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

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
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2. If | Fnet | ≠ 0 , assign an initial trial step size h proportional to 1/| Fnet |
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3. While | Fnet | ≠ 0 do
Calculate ∆f i (volume fraction change) and Err i (estimated error) by taking a Cash-
Karp Runge-Kutta step
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