Update pair_body_rounded_polygon docs

\documentstyle[12pt]{article}

\begin{document}

\begin{eqnarray*}

 F_n &=& k_n \delta_n - c_n v_n, \qquad \delta_n \le 0 \\

     &=& -k_{na} \delta_n - c_n v_n, \qquad 0 < \delta_n \le r_c \\

     &=& 0 \qquad \qquad \qquad \qquad \delta_n > r_c \\

 F_t &=& \mu k_n \delta_n - c_t v_t, \qquad \delta_n \le 0 \\

     &=& 0 \qquad \qquad \qquad \qquad \delta_n > 0

\end{eqnarray*}

\end{document}

.. index:: pair\_style body/rounded/polygon

.. index:: pair_style body/rounded/polygon

pair\_style body/rounded/polygon command

========================================

pair_style body/rounded/polygon command

=======================================

Syntax

""""""

.. parsed-literal::

.. code-block:: LAMMPS

   pair_style body/rounded/polygon c_n c_t mu delta_ua cutoff

.. parsed-literal::

   c_n = normal damping coefficient

   c_t = tangential damping coefficient

   mu = normal friction coefficient during gross sliding

""""""""

.. parsed-literal::

.. code-block:: LAMMPS

   pair_style body/rounded/polygon 20.0 5.0 0.0 1.0 0.5

   pair_coeff \* \* 100.0 1.0

   pair_coeff * * 100.0 1.0

   pair_coeff 1 1 100.0 1.0

Description

Style *body/rounded/polygon* is for use with 2d models of body

particles of style *rounded/polygon*\ .  It calculates pairwise

body/body interactions which can include body particles modeled as

1-vertex circular disks with a specified diameter.  See the :doc:`Howto body <Howto_body>` doc page for more details on using body

1-vertex circular disks with a specified diameter.  See the

:doc:`Howto body <Howto_body>` doc page for more details on using body

rounded/polygon particles.

This pairwise interaction between rounded polygons is described in

vertices and edges themselves.

This means that the specified cutoff in the pair\_style command is the

cutoff distance, r\_c, for the surface separation, \delta\_n (see figure

cutoff distance, :math:`r_c`, for the surface separation, :math:`\delta_n` (see figure

below).  This is the distance at which two particles no longer

interact.  If r\_c is specified as 0.0, then it is a contact-only

interact.  If :math:`r_c` is specified as 0.0, then it is a contact-only

interaction.  I.e. the two particles must overlap in order to exert a

repulsive force on each other.  If r\_c > 0.0, then the force between

repulsive force on each other.  If :math:`r_c > 0.0`, then the force between

two particles will be attractive for surface separations from 0 to

r\_c, and repulsive once the particles overlap.

:math:`r_c`, and repulsive once the particles overlap.

Note that unlike for other pair styles, the specified cutoff is not

the distance between the centers of two particles at which they stop

interacting.  This center-to-center distance depends on the shape and

size of the two particles and their relative orientation.  LAMMPS

takes that into account when computing the surface separation distance

and applying the r\_c cutoff.

and applying the :math:`r_c` cutoff.

The forces between vertex-vertex, vertex-edge, and edge-edge overlaps

are given by:

.. image:: Eqs/pair_body_rounded.jpg

   :align: center

.. math::

   F_n &= \begin{cases}

           k_n \delta_n - c_n v_n     &  \delta_n \le 0 \\

          -k_{na} \delta_n - c_n v_n  &  0 < \delta_n \le r_c \\

          0                           & \delta_n > r_c \\

          \end{cases} \\

   F_t &= \begin{cases}

          \mu k_n \delta_n - c_t v_t & \delta_n \le 0 \\

          0                          & \delta_n > 0

          \end{cases}

.. image:: JPG/pair_body_rounded.jpg

   :align: center

Note that F\_n and F\_t are functions of the surface separation \delta\_n

= d - (R\_i + R\_j).  In this model, when (R\_i + R\_j) < d < (R\_i + R\_j)

+ r\_c, that is, 0 < \delta\_n < r\_c, the cohesive region of the two

surfaces overlap and the two surfaces are attractive to each other.

Note that :math:`F_n` and :math:`F_t` are functions of the surface separation

:math:`\delta_n = d - (R_i + R_j)`.  In this model, when

:math:`(R_i + R_j) < d < (R_i + R_j) + r_c`, that is, :math:`0 < \delta_n < r_c`,

the cohesive region of the two surfaces overlap and the two surfaces are

attractive to each other.

In :ref:`Fraige <pair-Fraige>`, the tangential friction force between two

particles that are in contact is modeled differently prior to gross

via the :doc:`pair_coeff <pair_coeff>` command as in the examples above,

or in the data file read by the :doc:`read_data <read_data>` command:

* k\_n (energy/distance\^2 units)

* k\_na (energy/distance\^2 units)

* :math:`k_n` (energy/distance\^2 units)

* :math:`k_{na}` (energy/distance\^2 units)

Effectively, k\_n and k\_na are the slopes of the red lines in the plot

above for force versus surface separation, for \delta\_n < 0 and 0 <

\delta\_n < r\_c respectively.

Effectively, :math:`k_n` and :math:`k_na` are the slopes of the red lines in the plot

above for force versus surface separation, for :math:`\delta_n < 0` and

:math:`0 < \delta_n < r_c` respectively.

**Mixing, shift, table, tail correction, restart, rRESPA info**\ :

Restrictions

""""""""""""

These pair styles are part of the BODY package.  They are only enabled

if LAMMPS was built with that package.  See the :doc:`Build package <Build_package>` doc page for more info.