Materials Transactions, Vol. 46, No. 4 (2005) pp. 790 to 797 #2005 The Japan Institute of Metals
Stress-induced Martensitic Transformation and Superelasticity of Alloys: Experiment and Theory Victor A. L’vov1; * , Alexei A. Rudenko1 , Volodymyr A. Chernenko2 , Eduard Cesari3 , Jaume Pons3 and Takeshi Kanomata4 1
Department of Radiophysics, Taras Shevchenko University, Glushkov str.2, Build.5, Kiev 03127, Ukraine Institute of Magnetism, Vernadsky street 36-b, Kiev 03142, Ukraine 3 Department de Fı´sica, Universitat de les Illes Balears, Ctra. de Valldemossa km 7.5, E-07122, Palma, Spain 4 Faculty of Engineering, Tohoku Gakuin University, Tagajo 985-8537, Japan 2
The superelasticity of alloys undergoing a stress-induced martensitic transformation is studied in both experimental and theoretical way. Experimental stress–strain dependencies illustrating the different types of superelastic behavior are taken from Ni–Mn–Ga alloys typifying thermoelastic martensites and then modeled theoretically using a statistical approach to describe the growth of stress-induced martensite in the matrix of parent (austenitic) phase. A good agreement between the experimental and theoretical results is achieved whereby a physical interpretation of the different types of stress–strain dependencies is made and a quantitative evaluation of the main parameters controlling transformational behavior of the alloys is carried out. (Received November 9, 2004; Accepted February 28, 2005; Published April 15, 2005) Keywords: shape memory alloys (SMA), superelasticity, modeling
1.
Introduction
A large number of metallic and non-metallic materials undergoes a first-order transformation of the martensitic type from a high-symmetry (most often, cubic) austenitic phase to a low symmetry (tetragonal, orthorhombic or rhombohedral) martensitic phase. The martensitic transformation (MT) is characterized by the spontaneous shear deformation of crystal lattice1,2) and can be induced by cooling or mechanically stressing the specimen. When the latent heat of transformation is comparable with the energy of spontaneous strains the MT appears as thermoelastic.2–4) In this case the elastic strain energy is one of the main agents controlling the kinetics of MT and morphology of the resultant martensitic state. Due to the tendency to minimization of the total elastic energy of the specimen, the martensite possesses a spatial microstructure formed by elastically self-accommodated internally twinned domains. This type of microstructure stipulates the thermoelasticity of alloy, i.e. the high sensitivity of the morphology of martensitic phase to the temperature or stress variation. The thermoelasticity of martensites gives rise to the shape memory and superelasticity effects2,5–8) being the subjects of numerous studies and the basis for the related applications (see e.g. Refs. 8–10). The most clear way to study the superelastic behavior of alloys is the analysis of non-linear stress–strain dependencies obtained experimentally in the course of stress-induced MT-s.11) The variety of shapes of the stress–strain loops reflects, in principle, the variety of alloy properties and experimental conditions influencing the stressinduced martensitic transformation and superelasticity effect. However, the physical interpretation and theoretical modeling of the results of stress–strain tests of thermoelastic alloys is a very involved problem. Some of the existing theoretical models of stress–strain *Corresponding
author, E-mail address:
[email protected]
behavior in shape memory alloys are essentially thermodynamic and involve a complicate mathematical analysis of Gibbs potentials and constitutive equations for the different phases coexisting in the mixed austenitic–martensitic state (see Refs. 12–14) and references therein). The application of these models to the relatively large number of studied alloy systems has a very fragmentary character, because of the large mathematical processing needed and the difficulty for comparing results of different alloy systems.14) The present day situation in this field of scientific activity can be illustrated by the following fact: more than 120 experimental stress–strain loops are given and physically interpreted in the monograph Ref. 11 without any theoretical modeling, while only one theoretical stress–strain loop is compared with a experimental one through the theoretical paper of Ref. 14). Thus, despite of the intensive experimental study of superelasticity effect and numerous models explaining some of the individual features of stress–strain dependencies obtained for certain alloy specimens, an appreciable ‘‘gap’’ between the experimental studies and modeling of superelasticity phenomenon exists. Therefore, a consistent study of superelastic behavior of martensitic alloys providing allround theoretical description of a large number of experimental results continues to be of high interest. In this paper the experimental stress–strain dependencies illustrating different features of superelastic behavior of alloys are reported for the Ni–Mn–Ga alloy family typifying the thermoelastic martensites (see e.g. Refs. 9, 10, 15)). A new statistical model for the transformation behavior and superelasticity of martensitic alloys is developed. In contrast to the models elaborated in Refs. 12–14), the model developed in this paper do not involve any cumbersome mathematical analysis of Gibbs potentials of coexisting phases in the mixed austenitic–martensitic state. Instead of this, the model deals with the internal microstresses and uses a statistical description of the stress-induced martensitic nucleation, which is conceptually close to the physical
Stress-induced Martensitic Transformation and Superelasticity of Alloys: Experiment and Theory
3.
