A shape memory alloy helix model accounting for extension and torsion

In the present paper a mathematical model able to reproduce the mechanical response of a shape memory alloy (SMA) helical spring is presented. The proposed model is based on a simplified but effective theory of the helix that assumes small strains and large displacements. The kinematics of the helix and the related generalized strain measures are introduced, with stress resultants conjugated to them and the corresponding equilibrium equations of the curved beam. A specific constitutive law for this structural model is then presented for the SMA material, which accounts for the phase transformation due to the coupling of the shear and axial strains and to the effect of the temperature variation as well. Then, a numerical procedure, based on the backward Euler time integration algorithm and the prediction-correction technique for solving the nonlinear time step, is developed. Several numerical applications are presented to assess the reliability of the proposed SMA helix model and the implemented numerical procedure, and to investigate the response of the helix. Initially, the behavior of a circular SMA cross-section is studied, comparing the results obtained by the proposed modeling approach with the ones carried out by finite element analyses. Then, the helix structural element is considered and results are again compared with finite element outcomes. Finally, a sensitivity analysis is performed varying the pitch of the helix. •A mathematical model able to reproduce the mechanical response of a shape memory alloy (SMA) helical spring is presented.•The proposed model is based on a theory of the helix that assume small strains and large displacements.•A new constitutive law for this structural model is presented for the SMA material, accounting for shear and axial strains.•A numerical procedure, based on the backward time integration and the prediction-correction technique, is developed.•Numerical applications are presented to assess the reliability of the proposed model and to study the response of the helix.