A. B. Movchan

Achieving control of in-plane elastic waves

We derive the elastic properties of a cylindrical cloak for in-plane coupled shear and pressure waves. The cloak is characterized by a rank 4 elasticity tensor with spatially varying entries, which are deduced from a geometric transform. Remarkably, the Navier equations retain their form under this transform, which is generally untrue [G. W. Milton et al., N. J. Phys. 8, 248 (2006)]. The validity of our approach is confirmed by comparison of the analytic Green’s function in homogeneous isotropic elastic space against full-wave finite element computations in a heterogeneous anisotropic elastic region surrounded by perfectly matched layers.

Asymptotic models of dilute composites with imperfectly bonded inclusions

The asymptotic scheme for the analysis of dilute elastic composites, which includes circular inclusions with imperfect bonding at the interface is presented. Interface is characterized by a discontinuous displacement field across it, linearly related to the tractions. The problem of a linear-elastic, circular inclusion with generic loading condition at infinity is solved, and used to analyze effective elastic moduli of composite materials. Effects due to the interaction of a small circular defect and a crack are investigated. It is shown that interfacial stiffness has a strong effect on the crack path and therefore may be an important design parameter for composites.

Asymptotic Modelling of Adhesive Joints

An asymptotic approach is presented for modelling of adhesive joints in a layered structure. We consider the case when two elastic layers are bonded together by a thin and soft layer of glue. Differential equations for the displacements (including the displacement jump across the thin interface layer) in a compound thin beam are derived.

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.

Band-gap shift and defect-induced annihilation in prestressed elastic structures

Design of filters for electromagnetic, acoustic, and elastic waves involves structures possessing photonic/phononic band gaps for certain ranges of frequencies. Controlling the filtering properties implies the control over the position and the width of the band gaps in question. With reference to piecewise homogeneous elasticbeams on elastic foundation, these are shown to be strongly affected by prestress (usually neglected in these analyses) that (i) “shifts” band gaps toward higher (lower) frequencies for tensile (compressive) prestress and (ii) may “annihilate” certain band gaps in structures with defects. The mechanism in which frequency is controlled by prestress is revealed by employing a Green’s-function-based analysis of localized vibration of a concentrated mass, located at a generic position along the beam axis. For a mass perturbing the system, our analysis addresses the important issue of the so-called effective negative mass effect for frequencies within the stop bands of the unperturbed structure. We propose a constructive algorithm of controlling the stop bands and hence filtering properties and resonance modes for a class of elastic periodic structures via prestress incorporated into the model through the coefficients in the corresponding governing equations.

Sonic band gaps in PCF preforms: enhancing the interaction of sound and light

We study the localisation and control of high frequency sound in a dual-core square-lattice photonic crystal fibre preform. The coupled states of two neighboring acoustic resonances are probed using an interferometric set up, and experimental evidence is obtained for odd and even symmetry trapped states. Full numerical solutions of the acoustic wave equation show the existence of a two-dimensional sonic band gap, and numerical modelling of the strain field at the defects gives results that agree well with the experimental observations. The results suggest that sonic band gaps can be used to manipulate sound with great precision and enhance its interaction with light.

Eigenvalue problems for doubly periodic elastic structures and phononic band gaps

We consider the problem of elastic waves propagating in a two–dimensional array of circular cavities, taking rigorous account of coupling between shear and dilational waves. A technique, originally due to Rayleigh, is derived that involves an elegant identity between the singular and non–singular components of the stress fields in the array. This leads to an infinite linear system which can be truncated and solved in order to determine the complete structure of the propagating modes. Of particular interest is the possibility of exhibiting phononic band gaps, i.e. domains of frequency for which all propagating vibration in the material is suppressed.

Statics and dynamics of structural interfaces in elasticity

The concept of structural interface possessing finite width and joining continuous media is presented. Mechanical effects differentiating this model from conventional zero-thickness interfaces are explored for static and dynamic problems. In the static case, the thickness of the interface introduces an additional characteristic length, providing a parameter which can serve different design needs, for instance, may be employed to obtain neutral coated inclusions. In the dynamic case, a number of effects arise, related to the inertia of the interface. In particular, examples demonstrate that the design of inertial properties of the interface may be useful to produce sharp filters of elastic waves.

Three-dimensional dynamic perturbation of a propagating crack

Asymptotic formulae are derived for the perturbations to the stress intensity factors induced by a small three-dimensional dynamic perturbation of a propagating, nominally plane, crack. The crack propagates through an infinite isotropic uniform medium and full account is taken of the equations of elastodynamics. Previous work by the authors treated the case in which the edge of the crack was subject to an in-plane perturbation, so that the crack remained plane, but with a wavy edge. The present work completes the study, by allowing for out-of-plane perturbations.

Focussing bending waves via negative refraction in perforated thin plates

We propose a design of a periodically perforated thin plate leading to lensing effect of bending waves via negative refraction. To achieve this goal, we first analyze the band spectrum of the bi-harmonic operator for an array of freely vibrating square voids using both numerical (finite elements) and asymptotic methods. We then find some point in the reciprocal space where the acoustic dispersion surface displays a convex isofrequency contour shrinking with frequency. We finally demonstrate that a point force generating a bending wave above a finite array of 221 perforations displays an image underneath according to the Snell–Descartes inverted laws.