Nonlinear bending analysis of shape memory alloy beam considering both material and geometric nonlinearity effects

This study examined the nonlinear super-elastic bending of shape memory alloy beam considering the material and geometric nonlinearity effects that coupled together. Shape memory alloy properties change instantaneously at different points in the beam, while they are unknown at the same time. In other words, coupling of the governing and kinetic equations of the shape memory alloy beams together results in a more complicated analysis. In this study, the governing equations were extracted through using the Timoshenko beam theory and applying the principle of virtual work. For achieving this goal, von Karman strains were applied to consider large deflections. The Boyd-Lagoudas three-dimensional constitutive model and return mapping algorithm were also used for shape memory alloy modeling. Furthermore, in order to obtain the characteristics of finite element beam, the Galerkin weighted-residual method was used by developing the iterative nonlinear finite element model. Considering the different supporting conditions and forces for the shape memory alloy beam, this study examined their effects on the distribution of martensitic volume fraction, stress distribution, and changes in the location of the neutral axis. The obtained results revealed that as loading increases, the magnitude of martensitic volume fraction and the level of hysteresis increase, which in turn would result in reduction of the modulus of elasticity and the strength of the material and consequently increases the deflection of shape memory alloy beam. The findings suggested the necessity of nonlinear strain field in this modeling by which the stress distribution and volume fraction become asymmetric along the beam thickness. The results were presented in the forms of loading and unloading diagrams for different support and force conditions, and the martensitic volume fraction along the length and through the thickness of the shape memory alloy beam were also shown. To validate the proposed formulation, the results were compared with other experimental findings in this regard suggesting that there is an acceptable and satisfying level of agreement between them.