Gradient-index granular crystals: From boomerang motion to asymmetric transmission of waves

TL;DR

This paper introduces a gradient-index granular crystal capable of tunably controlling elastic wave propagation, demonstrating boomerang-like wave motion and asymmetric transmission through numerical and experimental methods, with potential applications in advanced wave control.

Contribution

It presents a novel gradient-index granular crystal that enables tunable control of elastic waves, including linear and nonlinear effects, through both numerical simulations and experiments.

Findings

Demonstrated boomerang-like wave motion in the crystal

Achieved asymmetric wave transmission at different crystal ends

Validated tunable wave control through experiments and simulations

Abstract

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 boomerang-like motion of 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 multi-scales.

Full text

Introduction.— The advent of phononic crystals and metamaterials in recent years have shown excellent possibilities to manipulate elastic waves in materials [Maldovan, 2013; Kadic et al., 2013; Hussein et al., 2014]. Several ingenious designs have been proposed to build exotic devices, e.g., diode [Li et al., 2004; Liang et al., 2009; Boechler et al., 2011a], cloak [Farhat et al., 2009], negative refraction metamaterial [Zhang and Liu, 2004; Zhu et al., 2014], energy harvester [Yang et al., 2004; Carrara et al., 2013], impact absorber [Daraio et al., 2006], flow stabilizer [Hussein et al., 2015] and topological lattice [Wang et al., 2015; Ma et al., 2019]. The key idea is to use one or more ingredients among structural periodicity [Brillouin, 1953], local resonances [Liu, 2000], nonlinear effects [Chong et al., 2017], etc., to achieve nontrivial dynamical responses. The underlying physics in these demonstrations could also open new ways to control mechanical vibrations at nanoscale by optomechanical [Eichenfield et al., 2009] and nanophononic metamaterials [Davis and Hussein, 2014]. Therefore, the need of exploring advanced material architectures that offer rich wave physics is ever growing.

In this context, granular crystals [Nesterenko, 2001; Sen et al., 2008] – systematic arrangement of granular particles – offer a unique advantage. These architectures mimic atomic lattice dynamics in the sense that the grains can be regarded as atoms that interact via a nonlinear interaction potential stemming from the nature of the contact. These crystals are highly tunable, and a plethora of wave physics can be demonstrated in the same system [Porter et al., 2015]. Control over wave propagation in this setting shows many technological advances, ranging from impact and blast protection [Kim et al., 2015] to micro-scale granular beds [Hiraiwa et al., 2016].

In this Letter, we present a gradient-index granular crystal that offers even further tunability in terms of manipulating both linear and nonlinear elastic waves. These granular particles are of a cylindrical shape, where simply by tuning the contact angles between them, a gradient in stiffness can be achieved [Khatri et al., 2012]. Gradient-index materials have been extensively studied in optics and acoustics for various purposes, such as rainbow trapping [Tsakmakidis et al., 2007], opening wide bandgaps [Kushwaha et al., 1998], waveguides [Kurt and Citrin, 2007; He et al., 2008], lens [Smith et al., 2005; Torrent and Sánchez-Dehesa, 2007; Horsley et al., 2014; Jin et al., 2019], beamwidth compressor Lin et al. [2009], wave concentration [Romero-García et al., 2013], and absorbers [Climente et al., 2012; Liang et al., 2014]. Gradient-index systems are unique as the gradual variation in material/structural properties enables control over wave speed and wave directions at the same time minimizing wave scattering.

Using the gradient-index granular crystal, here, we numerically and experimentally demonstrate capability of wave control in two fronts. For small-amplitude waves, the system follows linear dynamics, and therefore, we demonstrate frequency-dependent wave penetration into the system. This includes a boomerang-like motion of injected wave packet that returns back to the point of excitation without propagating through the whole crystal. This is similar to a mirage effect. For large-amplitude waves, we invoke nonlinear effects [Narisetti et al., 2010; R. and Gonella, 2013; Abedinnasab and Hussein, 2013], and we show that the system offers asymmetric wave transmission in two opposite directions. This leads to one-way energy transport as a result of the interaction of nonlinearity and spatial asymmetry [Yang et al., 2007; Liang et al., 2009; Boechler et al., 2011a; Devaux et al., 2015; Wu et al., 2018; Moore et al., 2018]. Remarkably, all these characteristics can be tuned simply by changing the stacking angles and controlling the wave amplitude in the system.

