MODELING OF NANO-BEAMS UNDER GENERAL LOADING CONDITIONS

Abstract

This thesis presents the analysis of bending, buckling, and post-buckling responses of a nano-beam under different end conditions. Both surface and non-local effects via Gurtin-Murdoch surface elasticity and Eringen non-local elasticity models are integrated into the classical Euler-Bernoulli beam theory to obtain a mathematical model capable of simulating the nano-scale influence and size-dependency of observed physical phenomena. The key governing equations for a deflected shape are formulated first within the context of large displacements and rotations using a classical elliptic integral technique and their linearized version is then established to form the eigen-value problem governing the buckling load. A conventional analytical procedure for eigen-hunt is employed to determine the exact buckling load whereas Newton-Raphson iterative scheme is adopted to solve a final system of nonlinear algebraic equations to obtain the deflected shape and other related quantities. Obtained results demonstrate that both the surface and non-local effects significantly influence the buckling load and bending and post-buckling responses of nano-beams. In particular, presence of those effects tends to reduce the overall stiffness of the beam and, as a result, decrease the buckling load for all cases considered. In addition, the predicted solutions exhibit strongly size-dependent and are significantly influenced by both the surface and non-local parameters (e.g., surface modulus of elasticity and residual surface tension) when the characteristic length of the beam is comparable to the intrinsic length of the material surface.