Application of Combined Nonlocal and Surface Elasticity Theories to Vibration Response of a Graded Nanobeam

This chapter investigates the combined nonlocal and surface effects on the free and forced vibration response of a graded nanobeam resting on a nonlinear elastic foundation. Material gradation is assumed to be through the beam depth according to a power-law model. Instead of adopting the customary choice of the geometrical central axis, this study uses instead the physical neutral axis to eliminate the stretching-bending coupling effect due to the unsymmetrical material variation. It is also shown that the choice of the physical neutral axis leads to the elimination of the quadratic nonlinearity from the equation of motion. The Euler–Bernoulli beam theory along with the von Kármán geometric nonlinearity is formulated while accounting for Eringen’s nonlocal elasticity differential model, determined neutral axis and surface effects. Under certain assumptions, a nonlinear fourth-order partial differential equation is derived and the Garlerkin technique with a single-mode approximation is used to obtain a second-order ordinary equation in the time domain with cubic nonlinearity. The Method of Multiple Scales (MMS) is used initially to derive the expression of the nonlinear natural frequency for the graded nanobeam accounting for both nonlocal and surface effects. Then, MMS is utilized to study the primary resonance of an externally forced nanobeam after obtaining its frequency response curve. The primary objective of this work is to investigate the effects of the nonlocal and surface effects parameters, power-law index and boundary conditions on the free and forced vibration response of the nanobeam.