Surface effect investigation for static bending of nanowires resting on elastic substrate using Timoshenko beam theory in tandem with the Laplace-Young equation

Amin Khajeansari,Gholam Hosein Baradaran

doi:10.1051/meca/2012010

Amin Khajeansari, Gholam Hosein Baradaran

Open Access

https://doi.org/10.1051/meca/2012010

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Journal: Mechanics & Industry	Publication Date: Jan 1, 2012
Citations: 2	License type: cc-by

Affiliation: Shahid Bahonar University of Kerman

Abstract

In the present study, an enriched continuum mechanics framework is employed to study the surface effects on bending behavior of silver nanowires (NWs) resting on elastic substrate. The Timoshenko beam theory and the Laplace-Young equation are employed to investigate static behavior of silver NWs lying on Winkler-Pasternak elastic substrate. Three types of boundary conditions are considered as doubly simply supported (S-S), doubly clamped (C-C) and cantilevered (C-F). Analytical solutions are obtained for NWs with surface crystallographic orientation of [001] subjected to a concentrated external force. By defining different normalized contact stiffness, extensive numerical results are carried out to study the influence of effective parameters such as substrate, surface, aspect ratio (L/D) and diameter on the stiffness of NWs. According to the obtained results, the effect of surface and its rate of variation on stiffness of NWs lying on Winkler and Winkler-Pasternak elastic foundation models are more significant in (C-F) type of boundary condition compared to the NWs without foundation. By increasing the modulus of elastic substrate, the effect of shear deformation increases which it is more considerable in (C-C) and (S-S) NWs resting on the Winkler-Pasternak and Winkler substrate models, respectively.

Highlights

In recent years, many methods and approaches have been implemented for studying the physical properties of nanostructures such as nanowires, nanotubes, nanoplates, nanocomposites, etc
The shear deformation effect is considered by assuming Timoshenko beam theory and an explicit solution is obtained for transverse displacement and stiffness of NWs with three different loads and boundary conditions: (a) cantilever (C-F) with a concentrated force at the free end, (b) supported (S-S) and (c) clamped (C-C) under concentrated forces at the middle point
The contact stiffness which is basically determined under static bending of NW, is defined as where Fi is a force prescribed at point i inducing a deflection vi

Summary

Introduction

Many methods and approaches have been implemented for studying the physical properties of nanostructures such as nanowires, nanotubes, nanoplates, nanocomposites, etc. A valuable enriched continuum mechanics model proposed by Gurtin and Murdoch [23] They introduced a non-classical phenomenon known as the surface/interface effect which is taken into account for prediction of the size dependent elastic properties of nanomaterials. Sharma et al [28, 29] have studied the sizedependent elastic state of non-homogenous nanomaterials containing inside inclusions The numerical approaches such as FEM and XFEM from the continuous models of nanowires and nonocomposites were proposed by Yvonnet et al [30, 31] to simulate the surface/interface effects. The shear deformation effect is considered by assuming Timoshenko beam theory and an explicit solution is obtained for transverse displacement and stiffness of NWs with three different loads and boundary conditions: (a) cantilever (C-F) with a concentrated force at the free end, (b) supported (S-S) and (c) clamped (C-C) under concentrated forces at the middle point

Formulation of the problem

Results and discussion

Determining the effective Young’s modulus of NWs and size-dependent behavior

Conclusion