Mode-converted ultrasonic scattering in polycrystals with elongated grains

Average grain size evaluation using scattering-induced attenuation of coda waves

Grain size is one of the key microstructural factors affecting the mechanical properties of polycrystalline metal materials. In this study, a novel method for grain size evaluation using ultrasonic coda waves is proposed. Different from conventional bulk wave methods that require a point-by-point scanning of the structure, the proposed method allows for a rapid evaluation of the average grain size of the whole part from a single inspection location using one-pass testing data. A piecewise energy attenuation function dealing with different attenuation mechanisms is proposed to obtain the effective attenuation coefficient of coda waves. A power-law model is constructed to correlate the effective attenuation coefficient with the average grain size. Ultrasonic testing on nickel-based superalloy plate specimens with different average grain sizes is performed for model calibration and method verification. The applicability and robustness of the proposed method are further validated using a realistic turbine disk specimen with an irregular shape and non-uniform grain sizes. Results show that the proposed method yields a reliable and accurate estimation of the average grain size with a maximum relative error less than 20 %.

Ultrasonic backscattering model for Rayleigh waves in polycrystals with Born and independent scattering approximations

This paper presents theoretical and numerical models for the backscattering of 2D Rayleigh waves in single-phase, untextured polycrystalline materials with statistically equiaxed grains. The theoretical model, based on our prior inclusion-induced Rayleigh wave scattering model and the independent scattering approximation, considers single scattering of Rayleigh-to-Rayleigh (R-R) waves. The numerical finite element model is established to accurately simulate the scattering problem and evaluate the theoretical model. Good quantitative agreement is observed between the theoretical model and the finite element results, especially for weakly scattering materials. The agreement decreases with the increase of the anisotropy index, owing to the reduced applicability of the Born approximation. However, the agreement remains generally good when weak multiple scattering is involved. In addition, the R-R backscattering behaviour of 2D Rayleigh waves is similar to the longitudinal-to-longitudinal and transverse-to-transverse backscattering of bulk waves, with the former exhibiting stronger scattering. These findings establish a foundation for using Rayleigh waves in the quantitative characterisation of polycrystalline materials.

Focal depth localization for highly focused transducers in isotropic materials

Focusing equations aim to define the point in a solid at which a transducer beam will reach a minimum cross section. The most commonly used focusing equation relies on a small angle assumption that inherently excludes sharply focused transducers with significant curvature. In this article, a revised focusing equation is proposed for normal and oblique incidence through a fluid-solid interface. The closed-form equation is derived using ray tracing approaches similar to the conventional expression but circumvents the paraxial approximation, extending the applicability to sharply focused probes. Both conventional and modified focusing equations are compared through normal and oblique incidence ray diagrams, and the proximity to the computationally derived geometric focus is explored. The proposed modification to the focusing equation generally results in a closer approximation to the geometric focus, a smaller beam cross section, and a greater time convergence when compared to the conventional focusing equation.

Characterization of microstructural anisotropy using the mode-converted ultrasonic scattering in titanium alloy

The mode-converted (Longitudinal to Transverse, L-T) ultrasonic scattering was utilized to characterize the microstructural anisotropy on three surfaces of samples cut from the low-scattering and high-scattering regions of a raw titanium alloy Ti-6Al-4V billet, respectively. The L-T ultrasonic measurements were performed in two perpendicular directions using two focused transducers with a 15 MHz center frequency in a pitch-catch configuration. The root mean square (RMS) of ultrasonic scattering was calculated for each L-T measurement and a Gaussian function was used to fit each RMS to determine the RMS amplitude. The ratio of RMS amplitudes for L-T measurements performed in two perpendicular directions was calculated to characterize the microstructural anisotropy on the measured surface of a sample. The results show that the amplitude of L-T ultrasonic scattering is highly dependent on the microstructural anisotropy. The microstructural isotropy was considered on the x-y planes of all samples, while the high anisotropy was seen on the x-z and y-z planes of all low-scattering and high-scattering samples. In addition, the microstructural anisotropy measured on the x-z planes of the low-scattering and high-scattering samples gradually increases and decreases, respectively, from the outside diameter (OD) to the centerline (CL) of the billet. The anisotropy measured on the y-z planes of the low-scattering samples slightly decreases and then increases towards the center, while the anisotropy measured on the y-z planes of the high-scattering samples continuously increases towards the center. The variation of microstructural anisotropy in the titanium alloy Ti-6Al-4V billet with duplex microstructure was quantified with the L-T ultrasonic method and the results agree well with micrographs shown in Ref. [18]. The mode-converted ultrasonic scattering method provides a NDE method to characterize microstructural anisotropy, which can be used as an NDE tool for quality control.

