- Email: [email protected]
Computer simulation of martensitic textures
COMPUTATIONAL MATERIALS SCIENCE
ELSEVIER
Computational
Materials
Science
10
(1998)16-21
Computer simulation of martensitic textures A. Saxena a-*, A.R. Bishop a, S.R. Shenoy b, T. Lookman’ a Los Alamos National Laboratory, Los Alamos, NM 87545, USA b ICTf: Trieste, Italy ’ University of Western Ontario, Canada
Abstract We consider a Ginzburg-Landau model free energy F(e, et, Q) for a (2D) martensitic transition, that provides a unified understanding of varied twin/tweed textures. Here F is a triple well potential in the rectangular strain (E) order parameter and quadratic e:, ei in the compressional and shear strains, respectively. Random compositional fluctuations n(r) (e.g. in an alloy) - A;)q(r)] in a “local-stress” model. We find that the compatibility condition are gradient-coupled to l, - - c, c(r)[(Az (linking tensor components t(r) and et(r), ez(r)), together with local variations such as interfaces or q(r) fluctuations, can drive the formation of global elastic textures, through long-range and anisotropic effective C--Einteractions. We have carried out extensive relaxational computer simulations using the time-dependent Ginzburg-Landau (TDGL) equation that supports our analytic work and shows the spontaneous formation of parallel twins. and chequer-board tweed. The observed microstructure in NiAl and Fe,Pdt_, alloys can be explained on the basis of our analysis and simulations. Copyright 0 1998 Elsevier Science B.V. Keywords:
Twinning;
Tweed; Time-dependent
Ginzburg-Landau;
Model A dynamics
1. Introduction A variety of minerals, ferroelectrics, ferroelastics, magnetoelastic materials, ceramics, Jahn-Teller materials, and most notably the shape memory alloys (e.g. NiTi, FePd, CuAuZn2) undergo a diffusionless, displacive (i.e. martensitic
[l]), weakly first order structural transition and usually exhibit twinning below the transition temperature
To (thermally
induced martensite) and even above To under external loading (stress induced martensite). In addition, these materials display a rich, stress-sensitive, family of elastic domain-wall patterns, both above and below To as observed in transmission electron microscopy (TEM) [2]. The role of surface (or interface) energy is very important in determining fine-scale features, e.g. twin width, near an austenite twinned-martensite interface (habit plane). For instance, the twins are stabilized by a long range, habit plane mediated, nonlocal elastic interaction [3] in these materials. * Corresponding
author.
0927-0256/98/$19.00 Copyright PII SO927-0256(97)00084-O
0 1998 Published
by Elsevier Science B.V. All rights reserved
A. Saxena et al. /ComputationalMaterialsScience 10 (1998) 16-21
17
These materials also exhibit transformation precursors that can occur up to hundreds of degrees above the transition temperature Z’,. Many types of pretransitional structures (or “mesoscopic textures”) have been observed in TEM including the so-called “tweed” (criss-cross pattern of twins) patterns [2]. The understanding of pattern formation in elastic materials is of much interest, with a variety of models and mechanisms invoked separately, for twins and tweed [3-71.
2. Model The modulated phases can be understood quite generally within a Ginzburg-Landau framework if, in addition to the traditional elasticity terms, one adds appropriate nonlinear and nonlocal (strain gradient) terms to the elastic energy functional [4]. The model presented below obtains tweed without explicit disorder, and as a stable pattern. formed by energy-lowering cross-gradient terms. The additional terms are quadratic in the strain and fourth order in strain gradients (arising from compositional disorder in the alloy), with all symmetry allowed terms consistently retained. The model synthesizes a variety of properties specific to these materials. It contains two important features: (a) a cross-derivative gradient term that favors domain wall crossing, and (b) the idea of hierarchical (e.g. Cayley tree) splitting of the domain walls from microscopic scales at the habit plane to macroscopic scales inside the tweed. In two dimensions we define the components of strain tensor (without “geometric nonlinearity”) by
eij=~(!!$+~),
i,j=x,Y.
