A Simplified Multivariant SMA Model Based on Invariant Plane ...

A <strong>Simplified</strong> <strong>Multivariant</strong> <strong>SMA</strong> <strong>Model</strong> <strong>Based</strong> on Invariant Plane
Nature of Martensitic Transformation
Xiujie Gao and L. Catherine Brinson
Department of Mechanical Engineering, Northwestern University
2145 Sheridan Road, Evanston IL 60208
ABSTRACT
The <strong>Multivariant</strong> model describing shape memory alloy constitutive behavior is further
developed in this paper for improved prediction of thermomechanical response. In the
original formulation of the <strong>Multivariant</strong> model, an interaction energy, representative of
the incompatibility of an inclusion to the matrix, was calculated by a micromechanics
model in which every variant group (inclusion) was embedded in the austenite phase with
numerous other inclusions. However, experiments show that martensite variants tend to
form large plates most of which have an invariant plane interface with the austenite and
reach the grain boundary. Hence, to better simulate material behavior, in the revised
model the micromechanical interaction energy is replaced by a small, constant term. The
predictions by the new model for different uniaxial tension directions on a single CuAlNi
crystal have excellent agreement with the experimental results. Furthermore, the counter-
intuitive results for a polycrystal CuZnAl under triaxial loading are also well captured by
the new model. As the revised model removes an iterative procedure for interaction
energy, calculations are simplified, making the multivariant model more suitable for
larger scale computations.
1. INTRODUCTION
Due to the nature of martensitic transformations with stress, research and applications of
shape memory alloys have often been one-dimensional in nature in order to obtain
maximal recoverable deformation. However, certain classes of applications, such as <strong>SMA</strong>
hybrid (self-healing) materials and many shape control or frequency control applications,
1

will subject the material to multi-dimensional stresses, partial loading and unloading,
reorientation of loading, as well as non-uniform temperature conditions. Thus an
understanding of <strong>SMA</strong> response to these non-simple loading conditions is needed. In
addition, better understanding of the directionality of <strong>SMA</strong> response in general 3-D stress
states will open the doors to a new set of potential uses.
There are several 3-D <strong>SMA</strong> models currently in the literature (Auricchio and Taylor,
1997; Boyd and Lagoudas, 1994; Boyd and Lagoudas, 1996; Graesser and Cozzarelli,
1994, Patoor, 1994 #26; Lagoudas et al., 1996; Lu and Weng, 1997; Sun and Hwang,
1993a; Sun and Hwang, 1993b). Predominantly, the existing continuum level models
approach <strong>SMA</strong> constitutive response by making use of macroscale plasticity theory
(often complete with backstresses, flow rules and yield functions). While it is appealing
to utilize the vast plasticity model literature and methodology, this approach must be
exercised with caution since the <strong>SMA</strong> deformation/stress response is based on different
underlying mechanisms than irreversible plastic response. To most accurately represent
<strong>SMA</strong> material behavior, the problem is best approached from the fundamental material
response up, accounting appropriately for the mechanisms which impart the unique
reversible, controllable nature to <strong>SMA</strong>s. Specifically, macroscopic response of <strong>SMA</strong> can
be built up from deformation at the variant level for single crystal behavior and
polycrystalline response can be based on individual grain (single crystal) response.
In recent years, several researchers have proposed <strong>SMA</strong> models based on a
micromechanics approach (Goo and Lexcellent, 1997; Huang and Brinson, 1998; Lu and
Weng, 1997; Patoor et al., 1994). These models use thermodynamics laws to describe the
transformation while using micromechanics to estimate the interaction energy due to the
phase transformation. All are based upon transformation strains of individual variants,
utilized in varying fashions in the micromechanics formulation. Most of the models
handle one variant only and will therefore not allow for loading involving reorientation of
martensitic variants. One exception is the <strong>Multivariant</strong> model by Huang and Brinson
(Huang and Brinson, 1998) and a further developed version by Gao et al. (Gao et al.,
2000), which not only accounts for all variants separately, but also includes the self-
2

accommodated grouping structure for the habit plane variants. Therefore, re-orientation,
i.e. transformation from one martensite variant to another martensite variant, is accounted
for. Although a wide array of thermomechanical response, including multiaxial loading,
can be simulated with the original <strong>Multivariant</strong> <strong>Model</strong> and the results are in good
qualitative agreement with the experimental data, a few difficulties in the <strong>Multivariant</strong>
model such as high transformation stress and long calculation time prevent it from being
practically used.
Observation of both stress and temperature induced martensitic transformation reveal that
martensite variants tend to form large plates, most of which have invariant plane
interfaces with the austenite and reach the grain boundary. Hence the original formulation
of the <strong>Multivariant</strong> model in which every variant group (inclusion) is embedded in the
austenite phase with numerous other inclusions is not physically representative of
material behavior and therefore predicts an interaction barrier to transformation that is too
large. In this paper, a simplified <strong>Multivariant</strong> model is developed in which the energy
contributed by the incompatibility of inclusion to the matrix is taken to be a small
constant.
Similar simplified models (with some variation) can be found in literature. However, no
systematic study or comparison to experiments with a model such as described here has
been performed. Patoor et al. (Patoor et al., 1993a) neglected the interaction between
martensite plates and parent phase in the single crystal simulation. However, only
forward transformations of orientation dependence etc. were simulated. In the polycrystal
case, no intergranular effect was taken into account, and thus stair-like curves were
obtained by averaging. Subsequent papers by Patoor et al. (Patoor et al., 1993b; Patoor et
al., 1994) use an interaction matrix to account for the interaction between plates and
austenite phase in single crystal case, and a self-consistent modulus predicting scheme to
obtain better averaging among grains in the polycrystal case. Other micromechanics
models (Lexcellent et al., 1996; Lu and Weng, 1997) use different methods to account for
the interaction energy, and these will be discussed in more detail in a later section. In
many cases, quantitative correlation with experimental results has been difficult to obtain.
3

We propose that a major obstacle in many of these models has been an overly large
estimation of interaction energy provided by the micromechanics methods.
In next section a summary of the original <strong>Multivariant</strong> model is given, then difficulties
for the <strong>Multivariant</strong> model are reviewed. The simplified <strong>Multivariant</strong> model is then
derived, motivated by reasonable interaction energies based on experimental observations
and implementations in previous micromechanical models. Finally, results predicted by
both the original and revised <strong>Multivariant</strong> models are compared to experimental results.
2. SUMMARY OF THE ORIGINAL MULTIVARIANT MODEL
In this section, the previously developed <strong>Multivariant</strong> <strong>Model</strong> will be briefly summarized
for both the single crystal and polycrystal forms. Although this introduction is intended to
be short, some details are still given so that readers can understand the basis of the
simplified model development. The readers are also referred to previous work on the
<strong>Multivariant</strong> <strong>Model</strong> (Gao et al., 2000; Huang et al., 2000; Huang and Brinson, 1998).
2.1. Single Crystal <strong>Model</strong> <strong>Based</strong> on Micromechanics
The <strong>Multivariant</strong> model begins with the complementary free energy of the material
n
ΨΣ ( , T, f ) = − [ ∆G + W + W − Σ E ]
(1)
ij
ch mech sur ij ij
the rate of change of which is set equal to the rate of change of the dissipation energy
dΨ | Σ ij ,T = dW d ≥ 0 (2)
For the chemical free energy, ∆G ch , a standard expression is used, linearly proportional
to temperature change; the surface energy, W sur , is small in comparison to other terms and
is neglected. The mechanical energy, W mech , is approximated by a micromechanics
technique in which the formation of martensitic variants in a shape memory alloy under
loading is viewed as a superposition of the external loading acting on a homogeneous
material and a set of transforming inclusions (see figure 1). In the simplified model, this
4

