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A reduced-order thermomechanical model and analytical solution for uniaxial shape memory alloy wire actuators

A lumped shape memory alloy (SMA) model is derived from the thermodynamic model of Chang et al (2006 Contin. Mech. Thermodyn. 18 83–118), using a set of simplifying assumptions, that reduces the system of partial differential equations for an SMA/bias spring actuator to a nonlinear, first-order, ordinary differential equation. Dimensionless state variables and parameters are defined that are useful for characterizing the actuator system and for studying its performance and scaling. A general analytical solution to the nonlinear differential equation governing phase transformation is found in terms of the Lambert function for a piecewise constant Joule heating input and a constant temperature convective environment. The analytical solution provides a useful and convenient tool for assessing the time-dependent, hysteretic response of this simple class of SMA actuators, with which design and optimization studies are performed.