An open problem in the non-destructive evaluation of fibre-reinforced laminate composites: establishing a validated quantitative inversion that maps locally measured ultrasonic shear-wave birefringence parameters — the attenuation-coefficient difference between warp and weft directions and the shear-moduli ratio — to microstructural damage descriptors such as matrix-crack density, crack size, and crack orientation distribution. Rakotonarivo, Payan, Moysan and Hochard (Composites Part B: Engineering 142, 2018, 287-292) demonstrated a single-transducer, contact, pulse-echo birefringence inversion that locally qualifies damage in one unbalanced woven glass/epoxy laminate, but the method stays qualitative: it cannot distinguish crack density from crack size, it has not been transferred to other laminate architectures, ply orientations, or damage states, and contact coupling (transducer pressure) perturbs amplitude-based attenuation estimates. Later work has produced only system-specific empirical correlations or continuum-level damage identification, so the general parameter-to-morphology mapping remains unresolved.

Background

Fibre-reinforced laminate composites accumulate damage as matrix micro-cracks, fibre-matrix debonds, and delaminations before final rupture, and this damage degrades stiffness and increases ultrasonic attenuation in a direction-dependent (anisotropic) way. A shear (transverse) ultrasonic wave propagating through the thickness of an anisotropic plate splits into two quasi-shear waves polarized along the principal anisotropy axes, which travel with different velocities and attenuation; this splitting is called shear-wave birefringence, the ultrasonic analogue of optical birefringence. For a woven glass/epoxy laminate probed through the thickness, the two relevant shear moduli are G13G_{13} and G23G_{23} (the shear moduli coupling the through-thickness direction 3 with the warp direction 1 and the weft direction 2, respectively), and the measurable birefringence quantities are the shear-moduli ratio Rg=G23/G13R_g = G_{23}/G_{13} and the attenuation-coefficient difference Δk\Delta k between waves polarized along warp and weft. Rakotonarivo, Payan, Moysan and Hochard (2018) built on earlier birefringence work of Rheinfurth, Fey, Allinger and Busse (International Journal of Fatigue 48, 2013, 80-89), who used birefringence as a correlation-level measure of fatigue damage in laminated composites. The 2018 method uses one piezoelectric contact transducer in pulse-echo mode (the same transducer emits the pulse and records the reflected echo), rotated over several polarization angles; a ply-by-ply propagator-matrix forward model (each ply is represented by a matrix that propagates the wave through it) is fitted to the full frequency content of the reflected signals by minimizing a normalized cost function, simultaneously estimating the average ply misalignment, RgR_g, and Δk\Delta k, with no reference measurement in a calibrated medium such as water. On pristine versus damaged specimens of the unbalanced woven glass/epoxy laminate (an unbalanced weave has different warp and weft construction, so the plate is in-plane anisotropic even when pristine) it found RgR_g rising from about 1.1 to about 1.16-1.2 with damage, and Δk\Delta k decaying from about 75 Np/m/MHz near the rupture zone to about −0.9-0.9 Np/m/MHz away from it (Np denotes the neper, the natural-logarithmic unit of amplitude ratio), with stable results over 10 repetitions on 10 different days. Related quantitative-ultrasonics results frame what is and is not known: Sayers (Ultrasonics 26, 1988, 73-77) inverted ultrasonic velocity data to recover the microcrack orientation distribution function in rocks (orientation only, not density or size); Hayford and Henneke (1979) modeled the correlation between damage and ultrasonic attenuation in composites; Olsen-Kettle (Composite Structures, 2018) identified the orthotropic continuum damage tensor from ultrasonic stiffness measurements, which quantifies a continuum damage variable rather than discrete crack descriptors. Independent measurements of matrix-crack density also exist in the cited literature, obtained by means other than birefringence: acousto-ultrasonic crack counting in carbon-fibre-reinforced polymer (Wang, Wu, Xiong, Zhang and Okabe, 2021, reference 7) and X-ray micro-computed-tomography calibration in filament-wound composite fuel tanks (Kim, Lee, Cho and Lee, 2024, reference 9). The 2018 study itself is explicit that its assessment is qualitative 'at a macroscopic scale': the variation of Δk\Delta k with position is attributed only 'probably' to 'larger density or larger size of micro-cracks', the density-versus-size ambiguity is not resolved, and the authors note that the pressure applied on the contact transducer can strongly affect measured amplitude and hence attenuation even though their tests were reproducible.

