Nonlinear vibration of nonlocal strain gradient functionally graded beam on nonlinear compliant substrate

Beams supported on compliant substrate are used in nanostructures. Due to atomistic dimensions, mechanics of nanostructures are governed by molecular interactions, modeled by the Nonlocal (NL) and Strain Gradient (SG) continuum model. Due to their slender nature (owing to significant strength/stiffness of nano-materials), nanostructures are liable to geometric nonlinearity. Additionally, the supporting substrate may be nonlinear. However, nonlinear forced vibration of nanostructures is not researched well. This study investigates the geometrically nonlinear vibration of NL-SG beams on nonlinear substrate with shear interactions. The higher-order curvature, von Karman nonlinearity are included along with nonlinear Pasternak model for the substrate. The depth-wise functional gradation of material properties is also included. The governing equations of motion are derived following the variational method. A two-step perturbation is employed to obtain a closed-form solution. The free vibration nonlinear frequency, response time history, and forced vibration transmissibility characteristics are presented. The influence of important parameters is illustrated. Specifically, the dominant role of nonlinear bending and substrate stiffness is noted. The effect of NL and SG interactions is shown to significantly influence the vibration behavior. However, the effect of functional gradation of material is shown to be minor.