## 2 Development of a Combined Size Effect Model for Rocks under Uniaxial Compression
### Establishment of a Combined Size Effect Model  
For rocks of different sizes under uniaxial compression, a macroscopic shear failure plane can be observed, and their angles are similar, as shown in Figure 3 [12]. Moreover, the stress-strain curve characteristics are also similar, as shown in Figure 4. This means that uniaxial compressive samples with different sizes may have the same failure mechanism. There is no doubt that rock samples contain a population of pre-existing flaws, and the global failure of a rock sample under compression occurs as soon as crack initiation is triggered from the weakest flaw. Then, the cracks propagate and coalesce to create a local damage zone, also known as a fracture process zone (FPZ). Therefore, a fracture mechanics model, which considers the weakest flaw and the FPZ, is introduced to characterize the size effect mechanism of the interior zone (Figure 5). Figure 4: Stress–strain curves of Gosford sandstone samples of different sizes under uniaxial compression: (**a**) 25 mm, (**b**) 50 mm, and (**c**) 96 mm [12]. Figure 3: Macroscopic failure characteristics of Gosford sandstone samples of different sizes under uniaxial compression [12]. In Figure 5, the length of the weakest flaw is \(c_{0}\). The angle between the extended line of the weakest flaw and the horizontal direction is \(\theta\). The normal stress \(\sigma_{\alpha}\) and shear stress \(\tau_{\alpha}\) on the surface of the weakest flaw are \(\sigma_{\alpha}=\sigma_{\rm i}\cos^{2}\!\theta\) and \(\tau_{\alpha}=\sigma_{\rm i}\cos\!\theta\) sin\(\theta\), respectively. The weakest flaw is closed, and the friction coefficient between the flaw surfaces is \(\mu\). When \(\tan\!\alpha\leq\mu\), frictional sliding along the weakest flaw will not occur, so the stress intensity factors (SIFs), \(K_{\rm I}\) and \(K_{\rm II}\), at the flaw tip are equal to 0. When \(\tan\!\alpha>\mu\), the flaw will exhibit the trend of sliding along the flaw surface; \(K_{\rm I}=0\) and \(K_{\rm II}>0\), and \(K_{\rm II}\) can be expressed as follows [79; 80; 81]:  
\[K_{\rm II}=Y(\alpha)\sqrt{\tau c_{0}}(\tau_{\theta}-\mu\sigma_{\theta})=Y( \alpha)\sigma_{\rm i}\sqrt{\tau c_{0}}cos^{2}\theta(tan\theta-\mu) \tag{6}\]  
where \(Y(\alpha)\) is a geometrical parameter and is related to the ratio of the weakest flaw length/sample diameter, that is, \(\alpha=c_{0}/D\). Based on linear elastic fracture mechanics (LEFM), the energy release rate at the weakest flaw tip can be estimated as follows [82]:  
\[G_{\rm II}=\frac{K_{\rm II}^{2}}{E}=[Y(\alpha)]^{2}\pi c_{0}(\tau_{\theta}-\mu \sigma_{\theta})=\sigma_{\rm i}[Y(\alpha)]^{2}\pi c_{0}\Big{[}cos^{2}\theta(tan \theta-\mu)\Big{]}^{2} \tag{7}\]  
where \(E\) is the Young's modulus of the material. \(g(\alpha=c_{0}/D)\) is defined as follows:  
\[g(\alpha)=\pi\Big{[}cos^{2}\theta(tan\theta-\mu)\Big{]}^{2}\alpha Y(\alpha) \tag{8}\]  
where \(g(\alpha)\) is the dimensionless energy release function and can be obtained using the finite element method. Thus, Equation (7) can be rewritten as follows:  
\[G_{\rm II}=\frac{\sigma_{\rm i}^{2}}{E}Dg(\alpha) \tag{9}\]  
In Figure 5, it is assumed that the energy release rate \(G_{\rm II}\) at the FPZ tip reaches \(G_{\rm IIC}\) (the mode II fracture energy of material) when the FPZ grows to a critical size c\({}_{\rm f}\). \(c_{\rm f}\) is a material property. At this time, the uniaxial compressive stress \(\sigma_{\rm i}\) attains its peak value \(\sigma_{\rm i\alpha}\), namely, the nominal UCS. Thus, the following assumption can be made:  
Figure 5: The size effect model of fracture mechanics considering the surface effect and the FPZ of the interior zone for a rock sample under uniaxial compression. \[G_{\rm II}\Big{(}\alpha+\frac{c_{\rm f}}{D}\Big{)}=\frac{\sigma_{\rm IC}^{2}}{E}Dg \Big{(}\alpha+\frac{c_{\rm f}}{D}\Big{)}=G_{\rm ITC} \tag{10}\]  
For intact samples, when the ratio of the weakest flaw length/sample diameter \(\alpha=c_{0}/D\to 0\), the first term of the dimensionless energy release rate function is zero [\(g(0)=g_{0}=0\)].