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\documentclass[12pt]{article}
\begin{document}
$$
F_{hz} = \sqrt{\delta} \sqrt{\frac{R_i R_j}{R_i + R_j}} F_{hk} =
\sqrt{\delta} \sqrt{\frac{R_i R_j}{R_i + R_j}}
\Big[ (k_n \delta \mathbf{n}_{ij} -
m_{\mbox{\scriptsize{eff}}} \: \gamma_n \mathbf{ v}_n) -
(k_t \delta\mathbf{ \Delta s}_t +
m_{\mbox{\scriptsize{eff}}} \: \gamma_t \mathbf{v}_t) \Big]
$$
\end{document}

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\documentclass[12pt]{article}
\begin{document}
$$
F_{hk} = (k_n \delta \mathbf{n}_{ij} -
m_{\mbox{\scriptsize{eff}}} \gamma_n\mathbf{ v}_n) -
(k_t \delta\mathbf{ \Delta s}_t +
m_{\mbox{\scriptsize{eff}}} \gamma_t \mathbf{v}_t)
$$
\end{document}

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\documentclass[12pt]{article}
\begin{document}
$$
F = f\left(\delta/d\right)\left(k_n \delta \mathbf{n}_{ij} -
m_{\mbox{\scriptsize{eff}}} \gamma_n\mathbf{ v}_n\right) +
f\left(\delta/d\right)\left(-k_t \delta\mathbf{ \Delta s}_t -
m_{\mbox{\scriptsize{eff}}} \gamma_t \mathbf{v}_t\right)
$$
\end{document}

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<CENTER><H3>LAMMPS Documentation
</H3></CENTER>
<CENTER>(21 May 2008 version of LAMMPS)
<CENTER>(6 Jan 2009 version of LAMMPS)
</CENTER>
<P>LAMMPS stands for Large-scale Atomic/Molecular Massively Parallel
Simulator.

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LAMMPS Documentation :c,h3
(21 May 2008 version of LAMMPS) :c
(6 Jan 2009 version of LAMMPS) :c
LAMMPS stands for Large-scale Atomic/Molecular Massively Parallel
Simulator.

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<HR>
<H3>pair_style gran/hertzian command
<H3>pair_style gran/hooke command
</H3>
<H3>pair_style gran/history command
<H3>pair_style gran/hooke/history command
</H3>
<H3>pair_style gran/no_history command
<H3>pair_style gran/hertz/history command
</H3>
<P><B>Syntax:</B>
</P>
<PRE>pair_style style Kn gamma_n xmu dampflag
</PRE>
<UL><LI>style = <I>gran/hertzian</I> or <I>gran/history</I> or <I>gran/no_history</I>
<UL><LI>style = <I>gran/hooke</I> or <I>gran/hooke/history</I> or <I>gran/hertz/history</I>
<LI>Kn = spring constant for particle repulsion (force/distance units)
<LI>Kn = spring constant for particle repulsion (see units below)
<LI>gamma_n = damping coefficient for normal direction collisions (1/time units)
<LI>gamma_n = damping coefficient for collisions in normal direction (see units below)
<LI>xmu = static yield criterion (unitless)
<LI>xmu = static yield criterion (unitless fraction between 0.0 and 1.0)
<LI>dampflag = 0 or 1 if tangential damping force is excluded or included
</UL>
<P><B>Examples:</B>
</P>
<PRE>pair_style gran/history 200000.0 0.5 1.0 1
<PRE>pair_style gran/history 200000.0 50.0 0.5 1
</PRE>
<P><B>Description:</B>
</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 = Ri + Rj, where Ri
and Rj are the radii of the two particles:
<P>The <I>gran</I> styles use the following formulas for the frictional force
between two granular particles, as described in <A HREF = "#Silbert">(Silbert)</A>
and in <A HREF = "#Zhang">(Zhang)</A>, 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.
</P>
<CENTER><IMG SRC = "Eqs/pair_granular.jpg">
<P>The two Hookean styles use this formula:
</P>
<CENTER><IMG SRC = "Eqs/pair_gran_hooke.jpg">
</CENTER>
<P>The 1st term is a normal force and the 2nd term is a tangential force.
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>The Hertzian style uses this formula:
</P>
<UL><LI>delta = d - r
<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>
<CENTER><IMG SRC = "Eqs/pair_gran_hertz.jpg">
</CENTER>
<P>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 <I>hooke/history</I> and <I>hertz/history</I>, but is not included
in pair style <I>hooke</I>. The tangential damping force term is included
in all three pair styles if <I>dampflag</I> is set to 1; it is not included
if <I>dampflag</I> is set to 0.
</P>
<P>The other quantities in the equations are as follows:
</P>
<UL><LI>delta = d - r = overlap distance of 2 particles
<LI>Kn = elastic constant for normal contact
<LI>Kt = elastic constant for tangential contact = 2/7 of Kn
<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>Delta St = tangential displacement vector between 2 spherical particles which is truncated to satisfy a frictional yield criterion
<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. 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>The Kn and gamma_n coefficients are specified as parameters to the
pair_style command. The interpretation and units for these
coefficients are different in the Hookean versus Hertzian formulas.
</P>
<P>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 Kn is in units of (force/distance). Note that this
push-back force is independent of absolute particle size (in the
monodisperse case) or 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),
Kt is in units of (force/distance), and gamma_t is in units of
(1/time).
</P>
<P>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 the specified Kn is in units of (force/area). The effects of
absolute particle size (monodispersity) and relative size
(polydispersity) are captured in the radii-dependent pre-factors.
When these pre-factors 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), Kt is in units of (force/area), and gamma_t is in
units of (1/time-distance).
</P>
<P>Note that in the Hookean case, Kn can be thought of as a spring
constant with units of force/distance. In the Hertzian case, Kn is
like a non-linear spring constant with units of force/area, and as
shown in the <A HREF = "#Zhang">(Zhang)</A> paper, Kn = 4G / (3(1-nu)) where nu =
the Poisson ratio, G = shear modulus = E / (1(1+nu)), and E = Young's
modulus. Thus in the Hertzian case Kn can be set to a value 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 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.
</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.
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.
</P>
<P>For granular styles there are no additional coefficients to set for
each pair of atom types via the <A HREF = "pair_coeff.html">pair_coeff</A> command.
@ -83,13 +129,12 @@ pairs of granular atom types. For example the command
<PRE>pair_coeff * *
</PRE>
<P>should be used if all atoms in the simulation interact via a granular
potential. If a granular potential is used as part of <A HREF = "pair_hybrid.html">pair_style
potential (i.e. one of the pair styles above is used). If a granular
potential is used as a sub-style of <A HREF = "pair_hybrid.html">pair_style
hybrid</A>, then specific atom types can be used in the
pair_coeff command to determine which atoms interact via a granular
potential.
</P>
<P>See the citation below for more discussion of granular potentials.
</P>
<HR>
<P><B>Mixing, shift, table, tail correction, restart, rRESPA info</B>:
@ -126,4 +171,8 @@ is only enabled if LAMMPS was built with that package. See the
<P><B>(Silbert)</B> Silbert, Ertas, Grest, Halsey, Levine, Plimpton, Phys Rev
E, 64, p 051302 (2001).
</P>
<A NAME = "Zhang"></A>
<P><B>(Zhang)</B> Zhang and Makse, Phys Rev E, 72, p 011301 (2005).
</P>
</HTML>

