## 3 Validation of the Combined Size Effect Model (CSE Model)
### Application to Three Types of Size Effects
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Type of Size** & **No.** & **Materials** & \(\sigma_{\infty}\)/MPa** & \(k\) & \(t\)/mm & \(a\) & \(D_{0}\) & \(m\) \\ \hline NSE & 1 & Coal & 3.63 & \(1.12\times 10^{-11}\) & \(4.986\times 10^{-3}\) & 0.003121 & 0.07395 & \(9.14\times 10^{-3}\) \\ ASE & 2 & Simulated granite & 297.9 & 0.5237 & 5.048 & \(1.14\times 10^{-6}\) & 1.787 & 14.53 \\ ASNSE & 3 & Gosford sandstone & 30.91 & 1.103 & 10.17 & 2.569 & 109.2 & 1.498 \(\times 10^{-2}\) \\  & 4 & Pilton sandstone & 54.92 & 2.311 & 7.986 & 10.08 & 113.1 & 3.995 \(\times 10^{-2}\) \\  & 5 & Pennant sandstone & 30.22 & 1.049 & 7.701 & 9.579 & 52.77 & 5.607 \(\times 10^{-3}\) \\  & 6 & Burringtonolite & 51.78 & 1.019 & 7.708 & 0.423 & 47.72 & 2.763 \(\times 10^{-3}\) \\  & & limestone & & & & & & \\  & 7 & Hollington sandstone & 6.216 & 2.807 & 7.074 & 1.296 & 387 & \(1.47\times 10^{-3}\) \\  & 8 & Gamber limestone & \(3.89\times 10^{-4}\) & 6505 & 14.61 & 3125 & 1428 & 0.1175 \\  & 9 & Artificial rock & 1.196 & 1.814 & 14.35 & \(2.641\times 10^{-2}\) & \(2.34\times 10^{-14}\) & \(2.368\times 10^{-4}\) \\  & 10 & High-strength & \(4.961\times 10^{-3}\) & 9344 & 11.57 & 427.4 & 178 & 0.1096 \\  & 11 & Limestone & 28.24 & 3.104 & 10.74 & 1.289 & 75.28 & 5.701 \(\times 10^{-3}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Fitting parameters of the CSE model for rocks under uniaxial compression. than or approximately equal to the test sample with the minimum size, and it is related to the processing methods, grain size, and internal homogeneity of rocks. It is noted in Table 2 that the values of \(t\) for all materials are greater than or equal to zero and less than or approximately equal to the test sample with the minimum size. The parameter \(k=\sigma_{\mathrm{s}}/\sigma_{\infty}\), and \(\sigma_{\mathrm{s}}\) is the nominal UCS of the surface layer. Owing to the surface layer thickness, \(t\) is assumed to be a constant, and \(\sigma_{\mathrm{s}}\) is supposed to be a size-independent constant. The magnitude of \(\sigma_{\mathrm{s}}\) is less than or equal to the nominal UCS of the interior zone of the rock. From Equation (15), the maximum nominal UCS of the interior zone of the rock \(\sigma_{\mathrm{icmax}}=\sigma_{\infty}\Big{(}1+\frac{a}{D_{0}}\Big{)}^{1/m}\) is obtained. Therefore, the maximum value of \(k=\frac{\sigma_{\mathrm{icmax}}}{\sigma_{\infty}}=\Big{(}1+\frac{a}{D_{0}} \Big{)}^{1/m}\), and \(k\geq 0\). It is noted in Table 2 that the parameter \(k\) falls within the specified range. The parameter \(a\) is a mixed parameter, which reflects the influence of the geometric characteristics, the nonlinear parameter \(m\), and the FPZ size \(c_{\mathrm{f}}\) on the size effect curve. Based on Equation (13), the magnitude of \(a\geq 0\). It is noted in Table 2 that the parameter \(a\) falls within the specified range. The parameter \(D_{0}\) is an empirical value, which is introduced to satisfy the finite value when \(D{\rightarrow}0\). Therefore, the condition that \(D_{0}\geq 0\) should be satisfied. It is noted in Table 2 that the parameter \(D_{0}\) falls within the specified range. The parameter \(m\) is used to reflect the nonlinearity of the normal size effect (NSE) of the interior zone of the rock; that is, the nominal UCS tends to rise as the sample size decreases. This means that the condition that \(D_{0}\geq 0\) should be satisfied. It is noted in Table 2 that the parameter \(m\) falls within the specified range. However, as shown in Table 2, the values of the fitting parameters \(\sigma_{\infty}\), \(k\), \(a\), \(D_{0}\), and \(m\) for the No. 8 and 10 materials deviate greatly from those of the other materials for the mixed type ASE-NSE. This is because the descending rate is too high during the descending stage of size effect for the No. 8 and 10 materials, and the experimental data cannot reflect the transition of the size effect curve from the descending stage to the flat stage.