# Image Location for Screw Dislocation—A New Point of View

- Jeng-Tzong Chen
^{1, 2}Email author, - Ying-Te Lee
^{1}, - Ke-Hsun Chou
^{1}and - Jia-Wei Lee
^{1}

**2010**:731741

https://doi.org/10.1155/2010/731741

© Jeng-Tzong 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 angle-based 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 complex-variable function and circle theorem. Dundurs [2] solved the screw dislocation with circular inclusion problem by using the image technique. Later, Sendeckyj [3] employed the complex-variable 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 complex-variable technique. Its extension to three-dimensional cases may be limited. A more general approach is nontrivial for further investigation.

In this paper, we will introduce the degenerate (or so-called separable) kernel for the angle-based fundamental solution ( ) for the screw dislocation instead of radial-basis one ( ) for the source singularity. By employing the degenerate kernel, the closed-form Green's function is expanded into the degenerate form. Also, the location of image point is found in a straightforward way. The two-dimensional Laplace exterior problems are solved. Finally, two examples were given to demonstrate the validity of the present method.

## 2. Degenerate (Separable) Kernel for the Angle-Based Fundamental Solution

*x*and the source point

*s*as shown in Figure 2. The strength of the fundamental solution is instead of unity. Here, we give the screw dislocation to replace the concentrated source and then the angle-based fundamental solution is used to substitute the radial-based one as shown in Figure 2. In order to fully capture circular geometry, we utilize the polar coordinates to replace the Cartesian coordinates. Therefore, the location of the screw dislocation

*s*and the position of field point

*x*are expressed as and , respectively, in the polar coordinate system. The position vector of screw dislocation point

*s*can be written as complex form, . Similarly, the field point

*x*can be expressed by in the complex plane as shown in Figure 2. By decomposing the into real and imaginary parts, we have

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 closed-form 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 closed-form 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 closed-form 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 closed-form 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 angle-based 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. NSC-98-2221-E-019-017-MY3 for National Taiwan Ocean University is gratefully appreciated. Thanks to Mr. S. R. Yu for preparing the figures.

## Authors’ Affiliations

## References

- Smith E:
**The interaction between dislocations and inhomogeneities-I.***International Journal of Engineering Science*1968,**6**(3):129-143. 10.1016/0020-7225(68)90012-8View ArticleGoogle Scholar - Dundurs J:
**Elastic interaction of dislocations with inhomogeneities.**In*Mathematical Theory of Dislocations*. Edited by: Mura T. American Society of Mechanical Engineers, New York, NY, USA; 1969:70-115.Google Scholar - Sendeckyj GP:
**Screw dislocations near circular inclusions.***Physica Status Solidi*1970,**30:**529-535.View ArticleGoogle Scholar - Sommerfeld A:
*Partial Differential Equations in Physics*. Academic Press, New York, NY, USA; 1949:xi+335.Google Scholar - Greenbreg MD:
*Application of Green's Functions in Science and Engineering*. Prentice-Hill, Upper Saddle River, NJ, USA; 1971.Google Scholar - Thomson W:
*Maxwell in His Treatise*.*Volume 1*. Oxford University Press, Oxford, UK; 1848.Google Scholar - Chen JT, Wu CS:
**Alternative derivations for the poisson integral formula.***International Journal of Mathematical Education in Science and Technology*2006,**37**(2):165-185. 10.1080/00207390500226028MathSciNetView ArticleGoogle Scholar - Chen JT, Shieh HC, Lee YT, Lee JW:
**Image solutions for boundary value problems without sources.***Applied Mathematics Computation*2010,**216:**1453-1468. 10.1016/j.amc.2010.02.048MathSciNetView ArticleMATHGoogle Scholar - Chen JT, Lee YT, Yu SR, Shieh SC:
**Equivalence between the Trefftz method and the method of fundamental solutions using the addition theorem and image concept.***Engineering Analysis with Boundary Elements*2009,**33:**678-688. 10.1016/j.enganabound.2008.10.003MathSciNetView ArticleMATHGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.