## 2 Determination of the Probabilistic Properties of \(G_{ic}\)  
In previous research [12], the standard deviation of \(G_{IC}\) was determined using an inverse analysis for a specific concrete mix design, where the mean value of \(G_{IC}\) was known. To further investigate the probabilistic properties of \(G_{IC}\), an analytical relation was proposed to establish a connection between the standard deviation, \(\sigma\), of \(G_{IC}\), and the degree of heterogeneity of the mesh element. In this work, the degree of heterogeneity, \(r_{e}\), is defined as follows:  
\[r_{e}=V_{e}/V_{a}, \tag{1}\]  
where \(V_{a}\) is the volume of the largest aggregate size present in the concrete, and \(V_{e}\) denotes the volume of the mesh element. The analytical relation proposed to establish this connection between \(\sigma\) and \(r_{e}\) is as follows:\[\sigma(G_{IC})=(A\ln(r_{e})+B)\times\mu(G_{IC}), \tag{2}\]  
where \(A=-8.538\); \(B=70.88\) and \(\mu(G_{IC})\) is the mean value of \(G_{IC}\). The relation described in Equation (2) has been proposed specifically for \(G_{IC}=1.25\times 10^{-4}\) MN/m and for the size of the largest aggregate equal to 12 mm (in terms of diameter). These concrete parameters were derived from experimental research, which enabled the determination of the intrinsic value of \(G_{IC}\)[29; 30]. It is crucial to note that in the model, the volume of finite elements must exceed the largest aggregate volume. Based on Equation (2), it is natural to propose the following relation for the coefficient of variation of \(G_{IC}\):  
\[\frac{\sigma(G_{IC})}{\mu(G_{IC})}=(A\ln(r_{e})+B), \tag{3}\]  
It is crucial to note that the use of Equation (2) does not lead to the determination of intrinsic values of \(\sigma(G_{IC})\). These values are inherently linked to the specific mechanical model proposed and the chosen type of finite elements, such as linear elements in the present case. Consequently, Equation (2) cannot be indistinctively applied within the framework of other mechanical models. From Equation (3), it can be observed that the coefficient of variation of \(G_{IC}\) becomes negligible when the degree of heterogeneity (\(r_{e}\)) reaches a value of 4000. Notably, it is important to reiterate that as \(G_{IC}\) is an intrinsic material property and, being size-effect independent, it is not dependent on the \(r_{e}\) in terms of its mean value. In previous research [29], focusing on the probabilistic properties of \(f_{t}\), which is dependent of the material heterogeneity degree and, therefore, can be expressed as \(f_{t}(r_{e})\); Equations (4) and (6) were proposed to evaluate the mean value, \(\mu(f_{t}(r_{e}))\), and coefficient of variation, \(\frac{c}{\mu}(f_{t}(r_{e}))\), respectively. \[\mu(f_{t}(r_{e}))=a(r_{e})^{-y}, \tag{4}\]  
in Equation (4), \(a=6.5\) MPa and \(y\) is provided by Equation (5), where \(f_{c}\) represents the concrete compressive strength in MPa and \(\alpha=1\) MPa. \[y=0.25-3.6\times 10^{-3}\bigg{(}\frac{f_{c}}{\alpha}\bigg{)}+1.3\times 10^{-5} \bigg{(}\frac{f_{c}}{\alpha}\bigg{)}^{2}, \tag{5}\]  
\[\frac{\sigma}{\mu}(f_{t}(r_{e}))=c(r_{e})^{-d}, \tag{6}\]  
in Equation (6), \(c=0.35\) and \(d\) are provided by Equation (7), as follows:  
\[d=4.5\times 10^{-2}+4.5\times 10^{-3}\bigg{(}\frac{f_{c}}{\alpha}\bigg{)}-1.8 \times 10^{-5}\bigg{(}\frac{f_{c}}{\alpha}\bigg{)}^{2}, \tag{7}\]  
The validity of Equations (4)-(7) has been confirmed for concrete with a compressive strength of \(f_{c}\leq 130\) MPa and a maximum aggregate size of 10 mm or larger. Consequently, by establishing a correlation between Equations (3) and (7) and Equations (4)-(7), it is feasible to estimate the coefficient of variation of \(G_{IC}\) as a function of \(r_{e}\) for concrete that satisfy the aforementioned criteria. This estimation can be accomplished by establishing an expression through algebraic manipulation using Equations (3)-(7).