## 3 Results  
### Test Results and Analyses of Small-Sized Granular Materials  
#### 3.1.1 Stress-Strain Characterization  
Figure 5 presents the stress-strain curves of small-sized glass bead (2-6 mm) specimens under four different levels of confining pressures.  
As shown in Figure 5, for granular materials with particle diameters ranging from 2 to 6 mm, the deviatoric stress (\(q\)) increases obviously with the increase of confining pressure (\(\sigma_{3}\)). Under low confining pressure (100 kPa), all specimens exhibit strain-hardening behavior without a pronounced peak shear stress (\(q_{\text{f}}\)). The axial strain (\(\epsilon_{1}\)) corresponding to \(q_{\text{f}}\) ranges from 1% to 3%, and the deviatoric stress tends to stabilize after the strain exceeds 3%. At higher confining pressures (200, 300, and 400 kPa), the stress-strain behavior varies with particle size, generally exhibiting an increasing trend of strain-softening as confinement increases. This trend is more pronounced for larger particle sizes. The axial strain at the \(q_{\text{f}}\) under higher confining pressures consistently remains within the range of 2% to 4%.  
Figure 5 illustrates that the shear stress values of glass bead specimen of varying sizes exhibit oscillations characterized by differing amplitudes. All curves begin to show  
Figure 5: Stress-strain curves of single-particle size specimens with different particle sizes under various confining pressures.  
shear stress oscillations prior to reaching the peak shear stress and persist until the end of the experiment. For specimens with the same particle size, the amplitude of these oscillations increases with rising confining pressure. Correspondingly, under the same confining pressure conditions, larger particle sizes are correlated with more pronounced oscillations in the strain curves.  
The observed stress-strain oscillation phenomenon can be attributed to the evolving inter-particle kinematics during shearing. As shearing progresses, the specimen's porosity decreases until a minimum is reached. Subsequently, changes in the relative positions of particles include sliding dislocation, particle rolling, and particle rearrangement. In this process, the particles are influenced not only by inter-particle frictional and interlocking forces, but also by the energy required for reorientation. These particle rearrangements lead to the oscillation in the macroscopic shear stress \(q\) value, a phenomenon often referred to as stick-slip behavior [33; 36; 37; 44]. Consequently, the frictional resistance experienced by particles increases, and the potential energy required for particles to slide or roll also increases. Macroscopically, this is manifested as more pronounced stick-slip phenomenon in the stress-strain curves.  
Figure 6 presents the peak shear stress (\(q_{\text{f}}\)) as a function of confining pressure (\(\sigma_{3}\)) for each particle size (\(d\)), derived from the stress-strain curves shown in Figure 5a-e. A positive correlation is observed between \(q_{\text{f}}\) and \(d\). The slope of the \(q_{\text{f}}\) vs. \(\sigma_{3}\) relationship varies slightly with \(d\), with the increase in \(q_{\text{f}}\) under high confining pressure being significantly greater than that observed under low confining pressure. This highlights the significant influence of particle size on the mechanical behavior of granular materials.