\(\renewcommand{\AA}{\text{Å}}\)

# pair_style gran/hooke command

Accelerator Variants: gran/hooke/omp

# pair_style gran/hooke/history command

Accelerator Variants: gran/hooke/history/omp, gran/hooke/history/kk

# pair_style gran/hertz/history command

Accelerator Variants: gran/hertz/history/omp

## Syntax

    pair_style style Kn Kt gamma_n gamma_t xmu dampflag keyword

- style = gran/hooke or gran/hooke/history or gran/hertz/history

- Kn = elastic constant for normal particle repulsion (force/distance units or pressure units - see discussion below)

- Kt = elastic constant for tangential contact (force/distance units or pressure units - see discussion below)

- gamma_n = damping coefficient for collisions in normal direction (1/time units or 1/time-distance units - see discussion below)

- gamma_t = damping coefficient for collisions in tangential direction (1/time units or 1/time-distance units - see discussion below)

- xmu = static yield criterion (unitless value between 0.0 and 1.0e4)

- dampflag = 0 or 1 if tangential damping force is excluded or included

- keyword = limit_damping

      limit_damping value = none
         limit damping to prevent attractive interaction

Note

Versions of LAMMPS before 9Jan09 had different style names for granular force fields. This is to emphasize the fact that the Hertzian equation has changed to model polydispersity more accurately. A side effect of the change is that the Kn, Kt, gamma_n, and gamma_t coefficients in the pair_style command must be specified with different values in order to reproduce calculations made with earlier versions of LAMMPS, even for monodisperse systems. See the NOTE below for details.

## Examples

    pair_style gran/hooke/history 200000.0 NULL 50.0 NULL 0.5 1
    pair_style gran/hooke 200000.0 70000.0 50.0 30.0 0.5 0
    pair_style gran/hooke 200000.0 70000.0 50.0 30.0 0.5 0 limit_damping

## Description

The gran styles use the following formulas for the frictional force between two granular particles, as described in (Brilliantov), (Silbert), and (Zhang), when the distance r between two particles of radii Ri and Rj is less than their contact distance d = Ri + Rj. There is no force between the particles when r > d.

The two Hookean styles use this formula:

\[F_{hk} = (k_n \delta \mathbf{n}_{ij} - m_{eff} \gamma_n\mathbf{ v}_n) - (k_t \boldsymbol{\Delta} \mathbf{s}_t + m_{eff} \gamma_t \mathbf{v}_t)\]

The Hertzian style uses this formula:

\[F_{hz} = \sqrt{\delta} \sqrt{\frac{R_i R_j}{R_i + R_j}} F_{hk} = \sqrt{\delta} \sqrt{\frac{R_i R_j}{R_i + R_j}} \Big[ (k_n \delta \mathbf{n}_{ij} - m_{eff} \: \gamma_n \mathbf{ v}_n) - (k_t \boldsymbol{\Delta} \mathbf{s}_t + m_{eff} \: \gamma_t \mathbf{v}_t) \Big]\]

In both equations the first parenthesized term is the normal force between the two particles and the second parenthesized term is the tangential force. The normal force has 2 terms, a contact force and a damping force. The tangential force also has 2 terms: a shear force and a damping force. The shear force is a “history” effect that accounts for the tangential displacement between the particles for the duration of the time they are in contact. This term is included in pair styles hooke/history and hertz/history, but is not included in pair style hooke. The tangential damping force term is included in all three pair styles if dampflag is set to 1; it is not included if dampflag is set to 0.

