Pulse mitigation in ordered granular structures: from granular chains to granular networks

Abstract

Ordered granular structures have garnered considerable attention across various fields due to their capacity to manipulate the transmission of mechanical energy and mitigate the adverse effects of impacts and vibrations. The ability to control wave propagation is crucial in the design of protective equipment, seismic isolation systems, aerospace vibroacoustic attenuation and shock-absorbing materials, among many other applications. Here, we delve into the myriad configurations of ordered granular systems: from one dimensional granular chains to granular chain networks, showcasing their significance for pulse mitigation. Given the unique behaviours that these granular structures can generate, they can be described as discrete or granular metamaterials. A detailed analysis of the wave behaviour in these structures is presented, encompassing the influence of heterogeneity, chain curvature, and dimensional complexity on energy dissipation. This discourse extends to encompass a comparison of analytical and numerical approaches used in the examination and application of these systems, along with an exploration of the implications of advances in manufacturing methods. Unlike other examinations, this comprehensive review underscores the multifaceted nature of our study, with a steadfast focus on their applicability to impact mitigation and wave control. We conclude with a summary on the current challenges and future outlook of engineered granular systems, emphasizing their transformative potential in safeguarding structures from dynamic forces and advancing the frontier of energy management technologies.

1 Introduction

Granular materials are a unique class of solids composed of discrete particles, such as sand, rice, or ball bearings, that interact through non-cohesive, dissipative contact forces [1]. Unlike traditional solids, granular materials have no tensile strength, and their mechanical behaviour is primarily dictated by the nonlinear, Hertzian force-displacement relationship between individual particles in compression. Specifically, when two spherical particles are compressed, the repulsive force increases with the relative displacement between them as $F = K\delta ^{3/2}$, where K is the stiffness coefficient that depends on particle shape and material properties, and $\delta$ is the overlap between the particles [2, 3]. If the interacting particles are not spherical, a similar power law but with a different exponent can often be used to describe the interaction. For example, when two cylindrical particles are in contact with their axes parallel, there is a linear relationship between the force and the displacement.

These nonlinear interactions give rise to intriguing and complex phenomena, including pattern formation [4], self-assembly [5], brazil nut effect [6] and nonlinear wave propagation behaviours unlike those found in continous media [7]. The way in which waves propagate through granular materials is highly dependent on the underlying particle structure. When the particles in granular materials like sand or rice are randomly arranged, they form a network of interconnected pathways known as force chains, which bear the bulk of the load and reorganize in unpredictable ways depending on the applied load [8]. On the other hand, when these particles are arranged in a neat, grid-like pattern, similar to the arrangement of atoms in a crystal, waves travel through the material in a predictable fashion and can be tailored for specific uses.

One particular type of wave that propagates through granular materials is a solitary wave, a spatially localized wave that propagates without any temporal evolution in shape or size when viewed in the reference frame moving with the group velocity of the wave. Their observation was first reported by John Scott Russell in 1834 [9]. Solitary waves are characterized by a non-linear relationship between their speed and amplitude. The solitary waves traveling through monomer granular chains, was first described by Nesterenko [10] and later studied by Nesterenko and Lazardi [11].

In a granular chain, the force-displacement characteristics of granular contacts are inherently nonlinear [12], which is manifested by the absence of a linear term in the force equation [13]. Consequently, the force needed to compress particles does not scale linearly with displacement. This nonlinearity leads to significantly different dynamics and wave propagation behaviours compared to systems adhering to Hooke’s Law, enabling phenomena such as solitary waves and what is known as a “sonic vacuum” [14]. In this “sonic vacuum,” when grains are in contact but not under static compression, the lack of harmonic potential terms implies that the system does not support oscillatory motion around an equilibrium position, inhibiting the propagation of sound, or any other linear wave. This phenomenon underpins the exceptional wave control capabilities of granular systems, as the energy from impacts remains localized rather than dispersing as in more common mediums. Moreover, the exploration of such nonlinear dynamics enhances our understanding of complex systems, providing insights into nonlinear science that have broad applications across physics and engineering.

The unique properties of granular systems make them ideal for use in the design of engineered material such as metamaterials. The ability to control the transmission of mechanical waves is essential in various fields, from materials science to earthquake engineering. Protective equipment design, seismic isolation, aerospace vibroacoustic attenuation and shock-absorbing material development all depend on the capacity to regulate the transmission of the pulses generated by impacts. The strong dependence on force transmission on particle arrangement allows for the design of predetermined wave propagation paths as well as changing the propagating pulse, either to lower its amplitude—mitigation—or shift its speed and frequency. Additionally, the nonlinear force-displacement relation allows for an added level of tunability through control of pre-compression.

Fig. 1

One-dimensional heterogeneous granular chains. A Disordered chain. B Tapered chain. C Decorated chain. D Chain with side decorations. Illustration from [15]. E Chain composed of Teflon O-rings (circles) and rigid cylinders (rectangles). Illustration from [16]

However, despite the established capabilities of granular metamaterials in manipulating mechanical waves, there is a critical gap in the literature regarding the relative effectiveness of different ordered granular structures for wave attenuation and control. This includes an understanding of how heterogeneity and geometry affect their on their wave-managing functions. This review aims to bridge this gap, providing an in-depth exploration of the state of the art in pulse mitigation within ordered granular systems, from granular chains to multidimensional configurations. Moreover, unlike previous works, this examination also critiques the diverse range of research methodologies applied to these systems, from analytical to numerical methods, enabling an evaluation of their comparative advantages and limitations. Additionally, it explores the impact of manufacturing methods on the field of research. In doing so, we underscore the essential role of ordered granular structures in enhancing or inhibiting the transmission of mechanical waves.

The review is organized into several sections. We begin by exploring the dynamic behaviours of disordered and heterogeneous granular chains, which offer significantly greater pulse mitigation than traditional homogeneous chains. Following this, we analyze the impact of structural anomalies, such as mass-in-mass units, on local resonances and their applications in wave filtering and energy management. Our discussion also extends to the study of configurations like curved chains that introduce novel wave reflection dynamics. We then assess the ability of granular chain networks and other complex arrangements to efficiently manage energy flow. The review continues with a critical synthesis of analytical, numerical, and manufacturing techniques applied in granular metamaterial research, emphasizing their strengths and limitations. We end with an overview of the present obstacles and future prospects for engineered granular systems, highlighting their great potential to protect structures against dynamic forces and control wave propagation. The ultimate aim is to provide a detailed insight into the potential and progress of pulse mitigation strategies based on ordered granular systems.

2 Disordered and inhomogenous granular chains

In this section, we will explore some of the different types of heterogeneity that can be incorporated into granular chains, such as disordered, tapered, and decorated chains. By investigating the properties and behaviour of these distinctive granular chains, we can gain a deeper understanding of their potential applications in fields such as vibration damping, energy harvesting, and shock absorption.

A granular chain that presents disorder (absence of periodicity) in any parameter of the particles that compose it—such as radius, mass, material—is called a disordered chain (see Fig. 1A). There are two types of disorders in granular chains: random disorder and controlled/ordered disorder. Random disorder is characterized by the randomness in distribution of particle positions and sizes, while controlled disorder involves intentional modifications to the chain for the purpose of manipulating its response to external stimuli. Physically constructed granular chains invariably exhibit imperfections, or ’random disorder,’ potentially leading to discrepancies between experimental observations and predictions from models that assume ’perfect’ particles. In most cases, the irregularities present in the chain are sufficiently small to be negligible. In this section we will focus on controlled disorder, and how it affects the wave propagation in granular chains.

Fig. 2

Force propagation in one-dimensional granular chains. Force vs time plots in decorated (above), tapered (middle) and size-optimized disordered (below) granular chains. Figure from [17]

The introduction of disorder is predicted to disrupt the propagation of solitary waves. This can be seen in a disordered chain where two separate decay regimes—exponential and power-law—are typically noted, as investigated by Manjunath et al. [18]. The first regime becomes apparent when a solitary wave moving in a monodisperse chain encounters a region of randomness, giving rise to a noisy broad pulse that shows a slower, power-law decay, which is the second regime. The rate at which the amplitude of the solitary wave decays exponentially is reliant on the level and nature of the randomness introduced. In their study, Manjunath et al. [18] explored different forms of randomness, including changes in mass, Young’s modulus, and radius. They found that a disorder in the size of the particles resulted in a markedly higher decay compared to the other factors examined. Interestingly, the power-law decay of the noisy pulse appears to be universal, meaning that it adheres to the same law regardless of the degree of randomness. This phenomenon mirrors what has been observed in dissipative chains [19, 20], where the initial pulse experiences an exponential decay, followed by a slower power-law decay of the subsequent pulse. However, unlike the decay of a noisy pulse, the power-law decay in dissipative chains isn’t universal and is influenced by the coefficient of viscosity.

