# Discrete Element Simple Shear Test Considering Particle Shape
## 2 Numerical Simulation
### Test Scheme and Variable Monitoring  
To investigate the effect of particle shape on the stress-strain relationship, nine groups of simple shear tests with various particle aspect ratios, numbered T01-T09, were modeled. The spherical particles were generated using the number T10 to prepare the samples as the control group, and the cell pressure was set at 200, 400, and 600 kPa. Table 2 shows the scheme's additional parameters. To calculate the deformation of the sample through the displacement, the laminar sidewall is modeled by rigid walls and the diameter is constant, the diameter of the sample is also a constant in the shear. The volumetric strain \(\varepsilon_{\mathrm{v}}\) can be written as Equation (1), \(\varepsilon_{\mathrm{i}}\) (i=-x, y, z) is the principal strain in the three-dimensional space, and \(h_{0}\) is the initial height of the sample. The stress can be obtained directly in PFC3D. \[\varepsilon_{\mathrm{v}}=\varepsilon_{x}+\varepsilon_{y}+\varepsilon_{z}= \varepsilon_{z}=\frac{\mathrm{dh}}{h_{0}} \tag{1}\]  
The microstructure of the sample is described by contact force and contact fabric. The contact force can be directly measured in PFC3D. The measurement of contact fabric is complicated, which can be quantified by fabric tensor and used to describe anisotropy. As in Oda [32], given the fabric tensor expression as shown in Equation (2), \(n\) is the unit vector of the contact normal on the integral interval \(\Omega=4\pi\), the physical meaning is shown in Figure 3a. \(n_{i}\) and \(n_{j}\) are the components of the unit vector \(n\) along three axes, \(E(n)\) is the probability density function of the spatial distribution of unit vector \(n\), and it satisfies \(\int_{\Omega}E(n)\mathrm{d}\Omega=1\). \[F_{ij}=\int_{\Omega}n_{i}n_{j}E(n)\mathrm{d}\Omega\quad\left(i,j=1,2,3\right) \tag{2}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Test Number** & **AR** & **Cell Pressure (kPa)** & **Clump Number** \\ \hline T01 & 1.3:1:1 & 200 & 16,536 \\ T02 & 1.6:1:1 & 200 & 22,088 \\ T03 & 1.9:1:1 & 200 & 27,310 \\ T04 & 1.3:1:1 & 400 & 16,536 \\ T05 & 1.6:1:1 & 400 & 22,088 \\ T06 & 1.9:1:1 & 400 & 27,310 \\ T07 & 1.3:1:1 & 600 & 16,536 \\ T08 & 1.6:1:1 & 600 & 22,088 \\ T09 & 1.9:1:1 & 600 & 27,310 \\ T10 & 1:1:1 & 200 & 1011 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Scheme of simple shear test. Figure 3: Contact normal unit vector diagram: (**a**) normal contact unit vector and (**b**) the angle between unit vector and coordinate axis. On the basis of Equation (2), [PERSON] [33] deduced the fabric expression of three-dimensional space (Equation (3)) which is used to describe fabric evolution in this paper, the derivation process is no longer described. Equation (3) describes the expression of orthogonal anisotropy, \(F_{1}\), \(F_{2}\), and \(F_{3}\) represent the fabric components in three orthogonal directions, it represents the difference of geomaterial properties in various directions. When \(F_{1}=F_{2}=F_{3}\), it shows that the material is isotropic and exhibits the same stress-strain characteristics when loaded in different directions. When the three components are different, the deformation in various directions is different under the same stress condition. To characterize this difference in theoretical calculation, different strain components are often obtained by combining with the fabric tensor, which reflects the anisotropy of geomaterials. Where \(a_{1}\) and \(a_{2}\) are the anisotropic amplitude parameters on the two orthogonal planes in the Cartesian coordinate system, and the values range from 0 to 1, the value of \(a_{i(i=1,2,3)}\) can be used to represent the difference of material properties in different directions and can measured through experiment. \(a\) is the anisotropic parameter of the transverse-isotropic fabric (Equation (4)), and the value ranges from 0 to 1. The fabric is isotropic when \(a=0\) and anisotropic when \(a\
eq 0\).