/D and -DID are the coefficient of linear expansion and the strain due to transformation which is often called the eigenstrain in micromechanics, respectively. The symbol ~ represents the volume fraction of the martensite phase. The volume fraction of the parent phase, the austenite phase actually, is, therefore, 1 - s~. The material parameters in the kinetics, b M, CM, bA and CA, can be determined from experiments. Their physical meaning will become clear later. Eq. (1) governs the thermomechanical behavior of the alloys, while Eq. (2) or (3) describes the progress of the martensitic transformation or the reverse transformation under a given thermomechanical process. The inequalities in Eqs. (2) and (3) determine which transformation, the martensitic or the reverse one, or none of them, pro-
K~ Tanaka et al. /Mechanics of Materials 19 (1995) 281-292
gresses at each instant of the thermomechanical loading. All material parameters in the equations can be determined by a series of simple experiments. Let us now consider minutely a reverse transformation process starting from the initial values ((r0, To, ~M0), where ~M0 stands for the fraction of martensite at the start of the reverse transformation. If the transformation kinetics (3) is integrated from the initial values to the current values (cr, T, ~) for constant material parameters bA and c A, one finally obtains an explicit form of the transformation kinetics;
283
Af A,
M, I
0
I
0.5
Mf
I
1
Volume fraction of residual martensite ~ 0 Fig. 1. Transformation start temperatures (schematic).
sc = ~M0 exp[bACA(To -- T ) + bA(O"-- o'0) ] .
(4)
The trace (o-0, T0) in the stress-temperature plane represents a line, the reverse transformation start line or the As-line, at which the reverse transformation starts. Now if we assume that the A~-line does not depend on the amount of residual martensite ~M0 at the start of the reverse transformation, and can be expressed by ~o =
CA( To - As),
(5)
where A s stands for the austenite start temperature under free stress, the final transformation kinetics can be written as ~: = ~:M0 e x p [ b A c A ( A s -- T ) + bAO"] .
(6)
It is, however, much more rational to assume that the A~-line depends on the amount of residual martensite through the form; cro = cA(To -- a(~M0)),
(7)
where a(~M o) represents the austenite start temperature that depends on the amount of residual martensite. The slope of the As-line, CA, is assumed to be constant. In this case, instead of Eq. (6), the following final transformation kinetics can be obtained by integrating Eq. (3): s¢ = ~:M0 exp[bACA(a(~MO) -- T ) + bA~r] .
(8)
The reverse transformation finish line, the Afline, can now be read as 1 (r = CAT-- 7---In(100SCM0) - CAOt(SeM0),
OA
(9)
where ~ = 0.01 is regarded as the completion of the reverse transformation following the usual understanding in metallurgy (Ashby and Jones, 1986). When the Af-line is assumed, for simplicity, to be independent of ~M0, the material parameter ba is determined by ln(100~:m0 ) bA = CA(A f _ a ( ~ M 0 ) ) ,
(10)
where Af stands for the austenite finish temperature under stress free conditions. The austenite start temperatures in both cases are schematically illustrated in Fig. 1. The previous analysis by the present authors was based on Eqs. (5) and (6), where the As-line is a fixed line in the stress-temperature plane, being independent of the amount of residual martensite. The broken line in Fig. 1 shows this case. Paskal and Monasevich (1981) have actually observed such behavior in a TiNi shape memory alloy. Amengual et al. (1994) have, on the other hand, very clearly shown in C u - Z n - A l and C u - A I - M n shape memory alloys that the austenite start temperature, and the martensite start temperature as well, is strongly dependent on the amount of preloading, strictly speaking, on ~M0 in our context, during the isostatic thermal loading. The present authors (Tanaka et al., 1993) have also observed that the same is true in an Fe-based polycrystalline shape memory alloy. This case corresponds to the solid curve in Fig. 1. The dependence of the A s- and Af-lines on the preloading
K. Tanaka et aL /Mechanics of Materials 19 (1995) 281-292
284
