## 3 Discharge Simulation  
### Constitutive Model  
The accuracy of the analysis depends on the constitutive model used to describe the behavior of the target material in the continuum approach. Therefore, selecting an appropriate constitutive model is the first step in interpreting the continuum approach. Several models for determining the composition of materials have been proposed: classical plastic theory [24; 25; 26], pole elastoplastic theory [27], and others. However, this analysis is considered to be suitable for expressing the properties of granular materials in the classical elastic-plastic model because micromechanics such as shear banding are not as important as bulk scale [28; 29].  
Thus, in this study, the mechanical behavior of the discharge of the material stored in the silo was simulated using the traditional Mohr-Coulomb elastoplastic model. This model comprises a linear isotropic elastic law that defines material behavior at small loads, a yield criterion that determines the transition of material to yield, and an unrelated plastic flow potential that determines the direction of flow after material yield. The exact description of the Mohr-Coulomb elastoplastic model can be found in [PERSON] 2018 [30], and it is briefly described below.  
The elastic behavior is determined by [PERSON]'s modulus E and [PERSON]'s ratio \(\
u\) in the law of isotropic elasticity as  
\[\sigma_{ij}\ =\ D_{ijkl}^{el}\epsilon_{kl}^{el} \tag{15}\]  
where \(\sigma_{ij}\) is the total stress, \(\epsilon_{kl}^{el}\) is the elastic strain, and \(D_{ijkl}^{el}\) is the fourth-order elastic tensor.  
The yield condition is  
\[R_{mc}q-p\tan\phi-c\ =\ 0 \tag{16}\]  
where  
\[R_{mc}=\frac{1}{\sqrt{3}\cos\varphi}\sin\Bigl{(}\Theta+\frac{\pi}{3}\Bigr{)}+ \frac{1}{3}\cos\Bigl{(}+\frac{\pi}{3}\Bigr{)}\tan\varphi \tag{17}\]  
Furthermore, \(p=-\frac{1}{3}trace\Bigl{(}\sigma_{ij}\Bigr{)}\) is the first invariant value of the stress representing the equivalent pressure, \(\mathrm{q}=\sqrt{\frac{3}{2}\Bigl{(}S_{ij}S_{ij}\Bigr{)}}\) is the Mises equivalent stress, and \(S_{ij}\) is the deviatoric stress tensor; \(\varphi\) and \(\mathrm{c}\) are the internal friction angle and the cohesive force of the material, respectively; \(\Theta\) is the deflection angle defined by \(\cos(3\Theta)=\left(r/q\right)^{3}\), where \(r\) is the third invariant of the bias stress that is defined as \(r=\left(\frac{9}{2}\Bigl{(}S_{ij}S_{jk}S_{ki}\Bigr{)}\right)^{1/3}\). The flow potential \(\mathrm{G}\) is selected based on the hyperbolic function in the meridian stress plane and the smooth elliptic function in the deflection stress plane. The mathematical expression is  
\[G=\sqrt{\left(\epsilon c|_{0}\tan\psi\right)^{2}+\left(R_{mm}q\right)^{2}}-p \tan\psi \tag{18}\]  
Figure 6: Comparison of tensile forces acting on the silo hotspot detector (Experimental vs. Analytical).  
where  
\[R_{mu\
u}=\frac{4\big{(}1-e^{2}\big{)}\cos^{2}\Theta+\big{(}2e-1\big{)}^{2}}{2(1-e ^{2})\cos\Theta+(2e-1)\sqrt{4(1-e^{2})\cos^{2}\Theta+5e^{2}-4e}}\frac{3-\sin \varphi}{6\cos\varphi} \tag{19}\]  
where \(\psi\) is the dilatancy angle of the material, \(c|_{0}\) is the initial cohesive yield stress, and \(\epsilon\) is a variable that characterizes eccentricity. In the analysis in this study, a fixed value of \(\epsilon=0.1\) was used. The \(e\) value was determined by the internal friction angle as \(\mathrm{e}=(3-\sin\varphi)/(3+\sin\varphi)\).  
The volumetric expansion of the material can be controlled by the expansion angle \(\psi\). Since the commonly used material has small expandability, the \(\psi\) value was set to \(5^{\circ}\) in this study. Furthermore, because it was assumed that there was no cohesive force, the cohesive force \(\mathrm{c}=0\). Other material property values are listed in Table 3.