While disordered chains offer some level of mitigation due to impedance mismatch, a completely random sequence isn’t the most efficient configuration for impact attenuation. Various researchers have suggested setups to enhance force attenuation in disordered chains [17, 21]. Specifically, Fraternali and colleagues used a Breeder Genetic Algorithm to explore the optimal design of composite granular protectors [17]. The results showed that the most advantageous arrangement included larger grains near the side where the impact enters the chain, smaller grains on the opposite side, and alternating sequences of smaller and larger grains in the middle (Fig. 2). Through a combination of analytical and numerical modelling, Hong [21] combined different granular sections, each having different contact forces and masses, to optimize impact attenuation. Hong demonstrated that a potent initial impulse acting on the granular system is divided into smaller solitary waves, each possessing less than 10% of the original energy when exiting the container [21].

Additional research related to highly nonlinear wave propagation and scattering in disordered granules has been carried out using insights from statistical mechanics, coupled with numerical simulations and experiments, to characterize the degree of disorder [22, 23]. In [22], Ponson et al. investigated a chain consisting of arrangements of dimers—two different types of beads in contact. They considered a dimer to have spin “up” if the heavier particle is closer to the direction of the incident wave. They characterized the degree of disorder in the chain based on the difference in the amount of “up” and “down” dimers, such that the disorder increased when there was an equal number of both types of dimers and was zero when only one type of dimer was present. It was found that under a certain limit, a solitary wave propagated along the chain with exponentially decaying amplitude. However, beyond a critical level of disorder, the wave amplitude decayed as a power law, and the wave transmission became insensitive to the degree of disorder.

2.1 Tapered chains

Tapered chains, which are one-dimensional sequences of grains progressively diminishing in size, are depicted in Fig. 1B. Originally suggested as a shock absorption method in [24], Sen and colleagues showed via numerical simulations that a chain made of spherical lead grains, with each successive grain decreasing in radius by a certain factor (q), could lower the force amplitude by 90% or more in 100 grain configurations due to the thermalization of the impulse. These simulations assumed the spheres were composed of lead and dismissed restitution and frictional losses. The first lead sphere had a radius of 0.15 m, and the initial velocity of the largest, leftmost grain was set at 0.1 m/s. The authors presented outcomes for a range of q values, from 0 to 0.01, and evaluated longer chains of 500 particles. Afterward, upon introducing adjustments to the dissipation model, they found that with appropriate conditions, the system could dissipate more energy when the gradient change is more abrupt, thus accomplishing the same reduction in force but in shorter chains consisting of 20 grains [25,26,27,28].

The classical solitary wave, recognized in a one-dimensional array of particles of identical size, rapidly loses its characteristic form in a tapered chain due to broken translation symmetry. It spreads throughout the grains in the chain and ultimately transforms into a series of smaller impulses. This mechanism was first experimentally validated by Nakagawa et al. [29], who noticed an exponential surge in bead velocity along the sequence when an impulse was applied at one end of the chain. Additional experiments were conducted by Melo and team [30], utilizing a linear reduction in particle size. A monodisperse grain array was placed just before the tapered chain, which allowed a fully formed solitary wave to reach the tapered chain. Their observations indicated that the momentum propagation along the tapered chain declined linearly, and the solitary wave’s flight time’s dependence on the position was slower than linear.

2.2 Decorated and stepped chains

A decorated chain refers to a one-dimensional composite array of grains formed by inserting smaller particles between larger grains in a monodisperse chain (as shown in Fig. 1C). This setup presents an edge over tapered chains in terms of energy dispersal capacity, especially for smaller arrangements. Doney and Sen conducted theoretical and numerical analyses comparing the shock absorption capabilities of both configurations (tapered and decorated) and a combined version: the decorated tapered chain (a tapered sequence with smaller grains interspersed between each larger bead) [28]. The study considered various tapering values—ranging from $q=0$ (monodisperse chain) to $q=0.1$—and fractional sizes of the interstitial spheres (for instance, $f=0.7$ and $f=0.3$). Using numerical simulations, they showed that under certain conditions ($f=0.3$ and $q=0.1$), extremely small decorated tapered chains consisting of just 5 grains could distribute energy so effectively that only about 10% of the total energy introduced into the system gets transmitted to the end with the initial pulse. Three years later, these authors experimentally confirmed that the improved energy dispersion provided by the decorated tapered chains is due to the larger grains retaining most of the input energy, rather than the smaller interstitial beads [31].

A similar setup to decorated chains is the granular step chains. These chains are composed of arrays of spheres, alternating and made from different materials. In 2006, Daraio and colleagues documented the first experimental observation of impulse energy confinement and the ensuing fragmentation of shock and solitary waves in a step chain [32]. This arrangement involved alternating groups of spheres, composed of high-modulus (stainless steel, 193 GPa) and low-modulus (PTFE, 1.46 GPa) materials, all of different masses but identical in size—a diameter of 4.76 mm. Their findings indicated that the confined energy was held within the “softer” segments of the composite chain and was gradually released in the form of weak, separate pulses over a prolonged duration. This alteration allowed for about a 5 times reduction in the amplitude of the impulse reaching the wall, compared to that observed in separate experiments with a uniform steel chain under the same impact conditions and particle count. The researchers demonstrated that this effect could be amplified by adjusting the precompression of the chain with a magnetically induced superimposed force.

In the study by Job et al. [34], experiments were performed on a stepped chain with a marked decrease in bead radius. The chain’s particles were made up of roll bearings from steel that was both high in carbon and chrome-hardened. The chains consisted in arrangements of 7 larger, 13 mm beads, followed by 25 and 50 beads of radii 6.5 mm. Through simple analytical arguments, the authors found that the unloading of compressive force at the edge of the chain exhibits a nearly exponential decrease. The characteristic time was primarily dependent on factors such as the masses of the individual grains and the mass of the striker. These findings were substantiated through numerical simulations and experimental evidence.

A contemporary approach to the decoration methods previously discussed is the integration of side beads, or additional granules placed laterally to the axial chain (as shown in Fig. 1D) [15, 35,36,37,38,39]. Pal and colleagues demonstrated through numerical simulations that monodisperse chains, when encircled by smaller beads under controlled radial precompression, have the capacity to either support solitary wave-like pulses or exhibit rapid decay of leading wave amplitudes [15]. Tapered chains decorated with side beads were also numerically explored by Machado and team [35], and were shown to possess exceptional impact mitigation capabilities even in very short chains. In subsequent work, the authors demonstrated that a compact, scalable system composed of five axial beads, given the right configurations [39], could decrease over 90% of the momentum amplitude of solitary pulses, delivering slightly improved mitigation compared to what Doney and Sen had forecasted [28] for decorated tapered chains also composed of 5 grains.

Non-reciprocal wave propagation constitutes a notable phenomenon in non-linear systems such as granular chains, wherein wave transmission is no longer symmetrical between source and receiver. This phenomenon has been predominantly studied in homogeneous granular chains [40] and continuous elastic metamaterials [41,42,43]. However, the study by Li and Rizzo [44] extends this understanding to granular chains with asymmetric potential barriers, akin to the previously mentioned fixed side decorations. They found that such barriers selectively reflect waves traveling in one direction while permitting those in the reverse direction to pass through. Their investigation, carried out through numerical simulations, varied the number and stiffness of the barriers, concluding that a greater number of stiffer barriers enhances unidirectional wave transmission compared to their softer counterparts. These characteristics hold promise for applications in devices such as acoustic diodes or for the purpose of wave rectification, where the directed propagation of waves is essential.

Fig. 3

Dimer cylindrical chain. A one-dimensional chain, which allows manipulation of elastic waves by changing the contact angles of cylinders. Image from [33]

Fig. 4

One-dimensional granular crystal with bimetallic beads. a Consists of two sub-structures—blue for stainless steel and gray for aluminum; b Four equivalent springs under different contact conditions—$k_1$, $k_3$, and $k_4$ obtained by Hertz’s contact law and $k_2$ given by Hooke’s law; c Spring-mass chain model. Figure from [17]

2.3 Non-spherical and other heterogeneous configurations

Granular chains where the contact interaction between composing particles varies along the chain have also been studied [33, 45, 46]. Figure  3 shows a chain of identical cylinders in a “dimer” configuration. Two different contact alternating angles were studied analytically, numerically, and experimentally [33]. Chaunsali et al. demonstrated that a device, composed of a 40 unit chain of 18 mm diameter cylinders, can provide two extremes of elastic wave propagation: near total transmission and substantial attenuation under impulse excitation, with over 40% loss in transmission. In a more recent study [46], a system that permits tunable adjustment of contact stiffness through variations in precompression between spheres was examined both theoretically and numerically. The authors devised an innovative one-dimensional extended granular crystal, which comprises two sub-lattices, each featuring two identical spheres half made of aluminum and half of stainless steel, as depicted in Fig. 4. One sub-lattice employed two kinds of nonlinear contact forms by altering the orientation of the second sphere, while the other sub-lattice used a thin cylinder to substitute the linear stiffness with the nonlinear form. Modifying the contact stiffness through precompression, resulting in band gap shifting. This crystal could then potentially function as a switch, where varying the precompression allows the system to alternate between permitting and blocking the propagation of specific vibrational modes using this band gap.

