Lamb modes having multiple zero-group velocity points

Abstract

This dissertation is concerned with Lamb modes having multiple zero-group velocity (ZGV) points. Lamb mode spectrum of an isotropic plate has been considered. As the symmetric Lamb mode S1 always exhibits negative group velocity in a narrow frequency domain, it means that this mode has a zero-group velocity. This novel behavior is explained analyti cally by examining the slope of each mode first in its initial state and then near its turning point. An infinite orthotropic plate is considered, having thickness 2h, and the sought symmet ric dispersion relation has been derived by employing the traction free boundary conditions. It is found that, in addition to modes with a single ZGV point, modes having two ZGV points exist in the spectrum of an orthotropic plate. These modes are infinite in number. A few modes possessing three such points also exist. This rigorous study shows the presence of Lamb modes with multiple ZGV points. A hollow cylindrical waveguide always supports the existence of ZGV Lamb modes. For the presence of Lamb modes, with multiple ZGV points, in a waveguide having cylindrical geometry, the spectrum of a transversely isotropic cylinder is considered. A large number of longitudinal as well as flexural modes exist with more than one ZGV points. The Lamb modes for an isotropic plate under incompressibility constraint is studied. It has been found that plateau region does not exist in the spectrum and all the modes start off with negative slope and the slope retains its sign till the end. This fact is explained analytically. Finally, the zero-group velocity Lamb modes are studied in an orthotropic plate, under the constraint of incompressibility. It is found that, existence of ZGV points critically depends on the parameter a = (c11 + c22 −2c12 −4c66)/c66. No mode with a ZGV point exists for materials having a > −1 and the opposite holds true if a < −1. Isotropic as well as transversely isotropic materials have a = 0, hence the incompressibility constraint will preclude anomalous modes in such materials.