We develop a framework for wave tailoring by altering the lattice network topology of a granular crystal consisting of spherical granules in contact. The lattice topology can alternate between two stable configurations, with the spherical granules of the lattice held in stable equilibrium in each configuration by gravity. Under impact, the first configuration results in a wave with rapidly decaying amplitude as it propagates along a primary chain, while the second configuration results in a solitary wave propagating along the primary chain with no decay. The mechanism to achieve such tunability is by having energy diverted to the granules adjacent to the primary chain in the first case but not the second. The tunable design of the proposed network is validated using both numerical simulations and experiments. In terms of potential applications, the proposed bistable lattice network can be viewed either as a wave attenuator or as a device that allows higher amplitude wave propagation in one direction than in the opposite direction. The lattice is analogous to a crystal phase transformation due to the change in atomic configurations, leading to the change in properties at the macroscale.

Introduction

Wave tailoring has diverse engineering applications and has been an active area of research over the past decade, primarily centered around the design of metamaterials with specific geometric and material properties to accomplish specific objectives. Granular materials have emerged as potential candidates for a metamaterial design because of their unique mechanical properties, very distinct from their continuum counterparts, arising from the nonlinear nature of the interparticle contact of granules, e.g., following Hertzian contact in the elastic case [1]. For instance, monodisperse chains of spheres support solitary waves when subjected to impulse loads [2,3]. Multiple applications exploiting solitary waves have been demonstrated in the last decade, ranging from acoustic rectifiers [4] to acoustic lenses [5]. Disordered granular media have applications such as the granular gripper [6], in which the effective stiffness of the granular packing can be varied significantly by altering its density.

Tuning of elastic waves has also recently been demonstrated numerically by, for instance, varying the stiffness of an elastic bar [7]. Granular lattices, which are ordered arrays of packed granules, support solitary waves for certain material properties. Jayaprakash et al. [8] demonstrated a new family of solitary waves in dimer chains, while Manjunath et al. [9] demonstrated plane solitary waves in higher dimensional lattices. Drawing upon these concepts of solitary waves in packed lattices, Pal and Geubelle [10] developed a mechanism to alter the response of a solitary wave propagating down a chain by applying precompression using side spheres. Daraio and coworkers [11] demonstrated logic gates using granular chains subjected to harmonic loading by applying precompression to alter the range of allowable frequencies.

We remark that all the above mechanisms require work or energy input to achieve tunable propagation—either applying precompression to deform the spheres or applying a harmonic force. In the present effort, we demonstrate a method to control waves, which requires negligible energy input to attain the desired change and is similar to operating a mechanical switch. We are interested in impact loadings which produce solitary waves in chains and develop a tunable one-way wave propagation chain by altering the lattice network topology in a granular chain. This is analogous to a phase change in a material at the microstructure level, which is typically associated with a rearrangement of atoms, and results in a change in material properties at the continuum level. Section 2 introduces the theoretical background of the concept, describes the associated numerical model, and the corresponding experimental setup for demonstrating the design. Section 3 presents the numerical and experimental results to validate the concept. Section 4 concludes the paper.

Tunable Wave Propagation: Concept and Methodology

Operating Principle: Wave Propagation in a Quasi-One-Dimensional (1D) Lattice.

We introduce a framework for wave tailoring based on changing the relative positions of granules, i.e., the lattice network, between the two configurations. The lattice is arranged in a plane, and the wave propagation is considered in the axial direction only, hence the name quasi-1D. The schematics of the two configurations of interest here are presented in Figs. 1(a) and 1(b) for both side and top views. The lattice consists of both a 1D primary chain of spheres (“axial spheres”) that are aligned with the impact source (red in Fig. 1), and “side spheres” which are symmetrically placed on the opposite sides of the selected axial spheres (blue in Fig. 1). The side spheres are placed at every other location simply for geometric reasons to avoid direct contact of side spheres with each other. The side spheres are confined between the two rigid parallel plates. The lattice attains two states based on its inclination relative to the impact direction. In the downstream configuration shown in Fig. 1(a), the lattice is inclined such that a pair of side spheres are in contact with the adjacent downstream axial sphere (defined as the direction of primary pulse propagation), and a gap exists between the pair of side spheres and the axial sphere upstream. In the upstream configuration shown in Fig. 1(b), the lattice is inclined such that a gap exists between a pair of side spheres and its adjacent axial sphere in the downstream direction. Switching between the two configurations can easily be achieved by slightly tilting the lattice in one of the two ways in a gravitational field, as illustrated in the lower part of Figs. 1(a) and 1(b). We remark here that the effect of gravity on the lattice dynamics is negligible considering that only a small inclination (about 2 deg as will be seen in the subsequent experiments) is needed to keep each configuration stable. Thus, the force due to gravity is about four orders of magnitude smaller than the magnitude of forces involved in the wave propagation. However, the gravity plays a key role in providing a force sufficient to ensure stability of both configurations in their static assembly.

