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

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.

Figure 1

Bistable element consisting of two elastic springs and a point mass, energy $\psi \left(u\right)$, and force $F\left(u\right)$.

Figure 2

Periodic chain of bistable elements.

Figure 3

Dispersion relation comparison.

Figure 4

Nonlinear spring force and approximations introduced for the three regimes for $d=1$.

Figure 5

Small-amplitude regime: numerical results compared to the linear Klein-Gordon solution.

Figure 6

Small-amplitude regime: $x-t$ contour diagram of the numerical solution; for comparison, the solid red line represents a positive characteristic of the theoretical solution.

Figure 7

Snapshots of the propagating wave (traveling from left to right) at different instances of time show the evolution of the envelope soliton. The sech-type envelope begins to form due to self-modulation, as the wave passes through the lattice.

Figure 8

Discrete Fourier transform of the spatial variation of the waveform at a chosen instant of time.

Figure 9

Comparison of the theoretical large-amplitude solution and numerically determined wave profile (for parameters $d=1$, $z_0=-93.2$, and $v=2.812$).

Figure 10

The $x-t$ contour diagram of the numerical solution for the large-amplitude regime. The red straight line is the best-fit line corresponding to the leading edge characteristic.

Figure 11

The characteristic curve corresponding to the leading edge of the wave is shown in comparison with the best-fit solution at $t=250$. The slope of the line determines the initial speed of the propagating wave. This speed is used to compute the exact solution in the kink soliton propagation. The kink slows down toward the end due to the energy radiated by the oscillatory tail.

Figure 12

Energy landscape $E\left(u\right)$ with positive and negative predeformation as well as without preloads.

Figure 13

The $x-t$ contour diagram of the numerical solution for a precompression ${\overline{u}}_0=0.03$. The kink characteristic is straighter than the characteristic without precompression.

Figure 14

(a) Wave profiles for ${\overline{u}}_0=0.3$ and $t_3>t_2>t_1$. (b) The $x-t$ contour diagram of the numerical solution for a precompression of ${\overline{u}}_0=0.3$.

Figure 15

Variation of the kink propagation velocity with precompression for an initial (normalized) velocity of the first node of $v_0=4$. The dotted line shows the characteristic sound speed ${\overline{c}}_0$ of the medium for comparison.