In addition to the configurations mentioned above, there are many other types of heterogeneous granular chains that have received special attention. For example, a certain chain composed of “soft” toroidal elements (Teflon O-rings) and rigid cylinders (see Fig. 1E) has been proposed and investigated due to its strong interaction force, $F \propto (\delta ^{3/2} + \delta ^{6})$, and its potential for high energy absorption [16, 47, 48]. Herbold and Nesterenko proposed this arrangement in their 2007 paper [16]. They conducted numerical and experimental studies on this double-power law system, which resulted in very high mitigation levels. They also observed that the initial impulse splits into two distinct and highly nonlinear solitary wave trains, a consequence of the system’s dual nonlinearity. Using experimentation, numerical simulations and theoretical analysis, Xu and Nesterenko [49] showed a dramatic increase in speed of sound and acoustic impedance, by three to four times under moderate force, was achievable using a similar metamaterial but made up of nitrile O-rings. Another type of composite was introduced by Chaunsali et al.: a tunable cylinder-based granular system that is functionally graded in its stiffness distribution in space by changing the contact angle [50]. They demonstrated through analytical, numerical, and experimental methods that the metamaterial could be tuned to accelerate or decelerate the impulse along the chain, without significant scattering. A similar behaviour was observed for the contact forces, which exhibited power-law scaling and reduced amplitude as the stiffness decreased. For the tested setup, which involved a chain of 200 cylindrical particles with radii of 9 mm, a reduction of nearly 70% was measured.

Fig. 5

Granular metamaterials with local resonators. A Diagrams of mass-with-mass (upper), and mass-in-mass (lower) granular chains. B Parts and assembly of a resonant unit cell (left), and schematic of the experimental setup to test the dynamic response of a single particle (right). Figure from [54]. C Granular chain with a single mass-with-mass defect in the from of an external ring resonator. Illustration from [55]. D Granular chain of beads with external ring resonators. Illustration and experimental setup from [52]

Heterogeneous granular chains enable the design of metamaterials with distinctive transmission characteristics for a variety of applications. Decorated chains, can be designed to absorb and gradually dissipate energy, and used in sound insulation or as protective devices against extreme pulses. Tapered chains, with their gradation in particle size, act as effective dampers for mechanical shocks. Stepped chains, featuring discrete changes in bead diameter, induce reflections that can mitigate seismic waves, beneficial in earthquake-resistant construction. The integration of side decoration or stationary barriers enables precise wave propagation control, such as non-reciprocal wave transfer, applicable in developing innovative waveguides or ensuring secure communication channels. Chains with variable contact angles between cylinders have demonstrated potential in controlling stress wave propagation under impact. These granular systems collectively contribute to the development of advanced metamaterials that enhance safety and functionality in various engineering contexts.

3 Granular chains with internal resonators

We have so far explored the current state of knowledge regarding the propagation and mitigation of pulses in granular chains, analyzing the strongly non-linear traveling waves excited by impact. In this section we will be discussing the so-called mass-in-mass [51] or mass-with-mass [52, 53] systems. The former setting involves an internal resonator within the chain, while in the latter case, the resonator is external to the bead (see Fig. 5).

The wave propagation in a chain of spherical elements, inclusive of a mass-in-mass defect in both linear and nonlinear regimes, was examined in [54] (Fig. 5B). Both theoretical and experimental findings revealed a broad band gap in the audible range. It was established that by manipulating the contact interaction between particles, the acoustic transmission could be adjusted, thereby filtering mechanical waves between 3 and 8.5 kHz and above 17 kHz. This frequency range can be fine-tuned by around 1 kHz using static precompression.

In another study, Kevrekidis et al. analyzed, numerically and analytically, the dynamic response of a granular chain of beads in a sonic vacuum, where a single mass-with-mass defect was incorporated [55]. The results indicated that the embedded harmonic oscillator could be employed to adjust the transmitted and reflected energy of a mechanical pulse by changing the mass ratio between the harmonic resonator and the bead. If the defect—the harmonic resonator mass—is smaller than the beads, the incoming solitary wave is largely unaffected, with only a small portion of its energy being reflected and trapped as localized oscillation. Conversely, for a high defect-to-bead mass ratio, the reflection is considerably more substantial than the transmission, with significant trapping occurring. Their numerical simulations also revealed that the energy trapped in the mass-with-mass defect exhibited a non-monotonic dependence on the mass ratio.

Fig. 6

Woodpile structure. Experimental setup (upper). Experimental and numerical space-time wave modulation results in woodpile woodpile crystals (lower). Images from [56]

The so-called “woodpile structure” is a frequently analyzed system. This structure consists of sequences of rod layers alternately oriented at 90 degrees. In such arrangements, the resonance stems from the bending of the rods, rather than compression or extension, facilitating the achievement of low-frequency resonances, especially in compact setups. An extensive study (encompassing experimental, numerical, and analytical methods) of a single column woodpile periodic structure revealed that this Hertzian, locally-resonant lattice offers an experimental model for the generation of traveling waves composed of a highly-localized solitary wave atop a low amplitude oscillatory tail [56] (Fig. 6). The system proved instrumental in manipulating stress waves to attain strong attenuation and modulation of high-amplitude impacts without depending on system damping. The investigated setup showed that the mass-in-mass/mass-with-mass analytical depiction of a granular chain with an internal resonator could be fine-tuned to closely replicate the experimental findings [57].

To summarize, the dynamics of wave propagation in granular chains with internal and external resonators can significantly influence sound wave transmission. By adjusting the frequency band gaps in the audible range, the acoustic properties can be fine-tuned, offering precise control over mechanical wave filtering. The behaviour of mechanical pulses is affected by the mass ratio between resonators and beads, determining the extent of energy reflection and trapping. Additionally, the woodpile structure enables low-frequency resonances and the manipulation of high-amplitude impacts, suggesting that granular chains can be engineered for specific wave management applications in acoustic engineering.

4 Curved chains

The propagation of solitary waves in granular chains can be controlled and mitigated by various methods, including introducing inhomogeneities or trapping energy in locally resonant structures. In particular, the use of curved chains, along with their interaction with the guides, has been discovered as an effective approach to mitigating the transmission of pulses.

Fig. 7

Curved granular chains. A Experimental setup of a curved (left) and straight (right) configuration of an homogeneous granular chain. Image from [58]. B Experimental setup of a curved (left) and straight (right) configuration of a diatomic chain composed of sphere-cylinder unit cells. Image from [59]. C Experimental setups of variable curved granular chains. Setup to study the effect of variable deflection angles (upper) and setup to study the effect of different radii of curvature at a given deflection angle (lower). Figures from [60]

In the research paper [59], 1D curved granular chains, both homogeneous and diatomic, were analyzed through both experimental and numerical methods. The homogeneous structure consisted of 20 stainless steel beads, while the diatomic chain was made up of 10 sphere-cylinder units (Fig. 7A, B). These chains were guided by four pre-curved rods fabricated from flexible polytetrafluoroethylene (PTFE) material. The researchers compared the energy filtration capabilities of the curved chains with their equivalent straight chains (of the same material and size), using the second norm energy (SNE) of the measured force profiles. They discovered that the transmitted energy amounts were similar for low-mass impacts, but started to diverge as the impact amplitude increased (with a large-mass striker). The curved chains demonstrated a higher efficiency in absorbing high-impact energy (forces exceeding 200 N), reducing energy up to five times more than the comparable straight granular chain. Particularly, for the diatomic configuration, both numerical and experimental results revealed that the linear frequency spectrum of the transmitted waves created pass- and stop-bands, illustrating the tunable, frequency- and amplitude-dependent filtering of incoming signals due to the interaction between granular particles and the flexible elastic medium. The researchers also validated the formation and propagation of highly nonlinear solitary waves and showed that wave propagation can be readily adjusted by altering the precompression or initial curvature of the granular chains. They also discovered that the wave transmission efficiency could be regulated by manipulating the imposed curvature and precompression of the granular crystal. The same authors reported similar findings in [58, 61].

Cai et al. investigated the attributes of highly nonlinear solitary waves (SWs) traveling in curved granular chains based on the curve angle and the radius of curvature, using experiments and numerical simulations [60] (Fig. 7C). The authors found that the attenuation of SWs was heavily dependent on chain deflection angles and curvatures. Remarkably, as a single pulse propagated through the bent chain, 71% of its initial force amplitude was conserved even under a 90$^{\circ }$ turn, and 50% of its force amplitude was conserved after a 180$^{\circ }$ turn. They also noticed that when SWs encountered a sharp turn with a radius smaller than a spherical particle’s diameter, secondary SWs were created due to the interaction with the guide rail. The measured transmission ratios for the tested curvatures were within the 40–70% range.

Overall, the studies suggest that curved chains can effectively control and mitigate the transmission of solitary waves in granular chains and that the wave transmission can be easily tuned by manipulating the curvature and precompression.

