## Materials and Methods
### MS-Schmidt and MMS Criteria for the Analysis of the Size Effect  
Two stress-based brittle fracture criteria for considering the size effect are evaluated in this section. These criteria are the mean stress criterion as defined by [PERSON] [55] (MS-Schmidt) and the modified mean stress (MMS) criterion. The basis in both cases is the same as that established by the classical mean stress (MS) criterion: the onset of fracture in the notched samples is achieved when the tangential stress averaged over a critical distance (d\({}_{\rm c}\)) attains a critical value of \((\sigma_{\Theta\theta})_{\rm c}\). For brittle materials, the value of \((\sigma_{\Theta\theta})_{\rm c}\) in the MS criterion is assumed to be the material tensile strength \(\sigma_{\rm u}\)[56; 57]. Following and simplifying the MS criterion, [PERSON] and [PERSON] [6; 8] stated that mode I fracture in notched conditions occurs when the notch stress intensity factor (NSIF) \(\rm K_{I}V,p\) reaches the notch fracture toughness \(\rm K_{I\rm c}V,p\). They also proposed a relation to predict the value of \(\rm K_{I\rm c}V,p\):  
\[\rm K_{I\rm c}V,p=(2\pi)^{1/2}\cdot(\sigma_{\Theta\theta})_{\rm c}\cdot d_{\rm c,V}/[(1/\lambda_{1})\cdot(d_{\rm c,V}\lambda^{1}-r_{0}\lambda^{1})+[n_{\Theta \theta}(0)\cdot(d_{\rm c,V}\
u^{\rm u1}-r_{0}\mu^{1})/(\mu_{1}\cdot r_{0}(\
u^ {\rm u1-\lambda 1}))], \tag{5}\]  
According to [8]:  
\[n_{\Theta\theta}(0)=q\cdot(\chi_{\rm d1}\cdot(1+\mu_{1})+\chi_{\rm c1})/\{4 \cdot(q-1)\cdot[1+\lambda_{1}+\chi_{\rm b1}\cdot(1-\lambda_{1})]\}, \tag{6}\]  
The eigenvalues \(\lambda_{1}\) and \(\mu_{1}\), as well as the values of \(\chi_{\rm b1}\), \(\chi_{\rm c1}\) and \(\chi_{\rm d1}\), depend on the notch opening angle (\(2\alpha\)) [54]. The critical distance d\({}_{\rm c,V}\) is an important parameter in Equation (5). RV notches follow Equation (7):  
\[d_{\rm c,V}=r_{0}+d_{\rm c}, \tag{7}\]  
where, again, \(r_{0}\) is the distance between the origin of the coordinate axes and the notch tip. According to [54], \(r_{0}\) can be acquired as follows:  
\[r_{0}=(\pi-2\alpha)/(2\pi-2\alpha)\cdot\rho, \tag{8}\]  
d\({}_{\rm c}\) in Equation (7) is the theoretical length of the fracture process zone (FPZ), based on the MS criterion, and can be obtained from [PERSON]'s formula [55] for cracked conditions:  
\[d_{\rm c-Schmidt}=(2/\pi)\cdot(K_{I\rm c}/\sigma_{\rm u})^{2} \tag{9}\]  
Figure 5 shows a schematic of d\({}_{\rm c,V}\) in RV notches. The MS criterion, in which the value of d\({}_{\rm c}\) is obtained from [PERSON]'s formula, will be denoted hereafter as the MS-Schmidt criterion. Figure 5: The critical distance d\({}_{\rm c,V}\) in RV notches according to the MS criterion. A second variant of the MS criterion for considering the size effect is the MMS criterion, which considers the higher order terms of the Williams series solution of the stress field when calculating the value of \(\mathrm{d_{c}}\) in a cracked specimen. More precisely, the tangential stress component is defined by taking into account the first three terms of the Williams series expansion in pure mode I loading [27]:  
\[\sigma_{\mathrm{\Theta\Theta}}=\mathrm{K_{I}}\cdot\{1+[3\cdot(A_{3}{}^{*}\cdot r )/(A_{1}{}^{*}\cdot R)]\}/(2\pi r)^{1/2}, \tag{10}\]  
where the parameters \(A_{1}{}^{*}\) and \(A_{3}{}^{*}\) are the dimensionless coefficients of the first and third terms of the Williams series expansion.