Image Location for Screw Dislocation—A New Point of View
 JengTzong Chen^{1, 2}Email author,
 YingTe Lee^{1},
 KeHsun Chou^{1} and
 JiaWei Lee^{1}
DOI: 10.1155/2010/731741
© JengTzong Chen et al. 2010
Received: 16 June 2009
Accepted: 18 January 2010
Published: 25 March 2010
Abstract
An infinite plane problem with a circular boundary under the screw dislocation is solved by using a new method. The anglebased fundamental solution for screw dislocation is expanded into degenerate kernel. Our method can explain why the image screw dislocation is required. Besides, the location of the image point can be obtained easily by using degenerate kernel after satisfying boundary conditions. Even though the image concept is required, the location of image point can be determined straightforwardly through the degenerate kernel instead of the method of reciprocal radii. Finally, two examples are demonstrated to verify the validity of the present method.
1. Introduction
The dislocation theory is essential for understanding many physical and mechanical properties of crystalline solids. Many researchers investigated the dislocation problems in the past years. Smith [1] successfully solved the problem of the interaction between a screw dislocation and a circular or elliptic inclusion contained within an infinite body subject to a uniform applied shear stress at infinity by using the complexvariable function and circle theorem. Dundurs [2] solved the screw dislocation with circular inclusion problem by using the image technique. Later, Sendeckyj [3] employed the complexvariable function in conjunction with the inverse point method to solve the problem of the screw dislocation near an arbitrary number of circular inclusions. Almost all above problems were solved by using the complexvariable technique. Its extension to threedimensional cases may be limited. A more general approach is nontrivial for further investigation.
In this paper, we will introduce the degenerate (or socalled separable) kernel for the anglebased fundamental solution ( ) for the screw dislocation instead of radialbasis one ( ) for the source singularity. By employing the degenerate kernel, the closedform Green's function is expanded into the degenerate form. Also, the location of image point is found in a straightforward way. The twodimensional Laplace exterior problems are solved. Finally, two examples were given to demonstrate the validity of the present method.
2. Degenerate (Separable) Kernel for the AngleBased Fundamental Solution
The real part ( ) is the fundamental solution of the source singularity while the imaginary part ( ) denotes the fundamental solution of the screw dislocation. For the exterior case , (2.1) can be expanded as follows:
Thus, the degenerate form for the fundamental solution of the screw dislocation, , can be expressed as
Similarly, we have
for the interior case. In Figure 2, the range of is defined between 0 and . To match the physical meaning and mathematical requirement, we modify the range of interest between and . Thus, the fundamental solution of the screw dislocation is expressed by
3. D Exterior Problem
where is the domain of interest and b is the Burger's vector which is equal to in this paper. The boundary condition on the circular boundary is the Dirichlet type
where a is the radius of the circular boundary and B is the circular boundary. By employing the image method, the image point is located outside the domain and the solution can be represented as follows:
where is the location of image point, c is a free constant, and
In order to match the boundary condition and the Burger's vector, first the sum of series is independent of . Therefore, we choose the collinear points and , that is, and we have
Finally, we can obtain the location of image point
Second, we found that c is equal to and the solution automatically matches the boundary condition and Burger's vector. The displacement field of the closedform Green's function can be obtained as below
Similarly, the Green's function in the other region ( ) is shown in Figure 6(b) and is expanded into
For comparison, the closedform solution of Smith's solution is expressed in terms of functions of complex variables
According to the successful experience of the Dirichlet boundary condition for the exterior problem, we extend our approach to the Neumann boundary condition, as shown in Figure 5(b),
In a similar way, we have the closedform Green's function for the Neumann boundary condition as
and the series form is expressed into two parts. For the domain ( ) as shown in Figure 6(a), the Green's function is expanded into
For the other domain as shown in Figure 6(b), we have
For comparison, the closedform solution of Smith's solution is expressed in terms of functions of complex variable
Comparison for the source or sink and the screw dislocation (Dirichlet B. C.).
Source or sink (Chen and Wu [7])  Screw dislocation (present paper)  




Exterior problem (Dirichlet B. C.) 


The closed form 



 
The series form 


Smith's solution 


Comparison for the source or sink and the screw dislocation (Neumann B.C.).
Source or sink (Chen and Wu [7])  Screw dislocation (present paper)  




Exterior problem (Neumann B. C.) 


The closed form 



 
The series form 


Smith's solution 


4. Conclusions
For the screw dislocation problem with circular boundaries, we have proposed a natural approach to construct the screw dislocation solution by using the degenerate kernel. The anglebased fundamental solution for screw dislocation was derived in terms of degenerate kernel in this paper. Based on this expression, the image location can be determined instead of using reciprocal radius. Two examples, including an infinite plane with a circular hole subject to the Dirichlet and Neumann boundary conditions, were used to demonstrate the validity of the present formulation.
Declarations
Acknowledgments
The financial support from the National Science Council under Grant no. NSC982221E019017MY3 for National Taiwan Ocean University is gratefully appreciated. Thanks to Mr. S. R. Yu for preparing the figures.
Authors’ Affiliations
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