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Semi-exact solution for nonlinear dynamic analysis of graded carbon nanotube-reinforced beam with graded shape memory wires

Abstract In this study, the nonlinear dynamical response of a graded carbon nanotube-reinforced beam with graded shape memory wire has been investigated regarding the instantaneous variations of martensite volume fraction and therefore material nonlinearity along the beam. The boundary conditions of the beam are considered as the clamped-clamped. Carbon nanotubes and shape memory wires have been arranged in various distributions of UD, and FG – A along the thickness of the beam. The governing differential equation of the beam is obtained by the use of Brinson’s relation to describe the behavior of the shape memory wire, considering Von Karman nonlinear strain and utilizing Hamilton's principle. The obtained nonlinear equation is solved using the semi-exact solution. For this purpose, Galerkin’s method was first used to isolate the spatial derivatives from the time derivatives and then the obtained equation was solved using Runge-Kutta method. A comparative study has been conducted between the present study and the reference paper to confirm the presented model. This comparison study shows an appropriate agreement between the results of this research and the previous studies. In this research, the influence of the certain parameters such as the considering the nonlinear effects, the volume fraction of carbon nanotube, the type of carbon nanotube distribution, the volume fraction of shape memory wire, and the type of its arrangement on the vibrational behavior of the beam and its loss factor is investigated. Some results show that, in case of the vibration of the beam by the initial excitement, the higher density of carbon nanotubes in the mid-layers and the higher density of the shape memory wires in the outer layers of the beam lead to the higher amount of loss factor. Also, the percentage of the reduction in the loss factor which occurs due to the reduction of the volume fraction of the shape memory wire is almost the same in the beams with different distribution and arrangement of carbon nanotubes and shape memory wire.