Nonlinear vibrations of gradient and nonlocal elastic nano-bars

Practical vibration-based nano-devices are normally subject to disturbances where intense vibrations reveal significant nonlinear characteristics. Efficient implementation of nonlinear nano-systems requires comprehensive knowledge of the nonlinear dynamics. Nonlinear vibration characterization of elastic nano-bars with large vibration amplitudes is rigorously examined in the present study. The fascinating concept of simulating long-range interactions can be realized in the framework of the nonlocal elasticity theory, and thus, nano-scale effects are taken into account in the framework of the stress-driven nonlocal integral elasticity. The equivalent nonlocal differential condition equipped with non-classical boundary conditions of constitutive type is consistently detected. For proper comparison sake, the strain gradient elasticity theory is selected due to its similarities in revealing the stiffening structural response at nano-scale. In simple structural schemes of technical interest, the space-time decoupled formulation is constructed applying the weighted residual Galerkin method which results in a strongly nonlinear ordinary differential equation with cubic and quadratic nonlinearities. Analytical approach for the nonlinear analysis of the system dynamics provides an effective tool for optimized design of vibration-based nano-devices. The homotopy analysis method is accordingly employed to analytically study the nonlinear vibration response of nano-bars and its efficiency and accuracy is verified in comparison with the multiple scales method. The conceived approach for the nonlinear vibration analysis of elastic nano-bars, therefore, provides a consistent methodology to tackle nonlinear dynamic phenomena in nano-mechanics.