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.