# Material vs. structure: Topological origins of band-gap truncation
resonances in periodic structures

Matheus I. N. Rosa 111Paul M. Rady Department of Mechanical Engineering Bruce
L. Davis 222Ann and H.J. Smead Department of Aerospace Engineering Sciences
Liao Liu 33footnotemark: 3 Massimo Ruzzene 44footnotemark: 455footnotemark: 5
<EMAIL_ADDRESS>Mahmoud I. Hussein 66footnotemark: 6777Department
of Physics<EMAIL_ADDRESS>University of Colorado Boulder, Boulder, CO, 80303

###### Abstract

While resonant modes do not exist within band gaps in infinite periodic
materials, they may appear as in-gap localized edge modes once the material is
truncated to form a finite periodic structure. Here, we provide an analysis
framework that reveals the topological origins of truncation resonances,
elucidating formally the conditions that influence their existence and
properties. Elastic beams with sinusoidal and step-wise property modulations
are considered as classical examples of periodic structures. Their non-trivial
topological characteristics stem from the consideration of a phason parameter
that produces spatial shifts of the property modulation while continuously
varying how the boundaries are truncated. In this context, non-trivial band
gaps are characterized by an integer topological invariant, the Chern number,
which is equal to the number of truncation resonances that traverse a band gap
as the phason is varied. We highlight the existence of multiple chiral edge
states that may be localized at opposite boundaries, and illustrate how these
can be independently tuned by modified boundary-specific phason parameters.
Boundary phasons modify the truncation of only one boundary at a time.
Furthermore, we show that the frequency location of a truncation resonance is
influenced by the modulation wavelength, modulation volume fraction, boundary
conditions, and number of cells comprising the finite structure, thus
quantifying its robustness to these factors. Non-topological in-gap resonances
induced by a defect are also demonstrated, with their frequency dependence on
the phason investigated to elucidate their contrast to truncation resonances.
A coupling between topological and non-topological modes is shown to be
possible when the defect is located at an edge. Finally, experimental
investigations on bi-material phononic-crystal beams are conducted to support
these findings. Our results provide a fundamental perspective on the
topological character of truncation resonances in periodic structures and how
this character relates to the underlying periodic material properties. The
tunability of these unique structural resonances through material-property
modulation may be exploited both in applications where in-gap resonances are
not desired, such as vibration attenuation and thermal conductivity reduction,
or where in-gap resonances provide a functional role, such as filtering,
waveguiding, energy harvesting, and flow control by phononic subsurfaces.

###### keywords:

Phononic materials , band-gap resonances , topological protection , phasons ,
experimental phononics

##  1 Introduction

The study of elastic wave propagation in a continuous periodic medium is a
classical problem in mechanics that can be traced back to Rayleigh in 1887
[1]. With the advent of composite materials, the interest in this problem
surged with early contributions in the 1950s [2] and 1960s [3] formulating
dispersion relations for wave propagation in laminated composites, and other
forms of periodic media [4, 5], followed by extension to multi-dimensional
composites in the 1970s [6]. The field re-emerged in the early 1990s with the
study of phononic crystals [7, 8] and the establishment of formal connections
with lattice dynamics in crystals [9], and gathered further pace with the rise
of acoustic and elastic metamaterials [10]. In all these studies, periodicity
is utilized enabling dynamic characterization by considering a representative
unit cell, as commonly done in condensed matter physics [11]. Calculating the
dispersion relation, or the band structure, using the Floquet/Bloch theorem
[12, 13] formally enforces the assumption of an extended medium with an
infinite number of unit cells. This is not only computationally rewarding, but
physically provides a fundamental description of the modal wave propagation
properties of the medium under investigation$-$removing any influence of
overall size and external boundary conditions. In this framework, the medium
under consideration is rendered a material with characteristic intrinsic
properties, such as band gaps (whose locations may be predicted analytically
[14, 15, 16, 17]) and other key features revealed by the nature of the band
structure. The thermal conductivity, for example, is an intrinsic material
property that is directly influenced by the band structure$-$determined by
analysis of only a single atomic-scale unit cell [18, 19]. Effective dynamic
properties, such as effective density and Young’s modulus [20], provide
another example of intrinsic material properties. On the other hand, unless a
medium practically comprises thousands or millions of unit cells (as in a bulk
crystal for example), realistic realizations are formed from a relatively
small finite number of unit cells, yielding a periodic structure, rather than
a material, with extrinsic properties. This is particularly the case in
engineering problems such as sound [21] and vibration [22] isolation, and
other similar applications [23, 24], and also the case in nanoscale thermal
transport [25] where unique dynamical properties emerge primarily from the
presence of finite size along the direction of transport.

###  1.1 Truncation resonances

A periodic structure in practice may still consist, in some cases, of a
relatively large but tractable number of unit cells, and in other cases, of
only a few unit cells along the direction of vibration transmission. The
number of cells impacts the degree of attenuation within a band gap [26].
However, the contrast between the material and structure behavior may not be
limited to only quantitative differences but also to fundamental qualitative
distinctions. One noticeable anomaly between the material and structure
responses is the possibility of existence of resonances inside band gaps,
i.e., resonance peaks in the frequency response function (FRF) of a finite
periodic structure that appear within band-gap frequency ranges of the
corresponding infinite periodic material. These resonances are often referred
to as truncation resonances [27, 28] because they emerge from the truncation
of a medium that is otherwise formed from an infinite number of unit cells.
These resonances are associated with mode shapes that localize at the
truncation junction, and are thus also commonly referred to as edge or surface
modes [29, 30, 31, 32, 33, 34, 35, 36, 37]. The presence of these modes has
been uncovered theoretically by Wallis [29] in his study of a finite discrete
diatomic chain of atoms with free ends. This followed the work of Born on
finite atomic chains [38] which was motivated by the study of the influence of
lattice vibrations on X-ray scattering. Recent studies extended Wallis’ theory
of finite discrete chains to more general conditions [39, 28, 40] and
experiments on chains of discrete-like coupled spheres validated the theory
[35].

The problem of truncation resonances in continuous periodic media$-$the focus
of this paper$-$has also been investigated extensively. Early studies examined
one-dimensional wave propagation in periodically layered/lamenated composites,
also referred to as superlattices. Existence conditions for truncation
resonances were derived for semi-infinite superlattices for out-of-plane [30,
32, 34] and in-plane [31, 33, 37] waves. It was shown that surfaces modes in
some instances may appear below the lowest bulk band, i.e., the band that
hosts conventional resonances. Investigations of the truncation phenomenon
were also done on finite layered phononic crystals examining transverse waves
[36, 41], on finite beam-based phononic crystals [42, 43] and locally resonant
elastic metamaterials [44, 45], and on rod-based phononic crystals [46, 47].
Among the factors that influence the frequency location of the truncation
resonances are the unit-cell symmetry and the boundary conditions [41, 43, 45,
46, 47]. When there is more than one layer in the unit cell, the number of
surface states increases [34, 37]. Techniques proposed for control of the
truncation resonances also include tuning of unit-cell spatial material
distribution or volume fraction [42], and the anomalous addition of a “cap
layer” [32, 34] or a “tuning layer” [48, 42] at the edge of the structure. A
cap layer is simply a homogeneous layer, whereas a tuning layer is a
purposefully truncated single unit cell. The concept of truncation resonances
is also relevant to other areas in applied physics such as photonic crystals
[49] and quantum lattices [41].

###  1.2 Connection to topological physics

The principle of a truncation resonance is fundamentally connected to the
periodic structure’s topological properties; this connection forms the core
focus of the present study. Inspired by the emergence of topological
insulators in condensed matter physics [50], classical analogues have been
developed in photonics [51] and phononics [52], demonstrating the features of
robust topological waves. In passive elastic materials, topological interface
modes are created by contrasting two materials with band gaps existing at the
same frequencies, but characterized by different topological invariants.
Examples include interface modes in one-dimensional (1D) structures [53, 54,
55, 56] in analogy to the Su-Schrieffer-Heeger model [57], and waveguiding
along interfaces in two-dimensional (2D) materials in analogy to the Quantum
Spin Hall Effect [58, 59] or to the Quantum Valley Hall Effect [60, 53, 61].
These effects rely on symmetry breaking by interfacing two domains whose unit
cells have opposite symmetries, which results in contrasting topological
properties in the reciprocal space. Hence, an actual interface between two
materials is required, which presents a contrast to the truncation resonances
we explore in this paper. We will show an intriguing connection that stems
from a stronger type of topological effect associated with the Quantum Hall
Effect (QHE) [62, 63]. The QHE manifests in 2D lattices of electrons under the
presence of a strong magnetic field, which leads to robust edge waves that
propagate along the boundaries of a finite sample (structure), without
backscattering at corners or defects. It is therefore sufficient to exploit
the interface between a single material medium and vacuum. However, such a
strong effect requires breaking time reversal symmetry, which in the quantum
case is achieved through the magnetic field. Emulating similar features on 2D
elastic materials is possible through active components that break time
reversal symmetry, such as rotating frames [64] or gyroscope spinners [65,
66]. An alternative that has emerged later, and which we adopt here, is to map
the QHE to 1D passive structures that have extended dimensionality emanating
from their parameter spaces [67, 68]. This has been achieved by using
patterned mechanical spinners [69], spring-mass lattices [70], acoustic
waveguides [71, 72], and continuous phononic crystals or elastic metamaterials
with modulations of inclusions such as ground springs [73], stiffeners [74],
and resonators [75, 76]. In these examples, edge states localized at the
boundaries of 1D periodic and quasi-periodic finite domains are observed to
appear in correspondence to non-zero topological inavariants called Chern
numbers. The boundary at which the localization occurs can be determined by a
phason parameter that is associated with spatial shifts in the medium’s
modulated properties. This feature leads to possibilities for topological
pumping by varying the phason parameter continuously along time [77, 78, 79]
or along a second spatial dimension [70, 80], inducing a transition of the
edge states from being localized at one boundary to the other. Thus, energy
can be ”pumped” between two boundaries of a system through a transition of a
topological edge state. The application of the field of topology to elastic
and acoustic material systems has been attracting much interest in recent
years [81, 52, 82].

