Two obliquely intersecting shear waves in a solid mix through elastic nonlinearity and generate a longitudinal wave at the sum frequency; when the interaction occurs at an imperfect interface (a partially contacting, weakly bonded 'kissing' bond), the interface adds its own nonlinearity, providing a sensitive probe of bond quality. Existing analytical models (Zhang, Nagy and Hassan, Ultrasonics 2016) treat the interface as a vanishing-thickness layer with quadratic elastic interfacial stiffness or, equivalently, a thin nonlinear interphase layer, and were verified only against finite-element simulations that required artificially high linear interfacial stiffness and assumed infinitely wide plane waves. The open problem is to extend the analytical and numerical modeling of non-collinear shear wave mixing at imperfect interfaces to realistic non-classical interfacial nonlinearity — the class of contact acoustic nonlinearity, characterized by friction, sticking and hysteresis and by rough-surface nonlinear contact mechanics — including finite beam and diffraction effects, and to validate quantitatively the predicted angular dependence and reflection-mode advantage on realistic bond imperfections such as graded diffusion bonds.

Background

When two monochromatic ultrasonic waves cross at an angle in a solid whose stress-strain relation is nonlinear, they interact and generate a scattered wave at the sum (or difference) frequency; this is called non-collinear wave mixing. For two vertically polarized shear (SV) waves in an isotropic solid, the mixing produces a longitudinal (pressure) wave at the sum frequency, with an amplitude proportional to the material's third-order elastic constants. Croxford, Wilcox, Drinkwater and Nagy (JASA 2009) introduced this method for nonlinear ultrasonic detection of plasticity and fatigue damage, exploiting the fact that the mixed frequency component is absent from the incident beams and therefore largely free of instrumental (system) nonlinearity.

A kissing bond is an imperfect adhesive or diffusion bond whose faces are in intimate mechanical contact but not properly joined: it is nearly transparent to conventional (linear) ultrasound, because its linear reflection is tiny, yet it is more compliant in tension than in compression. Such partially contacting interfaces exhibit contact acoustic nonlinearity (CAN): under an acoustic wave, the interface periodically opens and closes ('clapping'), and its tangential response involves frictional slipping, sticking, and hysteresis, so its stiffness depends on the stress amplitude and loading history rather than being a classical elastic constant. A standard mathematical idealization of an imperfect interface is a distribution of displacement discontinuity proportional to traction (an interfacial stiffness or compliance model), which represents an imperfect interface of vanishing thickness; a thin interphase layer model replaces the interface by a thin isotropic layer and homogenizes it in the quasi-static approximation.

Zhang, Nagy and Hassan (Ultrasonics 65 (2016) 165–176) developed two analytical models of non-collinear shear wave mixing at such an imperfect interface: a finite nonlinear interfacial stiffness model and its derivative, a thin nonlinear interphase layer model. Both assume classical quadratic elastic interfacial nonlinearity (the leading, first-order nonlinear approximation), and their predictions for the amplitude and direction of the reflected and transmitted mixed longitudinal waves were verified against COMSOL finite-element (FE) simulations. The models predict that the interface generates equal-amplitude nonlinear signals in reflection and transmission, so that detecting the reflected mixed wave removes the contaminating contribution of the surrounding material's own (bulk) nonlinearity — a reflection-mode advantage — and that misaligning the transducers from the bulk resonance condition suppresses the bulk signal, giving a characteristic angular dependence of the interface signature. These features make the technique attractive for grading bond quality in safety-critical aerospace joints, where linear methods lose sensitivity precisely for the best bonds.

The authors themselves listed the limitations that define the open problem. The quadratic elastic interfacial law is only a first-order approximation for tightly closed kissing bonds, whose real behavior is dominated by non-classical contact nonlinearity — friction, sticking, and hysteresis — which could modify the generation of the mixed longitudinal wave; the models neglect finite beam widths and diffraction, and the FE verification required artificially high (but finite) linear interfacial stiffnesses to keep the linear perturbation small. Precedents exist on both sides: Gusev (JASA 2002) showed theoretically that hysteretic quadratic nonlinearity changes the resonant selection rules of non-collinear interactions in bulk materials, and Blanloeuil, Croxford and Méziane (AIP Conf. Proc. 2014) demonstrated non-collinear mixing at a frictional contact interface experimentally and by FE with a unilateral contact law with Coulomb friction — but no analytical imperfect-interface mixing model of the Zhang–Nagy–Hassan type includes such non-classical interfacial laws.

A body of subsequent work addresses parts of this landscape without closing the gap. Rose, Blanloeuil, Veidt and Wang (J. Sound Vib. 468, 115078, 2020) derived analytical scattered mixed-wave amplitudes at a plane contact interface, but their traction law retains a quadratic normal response and an explicitly linear tangential response, so friction, sticking and hysteresis fall outside it, and their guided-wave companion (Blanloeuil et al., Ultrasonics 110, 106222, 2021) represents clapping and Coulomb sliding only by finite elements. Wang, Nagy, Carius, Kim and Jacobs (JASA 158, 3210–3225, 2025) extend the nonlinear-stiffness framework to five mixing modalities but keep the classical quadratic interfacial law, and Darmon and Yajioui (Acta Acustica 10, 70, 2026) model unilateral-contact friction for harmonic generation rather than non-collinear mixing. The finite-beam treatment of Rouwhorst, Gitman, Loendersloot and Yntema (NDT&E Int. 163, 103801, 2026) concerns bounded bulk media rather than the imperfect interface, and the experimental interface characterization of Carius, Nagy, Wang, Kim and Jacobs (NDT&E Int. 162, 103742, 2026) does not yet establish the full predicted angular dependence on graded bonds. No cited model combines a non-classical interfacial traction law with the Zhang–Nagy–Hassan non-collinear mixing framework, interface-specific finite-beam diffraction, and an explanation of the random reflected phase, so the problem stated above remains open.

Problem Statement

Develop a quantitatively predictive and experimentally validated description of non-collinear shear wave mixing at an imperfect interface beyond the classical quadratic elastic approximation. Concretely: (1) extend the analytical modeling framework of nonlinear interfacial stiffness (or the equivalent thin nonlinear interphase layer) so that the interfacial traction law captures non-classical contact acoustic nonlinearity — the class of behaviors characterized by unilateral contact with frictional slipping and sticking, hysteresis, and rough-surface nonlinear contact mechanics, not a conjunctive checklist of separately required mechanisms — and predict from it the amplitude, phase, frequency content, and scattering direction of the reflected and transmitted mixed longitudinal waves generated by two non-collinear incident shear waves, including how the resonant/phase-matching selection rules change relative to the quadratic case; (2) incorporate finite beam widths and diffraction into the interface-mixing prediction, removing the idealization of infinitely wide plane waves; (3) verify the extended model numerically (e.g., finite elements with explicit contact laws) without recourse to artificially high linear interfacial stiffness; and (4) validate, against experimental data on realistic bond imperfections such as diffusion-bonded or adhesively bonded specimens of graded quality — whether newly measured or drawn from published measurements — the predicted angular dependence of the interface signature and the reflection-mode advantage over transmission detection, including an explanation of the essentially random phase of the reflected nonlinear signature reported for high-quality diffusion bonds by Zhang, Nagy and Hassan's companion study (AIP Conf. Proc. 1706, 2016) and their thesis, which the quadratic model cannot capture. These components jointly serve one objective: a physically faithful, experimentally confirmed link between the interfacial contact state and the measured non-collinear mixing signature, as required for reliable nonlinear ultrasonic grading of kissing bonds. All four components are jointly posed and together constitute a single objective.

References

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