## 2 Development of a Combined Size Effect Model for Rocks under Uniaxial Compression
### Establishment of a Combined Size Effect Model
By approximating \(g(\alpha)\) with its Taylor series expansion at \(\alpha=0\) and retaining only the first three terms, Equation (10) can be rewritten as follows:  
\[\sigma_{\rm ic}=\left(\frac{EG_{\rm ITC}}{c_{\rm f}g_{0}^{\prime}+\frac{1}{2} g_{0}^{\prime\prime}\frac{c_{\rm f}}{D}}\right)^{1/2} \tag{11}\]  
where \(\langle x\rangle=\max\left(0,x\right)\), \(g_{0}^{{}^{\prime}}=[\mathrm{d}g(\alpha)/\mathrm{d}\alpha]_{\alpha=0}\), and \(g_{0}^{{}^{\prime\prime}}=[\mathrm{d}^{2}g(\alpha)/\mathrm{d}\alpha^{2}]_{ \alpha=0}\). \(g(\alpha)\) increases with increasing \(\alpha\), so \(g_{0}^{{}^{\prime}}>0\). Equation (11) can further be equivalently replaced by any binomial expansion that retains the first two terms. Considering that the limit of \(\sigma_{\rm ic}\) for \(D\to 0\) should be finite, Equation (11) can be rewritten as follows:  
Figure 4:0  
where \(m\) is a positive constant and reflects the nonlinearity of the size effect curve. \(D_{0}\) is an empirical value, which is introduced to satisfy the finite value \(\sigma_{\rm ic}\) when \(D\)\(\rightarrow\)\(0\). By defining \(\sigma_{\infty}\) and \(a\) as  
Figure 4:1  
one can simplify Equation (12) as follows:  
Figure 4:2  
where Equation (14) represents the size effect model of fracture mechanics (SEFM model). It is apparent that Equation (14) can only reflect the NSE; that is, \(\sigma_{\rm ic}\) decreases and converges to a constant \(\sigma_{\infty}\) with an increase in the sample size \(D\)[83; 84; 85; 86; 7, 8]. The parameter \(\sigma_{\infty}\) is the nominal UCS for a very large sample, and the FPZ size \(c_{\rm f}\) can be ignored compared to the sample size \(D\) in this case. \(a\) is a mixed parameter, which reflects the influence of the geometric characteristics (\(g_{0}^{{}^{\prime}}\) and \(g_{0}^{\prime\prime}\)), the nonlinear parameter \(m\), and the FPZ size \(c_{\rm f}\) on the size effect curve. When \(D=0\), the maximum nominal UCS \(\sigma_{\rm icmax}\) of the interior zone is determined as follows:  
Figure 4:3  
However, observations from various simulations and experiments have confirmed the existence of the mixed type ASE-NSE of nominal strength [11; 14; 61; 62; 63; 64; 65; 66; 68; 77; 78]. Furthermore, Figure 5 shows that the size effect of \(\sigma_{\rm nc}\) is impacted by both the surface effect and the FPZ of the interior zone. Hence, the nominal strength \(\sigma_{\rm nc}\) can be derived by inserting Equation (14) into Equation (4) as follows:  
\[\sigma_{\rm nc}= \begin{cases}\sigma_{\infty}k&D\leq 2t\\ \sigma_{\infty}\bigg{[}k\frac{4t}{D}\big{(}1-\frac{t}{D}\big{)}+\big{(}1-\frac {2t}{D}\big{)}^{2}\Big{(}1+\frac{a}{D+D_{0}}\Big{)}^{1/m}\bigg{]}&D>2t\end{cases}  \tag{16}\]where \(k=\sigma_{\rm s}/\sigma_{\infty}\). Equation (16) combines both the surface effect and FPZ of the interior zone; thus, Equation (10) represents a combined size effect model (CSE model). When \(t=0\), Equation (16) is simplified to the SEFM model. When \(a=0\) or \(m=\infty\), Equation (16) is simplified to the size effect model of surface effects [Equation (4)].