Analytical Model for a Superelastic SMA Beam

We propose an analytical model for a superelastic shape memory alloy (SMA) beam. The model considers reversible phase transformation between austenite and a single martensite variant driven by mechanical loading/unloading. In particular, we consider a cantilever beam subjected to a concentrated transverse force acting at the tip. The force is gradually increased from zero to a maximum value sufficient to cause complete transformation of the initially austenitic phase into martensite away from the beam core. The force is then gradually removed, resulting in complete strain recovery. In each stage of the loading/unloading process, an analytical relation is established between bending moment and curvature in terms of position along the axis of the beam. The model is compared to a uniaxial numerical beam model and to finite element analysis (FEA) results for the same beam in 3D, with very good agreement in each case. The moment-curvature relation is then integrated to obtain a nonlinear expression for the deflection and stress distribution in terms of position along the length of the beam. The expression is validated against 3D simulation results.