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

Abstract

Abstract. In the present study, various Higher-order Shear Deformation beam Theories (HSDTs) are applied in order to achieve the exact analytical solution to bending, buckling, and free vibration of Functionally Graded (FG) nanobeam lying on the Winkler and Pasternak elastic foundations. HSDTs are those in which the e ect of transverse shear strain is included. The displacement eld of these theories involves a quadratic variation of transverse shear strains and stresses; hence, this hypothesis leads to the diminishing of transverse shear stresses at the top and bottom surfaces of a beam. Thus, necessarily, there is no need to use a shear correction factor in the HSDTs. Nanobeam has been made of FG materials in which the properties of these materials are changed through the thickness direction of nanobeam according to the power-law distribution. Hamilton's principle is used to derive the equation of motions and the related boundary conditions of simply supported nanobeam. The present study shows that the stability and vibration behaviors of FG nanobeam are extremely dependent on the Winkler and Pasternak elastic foundation, gradient index, aspect ratio, and nonlocal parameter. The obtained results of the present study might be useful in the advanced eld of micro/nano electromechanical systems.

Finite Element Model and Size Dependent Stability Analysis of Boron Nitride and Silicon Carbide Nanowires/Nanotubes

“…They demonstrated the e ects of orthotropy ratio, side-tothickness ratio, and types of boundary conditions on the natural frequencies of plates. Later, in 2017, Rafaeinejad et al presented an analytical solution for bending, buckling, and free vibration of FG nanobeams [103]. Nanobeams were modeled resting on a doubleparameter Winkler-Pasternak elastic foundation, and results were obtained using di erent nonlocal higherorder shear deformation beam theories.…”

Studying buckling of composite rods made of hybrid carbon fiber/carbon nanotube reinforced polyimide using multiscale FEM

“…The nonlocal parameter of this theory can be determined based on experimental data or results of atomistic approaches like molecular dynamics simulations. The nonlocal theory has been applied to various problems such as buckling of microtubules and boron-nitride nanotubes embedded in an elastic medium [44,45] and bending/buckling/free vibration analyses of nanobeams [46,47]. The reader is also referred to [48{52] as some examples for the applications of strain gradient and couple stress theories.…”

Forced Vibration: Axially Functionally Graded Thin Beams on Elastic Foundation

“…A typical illustration of a dynamic loading condition is harmonic excitation, [1][2][3][4][5] which refers to forces that change harmonically throughout time. Pure harmonic excitation is less likely to happen in a working environment than periodic or other types of excitation [6][7][8][9][10], but it is important to understand how a system behaves under harmonic excitation to understand how the system will react to more general types of excitation. It is also acknowledged that a system may respond to harmonic stimuli in a periodic, non-harmonic manner.…”