The relative acoustical nonlinearity parameter measured from second-harmonic generation of ultrasonic guided waves rises as fatigue damage accumulates in fiber-reinforced composite laminates, but it is a scalar indicator of total nonlinearity. The experiments and isotropic crack simulations of Rauter, Lammering and Kühnrich (2016) left open how material anisotropy modifies second-harmonic generation by microcracks, how the contributions of horizontal (surface-parallel) cracks and invisible distributed matrix degradation can be separated quantitatively so that the laminate's remaining stiffness can be predicted, and how the method extends from small coupons to realistic structural components. Subsequent literature on nonlinear guided waves in composites reports substantial progress on anisotropic constitutive models, numerical schemes for guided-wave interaction with nonlinear damage, wave-mixing observables, and baseline-free detection, but no method that resolves damage mechanisms quantitatively or predicts residual stiffness; a 2025 review still describes higher-harmonic guided-wave techniques as evaluating only the total nonlinearity between transmitter and receiver. The problem remains open.

Background

Ultrasonic guided waves are elastic waves confined by the boundaries of a plate or shell; the modes used here are Lamb waves, whose particle motion lies in the sagittal plane and which satisfy traction-free conditions at the two plate surfaces. When a finite-amplitude elastic wave propagates in a weakly nonlinear solid, energy is transferred to the double frequency, the second harmonic; the growth of the second-harmonic amplitude A2A_2 relative to the squared fundamental amplitude A12A_1^2 defines the acoustic nonlinearity parameter, in practice measured as a relative parameter β′=A2/A12\beta' = A_2/A_1^2 normalized to a reference state. In an undamaged solid this nonlinearity stems from the material itself (intrinsic, lattice-level nonlinearity); in a damaged solid an additional, usually much stronger contribution arises at crack faces that alternately open, close, and touch under the passing wave, known as contact or breathing-crack nonlinearity. Fiber-reinforced polymer laminates used in aircraft and energy structures are stiffest along the fibers and equally compliant in all perpendicular directions, a material symmetry called transverse isotropy, and under cyclic (fatigue) loading they accumulate a characteristic sequence of micromechanisms: distributed matrix microcracking, crack-like surfaces parallel to the laminate plane (the horizontal cracks below, i.e., incipient delaminations), fiber/matrix debonding, and fiber breakage. Each mechanism reduces the laminate's remaining stiffness, meaning the elastic modulus still available to carry load; classical composite damage mechanics can predict modulus loss from the density of each crack type, but none of these micromechanisms is easily visible to conventional reflection-based ultrasonic inspection at early stages. Second-harmonic generation of guided waves is attractive precisely because the contact nonlinearity of incipient cracks is sensitive to damage that linear methods miss. The theory of second-harmonic generation of Lamb waves was developed for isotropic plates (de Lima and Hamilton 2003; Chillara and Lissenden 2012), including the internal-resonance or phase-matching conditions under which a primary mode and a double-frequency secondary mode travel at the same phase velocity so that the second harmonic accumulates coherently with propagation distance, the so-called cumulative effect. Anisotropic second-harmonic theory for the intrinsic material nonlinearity of plates also exists (Deng 2008). Against this background, Rauter, Lammering and Kühnrich (2016) measured β′\beta' with second-harmonic guided waves in fatigued composite coupons and simulated the crack nonlinearity in finite elements, showing that β′\beta' tracks fatigue progression, while their crack simulations used an isotropic material model although the laminate is transversely isotropic, and β′\beta' responded to both horizontal cracks and invisible matrix degradation with no way to tell the two apart. The limitations documented in that study, which are the source of the present problem, leave four linked gaps: the unquantified effect of material anisotropy on nonlinear wave propagation in the presence of microcracks; the absence of a method to separate quantitatively the contributions of horizontal cracks and of invisible matrix degradation in a single β′\beta' measurement, which is what a precise estimate of remaining stiffness would require; the confinement of the experiments to small coupons at fixed propagation distance; and the unknown influence of further damage types (fiber/matrix debonding, fiber breakage) on second-harmonic generation and their interplay with stiffness reduction.

Problem Statement

The problem is to turn second-harmonic guided-wave inspection of fatigued fiber-reinforced laminates into a quantitative, damage-mechanism-resolved characterization method. Concretely, for a transversely isotropic composite laminate accumulating fatigue damage, establish a predictive framework in which: (1) the forward problem is solved for anisotropic material behavior, quantifying how transverse isotropy modifies second-harmonic generation by microcracks with breathing-crack contact nonlinearity, replacing the isotropic simulations used so far; and (2) the inverse problem is solved, by a method that separates the quantitative contributions of horizontal (surface-parallel) cracks and of invisible distributed matrix degradation, and accounts for further mechanisms such as fiber/matrix debonding and fiber breakage, in the measured relative acoustical nonlinearity parameter β′\beta', using a single measurement or a minimal, justified set of guided-wave nonlinear observables, so that the remaining stiffness of the laminate can be estimated quantitatively. Together these would replace the scalar total-nonlinearity indicator with a mechanism-resolved, stiffness-predictive measure of the composite fatigue damage state. Extending the framework beyond the source's small laboratory coupons at fixed propagation distance to realistic structural components remains a further open direction.

References

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