Evaluating elongated grains with diffuse ultrasonic double scattering and rectangular transducers

Diffuse scattering of ultrasound by the microstructure of polycrystal specimens can be used to evaluate grain size and grain elongation. The existing diffuse scattering models mostly dealt with circular transducers whose symmetrical sound field is insensitive to the asymmetric elongated grain. The sound field of a rectangular transducer provides a new perspective for acquiring additional information. First, the existing single scattering response (SSR) and double scattering response (DSR) models are modified for a rectangular transducer, where the sound field of a rectangular transducer is equivalent to that of an elliptical transducer in the far-field. Therefore, an equivalent single Gaussian beam model is derived using amplitude-equivalent and beamwidth-equivalent coefficients. Then, the spatial correlation function of elongated grains is transformed into the wavenumber domain, giving rise to the SSR and DSR of a rectangular transducer that reveals the interaction effect of an asymmetric sound field and elongated grains on ultrasonic backscattering. The experimental results show that the sizes of elongated grains in a cold-rolled aluminum are evaluated as 1086 ± 8, 90 ± 4, and 10 ± 1 μm in the x, y, and z directions, where the exact values are 1184.2 ± 11.9, 80.7 ± 5.2, and 8.3 ± 0.5 μm according to metallographic measurements.

In situ characterization of laser-generated melt pools using synchronized ultrasound and high-speed X-ray imaging

Metal additive manufacturing is a fabrication method that forms a part by fusing layers of powder to one another. An energy source, such as a laser, is commonly used to heat the metal powder sufficiently to cause a molten pool to form, which is known as the melt pool. The melt pool can exist in the conduction or the keyhole mode where the material begins to rapidly evaporate. The interaction between the laser and the material is physically complex and difficult to predict or measure. In this article, high-speed X-ray imaging was combined with immersion ultrasound to obtain synchronized measurements of stationary laser-generated melt pools. Furthermore, two-dimensional and three-dimensional finite-element simulations were conducted to help explain the ultrasonic response in the experiments. In particular, the time-of-flight and amplitude in pulse-echo configuration were observed to have a linear relationship to the depth of the melt pool. These results are promising for the use of ultrasound to characterize the melt pool behavior and for finite-element simulations to aid in interpretation.

Ultrasonic mapping of hybrid additively manufactured 420 stainless steel

Metal hybrid additive manufacturing (AM) processes are suitable to create complex structures that advance engineering performance. Hybrid AM can be used to create functionally graded materials for which the variation in microstructure and material properties across the domain is created through a synergized combination of fully-coupled manufacturing processes and/or energy sources. This expansion in the engineering design and manufacturing spaces presents challenges for nondestructive evaluation, including the assessment of the sensitivity of nondestructive measurements to functional gradients. To address this problem, linear ultrasound measurements are used to interrogate 420 stainless steel coupons from three manufacturing methods: wrought, AM, and hybrid AM (directed energy deposition + laser peening). Wave speed, attenuation, and diffuse backscatter results are compared with microhardness measurements along the build/axial direction of the coupons, while microstructure images are used for qualitative verification. The ultrasound measurements compare well with the destructive measurements without any substantial loss in resolution. Furthermore, ultrasonic methods are shown to be effective for identification of the gradient and cyclic nature of the elastic properties and microstructure on the hybrid AM coupon. These results highlight the potential of ultrasound as an efficient and accessible nondestructive characterization method for hybrid AM samples and inform further nondestructive evaluation decisions in AM.

Evaluation of Structural Anisotropy in a Porous Titanium Medium Mimicking Trabecular Bone Structure Using Mode-Converted Ultrasonic Scattering

The mode-converted (longitudinal to transverse, L-T) ultrasonic scattering method was utilized to characterize the structural anisotropy of a phantom mimicking the structural properties of trabecular bone. The sample was fabricated using metal additive manufacturing from high-resolution computed tomography (CT) images of a sample of trabecular horse bone with strong anisotropy. Two focused transducers were used to perform the L-T ultrasonic measurements. A normal incidence transducer was used to transmit longitudinal ultrasonic waves into the sample, while the scattered transverse signals were received by an oblique incidence transducer. At multiple locations on the sample, four L-T measurements were performed by collecting ultrasonic scattering from four directions. The amplitude of the root mean square (rms) of the collected ultrasonic scattering signals was calculated for each L-T measurement. The ratios of rms amplitudes for L-T measurements in different directions were calculated to characterize the anisotropy of sample. The results show that the amplitude of L-T converted scattering is highly dependent on the direction of microstructural anisotropy. A strong anisotropy of the microstructure was observed, which coincides with simulation results previously published on the same structure as well as with the anisotropy estimated from the CT images. These results suggest the potential of mode-converted ultrasonic scattering methods to assess the anisotropy of materials with porous, complex structures, including trabecular bone.

