## 2 Implementation of the Algorithms of Fatigue and J-Integral in FEM  
The peridynamic theory utilizes a meshless discretization for material characterization. A body is subdivided into domains at the center of which exists a collocation, or material point. Each of these collocation points interacts with neighboring points within the horizon via an axial \"bond\" force that is coincident with the position vector in the current configuration. The interactions of these \"bonds\" can be described in existing Finite Element software by truss elements, and the material points can be represented by nodes. Truss elements are capable of transmitting only axial forces with no capability for shear or bending loads. This makes these elements well suited to represent the peridynamic force between neighboring collocation points. For truss elements, the conventional notion of strain in classical continuum mechanics is equivalent to stretch in the Peridynamic theory as examined by [PERSON] and [PERSON] [7]. By noting that the bond force density is force per volume squared, an equation can be established between Peridynamic stretch and strain in a truss as  
\[\epsilon=\frac{\delta}{L}=\frac{F}{AE}=\frac{F}{cV_{i}V_{j}} \tag{3}\]  
where \(\delta\) represents change in length in the axial direction, \(L\) and \(A\) are the respective length and cross sectional area of the truss, \(E\) is the [PERSON]'s modulus, \(F\) is the truss net force in each of the theories, and \(V_{i}\) and \(V_{j}\) represent the material volumes associated with the two end nodes of the truss. \(c\) is a micro-elastic modulus used in peridynamics, first described by [PERSON] and [PERSON] [7] as  
\[c=\frac{AE}{V_{i}V_{j}} \tag{4}\]  
Equation (4) is simplified by noting that in this study, a square 2D discretization is used resulting in \(V_{i}=V_{j}\). Because the distance between adjacent nodes can be described by the grid spacing, \(\Delta x\), \(V_{i}\) and \(V_{j}\) are then both equal to \(\Delta x^{2}t\) where \(t\) is the thickness. Then Equation (4) becomes  
\[c=\frac{AE}{\Delta x^{4}t^{2}} \tag{5}\]  
As the peridynamic bond force is described by the micromodulus \(c\), the individual choice of values for \(A\) and \(E\) is arbitrary. In this paper, the method proposed by [PERSON] and [PERSON] [7] was used to maintain physical units for \(A\) and \(E\)  
\[A=\Delta x^{2},\quad E=c\Delta x^{2}t^{2} \tag{6}\]  
By relating the strain energy with the classical continuum mechanics theory, for a uniform spatial discretization, \(c=18K/\pi\delta^{4}\) for 3D models and \(c=12K^{\prime}/\pi\delta^{3}t\) for 2D models where \(K\) and \(K^{\prime}\) are the bulk moduli for the 3D and 2D materials, respectively, [7; 17]. Due to the nonlocal nature of the peridynamic solution, a much greater number of elements are present than in a typical finite element solution. Depending on the horizon size, a single node will contain tens or possibly hundreds of elements which poses a difficulty for manual generation of the mesh. Therefore, a script was developed in Fortran to automate mesh generation. A subprogram first discretizes the geometry into smaller subdomains consisting of collocation points (nodes). While it is possible to spatially vary the density of the nodes, the program developed for this application utilizes a uniform grid, as shown in Figure 1. Following the definition of nodes, a second subprogram generates the truss element assemblage by identifying the family of nodes for each node given a specified horizon size. Truss elements are created once the family of each node has been determined. The subprogram also contains an algorithm to form a precrack by deleting elements crossing a specified plane (line in 2D), see the dashed bonds in Figure 1. A third subprogram groups elements based upon family data. The node family is determined for each node and used inEquation (7) to compute the required correction factor. This allows elements to be grouped based on specific material cards which account for varying stiffness near the boundaries.