Cantrell (2018) combined an ultrasonic distributed-spring model of adherend-adhesive interfaces with a physico-chemical model of molecular bonding, deriving a scaling equation that links the interfacial stiffness extracted from the phase of reflected ultrasonic tone-bursts to the fraction of intact interatomic bonds and thus, in principle, to the absolute tensile strength of the bond. The derivation was validated only in simulation, and the paper itself identified what remains unresolved: the inversion becomes ill-conditioned for the strongest bonds, requires material-specific molecular parameters whose uncertainties propagate unquantified, predominantly probes only the first interface encountered, and rests on untested modeling assumptions. The open problem is the experimental realization and full quantification of this inversion: validating the phase-to-stiffness-to-strength mapping on real bonded joints without destructive calibration, quantifying uncertainty propagation including the ill-conditioning of the inversion for the strongest bonds, characterizing both bonded interfaces, and assessing the robustness of the scaling relation under realistic variations in bond-length distribution and bonding chemistry. Later work has established empirical stiffness-strength correlations and inverse algorithms for interfacial stiffness, but no validated absolute-strength inversion, no uncertainty budget, and no both-interface protocol; a 2023 NASA Langley measurement-system analysis still states that no widely adopted NDE method measures interfacial bond strength.

Background

Adhesive bonding is the preferred method for joining lightweight composite and metal components in aerospace and automotive structures, but certification of bonded primary structure remains blocked by a long-standing nondestructive evaluation (NDE) gap: conventional ultrasonic inspection reliably detects gross disbonds (voids where the adhesive is separated from the adherend, the structural component being bonded), yet cannot measure how strong a bond is. The hardest case is the kissing bond — an interface in intimate mechanical contact that transfers little or no load because the molecular adhesion is weak — which is essentially invisible to standard pulse-echo testing. As a result, bond strength is currently assured by process control and destructive or semi-destructive proof testing rather than by inspection.

The physical basis for a quantitative ultrasonic approach is the distributed-spring interface model introduced by Tattersall (1973): an imperfect interface is represented as a layer of continuously distributed springs, characterized by an interfacial stiffness constant KK (stiffness per unit area, units N/m3^3) that ranges from zero for a fully fractured interface to infinity for perfect contact. When an ultrasonic wave (in Cantrell's work, a tone-burst, i.e. a short gated train of continuous waves at a single frequency) reflects from such an interface at normal incidence, both the amplitude and the phase γ\gamma of the reflection coefficient depend on KK, and the phase is by far the more sensitive observable. Phase can be measured with extraordinary precision using constant-frequency pulse phase-locked loop (CFPPLL) instrumentation, an ultrasonic technique that locks a transducer to a reference oscillator and tracks phase shifts to parts in 10910^9.

Cantrell had earlier developed a physico-chemical model of adhesive strength (2004, 2015) in which the interface consists of a finite number NN of discrete interatomic bonds drawn from a maximum possible number N0N_0 per unit area, each bond characterized by an interatomic force constant kk, a bond dissociation energy U0U_0, and an equilibrium separation r0r_0. The fraction of intact bonds is f=N/N0f = N/N_0, and the bond lengths are assumed to follow a Weibull distribution — a probability density commonly used for arrays of parallel brittle elements — with mean equal to the interface separation. The tensile strength follows by averaging the interatomic force over this distribution.

The 2018 work connects the two models. An ultrasonic inversion extracts KK from the measured reflection phase, and a scaling equation, Kab=f1−fN0kK_{ab} = \frac{f}{1-f}N_0 k, maps the continuum-model stiffness to the fraction of intact molecular bonds, so that the measured phase determines the absolute bond strength. A simulation on two aluminum adherends joined by an epoxy showed that the reflected phase is dominated by the first adherend-adhesive interface encountered by the tone-burst, changes sign at approximately K=3.5×1014K = 3.5\times10^{14} N/m3^3 (corresponding to roughly 10 kPa in the physico-chemical model, far below the 28 MPa maximum at N0k=9.6×1017N_0 k = 9.6\times10^{17} N/m3^3), and becomes decreasingly sensitive to KK as KK grows, since the derivative dγ/dK\mathrm{d}\gamma/\mathrm{d}K tends to zero as K→∞K \to \infty.

Follow-up experimental work by Haldren et al. (2019, 2022), using swept-frequency and constant-frequency phase measurements on joints whose cure was controlled by ultraviolet exposure, demonstrated that interfacial stiffness can indeed be extracted by inversion of the phase, and that destructively measured tensile strength correlates linearly with the extracted stiffness. What has not been achieved is the step that would make the method an absolute strength measurement: predicting strength from the molecular parameters through the scaling relation without calibrating against destructive tests, with quantified uncertainty, across the full strength range and both interfaces.

Problem Statement

Determine whether, and with what accuracy, the absolute tensile strength of an adhesive bond can be obtained nondestructively from the phase of ultrasonic tone-bursts reflected from the adherend-adhesive interfaces, by completing and validating the inversion chain proposed by Cantrell (2018): measured reflection phase γ\gamma, inverted to interfacial stiffness KK through the normal-incidence distributed-spring model, then mapped to the fraction of intact molecular bonds via the scaling Kab=f1−fN0kK_{ab} = \frac{f}{1-f}N_0 k and to tensile strength through the Weibull-averaged interatomic force. The chain is completed and validated jointly through: (i) experimental validation of the mapping on bonded specimens whose strengths are independently established by destructive testing, where the predicted strengths are computed from the molecular input parameters (bond density N0N_0, interatomic force constant kk, dissociation energy U0U_0, equilibrium distance r0r_0, Weibull parameter) without using those destructive tests to calibrate the prediction; (ii) quantification of how realistic uncertainties in the molecular parameters and in the phase measurement propagate to uncertainty in the inferred strength, including the ill-conditioning of the phase-to-stiffness inversion as K→∞K \to \infty that limits assessment of the strongest bonds; (iii) a measurement protocol that characterizes both the first and second adherend-adhesive interfaces; and (iv) assessment of the robustness of the scaling relation under realistic variations in bond-length distribution and bonding chemistry. The question is resolved by a determination of whether, and over what range of strengths, the inversion is reliable, including a negative determination.

References

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