## 2 Mathematical Model
### Coarse Grain DEM Model for Solid Phase
#### 2.1.2 Contact Force  
In case of a collision between two particles (treated as soft spheres), the contact force can be evaluated using a linear spring dashpot model, supplemented with a frictional slider [8]. Consider two colliding partners \(i\) and \(j\), the contact force \(\mathbf{F}_{cont,ij}\) is divided into a normal component \(\mathbf{F}_{cont_{n},ij}\) and a tangential component \(\mathbf{F}_{cont_{i},ij}\):  
\[\mathbf{F}_{cont_{n},ij}=-\kappa_{n}\delta_{n,ij}\mathbf{n}_{ij}-\eta_{n} \mathbf{v}_{n,ij} \tag{7}\]  
\[\mathbf{F}_{cont_{i},ij}=\left\{\begin{array}{ll}-\kappa_{t}\delta_{t,ij} \mathbf{t}_{ij}-\eta_{t}\mathbf{v}_{t,ij}&for\quad\left|\mathbf{F}_{cont_{i}, ij}\right|<\mu_{p}\mathbf{F}_{cont_{n},ij}\\ -\mu_{p}\left|\mathbf{F}_{cont_{n},ij}\right|\mathbf{t}_{ij}&for\quad\left| \mathbf{F}_{cont_{i},j}\right|\geq\mu_{p}\mathbf{F}_{cont_{n},ij}\end{array}\right. \tag{8}\]  
Here \(\kappa_{n}\) is the normal spring constant (contact stiffness), \(\delta_{n,ij}\) is the normal overlap between particles \(i\) and \(j\), \(\mathbf{n}_{ij}\) is the unit normal vector pointing from particle \(i\) to collision partner \(j\), \(\eta_{n}\) is the normal damping coefficient, \(\mathbf{v}_{n,ij}=(\mathbf{v}_{ij}\cdot\mathbf{n}_{ij})\mathbf{n}_{ij}\) is the relative normal velocity of particle \(i\) and collision partner \(j\), \(\kappa_{t}\) is the tangential spring constant, \(\delta_{t,ij}\) is the tangential overlap between particle \(i\) and collision partner \(j\), \(\mathbf{t}_{ij}\) is the unit tangential vector evaluated by \(\mathbf{v}_{t,ij}/\left|\mathbf{v}_{t,ij}\right|\), \(\eta_{t}\) is the tangential damping coefficient, \(\mathbf{v}_{t,ij}=\mathbf{v}_{ij}-\mathbf{v}_{n,ij}\) is the relative tangential velocity of particle \(i\) and collision partner \(j\), and \(\mu_{p}\) is the friction coefficient. The relative velocity at the contact point \(\mathbf{v}_{ij}\) is defined as follows:  
\[\mathbf{v}_{ij}=\mathbf{v}_{i}-\mathbf{v}_{j}-r_{p}(\boldsymbol{\omega}_{i}+ \boldsymbol{\omega}_{j})\times\mathbf{n}_{ij} \tag{9}\]  
where \(r_{p}\) and \(\boldsymbol{\omega}\) are the radius and angular velocity of the colliding bodies, respectively. The spring constant \(\kappa\) depends on the geometry and the elastic properties of the colliding bodies. We have:  
\[\kappa_{n}=\frac{\sqrt{2r_{p}}E_{p}}{3(1-\sigma_{p}^{2})} \tag{10a}\] \[\kappa_{t}=\frac{2\sqrt{2r_{p}}G_{p}}{2-\sigma_{p}}\sqrt{\delta_{n,ij}} \tag{10b}\]  
for the collisions between particles, and:  
\[\kappa_{n}=\frac{4\sqrt{r_{p}}/3}{(1-\sigma_{p}^{2})/E_{p}+(1-\sigma_{w}^{2}) /E_{w}} \tag{11a}\] \[\kappa_{t}=\frac{8\sqrt{r_{p}}G_{p}}{2-\sigma_{p}}\sqrt{\delta_{n,ij}} \tag{11b}\]  
for the collisions between particles and solid walls. Here \(E_{p}\) and \(\sigma_{p}\) are respectively the elasticity modulus and the Poisson ratio of the collisions between particles, and \(\sigma_{w}\) and \(E_{w}\) are those of the collisions between particles and solid walls. The equivalent shear modulus of the colliding bodies \(G_{p}\) in Equations (10) and (11) is given by:  
\[G_{p}=\frac{E_{p}}{2(1+\sigma_{p})} \tag{12}\]  
The damping coefficient \(\eta\) can be obtained by:  
\[\eta_{n}=2\sqrt{\overline{m}_{n}\kappa_{n}} \tag{13a}\] \[\eta_{t}=2\sqrt{\overline{m}_{t}\kappa_{t}} \tag{13b}\]  
where \(\overline{m}\) is the effective mass of the contact bodies evaluated by:  
\[m_{n}=\frac{m_{i}m_{j}}{m_{i}+m_{j}} \tag{14a}\]\[m_{t}=\frac{2}{7}\bigg{(}\frac{m_{i}m_{j}}{m_{i}+m_{j}}\bigg{)} \tag{14b}\]  
for the collisions between particles, and:  
\[m_{n}=m_{i} \tag{15a}\] \[m_{t}=\frac{2}{7}m_{i} \tag{15b}\]  
for the collisions between particles and solid walls. Now the contact force \(\mathbf{F}_{cont,i}\) acting on particle \(i\) can be expressed as the summation about all the particles and solid walls who collide with particle \(i\), i.e.,:  
\[\mathbf{F}_{cont,i}=\sum_{j}\mathbf{F}_{cont,ij}=\sum_{j}\Bigl{(}\mathbf{F}_ {cont_{n},ij}+\mathbf{F}_{cont_{p},ij}\Bigr{)} \tag{16}\]