last assumption will be revisited. With the micromechanics approach it can be shown that
the mechanical energy reduces to a stored elastic energy term plus an interaction energy
term, Eint ,
W mech = 1
∫
2V σ e
ijεijdV V
= 1
2 Σ −1
ijCijklΣkl − 1
5
∫
II tr
εij dV
2V σij � � � V�
� � �
where Σ ij is the macroscopic applied stress, σ ij , ε ij are the local stress and strain
(superscripts e and tr on the strain represent elastic and transformation strains
respectively), and C ijkl is the modulus of the material. In order to use the closed form
expression of the Eshelby tensor in the subsequent Eshelby inclusion analysis (Mura,
1987), C ijkl is taken to be isotropic and identical for austenite and martensite. In the new
model, since the Eshelby Tensor is no longer needed, the anisotropic effects of the
material’s crystal lattice can be easily captured in the single crystal case.
To calculate the original form of the interaction energy, the formation of martensitic
variants in self-accommodated groups of compatible variants was recognized, which
minimized the interaction energy compared to previous similar approaches. Using this
key concept, the interaction energy is approximated as a sum over G groups of M self-
accommodating variants each
where σ ij
g
Eint =− 1
2V σ ∫ ij
V
II tr
εij dV
E int
(3)
=− 1
G
g
g
∑ σij ε ij f
2
g (4)
g g
,ε ij , f are the average stress in an inclusion of group g, average
transformation strain of group g and the volume fraction of group g, respectively. The
n
transformation strain of each variant, εij (n=1,2, ... , GxM), is calculated from the basic
crystallographic information on the habit plane normal, n, invariant plane shear direction,
m, and magnitude of the invariant plane shear, g. For a given material:
tr 1
ε = gnm ( + nm)
(5)
ij
i j j i
2
Equation (4) can be shown to reduce to
g=1

where
g
and Sijkl can be written as
L
NM
G
G
1 g g m m g
Eint =− ε � σ − f � σ f
(6)
∑
ij ij
2 g=
1
m=
1
6
∑
ij
O
QP
g
g g g
� σ = C ( S ε − ε )
(7)
ij
ijkl klmn
mn
is the Eshelby tensor of group g. The final system of equations obtained from (2)
n
F
d n
tr E
F f B T T ij ij
F if f
f
n
Fext
F F
C n
n n
n
int
n
n
wall
fric
� int
=− ( − ) + − , �
∂
��������� 0
+ − >
0 Σ ε λ λ
< 0
∂
��� ���
∓
where superscript n represents the n th of the N=GxM variants. In the external driving
n
force Fext, B is a linearized factor of difference in the chemical free energies per unit
n
volume of the two phases near the thermodynamic equilibrium temperature T0. Fwall represents a boundary force applied on the system to keep the martensite fraction within
physical bounds ( 0 ≤ f n ≤ 1), and it only appears for a variant when the volume fraction
n C
of that variant is very close to 0 or 1. Ffric is a constant F
kl
6
(8)
if variant n is actively
transforming forward, and -F C
when transforming backward. The system of equations (8)
is solved numerically for a single constitutive point; a short description of the algorithm
can be found in the appendix.
Note that the formulation of the interaction energy imposes that an “inclusion” is a group
of variants; at the same time, however, the volume fractions for each individual variant
are tracked so that a final product of a single variant is possible.
2.2. Polycrystal <strong>Model</strong> <strong>Based</strong> on Micromechanics
The behavior of single and polycrystalline <strong>SMA</strong>s are quite different. Single crystal <strong>SMA</strong>s
are highly anisotropic and this fact has serious implications in both the modeling and
experimental response. An untextured polycrystalline material, however, consists of

many single crystal grains which when randomly oriented lead to an overall isotropic
response.
In the <strong>Multivariant</strong> <strong>Model</strong>, simulation of a polycrystalline specimen is conducted by
assuming that a number of randomly oriented single crystal grains are in the poly-
crystalline specimen. The single crystal model developed is used for each single crystal
grain and a self-consistent method (Budiansky and Wu, 1962; Kröner, 1961) is used to
account for the interaction between grains. The main difference between the original
<strong>Multivariant</strong> model and the simplified model proposed in this paper is at the single crystal
level (described in the next section). Therefore the method described here for the
polycrystalline formulation is applicable for both the original and the improved
<strong>Multivariant</strong> model.
We assume a homogeneous stress field at infinity, and that every grain is in turn under
the same external homogeneous stress field. Prior to any transformation, all grains see the
same overall stress field. However, since the grains are differently oriented some grains
will begin transformation at a lower external stress level. Therefore the transformation
strains in each grain will differ. Due to the incompatibility of these transformation strains
among the interacting grains, the internal stresses due to phase transformation also differ
and are calculated via an Eshelby inclusion technique.
γ
The actual transformation strain in grain γ is εij , which is
ε ij
N var
n
∑ εij (9)
γ = f n
n=1
where Nvar is the number of variants (same as N in single crystal case). If all grains have
the same transformation strain, the stress due to the incompatibility of transformation
strain is zero. To utilize the Eshelby's solution, we assume that the difference between the
actual transformation strain of each grain and the average transformation strain is the
“effective transformation strain” or eigenstrain. Thus the effective transformation strain
in grain γ is �ε γ
ij :
7

grain
γ γ 1
n
�εij = εij − ∑εij
N
grain n=
1
where Ngrain is the number of grains. The stress in grain γ due to incompatibility with
other grains can be calculated by using Eshelby's solution.
8
N
(10)
γ γ γ
� σ = C ( S � ε − � ε )
(11)
ij ijkl klmn mn kl
where Sklmn is the Eshelby tensor of spherical inclusion. The average stress in grain γ is
the summation of the external stress and the stress due to incompatibility, that is
γ γ
Σ = Σ + � σ
(12)
ij ij ij
Note that for the polycrystalline material, each grain is assumed to be isotropic to
simplify calculations, as the closed form solutions for the Eshelby tensor require isotropy.
Ongoing work is pursuing modifications to allow anisotropy in the polycrystalline model.
In all calculations shown here, a 10-grain specimen will be used to simulate a
polycrystalline material. These grains are randomly oriented to avoid texture effect in the
material. Earlier results show the difference between 10 and 100 grains simulations to be
minimal for random orientations (Huang et al., 2000). The <strong>Multivariant</strong> polycrystalline
response for a single constitutive point is calculated numerically, modifying the stress
state and temperature according to loading history and iterating to solution at each step
(see also appendix).
3. MODIFIED MULTIVARIANT MODEL
Temperature induced transformations, shape memory effect, pseudoelasticity and
ferroelasticity are all accounted for properly by the <strong>Multivariant</strong> model. In addition
multiaxial loading cases were also simulated and the results are in good qualitative
agreement with the experimental data. The reader is referred to previous papers showing
these results (Gao et al., 2000; Huang et al., 2000; Huang and Brinson, 1998). However,
close examination of results reveals a few difficulties in the <strong>Multivariant</strong> model:

1.Experimental data indicate that stress induced martensite in a single crystal is
formed by a single habit plane variant spanning the specimen, in contrast to the
Eshelby inclusion approach which assumes many inclusions in a matrix.
2.The transformation stresses remain a factor of two higher than experiments.
3.Anisotropy of austenite and martensite phases are not accounted for.
4.For polycrystalline materials, the iterations involved in solving the system of
equations for each grain become computationally expensive.
5.The self-accommodated grouping structure currently used is based on an 18R
structure and habit plane variants; it is oversimplified for B19′ (NiTi), 3R and 2H
(CuAlNi).
Among these issues, item 1 presents the fundamental insight leading to the simplified
<strong>Multivariant</strong> model. Items 2 and 4 are easily resolved by the simplified <strong>Multivariant</strong>
model as shown in the next section. Item 3 can be easily accommodated in the single
crystal case for the new model, but difficulty remains in the polycrystal case. Although
item 5 has not yet been addressed in either model, it will be easier to resolve in the
simplified model.
Returning to item 1 consider figure 2a which shows several thermally induced martensitic
habit plane variants in a CuAlNi single crystal. It is obvious that the martensite plates are
not only large in size but also often span the specimen. This phenomenon is also common
for stress induced martensite plates and figures 2b-2d show typical examples. Since the
habit plane is an invariant plane, the stress and strain fields of the austenitic phase around
the martensite plate are not disturbed by the transformation strain of the martensite plate.
Thus the original model formulation in which an interaction energy is approximated by
considering each martensite variant group as an inclusion in a many inclusion problem
(see figure 1) is physically inappropriate. In fact in many cases, such as formation of
CuAlNi γ 1 ′ (2H) phase from austenite, the undistorted habit plane indicates very small
interaction energy since the incompatibilities of the transformation strain are
accommodated by the finely twinned microstructure. Thus to modify the multivariant
model, we have taken the simplest possible approach and removed the micromechanics
9