When the downstream configuration (Fig. 1(a)) is impacted along the axial direction, a solitary wave propagates through the chain. The axial spheres moving forward do not contact the side spheres ahead as the wave propagates, and the side spheres play no role in the dynamics of the lattice. In contrast, when the upstream configuration (Fig. 1(b)) is subjected to the same impact, a wave with progressively decaying amplitude traverses through the lattice as the wave interacts both with the spheres along the axis and with the side spheres. After impact, the side spheres are in free flight and they eventually collide with the axial spheres adjacent to them in the downstream direction and there are local oscillations. The energy lost due to these local oscillations is similar to the oscillations observed in the wave propagation through dimer lattices [810] at the tail of the propagating wave where the smaller mass of a dimer unit cell may oscillate between the two larger masses.

Due to symmetry, there is a zero net force on the axial spheres in the lateral (vertical in Figs. 1(a) and 1(b)) direction, and the spheres only move axially. The two-dimensional lattice effectively behaves as a spring mass chain. Note that the difference between the two configurations is in the location of the side spheres along the chain. The key aspect in the design is a change in the lattice network topology, causing a change in the wave propagation behavior from a solitary wave to a rapidly decaying wave down the primary chain. We further remark here on the generality of the proposed framework. Indeed, other configurations may also be designed to change the lattice network and have the desired effect. For instance, moving the confining plates inward enough will induce a gap between the axial spheres so that there is a rapidly decaying wave along the chain. Moving the confining plates outward, so that the side spheres are no longer in contact with the axial spheres would allow a solitary wave to propagate through the chain. We conducted successful tests with these other configurations [12]. However, in this paper, we focus on the configurations presented in Fig. 1.

Numerical Model: Spring–Mass Model.

The spheres are modeled as point masses connected by nonlinear springs following the Hertzian contact law. The walls are assumed to be rigid, and all the surfaces are frictionless. As discussed earlier, the effect of gravity is small compared to the wave propagation forces, and it is neglected in the numerical model. All the interactions are thus through the mass centers, and there is no torque acting on the spheres which consequently are assumed to not rotate and to have only translational degrees-of-freedom. The forces generated are assumed to be small compared to the elastic limit of the spheres, and hence, no dissipation due to plastic deformations is considered. Let D, E, and ν denote, respectively, the diameter, modulus of elasticity, and Poisson's ratio of the main and side spheres (assumed identical, though they need not be). Then, the force F between the two identical spheres with a relative displacement α between their centers is 
F=[ED3(1ν2)]α3/2=kssα3/2
(1)
As noted earlier, the axial spheres move only in the axial direction and have one degree-of-freedom, while the side spheres have two degrees-of-freedom for the two in-plane displacement components. Let θ be the angle between the axial direction and the line joining the centers of a side sphere and its contacting axial sphere, and let [x]+=(max(x,0))3/2. The axial spheres are denoted by integer indices and the side spheres by fractional indices, whose index value is the average of indices of the two adjacent axial spheres. Let β  = 1 and β =−1 denote the lattice in the upstream and downstream configurations, respectively. The equation of motion of the axial sphere i is 
müi=kss([uiui+1]++[ui1ui]+)+2βcos(θ)kss[βuiβcos(θ)ui+β/2+βsin(θ)ui+β/2]+
(2)
where u is the axial displacement, v is the transverse displacement, m is the mass of the sphere, and kss is defined in Eq. (1). For the side spheres, the governing equations of motion are 
müi+β/2=cos(θ)kss[βui+1βcos(θ)ui+1/2+βsin(θ)ui+β/2]+
(3)
and 
mv̈i+β/2=sin(θ)kss[βui+1βcos(θ)ui+1/2+βsin(θ)ui+β/2]++kss[vi+β/2]+
(4)