5 Granular chain networks: pulse distribution

Granular chains have proven to be excellent metamaterials for acoustic applications, in particular wave control and pulse attenuation. So far we have been reviewing variations of these 1D granular systems, from homogeneous 1D straight chains, to decorated sequences, locally resonant mass-with-mass structures and finally curved chains. Given the richness of nonlinear dynamical phenomena found in one-dimensional structures, higher-dimensional nonlinear systems are expected to display a host of new dynamical effects. In this section we will discuss the propagation of waves in a particular case of such multidimensional systems: networks of 1D chains arranged in space.

Fig. 8

Granular chain networks. A 2D double Y-shaped structure. Illustration from [62]. B Y-shaped granular crystals with variable branch angles. Illustration from [66]. C 2D granular chain network. Image from [69]. D 3D granular chain network. Figures from [70]. E 2D granular chain network with discontinuous lateral boundaries. Image from [71]

In 2010, Daraio et al. published their findings from experimental and numerical studies on the propagation of highly nonlinear signals in a two-dimensional branched granular system [62]. The system under study involved a double Y-shaped guide that housed high- and low-modulus/mass chains of spheres in diverse configurations (see Fig. 8A). They used beads made from stainless steel, PTFE, and Parylene-C coated steel [63]. Their findings highlighted that the structure led to a quick division of the initial pulse, swift signal chaotization, and redirection and bending of impulses. The researchers also noticed pulse and energy trapping in the branches. An analytical examination of the same system, utilizing the binary collision approximation [64], was reported in [65].

The research in Fig. 8B presented a similar experimental setup involving a single Y-shaped granular crystal made of chains of spherical particles of different materials [66]. Stainless steel and aluminum beads were organized at varying branch angles. Asymmetrical structures were also created by modifying the angles of the left and right branches. Experimental results indicated that the amplitude and speed of reflected waves decreased as branch angles increased (with the sum of the branching angles reaching 90 degrees, no reflected wave was observed in the main stem). Additionally, the researchers offered analytical predictions based on the quasi-particle model [34, 67] and numerical simulations relying on the Discrete Element Method [68] interactions between the particles, which were congruent with experimental data.

Building on this research, Leonard et al. analyzed the propagation of stress waves through orderly two-dimensional networks of granular chains, with multiple levels, in [69] (Fig. 8C). The system was initially examined analytically using the quasi-particle approximation, which showed strong pulse mitigation properties. Results from simulations and experiments indicated that the amplitude of the primary pulse traveling through the system decayed exponentially with propagation distance, and the spatial structure of the transmitted wave displayed exponential localization along the incident wave’s direction. The reported transmission coefficients for the central branch (collinear with the incident pulse) ranged between 0.5 and 0.58 (for admissible branching angles between 30$^{\circ }$ and 45$^{\circ }$), while the values for the side branches varied between 0.25 and 0.36.

The same authors—Leonard et al.—reported their findings on the transmission of pulses in a similar network, but now on a 3-dimensional scale [70]. Taking inspiration from the naturally formed force chains in granular media, they engineered a 3D structured composite that exhibits an exponentially fast decay of the leading transmitted pulses. The system was studied analytically, experimentally and numerically. In the experiments, the granular networks were constructed from stainless steel sphere chains held in place by a polymer support channel structure. Experiments were performed for different levels of branching—1, 2 and 3—using branch angles of 40 degrees and granular chains made of 5 particles. Figure 8D shows an illustration of the 3-level granular network. The reported transmission coefficients for the central branch(es) were between 0.3 (for 1-level networks) and 0.15 (for 3-level networks), while the values for the lateral branches were much smaller (below 0.1). These networks exhibit reversible deformation, allowing them to return to their original shape after impacts within a certain threshold. However, what truly sets them apart is their exceptional functionality across all frequencies coupled with a low relative density, giving them a distinctive advantage over conventional impact mitigation materials like metal, rubber, or concrete.

A distinct system was examined analytically and numerically in [71]. The granular structure in question was a two-dimensional granular network made of two geometrically linked, orderly granular chains of spherical beads. These chains had spherical intruders placed between them and were subject to discontinuous lateral boundary conditions, as depicted in Fig. 8E. The authors noted intricate phenomena such as pulse propagation, localized motion, and the formation of propagation and attenuation zones due to nonlinear on-site potentials. These potentials were generated by the discontinuous lateral boundary conditions of the network, as well as by the highly nonlinear interactions between the heavy beads and the interstitial intruders. These transfers of energy, from low to high, facilitate an effective scattering mechanism for energy, which quickly attenuates axially propagating pulses and confines the impulsive energy close to its point of origination, preventing it from propagating to the network’s far field. The study reported high pulse mitigation values, indicating the system’s suitability for passive vibration and shock isolation.

In conclusion, the propagation of stress waves through granular chain networks has been extensively studied, revealing a wide range of complex phenomena and potential applications in the field of impact mitigation and passive vibration isolation. Various experimental and numerical investigations have been conducted on 2D and 3D granular systems, showing the rapid splitting, chaotization, and redirection of pulse signals in branched structures, as well as the exponential decay of transmitted pulses in ordered networks with interstitial intruders. The use of granular chain networks as structured materials for impact mitigation is particularly promising due to their reversible deformation, work at all frequencies, and low relative density.

However, their practical implementation faces challenges such as design complexity, manufacturing difficulties, questions surrounding long-term durability, rigorous testing requirements, and potential integration obstacles into existing product designs. Despite these hurdles, continued research shows potential for real-world applications in the future.

6 Multidimensional granular crystals

In the previous section we reviewed works related to the transmission of pulses through a certain type of multidimensional granular crystal—granular chain networks—however, there are other configurations that have been the focus of investigation [72,73,74,75,76,77,78,79,80]. We highlight some of the most interesting configurations below.

Fig. 9

Multidimensional granular crystals. A Dynamic load transfer in a 2D body-centered cubic assembly of elliptical particles. Figure from [81]. B Density plots at different time intervals corresponding to the amplitude of the particle velocity obtained as a function of spacial position in a 2D square granular crystal: under in-plane—between two edge particles—impact (upper) and out-of-plane impact—simultaneously impacting four central particles (lower). Results obtained from numerical simulations. Images from [82]. C Particle excitation in a 2D square granular crystal with interstitial particles inserted between the gaps. Figure from [83]. D Decorated two-dimensional granular crystals made with two types of granules: main and interstitial. Left and right correspond to tapering factors $q = 0$ and $q = 0.15$, respectively. Illustration from [84]. E Wave front shape evolution in a hexagonal granular crystal. The array was centrally impacted along the edge by a striker sphere. Figure from [85]. (F) 3D woodpile structure. Image from [86]

The simplest instance of a highly nonlinear multidimensional granular crystal is a uniform, uncompressed 2D square arrangement of touching elastic particles. The initial experimental examination of the dynamics of such a system was presented in [81], which used photoelastic elliptical disks excited by an explosive charge (Fig. 9A). The researchers analyzed the stress wave propagation in different geometric configurations of the system, finding that the main influences on wave propagation characteristics, such as load transfer path and load attenuation, are the contact normals and vectors connecting the particles’ centers of mass. Sadd et al. [87] reported a numerical approach to a similar structure using the Discrete Element Method (DEM [68]), with a focus on linking material microstructure to wave propagation behaviours.

The formation and movement of solitary waves (SWs) in 2D square granular crystals were studied for the first time, both experimentally and numerically, by Leonard et al. [82]. The system comprised uncompressed stainless steel spheres excited by impulsive loadings (Fig. 9B). The authors used a technique they developed, the miniature tri-axial accelerometers, to examine the stress wave properties in the array resulting from both an in-plane (between two edge particles) and out-of-plane impact (simultaneously impacting four central particles). The impulsive excitation was resolved into SWs traveling only down initially excited chains. The observed SWs exhibited similar (Hertzian) properties to the widely studied SWs supported by an uncompressed, uniform, one-dimensional chain of spheres.

The sensitivity of acoustic materials based on nonlinear granular networks to the presence of variation in particle geometry is a significant challenge for their experimental realization. Loss of contacts or local compression ultimately results in disordered energy transfer between the particles. Several studies have investigated the effects of imperfections in 2D granular crystals and their influence on the propagation of stress waves [72, 73, 88,89,90]. These studies found that defects tend to increase the velocity of wave propagation at a power law ratio of 1/4 to the maximum force, as opposed to the value of 1/6 predicted for Hertzian behaviour [91], for granular crystals under low precompression. In [92], the dynamic response of 2D square granular crystals, made of spherical and cylindrical particles, was experimentally and numerically characterized, showing that variations in packing geometry/composition can significantly affect the characteristics of propagating stress wave fronts.

A practical solution to the issue of lack of robustness in a square lattice was presented in [83, 93]. The approach involves the insertion of “intruder” particles into the gaps between particles in a host crystal, akin to a 2D extension of previously discussed side-decorated granular chains. This mechanism can be used to either redirect pulses [83] (Fig. 9C) or achieve nonlinear pulse equipartition [93] when intruders are not confined to a local area in the crystal. In [93], a line of interstitial particles (or intruders) was placed between two adjacent uncompressed chains of larger particles in a square packing of spheres. Experiments and numerical simulations demonstrated how the initial pulse propagating through one chain attenuates as the energy is transferred across the system until equipartition is reached.

