A comparison between different constitutive models in the analysis of the nonlinear dynamic behaviour of Shape Memory Alloys oscillators

The nonlinear dynamics of Shape Memory Alloys (SMA) oscillators has been studied by various authors since several years, by using different constitutive models for the restoring force: ranging from different kinds of internal variable models to drastically simplified polynomial models. Internal variable models are richer and can take into account most of the complex features of the SMA behavior, on the contrary, polynomial models are strongly approximate, being based on a Falk-type free energy that gives rise to a non-hysteretic non-monotonic force-displacement curve. While the two types of models give completely different results in the force-displacement-temperature space, in the nonlinear dynamics of SMA oscillators one is often interested in more global information like, for example, the determination of the maximum displacement over a periodic trajectory. Therefore one can wonder how much the fine details of the constitutive modelling actually influence the global dynamical information. To investigate this problem, in this work we compare two extreme cases of modelling: one internal variables model that takes into account hysteresis, different elastic properties of the phases, and asymmetry between tension and compression, versus a very simplified model without hysteresis which is qualitatively analogous to a polynomial one. Some sample trajectories are analysed and the whole dynamic picture is described and compared through the computation of frequency-response curves.