Results
3.1
Different experimental stress–strain loops for austenite Selected results of superelastic behavior of two Ni–Mn–Ga alloys, showing qualitatively different shapes of the stress– strain loops, are summarized below for the subsequent theoretical modeling. Figure 1(a) illustrates that the stress–strain dependencies
(a) 300 250 200 150 100 50 0
1
Single crystalline Ni51:2 Mn31:1 Ga17:7 and Ni49:4 Mn27:7 Ga22:9 alloys (denoted as S1 and S2, respectively) were grown by the Bridgman method using ingots prepared from the elements by induction melting in vacuum. The characteristic temperatures TM and TA of the forward and reverse martensitic transformations, respectively, were determined from calorimetric and low-field magnetic susceptibility measurements, while the martensitic structures were obtained by electron microscopy (Hitachi H600 TEM, 100 kV) and Xray diffraction (Siemens D5000, Cu K radiation). Alloy S1 exhibits a first transformation from parent to tetragonal martensite with c=a < 1 and 10-layer modulation at temperature TM1 ¼ 418 K, followed by an intermartensitic transformation to a tetragonal non-modulated martensite with c=a > 1 at temperature TM2 ¼ 350 K. A single reverse transformation occurs at TA ¼ 426 K. The lattice parameters of M2 martensite obtained at room temperature by X-ray diffraction are a ¼ 0:544 nm and c ¼ 0:660 nm. On its turn, alloy S2 shows a single martensitic transformation with characterisitc temperatures TM ¼ 274 K and TA ¼ 285 K. A five-layer modulated tetragonal structure of the martensitic
2
3
4
5
6
5
6
Strain, ε / %
Experimental Procedure
(b) 300
Stress, σ / MPa
2.
phase with c=a < 1 was observed for S2 alloy in the electron microscope at T ¼ 250 K. Parallelepiped-shaped samples of both alloys with 2 2 mm2 cross section and about 4 mm gauge length, with the longer axes being coincident with either h100i or h110i directions of austenite, were prepared for mechanical testing. Uniaxial compression tests at different constant temperatures were carried out in a mechanical testing machine ZWICK100 at a constant cross-head speed of 0.05 mm/min, corresponding to an average strain rate of 104 s1 .
Stress, σ / MPa
picture of the stress-induced martensitic nucleation substantiated in Ref. 16). According to this picture the growth of the martensite inside the mechanically loaded austenitic matrix has a step-wise character and can be thought of as the nucleation of small martensitic embryos in the neighboring spatial domains of the alloy specimen. A smooth character of the experimental stress–strain curves obtained for the different superelastic alloys is the evidence of the large number of nucleation events occurring during the loading/unloading cycle.16) Moreover, the smoothness of the stress–strain curves justifies the disregard of the burst-like nucleation events.16) A special attention is paid in the present paper to the problems of: (i) the influence of stress relaxation accompanying the nucleation of martensite on the superelasticity effect; (ii) the extraction of the ‘‘latent’’ spontaneous strain of MT and stiffness coefficient of martensite from the experimental stress–strain loops. The nontrivial character of the problem (ii) is elucidated very clearly. The comprehensive (both theoretical and experimental, both quantitative and qualitative) character of the studies and a step forward in the resolution of the problems (i), (ii) distinguish the present paper from the previous theoretical works dealing with superelasticity phenomenon (see Refs. 12–21) and references therein). Due to the simplicity of the underlying principles, the model enables the computation of theoretical stress–strain dependencies, their adjustment to experimental stress–strain loops, the physical interpretation of different features of these loops and the evaluation of main transformational characteristics of the alloys. The just mentioned characteristics of the model allows for its application to any shape memory alloy system, without requiring the previous knowledge of a large number of parameters, therefore making easier the comparison and physical interpretation in different alloy systems.
