Topologically protected edge states in mechanical metamaterials

CHAPTER THREE Topologically protected edge states in mechanical metamaterials Raj Kumar Pal, Javier Vila, Massimo Ruzzene1 School of Aerospace Engine...

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CHAPTER THREE

Topologically protected edge states in mechanical metamaterials Raj Kumar Pal, Javier Vila, Massimo Ruzzene1 School of Aerospace Engineering, Georgia Institute of Technology, Atlanta, GA, United States 1 Corresponding author: e-mail address: [email protected]

Contents 1. Introduction 2. Interface modes in a 1D lattice 2.1 Dispersion analysis for the periodic lattice 2.2 Band inversion and topological invariants 2.3 Analytical evaluation of the interface modes 2.4 Harmonic response of lattice with interface 3. Interface modes in a discrete two-dimensional hexagonal lattice 3.1 Dispersion analysis 3.2 Dispersion analysis of a finite strip and transient simulations 4. Topological valley modes in a continuous elastic hexagonal lattice 4.1 Configuration and material properties 4.2 Dispersion analysis 4.3 Experimental results 5. Conclusions Acknowledgments References

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Abstract Recent breakthroughs in condensed matter physics are opening new directions in band engineering and wave manipulation. Specifically, challenging the notions of reciprocity, time-reversal symmetry, and sensitivity to defects in wave propagation may disrupt ways in which mechanical metamaterials are designed and employed, and may enable totally new functionalities. Nonreciprocity and topologically protected wave propagation will have profound implications on how stimuli and information are transmitted within materials, or how energy can be guided and steered so that its effects may be controlled or mitigated. This chapter introduces one basic approach to generate topologically protected edge-bound wave propagation in mechanical metamaterials. The concept is based on breaking inversion symmetry within the geometry of a unit cell of a periodic media,

Advances in Applied Mechanics, Volume 52 ISSN 0065-2156 https://doi.org/10.1016/bs.aams.2019.04.001

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and in joining periodic assemblies that are inverted copies of each other. Such inversion leads to topologically different structures, as quantified by associated dispersion topological invariants. A nontrivial interface is thus produced which supports the propagation of defect and backscattering-immune edge states. The concept is first illustrated in a one-dimensional spring mass lattice, which is the simplest configuration that supports the considered broken inversion symmetry and the resulting interface bound modes. Next, the presentation is extended to a conceptual discrete two-dimensional hexagonal lattice, which provides the required symmetries for the nucleation of isolated Dirac points in reciprocal space with inverted topological invariants at the high symmetry points. This lattice forms the basis for the design of a continuous elastic hexagonal lattice, whose dispersion topology is investigated first numerically and then probed experimentally to demonstrate the existence of the predicted edge modes. The results shown for this continuous lattice demonstrate the effectiveness of the approach followed for the generation of topology inverted lattices and the production of nontrivial interfaces. Such procedure can be extended to a variety of structural configurations which can be exploited for designs of components that are capable of guiding elastic waves along predefined paths, or isolate vibrations to specific spatial locations.

1. Introduction Wave propagation in periodic media has been an active field of research for the past few decades. Energy transport by waves arise in multiple areas of physics as it relates to acoustic, elastic, electromagnetic, and electronic media. Unique phenomena like negative refraction, directional propagation, focusing, and cloaking have been pursued through careful engineering of the band structure, which is a unifying theme for exploration in this diverse set of physical domains. Recently, the advent of topological mechanics (Huber, 2016) has provided an effective framework for the pursuit of robust wave propagation which is protected against perturbations and defects. Topologically protected edge wave propagation was originally envisioned in quantum systems and has quickly evolved to other classical areas of physics, such as acoustics (Brendel, Peano, Painter, & Marquardt, 2018), photonics (Khanikaev et al., 2013; Lu, Joannopoulos, & Soljacic, 2014), mechanics (Mousavi, Khanikaev, & Wang, 2015; Pal & Ruzzene, 2017), as well as to coupled wave domains such as optomechanics (Peano, Brendel, Schmidt, & Marquardt, 2015). In all of these different domains, properties such as lossless propagation, existence of waves confined to a boundary or interface, immunity to backscattering and localization in the presence of defects and imperfections are related to band topology.

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This makes them classical analogues of topological insulators that support the propagation of topologically protected edge waves (TPEWs). There are two broad ways to realize topologically protected wave propagation in elastic media. The first one uses active components, thereby mimicking the quantum Hall effect. Changing of the parity of active devices or modulating the physical properties in time, for example, has shown to alter the direction and nature of edge waves (Nassar, Xu, Norris, & Huang, 2017; Swinteck et al., 2015). Examples include magnetic fields in biological systems (Prodan & Prodan, 2009), rotating disks (Nash et al., 2015), and acoustic circulators operating on the basis of a flow-induced bias (Khanikaev, Fleury, Mousavi, & Alu`, 2015). A second way uses solely passive components and relies on establishing analogues of the quantum spin Hall effect. These media feature both forward and backward propagating edge modes, which can be induced by an external excitation of appropriate polarization. The concept is illustrated in several studies by way of both numerical (He et al., 2016; Mousavi et al., 2015; Pal, Schaeffer, & Ruzzene, 2016) and experimental (Ningyuan, Owens, Sommer, Schuster, & Simon, 2015; S€ usstrunk & Huber, 2015) investigations, which involve coupled pendulums (S€ usstrunk & Huber, 2015), plates with two scale holes (Mousavi et al., 2015) and resonators (Pal et al., 2016), as well as electric circuits (Ningyuan et al., 2015). Numerous studies have also been conducted on localized nonpropagating deformation modes at the interface of two structural lattices (Chaunsali, Thakkar, Kim, Kevrekidis, & Yang, 2017; Pal & Ruzzene, 2017; Prodan, Dobiszewski, Kanwal, Palmieri, & Prodan, 2017). These modes depend on the topological properties of the bands, which in 1D lattices are characterized by the Zak phase as topological invariant (Xiao et al., 2015). In 2D and 3D lattices, several researchers have investigated the presence of floppy modes of motion due to nontrivial topological polarization and exploited these modes to achieve localized buckling and directional response (Kane & Lubensky, 2014; Paulose, Meeussen, & Vitelli, 2015; Rocklin, 2017; Rocklin, Chen, Falk, Vitelli, & Lubensky, 2016). In spite of the intense level of activity in this area, to the best of our knowledge, studies reporting on the experimental observation of TPEWs in continuous elastic media have so far been limited. Unique challenges in elastic systems exist due to their high modal densities, which complicate the analysis and design of the band structure and the effective achievement of nontrivial topologies. These also often lead to complex arrangements of materials and intricate connectivities that may be hard to realize in practice. A promising avenue in this regard is the use of valley degrees of freedom as originally envisioned in quantum systems