calculation of the interaction energy completely and replaced it by a small, constant
contribution to F C , the transformation resistance force. Now, equation (8) reduces to
d n
tr
F f B T T F if f
n
Fext
F F
C n
n
n
n
��������� wall
fric ���
� 0
=− ( − ) + Σ ε + λ − λ ∓ , � > < 0
0
ij ij
This change also allows us to incorporate the anisotropic effect into the single crystal
simulations as Eshelby tensor calculations are avoided. The numerical procedure for the
<strong>Simplified</strong> model is exactly same as that for the original <strong>Multivariant</strong> model.
A careful look at previous micromechanical model calculations also reveals
accommodation of the different models to the physically very small magnitude of an
interaction energy when calculations and comparisons to experimental data are made. A
few relevant models are reviewed here since this issue is not explicitly mentioned in
some of the original papers. In Table 1, several micromechanics models for <strong>SMA</strong>
constitutive response are listed, with references and expressions for the interaction energy
and interaction force used in the model. The last column lists pertinent features of each
model. Note that all of these models originally start with an integral expression for
interaction energy that is identical to that shown in equation (4) of this paper. From that
common starting point, different assumptions are made to enable realistic calculations.
The interaction force, Fint, the gradient of the interaction energy, is ultimately what is
used in calculating the transformation kinetics (as in equation (8) here).
Due to complexity in the evaluation of interaction energy, Patoor et al. (Patoor et al.,
1994; Patoor et al., 1996) replaced the original integral expression for Eint with a constant
interaction matrix H nm , representing the resistance to transformation (see table 1). Two
kinds of interaction are assumed: a weak H 1 corresponding to self-accommodated
variants, and strong H 2 for non-self-accommodated variants, with typical values provided
in Table 1. Note that due to the use of the interaction matrix, for non-zero values of H 1
and H 2 a strain-hardening effect is seen even in predictions for single crystal stress-strain
response (see figure 9 in (Patoor et al., 1996)). Unfortunately, experimental results on
single crystal (and polycrystal) materials at varying strain rates (Brinson et al., 2002;
10
n
6
(13)

Shaw and Kyriakides, 1995; Shaw and Kyriakides, 1997) have revealed that strain
hardening effects are due predominantly to temporary temperature changes of the
specimen with release of latent heat and therefore depend on specimen geometry,
surrounding fluid, and strain rate. The baseline prediction from a constitutive model for a
material point therefore should not include these strain hardening effects by default.
Thus, only for very small values of H i will the material prediction be appropriate.
Referring to Table 1, the model by Lexcellent and co-workers (Goo and Lexcellent, 1997;
Raniecki et al., 1992; Vivet and Lexcellent, 1998) and the model by Lu and Weng (Lu
and Weng, 1997) start with the same form for the interaction energy (the integral in
equation (4)) and both approximate the integral as a sum by considering the martensitic
plates to be transforming inclusions. The energy between martensitic plates and the
austenitic matrix for the two models is identical. Lexcellent and co-workers have an
additional term accounting for interaction between martensitic variants in the general
expression, but in application they assume only a single active variant such that this
second term drops out. Lu and Weng also in application of the model assume only a
single active variant and then perform the calculations based on that variant alone. The
original <strong>Multivariant</strong> model has a very similar expression for the interaction energy,
being also based on the transforming inclusion concept, but the exact form differs due to
the grouping of variants into self-accommodated clusters in the calculation.
In the original <strong>Multivariant</strong> model, this grouping of self-accommodated variants
dramatically decreased the interaction energy over what was calculated when considering
each inclusion to be a single variant. Since both the Lexcellent and Weng models take the
inclusion to be a single variant, there remains yet another difference to account for the
different magnitudes of the interaction force, seen in the 4 th column of Table 1. The form
and magnitudes of the Eshelby tensor, S, in each of these models is based on the aspect
ratio of the penny-shaped inclusion. In the original <strong>Multivariant</strong> model, the aspect ratio of
the inclusion/variant was selected to be approximately 0.01, based on experimental
observations (Saburi and Wayman, 1979). The images in Figure 2b-2d also support this
ratio for width to length of the martensitic plates formed. Lu and Weng, however,
11

explicitly take α→0 which enables a much smaller final interaction force; while not
stated in the paper, the aspect ratio for Vivet and Lexcellent can be calculated * to be
approximately 0.001 in order to obtain the value for Ws cited given the other material
constants provided.
Taken together, a trend is obvious: starting from the basic concept of treating the
martensitic variants as transforming inclusions in order to calculate an interaction energy
(a barrier to transformation), each of the micromechanical models has developed a
different “coping mechanism” to enable realistic calculations when compared with
experimental data. Furthermore, when the physical nature of the martensitic plate
formation, as illustrated in Figure 2, is viewed, it is obvious that the concept of a many-
inclusion problem (see figure 1) where each inclusion is surrounded by matrix material
and embedded in an infinite medium of additional inclusions and matrix material, is
incorrect. Since this latter criterion is the key to the Eshelby micromechanics approach,
we propose that discarding the interaction energy calculation by this micromechanics
method, as we have done in equation (13), is perhaps the most realistic strategy. It also
has additional advantages of practically improving the computation time required by
removing a level of iteration from the problem.
Finally, it should be mentioned that non-invariant plane martensitic transformation has
been found by Sun et al. (Sun et al., 1999; Sun et al., 1997) when austenite transforms to
β 1 ′ (pseudoelasticity) for a single CuAlNi crystal. In this case, there is a distortion of the
austenitic strain field near the A-M interface while the stress and strain fields away from
the interface are uniform. These results indicate that a slightly larger, but still small
interaction energy would be suitable for the β1→ β 1 ′ transformation. Appropriate means
to calculate an interaction energy in this non-invariant plane case will be considered in
* p p p
Using the parameters µ=12Gpa, ν=0.3, ε 11=
- 0.1174%, ε 12
=ε21 =6.646%, and zeros
for other transformation strain components, the authors calculated the value of
p p
LI ( − S)
ε ε . It is 4.16 and 0.416 Mpa for aspect ratio of 0.01 and 0.001 respectively.
12

the future. The same experiments also indicate that the A/M interface for the non-
invariant plane is macroscopically a straight line with an angle very close to the
prediction by phenomenological theory of martensitic transformation. Therefore, in our
calculations we will continue to use the phenomenological theory of martensitic
transformation for the values of transformation strain, equation (5), and we will ignore
the distortion near the interface in the calculation for a β to β 1 ′ (pseudoelasticity) for a
single CuAlNi.
4. RESULTS
In this section, the shape memory response under pure tension for the <strong>Simplified</strong> <strong>Model</strong>
and the <strong>Multivariant</strong> model will be compared first. Then the predicted orientation
dependence of both models will be compared to experimental results by Shield (Shield,
1995) on a Cu-13.95Al-3.93Ni (wt%) single crystal. Finally, triaxial loading simulation
results predicted by both models are compared to experimental results by Gall et al. (Gall
et al., 1998) for a polycrystalline Cu71Zn25Al4 (wt%) material with a volume change of -
0.3%.
Since not all parameters necessary for the simulation were readily available for the
experimental materials, some parameters (e.g. lattice parameters necessary to calculate �n,
�m and g for equation (5)) were found in the literature for materials with very similar
compositions. Table 2a contains parameters known for the materials used in the
experiments and table 2b lists other necessary parameters for similar materials attained
elsewhere in the literature. Although some parameters (e.g. B and F C obtained/used in
equation (13)) are tailored from the experimental materials, they are listed in table 2b
because they are not part of the experimental results. In the calculation of �n, �m and g for
equation (5) for Cu-14.0Al-4.2Ni (wt%), martensite phase lattice parameters were from
Otsuka and Shimizu (Otsuka and Shimizu, 1981). Since no parent phase parameter is
�
available, a 0 = 5.836 A
is used (Otsuka and Shimizu, 1974).
13

First examining the different pseudoelasticity response predicted by both models under
uniaxial tension, Figure 3 shows the comparison for single crystal Cu70.17Zn25.63Al4.2
(wt.%). The transformation stress predicted by the simplified <strong>Multivariant</strong> model is much
lower than that predicted by the original model. Note that experimental results on single
crystal Cu70.17Zn25.63Al4.2 (wt%) alloy by Goo and Lexcellent (Goo and Lexcellent, 1997),
and Lexcellent et al. (Lexcellent et al., 1996) are used to choose the parameters B and FC
used in the simulation and that the same values are used for both models; these
parameters can be used to tune the model to the experiment, but even values of zero,
which are not physically reasonable (and would result in loss of hysteresis and
temperature effects) could not bring the stress predictions of the original multivariant
model in line with experimental results. For example, setting F c to 0 only reduces the
critical transformation stress by approximately 30Mpa. Figure 4 shows the comparison
for the polycrystal pseudoelasticity effect on the same material. Again, the transformation
stress has dropped substantially to observed experimental values. In addition, since the
micromechanical interaction energy calculation is removed in the simplified model, the
computational time required to solve the system of equations has dropped by nearly two
orders of magnitude. For example, with non-optimized code on a HP 712/80 workstation,
a single crystal case such as Figure 3 completes in less than a minute, while a polycrystal
case such as in Figure 4 takes on the order of an hour. The decreased computation time
addresses concern 4 in the <strong>Multivariant</strong> model and renders the model suitable for further
practical use.
One major advantage to the <strong>Multivariant</strong> <strong>Model</strong> has been its good qualitative agreement
to more complicated experimental results. Thus, here we now examine orientation
dependence in single crystals, then a multiaxial loading case for polycrystals. Figure 5
shows results for uniaxial tension experiments performed on three single crystal
specimens of Cu-13.95Al-3.93Ni (wt%) by Shield (Shield, 1995). Specimen T1 had a
tensile axis of [2.43 1 0]; T2 was oriented 15 degree from [-1 -1 1] direction and had a [-1
-1 1.73] tensile axis direction; T3 had the [-1 -1 1] direction as its tensile axis.
14

