Characteristics of second harmonic generation of Lamb waves in nonlinear elastic plates

Abstract

This paper investigates the characteristics of the second harmonic generation of Lamb waves in a plate with quadratic nonlinearity. Analytical asymptotic solutions to Lamb waves are first obtained through the use of a perturbation method. Then, based on a careful analysis of these asymptotic solutions, it is shown that the cross-modal generation of a symmetric second harmonic mode by an antisymmetric primary mode is possible. These solutions also demonstrate that modes showing internal resonance-nonzero power flux to the second harmonic mode, plus phase velocity matching-are most useful for measurements. In addition, when using finite wave packets, which is the case in most experimental measurements, group velocity matching is required for a cumulative increase in the second harmonic amplitude with propagation distance. Finally, five mode types (which are independent of material properties) that satisfy all three requirements for this cumulative increase in second harmonic amplitude-nonzero power flux, plus phase and group velocity matching-are identified. These results are important for the development of an experimental procedure to measure material nonlinearity with Lamb waves.