In symmetrized form we write the area (et), shear (e2) and the rectangular (E) strains as et = $(&XX + +y),
e2 = Exy,
E =
&x
-
Eyy).
The above ideas are embodied in the following (dimensionless) elastic model Hamiltonian: H = Hbulk
H bulk
=
+ Hgrad
C[(S -
+ Ha + &win,
l)E? + E?(E? - 1)2] - C i
i
a H grad = 4 C[P.x~i)2 i
Htwin = ~~ i+j
Pia,
~.
Iri - rjl
+ (Vy~iJ21
+ f C[(o,2Ei)2 i
T - Tc
t = -,
To + (o,2Ci)21,
Tc
(3)
(3
Here Ei are dimensionless, scaled local rectangular (i.e. deviatoric) strains defined on the sites of a 2D square lattice; Pi and T ate dimensionless stress and scaled temperature in the @ (i.e. triple well) potential, respectively. T, denotes the temperature at which the shear modulus would soften completely, i.e. the elastic constants would satisfy Cl1 = Ct2. Of the three elastic gradient coefficients (a, b, LX),b and (Yare possibly modified by compositional fluctuations, and are necessarily positive. For a specific material these coefficients can be determined from the measured phonon dispersion data [4]. The gradient terms (Hgradand Hu) are evaluated using discrete derivatives on the lattice. For P = 0, Hbuk has three minima for 0 < ‘5 < 4, one minimum at 6 = 0 (pure austenite) for t > $, and two side minima (two pure martensitic variants) for t < 0. The range for stable tweed is 1 < r < 4. There
18
A. Saxena et al./Computationai
Materials
Science 10 (1998) 16-21
are three degenerate minima at r = 1. H twin represents the habit plane-mediated
long-range
elastic interaction
(of
strength u) which stabilizes twins below To [3]. ITS- rj ) denotes the distance between sites i and j on the square lattice.
3. Compatibility
condition
There is a connection
between compressional,
shear and rectangular
the strain tensor, i.e. are derivatives of the same underlying as a source term in the compatibility
displacement
strains, as they are different components
of
field U(T). The order parameter e(r) acts
equation, inducing variations in et (I), ez(r). The 2D compatibility
constraint,
that is satisfied at all times, is [5] A*e, (I) - 1/8A,Aye2(r)
= (A; - A+(r).
(6)
The validity of (6) as an identity can be seen, by rotating axes by 7r/4 to a preferred (primed) frame with displacements U - (L& + U’ )/1/2, Uy = (u: - ~:.)a, (6) becomes
and discrete derivatives
A, = (A: + Ak)/d,
A, = (AL - A>)/&.
ken
Ar2e, - &(A;*
- Ay)ez
with el = (ALU: + Al,u:.)/fi, then manifestly satisfied.
= 2A;A+. ez = (Aiu:
- A:.u1.)/2, c = (Aiui, + A$:)/&.
From the form (7) we see that for configurations
in the rotated frame; (7) is
for which Ai Ale = 0, the compressional
and shear strains can
minimize their costs by taking on their trivial solutions of zero. Conversely, Q = A: Ai,c # 0 configurations, at domain-wall
such as
crossings in tweed; or parallel twins ending at habit planes, will induce nonzero compressional/shear
strains, in general. The imposition of the compatibility constraint (implies et = et (E), e2 = e2(6)) leads to a dependence of the compressional and shear strain energy cost on the order parameter l . This means that the free energy contains compatibility-induced effective 6-e potentials in the bulk, and near habit planes. The simplified long-range
term (Eq. (5)) is derived from these considerations.