The other quantities in the equations are as follows:

- \(\delta\) = d - r = overlap distance of 2 particles

- \(K_n\) = elastic constant for normal contact

- \(K_t\) = elastic constant for tangential contact

- \(\gamma_n\) = viscoelastic damping constant for normal contact

- \(\gamma_t\) = viscoelastic damping constant for tangential contact

- \(m_{eff} = M_i M_j / (M_i + M_j) =\) effective mass of 2 particles of mass M_i and M_j

- \(\boldsymbol{\Delta} \mathbf{s}_t =\) tangential displacement vector between 2 particles which is truncated to satisfy a frictional yield criterion

- \(n_{ij} =\) unit vector along the line connecting the centers of the 2 particles

- \(V_n =\) normal component of the relative velocity of the 2 particles

- \(V_t =\) tangential component of the relative velocity of the 2 particles

The \(K_n\), \(K_t\), \(\gamma_n\), and \(\gamma_t\) coefficients are specified as parameters to the pair_style command. If a NULL is used for \(K_t\), then a default value is used where \(K_t = 2/7 K_n\). If a NULL is used for \(\gamma_t\), then a default value is used where \(\gamma_t = 1/2 \gamma_n\).

The interpretation and units for these 4 coefficients are different in the Hookean versus Hertzian equations.

The Hookean model is one where the normal push-back force for two overlapping particles is a linear function of the overlap distance. Thus the specified \(K_n\) is in units of (force/distance). Note that this push-back force is independent of absolute particle size (in the monodisperse case) and of the relative sizes of the two particles (in the polydisperse case). This model also applies to the other terms in the force equation so that the specified \(\gamma_n\) is in units of (1/time), \(K_t\) is in units of (force/distance), and \(\gamma_t\) is in units of (1/time).

The Hertzian model is one where the normal push-back force for two overlapping particles is proportional to the area of overlap of the two particles, and is thus a non-linear function of overlap distance. Thus Kn has units of force per area and is thus specified in units of (pressure). The effects of absolute particle size (monodispersity) and relative size (polydispersity) are captured in the radii-dependent prefactors. When these prefactors are carried through to the other terms in the force equation it means that the specified \(\gamma_n\) is in units of (1/(time*distance)), \(K_t\) is in units of (pressure), and \(\gamma_t\) is in units of (1/(time*distance)).

Note that in the Hookean case, \(K_n\) can be thought of as a linear spring constant with units of force/distance. In the Hertzian case, \(K_n\) is like a non-linear spring constant with units of force/area or pressure, and as shown in the (Zhang) paper, \(K_n = 4G / (3(1-\nu))\) where \(\nu =\) the Poisson ratio, G = shear modulus = \(E / (2(1+\nu))\), and E = Young’s modulus. Similarly, \(K_t = 4G / (2-\nu)\). (NOTE: in an earlier version of the manual, we incorrectly stated that \(K_t = 8G / (2-\nu)\).)

Thus in the Hertzian case \(K_n\) and \(K_t\) can be set to values that corresponds to properties of the material being modeled. This is also true in the Hookean case, except that a spring constant must be chosen that is appropriate for the absolute size of particles in the model. Since relative particle sizes are not accounted for, the Hookean styles may not be a suitable model for polydisperse systems.

Note

In versions of LAMMPS before 9Jan09, the equation for Hertzian interactions did not include the \(\sqrt{r_i r_j / (r_i + r_j)}\) term and thus was not as accurate for polydisperse systems. For monodisperse systems, \(\sqrt{ r_i r_j /(r_i+r_j)}\) is a constant factor that effectively scales all 4 coefficients: \(K_n, K_t, \gamma_n, \gamma_t\). Thus you can set the values of these 4 coefficients appropriately in the current code to reproduce the results of a previous Hertzian monodisperse calculation. For example, for the common case of a monodisperse system with particles of diameter 1, all 4 of these coefficients should now be set 2x larger than they were previously.

Xmu is also specified in the pair_style command and is the upper limit of the tangential force through the Coulomb criterion Ft = xmu*Fn, where Ft and Fn are the total tangential and normal force components in the formulas above. Thus in the Hookean case, the tangential force between 2 particles grows according to a tangential spring and dash-pot model until Ft/Fn = xmu and is then held at Ft = Fn*xmu until the particles lose contact. In the Hertzian case, a similar analogy holds, though the spring is no longer linear.

