## 2 Materials and Methods
### Statistical Model
#### 2.3.2 The Statistical Size Effect  
Modeling the relationship between the material strength and the critical volumes requires knowledge of the stress field in the analyzed structure. Considering the comparable dimensions of the notch and material bundles, the scale effect is, here, modeled at a mesoscopic level, necessitating the determination of the stress field and the critical volume at the mesoscale through the heterogeneous FEM introduced in Section 2.2.  
Following and adapting the statistical approach proposed in [29] for modeling the size effect in the fatigue response of components, the maximum equivalent stress in the component \(\sigma_{eq,\,max}\) is, here, the random variable. Through a linear elastic FEM, the volume of the part--in this case, the specimen--is discretized in sub-volumes characterized by the same equivalent stress, \(\sigma_{eq,\,i}\). The larger the equivalent stress, the higher the failure probability of the sub-volume. Furthermore, according to the weakest-link principle, the larger the sub-volume characterized by a specific equivalent stress, the higher the probability of failure. If the equivalent failure stress, \(\sigma_{f,i}\), of an individual element follows a Weibull distribution [1], the reliability of a single element is then:  
\[R_{\sigma_{f,i}}(\sigma_{eq,\,i})=\exp\left[-\left(\frac{\sigma_{eq,\,i}}{ \eta_{i}}\right)^{\beta}\right]\,, \tag{2}\]  
where \(\beta\) and \(\eta_{i}\) are the shape and scale parameters of the Weibull distribution, respectively. The dependence of the element reliability on its volume, \(v_{i}\), is modeled by expressing the scale parameter as:  
\[\eta_{i}=\alpha\,\,(v_{i})^{\gamma}, \tag{3}\]  
where \(\alpha\) and \(\gamma\) are two parameters, to be estimated from experimental results.  
Considering now a part or component, consisting of \(n\) elements, its reliability, \(R_{part}\), according to the weakest-link principle is:  
\[\begin{split}& R_{part}=P\Big{[}\Big{(}\sigma_{f,1}>\sigma_{eq, 1}\Big{)}\ and\ldots\ \Big{(}\sigma_{f,i}>\sigma_{eq,\ i}\Big{)}\ and\ldots\ \Big{(}\sigma_{f,n}>\sigma_{eq,\ n}\Big{)} \Big{]}=\\ &\quad=P\Big{[}\sigma_{f,1}>\sigma_{eq,\ 1}\Big{]}\cdot\ldots P \Big{[}\sigma_{f,i}>\sigma_{eq,\ i}\Big{]}\cdot\ldots P\Big{[}\sigma_{f,n}> \sigma_{eq,\ n}\Big{]}=\prod\limits_{i=1}^{n_{eq}}R_{\sigma_{f,i}}\big{(} \sigma_{eq,\ i}\big{)}\.\end{split} \tag{4}\]  
The ratio of each element's equivalent stress, \(\sigma_{eq,\ i}\), to the maximum, \(\sigma_{eq,\ max}\), in the entire component will be indicated, in the following, as:  
\[\sigma_{eq,\ ratio,i}=\frac{\sigma_{eq,\ i}}{\sigma_{eq,\ max}}. \tag{5}\]  
Substituting (2) in (4), applying a logarithmic transformation, and introducing the variable \(\sigma_{eq,\ ratio,i}\), as per (5), the probability of failure of the part, \(F_{part}\), becomes:  
\[F_{part}\big{(}\sigma_{eq,\ max}\big{)}=1-R_{part}=1-\exp\Bigg{[}-\big{(} \sigma_{eq,\ max}\big{)}^{\beta}\sum_{i=1}^{n_{eq}}\frac{\big{(}\sigma_{eq,\ ratio,i}\big{)}^{\beta}}{\alpha\ \big{(}v_{i}\big{)}^{\delta}}\Bigg{]}, \tag{6}\]  
where \(v_{i}\) and \(\sigma_{eq,\ ratio,i}\) are, respectively, the volume and the equivalent stress ratio in the \(i\)-th out of \(n\) elements, in which the component is subdivided; and \(\alpha\), \(\beta\), and \(\delta=\beta\cdot\gamma\) are three parameters.