On new nonlinearity in third-order elastic modulus for vibrational analysis of FG porous beam based on nonlocal strain gradient and surface energy by modified homotopy perturbation method

In the current study, by considering both material and geometry nonlinearity, the nonlinear free vibration of functionally graded porous nanobeams based on the nonlocal strain gradient theory is investigated. Generally, the third-order elastic modulus, von Karman strain-displacement relationship, is assumed to incorporate the nonlinearity into the differential equation of the problem. In this paper, for the first time, the nonlinear vibrational response of the nanobeam does not merely depend on the geometric nonlinearity. Also, material nonlinearity of the bulk and surface layers is taken into consideration. Governing equations and the related boundary conditions are obtained by utilizing Hamilton's principle. The nonlinear partial differential equations are reduced to the ordinary differential equations by employing the Galerkin's approach and are solved by using the modified homotopy perturbation method. The influence of parameters such as the length scale parameter, the porosity volume index, the power-law index, and material nonlinearity parameters of the bulk and surface layers on the time response and nonlinear frequency of functionally graded (FG) porous nanobeam is examined. Considering FG porous material for the nanobeam produces cubic and quantic nonlinear terms in the differential equation which leads to the creation of the drift phenomenon in the nonlinear free vibration of the nanobeam. Moreover, the coupled effect of the dimensionless length scale parameter and dimensionless nonlocal parameter on the nonlinear frequency of the nanobeam is also investigated comprehensively.