## 2 Bond-Based Peridynamic  
Peridynamic theory is a non-local formulation of the continuum media that uses spatial integration to describe the internal discretization of the material points. This characteristic became adequate for simulating the emergence of discontinuities produced by spontaneous initiation and fissure propagation. The bond-based formulation proposed by [8] is presented as the first kinematic field variable; the displacement or the strain field used in continuum mechanics released the necessary restriction to accomplish the continuum media hypothesis. In the bond-based formulation, each material point, \(\mathbf{X}\), can interact with other material points, \(\mathbf{X}^{\prime}\), inside a determined distance, \(\delta\), called the horizon, as presented in Figure 3. Considering the interaction between the two points \(\mathbf{X}\) and \(\mathbf{X}^{\prime}\) in the non-deformed configuration, the relative position is given for \(\mathbf{\xi}=\mathbf{X}-\mathbf{X}^{\prime}\). Where strain occurs in the region, the points \(\mathbf{X}\) and \(\mathbf{X}^{\prime}\) will suffer displacement, \(\mathbf{u}\), and \(\mathbf{u}^{\prime}\) having its new position determined for \(\mathbf{x}=\mathbf{X}+\mathbf{u}\) and \(\mathbf{X}^{\prime}=\mathbf{X}^{\prime}+\mathbf{u}^{\prime}\), where the relative displacement is \(\mathbf{\eta}=\mathbf{u}^{\prime}-\mathbf{u}\) and the relative position is \(\mathbf{\xi}+\mathbf{\eta}\). Figure 3: The interaction between two material points in the bond-based model. The arrow with blue color represents the horizon (\(\delta\)) and arrows with red color represents the distance between material points \(\mathbf{X}\) and \(\mathbf{X}^{\prime}\). Figure 2: (**a**) An example of device record vs. time. AE signal statistical density and the accumulated number of signals plotted on an axis in a log–log scale, where the \(b\)-value coefficient presented in Equation (1) can be seen. (**b**) The DPH index is represented in schematic form by the red curve. The motion equation within peridynamic theory up to [PERSON]'s second law is written as follows:  
\[\rho\ddot{u}(\mathbf{X},t)=\int_{H_{x}}\mathbf{f}(\mathbf{\eta},\mathbf{\xi},t)dV_{\mathbf{X}^{ \prime}}+\mathbf{b}(\mathbf{X},t),\qquad||\mathbf{X}^{\prime}-\mathbf{X}||\leq\delta, \tag{2}\]  
where \(\mathbf{f}\) is a force function exerted for the link between a pair of points, \(\mathbf{X}\) and \(\mathbf{X}^{\prime}\), and \(\mathbf{b}\) represents the body force in point \(\mathbf{X}\). The force function exerted for the bound must contain the material's constitutive information; then, we use the proposal presented in [8] for a micro-elastic material. \(\mathbf{f}\) must be derived for a scalar function, \(w(\mathbf{\xi},\mathbf{\eta})\); then,  
\[\mathbf{f}(\mathbf{\xi},\mathbf{\eta})=\frac{\partial w(\mathbf{\xi},\mathbf{\eta})}{\partial\mathbf{ \eta}}, \tag{3}\]  
where \(w(\mathbf{\xi},\mathbf{\eta})\) must represent the elastic strain energy for the volume stored in the material, and this can be defined as  
\[w(\mathbf{\xi},\mathbf{\eta})=\frac{c(\mathbf{\xi})s^{2}(\mathbf{\xi},\mathbf{\eta})|\mathbf{\xi}|}{2}. \tag{4}\]  
In Equation (4), the function \(c(\mathbf{\xi})\) represents the bound micro-module; it is a parameter related to the material, which can be considered as constant. \(s(\mathbf{\xi},\mathbf{\eta})\) represents the bound displacement and can be expressed in the following manner:  
\[s=\frac{|\mathbf{\xi}+\mathbf{\eta}|-|\mathbf{\xi}|}{|\mathbf{\xi}|}, \tag{5}\]  
where \(|\mathbf{\xi}|\) represents the distance between the material points linked by a bound in the non-deformed configuration, and \(|\mathbf{\xi}+\mathbf{\eta}|\) represents the distance between the same material points in the deformed configuration.