Attenuation and velocity of elastic waves in polycrystals with generally anisotropic grains: Analytic and numerical modeling

Better understanding of elastic wave propagation in polycrystals has interest for applications in seismology and nondestructive material characterization. In this study, a second-order wave propagation (SOA) model that considers forward multiple scattering events is developed for macroscopically isotropic polycrystals with equiaxed grains of arbitrary anisotropy (triclinic). It predicts scattering-induced wave attenuation and dispersion of phase velocity. The SOA model implements the generalized two-point correlation (TPC) function, which relates to the actual numeric TPC of simulated microstructure. The analytical Rayleigh and stochastic asymptotes for both attenuation and phase velocity are derived for triclinic symmetry grains, which elucidate the effects of the elastic scattering factors and the generalized TPC in different frequency regimes. Also, the computationally efficient far field approximation attenuation model is obtained for this case; it shows good agreement with the SOA model in all frequency ranges. To assess the analytical models, a three-dimensional (3D) finite element (FE) model for triclinic polycrystals is developed and implemented on simulated 3D triclinic polycrystalline aggregates. Quantitative agreement is observed between the analytical and the FE simulations for both the attenuation and phase velocity. Also, the quasi-static velocities obtained from the SOA and FE models are in excellent agreement with the static self-consistent velocity.

Generalized ultrasonic scattering model for arbitrary transducer configurations

Ultrasonic scattering in polycrystalline media is directly tied to microstructural features. As a result, modeling efforts of scattering from microstructure have been abundant. The inclusion of beam modeling for the ultrasonic transducers greatly simplified the ability to perform quantitative, fully calibrated experiments. In this article, a theoretical scattering model is generalized to allow for arbitrary source and receiver configurations, while accounting for beam behavior through the total propagation path. This extension elucidates the importance and potential of out-of-plane scattering modes in the context of microstructure characterization. The scattering coefficient is explicitly written for the case of statistical isotropy and ellipsoidal grain elongation, with a direct path toward expansion for increased microstructural complexity. Materials with crystallites of any symmetry can be studied with the present model; the numerical results focus on aluminum, titanium, and iron. The amplitude of the scattering response is seen to vary across materials, and to have varying sensitivity to grain elongation and orientation depending on the transducer configuration selected. The model provides a pathway to experimental characterization of microstructure with optimized sensitivity to parameters of interest.

Enhanced ultrasonic detection of near-surface flaws using transverse-wave backscatter

Diffuse ultrasonic backscatter measurements have been shown to enhance the detection capability of sub-wavelength flaws when combined with extreme value statistics. However, for a normal-incidence immersion measurement, a "dead zone" created by the ring-down of the front-wall echo will hide near-surface flaws. In this article, a pulse-echo transverse wave backscatter measurement is used to detect near-surface flaws under high gain. The approach is validated using a magnesium specimen with side-drilled holes. The confidence bounds of the grain noise from this specimen are given by a transverse-to-transverse scattering model, which takes the grain size distribution and the hexagonal crystal symmetry into account. The upper bound is then treated as a time-dependent threshold for the C-scan. Experiments show that the developed method has good performance for detecting sub-wavelength, near-surface flaws, and can suppress both missed detections and false positives.

Investigation of ultrasonic backscatter using three-dimensional finite element simulations

Theoretical models are commonly used to describe ultrasonic backscattering in polycrystalline materials. However, although a full multiple scattering formalism has been derived, due to the difficulty in evaluation, currently only the single and double scattering effects have been evaluated. Three-dimensional finite element (3D FE) models have recently been demonstrated to be capable of predicting ultrasonic attenuation in polycrystalline materials and thereby show great potential in overcoming this limitation. In this paper, the application of 3D FE models is extended to the backscatter problem. First, longitudinal-to-longitudinal backscattering amplitudes from single grains are predicted, where the setup and configuration of the finite element (FE) model are verified with an isotropic spherical inclusion for which an exact solution is available. Subsequently, backscatter in terms of the root-mean-square noise levels in two different pulse-echo scenarios is investigated; the first is an idealised configuration with plane wave transmission and point reception; the second represents a more realistic finite-size transducer acting with the same apodization in both transmission and reception. Comparisons of FE predictions and approximate theoretical solutions within a range of validity show good agreement; however, the results demonstrate that 3D FE is useful where the simple Independent Scatterer models break down. As computing power increases, 3D FE is an increasingly viable tool to further the understanding of wave propagation in polycrystalline materials.

