# Permeability Enhancement Induced by Fracture Shear Dilation During Close-Range Coal Seam Mining
## 2 Constitutive Model of Coal-Rock Mass
When the stress state reaches the yield surface, the plastic strain increment \(\mathrm{d_{ij}}^{\mathrm{P}}\) under the stress state can be determined by the plastic potential surface g. Plastic potential function is a surface in stress space that is perpendicular to the plastic strain increment. Plastic flow is defined by a flow potential that is associated on the cap and non-associated on the failure yield surface and transition yield surfaces. The potential function is described by means of two functions, \(\mathrm{g_{s}}\) and \(\mathrm{g_{t}}\), used to define shear plastic flow and tensile plastic flow, respectively. The non-associated flow rule is adopted with a shear plastic potential function defined as follows [36]:  
\[g_{\mathrm{s}}=\sigma_{1}-\sigma_{3}N_{\mathrm{\psi_{a}}}\]  
\[N_{\mathrm{\psi_{a}}}=\frac{1+\sin(\psi_{a})}{1-\sin(\psi_{a})} \tag{5}\]  
where \(\mathrm{\psi_{a}}\) is the dilatancy angle, \({}^{\circ}\). The tensile plastic potential function is:  
\[g_{\mathrm{t}}=\sigma_{3} \tag{6}\]  
Upon the onset of yielding, plastic deformation should conform to the consistency of the flow potential function [37]. According to the theory of plastic increment, the plastic increment can be obtained using the flow rule [37],  
\[\Delta\epsilon^{\mathrm{P}}=\dot{\lambda}\frac{\partial\mathrm{g}}{\partial \sigma_{ij}} \tag{7}\]  
Plastic strain is updated at each step when the plastic flow occurs. Equation (2) yields the following:  
\[\Delta\sigma_{ij}=\Big{(}E_{ijkl}\Delta\epsilon_{ij}-E_{ijkl}\Delta\epsilon_{ ij}^{\mathrm{P}}\Big{)}(1-D) \tag{8}\]  
Let the initial stress be \(\mathrm{\sigma_{i}}^{0}\), stress before plastic modification be \(\mathrm{\Delta\sigma_{i}}^{\mathrm{I}}\), stress after plastic modification be \(\mathrm{\sigma_{ij}}^{\mathrm{N}}\), and damage increment be \(\mathrm{\Delta D}\). Assuming that the coal is an isotropic material, Equation (8) leads to:  
\[\left\{ \begin{array}{c}\sigma_{1}^{N}=\Big{(}\Delta\sigma_{1}^{I}-\dot{ \lambda}_{\mathrm{s}}\big{(}\lambda\big{(}1-N_{\mathrm{\psi}}\big{)}+2G\big{)} \Big{)}(1-D)+\sigma_{1}^{0}(1-\mathrm{\Delta}D)\\ \sigma_{2}^{N}=\Big{(}\Delta\sigma_{2}^{I}-\dot{\lambda}_{\mathrm{s}}\big{(} \lambda\big{(}1-N_{\mathrm{\psi}}\big{)}\big{)}\Big{)}(1-D)+\sigma_{2}^{0}(1- \mathrm{\Delta}D)\\ \sigma_{3}^{N}=\Big{(}\Delta\sigma_{3}^{I}-\dot{\lambda}_{\mathrm{s}}\big{(} \lambda\big{(}1-N_{\mathrm{\psi}}\big{)}-2 GN_{\mathrm{\psi}}\big{)}\Big{)}(1-D )+\sigma_{3}^{0}(1-\mathrm{\Delta}D)\end{array} \right. \tag{9}\]  
The flow constant \(\mathrm{\lambda_{s}}\) is:  
\[\dot{\lambda}_{\mathrm{s}}=\frac{\mathrm{F_{s}}\big{(}\sigma_{1}^{\mathrm{I}}, \sigma_{3}^{\mathrm{I}}\big{)}}{\big{(}\lambda\big{(}1-N_{\mathrm{\psi}}\big{)} +2\mathrm{G}\big{)}-\big{(}\lambda\big{(}1-N_{\mathrm{\psi}}\big{)}-2\mathrm{G }N_{\mathrm{\psi}}\big{)}N_{\mathrm{\psi}}} \tag{10}\]  
The modified stress under tensile failure is:  
\[\left\{ \begin{array}{c}\sigma_{1}^{N}=\Big{(}\Delta\sigma_{1}^{I}-\dot{ \lambda}_{t}\lambda\Big{)}(1-D)+\sigma_{1}^{0}(1-\mathrm{\Delta}D)\\ \sigma_{2}^{N}=\Big{(}\Delta\sigma_{2}^{I}-\dot{\lambda}_{t}\lambda\Big{)}(1- D)+\sigma_{1}^{0}(1-\mathrm{\Delta}D)\\ \sigma_{3}^{N}=\Big{(}\Delta\sigma_{3}^{I}-\dot{\lambda}_{t}(\lambda-2G)\Big{)} (1-D)+\sigma_{1}^{0}(1-\mathrm{\Delta}D)\end{array} \right. \tag{11}\]  
The flow constant \(\mathrm{\lambda_{t}}\) is:  
\[\dot{\lambda}_{\mathrm{t}}=\frac{\mathrm{F_{t}}\big{(}\sigma_{1}^{\mathrm{I}}, \sigma_{3}^{\mathrm{I}}\big{)}}{\lambda-2G} \tag{12}\]The failure criterion established above can effectively determine the failure type of the element, and the corresponding plastic strain correction can be performed. Meanwhile, the change in the effective stress under the influence of damage is considered to obtain a reasonable stress value. The variables are updated with every iteration. A comparison between the simulated stress-strain curve and experimental data under uniaxial compression has been conducted to validate the proposed model [33].