791
250 200 150 100 50 0
1
2
3
4
Strain, ε / % Fig. 1 Experimental (a) and theoretical (b) stress–strain loops corresponding to the monotone (dashed lines) and non-monotone (solid lines) dependencies of the forward (top branches) and reverse (bottom branches) martensitic transformation of S1 alloy.
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corresponding to the forward and reverse martensitic transformations can be both monotone and non-monotone. The experimental curves shown in this figure are obtained for S1 specimen at T ¼ 460 K. The monotone stress–strain dependencies correspond to the compression/decompression of the specimen in [001] direction, while the non-monotone ones correspond to the stressing along [110]. The existence of intervals with d=d" < 0 obviously indicates the nonequilibrium character of the transient between the austenitic and martensitic phases, while the opposite inequality is inherent in the reversible thermodynamics of the elastic media. The experimental stress–strain loops obtained for S2 alloy under compression along [001] are shown in Fig. 2(a). These loops demonstrate different types of superelastic behavior: reversible deformation (in the sense of fully recoverable strain, solid line), irreversible deformation with residual strain (dashed line) and quasi-reversible deformation with residual stress (dot-and-dashed line). The reversible character of the deformation is usually considered as a definitive
(a) Stress, σ / MPa
80 60 40 20 0
1
2
3
4
5
Strain, ε / %
(b) 100
Stress, σ / MPa
lz Lz ly
lx
Ly
Lx Fig. 3 Schematic representation of a nucleus of martensitic phase in a column of the austenitic specimen.
feature of the superelastic martensitic alloys. Occurrence of residual strains, which frequently are observed experimentally, is considered as an imperfect superelastic behavior. In contrast, occurrence of residual stresses on the experimental curves tends rather to be considered as an artifact caused mainly by the temperature variation during the compression– decompression cycle than related to a real physical property. 3.2
100
0
Compressive force, z- axis
80
Theoretical model of superelastic behavior of austenite under compression The superelastic behavior of parallelepiped-shaped specimens of an alloy undergoing a stress-induced MT from cubic austenite is modeled below, taking into account the features of experimental procedure described above. During the compression (forward) cycle the deformation increases uniformly, being its absolute value " j"zz ðtÞj (the coordinate axis z is aligned with the compressive force and one of the edges of the specimen, as it is shown in Fig. 3). At some moments t ¼ tn the nucleation of tetragonal martensitic phase occurs in the different spatial domains of the specimen. The martensite nuclei are modeled by similar plane–parallel platelets with dimensions lx , ly and lz lx ; ly . The platelets are assumed to be coplanar with the shear planes corresponding to the maximal Schmid factor. The dimensions of the specimen are denoted as Lx , Ly , Lz and the values Ni ¼ Li =li are assumed to be integers. In this case, columns with dimensions lx ly Lz can be outlined inside the specimen (see Fig. 3). Elementary considerations show that at any moment the deformation is related to the number of martensitic nuclei nz in a certain column of the compressed specimen as
60
" ¼ Nz1 ½nz "ðmÞ þ ðNz nz Þ"ðaÞ ;
40
where "ðmÞ and "ðaÞ are the absolute values of deformations for the martensitic and austenitic domains in the column, respectively. The deformation of martensite can be subdivided in the elastic and transformation strains, while the deformation of austenite is completely elastic, i.e.
20 0
0
1
2
3
4
5
Strain, ε / % Fig. 2 Experimental (a) and theoretical (b) stress–strain loops recorded from alloy S2 at different temperatures corresponding to the reversible and irreversible cycles: T ¼ 301 K (upper loop), 290 K (middle loop) and 282 K (bottom loop).
