## 2 Development of a Combined Size Effect Model for Rocks under Uniaxial Compression
### Effect of Parameters on the Combined Size Effect Model  
According to Equation (16), the parameter \(\sigma_{\infty}\) can alter the magnitude of the locus of the CSE model without impacting its shape. Hence, Equation (16) is normalized by the parameter \(\sigma_{\infty}\) as follows:  
\[\frac{\sigma_{\rm nc}}{\sigma_{\infty}}=\left\{ \begin{matrix}k&D\leq 2t\\ k\frac{4t}{D}(1-\frac{t}{D})+\left(1-\frac{2t}{D}\right)^{2}\left(1+\frac{a}{D +D_{0}}\right)^{1/m}\end{matrix} \right\}&D>2t \tag{17}\]  
The parameters \(k,m,t,a\), and \(D_{0}\) in Equation (17) can influence changes in trends in the CSE model. Next, we explore these parameters' impacts to gain a thorough understanding of the CSE model. Figure 6 shows the effect of \(k\) on the CSE model with different values of \(m\). The small black box represents the size effect locus predicted by the CSE model when \(k=1300\) and \(k=1024\). The surface layer strength \(\sigma_{\rm s}\) is equal to the maximum nominal strength \(\sigma_{\rm icmax}\) [Equation (15)] of the interior zone when \(k=\sigma_{\rm s}/\sigma_{\infty}=\sigma_{\rm icmax}/\sigma_{\infty}=1024\). In Figure 6a, as the parameter \(k=\sigma_{\rm s}/\sigma_{\infty}\) increases from 0.1 to 1300, the size effect of \(\sigma_{\rm nc}/\sigma_{\infty}\) changes from the mixed type ASE-NSE to the NSE when \(m=0.1\). Furthermore, the sample size, which corresponds to the highest point of the mixed locus of NSE-ASE, decreases with increasing \(k\). The explanation for this is that the surface layer strength \(\sigma_{\rm s}\) gradually increases and becomes equal to, and then exceeds, the maximum nominal strength \(\sigma_{\rm icmax}\) of the interior zone with increasing \(k\); that is, the surface material gradually changes from being weaker than the material of the interior zone to being stronger. It can be seen in Figure 6a-d that the size effect of \(\sigma_{\rm nc}/\sigma_{\infty}\) changes from the mixed type ASE-NSE to the NSE or ASE with increasing \(m\). The explanation for this is that as \(m\) increases in Equation (14), \(1/m\) gradually approaches zero, causing the size effect of \(\sigma_{\rm ic}\) to diminish progressively. This implies that the size effect of \(\sigma_{\rm nc}/\sigma_{\infty}\) in Equation (17) transitions from being controlled by both the surface layer and the interior zone to being completely controlled by the surface layer with increasing \(m\). Figure 7 shows the effect of \(t\) on the CSE model with different values of \(k\) and \(m\). It can be seen in Figure 7a-d that the size effect model of fracture mechanics (SEFM model) can be obtained when \(t=0\) in Equation (17); that is, the CSE model can be degraded to the SEFM model. In Figure 7a, when \(m=0.1\) and \(k=0.1\), as the thickness \(t\) of the surface layer increases from 0 to 2, the size effect of \(\sigma_{\rm nc}/\sigma_{\infty}\) changes from the NSE to the ASE. This is because an increase in the thickness \(t\) of the surface layer increases the volume fraction \(f_{\rm s}\) of the surface layer, which, in turn, causes the size effect of \(\sigma_{\rm nc}/\sigma_{\infty}\) to change from being controlled by the interior zone to being mainly controlled by the surface layer. As shown in Figure 7a-c, as \(t\) increases, except for when \(t=0\), the size effect of \(\sigma_{\rm nc}/\sigma_{\infty}\) first changes from the ASE controlled by the weak surface layer to the mixed type ASE-NSE controlled by both the surface layer and the interior zone, and then to the NSE controlled by the interior zone. This means that \(k\) can have a significant effect on the size effect of \(\sigma_{\rm nc}/\sigma_{\infty}\). In Figure 7a,d, although the NSE of the interior zone when \(m=0.1\) is more significant than that when \(m=1\), the size effect locus still shows the ASE. The reason for this is that the sample has a weaker surface layer when \(k=0.1\). This indicates that \(m\) only has a small effect on the size effect of \(\sigma_{\rm nc}/\sigma_{\infty}\).