There are several notable features of these results. First, we see that the initial moduli of
all the specimens differ, as do the transformation stresses. Specimen T1 has the lowest
initial modulus, T3 the largest one, and T2 an intermediate modulus. The difference in
the initial modulus is consistent with the behavior of a cubic crystal (Nye, 1985; Shield,
1995) where the maximum and minimum Young’s moduli are in the and directions, respectively. The trend for magnitude of the transformation stress is similar
to the trend in the initial moduli (Shield, 1995) such that specimen T1 ([2.43 1 0]
direction) has the lowest transformation stress and specimen T3 ([-1 -1 1] direction) has
the largest transformation stress.
In order to verify the applicability of the single crystal model, we calculated three tensile
loading and unloading cases with identical orientations to those in Shield’s experiments
for the CuAlNi material listed in table 2. In the simulation the forward equilibrium
temperature T0 is approximated by (MS+Af)/2 (≈14 °C) while for the reverse
transformation equilibrium temperature T0′ is taken as (Mf+AS)/2 (≈ - 5 °C). T0 is the
metastable equilibrium temperature in transformation from austenite to martensite in
P→ M
which the difference in chemical free energy of the two phases, ∆Gc , is zero
(Kaufman and Cohen, 1958). During reverse transformation, the non-chemical free
M→ P
energy accumulated during the A→M transformation ( ∆Gnc , strain energy, surface
energy, etc) plays a role as a driving force for the transformation. The inverse role of this
non-chemical energy shifts the metastable equilibrium temperature from T0 to T0′ (Tong
M→P M→P and Wayman, 1974). Thus T0′ is where ∆G + ∆G
= 0 and is always less than AS.
c
The result predicted by the original <strong>Multivariant</strong> <strong>Model</strong> is shown in Figure 6, and that by
the simplified model is shown in Figure 7.
It is observed that both models predict the proper sequence of transformation stresses
compared to the experiments with T3 [-1 -1 1] having the highest transformation stress,
and T1 [2.43 1 0] the lowest. Moreover, the experimental transformation stresses have a
maximum to minimum ratio of 4.9, while the model predictions provide 4.0 and 4.2 for
15
nc

the original model and simplified models respectively. These results show that both
models are consistent with the main trends of stress-strain relationship in <strong>SMA</strong>s.
However, several significant discrepancies can be seen between the original model
predictions (Figure 6) and the experimental data (Figure 5). The magnitude of the
transformation stresses are nearly three times that of the experimental results; two of the
orientations are unable to recover the strain pseudoelastically upon unloading; and the
moduli differences are clearly missing due to the isotropy assumption. With a move to
the simplified model (Figure 7), the results are strikingly improved and very close to the
experimental results in terms of transformation stresses, strains and moduli. Note that the
parameters F c and B for the model were determined from the experimental result for T3
and there are no other free parameters in the model. Using values from the literature for
the anisotropic elastic constants (see Table 2b), the ratio of maximum to minimum
moduli for the three orientations shown is predicted to be 6.9, while the experimental
value is 5.9.
It should be mentioned that the original <strong>Multivariant</strong> model often predicted appearance of
multiple variants: for the simulations here the variants active in cases T1, T2 and T3 were
(17 & 18), (6 & 15) and (6, 8, 13, 15, 18, & 19) respectively. Experimentally however, a
final product of one variant is expected in most cases, although in directions with special
symmetry several variants are sometimes seen (Lexcellent et al., 1996). With the
simplified <strong>Multivariant</strong> model, the variants predicted in cases T1, T2 and T3 are (17 &
18), (6) and (8) respectively. Since all the materials simulated have 18R structure, the
variant order is same as Saburi and Wayman (Saburi et al., 1980) except here we identify
1(+) as 1, 1(-) as 2, …, and 6'(-) as 24. Angles between the tension direction and the
intersection direction of the habit plane and the sample surface are also measured in the
experiment and compared to prediction based on Type I or Type II twinning for a 2H
system (Shield, 1995). Since the material is at a temperature 17° higher than the Af
(23°C), the CuAlNi material should transform between the parent and a 18R martensite
phase which does not have twinning substructure (Otsuka and Shimizu, 1981; Sun et al.,
1999). Therefore using lattice parameters (table 2b), we calculated the habit plane
16

normal, invariant plane shear direction, magnitude of the shear and transformation strains
for the 18R system, shown in table 2b. The predicted angles between the tension direction
and the intersection direction of the habit plane and the sample surface are shown in table
3. Agreement between prediction and data is quite good with the exception of the highly
symmetric direction T3, where the transformation strain levels of several potential
variants are very close. The variants with largest transformation strain will show up in the
single crystal calculation since we are using a maximum work criterion and all variants
are under same stress state. The model predicts variant 8, with a habit plane angle very
different from the experimentally observed value, note, however, that a minor change in
lattice parameters can cause selections of variants 6,15,18 or19 (with angles very close to
the experiments). For such a symmetric loading direction, it is very likely that a small
amount of misalignment of or imperfection in the specimen may change the observed
variant from one to another, or high resolution microscopy during loading would reveal
presence of several, nearly equivalent variants.
There are only a few triaxial loading experiments which have been performed to
investigate the effect of three dimensional stress states on martensitic transformation in
<strong>SMA</strong>s. Jacobus et al. (Jacobus et al., 1996) investigated the effects of triaxial loading in a
polycrystalline Ni-Ti <strong>SMA</strong>. Lim and McDowell (Lim and McDowell, 1999) investigated
the response of a Ni-Ti <strong>SMA</strong> specimen under axial-torsional proportional and non-
proportional loading. Gall et al. (Gall et al., 1998) investigated triaxial loading in a
polycrystalline Cu71Zn25Al4 (wt%, ∆V/V ≈ -0.3%) <strong>SMA</strong> (shown in figures 8 and 9).
Under uniaxial loading, it was found that the compressive stress level required to
macroscopically trigger the transformation was 34% larger than the required tensile
stress. The triaxial tests produced effective stress-strain curves with critical effective
transformation stress levels in between the tensile and compressive results.
The effective stress-strain plots at a temperature above Af under a loading rate of dε/dt =
10 -4 s are shown in figure 8. Curves for the effective stress at the onset of transformation
vs. the hydrostatic stress at the onset of transformation are shown in figure 9. The
effective stress and effective strain are calculated by:
17

2
2
2
σ eff = [( σ11 − σ 22)
+ ( σ11 − σ 33)
+ ( σ 22 −σ
33)
]
2
/
2
2
2
εeff = [( ε11 − ε22) + ( ε11 − ε33) + ( ε22 −ε33)
]
3
/
where σnn and εnn are the n th principal stress and strain respectively. For the tests in
Figure 8 and 9 the applied stress state, the effective stress and the relative hydrostatic
pressure are shown in table 4.
We simulated the one dimensional and three dimensional loading paths listed in table 4
using both <strong>Multivariant</strong> models. The effective stress-strain curves similar to figure 8
predicted by the simplified model are plotted in figure 10. The effective stress at the onset
of transformation vs. the hydrostatic stress at the onset of transformation similar to figure
9 predicted by both models are shown in figure 11. One sees that the new model captures
the magnitude of transformation stress, and the trend of the effective transformation
stress with increasing of hydrostatic pressure. The counter-intuitive experimental results
of increasing transformation stress with increasing hydrostatic pressure for a material
with negative volume change are also well captured by the simplified model. The
compressive stress level required to macroscopically trigger the transformation was 16%
larger than the required tensile stress in the simulation.
4. DISCUSSION
The <strong>Multivariant</strong> model is one of the few that has been tested against complex
thermomechanical loading, and incorporates both variant level predictions as well as
continuum level response. As illustrated in the previous section, the simplified and
improved model provides excellent qualitative and quantitative agreement with
experimental data for sophisticated loading. Of the original list of 5 concerns with respect
to the original <strong>Multivariant</strong> model, the improved model has addressed the first four. 1) By
replacing the micromechanical interaction energy calculation, which assumed formation
of many different martensitic plates (inclusions), with a small constant interaction force
included in F C , the model now more closely resembles experimental observations. 2) This
18
2 1 2
2 1 2
(14)
(15)