In this paper, we provide a formal framework for the identification of the
topological character of truncation resonances in periodic structures, drawing
on concepts from the QHE. We consider a family of periodic elastic beams with
either sinusoidal or step-wise property modulations. The modulations offer key
parameters that expand the structure’s property space and allow us to readily
apply the concepts of topological band theory. In particular, the variation of
a periodic beam’s spectral properties with respect to the modulation
wavelength allows us to extract the Chern numbers of the band-gaps and
identify the locations of truncation resonances. Then, the phason parameters
associated with spatial shifts of the modulations further characterize the
truncation resonances as topological edge states spanning the band gaps. The
frequency dependence of the location of a truncation resonance on the phason
has recently been predicted, for periodic rods, by means of a closed-form
transfer-matrix-based mathematical formulation [47]. Here, we investigate, for
periodic flexural beams, the topological origins of this class of relations.
We show that the number of truncation resonances within a gap is equal to the
predicted Chern number, for any set of boundary conditions, although the
particular features of their branches as they traverse the gaps may vary. We
elucidate how additional boundary phason parameters can be defined,
formalizing the notion of the tuning layer [48, 42], to manipulate the edge
states localized at different boundaries independently. Furthermore, we
examine the convergence of the truncation resonant frequencies as a function
of the number of unit cells$-$a matter of significant practical importance
especially when this number is relatively small. The fundamental differences,
and the possibility of coupling, between truncation resonances and
corresponding non-topological defect-mode resonances are then investigated.
Next, we provide laboratory results using a bi-material phononic-crystal beam
as experimental validation of some of the key features of truncation
resonances and their association with topological theory. Finally, we use our
experiments to explore yet another important factor in the design space,
namely the role of the materials’ volume fraction within the unit cell in
influencing the frequency locations of the truncation resonances.

The paper is organized as follows: following this introduction, Section 2
provides a description of the considered periodic flexural beams and their
boundary truncation through phasons. Next, Section 3 develops the theory and
computational analysis to characterize the topological properties of
truncation resonances and those of non-topological defect resonances, and the
coupling of the two types of resonances, followed by Section 4 which provides
experimental results and further analysis. Finally, Section LABEL:Sec:Disc has
a general discussion on the key findings and their broader implications to
related areas of research, and Section 6 provides a closing summary and
outlines possible future research directions.

##  2 Modulated phononic-crystal beams: Truncation characterization by phasons

We consider elastic beams undergoing flexural motion described by transverse
displacement $w=w(x)$ and angle of rotation $\varphi=\varphi(x)$, where $x$ is
the axial position, as classical examples of 1D periodic materials or
structures. The properties of the beam are the Young’s modulus $E=E(x)$, shear
modulus $G=G(x)$, density $\rho=\rho(x)$, cross-sectional area $A=A(x)$, and
second moment of area $I=I(x)$. These properties are modulated in space as
illustrated in Fig. 1. Two scenarios are considered; in the first the Young’s
modulus is modulated according to a cosine function, i.e.
$E(x)=E_{0}[1+\alpha\cos(2\pi\theta x-\phi)]$, while other parameters remain
constant (Fig. 1(a)). This cosine-modulated phononic crystal (CM-PnC) serves
as an idealized continuous periodic waveguide used to illustrate the behavior
of interest in a simple setting. It is characterized by a unit cell of length
$a=1/\theta$, where $\alpha$ is the amplitude of the modulation with respect
to the mean value $E_{0}$ and $\theta$ may be viewed as the modulation
wavenumber. The second case corresponds to a beam modulated in a step-wise
fashion, which we refer to as step-wise modulated phononic crystal (SM-PnC).
It generically represents a periodic material of two alternating layers of
lengths $a_{1}$ and $a_{2}$, with different constituent material or
geometrical (e.g. cross-sectional area) properties. In this case, the material
or geometrical properties are modulated through a step-wise function of period
$a=1/\theta=a_{1}+a_{2}$ that takes two different values in the intervals of
length $a_{1}$ and $a_{2}$.

Figure 1: Elastic periodic beams with (a) sinusoidal and (b) step-wise
property modulation whose spatial distribution is defined by a phason $\phi$
or boundary phasons $\phi_{r}$ and $\phi_{l}$. A modulation characterized by
$\phi$ is a superposition of modulations characterized by $\phi_{r}$ and
$\phi_{l}$.

The appearance of in-gap resonances stems from the truncation of the
boundaries. The truncation details are here characterized by phason parameters
that are connected to non-trivial topological properties. The most natural
choice of the phason is simply the phase $\phi$ of the property modulations,
which rigidly shifts the modulation in space. Thus it results in a
simultaneous change of the local properties of the beam at both boundaries.
This is illustrated in the schematics of Fig. 1 for both the sinusoidal and
step-wise modulations. The blue boxes highlight the region of the modulations
selected to form the properties of the finite beams. From a given initial
configuration, a change in phason over the range $0<\phi<2\pi$ (higher values
of $\phi$ do not need to be considered due to the periodicity) can be
interpreted as simultaneously adding a segment of length $\phi a/2\pi$ to the
left boundary, while removing the same length from the right boundary. This
will naturally influence any vibration mode localized at either boundary. It’s
effect can be further understood as the superpostion of two independent
parameters which we call boundary phasons. A change in the right boundary
phason $\phi_{r}$ corresponds to removing a length $\phi_{r}a/2\pi$ from the
right boundary while keeping the left boundary unchanged, while a change in
the left boundary phason $\phi_{l}$ corresponds to adding a length
$\phi_{l}a/2\pi$ to the left boundary while keeping the right boundary
unchanged. Hence, changing the phason $\phi$ corresponds to changing both the
left and right boundary phasons by the same amount (as illustrated in the
figure). As we will show, the boundary phasons independently tune the
topological truncation resonances at their respective boundary, and their
superimposed effect leads to the variation of the resonances with respect to
the conventional phason $\phi$.

Herein, the flexural motion of the beam is modeled through Timoshenko theory
as governed by the following two coupled equations:

$\rho A\frac{\partial^{2}w}{\partial t^{2}}-q(x,t)=\frac{\partial}{\partial
x}\left[\kappa_{\mathrm{s}}AG\left(\frac{\partial w}{\partial
x}-\varphi\right)\right],$ (1a) $\rho I\frac{\partial^{2}\varphi}{\partial
t^{2}}=\frac{\partial}{\partial x}\left(EI\frac{\partial\varphi}{\partial
x}\right)+\kappa_{\mathrm{s}}AG\left(\frac{\partial w}{\partial
x}-\varphi\right),$ (1b)

where $\kappa_{\mathrm{s}}$ denotes the shear coefficient, and $t$ and
$q=q(x,t)$ represent time and the external forcing, respectively. Equations 1a
and 1b are combined to yield a single fourth-order partial differential
equation with only $w$ as the dependent variable [83]. In our investigation,
we consider three types of problems: a Bloch dispersion analysis problem for a
unit-cell representing an infinite material, an eigenvalue analysis problem
for a finite structure with arbitrary boundary conditions (BCs), and a
harmonic forced-response problem for a finite structure with arbitrary BCs. In
the first two problems, we set $q=0$ and

$w(x,t)=\hat{w}e^{i(\mu x-\omega t)},$ (2)

where $\omega$ denotes the frequency. In Eq.2, we set $0\leq\mu\leq\pi/a$ for
the Bloch dispersion problem, where $\mu=0$ is used for the finite periodic-
structure eigenvalue with arbitrary BCs. The results are obtained by a finite-
element discretization of the equations of motion. The implementation details
of these methods are omitted here for brevity since they are widely available
in the literature (for example, see Ref. [84]).

Motivated by the experimental portion of this work (see Section 4), we select
the following parameters. The SM-PnC consists of a bi-material beam composed
of alternating layers of Aluminum (Al) and the polymer acrylonitrile butadiene
styrene (ABS). These materials are selected due to the contrast of mechanical
properties leading to wide band gaps. Their properties are as follows: Young’s
moduli $E_{\mathrm{Al}}=68.9$ GPa and $E_{\mathrm{ABS}}=2.4$ GPa, shear moduli
$G_{\mathrm{Al}}=25.9$ GPa and $G_{\mathrm{ABS}}=0.872$ GPa, and densities
$\rho_{\mathrm{Al}}=2700$ kg/m3 and $\rho_{\mathrm{ABS}}=1040$ kg/m3,
respectively. While we will allow the unit-cell length to vary through the
$\theta$ parameter, the ABS polymer length filling fraction is fixed as
$a_{\mathrm{ABS}}/a=0.2$; this ratio will be changed only in Section 4.3. For
purposes of comparison, the properties of the CM-PnC are then chosen to make
it statically equivalent [26] to the SM-PnC by selecting a fixed density
$\rho_{0}=(0.2\rho_{\mathrm{ABS}}+0.8\rho_{\mathrm{Al}})$ and elastic modulus
modulation with a mean value of
$E_{0}=(0.2/E_{\mathrm{ABS}}+0.8/E_{\mathrm{Al}})^{-1}$. We consider a
Poisson’s ratio of $\nu=0.33$, which consequently determines the shear modulus
through the relation $G=E/(2(1+\nu))$. Throughout this paper, the CM-PnC
modulation amplitude is fixed at $\alpha=0.9$, and the beams have a square
cross-section geometry with side length $h=2.54$cm. The finite-element
analysis follows by discretizing the beams with linear Timoshenko beam
elements with a shear coefficient of 5/6. The beam element length varies
according to the case studied but does not exceed a maximum length of
$\bar{a}/100$, where $\bar{a}=203$ mm is the unit-cell size of the
experimental beams and is used as a reference unit-cell length throughout the
paper.

Figure 2 presents a comparison between the properties of the CM-PnC and SM-PnC
for the reference unit-cell size $\bar{a}=203$ mm, highlighting the contrast
between material and structure. Panels (a) and (b) display their dispersion
diagrams in a frequency range of interest from 0-9 kHz, which is a material
feature. Both CM-PnC and SM-PnC exhibit the same long-wave static limit that
approaches the dispersion of the homogenized beam with material property
constants $\rho_{0},E_{0}$ (dashed lines), but display different band-gaps
(shaded gray regions). In particular, the SM-PnC has wider gaps due to its
discrete nature and the contrast of both densities and elastic moduli, while
the CM-PnC has smaller gaps due to a fixed density and a continuous variation
of the elastic modulus only. On the right side of the dispersion diagrams, the
eigenfrequencies of representative finite beams with 15 unit cells and free-
free BCs are plotted as black dots, with $\phi=0.2\pi$ and $\phi=0.4\pi$
selected for the CM-PnC and SM-PnC beams, respectively. Truncation resonances
are observed to appear in band gaps, a feature which is unique to the
structure, non-existing at the material level. An arbitrary phason value is
chosen here to produce a large number of truncation resonances as an example,
but the behavior with the full range of $\phi$ will later be explored and
explained. The truncation resonances are localized at one of the two
boundaries of the finite beams, with selected mode shapes displayed in Figs.
2(c-f). By looking at such isolated cases (as has been largely done in
previous studies), there is no apparent reason or pattern pertaining to the
appearance of in-gap resonances, why are they localized at one boundary
instead of the other, and why these features can change by selecting different
BCs or different numbers of unit cells, etc. In the following sections, we
will shed light on all of these questions by illustrating the topological
character of in-gap truncation resonances associated with non-zero Chern
numbers, and consequently how they can be manipulated through the phason and
other parameters or design features.

