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

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
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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)