FOURTH-ORDER STRAIN GRADIENT BAR-SUBSTRATE MODEL WITH NONLOCAL AND SURFACE EFFECTS FOR THE ANALYSIS OF NANOWIRES EMBEDDED IN SUBSTRATE MEDIA

Abstract

This paper presents a new analytical bar-substrate model for the analysis of an isotropic and homogeneous nanowire embedded in an elastic substrate. A fourth-order strain gradient model based on a thermodynamic approach is employed to represent the small-scale effect (nonlocal effect) while the Gurtin-Murdoch continuum model based on the surface elastic theory is used to account for the size-dependent effect (surface energy effect). The proposed model is derived from the virtual displacement principle, leading to the governing differential equations and its associated natural boundary conditions. The analytical solutions of the sixth-order governing differential equation for the nanowire-substrate element are provided, and were employed in numerical simulations. Two numerical simulations are used to demonstrate the performance and to investigate the characteristics of the fourth-order strain gradient model on nanowire responses, when compared to the classical model and the second-order strain gradient model. The first simulation investigates the influences of nonlocal and surface effects on the responses of a nanowire embedded in an elastic substrate, while the second simulation study assessed the sensitivity of system stiffness on parameters in the nanowire-substrate model.