Dynamic stability and vibration of two-phase local/nonlocal VFGP nanobeams incorporating surface effects and different boundary conditions

Abstract

In the present study, it is shown that the fully nonlocal elasticity is failed to analyze the dynamic stability of Timoshenko nanobeams subjected to an axial load and, to solve this problem, for the first time, the two-phase local/nonlocal elasticity, as a paradox free form of nonlocal elasticity, is utilized to study the size-dependent dynamic stability and damping vibration of Viscoelastic Functionally Graded Porous (VFGP) Timoshenko nanobeams incorporating surface effects. Firstly, the governing equations, in presence of the axial and transverse displacements, are obtained through the Hamilton's principle. Next, to investigate the vibration and dynamic stability of nanobeams with different boundary conditions, the Generalized Differential Quadrature Method (GDQM) as well as Bolotin's method are utilized. After validation of the present results and formulation, to examine the influences of different parameters such as local phase fraction factor, nonlocal parameter, damping factor, FG index, volume fraction of porosities and surface effects in different boundary conditions, various benchmark results are presented. Furthermore, it is indicated that, against the fully nonlocal theory, using two-phase elasticity makes it possible to study the size dependent vibration and stability for several boundary conditions of Timoshenko nanobeams which are subjected to axial load.