## 2 Methodology  
### Peridynamics Damage Criterion of Coal-Rock Mass  
Peridynamics (PD), an emerging non-local mechanics theory proposed by Professor [PERSON] in 2000 that is continuously developing, can describe a series of mechanical behaviors of coal and rock masses from continuous to discontinuous and from micro- to macro-scales through the non-local theory [8; 9; 10]. This avoids the complexity of traditional discontinuous multi-scale models. In PD theory, the interaction force (also known as bond force) between material points x and x\({}^{\prime}\) within neighborhood radius \(\delta\) (horizon) is expressed in an integral form. The dynamic equation of the material point x can be written as  
\[\rho(x)\ddot{\mathrm{u}}(\mathrm{x},t)=\int_{H_{\mathrm{x}}}f(x^{\prime}, \mathrm{x},t)\,dV_{x^{\prime}}+\mathrm{b}(\mathrm{x},t) \tag{1}\]  
In the formula, \(\rho\) represents the mass density of the material;  
\(u\) represents the displacement of the material point;  
\(\ddot{\mathrm{u}}(\mathrm{x},t)=\frac{\partial^{2}\mathrm{u}(\mathrm{x},t)}{ \partial t^{2}}\) represents the acceleration of the material point;  
\(b\) (\(x\), \(t\)) represents the body force density of external loads;  
\(H_{\mathrm{x}}=H(\mathrm{x},\delta)=\{|x^{\prime}-\mathrm{x}|\leq\delta|x^{ \prime}\in\Omega\}\) denotes the set of material points in the spatial domain \(\Omega\) centered at the material point x (i.e., the near-field domain of the material point x), where \(\delta\) is the neighborhood radius;  
\(f(\mathrm{x}^{\prime},\mathrm{x},t)\) represents the interaction between the material points x and x\({}^{\prime}\), with the dimension of force density. The expression of f(x\({}^{\prime},\mathrm{x},t)\) is given by the following formula:  
\[f(\mathrm{x}^{\prime},\mathrm{x},t)=t(\mathrm{x}^{\prime},\mathrm{x},t)-t( \mathrm{x},\mathrm{x}^{\prime},t) \tag{2}\]  
\[t(\mathrm{x}^{\prime},\mathrm{x},t)= \tag{3}\]  
\[t(\mathrm{x},\mathrm{x}^{\prime},t)= \tag{4}\]  
In the formula, \(\underline{T}\) represents the vector force state, the quantities affecting this state are indicated within, and the mapping or action object of this state is shown within \(<>\). The force state acts on the relative position x\({}^{\prime}-\mathrm{x}\) between a point and any other point within its vicinity, resulting in a force density vector t. This form ensures the conservation of momentum for the entire system. Thus, dynamic Equation (1) for material point x can be expressed as  
\[ \begin{array}{rl}\rho(x)\ddot{\mathrm{u}}(\mathrm{x},t)=&\int_{H_{\mathrm{x }}}\{\underline{T}[\mathrm{x},t]\big{<}x^{\prime}-\mathrm{x}\}-\underline{T}[ \mathrm{x}^{\prime},t]\big{<}x-\mathrm{x}^{\prime}\big{>}\}dV_{x^{\prime}}\\ &+\mathrm{b}(\mathrm{x},t)\end{array}  \tag{5}\]  
Key-type PD theory can be regarded as a special case of state-type PD theory:  
\[t(\mathrm{x}^{\prime},\mathrm{x},t)=-\,t(\mathrm{x},\mathrm{x}^{\prime},t)= \frac{1}{2}f(\mathrm{x}^{\prime},\mathrm{x},t)=-\frac{1}{2}f(\mathrm{x}^{ \prime},\mathrm{x},t) \tag{6}\]  
Therefore, on the basis of Equation (5), the force vector state of the material point can be uniformly expressed as follows:  
\[\underline{T}[\mathrm{x},t]\big{<}x^{\prime}-\mathrm{x}\big{>}=t(\xi)\, \underline{M}(\underline{Y})\,\underline{M}=\frac{\underline{Y}}{|\underline{Y }|} \tag{7}\]  
\[\xi=\mathrm{x}^{\prime}-\mathrm{x}\quad\eta=\mathrm{u}^{\prime}-\mathrm{u} \tag{8}\]  
In the formula, \(t\) represents the magnitude of the force density; \(\underline{M}\) represents the direction of the force density; \(\underline{Y}\) represents the deformation; and \(\xi\) and \(\eta\), respectively, represent the initial position vector (the initial length of the bond) and the relative displacement vector of the material point. In the key-type PD theory, the force density between material points is equal in magnitude and opposite in direction, so the force density can be expressed as  
\[t\langle\xi\rangle=\left\{ \begin{array}{ll}ce&(\mathrm{when}\ |\mathrm{x}^{ \prime}-\mathrm{x}|\leq\delta)\\ 0&(\mathrm{otherwise})\end{array} \right.