4. Dynamical
simulations
The order parameter here, rectangular strain henceforth denoted 4, is not conserved; therefore the dynamics is We assume a Model A dynamics [8] for the evolution of 4 after a quench. The equation is given
purely relaxational. by
where r is the relative strength of elastic energy to thermal energy and { is a noise term. In this section we denote strain by 4 instead of E and the free energy density by f instead of H. Note that the time evolution for elastic systems must be treated carefully. Specifically, the final state depends on the damping mechanism [9] and inertial effects (acoustic waves) must be taken into account [IO]. The use of relaxation time approximation is justified since, instead of studying the time evolution per se, or details of the final state, our objective here is to demonstrate quditufively the emergence of twinning near the habit plane below the transition temperature, and of the tweed above the transition. Both of these features are expected to be independent of the damping mechanism.
A. Saxena et al. /Computational Materials Science 10 (1998) 16-21
19
The free energy F = j f ddr and the noise { satisfy the following fluctuation-dissipation relationship:
(C(r3t){(r’, t’)) = 2fs((r - r’)S(t - t’).
(9)
kx
ky
Fig. 1.
20
A. Saxena et al. /Computational Materials Science IO (1998) 16-21
Using the free energy from Eqs. (l)-(5) a@ -=--r
we obtain in two dimensions
(d = 2)
1
a2
(~~5-8Q+21~-p)-~~2~+$V48-~~aySB
-vf
at
[
#(r’)R(lr - r’l) ddr’ + <.
s
Next, we describe the results of TDGL simulations Hamiltonian
(Eqs. (l)-(5))
as a potential
(i.e. pattern formation)
for the deterministic
which are based on the above static
force term. We consider strain variables ei on an
N x N 2D square lattice (N = 96). The simulation results are shown in the left panel of Fig. 1 for representative 0.01, a = 2.4, j3 = 0.0001, r = 1. For a given material these parameters
parameter
are determined
values: a = 1, b = from the experimental
structural and phonon dispersion data [4]. Starting with the austenite phase, twins (T -C T’,) and a tweed pattern (for N To < T < T”ppr) emerge, respectively, as the system is cooled below To. The white and black regions correspond to the two regions with rectangular distortions (strain) in x and y directions. In the tweed case these are small local distortions whereas for twins they are the two martensitic variants. In Fig. 1 we have focused on the temperature range in which the tweed is “melting” discussed above and in [ 111.
into twins. The long-range
interaction
term Hmln was treated as
5. Stability of twin phases We study Eq. (10) in the Fourier space of spatial variable (for the following analysis, we ignore the noise): -
at
= Yk$k + h_
+ prsk,
(11)
where the linear dispersion relation yk = -I-
2~ + ;k2 + $(kj
and the nonlinear relation dominates
+ k;) - ;k;k;
+ B ’ - expk+-dk)
1
term DNL can be ignored for early time. The maximally the stability of early time dynamics.
(enclosed inside a “butterfly”
like figure) corresponding
,
(12)
unstable mode of the linear dispersion
The right panel in Fig. 1 depicts the region of stability to the texture in the left panel. The bottom figure in this
panel shows a diagonal line corresponding to a specific twin direction picked out by the system. The dots along the other diagonal are indicative of the mode associated with the other twinning direction becoming unstable.
6. Conclusion In conclusion we have studied the consequences of including the fourth order strain gradient terms, elastic compatibility condition and the role of compositional fluctuations in a Ginzburg-Landau model that may apply to a subclass of martensitic materials. Within this phenomenological model we described texture formation such as: (i) twins, (ii) tweed, and specifically (iii) the melting of tweed into twins. The model contains two key ingredients, namely a cross-derivative gradient term (arising from compositional disorder) that favors domain wall crossing, and the idea of hierarchical splitting of the domain walls from microscopic length scales at the habit plane to macroscopic scales inside the tweed (due to length-scale competition). An open issue in the present modeling context is to establish a criterion to systematically truncate the strain gradient expansion at a certain order.