has also been reported in an F e - M n - S i alloy (Robinson and McCormick, 1990) and an F e M n - S i - C r - N i - C o alloy (Tan and Yang, 1992). It should be noted that the thin solid line in the figure represents the shift in the austenite start temperature predicted by the thermodynamic theory of shape memory alloys (see, e.g., MOiler, 1989; MOiler and Xu, 1991; Raniecki et al. 1992). If a similar assumption, i.e. ~ o = C M ( TO --
~'(~A0)),
(11)
is imposed for the martensitic transformation with a constant material parameter C M , where /*(~:A0) stands for the martensite start temperature being dependent on the amount of residual austenite CA0 at the start of the martensitic transformation, the final transformation kinetics can be written as = 1 -~A0 exp[bMCM(/Z(~A0) -- T) + bMO"] . (12) When the martensite finish line or the Mf-line, determined from the condition £ = 0.99 (Ashby and Jones, 1986), is assumed to be immovable, its expression in the tr-T plane and the material parameter b M are given by o" = C M T --
1 ~M ln(100~:a0) -- CM/~(£A0) ,
ln(100sCn0) b M = CM(Mf _/Z(£A0)),
(13)
(14)
where Mf (and Ms, which appears later) stand for the martensite finish (and start) temperatures under free stress. The dependence of the martensite start temperature on the amount of residual austenite, £A0 = 1 - £M0, is schematically given in Fig. 1, where the solid line represents the theory to be developed here while the broken line and thin solid line correspond to the theory with a constant value of the martensite start temperature and the thermodynamic theory, respectively. The present paper is devoted to the consequences of Eqs. (7)-(10) and Eqs. (11)-(14). The behavior of shape memory alloys during thermal a n d / o r mechanical cyclic loadings is the topic to be investigated.
A comment on the transformation start lines: as Raniecki et al. (1992) have clearly discussed, the transformation start lines, both martensitic and reverse, can be derived from the driving force of the transformation (Kaufman and Hillert, 1992) as a condition of constrained equilibrium in continuum thermodynamics (Kestin and Rice, 1970; Rice, 1971). Their thermodynamic theory of transformation could, therefore, be extended rationally to include the effect of preloading by introducing additional terms into the driving force. Derivation of such a new driving force and its physical interpretation are a problem to be studied in thermomechanics, and in metallurgy as well (Segul et al., 1992).
3. Analysis of cyclic deformation/transformation During thermal a n d / o r mechanical cyclic loading the austenite/martensite phase interfaces travel forward and backward in the alloy. The dislocations are inevitably piled up cycle by cycle around the defects scattered in the alloy. From direct observations of the dislocations metallurgists have concluded that the microscopic residual stresses are induced there and are accumulated cycle by cycle (Miyazaki et al., 1986). They have also shown that the local residual stresses trigger off the formation and accumulation of the martensite in the vicinity of the defects, and that this local martensite does not take part in the subsequent martensitic/reverse transformations (Li and Ansell, 1983; Perkins and Muesing, 1983). In order to take into account these metallurgical phenomena observed during cyclic loading we introduce into the theory formulated in Section 2 the microscopic residual stress trir and the corresponding microscopic residual strain eir, both accumulated in the alloy due to cycling, as the internal variables. We also define as another internal variable the macroscopic volume fraction of the martensite phase, £ir, which is obtained by averaging the local martensite phase explained above over the whole specimen. This martensite is understood not to take part in the subsequent transformations.
K. Tanaka et al. / Mechanics of Materials 19 (1995) 281-292
The alloy is then assumed to have a proper condition to start the transformations, which depends on the stress and the temperature actually induced at the local material point considered and never changes even under cyclic loading. The alloy behavior is, therefore, still governed by Eq. (1) and Eqs. (7)-(10) or Eqs. (11)-(14), but the stress cr and the strain E must now be understood as the local values. The global stress ,~ and the global strain E measured in the macroscopic tests are proposed to be determined from the following compatibility formula:
285
"d
Number of cycles N Fig. 2. Accumulation of residual stress during cycling (schematic).
O" = ~¢ -[- Orir , e = E -
Eir.
