Nonlinear analysis of cantilever shape memory alloy beams of variable cross section

Cantilever beams, made of shape memory alloy (SMA), undergo much larger deflection in comparison to those made of other materials. Again, cantilever beams with reducing cross section along the span show much larger deflections compared to those of constant cross section beams. Analysis was conducted for such a cantilever beam with reducing cross-sectional area made of SMA, taking into account its highly nonlinear stress–strain curves. A computer code in C has been developed using the Runge–Kutta technique for the purpose of simulation. For rigorous analysis, the true stress–strain curves in tension as well as in compression have been used for the study. Moment–curvature and reduced modulus–curvature relations are obtained from the nonlinear stress–strain relations for different sections of the beam and used in the simulation. It is seen that load–deflection curves are initially linear but nonlinear and convex upward at a high load. Furthermore, the compressive stress in the beam is significantly higher than the tensile stress because of asymmetry. Interestingly, for the different cases considered, it is found that part of the SMA beam material may remain in the parent austenite phase, mixed phase or in the stress-induced martensitic phase. Importantly, it is found that more material can be removed from an SMA beam of uniform strength, originally designed without considering geometric nonlinearity and the effect of end-shortening. Comparison of the numerical results with the available theory shows very good agreement, verifying the soundness of the entire numerical simulation scheme.