## 2 A Numerical Solution Developed Using the Collocation Point Method
### A Dislocation near a Cylindrical Void  
The above problem formulation was developed for a void with planar/flat surfaces that are perpendicular to the three coordinate axes. If a void has curved surfaces instead, it can be a cylindrical void or a spherical void. This study focuses on a dislocation segment, or finite-length dislocation line, near a cylindrical void, as shown in Figure 4. The analytical solution for the stress field of this 3D problem does not exist, although the analytical solution of a 2D problem, i.e., the stress field arising from an infinite dislocation line near an infinitely-long cylindrical void, is presented in [19], where the dislocation line and cylinder axis are parallel to each other. Hence, the collocation method will be utilized to develop the numerical solution for the 3D problem. For the curved surface, the mesh is typically made up of structured or unstructured triangles. In this study, uniform or structured rectangular and triangular elements are used to mesh the cylinder for convenience. Figure 5 shows a cylindrical void meshed with triangular and rectangular elements. Figure 6 is the frontal view of the meshed cylinders in Figure 5.  
Figure 4: A dislocation line segment near a cylindrical void in a cubic crystal. Point P is an arbitrary field point in the material/crystal. Figure 5: A cylindrical void meshed with (**a**) triangular elements or (**b**) rectangular elements. Each element represents a dislocation loop, and the line sense of the dislocation is shown with the arrows around the loop. For the problem depicted in Figure 4, the mathematical formulation of the problem is not exactly the same as the one for the cubic void. For example, Equation (1) still holds for any collocation point (the center points of the mesh elements, i.e., generally prismatic dislocation loops). However, Equation (2) no longer holds and instead needs to be replaced by the following equation:  
\[ \begin{array}{l}T_{x}=0\\ T_{y}=0\\ T_{z}=0\end{array}  \tag{10}\]  
Equation (10) can be expressed for this general free-surface problem as:  
\[ \begin{array}{l}\sum\limits_{j=1}^{N}T_{x}^{j\shortdot{-}i}=-T_{x}^{AB \shortdot{-}i},\hskip 28.452756 pt\begin{array}{l}i=1,\,2,\ldots N\\ i=1,\,2,\ldots N\\ i=1,\,2,\ldots N\\ \sum\limits_{j=1}^{N}T_{z}^{j\shortdot{-}i}=-T_{z}^{AB\shortdot{-}i},\hskip 28.452756 pt \begin{array}{l}i=1,\,2,\ldots N\end{array}\end{array}\end{array}  \tag{11}\]  
where \(T_{\alpha}^{j\shortdot{-}i}(x,y,z)\) is the \(\alpha\) stress-traction component for loop j calculated at collocation point \(i\). Furthermore, \(T_{\alpha}^{AB\shortdot{-}i}\) is the \(\alpha\) stress-traction component of dislocation segment \(AB\) evaluated at collocation point \(i\). Here, \(N\) is the total number of loops or elements in the mesh. However, from continuum mechanics, the stress-traction components at a field point in space/material are related to the stress tensor components at the same point and to the components of a unit normal vector \(\overrightarrow{n}=\left(n_{x},n_{y},n_{z}\right)\) of a plane (the plane defining the traction) passing through the point as such:  
\[T_{\alpha}=\sigma_{\alpha\beta}n_{\beta} \tag{12}\]  
Expanding Equation (12), Equation (11) can be rewritten as:  
\[ \begin{array}{l}\sum\limits_{j=1}^{N}\left(\sigma_{xx}^{j\shortdot{-}i}n_{x} ^{i}+\sigma_{xy}^{j\shortdot{-}i}n_{y}^{i}+\sigma_{xz}^{j\shortdot{-}i}n_{z}^ {i}\right)=-T_{x}^{AB\shortdot{-}i},\hskip 28.452756 pt\begin{array}{l}i=1,\,2, \ldots N\\ \sum\limits_{j=1}^{N}\left(\sigma_{yx}^{j\shortdot{-}i}n_{x}^{i}+\sigma_{yy}^ {j\shortdot{-}i}n_{y}^{i}+\sigma_{yz}^{j\shortdot{-}i}n_{z}^{i}\right)=-T_{y}^ {AB\shortdot{-}i},\hskip 28.452756 pt\begin{array}{l}i=1,\,2,\ldots N \end{array}\end{array}\end{array}  \tag{13}\]  
Figure 6: Side view of the cylindrical void with (a) coarser and (b) denser meshes (note: the elements only increase along the perimeter of the cylinder and not along its length). any shape.