## 3 Validation of the Combined Size Effect Model (CSE Model)
### Comparison with Other Size Effect Models  
The SEFM model [Equation (14)], the size effect model of surface effects [Equation (4)], and the improved USEL (IUSEL) [14] were chosen for a comparative investigation. The equation for the IUSEL is as follows:  
\[\sigma_{\rm nc}=min\left[f_{\rm m}d^{(d_{\rm f-1})/2}\left(1+\frac{l}{D}\right)^ {1/2};\,f_{\rm m}\mathrm{D}_{l}^{(d_{\rm f-1})/2}\left(1+\frac{l}{D}\right)^{1 /2}\right] \tag{22}\]  
where \(f_{\rm m}\), \(D_{\rm i}\), \(l\), and \(d_{\rm f}\) are the parameters of the IUSEL.  
To ensure a fair comparison, we also used the fitting method to adjust the parameters of the SEFM model [Equation (14)], the size effect model of surface effects [Equation (4)], and the IUSEL, as outlined in Table 3. The CSE model [Equation (16)] is proposed by combining both the surface effect and the fracture process zone (FPZ); the IUSEL is created by integrating the modified SEL and MFSL through empirical methods.  
The SEFM model [Equation (14)] solely encompasses the NSE. The analysis in Section 2.1 indicates that the size effect model of surface effects can illustrate the NSE or the ASE, while the IUSEL [Equation (22)] is primarily employed to illustrate the mixed type ASE-NSE [14]. Thus, in this study, the SEFM model was solely utilized to illustrate the NSE, as depicted in Figure 11. Figure 12 presents the ASE predicted by the size effect model of surface effects. The IUSEL prediction is solely contrasted with the CSE model for the mixed type ASE-NSE and is illustrated in Figure 13. Figures 11-13 indicate that the collected data can be captured by the SEFM model, the size effect model of surface effects, and the IUSEL, respectively. Furthermore, the values of the determination coefficient \(R^{2}\) were computed to quantitatively evaluate the precision of the SEFM model, the size effect model of surface effects, the IUSEL, and the CSE model. As shown in Table 4, the \(R^{2}\) values for the NSE (No. 1) indicate that the CSE model has a greater prediction accuracy than the SEFM model. The \(R^{2}\) of the ASE (No. 2) determined by the CSE model is equal to that  
Figure 13: Comparison of the IUSEL and the CSE model with data on the mixed type ASE–NSE: (**a**) Gosford sandstone, (**b**) Pilton sandstone, (**c**) Pennnant sandstone, (**d**) Burrington oolite limestone, (**e**) Hollington sandstone, (**f**) Gambier limestone, (**g**) artificial rock, (**h**) high-strength gypsum, and (**i**) limestone.  
of the size effect model of surface effects. The \(R^{2}\) of the CSE model is close to that of the IUSEL for the mixed type ASE-NSE (No. 3-11).  
The IUSEL assumes that the ASE is mainly attributed to fractal characteristics [61]. Nevertheless, the reasons for the impact of fractal characteristics on the ASE remain uncertain. As a result, the IUSEL is incapable of elucidating the underlying mechanism of the mixed type ASE-NSE. Section 2.3 has clarified that the root cause of the ASE in the CSE model is the increased volume fraction \(f_{i}\) of the interior zone with increasing \(D\) [\(\sigma_{i}(f_{i})^{{}^{\prime}}>0\) in Equation (12)]. The reason underlying the NSE in the CSE model is the decreased volume fraction \(f_{s}\) of the surface layer [\(\sigma_{s}(f_{s})<0\) in Equation (18)] and the decreased interior zone strength \(\sigma_{i}\) [\((\sigma_{i})^{{}^{\prime}}f_{i}<0\) in Equation (18)] with increasing \(D\). Moreover, the competition among \(\sigma_{s}(f_{s})^{{}^{\prime}}<0\), \((\sigma_{i})^{{}^{\prime}}f_{i}<0\), and \(\sigma_{i}(f_{i})^{{}^{\prime}}>0\) in Equation (18) is the root cause of the mixed type ASE-NSE in the CSE model.