Nonlinear stability analysis of embedded restrained nanobeams using the stokes' transformation

Abstract

The geometrically nonlinear stability analysis of restrained nanobeams under an elastic medium is considered in the presented manuscript. The investigation is based on the nonlocal strain gradient elasticity theory, Euler-Bernoulli beam theory and geometrical nonlinearity which is a significant change in geometry. To the best of the authors' knowledge, the nonlinear stability response of a nanobeam with elastic boundary conditions and on an elastic foundation has not been presented before via the theory of nonlocal strain gradient. The aim of this paper is to fill this gap in the literature by offering a method that can give a solution under general elastic boundary conditions. Using a Fourier sine series, a Fourier coefficient suitable for constructing an eigenvalue problem is computed. The constructed problem has been treated in the transformed region with the help of the Stokes' transform, then the stability analysis is executed for arbitrary boundary conditions (rigid or restrained) based on the nonlocal strain gradient elasticity. The influences of various nonlinear parameters, deformable boundary conditions, small-scale parameters and elastic foundation parameters on the stability of the constrained nanobeam are studied. It is observed that the investigated variables produce significant changes in buckling loads.