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Research Papers

# Green Function for the Problem of a Plane Containing a Circular Hole With Surface Effects

[+] Author and Article Information
S. G. Mogilevskaya1

Department of Civil Engineering, University of Minnesota, 500 Pillsbury Drive S.E., Minneapolis, MN 55455mogil003@umn.edu

A. V. Pyatigorets, S. L. Crouch

Department of Civil Engineering, University of Minnesota, 500 Pillsbury Drive S.E., Minneapolis, MN 55455

It is sufficient to multiply $A−n$ and $C−(n+2)$ by $gn(z)$ and $gn+2(z)$, respectively, and sum up the corresponding series by using the power series $(1+z)q$(19).

1

Corresponding author.

J. Appl. Mech 78(2), 021008 (Nov 08, 2010) (9 pages) doi:10.1115/1.4002579 History: Received April 23, 2010; Revised August 17, 2010; Posted September 17, 2010; Published November 08, 2010; Online November 08, 2010

## Abstract

This paper presents the complex Green function for the plane-strain problem of an infinite, isotropic elastic plane containing a circular hole with surface effects and subjected to a force applied at a point outside of the hole. The analysis is based on the Gurtin and Murdoch [1975, “A Continuum Theory of Elastic Material Surfaces,” Arch. Ration. Mech. Anal., 57, pp. 291–323; 1978, “Surface Stress in Solids,” Int. J. Solids Struct., 14, pp. 431–440] model, in which the surface of the hole possesses its own mechanical properties and surface tension. Systematic parametric studies are performed to investigate the effects of both surface elasticity and surface tension on the distribution of hoop stresses on the boundary of the hole and on a line that connects the point of the applied force and the center of the hole.

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## Figures

Figure 1

Problem formulation

Figure 2

Case studies: (a) a force applied in the radial direction and (b) a force applied in the tangential direction

Figure 3

Variation of the normalized hoop stresses with the parameter η(χ=0, β=0.1, γ=0.5) along ((a) and (b)) the boundary of the hole and ((c) and (d)) the segment [R,ρ]

Figure 4

Variation of the normalized hoop stresses with the parameter χ(η=0, β=0.1, γ=0.5) along (a) the boundary of the hole and (b) the segment [R,ρ]

Figure 5

Variation of the normalized hoop stresses on the boundary of the hole with the parameter β(γ=0.5): (a) η=0, χ=0, and Im(β)=0; (b) η=0, χ=0.0245, and Im(β)=0; (c) η=−0.1289, χ=0, and Im(β)=0; (d) η=0, χ=0, and Re(β)=0; (e) η=0, χ=0.0245, and Re(β)=0; (f) η=−0.1289, χ=0, and Re(β)=0

Figure 6

Variation of the normalized hoop stresses on the boundary of the hole with the parameter γ(β=0.1): (a) η=0 and χ=0, (b) η=0 and χ=0.0245, and (c) η=−0.1289 and χ=0

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