Date of Award
Doctor of Philosophy (PhD)
Lamb waves are guided elastic waves in plates. These waves are dispersive, with the frequency and wavenumber related by the well-known Rayleigh-Lamb equation. Generally, at zero wavenumber the dispersion curves are parabolic and the waves are non-propagating thickness resonances. For distinct values of Poisson’s ratio, however, degeneracy can occur between thickness resonance modes. At these coincidence frequencies the dispersion curves become linear and in a three-dimensional representation the dispersion surface is shaped like a cone. This behavior is referred to as conical dispersion. Waves excited at coincidence frequencies maintain the infinite phase velocity associated with thickness resonances but transport energy at a finite group velocity. A unique characteristic of such waves is that they propagate with an infinite wavelength, resulting in uniform oscillation of the plate surface. Conical dispersion essentially decouples the spatial and temporal behavior of the wave field and produces a field that is static in space yet oscillating in time.
The focus of this thesis is to investigate Lamb waves with conical dispersion in homogenous isotropic plates. The mode shapes and energy transport along the plate are analyzed in order to elucidate the origin of conical dispersion for a specific degenerate case. The theoretical group velocity is derived based on the velocity of energy transport. Conical dispersion is measured in an aluminum plate by cooling the plate in order to tune Poisson’s ratio through the degenerate point. Linear dispersion at zero wavenumber is measured and found to agree with the theory. Waves excited near the degenerate frequency exhibited spatially uniform phase over the plate surface. Mode conversion upon encountering the free edge of the plate is studied. The mode converted field is found to propagate perpendicular to the plate edge, irrespective of angle of incidence. This behavior is demonstrated by focusing a mode converted field from a semi-circular edge in the plate. This peculiar type of lens will focus the field regardless of the location of the source on the plate. Experimental results also show that Lamb waves with conical dispersion flow around a hole in the plate without distortion. This phenomenon causes the hole in the plate to be hidden when observing just the long wavelength signal distal to the hole. All of the experimental results are confirmed by comparison with time domain finite element simulations. We propose that Lamb waves with conical dispersion allow the manipulation of elastic energy in novel ways and may find use in nondestructive evaluation of materials and development of acoustic devices.
Stobbe, David M., "Lamb Waves with Conical Dispersion at Zero Wavenumber" (2018). Mechanical Engineering Graduate Theses & Dissertations. 167.