## 3 Results
### Test Results and Analyses of Large-Sized Granular Materials
#### 3.2.3 Test Results and Analyses of Large-Sized Granular Materials
\begin{split} B_{t}&=S_{1}-S_{0}=\int_{d_{m}}^{d_{M}}[ F_{i}(d)-F_{0}(d)]d^{-1}\mathrm{d}d\\ B_{p}&=S_{2}-S_{0}=\int_{d_{m}}^{d_{M}}[F_{u}(d)-F_{0}(d)] d^{-1}\mathrm{d}d\\ \end{split}  \tag{4}\]  
\[F(d)=F(\Delta<d)=\int_{d_{m}}^{d}p(\Delta)\mathrm{d}\Delta \tag{5}\]  
In the equation, \(F(d)\) with subscripts \(i\), \(0\), and \(u\) correspond to the gradation curves of the current state, the initial state before experiment, and the limit state, respectively. \(\Delta\) is particles with diameters smaller than \(d\); \(P(\Delta)\) is the particle size distribution; \(P(\Delta)\mathrm{d}\Delta\) represents the probability that particles within the fraction \(\mathrm{d}\Delta\); \(d_{\mathrm{m}}\) and \(d_{\mathrm{M}}\) are the minimum and maximum particle diameters, respectively. The variation of \(B_{\mathrm{r}}\) with \(d\) in large-sized granular materials during the triaxial experiment at different confining pressures is shown in Figure 13.  
Figure 13 clearly demonstrates that for large-sized granular materials with particle diameters ranging from 15 mm to 30 mm, the particle breakage rate (\(B_{\mathrm{r}}\)) increases significantly with increasing confining pressure (\(\sigma_{3}\)) for specimens of the same particle size. Furthermore, the test results across four levels of confining pressure indicate that \(B_{\mathrm{r}}\) initially increases and then decreases with increasing particle diameter (d). No particle breakage was observed at a confining pressure of 100 kPa. Under higher confining pressure, the trend of \(B_{\mathrm{r}}\) with particle diameter is as follows: at high confining pressures (ranging from 300 kPa to 400 kPa), \(B_{\mathrm{r}}\) begins to decrease when \(d\) reaches 25 mm; at low confining pressure (200 kPa), \(B_{\mathrm{r}}\) also starts to decrease when \(d\) reaches 30 mm. The intrinsic deformation mechanisms of particle breakage are governed by a combination of particle sliding, rotation, abrasion, and breakage, all of which are energy-dissipating behaviors. In triaxial tests, as the axial strain of the specimen continuously increases, this is accompanied by particle breakage. \(B_{\mathrm{r}}\) is influenced by factors such as particle size and confining pressure. The breakage of particles inevitably leads to a rearrangement of the material structure, changes in inter-particle contact modes, and alterations in the morphology of contact surfaces. These changes result in a transformation of the material's strain characteristics. This helps to elucidate the intrinsic reasons for the transition of the strain curves for the 25 mm and 30 mm glass beads towards a softening characteristic under high confining pressures (300 kPa and 400 kPa), as shown in Figure 8.  
As particle size \(d\) increases from 15 to 20 mm, the breakage rate (\(B_{\mathrm{r}}\)) and internal friction angle (\(\varphi\)) both show the largest increase. When \(d\) grows from 20 to 25 mm, \(B_{\mathrm{r}}\) increase slows down compared to the 15-20 mm range, and \(\varphi\) barely changes with \(d\). However, as \(d\) rises from 25 to 30 mm, \(B_{\mathrm{r}}\) decreases while \(\varphi\) increases. This indicates  
Figure 13: Variation of \(B_{\mathrm{r}}\) with \(d\) in triaxial test of large-size granular materials under different \(\sigma_{3}\). that structural changes from particle breakage have a minimal impact on the strength of mono-sized glass-bead specimens. Instead, the variation in \(\varphi\) with size in large-particle specimens is mainly attributed to particle size itself. Analysis of the Influence and Mechanisms of Particle Size on the Strength Behavior of Cohesionless Soil