(15)
The evolution of the residual stress and strain are assumed to be governed by the evolution equations O;'ir = ( S - O'ir)///-' , • = ( n / A ) ( c r i r / A ) n - 1 . Orir -[- Eooir , Eir
(16)
where S, g, A and n are the material parameters to be determined from the experiments. The time derivative (denoted by the dot) is carried out with respect to the "intrinsic" time z which flows only when the transformations take place. Since the cycle dependence, not the time dependence, of the thermomechanical behavior is usually of interest in the shape memory alloys, the derivative should be carried out with respect to the "intrinsic" number of cycles. Eq. (16) 2 is generalized from the former equation presented by the same authors (Tanaka et al., 1992) so as to be consistent with any temporal transformation. The term ~ooir represents the case observed in an Fe-based polycrystalline shape memory alloy (Tanaka et al., 1993), in which the local residual strain still increases even after the accumulation of the local stress has almost stopped. If all the material parameters are assumed to be constant for simplicity, Eq. (16) 1 can be solved to be Orir= S[1 - exp( - T / u ) ] ,
(17)
which clearly shows the accumulation of the residual stress gi~ during cycling (cf. Fig. 2). The
material parameter S corresponds to the final limit value of the local residual stress, while u governs the "speed" of the accumulation. Evolution of the residual martensite phase due to local residual stress is, for simplicity, assumed to be given by ~ir = ~*O'ir
(18)
with a constant material parameter ~. Micromechanical verification of the assumptions (16) and (18) is the next theme to be investigated.
4. Numerical
illustrations
In order to show how the theory developed works, at least qualitatively, uniaxial deformation analyses are carried out under various thermomechanical loadings. Unless otherwise stated, the material data tabulated in Table 1 are used in the simulation for alloy A and alloy B. The "initial" transformation lines of the alloys are shown in Figs. 3 and 4. Although the data for the alloy A are partly taken from experiments in a Cu-based polycrystalline alloy (Eisenwasser and Brown, 1972) and for alloy B from experiments in an Fe-based polycrystalline alloy (Tanaka et al., 1993), the results given here should be understood to be a qualitative description of the behavior in shape memory alloys in general. Quantitative discussion for each practical alloy is another theme in the future.
K. Tanaka et al. /Mechanics o f Materials 19 (1995) 281-292
286
4.1. Full loops during isothermal cyclic loading
Table 1 Material parameters Alloy A D [MPa] /2 [MPa] 19 [ M P a / K ] M t [K] M s [K] A s [K] A f [K] cA [MPa/K] c M [MPa/K] S [MPa] v A [MPa] n -~ [ M P a - 1] E~ir
7.0× -7.0× -7.0× 233 243 253 263 1.5 1.5 40 1.5 2.0× 1.5 1.5 × 0
Alloy B 103 102 10 - 2
When a specimen is subjected to cyclic loading under isothermal condition, stress-strain hysteresis loops (i.e. pseudoelastic loops) are observed. According to the experimental observation in a TiNi alloy by Tobushi et al. (1991) the loop shifts to the larger strain side and the lower stress side with the number of cycles. Fig. 5 illustrates the stress-strain hystereses in alloy A at a test temperature T h = 303 K under the cyclic loading between 0 and 120 MPa, which fully covers the martensitic/reverse transformation zones. In the figure the elastic response above the martensite finish stress is omitted for simplicity. The hysteresis loop shifts with the number of cycles and tends gradually to a limit stationary loop. The characteristics, the monotonic decrease of the transformation start and finish stresses with the number of cycles, may be so rephrased phenomenologically that the transformation lines drawn in the global stress ,~ - temperature T plane shifts to the higher temperature side. The applied stress ,~ necessary to start the transformation decreases with the number of cyclessince the microscopic residual stresses accumulated during cycling elevates the stress state, as shown in Fig. 2.
1.7× 105 - 1.7x 104 -3.4 3 188 293 458 2.5 2.5 10 15 1.0× 103 1.0 1.5 × 10-3 a = 3.5 × 10-9 b = 30
103 10-3
Alloy A
60 M e M, A~
2[,
e~
30
0
Af
20
270
22(
4.2. Full loops during isostatic cyclic loading
Temperature T, K Fig. 3. Transformation lines (alloy A).
Alloy B 400 Mf
M s
/ 200 r r,o
0 100
300 Temperature T, K
Fig. 4. Transformation lines (alloy B).
7
5OO
The strain-temperature hysteresis loops are obtained when a specimen is subjected to a temperature cycling between the minimum temperature Train and maximum temperature Tm~x under a constant hold stress "~n. If the temperature range AT = Tmax - Tmin is wide enough to cover the whole martensitic/reverse transformation zones during the whole process of cycling, the full loops are obtained as given in Fig. 6 for the alloy A, where the results are shown for the two types of loading; Xh=19"5MPa,
Tmi. = 2 4 8 K ,
Xh = 49.4 MPa,
Tmin =
268 K,
Tm~=313K, Tm~x = 333 K.
