**Nonlinear dynamics from snapping instabilities**

Metamaterials are synethtic, engineered materials systems which oftentimes excel by properties not found in natural materials (including extreme or counterintuitive physical behavior). Instabilities present a particular avenue to create materials that may undergo reconfiguration. Current research in the field of mechanical metamaterials is primarily focused on their quasistatic performance (optimizing, e.g., stiffness, strength, reconfigurability, etc.), further the linear wave propagation characteristics (such as in phononic or photonic metmaterials) as well as combinations thereof (usually linear waves superimposed onto large pre-deformation possibly involving structural instabilities). Applications range from acoustic lenses and diodes to sound isolators and acoustic sensing, to acoustic cloaks and sonar stealth technologies. In contrast, research in nonlinear metamaterials made of periodic building blocks that undergo large deformation and possibly structural instability is still in its early stages.

In recent years, we have exploited structural instabilities to create nonlinear mechanical metamaterials which, unlike linear systems, permit multiple types of wave solutions and thus can be tuned to propagate a variety of waves having small or finite amplitudes. With our collaborators Professor Chiara Daraio at ETH Zurich as well as with Professors Katia Bertoldi, Jennifer Lewis and Jordan Raney at Harvard University, we have explored strategies to design such nonlinear systems, analyzed their performance theoretically, fabricated protoypes, and characterized their dynamic behavior experimentally.

*Figure: (a) bistable system consisting of elastic springs and masses to result in two stable equilibria (the curves illustrate the energy landscape and its derivative which represents the force on the element); (b) the periodic array of such bistable elements yields a tunable array for nonlinear elastic waves.*

Consider, e.g., the example system shown above, in which structural buckling and snapping leads to large nonlinear motion. The dynamics of the shown periodic elastic structure display four independent regimes of wave propagation: linear, weakly nonlinear, strongly nonlinear, and pre-compressed strongly nonlinear waves. For each regime, we have derived analytical solutions for the propagating wave and have characterized the wave profiles and speeds. At small amplitudes linear acoustic waves travel through the structure, whose dispersion relation can be controlled by the amount of precompression applied to the system. At moderate amplitudes envelope solitary waves transport localized energy packages through the array. At large amplitudes, a topological soliton (i.e., a transition wave) propagates down the chain by subsequently snapping each element (here, precompression can be utilized to not only alter the wave speed but also, in the presence of damping, to force the wave propagation to be unidirectional, resulting in nonlinear mechanical diodes).

*Figure: three regimes of wave propagation from linear phononic waves (on the left) to envelope solitary waves (in the middle) to transition waves (on the right) - click to zoom.*

Most interesting is the strongly-nonlinear transition wave whose energetics where further investigated theoretically. We derived a universal scaling law that shows how the speed of the propagating transition front is related to the difference in energy of the two stable equilibrium configurations, the damping in the system, and the shape of the propagating kink. Interestingly, the scaling law is linear despite the highly-nonlinear nature of the problem. Also, it is unaffected by the interactions between bistable elements and holds in the case of vanishing masses (diffusive/dissipative system). The latter is important because it implies that the derived relations also apply to domain wall motion in a variety of physical, chemical, and biological systems including, e.g., ferroelectric or ferromagnetic domain wall motion (the 1D theory also applies to wave fronts in higher dimensions).

*Figure: bistable asymmetric membranes elastically connected by repelling permanent magnets (force-displacement curves for magnets and membranes were determined quasistatically) and the resulting transition wave that propagates at constant speed (shown is the comparison of the wave profile from experiments and simulations) - click to zoom. Full study here.*

To test our theory, a macroscopic structure consisting of bistable membranes elastically connected by repelling permanent magnets was built and tested at ETH, which is shown above. The fabrication method renders the composite membranes inherently asymmetric, resulting in a non-symmetric energy potential (i.e., one equilibrium has higher energy than the other). As a consequence, we observed stable transition waves of constant propagation speed that propagate only unidirectionally, thus displaying the behavior of a mechanical diode. We note that despite the complex three-dimensional motion of the snapping membranes, the wave propagation can be described accurately by a one-dimensional model.

*Figure: experimental realization of transition waves in 3D-printed chains of bistable beams connected by elastic springs. (a) the chain propagates transition waves upon an initial impact from the left; (b) the wave moves with constant speed v and has width w, also shown are the two stable equilibrium configurations of each bistable beam element (having different levels of stored strain energy) - click to zoom. Full study here.*

Our experimental collaborators at Harvard University have fabricated a smaller-scale bistable chain shown above, using state-of-the-art 3D printing techniques. This system, made of polymeric materials, has high internal damping so that linear waves are attenuated almost instantanesouly. By contrast, our transition waves (powered by the energy release during each snapping event from higher to lower energy) propagate at constant speed over, in principle, infinite distances. The speed of the kink arises from the balance between energy release due to snapping and energy absorption due to damping. Again, diodic motion is observed (this time admitting both compressive and rarefaction waves, unlike in the magnetic chain without tensile strength of the connector magnets). Further, the same building principle was used to build elements of mechanical logic such as switches and diodes, as well as for wave tuning (e.g., to control the wave speed). Shown below are example systems.

*Figure: examples of elements of mechanical logic obtained from chains of bistable elements: (a) a mechanical diode transmits nonlinear transition waves only unidirectionally, (b) a mechanical analogs of AND and OR switches. Full study here.*

#### References:

- J. R. Raney, N. Nadkarni, C. Daraio, D. M. Kochmann, J. A. Lewis, K. Bertoldi. Stable propagation of mechanical signals in soft media using stored elastic energy, Proc. Natl. Acad. Sci., published online (2016).
- N. Nadkarni, A. F. Arrieta, C. Chong, D. M. Kochmann, C. Daraio.
Unidirectional Transition Waves in Bistable Lattices,
*Phys. Rev. Lett.***116**(2016), 244501. - N. Nadkarni, C. Daraio, R. Abeyaratne, D. M. Kochmann.
A universal energy transport law for dissipative and diffusive phase transitions,
*Phys. Rev. B***93**(2016), 104109. - N. Nadkarni, C. Daraio, D. M. Kochmann.
Dynamics of a periodic structure containing bistable elastic elements:
from elastic to solitary wave propagation,
*Phys. Rev. E***90**(2014), 023204. - M. J. Frazier, D. M. Kochmann. Band gap transmission in periodic, bistable mechanical systems, under review (2016).