Mathematical modelling of the dynamic response of metamaterial structures

Abstract

This thesis constitutes an exposition of the work carried out by the author whilst examining several physical problems under the broad theme of the dynamic response of structures. An outline of the thesis is provided in chapter one. Chapter two introduces some notation and preliminary results on general lattice equations. Chapter three examines the dispersive behaviour of non-classical discrete elastic lattice systems. In particular, the effect of distributing the inertial properties of the lattice over the elastic rods, in addition to at the junctions, is considered. It is demonstrated that the material properties in the long wavelength limit are not what one would expect from the static response of the lattice. The effect of various interactions on the dispersive properties of the triangular cell lattice is considered, including so-called truss, frame, and micro-polar interactions. Compact analytical estimates for the band widths are presented, allowing the design of structures possessing pass and/or stop bands at specific frequencies and in specified directions. The finite frequency response of several lattice structures is considered in chapter four. In particular, the dynamic anisotropy of both scalar and elastic lattices is examined. The resulting strongly anisotropic material response is linked, explicitly, to the dispersive properties of the lattice. A novel application of dynamic anisotropy to the focusing, shielding, and negative refraction of elastic waves using a flat discrete metamaterial lens'' is presented. Chapter five is devoted to the analysis, using the dynamic Green's function, of a finite rectilinear inclusion in an infinite square lattice. Several representations of the Green's function are presented, including expression in terms of hypergeometric functions, which are employed in deriving band edge expansions. It is shown that localised defect modes, characterised by displacements which decay rapidly away from the defect, can be initiated by reducing the mass of one or more lattice nodes, whilst ensuring that the mass of the nodes remains positive. For one- and three-dimensional multi-atomic lattices, there exists a bound on the contrast in mass between the defect and ambient lattice such that localised defect modes exist. However, it is shown that for the two-dimensional lattice, no such bound exists, provided that the masses remain positive. The analysis of a finite-sized defect region is accompanied by the waveguide modes that may exist in a lattice containing an infinite chain of point defects. A numerical simulation illustrates that the solution of the problem for an infinite chain can be used to predict the range of eigenfrequencies of localised modes for a finite but, sufficiently long, array of masses representing a rectilinear defect in a square lattice. Continuing with the theme of defects, chapter six examines response of a triangular thermoelastic lattice, with an edge crack under mode I loading. The response of the triangular lattice is compared with that of the corresponding continuum. The model is related to the phenomenon of thermal striping, which occurs when a structure is exposed to periodic variations in temperature. In the thermal striping regime, crack propagation is a fatiguing processes with the rate of crack growth being proportional to some power of the peak-to-peak amplitude of the stress intensity factor. An effective stress intensity factor'' for the lattice is introduced and it is demonstrated that, in the homogenised limit, the effective stress intensity factor'' is lower than the stress intensity factor of the continuum for sufficiently long cracks and low frequencies. Finally, chapter seven presents a detailed analysis of a non-singular square cloak for acoustic, out-of-plane shear elastic, and electromagnetic waves. The propagation of waves through the cloak is examined analytically and is complemented with a range of numerical illustrations. The efficacy of the regularised cloak is demonstrated and an objective numerical measure of the quality of the cloaking effect is introduced. The results presented show that the cloaking effect persists over a sufficiently wide range of frequencies. To illustrate further the effectiveness of the regularised cloak, a Young's 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. A significant advantage of this particular regularised square cloak is the straightforward connection with a discrete lattice. It is shown that an approximate cloak can be constructed using a discrete lattice structure. The efficiency of such a lattice cloak is analysed and several illustrative simulations are presented. It is demonstrated that cloaking can be achieved by using a relatively simple lattice, particularly in the low frequency regime. This discrete lattice structure provides a possible avenue toward the physical realisation of invisibility cloaks.