same change then removed a significant barrier to transformation and thereby lowered
transformation stresses to appropriate magnitudes. 3) Anisotropy can be fully accounted
for in the single crystal model, although work is still ongoing to bring that aspect into the
polycrystalline calculations; this change would be essential to properly modeling textured
materials. 4) Removing the interaction energy also eliminated one level of iterations in
the calculations and therefore significantly decreased computation time.
Concern 5 was related to the self-accommodated grouping structure and the applicability
of the model to habit plane variants only. While eliminating the interaction energy
removes much of the explicit grouping structure from the model, the transformation
strains (equation (5)) are still calculated based on crystallographic parameters for habit
plane variants. This equation is appropriate for materials such as CuZnAl with stacking
faults acting as the invariant plane shear prior to de-faulting (Miyazaki et al., 1984). It is
also appropriate for materials such as γ 1 ′ CuAlNi, and B19' NiTi with internally twinned
structure before detwinning of correspondence variants occurs. The conversion process
for a thermally induced martensitic structure of γ 1 ′ type is indicated after work by Saburi
(Saburi and Nenno, 1981) in Figure 12. Note that for clarity figure 12 ignores some finer
levels of twinning and conversion (intermediate steps) in the two middle steps. Similar
studies in microscale have not been performed for the stress-induced conversion process
when starting from austenite. However, it is now commonly accepted (Horikawa et al.,
1988; Leclercq and Lexcellent, 1996) that for such a material under loading, a distinct
habit plane forms first separating austenite from an internally twinned or faulted
martensitic structure (the habit plane variant); detwinning or de-faulting of the habit plane
variant then occurs after a higher critical load is reached resulting in a single
correspondence variant product. With further increases in load, reorientation or
conversion among correspondence variants can happen (as shown in the reorienting step
of figure 12 for the thermally induced case). Ongoing work to address these issues is
examining the stress induced detwinning/de-faulting and conversion process both
experimentally by in situ SEM and numerically by including a subgrouping scheme for
correspondence variants and direct calculation of their transformation strains, in an
19

approach similar to that pursued recently by Govindjee and Miehe (Govindjee and Miehe,
2001). Results will be presented in future work (Gao and Brinson, 2002).
Finally, in this paper we have simply removed the micromechanics calculation of the
interaction energy entirely, lumping any energy barrier due to incompatibilities of the
transformation strain with respect to the surrounding material into a constant resistance
force, F C . It is seen experimentally that in most cases, the stress-induced martensite forms
with a sharp habit plane with no distortion due to a finely twinned substructure in the
habit plane variant. Hence the incompatibility of the stress-induced variant is indeed quite
small and the assumption here is justified. However, as mentioned earlier, some recent
results have shown that a non-invariant plane relative to the austenite (Sun et al., 1999;
Sun et al., 1997) is possible, accompanied by a finite amount of distortion of the material
around the interface. In such cases, (for example, β to β 1 ′ in CuAlNi alloy) a different
calculation of the interaction energy due to the distorted region around the interface
would be appropriate. Future work should investigate this possibility and its influence on
<strong>SMA</strong> models.
5. CONCLUSION
Here we have presented a simplified <strong>Multivariant</strong> model for prediction of shape memory
alloy response to thermomechanical loading. While the original <strong>Multivariant</strong> model
performed very well providing qualitative agreement with complex experiments,
problems with transformation stress magnitude, material anisotropy and exact variant
prediction remained. In addition, due to the hierarchical iterative numerical approach,
calculation times for polycrystalline specimen response were prohibitive. <strong>Based</strong> on
experimental results showing that both thermally and stress induced martensitic variants
tend to form in large plates nearly spanning the single crystal (or grain), the original
calculation of an interaction energy, in which variants were represented as inclusions in a
many inclusion problem, was removed. It was also demonstrated here that similar
micromechanics approaches to <strong>SMA</strong> modeling have ultimately had to make significant
20

compromises in calculations of the interaction energy in order to achieve a small enough
value to provide realistic predictions. By removing the micromechanical calculation of
interaction energy, we achieved a model that is more physically based yet retains the
essential crystallographic and thermodynamic framework and still utilizes
micromechanics where appropriate in the multi-grain interactions for polycrystalline
samples. The simplified model performs extremely well for quantitative comparisons to
experimental data on non-simple loadings. In addition, the computation time is rapid,
making the model feasible for higher level studies.
Some further improvements to the model are under investigation, most notably work to
incorporate correspondence variant subunits and an effort to include anisotropy into the
polycrystalline calculations. With these additional improvements the <strong>Multivariant</strong> model
will be a powerful tool for both calculation of <strong>SMA</strong> constitutive response and material
level response and will provide a convenient test bed for simulating and understanding
<strong>SMA</strong> response to complex thermomechanical loading.
APPENDIX
The <strong>Multivariant</strong> model is solved numerically for a single constitutive point given some
initial conditions (the initial volume fractions of all variants, typically these start from
zeros). A brief description of the algorithm is given here and the reader is referred to
(Huang, 1997; Huang et al., 2000; Huang and Brinson, 1998) for more details.
Single Crystal Algorithm:
For each step of loading history (Temperature and Stress state)
i i i i i
1. Find net force for each variant Fnet = Fext + Fint + Ffric + Fwall
�
�
2. If | Fnet | ≠ 0 , assign an initial trial step size h proportional to 1/| Fnet |
�
3. While | Fnet | ≠ 0 do
Calculate ∆f i (volume fraction change) and Err i (estimated error) by taking a Cash-
Karp Runge-Kutta step
21

MaxErr = max( Err i )
If MaxErr > Tolerance
Else
Decrease h
Increase h
Advance f i by ∆f i
i i i i i
Find new net force for each variant F = F + F + F + F + F
End {If}
End {While}
Polycrystal Algorithm:
Each grain is treated as a single crystal and the single crystal algorithm above is used. In
each loading step, the program checks each grain for non-balanced driving forces (i.e.,
�
| Fnet | ≠ 0 ); if it finds a non-balanced grain, it calculates the new volume fractions of each
variant in that grain by one single or several Runge-Kutta steps. The key point is that
before iteration with other grains the total volume fraction change inside a grain should
not exceed certain amount (e.g., 10 -5 ). After all the grains are checked and the new
martensite fractions for non-balanced grains are obtained, the program uses the new
martensite fraction of each variant in each grain to calculate the stresses in each grain by
using (9) -.(12) Then the program checks all grains, advances the non-balanced grains for
one single or several Runge-Kutta steps, and then recalculates the stresses in each grain
again, repeating until all grains are balanced.
REFERENCES
Auricchio, F. and R.L. Taylor. 1997. "Shape-memory alloys: <strong>Model</strong>ing and numerical
simulations of the finite-strain superplastic behavior", Comput. Methods Appl. Mech.
Eng., 143(1-2): 175-194.
Boyd, J.G. and D.C. Lagoudas. 1994. "Thermomechanical response of shape memory
composites", J. Intell. Mater. Syst. Struct., 5: 333-346.
22
net
ext
int
fric
wall
0
wall