Note that we have not considered the side sphere contacting the sphere on the other side of the gap since the time scale of wave propagation is much smaller compared to the time of free flight and the time it takes for the side sphere to impact the next axial sphere downstream or upstream. The initial conditions are that the first sphere is given a prescribed velocity, and all the other spheres are at rest.

Experimental Setup.

The experimental setup consists of axial spheres that are placed on a grooved Teflon ramp that both confines the axial spheres to move axially and reduces friction with the substrate. The side spheres are placed at the same horizontal level outside the groove, and they are in contact with the two parallel plates, as illustrated in Fig. 1(c). Indeed, the side spheres can move both axially and laterally within the confines of the parallel plates. The axial spheres are restrained from moving laterally as they are constrained by the groove and impacted only along the axis. Furthermore, as noted earlier, the force due to the side spheres on the axial spheres is symmetric, and hence, the lateral component of force on the axial spheres is negligible. The inclination of the holder ramp is 2.2 deg, which is sufficient to achieve the desired configurations while minimizing the gravitational effects. The side spheres are in contact with the axial spheres, which hold them in equilibrium in the axial direction. As discussed earlier, the contact force at equilibrium that balances the weight of the side spheres is much smaller than the forces associated with the wave propagation and thus does not influence the dynamics of the spheres.

The axial and side spheres, obtained from Salem Specialty Ball Company, Canton, CT, are made of 440 C stainless steel and have a diameter of 9.53 mm, with material properties: Young's modulus 200 GPa, density 7670 kg/m3, and Poisson's ratio 0.3. Figure 2, a close-up of the loading area of the setup, shows that the Teflon holder rests between the adjustable aluminum walls lined with steel inserts. The steel inserts have been introduced to better match the rigid boundary conditions used in the numerical simulation. Pairs of side spheres are symmetrically placed along the sides of the primary chain such that each side sphere is in contact with one axial sphere from the primary chain and a sidewall. The axial sphere chain is impacted by another identical sphere released from a consistent height from a separate input ramp, seen in the left part of Fig. 2. The impact velocity of this sphere onto the first axial sphere is determined from the delay between the disruption of infrared light from two infrared emitters (also seen in Fig. 2). An impact velocity of 0.62 ± 0.15 m/s, low enough to ensure elastic deformations throughout, is used in all the experiments discussed subsequently.

Figure 3 shows the experimental downstream configuration, where the holder ramp is inclined (again at 2.2 deg) in the opposite direction such that the side spheres are in contact with only the downstream axial spheres. The axial chain is comprised of 21 total spheres, and a sphere with an embedded piezoelectric sensor is placed at the axial location of the 18th sphere (Fig. 3(b)) to read the reaction force generated in each case. The piezoelectric sensor, obtained from Steiner and Martins, Inc., Doral, FL, has a diameter of 7 mm and a thickness of 200 μm. A steel sphere is cut in half and machined appropriately to accommodate the sensor and its leads, and the thin sensor is then embedded in the sphere which is glued back together. The voltage read by this instrumented sphere is converted to force following the conservation of momentum calibration method outlined in Ref. [13]. The peak force at the 18th sphere is then recorded ten times for each tested configuration of 0, 3, and 5 sphere side pairs.

Figure 4 shows photographs from the experimental upstream configuration, in which the side spheres are only in contact with the upstream sphere of the primary chain. The impacting velocity was again 0.62 ± 0.15 m/s in all the experiments. The upstream configuration is composed of 18 axial spheres which are stopped from sliding along the incline by a steel support (Fig. 4(b)), that is, situated such that it does not interfere with the impact between the impacting sphere and the first sphere of the primary chain. Infrared detectors are placed at the end of the primary chain (Fig. 4(c)) to capture the departing (output) velocity of the last axial sphere in the primary chain. By comparing this output velocity with the input velocity, the solitary wave decay can be quantified. Note that it was not possible to capture the output velocity in the downstream configuration (as the motion of the last sphere is constrained for static equilibrium before impact). In this upstream configuration, 20 experiments were completed each for 0, 1, 2, 3, 4, and 5 side sphere pairs.