Experimental and numerical setup.—Our system is composed of 37 cylinders (with the length and the diameter equal to 18 mm) stacked vertically and pre-compressed by a free weight on top as shown in Fig. 1. We vertically align the cylinders using 3D-printed cylindrical enclosures (lower inset of Fig. 1). Each enclosure has one cylinder inside, and deliberate clearances are provided to restrict their rattling in rotation and to minimize any friction in the translational direction. The enclosures are assembled in series and can be rotated independently to dial in contact angles between neighboring cylindrical particles inside. The cylinders interact as per the Hertz contact law [Johnson, 1985], and therefore, linear (nonlinear) wave dynamics can be studied at small (large) dynamic excitations in comparison to the static pre-compressive force ($F_{0}=29.4$ N). We vary the contact angles ranging from 10*∘* to 90*∘* along the chain such that the contact stiffness varies linearly along the chain. 10*∘* represents a stiff side, whereas 90*∘* is a soft side. A piezoelectric actuator (Piezomechanik PSt 500/10/25 VS18) is placed at the bottom of the chain in contact with the first particle. The actuator excites the chain using a Gaussian wave packet with a specific central frequency. A function generator (Agilent 33220A) sends the input to the actuator via an amplifier (Piezomechanik LE 150/100 EBW). We measure velocity of each particle by a laser Doppler vibrometer (Polytec OFV-534) at $45^{\circ}$ through the delicately-designed holes in the enclosures. The point-by-point measurements of the particles are synchronized to reconstruct the wave field along the chain.

To investigate wave dynamics, we first numerically model the system by employing the discrete element method (upper inset of Fig. 1). Each fused-quartz cylinder (Young’s modulus $E=72$ GPa, Poisson’s ratio $\nu=0.17$, and density $\rho=2200$ kg/m3) is considered as a point mass having only one degree of freedom in the vertical direction. The interaction of $i$th and $(i+1)$th cylinders – making a contact angle $\alpha_{i}$ – is modeled as the following force-displacement law: $F=\beta(\alpha_{i})(\delta_{i}+u_{i}-u_{i+1})^{3/2}$. Here $\beta(\alpha_{i})$ is the contact stiffness coefficient, $u_{i}$ denotes the dynamic displacement of $i$th cylinder, and $\delta_{i}$ is the pre-compression due to the static force given to the system (see Supplementary Material [Sup, ] for the full expression of $\beta(\alpha_{i})$ and equations of motion). We neglect the gravitational force, because it is much smaller compared to $F_{0}$. We explore the linear wave dynamics of the system by studying modal response of the system. To this end, for small dynamical excitations, we can linearize our contact model such that contact stiffness $k_{\text{lin}}(\alpha_{i})=(3/2)\beta(\alpha_{i})^{2/3}F_{0}^{1/3}$.

Linear dynamics.—In Fig. 2A, we show modal frequencies of lossless gradient-index chain ($10^{\circ}\rightarrow 90^{\circ}$) in comparison to homogeneous chains ($10^{\circ}\rightarrow 10^{\circ}$ and $90^{\circ}\rightarrow 90^{\circ}$), i.e., uniform contact angle (thus stiffness $k_{\text{lin}}$) along the length. We observe that the eigen frequencies of the $10^{\circ}\rightarrow 10^{\circ}$ chain span upto a cutoff frequency about 17.78 kHz [$=(1/\pi)\sqrt{k_{lin}(10^{\circ})/m}$, where $m$ represents the mass of cylinders and $k_{lin}(10^{\circ})$ denotes linearized stiffness for $10^{\circ}$ contact]. Similarly, for the $90^{\circ}\rightarrow 90^{\circ}$ chain, we observe eigen frequencies cover the spectrum upto 11.97 kHz [$=(1/\pi)\sqrt{k_{lin}(90^{\circ})/m}$]. For the gradient-index chain, however, we observe eigen frequencies extend to about 17.78 kHz (i.e., the cutoff frequency for $10^{\circ}\rightarrow 10^{\circ}$ chain), but the curve has a portion that is concave upward starting from about 11.97 kHz (the cutoff frequency for $90^{\circ}\rightarrow 90^{\circ}$ chain). These modes are referred to as “gradons” in the previous literature [Xiao et al., 2006].

