Wave Propagation | Neel Nadkarni

Introduction

Wave propagation in solids or structures refers to the movement of a small disturbance through a solid medium, with the speed of this disturbance, known as the wave speed, being a defining property of the material or structure. This fascinating area of research has diverse applications, from designing speaker diaphragms to enabling non-destructive evaluations of structures and understanding seismic activity caused by earthquakes.

Within this field, acoustic metamaterials have emerged as a remarkable class of engineered materials, exhibiting novel wave propagation characteristics due to the carefully tailored design of their microstructures. When I began exploring this exciting sub-field, much of the research focused on creating microstructures to achieve unique wave phenomena in the linear elastic regime. This led me to ask the question, "Can the nonlinearities arising from large deformations be harnessed to design metamaterials with novel properties in the nonlinear regime?"

Harnessing Structural Instabilities

Structural instabilities refer to the phenomenon where a structure or system, under the application of a small incremental load, undergo large deformations due to their inability to sustain load-carrying capacity. These instabilities arise when the structure cannot maintain a stable equilibrium due to the applied loads, material properties, or geometry. Common occurrences of such instabilities are of the geometric kind like snap-through instability, buckling, and material instabilities such as softening, plasticity, or phase transitions. Mathematically, this phenomenon can be expressed as a "loss of convexity" of the energy functional of the system.

One of the simplest examples of a system that shows a structural instability, due to its geometry, is shown in the figure below. Consider an element consisting of two elastic springs connected at an angle in their undeformed state. If the energy stored in the system is plotted as a function of the horizontal displacement in the system, it would result in a bistable energy function as shown in the figure. The second minimum of the energy function is located as a mirror image of the first minimum across the vertical axis. Such a system has rich dynamics: (1) for small amplitude perturbations, a linear elastic wave propagates through the system, with the ability to tune dispersion relation by adjusting the pre-compression applied to the system. (2) At moderate amplitudes, envelope solitary waves emerge, transporting localized energy packets through the array. Such waves comprise a sinusoidal background wave modulated by a hyperbolic secant envelope, traveling at a distinct speed. (3) At large amplitudes, a transition wave propagates along the chain, sequentially snapping each element. In a pre-compressed spring chain, this high-amplitude transition wave disperses energy through background phonon radiation.

small amplitudes

medium amplitudes

large amplitudes

Experimental Demonstrations

The above analysis was conducted under the idealized assumption of a conservative system without dissipation. However, in real-world scenarios, dissipation plays a significant role, and must be accounted for to design a practical experimental system. When dissipation is introduced, an intriguing phenomenon emerges: the large-amplitude transition wave exhibits a "universal energy scaling" behavior. Specifically, the kinetic energy of the wave is always linearly proportional to its velocity and inversely proportional to the dissipation constant. Furthermore, for the wave to propagate, the energy of the transitioned state must be lower than the initial energy of the undeformed state. This theory was experimentally validated using a linear array, as shown below. In this setup, each member of the array consists of a bistable membrane with one stable state at a higher energy than the other, and the interaction between members is mediated by magnetic repulsion. It was demonstrated that transition waves propagate within this system, with the experimental results showing excellent agreement with the theoretical predictions. A video illustrating the transition wave propagation is provided below.

experimental lattice

transition wave propagation

force curve of bistable membrane

force curve of magnets

video showing the wave propagation within the lattice

After successfully showing that the design works in a large-scale system, we proceeded with miniaturizing it to advance the development of nonlinear metamaterials. Our collaborators at Harvard 3D printed a lattice structure made from polymeric material (shown below), and successfully demonstrated that the transition wave propagates within this smaller-scale system, with the energy transport law remaining valid. In contrast, this system, constructed from polymeric materials, exhibits high internal damping, causing linear waves to be attenuated almost immediately. Additionally, we developed mechanical logic gates, such as diodes, "and" and "or" gates and switches, to tune wave propagation. This form of logic could be leveraged to introduce feedback and control in soft autonomous systems such as soft robotics and programmable mechanical transistors.

References

N. Nadkarni, C. Daraio, D. M. Kochmann, "Dynamics of periodic mechanical structures containing bistable elastic elements: From elastic to solitary wave propagation.", Phys. Rev. E, vol. 90, p. 023204, 2014.
N.Nadkarni, C.Daraio, R.Abeyaratne, D.M.Kochmann, "Universal energy transport law for dissipative and diffusive phase transitions.", Phys. Rev. B, vol. 93, p. 104109, 2016.
N. Nadkarni, A. F. Arrieta, C. Chong, D. M. Kochmann, C. Daraio, "Unidirectional transition waves in bistable lattices.", Phys. Rev. Lett., vol. 116, p. 244501, 2016.
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. Nat. Acad. Sci., vol. 113, no. 35, p. 9722, 2016.