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Ferromagnetic Shape Memory Alloys (FSMAs) in the nickel manganese gallium system have been shown to exhibit large magnetically induced strains of up to 9.5% due to magnetically driven twin variant reorientation. In order for this strain to be reversible, however, an external restoring stress or magnetic field needs to be applied orthogonal to the field and hence the implementation of Ni-Mn-Ga in applications involves the use of electromagnets, which tend to be heavy, bulky and narrowband. In previous work at The Ohio State University a sample of Ni50Mn28.7Ga21.3 has been shown to exhibit reversible compressive strains of -4200 microstrain along its [001] direction when a magnetic field is applied along this same direction and no externally applied restoring force is present. This reversible strain is possible because of an internal stress field associated with pinning sites induced during manufacture of the crystal. This paper analyzes the switching between two variant orientations in the presence of magnetic fields (Zeeman energy) and pinning sites (pinning energy) through the formulation of a Gibbs energy functional for the crystal lattice. Minimization of the Gibbs free energy yields a strain kernel which represents the predicted behavior of an idealized 2-dimensional homogeneous single crystal with a single twin boundary and pinning site. While adequate, the kernel has limitations because it does not account for the following: (a) Ni-Mn-Ga consists of a large number of twin variants and boundaries, (b) the strength of the pinning sites may vary, and (c) the local and applied magnetic field will differ due to neighbor-to-neighbor interactions. These limiting factors are addressed in this paper by considering stochastic homogenization. Stochastic distributions are used on the interaction field and on the pinning site strength, yielding a phenomenological model for the bulk strain behavior of Ni50Mn28.7Ga21.3. The model quantifies both the hysteresis and saturation of the strain. Constrained optimization is used to determine the necessary parameters and an error analysis is performed to assess the accuracy of the model for various loading conditions.
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