Gradient-Index Granular Crystals: From Boomerang Motion to Asymmetric Transmission of Waves

We present a gradient-index crystal that offers extreme tunability in terms of manipulating the propagation of elastic waves. For small-amplitude excitations, we achieve control over wave transmission depth into the crystal. We numerically and experimentally demonstrate a boomeranglike motion of a wave packet injected into the crystal. For large-amplitude excitations on the same crystal, we invoke nonlinear effects. We numerically and experimentally demonstrate asymmetric wave transmission from two opposite ends of the crystal. Such tunable systems can thus inspire a novel class of designed materials to control linear and nonlinear elastic wave propagation in multiscales.

Figure 1

Experiment setup to investigate wave dynamics in a gradient-index granular crystal.

Figure 2

Modal frequencies and transmission of the gradient-index granular chain in comparison to homogeneous chains. (a) Modal frequencies of a gradient-index chain with contact angle $10°\to 90°$ along with the homogeneous chains with only 10° and 90°. (b)–(c) State-space-based wave transmission for the homogeneous chain with contact angle 10° and 90°. (d) The same for the gradient-index chain. (e) Frequency spectrum vs space obtained from full numerical simulations on the gradient-index chain under an impact excitation. The highlighted line in red represents analytically obtained local cutoff frequency.

Figure 3

Linear wave dynamics in the gradient-index chain ($10°\to 90°$) under a Gaussian-modulated wave excitation. (a)–(b) Numerically and experimentally obtained spatiotemporal velocity maps for the excitation centered at 7 kHz. 3D line plots (in blue) are superimposed to highlight velocity time history at certain locations along the chain. (c)–(d) The same at 14 kHz with the boomeranglike wave propagation. (e)–(f) The same at 24 kHz.

Figure 4

Nonlinear wave dynamics in the gradient-index granular crystal. We observe asymmetry in wave transmission with increasing excitation amplitude from 0.11 (left panel), 0.51 (middle), to $1.01\mu \mathrm{m}$ (right), at 13.5 kHz for two configurations: forward ($10°\to 90°$, top panel) and reverse ($90°\to 10°$, bottom).

Figure 5

Comparison of asymmetric wave transmission data obtained from experiments and numerical simulations for two configurations: forward ($10°\to 90°$) and reverse ($90°\to 10°$). The maximum velocity of the last (37th) particle from the actuator is compared with various excitation amplitude of the first particle from the actuator. Inset shows the experimentally obtained frequency content at the output of the chain. Two vertical gray lines denote the region of local cutoff frequencies in the chain. The arrow indicates the frequency shift from the input to the lowest local cutoff frequency. Double-sided arrow denotes the magnitude of asymmetry.