A. Saxena et al. /Computational MaterialsScience 10 (1998) 16-21
21
References [l] C.M. Wayman, Introduction to me Theory of Martensitic Transformations (Macmillan, London, 1964). [2] L.E. Tanner, A.R. Pelton and R. Gronsky, J. Phys. (Paris) 43 (1982) C4-169. [3] G.R. Barsch, B. Horovitz and J.A. Krumhansl, Phys. Rev. Lett. 59 (1987) 1251; B. Horovitz, G.R. Barsch and J.A. Krumhansl, Phys. Rev. B 43 (1991) 1021. [4] G.R. Barsch and J.A. Krumhansl, Phys. Rev. Lett. 53 (1984) 1069; Metallurg. Trans. A 19 (1988) 761; Proc. Int. Conf. on Martensitic Transformations (ICOMAT-92), eds. C.M. Wayman and J. Perkins (Monterey Institute of Advanced Studies, California, 1993) p. 53. [5] S. Kartha, T. Kast&n, J.A. Krumhansl and J.P. Sethna, Phys. Rev. Lett. 67 (1991) 3630; J.P. Sethna, S. Kartha, T. Kast?m and J.A. Krumhansl, Phys. Scripta T 42 (1992) 214; S. Kartha, J.A. Krumhansl, J.P. Sethna and L.K. Wickham, Phys. Rev. B 52 (1995) 803. [6] S. Semenovskaya and A.G. Khachaturyan, Phys. Rev. Lett. 67 (1991) 2223; Physica D 66 (1993) 205; S. Semenovskaya, Y. Zhu, M. Suenaga and A.G. Khachaturyan, Phys. Rev. B 47 (1993) 12 182. [7] A.M. Bratkovsky, E.K.H. Salje and V. Heine, Phase Trans. 52 (1994) 77; 55 (1995) 79. [8] PC. Hohenberg and B.I. Halperin, Rev. Modem Phys. 49 (1977) 435. [9] A.C.E. Reid and R.J. Gooding, Physica D 66 (1993) 180. [lo] F. Falk, J. Phys. C 20 (1987) 2501. [l l] C. Ro1andandR.C. Desai,Phys. Rev. B42(1990)6658;C. Sagui andR.C. Desai,Phys. Rev. E49 (1994)2225; C. Sagui,A.M. Somoza and R.C. Desai, Phys. Rev. E 50 (1994) 4865.
ELSEVIER
Computational
Materials
Science
10
(1998)16-21
Computer simulation of martensitic textures A. Saxena a-*, A.R. Bishop a, S.R. Shenoy b, T. Lookman’ a Los Alamos National Laboratory, Los Alamos, NM 87545, USA b ICTf: Trieste, Italy ’ University of Western Ontario, Canada
Abstract We consider a Ginzburg-Landau model free energy F(e, et, Q) for a (2D) martensitic transition, that provides a unified understanding of varied twin/tweed textures. Here F is a triple well potential in the rectangular strain (E) order parameter and quadratic e:, ei in the compressional and shear strains, respectively. Random compositional fluctuations n(r) (e.g. in an alloy) - A;)q(r)] in a “local-stress” model. We find that the compatibility condition are gradient-coupled to l, - - c, c(r)[(Az (linking tensor components t(r) and et(r), ez(r)), together with local variations such as interfaces or q(r) fluctuations, can drive the formation of global elastic textures, through long-range and anisotropic effective C--Einteractions. We have carried out extensive relaxational computer simulations using the time-dependent Ginzburg-Landau (TDGL) equation that supports our analytic work and shows the spontaneous formation of parallel twins. and chequer-board tweed. The observed microstructure in NiAl and Fe,Pdt_, alloys can be explained on the basis of our analysis and simulations. Copyright 0 1998 Elsevier Science B.V. Keywords:
Twinning;
Tweed; Time-dependent
Ginzburg-Landau;
Model A dynamics
1. Introduction A variety of minerals, ferroelectrics, ferroelastics, magnetoelastic materials, ceramics, Jahn-Teller materials, and most notably the shape memory alloys (e.g. NiTi, FePd, CuAuZn2) undergo a diffusionless, displacive (i.e. martensitic
[l]), weakly first order structural transition and usually exhibit twinning below the transition temperature
To (thermally
induced martensite) and even above To under external loading (stress induced martensite). In addition, these materials display a rich, stress-sensitive, family of elastic domain-wall patterns, both above and below To as observed in transmission electron microscopy (TEM) [2]. The role of surface (or interface) energy is very important in determining fine-scale features, e.g. twin width, near an austenite twinned-martensite interface (habit plane). For instance, the twins are stabilized by a long range, habit plane mediated, nonlocal elastic interaction [3] in these materials. * Corresponding
author.
