Nonlinear numerical assessment of damped oscillation of SMA Timoshenko curved beams under impulsive loading

Abstract

Due to the many applications of shape memory alloys (SMAs) to make the structures more intelligent, these materials are getting great attention of researchers. Meanwhile, the nonlinear dynamic analysis of curved beams made of SMAs has not been investigated so far. Therefore, this work focuses on a nonlinear dynamic analysis of SMA curved beams under transverse impulse loading taking into account the pseudo-elastic behavior of SMAs. It is worth noting that both material and geometrical nonlinearities of the SMA curved beam are considered in this study. In order to model the nonlinear behavior of SMAs, the Lagoudas model is employed and for the mathematical modeling of the curved beam the Timoshenko beam theory under the assumption of von Karman nonlinear strains is used. Then, by employing the Hamilton principle, the governing equations of the structure are extracted, while the nonlinear kinematic equations of SMAs are coupled with the governing equations of the curved beam. To solve these coupled nonlinear equations, the numerical technique of differential quadrature method (DQM) along with Newmark's time integration scheme is employed. In this regard, the return mapping algorithm in conjunction with the Newton–Raphson method is employed to solve the nonlinear terms of equations. The quick convergence and high accuracy of the proposed formulation are achieved by the analysis of different examples. After that, some novel results are presented by investigating the influence of different types of boundary conditions, radius of curvature, angle of curvature and thickness of beam on the transient damped response, hysteresis loops and also martensite phase transformation of the SMA curved beams.