Results and Discussion

Experiments and corresponding numerical simulations are conducted on both configurations to demonstrate the proposed concept. The experiments are repeated for various numbers of pairs of side spheres, while the input velocity is kept fixed at 0.62 ± 0.15 m/s. The numerical simulations are performed by taking the experimentally measured impact velocity as input. The system of equations described by Eqs. (2)(4) is solved using a fourth-order Runge–Kutta scheme with a time step of 10−8 s, which is sufficiently small compared to the time scale of solitary wave propagation through the lattice. Figure 5 illustrates both the numerical and experimental data for the velocity of the final sphere, normalized with the velocity of the impacting sphere, for the upstream configuration. The horizontal axis shows the number of side pairs in the setup, and the vertical axis shows the set of normalized velocities. The experimental data for each set of pairs of side spheres are in good agreement, demonstrating the repeatability of the experiment. The leading wave loses energy at each side sphere contact, resulting in progressive decay of the wave amplitude with increasing number of pairs of side spheres. The numerical results presented in Fig. 6 show the transfer of energy from the axial to the side spheres at each contact location. Initially, all of the energy is in the axial chain and then as the wave encounters side pairs, the energy is seen to partition between the axial chain and the side spheres. The velocities obtained from the corresponding numerical simulations are higher than the experimental measurements, though the trends are very similar. We note that some assumptions in our model may lead to higher predicted velocities: The contacts are assumed to be frictionless, and the walls are assumed to be perfectly rigid, i.e., no energy is radiated outward to the walls. Furthermore, the input velocity is measured about 2 cm before the impact point and may actually be lower than 0.62 ± 0.15 m/s at the impact point because of friction. Also friction between the side spheres and the wall can reduce the output velocity by a small amount. However, the trends are qualitatively similar. Note that from the 120 data points present in Fig. 5, two outliers exist for the case of four side pairs. In these two cases, the output to input velocity ratio is greater for the experiments than it is for the numerical model. This discrepancy is likely due to some of the side spheres not being in perfect contact with the axial chain. While each chain was visually inspected before each test to avoid such cases, the spheres are not uniform (tolerance:±0.0127 mm) which allows for a misalignment on rare occasions.

Figure 7 illustrates the peak forces at the same axial sphere (#18) for different numbers of side sphere pairs when the chain is in the downstream configuration. The forces are normalized by the force acting on the first sphere, which is calculated based on the impact velocity. In this case, a solitary wave propagates down the chain independent of the number of pairs of side spheres. The axial spheres do not interact with the side spheres as their displacement toward the impact direction is about 13 μm, which is much less than the gap between them and the side spheres. As before, the forces predicted by the numerical simulations are observed to be higher than those observed experimentally, likely because of the same aforementioned reasons. Figures 5 and 6 demonstrate that the behavior of this quasi-1D lattice is significantly different between the upstream and downstream configurations, thus, validating our proposed design of a solitary wave attenuator that is triggered by altering the lattice network topology.