0927-0256/98/$19.00 Copyright PII SO927-0256(97)00084-O
0 1998 Published
by Elsevier Science B.V. All rights reserved
A. Saxena et al. /ComputationalMaterialsScience 10 (1998) 16-21
17
These materials also exhibit transformation precursors that can occur up to hundreds of degrees above the transition temperature Z’,. Many types of pretransitional structures (or “mesoscopic textures”) have been observed in TEM including the so-called “tweed” (criss-cross pattern of twins) patterns [2]. The understanding of pattern formation in elastic materials is of much interest, with a variety of models and mechanisms invoked separately, for twins and tweed [3-71.
2. Model The modulated phases can be understood quite generally within a Ginzburg-Landau framework if, in addition to the traditional elasticity terms, one adds appropriate nonlinear and nonlocal (strain gradient) terms to the elastic energy functional [4]. The model presented below obtains tweed without explicit disorder, and as a stable pattern. formed by energy-lowering cross-gradient terms. The additional terms are quadratic in the strain and fourth order in strain gradients (arising from compositional disorder in the alloy), with all symmetry allowed terms consistently retained. The model synthesizes a variety of properties specific to these materials. It contains two important features: (a) a cross-derivative gradient term that favors domain wall crossing, and (b) the idea of hierarchical (e.g. Cayley tree) splitting of the domain walls from microscopic scales at the habit plane to macroscopic scales inside the tweed. In two dimensions we define the components of strain tensor (without “geometric nonlinearity”) by
eij=~(!!$+~),
i,j=x,Y.
In symmetrized form we write the area (et), shear (e2) and the rectangular (E) strains as et = $(&XX + +y),
e2 = Exy,
E =
&x
-
Eyy).
The above ideas are embodied in the following (dimensionless) elastic model Hamiltonian: H = Hbulk
H bulk
=
+ Hgrad
C[(S -
+ Ha + &win,
l)E? + E?(E? - 1)2] - C i
i
a H grad = 4 C[P.x~i)2 i
Htwin = ~~ i+j
Pia,
~.
Iri - rjl
+ (Vy~iJ21
+ f C[(o,2Ei)2 i
T - Tc
t = -,
To + (o,2Ci)21,
Tc
(3)
(3
Here Ei are dimensionless, scaled local rectangular (i.e. deviatoric) strains defined on the sites of a 2D square lattice; Pi and T ate dimensionless stress and scaled temperature in the @ (i.e. triple well) potential, respectively. T, denotes the temperature at which the shear modulus would soften completely, i.e. the elastic constants would satisfy Cl1 = Ct2. Of the three elastic gradient coefficients (a, b, LX),b and (Yare possibly modified by compositional fluctuations, and are necessarily positive. For a specific material these coefficients can be determined from the measured phonon dispersion data [4]. The gradient terms (Hgradand Hu) are evaluated using discrete derivatives on the lattice. For P = 0, Hbuk has three minima for 0 < ‘5 < 4, one minimum at 6 = 0 (pure austenite) for t > $, and two side minima (two pure martensitic variants) for t < 0. The range for stable tweed is 1 < r < 4. There
18
A. Saxena et al./Computationai
Materials
Science 10 (1998) 16-21
are three degenerate minima at r = 1. H twin represents the habit plane-mediated
long-range
elastic interaction
(of
strength u) which stabilizes twins below To [3]. ITS- rj ) denotes the distance between sites i and j on the square lattice.
