The Role of Frequency and Impedance Contrasts in Bandgap Closing and Formation Patterns of Axially-Vibrating Phononic Crystals

A bandgap, in solid-state physics, is an energy gap in the electronic band structure in which no electronic states exist [1]. Nearly four decades ago, the birth of photonic crystals gave way to photonic bandgaps, frequency ranges in which all optical modes are absent [2,3]. Several years later, phononic crystals (PnCs)—a class of periodic elasto-acoustic structures exhibiting forbidden wave propagation within given frequency regimes—extended the definition of bandgaps to the structural dynamics field [4]. Ever since, phononic bandgaps have played a central role in several engineering applications ranging from vibroacoustic control [5] and tunable materials [6], to topological mechanics [7] and nonreciprocal wave phenomena [8].

In its basic form, a PnC is a multi-layered composite where the layers self-repeat over an extended spatial domain. Rooted in the origins of periodic structure theory, studies depicting the unique wave propagation properties of PnCs predate the use of the term itself [9]. The most common one-dimensional PnC configuration involves two alternating materials (or a single material with alternating cross sections) forming a unit cell, often denoted as a diatomic or bi-layered PnC, in which bandgaps arise from Bragg scattering effects at the material (or geometric) interfaces. For an infinite medium, these Bragg bandgaps are a direct function of the structural periodicity and span one or more well-defined frequency ranges which can be predicted by a Bloch-wave analysis of the unit cell [10]. Increasing the number of unit cell layers (or atoms) gives rise to additional features which are uniquely defined by the sequencing and permutations of these individual layers [11]. Bandgap engineering, the science of manipulating phononic parameters within the infinite design space to achieve bandgaps of prescribed characteristics (e.g., bounds, location, attenuation level, targeted modes, directionality, and topological nature, among others) has significantly evolved [12]. In pursuit of such goal, studies have utilized geometric properties [13,14], material anisotropy [15,16], damping [17,18], viscoelasticity [19], inertance [20], pillared surfaces [21], topology optimization [22], and machine learning [23] as tunable knobs in an attempt to tune and achieve maximum control over the bandgap emergence process.

While the applications and utility of these bandgaps in novel and imaginative realizations of PnCs remain an active research area, especially with recent advances in manufacturing and fabrication, the physics underpinning the existence, formation mechanisms, and evolution of phononic bandgaps show intriguing phenomena which continue to be separately explored. Notable among these is the underlying connection between the dispersion relation of an infinite PnC relating the wavenumber of a wave to its frequency, and dictating the frequency-dependent phase and group velocities of a dispersive medium (of which a PnC is one) [24], and the structural resonances of a finite PnC where size and boundary effects become intrinsic to the dynamical problem [25,26]. This interplay between the mathematical description of infinite and finite media, and the ability to recover one from the other [27], was used to develop the theory of truncation resonances in finite PnCs by identifying a set of unique natural frequencies which avert dispersion branches at the infinite limit of the constitutive unit cell [28,29]. Furthermore, understanding the origination process of bandgaps in PnCs and the different ways in which wave attenuation manifests itself in finite periodic media has enabled phononic bandgaps to be artificially emulated in non-periodic lattices [30], or generated through radically different mechanisms such as inertial amplification [31,32].

Phononic bandgaps are accurately predicted from the conventional Bloch-wave analysis. However, PnCs made of solid continua exhibit a large number of bandgaps which vary in width, strength, and distribution, thus giving rise to the notion of “bandgap patterns.” As this work will show, these patterns are not random and bandgap arrangements in continuous PnCs are far from arbitrary. More importantly, bandgaps that obey certain conditions can be made to vanish (i.e., close by virtue of the preceding and following dispersion branches touching each other), thus rendering the mere existence of such bandgaps in phononic crystals not guaranteed. Instead of deploying numerical tools to seek bandgaps of desirable parameters, this work develops a generalized analytical framework which derives and unravels bandgap patterns and closing conditions in one-dimensional PnC rods undergoing longitudinal motion. This framework is then used to establish general rules which govern bandgap widths and folding frequencies, and connects deformational mode shapes of the culminating PnC to its constitutive layers. In doing so, we explain the conditions driving bandgap closing and connect them to physical origins in the context of the Bragg condition (for infinite media) and natural resonances (for finite counterparts). These developments are detailed for a bi-layered PnC and then generalized for a PnC of any number of layers by increasing the mathematical complexity, while retaining the fully-analytical nature of the model. The need to tailor phononic dispersion profiles have already been shown to play an instrumental role in metamaterial applications [33–36]. As such, we envision this mathematical development to be a future standard for designing bandgaps in PnCs with versatile and precisely targeted bandgap profiles.

