# New Perspectives on Excavation Disturbance Zones: Main Driving Forces
## 2 Numerical Methodology
### Humidity-Stress-Damage Coupling Model
\tag{5}\]  
where \(\gamma\) is defined in the same manner as tensile damage, and \(\varepsilon_{c0}\) is the maximum compressive principal strain, i.e., \(\varepsilon_{c0}=f_{c}/E_{0}\), and \(f_{c}\) represents the compressive strength of the element. The mechanical properties of the rock gradually weaken with humidity diffusion. Conversely, the weakened rock changes its diffusion properties. Simultaneously, the external load causes rock damage and further weakens the mechanical properties of the rock, thereby accelerating the diffusion of humidity. Coupling can be observed between humidity, stress, and damage. This coupling is mainly expressed using the diffusion coefficient and damage variable, whereas the relationship between the humidity diffusion coefficient and damage can be expressed as follows [45]:  
\[D_{h}=\left\{ \begin{array}{cc}D_{h0}&d=0\\ \eta D_{ht}&0<d<d_{c}\\ D_{hc}&d\geq d_{c}\\ \end{array} \right. \tag{6}\]where \(D_{h0}\) represents the initial humidity diffusion coefficient of rock; and \(d_{c}\) represents a critical damage value, i.e., when the damage reaches \(d_{c}\), the humidity diffusion coefficient instantaneously increases to a constant value \(D_{hc}\). Here, \(D_{hd}\) represents the humidity diffusion coefficient when the element is first damaged, and \(\eta\) represents the diffusion mutation coefficient. In addition, an increase in humidity causes the rock mass to swell; conversely, the rock mass exhibits a shrinking effect. Therefore, the constitutive law of the rock mass with the humidity effect can be expressed as follows [45]:  
\[\mathbf{\sigma}=\lambda\mathbf{I}t_{r}\mathbf{\varepsilon}+2\mu\mathbf{\varepsilon}-(3\lambda +2\mu)\mathbf{I}\mathbf{\varepsilon}_{\mathbf{h}} \tag{7}\]  
where \(\mathbf{I}\) is the unit vector; \(\lambda\) and \(\mu\) are Lame constants; and \(\mathbf{\varepsilon}_{\mathbf{h}}\) is the humidity swelling (shrinkage) strain, which can be expressed as a linear function of humidity, as follows:  
\[\mathbf{\varepsilon}_{\mathbf{h}}=\mathbf{\alpha}_{\mathbf{h}}(h_{0}-h_{t}) \tag{8}\]  
where \(\mathbf{\alpha}_{\mathbf{h}}\) is the humidity swelling (shrinkage) coefficient and \(h_{t}\) and \(h_{0}\) are the relative humidity values at time t and the initial time, respectively. The expressions above constitute the humidity-stress-damage coupling model, which is the core part of the RFPA\({}^{\text{2D}}\)-humid code. To verify the feasibility of the RFPA\({}^{\text{2D}}\)-humid code, [PERSON] et al. conducted experiments and compared the experimental results with the numerical results [8]. First, a creep test of the sandstone samples under a high-humidity environment was carried out in the laboratory. As shown in Figure 1, the dry sample was placed on the base of a sealed container with a high-stiffness stainless steel bottom and glass walls. Water was injected into the container until the top of the specimen was submerged, and then a constant load was applied to the specimen to conduct a uniaxial creep test. Thereafter, a numerical simulation was carried out under the same conditions as the creep test. The creep behaviors obtained from the experiment and numerical simulation are shown in Figure 1. As can be observed from the figure, the two curves are highly similar, indicating the reliability of the numerical method employed. In addition, using this numerical method, [PERSON] et al. conducted a simulation study on a case (headrace tunnel in this study), in accordance with the actual on-site conditions, and obtained the deformation curve of the tunnel surrounding rock with time [8]. Figure 2 presents a comparison of the results between the field measurement and numerical sim  
Figure 1: Creep behaviors of sandstone obtained from experimental and numerical results under the same conditions in a high-humidity environment [8]. ulation. As can be seen from the figure, there was good agreement between both sets of results, which further confirms the feasibility of the numerical method employed.