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like graphene bilayers (Xiao, Yao, & Niu, 2007; Zhang, Jung, Fiete, Niu, & MacDonald, 2011; Zhang, MacDonald, & Mele, 2013). The concept has also been adopted in classical areas such as photonics (Lu et al., 2014; Ma & Shvets, 2016), acoustics (Lu, Qiu, Ke, & Liu, 2016; Lu et al., 2017; Ye et al., 2017), and phononics (Brendel et al., 2018; Pal & Ruzzene, 2017). This chapter describes the concept of exploiting the valley degrees of freedom and the topological differences generated by symmetry inversions within the unit cell. The case of a 1D lattice is presented first, followed by the 2D extension on a discrete hexagonal lattice. This configuration leads to the design of a continuous hexagonal lattice, which is used to demonstrate experimentally TPEWs in continuous elastic media. The considered configuration consists of an elastic hexagonal lattice on which concentrated masses are attached at the sublattice sites. This provides a simple assembly that is characterized by the symmetry conditions sufficient to open a topologically nontrivial bandgap. The addition of masses at selected locations within a unit cell breaks the C3v symmetry inherent to the hexagonal geometry, while preserving the C3 symmetry. Exploiting the arrangement of masses conveniently leads to lattices that exhibit different topological properties of the bands. When two such lattices with different topological properties are joined together, TPEWs propagate along the shared interface. The outline of this chapter is as follows. Following this introduction, Section 2 presents the 1D lattice and the interface modes resulting from joining to mirror copies of the same periodic lattice. Section 3 describes the 2D extension on the discrete hexagonal lattice, while Section 4 presents the description of the continuous hexagonal lattice, along with its dispersion analysis, and the dispersion analysis of a finite strip containing an interface. Section 4 also describes the experimental setup, the estimation of the dispersion diagrams for the lattices, and results showing TPEWs for two different interfaces. Finally, Section 5 summarizes the main results of this study and presents potential future research directions.

2. Interface modes in a 1D lattice We begin by illustrating the existence and behavior of interface modes in a one-dimensional (1D) spring mass chain. This simple lattice is suitable to briefly describe the existence of localized modes at the interface of lattices that are characterized by distinct topological invariants. The relevant topologically invariant in this case is the known as the Zak phase (Xiao et al., 2015; Xiao, Zhang, & Chan, 2014).

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Fig. 1 Two sublattices (A) and (B), which are inverted copies of each other, are joined together. The interface supports support a localized mode in the bandgap frequencies.

The spring mass lattice model considered herein is displayed in Fig. 1. It consists of two sublattices, each with identical masses and having alternating springs with stiffness k1 and k2 and of an interface (or defect) mass connecting them. The interface mass is connected to springs with stiffness k1 on both sides. The unit cells on the right and left of this interface are inverted copies of each other. The governing equations for a unit cell p of the sublattice on the left of the interface is given by m€ ua, p + k2 ua, p ub, p + k1 ua, p ub, p1 ¼ 0 (1a) (1b) m€ ub, p + k2 ub, p ua, p + k1 ub, p ua, p + 1 ¼ 0 while for a unit cell p on the right sublattice the equations of motion are (2a) m€ ua, p + k1 ua, p ub, p + k2 ua, p ub, p1 ¼ 0 m€ ub, p + k1 ub, p ua, p + k2 ub, p ua, p + 1 ¼ 0: (2b) Finally, the governing equation for the motion of the interface mass is given by m€ uc, 0 + k1 ð2uc, 0 ub, 0 ub, 1 Þ ¼ 0 (3) The above equations are normalized by writing the spring constants as k1 ¼ k(1 + γ) and k2 ¼ k(1 γ), with γ being a stiffness parameter and k pﬃﬃﬃﬃﬃﬃﬃﬃ being the mean stiffness. A nondimensional time scale τ ¼ k=mt is introduced to express the equations in nondimensional form.

2.1 Dispersion analysis for the periodic lattice We first evaluate the dispersion relations for the infinite periodic lattice resulting form alternating the two stiffness constants k1, k2 in the absence

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of the interface. This is achieved by considering the governing equations for a representative unit cell, which can be expressed in the following nondimensional form: u€a, j + 2ua, j ð1 + γÞub, j ð1 γÞub, j1 ¼ 0, u€b, j + 2ub, j ð1 + γÞua, j ð1 γÞua, j + 1 ¼ 0: Imposing a plane wave solution of the form uj ¼ (ua, j, ub, j) ¼ A(μ)eiΩτ+iμj where Ω is the frequency and μ is the nondimensional wavenumber leads to the following eigenvalue problem Aa ð1 + γÞ ð1 γÞeiμ 2 Ω2 2 Aa ¼Ω : (4) Ab Ab ð1 + γÞ ð1 γÞeiμ 2 Ω2 The solution of this eigenvalue problem for wavenumber varying within the first irreducible Brillouin zone, i.e., μ 2 [0, π] leads to two dispersion branches, which are expressed as: qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Ω ¼ 2 2 + 2γ 2 + 2ð1 γ 2 Þ cosμ where the “” and “+” signs, respectively, define the acoustic and optical branch. The resulting dispersion diagram is presented in Fig. 2A, which compares the case of stiffness parameters γ ¼ 0 (green dotted curves) and γ ¼ 0.4 (black solid curves). For γ 6¼ 0, the lattice has a bandgap whose width is given by pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Ω2 2ð1 jγjÞ, 2ð1 + jγjÞ : Fig. 2B displays the frequencies bounding the bandgap between the acoustic and optical branches as the stiffness parameter γ varies. Note that these bounding frequencies are evaluated for μ ¼ π. The frequency on pﬃﬃﬃ the dashed (red) curve has an eigenvector ðAa , Ab Þ ¼ 1= 2ð1, 1ÞT , which can be characterized as a “symmetric” eigenvector. In contrast, the solid (blue) curve is associated with an antisymmetric eigenvector ðAa ,Ab Þ ¼ pﬃﬃﬃ 1= 2ð1, 1ÞT . Note that these eigenvector get inverted, i.e., the antipﬃﬃﬃ symmetric 1= 2ð1, 1ÞT becomes associated with the higher frequency, as γ varies from a negative value to a positive one. This phenomenon is called band inversion and has been previously exploited in electronic systems (Bernevig, Hughes, & Zhang, 2006; Hasan & Kane, 2010; Pankratov, Pakhomov, & Volkov, 1987) and continuous acoustic ones (Xiao et al., 2015) to obtain localized modes. These modes are localized at the interface of two lattices: one with γ > 0 and the other with γ < 0.

Topologically protected edge states in mechanical metamaterials

A

B

C

D

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Fig. 2 (A) Dispersion relations for lattices with γ ¼ 0 (dotted) and γ ¼ 0.5 (solid). (B) Limits of the bandgap showing band inversion as γ varies. Component 2 of eigenvector A(1)(μ) as μ varies from π to π. (C) γ ¼ 0.5 has a Zak phase π while (D) γ ¼ 0.5 has a zero Zak phase.

