## 5 Damage Constitutive Model  
As mentioned above, strain softening and ductile deformation are the dominant mechanical behaviors in the post-failure region. The deformation characteristics in the post peak zone are of great significance, especially in the construction and design process of mine pillars [30]. Previous studies indicated that damage mechanics is an effective way to study the fracture initiation and evolution process [31; 32]. Thus, in this paper, a damage constitutive model was proposed to study the mechanical behavior of the bimrocks with high RBPs. The uniaxial compression process can be considered a series of quasi-static processes [32], in which the initial and final states of the elastic energy are 0. Therefore, the mechanical energy absorbed  
Figure 8: Distribution curve of UCS of specimens with different RBPs. Figure 9: Failure modes of specimens with different rock contents. by the specimen (from the intact to failure) was used for damage evolution. The mechanical energy consumed per unit volume during the entire process of uniaxial compression was defined as the damage strain energy release rate [33; 34; 35]. Based on this concept, the ratio between mechanical energy consumption and the damage strain energy release rate per unit volume was defined as the damage done to the bimrocks. For any state of damage, it was assumed that the relation between the nominal stress and effective stress was as follows:  
\[\sigma=(1-D)^{\alpha}\overline{\sigma}, \tag{4}\]  
where \(\sigma\) is the nominal stress, \(\overline{\sigma}\) is the effective stress, and \(\alpha\) is the material constant. In the process of uniaxial compression, the deformation of the specimen is composed of two parts, elastic strain and strain caused by damage [36; 37], as follows:  
\[\varepsilon=(\varepsilon_{e}+\varepsilon_{d}), \tag{5}\]  
where \(\varepsilon\), \(\varepsilon_{e}\), and \(\varepsilon_{d}\) are the total strain, elastic strain, and damage strain of the damaged material. As the strain caused by damage does not contribute to the nominal stress, the constitutive equation of the welded bimrocks in any damage state should be as follows [38; 39]:  
\[\sigma=E(\varepsilon-\varepsilon_{d}), \tag{6}\]  
where \(E\) is the elastic modulus of the damaged material. Based on the experimental results, the process of elastic modulus deterioration and damage evolution was shown to have clear nonlinear characteristics. Thus, the energy equivalence hypothesis was adopted to describe the nonlinear mechanical behavior during the loading process, which assumes that the strain energy caused by nominal stress is equal to that induced by effective stress [40; 41]:  
\[W=\frac{1}{2}E\varepsilon_{e}^{2}=\frac{1}{2}\frac{\sigma^{2}}{E}=\frac{1}{2} \overline{\overline{E}}^{2}=\frac{1}{2}\overline{E}\overline{\varepsilon}_{e }^{2}=\overline{W}, \tag{7}\]  
where \(\overline{E}\) and \(\overline{\varepsilon}_{e}\) are the elastic modulus and elastic strain of the intact material, respectively. By combining Equation (4) and Equation (7), we can obtain the following equation:  
\[E=(1-D)^{2\alpha}\overline{E}=\varphi^{2\alpha}\overline{E}, \tag{8}\]  
where \(\varphi\) is defined as the continuity factor. With an increase in the damage, the irreversible damage strain grows accordingly. To determine the quantitative relation between the damage and irreversible deformation, it was assumed that the elastic strain and total strain satisfy the following:  
\[\varepsilon_{e}=\varepsilon-\varepsilon_{d}=(1-D)^{\beta}\varepsilon=\varphi^ {\beta}\varepsilon, \tag{9}\]  
where \(\beta\) is a material constant. Further, using Equation (6), Equation (8), and Equation (9) simultaneously, the following equation can be obtained:  
\[\sigma=(1-D)^{2\alpha+\beta}\overline{E}\varepsilon=\varphi^{2\alpha+\beta} \overline{E}\varepsilon. \tag{10}\]  
Equation (10) is the damage constitutive equation of welded bimrocks. To obtain the damage evolution equation, it is necessary to establish the internal relation between the total strain and damage.