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We investigate the existence of higher order topological localized modes in moir\'{e} lattices of bilayer elastic plates. Each plate has a hexagonal array of discrete resonators and one of the plates is rotated an angle ($21.78^\circ$) which results in a periodic moir\'{e} lattice with the smallest area. The two plates are then coupled by inter-layer springs at discrete locations where the top and bottom plate resonators coincide. Dispersion analysis using the plane wave expansion method reveals that a bandgap opens on adding the inter-layer springs. The corresponding topological index, namely fractional corner mode, for bands below the bandgap predicts the presence of corner localized modes in a finite structure. Numerical simulations of frequency response show localization at all corners, consistent with the theoretical predictions. The considered continuous elastic bilayered moir\'{e} structures opens opportunities for novel wave phenomena, with potential applications in tunable energy localization and vibration isolation.

We report the experimental observation of an elastic bound mode in the continuum (BIC) in a compact region of an architected beam. We consider a long slender beam with rigid masses attached at periodic intervals, with a compact segment bounded by four protruding side beams. The key idea is to seek a mode where the side beams move out-of-phase with the compact region, thereby nullifying the forces and moments outside this region and resulting in a bound mode. The structure is modeled using Euler-Bernoulli beam theory and the side beams are designed by imposing equilibrium constraints required for a BIC. Multiple BICs are found in the compact region, and for each BIC, we find a one-parameter family of BIC supporting side beam designs. The predictions are verified by three-dimensional finite element simulations, followed by their experimental observation using laser Doppler vibrometry in a macro-scale structure. Our approach allows to achieve BICs in an arbitrary sized compact region of the architected beam. Our findings may open avenues for confining elastic wave energy in compact regions for applications in sensors and resonators.

We analytically predict and numerically demonstrate the existence of a family of bound modes in the continuum (BICs) in bi-layered spring mass chains. A coupled array of such chains is then used to illustrate transversely bound waves propagating along a channel in a lattice. We start by considering the compact region formed by coupling two spring mass chains with defects and predict bound modes arising due to reflection symmetries in this region. Dispersion analysis of a waveguide consisting of an array of appropriately coupled bi-layered chains reveals the presence of a branch having bound modes in the passband. Finally, detailed numerical analyses verify the existence of a BIC and its propagation through the waveguide at passband frequencies without energy leakage. The framework allows to achieve BICs and their propagation, for any arbitrary size and location of the compact region. Such BICs open avenues for novel classes of resonators with extremely high $Q$ factors due to zero energy leakage and allow for guiding confined waves in structures without requiring bandgaps.

The introduction of structural defects in otherwise periodic media is well known to grant exceptional space control and localization of waves in various physical fields, including elasticity. Despite the variety of designs proposed so far, most of the approaches derive from contextual modifications that do not translate into a design paradigm due to the lack of a general theory. Few exceptions include designs endowed with topological dispersion bands, which, however, require changes over substantial portions of the structure. To overcome these limitations, here we introduce a new rationale based on real-space topology to achieve localized modes in continuous elastic media. We theoretically predict and experimentally demonstrate the spectral flow of a localized mode across a bulk frequency gap by modulating a single structural parameter at any chosen location in the structure. The simplicity and generality of this approach opens new avenues in designing wave-based devices for energy localization and control.

We investigate a family of quasiperiodic continuous elastic beams, the topological properties of their vibrational spectra, and their relation to the existence of localized modes. We specifically consider beams featuring arrays of ground springs at locations determined by projecting from a circle onto an underlying periodic system. A family of periodic and quasiperiodic structures is obtained by smoothly varying a parameter defining such projection. Numerical simulations show the existence of vibration modes that first localize at a boundary, and then migrate into the bulk as the projection parameter is varied. Explicit expressions predicting the change in the density of states of the bulk define topological invariants that quantify the number of modes spanning a gap of a finite structure. We further demonstrate how modulating the phase of the ground springs distribution causes the topological states to undergo an edge-to-edge transition. The considered configurations and topological studies provide a framework for inducing localized modes in continuous elastic structural components through globally spanning, deterministic perturbations of periodic patterns defined by the considered projection operations.

