## 1 Introduction  
Numerous advancements have been made in the field of structural mechanics over the last century. However, the current fracture mechanics formulated in partial differential equations (PDE) is plagued by limitations in describing the material response near a crack tip [1]. For example, linear elastic fracture mechanics is hindered by stress singularity at a crack tip or other discontinuity. Different methods were proposed to model fracture, such as using micromechanical models to represent the evolution of damage [2], the crack band models [3], the arbitrary local mesh replacement method [4], cohesive zone elements [5] and the extended finite element method (XFEM) with minimal remeshing [6]. However, these methods are all formulated based on the assumption of continuity or differentiability of the displacement field. A relatively new theory known as peridynamics has been developed to use a nonlocal theory to describe crack growth that may resolve issues associated with the classical continuum theory, such as the need for remeshing and external growth criteria. The peridynamics equilibrium equations are formulated in integro-differential equations as opposed to the differential equations in the classical theory where only continuum points in direct contact can exert a force on each other. Additionally, the peridynamic theory is a meshless numerical scheme in which a material body is divided into collocation points representing differential material volumes. A material volume exerts forces on all neighboring material volumes within a boundary of finite distance known as the horizon. These internal forcesare represented as \"bonds\" between collocation points. Damage then accumulates due to a critical load or cyclic load. The bonds are severed when damage reaches a critical level, allowing for a natural progression of macro damage (visible as a crack). As peridynamics is a nonlocal theory, it does not have the same computational efficiency as the traditional, local theory-based FEM. One way to improve on the computational costs of peridynamics is to pair the peridynamic theory with the existing FEM for feature and damage modeling. It was first shown in the study of [PERSON] and [PERSON] [7] that the bonds between collocation points can be properly modeled using truss elements. The FEM solutions for dynamic brittle fracture and ductile fracture by applying peridynamics have previously been reported by [PERSON] et al. [8] and [PERSON] et al. [9], respectively. Traditional equations of motion in fracture mechanics are obtained by the spatial derivative of the stress tensor, which is the \(T_{ij}\) term in Equation (1). \[\rho\ddot{u}_{i}=\frac{\partial T_{ij}}{\partial x_{j}}+b_{i} \tag{1}\]  
where \(\rho\) is the density, \(\dot{u}_{i}\) is the acceleration, \(b_{i}\) is the body force term. The peridynamic equation of motion shown in Equation (2) was first proposed by [PERSON] [10], who used an integro-differential equation to establish the internal force vectors. \[\rho(x)\ddot{u}_{i}(x,t)=\int_{Hx}f_{i}(q,x,t)dV_{q}+b_{i}(x,t) \tag{2}\]  
The \(f_{i}(q,x,t)\) term is the bond force density, which is a corollary to the traction vector acting on a subdomain. \(u_{i}\) is the \(i\) th component of the displacement. Different from the classical continuum theory, the peridynamic theory uses nonlocal interactions for force summation. Specifically, the integral represents the net force acting on a material point as described by all interacting material points exerting influence over a finite distance, opposing to the directly adjacent material points as used in the local theory. These interacting material points with the finite distance collectively make up a region known as the horizon. Analytical solutions of the peridynamic equation of motion are limited. [PERSON] et al. [11] developed an analytical solution for the static case of a 1D peridynamic bar under self-equilibrated loading. The dynamic response of a 1D bar subject to end loading was obtained in the study of [PERSON] et al.