To investigate further, we plot the wave transmission as a function of frequency for all the aforementioned configurations in Figs. 2B-D. For this, we use state-space approach to calculate the ratio of output force (felt by the upper mass) to input force (by the lower actuator) [Boechler et al., 2011b]. It is evident that $10^{\circ}\rightarrow 10^{\circ}$ and $90^{\circ}\rightarrow 90^{\circ}$ homogeneous chains have pass bands from 0 kHz to their respective cutoff frequencies, whereas the gradient chain shows a pass band with decreasing transmission in the frequency range marked by the double-sided arrow in Fig. 2D, which corresponds to the region with the concave upward trend in Fig. 2A. We show in the inset a mode shape for a frequency in this region. Since its modal amplitude dominate the chain only partially, we can explain why the wave transmission decreases in this region.

We verify this argument further by performing full numerical simulation with a small-amplitude impulse excitation given to the chain (Runge-Kutta solver with 0.01 m/s of initial velocity to the first particle). We then perform the fast Fourier transformation (FFT) on the velocity time history of each particle to plot frequency spectrum along the length of the chain as shown in Fig. 2E. We observe that the wave transmission is only partial along the chain in the frequency range mentioned above. As the input frequency increases in this region, the transmission is more limited to the front end of the chain. Therefore, we can interpret linear dynamics in this gradient-index chain as if the system has spatially-varying “local” cutoff frequency. Analytical expression of such a local cutoff frequency can be mathematically expressed as $f_{c,i}=(1/\pi)\sqrt{k_{lin}(\alpha_{i})/m}$), which shows an excellent fit with the numerical results shown Fig. 2E. We thus conclude that the our gradient-index chain would have three regions of wave transmission. From 0 kHz to 11.97 kHz, there is a pass band; from 11.97 kHz to 17.78 kHz, there is a quasi stop band, i.e., wave transmission upto a fraction of the chain; and for frequencies above 17.78 kHz, there is a stop band.

With the understanding of the three different regions of wave transmission in our gradient-index chain, we now send Gaussian-modulated waveforms centered at frequencies residing in these three regions. We numerically and experimentally show how the wave packet propagates along the chain when sent from the stiffer side ($10^{\circ}$). In Figs. 3A and B, we show spatiotemporal evolution of a wave packet at 7 kHz obtained numerically and experimentally. As the frequency falls in the region of full transmission, we clearly observe that the wave packet is transmitted to the other end of chain. A significant decay in amplitude, however, is due to the damping in the experiments, which is modeled in simulations as well (Supplementary Material [Sup, ]). In Figs. 3C and D, we show spatiotemporal evolution of a wave packet at 14 kHz, which lies in the partial wave transmission region. Evidently, the wave packet slows down as it propagates along the chain. It stops at a spatial location and then turns back to the front of the chain. This is analogous to boomerang motion, which we could successfully capture in our experiments. This boomerang motion typically involves wave amplification near the turning location (Supplementary Material [Sup, ]). Lastly, the wave sent at 24 kHz in the stop band does not propagate along the chain at all and is confined to the left end (Figs. 3E and F). In this way, we have demonstrated that our gradient-index system offers a great control over the penetration depth of the wave packet as a function of its frequency.

