## 2 A Numerical Solution Developed Using the Collocation Point Method  
### A Dislocation near a Cubic Void  
The aim of the collocation point method is to solve ordinary differential equations, partial differential equations, differential-algebraic equations, and integral equations by enforcing the solution at a set of collocation points [25]. Figure 1 shows a dislocation line of finite length, or a Frank-Read dislocation source, near an internal cubic void. In this case, the known stress field due to a finite dislocation segment (and provided for an infinite homogeneous material) [26] will violate the traction-free or zero-traction boundary condition of such planar or flat void surfaces. This stress-traction \(\overset{\rightarrow}{T}\) condition can be stated at every collocation point on the free surface as:  
\[\overset{\rightarrow}{T}=\boldsymbol{\sigma}\overset{\rightarrow}{n}= \overset{\rightarrow}{0} \tag{1}\]  
where \(\boldsymbol{\sigma}\) is the state of stress at the collocation point, and \(\overset{\rightarrow}{n}\) is the unit normal vector of a plane passing through the surface collocation point in space (where the stress-traction vector is determined). The last vector equation translates or equates to three scalar equations: \(T_{x}=T_{y}=T_{z}=0\). Equation (1) also translates, for every free surface point in this case, to:  
\[\sigma_{xi}=\sigma_{yi}=\sigma_{zi}=0 \tag{2}\]  
where \(i\) represents the direction of the unit normal vector, and (\(i=1\),0,0), (\(i=0,1,0\)), or (\(i=0,0,1\)) are the unit normal vectors pointing in the \(x\)-axis, \(y\)-axis, or \(z\)-axis, respectively. When \(\overset{\rightarrow}{n}=(0,0,1)\), Equation (2) becomes:  
\[\sigma_{xz}=\sigma_{yz}=\sigma_{zz}=0 \tag{3}\]  
which is the zero stress-traction condition, or free surface condition, on the top surface in Figure 1.  
Analytical solutions for such a problem, i.e., a dislocation segment, or a finite-length dislocation line, near an internal void have not been found before. Therefore, one must rely on numerical solutions or methods, such as the collocation point method. The collocation point method starts with discretizing the free surfaces of the cubic void shown in Figure 1 by using a mesh (with rectangular elements or triangular elements): see Figure 2. In this scenario, the collocation points are the centers of each mesh element. To satisfy the zero-traction condition on the free surfaces, an extra source of self-equilibrated stress is needed to annul the unphysical traction stemming from the dislocation segment (which is given  
Figure 1: A dislocation line segment near a cubic void inside a cubic crystal. Point P is an arbitrary field point in the material/crystal. for an infinite medium). Each element is considered as a Volterra prismatic dislocation loop (a rectangular or triangular dislocation loop). Dislocation loops are chosen as the surface elements because each dislocation loop is a self-equilibrated source of stress. The sum of the stress fields of the dislocation loops or the surface elements is the complementary solution, while the stress field arising from the real dislocation line/source is the original solution; the residual is the difference between these two solutions. The residual is enforced to be zero at the collocation points, i.e., by solving the system of Equations (4) or (11) in the next sub-section. As mentioned earlier in this section, zero-traction conditions need to be enforced on the free surfaces. Let us focus on one of the planar free surfaces, as shown in Figure 3. According to Equation (3), \(\sigma_{xz}=\sigma_{yz}=\sigma_{zz}=0\). These three stress components are annulled on a set of \(N\) collocation points representing the centers of the dislocation loops padding the surface. Mathematically, the problem can be expressed as follows:  
\[