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.

在本文中，提出了一种能够再现形状记忆合金（SMA）螺旋弹簧机械响应的数学模型。所提出的模型基于螺旋的简化但有效的理论，该理论假定应变小且位移大。介绍了螺旋线的运动学和相关的广义应变测度，并结合了应力合成和弯曲梁的相应平衡方程。然后针对SMA材料给出了针对该结构模型的特定本构定律，该定律解释了由于剪切应变和轴向应变的耦合以及温度变化的影响而引起的相变。然后，基于反向欧拉时间的数值过程开发了求解非线性时间步长的集成算法和预测校正技术。提出了几个数值应用程序，以评估所提出的SMA螺旋模型和已实施的数值程序的可靠性，并研究螺旋的响应。最初，研究了圆形SMA横截面的行为，将所提出的建模方法所获得的结果与有限元分析所获得的结果进行了比较。然后，考虑螺旋结构元素，并将结果再次与有限元素结果进行比较。最后，执行灵敏度分析以改变螺旋的螺距。