A similar study was carried out by Machado and Sen [84]. The article presents a numerical study of the behaviour of uncompressed granular crystals under sinusoidal excitation within the frequency region 1–100 kHz. The authors observed that the incident signals could either be transmitted or filtered, depending on the driving amplitude, and demonstrated the use of square tapered crystals with decoration to enhance the low—frequency filtration properties of granular systems (Fig. 9E). Additionally, they found that a crystal thickness of 4 grains could attenuate 80% or more of the input force.

There are several studies that describe the dynamic response of 2D hexagonal networks under different loading conditions [85, 87, 89, 94]. One example is the study conducted by Sadd et al. [87], which numerically investigated wave speed and pulse attenuation in materials composed of elliptical particles, both in regular and random assemblies. In another study, Leonard et al. [85] experimentally, numerically and analytically analyzed another type of such structures, namely a two-dimensional hexagonal packing of uncompressed stainless steel spheres excited by localized impulsive charges (Fig. 9E). The authors provided scaling relationships characterizing the directional power law decay of wave velocity for various angles of impact (note that the hexagonal lattice does not support constant velocity traveling waves). They also discussed the effects of weak disorder on the directional amplitude decay rates and concluded that the level of disorder present in experiments does not cause significant deviation of the propagating wave structure from the predicted system response.

To progress towards the industrial application of granular structures, it is crucial to investigate these structures using 3D geometries. However, the dynamic response of 3D granular crystals is highly complex, presenting a challenge for the research community and consequently receiving less attention compared to their 1D and 2D counterparts [78, 95, 96]. Despite this, there is promising news as studies have demonstrated that the properties of 1D systems can be scaled to predict the behaviour of more intricate 2D and 3D granular crystals [95].

Three-dimensional granular composites made of non-spherical particles have also been investigated. In [86], Kim et al. studied a system consisting of a 3D woodpile metamaterial composed of vertically stacked thin cylindrical rods made of stainless steel, with 2 rods per layer (Fig. 9F). The authors demonstrated through numerical and experimental studies that the woodpile system can effectively localize, modulate, and attenuate propagating nonlinear waves, without relying on material damping. Consistent with the findings of previous studies on one-dimensional configurations [33, 97], as discussed in earlier sections, the transmission properties of the assembled woodpile structures can be adjusted by varying the layup angle. As a result, they exhibit both high damping and high stiffness simultaneously, where the damping is not due to inherent material damping of the rods but rather arises from the contact changes. For instance, the authors showed that changing the layup angle from 90$^{\circ }$ to 45$^{\circ }$ reduced the maximum total force transmitted through the woodpile by more than a half.

In summary, multidimensional granular crystals have attracted interest because of their ability to modulate elastic wave propagation through configurations like square and hexagonal arrays and respond to defects and particle variation. These systems’ ability to manage energy transfer and influence wave speed through structural design has important implications for acoustic and mechanical engineering. However, challenges persist, particularly the sensitivity to imperfections that can disrupt energy transfer, and the complexity of modelling the dynamic response of 3D configurations. Nonetheless, ongoing research suggests their potential for practical applications is considerable.

7 Research methods: development and impact

In this section, we delve into the evolution of scientific study in recent years, tracing the trajectory from Nesterenko’s pioneering work on solitary waves to the cutting-edge granular structures employed for pulse mitigation today. Our objective is to provide a comprehensive understanding of the primary research methods and directions pursued within the scope of granular metamaterials, while also shedding light on the associated limitations and challenges.

7.1 A brief history of research on wave propagation in ordered granular structures

The exploration of wave propagation in ordered granular structures traces its roots back to the groundbreaking work of Nesterenko in 1984 [10]. This seminal study focused on investigating nonstationary, nonlinear perturbations in one-dimensional granular systems, considering the Hertzian-like interactions between neighboring granules. Nesterenko employed a continuum equation of a nonlinear chain of oscillators, being a long-wave approximation, to describe the mechanical wave propagation in the system (see 7.2 Analytical Methods). Although experimental verification was not conducted in this particular study, subsequent research by Lazaridi and Nesterenko in 1985 [11] confirmed the existence of a new type of solitary waves (SW) in one-dimensional granular media. These waves are now commonly referred to as Nesterenko solitary waves, a term coined by Coste, Falcon, and Fauve in 1997 [99].

Nesterenko’s pioneering study ignited a flame of interest, turning this area into an active and vibrant domain of both theoretical and experimental research. Over the last few decades, numerous papers—over 500—have been dedicated to exploring the dynamics of strongly nonlinear waves in granular structures.

Among the notable earlier contributions, Chatterjee’s asymptotic approach stands out for deriving a highly accurate description of the traveling wave, surpassing Nesterenko’s approximate solution [100]. On the experimental front, Job et al. developed a non-intrusive and reliable method to investigate the propagation and reflection of SW at walls [101]. Their work revealed that the characteristics of SW reflection depend on the mechanical properties of the walls. Looking further ahead, the introduction of modern numerical methods revolutionized the study of pulse transmission in granular chains. Fraternali, for instance, employed evolutionary algorithms to search for an optimal design of composite granular protectors [17] (for more details on the article, refer to Sect. 2).

The scope of research has also extended beyond the confines of 1D granular systems and linear chains. As detailed in this review, granular metamaterials of diverse types have garnered significant attention, ranging from disordered chains to curved chains, locally resonant structures, granular networks, and multidimensional granular crystals. Moreover, recent advancements have introduced various novel approaches, such as investigating the impact response of 2D granular crystals submerged in different fluids [98] (Fig. 10), exploring wave dynamics in 2D finite granular crystals comprising polyurethane cylinders under low-velocity conditions [80], and delving into granular micromechanics [102].

Fig. 10

2D granular crystals submerged in different fluids. Time evolution of velocities of the grains in the rightmost column starting with grain in 1st row to the 13th row (upper). Experimentally obtained coefficient of attenuation for the crystals under various fluids. Image from [98]

7.2 Analytical methods: transitioning from the continuous approach to the discrete p-Schrödinger equation

In recent years, numerous theoretical approaches have emerged to examine wave propagation in ordered granular structures. From Nesterenko’s continuous approximation to the discrete p-Schrödinger equation, this section provides a review of these approaches, including their strengths and limitations, as well as their evolution over time.

In the majority of granular chain analyses, the Hertz law is used to characterize the interaction between grains. However, this law is subject to certain conditions, such as the requirement for the maximum stress to remain below the elastic limit, the length of the contact surface to be significantly less than the particle radius of curvature, and the problem’s characteristic time to be substantially longer than the elastic grain oscillation period (long wave limit). The Hertzian power index, which portrays how the repulsive potential scales with a material’s deformation, is dependent on the contact particles’ geometry. For spherical particles, the exponent stands at 5/2, whereas for cylinders with their axes in parallel, the exponent is 2. Note that the force-displacement power index is 3/2 and 1, respectively, as noted before, as the force is the time derivative of the potential. The Hertz law is used as the basic principle for determining contact forces in most of the methods discussed in this section.

In his seminal paper of 1983, Nesterenko aimed to study pulse propagation in granular chains, taking into account the unique properties of these materials. To achieve this, he adopted the continuous approach, recognizing that the discrete nature of granular chains could not be described by conventional wave equations. Nesterenko treated the chain as an elastic medium and derived a set of equations to describe the motion of each particle in the chain.

He considered two scenarios: strongly and weakly precompressed chains. In the case of strong precompression, the chain was assumed to be weakly disturbed, meaning that its deformations were small in comparison to the precompression and that the grains would never lose contact:

$$\begin{aligned} \frac{|u_{k-1}-u_k|}{\delta } \ll 1 \end{aligned}$$

(1)

where $u_k$ is the position of grain with index k and $\delta$ is the precompression. Nesterenko found that for this scenario, the solitary waves propagate at a speed proportional to the square root of the compression rate, meaning that they could travel faster in a strongly precompressed chain than in an uncompressed chain. Furthermore, the solitary waves were found to be highly stable, retaining their shape over long distances.

On the other hand, if the disturbances along the chain significantly outweighed the precompression, the precompression was deemed to be weak:

$$\begin{aligned} \frac{|u_{k-1} - u_k|}{\delta } \gg 1 \end{aligned}$$

(2)

In a weakly precompressed chain, energy propagation is a highly nonlinear phenomenon, with linear waves being insignificant compared to the nonlinear waves. Nesterenko found that in this scenario, the solitary waves propagate at a speed proportional to the compression rate, resulting in a much slower speed of the waves in a weakly precompressed chain compared to a strongly precompressed chain. Furthermore, the solitary waves in a weakly precompressed chain were found to be highly unstable, tending to lose its shape and amplitude over short distances.