Boyd, J.G. and D.C. Lagoudas. 1996. "A thermodynamical constitutive model for shape
memory materials. Part I. The monolithic shape memory alloy", Int. J. Plast., 12(6):
805-842.
Brinson, L.C., I. Schmidt and R. Lammering. 2001. "Micro and macromechanical
investigations of transformation behavior of a polycrystalline NiTi shape memory
alloy using in situ optical microscopy", Acta Metall. Mater., submitted.
Brinson, L.C., I. Schmidt and R. Lammering. 2002. "Micro and macromechanical
investigations of CuAlNi single crystal and CuAlMnZn polycrystalline shape memory
alloys", J. Intell. Mater. Syst. Struct., to appear.
Budiansky, B. and T.T. Wu. 1962. "Theoretical prediction of plastic strains of
polycrystals", Fourth U. S. National Congress of Applied Mechanics, Berkeley, CA,
Proceedings of the Fourth U. S. National Congress of Applied Mechanics, R.M.
Rosenberg, ed. New York: ASME, pp.1175-1185.
Gall, K., H. Sehitoglu, H.J. Maier and K. Jacobus. 1998. "Stress-induced martensitic
phase transformations in polycrystalline CuZnAl shape memory alloys under different
stress states", Metall. Trans. A, 29A(Mar.): 765-773.
Gao, X. and L.C. Brinson. 2002. "A 3-D two-tier multivariant <strong>SMA</strong> model based on
hierachical structural characteristics of martensitic transformation",, manuscript in
preparation.
Gao, X., M. Huang and L.C. Brinson. 2000. "A multivariant micromechanical model for
<strong>SMA</strong>s Part 1: Crystallographic issues for single crystal model", Int. J. Plast., 16(10-
11): 1345-1369.
Goo, B.C. and C. Lexcellent. 1997. "Micromechanics-based modeling of two-way shape
memory effect of a single crystalline shape memory alloy", Acta Metall. Mater., 45(2):
727-737.
Govindjee, S. and C. Miehe. 2001. "A multi-variant martensitic phase transformation
model: formulation and numerical implementation", Comput. Methods Appl. Mech.
Eng., 191: 215-38.
Graesser, E.J. and F.A. Cozzarelli. 1994. "A proposed three-dimensional constitutive
model for shape memory alloys", J. Intell. Mater. Syst. Struct., 5(Jan.): 78-89.
Hellwege, K.H. and A.M. Hellwege. 1984. "Elastic, Piezoelectric, Pyrelectric,
Peizooptic, Electrooptic Constants, and Nonlinear Dielectric Susceptibilities of
Crystals." Landolt-Bornsterin: Numerical Data and Functional Relationships in
Science and Technology: New Series: Group III: Crystal and Solid State Physics, K.H.
Hellwge and O. Magdelung, eds., New York, NY: Springer-Verlag, Vol:18: 5-5.
Horikawa, H., S. Ichinose, K. Morii, S. Miyazaki and K. Otsuka. 1988. "Orientation
dependence of β1 to β'1 stressed-induced martensitic transformation in a Cu-Al-Ni
alloy", Metall. Trans. A, 19A: 915-923.
Huang, M. 1997. "A <strong>Multivariant</strong> <strong>Model</strong> for Shape Memory Alloys", Ph. D.,
Northwestern University, Evanston, IL.
23

Huang, M., X. Gao and L.C. Brinson. 2000. "A multivariant micromechanical model for
<strong>SMA</strong>s Part 2: Polycrystal model", Int. J. Plast., 16(10-11): 1371-1390.
Huang, M.S. and L.C. Brinson. 1998. "A multivariant model for single crystal shape
memory alloy behavior", J. Mech. Phys. Solids, 46(8): 1379-1409.
Jacobus, K., H. Sehitoglu and M. Balzer. 1996. "Effect of stress state on the stressinduced
martensitic phase transformation in polycrystalline Ni-Ti alloy", Metall.
Trans. A, 27A(Oct.): 3066-3073.
Kaufman, L. and M. Cohen. 1958. "Thermodynamics and kinetics of martensitic
transformations", Prog. Met. Phys., 7: 165-246.
Kröner, E. 1961. "Zur plastischen verformung des vielkristalls", Acta Metall., 9: 155-161.
Lagoudas, D.C., Z. Bo and M.A. Qidwai. 1996. "A unified thermodynamic constitutive
model for <strong>SMA</strong> and finite element analysis of active metal matrix composites", Mech.
Compos. Mater. Struct., 3: 153-179.
Leclercq, S. and C. Lexcellent. 1996. "A General macroscopic description of the
thermomechanical behaviors of shape memory alloys", J. Mech. Phys. Solids, 44(6):
953-980.
Lexcellent, C., B.C. Goo, Q.P. Sun and J. Bernardini. 1996. "Characterization,
thermomechanical behavior and micromechanical-based constitutive model of shapememory
Cu-Zn-Al single crystals", Acta Metall. Mater., 44(9): 3773-3780.
Lim, T.J. and D.L. McDowell. 1999. "Mechanical behavior of an Ni-Ti shape memory
alloy under axial-torsional proportional and nonproportional loading", Trans. Metall.
Soc. AIME, 121(1): 9-18.
Lu, Z.K. and G.J. Weng. 1997. "Martensitic transformation and stress-strain relations of
shape-memory alloys.", J. Mech. Phys. Solids, 45(11/12): 1905-1928.
Miyazaki, S., S. Kimura, K. Otsuka and Y. Suzuki. 1984. "The habit plane and
transformation strains associated with the martensitic transformations in Ti-Ni single
cystals", Scr. Metall., 18: 883-888.
Mura, T., 1987. Micromechanics of Defects in Solids, 2nd, rev. ed., Boston, MA: Kluwer
Academic Publishers.
Nye, J.F., 1985. Physical Properties of Crystals: Their Representation by Tensors and
Matrices, Repr. with corrections and new material ed., Oxford, UK: Clarendon Press.
Otsuka, K. and K. Shimizu. 1974. "Morphology and crystallography of thermoelastic Cu-
Al-Ni martensite analyzed by the phenomenological theory", Trans. Jpn. Inst. Met.,
15(2): 103-108.
Otsuka, K. and K. Shimizu. 1981. "Stress-induced martensitic transformations and
martensite-to-martensite transformations", An International Conference on Solid to
Solid Phase Transformations, Pittsburgh, PA, Proceedings of An International
Conference on Solid to Solid Phase Transformations, H.I. Aaronson, D.E. Laughlin,
R.F. Sekerka et al., eds., Warrendale, Pa. : Metallurgical Society of AIME, pp.1267-
1286.
24

Otsuka, K. and C.M. Wayman, 1998. Shape Memory Materials, New York, NY:
Cambridge University Press.
Patoor, E., M.O. Bensalah, A. Eberhardt and M. Berveiller. 1993a. "Thermomechanical
behaviour determining of shape memory alloys using a thermodynamical potential
optimizing", Rev. Metall., 90: 1587-1592.
Patoor, E., A. Eberhardt and M. Berveiller. 1993b. "Micromechanical <strong>Model</strong>ing of
Superelasticity in Shape Memory Alloys." Pitman Research Notes in Mathematics
Series: Progress in Partial Differential Equations: the Metz Surveys 2, M. Chipot, ed.,
New York, NY: J. Wiley, Vol:296: 38-54.
Patoor, E., A. Eberhardt and M. Berveiller. 1994. "Micromechanical modeling of the
shape memory behavior", International Mechanical Engineering Congress and
Exposition, Chicago, IL, Mechanics of phase transformations and shape memory
alloys: presented at 1994 International Mechanical Engineering Congress and
Exposition, L.C. Brinson and B. Moran, eds., New York, N.Y. : ASME, AMD - vol.
189; PVP - vol. 292, pp.23-37.
Patoor, E., A. Eberhardt and M. Berveiller. 1996. "Micromechanical modeling of
superelasticity in shape memory alloys", MECAMAT '95 International Seminar on
Mechanics and Mechanisms of Solid-Solid Phase Transformations, La Bresse, France,
J. Phys. IV, Colloq. (France), C. Lexcellent, ed. Editions de Physique, vol.6, no.C1,
pp.277-292.
Raniecki, B., C. Lexcellent and K. Tanaka. 1992. "Thermodynamic models of
pseudoelastic behaviour of shape memory alloys", Arch. Mech., 44(3): 261-284.
Saburi, T. and S. Nenno. 1981. "The shape memory effect and related phenomena", An
International Conference on Solid to Solid Phase Transformations, Pittsburgh, PA,
Proceedings of An International Conference on Solid to Solid Phase Transformations,
H.I. Aaronson, D.E. Laughlin, R.F. Sekerka et al., eds., Warrendale, Pa. :
Metallurgical Society of AIME, pp.1455-1479.
Saburi, T. and C.M. Wayman. 1979. "Crystallographic similarities in shape memory
martensites", Acta Metall., 27: 979-995.
Saburi, T., C.M. Wayman, K. Takata and S. Nenno. 1980. "The shape memory
mechanism in 18R martensitic alloys", Acta Metall., 28: 15-32.
Shaw, J.A. and S.K. Kyriakides. 1995. "Thermomechanical aspects of NiTi", J. Mech.
Phys. Solids, 43(8): 1243-1281.
Shaw, J.A. and S.K. Kyriakides. 1997. "On the nucleation and propagation of phase
transformation fronts in a NiTi alloy", Acta Metall. Mater., 45(2): 683-700.
Shield, T.W. 1995. "Orientation dependence of the pseudoelastic behavior of single
crystals of Cu-Al-Ni in tension", J. Mech. Phys. Solids, 43(6): 869-895.
Sun, Q.P. and K.C. Hwang. 1993a. "Micromechanics modeling for the constitutive
behavior of polycrystalline shape memory alloys-I. Derivation of general relations", J.
Mech. Phys. Solids, 41(1): 1-17.
25