"ðmÞ ¼ "M þ zz =SðmÞ ;
"ðaÞ ¼ zz =SðaÞ ;
ð1Þ
ð2Þ
where zz is the absolute value of the mechanical stress applied to the column cross-section, "M is the absolute value of the spontaneous lattice distortion in z direction, SðmÞ and SðaÞ are the stiffness coefficients of martensite and austenite, respectively. In the case of uniform uniaxial stressing, the
Stress-induced Martensitic Transformation and Superelasticity of Alloys: Experiment and Theory
equations for strains "ðmÞ and "ðaÞ involve the same stress value zz . The deformation of the specimen is expressed, therefore, as SðaÞ ðaÞ 1 ðaÞ " ¼ Nz nz "M þ ðmÞ " þ ðNz nz Þ" : ð3Þ S The deformation of the column containing nz martensitic nuclei is the sum of the spontaneous deformation of the martensitic domain ðnz =Nz Þ"M and the elastic strain " ðnz =Nz Þ"M . An effective stiffness S of the column can be defined as the ratio between the applied stress and elastic part of deformation, i.e. zz ¼ S½" ðnz =Nz Þ"M :
the effective
SðaÞ SðmÞ Nz þ ðSðaÞ SðmÞ Þnz ð"Þ
ð5Þ
Using eq. (3) and the condition zz ¼ S " stiffness can be expressed as Sð"Þ ¼
SðmÞ Nz
ð4Þ
ðaÞ ðaÞ
As S, ", Nz and "M are positive, eq. (4) shows that the stress is a decreasing function of nz (i.e. ð"; nz þ 1Þ ð"; nz Þ < 0), and hence, the appearance of each new nucleus is accompanied by stress relaxation. A simple physical explanation for this result is the following: the compression of the specimen in z direction leads to the nucleation of the martensite variant whose lattice is contracted along z; this contraction is compensated by the elastic expansion of the outward (with respect to arising martensite nucleus) part of the column, which is an evidence of the stress reduction. It should be emphasized that the stress relaxation causes the deviation of stress–strain curve downwards from the curve ¼ "Sð"Þ and the appearance of the plateau visualizing the stress-induced MT. As far as the appearance of the first nucleus in a certain column is accompanied by the stress relaxation, the following nucleation in this column becomes unlikely until the moment when every column will contain one nuclei. Disregarding the fact that the coordinates z1 of the nuclei may be different for the different columns, this moment can be conventionally referred to as the moment of completion of the first martensitic layer. Later on, the layer-wise martensite formation will continue up to the finish of the martensitic transformation (in the case of complete MT) or compression cycle (in the case of partial MT). It means that nz is a stepwise function of the deformation and for each value of " the value of this function is equal to the integer part of the fractional number nð"Þ=Nx Ny . Thus, nz ð"Þ ¼ trunc½nð"Þ=Nxy ;
ð6Þ
where nð"Þ is the number of nuclei in the whole specimen, Nxy ¼ Nx Ny is the maximal number of nuclei in the layer, the function ‘‘trunc’’ truncates the fractional part of its argument. As long as the formation of the nth z layer lasts, the number of columns with nz nuclei is equal to n ðnz 1ÞNxy , while the number of columns with nz 1 nuclei equals to nz Nxy n [here n ¼ nð"Þ, nz ¼ nz ð"Þ]. Therefore, the average stress hi measured in the course of the compression cycle is 1 hi ¼ Nxy f½n ðnz 1ÞNxy zz ðnz Þ
793
þ ðnz Nxy nÞzz ðnz 1Þg;
ð7Þ
where the values zz ðnz Þ and zz ðnz 1Þ should be determined from the eqs. (4) and (5). The eqs. (4)–(7) enable the computation of stress–strain curves characterizing the superelastic behavior of an alloy provided that the function nð"Þ expressing the evolution of the total number of the martensitic nuclei in the specimen is given. Generally speaking, different approaches to the determination of this function can be used. As far as the nð"Þ function is essentially similar to the volume fraction of martensite, it may be deduced (at least in principle) from thermodynamic considerations and/or balance between the elastic and surface energies of the nuclei (see, e.g., Ref. 22). However, this approach to the problem solution is rather complicate and cannot be immediately incorporated into the framework of the model under consideration. To avoid these difficulties, a simple statistical approach to the modeling of nð"Þ function was proposed recently in our preliminary communication.23) In this approach, the martensitic transformation is modeled with help of the Gaussian probability densities 2 !2 3 1 1 " "ðf,rÞ c 5; Pf,r ð"Þ ¼ ðf,rÞ pffiffiffiffiffiffi exp4 ð8Þ 2 "0 2 "ðf,rÞ 0 The probability distributions Pf ð"Þ and Pr ð"Þ characterize the rate of formation/disappearance of martensite during