Some of the limitations of Nesterenko’s continuous approach include assumptions of homogeneity and lack of friction or viscoelastic damping in the chain, and applicability only in the long-wave limit. Additionally, the discrete nature of granular chains is not taken into account in the model. Despite these limitations, the continuous approach remains a widely adopted reference for the study of pulse propagation in granular chains. The model has been adapted and modified to study various configurations, including cylindrical particles [97], ellipsoidal chains [103], toroidal particles [49], chains with friction and viscoelastic damping [104, 105], local resonators [52, 53, 106], curved chains [61], and multidimensional granular crystals [107]. The approach’s simplicity and ability to provide accurate results have made it a valuable tool for researchers and engineers [108].

In 1998, Chatterjee introduced a new asymptotic solution for the study of solitary wave propagation in granular chains, which is valid near the wave crest [100]. The asymptotic approximation provides a more accurate description of the wave compared to Nesterenko’s continuous approach, and has since been applied to various types of granular chains, including cylindrical chains [33, 109], dimer chains [110, 111], and local resonators [112]. This method is based on a perturbation analysis of the equations of motion of the granular chain, providing insights into the interparticle forces and interactions that drive wave propagation. It can also be applied to chains with different grain size distributions and interparticle interactions, making it a versatile tool for the study of granular chains. That said, one of the main breakthroughs of Chatterjee’s work is in establishing asymptotically a super-exponential (double-exponential) decay of Nesterenko solitons, an important finding that sheds light on an important property of compact-like solitary waves of uncompressed granular media. This pioneering study has been supported by more advanced analytical studies, including English and Pego’s work years later that validated the double-exponential asymptotic decay of solitary wave pulses in a bead chain system [113]. That said, the asymptotic approximation is a complex method, which may not always accurately capture the nonlinear phenomena present in granular chains. Despite its limitations, Chatterjee’s asymptotic approximation remains a valuable tool for the study of wave propagation in granular chains.

An alternative method for investigating wave propagation in granular chains is the hard-sphere approximation, also known as the independent collision model [17, 25, 30, 114]. This model postulates that a pulse advances via a series of binary interactions between an inbound grain and a stationary one. The resting grain’s final velocity, ascertained using energy and momentum conservation laws, is subsequently used as the initial velocity for the inbound grain in the ensuing collision. Despite its straightforwardness, this approximation has its limitations as it fails to consider the power-law potential, resulting in an inability to differentiate between pulse propagation in chains with varying interactions.

A further development of the hard-sphere approximation is the binary collision approximation (BCA) [64]. This model builds on the hard-sphere model, with the key insight that collisions are not strictly independent. It defines the residence time of a pulse on a grain as the interval from the onset of the collision to the moment the target grain surpasses the speed of the colliding grain. The velocity of the pulse is then identified as the reciprocal of the residence time.

A comparative study between the continuous model and BCA was conducted by Rosas and Lindberg [104]. They found that both methods exhibit qualitative similarities, however, the continuous model performs better quantitatively for softer potentials, while the binary collision approximation is more appropriate for the study of chains with harder potentials.

The hard-sphere approximation and BCA have been used to study the wave propagation in various configurations of granular chains, including cylindrical chains [50], decorated and tapered chains [111, 115], local resonators [57], and granular networks [65, 116]. Efforts have been made to extend this approach to different geometries and incorporate modifications to account for unique features of the system under investigation. One of these cases involves the addition of a third element to the model, known as ternary collision approximation, as introduced by Kim et al. in their study of a system where three elements were involved in the collision: two particles and a local resonator [56]. The method has also been extended to investigate two-dimensional hexagonal packings [85], where 4 particles were considered, and the symmetry was used to reduce the degrees of freedom. Building upon the previous models, an advanced amplitude mapping technique was introduced to evaluate the effects of localized dissipative and non-dissipative perturbations on granular chains [117, 118].

The quasi-particle approximation is another widely used approach for studying wave propagation in granular chains. This approximation, first proposed by Nesterenko in 1994 [12, 67], replaces a solitary wave with an effective particle that has the same energy and momentum as the wave. This model was later refined by Job et al. [34].

Despite its limitations, such as being based on a linear theory that may not accurately describe nonlinear phenomena and not providing detailed information about grain interactions, the quasi-particle approximation has several advantages. Firstly, it is a simple and intuitive model that reduces the complex interactions between grains to a small number of coupled equations of motion for the quasi-particles. Secondly, it has been found to be a useful tool for predicting the formation and stability of solitary waves, as well as the response of granular chains to external perturbations [34]. Additionally, the flexibility of the approximation allows for the inclusion of various types of interactions between the grains, including long-range interactions and interactions with external fields [103]. The simplicity, adaptability and predictive power of the quasi-particle approximation have made it a valuable tool in the study of granular networks, as demonstrated by its application in several studies [69, 70, 103].

More recent studies on wave propagation in granular chains involve the use of the Nonlinear Schrödinger equation (NLS) [119] and the Discrete p-Schrödinger equation (DpS) [120]. The Nonlinear Schrödinger equation (NLS) is a partial differential equation that describes the behaviour of waves in a nonlinear medium, typically formulated as:

$$\begin{aligned} i\varPsi _t + \frac{1}{2}\varPsi _{xx} - \kappa |\varPsi |^2\varPsi = 0, \end{aligned}$$

(3)

where $\varPsi$ represents the wave function, and $\kappa$ depends on the nonlinearity.

The DpS equation was derived by Guillaume James as an approximation for small amplitude nonlinear waves in Newton’s cradle [121]. The derivation involves formally deriving a spatially discrete modulation equation for these waves, which consists of a discretization of a generalized Schrödinger equation with p-Laplacian. The value of p in the DpS equation depends on the exponent of Hertz’s contact force. The DpS equation captures the fully-nonlinear and unilateral interactions between beads, yielding a nonstandard modulation equation that accurately describes the behaviour of the system in the weakly nonlinear regime.

The NLS equation is able to capture several important features of the dynamics, including explicit periodic traveling wave solutions, standing wave solutions, and spatially localized breather solutions [122]. It is also able to reproduce modulational instabilities, the existence of static and traveling breathers (spatially localized, time-periodic oscillations), and the repulsive or attractive interactions of these localized structures. Both models have been used in the study of locally resonant granular chains: NLS [53, 106, 112] and DpS [121, 123,124,125].

The NLS and DpS equations are cornerstone models in the study of wave dynamics in nonlinear systems, such as granular crystals. These models offer asymptotic approximations for the dynamics of amplitude envelopes of low-amplitude waves, providing essential insights into the complex behaviour of these systems. In the field of granular media, these models have been employed to describe breather dynamics in Fermi-Pasta-Ulam (FPU) lattices [126] representing chain of beads interacting via Hertz’s contact forces.

The NLS equation is adept at describing the interaction and evolution of wave packets in settings where nonlinearity plays pivotal roles, such as in granular chains under pre-compression. Chong et al. [127] present a study on the presence, stability, and bifurcation of dark breather families within a one-dimensional uniform bead chain under static load. They derive a defocusing NLS equation for frequencies near the phonon band edge to set up initial conditions for numerical analysis. Key findings include the discovery of large amplitude solutions evolving from smaller ones predicted by the NLS equation, and a nonlinear instability previously unobserved in classical FPU lattices. For reference, dark breathers are localized nonlinear wave phenomena characterized by a localized region of low amplitude oscillations surrounded by a background of higher amplitude oscillations. They are the opposite of bright breathers, which feature a localized region of high amplitude oscillations against a calm background [128].

On the other hand, the DpS equation is tailored to address specific types of nonlinear dynamics that may not be directly covered by the NLS model. It is particularly suited for systems where the standard assumptions of the NLS model might not hold, such as uncompressed granular crystals mounted on elastic foundations, such as Newton’s cradle under the influence of gravity [121]. Two years later [123], investigated the occurrence of breathers in uncompressed FPU lattices with Hertzian contact interactions, initially proving their nonexistence. When introducing an on-site potential, akin to a Newton’s cradle influenced by gravity, breathers are shown to exist through numerical methods and a simplified DpS equation model. Stability and the ability for breathers to move under slight disturbances are confirmed via spectral analysis. Numerical simulations reveal that traveling breathers can be excited by minor perturbations at the chain’s start, accurately modelled by the DpS equation for specific physical parameters in granular chains with stiff local oscillators. However, breather propagation can be blocked or stopped in certain conditions. With soft on-site potentials, energy tends to stay near the boundary, creating a surface mode, while hard potentials with significant initial excitations produce a “boomeron”-a breather that reverses direction spontaneously. Additionally, precompression enhances dispersion, leading to either a lack of traveling breathers over long periods or the creation of a “nanopteron,” marked by a significant wave train around the localized excitation, depending on the parameters.

