A shape memory alloy (SMA) actuator typically has to operate for a large number of thermomechanical cycles due to its application requirements. Therefore, it is necessary to understand the cyclic behavioral response of the SMA actuation material and the devices into which they are incorporated under extended cycling conditions. The present work is focused on the nature of the cyclic, evolutionary behavior of two widely used SMA actuator material systems: (1) a commercially available Ni49.9Ti50.1, and (2) a developmental high-temperature Ni50.3Ti29.7Hf20 alloy. Using a recently developed general SMA modeling framework that utilizes multiple inelastic mechanisms, differences and similarities between the two classes of materials are studied, accounting for extended number of thermal cycles under a constant applied tensile/compressive force and under constant applied torque loading. From the detailed results of the simulations, there were significant qualitative differences in the evolution of deformation responses for the two different materials. In particular, the Ni49.9Ti50.1 tube showed significant evolution of the deformation response, whereas the Ni50.3Ti29.7Hf20 tube stabilized quickly. Moreover, there were significant differences in the tension-compression-shear asymmetry properties in the two materials. More specifically, the Ni50.3Ti29.7Hf20 tube exhibited much higher asymmetry effects, especially at low stress levels, compared to the Ni49.9Ti50.1. For both SMA tubes, the evolution of the deformation response under thermal cycling typically exhibited regions of initial transients, and subsequent evolution.