the forward/reverse transformations, respectively, where "ðfÞ 0 and "ðrÞ are the standard parameters characterizing the width of 0 ðrÞ Gauss distributions and "ðfÞ and " are the peak positions of c c the distributions. These distributions are introduced in a phenomenological way to take into account that the formation/disappearance of martensite during the forward/ reverse transformation effectively goes in a finite interval of strain, which can be conventionally expressed by the inequalities "ðf,rÞ "ðf,rÞ < " < "ðf,rÞ þ "ðf,rÞ 0 0 . Thus, the strain c c ðfÞ ðrÞ values "c and "c correspond to the maximal rates of martensite and austenite formation during the forward and ðrÞ reverse branches, respectively, while "ðfÞ 0 and "0 characterize the range of strains where the forward and reverse MT-s occur. The number of martensitic nuclei is expressed as 2 " 3 Z nð"Þ ¼ trunc4N Pf ðxÞdx5 ð9Þ 0
for the compression cycle and
2
nð"Þ ¼ nð"max Þ trunc4N
"Zmax
3 Pr ðxÞdx5
ð10Þ
"
for decompression one, where N ¼ Nx Ny Nz is the maximal number of nuclei in the specimen and "max is the deformation value in the finish point of the compression cycle. The plot of the value nð"Þ=N presents the volume fraction of martensite as a function of strain and the distribution eq. (8) corresponds to the simplest case of the sigmoid curve, which is symmetric with respect to the point with coordinate " ¼ "c . It should be emphasized, however, that the Gaussian form of probability distributions is not rigorously substantiated and other
794
V. A. L’vov et al. Table 1 Values of the parameters leading to the stress–strain loops of Fig. 1(b).
600
Parameter
Saε
Smε
400
Stress relaxation
Stress, σ / MPa
500
300 200 (f)
εc
100
εM
0 0
2
4
6
Strain, ε / % Fig. 4 Stress–strain dependence characterizing the stress-induced martensitic transformation (solid line) in comparison with the function ¼ "Sð"Þ (dot and dashed line) computed for SðaÞ ¼ 5 GPa, SðmÞ ¼ 10 GPa, "M ¼ 4% and "cðf Þ ¼ 3:5%.
distributions may also be used in the framework of the model. In this sense, the probability distributions can be considered as the characteristic functions featuring the stress-induced MT-s in the different alloy systems. The main physical values involved in the theoretical model of superelastic behavior of thermoelastic martensites are presented in Fig. 4. This figure explicitly illustrates the role of stress relaxation in the formation of plateau in the stress– strain curve characterizing forward martensitic transformation: the function ¼ "Sð"Þ accounts for the variation of the stiffness of the austenite-martensite mixture during the martensitic transformation but disregards the appearance of spontaneous deformation and its related stress relaxation; the stress–strain dependence shown by the solid line demonstrates the appearance of plateau-like region caused by the two factors mentioned above. 3.3
Qualitative modeling of the different superelastic loops The martensitic phase typically arises inside the austenitic matrix in the form of small platelets with lz lx ; ly . As an example, we used the values Nz ¼ 150, Nxy ¼ 100 for all computations presented below. In view of the qualitative character of the results reported in this section, the values of the model parameters were chosen with the purpose to illustrate the main features of experimental stress–strain loops shown in Sec. 3.1. The possibility of the quantitative description of experimental results and accurate determination of the model parameters will be substantiated in the next section. The computations show that the slopes of the plateaus in the stress–strain curves are controlled by both the parameters of Gauss distributions "ðf,rÞ and the difference in the stiffness 0 coefficients of the two phases: positive slopes are observed when "ðf,rÞ values are rather large and/or SðmÞ > SðaÞ while 0
"max SðaÞ (GPa) SðmÞ (GPa)
k ½001 5.7 6 35
k ½110 4.5 12.5 7.5
"ðcf Þ (%)
4.5
2.6
"ð0f Þ (%)
1.5
0.6
"ðcrÞ (%)
4
1.9
"ð0rÞ (%)
1.7
0.35
"M (%)
5
1.25
