## 6 Damage Pattern and Re-Load Damage Mechanism
### Reload Damage Mechanism of Sandstone before Peak Unloading  
Studies have shown that the macroscopic damage of rock mechanical properties manifests on the micro and fine upper scales as the sprouting, expansion, and penetration of pores or microcracks [22; 23]. A large number of microfractures exist inside the sandstone  
Figure 11: Re-bearing and uniaxial damage characteristics of pre-peak unloading sandstone: (1) newborn primary crack; (2) newborn secondary crack; (3) secondary primary crack; (4) secondary secondary crack. after peak front unloading, which will cause the closure of microfractures under compressive stress during uniaxial reloading, and the contact of crack surfaces with each other will produce frictional restraint to impede the slip between crack surfaces. Suppose there is a crack 2a in length within the infinite rock slab, with the upper and lower edges subjected to a uniform uniaxial compressive stress \(\sigma_{1}\), and the angle between the central crack direction and the direction of action of \(\sigma 1\) is \(\beta\) (called the crack angle), as shown in Figure 12.  
According to the pseudo-force method [24], while considering the frictional constraint effect on the crack surface, the force of the far-field stress on the crack surface can be calculated as:  
\[P=\sigma_{1}\sin^{2}\beta \tag{8}\]  
\[Q=\sigma_{1}\sin\beta\cos\beta \tag{9}\]  
\[\tau=fP \tag{10}\]  
\[\tau_{ef}=\left\{ \begin{matrix}Q-\tau&(|\tau|<|Q|)\\ 0&(|\tau|>|Q|)\end{matrix} \right. \tag{11}\]  
where \(P\) is the normal pseudo-force acting on the crack surface; \(Q\) represents the tangential pseudo-force acting on the crack surface; \(\tau\) denotes the sliding friction between the upper and lower surfaces of the crack; \(f\) refers to the friction coefficient on the crack surface, and \(\tau_{\text{ef}}\) is the effective shear stress. Using the Muskhelishvili complex potential theory, the basic solution for the cracked surface subjected to pseudo-forces can be obtained as [25]:  
\[ \begin{split}\varphi(z)&=\frac{z[-P+i(Q-\tau)]}{2 \sqrt{z^{2}-a^{2}}}\\ \psi(z)&=\frac{z[-P-i(Q-\tau)]}{2\sqrt{z^{2}-a^{2}}} \end{split}  \tag{12}\]  
where \(\varphi(\text{z})\) and \(\psi(\text{z})\) represent complex potential functions in the plane; \(z\) denotes the independent variable, indicating the length from any point on the crack to the endpoint; a stands for the crack half-length, and \(i\) is a constant. The stress intensity factor can be calculated by the following equation:  
\[K=K_{I}-iK_{II}=2\sqrt{2\pi}\lim_{z\rightarrow\pm a}\left[\sqrt{z\pm a}\varphi (z)\right]=[-P+i(Q-\tau)]\sqrt{\pi a} \tag{13}\]  
Figure 12: Mechanical analysis model of closed cracks under uniaxial compressive stress. where \(K_{\rm I}\) and \(K_{\rm II}\) are the stress intensity factors of type I and type II crack tips, respectively. From Equation (13), the crack tip stress intensity factors \(K_{\rm I}\) and \(K_{\rm II}\) are:  
\[K_{{}_{I}}=-P\sqrt{\pi a} \tag{14}\]  
\[K_{{}_{II}}=-(Q-\tau)\sqrt{\pi a} \tag{15}\]  
The planar composite crack stress component problem is put into polar coordinates and solved as shown in Figure 13. We then have:  
\[\left. \begin{array}{l}\sigma_{\tau}=\frac{1}{2\sqrt{2\pi\tau}}\cdot\left[K_ {{}_{I}}(3-\cos\theta)+K_{{}_{II}}(3\cos\theta-1)\sin\frac{\theta}{2}\right] \\ \sigma_{\theta}=\frac{1}{2\sqrt{2\pi\tau}}\cdot\cos\frac{\theta}{2}[K_{{}_{I} }(1+\cos\theta)-3K_{{}_{II}}\sin\theta]\\ \tau_{\theta}=\frac{1}{2\sqrt{2\pi\tau}}\cdot\cos\frac{\theta}{2}[K_{{}_{I}} \sin\theta+K_{{}_{II}}(3\cos\theta-1)]\end{array} \right\} \tag{16}\]  
where \(\theta\) and \(r\) are the angle and distance between a point of the crack and the polar coordinates, respectively; and \(\sigma_{\theta}\), \(\sigma_{\rm r}\), and \(\tau_{\rm r\theta}\) represent the circumferential stress, radial stress, and shear stress, respectively. Numerous experimental results show that the direction of composite crack cracking is the direction where the circumferential stress \(\sigma_{\theta}\) reaches its maximum value.