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<H3>gran rolling_friction epsd3
</H3>
<P><B>Syntax:</B>
</P>
<PRE>rolling_friction epsd3
</PRE>
<P><B>Description:</B>
</P>
<P>This model can be used as part of <A HREF = "pair_gran.html">pair gran</A> and
<A HREF = "fix_wall_gran.html">fix wall/gran</A>
</P>
<P>The elastic-plastic spring-dashpot (EPSD) model (see Ai et al.) adds an additional
torque contribution, equal to
</P>
<CENTER><IMG SRC = "Eqs/pair_gran_hooke_eqEpsd1.gif">
</CENTER>
<P>where the torque due to the spring M_rk is calculated as
</P>
<CENTER><IMG SRC = "Eqs/pair_gran_hooke_eqEpsd2.gif">
</CENTER>
<P>Here k_r denotes the rolling stiffness and dtheta_r is the incremental relative
rotation between the particles. The spring torque is limited by the full mobilisation
torque M_rm that is determined by the normal force F_n and the coefficient of rolling
friction (rmu) (compare the <A HREF = "gran_rolling_friction_cdt.html">CDT model</A>).
</P>
<P>The rolling stiffness k_r is computed using the "coeffRollingStiffness" prefactor,
that needs to be defined by the user (see below). k_r is computed from
k_r = coeffRollingStiffness*k_n*rmu*rmu*reff*reff.
</P>
<P>The viscous damping torque M_rd is implemented as
</P>
<CENTER><IMG SRC = "Eqs/pair_gran_hooke_eqEpsd3.gif">
</CENTER>
<P>where in the current implementation the damping is disabled in case of full mobilisation
(f = 0). The damping coefficient C_r may be expressed as:
</P>
<CENTER><IMG SRC = "Eqs/pair_gran_hooke_eqEpsd4.gif">
</CENTER>
<P>Here I_i/j is the moment of inertia and m_i/j is the mass of the particles i and j,
respectively.
</P>
<P>The coefficient of rolling friction (rmu) must be defined as
</P>
<PRE>fix id all property/global coefficientRollingFriction peratomtypepair n_atomtypes value_11 value_12 .. value_21 value_22 .. .
(value_ij=value for the coefficient of rolling friction between atom type i and j; n_atomtypes is the number of atom types you want to use in your simulation)
</PRE>
<P>This coefficient rmu is equal to the rmu as defined in the <A HREF = "gran_rolling_friction_cdt.html">CDT model</A>.
In addition to rmu, eta_r is the required material property that must be defined as
</P>
<PRE>fix id all property/global coefficientRollingViscousDamping peratomtypepair n_atomtypes value_11 value_12 .. value_21 value_22 .. .
(value_ij=value for the coefficient of rolling friction between atom type i and j; n_atomtypes is the number of atom types you want to use in your simulation)
</PRE>
<P>The "coeffRollingStiffness" prefactor needs to be defined by the user as
</P>
<PRE>fix id all property/global coeffRollingStiffness scalar value
</PRE>
<P>Please see Ai et al., 2011 for discussion.
</P>
<P>IMPORTANT NOTE: You have to use atom styles beginning from 1, e.g. 1,2,3,...
</P>
<P><B>(Ai)</B> Jun Ai, Jian-Fei Chen, J. Michael Rotter, Jin Y. Ooi, Powder Technology, 206 (3), p 269-282 (2011).
</P>
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