negative slopes are inherent to the smaller values of "0ðf,rÞ and/ or to the case SðmÞ < SðaÞ This conclusion is illustrated in Fig. 1(b). The stress–strain loops presented in this figure were computed using the values presented in Table 1. The values presented in Table 1 were determined approximately, to provide a qualitative description of experimental stress–strain loops. First of all, the experimental value "max was taken. Then, the values of stiffness coefficients were estimated from the experimentally observed slopes of the initial and final regions of stress–strain curves. Finally, the values of transformation strain and parameters of the Gauss distribution providing semi-quantitative agreement between the theoretical and experimental results were chosen. Some additional comments concerning the data of Table 1 are appropriate here: (i) A substantial difference SðmÞ ð½100Þ–SðaÞ ð½100Þ arises from the obvious difference of the initial and final slopes of stress–strain curves. The comparatively small value of the initial slope is caused by the temperature destabilization of austenite and relevant softening of the shear elastic modulus in the vicinity of MT temperature. The SðaÞ ð½110Þ coefficient is double of SðaÞ ð½100Þ in agreement with the ordinary relationships of the elasticity theory. The small slope of the final part of experimental stress–strain curve taken during the compression in [110] and the relevant small value SðmÞ ð½100Þ ¼ 7:5 GPa are not clear at present. (ii) In accordance with the argumentation presented in Ref. 15), the value of the transformation strain "M induced by compression along [110] direction is about three times smaller than in the case of compression along [001]. This fact naturally causes a substantial reduction of all critical strain values involved in the model, and therefore, the parameters "ðf,rÞ characteriz0 ing the compression in [110] are few times smaller than in the case of compression in [001] direction. The main conclusion from the Fig. 1 and Table 1 is the following: the relatively large values of the stiffness of martensite SðmÞ and parameters "ðf,rÞ inherent to the case of 0 compression along [001] result in the monotone stress–strain dependencies, while the reduction of "ðf,rÞ in combination 0 with the inequality SðmÞ < SðaÞ inherent to the case of compression along [110] results in the pronounced negative slopes of the plateaus at both forward and reverse stress– strain curves (Fig. 1b). It should be emphasized, however, that the ascertainment of the relationships between the model
Stress-induced Martensitic Transformation and Superelasticity of Alloys: Experiment and Theory Table 2 Values of the parameters leading to the stress–strain loops of Fig. 2(b). Upper loop
4.5
4
2
4
SðmÞ (GPa)
26
26
26
"ðcf Þ (%)
3
3
3
"ð0f Þ "cðrÞ "0ðrÞ
(%)
1.11
1.11
1.11
(%)
1.89
2.61
2.81
(%)
1.28
1.28
1.15
"M (%) "fin
4 3.5
4 0
4 0
vðf Þ
0.95
0.90
0.82
vðrÞ
0.09
0.90
0.85
3.4
parameters for the different directions of compression is a separate problem. Figure 2(b) depicts the results of theoretical modeling of the compression–decompression cycles for the compressive force applied in [001] direction. The loops computed for the reversible cycle (solid line), irreversible cycle with residual strain (dash line) and quasi-reversible cycle with residual stress (dash and dot line) show a good qualitative agreement with the experimental curves presented in Fig. 2(a). The values of model parameters used for computations are presented in Table 2 including the numerical values of the volume fractions, vðfÞ , vðrÞ , of the specimen transformed during the forward and reverse transformation cycles. These values were computed from the obvious expressions v
ðfÞ
"Zmax
Pf ðxÞdx;
¼ 0
v
ðrÞ
"Zmax
Pr ðxÞdx;
¼
ð11Þ
"fin
where "fin is the final strain value reached after the decompression of the specimen. Despite the qualitative character of modeling, the stiffness coefficient of martensite SðmÞ ¼ 26 GPa shown in the Table 2 is close enough to the value 35 GPa shown in Table 1. The SðaÞ values presented in Table 2 are very small due to the fact that the stress-induced MT is experimentally observed in the range of low stresses (see Fig. 2(a)), i.e. at temperatures close to TM where the austenite is softened. Table 2 and Fig. 2 show that residual strain occurs after the compression–decompression cycle with vðfÞ > vðrÞ , the reversible cycle is characterized by the equality vðfÞ ¼ vðrÞ and the residual stress occurs when vðfÞ < vðrÞ . The last inequality is possible only if a small fraction of hardly deformable martensite initially exists in the specimen before the compression. This fraction is assumed to be more stable when the temperature is close to MT temperature region and gradually losing its stability on heating. In such a case, the strain energy stored during the forward MT is able to transform this martensitic fraction into austenitic state during the reverse MT if the temperature of the specimen is large enough. Thus, the theoretical model predicts the possibility of observation of cycles with residual stress in the mixed austenitic–martensitic state even in the absence of the
