Thermo-mechanical modelling of a superelastic shape-memory wire under cyclic stretching–bending loadings

Experimental investigations on superelastic shape-memory alloy wires show a significant dependency of the load–displacement response on the loading–unloading rate, coupled with a non-negligible oscillation of the wire temperature. Motivated by this observation, the present work proposes a thermo-mechanical model, suitable to study the response of a superelastic shape-memory alloy wire under cyclic stretching–bending loadings. The model is based on the kinematical assumptions that sections which are plane in the undeformed configuration remain plane also in the deformed configuration and that the wire shear deformation is negligible. As a consequence, we need to introduce only a one-dimensional constitutive stress–strain equation; this relation involves a scalar internal variable, representing the martensite fraction, variable for which evolutionary equations are proposed. The whole model is then thermo-mechanically coupled due to the presence of internal heat sources in the form of phase-transition latent heat and of mechanical dissipation. The paper also addresses a numerical scheme for the integration, in time, of the model and the algorithmic solution of the corresponding time-discrete thermo-mechanical problem. Finally, results from numerical simulations are presented to show the response of a wire under cyclic stretching and bending loading conditions imposed at different strain-rates.