Figure 2: Material versus structure properties. Dispersion diagrams (material)
for the CM-PnC and the SM-PnC models are displayed in (a-b) as solid lines,
while dashed lines correspond to the homogenized beam dispersion. Band-gap
frequency ranges are shaded grey. A finite structure with 15 unit cells
exhibits in-gap truncation resonances as illustrated alongside the dispersion
diagrams, with selected mode shapes displayed in (c-f). For both models, the
unit-cell length is $\bar{a}=203$ mm.

##  3 Topological properties of modulated phononic-crystal beams

In this section we develop the theoretical tools for the topological
characterization truncation resonances by examining their behavior inside band
gaps. We begin by investigating the effect of the modulation wavenumber
$\theta$, which allows us to extract the topological invariants (Chern
numbers). We then show how the Chern numbers are related to in-gap truncation
resonances through the variation of the phason parameters. We also study the
effect of the number of unit cells comprising the finite structure on the
convergence of the truncation resonance frequencies. Finally, we provide a
comparison between topological truncation resonances and non-topological
defect resonances, highlighting their key differences and demonstrating the
possibility of their coupling as a defect is moved towards a boundary.

###  3.1 Topological characterization by the Chern number

In principle, the Chern number characterizes the topology of a vector field
defined over a two-dimensional torus. For 2D periodic materials the torus is
composed of two orthogonal wavenumber coordinates $\kappa_{x}$ and
$\kappa_{y}$ and describes the reciprocal space Brillouin zone [85, 58, 59,
53, 61]. For 1D modulated materials such as the considered beams, the phason
$\phi$ serves as an additional dimension and replaces the missing wavenumber
component to form a torus based on $\kappa$ and $\phi$. [70]. The eigenvector
field is the Bloch mode displacement $\hat{w}_{n}(\kappa,\phi)$ corresponding
to the $n_{\text{th}}$ band defined over the torus
$(\kappa,\phi)\in\mathbb{T}^{2}=[0,2\pi]\times[0,2\pi]$, recalling that the
dispersion is $2\pi$-periodic in both $\phi$ and $\kappa$, with $\kappa=\mu a$
defined as the non-dimensional wavenumber. Due to the continuous nature of the
beams, the dispersion frequency bands are invariant with $\phi$, which only
produces a shift in the choice of the unit cell. However, the variation of
$\phi$ produces changes in Bloch eigenvectors, which may reflect in non-
trivial topological properties. The Chern number $C_{n}$ for the
$n_{\text{th}}$ band is defined as

$C_{n}=\dfrac{1}{2\pi i}\int_{\mathcal{D}}\beta_{n}\;d\mathcal{D},$ (3)

where $\mathcal{D}=\mathbb{T}^{2}$, $\beta_{n}=\nabla\times\mathbf{A}_{n}$ is
called the Berry curvature, and
$\mathbf{A}_{n}=\hat{w}_{n}^{*}\cdot\nabla\hat{w}_{n}$ is the Berry
connection, with $()^{*}$ denoting a complex conjugate. The Chern number is an
integer that quantifies the topological properties of the bands; these are
robust to small perturbations in the system’s unit cell as long as these
perturbations do not close the gaps separating the bands. Among other
features, the Chern number is related to discontinuities (or vorticities) in
the eigenvector field [85], localization of the Berry curvature [53], and to
phase accumulation of the Bloch modes along cyclic paths in the torus
Brillouin zone [62, 70].

Of particular relevance to the present work is the bulk-boundary
correspondence principle that relates the existence of in-gap edge states in
finite systems to the Chern numbers [86]. This is done through the computation
of a gap label $C_{g}$ given by the summation of the Chern numbers of the
bands below the gap, i.e. $C_{g}^{(r)}=\sum_{n=1}^{r}C_{n}$, which is equal to
the number of truncation resonances found inside such gap when the phase
$\phi$ varies in an interval of $2\pi$ (see Section 3.2 for more details).
However, the computation of the Chern number as given by Eq. (3) is often
challenging due to phase or gauge ambiguities [87]. Furthermore, it has to be
done for each $\theta$ value that defines a different unit-cell size (see, for
example, Refs. [70, 80]). Here, we take an alternative, and more generic,
approach that produces the gap labels $C_{g}$ without direct computation of
the band Chern numbers $C_{n}$, and for all $\theta$ values at once. Such
approach relies on density of states computations based on the spectral
variation with $\theta$, which has been developed using mathematical
principles of K-theory in the context of periodic and aperiodic topological
insulators [88, 89], and later extended to quasi-periodic acoustic/elastic
metamaterials [71, 72, 73, 74, 75, 76]. This approach has not yet been
extended to continuous elastic periodic waveguides such as the beams studied
here.

####  3.1.1 Extraction of the Chern number by varying the modulation
wavenumber

To begin, we investigate the variation of the beams’ spectral properties as a
function of the modulation wavelength $\theta$. The procedure relies on a
large finite structure of fixed size $L=100\bar{a}$, and the computation of
its eigenfrequencies under periodic boundary conditions (PBCs). The results
are illustrated in Fig. 3(a,b) for the CM-PnC and SM-PnC configurations, where
the eigenfrequencies are plotted as a function of $\theta$ as black dots. In
the computation, the considered range of $\theta$ is discretized in intervals
of $\Delta\theta=1/L$, i.e., $\theta_{n}=n/L$, such that each considered
structure has an integer number $n$ of unit cells. By doing so, the resulting
eigenfrequencies sample the Bloch dispersion bands defined for the considered
$\theta$ value, and no frequencies are found inside the gaps due to the PBCs
and the ”perfect” periodicity emanating from an integer number of unit cells
[73]. The resulting spectrum provides a map for the location of the bands
(black regions) and band gaps (white regions) as a function of $\theta$, and
consequently of unit-cell length $a=1/\theta$. We note that SM-PnC produces a
more complex spectrum (Fig. 3(b)) with a larger number of gaps when compared
to CM-PnC (Fig. 3(a)), in particular for lower values of $\theta$ as
illustrated in the zoomed view of Fig. 3(c).

Figure 3: Eigenfrequencies of finite beam with $L=100\bar{a}$ and PBCs for (a)
sinusoidal and (b) step-wise modulation, with zoomed view in (c). Black dots
represent eigenfrequencies while white areas denote band gaps. The
corresponding IDS plots are displayed in the bottom panels (d-f), where
selected fitted lines have colors corresponding to the gaps marked and labeled
in (a-c).

The band-gap Chern numbers can be extracted by computing the Integrated
Density of States (IDS) of the spectrum. It is defined as

$\text{IDS}(\theta,f)=\lim_{L\to\infty}\dfrac{\sum_{n}[f_{n}\leq f]}{L},$ (4)

where $[\cdot]$ denotes the Iverson Brackets, which provides a value of $1$
whenever the argument is true. In simple terms, for a given $\theta$ and
frequency $f$, the IDS is the summation of all the eigenfrequencies below $f$,
normalized by the structure size $L$. It theoretically converges as the
structure size tends to infinity, but it is practically sufficient to consider
large structures such as the one with $L=100\bar{a}$ considered in our
investigation. The IDS is displayed for the CM-PnC medium in Fig. 3(d), and
for the SM-PnC medium in Fig. 3(e) with a zoomed view for the lower $\theta$
range in (f). In this representation, the $z$-axis and the associated colormap
represent frequency $f$ as a function of IDS and $\theta$. The insets in (d,e)
illustrate the 3D views highlighting sharp discontinuities in the surface
plot, which are visualized as straight lines in the top view colormaps. Each
straight line is associated with a band gap and occurs since the IDS does not
change inside the gap. Hence, a jump in frequency (color) occurs as the IDS
changes from the last mode before the gap to the first mode right after the
gap. According to the theory [71], and confirmed by our findings, the
variation of the IDS with $\theta$ inside the gaps identify straight lines
expressed as

$\text{IDS}(f)=n_{0}+C_{g}\theta,$ (5)

with the gap Chern number $C_{g}$ corresponding to the slope. The lines of the
most prominent gaps in Fig. 3 are fitted and overlaid to the IDS plots,
allowing the extraction of the Chern gap labels from the slopes as marked in
the top panels, with different colors used to represent different gaps. These
gap labels are defined generically for any $\theta$ value that defines the
band gap, and are related to the truncation resonances as described in the
following section.

###  3.2 Topological edge states and their control by phasons

The non-zero Chern gap labels indicate the presence of in-gap edge states
existing for structures with truncated boundaries, i.e., the truncation
resonances. Their properties are illustrated in Figs. 4 and 5 for the CM-PnC
and SM-PnC configurations, respectively. The figures display the frequencies
of a finite structure of fixed length $L=15\bar{a}$ as a function of
modulation wavenumber $\theta$ and phase $\phi$, for different BCs such as
free-free and pinned-pinned. The frequencies are color-coded according to a
localization factor $p$ to identify modes localized at the boundaries, which
is defined as

$p=\dfrac{\int_{\mathcal{L}_{r}}|w|dx-\int_{\mathcal{L}_{l}}|w|dx}{\int_{\mathcal{L}}|w|dx},$
(6)

where $\mathcal{L}$ denotes the domain of the beam, and $\mathcal{L}_{r}$ and
$\mathcal{L}_{l}$ correspond to a smaller portion of length $0.15L$ at the
right and left boundaries, respectively. With this definition, positive (red)
and negative (blue) $p$ values indicate modes localized at the right and left
boundary, respectively, while values that are close to zero (black) indicate
non-localized bulk modes.

The left panels in Figs. 4 and 5 display the eigenfrequencies of the finite
beam as a function of $\theta$, for different BCs as illustrated by the
schematics. The spectra are overall similar to the bulk spectra exhibited in
Fig. 3, with black regions also defining the bulk bands, but with additional
modes appearing inside the band gaps. These modes are the topological edge
states, corresponding to the truncation resonances which are localized at one
of the boundaries of the beam. The modes localized at the right boundary (red)
traverse the band gaps multiple times as they migrate from the band above to
the band below their respective gaps. Although not the focus of the present
investigation, this behavior stems from the positive gap labels $C_{g}>0$ and
can be explained by density of states arguments [73]. Furthermore, the modes
localized at the left boundary (blue) do not migrate between bands and instead
remain inside the gap for the considered range of $\theta$. The different
behavior between left- and right-localized modes occur due to the way the
finite structure is constructed, where the change in $\theta$ produces a
qualitative change at the right boundary (the modulation is truncated at
different places for different $\theta$), but not of the left boundary (the
modulation is always truncated at the same place).

