One-dimensional constitutive SMA model with two martensite variants: analytical and numerical solutions

This paper deals with the one-dimensional modeling of a shape memory alloy (SMA) in order to reproduce the special thermo-mechanical response of the material. In particular, a new one-dimensional SMA model is developed, based on the introduction of three phases: tensile martensite, compressive martensite and austenite; the ability to reproduce the pseudo-elastic and shape memory effects; the different behavior in tension and compression; the different elastic properties of the three phases; and the reorientation process of the martensite. The model assumes the martensite volume fractions as internal variables, whose evolution is governed by stress and temperature. New analytical solutions are developed to solve the axial problem, which take into consideration the axial loads and thermal processes. Furthermore, a numerical procedure is developed in order to time-integrate the kinetic laws that rule the evolution of the internal variables during the phase transformations and a 2-node finite element beam is implemented. Considering the constant and non-constant elastic properties for the three SMA phases, some applications are presented in order to verify the effectiveness of the proposed model and the analytical and numerical solutions. Comparisons with experimental data related to the axial response of SMA wires and lamina are carried out. Finally, a numerical application concerning an SMA lamina is presented, which is subjected to bending load at high temperatures inducing the pseudo-elastic effect in the material.