A nonlinear finite element method for analyzing the bending behavior of functionally graded shape memory alloys under the loading process

Based on the Euler–Bernoulli beam element theory, a nonlinear finite element method is proposed to solve the mechanical response of functionally graded shape memory alloys (FG-SMAs) three-point bending beam considering the tension–compression asymmetry under the loading process. The Auricchio’s shape memory alloy (SMA) constitutive model was used to describe the phase transformation relationship of SMA. The constitutive model of the power exponent distribution law of FG-SMA was developed by using the theory of composite mechanics. Using the virtual work principle, the bending beam equilibrium equation of FG-SMA was developed. The finite element method and Newton–Raphson method were used to solve the equilibrium equation of the three-point bending beam of FG-SMA. The accuracy of the proposed method was verified by comparing the present results obtained by the nonlinear finite element method with the existing literature. Then three-point bending beam analysis of FG-SMA was carried out. The results show that the increase in the graded index and tensile asymmetry coefficient increases the stiffness of the beam. When the SMA enters phase transformation, the increase in temperature will reduce the deflection of the beam. In addition, the strain variation in the mid-span section near pure shape memory alloy layer is more obvious than that near pure ceramic layer due to temperature. The compressive axial load is more likely to cause beam bending than tensile axial load.