Topological protection offers unprecedented opportunities for wave manipulation and energy transport in various fields of physics, including elasticity, acoustics, quantum mechanics and electromagnetism. Distinct classes of topological waves have been investigated by establishing analogues with the quantum, spin and valley Hall effects. We here propose and experimentally demonstrate the possibility of supporting multiple classes of topological modes within a single platform. Starting from a patterned elastic plate featuring a double Dirac cone, we create distinct topological interfaces by lifting such degeneracy through selective breaking of symmetries across the thickness and in the plane of the plate. We observe the propagation of a new class of heterogeneous helical-valley edge waves capable of isolating modes on the basis of their distinct polarization. Our results show the onset of wave splitting resulting from the interaction of multiple topological equal-frequency wave modes, which may have significance in applications involving elastic beam-splitters, switches, and filters.

We investigate the dispersion topology of elastic lattices characterized by spatial stiffness modulation. The modulation is defined by the sampling of a two-dimensional surface, which provides the lattices with topological properties that are usually attributed to two-dimensional crystals. We show that the cyclic variation of the phase of the stiffness modulation leads to a Berry phase accumulation for the Bloch eigenmodes, which is characterized by integer valued Chern numbers for the bands and associated gap labels. The resulting non-trivial gaps are spanned by edge modes localized at one of the boundaries of the considered 1D lattices. The edge mode location is governed by the phase of the stiffness distribution, whose spatial modulation drives these modes to transition from one edge to the other, through a bulk state that occurs when the corresponding dispersion branch touches the bulk bands. This property enables the implementation of a topological pump that is obtained by stacking and coupling a family of modulated 1D lattices along a second spatial dimension. The gradual variation of the stiffness phase modulation drives the adiabatic transition of the edge states, which transition from their localized state at one boundary, to a bulk mode and, finally, to another localized state at the opposite boundary. Similar effects were illustrated under the assumption of quasiperiodic modulation of the lattice interactions. We here demonstrate that a topological pump can be achieved also in periodic media, and illustrate this for the first time in elastic discrete lattices.

This paper investigates the dynamic properties of one, two and three-dimensional tensegrity-based periodic structures introduced in Rimoli and Pal, Comp. B, 2017, which are here termed as tensegrity beams, plates and solids, respectively. We study their linear wave propagation properties and show that in each case, these properties can be significantly altered by the prestrain in the cables. As the prestrain is varied, we observe jumps in the wave velocities at two critical prestrain values, which define transitions between the three distinct phases of these structural assemblies. At low cable prestrains, the wave speeds are zero as the lattices have zero effective stiffness. At moderate prestrains, the wave speed is nonzero and finally, at prestrain levels where the bars buckle, the wave speed decreases to a lower value. Dispersion analysis on these beams, plates and solids reveal unique properties such as very low wave velocities compared to their constituent material and the existence of flat bands at low frequencies. Furthermore, we find that shear waves travel faster than longitudinal waves in tensegrity solids in a range of cable prestrains. Finally, we verify the key observations through detailed numerical simulations on finite tensegrity solids., Comment: 11 pages, 6 figures

The paper investigates localized deformation patterns resulting from the onset of instabilities in lattice structures. The study is motivated by previous observations on discrete hexagonal lattices, where the onset of non-uniform, quasi-static deformation patterns was associated with the loss of convexity of the interaction potential, and where a variety of localized deformations were found depending on loading configuration, lattice parameters and boundary conditions. These observations are here conducted on other lattice structures, with the goal of identifying models of reduced complexity that are able to provide insight into the key parameters that govern the onset of instability-induced localization. To this end, we first consider a two-dimensional square lattice consisting of point masses connected by in-plane axial springs and vertical ground springs. Results illustrate that depending on the choice of spring constants and their relative values, the lattice exhibits in-plane or out-of plane instabilities leading to folding and unfolding. This model is further simplified by considering the one-dimensional case of a spring-mass chain sitting on an elastic foundation. A bifurcation analysis of this lattice identifies the stable and unstable branches and illustrates its hysteretic and loading path-dependent behaviors. Finally, the lattice is further reduced to a minimal four mass model which undergoes a folding/unfolding process qualitatively similar to the same process in the central part of a longer chain, helping our understanding of localization in more complex systems. In contrast to the widespread assumption that localization is induced by defects or imperfections in a structure, this work illustrates that such phenomena can arise in perfect lattices as a consequence of the mode-shapes at the bifurcation points., Comment: 28 pages, 13 figures