3. Compatibility
condition
There is a connection
between compressional,
shear and rectangular
the strain tensor, i.e. are derivatives of the same underlying as a source term in the compatibility
displacement
strains, as they are different components
of
field U(T). The order parameter e(r) acts
equation, inducing variations in et (I), ez(r). The 2D compatibility
constraint,
that is satisfied at all times, is [5] A*e, (I) - 1/8A,Aye2(r)
= (A; - A+(r).
(6)
The validity of (6) as an identity can be seen, by rotating axes by 7r/4 to a preferred (primed) frame with displacements U - (L& + U’ )/1/2, Uy = (u: - ~:.)a, (6) becomes
and discrete derivatives
A, = (A: + Ak)/d,
A, = (AL - A>)/&.
ken
Ar2e, - &(A;*
- Ay)ez
with el = (ALU: + Al,u:.)/fi, then manifestly satisfied.
= 2A;A+. ez = (Aiu:
- A:.u1.)/2, c = (Aiui, + A$:)/&.
From the form (7) we see that for configurations
in the rotated frame; (7) is
for which Ai Ale = 0, the compressional
and shear strains can
minimize their costs by taking on their trivial solutions of zero. Conversely, Q = A: Ai,c # 0 configurations, at domain-wall
such as
crossings in tweed; or parallel twins ending at habit planes, will induce nonzero compressional/shear
strains, in general. The imposition of the compatibility constraint (implies et = et (E), e2 = e2(6)) leads to a dependence of the compressional and shear strain energy cost on the order parameter l . This means that the free energy contains compatibility-induced effective 6-e potentials in the bulk, and near habit planes. The simplified long-range
term (Eq. (5)) is derived from these considerations.
4. Dynamical
simulations
The order parameter here, rectangular strain henceforth denoted 4, is not conserved; therefore the dynamics is We assume a Model A dynamics [8] for the evolution of 4 after a quench. The equation is given
purely relaxational. by
where r is the relative strength of elastic energy to thermal energy and { is a noise term. In this section we denote strain by 4 instead of E and the free energy density by f instead of H. Note that the time evolution for elastic systems must be treated carefully. Specifically, the final state depends on the damping mechanism [9] and inertial effects (acoustic waves) must be taken into account [IO]. The use of relaxation time approximation is justified since, instead of studying the time evolution per se, or details of the final state, our objective here is to demonstrate quditufively the emergence of twinning near the habit plane below the transition temperature, and of the tweed above the transition. Both of these features are expected to be independent of the damping mechanism.
A. Saxena et al. /Computational Materials Science 10 (1998) 16-21
19
The free energy F = j f ddr and the noise { satisfy the following fluctuation-dissipation relationship:
(C(r3t){(r’, t’)) = 2fs((r - r’)S(t - t’).
(9)
kx
ky
Fig. 1.
20
A. Saxena et al. /Computational Materials Science IO (1998) 16-21
Using the free energy from Eqs. (l)-(5) a@ -=--r
we obtain in two dimensions
(d = 2)
1
a2
(~~5-8Q+21~-p)-~~2~+$V48-~~aySB
-vf
at
[
#(r’)R(lr - r’l) ddr’ + <.
s
Next, we describe the results of TDGL simulations Hamiltonian
(Eqs. (l)-(5))
as a potential
(i.e. pattern formation)
for the deterministic
which are based on the above static
force term. We consider strain variables ei on an
N x N 2D square lattice (N = 96). The simulation results are shown in the left panel of Fig. 1 for representative 0.01, a = 2.4, j3 = 0.0001, r = 1. For a given material these parameters
parameter
are determined
values: a = 1, b = from the experimental
structural and phonon dispersion data [4]. Starting with the austenite phase, twins (T -C T’,) and a tweed pattern (for N To < T < T”ppr) emerge, respectively, as the system is cooled below To. The white and black regions correspond to the two regions with rectangular distortions (strain) in x and y directions. In the tweed case these are small local distortions whereas for twins they are the two martensitic variants. In Fig. 1 we have focused on the temperature range in which the tweed is “melting” discussed above and in [ 111.
into twins. The long-range
interaction
term Hmln was treated as
5. Stability of twin phases We study Eq. (10) in the Fourier space of spatial variable (for the following analysis, we ignore the noise): -
at
= Yk$k + h_
+ prsk,
(11)
where the linear dispersion relation yk = -I-
2~ + ;k2 + $(kj
and the nonlinear relation dominates
+ k;) - ;k;k;
+ B ’ - expk+-dk)
1
term DNL can be ignored for early time. The maximally the stability of early time dynamics.
