Description of isothermal localization phenomena in Shape Memory Alloys bars by means of a variational approach

While the basic properties of Shape Memory Alloys (SMA) were known since 1960s, Shaw and Kyriakides first noticed the occurrence of non-homogeneous strain fields associated with the localization of the phase transformations at the macroscopic scale in NiTi superelastic wires and, nowadays, localizations of the phase transformations in SMA wires are well known.Several experimental and theoretical studies appeared in the last years on this type of phenomena. 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 the 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 a second internal length of the material. 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 of spatio-temporal evolution of the transformation domains.