2.2 Band inversion and topological invariants To further shed light on the topological properties of the eigensolutions, we examine the eigenvectors of lattices with γ > 0 and γ < 0. In particular, we study how they vary with wavenumber μ over the first Brillouin zone. Observe that the matrix in Eq. (4) gives the same eigenvalues under the transformation γ !γ but the eigenvectors are different. Indeed, note that the transformation γ !γ may be achieved by simply reversing the direction of the lattice basis vector. An alternate way is to simply translate the unit cell by one mass to the right or left and relabel the masses appropriately. Both of these operations correspond to changes in gauge, which modify the

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eigenvectors, thereby changing the topology of the vector bundle associated with the solution of the above eigenvalue problem. We characterize the topology of this vector bundle by evaluating the Zak phase (Zak, 1989) for the bands. This quantity can be considered as a special case of Berry phase (Xiao et al., 2015; Zak, 1989) to characterize the band topology in 1D periodic media. The Zak phase associated with band m is given by: Z πh i Z¼ iðAðmÞ ÞH ðμÞ ∂μ AðmÞ ðμÞ dκ, (5) π

H

where () denotes the Hermitian. For numerical calculations, we use an equivalent discretized form of Eq. (5) given by (Xiao et al., 2015)

N 1 X n+1 H n Zak (6) θ ¼ Im ln Am π Am π : N N n¼N The Zak phase of both the acoustic and optical bands takes the values Z ¼ 0 and Z ¼ π for the lattices with γ > 0 and γ < 0, respectively. Indeed, it should be noted that since the Zak phase is not gauge invariant (Atala et al., 2013), the choice of coordinate reference and unit cell must remain the same for computing this quantity. To understand the meaning of the Zak phase, we show the behavior of the acoustic mode eigenvector for both γ > 0 and γ < 0 lattices. For consistent representation, a gauge is fixed such that the eigenvector has magnitude 1 and its first component is real and positive, ð1Þ

i.e., at zero angle in the complex plane. The second component Ab of the eigenvector is displayed in the complex plane for μ varying from π to π, see Fig. 2C and D. This component of the eigenvector forms a loop as the wavenumber μ varies from π to π. When γ > 0, this eigenvector loop does not enclose the origin and is associated with to a Zak phase equal to 0. On the other hand, the acoustic band of a lattice with γ < 0 has a Zak phase of Z ¼ π and its eigenvector loop Ab(μ) encloses the origin.

2.3 Analytical evaluation of the interface modes Combining to lattices with opposite Zak phase as in Fig. 1 leads to a nontrivial interface, which is associated with modes localized at the interface. We seek the values of the frequencies for which the above linear chain admits a localized mode solution at the interface and derive explicit expressions for their values and for their corresponding mode shapes. For this purpose, we consider a finite lattice having N unit cells on either side of the

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interface, with N large enough so that boundary effects are negligible in the dynamics of the interface mass. The unit cells are indexed from p ¼ N to N, with the interface mass being at position p ¼ 0. We investigate the dynamics of this lattice in the bandgap frequencies. We thus impose a solution of the form up(t) ¼ upeiΩτ, where p denotes the cell index, and where pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Ω 2 ½ 2ð1 jγjÞ, 2ð1 + jγjÞ or Ω > 2: A similar solution is also imposed on the interface mass. To relate the displacements in two neighboring cells p 1 and p on the right of the interface (p > 0), we rewrite the governing equations for the masses at the lattice sites bp1 and ap as ð2 Ω2 Þua, p ð1 + γÞub, p ð1 γÞub, p1 ¼ 0

(7a)

ð2 Ω Þub, p1 ð1 γÞua, p ð1 + γÞua, p1 ¼ 0

(7b)

2

Rearranging the terms in the above equation yields a relation between the displacements of adjacent unit cells on the right side of the interface. In nondimensional form, this relation is expressed using a transfer matrix T as 0 1 γ+1 2 Ω2 B γ 1 C ua ua u 1 γ C ¼B ¼T a : ub p @ 2 Ω2 ð2 Ω2 Þ2 ðγ 1Þ2 A ub p1 ub p1 1γ 1 γ2 (8) From the above relation, the displacement at unit cell p ¼ N may be written in terms of the displacement at the interface unit cell (p ¼ 0), as uN ¼TNu0. Note that the vector u0 has components u0 ¼ (uc,0, ub,0)T. We now solve for the frequencies and corresponding mode shapes at which this lattice supports localized modes. We seek solutions which are localized at the interface and decay away from it, i.e., kuNk! 0 as N becomes large. The solution procedure involves seeking eigensolutions of the transfer matrix T which satisfy the decay condition. For a mode localized at the interface, the displacement should decay away from the interface, i.e., kuNk! 0 as N ! ∞. To make further progress, we let T be diagonalizable and (λi, ei) be its eigenvalue-vector pairs. Then, kTNuk! 0 as N ! ∞ with a

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nontrivial solution u 6¼ 0 if and only if u is in the subspace spanned by the eigenvectors ei whose corresponding eigenvalues satisfy jλij < 1. To prove this statement, let us denote by fi the subset of eigenvectors of T with associated eigenvalues jλij < 1 and gj the eigenvectors with jλjj 1. P P If u ¼ αi f i , then T N u ¼ αi λi N f i and hence its norm goes to zero as N increases. We prove the “only if” part by contradiction. PAssumePthat u is not in the fi subspace as required. We may write u ¼ i αi f i + j βj gj . P P Then T N u ¼ i αi λi N f i + j βj λj N gj . Since there is a nonzero βj by assumption, the norm of this vector does not converge to 0 as N ! ∞, which completes the proof. Note that the product of the eigenvalues of the transfer matrix T is unity since detðTÞ ¼ 1. In the bandgap frequencies, the eigenvalues of T are real and distinct, hence exactly one eigenvalue satisfies jλij < 1. The eigenvector corresponding to this eigenvalue is ! 2 ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2Þ2 ð1 + γÞ 2ðΩq e¼ : (9) ðΩ2 2Þ2 + 4γ + Ω ðΩ2 4ÞððΩ2 2Þ2 4γ 2 Þ The statement above implies that a localized mode arises if the displacement u is a scalar multiple by the eigenvector e, i.e., e ¼ su0, with s being a scaling factor and u0 ¼ (uc,0, ub,0) having the displacement components of the unit cell at the interface. Let us now derive an expression for u0 from the governing equation of the interface mass. It may be rewritten as Ω2 (10) 2 1 uc, 0 ¼ ðub, 0 + ub, 1 Þ: 2ð1 + γÞ Since the localized mode is nonpropagating and the lattices on either side of the interface mass are identical, symmetry conditions lead to the following relation between the masses adjacent to the interface mass jub, 0 j ¼ jub, 1 j:

(11)

The above condition may be rewritten as ub,1 ¼ e ub,0. Substituting this into Eq. (10), the displacement u0 may be written as 0 iθ 1 e cosθ Ω2 A: (12) u0 ¼ @ 1 2ð1 + γÞ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Note that in a bandgap frequency, i.e., for Ω 2 ½ 2ð1 jγjÞ, 2ð1 + jγjÞ or Ω > 2, the argument of the square root in Eq. (9) is positive when Ω is in 2iθ

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the bandgap frequencies and the components of e are real. The condition e ¼ cu0 implies θ ¼ nπ=2,n 2  and eiθ cos θ 2 f0,1g. Applying this condition (e1/uc,0 ¼ e2/ub,0 ¼ c) to the two cases separately allows us to solve for the frequencies Ωi of the localized modes. Specifically, θ ¼ π/2 leads pﬃﬃﬃ to Ω ¼ 2, while θ ¼ 0 leads to the following equation ðΩ2 2ð1 + γÞÞðΩ2 2ð1 γÞÞðΩ2 4Þ ¼ 4γ 2 Ω2 :

(13)

