## 4 Numerical Analyses Using Peridynamic Model  
The input data for the peridynamic model consist of the mechanical properties, geometry, and boundary conditions, as depicted in Figure 5b. The mechanical properties that characterize the model are (i) Young modulus (\(E\)) and (ii) critical stretch (\(s_{c}\)), as defined in Equation (9), (iii) fracture toughness (\(G_{f}\)), and (iv) material density (\(\rho\)). The computational implementation of the peridynamic model used is based on the Fortran code described in the book of [PERSON] and [PERSON] [29]. In this model, critical stretch, \(s_{c}\), is computed using Equation (9), and it is defined as a function of the parameters \(E\), \(G_{f}\), and the horizon, \(\delta\), which is a material characteristic length. The horizon represents the number of nodes that will be connected through bounds, as depicted in Figure 4. The meaning of this material characteristic length is discussed in [30; 31] and in the context of  
Figure 5: Information about the test performed: (**a**) the fixer dispositive used in the test, (**b**) the boundary conditions used in the test, and (**c**) the dimensions that characterize the test specimen. (**d**) Failure specimen. the discrete element methods in [32; 33]. Material strength is linked to critical stretch using the simple relation \(\sigma_{c}=Es_{c}\). Some additional observations are pointed out below:  
1. In the peridynamic context, the characteristic length of the material indicates the interconnection length between neighboring nodes;
2. A good relation between the horizon and the discretization level (\(\Delta\)) for numerical reasons is considered by [9] to be \(\delta=3.015\Delta\). This relation is used in the present work;
3. Toughness, \(G_{f}\), is considered a random field regarding the mean, variation coefficient, Weibull distribution, and the correlation length of a random space field. Details about the implementation of this random field are explained in [27];
4. The research group developed proposals that allow for establishing a link between the independent material characteristic length and discretization. In [33], replacing the constitutive law for the bound with a bi-linear law is proposed, which allows for defining the material characteristic length independent of the discretization level. This possibility was not used in the simulation implemented in the present paper. The adopted material parameters are presented in Table 1. In previous publications carried out by the research group, the peridynamic approach was utilized to duplicate experimental tests (see, for example, [33]). The focus of the present study is not to duplicate or calibrate any specific test performed. The flexibility of the support produced high variability in the experimental results, making it difficult to objectively calibrate the model. Therefore, for this work, values that could be considered typical of a quasi-brittle material were adopted: \(E=17\,\mathrm{GPa}\), \(G_{f}=70\,\mathrm{N/m}\), and \(\sigma_{t}=10\,\mathrm{MPa}\). With these material parameters, it is possible to use Equation (9) to obtain a material characteristic length of \(\delta=10.21\,\mathrm{mm}\). By using the recommendations of [9], the discretization adopted was \(\Delta=3.4\,\mathrm{mm}\). The coefficient of variation for the \(G_{f}\) random field is (\(CV(G_{f})=50\%\)). This variation coefficient that is defined over the bound, in fact, represents a lesser value of variation for the volumetric control, as discussed by [33]. Figure 6 illustrates the geometry and boundary conditions considered. It was built using a prism of \(30\times 30\times 15\) node lines corresponding to the \(x,y\), and \(z\) directions, respectively. The inclined pre-notch is represented with the removal of the bound (disconnection between nodes) that crosses the crack, as is illustrated in Figure 6c. The boundary condition was applied to three lines of nodes above and below the edge of the specimen; that is, they were applied to six lines of nodes in the y direction.