# calculation of gamma_n for Hertzian soft-sphere model

I am using the granular package and have a question regarding the calculation of gamma_n.

I am using the function gamma_n = (-2/tc)ln(en)

where tc is the contact time, and en is the restitution coefficient.

I believe this function gives units of 1/time, however the notes show that in the Hertzian model, the coefficient requires units of 1/(time*distance). How can I adjust my calculated value to take into account the distance unit?

Thanks,

Tim

I am using the granular package and have a question regarding the calculation of gamma_n.

I am using the function gamma_n = (-2/tc)ln(en)

This relationship only holds for the linear spring dash-pot(Hookean) model and is the reason you are getting inconsistent units. Without going into too much details not specific to LAMMPS, here is my advice. Look into how this relationship is derived, i.e. you get a solution to a linear second-order differential equation. You can then invert the solution to get the contact time and from that use the velocity solution to get the coefficient of restitution. After that you have all you need to show your relationship is true.

That being said, you will find that if this same analysis is applied to the Hertzian (Kuwabara-Kono) model, you will get VERY different answers, and you’ll need a lot more mathematical firepower. (See Pöschel and Schwager, Phys Rev E 1998, KT of Granular Gases - Brilliantov and Pöschel) You’ll find that the form of the coefficient of restitution is also more complex(even non-linear) than just a constant preserved by the dynamics.

where tc is the contact time, and en is the restitution coefficient.

I believe this function gives units of 1/time, however the notes show that in the Hertzian model, the coefficient requires units of 1/(time*distance). How can I adjust my calculated value to take into account the distance unit?

A better route than augmenting an invalid relationship (and easier than applying what I said above) would be to non-dimensionalize your system yourself (could probably even find ref’s that do it if you’re so inclined), and go from there as its not hard to extract this behavior numerically(binary collision set-up). With so many parameters you should be doing this anyway.

Thanks Eric,

I’m just trying to get my head around everything you’ve said, but in regard to making a numerical model - I have created an FEM like analysis to give a prediction for the contact time and restitution coefficient. It’s basically a steel sphere hitting into a steel plate, as this is scenario I’m interested in. From this I can get the approximate contact time (when the velocity of the sphere stops changing) and the restitution coefficient (Final Vout/Vin).

Are you saying it is possible to gather an approximate value for the gamma_n constant from this simulation too?

If you take a look at the Brilliantov book(early chapters), you can get an approximate expression for the coefficient of restitution and t_c in the limit of low velocities as a function of gamma_n and impact velocity (low is a relative term, you need more terms if the velocity is sufficiently high). But, numerically you should be able to ‘map’ gamma_n to your system of interest. You could do it knowing t_c, but ideally you would know the coefficient of restitution as a function of impact velocity, which would make your matching job much easier.

I’d suggest taking a look at the book.

Has anyone else tried to use the Hertzian case and calculate a value for gamma_n? I feel like this should be a common step for anyone doing granular analysis.

Eric, I think I found the reference you are talking about: Kinetic Theory of Granular Gases, Nikolai V. Brilliantov and Thorsten Pöschel . I’m not sure which is the relation you’re talking about though. It is quite in-depth, and it will take me some time to understand what’s being expressed there.

I have experimental data showing a range of impacting / rebounding velocities. So I could probably find the coefficient of restitution as a function of impact velocity. The magnitude of impact is in the order of 1 - 10 m/s.

I’m not sure what you mean by ‘map’ gamma_n to my system? I can plug in values of gamma_n and generate similar results to the experimental data, however, I want to expand this simulation to other materials, so I need a relationship to work from.

Thanks,

Tim

Has anyone else tried to use the Hertzian case and calculate a value for gamma_n? I feel like this should be a common step for anyone doing granular analysis.

Eric, I think I found the reference you are talking about: Kinetic Theory of Granular Gases, Nikolai V. Brilliantov and Thorsten Pöschel . I’m not sure which is the relation you’re talking about though. It is quite in-depth, and it will take me some time to understand what’s being expressed there.

I have experimental data showing a range of impacting / rebounding velocities. So I could probably find the coefficient of restitution as a function of impact velocity. The magnitude of impact is in the order of 1 - 10 m/s.

I’m not sure what you mean by ‘map’ gamma_n to my system? I can plug in values of gamma_n and generate similar results to the experimental data, however, I want to expand this simulation to other materials, so I need a relationship to work from.

Pgs. 21-29 you’ll find your relation; coefficient of restitution dependent on gamma_n and impact velocity.

Thanks Eric,

I’ve looked further into the reference you suggested and just want to check a few things with you. I’ve not quite found the solution just yet…

In that section of his book, Equation 3.4 shows the dissipative force, which I assume is equal to the dissipative normal component of the hertzian function in Lammps. Then equating the components, gamma_n becomes equal to AY/(m_eff*(1-v)^2).

So then down to equation 3.20, it shows that the restitution coefficient is a function of A, some further material parameters, p, and the initial impact velocity, g. It then gives an actual equation to use in the form of equation 3.22, providing the first 4 coefficients (Ci) for this expansion (equation 3.24).

However, it is not clear the formula for A. The identity for A is given in equation 3.5, where it is a function of Poisson’s ratio, the Young’s modulus, and the viscous constants(?) n1 and n2. I’m not familiar with these latter two. I had a look at his 1996 papers, and he doesn’t actually explain anywhere how they were determined or calculated. Are these generally accepted material parameters that I’m unaware of? They seem related to the elastic constants he uses, E1 and E2, which he has a formula for, but not n1 and n2.

Given that I have an estimate for the restitution coefficient, I tried solving the expansion in equation 3.22 for the A value, but this proved quite difficult, and I haven’t been able to solve it yet… I used the solver function in excel.