## 2 Methods  
### State-Based Peridynamic Theory  
A brief overview of state-based peridynamic theory is presented in the following paragraphs. An extended overview can be found in [27; 28; 29]. A peridynamic body consists of some number of nodes each uniquely described by its volume \(V_{i}\), density \(\rho_{i}\), and position vector in the reference configuration \(\mathbf{x}_{i}\). An example of a 2D body is shown in Figure 2. Node \(\mathbf{x}_{i}\) interacts with other nodes \(\mathbf{x}_{j}\) through bonds (relative position vectors) \(\mathbf{\xi}_{ij}=\mathbf{x}_{j}-\mathbf{x}_{i}\). These interactions are limited to a range called the horizon \(\delta\). Nodes \(\mathbf{x}_{j}\) that are connected to \(\mathbf{x}_{i}\) are called the family of \(\mathbf{x}_{i}\), \(\mathbf{H}_{\mathbf{x}_{i}}\). When a body deforms, node \(\mathbf{x}_{i}\) experiences displacement \(\mathbf{u}_{i}\) and moves to its deformed position \(\mathbf{y}_{i}=\mathbf{x}_{i}+\mathbf{u}_{i}\). The bond in the deformed configuration is \(\mathbf{y}_{j}-\mathbf{y}_{i}\). This deformation creates a bond force density vector \(\mathbf{t}_{ij}\) that depends on the collective deformation of all nodes in \(\mathbf{H}_{\mathbf{x}_{i}}\) and an opposite bond force density vector \(\mathbf{t}_{ji}\) that depends on the collective deformation of \(\mathbf{H}_{\mathbf{x}_{j}}\). Bond forces are force densities (force per volume), not stresses (force per area), because each node describes some volume. The bond deformation vectors are stored in an array called the deformation state  
\[\mathbf{Y}_{\mathbf{x}_{i}}=\left\{ \begin{array}{c}\mathbf{y}_{1}-\mathbf{y}_{i}\\ \vdots\\ \mathbf{y}_{n}-\mathbf{y}_{i}\end{array} \right\}, \tag{1}\]  
similarly, the force density vectors are stored in an array called the force state  
\[\mathbf{T}_{\mathbf{x}_{i}}=\left\{ \begin{array}{c}\mathbf{t}_{i1}\\ \vdots\\ \mathbf{t}_{in}\end{array} \right\}. \tag{2}\]The bond force density vectors are computed using bond deformations:  
\[T(\mathbf{x}_{i})=T(\mathbf{Y}(\mathbf{x}_{i})), \tag{3}\]  
where the function \(T(\mathbf{x}_{i})\) is a material model. It is common to state \(T(\mathbf{x}_{i})\big{<}\mathbf{x}_{j}-\mathbf{x}_{i}\big{>}\) or \(T(\mathbf{x}_{i})\big{<}\mathbf{\tilde{x}}_{ij}\big{>}\) and when referring to the force density vector \(\mathbf{t}_{ij}\) in a bond \(\mathbf{\tilde{\xi}}_{ij}=\mathbf{x}_{j}-\mathbf{x}_{i}\), and similarly for deformation state and deformed bond vectors. The peridynamic equation of motion in the integral form is  
\[\rho(\mathbf{x}_{i})\ddot{\mathbf{u}}(\mathbf{x}_{i},\ \mathbf{t})=\int\limits_{\mathbf{H}_{\mathbf{x} _{i}}}\big{(}T(\mathbf{x}_{i})\big{<}\mathbf{x}_{j}-\mathbf{x}_{i}\big{>}-\mathbf{T}\big{(}\bm {x}_{j}\big{>}\big{<}\mathbf{x}_{i}-\mathbf{x}_{j}\big{>}\big{)}dV_{\mathbf{x}_{j}}+\mathbf{b}( \mathbf{x}_{i}) \tag{4}\]  
where \(\rho(\mathbf{x}_{i})\) is the density, \(\ddot{\mathbf{u}}(\mathbf{x}_{i},\ t)\) is the acceleration, and \(\mathbf{b}(\mathbf{x}_{i})\) is the external force density. The contribution of a bond to the force density at a node can be weighed using an influence function \(\omega(\mathbf{x}_{i})\). They have been introduced in [16], and their role is further explored in [30]. The value of an influence function can depend on the length, direction, or other bond properties. It can also be used to introduce damage; remove the interaction between two nodes by setting the influence function to 0, i.e., break the bond, when some damage criterion is reached. The simplest damage criterion could be the critical stretch, in which a bond breaks when it is stretched past some critical value \(s_{c}\):  
\[\omega(\mathbf{x}_{i})=\left\{ \begin{array}{ll}1,&if\ s_{ij}<\ s_{c}\\ 0,&if\ s_{ij}\geq s_{c}\end{array}