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224 lines
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<CENTER><A HREF = "http://lammps.sandia.gov">LAMMPS WWW Site</A> - <A HREF = "Manual.html">LAMMPS Documentation</A> - <A HREF = "Section_commands.html#comm">LAMMPS Commands</A>
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<H3>pair_style gran/hooke command
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</H3>
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<H3>pair_style gran/hooke/history command
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</H3>
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<H3>pair_style gran/hertz/history command
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</H3>
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<P><B>Syntax:</B>
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</P>
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<PRE>pair_style style Kn Kt gamma_n gamma_t xmu dampflag
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</PRE>
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<UL><LI>style = <I>gran/hooke</I> or <I>gran/hooke/history</I> or <I>gran/hertz/history</I>
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<LI>Kn = elastic constant for normal particle repulsion (force/distance units or pressure units - see discussion below)
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<LI>Kt = elastic constant for tangential contact (force/distance units or pressure units - see discussion below)
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<LI>gamma_n = damping coefficient for collisions in normal direction (1/time units or 1/time-distance units - see discussion below)
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<LI>gamma_t = damping coefficient for collisions in tangential direction (1/time units or 1/time-distance units - see discussion below)
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<LI>xmu = static yield criterion (unitless fraction between 0.0 and 1.0)
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<LI>dampflag = 0 or 1 if tangential damping force is excluded or included
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</UL>
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<P>IMPORTANT NOTE: Versions of LAMMPS before 9Jan09 had different style
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names for granular force fields. This is to emphasize the fact that
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the Hertzian equation has changed to model polydispersity more
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accurately. A side effect of the change is that the Kn, Kt, gamma_n,
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and gamma_t coefficients in the pair_style command must be specified
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with different values in order to reproduce calculations made with
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earlier versions of LAMMPS, even for monodisperse systems. See the
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NOTE below for details.
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</P>
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<P><B>Examples:</B>
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</P>
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<PRE>pair_style gran/hooke/history 200000.0 NULL 50.0 NULL 0.5 1
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pair_style gran/hooke 200000.0 70000.0 50.0 30.0 0.5 0
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</PRE>
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<P><B>Description:</B>
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</P>
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<P>The <I>gran</I> styles use the following formulas for the frictional force
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between two granular particles, as described in
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<A HREF = "#Brilliantov">(Brilliantov)</A>, <A HREF = "#Silbert">(Silbert)</A>, and
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<A HREF = "#Zhang">(Zhang)</A>, when the distance r between two particles of radii
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Ri and Rj is less than their contact distance d = Ri + Rj. There is
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no force between the particles when r > d.
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</P>
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<P>The two Hookean styles use this formula:
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</P>
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<CENTER><IMG SRC = "Eqs/pair_gran_hooke.jpg">
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</CENTER>
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<P>The Hertzian style uses this formula:
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</P>
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<CENTER><IMG SRC = "Eqs/pair_gran_hertz.jpg">
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</CENTER>
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<P>In both equations the first parenthesized term is the normal force
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between the two particles and the second parenthesized term is the
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tangential force. The normal force has 2 terms, a contact force and a
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damping force. The tangential force also has 2 terms: a shear force
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and a damping force. The shear force is a "history" effect that
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accounts for the tangential displacement between the particles for the
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duration of the time they are in contact. This term is included in
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pair styles <I>hooke/history</I> and <I>hertz/history</I>, but is not included
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in pair style <I>hooke</I>. The tangential damping force term is included
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in all three pair styles if <I>dampflag</I> is set to 1; it is not included
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if <I>dampflag</I> is set to 0.
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</P>
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<P>The other quantities in the equations are as follows:
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</P>
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<UL><LI>delta = d - r = overlap distance of 2 particles
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<LI>Kn = elastic constant for normal contact
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<LI>Kt = elastic constant for tangential contact
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<LI>gamma_n = viscoelastic damping constant for normal contact
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<LI>gamma_t = viscoelastic damping constant for tangential contact
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<LI>m_eff = Mi Mj / (Mi + Mj) = effective mass of 2 particles of mass Mi and Mj
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<LI>Delta St = tangential displacement vector between 2 spherical particles which is truncated to satisfy a frictional yield criterion
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<LI>n_ij = unit vector along the line connecting the centers of the 2 particles
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<LI>Vn = normal component of the relative velocity of the 2 particles
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<LI>Vt = tangential component of the relative velocity of the 2 particles
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</UL>
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<P>The Kn, Kt, gamma_n, and gamma_t coefficients are specified as
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parameters to the pair_style command. If a NULL is used for Kt, then
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a default value is used where Kt = 2/7 Kn. If a NULL is used for
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gamma_t, then a default value is used where gamma_t = 1/2 gamma_n.
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</P>
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<P>The interpretation and units for these 4 coefficients are different in
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the Hookean versus Hertzian equations.
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</P>
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<P>The Hookean model is one where the normal push-back force for two
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overlapping particles is a linear function of the overlap distance.
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Thus the specified Kn is in units of (force/distance). Note that this
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push-back force is independent of absolute particle size (in the
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monodisperse case) and of the relative sizes of the two particles (in
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the polydisperse case). This model also applies to the other terms in
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the force equation so that the specified gamma_n is in units of
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(1/time), Kt is in units of (force/distance), and gamma_t is in units
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of (1/time).
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</P>
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<P>The Hertzian model is one where the normal push-back force for two
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overlapping particles is proportional to the area of overlap of the
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two particles, and is thus a non-linear function of overlap distance.