Starting with the most general case, we consider a continuum PnC in the form of a one-dimensional rod which consists of self-repeating unit cells, where each cell is comprised of L material layers, as depicted in Fig. 1(a). In this work, we exclude any flexural and torsional waves, and focus on longitudinal motion described by the continuous function u(x,t). In the proposed PnC, each layer has unique geometrical and mechanical properties that do not necessarily match the rest. The sth layer of the unit cell has a mechanical impedance $zs=AsEsρs$ and a sonic speed $cs=Es/ρs$⁠, where E_s, ρ_s, and A_s are the elastic modulus, density, and cross-sectional area, respectively (⁠$s=1,2,…,L$⁠). The lumped parameter (spring-mass) description of this model is commonly referred to as a polyatomic PnC [11], with each layer within the unit cell denoted as an “atom.” The unit cell’s total length is $ℓ=∑sℓs$⁠, which is analogous to the lattice constant of a one-dimensional polyatomic PnC.

As can be observed from the right panel of Fig. 2, bandgap width profiles exhibit a wave-like behavior for all considered values of α, which perfectly repeats itself. Additionally, these profiles are noted to be mirror-symmetric around the closing points. Finally, we emphasize that the order of the bandgaps that close is always related to α, except for the special case of α = 0. Specifically, the order of closed bandgaps is equal to multiples of α_d if both numerator and denominator are odd, while equal to twice the multiples of α_d otherwise.

Following this discussion of the role of rational α values in bandgap closure, it is imperative to understand the different consequences of PnCs with rational and irrational α values that are close in magnitude. Consider two bi-layered unit cells of an identical impedance contrast β = −0.75 with α = 2/3 for the first, which is the rational value that corresponds to the dispersion diagram shown in Fig. 2(d), and α = 2/π for the second, which is the irrational value used to construct the dispersion diagram shown in Fig. 1(b). The bandgap widths for the first 120 bandgaps of both PnCs are computed in Fig. 3. It is immediately noticed that the rational (α = 2/3) case maintains a perfectly periodic pattern of bandgap widths that repeats itself every six bandgaps (which is twice α_d as explained earlier). On the other hand, the bandgap widths corresponding to the irrational (α = 2/π) case clearly move further away from zero as the bandgap number increases, indicating the absence of a bandgap closing pattern due to α not being an exact rational number. Despite the absence of a bandgap closing pattern, Fig. 3(b) still shows a near-periodic profile with a period of 11 bandgaps. This is understandable because the closest rational approximation of α = 2/π ≈ 2/(22/7) ≈ 7/11 (using the known approximation of π) reveals that this system should closely mimic one which exhibits bandgap closing at multiples of α_d = 11.

The rightmost panel of Fig. 4(b) shows the mode shapes for the case of |α| = 1/3 and β = −0.75. Here, we chose ℓ₁/ℓ = 0.6 and thus ℓ₂/ℓ = 0.4. The two modes derived earlier are normalized such that the maximum amplitude is unity and they are plotted at the three frequencies corresponding to bandgap closings within the range Ω ∈ (0, 6π), namely 3π/2, 3π, and 9π/2. The spatial frequency of the mode shapes is controlled by the value of n_1,2. As can be inferred from Eq. (27), flipping the sign of α switches the values of n₁ and n₂. This is graphically shown in Fig. 4(b) where the deformation shape of layer 1 at α = 1/3 becomes identical to that of layer 2 at α = −1/3, and vice versa, at any of the three bandgap closing frequencies shown (understandably, the deformation shape spans a shorter or larger distance when the sign of α is swapped to accommodate for the different lengths of the individual layers). For validation, all of the analytically-obtained results shown in Fig. 4(b) are verified via a finite element model implementing two-node rod elements [43], and shown as dashed lines in all the plotted mode shapes.

The leftmost panel of Fig. 4(b) shows several unique features of the dispersion diagrams of the bi-layered PnC and its two constitutive layers. The latter are given by two sets of folded lines described by Ω(1 ± α) = q (black and blue dashed lines). The red dashed lines indicate the locations at which the two sets fold at the same frequency, indicating a bandgap closing of the bi-layered PnC as shown. Finally, it can also be shown that n_1,2 indicate precisely the number of folded lines that the dispersion relations of the individual layers have up to each bandgap closing of the bi-layered unit cell. For example, consider the first bandgap closing at Ω = 3π/2. The dispersion relation Ω(1 + α) = q before and up to that frequency consists of exactly two folded lines, while Ω(1 − α) = q consists of one folded line, indicating values of n₁ = 2 and n₂ = 1. Using these values of n_1,2, it immediately follows that α = 1/3 by using Eq. (30), as expected.

The observations drawn in Secs. 3.2 and 3.3 regarding rational and irrational values of α are in fact independent of the chosen value of β. As a demonstration, Fig. 5 shows the width ΔΩ of the first six bandgaps over the entire range of α and β values. The following observations can be made:

1. Confirming the bandgap closing rules observed in Fig. 2, the number of the zero-width bandgap is directly related to the value of α. For example, the fourth bandgap (which is even-numbered) closes at α = ±1/2 as expected, with the closed bandgap number being twice the denominator value (α_d = 2). Similarly, the fifth bandgap (which is odd-numbered) closes at α = ±1/5 and α = ±3/5, which are both ratios of odd numbers, and with the closed bandgap number matching the denominator value (α_d = 5). Finally, the sixth bandgap closes at both α = ±1/3 and α = ±2/3, which represent rational values of identical and different numerator–denominator parity, respectively.

