# A Dynamic Numerical Simulation on the Grouting Timing in Retained Rib of Pillarless Mining
## 2 The Principle of the Dynamic Numerical Simulation
### Surrounding Rock Constitutive Model
#### 2.2.2 Constitutive Model of Coal and Fragmented Coal  
In the Mohr-Coulomb model, it is assumed that the cohesion, internal friction angle, tensile strength and shear expansion property will not change after yielding. In fact, previous studies have shown that coal shows a significant strain-softening characteristic after yielding [35; 36]. The strain-softening constitutive model can better simulate the roadway surrounding rock stability with the high mining-induced stress in deep mines. Users can define the cohesion, internal friction angle, tensile strength and shear expansion property after the peak of the coal. The yield criterion is the same as the Mohr-Coulomb model [24; 37]:  
\[\left\{\begin{array}{l}f^{s}=\sigma_{1}-\sigma_{3}N_{p}+2c\sqrt{N_{p}}\\ f^{t}=\sigma^{t}-\sigma_{3}\end{array}\right. \tag{7}\]  
The reference of each variable in the equation remains unchanged through this study.  
The failure process of coal under pressure shows the characteristics of strain softening, which was manifested by the decrease in cohesion and internal friction angle [38]. For the roadway surrounding rock in a fragmented rheological state, under the circumstance that the supplementary support is not provided in time, the roadway tends to soon enter into a state of instability. Grouting can cement the fractured surrounding rock into a whole body with the bearing capacity, which is equivalent to improving the cohesion and internal friction angle of the surrounding rock. The increased ratio of the two is determined by the rock type, failure state, grouting material [39] and grouting parameters. Existing studies suggest that the lower surrounding rock strength and the higher the fracture degree, the more the generalized rock strength will increase after grouted. At the water-cement ratio of 0.8, the strength of the grouting geocomposite can be restored to 31.43%, 57.11% and 85.51% of the strength of intact coal after being grouted with P.O42.5 Portland cement, ultra-fine cement and MasterRoc MP 364, which have 2.55, 2.15 and 5.28 times the residual strength of fragmented coal, respectively [40].  
In the FLAC3D, the strain-softening constitutive model allows users to express the internal friction angle, cohesion, and dilatancy angle of the frail material as a piecewise function of shear plastic strain [24], as shown in Figure 2. \(\epsilon^{ps}\) represents the damage degree of the rock behind the peak. With the increase in \(\epsilon^{ps}\), the strength of the roadway surrounding rock gradually decreases.  
The piecewise function of the cohesion and internal friction angle of the surrounding rock versus the shear plastic strain can be used in FLAC3D to simulate the improvement of the overall strength of the fractured surrounding rock by grouting. For the low-strength surrounding rocks, the post-peak strain softening mainly shows a decrease in cohesion. From the peak strength to the residual strength, the cohesion is reduced by about 50%, while the internal friction angle basically stays unchanged [33]. According to the research of [PERSON] and [PERSON] [38; 41], the coal strain-softening parameters in the numerical simulation are shown in Figure 2.