Abstract Specimens of magnetostrictive materials can transform a variation of their stress-induced size into a variation of a produced magnetic field and vice versa. These phenomena are utilized in magnetostrictive energy harvesters, vibration sensors, etc. Mathematical models of magnetostrictive materials vary from complex hysteretic models to relatively simple non-hysteretic models. In this paper, three mathematical models of Galfenol are considered, namely a non-hysteretic model, a non-hysteretic model with a feedback loop, and a model where the hysteresis is represented by the Preisach operator with a simplified Preisach density function. The parameters of these models are identified from measured magnetic and magneto-elastic curves. All the models are applicable in technical praxis. The output of the non-hysteretic model with a feedback loop best fits the measured data and, to some extent, reproduces fine features of magneto-elastic curves.