## 3 Validation of the Combined Size Effect Model (CSE Model)  
### Application to Three Types of Size Effects  
The CSE model's predictive ability was validated using experimental and simulated data for the NSE, the ASE, and the mixed type ASE-NSE, sourced from fourteen different kinds of rocks under uniaxial compression. Table 1 presents a summary of the three types  
Figure 10: The effect of \(D_{0}\) on the CSE model [Equation (17)] when (**a**) \(k=0.1\) and \(m=0.1\); (**b**) \(k=1.0\) and \(m=0.1\); (**c**) \(k=2.0\) and \(m=0.1\); and (**d**) \(k=0.1\) and \(m=1.0\). of size effects, the data bulk, and the data origin. Additionally, to assess the validity of different size effect models, the dataset needs to be sufficiently large to avoid parameter overfitting in the CSE and other models. It took a considerable amount of time to gather the data; therefore, this dataset is shared on ResearchGate ([[https://www.researchgate.net/publication/381312408](https://www.researchgate.net/publication/381312408)]([https://www.researchgate.net/publication/381312408](https://www.researchgate.net/publication/381312408))) for the benefit of other researchers later on. The six parameters \(\sigma_{\infty}\), \(k\), \(t\), \(a\), \(D_{0}\), and \(m\) need to be determined in the CSE model [Equation (16)]. It is well known that more data can forecast the size effect locus with higher accuracy. Therefore, all the gathered data were utilized to fit the material parameters in the CSE and other models by maximizing the determination coefficient \(R^{2}\). The CSE model's predictive performance was evaluated through the use of three types of size effect data. The parameters \(\sigma_{\infty}\), \(k\), \(t\), \(a\), \(D_{0}\), and \(m\) in the CSE model [Equation (16)] were calculated using the fitting method, and the results are given in Table 2. Based on Equation (13), given that \(\sigma_{\infty}\geq 0\) and \(a\geq 0\), it is obvious that \(k\geq 0\), \(t\geq 0\), \(D_{0}\geq 0\), and \(m\geq 0\) should be satisfied in this study. The parameter \(\sigma_{\infty}\) is the nominal UCS for a very large sample, and the FPZ size \(c_{\rm f}\) can be ignored in this case. Therefore, the magnitude of \(\sigma_{\infty}\) is less than or equal to the nominal UCS of the test sample with the largest size. Based on Equation (13), \(\sigma_{\infty}\geq 0\). It can be seen from Table 2 that the values of \(\sigma_{\infty}\) for all materials are greater than zero and less than the nominal UCS of the sample with the largest size. The parameter \(t\) is the surface layer thickness and is assumed to be a constant regardless of the sample size \(D\) for specific materials. Therefore, the magnitude of \(t\) is less \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Type of Size Effect** & **No.** & **Materials** & **Data Bulk** & **Data Origin** \\ \hline NSE & 1 & Coal & 12 & [PERSON] [88] \\ ASE & 2 & Simulated granite & 7 & [PERSON] et al. [89] \\ ASE–NSE & 3 & Gosford sandstone & 8 & [PERSON] et al. [61] \\  & 4 & Pilton sandstone & 8 & [PERSON] [11] \\  & 5 & Pennant sandstone & 8 & [PERSON] [11] \\  & 6 & Burringtonolite limestone & 8 & [PERSON] [11] \\  & 7 & Hollington sandstone & 8 & [PERSON] [11] \\  & 8 & Gamber limestone & 9 & [PERSON] et al. [14] \\  & 9 & Artificial rock & 9 & [PERSON] et al. [14] \\  & 10 & High-strength gypsum & 9 & [PERSON] [68] \\  & 11 & Limestone & 9 & [PERSON] [68] \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The collected data on the NSE, the ASE, and the mixed type ASE–NSE for rocks under uniaxial compression.