Nonlinear dynamics.—We now investigate wave dynamics for larger amplitudes by invoking nonlinear effects. In particular, we consider the frequency regime that offers partial wave transmission, the uniqueness of this gradient-index chain, and then increase wave amplitude to assess transmission characteristics of the system. We send a Gaussian-modulated pulse centered at 13.5 kHz from the two opposite ends and numerically monitor wave transmission as shown in Fig. 4. We quantify wave transmission as the ratio of the maximum velocity of the last particle to that of the first particle. Viscous damping is ignored here. For small-amplitude excitations, the forward configuration ($10^{\circ}\rightarrow 90^{\circ}$) shows boomerang wave motion and returns back without reaching the other end as predicted earlier. However, upon increasing the wave amplitude, we see significant rise in wave transmission through the chain due to wave leakage as seen in the upper panels of Fig. 4. In contrast, for the reverse configuration ($90^{\circ}\rightarrow 10^{\circ}$), the wave does not penetrate the bulk of the chain and remains localized near the excitation point as seen in the bottom panels of Fig. 4. Upon increasing the wave amplitude, the localization still persists, and there is not a significant rise in the wave transmission.

This amplitude-dependent asymmetric wave transmission can be understood as the interplay between nonlinearity and spatial gradient in the system. Looking back at the eigenmode (“gradon”) plotted in the inset in Fig. 2D, when we excite the system from the stiffer side ($10^{\circ}$), the presence of larger modal amplitude contributes to invoking nonlinear effects (such as frequency shifts) easily with an increased excitation amplitude. However, when we excite the system from the soft side ($90^{\circ}$), nonlinear effects become substantially suppressed, similar to the mechanism observed in thermal systems [Yang et al., 2007]. By further investigating this phenomenon in the frequency spectra for both small and large excitation amplitudes, we observe that the enhancement of the wave transmission in the forward configuration is due to the gradual frequency softening and spatial extension of nonlinear mode (Supplementary Material [Sup, ]). We note that this mechanism is different from those relying on harmonic generation [Liang et al., 2009], bifurcation [Boechler et al., 2011a], or self-demodulation [Devaux et al., 2015].

Next, we experimentally demonstrate the asymmetric wave transmission in our gradient-index chain. We send a Gaussian-modulated pulse used in Fig. 4 from the actuator to the two different configurations: forward ($10^{\circ}\rightarrow 90^{\circ}$) and reverse ($90^{\circ}\rightarrow 10^{\circ}$), and measure wave transmission. In Fig. 5, we show the experimental evidence of asymmetric transmission in our system when the excitation amplitude is increased. The numerical simulation, which also includes the effect of viscous damping, follows the experimental data with a decent agreement. The inset highlights the aforementioned frequency shift governed by the local cutoff frequencies, and thereby leading to asymmetric wave transmission of about 15 dB. Note that the excitation range in the experiments is narrower than that in the simulations due to the limitation of our piezoelectric stack actuator. However, the asymmetric transmission is clearly verified within the range covered.

Conclusion.—We have proposed a highly tunable gradient-index system that is made of cylindrical granules. The contact interaction allows us to easily maintain a stiffness gradient along the chain. Due to the nonlinear Hertz contact law, the system is further tunable by the amplitude of wave excitation. For small amplitudes, the system follows linear dynamics and shows three distinctive frequency regions of wave transmission. These are a stop band, a pass band, and a quasi stop band that allows waves to penetrate only to a fraction of the system and then return back to the point of excitation. We experimentally demonstrate such a boomerang motion. For high amplitude excitations, we invoke nonlinear effects (gradual frequency softening) in the system, and demonstrate that the same system supports asymmetric wave transmission, leading to a rapid enhancement of transmission from one end to the other. Therefore, this contact-based tunable system can inspire novel class of systems to manipulate the flow of elastic energy for engineering applications, e.g., impact mitigation, vibration filtering, energy harvesting, and even mechanical logic gates. The underlying nonlinear mechanism of our system can also stimulate future studies in other domains such as plasmonics and photonics.