Model for strain and magnetization in magnetic shape-memory alloys

The large magnetic-field-induced strains observed in martensitic phases based on ${\mathrm{Ni}}_{\mathrm{2}}\mathrm{MnGa}$ and in other magnetic shape memory alloys are believed to arise from a process of twin-boundary motion rather than magnetostriction. The dependence of strain on magnetization, $\mathrm{e(M),}$ generally shows a large component that is linear (rather than quadratic) in $M$ below saturation (quadratic dependence being typical of magnetostrictive strain). A simple phenomenological model for the magnetization process and field-induced strain by twin-boundary and phase-boundary motion is proposed for both the strong and weak anisotropy cases. The model is shown to account for the nearly linear dependence of strain on magnetization in the martensitic phases of these materials. It shows the field dependence of the magnetization and strain to be functions of an effective stiffness constant, $\mathrm{C,}$ the transformation strain, $e_0,$ and the magnetic anisotropy of the martensitic phase, $K_u,$ through two reduced field parameters, $h_e{\mathrm{=M}}_s{\mathrm{H/Ce}}_0^2$ and $h_a{\mathrm{=M}}_s{\text{H/2K}}_u.$ The model also accounts for the magnetization remanence and the nonlinear field dependence closer to saturation (which produces little strain). The curvature observed in $\mathrm{e(H)}$ at very low fields is not described by this two-variant model and may be related to the fact that more variants exist which respond to the field with a distribution of susceptibilities.