- Email: [email protected]
To simulate the behavior of shape memory alloys under thermomechanical cycling
Materials Science and Engineering A365 (2004) 298–301
To simulate the behavior of shape memory alloys under thermomechanical cycling J.J. Zhu b , Q.Y. Liu a,∗ , W.M. Huang a , K.M. Liew b a
b
Center for Mechanics in Micro-Systems, School of Mechanical and Production Engineering, Nanyang Technological University, Singapore 639798, Singapore Center for Advanced Numerical Engineering Simulations, School of Mechanical and Production Engineering, Nanyang Technological University, Singapore 639798, Singapore
Abstract In this paper, a thermodynamic constitutive model previously proposed by the authors was further developed in order to be applicable for the simulation of thermomechanical behavior of shape memory alloys under different loading conditions. The volume fractions of austenite and martensite variants were used as internal variables to describe the microstructural evolution. The model was used to investigate the behavior of a round tube Cu–Al–Zn–Mn polycrystalline shape memory alloy under various mechanical loads, including non-proportional loading, torsion cycling in opposite directions and the shape memory effect. The non-proportional loading simulation results were compared with experimental results reported in the literature. © 2003 Published by Elsevier B.V. Keywords: Phase transformation; Martensite; Constitutive model; Shape memory alloy; Shape memory effect; Super-elasticity
1. Introduction
2. Micro-mechanical model
Due to the unique shape memory phenomenon, the potential applications of shape memory alloys (SMAs) are enormous. After a few decades of research, some SMA products are already available in the market [1]. More novel applications are currently under development. SMA modeling has attracted much attention because of its potential to provide a robust design tool for engineers. Examples of such work are [2–6]. However, most of the current models are not applicable for complex thermomechanical loading cases. Non-proportional loading and the shape memory effect are typical cases that most of the current models have difficulty with. In this paper, the thermodynamic constitutive model previously proposed by [7] is further developed to apply to SMAs under (1) non-proportional loading, (2) cyclic loading, and (3) the shape memory effect. Firstly a basic micro-mechanical model is presented. Afterwards, a simplified model is used to simulate the behavior of non-textured polycrystalline SMAs which is compared with reported experimental results (if applicable).
Fig. 1 shows a polycrystalline SMA. If no external load is applied, at a temperature which is higher than austenite finish temperature Af , the whole material is entirely in austenite phase. In the representative volume element (RVE), it is possible to divide all grains into N categories according to the grain orientation. The total volume fraction of grains with orientation i is given by ci . In the martensite phase, there are n possible lattice correspondence variants (LCVs). Corresponding to a given grain orientation i, the martensite phase may be divided into n groups based on LCV. Assume the volume fraction of the k-th group (with orientation i) tr . Note is zik and its phase transformation eigenstrain is Eik tr that k = 0 represents austenite, in which Ei0 = 0. Then the macroscopic Gibbs free energy function may be expressed by [7], N ρt G = ci zi0 (uA 0 + CV (T − T0 )) i=1
+
n
zik (uM 0
k=1
∗
Corresponding author. Tel.: +65-7904051; fax: +65-7911975. E-mail address: liu [email protected] (Q.Y. Liu).
0921-5093/$ – see front matter © 2003 Published by Elsevier B.V. doi:10.1016/j.msea.2003.09.039
+ CV (T
− T0 )) + fis
N T −T ci zi0 hA + C ln V 0 T0 i=1
J.J. Zhu et al. / Materials Science and Engineering A365 (2004) 298–301
299
HPVs (with orientation i) is z∗il . The relationship between the volume fraction of HPV z∗il and the volume fraction of LCV zik may be expressed by zik =
H
νkl z∗il
(k = 0, 1, 2, . . . , n)
(7)
l=1
where vkl is stoichiometric coefficient corresponding to the volume fraction of HPV z∗il . The phase transformation eigenstrain corresponding to volume fraction z∗il of HPV is Eil∗ =
n
tr νkl Eik
(8)
k=1
Fig. 1. Scale illustration. The gray areas in the middle stand for grains with the same (i-th) orientation in austenite phase. The black areas in the right represent martensite with the same type (k-th) of lattice corresponding variant.
