Second order harmonic guided wave mutual interactions in plate: Vector analysis, numerical simulation, and experimental results

Abstract

The extraordinary sensitivity of nonlinear ultrasonic waves to the early stages of material degradation makes them excellent candidates for nondestructive material characterization. However, distinguishing weak material nonlinearity from instrumentation nonlinearity remains problematic for second harmonic generation approaches. A solution to this problem is to mix waves having different frequencies and to let their mutual interaction generate sum and difference harmonics at frequencies far from those of the instrumentation. Mixing of bulk waves and surface waves has been researched for some time, but mixing of guided waves has not yet been investigated in depth. A unique aspect of guided waves is their dispersive nature, which means we need to assure that a wave can propagate at the sum or difference frequency. A wave vector analysis is conducted that enables selection of primary waves traveling in any direction that generate phase matched secondary waves. We have tabulated many sets of primary waves and phase matched sum and difference harmonics. An example wave mode triplet of two counter-propagating collinear shear horizontal waves that interact to generate a symmetric Lamb wave at the sum frequency is simulated using finite element analysis and then laboratory experiments are conducted. The finite element simulation eliminates issues associated with instrumentation nonlinearities and signal-to-noise ratio. A straightforward subtraction method is used in the experiments to identify the material nonlinearity induced mutual interaction and show that the generated Lamb wave propagates on its own and is large enough to measure. Since the Lamb wave has different polarity than the shear horizontal waves the material nonlinearity is clearly identifiable. Thus, the mutual interactions of shear horizontal waves in plates could enable volumetric characterization of material in remote regions from transducers mounted on just one side of the plate.

Mixed third harmonic shear horizontal wave generation: interaction between primary shear horizontal wave and second harmonic Lamb wave

“…Specifically, to identify the mixed mechanism, a gel test is performed by placing a layer of gel on the SH wave propagating path. The gel allows an effective dissipation of the Lamb waves (out-ofplane particle motion) without, in principle, affecting the SH waves (in-plane particle motion) [25]. If the mixed 3rd harmonic SH waves (due to the mutual interaction between the 1st SH waves and their generated 2nd Lamb waves) do exist in the system, the overall third harmonic response of the system is expected to be changed by the presence of the gel.…”

Nonlinear Lamb wave mixing for assessing localized deformation during creep

“…Now, the simple extension from linear dynamic elasticity to nonlinear equation requires the introduction of third order elastic constants-l, m, and n (Murnaghan's constants considering the materials to be hyperelastic [17]). The nonlinear equation of motion for the displacement field u for an isotropic elastic material without any viscous dissipation takes the form [18].…”

“…(6) into nonlinear part of eqns. (4) and (5) These terms correspond to possible second order harmonic waves at 2k 1 ) and (2ω 2 , 2k 2 ) which are the sum and difference combinational harmonics respectively [18] along with harmonics of individual fundamental wave. Nonlinear acoustic parameter can be expressed as 4A ω1þω2= ABk 1 k 2 x [15].…”

Signal Strength Enhancement of Magnetostrictive Patch Transducers for Guided Wave Inspection by Magnetic Circuit Optimization

“…Guided waves, which are ultrasonic waves restricted by the boundaries of the solid waveguide during propagation [1], have been applied in the inspection of many industrial structures, such as pipes [2,3], rods [4], plates [5][6][7] and rails [8][9][10] with the advantages of long distance, full coverage and low cost [11]. The conversions between the electrical energy and mechanical energy are mainly involved in guide wave inspection, which are accomplished by the transducers as the key parts in the inspection system.…”