## 4 Conclusions  
In this study, a size effect model of surface effects was developed by analyzing how the free surface and surface flaws impact the nominal uniaxial compressive strength (UCS) of rocks under compression. Following this, a combined size effect model (CSE model) was formulated by integrating the surface effect size model and the size effect model of \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Type of Size Effect**} & \multirow{2}{*}{**No.**} & \multirow{2}{*}{**Materials**} & **SEFM Model, Size Effect Model of Surface Effects, or IUSEL** & **CSE Model** \\ \hline NSE & 1 & Coal & 0.9735 & 0.9952 \\ ASE & 2 & Simulated granite & 0.9972 & 0.9972 \\ ASE–NSE & 3 & Gosford sandstone & 0.9960 & 0.9858 \\  & 4 & Pilton sandstone & 0.9330 & 0.9653 \\  & 5 & Pennant sandstone & 0.9758 & 0.9521 \\  & 6 & Burringtonolite limestone & 0.9929 & 0.7326 \\  & 7 & Hollington sandstone & 0.9844 & 0.9799 \\  & 8 & Gambler limestone & 0.9924 & 0.94 \\  & 9 & Artificial rock & 0.9670 & 0.9351 \\  & 10 & High-strength gypsum & 0.9068 & 0.9653 \\  & 11 & Limestone & 0.9277 & 0.8524 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The determination coefficient \(R^{2}\) of the SEFM model, the size effect model of surface effects, the IUSEL, and the CSE models for rocks under uniaxial compression. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Type of Size Effect**} & \multirow{2}{*}{**No.**} & \multirow{2}{*}{**Materials**} & \multicolumn{2}{c}{**Model Parameters**} \\ \hline  & & & \(\sigma_{\infty}\)/MPa & \(a\) & \(D_{0}\) & \(m\) \\ NSE & 1 & Coal & 3.59 & 0.000161 & 0.09978 & 0.000473 \\  & & & \(\sigma_{s}\)/MPa & \(\sigma_{i}\) & \(t\) & - \\ ASE & 2 & Simulated granite & 156.000 & 297.900 & 5.048 & - \\  & & \(f_{\rm m}\)/MPa & \(D_{\rm i}\) & \(l\) & \(d_{\rm f}\) \\ ASE–NSE & 3 & Gosford sandstone & 3.057 & 63.32 & 25.78 & 2.346 \\  & 4 & Pilton sandstone & 7.602 & 56.38 & 144.8 & 2.281 \\  & 5 & Pennant sandstone & 0.7611 & 48.88 & 242.6 & 3.02 \\  & 6 & Burringtonolite limestone & 1.292 & 47.61 & 268.4 & 2.997 \\  & 7 & Hollington sandstone & 1.488 & 55.85 & 40.49 & 2.435 \\  & 8 & Gambler limestone & 0.005379 & 124.1 & 37,560 & 2.561 \\  & 9 & Artificial rock & 0.0595 & 51 & 385 & 2.576 \\  & 10 & High-strength gypsum & 0.05191 & 48.32 & 472,400 & 2.218 \\  & 11 & Limestone & 3.029 & 56 & 222.3 & 2.428 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Fitting parameters of the SEFM model, the size effect model of surface effects, and the IUSEL for rocks under uniaxial compression. -fracture mechanics (SEFM model) associated with the FPZ for rocks. Parametric studies were undertaken to gain deeper insight into the CSE model. Validation of the CSE model was conducted by comparing it with the collected data. The CSE model's predictive performance was contrasted with that of other size effect models. The key findings are as described below. The size effect model of surface effects can describe the ASE of nominal strength. If the strength of the surface layer is greater than that of the interior zone, the NSE of nominal strength can be predicted by the size effect model of surface effects. The parametric studies indicated that the CSE model is able to illustrate the NSE, the ASE, and the mixed type ASE-NSE, and their underlying mechanisms. The validation study demonstrated that the CSE model accurately predicts both the ASE and the NSE, as well as effectively captures the mixed type ASE-NSE. In addition, the fitting parameters of the CSE model are within the specified range. It was observed that the determination coefficient \(R^{2}\) of the CSE model is greater than that of the SEFM model for the NSE, equivalent to that of the size effect model of surface effects for the ASE, and close to that of the IUSEL for the mixed type ASE-NSE. Although the CSE model can capture the NSE, the ASE, and the mixed type ASE-NSE, there are still limitations.