Note that θ ¼ 0 implies ub,0 ¼ ub,1. From the transfer matrix expression, we note that the mode shape is indeed antisymmetric about the interface mass. In contrast, θ ¼ π/2 results leads to ub,0 ¼ ub,1 and uc,0 ¼ 0. In this case the mode shape is symmetric about the interface mass. Eq. (13) leads to the following expressions for the frequencies which support localized solutions qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ γ < 0 : Ω ¼ 3 1 + 8γ 2 , Antisymmetric mode pﬃﬃﬃ (14) γ > 0 : Ω ¼ q2ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ , Symmetric mode pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ γ > 0 : Ω ¼ 3 + 1 + 8γ 2 , Antisymmetric mode: Substituting the frequencies Ω into the eigenvectors in Eq. (9), taking appropriate signs under the square root and checking the condition e ¼ cu0 show that the first solution is valid when γ < 0, and the other two solutions are valid when γ > 0. The displacement components of the interface unit cell for these localized modes are given by uc, 0 e1 e¼ ¼ : (15) u b, 0 e2 from which the displacement up of unit cell p can be obtained by using the relation up ¼Tpu0. Note that the first and second solutions give frequencies which are localized in the bandgap between the acoustic and optical branches, while the third frequency is above the optical branch. Furthermore, the first and third frequencies are associated with antisymmetric mode shapes where the unit cells on both sides of the interface are in phase, while the second frequency is associated with a symmetric mode shape, with the interface mass being at rest, while the unit cells on both sides have a phase difference of π.

2.4 Harmonic response of lattice with interface We verify our analytical predictions by numerically computing the forced harmonic response of a finite chain having 60 unit cells with an interface

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at the center, see Fig. 1. The stiffness parameter is set to γ ¼ 0.4. The governing equations for the lattice may be written in matrix form as M q€ðτÞ + KqðτÞ ¼ f ðτÞ. We seek the forced vibration response of the lattice when subjected to an external force f cosΩτ. Imposing a solution ansatz of the form q(τ) ¼ qeiΩτ, the governing equation reduces to (16) K Ω2 M q ¼ f : Fig. 3A displays the natural frequencies Ω of this chain, obtained by solving the eigenvalue problem that arises by setting f ¼ 0. The figure illustrates A

B

Fig. 3 (A) Natural frequencies of a finite chain exhibiting interface modes (red circles) in the bandgap frequencies. (B) Frequency response function (red, solid) showing the interface mode within the bandgap, in agreement with analytical predictions (blue, dotted). Interface modes are absent in a regular chain with all identical unit cells (black, dashed).

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the presence of a bandgap between the acoustic and optical modes. Furthermore, there is an interface mode in the bandgap at frequency pﬃﬃﬃ Ω ¼ 2, which matches exactly with the analytical solution of Ωi for γ > 0 in Eq. (14). Analogous results are obtained for the lattice with γ < 0, consistent with the analytical expressions for the localized mode frequencies and shapes. To illustrate the dynamic behavior of this lattice, we compute the harmonic response by imposing a displacement ub, 30 ¼ cos ðΩτÞ to the mass at the left boundary p ¼ N. The other end of the chain is free and the frequency response is normalized to the excitation amplitude, which is unity in our study. We also consider a lattice that has no interface and comprises 60 identical unit cells (regular chain), in which edge modes are not expected. Fig. 3B displays the displacement amplitude of the center mass for both lattices obtained by solving Eq. (16) with appropriate displacement boundary conditions and for f ¼ 0 over the frequency range comprising the bandgap. In the bandgap frequency range, the regular lattice with all identical unit cells does not support any localized modes (black dashed line), whereas an interface leads to the localized mode (red solid line) at the frequency predicted by the analytical solution (Eq. (14)—blue dotted line).

3. Interface modes in a discrete two-dimensional hexagonal lattice We now extend the study to a two-dimensional (2D) discrete lattice, whereby topologically protected edge modes in a nontrivial bandgap are obtained by breaking inversion symmetry within the unit cell. The considered hexagonal lattice consists of point masses at nodes connected by linear springs (Fig. 4A). Each unit cell contains two different masses, respectively, equal to ma ¼ (1 + β) m and mb ¼ (1 β) m, so that inversion symmetry of the lattice is broken when β 6¼ 0, while C3 symmetry (rotation by 2π/3) is always preserved. Each mass has one degree of freedom corresponding to its out-of-plane motion, while the springs provide a force proportional to the relative motion of connected masses through a constant k. The governing equations for the masses in unit cell p, q are ma u€ap, q + k 3uap, q ubp, q ubp, q1 ubp1, q ¼ 0, (17a) mb u€bp, q + k 3ubp, q uap, q uap, q + 1 uap + 1, q ¼ 0: (17b)

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A

mb ma

B 2.5 2

Ω

1.5 1 M

0

K

Γ

0.5

M

κ

K

Γ

Fig. 4 (A) Schematic of a 2D hexagonal lattice having distinct masses in a unit cell, resulting in broken inversion symmetry, but preserved C3 symmetry. (B) Dispersion diagrams along the IBZ for unit cell with equal masses (β ¼ 0) (dashed lines) and unit cell having dissimilar masses (solid lines) (β ¼ 0.2). A bandgap opens due to broken inversion symmetry in the latter case.

3.1 Dispersion analysis We seek for plane harmonic waves in the form up,q ¼ u0ei(ωt+κrp,q), where rp,q ¼ pa1 + qa2 defines the position of the cell p, q in terms of the lattice vectors a1 ¼ a½1 0,a2 ¼ a½ cos ðπ=3Þ, sinðπ=3Þ (where a ¼ 1 for simplicity), while κ ¼ κ 1g1 + κ 2g2 is the wave vector expressed in the basis of the reciprocal lattice vectors g1, g2. Substituting this expression into the governing equations leads to the following eigenvalue problem to be solved in terms of frequency for an assigned wave vector κ

Topologically protected edge states in mechanical metamaterials

Ω

2

1+β 0 0 1β

161

ua 3 1 eiκ a1 eiκ a2 ua ¼ : ub ub 1 eiκ a1 eiκ a2 3 (18)

where the nondimensional frequency Ω2 ¼ ω2m/k is introduced for convenience. The equivalence with lattices which exhibit edge modes in quantum, photonic or acoustic systems (Chen, Zhao, Chen, & Dong, 2017; Lu et al., 2016; Ma & Shvets, 2016; Xiao et al., 2007), can be illustrated by considering the change of variables v ¼ Pu, where u ¼ [ua, ub], and pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

1 + β pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 0 , (19) P¼ 0 1β which effectively corresponds to a stretching of coordinates. Premultiplying both sides of the eigenvalue problem in Eq. (18) by P1gives HðκÞvðκÞ ¼ P 1 KðκÞP 1 vðκÞ ¼ Ω2 vðκÞ: where matrix H(κ) can be written as " pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ # 3=1 + β d ðkÞ= 1 β2 HðkÞ ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 3=1 β dðkÞ= 1 β2

1 0 1 0 0 d ðkÞ 6 1 6β p ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ + , ¼ 1 β2 0 1 1 β2 0 1 1 β2 dðkÞ 0 (20) where d(k) ¼ 1 eiκa1 eiκa2, and d*(k) is the complex conjugate. The first term in the above expression is a constant times the identity matrix. Its sole effect is to translate the dispersion bands upward or downward, without affecting their topology. The second term is similar to the effective mass Hamiltonian of graphene (Kane & Mele, 2005) and leads to a Dirac cone in the absence of additional interaction terms. Finally, the last term is the result of breaking the inversion symmetry and vanishes when the masses are equal (β ¼ 0). Fig. 4B displays the dispersion diagram for two types of unit cells along the edges of the irreducible Brillouin zone (IBZ), sketched as inset in the figure. The dispersion diagram of a unit cell with identical masses β ¼ 0, i.e., ma ¼ mb ¼ 1, is represented by the dashed lines. A Dirac cone is

