Magnetic field within a magnetic shape memory alloy and an equivalent uniform applied magnetic field for model input

Magnetic shape memory alloys (MSMAs) exhibit recoverable strains of up to 10% due to reorientation of their martensitic tetragonal unit cell. A stress or magnetic field applied to the material will cause the short side of the unit cell (which is approximately aligned with the magnetic easy axis) to align with the input to the material, resulting in an apparent plastic strain. This strain can be fully recovered by an applied stress or magnetic field in a perpendicular direction. When the martensitic variants reorient, twin boundaries, which separate the different variants, form and move throughout the specimen. A number of models have been proposed for MSMAs and many of these models are homogenized, i.e. the models do not account for twin boundaries, but rather account for the volume fraction of material in each variant. These types of models often assume that the MSMA is subject to a uniform field so that there is no appreciable difference in the volume fraction of variants in each location. In this work, we address the issue of how these models can be used when the field is not uniform. In particular, we look at the experiments from Feigenbaum et al., in which a MSMA trained to accommodate three variants, was subject to 3-dimensional magneto-mechanical loading. Due to experimental constraints, the field applied to the MSMA was not uniform. In this work, to understand the actual field distribution during experiments, we performed a high-resolution 3- dimensional finite element analysis (FEA) of the magnetic field experienced by the MSMA sample. The FEA allowed us to determine how non-uniform the experimentally applied field was and the differences between the applied field and the field experienced by the MSMA. Furthermore, we use the FEA to determine the average field experienced by the MSMA, and identify an equivalent uniform applied field that could serve as input for the model. For the latter, we seek a uniform magnetic field which gives similar magnetic field within the MSMA specimen as the true experimental conditions.