## 2 Development of a Combined Size Effect Model for Rocks under Uniaxial Compression
### Effect of Parameters on the Combined Size Effect Model
Here, the influence of the surface layer thickness on the CSE model is qualitatively verified. [PERSON] et al. investigated the size effect of the nominal UCS of wastewater concrete under freeze-thaw cycles [PERSON] et al. (2017), and the results are shown in Figure 8. It was found that the size effect of the nominal UCS of wastewater concrete changes from the NSE to the ASE as the number of freeze-thaw cycles increases from \(N=0\) to \(N=50\) (Figure 8). There is no doubt that the freeze-thaw cycles can gradually erode the wastewater concrete from the sample surface to the interior zone and weaken the compressive strength of the surface layer. Moreover, with an increase in the number of freeze-thaw cycles, the thickness of the weak concrete surface layer increases gradually. In Figure 8a, when the wastewater concrete samples are not subjected to freeze-thaw cycles (\(N=0\)), these samples do not have a weak surface layer caused by the freeze-thaw cycles. Therefore, the size effect of the nominal UCS is completely controlled by the interior zone and represents the NSE phenomenon, as shown in Figure 8a. In Figure 7a-d, when the thickness of the weak surface layer is equal to 0 (\(t=0\)), the NSE can be predicted by the CSE model. In Figure 8b, when the wastewater concrete samples are subjected to 50 freeze-thaw cycles (\(N=50\)), the samples have a weak surface layer caused by the freeze-thaw cycles. Thus, the size effect of the nominal UCS is completely controlled by the weak surface layer and represents the ASE phenomenon, as shown in Figure 8b. In Figure 7a,d, \(k=\sigma_{\mathrm{s}}/\sigma_{\infty}=0.1\) indicates that the sample has a weak surface layer. When the thickness of the weak surface layer is not equal to 0 [\(t=0.5\), \(1.0\), \(1.5\), \(2.0\) in Figure 7a,d], the ASE can be predicted by the CSE model. It is obvious that when \(N=0\) and \(N=50\), as shown in Figure 8, the size effect of the nominal UCS of wastewater concrete can be predicted by the CSE model. Moreover, due to the lack of a quantitative relationship between the number of freeze-thaw cycles and the weak surface layer thickness caused by the freeze-thaw cycles, it is only qualitatively proven that the CSE model can predict the influence of a weak surface layer thickness on the size effect of the nominal UCS. Figures 9 and 10 show the effects of \(a\) and \(D_{0}\) on the CSE model with different values of \(k\) and \(m\). In Figure 9a, when \(m=0.1\) and \(k=0.1\), because the interior zone's NSE grows  
Figure 8: Size effect of compressive strength of wastewater concrete under different numbers of freeze–thaw cycles: (**a**) \(N=0\) and (**b**) \(N=50\)[87]. Figure 7: The effect of \(t\) 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 \(m=1.0\). as \(a\) increases, the ASE of nominal strength gradually weakens for the samples with a weak surface layer. It can be seen in Figure 9a-c that \(k\) and \(a\) can have a significant impact on the size effect of \(\sigma_{\rm nc}/\sigma_{\infty}\). In Figure 9d, because the NSE of the interior zone when \(m=1\) is weak, the parameter \(a\) has almost no effect on the ASE curves. The influence of \(D_{0}\) on the CSE model is negligible, as illustrated in Figure 10a-d. As \(k\) increases, as shown in Figures 9a-c and 10a-c, the ASE is initially controlled by the weak surface layer, then shifts to the mixed type ASE-NSE controlled by both the surface layer and interior zone, and ultimately transitions to the NSE being controlled by the interior zone. In Figure 9a,d and Figure 10a,d, because the samples have a weaker surface layer when \(k=0.1\), the size effect curves show the ASE when \(m=0.1\) and \(m=1\). In Figures 9 and 10, when \(t=2.0\), it can be determined from Equation (17) that \(\sigma_{\rm nc}/\sigma_{\infty}=k\) when \(D\leq 2t=4.0\). Therefore, the size effect of \(\sigma_{\rm nc}/\sigma_{\infty}\) is constant when \(D\leq 2t=4.0\).