Energy Approach for Nanoscale Beams with Surface Effects

Abstract

Nanoscale structures are the key components of NEMS-based sensor and actuator technology. They have a high surface area to volume ratio. Consequently, surface energy has a significant effect on the response of nanoscale structures, and is attributed to their size-dependent behaviour. In this paper, uniaxial buckling and transverse vibration of nanoscale beams are studied using Rayleigh’s energy method (Washizu, Variational principles in continuum mechanics. Department of Aeronautical Engineering Report, University of Washington, Seattle, WA, 1962). Surface energy is incorporated using Gurtin–Murdoch surface elasticity theory (Gurtin ME, Murdoch, Arch Rat Mech Anal 57(4):291–323; Gurtin ME, Murdoch AI, Arch Rat Mech Anal 59:389–390, 1975). Closed–form analytical solutions for critical compressive force and natural frequencies of free vibration of nanoscale beams under various boundary conditions (simply-supported, fixed-free and fixed-fixed ends) are derived using Rayleigh’s energy method based on Rayleigh quotient. The solutions from the energy method are shown to be in good agreement with those obtained using exact theory developed by the authors, previously (Liu C, Rajapakse RKND, IEEE Trans Nanotechnol 9(4):422–431, 2010; Liu C, Rajapakse RKND, Srikantha Phani A, J Appl Mech 78(3), 031014, 2011. Selected numerical results are presented for aluminum and silicon beams with [100] surfaces to demonstrate their salient response features. It is shown that both surface elasticity and surface residual stress influence the critical loads and natural frequencies. Their effects will become more pronounced with decreasing thickness of the beam. The influence of surface energy is shown to depend upon the boundary conditions. The analytical solution for natural frequency is further employed to fit the experimentally measured natural frequencies of GaAs fixed-free beams reported by Lagowski and his coworkers (Lagowski J, Gatos HC, Sproles ES Jr, Appl Phys Lett 26:493, 1975). The present formulation offers an explanation for the dependence of the experimentally observed first natural frequency of a GaAs fixed-free beam specimen. A technique for the determination of surface elastic constants and surface residual stress by measuring the natural frequency of free vibration is then proposed. The energy method incorporating surface effects provides an efficient approximation in cases where an equilibrium solution would not be practical due to variable geometry and material properties, and it can be applied to investigate the buckling and vibration of nanoscale beams encountered in the NEMS device design.