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TECHNICAL PAPERS

# Analytical Solution for Size-Dependent Elastic Field of a Nanoscale Circular Inhomogeneity

[+] Author and Article Information
L. Tian

Department of Mechanical Engineering, The University of British Columbia, Vancouver, BC, Canada V6T 1Z4

R. K. N. D. Rajapakse1

Department of Mechanical Engineering, The University of British Columbia, Vancouver, BC, Canada V6T 1Z4rajapakse@mech.ubc.ca

1

Corresponding author.

J. Appl. Mech 74(3), 568-574 (May 30, 2006) (7 pages) doi:10.1115/1.2424242 History: Received November 15, 2005; Revised May 30, 2006

## Abstract

Two-dimensional elastic field of a nanoscale circular hole/inhomogeneity in an infinite matrix under arbitrary remote loading and a uniform eigenstrain in the inhomogeneity is investigated. The Gurtin–Murdoch surface/interface elasticity model is applied to take into account the surface/interface stress effects. A closed-form analytical solution is obtained by using the complex potential function method of Muskhelishvili. Selected numerical results are presented to investigate the size dependency of the elastic field and the effects of surface elastic moduli and residual surface stress. Stress state is found to depend on the radius of the inhomogeneity/hole, surface elastic constants, surface residual stress, and magnitude of far-field loading.

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## Figures

Figure 1

Nanoscale inhomogeneity in an infinite matrix

Figure 2

Variation of stress concentration factor with loading ratio, a, for a hole without residual surface stress (R0=5nm)

Figure 3

Variation of stress concentration factor with radius of hole and KS for a hole without residual surface stress (a=0)

Figure 4

Circumferential variation of normalized hoop stress along hole surface (R0=5nm, a=0)

Figure 5

Variation of normalized hoop and radial stresses along the x1 direction for different KS (R0=5nm, a=0)

Figure 6

Variation of normalized hoop and radial stress components along the x1 direction for a hole with residual surface stress (τ0≠0, a=0)

Figure 7

Variation of normalized interfacial hoop stress at θ=0 with loading ratio a for an inhomogeneity with R0=5nm (solid line for matrix and dash line for inhomogeneity)

Figure 8

Variation of normalized interfacial hoop stress at θ=0 with inhomogeneity radius and KS (a=0; solid line for matrix and dash line for inhomogeneity)

Figure 9

Variation of normalized interfacial hoop stress due to an eigenstrain ε* with inhomogeneity radius (solid line and left Y axis for matrix, and dash line and right Y axis for inhomogeneity)

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