2. Including the limiting case of |α| = 1, the number of times a given bandgap closes is equal to its number plus one. For instance, the third bandgap closes four times at α = ±1/3 and α = ±1.

3. The special case of α = 0 results in the closing of all even-numbered bandgaps, further confirming the result of Fig. 2(a).

4. The special case of β = 0 forces all bandgaps to close regardless of the value of α (as reported in Ref. [39]).

It is also of interest to understand how bandgap limits behave as the value of β changes at a given α, as shown in Figs. 6(a) and 6(b). Bragg bandgaps initiate with a non-zero impedance contrast β at frequencies at which the linear dispersion relation folds within the irreducible Brillouin zone, as shown in Fig. 6(a), and grow in width (ΔΩ) with higher contrast values. As the contrast β approaches the limiting value of unity, the dispersion branches become flat. The growth of ΔΩ with increasing magnitude of β is further emphasized in Fig. 6(b), and is shown to be symmetric about β = 0. It can be seen that even or odd-numbered bandgaps close when the two solutions of Eq. (29) match regardless of the value of β. These closings are denoted with dashed lines in Fig. 6(b).

The behavior is quite different when observing the evolution of bandgap limits with a varying α at specific values of β, which is depicted in Fig. 6(c). As the value of α changes, the locus of the bandgap limits oscillates in a manner which increases at higher frequencies. Furthermore, the amplitude (i.e., frequency width) of these oscillations grows as the value of β increases. These oscillatory profiles have nodal points at the locations where the bandgap limit curves intersect, which represent rational values of α. At such nodes, the curves corresponding to the Bragg condition established in Eq. (29), and shown as dotted black lines, also intersect, thus constituting the requisite condition for bandgap closing.

Finally, for completeness, we point out that the existence of two identical layers in a multi-layered unit cell of a PnC rod affects the calculation of α_j/α_d in Eq. (42). This is because of a hidden common factor that exists between the chosen vibrational modes that are intended to be matched. When faced with such a special case, the sum of the modes of the identical layers can be calculated and they can then be treated as a single continuous layer, even if they are not adjacent to one another. For instance, consider a four-layered unit cell made of aluminum-steel-aluminum-brass. We choose n₁ = 1, n₂ = 8, n₃ = 3, and n₄ = 4 and then calculate n′ = n₁ + n₃ = 4 for a collective mode for the aluminum layers. As such, the common factor between n₂, n₄, and n′ is 4, and, as a result, $αd=∑sns/4$⁠.

The qualitative and quantitative criteria governing bandgap formation, distribution, and closing conditions were established in a generalized class of rod-based PnCs undergoing longitudinal deformations. A transfer matrix-based approach was used to generate the wave dispersion profiles and develop expressions for bandgap limits and frequencies of maximum attenuation. By implementing two non-dimensional contrast parameters, a frequency contrast α and an impedance contrast β, which stem from the parameters of the PnC’s constitutive layers, the conditions that lead to diminishing bandgaps were derived in closed form, showing that α being a rational number is a necessary condition for bandgap closing in bi-layered PnCs. Furthermore, it was shown that, depending on the parity of the integer numerator and denominator values of α, the pattern and frequency location of the bandgap closing can be predicted as a function of the rational number α. It was found that the bandgap widths ΔΩ of a PnC with a rational α exhibit a periodic profile, which perfectly repeats itself every time a bandgap closes. This pattern was correlated to the resonances of the individual layers of the PnC, and it was proven that matching the natural frequencies of the individual layers (if they were to be treated as stand-alone entities) forces a bandgap to close at the same frequencies. An additional connection was made between bandgap closing criteria and the physics underlying the mode shapes of the individual layers forming the PnC unit cell at different boundary conditions.

The conclusions drawn from the bi-layered case were generalized to a PnC comprised of three or more layers, where it was similarly shown that the dispersion branches of the multi-layered PnC exhibit mirror symmetry about the frequencies at which bandgaps close. In fully analytical terms, it was also proven that the resonance matching condition for bandgap closing is independent of the number of layers forming the PnC’s unit cell. Resolving the patterns and formation mechanisms of multi-bandgap dispersion profiles is particularly useful for a wide array of new and exciting topics, given the rising interest in exciting applications that require an understanding of how bandgaps close (e.g., topological transition). In tandem, the need to finely tailor phononic band structures remains highly critical for a broad range of elasto-acoustic metamaterials. As such, the developments established herein provide a great asset for bandgap engineering in future configurable and tunable PnCs.

The authors acknowledge the support of this work by the US National Science Foundation through CMMI research award no. 1847254 (CAREER).

There are no conflicts of interest.

The authors attest that all data for this study are included in the paper.