This work investigates the effect of nonlinearities on topologically protected edge states in one and two-dimensional phononic lattices. We first show that localized modes arise at the interface between two spring-mass chains that are inverted copies of each other. Explicit expressions derived for the frequencies of the localized modes guide the study of the effect of cubic nonlinearities on the resonant characteristics of the interface which are shown to be described by a Duffing-like equation. Nonlinearities produce amplitude-dependent frequency shifts, which in the case of a softening nonlinearity cause the localized mode to migrate to the bulk spectrum. The case of a hexagonal lattice implementing a phononic analogue of a crystal exhibiting the quantum spin Hall effect is also investigated in the presence of weakly nonlinear cubic springs. An asymptotic analysis provides estimates of the amplitude dependence of the localized modes, while numerical simulations illustrate how the lattice response transitions from bulk-to-edge mode-dominated by varying the excitation amplitude. In contrast with the interface mode of the first example studies, this occurs both for hardening and softening springs. The results of this study provide a theoretical framework for the investigation of nonlinear effects that induce and control topologically protected wave-modes through nonlinear interactions and amplitude tuning., Comment: 17 pages, 13 figures

A long standing postulate in crystal plasticity of metals is that yielding commences once the resolved shear stress on a slip plane reaches a critical value. This assumption, known as Schmid law, implies that the onset of plasticity is independent of the normal stress acting on the slip plane. We examine the validity of this assumption in single crystal perfect lattices at zero temperature by subjecting them to a wide range of combined normal and shear stresses and identifying the onset of plasticity. We employ phonon stability analysis on four distinct single crystal metals and identify the onset of plasticity with the onset of an instability. Our results show significant dependence of yielding on the normal stress, thereby illustrating the necessity of considering non-Schmid effects in crystal plasticity. Finally, contrary to the common assumption that instabilities in single crystals are of long wavelength type, we show that short wavelength instabilities are abundant in the nucleation of defects for a wide range of loading conditions.

We present a procedure for the systematic estimation of the dispersion properties of linear discrete systems with periodic time-varying coefficients. The approach relies on the analysis of a single unit cell, making use of Bloch theorem along with the application of a harmonic balance methodology over an imposed solution ansatz. The solution of the resulting eigenvalue problem is followed by a procedure that selects the eigen-solutions corresponding to the ansatz, which is a plane wave defined by a frequency-wavenumber pair. Examples on spring-mass superlattices demonstrate the effectiveness of the method at predicting the dispersion behavior of linear elastic media. The matrix formulation of the problem suggests the broad applicability of the proposed technique. Furthermore, it is shown how dispersion can inform about the dynamic behavior of time-modulated finite lattices. The technique can be extended to multiple areas of physics, such as acoustic, elastic and electromagnetic systems, where periodic time-varying material properties may be used to obtain non-reciprocal wave propagation.

We investigate elastic periodic structures characterized by topologically nontrivial bandgaps supporting backscattering suppressed edge waves. These edge waves are topologically protected and are obtained by breaking inversion symmetry within the unit cell. Examples for discrete one and two-dimensional lattices elucidate the concept and illustrate parallels with the quantum valley Hall effect. The concept is implemented on an elastic plate featuring an array of resonators arranged according to a hexagonal topology. The resulting continuous structures have non-trivial bandgaps supporting edge waves at the interface between two media having different topological invariants. The topological properties of the considered configurations are predicted by unit cell and finite strip dispersion analyses. Numerical simulations on finite structures demonstrate edge wave propagation for excitation at frequencies belonging to the bulk bandgaps. The considered plate configurations define a framework for the implementation of topological concepts on continuous elastic structures of potential engineering relevance., Comment: 28 pages, 13 figures

We propose a framework to realize helical edge states in phononic systems using two identical lattices with interlayer couplings between them. A methodology is presented to systematically transform a quantum mechanical lattice which exhibits edge states to a phononic lattice, thereby developing a family of lattices with edge states. Parameter spaces with topological phase boundaries in the vicinity of the transformed system are illustrated to demonstrate the robustness to mechanical imperfections. A potential realization in terms of fundamental mechanical building blocks is presented for the hexagonal and Lieb lattices. The lattices are composed of passive components and the building blocks are a set of disks and linear springs. Furthermore, by varying the spring stiffness, topological phase transitions are observed, illustrating the potential for tunability of our lattices., Comment: 10 pages, 10 figures