# The Hysteretic Spring Model

Once this stress is exceeded, the particles behave as though undergoing plastic deformation. The result is that large overlaps are achievable without excessive forces acting upon them, thus representing a compressible material. Hysteretic Spring normal force calculation is based on the Walton-Braun theory described in the following references (Walton and Braun 1986), (Walton and Braun 1986).

The following figure shows a schematic of the Hysteretic Spring contact model force displacement relationship.

The quantity δ_{0} represents the residual ‘overlap’ that remains because of the plastic deformation (i.e., flattening) in the contact region. The exact location of each old contact spot is not ‘remembered,’ so that particles become like new undeformed spheres once they separate. Subsequent contacts will load along a new loading path with slope, K_{1}. Any reloading before separation follows the slope K_{2} until the original loading curve K_{1} is reached, whereupon the lower slope is followed until unloading occurs.

In particular, the normal force, F_{N}, is defined as:

where K_{1} and K_{2} are the loading and unloading stiffness respectively, δ_{n}, is the normal overlap and δ_{0} is the residual overlap. Loading stiffness, K_{1}, is related to the yield strength of each material participating in contact, Y_{1} and Y_{2}, as follows (Walton O., 2006):

Here, R* is the equivalent radius of the two particles participating in the contact. The following expression for coefficient of restitution:

allows for the determination of K>_{2} from K_{1} (Walton O. , 2006). Residual overlap is updated every Time Step according to the following rule:

The main energy dissipation mechanism is due to the difference in spring stiffness between loading and unloading phases. The need for an additional velocity dependent dissipation mechanism, to suppress possible undamped low amplitude oscillations, was suggested in (Walton O. , 2006). In EDEM's Hysteretic Spring model, this mechanism was implemented in a way similar to the linear spring normal damping force scaled by a dimensionless user set parameter ‘Damping Factor’, b_{n}, as follows:

Here k is either K_{1} or K_{2}. The rest of the variables and factors in the above expression are explained in the Linear Spring contact model section.

Since EDEM's Hysteretic Spring model for the normal force can be considered as an extension of linear spring model for normal force it makes sense to use tangential components of linear spring model for tangential forces. In particular, more general form of linear spring elastic tangential force is used in Hysteretic Spring model in which the force depends on user set stiffness factor, γ_{t}, as follows:

Unlike for the normal component, the tangential damping force is not scaled by the damping factor.

For particles, a default value for the Yield Strength is estimated from the Young’s Modulus E and Radius of the particle R according to the algorithm suggested in (Walton O. , 2006). First, the possible default value is calculated as follows:

If this value is less than 100Pa, it is replaced with a value of 0.003 * E.

For geometries, the default value is set to 1e+09 Pa.

Interaction | Configurable Parameters | Position |

Particle to Particle, Particle to Geometry |
Click + to add particle-particle or particle-geometry interactions. Set the following interaction parameters:Damping factorThis dimensionless parameter controls the amount of velocity dependent damping. Without this additional damping, small vibrations of particles can persist for a long time. Setting this parameter to zero suppresses only the velocity-dependent damping without affecting the main energy loss mechanism due to the nature of the linearized elasto-plastic model. The recommended value for this parameter is zero when modeling “aggressive” compression of materials and small non-zero value for small stain compression regimes. Stiffness factorThis dimensionless parameter is defined as the ratio of tangential stiffness to normal loading stiffness and used in calculation of tangential forces at the contact. According to the literature, the typical value of this parameter is in the range between 0.7 and 1. Also set the yield strength in units of Pa for each particle or geometry that has an interaction. EDEM offers a reasonable default value for this parameter estimated from material Shear Modulus. You can, however, overwrite this default value. |
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- Select the required category from the Interaction pulldown menu in the Physics section of the Creator.
- Click the
**+**dropdown and then select**Hysteretic****Spring**. - Click the configuration icon to define contact model parameters.

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