calculation of gamma_n for Hertzian soft-sphere model

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.

n1 (in refers to the coefficient of viscosity, while n2 refers to the coefficient of bulk viscosity, both are encountered in, and fundamental to, fluid mechanics/rheology. You’d have to ask an experimentalist outside of the LAMMPS community how you’d measure/isolate this for a solid granular particle. No theories I know of are able to quantitatively predict this behavior for a solid from first principles.

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.

I agree, this is a difficult task, which is why I’ve never done it for materials I’ve worked with. Not to mention the method you are looking at only works if viscous type material behavior is the source of inelasticity in collisions. In principle, there are many possibilities(and many more proposed models) specific to a system you consider.