The loops shift to the higher strain and higher temperature sides, and finally converge to a limit
287
K. Tanaka et al. / Mechanics of Materials 19 (1995) 281-292
Alloy A 1 2 0 [ Th = 303 K
M
60
0
0.015
0.03
prior to the tests to have a limit stationary loop are always used for the experimental study of the subloops. The coupling of the cyclic effect and the incomplete transformations will be discussed in Section 4.4. Employing the assumption for the transformation start temperatures in Eqs. (7) and (11) for alloy A;
=
1[
{ms+
(As-ms)(1 --*n0)}
Strain E Fig. 5. Stress-strain hysteresis during cyclic isothermal loading (alloy A).
stationary loop. The position of the loops in the strain-temperature plane depends on the hold stress.
4.3. Subloops
Alloy A 0.02
=
_ . ~ ~ 9 . 4
M~a
0.01
0 240
(
Ms +
ln,,0, /1
'
1[
a(SCM0) = ~ {As + (Ms - A s ) ( 1 - ~:M0)}
+ A s + (A s - A f ) ln(lO0)
If the range of the cyclic thermomechanical loading does not fully cover the transformation zones, the martensitic a n d / o r reverse transformations are not complete in each cycle, resulting in subloops. To simplify the discussion the cyclic effect discussed in Section 3 is not taken into account for the moment. In other words, the alloy behavior with a limit stationary loop is considered here. Actually speaking, the specimens trained
m
+
I
Xh = 19.5 MPa I I 290
340
Temperature T, K Fig. 6. Strain-temperature hysteresis during cyclic isostatic loading (alloy A).
which corresponds to the thick solid curves in Fig. 2, the strain-temperature subloops in Figs. 7 and 8 are obtained for the hold stress 2h = 22.5 MPa. Fig. 7 represents the results for the isostatic loading with the different values of Tmin between Me and M s and a constant Tm~, = 278 K which is equal to the reverse transformation finish temperature under "~h- The martensitic transformation is incomplete in the process of cooling while the reverse transformation completes in each cycle. Fig. 8, on the other hand, shows the cases in which the reverse transformation stops on the way while the martensitic transformation always completes. The martensite and austenite start temperatures determined from the figures, which actually corresponds to Eq. (19), are shown by the broken lines in Fig. 9. The figure qualitatively explains the data of Amengual et al. (1994). For the same alloy the stress-strain subloops given in Figs. 10 and 11 are calculated at Th = 268 K when the martensitic and reverse transformation is incomplete, respectively, in the process of the isothermal loading-unloading. The transformation start stress in Figs. 10 and 11 is plotted with the broken lines in Fig. 12.
288
K. Tanaka et al. /Mechanics of Materials 19 (1995) 281-292
Alloy A 0.016 ~ m .~
0.008
Alloy A
]
~
2
60
~ = 22.5 MPa 7
8
K
-
Ix,
M
0 240
1
I 260
30
L
I
I
280
I
0.01
Temperature T, K
0.02
Strain E
Fig. 7. Strain-temperature subloops under an incomplete martensitic transformation (alloy A).
Fig. 10. Stress-strain subloops under incomplete martensitic transformation (alloy A).
Alloy A
Alloy A 0.016 _
=
60
] ~
~ = 22.5 MPa
t~
~4
0.008 -
30
re]
0 240
I
I 260
S I
I
T h = 268 K I
280
Fig. 8. Strain-temperature subloops under an incomplete reverse transformation (alloy A).
Fig. 11. Stress-strain subloops under incomplete reverse transformation (alloy A).
Alloy A
~
0.008 r/'J
0 240
f
Alloy A 60 5 dPa
M
30
?=
I
260
0.02
Strain E
Temperature T, K
0.016
I
0.01
I
280
Temperature T, K Fig. 9. Transformation start temperature (alloy A).
0
I
0.01
I
0.02
Strain E Fig. 12. Transformation start stress (alloy A).
K Tanaka et aL/Mechanics of Materials 19 (1995) 281-292
To show the stress-strain-temperature response predicted by the thermodynamic theory (Miiller and Xu, 1991; Raniecki et al., 1992; Brandon and Roger, 1992), which corresponds to the thin straight line in Fig. 2, is a simple task.
