Two simple and fully analytical models are presented for a SMA bar or wire of circular cross section subjected to fully-reversed cyclic torsional loading, by taking into consideration the reorientation of the martensitic variants occurring during unloading and reverse loading. The process is assumed to take place at constant temperature between the start temperatures of the martensitic and austenitic transformations. In the first part of this work, the same shear modulus is taken both for Martensite and Austenite. In the second part, different elastic shear moduli are considered for the two phases. The volume fractions of both positive and negative twisted Martensite are assumed to evolve linearly with the shear stress. The bar is initially in a state of Austenite. As the applied torque is increased the martensitic transformation starts from the outer surface and then it extends inwards. If the maximum applied torque is large enough, then the complete Martensitic transformation takes place in the outer region of the cross section. During unloading and subsequent reverse loading the martensitic reorientation process may occur starting from the boundary between the fully martensitic outer region and the intermediate transforming region. Particular attention is focused on modeling the unloading and reverse loading processes. At each stage, the radial distributions of shear stress and Martensite variant are calculated analytically. A closed form relation between the applied torque and the angle of twist is presented for the entire process in the case of equal shear moduli, and only for the loading and elastic unloading stages in the case of different shear moduli. The approach is then validated against analytical, numerical and experimental results available in the literature for the direct loading–unloading process. Application to the seismic response of dissipative systems based on SMA helical springs is also envisaged.

通过考虑在卸载和反向加载过程中发生的马氏体变体的重新定向，为圆形截面的SMA棒材或线材提供了两个简单而完整的分析模型，该模型承受了完全反向的循环扭转载荷。假定该过程在马氏体和奥氏体转变的起始温度之间的恒定温度下进行。在这项工作的第一部分中，马氏体和奥氏体的剪切模量相同。在第二部分中，考虑了两个相的不同弹性剪切模量。假定正扭曲马氏体和负扭曲马氏体的体积分数均随剪切应力线性变化。钢筋最初处于奥氏体状态。随着施加扭矩的增加，马氏体相变从外表面开始，然后向内延伸。如果施加的最大扭矩足够大，则在横截面的外部区域将发生完整的马氏体相变。在卸载和随后的反向加载期间，马氏体的重新定向过程可能从完全马氏体的外部区域和中间转变区域之间的边界开始。特别要注意的是对卸载和反向加载过程进行建模。在每个阶段，均会通过解析计算出切应力和马氏体变体的径向分布。在剪切模量相等的情况下，整个过程中所施加的扭矩和扭转角之间呈闭合关系，并且仅在剪切模量不同的情况下用于加载和弹性卸载阶段。然后，根据文献中直接加载-卸载过程中可用的分析，数值和实验结果对这种方法进行了验证。还设想将其应用于基于SMA螺旋弹簧的耗散系统的地震响应。