# New Perspectives on Excavation Disturbance Zones: Main Driving Forces
## 2 Numerical Methodology
### Humidity-Stress-Damage Coupling Model  
The movement of water in a porous material generally conforms to [PERSON]'s law [40, 41, 42]. The use of the diffusion equation to describe the behavior of humidity diffusion can more accurately reflect the variation in humidity in the surrounding rock with respect to time and space. If the influence of heat and gravity on moisture is not considered, the humidity diffusion equation can be expressed as follows:  
\[\frac{\partial h}{\partial t}=\
abla(D_{h}\;\
abla h) \tag{1}\]  
where \(h\) is the relative humidity, \(D_{h}\) is the humidity diffusion coefficient, and \(t\) represents time. The humidity variable \(h\) obtained using Equation (1) is a function of space and time. Equation (1) can compensate for the deficiency in the existing theoretical models that cannot describe the humidity changes in rock masses. Although water affects the deformation and fracture of rock under loading, the mechanical effect of water on rock is considered by the principle of effective stress. However,the influence of water on the physical properties of rocks (such as the strength and elastic modulus) has not received sufficient attention, and an understanding of the influence of humidity on the mechanical properties of rock is relatively lacking. Consequently, the influence of humidity on the strength and elastic modulus was comprehensively considered in this study. The relationships between these two parameters and humidity can be expressed as follows [43,44]:  
\[E=\left\{ \begin{array}{cc}E_{0}&h<h_{cri}\\ E_{0}\Big{(}1-\frac{h-h_{cri}}{1-h_{cri}}\omega\Big{)}&h\geq h_{cri}\\ \end{array} \right. \tag{2}\]  
\[\sigma_{c}=\left\{ \begin{array}{cc}\sigma_{c0}&h<h_{cri}\\ \sigma_{c0}\Big{(}1-\frac{h-h_{cri}}{1-h_{cri}}\xi\Big{)}&h\geq h_{cri}\\ \end{array} \right. \tag{3}\]  
where \(E\) and \(E_{0}\) are the current and initial elastic moduli of rock, respectively; \(\sigma_{c}\) and \(\sigma_{c0}\) are the current and initial strengths of rock, respectively; \(h\) represents the current relative humidity and \(h_{cri}\) represents the critical humidity value. That is, when \(h\) is higher than \(h_{cri}\), the weakening effect of humidity on the material properties should be considered, otherwise, it can be neglected. In addition, \(\omega\) and \(\xi\) are the degradation coefficients of the elastic modulus and strength with humidity variable \(h\), respectively. Damage occurs when an external load is applied to a rock. The RFPA2D-humid code classifies the damage in a meso-element as tensile or shear damage. The maximum tensile stress criterion is used as the threshold for tensile damage, and the Mohr-Coulomb criterion is used as the threshold for shear damage. Tensile damage occurs when the tensile stress applied to an element exceeds its maximum tensile strength. In this case, the corresponding damage variable d can be expressed as follows [45,46]:  
\[d=\left\{ \begin{array}{cc}0&\varepsilon<\varepsilon_{t0}\\ 1-\frac{\gamma\varepsilon_{t0}}{\varepsilon}&\varepsilon_{t0}\leq\varepsilon <\varepsilon_{tu}\\ 1&\varepsilon\geq\varepsilon_{tu}\\ \end{array} \right. \tag{4}\]  
where \(\gamma\) is the residual strength coefficient of the element (0-1), i.e., \(\lambda=f_{tr}/f_{t};f_{t}\) represents the tensile strength of the element; \(f_{tr}\) represents the residual strength of the damaged element; and \(\varepsilon_{t0}\) represents the ultimate elastic strain, i.e., \(\varepsilon_{t0}=f_{t}/E_{0}\). Moreover, \(\varepsilon_{tu}\) represents the maximum tensile strain. Shear damage occurs when the shear stress of an element is higher than its shear strength. In this case, the corresponding damage variable can be defined as follows [45,46]:  
\[d=\left\{ \begin{array}{cc}0&\varepsilon<\varepsilon_{c0}\\ 1-\frac{\gamma\varepsilon_{c0}}{\varepsilon}&\varepsilon\geq\varepsilon_{c0} \\ \end{array} \right.