Analytical investigation of internally resonant second harmonic lamb waves in nonlinear elastic isotropic plates
Mueller, Martin Fritz
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This research deals with the second harmonic generation of Lamb waves in nonlinear elastic, homogeneous, isotropic plates. These waves find current applications in the field of ultrasonic, nondestructive testing and evaluation of materials. The second harmonic Lamb wave generation is investigated analytically in order to provide information on suitable excitation modes maximizing the second harmonic amplitude. Using an existing solution for the problem of second harmonic generation in wave guides, the solution is explained for the plate and examined as to the symmetry properties of the second harmonic wave, since published results are contradictory. It is shown that the cross-modal generation of a symmetric secondary mode by an antisymmetric primary mode is possible. Modes showing internal resonance, whose conditions are nonzero power flux from the primary wave and phase velocity matching, are shown to be most useful for measurements. In addition, group velocity matching is required. A material-independent analysis of the linear Lamb mode theory provides mode types satisfying all three requirements. Using the example of an aluminum plate, the found internally resonant modes are evaluated with regard to the rate of second harmonic generation and practical issues such as excitability and ease of measurement. Pros and cons of each mode type are presented.