## 2 Development of a Combined Size Effect Model for Rocks under Uniaxial Compression
### Effect of Parameters on the Combined Size Effect Model
Moreover, it can be seen that \(m\) has little influence on the CSE model by comparing Figure 10a,d. As mentioned above, it can be concluded from Figures 6, 7, 9 and 10 that the parameter \(k\) greatly influences the CSE model, while the influence of the parameter \(m\) on the CSE model gradually decreases with an increase in the parameter \(k\). The influence of the parameters \(t\) and \(a\) on the CSE model is weaker than that of the parameters \(k\) and \(m\), and the influence of the parameter \(D_{0}\) on the CSE model is negligible. Here, the reasons for the variation in the CSE model are further clarified. The derivative of Equation (1) with respect to \(D\) is determined as follows:  
\[(\sigma_{\rm nc})^{\prime}=\sigma_{\rm s}(f_{\rm s})^{\prime}+(\sigma_{\rm ic} )^{\prime}f_{\rm i}+\sigma_{\rm ic}(f_{\rm i})^{\prime} \tag{18}\]  
Figure 9: The effect of \(a\) 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 \(m=1.0\). where \(\langle f_{\rm s}\rangle^{{}^{\prime}},\langle f_{\rm i}\rangle^{{}^{\prime}}\), and \(\langle\sigma_{\rm ic}\rangle^{{}^{\prime}}\) are the derivatives of \(f_{\rm s}f_{\rm i}\), and \(\sigma_{\rm ic}\) with respect to \(D\). The values of \(\langle f_{\rm s}\rangle^{{}^{\prime}}\), \(\langle f_{\rm i}\rangle^{{}^{\prime}}\), and \(\langle\sigma_{\rm ic}\rangle^{{}^{\prime}}\) are obtained based on Equations (2) and (14) and are calculated as follows:  
\[\langle f_{\rm s}\rangle^{{}^{\prime}}=-\frac{4t}{D^{2}}\bigg{(}1-\frac{2t}{D }\bigg{)} \tag{19}\]  
\[\langle f_{\rm i}\rangle^{{}^{\prime}}=-(f_{\rm s})^{{}^{\prime}}=\frac{4t}{D ^{2}}\bigg{(}1-\frac{2t}{D}\bigg{)} \tag{20}\]  
\[\langle\sigma_{\rm ic}\rangle^{{}^{\prime}}=-\frac{\sigma_{\infty}a}{m(D+D_{0 })^{2}}\bigg{(}1+\frac{a}{D+D_{0}}\bigg{)}^{1/m-1} \tag{21}\]  
In Equations (19)-(21), \((1-2t/D)<0,a>0,m>0,\sigma_{\infty}>0\), and \(D_{0}>0\), so \(\langle f_{\rm s}\rangle^{{}^{\prime}}<0\), \(\langle f_{\rm i}\rangle^{{}^{\prime}}>0\), and \(\langle\sigma_{\rm ic}\rangle<0\) in Equations (2) and (14), it is obvious that \(\sigma_{\rm s}\langle f_{\rm s}\rangle^{{}^{\prime}}<0\), \(\langle\sigma_{\rm ic}\rangle^{{}^{\prime}}f_{\rm i}<0\), and \(\sigma_{\rm ic}\langle f_{\rm i}\rangle^{{}^{\prime}}>0\) in Equation (18). This implies that the root cause of the ASE in the CSE model is the increased volume fraction \(f_{\rm i}\) of the interior zone with increasing \(D\)\(\langle\sigma_{\rm ic}\langle f_{\rm i}\rangle^{{}^{\prime}}>0\rangle\). The reason underlying the NSE of the CSE model is the decreased volume fraction \(f_{\rm s}\) of the surface layer \([\sigma_{\rm s}\langle f_{\rm s}\rangle^{{}^{\prime}}<0]\) and the decreased interior zone strength \(\sigma_{\rm ic}\)\([\langle\sigma_{\rm ic}\rangle^{{}^{\prime}}f_{\rm i}<0]\) with increasing \(D\). Moreover, the mixed type ASE-NSE in the CSE model stems from the competition among \(\sigma_{\rm s}\langle f_{\rm s}\rangle^{{}^{\prime}}<0\), \(\langle\sigma_{\rm ic}\rangle^{{}^{\prime}}f_{\rm i}<0\), and \(\sigma_{\rm ic}\langle f_{\rm i}\rangle^{{}^{\prime}}>0\).