(enclosed inside a “butterfly”
like figure) corresponding
,
(12)
unstable mode of the linear dispersion
The right panel in Fig. 1 depicts the region of stability to the texture in the left panel. The bottom figure in this
panel shows a diagonal line corresponding to a specific twin direction picked out by the system. The dots along the other diagonal are indicative of the mode associated with the other twinning direction becoming unstable.
6. Conclusion In conclusion we have studied the consequences of including the fourth order strain gradient terms, elastic compatibility condition and the role of compositional fluctuations in a Ginzburg-Landau model that may apply to a subclass of martensitic materials. Within this phenomenological model we described texture formation such as: (i) twins, (ii) tweed, and specifically (iii) the melting of tweed into twins. The model contains two key ingredients, namely a cross-derivative gradient term (arising from compositional disorder) that favors domain wall crossing, and the idea of hierarchical splitting of the domain walls from microscopic length scales at the habit plane to macroscopic scales inside the tweed (due to length-scale competition). An open issue in the present modeling context is to establish a criterion to systematically truncate the strain gradient expansion at a certain order.
A. Saxena et al. /Computational MaterialsScience 10 (1998) 16-21
21
References [l] C.M. Wayman, Introduction to me Theory of Martensitic Transformations (Macmillan, London, 1964). [2] L.E. Tanner, A.R. Pelton and R. Gronsky, J. Phys. (Paris) 43 (1982) C4-169. [3] G.R. Barsch, B. Horovitz and J.A. Krumhansl, Phys. Rev. Lett. 59 (1987) 1251; B. Horovitz, G.R. Barsch and J.A. Krumhansl, Phys. Rev. B 43 (1991) 1021. [4] G.R. Barsch and J.A. Krumhansl, Phys. Rev. Lett. 53 (1984) 1069; Metallurg. Trans. A 19 (1988) 761; Proc. Int. Conf. on Martensitic Transformations (ICOMAT-92), eds. C.M. Wayman and J. Perkins (Monterey Institute of Advanced Studies, California, 1993) p. 53. [5] S. Kartha, T. Kast&n, J.A. Krumhansl and J.P. Sethna, Phys. Rev. Lett. 67 (1991) 3630; J.P. Sethna, S. Kartha, T. Kast?m and J.A. Krumhansl, Phys. Scripta T 42 (1992) 214; S. Kartha, J.A. Krumhansl, J.P. Sethna and L.K. Wickham, Phys. Rev. B 52 (1995) 803. [6] S. Semenovskaya and A.G. Khachaturyan, Phys. Rev. Lett. 67 (1991) 2223; Physica D 66 (1993) 205; S. Semenovskaya, Y. Zhu, M. Suenaga and A.G. Khachaturyan, Phys. Rev. B 47 (1993) 12 182. [7] A.M. Bratkovsky, E.K.H. Salje and V. Heine, Phase Trans. 52 (1994) 77; 55 (1995) 79. [8] PC. Hohenberg and B.I. Halperin, Rev. Modem Phys. 49 (1977) 435. [9] A.C.E. Reid and R.J. Gooding, Physica D 66 (1993) 180. [lo] F. Falk, J. Phys. C 20 (1987) 2501. [l l] C. Ro1andandR.C. Desai,Phys. Rev. B42(1990)6658;C. Sagui andR.C. Desai,Phys. Rev. E49 (1994)2225; C. Sagui,A.M. Somoza and R.C. Desai, Phys. Rev. E 50 (1994) 4865.