Analysis of localization phenomena in Shape Memory Alloys bars by a variational approach

Localization of the phase transformations in Shape Memory Alloys (SMA) wires are well known. Several experimental and theoretical studies appeared in the last years. In this work the problem is addressed by means of a variational approach within the framework of the modeling of rate-independent materials by the specification of a non-local free energy and a dissipation function, focusing attention on the basic case of isothermal conditions. General expressions are given for a rather broad class of models, whereas a simple model is studied in detail. A full stability analysis of both homogeneous and non-homogeneous solutions is carried out analytically, showing that stable non-homogeneous solutions have necessarily to occur if the bar is longer than an internal length determined by the constitutive parameters. The analysis also shows that snap-back phenomena may occur both in the nucleation and the coalescence phase, depending on another material length which is also function of the number of transformation fronts. This helps to explain why the second stress drop associated to coalescence is much more difficult to observe experimentally. Closed form expressions are given for the phase fraction profiles of both single and multiple localizations as well as nucleation and propagation stresses. A comparison between the prediction of the model with experimental data finally shows a good agreement both in terms of global response and in the spatio-temporal evolution of the transformation domains.