Frequency Dependent Strain-Field Hysteresis Model for Ferromagnetic Shape Memory Ni–Mn–Ga

We quantify the relationship between magnetic fields and strains in dynamic Ni-Mn-Ga actuators. As a result of magnetic field diffusion and structural actuator dynamics, the strain-field relationship changes significantly relative to the quasistatic response as the magnetic field frequency increases. We model the magnitude and phase of the magnetic field inside a Ni-Mn-Ga sample as a 1-D magnetic diffusion problem with applied dynamic fields known on the surface of the sample, from which we calculate an averaged or effective field. We use a continuum thermodynamics constitutive model to quantify the hysteretic response of the martensite volume fraction due to this effective magnetic field. We postulate that the evolution of volume fractions with effective field exhibits a zero-order response. To quantify the dynamic strain output, we represent the actuator as a lumped-parameter, single-degree-of-freedom resonator with force input dictated by the twin-variant volume fraction. This results in a second-order, linear ordinary differential equation whose periodic force input is expressed as a summation of Fourier series terms. The total dynamic strain output is obtained by superposition of strain solutions due to each harmonic force input. The model accurately describes experimental measurements at frequencies up to 250 Hz.