Figure 4: Eigenfrequencies of finite CM-PnC structure with length
$L=15\bar{a}$ and free-free (top) or pinned-pinned (bottom) BCs. The left
panels (a,d) display the variation of the eigenfrequencies with $\theta$,
while the middle (b,e) and right (c,f) panels display the variation with
$\phi$ for the selected $\theta$ values highlighted as vertical dashed green
lines in (a,d). The frequencies are color-coded according to the polarization
$p$, and the gap labels $C_{g}$ are added for reference. Figure 5:
Eigenfrequencies of the finite SM-PnC structure with length $L=15\bar{a}$ and
free-free (top row) or pinned-pinned (middle row) BCs. The left panels (a,d)
display the variation of the eigenfrequencies with $\theta$, while (g)
displays a zoom of (d) in the low $\theta$ range. The middle (b,e,h) and right
(c,f,i) panels display the variation with $\phi$ for the selected $\theta$
values highlighted as vertical dashed green lines in (a,d,g). The frequencies
are color-coded according to the polarization $p$, and the gap labels $C_{g}$
are added for reference.

The gap label $C_{g}$ dictates the number of left- and right-localized edge
modes that span the band gap as the phason $\phi$ varies within an interval of
$2\pi$, for a fixed $\theta$ value. This is illustrated for selected $\theta$
values (marked as vertical dashed green lines) in the middle and right panels
of Figs. 4 and 5, which display the variation of the eigenfrequencies with the
phason $\phi$. As previously mentioned, variations of $\phi$ do not affect the
frequencies of the dispersion bands, and therefore the boundaries of the band
gaps (material property) remain unchanged with $\phi$. However, the phason
influences how both boundaries of a finite structure are truncated (Fig. 1),
and its variation causes the eigenfrequency branches of the truncation
resonances to traverse the gaps. The first selected value
$\theta_{1}=1/\bar{a}$ corresponds to the modulation wavelength for the
reference unit-cell size $\bar{a}$. In the CS-PnC case (Figs. 4(b,e)), this
unit-cell size produces two small gaps with Chern labels $C_{g}=1$ and
$C_{g}=2$, which were extracted from the procedure in Fig. 3. For both types
of BCs (free-free in (b) and pinned-pinned in (e)), one left- and one right-
localized edge state traverse the first gap, and two edge states traverse the
second gap, as the phason $\phi$ varies from $0$ to $2\pi$. In the SM-PnC case
(Figs. 5(b,e)), the choice $\theta_{1}=1/\bar{a}$ corresponds to the case
investigated in the experimental section of this paper (see Section 4), which
produces three band-gaps with $C_{g}$ values ranging from $1$ to $3$.
Regardless of the type of boundary condition, the number of left- and right-
localized edge modes spanning the band gaps is equal to the corresponding
Chern gap label. In addition, the gap label sign is related to the direction
the edge modes cross the gap [89]. A positive $C_{g}>0$ indicates that
$|C_{g}|$ left-localized branches will cross the gap from the lower band to
the upper band, and an equal number of right-localized states will cross from
the upper band to the lower band. Although no examples are found in this
paper, a negative sign $|C_{g}|<0$ produces transitions in opposite directions
[70]. Also note that the eigenfrequencies have a periodic behavior with
$\phi$, and are actually continuous at $\phi=0=2\pi$. Therefore a few branches
of the truncation resonances traverse the gap through that point; for example,
see the second right-localized mode in the second gap of Fig. 5(e). Indeed,
the phason variable $\phi$ defines a continuous ring, with no start or ending
point, with the beginning and end at $\phi=0$ and $\phi=2\pi$, respectively,
being arbitrary choices for the plots.

Other examples are shown to demonstrate the generality of the approach and
give more insights into the behavior of the edge states. The case of
$\theta_{2}=2.5/\bar{a}$ (panels (c,f) in Figs. 4 and 5) corresponds to a
unit-cell size 2.5 times smaller than the reference $\bar{a}$, and therefore
the finite length $L=15\bar{a}$ now comprises $37.5$ unit cells. Even without
an integer number of unit cells, the number of edge sates inside each gap
matches the corresponding gap labels, for both CS-PnC and SM-PnC, and both
types of BCs considered. In fact, this behavior is general and holds for any
arbitrary $\theta$ value. The last row in Fig. 5 focuses on the lower $\theta$
range, where the SM-PnC features additional gaps with higher Chern gap labels.
The examples $\theta_{3}=2$ m-1 and $\theta_{4}=3$ m-1 correspond to unit cell
sizes of $0.5$m and $0.33$m respectively, and form finite structures with 6.09
and 9.135 unit cells for the fixed length $L=15\bar{a}$. They feature gap
labels as high as $C_{g}=8$, and the behavior of the edge states spanning the
gaps with $\phi$ is in agreement with the extracted gap labels, again even
without an integer number of unit cells. Among many edge states, two
transitions experienced by the modes as a function of $\phi$ are highlighted
by thicker lines and dots in Fig. 4(f) and in Fig. 5(h), and have their mode
shape variation displayed in Figs. 6(a,b) respectively. These examples
illustrate a transition between a right- and left-localized mode that occurs
as a function of $\phi$, with an intermediate state as a non-localized bulk
mode when the eigenfrequency branch tangentially approaches the boundary of
the gap. This type of transitions have been exploited for topological pumping
applications, where the phason $\phi$ is varied along an additional spatial
[70, 80] or temporal [77, 78, 79] dimension to induce a migration of localized
modes between two boundaries.

Figure 6: Examples of mode shape transitions as a function of phason $\phi$
for the (a) CM-PnC and (b) SM-PnC structures, corresponding to the branches
highlighted in Fig. 4(f) and Fig. 5(h), respectively.

These results reveal that the truncation resonances are in fact topological
edge states that traverse the band gaps for variations of the phason $\phi$.
The number of truncation resonances that traverse a gap is equal to the
corresponding gap label $C_{g}$. This holds true for any set of BCs, although
the particular shape of the branches of the edge states as they traverse the
gap may be different. In addition, while the number of in-gap resonances can
be predicted, one cannot guarantee the existence of truncation resonances for
a particular phason value $\phi$, but only that $|C_{g}|$ branches will
traverse the gap when $\phi$ varies in an interval of $2\pi$. For example, the
finite structure considered in Fig. 2(a) correspond to a phason value
$\phi=0.2\pi$, which intersects both the right- and left-localized edge state
branches of Fig. 4(b), and therefore one resonance localized at each boundary
is found in this case. In contrast, for a phason value $\phi=\pi$, the same
gap in Fig. 4(b) does not exhibit any edge states, and therefore no truncation
resonances would be found. Similarly, the modes I and II in Fig. 2(b) are
intersections of the left- and right-localized edge state branches in the
first and third gap of Fig. 5(b), respectively, for $\phi=0.4\pi$, while other
phason choices would define different truncation resonances or the their
absence. Therefore, to better understand the behavior of the truncation
resonances one needs to consider the entire family of structures defined for
variations of $\phi$, instead of separately considering particular cases.

####  3.2.1 Boundary phasons

As described, the phason $\phi$ simultaneously modifies the properties of both
boundaries of a finite structure (Fig. 1), and therefore influence the
truncation resonances localized at both boundaries. A higher degree of control
over the truncation resonances is achieved by using the right- and left-
boundary phasons introduced in Fig. 1, which modify only one boundary at a
time. This is equivalent to adding a tuning layer at one end of the structure
as done in Refs. [48, 42]. The effect of boundary phasons is demonstrated in
Fig. 7, which repeats the eigenfrequency variation with $\phi$ of Fig. 3(f)
and Fig. 4(d) in the left panels, and compares them to the the variation as a
function of right-boundary phason $\phi_{r}$ and left-boundary phason
$\phi_{l}$ displayed in the middle and right panels, respectively. The plots
clearly show evidence of how the boundary phason only causes the edge states
localized at the corresponding boundary to traverse the gap, while the
superimposed effect of both boundary phasons lead to the effect caused by the
phason $\phi$. Indeed, as $\phi_{r}$ varies (Figs. 7(b,e)), only the right-
localized modes traverse the gaps, producing the same branches as the ones in
Figs. 7(a,d). Any left-localized modes that were defined for $\phi=0$ (the
starting point) appear as roughly flat bands inside the gap, since the left
boundary is not changing with $\phi_{r}$. A similar effect is observed for the
variation with $\phi_{l}$ in Figs. 7(c,f). For a structure that has a
sufficient number of unit cells (i.e., has reached convergence as described
Section 3.3 to follow), the right- and left-localized edge states form a set
of decoupled chiral bands [89], the number of which corresponds to the gap
label magnitude $|C_{g}|$ and whose slopes are associated with the gap label
sign.

###  3.3 Effect of number of unit cells on frequency convergence of
topological truncation resonances

Next, we investigate the effect of the number of unit cells on the behavior of
the truncation resonances. As shown earlier, truncation modes exhibit an
exponential decay away from the boundary since their frequency lies inside a
band gap, and therefore correspond to a complex wave number. For structures
with a large number of unit cells, the in-gap truncation modes are only mildly
affected by further addition of unit cells since their displacement tend to
zero away from the boundary. In that scenario, a further increase in number of
unit cells will produce a larger number of bulk modes, while the branches of
the edge states spanning the band gaps with $\phi$ will remain the same.
However, for structures with a small number of unit cells, the truncation
resonances are more likely to be influenced by the opposing edge and by other
effects such as mode coupling and veering with bulk modes or another edge
state.

This behavior and the convergence with the number of unit cells is elucidated
by the results of Fig. 8. The SM-PnC structure with $\theta_{1}=1/\bar{a}$ is
chosen to exemplify these features, with the first and second row
corresponding to free-free and pinned-pinned BCs, respectively. The panels
(a,d) display the variation of the eigenfrequencies with $\phi$ for a
structure with 5 unit cells, while the right panels (c,f) correspond to a
larger structure comprising 15 cells. In the middle panels (b,e), the
variation of the frequencies with the number of unit cells is displayed for
the fixed phason value highlighted by the vertical dashed-line intersections
in the other panels. Overall, the number of bulk modes increase with the
number of unit cells as expected, and the edge state branches traversing the
gaps are similar but exhibit small differences. These differences are
amplified for phason values that are close to mode couplings as illustrated in
the top row. At the selected phason value, there is a strongly coupled avoided
crossing between the right- and left-localized edge states for the case with 5
unit cells shown in (a), and therefore the eigenfrequencies defined for that
phason value are more separated when compared to the structure shown in (c)
with 15 cells and without the avoided crossing. Therefore, the frequencies of
the edge states for this phason value vary as a function of the number of unit
cells and converge to a fixed value at approximately 10 unit cells as
illustrated in Fig. 8(b). In contrast, in the case of the bottom row with
pinned-pinned BCs, the chosen phason value intersects the edge state mode and
an adjacent mode that is well isolated, and therefore the truncation frequency
converges quicker at around four unit cells. These results illustrate that
while convergence is always achieved, the required number of unit cells may
vary between different structures depending on the BCs and the presence of
coupling effects at the phason value of interest.

