Nonlinear Quasi-static Analysis of SMA Curved Beams

Considering many applications of Shape Memory Alloys (SMAs) in order to make structures smarter, a quasi-static analysis of curved beams made of SMA has yet to be investigated. Therefore, a quasi-static analysis of curved beams made of SMA under constant transverse force was investigated in this research. Hence, the 3D Hernandez-Lagoudas model was used to express the nonlinear properties of an SMA curved beam. For the mathematical modeling of the SMA curved beam, the Timoshenko beam theory was applied considering the nonlinear strains. Then, the governing equations were extracted using Hamilton’s principle. The kinematic equations of SMAs were coupled with the governing equations of the beam, which made the analysis more complex. Differential quadrature method (DQM) in conjunction with the Newton–Raphson method was utilized to solve the nonlinear governing equations. DQM is a new numerical method based on the transformation of differential governing equations into algebra using weighting coefficients. The main objective of this paper is to obtain the martensite volume fraction, stress, and strain for all points of the beam. To validate this paper, results of static bending of a straight beam made of SMA using DQM were confirmed with other researchers’ works. Numerical results showed that with the increase in the curvature radius of the curved beam, the area of the hysteresis loop was increased, indicating a decrease in the strength of the structure and, consequently, an increase in the
static deflection. In addition, some new outcomes were studied for various boundary conditions. In the case of a clamped-clamped curved beam, it was concluded that at the beam’s ends, the martensite volume fraction had the highest value and as it approached the center of the beam, this value decreased.