A 2019 review by Li and Cho (AIP Conf. Proc. 2102, 020002) posed the measurement-side obstacles that still separate nonlinear ultrasonic guided-wave nondestructive evaluation (NDE) from being a quantitative, field-usable technique for early-stage, otherwise invisible micro-damage in plates, pipes, and composite laminates. Four linked issues were left unresolved: (1) among multiple phase-matched primary/secondary mode pairs satisfying the internal-resonance conditions, no general algorithm ranks candidates by expected experimental sensitivity (excitation purity, modal power flux, attenuation, mode crowding); (2) instrumentation- and couplant-induced nonlinearities share the spectral signatures of damage, and no robust field procedure separates them; (3) the usable range of propagation distance, frequency, and geometry over which cumulative second-harmonic growth remains measurable is unquantified across materials and defect types; (4) nonlinear guided-wave tomography based on signal-difference coefficients shows spurious features of unresolved origin and lacks ground-truth validation.

Background

Ultrasonic guided waves are elastic waves confined by the boundaries of a plate, pipe, or laminate; in plates the fundamental solutions are the symmetric (S) and antisymmetric (A) Lamb modes, which are dispersive (their speeds depend on frequency) and occur in crowds of overlapping modes. Because a guided wave interrogates large or partly inaccessible structures from one transducer position, guided-wave nondestructive evaluation (NDE) is a standard inspection tool, and quantitative NDE (QNDE) seeks to measure damage severity, not merely detect it. Linear guided waves respond weakly to early-stage 'invisible' micro-damage — micro-cracks, micro-corrosion, thermal or fatigue microstructural degradation — because such damage produces little linear scattering. The material's acoustic nonlinearity offers higher sensitivity: a finite-amplitude wave generates second harmonics, and two co-propagating primary waves generate mixed components at the sum and difference frequencies fa±fbf_a \pm f_b, so the ratio A2/A12A_2/A_1^2 (second-harmonic amplitude over squared fundamental amplitude) or the mixed amplitudes serve as damage indicators.

In a dispersive multimode waveguide these effects become measurable only under the internal-resonance (phase-matching) conditions analyzed for plates by Deng (1999) and de Lima and Hamilton (2003): the wavenumber of the double-frequency mode must equal twice the fundamental wavenumber (phase-velocity matching), the group velocities must be aligned so that energy is transferred continually along the propagation path (producing cumulative growth of the harmonic with distance), and the mode-to-mode power flux must be nonzero. Several mode pairs in a typical plate or pipe spectrum can satisfy these conditions at once while differing greatly in practical usefulness — in how purely each can be excited, how much power flows between the modes, how strongly the modes are attenuated, and how crowded the surrounding spectrum is. The 2019 review by Li and Cho, which summarized phase-matched measurements in an Al-6063 pipe (the L(0,6)L(0,6)–L(0,10)L(0,10) pair at 1.68 MHz), in a [0]6[0]_6 carbon/epoxy laminate (the S1–S2 pair at 2.25 MHz), and a tomographic reconstruction of micro-corrosion built on the authors' earlier Ultrasonics (2016) work, acknowledged that 'the clear guideline for choosing the better phase-matched modes is still obscure'.

Two further practical obstacles are recognized in the same review. First, the measurement chain itself — couplant between transducer and structure, sensor grounding, analog-to-digital (A/D) conversion, and the finite reproducibility of gated tone-burst excitation — generates nonlinear frequency-converted components with the same spectral content as damage-induced ones, so the measured harmonic cannot automatically be attributed to the structure. Second, material attenuation (damping) limits the propagation distance over which cumulative second-harmonic growth is usable, so the useful measurement window in distance, frequency, and geometry must be quantified for each material and damage type. Finally, the review's tomography implementation modifies the RAPID probabilistic reconstruction algorithm — a standard guided-wave tomography scheme whose damage index is a signal-difference coefficient (SDC), a normalized measure of the change between the current and baseline signals — by feeding it nonlinear frequency-converted features; the resulting tomograms showed spurious SDC variations not corresponding to the defect, of unresolved origin in measurement system noise and reconstruction regularization.

Problem Statement

Remove the remaining measurement-side obstacles that keep nonlinear guided-wave NDE from being a quantitative, field-usable technique for early-stage invisible micro-damage — the single objective that unifies the four issues the source review left unresolved. Four linked results: (i) a general, experimentally validated selection algorithm that, for a given waveguide, ranks candidate phase-matched primary/secondary mode (or frequency) pairs by expected experimental sensitivity, accounting for realistic excitation purity, modal power flux, attenuation, and mode crowding, so that mode choice no longer rests on case-by-case tuning; (ii) a robust procedure separating damage-induced nonlinearity from instrumentation- and couplant-induced nonlinearity and noise under field conditions (couplant variability, sensor grounding, A/D conversion, finite reproducibility of gated excitation); (iii) a quantification of the usable range — propagation distance, frequency, and geometry — over which cumulative second-harmonic (or mixed-frequency) growth remains measurable, across materials and defect types; and (iv) a nonlinear guided-wave tomographic reconstruction in which spurious signal-difference-coefficient features are understood and suppressed and which is validated against ground truth across varied defect morphologies and complex structural geometries. All four results are jointly required to resolve this unified objective — a solution addressing only a subset does not by itself resolve it — and each must be established for the general class of plate, pipe, and composite-laminate waveguides, not for a single material, geometry, or defect type. The required breadth and validation depth are bounded: a result counts as established once it is demonstrated with experimental validation on at least one representative specimen from each of the plate, pipe, and laminate classes and is accompanied by a stated, defensible argument for transfer to other materials, defect types, and geometries.

References

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