Figure 7: Eigenfrequency variation as a function of phason $\phi$ (a,d),
right-boundary phason $\phi_{r}$ (b,e) and left-boundary phason $\phi_{l}$
(c,f) for finite beam with $L=15\bar{a}$ and pinned-pinned BCs. The top row
consists of a CM-PnC structure with $\theta_{2}=2.5/\bar{a}$ while the bottom
row consists of a SM-PnC structure with $\theta_{3}=2$ m-1. Figure 8:
Eigenfrequency variation with $\phi$ for structure with $\theta_{1}=1/\bar{a}$
comprising 5 cells (a,d) and 15 cells (c,f). The middle panels (b,e) show the
variation with the number of unit cells for the fixed phason values
highlighted as vertical dashed lines in the other panels. Top and bottom rows
correspond to free-free and pinned-pinned BCs, respectively. Band-gap
frequency ranges are shaded grey.

###  3.4 Topological truncation resonance versus non-topological defect
resonance

Truncation resonances, with their topological character, are not the only type
of resonances that appear due to truncation or breakage of symmetry in a
periodic medium. Another type of resonance, that is also of localized nature,
is that associated with defect modes [90, 91, 92]. Under the developed
framework, the band gaps characterized by non-zero Chern labels are guaranteed
to support $|C_{g}|$ truncation resonances spanning the gaps as a function of
phason or boundary phason parameters. Although we do not present an example in
this paper, in some cases a band gap may be characterized by $C_{g}=0$, which
is referred to as a topologically trivial band gap. In this case, the presence
of in-gap resonances is not guaranteed, although they may appear. Since there
is no topological explanation or origin to their appearance, these truncation
resonances are usually categorized as defect modes. One example can be found
in reference [75], where a central trivial gap with $C_{g}=0$ does not exhibit
in-gap resonances under pinned-pinned BCs (Fig. 2a), but exhibits truncation
resonances under clamped-free BCs (Fig. 3a). Note that the truncation
resonances in this second case do not traverse the band gap, which is a key
feature expected from topological modes as we highlight in this work.

We here illustrate another important scenario where a physical defect is
introduced to a finite structure in order to create an in-gap resonance,
although in this case a non-topological resonance as we will show. As an
example, we consider a finite SM-PnC structure comprising 15 unit cells with
$\theta_{1}=1/\bar{a}$, and introduce a defect initially located at the 8th
unit cell by ”skipping” the ABS portion within this unit cell, making it
entirely out of aluminum. The results displayed in Fig. 9 show the variation
of the eigenfrequencies with $\phi$, with the defect unit cell highlighted in
the schematics at the top and identified by the larger white segment, which
represents aluminum. As the phason varies, material is added to the left
boundary and removed from the right boundary (Fig. 1), which causes the defect
to continuously drift towards the right boundary. The defect moves by one unit
cell with every change in $2\pi$; these increments are marked by the vertical
dashed lines in the figure. After a change in phason of $14\pi$, the defect is
at the last unit cell, and finally for $16\pi$ it exists the structure and a
perfect periodic domain is restored. In a defect-free structure, the variation
of the eigenfrequencies with $\phi$ is trivially periodic in intervals of
$2\pi$. With the inclusion of the defect, additional modes are found inside
the gaps and co-exist with the truncation resonances. The interplay between
the in-gap defect mode and the truncation resonances is highlighted by the
selected mode shapes displayed in the bottom panels. In the initial
configuration, the in-gap defect resonance is localized at the center (8th
unit cell) of the structure and is completely decoupled from the truncation
resonances, as evidenced by the plots in stage I. As the phason varies, the
trajectory of the defect modes remain almost flat inside the gaps, in sharp
contrast to the behavior of the topological states which transverse the gaps.
Indeed, the truncation resonances exhibit the expected periodic behavior as
their branches traverse the gaps in a pattern that repeats periodically in
intervals of $2\pi$. However, as the defect physical position approaches the
right boundary, the in-gap defect modes progressively couple with the
truncation resonances localized at the right boundary, this is seen in all
three gaps viewed in the figure. Focusing on the third band gap as an example,
the frequency curves in stage II exhibit a weak coupling, while in stage III a
larger coupling is observed causing an avoided crossing with relatively strong
repulsion between the defect and truncation resonances. As the defect moves
within the last unit cells (13th-15th), it slowly transforms to capture,
itself, the characteristics of a truncation resonance localized at the right
boundary, with a mode shape example displayed for stage IV. At this last
stage, the branches of the right-localized truncation resonances are very
different from the periodic pattern of the perfect periodic structure, since
they are created by a truncation near a defect.

These results highlight key differences between the truncation resonances and
defect modes. The defect mode defines a flat branch inside the gap as a
function of $\phi$, until it starts to couple with the topological truncation
resonances$-$which happens as the position of the defects nears the boundary.
It is interesting to note that when the coupling takes place, the shape of the
coupled truncation resonance branch changes as it traverses the gap. However,
the counting principle given by the gap Chern labels is still valid. This can
be verified as in every interval of $\phi=2\pi$, there is a net number of 1, 2
and 3 right-localized modes transversing the first, second, and third gap,
respectively. Therefore, the truncation resonances retain this key topological
property even with the interference of a defect at the boundary. We should
also stress that the topological classification of an in-gap mode is always
relative to a given set of parameters. The defect mode introduced here is non-
topological in the context of the phason degree of freedom, which causes it to
remain confinded inside the gap as a flat band. However in some cases this
type of defect mode might find a topological classification under a different
set of parameters and analysis framework [93].

Figure 9: Eigenfrequencies as a function of phason $\phi$ for a finite SM-PnC
structure with $\theta_{1}=1/\bar{a}$, $L=15\bar{a}$ and a defected unit cell.
The location of the defect changes by one unit-cell increments with every
change of $2\pi$ in $\phi$, as marked by the vertical dashed lines and
illustrated in the top schematics. Band-gap frequency ranges are shaded grey.
Selected mode shapes are displayed in the bottom panels, whose colors
correspond to the polarization of the mode, with dashed and solid lines
representing the mode with open and closed circle markers, respectively.

##  4 Experimental investigation of modulated phononic crystal beams

###  4.1 Experimental set-up and measurements

For the experimental investigation, we focus on the SM-PnC beam structure,
again composed of alternating layers of Al and ABS with a ratio of layer
lengths of 4:1 (Al:ABS) for the baseline unit-cell configuration. The unit-
cell length and cross-sectional area are selected as
$\bar{a}=203\leavevmode\nobreak\ \text{mm}$ and $A=645\leavevmode\nobreak\
{\text{mm}^{2}}$, except in Section 4.3 where the unit-cell length is varied.
The values of these geometric parameters are chosen to allow for the
generation of several band gaps below 9 kHz for practical reasons; however,
all conclusions are scale invariant and hence applicable to periodic
structures that are orders of magnitude smaller in size (with the limit that
they are appropriately represented by continuous models). In this section, we
show additional FE results for direct comparison with the experiments, where
we use the same FE model details as in Section 3 with specifically 100 finite
elements being used per unit-cell. For our experimental set-up, a set of Al
and ABS solid blocks were fabricated and connected to each other by an
adhesive to form the periodic structure. The test articles were suspended
using thin nylon wires to simulate free-free BCs as depicted in Fig. 10(a).

Figure 10: Experimental validation: (a) Photograph of the experimental setup
showing a 5-unit-cell SM-PnC beam structure consisting of layers of Aluminum
and ABS polymer with a ABS volume fraction of $20\%$ and
$\bar{a}=203\leavevmode\nobreak\ \text{mm}$. The structure was excited on the
far left side (on the first ABS polymer layer) with a force hammer and
measured with an accelerometer on the other far end. (b) Frequency band
diagram of the infinite (material) constituent of the SM-PnC beam and (c)
corresponding FRF response of the finite structure. Inset: Resonance frequency
(thin solid lines, theory; dots, experiment) versus unit-cell length $a$ for
the 5-unit-cell periodic beam structure. (d) Corresponding resonance frequency
(solid lines, theory; dots, experiment) versus left boundary phase (i.e.,
length of a tuning layer attached at the far left end). At $\phi_{\rm
l}=0.4\pi$, the tuning layer transitions from ABS to Al. At $\phi_{\rm
l}=2\pi$, the tuning layer is a full regular unit cell and the total structure
is rendered a 6-unit-cell structure. In (a), the solid lines represent
propagation modes, and the dashed lines represent attenuation modes. Band-gap
frequency ranges are shaded grey.

First we show the complex band structure of the unit cell, which is shown in
Fig. 10(b)$-$the real part of which is identical to Fig. 2(b). This
calculation shows that three relatively large band gaps exist between 0 and 9
kHz. Figure. 10(c) shows a corresponding FRF obtained theoretically (solid
line) and experimentally (dashed line) for a 5-unit-cell version of the
structure, in which the “input” force excitation and the “output” displacement
evaluation are at the extreme ends. For the experimental results, the test
article was excited at the tip of the structure using a force hammer. The
impulse forcing data $F$ from the force hammer was used in conjunction with
the response data $U$ obtained by a sensing accelerometer connected at the
other end of the structure, to generate the receptance $U/F$ over the
frequency range 0-9 KHz. The amplitude of the experimental response was
calibrated to match the average of all theoretical data points over the 0-9
KHz frequency range. An excellent correlation is observed between the
theoretical and experimental FRF curves. It can be seen, however, that the
correlation generally degrades at higher frequencies along with an increasing
level of noise. This is due to the difficulty of stimulating high frequencies
with a force hammer as well as the reduced resolution when using a constant
sampling rate over all frequencies.

