Propagation and filtering of elastic and electromagnetic waves in piezoelectric composite structures

In this article we discuss the modelling of elastic and electromagnetic wave propagation through one- and two-dimensional structured piezoelectric solids. Dispersion and the effect of piezoelectricity on the group velocity and positions of stop bands are studied in detail. We will also analyze the reflection and transmission associated with the problem of scattering of an elastic wave by a heterogeneous piezoelectric stack. Special attention is given to the occurrence of transmission resonances in finite stacks and their dependence on a piezoelectric effect. A 2D doubly-periodic piezoelectric checkerboard structure is subsequently introduced, for which the dispersion surfaces for Bloch waves have been constructed and analysed, with the emphasis on the dynamic anisotropy and special features of standing waves within the piezoelectric structure.Comment: 24 pages, 18 figures, 3 tables. Preprint version of a research article, accepted for publication in "Mathematical Methods in the Applied Science (2016)

Uniform asymptotic formulae for Green's kernels in regularly and singularly perturbed domains

Asymptotic formulae for Green's kernels $G_\epsilon({\bf x}, {\bf y})$ of various boundary value problems for the Laplace operator are obtained in regularly perturbed domains and certain domains with small singular perturbations of the boundary, as $\epsilon \to 0$. The main new feature of these asymptotic formulae is their uniformity with respect to the independent variables ${\bf x}$ and ${\bf y}$.Comment: 9 page

Mesoscale asymptotic approximations to solutions of mixed boundary value problems in perforated domains

We describe a method of asymptotic approximations to solutions of mixed boundary value problems for the Laplacian in a three-dimensional domain with many perforations of arbitrary shape, with the Neumann boundary conditions being prescribed on the surfaces of small voids. The only assumption made on the geometry is that the diameter of a void is assumed to be smaller compared to the distance to the nearest neighbour. The asymptotic approximation, obtained here, involves a linear combination of dipole fields constructed for individual voids, with the coefficients, which are determined by solving a linear algebraic system. We prove the solvability of this system and derive an estimate for its solution. The energy estimate is obtained for the remainder term of the asymptotic approximation.Comment: 20 pages, 8 figure

Green's kernels for transmission problems in bodies with small inclusions

The uniform asymptotic approximation of Green's kernel for the transmission problem of antiplane shear is obtained for domains with small inclusions. The remainder estimates are provided. Numerical simulations are presented to illustrate the effectiveness of the approach.Comment: 39 pages, 6 figure

Edge waves and localisation in lattices containing tilted resonators

The paper presents the study of waves in a structured geometrically chiral solid. A special attention is given to the analysis of the Bloch-Floquet waves in a doubly periodic high-contrast lattice containing tilted resonators. Dirac-like dispersion of Bloch waves in the structure is identified, studied and applied to wave-guiding and wave-defect interaction problems. The work is extended to the transmission problems and models of fracture, where localisation and edge waves occur. The theoretical derivations are accompanied with numerical simulations and illustrations

Localised bending modes in split ring resonators

International audienceWe present band diagrams for in-plane elastic waves propagating within a doubly periodic structure involving split ring resonators (SRR). From the Navier equations we derive that the lowest resonant frequencies are associated with localised bending modes solutions of a fourth-order differential equation in thin-bridges, which are responsible for the appearance of a low frequency stop band. Potential applications lie in the design of earthquake resistant systems