Applying Eringen's nonlocal elasticity theory for analyzing the nonlinear free vibration of bidirectional functionally graded Euler-Bernoulli nanobeams

The present study develops a size-dependent Euler-Bernoulli beam model to analyze the nonlinear free vibration of a bidirectional functionally graded (BFG) nanobeam with immovable ends, based on the nonlocal theory. In order to eliminate the stretching and bending coupling caused by the asymmetrical material variation along the thickness, the problem is formulated relative to the physical neutral surface. By using Hamilton's principle, the underlying equations of motion, as well as the corresponding boundary conditions, of the problem are obtained. The differential quadrature method together an iterative algorithm has been used for determining the nonlinear vibration frequencies of the BFG nanobeams. The precision of the present formulation is evaluated through comparing the nonlinear vibration frequencies calculated using the proposed method with frequencies available in previous studies. Moreover, a number of illustrative numerical examples are also presented as a means to investigate the influences of the nonlocal parameters, vibration amplitude, material property gradient index, and boundary conditions on the nonlinear frequency ratio (i.e., the ratio of nonlinear frequency to linear frequency). It is found that the nonlinear vibration frequencies are greater than the linear frequencies for the same amplitude of the nonlinear oscillator.