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 localized compression pulses (solitary waves) or 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 [22], 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-Lipschitzian 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 reference [8], we analyze the discrete $p$-Schrödinger equation, an envelope equation that describes small oscillations in a Newton's cradle. In the limit when the exponent of the contact force lies slightly above unity, we derive three different continuum limits of the model which allow us to approximate the profiles of traveling breather solutions. One model consists of a logarithmic nonlinear Schrödinger equation which leads to a Gaussian approximation, and the two other are fully nonlinear degenerate Schrödinger equations which provide compacton approximations. These approximations are numerically validated by Newton-type computations. In the opposite (vibroimpact) limit when the exponent of the contact force is large, we obtain an analytical approximation of solitary waves in the form of a compacton.