Since the numerical simulations can provide a more detailed history of the force transfer in the system, we further analyze the resulting wave and sphere motions in the lattice based on the simulation results. Figure 8 displays the contact force profile obtained from the numerical simulations on the last axial sphere for all the side sphere configurations used here (0, 1, 2, 3, 4, and 5 side pairs). As discussed earlier, the case with zero side spheres is identical to the lattice in the downstream configuration, since the side spheres do not interact with the axial spheres during the wave propagation. The decrease in the force for the zero side sphere case is due to the fact that the impact energy is localized in the impacting bead and the first axial bead, while the energy associated with the solitary wave is distributed over five particle diameters [2]. Two key observations can be made from Fig. 8 for the upstream configuration: the peak force amplitude decreases rapidly, and the wave arrival time increases with increasing number of side sphere pairs. The pulse with lower amplitude travels slower, though the shape of the pulse is similar in all the cases. Indeed, both these features are well-known characteristics of solitary waves: their shape is independent of the wave amplitude, while their velocity depends on the peak force [2]. Figure 8(b) displays the velocity profiles of the 15th axial sphere in the lattice for various side sphere configurations. Also shown is the velocity profile of the impacting sphere. Similar to the force profiles, the sphere velocity decreases, and the wave arrival time increases as the number of side spheres increase. Note that solitary waves form in the regions after the side spheres, and they are nondispersive. Thus, the trends displayed here do not vary as the number of axial spheres is increased. Indeed, by having a lattice design with side spheres only in a localized region, along the chain, the solitary wave properties can be altered. Figure 9 displays the respective values of contact force and bead velocity for the chain in the downstream configuration. As discussed earlier, the side spheres do not interact with the propagating wave and the response is independent of the number of side sphere pairs. Indeed, both the force and velocity profiles match with the case of zero side sphere pairs in the upstream configuration. Note that the dynamics in the downstream configuration is thus identical to a solitary wave traversing in a monodisperse chain. Finally, a comparison of Figs. 7 and 8 illustrates the tunable nature of wave propagation along the chain by changing the configuration from upstream to downstream. Furthermore, note that an upstream configuration in one direction is identical to the downstream configuration for a wave traveling in the opposite direction.

By having a lattice network with only a few side spheres, we can substantially alter the wave propagation down the chain. Indeed, there are two ways of looking at our system: as a device that permits higher amplitude propagation in one direction than the other and as an attenuator. The downhill configuration endeavors to realize a closed circuit and transforming to the uphill configuration corresponds to an open circuit. Thus, the mechanism to alter between configurations can be viewed as a device for the management of mechanical impulses. We also remark here that the difference between the uphill and downhill configurations is in their lattice network topology, and the transformation is analogous to a phase transformation in materials from one atomic configuration to another, resulting in changes in effective material properties at the macroscale.

Conclusions

This work introduced a framework for wave tailoring by altering the lattice network topology of granular lattices. The proposed framework was validated using both numerical simulations and experimental measurements. In the upstream configuration, the primary wave amplitude decreased along the chain due to the presence of side spheres, while in the downstream configuration, a (fixed-amplitude) solitary wave propagated along the chain. Energy “leaking” via the side spheres is responsible for the amplitude decay in the upstream configuration, and thus, the more side pairs are present the more significant the decay. With five side pairs, a decay of about 70% was achieved and almost complete extinction would be possible with more side pairs. Furthermore, since a given side pair may be placed adjacent to any axial sphere and the side pairs may be placed nonconsecutively, it is possible to tailor regions with different solitary wave properties within a granular lattice. In practice, switching between the two configurations, decaying and steady, can be done by exploiting gravity and a small tilt of the primary chain in one or the other direction. The experimental data and numerical simulations were in good agreement, and they validate the proposed attenuator design framework. This framework of altering the lattice network topology, in this case by gravity, can clearly be extended to alternate designs and more complex mechanisms. The device presented here can also be used either as a one way propagation device or as a logic element in a mechanical circuit. The change in configurations may alternately be viewed as a phase transformation, and our work demonstrates the potential to develop a family of devices for wave tailoring using granular crystals.

Acknowledgment

This work was funded by the U.S. Army Research Office (ARO) MURI Grant No. W911 NF-09-1-0436. Dr. David Stepp is the grant monitor. We appreciate the useful contributions Dr. Owen Kingstedt made to the experimental setup.

Nomenclature

     
  • D =

    sphere diameter

  •  
  • E =

    modulus of elasticity

  •  
  • F =

    contact force between two spheres

  •  
  • i =

    index to denote sphere location

  •  
  • kss =

    effective stiffness between two identical spheres

  •  
  • m =

    mass of the sphere

  •  
  • u =

    axial displacement

  •  
  • v =

    transverse displacement

  •  
  • α =

    relative displacement between sphere centers

  •  
  • β =

    index to distinguish between upstream and downstream configurations

  •  
  • θ =

    angle between the axial direction and the line joining the centers of a side sphere and its contacting axial sphere

  •  
  • ν =

    Poisson's ratio

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