Sensitivity Analysis of Shape Memory Alloy Shells

Shape memory alloys (SMAs) are active materials with a high power density capable of producing comparatively large actuation strains and stresses. However, designing effective multi-dimensional SMA actuators is a challenging task, due to the complex behavior of the material and the fact that often electrical, thermal and mechanical aspects have to be considered simultaneously. For this reason, interest in the application of systematic computational design approaches, such as design optimization techniques, to the design of SMA structures is increasing. To enable efficient SMA design optimization procedures, the availability of sensitivity information is crucial.

This paper presents the formulation and computation of design sensitivities of SMA shell structures using the direct differentiation method, in a steady state electro-thermo-mechanical finite element context. Semi-analytical and refined semi-analytical approaches are considered. The SMA constitutive model used in this study is particularly intended for design optimization of SMA structures and actuators. In contrast to the majority of SMA models, the formulation of this model is history-independent, which simplifies the sensitivity analysis considerably. The model is specifically aimed at the description of the superelastic behavior of NiTi alloys, based on the R-phase transformation. This behavior is characterized by its negligible hysteresis, which is very attractive for actuator applications.

This research is aimed at SMA shell structures, which can generate large actuator displacements through bending. The most general case of actuation is considered, where the SMA effects are initiated by temperature changes induced by Joule heating. This implies that a coupled electrothermo- mechanical finite element analysis is required. As a consequence, the sensitivity analysis includes coupling terms between three different physical fields.

The most challenging aspect of this work lies in particular in the fact that the constitutive model in the considered plane stress setting contains implicit equations, which lead to complications in the sensitivity analysis. This problem manifests itself in the thermo-mechanical sensitivity coupling terms and in sensitivities of derived mechanical responses such as stresses or equivalent strains. Solutions for this difficulty based on finite difference and analytical approaches are discussed. Finally, the accuracy of the sensitivities is evaluated on a representative finite element model of a miniature gripper, as a function of relative design perturbations in thickness, shape, loading and material parameters. A comparison is given between results obtained by finite-difference, semi-analytical and refined semi-analytical methods.