## 4 Strength Theory of Primary Coal-Rock Combination  
To facilitate understanding of the changes in composite strength, we further analyze the interface effects in the primary coal-rock combination. Due to the disparity in Poisson's ratios between coal and rock, there exists a radial force at the coal-rock interface. In the coal-rock combination, the radial force follows a relationship of action and reaction, with the radial force in coal samples directed radially towards the center of the interface, exerting a restraining effect on the deformation and failure of the coal proportion at the interface. Therefore, within a certain range of the coal-rock interface, the coal proportion experiences a \"triaxial\" stress state [2].  
By conducting a micro-element analysis of the interface in the primary coal-rock combination, considering deformation continuity and static equilibrium, we derive the following equations [8]:  
\[\sigma_{r}^{R}=\sigma_{r}^{C}=\frac{E_{R}\mu_{C}-E_{C}\mu_{R}}{E_{R}(1+\mu_{C} )-E_{C}(1+\mu_{R})}\sigma_{\theta} \tag{7}\]  
where \(\sigma_{r}^{R}\) and \(\sigma_{r}^{C}\), respectively, denote the radial stresses experienced by the rock and coal bodies; \(\sigma_{\theta}\) represents the axial stress; \(E_{R}\) and \(E_{C}\) denote the elastic moduli of the rock and coal bodies, respectively; and \(\mu_{R}\) and \(\mu_{C}\) represent the [PERSON]'s ratios of the rock and coal, respectively. The schematic diagram illustrating the force distribution in the composite is shown in Figure 11.  
According to the Drucker-Prager plasticity criterion, it can be derived that:  
\[f=\alpha I_{1}+\sqrt{J_{2}}-K=0\]where \(I_{1}=\sigma_{1}+\sigma_{2}+\sigma_{3}\) represents the first invariant of stress; \(J_{2}=\left[(\sigma_{1}-\sigma_{2})^{2}+(\sigma_{2}-\sigma_{3})^{2}\right.\)  
\(\left.+(\sigma_{3}-\sigma_{1})^{2}\right]/6\) denotes the second invariant of stress deviation; and \(\alpha\), K stands for the experimental constant, \(\alpha=\frac{2\sin\varphi}{\sqrt{3}(3-\sin\varphi)}\), \(K=\frac{6\cos\varphi}{\sqrt{3}(3-\sin\varphi)}\).  
When \(\sigma_{2}=\sigma_{3}\), Equation (8) can be simplified to:  
\[\sigma_{1}=\frac{3-5\sin\varphi}{3+\sin\varphi}\sigma_{3}+\frac{6\cos\varphi}{ 3+\sin\varphi} \tag{9}\]  
When \(\sigma_{2}=\sigma_{3}=0\), the strength of the coal-rock mass away from the coal-rock interface can be obtained as:  
\[\sigma_{ucs}=\frac{6\cos\varphi}{3+\sin\varphi} \tag{10}\]  
Derived from Equations (7) and (9), the strength of the coal body at the interface can be obtained as:  
\[\sigma_{CSR}=\frac{\sigma_{ucs}}{1-\alpha\lambda} \tag{11}\]  
where \(0<\alpha<1\), \(0<\lambda<1\); and \(\alpha=\frac{3-5\sin\varphi}{3+5\sin\varphi}\), \(\lambda=\frac{E_{BIC}-E_{C}\mu_{R}}{E_{R}(1+\mu_{C})-E_{C}(1+\mu_{R})}\).  
In summary, during uniaxial compression, with the axial stress loading, the coal and rock components of the primary coal-rock composite undergo radial deformation. However, due to the different Poisson ratios, a trend of relative deformation arises, thus generating interface effects acting on the coal-rock mass. Influenced by the interface effects, the coal body at the interface generates horizontal tensile stress, promoting deformation of the rock and coal at the interface. Meanwhile, the rock body at the interface generates horizontal compressive stress, restricting the deformation of the rock and coal at the interface. Moreover, positions closer to the interface experience stronger limitations due to the interface effects.