Flaw detection with ultrasonic backscatter signal envelopes

Ultrasound is a prominent nondestructive testing modality for the detection, localization, and sizing of defects in engineering materials. Often, inspectors analyze ultrasonic waveforms to determine if echoes, which stem from the scattering of ultrasound from a defect, exceed a threshold value. In turn, the initial selection of the threshold value is critical. In this letter, a time-dependent threshold or upper bound for the signal envelope is developed based on the statistics governing the scattering of ultrasound from microstructure. The utility of the time-dependent threshold is demonstrated using experiments conducted on sub-wavelength artificial defects. The results are shown to enhance current nondestructive inspection practices.

Enhanced Ultrasonic Flaw Detection using an Ultra-high Gain and Time-dependent Threshold

In an attempt to improve the ultrasonic testing capability of a conventional C-scan system, a flaw detection method using an ultra-high gain is developed in this paper. A time-dependent threshold for image segmentation is applied to identify automatically material anomalies present in the sample. A singly-scattered response (SSR) model is used with extreme value statistics to calculate the confidence bounds of grain noise. The result is a time-dependent threshold associated with the grain noise that can be used for segmentation. Ultrasonic imaging experiments show that the presented method has advantages over a traditional fixed threshold approach with respect to false positives and missed flaws. The results also show that a low gain is adverse to the detection of micro-flaws with subwavelength dimensions. The forward model is expected to serve as an effective tool for the probability of detection (POD) of flaws and the inspection of coarse-grained materials in the future.

Ultrasonic backscatter from elongated grains using line focused ultrasound

Ultrasonic backscattering from polycrystalline materials with elongated grains is investigated. A normal incident line-focus transducer is employed such that refracted longitudinal and transverse waves are focused within the polycrystal and scatter at grain boundaries back to the transducer. A ray-based scattering model is developed to explain the dependence of the statistics of scattering measurements on grain elongation. The spatial variance of measured scattered signals from Al alloy (7475-T7) is compared to the model. This work promotes the ultrasonic backscatter technique for monitoring grain elongation of metals using one transducer with access to a single sample face.

Transverse-to-transverse diffuse ultrasonic scattering

Ultrasonic scattering occurs when elastic waves interact with interfaces within heterogeneous media. Diffuse ultrasonic backscatter measurements are used to capture the effective grain scattering within a polycrystal for extracting microstructural information. Recently, a mode-conversion scattering model was developed to describe the longitudinal-to-transverse ultrasonic scattering within polycrystalline materials and successfully applied to determine the material spatial correlation length L by fitting experimental results with the theoretical model. The mode-conversion model may allow additional microstructural information, such as grain shape, to be assessed. In this article, a theoretical extension of the previous mode-conversion ultrasonic scattering model is presented. The transverse-to-transverse (T-T) scattering can be measured by an experimental configuration with both source and receiving transducers oriented at angles between the first and second critical angles, including pitch-catch and pulse-echo measurements. The model is used to determine the correlation length from a sample of 1040 steel through pulse-echo T-T scattering measurements using 7.5 and 10 MHz transducers. The results show that the derived T-T model works well for lower frequencies but the results for higher frequencies reveal deficiencies in the model.

Iterative solution to bulk wave propagation in polycrystalline materials

This article reevaluates two foundational models for bulk ultrasonic wave propagation in polycrystals. A decoupling of real and imaginary parts of the effective wave number permits a simple iterative method to obtain longitudinal and shear wave attenuation constants and phase velocity relations. The zeroth-order solution is that of Weaver [J. Mech. Phys. Solids 38, 55-86 (1990)]. Continued iteration converges to the unified theory solution of Stanke and Kino [J. Acoust. Soc. Am. 75, 665-681 (1984)]. The converged solution is valid for all frequencies. The iterative method mitigates the need to solve a nonlinear, complex-valued system of equations, which makes the models more robust and accessible to researchers. An analysis of the variation between the solutions is conducted and is shown to be proportional to the degree of inhomogeneity in the polycrystal.