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observed at the K point, which features sixfold symmetry. The solid curves correspond instead to the dispersion diagram for a lattice with β ¼ 0.2 (ma ¼ 0.8 and mb ¼ 1.2) for which a bandgap opens at the K point. The dispersion is characterized by broken inversion symmetry, and preserved C3 symmetry. The case obtained with values of masses interchanged, i.e., β ¼ 0.2 (ma ¼ 1.2 and mb ¼ 0.8) would appear identical, although a band inversion would occur with eigenvectors associated with the corresponding frequencies flipped. To illustrate the analogy with systems exhibiting quantum valley Hall effect, the above Hamiltonian is expressed in the basis of an extended vector, combining the displacements at the K and K0 valley points. These points have Dirac cones in the presence of inversion symmetry (β ¼ 0). We consider an extended state vector ψ ¼ [UK, UK0 ] combining both valleys. Let τi and σ i be Pauli matrices associated with the valley and the unit cell degrees of freedom. The effective Hamiltonian near the Dirac points in this extended basis can then be expressed as 1 6 6β p ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ τ0 σ 0 τ0 σ z : δkx τz σ x + δky τ0 σ y + H0 ðδkÞ ¼ 1 β2 1 β2 1 β2 (21) Alternately, the nontrivial nature of the bands given by the solution of the eigenvalue problem in Eq. (18) can be characterized by computing the associated topological invariant, which in this case is the valley Chern number (Ma & Shvets, 2016; Xiao et al., 2007) obtained by integrating the Berry phase over an area equivalent to half of the Brillouin zone, surrounding one of the valleys (K or K0 ). We express the eigenvalue relation in Eq. (18) as Ω2m Mm ¼ Km with Ωm being the frequency associated with the eigenvector m. Note that the eigenvectors are normalized to satisfy P I ¼ jnihMjnj, where the bracket notation hajbi ¼ p a*p bp denotes the inner product of the vectors a, b. The Berry curvature of a band at wave vector κ having eigenvector m is given by B(κ)dkxdky ¼ ihdmjMjdmi, where d is the exterior derivative operator. Differentiating the above eigenvalue relation with respect to κs, with s ¼ {x, y}, and premultiplying by the eigenvector n leads to the following identity: ∂m hnj∂K=∂κs jmi ¼ : njMj (22) ∂κs Ω2m Ω2n We compute the Berry curvature at κ by considering an equivalent expression, given in component form as

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∂m ∂m c:c BðκÞ ¼ ihdmjMjdmi ¼ i jMj ∂κ x ∂κ y N X ∂m ∂m ¼i c:c: jMjn njMj ∂κx ∂κy n¼1, n6¼m ∂K ∂K mj jn nj jm c:c: N X ∂κ x ∂κ y ¼i : 2 2 Ωm Ω2n n¼1, n6¼m

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where c.c. denotes the complex conjugate and N denotes the number of eigenmodes. Note that the summation in the last step reduces to a single term as we only have two bands for the considered discrete hexagonal lattice. Fig. 5 displays the Berry curvature over the entire Brillouin zone. It is localized at the K and K0 points, with opposite signs at those points. Furthermore, since the system is time reversal invariant, the total Berry curvature over the entire band is zero. The valley Chern number is then computed by integrating the Berry curvature over a small region near the K, K 0 point as: Z Cν ¼ ð1=2πÞ BðκÞdκ ν

0

where ν ¼ K, K denotes the valley type. Evaluation of the Chern number reveals that for β > 0, i.e., ma > mb, the lower band is characterized by

Fig. 5 Berry curvature over the first Brillouin zone (dashed line) is localized at the K and K0 points. It has opposite signs at these points.

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Cν ¼ ()1/2 at K (K 0 ) valleys, while opposite signs are found for the upper band. In contrast, the values are reversed, i.e., Cν ¼ ()1/2 at K 0 (K) valleys for the lattice with β < 0, i.e., ma < mb. This demonstrates how breaking inversion symmetry by varying the relative masses of the two resonators provides distinct valley Chern numbers for the bands. As discussed in numerous studies on quantum and photonic systems, at the interface between two lattices with distinct valley Chern numbers, bulk boundary correspondence guarantees the presence of topologically protected localized modes (Xiao et al., 2007).

3.2 Dispersion analysis of a finite strip and transient simulations Next, we analyze the dynamic behavior at an interface between two hexagonal lattices with mass parameters + β and β. The two lattices thus have the same bulk band structure. Similar to the 1D case, we label a unit cell of type A or B when β > 0 or β < 0, respectively. Fig. 6A displays a schematic A

Interface

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Fig. 6 (A) Schematic of unit cell considered for the dispersion analysis of a strip with an “L” interface (light masses are denoted as red, filled circles). (B) Dispersion diagram corresponding to an “L” interface and (C) to an “H” interface (bulk modes—solid black lines, interface edge modes—dashed red lines, localized modes at the ends of the strip—blue solid line).

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of part of the strip, along with the interface and a part of the unit cell. The considered strip is infinite along a2 and finite along a1 with 16 unit cells of each type on each side of the interface. As in the 1D case, two types of interfaces can be constructed: one that connects two light masses, and one that consists of two heavy masses. These two interfaces are denoted as “light” (L) and “heavy” (H), respectively. An example of an “L” interface is illustrated in Fig. 6A, where the light masses are denoted as solid red circles, while the heavy masses are denoted as empty circles. The band structure is evaluated by constraining the displacement of the masses at the left and right boundaries. Fig. 6B displays the band diagrams for an “L” interface and for ma ¼ 1.0 and mb ¼ 1.5 (or equivalently, m ¼ 1.25 and β ¼ 0.2). The diagram features two sets of bulk modes (black solid lines), along with two sets of modes within the bulk bandgap. The modes within the bandgap are localized modes either at the fixed ends or at the interface. The type of boundary where the mode localizes can be determined by examining the corresponding eigenvectors, which decay rapidly away from the boundary. The blue (thick solid) curve is associated with two overlapping frequencies corresponding to a localized mode at each end of the strip. Note that the two ends are locally identical, which results in this degeneracy of localized modes. The branch denoted by the dashed red line is a single mode localized at the interface. In addition, a third branch is observed above the bulk optical band, also represented by a red dashed line in the figure, which is also localized at the interface. However, note that it is hard to excite this mode in practice as the frequency it spans also has a wide spectrum of bulk bands. We now consider a strip characterized by a “H” type interface, i.e., with two heavy masses adjacent to each other at the interface. The mass m is kept the same, while β is reversed in sign compared to the previous case. Fig. 6C displays the dispersion diagram for this interface, having a different set of localized modes within the bulk bandgap. Here the mode localized at the interface, shown again in dashed red (light) color starts from the bulk acoustics band, in contrast with the previous case. Furthermore, the two localized modes at the boundaries also have different frequencies, as there are now light masses at each boundary. Note that there is an additional interface localized mode which now appears below the acoustic band. To verify the observations based on the above dispersion analysis, we conduct transient simulations on a finite lattice of 32 32 unit cells. Cell types A and B are separated by a zig-zag interface of the “L” type (see Fig. 7A). The excitation is a 30-cycles sinusoidal force of frequency Ωe ¼ 1.5, modulated by a Hanning window and applied to one of the interface masses along the lower

