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.

Review of nonlinear ultrasonic guided wave nondestructive evaluation: theory, numerics, and experiments

“…This finding, which is explained further in Ref. 40, is in direct conflict with the reasonable argument presented by Muller et al 25 that the primary and secondary wave packets must travel together in order for energy transfer to occur. Our explanation is that the higher harmonic is generated by the material nonlinearity with the nonlinear surface traction f surf n and nonlinear body force f vol n terms [Eq.…”

“…Here, m, n denote arbitrary positive integers. In addition to the modes listed in Table 3, other modes 25 like the quasi-Rayleigh modes and (high-frequency) guided wave modes near the transverse wave speeds are also capable of second-harmonic generation.…”

“…• Phase-matching condition 19,[23][24][25][26] The phase-matching condition is satisfied if and only if there exists a propagating guided wave mode in the plate at (2ω, 2k), where (ω, k) corresponds to the frequency and wavenumber of the primary mode. To proceed with the analysis, we first start with the following dispersion relations E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 2 4 ; 6 3 ; 4 3 …”

“…• Nonzero power flux criterion 19,22,[25][26][27] To analyze nonzero power flux criterion, we adopt a parity analysis in terms of the displacement gradient (H). First, we begin by observing that the displacement field and the corresponding displacement gradient in the plate for different modes are of the following form given in Table 1.…”

Numerical studies of nonlinear ultrasonic guided waves in uniform waveguides with arbitrary cross sections

“…6,7 Two conditions for the generation of the cumulative second harmonic wave were summarized using perturbation method and modal analysis: (1) phase velocity matching; (2) non-zero power transfer from the primary to the second harmonic wave. 8 Müller et al 9 obtained analytical asymptotic solutions to Lamb waves and five mode types for the cumulative increase in the amplitude of the second harmonic wave with the propagation distance. The theoretical analysis was extended from plate like structures to cylindrical rod and pipe structures by de Lima et al 10 and Chillara et al 11 More recently, Liu et al demonstrated a method for the selection of primary modes with respect to the internal resonance, 12 also extended the work to higher order harmonics.…”