289
Alloy A 0.016
[-O e~
0.008
4. 4. Coupling of cyclic effect and incomplete transformations Eh = 19.5 MPa The alloy response due to incomplete transformations during cyclic loading has of course to be analyzed as a problem of subloops, but by taking account of the cyclic effect formulated in Section 3. If we regard the M s, Mf, A s and Af temperatures in Eq. (19) be the transformation start temperatures determined for the fully hysteresis loop in each cycle, the whole discussion on the subloops in Section 2 can cooperate without any problem with the theory on the cyclic deformation developed in Section 3. Some illustrative examples are given below for alloy A. When the test temperatures Th is low enough during isothermal loading, the reverse transformation lines cross over the test temperature on the way of cyclic loading. Fig. 13 shows the alloy response in such a case mechanically cycled at T h = 268 K (note Af = 263 K). As the cycle progresses the stress-strain hysteretic behavior gradually transfers from the full pseudoelasticity explained in Section 4.1 to the partial pseudoelasticity in which the strain recovers only partly
Alloy A 60 T. = 268 K
M
30
r~ 0
0.01
0.02
Strain E Fig. 13. Stress-strain hysteresis under a test temperature just above Af (alloy A).
0 240
I
Tm~ = 283 K I
I
270
300
Temperature T, K Fig. 14. Strain-temperature hysteresis for an upper limit temperature just above Af (alloy A).
during unloading due to the incomplete reverse transformation. Subloops are then formed under the condition of decreasing ~M0 cycle by cycle. It should be noted that the martensitic transformation is always complete during loading since the upper limit stress of cycling is high enough. The strain range of the hysteresis loop quickly decreases with cycling, due partly to the cyclic effect but mainly to the decrease of the range in reverse transformation. The transition stops just when the A s reaches the test temperature. The alloy behavior exhibits the shape memory effect because it does not undergo the reverse transformation in the process of unloading. The elastic loading-unloading of the martensite phase is finally observed for the subsequent cycling. This phenomenon has been reported in a Cu-based alloy (Vacher and Lexcellent, 1990). Let us next consider the behavior of the alloy subjected to isostatic loading in which Tma~ is just above the austenite finish temperature while Tmin is low enough than the martensite finish temperature. The strain-temperature hysteretic behavior for alloy A is given in Fig. 14 for the isostatic loading path in Fig. 3 under -~n = 19.5 MPa with Tmin = 248 K and Tmax = 283 K. As explained in Section 4.2, the full loop shifts to the higher strain and higher temperature sides at the first stage of cycling. After the austenite finish temperature reaches a n d / o r passes Tm~x, the reverse transformation is always incomplete during heat-
K. Tanaka et al. /Mechanics of Materials 19 (1995) 281-292
290
ing. The strain recovery during heating becomes, therefore, more imperfect as the cycle progresses. This is the reason why the lower branch of the hysteresis loops goes up quickly after the 4th cycle while the upper branch still continues to rise slowly due to the cyclic effect. The hysteresis finally converges to a linear thermal c o n t r a c t i o n / expansion line when the austenite start temperature reaches Tmax. The phenomenon have actually been observed in a TiNi alloy by Tobushi et al. (1991).
Alloy B 450 I Ema~=350MPa
M
Th =288K
250
50 288
4.5. Hysteresis under thermomechanical loading 388 The Fe-based shape memory alloys usually have wide transformation zones, as shown in Fig. 4, compared to the other TiNi or Cu-based shape memory alloys; the value of Ms - My and Af - A ~ are of the order of 102 K in the usual case. And the martensitic transformation zone and the austenite transformation zone are apart to each other compared with these alloys. In the Fe-based alloys, therefore, isothermal pseudoelasticity is difficult to observe. The thermomechanical loading illustrated in Fig. 15 is considered to realize the cyclic martensitic/reverse transformations, each cycle of which is composed of a mechanical loading up t o "~max at the test temperature T h followed by a thermal loading up to Tmax under the hold stress Xh. Fig. 16 shows the stress-strain-temperature hysteresis loops in alloy B during thermomechan-
;0 MPa : 473 K
E
¢.