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Thus Kn has units of force per area and is thus specified in units of
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(pressure). The effects of absolute particle size (monodispersity)
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and relative size (polydispersity) are captured in the radii-dependent
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pre-factors. When these pre-factors are carried through to the other
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terms in the force equation it means that the specified gamma_n is in
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units of (1/(time*distance)), Kt is in units of (pressure), and
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gamma_t is in units of (1/(time*distance)).
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</P>
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<P>Note that in the Hookean case, Kn can be thought of as a linear spring
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constant with units of force/distance. In the Hertzian case, Kn is
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like a non-linear spring constant with units of force/area or
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pressure, and as shown in the <A HREF = "#Zhang">(Zhang)</A> paper, Kn = 4G /
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(3(1-nu)) where nu = the Poisson ratio, G = shear modulus = E /
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(2(1+nu)), and E = Young's modulus. Similarly, Kt = 8G / (2-nu).
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Thus in the Hertzian case Kn and Kt can be set to values that
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corresponds to properties of the material being modeled. This is also
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true in the Hookean case, except that a spring constant must be chosen
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that is appropriate for the absolute size of particles in the model.
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Since relative particle sizes are not accounted for, the Hookean
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styles may not be a suitable model for polydisperse systems.
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</P>
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<P>IMPORTANT NOTE: In versions of LAMMPS before 9Jan09, the equation for
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Hertzian interactions did not include the sqrt(RiRj/Ri+Rj) term and
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thus was not as accurate for polydisperse systems. For monodisperse
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systems, sqrt(RiRj/Ri+Rj) is a constant factor that effectively scales
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all 4 coefficients: Kn, Kt, gamma_n, gamma_t. Thus you can set the
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values of these 4 coefficients appropriately in the current code to
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reproduce the results of a previous Hertzian monodisperse calculation.
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For example, for the common case of a monodisperse system with
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particles of diameter 1, all 4 of these coefficients should now be set
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2x larger than they were previously.
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</P>
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<P>Xmu is also specified in the pair_style command and is the upper limit
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of the tangential force through the Coulomb criterion Ft = xmu*Fn,
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where Ft and Fn are the total tangential and normal force components
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in the formulas above. Thus in the Hookean case, the tangential force
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between 2 particles grows according to a tangential spring and
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dash-pot model until Ft/Fn = xmu and is then held at Ft = Fn*xmu until
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the particles lose contact. In the Hertzian case, a similar analogy
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holds, though the spring is no longer linear.
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</P>
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<P>For granular styles there are no additional coefficients to set for
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each pair of atom types via the <A HREF = "pair_coeff.html">pair_coeff</A> command.
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All settings are global and are made via the pair_style command.
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However you must still use the <A HREF = "pair_coeff.html">pair_coeff</A> for all
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pairs of granular atom types. For example the command
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</P>
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<PRE>pair_coeff * *
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</PRE>
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<P>should be used if all atoms in the simulation interact via a granular
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potential (i.e. one of the pair styles above is used). If a granular
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potential is used as a sub-style of <A HREF = "pair_hybrid.html">pair_style
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hybrid</A>, then specific atom types can be used in the
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pair_coeff command to determine which atoms interact via a granular
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potential.
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</P>
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<HR>
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<P><B>Mixing, shift, table, tail correction, restart, rRESPA info</B>:
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</P>
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<P>The <A HREF = "pair_modify.html">pair_modify</A> mix, shift, table, and tail options
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are not relevant for granular pair styles.
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</P>
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<P>These pair styles write their information to <A HREF = "restart.html">binary restart
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files</A>, so a pair_style command does not need to be
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specified in an input script that reads a restart file.
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</P>
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<P>These pair styles can only be used via the <I>pair</I> keyword of the
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<A HREF = "run_style.html">run_style respa</A> command. They do not support the
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<I>inner</I>, <I>middle</I>, <I>outer</I> keywords.
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</P>
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<HR>
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<P><B>Restrictions:</B> none
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</P>
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<P>All the granular pair styles are part of the "granular" package. It
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is only enabled if LAMMPS was built with that package. See the
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<A HREF = "Section_start.html#2_3">Making LAMMPS</A> section for more info.
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</P>
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<P>These pair styles require that atoms store torque and angular velocity
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(omega) as defined by the <A HREF = "atom_style.html">atom_style</A>. They also
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require a per-particle radius is stored. The <I>granular</I> atom style
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does all of this.
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</P>
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<P>This pair style requires you to use the <A HREF = "communicate.html">communicate vel
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yes</A> option so that velocites are stored by ghost
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atoms.
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</P>
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<P><B>Related commands:</B>
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</P>
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<P><A HREF = "pair_coeff.html">pair_coeff</A>
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</P>
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<P><B>Default:</B> none
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</P>
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<HR>
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<A NAME = "Brilliantov"></A>
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<P><B>(Brilliantov)</B> Brilliantov, Spahn, Hertzsch, Poschel, Phys Rev E, 53,
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p 5382-5392 (1996).
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</P>
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<A NAME = "Silbert"></A>
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<P><B>(Silbert)</B> Silbert, Ertas, Grest, Halsey, Levine, Plimpton, Phys Rev
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E, 64, p 051302 (2001).
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</P>
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<A NAME = "Zhang"></A>
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<P><B>(Zhang)</B> Zhang and Makse, Phys Rev E, 72, p 011301 (2005).
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</P>
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</HTML>
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