Quantitative theoretical analysis of the stress-induced MT in Ni–Mn–Ga alloy The irregular form of the experimental stress–strain loops presented in Fig. 2(a) a priori shows that they cannot be accurately approximated by the theoretical curves as far as the Gauss-like statistical distributions (8) are used for computations. Therefore, more attention should be focused on the alloys with smooth stress–strain characteristics. Smooth stress–strain dependencies were experimentally observed for S1 specimen in the course of compression along [001] direction at different temperatures. As examples, the experimental stress–strain loops obtained at 476, 460 and 434 K are shown by solid lines in Fig. 5. The theoretical curves fitted to experimental dependencies are shown in the figure by dashed lines. The parameters of statistical distributions and physical characteristics of the alloy evaluated as a result of the fitting procedure are presented in Table 3. Several important conclusions about the stress-induced MT and alloy properties can be deduced from Fig. 5 and the data presented in Table 3: 3.4.1 Volume fractions of transformed martensite In view of the reversible character of deformation cycles, the volume fractions of martensite transformed during the forward and reverse MT-s are practically equal to each other (vðfÞ vðrÞ ). The volume fractions of transformed martensite are different from 1, and therefore, all observed transformation cycles are not complete. 3.4.2 Stiffness coefficients The stiffness of austenite SðaÞ decreases with decreasing temperature, showing the well known softening of austenitic phase when approaching TM . In contrast, the stiffness of martensite SðmÞ decreases on heating which is an evidence of 476 K 460 K 434 K 300 250 200 150 100 50 0 0
1
2
3
4
5
434 K
4.9 0.8
460 K
"max SðaÞ (GPa)
476 K
Middle loop
temperature effect mentioned in the Section 3.1. However, the experimental observation of the theoretically predicted residual stress needs special measurements carried out in the temperature region corresponding to the two-phase state; it means that the direct comparison between the upper curves presented in Figs. 2(a) and 2(b) can not be done.
Stress, σ / MPa
Bottom loop
Parameter
795
6
Strain, ε / % Fig. 5 Experimental (solid lines) and theoretical (dashed lines) stress– strain characteristics of S1 specimen. Theoretically determined slopes corresponding to the elastic deformations of martensitic phase for different temperatures are shown by the dash-dot lines.
796
V. A. L’vov et al.
Table 3 Values of the parameters leading to the theoretical stress–strain loops of Fig. 5. Parameter/T vðf Þ Sð"max Þ (GPa) SðmÞ (GPa)
434 K
460 K
0.87
0.82
25 49
476 K 0.65
19 33
17 30
SðaÞ (GPa)
5.9
6.4
"M (%)
5.67
5.00
4.17
"ðcf Þ (%)
3.53
4.25
3.93
"ð0f Þ (%)
1.95
1.56
1.40
"cðrÞ (%)
3.11
3.67
3.54
"0ðrÞ (%)
2.01
1.65
1.61
9.6
Transformation strain, εΜ / %
the temperature destabilization of the martensitic phase. This is again consistent with the experimental evolution of the elastic modulus in martensitic state.24) The maximal stiffness values Sð"max Þ attained during the compression cycles are very different from the stiffness of martensitic phase due to the partial character of stress-induced MT and the great difference in stiffness coefficients of two phases. 3.4.3 The spontaneous strain of MT The spontaneous strain, accompanying MT is a decreasing function of the temperature. In the case of compression in [001] direction, the stress-induced martensitic phase is tetragonal with c=a < 1.15) For such structure, the absolute value of MT strain is expressed through the lattice parameters as "M ða0 cÞ=a0 (a0 is the lattice parameter of the parent phase) and therefore, the information about the temperature dependence of lattice parameters of martensitic phase can be obtained from the stress–strain loops even for the temperature range where this lattice is unstable in the absence of mechanical stress. The ratio ða0 cÞ=a0 for the unstressed martensite arising on cooling the specimen can be approximately determined by the extrapolation of the "M values presented in Table 3 to the temperature of martensitic transformation. For the S1 specimen the linear extrapolation results in a value of about 6.3%, as it is indicated by dashed lines in Fig. 6. If we take the experimental values of the lattice parameters of the same martensitic and austenitic
6.4 6.0 5.6 5.2 4.8 4.4 4.0 410
420
430
440
450
460
470
480
Temperature, T / K Fig. 6 Plot of the transformation strain shown in Table 3 as a function of temperature.
