SEED: Cnoidal Wave Propagation in an Elastic Metamaterial

figure one

Geometry of metamaterial with rotating subunits and accelerometer locations.

Cnoidal waves are nonlinear waves that have been observed broadly in numerous physical contexts, including shallow water, geophysical fluid dynamics, ion plasmas, nonlinear optics and geomaterials. We have recently shown that cnoidal waves can also be observed in elastic solids [1].

This work is of importance to the growing community of  researchers who study nonlinear dynamics in structured media  (“mechanical metamaterials”), in which the understanding of nonlinear waves is central to the control of stress wave propagation in applications as varied as protective materials, acoustics, vibration mitigation, and aerospace. The past decade has seen advances in fabrication techniques that have led to nearly arbitrary control of geometry in mechanical metamaterials. The  dynamic behavior of these materials has been of significant interest, but primarily from the vantage point of linear phenomena, such as the phonon spectrum and band gaps. More recently, a few studies have appeared in which  the propagation of solitary waves, or solitons, in nonlinear media is explored. Here, we  show analytically, numerically, and experimentally that cnoidal waves can propagate in an elastic metamaterial. Crucially, this generalizes the results of many prior works, since both phonon spectra and solitary waves are merely cnoidal wave solutions at opposite distinct limits.

figure 2

Sharp peaks and broad troughs in the angular deflection plots as a function of
time are a signature of cnoidal waves. The linear acceleration corresponding to the angular deflection on the left is periodic, but not sinusoidal [1].

We also have work in review on the effect of material anisotropy on impulse propagation in a similar system [2].

[1] Mo, C., Singh, J., Raney, J.R., Purohit, P. K.,”Cnoidal wave propagation in an elastic metamaterial”, in review Phys. Rev. (2019).

[2] Deng, B., Mo, C., Tournat, V., Bertoldi, K., Raney, J.R.,”Focusing and mode separation of elastic vector solitons in a 2D soft mechanical metamaterial”, in review Phys. Rev. Lett. (2019).



At Philly Materials Day 2019 Raney designed experiments to illustrate the importance of nonlinear mechanical behavior in materials and structures. Our booth was titled “The Utility of Instability” and explained concepts such as snap-through, multistability, and material dynamics to a diverse audience of K-12 students and families from the local area. We are also organizing technical symposia related to these topics at conferences such as Mach, SES, and IMECE.

figure 2

Outreach at Philly Materials Day, February 2019