Dissipative Solitary Waves in Granular Crystals

My third paper in Physical Review Letters was officially published yesterday.

Title: Dissipative Solitary Waves in Granular Crystals

Authors: R. Carretero-Gonzalez, D. Khatri, Mason A. Porter, P. G. Kevrekidis, and C. Daraio

Abstract: We provide a quantitative characterization of dissipative effects in one-dimensional granular crystals. We use the propagation of highly nonlinear solitary waves as a diagnostic tool and develop optimization schemes that allow one to compute the relevant exponents and prefactors of the dissipative terms in the equations of motion. We thereby propose a quantitatively accurate extension of the Hertzian model that encompasses dissipative effects via a discrete Laplacian of the velocities. Experiments and computations with steel, brass, and polytetrafluoroethylene reveal a common dissipation exponent with a materialdependent prefactor.


In English, the point of this paper is as follows: The common model for the dynamics of granular crystals (otherwise known as chains of beads) is a conservative model. It's not that anybody thinks that other effects aren't relevant; rather, nobody knows how to incorporate them correctly, and it's a not a simple matter of getting something mechanistic through measurement of response curves. Trying to get a correct first-principles derivation of effects such as viscoelasticity and plasticity in order to improve the state-of-the-art models is a difficult and important endeavor. A few papers have recently appeared that insert an ad hoc dissipation term based on a linear dashpot (i.e., using a basic dx/dt type of term, as one sees in textbooks for damped harmonic oscillators), and our paper uses fitting between numerics and experiments to show that this is bounded away from being correct. (The exponent for a [dx/dt]^m type term appears to be around m = 1.75 to get the best fit to the amount of dissipation one actually sees experimentally.) Of course, our paper is not a first-principles by any stretch of the imagination. What we are now doing in this direction is to look at (1) highly plastic situations (by basically firing bullets at the chain of beads to obtain large irreversible deformations) and (2) highly viscoelastic situations in order to examine "limiting cases" to try to improve the prevailing conservative model there. We will then do the same type of fitting as in the just-published paper and ideally we'll get a sensible term that we can try to obtain with a derivation by generalizing some relevant papers that propose some functional forms that have some chance of being reasonable but haven't yet been tested in the lab for these systems.