# Virtual Prototyping of Bulk Material Preparation Devices in Mining Using Multiphysics Simulations
## 2 Materials and Methods
### Contact Models  
Particles interact with each other through the so-called contact models that describe mutual acting forces and force moments. Changes in the state of the system over time are calculated numerically using the so-called explicit integration methods. The forces acting on the system may lead to static equilibrium (in which there is no particles movement) or cause the flow of bulk material (particle movement). If the contact model is assumed, describing the elastic interaction between particles, their mutual behavior uses the so-called \"soft contact\", in which a specific set stiffness of a particle in the normal direction describes the contact force of the connection between the adjacent particles. As a result, each particle may overlap with each other near the contact point, reflecting the compressibility of the material. In software tools based on DEM, it is possible to define various contact models that reflect the types of physical interactions between molecules [51]. The following contact models should be mentioned here:  
* Hertz-Mindlin (no slip)--the default model due to its accurate and efficient calculation of force values. In this model, the normal component of the force is based on [PERSON]'s contact theory [52]. The tangential force model is based on the work of [PERSON] [53; 54]. Both normal and tangential forces have damping components, where the damping coefficient is related to the coefficient of restitution, which is discussed in [55]. The tangential friction force follows the model of [PERSON]'s friction law [56]. Rolling friction is implemented as a contact-independent directional constant torque model [57]. * Hertz-Mindlin with JKR (Johnson-Kendall-Roberts)--cohesive contact model that takes into account the impact of Van der Waals forces in the contact zone and allows for modelling of highly adhesive systems such as dry powders or wet materials. In this model, implementation of the normal elastic contact force is based on the Johnson-Kendall-Roberts theory [58]. * Hysteretic spring contact--taking into account plastic deformations in the contact mechanics equations, as a result of which the particles behave elastically up to a predefined stress. Once this stress is exceeded, the particles behave as if they were undergoing plastic deformation. As a result, it is possible to obtain large contact surfaces between particles without excessive forces acting on them, which represents a compressible material. The calculation of the normal force is based on the Walton-Braun theory, presented in [59; 60]. * Damped linear spring compression force model, based on the work of [56]. A linear spring with stiffness \(k\) is connected in parallel with a damper with coefficient \(c\). * EEPA elastic-plastic adhesion model (Edinburgh Elasto-Plastic Adhesion Model)--takes into account the dependence of past interactions, which is crucial, and the characteristic behaviors of cohesive solids. The flow behavior and transfer characteristics of cohesive granular solids are strongly dependent on the prior consolidation stress in the given body. The contact model includes a nonlinear spring model with hysteresis, which takes into account the elastic-plastic contact deformation and the adhesion force component. It has been assumed that the peel force (adhesion) increases with an increase in the plastic contact area [61; 62]. Moreover, particles can be combined into groups to create more complex and larger granular structures; for example, rocks and stones. Taking into account the heterogeneity of grains (different particle sizes) and the presence of internal cracks (through the appropriate distribution of the parameters of permanent connections), the material crushing process can be simulated. This movement is the result of external forces and contact forces between adjacent particles and between particles and external space constraints, e.g., vessel walls.