## 2 Theoretical Background
### Peridynamics Theory and Peridynamics Differential Operator  
The peridynamics (PD) theory [26; 43] reformulates the classical continuum mechanics theory by including spatial integrals instead of spatial derivatives in its equations of motion. Owing to this feature, the PD model enables the possibility to effectively simulate discontinuities (e.g., cracks, defects) without the need of any additional models. In the PD model the domain is discretized into a square grid of equally spaced material points (PD grid spacing \(\Delta x\), see Figure 4) and each material point \(x\), defined by the position vector \(\mathbf{x}\), interacts with the material points \(x^{\prime}\) in the interaction range \(H_{x}\) called PD horizon, which is a measure of the material non-local behavior defining the range of interaction of the material points. The horizon size \(\delta\) is, usually expressed as a radius equal to a multiple of the PD grid spacing \(\delta=m\Delta x\) (Figure 4). According to studies on the parameter \(m\)[44], a PD horizon of about 3-\(\Delta x\) ensures the PD model stability and convergence in most cases. The PD model grid size \(\Delta x\) should be selected to evaluate the local effects with suitable accuracy, and can be found based on results of convergence analysis. The interaction between the PD points \(x\) and \(x^{\prime}\) is called PD bond, and the deformation of the body (displacement vectors \(\mathbf{u}\) and \(\mathbf{u^{\prime}}\) of points \(x\) and \(x^{\prime}\)) causes deformation of the PD bonds and thus forces \(\mathbf{f}\) and \(\mathbf{f^{\prime}}\) acting on the PD bonds between the points \(x\) and \(x^{\prime}\). Due to formulation of bond forces \(f\) and \(f^{\prime}\) being opposite in direction and of equal magnitudes (bond-based PD), interactions of a material point to other points are limited in the range of the PD horizon and it results that the [PERSON]'s ratio can only be 0.25 for 3D case and 0.33 for plane stress case [45]. This limitation is released by including new mathematical operators-deformation and forces states (state-based PD) in the PD theory resulting that the PD bond forces are not equal in magnitude (PD bond force depends on the current bond deformation and the deformation of all other bond connecting the same point). The PD states map undeformed horizon points on any deformed configuration of \begin{table}
\begin{tabular}{c c c} \hline \hline
**Specimens Preparation** & **Experiment** & **Post Processing** \\ \hline \multirow{4}{*}{Speckle size (randomness, contrast, etc.)} & Calibration & Subset size \\  & Camera resolution & Step size \\ \cline{1-1}  & Camera focus & Subset shape function \\ \cline{1-1}  & Lightning & Displacement sub-pixel \\ \cline{1-1}  & Camera noise & interpolation \\ \hline \hline \end{tabular}
\end{table} 
Table 1: DIC parameters determining the measurements accuracy. the horizon (detailed description of the PD states can be found in [45]). The state-based PD equation of motion is written as:  
\[\rho\ddot{\mathbf{u}}(\mathbf{x},t)=\int_{H_{x}}\big{\{}\mathbf{\underline{T}}[\mathbf{x},t] \big{<}\mathbf{x}^{\prime}-\mathbf{\underline{x}}\big{>}-\mathbf{\underline{T}}[\mathbf{x}^{ \prime},t]\big{<}\mathbf{x}-\mathbf{x}^{\prime}\big{>}\big{>}dV_{x^{\prime}}+\mathbf{b}(\bm {x},t), \tag{2}\]  
where \(\rho\) is the material density, \(\mathbf{b}(\mathbf{x},\,t)\)--body forces, and \(V_{x}\)--volume of the point \(x\). \(\mathbf{\underline{T}}[\mathbf{x},t]\big{<}\mathbf{\underline{\zeta}}\big{>}\) is a force state valued function of position \(x\) and time \(t\) and operating on the vector \(\xi=\mathbf{x}^{\prime}-\mathbf{x}\) (Figure 4). The classical continuum mechanics concepts of strains and stress are not included in the PD theory, their PD equivalents being the PD bond stretch and the PD bond force. However, retrieving the conventional strain quantities from the PD formulation is possible based on differentiation of the integral PD equations of deformation.