## 3 Validation  
### Experimental Test Chosen for the Validation  
To validate the proposed strategy for estimating the coefficient of variation of \(G_{IC}\), a structural problem that has been previously studied by the author [29; 30] was selected to be simulated. This specific experiment was previously used to determine Equation (2) for a regular concrete mix design [12; 21], defined as Concrete 1. However, for this validation process, a high-strength concrete mix design was employed, defined as Concrete 2. Details regarding the compositions of both concrete mixtures can be found in Table 1, while the properties of Concrete 2 are outlined in Section 3.3. The test entails inducing a macro crack propagation in a large double cantilever beam (DCB) concrete specimen. Widely acknowledged as a method to measure Mode I fracture toughness in unidirectional composites under both static and cyclic loading conditions [31; 32], it involves applying a tensile load normal to the specimen's notch surface.  
In the field of LEFM, one of the major challenges lies in experimentally determining the critical fracture energy (\(G_{IC}\)) of concrete. It has been widely acknowledged that such tests need to be conducted on large concrete specimens to obtain accurate results [6; 29; 30; 33; 34; 35; 36; 37; 38; 39; 40]. This is primarily due to the higher degree of homogeneity achieved in concrete with larger aggregate sizes. Consequently, the process zone at the tip of the propagating macrocrack, which is crucial for determining \(G_{IC}\), extends to approximately 30 cm [30].  
The distinguishing feature of this double cantilever beam (DCB) specimen lies in its considerable dimensions: 3.5 m length, 1.1 m width, and 0.3 m thickness, rendering it suitable for simulation purposes. The specimen's geometric details and applied loading conditions are depicted in Figure 1. During the test, crack propagation occurred from the bottom to the top. The load application point (P) was positioned 0.175 m from the beam's lower side, where the crack opening measurements were taken. Initially, section thinning was employed to guide the crack and maintain it in the median plane. However, this method was found to be inadequate, leading to the introduction of longitudinal prestressing through post-tension using multiple cables. The value of the applied prestressing force was 1230 KN.  
An interesting aspect of the experimental study developed by [30] was the evaluation of the process zone using acoustic emission techniques. This assessment revealed that the process zone had dimensions of approximately 30 cm in length and 12 cm in width, estimating the volume of the process zone (\(V_{pz}\)) at around 3600 cm\({}^{3}\). The size of this process zone was associated with a maximum aggregate size of 12 mm, resulting in a maximum aggregate volume (\(V_{a}\)) of 1.13 cm\({}^{3}\). Furthermore, the determination of \(G_{IC}\) in [30] provided a mean value of \(\mu(G_{IC})=1.25\times 10^{-4}\) MN/m and a standard deviation of \(\sigma(G_{IC})=0.073\times 10^{-4}\) MN/m. Remarkably, these experimental findings aligned with the theoretical values discussed in Section 2, specifically indicating a standard deviation of \(G_{IC}\) reaching zero for a \(r_{e}\) approximately equal to 4000.  
\begin{table}
\begin{tabular}{l c c} \hline \hline \multirow{2}{*}{**Constituent**} & **Concrete 1** & **Concrete 2** \\ \cline{2-3}  & & **Quantity** \\ \hline Aggregate 1 (4–12 mm) & 1105 kg/m\({}^{3}\) & - \\ Aggregate 2 (5–20 mm) & - & 1265 kg/m\({}^{3}\) \\ Sand (0–5 mm) & 700 kg/m\({}^{3}\) & 652 kg/m\({}^{3}\) \\ Cement & 400 kg/m\({}^{3}\) & 421 kg/m\({}^{3}\) \\ Water & 190 L/m\({}^{3}\) & 112 L/m\({}^{3}\) \\ Silica fume & - & 42.1 kg/m\({}^{3}\) \\ Superplasticizer (dry powder) & - & 7.6 kg/m\({}^{3}\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Description of the mixtures used to determine Equation (2) and perform the validation.