An analytical solution for bending, buckling, and free vibration of FG nanobeam lying on Winkler-Pasternak elastic foundation using different nonlocal higher order shear deformation beam theories

In the present study, various higher order shear deformation beam theories (HSDTs) applied in order to achieve an the exact analytical solution of bending, buckling, and free vibration of functionally graded (FG) nanobeam lying on the Winkler and Pasternak elastic foundations. HSDTs are thosein which the effect of transverse shear strains is included.The displacement field of these theories involves a quadratic variation of transverseshear strains and stresses, hence this hypothesis leads to the diminishing of transverse shearstresses at the top and bottom surfaces of a beam. Thus, necessarily there is noneed to use shear correction factors in the HSDTs. Nanobeam has been made of FG materials in which the properties of this material are changed through in the thickness of nanobeam according to the power law distribution. Hamilton`s principle is used to derive equation of motions and the related boundary conditions of simply supported nanobeam. The present study shows that the stability and vibration behavior of FG nanobeam are extremely dependent on the Winkler and Pasternak elastic foundation, gradient index, aspect ratio, and nonlocal parameter. For controlling the stability and vibration of nano-structure, it can be considered that the Winkler-Pasternak foundation as a smart medium gathered with nanobeam. The obtained results of the present study might be useful in the advanced field of micro/nano electromechanical systems.