A nonlocal Layerwise theory for free vibration analysis of nanobeams with various boundary conditions on Winkler-Pasternak foundation

In this study, a nonlocal Layerwise theory is presented for free vibration analysis of nanobeams resting on an elastic foundation. Eringen's nonlocal elasticity theory is used to consider the small-scale effect on behavior of nanobeam. The governing equations are obtained by employing Hamilton's principle and Layerwise theory of beams and Eringen's nonlocal constitutive equation. The presented theory takes into account the in-plane and transverse normal and shear strain in the modeling of the nanobeam and can predict more accurate results. The governing equations of the beam are solved by Navier's method for Simple-Simple boundary conditions and semi-analytical methods to obtain the natural frequency for various boundary conditions including Clamped-Simple (C-S), Clamped-Clamped (C-C) and Free-Free (F-F) boundary conditions. Predictions of the present theory are compared with benchmark results in the literature. Effects of nonlocal parameter, Pasternak shear coefficient, Winkler spring coefficient, boundary conditions, and the aspect ratio on the free vibration of nanobeams are studied. The flexural mode and thickness mode natural frequencies of the nanobeam are predicted. It is shown that the predictions of present method are more accurate than the equivalent single layer theories. The theoretical developments and formulation presented herein should also be served to analyze the mechanical behavior of various nanostructures with various loading and boundary conditions.