+
n
zik
hM 0
+ CV ln
k=1
T T0
N
n tr − : ci zik Eik − 21 : M : i=1
(1)
where (2)
Therefore, E = −ρt
N
n
i=1
k=1
∂G tr ci zik Eik =M :+ ∂
(3)
If ik is the thermodynamic driving force conjugate for internal variable zik , ∂G tr = : Eik + (hT − u) + A(1 − 2zi0 ) ik = −ρt ∂zik
(9)
˙ il = Π ˙ c± , Π il
(l = 1, 2, . . . , H)
(10)
where “+” stands for the forward transformation, and “−‘’ for the reverse transformation. Πilc+ > 0 and Πilc− < 0 correspond to the critical thermodynamic driving forces for the forward transformation and the reverse transformation, respectively. The evolution equation for further phase transformation may be expressed as 1 c+ ˙ ˙ Πil = Πil = 2 λ + µ z˙ ∗ zi0 il 1 c− ˙ ˙ Πil = Πil = 2 λ + µ ∗ z˙ ∗il zil
(for forward transformation)
(for reverse transformation) (11)
(4) where A h = hM 0 − h0 A u = uM 0 − u0
∗ ˙ il = : Eiil Π + (hT − u) + A(1 − 2z0 )
From the experimental results reported in [9,10], the critical condition for the start of phase transformation may be expressed by:
k=0
fis = Azi0 (1 − zi0 )
The thermodynamic driving force conjugate to z∗il is
(5)
Here, u is the latent heat for phase transformation, and the phase equilibrium temperature Teq is u (6) T eq = h It is well known that habit plane variant (HPV) is not always the minimum sub-unit that composes the bulk martensite. In some SMAs, for instance CuZnAl, HPV is LCV. But in some other SMAs, such as TiNi and CuAlNi, HPV is composed of two twin-related LCVs (refer to [7,8] for details). Therefore, the relationship between HPV and LCV must be provided. For any given orientation i, the martensite also may be divided into H possible groups based on habit plane variants (HPVs). Assume the volume fraction of the l-th group
Here, λ and µ are introduced to describe hardening behavior. In the forward transformation, Πilc+ increases, while in the reverse transformation, Πilc− decreases. In the forward transformation, as soon as nucleation starts, Πilc− immediately returns to its maximum value (−Π0 ). When the reverse transformation starts, Πilc+ returns to its minimum value (Π0 ) instantly. Here, Π0 , λ and µ are non-negative material constants. For a material with apparent hardening behavior, the following relationship holds, λ>A
(12)
From Eqs. (11) and (12), we can calculate the volume fraction of each HPV. The eigenstrain in phase transformation may be calculated by: Etr =
N H ci z∗il Eil∗ i=1
l=1
(13)
300
J.J. Zhu et al. / Materials Science and Engineering A365 (2004) 298–301
3. Simulation
0.0175
So far, the formulas governing the martensite transformation have been established. In theory, the model presented above can be used for any kind of SMAs under any possible loading case. For non-textured polycrystal SMAs further simplification can be made by assuming
0.0150
(a) The volume fraction of each orientation group is the same, i.e. ci = 1/N (i = 1, 2, . . . , N). (b) The interaction among HPVs is small, each orientation group has H possible phase transformation systems, and they are independent of each other.
0.0075
Based on the simplified model, the behavior of a round tube Cu–Al–Zn–Mn polycrystalline SMA under three loading cases are studied to show the capability of the proposed model. In the first case non-proportional isothermal loading is simulated and compared with the experimental results. The second examines shear cycling and the third the shape memory effect. Case study 1: isothermal loading. The pseudoelastic deformation behavior of a round tube Cu–Al–Zn–Mn polycrystalline SMA under various tension and/or torsion loading conditions was studied experimentally by [11]. The material properties, which include the elastic modulus (E), shear modulus (G) and four transformation temperatures are summarized in Table 1. Other parameters used in the model may be determined by either Table 1 [7] or from the experimental results of [11]. The testing temperature of 285 K, was higher than Af . Therefore, the materials showed superelastic behavior, i.e. the material was able to fully recover its original shape after the applied load was removed. The equivalent shear stress and equivalent shear strain from [11] are used to compare the simulation with the measured results. Fig. 2 shows the results of loading under stress control, while Fig. 3 shows the results for strain control. Case study 2: shear cycling. In the simulation the sample was cooled from 285 K down to 243.5 K, which is between As and Ms . The initial phase of the sample was austenitic, however both austenite and martensite are stable at this temperature. The applied cyclic shear stress started at 0 MPa (original point) and then varied between 360 and −360 MPa. The result in terms of shear strain versus shear stress is plotted in Fig. 4. At the beginning, it deformed elastically, once the applied stress exceeded a critical value, the martensitic transformation occurred, which was accompanied by a significant shear strain. Upon unloading from 360 to 0 MPa,
γ
0.0125 0.0100 Simulation Expriment
0.0050 0.0025
ε
0.0000 0.000
0.005
E (Gpa)
G (GPa)
Ms (K)
Mf (K)
As (K)
Af (K)
53.0
19.5
239
223
248
260
0.015
Fig. 2. Strain path corresponding to imposed stress path.