488 0
0.0l
0.02
0.03
0.04
--] 0.05
Strain E Fig. 16. Stress-strain-temperature hysteresis during thermomechanical loading (alloy B).
ical cyclic loading characterized by T h -- 288 K,
Xmax =
350 MPa,
X h =
50 MPa,
Tma~ = 473 K. The transformation kinetics in alloy B is heuristically, as determined from the experimental data (Tanaka et al., 1993), so that Eq. (12) is replaced by = 1 - 1.05~:A0/{1 + exp[bMCM(/X(~A0 ) -- T ) +bM~r + In 0.05]}
(20)
for the martensitic transformation, and Eq. (8) by Zma
= 1.05~M0/{1 + exp[bACA(a(,~MO ) -- T) +bA~r + In 0.051}
,.a
Zh
//-Th
(21)
for the reverse transformation. The strain Eo~ir in the evolutional Eq. (16) 2 is expressed as T~
Temperature T Fig. 15. Thermomechanicalcyclicloading (schematic).
e=ir = a ( N h + b)
(22)
in order to take into account the effect of the hold stress Nh. The value of the material parameters a and b is tabulated in Table 1.
1(2 Tanaka et al. / Mechanics of Materials 19 (1995) 281-292
Alloy B 0.004 - -
_
0.002
0
20
291
isothermal stress-strain hysteresis and the isostatic strain-temperature hysteresis during cycling, and their convergence to a limit loop. The formation of subloops due to the incomplete transformation was also well demonstrated, both in the isothermal and isostatic cases. The theory was successfully generalized to describe the hysteretic behavior in an Fe-based shape memory alloy during thermomechanical loading.
40
Number of cycles N Fig. 17. Change in residual strain during thermomechanical loading (alloy B).
It should be noted that the martensitic transformation is always incomplete in each cycle while the reverse transformation is complete in the thermal loading process. A limit stationary loop is not obtained, but the final loop continues shifting without changing its shape to the higher strain side by a certain amount cycle by cycle. The hold stress Zh is an important factor for predicting the progress of strain, as shown in Fig. 17 for the residual strain ER defined in the figure. These phenomena have actually been found in an Febased polycrystalline shape memory alloy (Tanaka et al., 1993).
5. Conclusions A phenomenological theory has been developed to describe the uniaxial deformation in shape memory alloys during cyclic thermal a n d / o r mechanical loading. The cyclic effect on the hysteretic behavior and the effect of the prior incomplete transformation on the subloops were the topics to be investigated. In order to characterize the microscopic processes progressing in the alloys during cyclic loading the local residual stress, local residual strain and the volume fraction of the martensite phase which takes no part in the subsequent transformations, were introduced as the internal variables. The simulation by means of the theory developed has clearly explained the shift of the
Acknowledgements Part of this work was financially supported by the Amada Foundation for Metal Work Technology, Japan as well as by the Grant-in-Aid for Scientific Research (No. 04550093) through the Ministry of Education, Science and Culture, Japan. One of the authors (K. Tanaka) is grateful for the travel grant from the Japan Society for Promotion of Sciences, which enabled the authors to complete this work.
References Ashby, M.F. and D.R.H. Jones (1986), Engineering Materials 2: An Introduction to Microstructures, Processing and Design, Ch. 8, Pergamon Press, New York. Amengual, A., E. Cesari and C. Segul (1993), Subloop behaviour in the thermoelastic martensitic transformations, in: C.M. Wayrnan and J. Perkins, eds., Proc. ICOMAT 92, Monterey Institute for Advanced Studies, Monterey, p. 377. Brandon, D. and R.C. Rogers (1992), Constitutive laws for pseudo-elastic materials, J. Intel. Mater. Syst. Struct. 3, 255. Contardo, L. and G. Gu~nin (1990), Training and two way memory effect in C u - Z n - A I alloy, Acta Metall. Mater. 38, 1267. Duerig, T.W., K.N. Melton, D. St6ckel and C.W. Wayman, eds. (1990), Engineering Aspects of Shape Memory Alloys, Butterworth-Heinemann, London. Eisenwasser, J.D. and L.C. Brown (1972), Pseudoelasticity and the strain-memory effect in C u - Z n - S n alloy, Metall. Trans. 3, 1359. Falk, F. (1989), Pseudoelastic stress-strain curves of polycrystalline shape memory alloys calculated from single crystal data, Int. J. Eng. Sci. 27, 277. Fischer, F.D. and K. Tanaka (1992), A micromechanical model for the kinetics of martensitic transformation, Int. Z Solids Struct. 29, 1723.