###  4.2 Effects of modulation wavenumber, boundary phasons, and number of
unit cells by experiment

In Fig. 5(a), we have shown the effect of the modulation wavenumber (i.e.,
unit-cell length) on the locations of the truncation resonances. Here we
repeat our computational investigation focusing on the range $0.18\leq a\leq
0.22$ m and overlay the data of the experimental case of $a=0.2$ m
$(\theta=5)$. The results, which are shown in the inset of Fig. 10(c),
indicate very good agreement between theory and experiments. Another approach
that keeps the unit-cell geometric configuration intact is the addition of a
single tuning layer (or a partial unit-cell) at the end of the finite periodic
structure [48, 42], as demonstrated in Section3.2.1. As illustrated in Fig. 1,
the addition of a tuning layer corresponds to the application of a boundary
phason $\phi_{l}$. The material and geometrical configuration of the tuning
layer should be chosen such that it would generally form a physically cropped
unit-cell, i.e., it would form a partial unit-cell when its length is less
than $a$ and a full unit-cell when its length is $a$. Figure 10(d) displays a
plot of the resonant frequencies as a function of the length of the tuning
layer, denoted by $l_{\text{TL}}$ and ranging from $l_{\text{TL}}$ = 0
($\phi_{l}=0$, 5 unit-cells) to $l_{\text{TL}}=a$ ($\phi_{l}=2\pi$, 6 unit-
cells) for the same baseline design of Fig.10(a)$-$this corresponds partially
to the results shown in Fig. 5(b) but now with the addition of experimental
data points. With the addition of a tuning layer, band-gap resonances rapidly
traverse the band gaps. However, once they reach the band-gap boundaries they
behave like regular structural resonances (bulk modes) with slower levels of
variation as a function of $l_{\text{TL}}$.

Figure 11: Frequency response function comparison for the finite SM-PnC beam
structure with different number of unit cells. The results show a truncation
resonance in the second band gap. Compared to the baseline case of 5 unit
cells, the tructioan resonance is observed to experience negligible shift in
frequency for (a) a 6 unit-cell structure and (b) a 15 unit-cell structure.
Strong spatial attenuation in displacement amplitude across the structure is
observed as the number of unit cells is increased. These results are for the
same unit-cell configuration considered in Fig. 10. Band-gap frequency ranges
are shaded grey. Figure 12: Experimental validation: Resonance frequency (thin
solid lines, theory; dots, experiment) versus ABS length-fraction for the
5-unit-cell SM-PnC beam structure with $\bar{a}=203\leavevmode\nobreak\
\text{mm}$. The experimental data points correspond to an ABS length-fraction
of 0.1, 0.15, 0.2, 0.25 and 0.3, respectively. The thick solid lines represent
the band-gap boundaries for the corresponding infinite periodic materials.

Given the localization nature of truncation resonances, the measured amplitude
at the far end of the SM-PnC structure is expected to be less than at the edge
where the mode is localized and where the excitation is applied. In Fig. 11,
we show using both theory and experiment an FRF comparison between 5- and
6-unit-cell structures in (a) and 5- and 15-unit-cell structures in (b). A
truncation resonance peak clearly exists inside the second band gap. We also
observe a stronger attenuation from edge-to-edge as the number of unit cells
(and total structure length) increases. As for the effect of the number of
unit cells on the frequency of the truncation resonance, we note that there is
a negligible shift from 5 to 15 unit cells. These results are to be compared
with the eigenfrequency versus phason plot shown in Fig. 8(b) for free-free
BCs. It is shown in that figure that beyond 5 unit cells, the change in the
frequency of the truncation resonances become negligible. In contrast, the
frequencies of the conventional resonances demonstrate substantial shifts, as
shown in both Fig. 8(b) and Fig. 11. We also observe in Fig. 11(b) that while
the amplitude of the truncation resonance peak drops significantly as the
number of unit cells is increased from 5 to 15, the amplitudes of all the
conventional resonances do not experience any noticeable drops.

###  4.3 Effect of unit-cell material volume fraction by experiment

In addition to property modulation wavenumber and phasons, an alternative
approach for controlling the frequency locations of truncation resonances is
alternation of the unit-cell design, e.g., by changing its material
composition and/or spatial distribution or its geometry. This can result in
achieving a total exit of a truncation resoance from a band-gap frequency
range, as illustrated in Fig. 12 for a 5-unit-cell SM-PnC structure, which
shows that when $a_{\text{ABS}}/a$ is set to 0.25 or higher, no in-gap
resonances appear in any of the three gaps covered by both computation and
experiment. In this figure, we consider the full range $a_{\text{ABS}}/a$,
which at one extreme ($a_{\text{ABS}}/a=0$) represents a homogenous Al beam,
and at the other extreme ($a_{\text{ABS}}/a=1$) represents a beam composed of
only ABS polymer. This figure also allows us to examine the sensitivity of the
truncation resonances’ frequencies to smooth variations in the material volume
fraction. It can be seen that the truncation resonances are noticeably more
sensitive to varying the unit-cell layer dimensions than the conventional
resonances. Once they exit the band gaps however, these unique resonances
become less sensitive to varying $a_{\text{ABS}}/a$, and their sensitivity
becomes similar to that of the conventional resonances.

##  5 Further reflection on the material vs. structure theme

The distinction and interconnection between a material and a structure may be
examined and classified at various levels. A basic distinction is that of
intrinsic versus extrinsic properties or characteristics, e.g., the Young’s
modulus and density being intrinsic material properties in contrast to the
stiffness and total mass as extrinsic structural characteristics. The
distinction may also be made based on physical response. In this context, an
elementary classification may be based on the behavior of static deformation,
such as the length scale of deformation or spatial span of tangible force
interactions. For example, consider a lattice configuration of beams forming a
truss that lies at the core of a larger structural frame. If the length scale
of deformation at, say, the center of the core is much larger than the
individual beam elements and negligible force interaction occurs with the
boundaries formed by the frame, then this deformation may be viewed as a form
of material behavior. On the other hand, if the length scale of the
deformation is on the order of the beam elements, and non-negligible
interaction occurs with the boundaries, then the “periodic network of beams
behaves as a structure, such as a frame in a building or a truss in a bridge
[94].”

In this work, we have addressed the material-versus-structure correlation
problem at a more fundamental level; that is, by examining the characteristics
pertaining to finite size in comparison to the properties associated with
idealized infinite size, and doing so from a topological elastodynamics
perspective. Here, the dispersion curves represent material properties and the
natural frequencies represent structural characteristics. In this context,
finite size along the direction where the physical phenomenon of interest
takes effect (in this case, wave propagation) is what distinguishes the
material versus structure character. Finite dimensions in other lateral
dimensions (such as the thickness of a beam, for exmaple) may play a
significant role in altering the material properties or structural
characteristics, but not in altering the classification of material versus
structure. As a periodic material is truncated, and rendered a structure, both
bulk and truncation resonances emerge; the latter being intimately connected
to the nature of the truncation. This investigation focuses specifically on
this aspect.

##  6 Conclusions

In this paper, we have investigated using theory and experiments the
fundamental question of the relation and interplay between material and
structure. We provided a formal connection between topological physics and
truncation resonances in finite periodic structures. Periodic structures can
be understood and topologically characterized using property modulation
parameters such as the modulation wavelength $\theta$ and phason $\phi$. These
parameters expand the physical space and allow for a rigorous study of the
nature of truncation resonances.

The Chern number is a material property obtained from unit-cell analysis,
considering a large number of unit cells with periodic boundary conditions
applied. It allows us to predict the behavior of a periodic medium through the
bulk-boundary correspondence principle, which in fact is itself a
manifestation of the interconnection between the notion of a material and a
structure, originated in the quantum realm$-$which we bring here to elastic
media. In the QHE theory, for example, the Chern number is a material
invariant that predicts the existence of edge currents propagating along the
edges of truncated finite samples. Similarly, for our elastic structures, the
gap labels predict the number of truncation resonances that span a band gap as
$\phi$ is varied for a finite structure with any prescribed BCs.

We have shown that the existence of in-gap truncation resonances cannot be
guaranteed for any $\phi$ and that the topological character is understood
only when sweeping through $\phi$. This brings a more comprehensive
perspective rather than analysing particular truncation cases, and provides a
methodology for designing for truncation resonances or their absence. The
boundary phasons, which is a concept we introduce in this work, provide an
additional tool to control truncation resonances, albeit at different
boundaries independently. We have also investigated the effect of the number
of unit cells in a finite structure, elucidating that the left- and right-
boundary phasons become independent only when a sufficient number of unit
cells is present. We similarly demonstrated that the frequency location of
truncation resonances converge only when the structure is comprised of a
sufficiently large number of unit cells, at least five cells in most cases.
Mode couplings$—$whose locations are influenced by the boundary conditions
among other factors$—$impact the rate of convergence of the truncation
resonances. The impact of the unit-cell constituent material composition was
also studied, showing that a truncation resonance may be forced to exit a band
gap with an appropriate choice of material volume fraction.

We have also examined another important type of localized mode in finite
structures, the defect mode. We have shown it to be non-topological, since it
remains flat with change of $\phi$ inside the band gap unless it couples with
a truncation resonance. In a perfect “undefected” periodic structure, there
can only be one mode localized at each boundary for any given phason value. By
coupling with a defect, it is possible to have two modes localized at the same
boundary for a given structure, living inside a band gap, with different
frequencies.

This study, we expect, will inspire future work on multiple fronts. For
example, similar principles may be extended to 2D and 3D periodic structures
and their truncation resonances, which may manifest as localized modes at
points, edges, and surfaces, having connections to topological physics and
possibly to higher-order Chern numbers and higher-order topological modes
(such as corner modes). Another domain of potential applicability is coiled
phononic crystals for space saving [95]. A further angle to be explored in the
question of material versus structure is the static regime, where similar
connections may be established for topological floppy modes [96]. Other areas
to be investigated are the interplay with nonlinearities [97], the
applicability to damage mechanics such as the effect of number of unit cells
on the fracture toughness [98], and the role of size effects in nanoscience
where small finite dimensions have profound impact on thermal transport [25]
and other physical properties. Implications to quasiperiodic media [99, 73,
74, 75, 76] or nonperiodic media described statistically by representative
volume elements may also be explored. Finally, the framework presented for
connecting between topology and truncation may potentially be applied to
finite systems in other branches of physics, such as photonics [49] and
quantum mechanics [41].

## Acknowledgement

The authors acknowledge the students Andrew S. Tomchek and Edgar A. Flores for
their assistance in conducting the experiments.