its deformation behavior was purely elastic, and no reverse transformation occurred since the temperature was below As . Subsequent loading in the opposite direction (from 0 to −360 MPa) resulted in different martensite variants. Case study 3: the shape memory effect. At a low temperature, after severe “plastic” deformation, a shape memory alloy is able to recover the original shape upon heating. This is the shape memory effect. In the third simulation the material was cooled to 243.5 K from 285 K (>Af ) in stress free condition, i.e. the same as that in case study-2. A tensile stress up to 400 MPa was gradually applied, the martensitic transformation occurred after the applied stress exceeded the phase transformation start stress (Fig. 5). The behavior during unloading was elastic, and a residual phase transformation strain was observed. When the sample was heated to a temperature above As , the reverse transformation occurred. 300 250
2
200
τ[MPa]
150 1
Simulation Experiment
100 50
σ [MPa]
0 -50
Table 1 Parameters for Cu–Al–Zn–Mn [11]
0.010
-100 -50
3
4
0
50
100 150 200 250 300 350
Fig. 3. Stress path corresponding to imposed strain path.
J.J. Zhu et al. / Materials Science and Engineering A365 (2004) 298–301
301
400
T=243.5(K)
τ [MPa]
200
0
-200
-400 -0.04
-0.02
0.00
0.02
0.1
γ Fig. 4. Shear cycling.
T [K]
Heating Cooling
280
270
plex thermomechanical loading is presented. This model can describe the behavior of SMAs under proportional/ non-proportional mechanical loading, cyclic thermamechanical loading, super-elasticity and the shape memory effect. Only one set of material parameters, which can be easily obtained by standard tests, is required for modeling. The simulation results of a round tube Cu–Al–Zn–Mn polycrystalline SMA agreed well with the experiment results.
Hea ting
260
Unloadin g 250 0.04 0.03
Loading
0 100
0.02
σ [M
200
Pa]
0.01 300
Acknowledgements
0.00
ε
Financial support from NSTB and EDB through CMMS of Nanyang Technological University is acknowledged.
400
Fig. 5. The shape memory effect.
Further heating caused a full recovery to the original shape. Above Af , the material is entirely in austenite phase in the case without external stress. In the subsequent cooling from 285 to 243.5 K, there was no phase change and no shape change, since 243.5 K is still above Ms . The simulation corresponded well with the experiment results.
4. Conclusions In this paper, a thermomechanical constitutive model for single crystalline and polycrystalline SMAs under com-
References [1] Funakubo, H., Shape Memory Alloys, Gorden and Breach Science Publishers, 1987. [2] B.C. Goo, C. Lexcellent, Acta Mater. 45 (1997) 727. [3] W. Huang, Acta Mater. 47 (1999a) 2769. [4] W. Huang, J. Intell. Mater. Sys. Struct. 10 (1999) 221. [5] J.G. Boyd, D.C. Lagoudas, Int. J. Plasticity 12 (1996) 805. [6] W.M. Huang, J.J. Zhu, Mech. Mater. 34 (2002) 521. [7] J. Zhu, N. Liang, W. Huang, K.M. Liew, Z. Liu, Int. J. Solids Struct. 39 (2002) 741. [8] T. Saburi, S. Nenno, in: H.I. Aaronson (Ed.), Proceedings of the International Conference on Solid to Solid Phase Transformation, 1982, p. 1455. [9] S. Fu, Y. Huo, I. Müller, Acta Mech. 99 (1993) 1. [10] Y. Huo, I. Müler, Count. Mech. Thermodyn. 5 (1993) 163. [11] P. Sittner, Y. Hara, M. Tokuda, Metal. Mater. Trans. 26A (1995) 2923.