In summary, the NLS and DpS equations have significantly advanced the study of discrete breathers in granular chains, addressing systems both with and without compression. The NLS equation excels in modelling compressed granular chains, capturing the essence of nonlinear wave propagation. It has underscored complex wave phenomena in such settings, such as the dynamics of dark breathers. The DpS equation, on the other hand, is particularly effective for exploring the dynamics of uncompressed granular crystals mounted on elastic foundations. It offers insights into the existence, stability, and propagation of breathers in scenarios where traditional NLS assumptions may not apply. Both models have enriched our understanding of nonlinear systems, each highlighting unique aspects of wave dynamics and interaction in granular media, demonstrating the intricate balance between nonlinearity, discreteness, and external forces in shaping wave behaviour (Fig. 11).

Fig. 11

Extraction of force-chain network architecture in granular materials using community detection. Sample community-detection results for a single granular experiment. Image from [129]

In the study of ordered granular systems, it is crucial to highlight the supportive role of various semi-analytical methods that build upon and enhance the previously discussed analytical framework. The Padé approximation notably augments this analysis by representing a function through the ratio of two polynomials, thereby providing a more expansive representation over the function’s domain than traditional polynomial series. As the most suitable rational function approximation of its respective order, the Padé approximation has proven effective in estimating soliton solutions in granular lattices under conditions of acoustic vacuum [24, 130, 131]. Furthermore, English and Pego’s iterative method has contributed a distinct approach to formulating precise solitary wave solutions in a contiguous array of beads lacking initial compression [113].

Complementing these, the WKB (Wentzel-Kramers-Brillouin) approximation stands out as another semi-analytical method, predominantly employed to approximate solutions to linear differential equations with spatially variable coefficients. Its application extends to the analysis of phenomena such as nonlinear resonance in dimer granular chains, where its utility has been demonstrated in several studies [109,110,111].

7.3 Numerical methods: exploring the influence of enhanced computational capacity and AI

In order to delve deeper into the intricacies of pulse propagation in granular chains, it is necessary to supplement theoretical approaches such as the quasi-particle approximation and Chatterjee’s asymptotic approximation with numerical simulations. Despite the valuable insights provided by these theoretical approaches, they have limitations in their accuracy in capturing nonlinearities and providing a complete picture of inter-grain interactions. This section will provide a comprehensive overview of the numerical methods used in studying pulse propagation in granular chains, highlighting some of their strengths and limitations.

Arguably, the most widely used numerical method for the study of granular particle dynamics is the Discrete Element Method (DEM), first introduced by Cundall and Strack in 1979 [68]. This computational method models the interactions between individual particles in a granular system, enabling the simulation of granular material behaviour. The method involves numerical integration of the equations of motion for each particle, taking into account interactions such as collisions and friction, providing a detailed understanding of the physical coordinates of the system at any point in time. The output of the DEM is fully determined by its initial conditions, making the method deterministic. Its versatility and accuracy have made the method widely used in studies of granular materials, including single particle behaviour [132], large scale simulations of granular flows (more than 200 000 particles) [133], and granular materials [134, 135].

In the study of wave propagation in granular chains, the DEM has been used to model the behaviour of various granular systems. This includes monodisperse spherical chains [101], stepped chains [32, 34], dimer chains [33], decorated chains [39], tapered chains [26], disordered chains [18, 23], curved chains [136], cylindrical particles [97], elliptical particles [103], local resonators [57], granular networks [69,70,71], multidimensional granular crystals [79, 95], and particle dampers [137,138,139] (mechanical devices used to mitigate vibrations and shock loads in a system).

There are multiple software options available for conducting DEM simulations. Commercial packages such as EDEM [140], Bulk Flow Analyst [141], and Ansys Rocky [142] provide advanced features and capabilities for simulating the behaviour of granular materials, making them suitable for both research and industrial applications. On the other hand, non-commercial and open-source packages like LAMMPS [143] and LIGGGHTS [144] are available for those seeking more flexible and customizable options. These software options offer a range of possibilities for studying granular systems and provide a valuable resource for the research and practical work in this field.

The Discrete Element Method has limitations such as computational intensity, which increases with the number of particles being tracked and their interactions. When complexity is added to the simulation such as using non-spherical particle shapes, moving boundaries, contact force models, and secondary forces like cohesion and coupling with CFD [145] computational time increases. Despite these challenges, the processing speed of DEM simulations is improving with advancements in computing power. Validation of DEM results can also be difficult in systems where experimental measurements are hard or costly. Using spherical shapes for particle representation in DEM models may simplify model development but can fail to accurately reflect mechanical interlocking in non-spherical particles [146]. Additionally, this numerical method is not optimized for modelling continuous media and fluid dynamics, and as a result, it can be an inefficient approach for such simulations. In DEM modelling, numerous parameters such as material properties and contact parameters need to be precisely specified. These parameters have a substantial impact on the simulation outcomes, yet they can be challenging to measure or estimate with accuracy in real-world systems.

Another commonly used numerical method in the study of wave propagation in granular chains is the Finite Element Method (FEM) [147]. This is a well-established numerical approach for solving partial differential equations in two or three dimensions, that unlike DEM, focuses on the modelling of continuous media. This method is particularly useful in the study of pulse propagation through granular chains as it provides a comprehensive simulation and analysis of energy or signal movement through the granular material. FEM discretizes a complex system into smaller, simpler parts called finite elements that are connected at nodes to form a mesh. Equations governing the behaviour of each element are then formulated and assembled into a system of equations, which is solved using numerical methods.

FEM’s ability to accommodate non-linear material properties, a common characteristic in granular materials and mechanics, makes it a useful tool in understanding pulse propagation through granular chains. This allows for more detailed design and optimization of these systems, and also accounts for the effect or interaction between particles and their containing or guiding structure. Additionally, FEM’s extensive usage in the study of wave propagation in granular chains has generated a wealth of knowledge that can serve as a foundation for future research. It has been used in the investigation of monodisperse spherical chains [108], cylindrical particles [86], local resonators [54], multidimensional granular crystals [79] and particle dampers [148, 149].

There are several software programs available for implementing Finite Element Method simulations. Some of the popular commercial FEM software programs include Ansys [150], Simulia [151] and COMSOL Multiphysics [152]. There are also open-source FEM software programs, such as Elmer [153, 154] and GetFEM++ [155].

The Finite Element Method for pulse propagation in granular chains faces limitations including high computational cost for complex and nonlinear models, difficulty in modelling discontinuous behaviour, potential errors from element choice and material model formulation, and computational intensity for large-scale models requiring specialized hardware/software.

A prevalent methodology for investigating pulse propagation in granular chains is the coupling of the Discrete Element Method (DEM) and Finite Element Method (FEM) [56, 58, 61]. This approach harnesses the advantages of both techniques, with the DEM being used to model the two-dimensional particle interactions in granular materials, while the FEM is used for quantifying the dynamics of linear elastic guides, breaking them down into Bernoulli-Euler beam elements. The derived combined DEM and FEM model effectively encapsulates the axial and tangential interactions between particles, inclusive of the dissipative elements, offering a more thorough comprehension of pulse propagation in granular chains.

The Finite Difference Method (FDM) [156] is a numerical approach used in the examination of granular systems, although to a limited extent when compared to the more popularly used FEM. Unlike FEM, which divides the domain into smaller subdomains and represents the solution using a set of basis functions, FDM divides the domain into a grid of discrete points and approximates the solution and its derivatives using finite differences.

While FEM is known to be more flexible and accurate, it comes at the cost of increased computational intensity. On the other hand, FDM is relatively simpler to implement, albeit being less flexible and less accurate.

Another numerical method used in the study of wave propagation in granular crystals is the Finite Difference Time-Domain Method (FDTD) [157]. This method divides the simulation into a series of space steps, and has been paired with the Discrete Element Method (DEM) to investigate the damping effect of bending vibration of a plate caused by granular materials [139].

A more recent method that builds upon the principles of FEM is the Particle Finite Element Method (PFEM) [158]. PFEM divides the solution domain into a series of particles and approximates the solution by tracking the movement of these particles over time. It has been used in the study of compaction of composite mixtures of soft and hard micro-/nano-sized particles [159].

Fig. 12

3D-printed supporting structures for ordered granular systems. A Supporting channel structure for granular chain network. Image from [160]. B 3D architecture of helical granular chains embedded in a soft medium. Image from [59]. C 3D-printed enclosures that provide support for the cylinders and enable independent rotation. Image from [50]

Recently, there has been a growing interest in incorporating machine learning into the study of pulse propagation in granular crystals. One of these innovative approaches is the use of Evolutionary Algorithms (EA) [161]. EA is a class of optimization algorithms that are inspired by the biological evolution process, where candidate solutions are treated as individuals in a population and undergo processes like reproduction, mutation, recombination, and selection to evolve towards better solutions.

Fraternali, Porter, and Daraio employed Evolutionary Algorithms (EA) in their exploration of the optimal design of composite granular protectors [17]. Their research used the Breeder Genetic Algorithm [162] to optimize the topology, size, and material properties of one-dimensional composite granular chains. The effectiveness of prospective solutions was determined by calculating the force transmitted from the protector to a wall under impact conditions, using a time-discretization method of Hamilton’s equations of motion. For computational efficiency, the hard-sphere model was used to represent the interactions between neighboring beads, given that the optimization process requires a significant number of simulations. Dissipative effects, typically neglected in literature [163], were not taken into account.

