Dynamics of periodic mechanical structures containing bistable elastic elements: From elastic to solitary wave propagation

Abstract

We investigate the nonlinear dynamics of a periodic chain of bistable elements consisting of masses connected by elastic springs whose constraint arrangement gives rise to a large-deformation snap-through instability. We show that the resulting negative-stiffness effect produces three different regimes of (linear and nonlinear) wave propagation in the periodic medium, depending on the wave amplitude. At small amplitudes, linear elastic waves experience dispersion that is controllable by the geometry and by the level of precompression. At moderate to large amplitudes, solitary waves arise in the weakly and strongly nonlinear regime. For each case, we present closed-form analytical solutions and we confirm our theoretical findings by specific numerical examples. The precompression reveals a class of wave propagation for a partially positive and negative potential. The presented results highlight opportunities in the design of mechanical metamaterials based on negative-stiffness elements, which go beyond current concepts primarily based on linear elastic wave propagation. Our findings shed light on the rich effective dynamics achievable by nonlinear small-scale instabilities in solids and structures.

Stable propagation of mechanical signals in soft media using stored elastic energy

“…1D), a sufficiently large displacement applied to any of the bistable elements can cause the displaced element to transition states, producing a nonlinear transition wave that propagates indefinitely outward from the point of initiation with constant speed and shape. This is due to both (i) an equilibrium between dispersive and nonlinear effects of the periodic structure (18) and (ii) a release of energy that equals the effects of dissipation as, stimulated by the wavefront, each of the bistable elements along the chain transitions from its higher-to lower-stable energy state (i.e., from x = x s0 to x = x s1 = 0).…”

“…The damping intrinsic to the soft materials removes all signals except the desired transition wave, which therefore propagates with high fidelity, predictability, and controllability. Furthermore, as observed for nondissipative (18) or minimally dissipative systems (19) made from stiff materials, a series of interacting bistable units can transmit nondispersive transition waves. By contrast, the proposed architecture is capable of propagating stable waves with constant velocity over arbitrary distances, overcoming both dissipative and dispersive effects, despite the soft, dissipative material of which it is composed.…”

“…Dispersion can be controlled or eliminated through nonlinear effects produced via the control of structure in the medium (12). For example, periodic systems based on Hertzian contact (13)(14)(15), tensegrity structures (16), rigid bars and linkages (17), and bistable elastic elements (18) can behave as nondispersive media, with the nonlinearity of their local mechanical response canceling out the tendency for the signal to disperse at sufficiently large amplitudes. However, dissipation is still an overarching problem.…”

Exploiting Microstructural Instabilities in Solids and Structures: From Metamaterials to Structural Transitions

“…2(a), which results in a chain of particles grounded nonlinearly and in a bistable onsite potential [247]. Again, small, moderate, and large excitations of the chain will result in, respectively, linear, weakly nonlinear, and strongly nonlinear wave motion [248]. To treat the problem more generally, consider an infinite 1D chain of particles at positions x ¼ fx À1 ; …; x 1 g with x j ¼ ja and particle spacing a.…”

“…Under infinitesimalamplitude loading, linear waves propagate in the classical acoustic sense. When excited by moderate and large amplitudes, the chain propagates weakly nonlinear solitons or strongly nonlinear transition waves [248].…”