Models for normal damping in granular interactions
The normal force is augmented by a damping term of the following general form:
Here, \(\mathbf{v}_{n,rel} = (\mathbf{v}_j - \mathbf{v}_i) \cdot \mathbf{n}\ \mathbf{n}\) is the component of relative velocity along \(\mathbf{n}\), where \(\mathbf{n}\) is the unit vector along the direction connecting the two particle centers, discussed here.
Different damping models result in different expressions for \(\eta_n\):
velocity damping model
Parameters: none beyond the normal model damping coefficient \(\eta_{n0}\).
Example:
pair_style granular
pair_coeff * * hooke 1000.0 50.0 tangential linear_nohistory 1.0 0.4 damping velocity
For damping velocity, the normal damping is simply equal to the user-specified damping coefficient in the normal model:
Here, \(\eta_{n0}\) is the damping coefficient specified for the normal contact model, in units of mass/time.
mass_velocity damping model
Parameters: none beyond the normal model damping coefficient \(\eta_{n0}\).
Example:
pair_style granular
pair_coeff * * hooke 1000.0 50.0 tangential linear_history 500.0 1.0 0.4 damping mass_velocity
For damping mass_velocity, the normal damping is given by:
Here, \(\eta_{n0}\) is the damping coefficient specified for the normal contact model, in units of 1/time and \(m_{eff} = m_i m_j/(m_i + m_j)\) is the effective mass. Use damping mass_velocity to reproduce the damping behavior of pair gran/hooke/*.
viscoelastic damping model
Parameters: none beyond the normal-model damping coefficient \(\eta_{n0}\).
Example:
pair_style granular
pair_coeff * * hertz 1000.0 50.0 tangential mindlin 1000.0 1.0 0.4 damping viscoelastic
The damping viscoelastic model is based on the viscoelastic treatment of (Brilliantov et al), where the normal damping is given by:
Here, a is the contact radius, given by \(a =\sqrt{R\delta}\) for all models except jkr, for which it is given implicitly according to \(\delta = a^2/R - 2\sqrt{\pi \gamma a/E}\). For damping viscoelastic, \(\eta_{n0}\) is in units of 1/(time*distance).
tsuji damping model
Parameters: none beyond the normal-model restitution coefficient \(e\).
Example:
pair_style granular
pair_coeff * * hertz/material 1.0e8 0.3 0.3 tangential mindlin NULL 1.0 0.4 damping tsuji
The tsuji model is based on the work of (Tsuji et al). Here, the damping coefficient specified as part of the normal model is interpreted as a restitution coefficient \(e\), assuming the normal force is Hertzian. The damping constant \(\eta_n\) is given by:
where \(k_{nd}\) is an effective harmonic stiffness equal to the ratio of the normal force to the overlap. For example, \(k_{nd} = 4/3Ea\) for a Hertz contact model based on material parameters with \(a\) being the contact radius of \(\sqrt{\delta R}\). For Hooke, \(k_{nd}\) is simply the spring constant or \(k_{n}\). This damping model is not compatible with cohesive normal models such as JKR or DMT. The parameter \(\alpha\) is related to the restitution coefficient e according to:
The dimensionless coefficient of restitution \(e\) specified as part of the normal contact model parameters should be between 0 and 1, but no error check is performed on this. Using this damping model with normal contact models other than Hertz is possible, but the resulting coefficient of restitution is not likely to accurately match the specified value.
Changed in version 2Sep2026.
This numerical solution is from (Marshall, 2009) where the factor of \(\sqrt{2}\) arises from a difference in convention from Tsuji when defining \(\alpha\) using either the mass vs. effective mass. This factor was missing in earlier versions of LAMMPS.
coeff_restitution damping model
Parameters: none beyond the normal-model restitution coefficient \(e\).
Example:
pair_style granular
pair_coeff * * hertz/material 1.0e8 0.3 0.3 tangential mindlin NULL 1.0 0.4 damping coeff_restitution
The coeff_restitution model is useful when a specific normal coefficient of restitution \(e\) is required. It operates much like the Tsuji model, but the normal coefficient of restitution \(e\) is calculated using a different expression (below) and is designed to work with Hooke and Hertz. Following the approach of (Brilliantov et al), when using the hooke normal model, coeff_restitution then calculates the damping coefficient as:
where \(k_{nd}\) is the same stiffness defined in the above Tsuji model. For any other normal model, e.g. the hertz and hertz/material models, the damping coefficient is:
Since coeff_restitution accounts for the effective mass, effective radius, and pairwise overlaps (except when used with the hooke normal model) when calculating the damping coefficient, it accurately reproduces the specified coefficient of restitution for both monodisperse and polydisperse particle pairs. This damping model is not compatible with cohesive normal models such as JKR or DMT.
mdr class of damping models
Parameters: \(d_{type}\)
Example:
pair_style granular
pair_coeff * * mdr 5.0e6 0.4 1.9e5 2.0 0.5 0.5 tangential linear_history 940.0 1.0 0.7 damping mdr 1
The mdr damping class contains multiple damping models that can be toggled between by specifying different integer values for the \(d_{type}\) input parameter. This damping option is only compatible with the normal mdr contact model.
Setting \(d_{type} = 1\) is the suggested damping option. This specifies a damping model that takes into account the contact stiffness \(k_{mdr}\) calculated by the normal mdr contact model to determine the damping coefficient:
where \(k_{mdr}\) is proportional to contact radius \(a_{mdr}\) tracked by the normal mdr contact model:
In this case, \(\eta_{n0}\) is simply a dimensionless coefficient that scales the overall damping coefficient.
The other supported option is \(d_{type} = 2\), which defines a simple damping model similar to the velocity option:
but has additional checks to avoid non-physical damping after plastic deformation.
The total normal force is computed as the sum of the elastic and damping components:
References
(Brilliantov et al, 1996) Brilliantov, N. V., Spahn, F., Hertzsch, J. M., & Poschel, T. (1996). Model for collisions in granular gases. Physical review E, 53(5), 5382.
(Tsuji et al, 1992) Tsuji, Y., Tanaka, T., & Ishida, T. (1992). Lagrangian numerical simulation of plug flow of cohesionless particles in a horizontal pipe. Powder technology, 71(3), 239-250.