Asymptotic analysis of discrete and continuous periodic media

Abstract

Mechanical mass-spring networks have long acted to motivate, and gain qualitative intuition, in solid-state physics, continuous media containing periodic arrays of inclu- sions such as phononic crystals, and more recently in metamaterials. While in some cases an exact or approximate analogy between the continuous model and its discrete representation can be systematically drawn, more often such analogies are introduced heuristically to aid interpretation with the lumped parameters estimated and accepted as qualitative. This thesis builds towards making the analogy exact; we first look at the discrete masses and springs lattices and apply multiple-scales methods directly to Green’s function integrals to extract the behaviour near critical frequencies. The features we uncover, and the asymptotics, are generic for many lattice structures. We then identify and study a new class of materials, two- and three- dimensional phononic crystals formed by closely spaced rigid cylinders or interconnected perforated boxes, respectively, and show that such materials constitute a versatile and tuneable family of subwavelength metamaterials. Intuitively, the voids and narrow gaps that characterise the crystals form an interconnected network of Helmholtz-like resonators. We use this intuition to argue that these continuous phononic crystals are in fact asymptotically equivalent, at low frequencies, to discrete mass-spring networks whose lumped param- eters we derive explicitly. The crystals are tantamount to metamaterials as their entire acoustic branch is squeezed into a subwavelength regime where the ratio of wavelength to period scales like the ratio of period to gap width raised to the power 1/4 in two dimensions and 1/2 in three dimensions; at yet larger wavelengths we accordingly find a comparably large effective refractive index. The fully analytical dispersion relations predicted by the discrete models yield dispersion curves that agree with those from finite-element simulations of the continuous crystals.