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Fig. 7 (A) Lattice schematic. Transient simulation illustrating backscattering suppressed wave propagation along a zig-zag channel at 3 time instants: (B) t ¼ 106, (C) t ¼ 166, and (D) t ¼ 240.

right boundary identified by an arrow in the schematic. The response of the lattice is evaluated through numerical integration of the equation of motion for the finite system considered. Fig. 7 displays the amplitude of the displacements in the lattice at three distinct time instants. Initially, at t ¼ 106 in Fig. 7B, the wave travels along the straight portion of the interface and does not propagate into the interior or along the boundaries of the lattice. This solution is consistent with the dispersion analysis which predicts only a localized interface mode at frequency Ωe. As time progresses (t ¼ 166 in Fig. 7C),

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the wave bends around the zig-zag edges without any backscattering. The wave is immune to localization and experiences negligible backscattering even as it navigates multiple bends as clearly shown from the displacement amplitude contours at t ¼ 240 (Fig. 7D). After the wave hits the lattice boundary, it reflects and traverses in the opposite direction, as it is not immune to backscattering at the boundary, where mode hybridization occurs. We note that this response is consistent with the behavior observed for other classical analogues of the quantum Hall effect, which are also immune to backscattering only in the presence of a certain class of defects that do not cause the modes at the two valleys to hybridize.

4. Topological valley modes in a continuous elastic hexagonal lattice We finally report on the experimental observation of topologically protected edge waves in a continuous 2D elastic hexagonal lattice. The lattice is designed to feature K point Dirac cones that are well separated from the other numerous elastic wave modes characterizing this continuous structure. We exploit the arrangement of localized masses at the nodes to break mirror symmetry at the unit cell level, which opens a frequency bandgap. The resulting nontrivial band structure supports topologically protected edge states along the interface between two realizations of the lattice obtained through mirror symmetry. Detailed numerical models support the investigations of the occurrence of the edge states, while their existence is verified through full-field experimental measurements. The test results show the confinement of the topologically protected edge states along predefined interfaces and illustrate the lack of significant backscattering at sharp corners. Experiments conducted on a trivial waveguide in an otherwise uniformly periodic lattice reveal the inability of a perturbation to propagate and its sensitivity to backscattering, which suggests the superior waveguiding performance of the class of nontrivial interfaces investigated herein.

4.1 Configuration and material properties The characteristics of the conceptual lattice summarized above guide the design of the continuous hexagonal elastic lattice of Fig. 8. The lattice is fabricated out of a square acrylic panel of side 308.4 mm and thickness of 1.59 mm. The side of the each hexagon measures L ¼ 10.7 mm, while the width of the beams is w ¼ 3.2 mm. The masses consist of cylindrical

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Fig. 8 Experimental lattice with added masses at the sublattice sites. Insets show the FE discretization of the unit cell for numerical study for configurations defined by γ ¼ 0 and γ ¼ 1, as well as the first and irreducible Brillouin zone (red line) for the lattice.

nickel-plated neodymium magnets (ρc ¼ 7400 kg/m3, Ec ¼ 41 GPa and νc ¼ 0.28) of height 1.5 mm and diameter 3.2 mm. The material properties of acrylic are: density ρ ¼ 1190 kg/m3, Young’s modulus E ¼ 3.2 GPa, Poisson’s ratio ν ¼ 0.35. The lattice is generated by a set of lattice vectors pﬃﬃﬃ pﬃﬃﬃ pﬃﬃﬃ pﬃﬃﬃ a1 ¼ 3L ½ 3=2, 1=2 and a2 ¼ 3L ½ 3=2, 1=2. Unit cell mirror symmetry is broken by adding cylindrical masses at selected sublattice sites a, b, in analogy with the discrete lattice in Fig. 4A. The mass added by the cylinders is defined, respectively, as ma ¼ ðjγj + γ Þmc , mb ¼ ðjγj γ Þmc , where mc denotes the mass of one cylinder and γ is the parameter chosen to define both magnitude and location of the added mass at each site. Of note is the fact that the lattice is a continuous structure that is characterized by inherent mass properties defined by the density of the material. Therefore the terms ma, mb denote the added mass at the sublattice sites. An even number of cylinders are added at each location in order to preserve symmetry in the thickness direction, and for practical purposes in the experimental implementation of the concept, whereby attracting magnetic cylinders are clamped at the desired location. Hence, values γ > 0 describe the addition of masses at site a, while γ < 0 corresponds to an added mass in b. In addition, the case γ ¼ +1(1) describe the addition of two cylinders in a(b), and finally when γ ¼ 0 corresponds to the case of no additional masses.

4.2 Dispersion analysis 4.2.1 Unit cell The dispersion properties of the lattice are estimated based on the Finite Element (FE) discretization of the lattice modeled as a three-dimensional

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continuous solid. In the model, the motion of each material point within the domain at the generic location x, y, z is governed by the standard equations of linear elasticity for an isotropic medium (Achenbach, 2012), ρ€ u ½ðλ + μÞrðr uÞ + μr2 u ¼ 0,

(25)

where u(x, y, z) ¼ uxi + uzj + uzk is the displacement vector and i, j, k denote the unit vectors along the x, y, z directions. Also, λ, μ are the Lame constants of the solid. For reference, in here and in the remainder of the paper, the x, y directions define the plane of the lattice, while the z direction is the thickness, or out-of-plane, direction. The solid is patterned according to the hexagonal topology considered herein and is discretized for analysis within the COMSOL Multiphysics environment. Each unit cell is discretized using around 20,000 second-order tetrahedral elements which produces the mesh shown in the insets of Fig. 8. Upon discretization, imposing a plane wave solution along with the enforcement of Floquet–Bloch conditions (Hussein, Leamy, & Ruzzene, 2014) to the unit cell degrees of freedom leads to a linear eigenvalue problem that is solved in terms of frequency for a wave vector varying along the edge of the first irreducible Brillouin zone for the lattice under consideration. Results for lattices characterized by γ ¼ 0 and γ ¼ 1 are shown in Fig. 9A and B. The dispersion analysis predicts multiple wave modes that correspond to the numerous degrees of freedom in the considered FE unit cell model. Each node has 3 degrees of freedom, which is reflected by the three branches emanating from the Γ point at zero frequency. In the long wavelength limit, the lattice approaches the behavior of a thin plate, and is characterized by a flexural, or out-of-plane, mode, and two in-plane modes which can be described as “shear” and “longitudinal”-like. Consistently with its flexural nature, the out-of-plane mode is characterized by a parabolic dispersion branch at long wavelengths, and it is loosely coupled with the in-plane modes. This makes its identification based on the eigenvector components relatively simple (see blue, thick line in Fig. 9). In contrast, the modes that are mostly in-plane polarized, i.e., that are associated with eigenvectors where ux, uy ≫ uz (red lines in Fig. 9), are significantly more difficult to differentiate from one another. However, their distinct representation goes beyond the scope of the work, which focuses on the out-of-plane mode. In analogy with the discrete lattice, the hexagonal lattice with no masses attached (γ ¼ 0) has C3v symmetry. In addition, a Dirac point at the frequency identified by the horizontal dashed line is observed for the outof-plane branches at the K point of the reciprocal lattice space (Fig. 9A).

