# The Size Effect of Shear Bands in Dense Sands--A Discrete Element Analysis
## 4 Explanations of the Size Effect
### Explanations Based on the Characteristics of the Shear Band  
During the test, it was observed that deformation was concentrated at shear bands, as plotted in Figure 7. It is noted that as the specimen size increased, the width of the basic shear band roughly remained unchanged. Accordingly, the proportion of the shear band decreased with the specimen size. That is to say, a larger specimen is more likely to fail because of a narrower shear band.  
For a quantitative evaluation, the proportion of the shear band area (i.e., the ratio of the shear band to the initial specimen) against different isotropic confining stresses is plotted in Figure 8. The results show that:  
Figure 7: The specimen tends to give a cross-type shear band at a lower isotropic confining stress and a single-type shear band at a larger isotropic confining stress. (**a**) Confining stress 100 kPa; (**b**) confining stress 500 kPa; (**c**) confining stress 900 kPa.  
* The area ratio gradually decreases as specimen size increases, although the width of the shear band increases with the specimen size.
* The degree of the decrease is related to the level of isotropic confining stress. At a higher isotropic confining stress level, the proportion of the shear band area of specimens tends to be the same. In detail, it can be seen that the width of the shear band is about 8-24 times the particle size, and as the isotropic confining stress increases, the width of the shear band gradually decreases and eventually stabilizes at a certain level.  
The evolution of the shear band is also investigated. The evolutions of the displacement field of the smallest specimen (S) and the largest specimen (XL) are plotted in Figure 9. It is seen that as the shear band was generated, a relatively loose core area with high porosity formed at the center of the specimen. As the axial strain continued to increase, the shear band gradually developed towards both sides, the core area gradually disappeared, and the deformation became more concentrated in a narrow band area. Finally, stable shear bands were observed. The angles of the shear band of the two specimens were basically the same. The development of the shear band resulted in a sliding surface, and the particle structure of the soil near the sliding surface rapidly arranged and changed.  
Figure 8: The statistical results of the width of shear bands. (**a**) The width of the shear band; (**b**) the area ratio of the shear band.  
Figure 9: The evolution process of the shear band at the isotropic confining stress of 900 kPa: (**a**) specimen S; (**b**) specimen XL.  
As to the influence of the isotropic confining stress, both the cross-type and the single-type shear bands can be observed in Figure 7 (this is also seen in, e.g., Alshibli and Sture [6]). One important finding is that the specimen tends to give cross-type shear bands at a low isotropic confining stress level. This could be a possible reason that the size effect is influenced by the magnitude of isotropic confining stress.  
The influence of isotropic confining stress on the form of shear bands is analyzed thereby. During the shearing process, the specimen was first compressed and compacted. The soil near the loading plate gradually formed a rigid core. For a lower-level isotropic confining stress, the lateral isotropic confining stress cannot fully limit the deformation of the soil on both sides. The core area compressed the soil on both sides, causing the soil on both sides to detach from the core area. For a higher isotropic confining stress, the soil on both sides tightly adhered to the wedge-shaped rigid core area, limiting lateral deformation and increasing soil strength. As the shear process progressed, the internal soil of the specimen reached the ultimate stress state, and the upper and lower parts of the specimen experienced relative horizontal displacement. Subsequently, relative sliding occurred along the weak failure surface with a certain angle, resulting in a single-type shear band.