High-frequency asymptotics for microstructured thin elastic plates and platonics

Abstract

We consider microstructured thin elastic plates that have an underlying periodic structure, and develop an asymptotic continuum model that captures the essential microstructural behaviour entirely in a macroscale setting. The asymptotics are based upon a two-scale approach and are valid even at high frequencies when the wavelength and microscale length are of the same order. The general theory is illustrated via one-and two-dimensional model problems that have zero-frequency stop bands that preclude conventional averaging and homogenization theories. Localized defect modes created by material variations are also modelled using the theory and compared with numerical simulations.

‘Parabolic’ trapped modes and steered Dirac cones in platonic crystals

“…The final column gives the coefficient of |k − k D2 | in the contribution to the DOS from each cone. 12) and, as the product of the areas of the Wigner Seitz cell and the Brillouin zone is 4π 2 in two dimensions, 13) where γ i represents the contribution to the DOS from the ith band surface. These are given by 14) and |v g | = 0 at t = 0; however, the DOS is not unbounded as the length of the isofrequency contour at k = k D1 is zero.…”

“…The area of high-frequency homogenization for waves in structured media has been recently developed in the papers by Craster and co-workers [9][10][11][12][13][14]. These studies include a wide range of formulations for waves in lattices as well as structured continuous media.…”

“…These studies include a wide range of formulations for waves in lattices as well as structured continuous media. In particular, the high-frequency homogenization approach has been applied to flexural waves in [12,14], which has provided a valuable insight and analytical description for waves at frequencies in the neighbourhood of stop bands observed on the dispersion diagrams. The high-frequency homogenization approach has also covered the cases of dynamic anisotropy, corresponding to saddle points on the dispersion surfaces, where directional preference is observed along two characteristics constructed for the homogenization approximation [13,14].…”

Front Matter

“…The elastic waves carry the heat (phonons). It is worth remarking that, subsequent to pioneering work by Bensoussan, Lions, and Papanicolaou in Chapter 4 of their book [14], there has been a resurgence of interest in Foreword xvii high-frequency homogenization at stationary points in the dispersion diagram, which may be local minima or maxima, or even saddle points [15][16][17][18][19][20][21]. The wave is a modulated Bloch wave and modulation satisfies appropriate effective equations.…”

Steering in-plane shear waves with inertial resonators in platonic crystals

“…The zero frequency band gaps in arrays of infinite conducting wires [12] and in pinned plates [13], lensing in platonic crystals [14] or low frequency band gaps in locally resonant sonic crystals with stiff pillars [15,16], heavy pillars combined with soft rubber [17] or Helmholtz-like resonators [18] are examples of those.…”