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A

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Fig. 9 Dispersion diagrams: (A) lattice without masses, γ ¼ 0; (B) lattice with masses, γ ¼ 1 (black thin dashed line: location of Dirac point, blue thick lines: out-of-plane wave mode, red thin lines: shear polarized modes). (C) Variation of bandgap bounding frequencies ω+(red line), ω (blue line) at K as a function of γ showing band inversion, and (D) Phase of corresponding eigenfunctions of the first two out-of-plane modes at K for γ ¼ 1 and γ ¼ 1.

Adding two cylindrical masses at the a site (γ ¼ 1) breaks mirror symmetry, and produces a bandgap (see Fig. 9B). The bounding frequencies of the gap, denoted ω+ and ω are tracked as a function of γ, which produces the plot of Fig. 9C, where a band inversion is observed. The eigenfunctions U associated to these eigenvalues for γ ¼ 1 and γ ¼ 1 are depicted in Fig. 9D, which also illustrate the predominantly out-of-plane polarization of these modes. We observe that, while the eigenvalues are preserved under the transformation γ !γ due to time reversal symmetry, the eigenfunctions feature different polarizations that reflect the mirror-symmetry relations of the corresponding unit cells. The transformation γ !γ may be achieved by simply reversing the direction of the lattice basis vectors, so the positions of the sublattice sites a and b are switched. Due to the broken C3v mirror symmetry, a reflection changes the eigenfunctions and thereby the bands topology (Brendel et al., 2018). Let us examine in detail the phase of the

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eigenfunctions U at K point. Note that for γ < 0 the eigenfunction of the first mode has clockwise polarization and the eigenfunction of the second one has counter-clockwise polarization, whereas the opposite is observed for γ > 0. Fig. 9C shows that the bands are inverted when γ changes sign (at γ ¼ 0). Furthermore, the K0 points have opposite polarizations to the K points due to time reversal symmetry. The change in polarization across γ ¼ 0 suggests that lattices with γ > 0 and γ < 0 have opposite valley Chern numbers (F€ osel, Peano, & Marquardt, 2017; Pal & Ruzzene, 2017), and that TPEWs are expected to exist along an interface between a lattice with γ > 0 and a lattice with γ < 0 at frequencies within the common bandgap. 4.2.2 Finite strip The dispersion analysis of a strip including a finite number of cells and an interface is conducted to evaluate the existence of edge and interface modes. The study is based on the FE model previously considered for the unit cell, extended to include the finite strip assembly of Fig. 10A. The strip consists of 10 unit cells of the type γ ¼ 1 and 10 unit cells with γ ¼ 1, which, as discussed in the previous section, are characterized by different topologies. The top and bottom boundaries of the strip are considered free, which is a choice that does not affect the existence of the interface mode. The corresponding dispersion diagram is presented in Fig. 10B, where again, the out-of-plane modes are easily distinguished from the in-plane polarized ones through the evaluation of the displacement components of the eigenvectors, and are A

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denoted by the thick blue lines. In addition to these bulk modes, a mode localized at the interface (black dotted line) appears in the bandgap. One eigenvector corresponding to the interface mode is displayed in Fig. 10C, which shows the displacement field of the edge mode for κ x ¼ 0.7π and frequency f ¼ 3 kHz. The dispersion study shows that in spite of the large number of modes present and of the fact that the bandgap is only associated with the out-of-plane modes, and is therefore only partial, the topological differences between the two lattices still lead to a topologically protected interface modes.

4.3 Experimental results The numerical simulations described in the previous section guide the design and experimental characterization of the considered hexagonal lattice. The lattice is cut out of an acrylic panel according to the dimensions described in Section 4.1. In the experiments, the lattice is held in a vertical position by a vice that clamps its lower left corner. In addition, commercially adhesive putty tape is added along the boundaries for absorption of incoming waves and to minimize reflections which may affect the visualization of the propagation of the interface modes. Wave motion in the lattice is induced by PZT discs bonded at selected locations, and driven by a voltage signal generated by a signal generator upon amplification. The PZT discs are bonded to the top surface of the lattice. When the voltage is applied, they induce a distribution of shear stresses at the bonded interface, which generates both the in-plane and out-of-plane motion of the lattice. Full-field response of the lattice is recorded through a scanning Laser Doppler vibrometer (SLDV), which measures the out-of-plane velocity of points belonging to a predefined measurement grid. Given the SLDV limitation to measurements of the out-of-plane motion only, no contribution from in-plane modes is observed in the measured responses. While the equipment records one point at a time, repeating the excitation to record the response at every measurement location and the tracking of the phase between subsequent measurements allow the recording of the full-field wave motion of the lattice. The measurements include 7 points along the side L of the hexagon, so that a total of 3670 points are recorded over the entire lattice. After recording, the wavefield data are interpolated on a regular rectangular grid that includes 100 points along the horizontal (x) and vertical (y) extent of the measurement domain. The excitation consists of broadband frequency pulses that cover the frequency range of interest, which is up to 12 kHz.

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This is achieved through modulated sinusoidal pulses and their superposition, or half-cycle pulses whose duration defines the frequency bandwidth of each individual excitation. 4.3.1 Estimation of the dispersion properties The measurement and their subsequent interpolation produce a data set in the form of a matrix w(x, y, t) that describes the evolution of the deflection of the lattice in time. The matrix of the experimental results is analyzed in Fourier space by performing a three-dimensional Fourier transformation (3D-FT), which gives (Michaels, Michaels, & Ruzzene, 2011): wðκ ^ x ,κ y ,ωÞ ¼ F 3D ½wðx, y, tÞ The resulting quantity describes the spectral content, both in terms of frequency as well as reciprocal space, of the recorded wavefield. Cross sections along defined wave paths C, i.e., wðκj ^ C , ωÞ, illustrate the spectral content as a function of frequency for the wave vector varying along specific lattice directions, while evaluation at one frequency ω0, i.e., wðκ ^ x ,κ y ,ω0 Þ, illustrates the distribution of energy in the reciprocal wavenumber space at that frequency. These maps provide direct visualization of the dispersion characteristics of the domain of interest and are therefore used to validate the numerical predictions for the lattice under consideration. We first verify experimentally the dispersion diagrams of the acrylic hexagonal lattice with γ ¼ 0 configuration (no masses attached). The elastic lattice is excited at its center using a PZT disc that applies a pulse of 40 μs to excite frequencies up to 12.5kHz. Fig. 11A displays the magnitude jwðκ ^ x ,κ y ,ω0 Þj of the 3D-FT at a frequency of f0 ¼ ω0/(2π) 3.75 kHz, which is close to the Dirac cone frequency identified by the numerical study of dispersion (see Fig. 9A). The contours correspond to the magnitude jwj, ^ which is not of particular interest here. Most relevant is their location: they are localized at the high symmetry points and effectively illustrate a condition that defines a Dirac point. For reference, the boundaries of the first irreducible Brillouin zone are shown along with the points defining its boundary. pﬃﬃﬃ The size of the zone is defined by a lattice vector a ¼ 3L 18:4 mm, which corresponds to the magnitude of the wave vector at the K point of κ 226 rad/m. Next, results are presented in terms of frequency/wavenumber content by considering a cross section of the 3D-FT along the path C : Γ K Γ for γ ¼ 0 (no masses added). Fig. 11B illustrates the dispersion branches detected during the experiments, which compare very well with the

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COMSOL predictions (red solid line). The case of γ ¼ 1 is then tested and Fig. 11C displays the results in the frequency/wavenumber domain. An opening of the bandgap at the K point is observed as predicted by the COMSOL simulations, again represented by the solid red line superimposed to the contours. 4.3.2 Experimental observation of topologically protected interface waves The dispersion studies and the experimental setup developed allow the investigation of the existence of topologically protected modes at the interface of lattices consisting of unit cells that are inverted copies of each other (γ ¼ 1 and γ ¼ 1). This is easily achieved by placing magnetic cylinders as added masses at the selected locations, so that a variety of interfaces can be introduced and tested.

