## 2 Review on KCC Model
### Damage Factor in the KCC Model  
After reaching the initial yield surface, but before reaching the maximum strength surface, the current surface can be obtained as a linear interpolation between yield surface \(\Delta\sigma_{y}\) and maximum strength surface \(\Delta\sigma_{m}\):  
\[\Delta\sigma=\eta(\Delta\sigma_{m}-\Delta\sigma_{y})+\Delta\sigma_{y} \tag{10}\]  
After reaching the maximum strength surface, the current failure is interpolated between the maximum strength surface \(\Delta\sigma_{m}\) and the residual strength surface \(\Delta\sigma_{r}\), which is similar to the above computation:  
\[\Delta\sigma=\eta(\Delta\sigma_{m}-\Delta\sigma_{r})+\Delta\sigma_{r} \tag{11}\]  
where \(\eta\) varies from zero to one depending on the accumulated effective plastic strain parameter \(\lambda\). The value of \(\eta\) normally starts at zero and increases to unity at \(\lambda=\lambda_{m}\), then decreases back to zero at some larger value of \(\lambda\). \(\lambda_{m}\) is the plastic strain at maximum strength surface. The accumulated effective plastic strain \(\lambda\) can be expressed as follows:  
\[\lambda=\int_{0}^{\overline{p}}\frac{d\overline{e}^{p}}{r_{f}[1+p/(r_{f}f_{i} )]^{b_{1}}}\quad\text{for}\quad p\geq 0 \tag{12}\]  
\[\lambda=\int_{0}^{\overline{p}}\frac{d\overline{e}^{p}}{r_{f}[1+p/(r_{f}f_{i} )]^{b_{2}}}\quad\text{for}\quad p<0 \tag{13}\]  
where \(f_{i}\) is the quasi-static tensile strength, \(d\overline{e}^{p}\) is effective plastic strain increment and \(r_{f}\) is the dynamic increase factor (DIF) of the material under dynamic loading. The damage factors \(b_{1}\) and \(b_{2}\) define the softening behaviour due to compression (\(P\geq 0\)) and tension (\(P<0\)), respectively. Parameter \(b_{1}\) can be determined by considering compressive energy \(G_{c}\) (area under the compressive stress-strain curve) obtained from the uniaxial compression test in single element simulation. It is obtained iteratively until the area under the stress-stain curve from single element simulation coincides with \(G_{c}/h\), where \(h\) is the element size. The damage factor \(b_{2}\) in Equation (13) is related to tensile softening of the material and determined from experimental data. The fracture energy \(G_{f}\) can be obtained from the uniaxial  
Figure 1: Strength surfaces for the Karagozian and Case concrete model (KCC) material model [14].  
tensile test or three-point notched beam test. Then, the single element simulation of uniaxial tensile test is employed to obtain the stress-strain curve from the numerical analysis. Hence, the value of \(b_{2}\) can be obtained until the area under the tensile stress-stain curve from a single element coincides with \(G_{f}/w_{c}\), where \(w_{c}\) is the localization width, and typically, \(w_{c}\) is normally taken as one- to six-times the maximum aggregate size [14].  
Based on Equations (10) and (11), the stress softening factors \(\eta\) and \(\lambda\) are governed by the accumulation of effective plastic strain. However, when the stress path is very close to the negative hydrostatic pressure axis, i.e., isotropic tension, wherein the hydrostatic pressure would decrease from zero to \(-f_{t}\), where no deviatoric stress occurred, no damage accumulation would occur based on these equations. However, in such concrete-like materials, damage cannot be avoided even at this state. Therefore, the above condition has to be modified by including pressure-softening effects near or after tensile failures. In this case, a volumetric damage increment is calculated and added to the total damage factor \(\lambda\) whenever the stress path is close to the triaxial tensile path.