</PRE>

<UL><LI>style = <I>gran/hertzian</I> or <I>gran/history</I> or <I>gran/no_history</I> 

<LI>Kn = spring constant for particle repulsion 

<LI>Kn = spring constant for particle repulsion (force/distance units) 

<PRE>  (mg/d units where m is mass, g is the gravitational constant, d is diameter of a particle) 

</PRE>

<LI>gamma_n = damping coefficient for normal direction collisions (sqrt(g/d) units) 

<LI>gamma_n = damping coefficient for normal direction collisions (1/time units) 

<LI>xmu = static yield criterion 

<LI>xmu = static yield criterion (unitless) 

<LI>dampflag = flag (0/1) for whether to (no/yes) include tangential damping 

<LI>dampflag = 0 or 1 if tangential damping force is excluded or included 

</UL>

<P><B>Examples:</B>

</P>

</P>

<P>The <I>gran</I> styles use the following formula <A HREF = "#Silbert">(Silbert)</A> for

frictional force between two granular particles that are a distance r

apart when r is less than the contact distance d.

apart when r is less than the contact distance d = Ri + Rj, where Ri

and Rj are the radii of the two particles:

</P>

<CENTER><IMG SRC = "Eqs/pair_granular.jpg">

</CENTER>

<P>The 1st term is a normal force and the 2nd term is a tangential force.

The other quantities are as follows:

The normal force has 2 parts: a contact force and a damping force.

The tangential force also has 2 parts: a shear force and a damping

force.  The shear force is included in pair styles <I>history</I> and

<I>Hertzian</I>, but is not included in pair style <I>no_history</I>.  The

tangential damping force is not included if <I>dampflag</I> is set to 0.

The other quantities in the equation are as follows:

</P>

<UL><LI>delta = d - r

<LI>f(x) = 1 for Hookean contacts used in pair styles <I>history</I> and <I>no_history</I>

<LI>f(x) = sqrt(x) for pair style <I>hertzian</I>

<LI>f(x) = 1 for Hookean contact used in pair styles <I>history</I> and <I>no_history</I>

<LI>f(x) = sqrt(x) for Hertzian contact used in pair style <I>hertzian</I>

<LI>Kn = elastic constant for normal contact

<LI>Kt = elastic constant for tangential contact = 2/7 of Kn

<LI>gamma_n = viscoelastic constants for normal contact

<LI>gamma_t = viscoelastic constants for tangential contact = 1/2 of gamma_n

<LI>gamma_n = viscoelastic damping constant for normal contact

<LI>gamma_t = viscoelastic damping constant for tangential contact = 1/2 of gamma_n

<LI>m_eff = Mi Mj / (Mi + Mj) = effective mass of 2 particles of mass Mi and Mj

<LI>Delta St = tangential displacement vector between the 2 spherical particles       which is truncated to satisfy a frictional yield criterion

<LI>n = a unit vector along the line connecting the centers of the 2 particles

<LI>n_ij = unit vector along the line connecting the centers of the 2 particles

<LI>Vn = normal component of the relative velocity of the 2 particles

<LI>Vt = tangential component of the relative velocity of the 2 particles 

</UL>

<P>The Kn and gamma_n coefficients are set as parameters to the

pair_style command.  Xmu is also specified which is the upper limit of

the tangential force through the Coulomb criterion Ft = xmu*Fn.  The

tangential force between 2 particles grows according to a tangential

spring and dash-pot model until Ft/Fn = xmu and then is held at Ft =

Fn*xmu until the particles lose contact.

pair_style command.  You can also think of Kn as being in mg/d units

where m is mass, g is the gravitational constant, and d is the

characteristic diameter of a particle.

</P>

<P>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.

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.

</P>

<P>For granular styles there are no individual atom type coefficients

that can be set via the <A HREF = "pair_coeff.html">pair_coeff</A> command.  All

pair_style style Kn gamma_n xmu dampflag :pre

style = {gran/hertzian} or {gran/history} or {gran/no_history} :ulb,l

Kn = spring constant for particle repulsion :l

  (mg/d units where m is mass, g is the gravitational constant, d is diameter of a particle) :pre

gamma_n = damping coefficient for normal direction collisions (sqrt(g/d) units) :l

xmu = static yield criterion :l

dampflag = flag (0/1) for whether to (no/yes) include tangential damping :l,ule

Kn = spring constant for particle repulsion (force/distance units) :l

gamma_n = damping coefficient for normal direction collisions (1/time units) :l

xmu = static yield criterion (unitless) :l

dampflag = 0 or 1 if tangential damping force is excluded or included :l,ule

[Examples:]

The {gran} styles use the following formula "(Silbert)"_#Silbert for

frictional force between two granular particles that are a distance r

apart when r is less than the contact distance d.

apart when r is less than the contact distance d = Ri + Rj, where Ri

and Rj are the radii of the two particles:

:c,image(Eqs/pair_granular.jpg)

The 1st term is a normal force and the 2nd term is a tangential force.

The other quantities are as follows:

The normal force has 2 parts: a contact force and a damping force.

The tangential force also has 2 parts: a shear force and a damping

force.  The shear force is included in pair styles {history} and

{Hertzian}, but is not included in pair style {no_history}.  The

tangential damping force is not included if {dampflag} is set to 0.

The other quantities in the equation are as follows:

delta = d - r

f(x) = 1 for Hookean contacts used in pair styles {history} and {no_history}

f(x) = sqrt(x) for pair style {hertzian}

f(x) = 1 for Hookean contact used in pair styles {history} and {no_history}

f(x) = sqrt(x) for Hertzian contact used in pair style {hertzian}

Kn = elastic constant for normal contact

Kt = elastic constant for tangential contact = 2/7 of Kn

gamma_n = viscoelastic constants for normal contact

gamma_t = viscoelastic constants for tangential contact = 1/2 of gamma_n

gamma_n = viscoelastic damping constant for normal contact

gamma_t = viscoelastic damping constant for tangential contact = 1/2 of gamma_n

m_eff = Mi Mj / (Mi + Mj) = effective mass of 2 particles of mass Mi and Mj

Delta St = tangential displacement vector between the 2 spherical particles \

      which is truncated to satisfy a frictional yield criterion

n = a unit vector along the line connecting the centers of the 2 particles

n_ij = unit vector along the line connecting the centers of the 2 particles

Vn = normal component of the relative velocity of the 2 particles

Vt = tangential component of the relative velocity of the 2 particles :ul

The Kn and gamma_n coefficients are set as parameters to the

pair_style command.  Xmu is also specified which is the upper limit of

the tangential force through the Coulomb criterion Ft = xmu*Fn.  The

tangential force between 2 particles grows according to a tangential

spring and dash-pot model until Ft/Fn = xmu and then is held at Ft =

Fn*xmu until the particles lose contact.

pair_style command.  You can also think of Kn as being in mg/d units

where m is mass, g is the gravitational constant, and d is the

characteristic diameter of a particle.

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.

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.

For granular styles there are no individual atom type coefficients

that can be set via the "pair_coeff"_pair_coeff.html command.  All