To simulate the behavior of shape memory alloys under thermomechanical cycling J.J. Zhu b , Q.Y. Liu a,∗ , W.M. Huang a , K.M. Liew b a
b
Center for Mechanics in Micro-Systems, School of Mechanical and Production Engineering, Nanyang Technological University, Singapore 639798, Singapore Center for Advanced Numerical Engineering Simulations, School of Mechanical and Production Engineering, Nanyang Technological University, Singapore 639798, Singapore
Abstract In this paper, a thermodynamic constitutive model previously proposed by the authors was further developed in order to be applicable for the simulation of thermomechanical behavior of shape memory alloys under different loading conditions. The volume fractions of austenite and martensite variants were used as internal variables to describe the microstructural evolution. The model was used to investigate the behavior of a round tube Cu–Al–Zn–Mn polycrystalline shape memory alloy under various mechanical loads, including non-proportional loading, torsion cycling in opposite directions and the shape memory effect. The non-proportional loading simulation results were compared with experimental results reported in the literature. © 2003 Published by Elsevier B.V. Keywords: Phase transformation; Martensite; Constitutive model; Shape memory alloy; Shape memory effect; Super-elasticity
1. Introduction
2. Micro-mechanical model
Due to the unique shape memory phenomenon, the potential applications of shape memory alloys (SMAs) are enormous. After a few decades of research, some SMA products are already available in the market [1]. More novel applications are currently under development. SMA modeling has attracted much attention because of its potential to provide a robust design tool for engineers. Examples of such work are [2–6]. However, most of the current models are not applicable for complex thermomechanical loading cases. Non-proportional loading and the shape memory effect are typical cases that most of the current models have difficulty with. In this paper, the thermodynamic constitutive model previously proposed by [7] is further developed to apply to SMAs under (1) non-proportional loading, (2) cyclic loading, and (3) the shape memory effect. Firstly a basic micro-mechanical model is presented. Afterwards, a simplified model is used to simulate the behavior of non-textured polycrystalline SMAs which is compared with reported experimental results (if applicable).
Fig. 1 shows a polycrystalline SMA. If no external load is applied, at a temperature which is higher than austenite finish temperature Af , the whole material is entirely in austenite phase. In the representative volume element (RVE), it is possible to divide all grains into N categories according to the grain orientation. The total volume fraction of grains with orientation i is given by ci . In the martensite phase, there are n possible lattice correspondence variants (LCVs). Corresponding to a given grain orientation i, the martensite phase may be divided into n groups based on LCV. Assume the volume fraction of the k-th group (with orientation i) tr . Note is zik and its phase transformation eigenstrain is Eik tr that k = 0 represents austenite, in which Ei0 = 0. Then the macroscopic Gibbs free energy function may be expressed by [7], N ρt G = ci zi0 (uA 0 + CV (T − T0 )) i=1
+
n
zik (uM 0
k=1
∗
Corresponding author. Tel.: +65-7904051; fax: +65-7911975. E-mail address: liu [email protected] (Q.Y. Liu).
0921-5093/$ – see front matter © 2003 Published by Elsevier B.V. doi:10.1016/j.msea.2003.09.039
+ CV (T
− T0 )) + fis
N T −T ci zi0 hA + C ln V 0 T0 i=1
J.J. Zhu et al. / Materials Science and Engineering A365 (2004) 298–301
299
HPVs (with orientation i) is z∗il . The relationship between the volume fraction of HPV z∗il and the volume fraction of LCV zik may be expressed by zik =
H
νkl z∗il
(k = 0, 1, 2, . . . , n)
(7)
l=1
where vkl is stoichiometric coefficient corresponding to the volume fraction of HPV z∗il . The phase transformation eigenstrain corresponding to volume fraction z∗il of HPV is Eil∗ =
n
tr νkl Eik
(8)
k=1
Fig. 1. Scale illustration. The gray areas in the middle stand for grains with the same (i-th) orientation in austenite phase. The black areas in the right represent martensite with the same type (k-th) of lattice corresponding variant.
