Influence of third-order elastic modulus as material nonlinearity on free and forced vibration of a nonlocal nanobeam rested on viscoelastic medium

In this study, the nonlinear free and forced vibrations of a nanobeam resting on a viscoelastic medium have been investigated. The quadratic material nonlinearity, which is neglected in the literature, is included in the stress-strain relation, and then using nonlocal elasticity theory, the nonlinear vibration of the nanobeam considering both material and geometrical nonlinearities is investigated. To this end, the governing equation of motion is extracted using the Euler-Bernoulli beam theory and utilizing Hamilton's principle. Applying Galerkin's method, the nonlinear differential equation of the nanobeam is obtained. The cubic and quantic nonlinearities in the governing differential equation are due to geometrical and material nonlinearities, respectively. Using Modified Homotopy Perturbation Method, the nonlinear differential equation is solved, and the nanobeam time response and frequency are obtained. Also, the frequency response of nanobeam in the presence of a viscoelastic medium and harmonic excitation is obtained. The results illustrate that material nonlinearity has an important effect on the free and forced vibration responses of the system. For validation, the obtained results of nanobeam are compared with the results obtained from the fourth-order of Runge Kutta numerical method and previous research.