## References

  * Rayleigh [1887] J. W. S. Rayleigh,  Xvii. on the maintenance of vibrations by forces of double frequency, and on the propagation of waves through a medium endowed with a periodic structure,  Philos. Mag. 24 (1887) 145–159.
  * Rytov [1956] S. M. Rytov,  Acoustical properties of a thinly laminated medium,  Sov. Phys.-Acoust. 2 (1956) 68–80.
  * Sun et al. [1968] C. T. Sun, J. D. Achenbach, G. Herrmann,  Continuum theory for a laminated medium,  ASME J. Appl. Mech. 35 (1968) 467–475.
  * Heckl [1964] M. Heckl,  Investigations on the vibrations of grillages and other simple beam structures,  J. Acoust. Soc. Am. 36 (1964) 1335–1343.
  * Mead [1970] D. J. Mead,  Free wave propagation in periodically supported, infinite beams,  J. Sound Vib. 11 (1970) 181–197.
  * Nemat-Nasser et al. [1975] S. Nemat-Nasser, F. C. L. Fu, S. Minagawa,  Continuum theory for a laminated medium,  Int. J. Solids Struct. 11 (1975) 617–642.
  * Sigalas and Economou [1993] M. Sigalas, E. N. Economou,  Band structure of elastic waves in two dimensional systems,  Solid State Commun. 86 (1993) 141–143.
  * Kushwaha et al. [1993] M. S. Kushwaha, P. Halevi, L. Dobrzynski, B. Djafari-Rouhani,  Acoustic band structure of periodic elastic composites,  Phys. Rev. Lett. 71 (1993) 2022–2025.
  * Maradudin [1969] A. A. Maradudin, Lattice Dynamics, W. A. Benjamin, California, 1969.
  * Liu et al. [2000] Z. Liu, X. Zhang, Y. Mao, Y.-Y. Zhu, Z. Yang, C. T. Chan, P. Sheng,  Locally resonant sonic materials,  Science 289 (2000) 1734–1736.
  * Kittel [1976] C. Kittel, Introduction to Solid State Physics, Wiley, New York, 1976.
  * Bloch [1883] F. Bloch,  Sur les equations différentielles linéaries à coefficients périodiques,  Ann. Sci. Ecole Norm. S. 12 (1883) 47–88.
  * Bloch [1929] F. Bloch,  Über die quantenmechanik der elektronen in kristallgittern,  Z. Phys. 52 (1929) 555–600.
  * Mead [1975a] D. J. Mead,  Wave propagation and natural modes in periodic systems: I. mono-coupled systems,  J. Sound Vib. 40 (1975a) 1–18.
  * Mead [1975b] D. J. Mead,  Wave propagation and natural modes in periodic systems: Ii. multi-coupled systems, with and without damping,  J. Sound Vib. 40 (1975b) 19–39.
  * Xiao et al. [2011] Y. Xiao, B. R. Mace, J. Wen, X. Wen,  Formation and coupling of band gaps in a locally resonant elastic system comprising a string with attached resonators,  Phys. Lett. A 375 (2011) 1485–1491.
  * Raghavan and Phani [2013] L. Raghavan, A. S. Phani,  Local resonance bandgaps in periodic media: Theory and experiment,  Journal of the Acoustical Society of America 134 (2013) 1950–1959.
  * Callaway [1959] J. Callaway,  Model for lattice thermal conductivity at low temperatures,  Phys. Rev. 113 (1959) 1046–1051.
  * Holland [1963] M. G. Holland,  Analysis of lattice thermal conductivity,  Phys. Rev. 132 (1963) 2461–2471.
  * Nemat-Nasser et al. [2011] S. Nemat-Nasser, J. R. Willis, A. Srivastava, A. V. Amirkhizi,  Homogenization of periodic elastic composites and locally resonant sonic materials,  Phys. Rev. B 83 (2011) 104103.
  * Sánchez-Pérez et al. [1998] J. V. Sánchez-Pérez, D. Caballero, R. Mártinez-Sala, C. Rubio, J. Sánchez-Dehesa, F. Meseguer, J. Llinares, F. Gálvez,  Sound attenuation by a two-dimensional array of rigid cylinders,  Phys. Rev. Lett. 80 (1998) 5325.
  * Vasseur et al. [2001] J. O. Vasseur, P. A. Deymier, B. Chenni, B. Djafari-Rouhani, L. Dobrzynski, D. Prevost,  Experimental and theoretical evidence for the existence of absolute acoustic band gaps in two-dimensional solid phononic crystals,  Phys. Rev. Lett. 86 (2001) 3012.
  * Hussein et al. [2014] M. I. Hussein, M. J. Leamy, M. Ruzzene,  Dynamics of phononic materials and structures: Historical origins, recent progress, and future outlook,  Appl. Mech. Rev 66 (2014) 040802.
  * Jin et al. [2021] Y. Jin, Y. Pennec, B. Bonello, H. Honarvar, L. Dobrzynski, B. Djafari-Rouhani, M. I. Hussein,  Physics of surface vibrational resonances: pillared phononic crystals, metamaterials, and metasurfaces,  Reports on Progress in Physics 84 (2021) 086502.
  * Maruyama [2002] S. Maruyama,  A molecular dynamics simulation of heat conduction in finite length swnts,  Physica B: Condensed Matter 323 (2002) 193–195.
  * Hussein et al. [2006] M. I. Hussein, G. M. Hulbert, R. A. Scott,  Dispersive elastodynamics of 1D banded materials and structures: analysis,  J. Sound Vib. 289 (2006) 779–806.
  * Hussein et al. [2015] M. I. Hussein, S. Biringen, O. R. Bilal, A. Kucala,  Flow stabilization by subsurface phonons,  Proc. R. Soc. A 471 (2015) 20140928.
  * Al Ba’ba’a et al. [2019] H. Al Ba’ba’a, M. Nouh, T. Singh,  Dispersion and topological characteristics of permutative polyatomic phononic crystals,  Proc. R. Soc. A 475 (2019) 20190022.
  * Wallis [1957] R. F. Wallis,  Effect of free ends on the vibration frequencies of one-dimensional lattices,  Phys. Rev. 105 (1957) 540.
  * Camley et al. [1983] R. E. Camley, B. Djafari-Rouhani, L. Dobrzynski, A. A. Maradudin,  Transverse elastic waves in periodically layered infinite and semi-infinite media,  Phys. Rev. B 27 (1983) 7318.
  * Djafari-Rouhani et al. [1983] B. Djafari-Rouhani, L. Dobrzynski, O. H. Duparc, R. Camley, A. Maradudin,  Sagittal elastic waves in infinite and semi-infinite superlattices,  Phys. Rev. B 28 (1983) 1711–1720.
  * El Boudouti et al. [1993] E. H. El Boudouti, B. Djafari-Rouhani, E. M. Khourdifi, L. Dobrzynski,  Surface and interface elastic waves in superlattices: Transverse localized and resonant modes,  Phys. Rev. B 48 (1993) 10987.
  * El Boudouti et al. [1995] E. H. El Boudouti, B. Djafari-Rouhani, A. Nougaoui,  Surface elastic waves in superlattices: Sagittal localized and resonant modes,  Phys. Rev. B 51 (1995) 13801.
  * El Boudouti et al. [1996] E. H. El Boudouti, B. Djafari-Rouhani, A. Akjouj, L. Dobrzynski,  Theory of surface and interface transverse elastic waves in $N$-layer superlattices,  Phys. Rev. B 54 (1996) 14728.
  * Hladky-Hennion et al. [2005] A.-C. Hladky-Hennion, G. Allan, M. de Billy,  Localized modes in a one-dimensional diatomic chain of coupled spheres,  J. Appl. Phys. 98 (2005) 054909.
  * Ren and Chang [2007] S. Y. Ren, Y.-C. Chang,  Theory of confinement effects in finite one-dimensional phononic crystals,  Phys. Rev. B 75 (2007) 212301.
  * Darinskii and Shuvalov [2018] A. N. Darinskii, A. L. Shuvalov,  Surface acoustic waves on one-dimensional phononic crystals of general anisotropy: Existence considerations,  Phys. Rev. B 98 (2018) 024309.
  * Born [1942] M. Born,  Lattice dynamics and x-ray scattering,  P. R. Soc. London 54 (1942) 362–376.
  * Al Ba’ba’a et al. [2017] H. Al Ba’ba’a, M. Nouh, T. Singh,  Pole distribution in finite phononic crystals: Understanding Bragg-effects through closed-form system dynamics,  J. Acoust. Soc. Am 142 (2017) 1399–1412.
  * Bastawrous and Hussein [2022] M. V. Bastawrous, M. I. Hussein,  Closed-form existence conditions for bandgap resonances in a finite periodic chain under general boundary conditions,  J. Acoust. Soc. Am. 151 (2022) 286–298.
  * Ren and Chang [2010] S. Y. Ren, Y.-C. Chang,  Surface states/modes in one-dimensional semi-infinite crystals,  Ann. Phys. (N. Y). 325 (2010) 937–947.
  * Davis et al. [2011] B. L. Davis, A. S. Tomchek, E. A. Flores, L. Liu, M. I. Hussein,  Analysis of periodicity termination in phononic crystals,  Proceedings of ASME 2011 International Mechanical Engineering Congress and Exposition; Paper No. IMECE2011-65666 8 (2011) 973–977.
  * Hvatov and Sorokin [2015] A. Hvatov, S. Sorokin,  Free vibrations of finite periodic structures in pass-and stop-bands of the counterpart infinite waveguides,  J. Sound Vib. 347 (2015) 200–217.
  * Xiao et al. [2013] Y. Xiao, J. Wen, D. Yu, X. Wen,  Flexural wave propagation in beams with periodically attached vibration absorbers: band-gap behavior and band formation mechanisms,  J. Sound Vib. 332 (2013) 867–893.
  * Sangiuliano et al. [2020] L. Sangiuliano, C. Claeys, E. Deckers, W. Desmet,  Influence of boundary conditions on the stop band effect in finite locally resonant metamaterial beams,  J. Sound Vib. 473 (2020) 115225.
  * Carneiro Jr et al. [2021] J. Carneiro Jr, M. Brennan, P. Gonçalves, V. Cleante, D. Bueno, R. Santos,  On the attenuation of vibration using a finite periodic array of rods comprised of either symmetrical or asymmetrical cells,  J. Sound Vib. 511 (2021) 116217.
  * Al Ba’ba’a et al. [2022] H. Al Ba’ba’a, C. L. Willey, V. W. Chen, A. T. Juhl, M. Nouh,  Theory of truncation resonances in continuum rod-based phononic crystals with generally asymmetric unit cells,  arXiv:2211.01423v1 (2022).
  * Hussein et al. [2007] M. I. Hussein, G. M. Hulbert, R. A. Scott,  Dispersive elastodynamics of 1D banded materials and structures: Design,  J. Sound Vib. 307 (2007) 865–893.
  * Yeh et al. [1978] P. Yeh, A. Yariv, A. Y. Cho,  Optical surface waves in periodic layered media,  Appl. Phys. Lett. 32 (1978) 104–105.
