The Linear Elastic Bonding contact model (LEBM) consists of elastic bonds, providing
        a distinct type of behavior than the Bonding V2 model.
        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, considers both normal and
            tangential (or shear) force components, with the normal direction always acting along
            the line of centers of the two contacting elements and the tangential direction always
            acting perpendicular to this.
        The contact force F between a pair of contacting elements is decomposed into both normal
            and tangential components, Fn and Ft, respectively,
            as:
             
        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:
                 
            Where Gi is the Shear modulus and νi the Poisson’s ratio. A
                normal spring stiffness term is calculated for the bond associated with each element
                i as:
                 
            Where ri is the physical radius of element i. A single value for the
                overall normal stiffness of the bond kn is defined as:
                        
                     The corresponding tangential stiffness kt
                is calculated as:
                     
            With these bond stiffnesses defined, the associated normal contact force of the bond
                is defined as:
                     
            Where δn is the normal overlap of the contact.
            This normal contact force is then damped as follows:
                 
            Where vnrel is the magnitude of the normal component of the
                relative velocity between the two contacting elements and cn is the
                damping coefficient defined as:
                     
            Where
             
            e is the coefficient of restitution, m* is the equivalent
                    mass:
                         
            and mi is the mass of element i:
            The magnitude of the tangential contact force is calculated as:
                 
            Where δt 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:
                        
                     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:
                    
                 and the tangential contact force is now calculated by
            first calculating a trial state: 
                    
                 and subtracting the friction force as given by the
                Coulomb Friction law:
                        
                     Where μ is the Static Coefficient of
                Friction.
The actual tangential contact force Ft is then calculated
                    as:
                        
                     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 for which you would like to set up
                            bonds. 
 
                        - 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 V2 model, the bond properties are
                post-processed by examining the characteristics of the associated contacts. For bond
                visualization, it is necessary to visualize 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:
                            
                                    
                                 
                        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. 
                
Note: To activate the reporting of the following custom
                    properties, you must first set the environment variable
                        EDEM_ADDITIONAL_CUSTOM_PROPERTIES equal to 1.