292
IC Tanaka et al. / Mechanics of Materials 19 (1995) 281-292
Funakubo, H., ed. (1984), Shape Memory Alloys, Gordon and Breach, New York. Kaufman, L. and M. HiUert (1992), Thermodynamics of martensitic transformations, in: G.B. Olson and W.S. Owen, eds., Martensite, ASM International, Ch. 4. Kestin, J. and J.R. Rice (1970), Paradoxes in the application of thermodynamics to strained solids, in: E.B. Stuart, B. Gal-Or and A.J. Brainard, eds., A Critical Review of Thermodynamics, Mono Book Corp., Baltimore, p. 275. Li, J.C. and G.S. Ansell (1983), The effect of thermal cycling on the thermoelastic martensitic transformation in a C u Z n - A I alloy, Metall. Trans. A 14A, 1293. Miyazaki, S. and K. Otsuka (1989), Development of shape memory alloys, ISIJ Int. 29, 353. Miyazaki, S., T. Imai, Y. Igo and K. Otsuka (1986), Effect of cyclic deformation in the pseudoelasticity characteristics of TiNi alloys, Metall. Trans. A 17A, 115. Miiller, I. (1989), On the size of the hysteresis in pseudoelasticity, Contin. Mech. Thermodyn. 1, 125. Miiller, I. and H.B. Xu (1991), On the pseudo-elastic hysteresis, Acta Metall. Mater. 36, 263. Paskal, Yu.I. and L.A. Monasevich (1981), Hysteresis features of the martensitic transformation of Titanium Nicklide, Phys. Met. Metall. 53, 95. Patoor, E., A. Eberhardt and M. Berveiller (1987), Potentiel pseudoelastique et plasticite de transformation martensitique dans les mono- et polycristaux metalliques, Acta Metall. 35, 2779. Perkins, J., ed. (1975), Shape Memory Effects inAlloys, Plenum Press, New York. Perkins, J. and W.E. Muesing (1983), Martensitic transformation cycling effects in Cu-Zn-AI shape memory alloys, Metall. Trans. A 14A, 33. Raniecki, B., C. LexceUent and K. Tanaka (1992), Thermodynamic model of pseudoelastic behaviour of shape memory alloys, Arch. Mech. 44, 261. Rice, J.R. (1971), Inelastic constitute relations for solids: an internal variable theory and its application to metal plasticity, J. Mech. Phys. Solids 19, 433.
Robinson, J.S. and P.G. McCormick (1990), Shape memory in an F e - M n - S i alloy, Mater. Sci. Forum 56-58, 649. Sade, M., K. Halter and E. Hornbogen (1988), The effect of thermal cycling on the transformation behaviour of F e Mn-Si shape memory alloys, Z. MetaUkde. 79, 487. Segul, C., E. Cesari and J. Pons (1992), Phenomenological modelling of the hysteresis loop in thermoelastic martensitic transformations, Mater. Trans. JIM 33, 650. Tan, Shiming and Shiwei Yang (1992), Effect of pre-strain on shape memory behaviour of an Fe-Mn-Si-Cr-Ni-Co alloy, Scripta Metall. Mater. 27, 229. Tanaka, K. (1990), A phenomenological description on thermomechanical behavior of shape memory alloys, J. Pressure Vessel Technol. 112, 158. Tanaka, K. (1991), Analysis of recovery stress and cyclic deformation in shape memory alloys, in: O. Briiller, V. Mannl and J. Najar, eds., Advances in Continuum Mechanics, Springer, Berlin-Heidelberg-New York, p. 441. Tanaka, K., T. Hayashi, Y. Itoh and H. Tobushi (1992), Analysis of thermomechanical behavior of shape memory alloys, Mech. Mater. 13, 207. Tanaka, K., T. Hayashi, Y. Aida and H. Toshushi (1993), Thermomechanical behavior of an F e - C r - N i - M n - S i polycrystalline shape memory alloy, J. Intel. Mater. Syst. Struct. 4, 567. Tobushi, H., H. Iwanaga, K. Tanaka, T. Hori and T. Sawada (1991), Deformation behavior of TiNi shape memory alloy subjected to variable stress and temperature, Contin. Mech. Thermodyn. 3, 79. Vacher, P. and C. Lexcellent (1990), Comportent pseudo61astique d'un alliage polycrystallin Cu Zn A1, ~ m6moire de forme, Rev. Phys. AppL 25, 783. Wei, Zhongguo and Dazhi Yang (1988), On the hysteresis loops and characteristic temperatures of thermoelastic martensitic transformations, Scr. Metall. 22, 1245. Xu, Huibin and Shusong Tan (1991), Relation between temperature hysteresis and recovery strain in a CuZnAI alloy, Scr. Metall. Mater 25, 1507.