# Permeability Enhancement Induced by Fracture Shear Dilation During Close-Range Coal Seam Mining
## 2 Constitutive Model of Coal-Rock Mass  
The coal-rock mass is a natural geological material with many types of joints, fissures, and micro-defects. However, the development and evolution of microcracks have an evident influence on the mechanical properties of materials, which can lead to damage to the materials themselves. In the mechanical sense of solid materials, damage refers to the generation and propagation of micropores or microcracks. To investigate the damage process, it is common practice to take a representative elementary volume (REV) inside the material that is sufficiently large relative to the micro-fissures and micro-cracks, and sufficiently small relative to the entire material. When geological materials undergo plastic deformation, a dilatancy effect will occur. As the dilatancy behavior is mainly due to the initiation and expansion of microcracks, the microcracks can be considered to be uniformly distributed on the entire material at the REV scale. In continuum damage mechanics, the elastic modulus of the element is assumed to be degraded gradually as damage progresses. The damage softening of the coal body is due to the volume expansion caused by the development of micro-fractures, and is in line with the damage law of geological materials in the form of an exponential function. The stress-strain curve of each element is considered to be linearly elastic before the given damage threshold is reached. Therefore, combined with previous studies, the following damage evolution equation is proposed [33]:  
\[D=1-\exp\left(-\left(\frac{\epsilon^{\mathrm{V}}-\epsilon_{\mathrm{I}}^{ \mathrm{V}}}{F}\right)^{\mathrm{m}}\right) \tag{1}\]  
where \(\epsilon^{\mathrm{V}}\) is the volumetric strain; \(\epsilon_{\mathrm{I}}^{\mathrm{V}}\) is the volumetric strain threshold; i.e., the damage starts when the volumetric strain reaches this threshold, which reflects a reasonable starting point of the damage of rock [34]; F is the rock strength parameter, reflecting the average strength based on macroscopic statistics; m is a constant, reflecting the concentration degree of the coal strength distribution. In this study, the element becomes damaged when the stress in the element satisfies the strength criterion. The element and its damage are assumed to be isotropic. This model alone can predict the non-linear deformation of a quasi-brittle rock. As the influence of damage on the strain is reflected in the effective stress, the constitutive relation of rock damage can be established according to the strain equivalence hypothesis of [PERSON] [35]. It can be written as follows:  
\[\sigma_{ij}=E_{ijkl}\big{(}\epsilon_{ij}-\epsilon_{ij}^{p}\big{)}(1-D) \tag{2}\]  
where \(\sigma_{\mathrm{ij}}\) is the stress; E\({}_{\mathrm{ijkl}}\) is the elastic modulus; \(\epsilon_{\mathrm{ij}}\) is the overall strain; \(\epsilon_{\mathrm{ij}}\)P is the plastic strain. The failure of a rock mass is mainly in the form of shear and tension. The Mohr-Coulomb criterion is generally adopted for the shear strength of coal bodies with multiple fractures; thus, the basic form of the shear failure criterion can be written as:  
\[F_{\mathrm{s}}=\sigma_{1}-\sigma_{3}N_{\phi_{a}}+2c\sqrt{N_{\phi_{a}}} \tag{3}\]  
\[N_{\phi_{a}}=\frac{1+\sin(\phi_{a})}{1-\sin(\phi_{a})}\]  
The tensile failure criterion is:  
\[F_{\mathrm{t}}=\sigma_{3}-\sigma^{\mathrm{t}} \tag{4}\]  
where \(\sigma_{1}\) and \(\sigma_{3}\) are the maximum and minimum principal stresses, respectively, MPa; \(\Phi_{\mathrm{a}}\) is the angle of friction, \({}^{\circ}\); \(c\) is the cohesion, MPa; \(\sigma_{\mathrm{t}}\) is the tensile strength, MPa. In the plastic theory, a rule to determine the relationship between the components of plastic strain increment--that is, to specify the direction of plastic strain increment--is called the flow rule.