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Research Papers

# Elastic Displacement and Stress Fields Induced by a Dislocation of Polygonal Shape in an Anisotropic Elastic Half-Space

[+] Author and Article Information
H. J. Chu

Research Group of Mechanics, Yanzhou University, Yangzhou 225009, China;  Department of Civil Engineering, University of Akron, Akron, OH 44325

E. Pan1

Department of Civil Engineering, University of Akron, Akron, OH 44325pan2@uakron.edu

J. Wang

Materials Science and Technology Division, Los Alamos National Laboratory, Los Alamos, NM 87545

I. J. Beyerlein

Theoretical Division, Fluid Dynamics and Solid Mechanics Division, Los Alamos National Laboratory, Los Alamos, NM 87545

1

Corresponding author.

J. Appl. Mech 79(2), 021011 (Feb 24, 2012) (9 pages) doi:10.1115/1.4005554 History: Received December 02, 2010; Revised April 25, 2011; Posted January 30, 2012; Published February 13, 2012; Online February 24, 2012

## Abstract

The elastic displacement and stress fields due to a polygonal dislocation within an anisotropic homogeneous half-space are studied in this paper. Simple line integrals from 0 to π for the elastic fields are derived by applying the point-force Green’s functions in the corresponding half-space. Notably, the geometry of the polygonal dislocation is included entirely in the integrand easing integration for any arbitrarily shaped dislocation. We apply the proposed method to a hexagonal shaped dislocation loop with Burgers vector along [$1¯$ 1 0] lying on the crystallographic (1 1 1) slip plane within a half-space of a copper crystal. It is demonstrated numerically that the displacement jump condition on the dislocation loop surface and the traction-free condition on the surface of the half-space are both satisfied. On the free surface of the half-space, it is shown that the distributions of the hydrostatic stress (σ11  + σ22 )/2 and pseudohydrostatic displacement (u1  + u2 )/2 are both anti-symmetric, while the biaxial stress (σ11  − σ22 )/2 and pseudobiaxial displacement (u1  − u2 )/2 are both symmetric.

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## Figures

Figure 1

Geometry of an arbitrarily shaped dislocation loop S in an anisotropic, elastic, and half-space E3 /2. The vector b denotes the Burgers vector of the dislocation, which is equal to the displacement jump across the dislocation surface. The vector n+ denotes the normal of the surface S+ towards S− , and n−  = −n+ .

Figure 2

Geometry of a triangular dislocation with corners P1 , P2 , P3 with respect to the global coordinate system (O; x1 ,x2 ,x3 ) and local coordinate system (x0 ; ξ1 , ξ2 , ξ3 ) where h = ξ2 (P1 ), l1  = −ξ1 (P2 ), l2  = ξ1 (P3 ). Lengths l1 and l2 can also be negative.

Figure 3

Geometry of a regular hexagon dislocation where (O; x1 ,x2 ,x3 ) is the global coordinate system and (P0 ; η1 , η2 , η3 ) the local dislocation coordinate system

Figure 4

Contour map of the normalized local displacement component u1 /b in local coordinates η1 /a and η2 /a induced by a regular hexagonal dislocation, where b is the magnitude of the Burgers vector

Figure 5

Contour map of the normalized local displacement component u2 /b in local coordinates η1 /a and η2 /a induced by a regular hexagonal dislocation, where b is the magnitude of the Burgers vector

Figure 6

Contour map of the normalized local displacement component u3 /b in local coordinates η1 /a and η2 /a induced by a regular hexagonal dislocation, where b is the magnitude of the Burgers vector

Figure 7

Contour map of (σ11 +σ22 )/2 normalized according to Eq. 43 in global coordinates, where the coordinates are normalized by the side length a of the hexagonal dislocation with Burgers vector of magnitude b

Figure 8

Contour map of (σ11 –σ22 )/2 normalized according to Eq. 43 in global coordinates, where the coordinates are normalized by the side length a of the hexagonal dislocation with Burgers vector of magnitude b

Figure 9

Contour map of the normalized stress component σ12 (normalized according to Eq. 43) in global coordinates, normalized by the side length a of the hexagonal dislocation with Burgers vector of magnitude b

Figure 10

Contour map of the normalized displacement (u1  + u2 )/2b in the normalized global coordinates x1 /a and x2 /a induced by a regular hexagonal dislocation with Burgers vector of magnitude b

Figure 11

Contour map of the normalized displacement (u1 −u2 )/2b in the normalized global coordinates x1 /a and x2 /a induced by a regular hexagonal dislocation with Burgers vector of magnitude b

Figure 12

Contour map of the normalized displacement u3 /b in the normalized global coordinates x1 /a and x2 /a induced by a regular hexagonal dislocation with Burgers vector of magnitude b

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