Sun, Q.P. and K.C. Hwang. 1993b. "Micromechanics modeling for the constitutive
behavior of polycrystalline shape memory alloys-II. Study of the individual
phenomena", J. Mech. Phys. Solids, 41(1): 19-33.
Sun, Q.P., T.T. Xu and X. Zhang. 1999. "On deformation of A-M interface in single
crystal shape memory alloys and some related issues", Trans. ASME, J. Eng. Mater.
Technol., 121(Jan.): 38-43.
Sun, Q.P., X. Zhang and T.T. Xu. 1997. "Some recent advances in experimental study of
shape memory alloys", IUTAM Symposium on Macro- and Micro- Aspects of
Thermoplasticity, Bochum, Germany, Solid Mechanics and Its Application, O.T.
Bruhns and E. Stein, eds., Vol. 62, pp.407-416.
Tong, H.G. and C.M. Wayman. 1974. "Characteristic temperatures and other properties
of thermoelastic martensites", Acta Metall., 22(July): 887-896.
Vivet, A. and C. Lexcellent. 1998. "Micromechanical modeling for tension-compression
pseudoelastic behavior of AuCd single crystals", Eur. Phys. J. Appl. Phys., 4(2): 125-
132.
Zhu, W., B. Jiang and Z. Xu. 1986. "Crystallography of martensitic transformation in a
Cu-26Zn-4Al shape memory alloy", Acta Metall. Sin. (Chin. Ed.), 22(3): 229-236.
26

Table 1. Interaction energy and force comparison between different models
chem surf int
Helmholtz’s free energy: Ψ = ∆G
+ W + W + �
Author Interaction energy
W int Interaction
=
1 II p
−
2V z σ ij ε ijdv
v
force
int
int W
Fi
=−
f i
∂
Magnitude
Features
∂
(unit volume)
int
Fi (Mpa)
Patoor et 1
al. (Patoor ∑ Hnm fnfm 2 nm ,
et al., 1994;
Patoor et
al., 1996)
−∑ Hnm fm
m
H 1 µ
= ≈40
1000
H 2 µ
= ≈270
150
Lu and
Weng
(Lu and
Weng,
1997)
Vivet and
Lexcellent
(Vivet and
Lexcellent,
1998)
Huang and
Brinson
(Huang et
al., 2000;
Huang and
Brinson,
1998)
1
c1( 1−
c1) L
2
p p
( I − S)
ε ε
(only 1 variant
considered)
N
∑
s=
1
−
N
f ( 1−
f ) W
s s s
N
∑ ∑
f f W
s t st
s=
1 s= 1, t= 1,
t≠s 1 p p
Ws= ε L( I −S)
ε
2
G
1
p
− σ ε f =
1
∑
2 g=
1
G
∑
2 g=
1
G
∑
m=
1
g g
p
p
ε [ LI ( − S)
ε
g
p
− f L( I −S)
ε ] f
g
m m
g
g
1
− −
2
−
1 2 ( c1) L
p p
( I S)
ε ε
−( 1−2f ) W
s s
(simplified form
when only one
variant is
considered; same
as Lu and Weng
above)
− ∂W
∂
f ( gn , )
Gao and No explicit form. Small constant,
Brinson
(this paper)
lumped into Fc
int
< 1
W s ≈ 0.468
(Interaction
between
austenite and
martensite)
–40 to 60
when µ=40
-12 to 18
when µ=12
~1
note Fc=4
typically
27
A multivariant model
Interaction matrix H as approximation:
H 1 for self-accommodating variants
H 2 for non-self-accommodating ones
Resistance force of F int increases with
increased volume fraction, causing a
continuous strain hardening effect
Only values of H an order of magnitude
lower than listed will produce a more
appropriate non-strain-hardening response.
Good qualitative agreement to uniaxial
experimental results.
A single variant model
Analytical solution just for one variant
Resistance force of F int decreases with
increased volume fraction, becoming
actually a driving force for f >0.5
Aspect ratio of thin oblate was taken to be 0,
reducing the magnitude of F int
Non-linear dependence on f of dissipation
Good agreement to uniaxial experimental
results for plates of aspect ratio 0
A single variant model
Analytical solution just for one variant
Resistance force of F int decreases with
increased volume fraction, becoming
actually a driving force for f >0.5
Very thin oblate (a3/a1≈0.001) → lower Ws
Very good agreement to uniaxial
experimental results
A multivariant model
Analytical solution without additional
approximation
Self-accommodating groups of variants
without interaction among themselves
Plate aspect ratio: a3/a1=0.01
Long time of calculation
Good qualitative agreement to uni- or multiaxial
experimental results
A multivariant model
<strong>Based</strong> on invariant plane nature of
martensitic transformation
Accounts for anisotropy in single crystal case
Very fast calculation
Excellent agreement to uni- or multi- axial
experimental results

Table 2. Not all parameters necessary for the simulation were readily available for the experimental
materials. Therefore some parameters were found in the literature for materials with very similar
compositions. Table 2a contains parameters known for the materials used in the experiments and table 2b
lists other necessary parameters for similar materials attained elsewhere in the literature. Original sources
are cited in the tables.
Experimental
Materials (wt%)
Cu-13.95Al-3.93Ni
(Shield, 1995) in Fig.
5
Cu70.17Zn25.63Al4.2
(Goo and Lexcellent,
1997; Lexcellent et al.,
1996) in Fig. 3
Cu71Zn25Al4
(Gall et al., 1998) in
Fig. 8, 9
�n
�m
g
volume change
Transformation
Temperatures
Mf= - 13 °C, MS=5 °C
AS=3 °C, Af=23 °C
Mf= 12.5 °C, MS=22.5 °C
AS=23.25 °C, Af=30 °C
Mf= - 32 °C, MS= - 10 °C
AS= - 15 °C, Af=5 °C
0.6575, -0.7376, -0.1536
-0.7256, -0.6737, -0.1403
0.1903
- 0.79%
(Calculated by the authors using
lattice parameters for Cu-14.0Al-
4.2Ni (wt.%, β1 ′ 18R) (Otsuka and
Shimizu, 1974; Otsuka and Shimizu,
1981))
-0.1489 0.7223, 0.6754
-0.1350, 0.6550, -0.7434
0.19386
- 0.17%
(Recalculated by the authors using
lattice parameters for Cu70Zn26Al4
(wt.%, 18R) (Zhu et al., 1986))
Table 2a
Equilibrium Test Volume Properties
Temperatures Temperature change used for
T0=(MS+Af)/2=14 °C
T0′=(Mf+AS)/2= - 5 °C
40 °C N/A Fig. 6, 7
T0=(MS+Af)/2=26 °C
T0′=(Mf+AS)/2= 18 °C
T0=(MS+Af)/2= - 3 °C
T0′=(Mf+AS)/2= - 24 °C
Table 2b
Elastic constants
C11 = 142.6 (Gpa)
C12 = 127.4 (Gpa)
C44 = 97 (Gpa)
(Hellwege and Hellwege, 1984)
or
µ=35 (Gpa)
ν=0.3
(Otsuka and Wayman, 1998; Shield,
1995))
µ=26 (Gpa)
ν=0.3
(Goo and Lexcellent, 1997; Lexcellent
et al., 1996)
28
62 °C N/A Fig. 3, 4
25 °C - 0.3% Fig.
10 & 11
B (MPaK -1 )
F C (MPa)
0.30
4.1
(tailored for
(Shield, 1995))
0.21
1.7
(tailored for
(Goo and
Lexcellent, 1997;
Lexcellent et al.,
1996))
Properties
used for
All CuAlNi
calculations
Fig. 6 & 7
All CuZnAl
calculations
Fig. 3, 4, 10 &
11