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doc/pair_gran.txt
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@ -6,65 +6,111 @@
:line
pair_style gran/hertzian command :h3
pair_style gran/history command :h3
pair_style gran/no_history command :h3
pair_style gran/hooke command :h3
pair_style gran/hooke/history command :h3
pair_style gran/hertz/history command :h3
[Syntax:]
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 (force/distance units) :l
gamma_n = damping coefficient for normal direction collisions (1/time units) :l
xmu = static yield criterion (unitless) :l
style = {gran/hooke} or {gran/hooke/history} or {gran/hertz/history} :ulb,l
Kn = spring constant for particle repulsion (see units below) :l
gamma_n = damping coefficient for collisions in normal direction (see units below) :l
xmu = static yield criterion (unitless fraction between 0.0 and 1.0) :l
dampflag = 0 or 1 if tangential damping force is excluded or included :l,ule
[Examples:]
pair_style gran/history 200000.0 0.5 1.0 1 :pre
pair_style gran/history 200000.0 50.0 0.5 1 :pre
[Description:]
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 = Ri + Rj, where Ri
and Rj are the radii of the two particles:
The {gran} styles use the following formulas for the frictional force
between two granular particles, as described in "(Silbert)"_#Silbert
and in "(Zhang)"_#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.
:c,image(Eqs/pair_granular.jpg)
The two Hookean styles use this formula:
The 1st term is a normal force and the 2nd term is a tangential force.
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:
:c,image(Eqs/pair_gran_hooke.jpg)
delta = d - r
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}
The Hertzian style uses this formula:
:c,image(Eqs/pair_gran_hertz.jpg)
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
Kn = elastic constant for normal contact
Kt = elastic constant for tangential contact = 2/7 of Kn
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 \
Delta St = tangential displacement vector between 2 spherical particles \
which is truncated to satisfy a frictional yield criterion
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. 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.
The Kn and gamma_n coefficients are specified as parameters to the
pair_style command. The interpretation and units for these
coefficients are different in the Hookean versus Hertzian formulas.
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 Kn is in units of (force/distance). Note that this
push-back force is independent of absolute particle size (in the
monodisperse case) or 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),
Kt 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 the specified Kn is in units of (force/area). The effects of
absolute particle size (monodispersity) and relative size
(polydispersity) are captured in the radii-dependent pre-factors.
When these pre-factors 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), Kt is in units of (force/area), and gamma_t is in
units of (1/time-distance).
Note that in the Hookean case, Kn can be thought of as a spring
constant with units of force/distance. In the Hertzian case, Kn is
like a non-linear spring constant with units of force/area, and as
shown in the "(Zhang)"_#Zhang paper, Kn = 4G / (3(1-nu)) where nu =
the Poisson ratio, G = shear modulus = E / (1(1+nu)), and E = Young's
modulus. Thus in the Hertzian case Kn can be set to a value 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 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.
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.
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.
For granular styles there are no additional coefficients to set for
each pair of atom types via the "pair_coeff"_pair_coeff.html command.
@ -75,13 +121,12 @@ pairs of granular atom types. For example the command
pair_coeff * * :pre
should be used if all atoms in the simulation interact via a granular
potential. If a granular potential is used as part of "pair_style
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"_pair_hybrid.html, then specific atom types can be used in the
pair_coeff command to determine which atoms interact via a granular
potential.
See the citation below for more discussion of granular potentials.
:line
[Mixing, shift, table, tail correction, restart, rRESPA info]:
@ -116,3 +161,6 @@ is only enabled if LAMMPS was built with that package. See the
:link(Silbert)
[(Silbert)] Silbert, Ertas, Grest, Halsey, Levine, Plimpton, Phys Rev
E, 64, p 051302 (2001).
:link(Zhang)
[(Zhang)] Zhang and Makse, Phys Rev E, 72, p 011301 (2005).