# A New Shear Strength Model with Structural Damage for Red Clay in the Qinghai-Tibetan Plateau
## 4 Discussion
### Extended Mohr-Coulomb Strength Model  
The shear strength of red clay deteriorated after FTCs, and the deterioration process of the shear strength must be accompanied by structural damage. The NMR test results can accurately reflect the alterations in the microstructure of red clay during the FTC process. The damage factor was established through the microscopic test results, and then the shear strength was characterized at the macroscale. Therefore, an extended Mohr-Coulomb strength criterion was proposed, considering the structural damage caused by the action of FTCs. Under the triaxial compression condition, the expression of the Mohr-Coulomb strength criterion is  
\[q=c\frac{6\cos\varphi}{3-\sin\varphi}+Mp \tag{6}\]  
where \(q\) is the deviator stress, the difference between the axial stress \(\sigma_{1}\) and radial stress \(\sigma_{3}\), \(c\) is the cohesion; \(\varphi\) is the internal friction; \(M\) is the critical state stress ratio, which can be obtained through the friction angle \(\varphi\); \(p\) is the mean stress. \[q=\sigma_{1}-\sigma_{3} \tag{7}\]  
\[p=(\sigma_{1}+2\sigma_{3})/3 \tag{8}\]  
\[\sigma_{1}=\sigma_{3}(1+\sin\varphi)/(1-\sin\varphi) \tag{9}\]  
\[M=\frac{q}{p}=\frac{\sigma_{1}-\sigma_{3}}{(\sigma_{1}+2\sigma_{3})/3}=\frac {6\sin\varphi}{3-\sin\varphi} \tag{10}\]  
where \(\sigma_{1}\) is the axial stress; \(\sigma_{3}\) is the radial stress. In the triaxial shear test with invariable confining pressure, the radial stress \(\sigma_{3}\) is constant. By substituting Equations (7) and (8) into Equation (6), the expression of the Mohr-Coulomb strength criterion can be expressed as Equation (11). \[q=\frac{3}{3-M}\bigg{(}c\frac{6\cos\varphi}{3-\sin\varphi}+M\sigma_{3}\bigg{)} \tag{11}\]  
The microstructure of red clay is altered due to the effect of FTCs. The ability of specimens to resist shear displacement weakens, causing a sharp decrease in cohesion. Therefore, the expression of cohesion of red clay considering the microstructure damage caused by FTCs is established as Equation (12). \[c(\eta)=c_{0}(1-A\eta) \tag{12}\]  
Figure 10: The fitting curve of \(\eta\) with the number of freeze–thaw cycles. where \(c_{0}\) indicates the cohesion of the specimens that have not undergone the FTC; \(A\) is an introduced coefficient, which is a constant, representing the evolution rate of cohesion with the structural damage. Similarly, due to the increase in macropores in the soil, the contact area of soil particles decreases, causing a reduction in the internal friction angle. Therefore, the expression of the internal friction angle of red clay considering structural damage caused by FTCs is established as Equation (13). \[\varphi(\eta)=\varphi_{0}\Big{[}(1+b)-be^{B\eta}\Big{]} \tag{13}\]  
where \(\varphi_{0}\) indicates the internal friction angle of the specimens that have not undergone the FTC; \(B\) and \(b\) are two introduced coefficients, which are constants. \(B\) represents the evolution rate of the internal friction angle with structural damage caused by FTCs; \(b\) is the compensation coefficient, ensuring that the internal friction angle degenerates to \(\varphi_{0}\) when \(\eta=0\). By substituting Equations (12) and (13) into Equation (11), the shear strength model considering structural damage caused by FTCs is obtained as follows. \[q(\eta)=\frac{3}{3-M(\eta)}\Big{(}c(\eta)\frac{6\cos\varphi(\eta)}{3-\sin \varphi(\eta)}+M(\eta)\sigma_{3}\Big{)} \tag{14}\]  
In practice, coefficient \(A\) can be calibrated through the measured cohesion, and coefficients \(b\) and \(B\) can be calibrated through the measured internal friction angle. \(c_{0}\) and \(\varphi_{0}\) of the specimens are 48.2 kPa and 8.25\({}^{\circ}\), respectively. The results of parameter calibration are displayed in \[q=\sigma_{1}-\sigma_{3} \tag{7}\]0  
Figure 12 shows the contrast between the predicted and the measured shear strength values of the specimens after different numbers of FTCs. As the structural damage intensities, the shear strength of red clay gradually declines.