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 the potential theory, it is well known that the image method can solve potential problems when the fundamental solution is known. The image point was found in a semiinverse method a priori through the reciprocal radii in the Sommerfeld's book [4] as shown in Figure 1. Sommerfeld and Greenberg [5] both utilized the concept of reciprocal radii of Thomson [6] to derive the Poisson integral formula. It is important to find where the location of image point is. However, we do not find a natural and logical way about how to determine the location of image point in the literature until an alternative way proposed by Chen and Wu [7]. Chen and Wu derived the location of image point for a source singularity in a straightforward way through the use of degenerate kernel and proved an alterative way to derive the Poisson integral formula. The fundamental solutions of source singularity are expanded into degenerate kernels for constructing the Green's function. Since the degenerate kernel separates the source and field points for the closedform fundamental solution, it plays an important role in studying the image location [8, 9]. To determine the location of image point for screw dislocation in a straightforward way is the main concern of the present paper.
Figure 1. Method of reciprocal radii.
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
It is well known that the fundamental solution of twodimension Laplace problem with a concentrated source is , where is the distance between the field point 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 anglebased fundamental solution is used to substitute the radialbased 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
Figure 2. Sketch of the concentrated source and the screw dislocation.
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
where the superscripts and denote the interior and exterior cases, respectively. It is noted that the denominator in (2.5) involves the larger argument to ensure the series convergence. The displacement contour of the screw dislocation in the four quadrants by using (2.5) is shown in Figures 3(a)–3(d). When the screw dislocation locates at the four quadrants, there are certain areas falling outside the range between and . We subtract , where the value is greater than to ensure the value in the range. Similarly, we add , where the value is smaller than . When the response falls in the defined range, Figure 4 shows the displacement contour for the screw dislocation. To the authors' best knowledge, the degenerate kernel for the anglebased fundamental solution was not found in the literature.
Figure 3. Screw dislocation in (a) the first, (b) the second, (c) the third, and (d) the forth quadrant without modification.Screw dislocation in the first quadrant without modification Screw dislocation in the second quadrant without modification Screw dislocation in the third quadrant without modification Screw dislocation in the forth quadrant without modification
Figure 4. Screw dislocation in the first quadrant after modification (, ).
3. D Exterior Problem
For the problem of an infinite plane problem with a circular boundary under the screw dislocation as shown in Figure 5(a), the function of displacement field satisfies
Figure 5. 2D exterior problem (a) Dirichlet boundary condition and (b) Neumann boundary condition.
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
For the domain () as shown in Figure 6(a), the Green's function is expanded into
Figure 6. Green's function of (a) the inner domain () and (b) the outer domain () for the exterior problem.
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
where and denote the complex function and shear modulus, respectively, denotes the conjugate of the position vector of the screw dislocation, and denotes the real part. Figures 7(a) and 7(b) show the contour of displacement field by using the Smith's method [1] and the present approach, respectively. Good agreement is made.
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
Figures 8(a) and 8(b) show the contour of displacement field by using the Smith's method [1] and the present approach, respectively. It is found that the result of the present approach is acceptable. Based on the image method, it is a straightforward, logical, and natural way to find that the location of image point is (). We summarize the result of our approach for the screw dislocation and compare with those of Chen and Wu [7] for the source case in Tables 1 and 2.
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.
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.
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