## 2 Review on KCC Model
### Strength Surface in KCC Model  
The KCC model decouples stress into the hydrostatic pressure and deviatoric stress as shown in Equation (1):  
\[\sigma_{ij}=s_{ij}+\frac{1}{3}\sigma_{ii}\delta_{ij} \tag{1}\]  
where \(\sigma_{ij}\) is the stress tensor, \(s_{ij}\) is the deviatoric stress tensor and \(\sigma_{ii}\) is the hydrostatic pressure tensor. It should be noted that stress is positive in tension, and pressure is positive in compression. The hydrostatic pressure is related to the volumetric change of material, while the deviatoric stress is related to shear resistance of the material and is usually expressed by the second invariant of the deviatoric stress tensor, \(J_{2}\):  
\[J_{2}=\frac{1}{2}s_{ij}s_{ii}=\frac{s_{1}^{2}+s_{2}^{2}+s_{3}^{2}}{2} \tag{2}\]  
The KCC model has three independent strength surfaces: maximum strength surface, yield surface and residual strength surface, which are shown in Figure 1. The general formation of strength surfaces can be written as:  
\[\Delta\sigma=\sqrt{3J_{2}}=f(p,J_{2}) \tag{3}\]  
in which \(\Delta\sigma\) is the principal stress difference and \(p\) is the hydrostatic pressure. Usually, the above Equation (3) refers to the compressive meridian. The whole failure curve can be obtained through rotation of the compressive meridian around the hydrostatic pressure axis by multiplying \(r_{3}(\theta_{L})\), which has the formation:  
\[\Delta\sigma=r_{3}(\theta_{L})\cdot\sqrt{3J_{2}}=f(p,J_{2},J_{3}) \tag{4}\]  
\[r_{3}(\theta_{L})=\frac{r}{r_{c}}=\frac{2\left(1-\psi^{2}\right)\cos\theta_{L }+\left(2\psi-1\right)\sqrt{4\left(1-\psi^{2}\right)\cos^{2}\theta_{L}+5\psi^ {2}-4\psi}}{4\left(1-\psi^{2}\right)\cos^{2}\theta_{L}+\left(1-2\psi\right)^{2}} \tag{5}\]  
where \(\psi=r_{t}/r_{c}\) and \(r_{t}\) and \(r_{c}\) are the radius of tensile and compressive meridian, respectively. According to Equation (5), it can be found that the \(r_{3}(\theta_{L})\) depended on \(\psi\) and \(\theta_{L}\). The parameter \(\psi\) in turn relies on the hydrostatic pressure. For the concrete-like material, the value of \(\psi\) varies from \(\frac{1}{2}\) at negative (tensile) pressures to unity at high compressive pressures [14]. The value of Lode angle \(\theta_{L}\) can be obtained from:  
\[\cos\theta_{L}=\frac{\sqrt{3}}{2}\frac{s_{1}}{\sqrt{J_{2}}}\quad\text{or} \quad\cos 3\theta_{L}=\frac{3\sqrt{3}}{2}\frac{J_{3}}{j_{2}^{3/2}} \tag{6}\]  
During the initial increase of hydrostatic pressure \(p\), the deviatoric stresses \(\Delta\sigma\) remains in the elastic region until the yield surface is reached. Deviatoric stress can be further developed until the maximum strength surface is reached, and the material will subsequently start to fail. After failure is initiated, the material will gradually lose its load-carrying capacity and reaches its residual strength surface. The formations of these three surfaces are also given in Equations (7)-(9).  
\[\text{Yield surface }\Delta\sigma_{y}=a_{0y}+\frac{p}{a_{1y}+a_{2y}p} \tag{7}\]  
\[\text{Maximum strength surface }\Delta\sigma_{m}=a_{0}+\frac{p}{a_{1}+a_{2}p} \tag{8}\]  
\[\text{Residual strength surface }\Delta\sigma_{r}=\frac{p}{a_{1f}+a_{2f}p} \tag{9}\]where the eight parameters, namely, \(a_{0},a_{1},a_{2},a_{1f},a_{2f},a_{0y},a_{1y}\) and \(a_{2y}\), for three surfaces can be obtained from the experimental data (e.g., triaxial compression test, biaxial compression test or uniaxial tension/compression test) [14].