Table 3. Measured and predicted angle between the tension direction and the intersection direction of the
habit plane and the sample surface for figure 5 and figure 7. See text for details.
Specimen
A1-T1b
[2.43 1 0]
A1-T2b
[-1 -1 1.73]
A1-T3b
[-1 -1 1]
Experimentally
measured angle
for initial
interface
The axial strains of possible active variants
along tensile direction from
phenomenological calculation for β1′ (18R)
46.8° V17: 0.0954020
V18: 0.0954020
21.4° V6: 0.0599651
V15: 0.0599475
9.8°
29
Calculated angle
for initial
interface
48.402386°
same as above
24.997863°
same as above
Variants picked
up by model
√
√
√
V6: 0.0228394 9.686614°
V8: 0.0228493 55.529479° √
V13: 0.0227965 55.529479°
V15: 0.0228120 9.686614°
V18: 0.0228072 8.748870°
V19: 0.0228325 8.748870°
Table 4. Uniaxial (1 and 3) and 3D (2 and 4 through 7) Stress State. Applied to the Polycrystalline CuZnAl
Specimens.
Test Number and
Description
#1
Pure tension
#2
Zero hydrostatic
stress
#3
Pure compression
#4
Triaxial compression
#5
Triaxial compression
#6
Triaxial compression
#7
Pure hydrostatic
stress
Applied Stress State Effective
Stress σ eff
L
M
O
P
σ
σ ij =
0 0
M0
0 0
NM
P
0 0 0QP
L2σ0
0
σ ij = M
O
M 0 −σ0P
NM
P
0 0 −σQP
σ
σ ij =
L−
0 0
M
O
M 0 0 0P
NM
P
0 0 0QP
σ
σ ij σ
σ
=
L−3
0 0
M
O
M 0 − 0P
NM
P
0 0 − QP
σ
σ ij σ
σ
=
L−2
0 0
M
O
M 0 − 0P
NM
P
0 0 − QP
σ
σ ij σ
σ
=
L−17
. 0 0
M
O
M 0 − 0P
NM
P
0 0 − QP
σ
σ ij σ
σ
=
L−
0 0
M
O
M 0 − 0 P
0 0 −
NM
QP
Relative
Hydrostatic
Pressure σ h
. σ eff
σ +033
3σ −000 . σ eff
σ −033 . σ eff
2σ −083 . σ eff
σ −133 . σ eff
0.7σ −176 . σ eff
0 −∞σ eff

Figure 1. Typical micromechanics assumption on martensitic variants: an inclusion (most models include
one variant only while a few include a group of variants) is embedded in an infinite matrix together with
numerous other inclusions.
c
b
a
3.5mm
Figure 2. Temperature-induced martensites (a) and stress-induced martensites (b-d). a) In a single CuAlNi
crystal: two martensite plates are seen, one toward the end of the narrow section and the other near the hole.
Each martensite plate is composed of two HPVs; b) single CuAlNi crystal under uniaxial tension, strain
rate 10 -4 s -1 used: one large martensite band seen at the right edge (specimen width 3mm); c) a section of
same single crystal loaded at strain rate of 1 s -1 showing evenly spaced martensitic plates spanning the
specimen; and d) NiTi polycrystal under uniaxial tension, where martensitic plates are seen to typically
span the entire width of the grains.
30
β1
10 −4 m
d

Stress σ 11 (x10 6 Pa)
600
500
400
300
200
100
0
0
10
Single crystal pseudoelasticity
Simple constant interaction
Micromechanics interaction
Experimental result
20
30
31
40
Strain ε 11 (x10 -3 Pa)
Figure 3. Single crystal pseudoelastic response predicted by the original <strong>Multivariant</strong> model with
Micromechanics interaction and the simplified <strong>Multivariant</strong> <strong>Model</strong> with simple constant interaction for a
Cu70.17Zn25.63Al4.2 (wt%) alloy. The test temperature is at 62 °C, T0= 26 °C, and T0′= 18 °C. Angles between
tensile axis and [100], [010], and [001] of the single crystal are 46.16°, 70.68°, and 50.13° respectively.
Experimental result is from (Goo and Lexcellent, 1997; Lexcellent et al., 1996).
Stress σ 11 (x10 6 Pa)
600
500
400
300
200
100
0
0
10
20
30
50
40
60
Polycrystal pseudoelasticity
Simple constant interaction
Micromechanics interaction
Strain ε (x10 11 -3 Pa)
Figure 4. Polycrystal pseudoelastic response predicted by the original <strong>Multivariant</strong> model with
Micromechanics interaction and the simplified <strong>Multivariant</strong> <strong>Model</strong> with simple constant interaction for a
Cu70.17Zn25.63Al4.2 (wt%) alloy. The test temperature is at 62 °C, T0= 26 °C, and T0′= 18 °C (Goo and
Lexcellent, 1997; Lexcellent et al., 1996).Note that a volume fraction of martensite when averaged over all
grains is approximately 50% at the stress level of 380Mpa in the simplified model. <strong>Based</strong> on recent
experimental observations, such a state is reasonable in polycrystalline materials at the onset of significant
permanent plastic deformation and the stop of continued martensite formation (Brinson et al., 2001).
70
50

Figure 5: The stress-strain curves for fully austenitic specimens A1-T1b, A1-T2b and A1-T3b at 40 °C of a
Cu-13.95Al-3.93Ni (wt%) single crystal (after (Shield, 1995)).
Stress σ 11 (x10 9 Pa)
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0
0.00
0.02
T3
T2
0.04
Isotropic PE with
Micromechanics interaction
[-1 -1 1] or T3
[-1 -1 1.73] or T2
[2.43 1 0] or T1
T1
Strain ε11 Figure 6: Results predicted by the original <strong>Multivariant</strong> <strong>Model</strong> with interaction for a Cu-13.95Al-3.93Ni
(wt%) single crystal under different uniaxial loading directions. The test temperature is at 40 °C,
T0=(MS+Af)/2=14 °C, and T0′=(Mf+AS)/2= - 5 °C. See text for details.
32
0.06
0.08
0.10

Stress Σ 11 (x10 6 Pa)
600
500
400
300
200
100
0
0.00
T3
0.02
T2
Anisotropic PE with
small constant interaction
[-1 -1 1] or T3
[-1 -1 1.73] or T2
[2.43 1 0] or T1
0.04
Strain ε11 Figure 7: Results predicted by the simplified <strong>Multivariant</strong> <strong>Model</strong> without interaction for a Cu-13.95Al-
3.93Ni (wt%) single crystal under different uniaxial loading directions. The test temperature is at 40 °C,
T0=(MS+Af)/2=14 °C, and T0′=(Mf+AS)/2= - 5 °C. See text for details.
Figure 8: Effective stress-strain plots for a polycrystalline Cu71Zn25Al4 (wt%) <strong>SMA</strong> with a volume change
of - 0.3% at a temperature of 25 °C. Each number corresponds to a 1D or 3D loading shown in table 4
(after Gall et al. (Gall et al., 1998)).
33
0.06
T1
0.08
0.10

Figure 9: Effective stress at the onset of transformation vs. the hydrostatic stress at the onset of
transformation for a polycrystalline Cu71Zn25Al4 (wt%) <strong>SMA</strong> with a volume change of - 0.3% at a
temperature of 25 °C. Each number corresponds to a 1D or 3D loading in table 4 (after Gall et al. (Gall et
al., 1998)).
Effective Stress (x10 6 Pa)
400
300
200
100
0
0
5
10 15 20
Effective Strain (x10 -3
)
34
Pure Tension (#1)
Zero Hydrostatic Pressure (#2)
Pure Compression (#3)
Triaxial Compression (#4)
Triaxial Compression (#5)
Triaxial Compression (#6)
Figure 10: <strong>Simplified</strong> <strong>Multivariant</strong> model predictions for effective stress-strain plots for a polycrystalline
Cu71Zn25Al4 (wt.%) with a volume change of - 0.3%. The test temperature is at 25 °C with the equilibrium
temperature T0= - 3 °C, and T0′= - 24 °C. Each one of the curves corresponds to a 1D or 3D loading shown
in table 4. See text for details.
25
30

Effective Trans. Stress (x10 6 Pa)
350
300
250
200
150
100
50
σ eff Y offset
0.100%
0.050%
0.025%
With Micromechanics Interaction
Pure Tension (#1) Zero Hydrostatic (#2)
Pure Compression (#3) Triaxial Compression (#4)
Triaxial Compression (#5) Triaxial Compression (#6)
With Simple Constant Interaction
-300 -200 -100 0 100
Hydrostatic Pressure (x10 6
Pa)
Figure 11: <strong>Model</strong> predicted effective stress at the onset of transformation vs. the hydrostatic stress at the
onset of transformation for a polycrystalline Cu71Zn25Al4 (wt%) <strong>SMA</strong> with a volume change of - 0.3%. The
test temperature is at 25 °C with the equilibrium temperature T0= - 3 °C, and T0′= - 24 °C. Each number
corresponds to a 1D or 3D loading shown in table 4. See text for details.
Figure 12: Conversion process during loading of a thermally induced 2H (γ1′) <strong>SMA</strong> material at low
temperature. Note that the final product after loading is not a single habit plane variant, but a
correspondence variant (a detwinned habit plane variant). After work by Saburi and Nenno (Saburi and
Nenno, 1981) where some information about the intermediate products are known, however the details of
the loading and specimen orientation are not given.
35