Linear Elastic Bonding Model (LEBM)

The Linear Elastic Bonding contact model (LEBM) consists of elastic bonds, providing a different type of behavior than the Bonding and Bonding V2 models.

The theory presented in the articles Comparative Study of Different Discrete Element Models and Evaluation of Equivalent Micromechanical Parameters by J. Rojek et al. (2012) and Advances in the Development of the Discrete Element Method for Excavation Processes by C.A. Labra (2012) served as the foundation for the model's implementation in EDEM. It is recommended that you refer to the aforementioned paper and thesis for an in-depth description of the model.

The behavior of the LEBM is determined by the contact information for a certain pair of elements, either particle-particle or particle-geometry. The LEBM, like many other EDEM contact models, takes into account both normal and tangential (or shear) force components, with the normal direction always acting perpendicular to the line of centers of the two contacting elements and the tangential direction always acting parallel to this.

The contact force F between a pair of contacting elements is decomposed into both normal and tangential components, F_n and F_t, respectively, as:

F = F_{n} + F_{t} = F_{n} n + F_{t}

Where n is the unit vector along the line of centers of the two elements.

Calculating the force with active bonds

Consider the constitutive equations when the bonds are present and active, for a given pair of elements (where subscripts 1 and 2 indicate the different elements).

The following are calculated on a per-Time Step basis and adjusted incrementally.

The Young's modulus for element i is calculated by:

E_{i} = 3 G_{i} (1 - 2 v_{i})

Where G_i is the Shear modulus and ν_i the Poisson’s ratio. A normal spring stiffness term is calculated for the bond associated with each element i as:

K_{n i} = 4 E_{i} r_{i}

Where r_i is the physical radius of element i. A single value for the overall normal stiffness of the bond k_n is defined as:

k_{n} = \frac{K_{n_{1}} K_{n_{2}}}{K_{n_{1}} + K_{n_{2}}}

The corresponding tangential stiffness k_t is calculated as:

k_{t} = k_{n} (\frac{2 v_{1} v_{2}}{v_{1} + v_{2}})

With these bond stiffnesses defined, the associated normal contact force of the bond is defined as:

F_{n} = k_{n} δ_{n}

Where δ_n is the normal overlap of the contact.

This normal contact force is then damped as follows:

F_{n}^{d} = c_{n} v_{n}^{r e l}

Where v_n^rel is the magnitude of the normal component of the relative velocity between the two contacting elements and c_n is the damping coefficient defined as:

c_{n} = 2 α \sqrt{m^{*} k_{n}}

Where

α = \frac{- \ln e}{\sqrt{π^{2} + \ln^{2} e}}

e is the coefficient of restitution, m^* is the equivalent mass:

m * = \frac{m_{1} m_{2}}{m_{1} + m_{2}}

and mⁱ is the mass of element i:

The magnitude of the tangential contact force is calculated as:

F_{t} = k_{t} δ_{t}

Where δ_t is the tangential overlap of the contact. The tangential contact force component is not damped.

Calculating the force with inactive bonds

Consider the constitutive equations and associated failure criteria for when the bonds break or if they were never created.

The failure criteria for the bonds can be defined as follows:

\begin{array}{l} F_{n} \leq R_{n} (t e n s i o n o n l y) \\ F_{t} \leq R_{t} \end{array}

Where R_n and R_t are the normal and Shear strengths (defined by the user). The bond will break under one of the following conditions:

The bond is under tension and the magnitude of the normal component of the contact force exceeds the defined normal strength.

The bond is under either tension or compression and the magnitude of the tangential component of the contact force exceeds the defined shear strength.

When under compression, bonds cannot break because of an excessive normal contact force. This is what gives the model its elastic behavior.

In the absence of any bonds - if they were never formed or previously met one of the failure criteria and broke, the normal contact force is calculated only when the contacting elements are compressed.

In this instance, the normal contact force and associated damping force remain the same as defined previously:

F_{n}^{d} = c_{n} v_{n}^{r e l}

and the tangential contact force is now calculated by first calculating a trial state:

F_{t}^{t r i a l} = F_{t}^{p r e v} - k_{t} v_{t}^{r e l} Δ t

and subtracting the friction force as given by the Coulomb Friction law:

ε = |F_{n}^{t r i a l}| - μ |F_{n}|

Where μ is the Static Coefficient of Friction.

The actual tangential contact force F_t is then calculated as:

F_{t} = \{\begin{array}{l} F_{t}^{t r i a l}, ε \leq 0 \\ μ |F_{n}| \frac{F_{t}^{t r i a l}}{|F_{t}^{t r i a l}|}, ε > 0 \end{array}\}

The model yields the following force-displacement behavior:

Note:

After you have selected a list of all possible interactions for Particle to Particle or Particle to Geometry contacts, select the interaction you would like to set up bonds for.
Specify values for Normal Strength and Shear Strength of the formed bonds.
Once you have specified values for the interaction, click OK and repeat the same steps for other interactions you would like to set up bonds for.

Post Processing

As with EDEM's Bonding and Bonding V2 models, bond attributes are post-processed by examining the characteristics of the associated contacts rather than using EDEM's heritage bond properties. For bond visualization, it is necessary to see the relevant bonds and apply the appropriate coloring.

After a simulation has run with the Linear Elastic Bonding Model enabled, you can post process the behavior of the model using the following custom properties:

Contact custom properties

customInitialOverlap – Indicates the initial value of the overlap of the elements in contact before the bond is formed.
BondStatus – Indicates the property that determines whether the bond exists or not. A value of 1.0 indicates a successful bond has been formed and is currently active. A value of -1.0 indicates that the bond has been broken. A value of 0.0 indicates a bond was never formed.
customShearForce – Indicates the Shear force acting on the contact.

Particle custom properties

customBondsInitial – Indicates the number of bonds that an element has after the initial bond formation time.
customBondsDamage – Indicates the metric for the damage to the bonds for a given particle, calculated as:
$1.0 - \frac{c u r r e n t n u m b e r o f b o n d s f o r t h e p a r t i c l e}{c u s t o m B o n d s I n i t i a l}$
A value of 0.0 indicates all initial bonds are still formed. A value of 1.0 indicates all initial bonds are now broken.

Simulation custom properties

customTotalBonds – Indicates the total number of bonds in the simulation at the current Time Step. This will be at its maximum value the Time Step after the bonds are formed and will be decremented as bonds are broken.
customBrokenBondsNormal – Indicates the total number of bonds in the simulation that have broken due to excessive normal force.
customBrokenBondsShear – Indicates the total number of bonds in the simulation that have broken due to excessive Shear force.