Nonlinear analysis of superelastic shape memory alloy cantilever beam of variable cross-section

Abstract

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 larger deflections compared to those of constant cross-section beams. Furthermore, the degree of variability/complexity will further increase if the material or physical nonlinearity is involved, typically for an SMA beam. That takes such a study in the domain of geometric nonlinearity together with material nonlinearity. Problems of physical and geometric nonlinearities are always challenges for the engineers. Analysis was conducted for such a canti lever beam with reducing cross-sectional area, made of SMA with highly nonlinear stress-strain curves. Initially, experiments were conducted for stainless steel cantilever beams theoretically of uniform strenb'lh, with nonlinear stress-strain curves. In addition to the experiment, a computer code in 'C++' has been developed using the Runge-Kulla technique for the purpose of simulation. Et1ective modulus-curvature relations obtained from the nonlinear stress-strain relations for dit1erent sections of the beam that are used j()r the analysis. Nonlinear analysis shows the stresses are not that high as predicted by ideal theories. Moreover, the tensile and compressive stresses are slightly different in. magnitude and both decrease along the span. Experimental load-deflection curves are found to be initially linear but, nonlinear and convex upward at a high load. Comparison of the numerical results with the available experimental results and theory shows" excellent agreement verifying the soundness of the entire numerical simulation scheme. Next the same computer code has been used for the purpose of simulation for SMA beam but with SMA's stress-strain data. Moment-curvature and effective moduluscurvature relations are obtained from the highly nonlinear stress-strain relations for different sections of the beam .. For rigorous analysis, the true stress-strain curves in tension as well as in compression have been used for the study. It is seen that nonlinear stress-strain curve governs the response of the beam. Moreover, load-deflection curves are initially linear but, nonlinear and convex upward at a high load. It is found that more material can be removed from an SMA beam of uniform strength, originaily designed without considering geometric nonlinearity and the effect of end-shortening. Furthermore, the compressive stress IS significantly higher than the tensile stress because of asymmetry in stress-strain relations. If 'end-shortening' is considered, stress falls along the span. Interestingly, for different cases considered, it is found that the beam material may remain in the parent austenite phase, mixed phase or in the stress induced martensitic phase.