# The Size Effect of Shear Bands in Dense Sands--A Discrete Element Analysis
###### Abstract  
The localization of deformation in shear bands is a fundamental phenomenon in granular materials like soil. In this study, we focus on the characteristics of shear bands, particularly the size effect, by implementing biaxial discrete element method (DEM) modeling. Firstly, we describe the establishment of the biaxial experimental model with dense sands. Then, we implement analyses of specimens with different sizes and find that there is a clear size effect in the stress-strain curve after the peak strength point, and there is less of a size effect in the angle of the shear band; the angle is consistent with [PERSON]'s theory. Finally, the reason for the size effect is analyzed using the width of the shear band and the porosity inside the shear band. As the specimen size increases, the ratio between the shear band area and the whole specimen decreases. This effect reduces as the isotropic confining stress increases. The difference in the proportion of the shear band area mainly causes the size effect that affects the specimen deformation characteristics. We also find that with the increase in isotropic confining stress, the type of shear band gradually changes from cross-type to single-type. Our study provides valuable insights into understanding the behavior of granular materials. shear band; size effect; biaxial test; dense sand; discrete element method +
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however, a single well-defined shear band rarely develops [6]. Shear band formation and characteristics are influenced by the density, the isotropic confining stress, the particle size, the specimen shape, etc. [PERSON] and [PERSON] [6] conducted biaxial experiments and found that the failure of the specimen's mode can be characterized by two distinct and opposite shear bands. Also, they suggested that measured dilatancy angles increase as the sand angularities and sizes increase. [PERSON] et al. [8] found that grain size distribution can greatly affect shear banding characteristics; there is no simple relationship between the characteristics of localization and the mean grain size or degree of uniformity. [PERSON] and [PERSON] [9] applied true triaxial tests and revealed that the peak failure is caused by shear banding in this middle range of \(b\) values. [PERSON] et al. [10] implemented a series of biaxial tests and concluded that shear bands would give rise to a large void ratio, low coordination number, and high particle sliding and rotation compared with other parts of the specimen. The detection of shear bands' characteristics is not easy using conventional sensors. Accordingly, researchers have sought help from non-destructive techniques (such as digital image correlation (DIC) [2] and X-ray tomography [11]) and numerical simulation techniques (such as the finite element method (FEM) [12; 13; 14; 15] and discrete element method (DEM) [16; 10; 17; 18]). In general, numerical simulation can significantly reduce the costs associated with physical experiments and can be easily adjusted to test different variables and conditions [19]. In the simulation of shear bands, the FEM usually has a lower computational cost. However, the FEM needs some mathematical description to deal with inhomogeneities [20]; otherwise, the solution can depend largely on the mesh used in the simulation. To obtain a solution for this problem, often, an internal length scale is added to enhance the continuum mechanics formulation of the constitutive relationship. Typical theories include the Cosserat theory [21], strain gradient theory [22], etc. However, since the internal length scale may not correspond to the size of the grain, it is hard to calibrate. Furthermore, these methods are not robust enough to be employed to solve problems of engineering interest [20]. In comparison, the DEM naturally handles large deformations, movement, and fragmentation of materials, making it ideal for simulations of shear bands, in which a significant strain concentration occurs.