Note

Normally, xmu should be specified as a fractional value between 0.0 and 1.0, however LAMMPS allows large values (up to 1.0e4) to allow for modeling of systems which can sustain very large tangential forces.

The effective mass m_eff is given by the formula above for two isolated particles. If either particle is part of a rigid body, its mass is replaced by the mass of the rigid body in the formula above. This is determined by searching for a fix rigid command (or its variants).

For granular styles there are no additional coefficients to set for each pair of atom types via the pair_coeff command. All settings are global and are made via the pair_style command. However you must still use the pair_coeff for all pairs of granular atom types. For example the command

    pair_coeff * *

should be used if all atoms in the simulation interact via a granular potential (i.e. one of the pair styles above is used). If a granular potential is used as a sub-style of pair_style hybrid, then specific atom types can be used in the pair_coeff command to determine which atoms interact via a granular potential.

If two particles are moving away from each other while in contact, there is a possibility that the particles could experience an effective attractive force due to damping. If the limit_damping keyword is used, this option will zero out the normal component of the force if there is an effective attractive force.

Styles with a gpu, intel, kk, omp, or opt suffix are functionally the same as the corresponding style without the suffix. They have been optimized to run faster, depending on your available hardware, as discussed on the Accelerator packages page. The accelerated styles take the same arguments and should produce the same results, except for round-off and precision issues.

These accelerated styles are part of the GPU, INTEL, KOKKOS, OPENMP, and OPT packages, respectively. They are only enabled if LAMMPS was built with those packages. See the Build package page for more info.

You can specify the accelerated styles explicitly in your input script by including their suffix, or you can use the -suffix command-line switch when you invoke LAMMPS, or you can use the suffix command in your input script.

See the Accelerator packages page for more instructions on how to use the accelerated styles effectively.

## Mixing, shift, table, tail correction, restart, rRESPA info

The pair_modify mix, shift, table, and tail options are not relevant for granular pair styles.

These pair styles write their information to binary restart files, so a pair_style command does not need to be specified in an input script that reads a restart file.

These pair styles can only be used via the pair keyword of the run_style respa command. They do not support the inner, middle, outer keywords.

The single() function of these pair styles returns 0.0 for the energy of a pairwise interaction, since energy is not conserved in these dissipative potentials. It also returns only the normal component of the pairwise interaction force. However, the single() function also calculates 10 extra pairwise quantities. The first 3 are the components of the tangential force between particles I and J, acting on particle I. The fourth is the magnitude of this tangential force. The next 3 (5-7) are the components of the relative velocity in the normal direction (along the line joining the 2 sphere centers). The last 3 (8-10) the components of the relative velocity in the tangential direction.

These extra quantities can be accessed by the compute pair/local command, as p1, p2, …, p10.

## Restrictions

All the granular pair styles are part of the GRANULAR package. It is only enabled if LAMMPS was built with that package. See the Build package page for more info.

These pair styles require that atoms store torque and angular velocity (omega) as defined by the atom_style. They also require a per-particle radius is stored. The sphere atom style does all of this.

This pair style requires you to use the comm_modify vel yes command so that velocities are stored by ghost atoms.

These pair styles will not restart exactly when using the read_restart command, though they should provide statistically similar results. This is because the forces they compute depend on atom velocities. See the read_restart command for more details.

Accumulated values for individual contacts are saved to to restart files but are not saved to data files. Therefore, forces may differ significantly when a system is reloaded using A read_data command.

## Related commands

pair_coeff

## Default

none

(Brilliantov) Brilliantov, Spahn, Hertzsch, Poschel, Phys Rev E, 53, p 5382-5392 (1996).

(Silbert) Silbert, Ertas, Grest, Halsey, Levine, Plimpton, Phys Rev E, 64, p 051302 (2001).

(Zhang) Zhang and Makse, Phys Rev E, 72, p 011301 (2005).