Raj Kumar Pal

Edge waves in plates with resonators: an elastic analogue of the quantum valley Hall effect

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 with different topological invariants. The topological properties of the considered configurations are predicted by unit cell and finite strip dispersion analyses. Numerical simulations 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.

Helical edge states and topological phase transitions in phononic systems using bi-layered lattices

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.

Observation of topological valley modes in an elastic hexagonal lattice

We report on the experimental observation of topologically protected edge waves in a two-dimensional 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. This produces a non-trivial band structure that 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 pre-defined 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 non-trivial interfaces investigated herein.

Effect of large deformation pre-loads on the wave properties of hexagonal lattices

We study linear wave propagation in nonlinear hexagonal lattices capable of undergoing large deformations, under different levels of pre-load. The lattices are composed of a set of masses connected by linear axial and angular springs, with the nonlinearity arising solely from geometric effects. By applying different levels of pre-load, the small amplitude linear wave propagation response can be varied from isotropic to highly directional. Analytical expressions for the stiffness of a unit cell in the deformed configuration are derived and they are used to analyze the dispersion surfaces and group velocity variation with pre-load. Numerical simulations on finite lattices demonstrate the validity of our unit cell predictions and illustrate the wave steering potential of our lattice.

A Bloch-based procedure for dispersion analysis of lattices with periodic time-varying properties

Abstract 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.

Impact Response of Elasto-Plastic Granular Chains Containing an Intruder Particle

Wave propagation in homogeneous granular chains subjected to impact loads causing plastic deformations is substantially different from that in elastic chains. To design wave tailoring materials, it is essential to gain a fundamental understanding of the dynamics of heterogeneous granular chains under loads where the effects of plasticity are significant. In the first part of this work, contact laws for dissimilar elastic–perfectly plastic spherical granules are developed using finite element simulations. They are systematically normalized, with the normalizing variables determined from first principles, and a unified contact law for heterogeneous spheres is constructed and validated. In the second part, dynamic simulations are performed on granular chains placed in a split Hopkinson pressure bar (SHPB) setup. An intruder particle having different material properties is placed in an otherwise homogeneous granular chain. The position and relative material property of the intruder is shown to have a significant effect on the energy and peak transmitted force down the chain. Finally, the key nondimensional material parameter that dictates the fraction of energy transmitted in a heterogeneous granular chain is identified. [DOI: 10.1115/1.4028959]

A study of deformation localization in nonlinear elastic square lattices under compression

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 a variety of localized deformations were found depending on loading configuration, lattice parameters and boundary conditions. These studies are conducted on other lattice structures, with the objective of identifying and investigating minimal models that exhibit localization, hysteresis and path-dependent behaviour. 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, which may be considered as a discrete description of an elastic membrane supported by an elastic substrate. Results illustrate that, depending on the relative values of the spring constants, the lattice exhibits in-plane or out-of-plane instabilities leading to localized deformations. 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 sheds light on the mechanism of transition from affine deformation to global or diffuse deformation to localized deformation. Finally, the lattice is further reduced to a minimal four-mass model, which exhibits a deformation qualitatively similar to that in the central part of a longer chain. 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. This article is part of the theme issue ‘Nonlinear energy transfer in dynamical and acoustical systems’.

Tunable Wave Propagation in Granular Crystals by Altering Lattice Network Topology

We develop a framework for wave tailoring by altering the lattice network topology of a granular crystal consisting of spherical granules in contact. The lattice topology can alternate between two stable configurations, with the spherical granules of the lattice held in stable equilibrium in each configuration by gravity. Under impact, the first configuration results in a wave with rapidly decaying amplitude as it propagates along a primary chain, while the second configuration results in a solitary wave propagating along the primary chain with no decay. The mechanism to achieve such tunability is by having energy diverted to the granules adjacent to the primary chain in the first case but not the second. The tunable design of the proposed network is validated using both numerical simulations and experiments. In terms of potential applications, the proposed bistable lattice network can be viewed either as a wave attenuator or as a device that allows higher amplitude wave propagation in one direction than in the opposite direction. The lattice is analogous to a crystal phase transformation due to the change in atomic configurations, leading to the change in properties at the macroscale. [DOI: 10.1115/1.4034820]

Boundary modes in quasiperiodic elastic structures

Topological metamaterials are a new class of materials that support topological modes such as edge modes and interface modes, which are commonly immune to scattering and imperfections. This novelty has been the subject of extensive research in many branches of physics such as electronics, photonics, phononics, and acoustics. The nontrivial topological properties related to the presence of topological modes are tipically found in periodic media. However, it was recently demonstrated that structures called quasicrystals may also exhibit nontrivial topological behavior attributed to dimensions higher than that of the quasicrystal. While quasiperiodicity has received a lot of attention in the fields of crystallography and photonics, research into quasiperiodic elastic structures has been scarce. In this paper, we show how the concepts of quasiperiodicity may be applied to the design of topological mechanical metamaterials. We start by investigating the boundary modes present in quasiperiodic 1D phononic lattices. These modes have the interesting property of being localized at either one of the two different boundaries depending on the value of an additional parameter, which is remnant of the higher dimension. A smooth variation of this parameter in either time or a spatial dimension can lead to a robust transfer of energy between two sites of the structure. We present an idealized mechanical system composed by an array of coupled rods that may be used as a platform for realizing this kind of robust transfer of energy. These are preliminary investigations into a entirely new class of structures which may lead to novel engineering applications.