Hi all,
I'm hoping someone familiar with granular models might be able to help
me understand the following.
The LAMMPS manual page for "pair_style gran/..."
(http://lammps.sandia.gov/doc/pair_gran.html) describes the various
models used in LAMMPS for granular dynamics simulations, in particular
a model based on Hertzian contact forces. The model given in the
LAMMPS manual most closely resembles that used in the paper by Silbert
et al. which is cited at the bottom of that same manual page.
However, it seems that the damping term used in that model doesn't
seem to agree with that used in Brilliantov et al. cited on the same
page, nor the models cited in some other articles, in particular those
of Brilliantov, Pöschel, and others, or similar models given in the
recent book by Pöschel and Schwager. These all seem to use a model
with a damping term of \+A\\sqrt\{\\delta\}\\frac\{d\\delta\}\{dt\}. The model
in LAMMPS uses a similar damping term, but scaled by the effective
particle mass.
So the right question might be, why scale by effective mass, or why
not? The paper by Brilliantov et al. suggests that the constant A
in their damping term is related to the elastic and viscous properties
of the material, with no mention of scaling by effective mass. I
should also note that, having performed simple simulations with both
the model in LAMMPS and some quick and dirty coding of the other
model, I see qualitative differences between them. In particular, the
particle size dependence of the coefficient of restitution (COR) seems
to be different: LAMMPS predicts that increasing particle size (with
constant density) leads to decreasing COR, and the other model
suggests the opposite. What I see experimentally with (low speed
collisions of small metallic spheres) suggests increasing particle
size increases COR, as in the other model.
In any case, if anyone has some insight into the models above (or the
actual physics of the problem), it would be quite helpful.
Thanks,
Jason
I'm forwarding this to a couple granular folks
here at Sandia who helped formulate the specific
model that is within LAMMPS. And to Leo Silbert at SIU.
Any of them may wish to comment.
Steve
Scaling by effective particle mass is simply a convention. It is
often used when studying 2nd order linear ODEs to reduce the amount of
terms appearing in the solution. This is often referred to as the
damping ratio, where as the non-scaled damping term is the damping
coefficient. Obviously, if you keep both terms the same and then
alter the mass, the effect will be different because the damping
coefficient will be the same in one and change in the other. However,
this also changes the effect of the spring, so the coefficient of
restitution changes.
The damping term you refer to from Brilliantov, is identical to the
Hertz model provided in lammps, which is more often referred to as the
Kuwabara Kono model in literature. There the CoR is not a constant,
but depends on velocity. Also remember, that these are phenomological
models for a give geometry, which only tries to approximate observed
behavior of real systems. More specifically to satisfy the
observation that for many spherical particles the CoR scales as V^1/5.
To the best of my knowledge these models have not been vigorously
proven to be predictable from material properties alone.
Scaling by effective particle mass is simply a convention. It is
often used when studying 2nd order linear ODEs to reduce the amount of
terms appearing in the solution. This is often referred to as the
damping ratio, where as the non-scaled damping term is the damping
coefficient. Obviously, if you keep both terms the same and then
alter the mass, the effect will be different because the damping
coefficient will be the same in one and change in the other. However,
this also changes the effect of the spring, so the coefficient of
restitution changes.
The damping term you refer to from Brilliantov, is identical to the
Hertz model provided in lammps, which is more often referred to as the
Kuwabara Kono model in literature. There the CoR is not a constant,
but depends on velocity. Also remember, that these are phenomological
models for a give geometry, which only tries to approximate observed
behavior of real systems. More specifically to satisfy the
observation that for many spherical particles the CoR scales as V^1/5.
To the best of my knowledge these models have not been vigorously
proven to be predictable from material properties alone.
Thanks for the response. You've confirmed some of what I suspected
about the mass scaling, but not being very familiar with the
experimental results from the literature myself, I'm still left
wondering, is the force model in LAMMPS reasonable to use for a
polydisperse system? Obviously, if I only wanted to model a system
with identical particles, then this is a moot point. But I'm more
interested in modeling a system with some distribution of sizes. If
the "V" in your expression below stands for volume, then I think the
model in LAMMPS gives the wrong trend, because the higher mass
particles would be subjected to stronger damping.
Jason,
I will assume that we are talking about the Hertzian model included in
Lammps. The Kuwabara Kono [1][2] model is an extension to the
Hertzian model, which is supposed to apply to arbitrary spherical
contacting geometries, i.e. polydispersity should not be a problem.
The 'V' I was referring to is, in fact, the relative normal velocities
between colliding granules.
To touch on how reasonable the model is, depends on what system you
want to simulate. The KK model was originally formulated for metals,
and so things like polymers wouldn't necessarily respond in a similar
fashion because linear elasticity isn't a good assumption for an
underlying constitutive model. The damping term in this model is, as
I wrote before, phenomelogical, and there are actually many possible
candidate damping models depending on your choice of material. As an
example, granite and other rocks have shown to produce a better fit to
data using the damping term, A\delta^{1/4}frac{d\delta}{dt} [3]. If
you provide a little more information about your system, I may be able
to provide more specific literature. Good luck.
[1] Kuwabara G, Kono K, Restitution Coefficient in a Collision between
Two Spheres. Japanese Journal of App. Phys. 26 (1987) 1230-1233.
[2] Stevens A.B. Hrenya C. M. Comparison of soft-sphere models to
measurements of collision properties during normal impacts. Powder
Tech. 154 (2005) 99-109.
[3] Ji S. Shen H. Effect of Contact Force Models on Granular Flow
Dynamics. Journal Eng. Mech. 132 (2006) 1252.
Jason,
I will assume that we are talking about the Hertzian model included in
Lammps. The Kuwabara Kono [1][2] model is an extension to the
Hertzian model, which is supposed to apply to arbitrary spherical
contacting geometries, i.e. polydispersity should not be a problem.
The 'V' I was referring to is, in fact, the relative normal velocities
between colliding granules.
To touch on how reasonable the model is, depends on what system you
want to simulate. The KK model was originally formulated for metals,
and so things like polymers wouldn't necessarily respond in a similar
fashion because linear elasticity isn't a good assumption for an
underlying constitutive model. The damping term in this model is, as
I wrote before, phenomelogical, and there are actually many possible
candidate damping models depending on your choice of material. As an
example, granite and other rocks have shown to produce a better fit to
data using the damping term, A\delta^{1/4}frac{d\delta}{dt} [3]. If
you provide a little more information about your system, I may be able
to provide more specific literature. Good luck.
[1] Kuwabara G, Kono K, Restitution Coefficient in a Collision between
Two Spheres. Japanese Journal of App. Phys. 26 (1987) 1230-1233.
[2] Stevens A.B. Hrenya C. M. Comparison of soft-sphere models to
measurements of collision properties during normal impacts. Powder
Tech. 154 (2005) 99-109.
[3] Ji S. Shen H. Effect of Contact Force Models on Granular Flow
Dynamics. Journal Eng. Mech. 132 (2006) 1252.