Section: New Results

Nonlinear waves in granular chains

Participants : Guillaume James, Bernard Brogliato, Kirill Vorotnikov.

Granular chains made of aligned beads interacting by contact (e.g. Newton's cradle) are widely studied in the context of impact dynamics and acoustic metamaterials. In order to describe the response of such systems to impacts or vibrations, it is important to analyze different wave effects such as the propagation of compression waves (solitary waves or fronts) or localized oscillations (traveling breathers), or the scattering of vibrations through the chain. Such phenomena are strongly influenced by contact nonlinearities (Hertz force), spatial inhomogeneities and dissipation.

In the work [8], we analyze the Kuwabara-Kono (KK) model for contact damping, and we develop new approximations of this model which are efficient for the simulation of multiple impacts. The KK model is a simplified viscoelastic contact model derived from continuum mechanics, which allows for simpler calibration (using material parameters instead of phenomenological ones), but its numerical simulation requires a careful treatment due to its non-Lipschitz character. Using different dissipative time-discretizations of the conservative Hertz model, we show that numerical dissipation can be tuned properly in order to reproduce the physical dissipation of the KK model and associated wave effects. This result is obtained analytically in the limit of small time steps (using methods from backward analysis) and is numerically validated for larger time steps. The resulting schemes turn out to provide good approximations of impact propagation even for relatively large time steps.

In addition, G.J. has developed a theoretical method to analyze impacts in homogeneous granular chains with KK dissipation. The idea is to use the exponent $\alpha$ of the contact force as a parameter and derive simpler dynamical equations through an asymptotic analysis, in the limit when $\alpha$ approaches unity and long waves are considered. In that case, different continuum limits of the granular chain can be obtained. When the contact damping constant remains of order unity, wave profiles are well approximated by solutions of a viscous Burgers equation with logarithmic nonlinearity. For small contact damping, dispersive effects must be included and the continuum limit corresponds to a KdV-Burgers equation with logarithmic nonlinearity. By studying traveling wave solutions to these partial differential equations, we obtain analytical approximations of wave profiles such as compression fronts. We observe that these approximations remain meaningful for the classical exponent $\alpha =3/2$. Indeed, they are close to exact wave profiles computed numerically for the KK model, using both dynamical simulations (response of the chain to a compression by a piston) and the Newton method (computation of exact traveling waves by a shooting method). In addition, in analogy with the Rankine-Hugoniot conditions for hyperbolic systems, we relate the asymptotic states of the KK model (for an infinite granular chain) to the velocity of a propagating front. These results are described in an article in preparation.