2.2.2. Particle Motion

The governing equations of particle motion are the same as those described in Section 2.1.2. Equation (12) can be extended to include particle–particle interaction force (*FDEM*)

$$m\_p \frac{d\upsilon}{dt} = F\_d + F\_b + F\_\mathcal{S} + F\_{\mathcal{P}\mathcal{S}} + F\_{\text{vm}} + F\_{mI} + F\_{\text{DEM}}.\tag{21}$$

Particle–particle interaction is modelled according to DEM. Deformations of particles during collisions are accounted for by introducing the collision overlap between particles instead of collision deformations, as shown in Figure 1. Contact between colliding particles is modelled with the Hertz–Mindlin model [27,28], whereby the particle interaction is represented by a spring–dashpot system, as shown in Figure 2.

The particle interaction force is split into the normal and tangential components, where normal is defined at the point of contact between particles.

$$F\_{\rm DEM} = F\_{\rm DEM,n} \mathfrak{n} + F\_{\rm DEM,t} \mathfrak{t}.\tag{22}$$

The normal and tangential components of the interaction force are expressed in terms of the spring–dashpot system as the spring force and the damping force.

**Figure 1.** Particle overlap definition.

**Figure 2.** Hertz–Mindlin spring and dashpot system.

$$F\_{\rm DEM,n} = F\_{\rm n,spring} + F\_{\rm n, damping} = \frac{4}{3} E^\ast \sqrt{R^\ast} \delta\_{\text{n}}^{\frac{3}{2}} - 2 \sqrt{\frac{5}{6}} \beta \sqrt{S\_n m^\ast} v\_{rel, n\prime} \tag{23}$$

$$F\_{DEM,t} = F\_{t,spring} + F\_{t,damping} = 8G^\* \sqrt{R^\* \delta\_n} \delta\_t - 2\sqrt{\frac{5}{6}} \beta \sqrt{S\_l m^\*} v\_{rel,t'} \tag{24}$$

where *E*\* is the equivalent Young's modulus, *G*\* is the equivalent shear modulus, *R*\* is the equivalent radius, *δ<sup>n</sup>* is the overlap in the normal direction, *δ<sup>t</sup>* is the overlap in the tangential direction, *β* is a constant, *Sn* is the normal stiffness, *St* is the tangential stiffness, *m*\* is the equivalent mass, *vrel*,*<sup>n</sup>* is the normal component of the relative velocity and *vrel*,*<sup>t</sup>* is the tangential component of the relative velocity. These are defined as

$$\frac{1}{E^{\star}} = \frac{1-\nu\_1^2}{E\_1} + \frac{1-\nu\_2^2}{E\_2},\tag{25}$$

$$\frac{1}{G^\*} = \frac{2(1+\nu\_1)\left(1-\nu\_1^2\right)}{E\_1} + \frac{2(1+\nu\_2)\left(1-\nu\_2^2\right)}{E\_2},\tag{26}$$

$$\frac{1}{R^\*} = \frac{R\_1 + R\_2}{R\_1 R\_2},\tag{27}$$

$$\beta = \frac{-\ln \varepsilon}{\sqrt{\ln^2 \varepsilon + \pi^2}},\tag{28}$$

$$S\_n = 2E^\star \sqrt{R^\star \delta\_n} \tag{29}$$

$$S\_{\mathbb{H}} = 8G^\* \sqrt{R^\* \delta\_{\mathbb{H}}} \tag{30}$$

$$m\_1 m\_2$$

$$m^\* = \frac{m\_1 m\_2}{m\_1 + m\_2}'\tag{31}$$

$$
\upsilon\_{rel,n} = \upsilon\_{1,n} + \upsilon\_{2,n} \tag{32}
$$

$$
v\_{rel,t} = v\_{1,t} + v\_{2,t} \tag{33}$$

where *ν* is the Poisson ratio, *E* is the Young's modulus, *R* is the particle radius, *e* is the coefficient of restitution, *m* is the particle mass, *vn* is the particle normal velocity and *vt* is the particle tangential velocity. Subscripts 1 and 2 refer to particle 1 and particle 2, respectively, in collision, as shown in Figure 1. Normal and tangential velocities of particles are observed according to the local collision coordinate system, as shown in Figure 1.
