A Levy Type Solution for Free Vibration Analysis of a Nano-Plate Considering the Small Scale Effect

Abstract

Experimental results show that as length scales of a material are reduced, the influences of long-range interatomic and intermolecular cohesive forces on the mechanical properties become prominent and cannot be neglected. It is well known that surfaces and interfaces in nano structures behave differently from their bulk counterparts. For nanostructures with size less than 100nm, the surface to volume ratio is significant and the effective properties are altered by surface and nonlocal effects. Therefore, at nanolength scales, size effects often become prominent, the causes of which need to be explicitly addressed especially with an increasing interest in the general area of nanotechnology (Sharma et al., 2003). Due to the vast computational expenses of nano-structures analyses when using atomic lattice dynamics and molecular dynamic simulations, there is a great interest in applying continuum mechanics for analysis of nano-structures. Classical continuum elasticity, which is a scale free theory, cannot predict the size effects. Nonlocal continuum mechanics allows one to account for the small length scale effect that becomes significant when dealing with microstructures or nanostructures. It has been showed that it is possible to represent the integral constitutive relations of nano-structures in an equivalent differential form (Eringen, 1983). Eringen presented a nonlocal elasticity theory to account for the small scale effect by specifying the stress at a reference point is a functional of the strain field at every point in the body. Since then, many studies have been carried out nonlocal theory of elasticity for bending, buckling and vibration analyses of nano-structures. Small scale effect on static deformation of microand nano-rods or tubes is revealed through nonlocal Euler–Bernoulli and Timoshenko beam theories by Wang and Liew (2007). Li and Wang (2009) investigated a theoretical treatment of Timoshenko beams, in which the influences of shear deformation, rotary inertia, and scale coefficient are taken into account. Murmu and Pradhan (2009a) studied vibration response of nano cantilever considering nonuniformity in the cross sections using nonlocal elasticity theory. Although graphite sheet has many superior properties, such as low electrical and thermal conductivities normal to the sheet but high electrical and thermal conductivities in the plane of the sheet, relatively little research have been reported in the literature for mechanical analyses of graphene sheets.