Modified couple stress flexure mechanics of nanobeams

Abstract

In view of numerous contributions accessible in the literature associated with the modified couple stress theory, its deficiency in precise description of the material response of continua with nano-structural features is investigated. The flexure mechanics of nano-scale beams in the framework of the modified couple stress theory is reassessed applying a consistent variational scheme. The physically motivated constitutive laws of the stress resultant fields, flexural moment and shear force, associated with the modified couple stress beam are introduced. The well-established differential conditions of equilibrium equipped with the proper mathematical form of the higher-order boundary conditions are reinstated. The exact analytical solutions of the kinematic fields of the nano-sized beam are derived by direct solving of the governing coupled differential equations. The flexure response of the modified couple stress beam is numerically illustrated wherein the effects of the symmetric rotation gradients along with the transverse shear deformation are examined. Closed-form analytical formulae of the elastic modulus of Carbon Nanotubes are detected and thoroughly discussed. Serious suspicions on the applicability of the modified couple stress beam model in accurately capturing the size-effects in nano-materials are emerged as evinced by rigorous examination of the elastic modulus of CNTs.