  * Hasan and Kane [2010] M. Z. Hasan, C. L. Kane,  Colloquium: topological insulators,  Reviews of Modern Physics 82 (2010) 3045.
  * Ozawa et al. [2019] T. Ozawa, H. M. Price, A. Amo, N. Goldman, M. Hafezi, L. Lu, M. C. Rechtsman, D. Schuster, J. Simon, O. Zilberberg, et al.,  Topological photonics,  Reviews of Modern Physics 91 (2019) 015006.
  * Miniaci and Pal [2021] M. Miniaci, R. Pal,  Design of topological elastic waveguides,  Journal of Applied Physics 130 (2021) 141101.
  * Pal and Ruzzene [2017] R. K. Pal, M. Ruzzene,  Edge waves in plates with resonators: an elastic analogue of the quantum valley Hall effect,  New Journal of Physics 19 (2017) 025001.
  * Chaunsali et al. [2017] R. Chaunsali, E. Kim, A. Thakkar, P. G. Kevrekidis, J. Yang,  Demonstrating an in situ topological band transition in cylindrical granular chains,  Physical Review Letters 119 (2017) 024301.
  * Wang et al. [2020] W. Wang, Y. Jin, W. Wang, B. Bonello, B. Djafari-Rouhani, R. Fleury,  Robust fano resonance in a topological mechanical beam,  Physical Review B 101 (2020) 024101.
  * Yin et al. [2018] J. Yin, M. Ruzzene, J. Wen, D. Yu, L. Cai, L. Yue,  Band transition and topological interface modes in 1D elastic phononic crystals,  Scientific reports 8 (2018) 1–10.
  * Su et al. [1979] W. Su, J. Schrieffer, A. J. Heeger,  Solitons in polyacetylene,  Physical Review Letters 42 (1979) 1698.
  * Süsstrunk and Huber [2015] R. Süsstrunk, S. D. Huber,  Observation of phononic helical edge states in a mechanical topological insulator,  Science 349 (2015) 47–50.
  * Miniaci et al. [2018] M. Miniaci, R. Pal, B. Morvan, M. Ruzzene,  Experimental observation of topologically protected helical edge modes in patterned elastic plates,  Physical Review X 8 (2018) 031074.
  * Torrent et al. [2013] D. Torrent, D. Mayou, J. Sánchez-Dehesa,  Elastic analog of graphene: Dirac cones and edge states for flexural waves in thin plates,  Physical Review B 87 (2013) 115143.
  * Vila et al. [2017] J. Vila, R. K. Pal, M. Ruzzene,  Observation of topological valley modes in an elastic hexagonal lattice,  Physical Review B 96 (2017) 134307.
  * Thouless et al. [1982] D. J. Thouless, M. Kohmoto, M. P. Nightingale, M. den Nijs,  Quantized Hall conductance in a two-dimensional periodic potential,  Physical Review Letters 49 (1982) 405.
  * Klitzing et al. [1980] K. v. Klitzing, G. Dorda, M. Pepper,  New method for high-accuracy determination of the fine-structure constant based on quantized Hall resistance,  Physical Review Letters 45 (1980) 494.
  * Wang et al. [2015a] Y.-T. Wang, P.-G. Luan, S. Zhang,  Coriolis force induced topological order for classical mechanical vibrations,  New Journal of Physics 17 (2015a) 073031.
  * Wang et al. [2015b] P. Wang, L. Lu, K. Bertoldi,  Topological phononic crystals with one-way elastic edge waves,  Physical Review Letters 115 (2015b) 104302.
  * Nash et al. [2015] L. M. Nash, D. Kleckner, A. Read, V. Vitelli, A. M. Turner, W. T. Irvine,  Topological mechanics of gyroscopic metamaterials,  Proceedings of the National Academy of Sciences 112 (2015) 14495–14500.
  * Kraus and Zilberberg [2016] Y. E. Kraus, O. Zilberberg,  Quasiperiodicity and topology transcend dimensions,  Nature Physics 12 (2016) 624.
  * Prodan [2015] E. Prodan,  Virtual topological insulators with real quantized physics,  Physical Review B 91 (2015) 245104.
  * Apigo et al. [2018] D. J. Apigo, K. Qian, C. Prodan, E. Prodan,  Topological edge modes by smart patterning,  Physical Review Materials 2 (2018) 124203.
  * Rosa et al. [2019] M. I. Rosa, R. K. Pal, J. R. Arruda, M. Ruzzene,  Edge states and topological pumping in spatially modulated elastic lattices,  Physical Review Letters 123 (2019) 034301.
  * Apigo et al. [2019] D. J. Apigo, W. Cheng, K. F. Dobiszewski, E. Prodan, C. Prodan,  Observation of topological edge modes in a quasiperiodic acoustic waveguide,  Physical Review Letters 122 (2019) 095501.
  * Ni et al. [2019] X. Ni, K. Chen, M. Weiner, D. J. Apigo, C. Prodan, A. Alù, E. Prodan, A. B. Khanikaev,  Observation of hofstadter butterfly and topological edge states in reconfigurable quasi-periodic acoustic crystals,  Communications Physics 2 (2019) 55.
  * Pal et al. [2019] R. K. Pal, M. I. Rosa, M. Ruzzene,  Topological bands and localized vibration modes in quasiperiodic beams,  New Journal of Physics 21 (2019) 093017.
  * Gupta and Ruzzene [2020] M. Gupta, M. Ruzzene,  Dynamics of quasiperiodic beams,  Crystals 10 (2020) 1144.
  * Xia et al. [2020] Y. Xia, A. Erturk, M. Ruzzene,  Topological edge states in quasiperiodic locally resonant metastructures,  Physical Review Applied 13 (2020) 014023.
  * Rosa et al. [2021] M. I. Rosa, Y. Guo, M. Ruzzene,  Exploring topology of 1D quasiperiodic metastructures through modulated lego resonators,  Applied Physics Letters 118 (2021) 131901.
  * Grinberg et al. [2020] I. H. Grinberg, M. Lin, C. Harris, W. A. Benalcazar, C. W. Peterson, T. L. Hughes, G. Bahl,  Robust temporal pumping in a magneto-mechanical topological insulator,  Nature communications 11 (2020) 1–9.
  * Chen et al. [2019] H. Chen, L. Yao, H. Nassar, G. Huang,  Mechanical quantum Hall effect in time-modulated elastic materials,  Physical Review Applied 11 (2019) 044029.
  * Xia et al. [2021] Y. Xia, E. Riva, M. I. Rosa, G. Cazzulani, A. Erturk, F. Braghin, M. Ruzzene,  Experimental observation of temporal pumping in electromechanical waveguides,  Physical Review Letters 126 (2021) 095501.
  * Riva et al. [2020] E. Riva, M. I. Rosa, M. Ruzzene,  Edge states and topological pumping in stiffness-modulated elastic plates,  Physical Review B 101 (2020) 094307.
  * Ma et al. [2019] G. Ma, M. Xiao, C. Chan,  Topological phases in acoustic and mechanical systems,  Nature Reviews Physics 1 (2019) 281–294.
  * Huang et al. [2021] H. Huang, J. Chen, S. Huo,  Recent advances in topological elastic metamaterials,  Journal of Physics: Condensed Matter (2021).
  * Liu and Hussein [2012] L. Liu, M. I. Hussein,  Wave motion in periodic flexural beams and characterization of the transition between Bragg scattering and local resonance,  Journal of Applied Mechanics 79 (2012).
  * Phani et al. [2006] A. S. Phani, J. Woodhouse, N. A. Fleck,  Wave propagation in two-dimensional periodic lattices,  Journal of the Acoustical Society of America 119 (2006) 1995–2005.
  * Hatsugai [1997] Y. Hatsugai,  Topological aspects of the quantum Hall effect,  Journal of Physics: Condensed Matter 9 (1997) 2507.
  * Hatsugai [1993] Y. Hatsugai,  Chern number and edge states in the integer quantum Hall effect,  Physical Review Letters 71 (1993) 3697.
  * Fukui et al. [2005] T. Fukui, Y. Hatsugai, H. Suzuki,  Chern numbers in discretized Brillouin zone: efficient method of computing (spin) Hall conductances,  Journal of the Physical Society of Japan 74 (2005) 1674–1677.
  * Bellissard [1992] J. Bellissard,  Gap labelling theorems for schrödinger operators,  in: From number theory to physics, Springer, 1992, pp. 538–630.
  * Prodan and Schulz-Baldes [2016] E. Prodan, H. Schulz-Baldes,  Bulk and boundary invariants for complex topological insulators,  K (2016).
  * Sigalas [1997] M. M. Sigalas,  Elastic wave band gaps and defect states in two- dimensional composites,  Journal of the Acoustical Society of America 101 (1997) 1256–1261.
  * Torres et al. [1999] M. Torres, F. R. Montero de Espinosa, D. Garcia-Pablos, N. Garcia,  Sonic band gaps in finite elastic media: Surface states and localization phenomena in linear and point defects,  Physical Review Letters 82 (1999) 3054–3057.
  * Khelif et al. [2003] A. Khelif, B. Djafari-Rouhani, J. O. Vasseur, P. A. Deymier,  Transmission and dispersion relations of perfect and defect-containing waveguide structures in phononic band gap materials,  Physical Review B 68 (2003) 024302.
  * Miniaci et al. [2021] M. Miniaci, F. Allein, R. K. Pal,  Spectral flow of a localized mode in elastic media,  arXiv preprint arXiv:2111.09021 (2021).
  * Phani and Hussein [2017] A. S. Phani, M. I. Hussein, Dynamics of Lattice Materials, John Wiley & Sons, 2017.
  * Willey et al. [2022] C. L. Willey, V. W. Chen, D. Roca, A. Kianfar, M. I. Hussein, A. T. Juhl,  Coiled phononic crystal with periodic rotational locking: Subwavelength Bragg band gaps,  Physical Review Applied 18 (2022) 014035.
  * Kane and Lubensky [2014] C. L. Kane, T. C. Lubensky,  Topological boundary modes in isostatic lattices,  Nature Physics 18 (2014) 39–45.
  * Zhou et al. [2022] D. Zhou, D. Rocklin, M. Leamy, Y. Yao,  Topological invariant and anomalous edge modes of strongly nonlinear systems,  Nature Communications 13 (2022) 1–9.
  * Shaikeea et al. [2022] A. Shaikeea, H. Cui, M. O’Masta, X. R. Zheng, V. S. Deshpande,  The toughness of mechanical metamaterials,  Nature Materials 21 (2022) 297–304.
  * Liu et al. [2004] G.-T. Liu, T.-Y. Fan, R.-P. Guo,  Governing equations and general solutions of plane elasticity of one-dimensional quasicrystals,  International Journal of Solids and Structures 41 (2004) 3949–3959.