On a different tangent, Basset et al. [164] pursued an approach that centered on examining the impact of network topology on sound propagation in granular materials. They perceived these materials as networks where the nodes (particles) were linked by weighted edges based on contact forces. An extensive investigation was conducted on various network metrics to determine their aptitude in describing sound propagation. Aspects such as nodes, number of edges, and global efficiency were found to contribute to the understanding of particle curvature, domain, and system-level structures in granular media. The authors showcased that network analysis is beneficial for studying granular materials as it aids in identifying characteristic size scales and mesoscale structures that sound prefers to travel through. They discovered that particles with strong connections had larger-amplitude signals passing through them, and a weighted network predicted sound propagation better, implying the significance of the force-chain structure of the granular material in sound propagation. Both system-level and local-level network features were deemed necessary to comprehend how sound travels through a granular material.

In a subsequent work [129], they used Community Detection [165] to unravel the force-chain network architecture in granular materials. Community detection is a process that identifies subgroups of nodes in a network that have denser connections to each other than to other nodes in the network. This procedure aims to expose meaningful divisions or clusters in a network and improve understanding of the relationships and structure.

In more recent research, Chen et al. [166] presented two novel approaches to discovering metamaterials using machine learning, termed shape-frequency features and unit-cell templates. These techniques allow for the discovery of 2D metamaterials with user-specified frequency band gaps by using rule-based conditions on metamaterial unit-cells. The shape-frequency features approach computes the frequency of occurrence of a pattern in the unit cell, while the unit-cell template approach captures a global pattern related to the target properties. Both methods offer interpretable reasoning processes, generalize well across various design spaces, and provide design flexibility.

Over the years, specialized numerical approaches tailored for specific applications have also emerged. A notable example of recent progress involves algorithms that incorporate successive iterations and interpolations, facilitating precise modelling of the highly nonlinear and non-smooth scattering of impeding pulses from granular chains to flexible boundaries [167].

It is worth noting that despite the introduction of new and diverse computational methods, the Discrete Element Method and the Finite Element Method remain the most widely used numerical techniques in the study of granular systems. However, over the years, both methods have undergone significant advancements and refinements, tailored to specific applications. From the initial improvements in the frictional contact law [168, 169] to up-to-date advancements [170, 171], these developments, coupled with the significant growth in computational capacities, have facilitated more accurate investigations of larger and more realistic granular systems. To provide perspective, current DEM simulations have been able to model systems with up to 100 billion particles [172].

7.4 Manufacturing methods: assessing the impact of additive manufacturing

Additive manufacturing (AM), has profoundly transformed the manufacturing landscape, offering unprecedented capabilities for fabricating intricate and customized structures [173]. Within the scope of granular metamaterials, AM presents unique opportunities for designing and producing granular structures with tailored properties to suit specific applications [174]. Here, we delve into the implications of the advancements in AM technology, exploring their impact on the study and practical implementation of granular metamaterials.

The primary application that comes to mind is the implementation of customized guides to mold the ordered granular structure. Previous sections have explored several of these adaptations. For instance, in the design of Leonard’s 3D granular chain network [70], the supporting channel structure was fabricated through 3D printing (Fig. 12A). The components were produced modularly and subsequently assembled into networks of different sizes. AM technology was also employed to design a 3D architecture of helical granular chains embedded in a soft medium (Fig. 12B) [59]. Similarly, Chaunsali et al. [50] used AM to fabricate 3D-printed enclosures that provide support for the cylinders and enable independent rotation, thereby allowing for adjustment of their contact angles (Fig. 12C).

Beyond the production of custom support structures for granular systems, additive manufacturing is also used for granular particles. A study by Howard et al. [175] specifically investigates the use of 3D printing for controlling design parameters related to the shape and size of grains. The authors establish that grain shape is a critical factor in achieving high-performance, customized actuation behaviour, and it can be harnessed to enhance the scope and effectiveness of granular grippers in various application contexts.

The intricate structures required for mass-in-mass defects are another application for this technology. In [54], AM was employed to fabricate the elastic structure embedded within the beads that connected the external and internal masses, as illustrated in Fig. 5B. This approach allowed for the precise customization of the internal structure, facilitating the desired wave propagation characteristics and enabling the study of wave dynamics in the locally resonant systems (Fig. 13).

Fig. 13

3D printed grains of various shapes and sizes. Arrangements in groups of 4. Image from [175]

8 Summary

This review highlights the potential of granular metamaterials to effectively mitigate pulses and shape and control wave propagation through heterogeneous granular chains, resonant chains, curved chains, granular chain networks, and multidimensional granular crystals. Disordered and heterogeneous granular chains have been shown to attenuate the amplitude of the transmitted pulse by thermalization and deconstruction of the wave in a train of smaller pulses. By appropriately setting the size and material of the particles, high mitigation can be achieved. Resonant chains, on the other hand, exhibit wave filtering and mechanical energy trapping properties. Curved chains, showcase unique wave reflection properties due to the particle-guide interaction, and the wave transmission can be easily tuned by manipulating curvature and precompression. Granular chain networks achieve exponential stress mitigation by distributing the pulse energy. Multidimensional granular crystals, meanwhile, show highly controllable wave propagation properties depending on the particle arrangement.

The granular structures reviewed here have diverse applications, including pulse mitigation, impact energy absorption, and vibration isolation. Heterogeneous granular chains and granular chain networks hold promise for mitigating earthquake impacts on buildings and infrastructure, while resonant chains and curved chains offer opportunities for designing materials with specific filtering and reflection properties. Additionally, multidimensional granular crystals show promise in fields like optics, where precise control over wave propagation is of utmost importance.

9 Current challenges

One of the key challenges in the field lies in the development of more sophisticated analytical and numerical models that can accurately capture the complex dynamics and interactions within granular metamaterials. While significant progress has been made in this area, there is still a need for improved models that can account for the nonlinear and non-smooth behaviour exhibited by granular systems under various loading conditions and environments, not limited to single, specific applications. The development of advanced computational techniques, including machine learning, holds considerable promise for enhancing the accuracy and efficiency of these models.

Another challenge is translating research findings into practical applications. For instance, while studies on wave propagation in ordered granular structures have provided potential solutions for seismic wave mitigation, the real-world application in earthquake-resistant building design involves complex variables not present in controlled laboratory settings such as varying soil conditions, architectural constraints, cost limitations, and regulatory compliance. Similarly, in the development of protective gear like helmets or body armor, laboratory insights on impact absorption face real-world complexities such as the variability of impact scenarios, ergonomic design, material durability in diverse environments, and the demands of mass production for consistent quality. These cases illustrate the multifaceted challenges that must be navigated to move from the research stage to a market-ready product.

The scalability and reproducibility of manufacturing procedures for granular metamaterials also present significant challenges. Although additive manufacturing has proven its capacity to fabricate intricate, tailor-made granular structures, a pressing need exists to evolve scalable manufacturing techniques that can cater to the requirements of large-scale production. Notably, the inherent variability in the manufacturing process could introduce inconsistencies in the resulting structures, potentially impacting their performance. Careful quality control and process standardization are necessary to minimize this variability and optimize the efficiency of these materials.

Moreover, it is crucial to ensure the reproducibility of these structures in order to achieve reliable and uniform performance. Beyond this, it is important to consider how these granular metamaterials can be integrated into existing systems. Their multifunctionality should be explored—for instance, an impact pulse reducing chain system may also need to fulfil fire safety standards and structural requirements. An integrated approach to design and production could significantly broaden the potential applications and utility of these metamaterials.

10 Future outlook

Looking towards the future, there are several exciting prospects for the field of wave propagation in ordered granular structures. The continued exploration of novel materials and geometries, combined with advancements in manufacturing technologies, opens up opportunities for designing granular metamaterials with tailored properties for particular applications. For instance, the construction industry may see the integration of these materials in shock-absorbing layers for buildings in earthquake-prone areas, dramatically enhancing structural resilience. Similarly, the automotive sector could benefit from granular metamaterials in the design of car components where controlled energy absorption is crucial for improving crashworthiness.

Furthermore, medical devices might employ granular metamaterials for new types of protective equipment or in prosthetics, where the materials’ unique properties could lead to better shock absorption and comfort for the user. In sports, industry and emergency response, enhanced safety gear with wave-manipulating materials could reduce the risk and severity of injuries.

The integration of sensing and actuation capabilities within granular structures can further enhance their functionality and enable the development of smart and adaptive systems. Imagine wearable technologies with granular protectors that can dynamically adjust their shape and properties for protection in high-risk activities. In a similar vein, ordered granular elements in architecture could modulate their acoustic properties for noise cancellation or enhancement, reacting to environmental changes. Moreover, the precise wave manipulation capabilities of these materials offer exciting possibilities for advanced signal processing, opening avenues for breakthroughs in wave-based computing and neural networks. These examples represent just a glimpse of the potential applications that could transform industries and everyday life.