Problem Statement

Establish a quantitative and experimentally validated mapping between the ultrasonic shear-wave birefringence parameters measured locally in a laminate composite plate — the attenuation-coefficient difference Δk\Delta k between waves polarized along the warp and weft directions and the shear-moduli ratio Rg=G23/G13R_g = G_{23}/G_{13}, as extracted by contact pulse-echo birefringence inversion — and the microstructural damage morphology of the same material volume, namely matrix-crack density and crack size, with crack-orientation distribution an optional secondary aspect of the mapping. The source work leaves this mapping qualitative: it cannot distinguish crack density from crack size, attributing the variation of Δk\Delta k with position only 'probably' to 'larger density or larger size of micro-cracks', so a quantitative mapping must resolve or explicitly quantify this density-versus-size non-uniqueness. The source work also documents the limitations that frame the difficulty of validating any such mapping: damage was assessed against proximity to a rupture zone rather than against independent ground truth on the inspected volume, the method was demonstrated on a single unbalanced woven glass/epoxy laminate and not transferred to other laminate architectures, ply orientations, or damage states, and contact coupling (transducer pressure) perturbs amplitude-based attenuation estimates. This quantitative mapping must resolve or explicitly quantify the density-versus-size non-uniqueness: the crack-density and crack-size values it returns must be fixed by evidence independent of the Δk\Delta k and RgR_g trends themselves, and must also be consistent with the measured Δk\Delta k and RgR_g versus damage-state data reported in the source work (reference 1) and the related birefringence studies (references 2 and 3). Reproducing the reported Δk\Delta k and RgR_g trends alone does not by itself determine the absolute crack-density or crack-size values. Transfer to other laminate architectures, ply orientations, and damage states, and robustness of the attenuation-based part of the mapping to contact-coupling (transducer-pressure) variability, are context that frames the difficulty of the mapping rather than parts of the mapping itself.

References

  1. S. T. Rakotonarivo, C. Payan, J. Moysan, C. Hochard, Local damage evaluation of a laminate composite plate using ultrasonic birefringence of shear wave, Composites Part B: Engineering 142 (2018) 287-292, DOI 10.1016/j.compositesb.2018.01.006, https://doi.org/10.1016/j.compositesb.2018.01.006
  2. M. Rheinfurth, P. Fey, S. Allinger, G. Busse, Ultrasonic birefringence as a measure of mechanically induced fatigue damage in laminated composites, International Journal of Fatigue 48 (2013) 80-89, DOI 10.1016/j.ijfatigue.2012.10.005, https://doi.org/10.1016/j.ijfatigue.2012.10.005
  3. P. Fey, M. Kreutzbruck, Quantification of fatigue state in CFRP using ultrasonic birefringence, AIP Conference Proceedings 1706, 050007 (2016), DOI 10.1063/1.4940506, https://doi.org/10.1063/1.4940506
  4. C. M. Sayers, Inversion of ultrasonic wave velocity measurements to obtain the microcrack orientation distribution function in rocks, Ultrasonics 26 (1988) 73-77, DOI 10.1016/0041-624x(88)90057-1, https://doi.org/10.1016/0041-624X(88)90057-1
  5. D. T. Hayford, E. G. Henneke, A Model for Correlating Damage and Ultrasonic Attenuation in Composites, in Fifth Conference on Composite Materials: Testing and Design, ASTM Special Technical Publication 674, ASTM International (1979), DOI 10.1520/stp36910s, https://doi.org/10.1520/stp36910s
  6. L. Olsen-Kettle, Quantifying the orthotropic damage tensor for composites undergoing damage-induced anisotropy using ultrasonic investigations, Composite Structures 204 (2018) 701-711, DOI 10.1016/j.compstruct.2018.07.096, https://doi.org/10.1016/j.compstruct.2018.07.096
  7. R. Wang, Q. Wu, K. Xiong, H. Zhang, Y. Okabe, Evaluation of the matrix crack number in carbon fiber reinforced plastics using linear and nonlinear acousto-ultrasonic detections, Composite Structures 255 (2021) 112962, DOI 10.1016/j.compstruct.2020.112962, https://doi.org/10.1016/j.compstruct.2020.112962
  8. R. Czichos, Y. Bernhardt, J. Dittmann, P. Middendorf, M. Kreutzbruck, Experimental and numerical investigations on the micro-damage behaviour of glass fibre-reinforced plastics, Composites Part C: Open Access 10 (2023) 100341, DOI 10.1016/j.jcomc.2022.100341, https://doi.org/10.1016/j.jcomc.2022.100341
  9. K. Kim, J. Lee, Y. Cho, J. Lee, Structural health monitoring of type 4 composite fuel tank based on correlation between ultrasonic attenuation and crack density, Advanced Composite Materials 34 (4) (2025) 707-724, published online 4 September 2024, DOI 10.1080/09243046.2024.2398289, https://doi.org/10.1080/09243046.2024.2398289