+
n
zik
hM 0
+ CV ln
k=1
T T0
N
n tr − : ci zik Eik − 21 : M : i=1
(1)
where (2)
Therefore, E = −ρt
N
n
i=1
k=1
∂G tr ci zik Eik =M :+ ∂
(3)
If ik is the thermodynamic driving force conjugate for internal variable zik , ∂G tr = : Eik + (hT − u) + A(1 − 2zi0 ) ik = −ρt ∂zik
(9)
˙ il = Π ˙ c± , Π il
(l = 1, 2, . . . , H)
(10)
where “+” stands for the forward transformation, and “−‘’ for the reverse transformation. Πilc+ > 0 and Πilc− < 0 correspond to the critical thermodynamic driving forces for the forward transformation and the reverse transformation, respectively. The evolution equation for further phase transformation may be expressed as 1 c+ ˙ ˙ Πil = Πil = 2 λ + µ z˙ ∗ zi0 il 1 c− ˙ ˙ Πil = Πil = 2 λ + µ ∗ z˙ ∗il zil
(for forward transformation)
(for reverse transformation) (11)
(4) where A h = hM 0 − h0 A u = uM 0 − u0
∗ ˙ il = : Eiil Π + (hT − u) + A(1 − 2z0 )
From the experimental results reported in [9,10], the critical condition for the start of phase transformation may be expressed by:
k=0
fis = Azi0 (1 − zi0 )
The thermodynamic driving force conjugate to z∗il is
(5)
Here, u is the latent heat for phase transformation, and the phase equilibrium temperature Teq is u (6) T eq = h It is well known that habit plane variant (HPV) is not always the minimum sub-unit that composes the bulk martensite. In some SMAs, for instance CuZnAl, HPV is LCV. But in some other SMAs, such as TiNi and CuAlNi, HPV is composed of two twin-related LCVs (refer to [7,8] for details). Therefore, the relationship between HPV and LCV must be provided. For any given orientation i, the martensite also may be divided into H possible groups based on habit plane variants (HPVs). Assume the volume fraction of the l-th group
Here, λ and µ are introduced to describe hardening behavior. In the forward transformation, Πilc+ increases, while in the reverse transformation, Πilc− decreases. In the forward transformation, as soon as nucleation starts, Πilc− immediately returns to its maximum value (−Π0 ). When the reverse transformation starts, Πilc+ returns to its minimum value (Π0 ) instantly. Here, Π0 , λ and µ are non-negative material constants. For a material with apparent hardening behavior, the following relationship holds, λ>A
(12)
From Eqs. (11) and (12), we can calculate the volume fraction of each HPV. The eigenstrain in phase transformation may be calculated by: Etr =
N H ci z∗il Eil∗ i=1
l=1
(13)
300
J.J. Zhu et al. / Materials Science and Engineering A365 (2004) 298–301
3. Simulation
0.0175
So far, the formulas governing the martensite transformation have been established. In theory, the model presented above can be used for any kind of SMAs under any possible loading case. For non-textured polycrystal SMAs further simplification can be made by assuming
0.0150
(a) The volume fraction of each orientation group is the same, i.e. ci = 1/N (i = 1, 2, . . . , N). (b) The interaction among HPVs is small, each orientation group has H possible phase transformation systems, and they are independent of each other.
0.0075
Based on the simplified model, the behavior of a round tube Cu–Al–Zn–Mn polycrystalline SMA under three loading cases are studied to show the capability of the proposed model. In the first case non-proportional isothermal loading is simulated and compared with the experimental results. The second examines shear cycling and the third the shape memory effect. Case study 1: isothermal loading. The pseudoelastic deformation behavior of a round tube Cu–Al–Zn–Mn polycrystalline SMA under various tension and/or torsion loading conditions was studied experimentally by [11]. The material properties, which include the elastic modulus (E), shear modulus (G) and four transformation temperatures are summarized in Table 1. Other parameters used in the model may be determined by either Table 1 [7] or from the experimental results of [11]. The testing temperature of 285 K, was higher than Af . Therefore, the materials showed superelastic behavior, i.e. the material was able to fully recover its original shape after the applied load was removed. The equivalent shear stress and equivalent shear strain from [11] are used to compare the simulation with the measured results. Fig. 2 shows the results of loading under stress control, while Fig. 3 shows the results for strain control. Case study 2: shear cycling. In the simulation the sample was cooled from 285 K down to 243.5 K, which is between As and Ms . The initial phase of the sample was austenitic, however both austenite and martensite are stable at this temperature. The applied cyclic shear stress started at 0 MPa (original point) and then varied between 360 and −360 MPa. The result in terms of shear strain versus shear stress is plotted in Fig. 4. At the beginning, it deformed elastically, once the applied stress exceeded a critical value, the martensitic transformation occurred, which was accompanied by a significant shear strain. Upon unloading from 360 to 0 MPa,
γ
0.0125 0.0100 Simulation Expriment
0.0050 0.0025
ε
0.0000 0.000
0.005
E (Gpa)
G (GPa)
Ms (K)
Mf (K)
As (K)
Af (K)
53.0
19.5
239
223
248
260
0.015
Fig. 2. Strain path corresponding to imposed stress path.
