## 2 Model Description  
In this section, all formulas are described in triaxial conditions, where \(\mathrm{p}^{\prime}=\left(\sigma_{1}^{\prime}+2\sigma_{3}^{\prime}\right)/3\) and \(\mathrm{q}=\sigma_{1}^{\prime}-\sigma_{3}^{\prime}\). Generally speaking, sand deformation could be caused by grain sliding and sand crushing. However, within the scope of the present work, it is supposed that the rupture of sand specimens is solely caused by the relative displacement between soil grains. Under this assumption, the yield criterion surface appears as an open wedge with \(\mathrm{p}^{\prime}\)-axis in the \(\mathrm{p}^{\prime}-\mathrm{q}\) space with its apex at the origin [4; 9]:  
\[\mathrm{f}=\mathrm{q}-\eta\mathrm{p}^{\prime}=\frac{\mathrm{q}}{\mathrm{p}^{ \prime}}-\eta=0 \tag{2}\]  
In the framework of elasto-plasticity theory and the range of small deformations and rotations, the mechanical behaviour of sand could be divided into elastic and plastic parts, namely:  
\[\dot{\epsilon_{\mathrm{q}}}=\dot{\epsilon}_{\mathrm{q}}^{\mathrm{e}}+\dot{ \epsilon}_{\mathrm{q}}^{\mathrm{P}} \tag{3}\]  
\[\dot{\epsilon_{\mathrm{v}}}=\dot{\epsilon}_{\mathrm{v}}^{\mathrm{e}}+\dot{ \epsilon}_{\mathrm{v}}^{\mathrm{P}} \tag{4}\]  
In the elastic part, the shear modulus \(\mathrm{G}\) is expressed by the following empirical equation [4; 10]:  
\[\mathrm{G}=\mathrm{G}_{0}\frac{(2.97-\mathrm{e})^{2}}{(1+\mathrm{e})}\sqrt{ \mathrm{p}^{\prime}\mathrm{P}_{\mathrm{a}}} \tag{5}\]  
where \(\mathrm{e}\) is the current void ratio, and \(\mathrm{p}_{\mathrm{a}}\) is the atmospheric pressure for normalization (\(\approx 100\) kPa). Note that the unit of the elastic shear modulus \(\mathrm{G}\) is given by the unit of mean effective stress \(\mathrm{p}^{\prime}\), which causes \(\mathrm{G}_{0}\) to be a dimensionless model constant. On the basis of elasticity theory, the bulk modulus \(\mathrm{k}\) is equal to  
\[\mathrm{k}=\mathrm{G}\frac{2(1+\mathrm{v})}{3(1-2\mathrm{v})} \tag{6}\]  
where \(\mathrm{v}\) is the [PERSON]'s ratio, which is considered constant and independent of effective confining pressure and the density index. By applying the loading index \(\mathrm{L}\)[11], the plastic deviatoric strain increment can be expressed as  
\[ \begin{split}\dot{\epsilon}_{\mathrm{q}}^{\mathrm{P}}=\mathrm{L} \cdot\frac{\partial\dot{\epsilon}}{\partial\mathrm{q}}=\mathrm{L}\cdot 1=\mathrm{L}=\frac{\frac{\partial\dot{\epsilon}}{ \partial\dot{\epsilon}}\dot{\mathrm{p}^{\prime}}+\frac{\partial\dot{\epsilon} }{\partial\dot{\epsilon}}\dot{\mathrm{q}}}{\mathrm{K}_{\mathrm{P}}}=\frac{ \dot{\mathrm{q}}-\eta\dot{\epsilon}^{\prime}}{\mathrm{K}_{\mathrm{P}}}\\ =\frac{\mathrm{p}^{\prime}\cdot\left(-\frac{\mathrm{q}}{(\dot{ \epsilon}^{\prime})^{2}}\dot{\epsilon}^{\prime}+\frac{\mathrm{q}}{\mathrm{p}^ {\prime}}\dot{\mathrm{q}}\right)}{\mathrm{K}_{\mathrm{P}}}=\frac{\mathrm{p}^{ \prime}\dot{\eta}}{\mathrm{K}_{\mathrm{P}}}\end{split}  \tag{7}\]  
where \(\dot{\eta}=\frac{\partial\eta}{\partial\dot{\epsilon}^{\prime}}\dot{\mathrm{p} ^{\prime}}+\frac{\partial\eta}{\partial\dot{\epsilon}}\dot{\mathrm{q}}=-\frac{ \mathrm{q}}{(\dot{\mathrm{p}^{\prime}})^{2}}\dot{\mathrm{p}^{\prime}}+\frac{ 1}{\dot{\mathrm{p}^{\prime}}}\dot{\mathrm{q}}\) and \(\mathrm{K}_{\mathrm{P}}\) is the plastic hardening modulus. As mentioned earlier, the key issue of describing the deformation of sand is the implantation of the state parameter into the dilatancy law \(\mathrm{d}=\dot{\epsilon}_{\mathrm{v}}^{\mathrm{P}}/\dot{\epsilon}_{\mathrm{q}}^ {\mathrm{P}}\)[9]:  
\[d=\dot{\epsilon}_{\mathrm{v}}^{\mathrm{P}}/\dot{\epsilon}_{\mathrm{q}}^{ \mathrm{P}}=\psi^{\mathrm{n}}-\frac{\eta}{\mathrm{M}} \tag{8}\]  
where \(\mathrm{n}\) is a model dimensionless constant, and \(\mathrm{M}\) is the stress ratio at critical state.