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Fig. 12 (A) Line interface (red dots indicate two masses attached at the sublattice sites and cyan dots denote locations where no masses are added) with γ ¼ ()1 on the left (right). (B) and (C) Snapshots of measured wave motion at two instants of time. Excitation is a 11 cycles tone burst at 3 kHz (contours are normalized by the displacement amplitude of 3 108 m).

We first investigate the straight line interface shown in Fig. 12A. The cylindrical masses are placed so that the unit cells to the left of the interface have γ ¼ 1 and those to the right have γ ¼ 1. Note that this interface has zero width, and the green lines in the figure only indicate the path that TPEWs are expected to follow. The structure is excited at the location shown with a tone burst signal of 11 cycles at a frequency of 3 kHz. The considered signal has a bandwidth of approximately 1 kHz, and therefore excites a relatively broad frequency range which falls entirely within the bandgap. Fig. 12B and C display time snapshots of the measured

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out-of-plane displacement by plotting the contours of the interpolated wavefield. As indicated above, the displacement field shown is an interpolation over a rectangular grid of the measured response in points belonging to the lattice. This produces a continuous representation which facilitates visualization of the results along with the overlay of the geometry of the lattice represented as thin black lines. The contours are normalized by the maximum displacement amplitude of 3 108 m. The figures illustrate how the induced out-of-plane wave travels along the interface, and has limited penetration into the bulk. The results also show that the amplitude decays below 10% of the original value in approximately 4 unit cells, which is a number consistent with observations from similar investigations presented in the literature (Lu et al., 2017). The number of unit cells in our lattice (18 by 16) and their size was set according to convenience of fabrication and testing and is quite limited. Future designs may be scaled in order to include a larger number of units that may lead to a reduction in the lateral spreading of the interface modes. The rate of spatial decay may affect the results in Fig. 12A due to the proximity of the source from the boundary. The excited in-plane waves, not measured in the experiment, do not exhibit a bandgap at the targeted frequencies, and therefore are allowed to travel on both sides of the interface. Of interest is the fact that the interface mode (for out-of-plane waves) is still observable after it reaches the boundary opposite to the excitation location, although the amplitude is reduced by material dissipation that is particularly noticeable in the acrylic substrate utilized for the tests. The choice of acrylic as material for the experiments is driven by considerations of convenience of fabrication and cost. In further studies, the use of metallic lattices, such as aluminum for example, will be considered for investigations of effects such as extent of propagation and attenuation along the interface, and interactions with boundaries and defects. These interactions are not studied as part of this work, but are important aspects of follow-on investigations for the characterization of the robustness of this class of modes. A second example considers a N-shaped zero width interface with segments parallel to the lattice vectors (Fig. 13A). The objective is to observe the behavior of the wave in the presence of 120 degrees corners along the interface. The cylindrical masses are attached so that the unit cells to the left top of the interface have γ ¼ 1 and those to the right bottom have γ ¼ 1. The results in Fig. 13B and C show the propagation of the wave along the N-shaped topological interface, and a limited propagation into the bulk. Furthermore, the wave manages the 120 degrees turn, illustrating the ability

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Fig. 13 (A) A nontrivial N-shaped interface (red dots indicate two masses attached at the sublattice sites and cyan dots denote locations where no masses are added) with γ ¼ () 1 on the left (right). (B) and (C) Snapshots of measured wave motion at two instants of time. Excitation is a 11 cycles tone burst at 3 kHz (contours are normalized by the displacement amplitude of 1.5 108 m).

to change direction with limited backscattering. The contours are normalized by the displacement amplitude 1.5 108 m. We compare the results above to the case of a waveguide obtained by removing a unit cell from an otherwise periodic domain, which leads to a trivial (nontopological) interface. To this end, masses are attached so that all unit cells have γ ¼ 1. The N-interface within one lattice type is thus obtained by simply removing a line of masses, as illustrated in Fig. 14A, which corresponds to a nonzero width interface. The resulting lattice

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Fig. 14 (A) N-shaped trivial interface (red dots indicate two masses attached at the sublattice sites and cyan dots denote locations where no masses are added) with γ ¼ 1 everywhere. (B) and (C) Snapshots of measured wave motion at two instants of time. Excitation is a 11 cycles tone burst at 3 kHz (contours are normalized by the displacement amplitude of 1.5 108 m).

response in Fig. 14B and C, illustrates the limited ability of the wave to enter and propagate along the interface. The induced motion of the lattice appears to remain localized in the vicinity of the excitation point, and eventually decays as a result of material dissipation. Comparing the amplitude of transmitted waves, we conclude that the amount of energy traveling through the interface is much lower in this case, in spite of the nonzero width of the interface. This test is here based on the fact that for this nonzero width interface, the dispersion properties are those of a lattice without added masses for

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which the flexural bandgap does not exist. Thus the goal is here to illustrate that removal of a single row from a periodic assembly is not sufficient to guarantee propagation.

5. Conclusions This chapter illustrates how localized modes can be induced at the interface or boundaries of both one and two-dimensional lattices. In the one-dimensional case, we consider a lattice of point masses connected by alternating springs. We show that a mode exists in the bandgap and it is localized at the interface between two lattices which are inverted copies of each other. We derive explicit expressions for the frequencies of the localized modes for various interface types and their associated mode shapes. The chapter also illustrates the existence of interface modes within the bandgap of a two-dimensional elastic hexagonal lattice and the propagation of TPEWs exploiting a mechanical analogue of the quantum valley Hall effect. This phenomenon allows creating a simple and robust waveguide for elastic waves in a wide range of frequencies. Guided by studies on conceptual lattices and numerical simulations, experiments are presented that predict the dispersion properties of the considered hexagonal lattices and to explore the existence of TPEWs along predefined interfaces. The difference in propagation along nontrivial interfaces is also illustrated through an experiment that reveals the difference of modes of propagation endowed with topological protection from those that are obtained by introducing a line defect in an otherwise periodic assembly. The experimental configurations illustrated herein are suitable for potential implementation of the concept to phononic systems and structural components, and could be further utilized to investigate the sensitivity of these configurations to a variety of defect and interface configurations. The presented results suggest the application of these concepts in several technological fields of interest, including mass and damage detection, energy harvesting, vibration isolation, and information transmission and processing.

Acknowledgments The authors are indebted to the US Army Research Office (Grant number W911NF1210460), the US Air Force Office of Scientific Research (Grant number FA9550-13-1-0122) and the National Science Foundation (EFRI Award 1741685) for the financial support.

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Topologically protected edge states in mechanical metamaterials

Topologically protected edge states in mechanical metamaterials

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