N. V. Movchan

Bloch-Floquet bending waves in perforated thin plates

This paper presents a mathematical model describing propagation of bending waves in a perforated thin plate. It is assumed that the holes are circular and form a doubly periodic square array. A spectral problem for the biharmonic operator is formulated in a unit cell containing a single defect, and its analytical solution is constructed using a multipole method. The overall system for the coefficients in the multipole expansion is then solved numerically. We generate dispersion diagrams for the two cases where the boundaries of holes are either clamped or free. We show that in the clamped case, there is a total low-frequency band gap in the limit of inclusions of zero radius, and give a simple formula describing the corresponding band diagram in this limit. We show that in the free-edge case, the band diagram of the vibrating plate is much closer to that of plane waves in a uniform plate than for the clamped case.

Transformation elastodynamics and cloaking for flexural waves

The paper addresses an important issue of cloaking transformations for fourth-order partial differential equations representing flexural waves in thin elastic plates. It is shown that, in contrast with the Helmholtz equation, the general form of the partial differential equation is not invariant with respect to the cloaking transformation. The significant result of this paper is the analysis of the transformed equation and its interpretation in the framework of the linear theory of pre-stressed plates. The paper provides a formal framework for transformation elastodynamics as applied to elastic plates. Furthermore, an algorithm is proposed for designing a broadband square cloak for flexural waves, which employs a regularised push-out transformation. Illustrative numerical examples show high accuracy and efficiency of the proposed cloaking algorithm. In particular, a physical configuration involving a perturbation of an interference pattern generated by two coherent sources is presented. It is demonstrated that the perturbation produced by a cloaked defect is negligibly small even for such a delicate interference pattern.

Dispersion and localization of elastic waves in materials with microstructure

This paper considers the interaction of elastic waves with materials with microstructure. The paper presents a mathematical model of elastic waves within a lattice system incorporating rotational motions and interaction between different lattice elements through elastic links. The waves are dispersive and the lattice system itself is heterogeneous, i.e. the elastic stiffness and/or mass are non-uniformly distributed. For such systems, one can identify stop bands, representing the intervals of frequencies of waves, which become evanescent and cannot propagate through the structure. Filtering properties of such lattices are studied in this paper. Defect modes are created by removing a periodic array of elastic links, which leads to localization within a macro-cell. Special attention is given to the evaluation of the effective group velocities and to the study of standing waves within the system. Analytical estimates are accompanied by numerical simulations and analysis of dispersion surfaces. We also consider an example showing the focusing and the creation of an image point by a flat elastic ‘lens’ formed from a finite micropolar lattice system.

Two-dimensional phononic crystals and scattering of elastic waves by an array of voids

We study the diffraction of plane elastic waves by a grating of circular holes in an elastic matrix, using a multipole formulation to determine scattering matrices, for both the scalar (out–of–plane shear) and vector (plane–strain) problems. We use these matrices in a recurrence procedure to analyse the reflection and transmission properties of a stack containing an arbitrary number of such gratings. We establish the form of symmetry properties that may be used to verify the elastodynamic scattering matrices and comment on the filtering properties of grating stacks and their relationship to phononic band diagrams.

Making waves round a structured cloak: lattices, negative refraction and fringes

Using the framework of transformation optics, this paper presents a detailed analysis of a non-singular square cloak for acoustic, out-of-plane shear elastic and electromagnetic waves. Analysis of wave propagation through the cloak is presented and accompanied by numerical illustrations. The efficacy of the regularized cloak is demonstrated and an objective numerical measure of the quality of the cloaking effect is provided. It is demonstrated that the cloaking effect persists over a wide range of frequencies. As a demonstration of the effectiveness of the regularized cloak, a Youngs double slit experiment is presented. The stability of the interference pattern is examined when a cloaked and uncloaked obstacle are successively placed in front of one of the apertures. This novel link with a well-known quantum mechanical experiment provides an additional method through which the quality of cloaks may be examined. In the second half of the paper, it is shown that an approximate cloak may be constructed using a discrete lattice structure. The efficiency of the approximate lattice cloak is analysed and a series of illustrative simulations presented. It is demonstrated that effective cloaking may be obtained by using a relatively simple lattice structure, particularly, in the low-frequency regime.

Wave scattering by platonic grating stacks

We address the problem of scattering of flexural waves obeying the biharmonic equation by a stack of a finite number of gratings. We express the solution of the scattering problem for a single grating in terms of reflection and transmission matrices, incorporating the effects of both propagating and evanescent incident waves. The plane wave expansion coefficients above and below the grating are linked to multipole coefficients within the grating using the grating sums and the Rayleigh identities. We derive the recurrence procedure giving the reflection and transmission matrices of the stack in terms of those of individual layers. Trapped waves between a pair of gratings are investigated.

Dynamic anisotropy and localization in elastic lattice systems

This paper presents an analytical study and numerical simulations concerning the dynamic anisotropy of lattice systems in vector problems of elasticity. Connections are made with models of optics involving interaction of light with a small aperture and aberration effects. Special attention is given to standing waves possessing directional localization of different kinds. We analyze a special class of waveforms corresponding to saddle points on the dispersion surfaces. Furthermore, a modeling algorithm is developed to design a structured slab of finite thickness, which possesses focusing properties for waves within a certain frequency range.

Asymptotics of eigenfrequencies in the dynamic response of elongated multi-structures

The paper addresses a mathematical model describing the dynamic response of an elongated bridge supported by elastic pillars. The elastic system is considered as a multi-structure involving subdomains of different limit dimensions connected via junction regions. Analytical formulae have been derived to estimate eigenfrequencies in the low frequency range. The analytical findings for Bloch–Floquet waves in an infinite periodic structure are compared with the finite element numerical computations for an actual bridge structure of finite length. The asymptotic estimates obtained here have also been used as a design tool in problems of asymptotic optimization.

'Parabolic' trapped modes and steered Dirac cones in platonic crystals.

This paper discusses the properties of flexural waves governed by the biharmonic operator, and propagating in a thin plate pinned at doubly periodic sets of points. The emphases are on the design of dispersion surfaces having the Dirac cone topology, and on the related topic of trapped modes in plates for a finite set (cluster) of pinned points. The Dirac cone topologies we exhibit have at least two cones touching at a point in the reciprocal lattice, augmented by another band passing through the point. We show that these Dirac cones can be steered along symmetry lines in the Brillouin zone by varying the aspect ratio of rectangular lattices of pins, and that, as the cones are moved, the involved band surfaces tilt. We link Dirac points with a parabolic profile in their neighbourhood, and the characteristic of this parabolic profile decides the direction of propagation of the trapped mode in finite clusters.

Convergence properties and flat bands in platonic crystal band structures using the multipole formulation

We present converged band diagrams for Bloch–Floquet bending waves in a thin elastic plate containing a square array of circular perforations, using the multipole formulation developed by Movchan et al. (2007) and applied in the situation where the perforations are no longer considered to be small in comparison with the lattice pitch. We give tables of converged frequencies and the number of multipoles necessary to achieve them, for a range of radii of the perforations for both clamped-edge and free-edge boundary conditions. We find that the larger filling fraction leads to extremely flat bands within the band structure; this can be explained by considering the energy of the vibrational modes. We derive the energy balance relation as well as convenient expressions for the group velocity of eigenmodes, which reveal the interplay between the Helmholtz and the modified Helmholtz components of the eigenfield.