## 3 Results
### Analysis of Energy Evolution Law  
The essence of specimen deformation and failure lies in the development of microscopic cracks within the specimen under the action of external loads, progressing from disorder to order and eventually forming macroscopic cracks along a certain direction, leading to overall instability and failure of the specimen. From an energy perspective, this can be explained as the sudden release of energy. Therefore, in order to delve into the deformation and failure characteristics of specimens, it is necessary to study their energy conversion mechanisms [6]. The damage and failure of coal-rock mass represent a state instability phenomenon driven by energy. Investigating the energy variation patterns during the failure process of primary coal-rock combinations with varying coal-rock ratios and their correlation with strength and overall failure is advantageous for elucidating the fundamental characteristics of strength evolution and overall failure of primary coal-rock combinations under loading conditions. Throughout the uniaxial compression process, internal fissures within the coal-rock mass undergo compaction, microcrack initiation, expansion, and eventual penetration. This entire deformation and failure process is accompanied by the accumulation, storage, and vigorous release of energy. As axial stress is applied, continual work is performed on the specimen, transferring energy to the coal-rock primary coal-rock combination. With the progression of loading, the energy of the specimen ultimately transforms into two distinct components: (1) Irreversible dissipative energy dissipated in the form of crack development due to the loss of particle restraint in the specimen material. (2) Reversible elastic strain energy stored internally in the specimen as work carried out by external forces, manifesting as reversible deformation. According to the first law of thermodynamics, assuming no heat exchange with the surroundings during external force loading, the total energy input from the surroundings is as follows:  
\[U=U_{e}+U_{d} \tag{1}\]  
where \(U\) represents the total energy exerted on the coal-rock composite specimen by external forces; \(U_{e}\) denotes the reversible elastic strain energy; and \(U_{d}\) signifies the irreversible dissipative energy. As shown in Figure 9, the input total energy \(U\) during the stress loading process is the enveloped area under the stress-strain curve of the specimen, while the shaded area under the stress-strain curve represents the reversible elastic strain energy \(U_{e}\). The dissipated energy during the uniaxial compression process of the specimen is the area under the stress-strain curve, excluding the shaded region. Figure 9: Elastic strain energy and dissipation energy relationship curve. Thus, the formula for calculating the total energy of the uniaxial compression specimen is as shown in Equation (2):  
\[U=\int_{0}^{\varepsilon_{c}}\sigma_{\text{i}}d\varepsilon \tag{2}\]  
In the triaxial stress state, the elastic strain energy \(U_{e}\) can be expressed as:  
\[U_{e}=\frac{1}{2E_{t}}\Big{[}{\sigma_{1}}^{2}+2{\sigma_{3}}^{2}-u^{t}\Big{(}2 \sigma_{1}\sigma_{3}+{\sigma_{3}}^{2}\Big{)}\Big{]} \tag{3}\]  
Under uniaxial compression conditions, this simplifies to:  
\[U_{e}=\frac{{\sigma_{c}}^{2}}{2E_{t}} \tag{4}\]  
where \(\sigma_{i}\) is the stress at any point; \(\sigma_{c}\) is the peak stress; \(\varepsilon_{c}\) is the peak strain; and \(E_{t}\) is the unloading modulus, which for convenience in calculation is generally replaced by the elastic modulus \(E_{0}\). According to the law of energy conservation, the dissipated energy \(U_{d}\) is:  
\[U_{d}=U-U_{e}=\int_{0}^{\varepsilon_{c}}\sigma_{i}d\varepsilon-\frac{{\sigma_ {c}}^{2}}{2E_{0}} \tag{5}\]  
Usually, the slope of the tangent line or the secant line of the stress-strain data near 50% of the peak strength is taken as the dissipated energy. In this paper, the slope of the tangent line is taken.