its deformation behavior was purely elastic, and no reverse transformation occurred since the temperature was below As . Subsequent loading in the opposite direction (from 0 to −360 MPa) resulted in different martensite variants. Case study 3: the shape memory effect. At a low temperature, after severe “plastic” deformation, a shape memory alloy is able to recover the original shape upon heating. This is the shape memory effect. In the third simulation the material was cooled to 243.5 K from 285 K (>Af ) in stress free condition, i.e. the same as that in case study-2. A tensile stress up to 400 MPa was gradually applied, the martensitic transformation occurred after the applied stress exceeded the phase transformation start stress (Fig. 5). The behavior during unloading was elastic, and a residual phase transformation strain was observed. When the sample was heated to a temperature above As , the reverse transformation occurred. 300 250
2
200
τ[MPa]
150 1
Simulation Experiment
100 50
σ [MPa]
0 -50
Table 1 Parameters for Cu–Al–Zn–Mn [11]
0.010
-100 -50
3
4
0
50
100 150 200 250 300 350
Fig. 3. Stress path corresponding to imposed strain path.
J.J. Zhu et al. / Materials Science and Engineering A365 (2004) 298–301
301
400
T=243.5(K)
τ [MPa]
200
0
-200
-400 -0.04
-0.02
0.00
0.02
0.1
γ Fig. 4. Shear cycling.
T [K]
Heating Cooling
280
270
plex thermomechanical loading is presented. This model can describe the behavior of SMAs under proportional/ non-proportional mechanical loading, cyclic thermamechanical loading, super-elasticity and the shape memory effect. Only one set of material parameters, which can be easily obtained by standard tests, is required for modeling. The simulation results of a round tube Cu–Al–Zn–Mn polycrystalline SMA agreed well with the experiment results.
Hea ting
260
Unloadin g 250 0.04 0.03
Loading
0 100
0.02
σ [M
200
Pa]
0.01 300
Acknowledgements
0.00
ε
Financial support from NSTB and EDB through CMMS of Nanyang Technological University is acknowledged.
400
Fig. 5. The shape memory effect.
Further heating caused a full recovery to the original shape. Above Af , the material is entirely in austenite phase in the case without external stress. In the subsequent cooling from 285 to 243.5 K, there was no phase change and no shape change, since 243.5 K is still above Ms . The simulation corresponded well with the experiment results.
4. Conclusions In this paper, a thermomechanical constitutive model for single crystalline and polycrystalline SMAs under com-
References [1] Funakubo, H., Shape Memory Alloys, Gorden and Breach Science Publishers, 1987. [2] B.C. Goo, C. Lexcellent, Acta Mater. 45 (1997) 727. [3] W. Huang, Acta Mater. 47 (1999a) 2769. [4] W. Huang, J. Intell. Mater. Sys. Struct. 10 (1999) 221. [5] J.G. Boyd, D.C. Lagoudas, Int. J. Plasticity 12 (1996) 805. [6] W.M. Huang, J.J. Zhu, Mech. Mater. 34 (2002) 521. [7] J. Zhu, N. Liang, W. Huang, K.M. Liew, Z. Liu, Int. J. Solids Struct. 39 (2002) 741. [8] T. Saburi, S. Nenno, in: H.I. Aaronson (Ed.), Proceedings of the International Conference on Solid to Solid Phase Transformation, 1982, p. 1455. [9] S. Fu, Y. Huo, I. Müller, Acta Mech. 99 (1993) 1. [10] Y. Huo, I. Müler, Count. Mech. Thermodyn. 5 (1993) 163. [11] P. Sittner, Y. Hara, M. Tokuda, Metal. Mater. Trans. 26A (1995) 2923.