Cosserat modeling for deformation configuration of shape memory alloy unimorph actuators

Abstract

Shape memory alloys (SMAs) can contract their length via a crystalline phase transition that is dependent upon their temperature and stress state. SMAs have been used as linear micro-actuators due to their high strength to weight ratio and compact structure. However, the relatively low linear contraction ( $\mathrm{~}$ 4%–5% in length) limits their use. To remedy this, the SMA can be offset from a passive structure, which acts to magnify the deformation. The resulting amount of deformation depends upon the material properties and geometry of both the SMA and the passive structure. In this work, geometrically exact beam theory (also known as Cosserat theory) is coupled with SMA constitutive relations to model the maximum deformation configuration of these actuators. Four of these actuators of various lengths were fabricated and tested to verify the model. For the four actuators tested, the mean squared error between the experimental results and the Cosserat model were between 0.0702 mm (0.1% error) for the shortest actuator (66 mm in length) and 3.59 mm (2.7% error) for the longest actuator (135 mm in length). These results show that the closed form solution derived for this Cosserat beam model can accurately model the deformation of these active structures.