# The Hertz-Mindlin with JKR Version 2 Model

For very small particles (smaller than 100 µm), van der Waals forces become significant and particles tend to stick to each other. The Johnson-Kendal-Roberts theory of adhesion (also known as the JKR Model) is typically used for calculating the contact forces acting on elastic and adhesive particles and assumes that the attractive forces are short range. To date, a large number of studies on many different particulate systems have been reported in the literature using the JKR model. However, most of them use a simplified version of the JKR model that depends on the surface energy of the particles involved in the contact and also for cases where materials of the same type are involved too. The Hertz-Mindlin with JKR Version 2 (JKR V2) model implemented in EDEM captures the behavior of multiple materials and uses a more accurate implementation of the JKR theory. It calculates the additional work required to break the contact (adhesion) after physical detachment of particles, thus making it applicable to contacts involving very small particles.

The model is suitable for elastic and adhesive systems and the force-overlap response of the JKR model is shown in the following figure. It states that when two elastic and adhesive spheres approach each other, the force acting on the spheres is zero (from A to B). The DEM contact between these two spheres is established when they physically come into contact (B), and the normal contact force immediately drops to *8/9 f _{c}*, where f

_{c}is the pull-off fore due to the presence of the van der Waals attractive forces (C). Upon loading the two spheres, the normal contact force follows the trend from C to D. During the recovery stage (unloading), the stored elastic energy is released and is converted into kinetic energy which causes the spheres to move in the opposite direction. All the work done during the loading stage is recovered when the contact overlap becomes zero (C). However, at this point, the spheres remain adhered to each other and further work is required to separate the two spheres (within the area highlighted in red). In order to break the contact, a minimum force equal to the pull-off force (E) is required and the contact breaks at F as shown.

In order to account for the work of adhesion, the EDEM contact radius needs to be activated and set to be greater than the physical radius of the particle as:

where r is the physical particle radius and α_{f} is the relative approach where the contact breaks. The range of values where the EDEM contact radius can be considered valid should be defined prior to the simulation by using Equation (3) and (5):

**EDEM Creator** > **Bulk Material** > **Particle**. The contact radius being greater than the physical radius allows the influence of a negative overlap in the force calculations.

*Figure 1:Loading and Unloading Behavior of JKR Contact Model.*

The normal contact force (or adhesion force) in the JKR V2 model is defined as:

where E*, R*, Γ and *a* are the relative elasticity, relative radius, interfacial surface energy (also known as work of adhesion) and contact radius (as described in Thornton, 2015), respectively. This is not the same as the EDEM contact radius. In this implementation of the model, the adhesive force depends on the interfacial surface energy and the relative approach (negative) at which the contact breaks. These are defined by the following generalized equations:

here γ_{1} and γ_{2} are the surface energies of the two spheres and γ_{1,2} is the interfacial surface energy. For the special case where two spheres of the same material come into contact, the interfacial surface energy is zero γ_{1,2} = 0 , and the interfacial surface energy becomes Γ = 2γ.

The relative approaching distance, α, (also known as contact overlap) and the pull-up force are defined by the following equations:

where a is the normal overlap between particles. For contacts between spheres of the same material, then Γ = 2γ and so equations 1 and 4 can be rewritten as:

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