"LIGGGHTS WWW Site"_liws - "LAMMPS WWW Site"_lws - "LIGGGHTS Documentation"_ld - "LIGGGHTS Commands"_lc :c

:link(liws,http://www.cfdem.com)
:link(lws,http://lammps.sandia.gov)
:link(ld,Manual.html)
:link(lc,Section_commands.html#comm)

:line

gran model hooke :h3

[Syntax:]

model hertz keyword values :pre
zero or more keyword/value pairs may be appended :l
  {tangential_damping} values = 'on' or 'off'
    on = activates tangential damping
    off = no tangential damping
  {viscous} = 'stokes' or 'off'
    viscous = restitution coefficient varies with a local Stokes number of the particle
    off = no modification to the restitution coefficient  :pre

[LIGGGHTS vs. LAMMPS Info:]

This part of "pair gran"_pair_gran.html and 
"fix wall/gran"_fix_wall_gran.html
is not availabe in LAMMPS.

[Description:]

This granular model uses the following formula for the frictional force between two granular 
particles, 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: 

:c,image(Eqs/pair_gran_html_60b8ced2.png)

In the first term is the normal force between the two particles and the second term is the 
tangential force. The normal force has 2 terms, a spring 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 ("tangential overlap") 
between the particles for the duration of the time they are in contact.
This term is controlled by the "tangential model"_Section_gran_models.html in action

Keyword {tangential_damping} can be used to eliminate the second part of the force in 
tangential direction. 

The quantities in the equations are as follows:

delta_n = d - r = overlap distance of 2 particles  :ulb,l
k_n = elastic constant for normal contact :l
k_t = elastic constant for tangential contact :l
gamma_n = viscoelastic damping constant for normal contact :l
gamma_t = viscoelastic damping constant for tangential contact :l
delta_t = tangential displacement vector between 2 spherical particles which is truncated to satisfy a frictional yield criterion :l
rmu = coefficient of rolling friction :l
contactradius = contact radius, equal to particle radius - 0.5 * delta_n :l
v_n = normal component of the relative velocity of the 2 particles :l
v_t = tangential component of the relative velocity of the 2 particles :l
w_r = relative rotational velocity of the 2 particles :l
:ule

The Kn, Kt, gamma_n, and gamma_t coefficients are calculated as follows from the material properties:

:c,image(Eqs/pair_gran_html_147eaf38.png)

:c,image(Eqs/pair_gran_html_m225ba7de.png)

To define those material properties, it is mandatory to use multiple "fix property/global"_fix_property.html commands:

fix id all property/global youngsModulus peratomtype value_1 value_2 ...
    (value_i=value for Youngs Modulus of atom type i)
fix id all property/global poissonsRatio peratomtype value_1 value_2 ...
    (value_i=value for Poisson ratio of atom type i)
fix id all property/global coefficientRestitution peratomtypepair n_atomtypes value_11 value_12 .. value_21 value_22 .. .
    (value_ij=value for the coefficient of restitution between atom type i and j; n_atomtypes is the number of atom types you want to use in your simulation)
fix id all property/global coefficientFriction peratomtypepair n_atomtypes value_11 value_12 .. value_21 value_22 .. .
    (value_ij=value for the (static) coefficient of friction between atom type i and j; n_atomtypes is the number of atom types you want to use in your simulation) :pre

IMPORTANT NOTE: You have to use atom styles beginning from 1, e.g. 1,2,3,...

The "characteristic impact velocity" is additionally used for hooke: 

fix id all property/global characteristicVelocity scalar value
    (value=value for characteristic impact velocity) :pre

IMPORTANT NOTE: You have to use atom styles beginning from 1, e.g. 1,2,3,...

[Viscous model:]

Using option {viscous} = stokes adapts the coefficient of restitution as proposed by
"(Legendre)"_#Legendre, {viscous} = off performs no modification.

One has to provide the 3 peratomtypepair parameters via a "fix property/global"_fix_property.html
command needed for the viscous damping:

fix id all property/global FluidViscosity peratomtypepair n_atomtypes value_11 value_12 .. value_21 value_22 .. .
    (value_ij=value for fluid viscosity between atom type i and j; n_atomtypes is the number of atom types you want to use in your simulation)
fix id all property/global CriticalStokes peratomtypepair n_atomtypes value_11 value_12 .. value_21 value_22 .. .
    (value_ij=value for critical Stokes number between atom type i and j; n_atomtypes is the number of atom types you want to use in your simulation)
fix id all property/global MaximumRestitution peratomtypepair n_atomtypes value_11 value_12 .. value_21 value_22 .. .
    (value_ij=value for maximum coefficient of restitution between atom type i and j; n_atomtypes is the number of atom types you want to use in your simulation) :pre

IMPORTANT NOTE: You have to use atom styles beginning from 1, e.g. 1,2,3,...

The coefficient of friction cof is the upper limit of the tangential force through 
the Coulomb criterion Ft = cof*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 = cof and is then held at Ft = Fn*cof until the particles lose contact. 
In the Hertzian case, a similar analogy holds, though the spring is no longer linear.

[Restrictions:]

If using SI units, youngsModulus must be > 5e6
If using CGS units, youngsModulus must be > 5e5
When using {viscous}, FluidViscosity has to be  > 0

[Default:] 

{viscous} = 'off'
{tangential_damping} = 'on'

:link(Legendre)
[(Legendre)] Legendre, Daniel and Guiraud. Phys. Fluids 17, 097105 (2005).

[(Di Renzo)] Alberto Di Renzo, Francesco Paolo Di Maio, CES, 59 (3), p 525–541 (2004).

[(Ai)] Jun Ai, Jian-Fei Chen, J. Michael Rotter, Jin Y. Ooi, Powder Technology, 206 (3), p 269-282 (2011).

[(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). 

