# Macro-Meso Mechanical Behavior of Loose Sand under Multi-Directional Cyclic Simple Shear Tests
## 3 Simulation Results
### Fabric  
The plastic flow deformation of sand under shearing will lead to the adjustment of particle position by sliding and rolling, which provides better support for external loads and leads to anisotropy in sand. In recent years, great progress has been made in experimental techniques for detecting particle size information, such as X-ray tomography and MRI imaging [27]. However, there is no effective and accurate method to quantify the anisotropic evolution in a shearing process on a meso-scale. Numerical simulation provides a meso-scale tool to study anisotropy. However, most studies used two-dimensional numerical models [28], or particle contact parameters without calibration. In this study, a three-dimensional DEM model was used to quantify the anisotropy of samples by analyzing the contact-based fabric tensor. The contact-based fabric tensor is a mathematical statistic of the direction of each contact unit normal vector, which can quantify the structural anisotropy. In this study, the contact-based fabric tensor (\(F_{ij}\)) proposed by [PERSON] [29] and [PERSON] [30] was used to quantify the direction of contact normal:  
\[F_{ij}=\frac{1}{N_{C}}\sum_{k=1}^{N_{C}}n_{i}^{(k)}n_{j}^{(k)}=\int_{V}E(V)n_{ i}n_{j}dV \tag{3}\]  
\[= \begin{pmatrix}F_{\mathrm{xx}}&F_{\mathrm{yy}}&F_{\mathrm{zz}}\\ F_{\mathrm{yx}}&F_{\mathrm{yy}}&F_{\mathrm{yz}}\\ F_{\mathrm{xx}}&F_{\mathrm{zy}}&F_{\mathrm{zz}}\end{pmatrix}  \tag{4}\]  
where \(n_{i}\) is the \(i\) component of the contact normal, \(V\) is the representative elemental volume, and \(E(V)\) is the directional distribution function of contact normal [31], which is usually expressed by its second-order Fourier expansion:  
\[E(V)=\frac{1}{4\pi}\big{[}1+a_{ij}n_{i}n_{j}\big{]} \tag{5}\]  
where \(a_{ij}\) is the second-order anisotropic tensor. Substituting Equation (5) with Equation (3), the expression of the anisotropic tensor \(a_{ij}\) can be obtained:  
\[a_{ij}=\frac{15}{2}\bigg{(}F_{ij}-\frac{1}{3}\delta_{ij}\bigg{)} \tag{6}\]  
where \(\delta_{ij}\) is the Kronecker constant [32]. If \(a_{ij}>0\), it means that the density of contact in the sample is higher than that in the isotropic sample. Since the anisotropic tensor \(a_{ij}\) is a deviatoric tensor, in order to avoid the impact of changes in axis, its deviatoric invariant is usually used to quantify the degree of the anisotropy of samples [33]:  
\[a_{c}=\sqrt{\frac{3}{2}a_{ij}a_{ij}} \tag{7}\]  
In this DEM simulation, the fabric tensor of a sample after consolidation was:  
\[F_{ij},0= \begin{pmatrix}0.331&-0.002&-0.012\\ -0.002&0.334&0.002\\ -0.012&0.002&0.334\end{pmatrix}  \tag{8}\]  
In Equation (8) \(F_{\mathrm{xx}}\), \(F_{\mathrm{yy}}\), and \(F_{\mathrm{zz}}\) were close to \(1/3\), while the other fabric tensors were close to \(0\). Figure 20 showed the evolution of components of fabric tensor in the SL-0 and FIG8-0. It can be observed that in uni-directional cyclic simple shear tests, the diagonal components \(F_{\mathrm{xx}}\), \(F_{\mathrm{yy}}\), and \(F_{\mathrm{zz}}\) were close to \(1/3\), while the non-diagonal components \(F_{\mathrm{xy}}\), \(F_{\mathrm{xz}}\), andwere close to 0. But \(F_{\rm xz}\) and \(F_{\rm yz}\) gradually increased during shearing, while the change in \(F_{\rm xy}\) can be ignored. By integrating Equations (4), (7), and (8), one can obtain the deviatoric invariant of fabric anisotropy tensors, also called anisotropy invariant, which can quantify the anisotropy of the contact network, which is:  
\[ \begin{array}{ll}a_{c}&=\frac{15}{2}\sqrt{\frac{3}{2}\left(2F_{\rm xz}^{2}+2 F_{\rm yz}^{2}+2F_{\rm xy}^{2}+\left(F_{\rm xx}-\frac{1}{3}\right)^{2}+\left(F_{ \rm yy}-\frac{1}{3}\right)^{2}+\left(F_{\rm zz}-\frac{1}{3}\right)^{2}\right) }\\ &=\frac{15\sqrt{3}}{2}\sqrt{F_{\rm xz}^{2}+F_{\rm yz}^{2}}\end{array}  \tag{9}\]  
Figure 21 showed the evolution of \(F_{\rm xy}\), \(F_{\rm xz}\), and \(F_{\rm yz}\) with the number of cycles in representative samples.