# Experimental Study on a Granular Material-Filled Lining in a High Ground-Stress Soft-Rock Tunnel
## 4 Study of the Mechanical Properties of Different Filling Materials
### Analysis of the Load Reduction Law of Compressible Granular Filling Materials
#### 4.2.2 Analysis of the Load Reduction Rate at Each Stage  
(1) Load reduction rate rapid improvement section (oa)  
Before compressive strain = 0.03, all filling materials reached a peak load reduction rate. At this stage, load reduction was the fastest. The order of load reduction rate of different granular fillings was ceramsite < 4 mm soil < 8 mm soil < coarse sand < fine sand.  
Accordingly, particle dislocation occurred in each particle-filled material at this stage. The filling material force skeleton was not completely formed. The upper load was dissipated due to particle dislocation, and the load reduction rate increased rapidly. Fine sand reached peak load reduction rate most quickly due to its small particle size and easy dislocation.  
(2) Steady-state peak load rate reduction section (ab)  
After the particle filling load reduction rate reached its peak value, when the compressive strain increased greatly, the load reduction rate remained stable. This means it had the maximum load reduction effect in steady state. In order to facilitate the analysis of the maximum reduction effect of steady state, the evaluation index S was introduced.  
\(S\) is the area enclosed by the steady-state peak load reduction rate and the \(X\)-axis compressive strain, which is used to represent the ability of the particle filling to retain the energy generated by the compressive strain to the greatest extent, that is, the ability of the particle to absorb the maximum elastic potential energy. The larger the \(S\), the better the particle's ability to absorb maximum elastic potential energy, and vice versa. Formula (6) shows the evaluation index \(S\).  
\[\begin{split}& S=\int_{a}^{b}(\varepsilon_{b}-\varepsilon_{a}) \eta_{\max}dx=\int_{a}^{b}\Delta\varepsilon_{ba}\Big{(}\frac{\Delta E}{F_{i}} \Big{)}_{\max}dx\\ &=\int_{a}^{b}\frac{\Delta\eta_{ba}}{R_{0}}\frac{\Delta E}{F_{i}} dx=\int_{a}^{b}\Big{(}\frac{\Delta E}{E}\Big{)}_{\max}dx\end{split} \tag{6}\]  
In the formula, \(\varepsilon_{a}\) and \(\varepsilon_{b}\) represent the compressive strain at points a and b, \(\Delta\varepsilon_{ba}\) represents the difference between the compressive strain at points a and b. \(\eta_{\max}\) represents  
Figure 11: Compressive strain–load shedding rate law.  
the peak value (maximum value) of the reduction load rate; \(\Delta E\) represents the elastic potential energy absorbed by the particles, and \(E\) represents the total elastic potential energy of the granular materials.  
Table 2 shows that 8 mm soil had the strongest ability to absorb elastic potential energy. The reason is that 8 mm soil particles are larger and the internal pores are more numerous. At this stage, the force skeleton is basically formed. The load reduction rate remains unchanged after reaching the peak, so the ability to absorb elastic potential energy is stronger. The order of the granular filling's ability to retain elastic potential energy from strong to weak was 8 mm soil > 4 mm soil > coarse sand > fine sand > ceramsite.  
(3) Drop section of the load reduction rate (bc)  
At this stage, the load reduction rate of the particles decreases, i.e., the ability of the particles to reduce the load diminishes until the load reduction rate is minimized. At this time, the compressive strain of the particles reaches maximum, the porosity is the lowest, and the particles are almost incompressible.  
In summary, fine sand had the fastest speed to reach the peak load reduction rate. In the early stages of loading, fine sand had a better load reduction effect under the small compressive strain. The 8 mm soil had a strong ability to absorb elastic potential energy. When the compressive strain changed greatly, 8 mm soil had a significant load reduction effect, and the overall load reduction effect was the most efficient.