phases measured for a similar alloy in Ref. 25); a ¼ 0:60 nm, c ¼ 0:54 nm, a0 ¼ 0:58 nm, the corresponding transformation strain is 6.9%, which is close to the extrapolated value. In addition, as the thermal contraction of austenite is rather small,26) the slope of the line shown in Fig. 6, 0.03%/K, arises mainly from the temperature increase of the lattice parameter c of the martensitic phase. The obtained value is again in agreement with diffraction data: a slope of about 0.03%/K is deduced from the cðTÞ dependence reported in Ref. 26) upon cooling a Ni2 MnGa specimen in the temperature range TM 50 K < T < TM . Therefore, the "M ðTÞ dependence presented in Fig. 6 fits rather well with the prolongation of cðTÞ dependence measured for unstressed martensite (by X-ray diffraction method) to the region of stress–temperature diagram corresponding to T > TM and 6¼ 0. It should be emphasized that the numerical values of both SðmÞ and "M cannot be obtained from the stress–strain curves presented in Fig. 5 in a simple graphical way. This statement is obviously illustrated in Fig. 5 by the dash-dot straight lines, which present the elastic stressing of pure martensitic phase defined as SðmÞ ð" "M Þ. The stiffness of martensitic phase is usually evaluated from the slope of the tangent line to the stress–strain curve in its final point. The MT strain is assumed to be equal to the coordinate of the intersection point of tangent line with the strain axis. However, it is seen very clearly that the lines presenting the function SðmÞ ð" "M Þ at the different temperatures are not tangential to the stress– strain curves. This fact results from the partial character of experimental stress–strain cycles. The partial character of the cycles is, in turn, hardly visible (at least, for T ¼ 434 K). Therefore, only the accurate theoretical treatment of experimental data enables the determination of the volume fraction of transformed martensite, spontaneous strain of MT and stiffness of martensite from the partial stress–strain cycles. 4.
Discussion and Conclusions
A consistent examination of the superelastic behavior of martensitic alloys makes evident that extraction of basic information about the martensitic transformation cycles and stress-induced martensitic phases from the experimental stress–strain curves is a complicate problem. In particular, the determination of the volume fraction of transformed alloy and the decomposition of the total deformation of the mixed (martensitic–austenitic) state into the physically different components presents serious difficulties. An advance in the resolution of the problem can be achieved by the theoretical modeling of stress–strain dependencies. A comparison of the theoretical stress–strain loops with the experimental ones leads to the following conclusions: (1) The appearance of the ‘‘plateau’’ at the stress–strain curve characterizing the superelastic behavior of martensitic alloy can be physically interpreted as the result of stress relaxation accompanying the nucleation and growth of martensite in the austenitic matrix. (2) The stress relaxation is most pronounced in the case of the abrupt stress-induced transformation of the sufficiently rigid austenite into the comparatively soft martensite; in this case the non-monotone stress–strain
Stress-induced Martensitic Transformation and Superelasticity of Alloys: Experiment and Theory
dependence is observed. (3) An accurate fit of the theoretical stress–strain curves to the experimental ones enables the evaluation of the volume fraction of alloy transformed during the compression–decompression cycle, ‘‘latent’’ spontaneous strain of martensitic transformation and stiffness coefficient of the stress-induced martensitic phase from the data obtained for both complete and partial transformation cycles. It is worth noting, however, that an additional study of the unusual stress–strain dependencies is needed for their unequivocal treatment. Moreover, the present version of the theoretical model enables an accurate theoretical evaluation of physical parameters of martensite only in the case when the experimental stress–strain curves can be satisfactorily approximated with help of Gauss-like statistical distribution of critical strain values. On the one hand, the applicability of this distribution is justified by the Entropy Principle and successful modeling of magnetic and magneto-mechanical properties of Ni–Mn–Ga alloys.27,28) On the other hand, it is quite obvious that the Gauss-like form of statistical distribution is not universal; for example, this distribution is not applicable to the Cu–Al–Ni alloys in compression mode, as will be reported in a next publication. Acknowledgements VAC is grateful to Dept. de Fı´sica, UIB, and Faculty of Engineering, Tohoku Gakuin University for support of his stays there. Partial financial support from DGI (project MAT 2002-00319) is acknowledged. REFERENCES 1) J. W. Christian: The Theory of Transformations in Metals and Alloys, Part I (Pergamon, Oxford, 2002) pp. 13–16. 2) L. Delaey: Diffusionless transformations, in Phase Transformations in Materials, edited by G. Kostorz (WILEY-VCH Verlag GmbH, Weinheim, Germany, 1991) pp. 339–404.
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