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Research Papers

# Surface Green Function With Surface Stresses and Surface Elasticity Using Stroh’s Formalism

[+] Author and Article Information
Hideo Koguchi

Department of Mechanical Engineering, Nagaoka University of Technology, 1603-1 Kamitomioka, Nagaoka, Niigata 940-2188, Japankoguchi@mech.nagaokaut.ac.jp

J. Appl. Mech 75(6), 061014 (Aug 21, 2008) (12 pages) doi:10.1115/1.2967893 History: Received May 03, 2007; Revised June 02, 2008; Published August 21, 2008

## Abstract

In the present paper, surface Green functions in an anisotropic elastic half-domain subjected to a concentrated force and a line force are derived using Stroh’s formalism considering surface stress and surface elasticity. Formulation of the boundary condition based on Stroh’s formalism is presented and is used to derive the surface Green functions. The displacement field far from the surface is affected only slightly by the surface stress and elasticity. However, the stress field is influenced to a somewhat greater degree by the surface stress and elasticity. The influence of the mechanical properties of the surface on the distributions of displacement and stress near the surface is investigated for various values of surface elastic modulus and surface stress. The surface stress and surface elasticity affect the displacements and stresses, respectively, in different manners. Displacement fields in molecular dynamics are compared with those in the Green function, and it is shown that the results are in fair agreement.

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Copyright © 2008 by American Society of Mechanical Engineers
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## Figures

Figure 11

Comparison of the displacement in MD and the surface Green function considering surface stresses and surface elasticity. (The solid line is the displacement at z=−0.05nm.) (a) Displacement ux; (b) displacement uz.

Figure 5

Distributions of displacement and stress at y=0 for fx=1.0nN. (a) displacement ux (surface stress and elasticity: solid line, Max: 8.27×10−3nm, no surface stress or elasticity: dashed line, Max: 1.16×10−2nm). (b) Displacement uz (surface stress and elasticity: solid line, Min: −1.34×10−3nm, Max: 1.97×10−3nm, no surface stress or elasticity: dashed line, Min: −3.72×10−3nm, Max: 4.98×10−3nm). (c) Stress σxz (surface stress and elasticity: solid line, Min: −0.0053Pa, Max: 9.30Pa, no surface stress or elasticity: dashed line, Min: −1.60Pa, Max: 9.60Pa). (d) Stress σzz (surface stress and elasticity: solid line, Min: −4.0Pa, Max: 6.38Pa, no surface stress or elasticity: dashed line, Min: −31.0Pa, Max: 40.0Pa).

Figure 6

Distribution of displacements and stresses for various m values of surface stress along the x-axis at y=0 and z=−0.05nm for fx=1.0nN. (a) Displacement ux; (b); displacement uz; (c); stress σxz; (d) Stress σzz.

Figure 7

Distribution of displacements and stresses for various m values of surface elasticity along the x-axis at y=0 and z=−0.05nm for fx=1.0nN. (a) Displacement ux; (b) displacement uz; (c) stress σxz; (d) stress σzz.

Figure 8

Distribution of displacement, ux, for fz=1nN against the distance from the surface. (a) Displacement, uz, for various m values of surface stress; (b) Displacement, uz, for various m values of surface elasticity.

Figure 9

Model for MD analysis with an atom on the surface. The dimensions of this model are a=14.74nm, b=14.994nm, c=10.4995nm, and d=0.27nm. The total number of atoms is 186,481.

Figure 10

Comparison of displacements for fz=1.21nN in the z=−0.05nm plane

Figure 1

Layer decomposition of the two components of the surface stress tensor

Figure 2

Distributions of displacement and stress at y=0 for fz=1.0nN. (a) Displacement ux (surface stress and elasticity: solid line, Min: −1.37×10−3nm, Max: 2.54×10−3nm, no surface stress or elasticity: dashed line, Min: −1.50×10−3nm, Max: 2.70×10−3nm). (b) Displacement uz (surface stress and elasticity: solid line, Max: 2.22×10−2nm, no surface stress or elasticity: dashed line, Max: 2.46×10−2nm). (c) Stress σxz (surface stress and elasticity: solid line, Min: −13.9Pa, Max: 15.5Pa, no surface stress or elasticity: dashed line, Min: −12.4Pa, Max: 14.1Pa). (d) Stress σzz (surface stress and elasticity: solid line, Min: −0.03Pa, Max: 53Pa, no surface stress or elasticity: dashed line, Min: −0.04Pa, Max: 64.4Pa).

Figure 3

Distribution of displacements and stresses for various m values of surface stress along the x-axis at y=0 and z=−0.05nm for fz=1.0nN. (a) displacement ux; (b) displacement uz; (c) stress σxz; (d) stress σzz.

Figure 4

Distribution of displacements and stresses for various m values of surface elasticity along the x-axis at y=0 and z=−0.05nm for fz=1.0nN. (a) Displacement ux; (b) displacement uz; (c) stress σxz; (d) stress σzz.

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