Effect of texture and grain shape on ultrasonic backscattering in polycrystals

Scattering of elastic waves by a sphere with orthorhombic anisotropy and application to polycrystalline material characterization

Scattering of elastic waves by an anisotropic sphere with orthorhombic symmetry inside an isotropic medium is studied and applied to characterization of polycrystalline materials with anisotropic grains. For a single sphere the waves in the isotropic surrounding are expanded in the spherical vector wave functions. Inside the sphere, the elastodynamic equations are first transformed to spherical coordinates and the displacement field is expanded in terms of the vector spherical harmonics in the angular directions and power series in the radial direction. The governing equations inside the sphere give recursion relations among the expansion coefficients in the power series. The boundary conditions on the sphere then determine the relation among the scattered wave expansion coefficients and those of the incident wave, expressed as the transition (T) matrix. For low frequencies the elements of the T matrix are obtained in explicit form. According to the theory of Foldy the T matrix elements of a single sphere are used to study attenuation and phase velocity of polycrystalline materials, explicitly for low frequencies. Comparisons of the present method with previously published results and recent FEM results show a good correspondence for low frequencies. The present approach shows a better agreement with FEM for strongly anisotropic materials in comparison with other published methods.

Spatial and directional characterization of wire and arc additive manufactured aluminum alloy using phased array ultrasonic backscattering method

Pores, grains, or textures can collectively cause microstructural inhomogeneity and anisotropy in metallic materials fabricated by additive manufacturing. In this study, a phased array ultrasonic method is developed to characterize the inhomogeneity and anisotropy of wire and arc additively manufactured components by performing both beams focusing and steering. Two backscattering features, i.e., the integrated backscattering intensity and the root mean square of the backscattering signals, are employed to quantify the microstructural inhomogeneity and anisotropy, respectively. An experimental investigation is performed using an aluminum sample fabricated by wire and arc additive manufacturing. The ultrasonic measurements, performed on wire and arc additive manufactured 2319 aluminum alloy, show that the sample is inhomogeneous and weakly anisotropic. Metallography, electron backscatter diffraction, and X-ray computed tomography are used to verify the ultrasonic results. An ultrasonic scattering model is used to identify the influence of grains on the backscattering coefficient. Compared with a wrought aluminum alloy, the complex microstructure in additively manufactured material significantly influence the backscattering coefficient, and the presence of pores cannot be neglected in ultrasonic-based nondestructive evaluation for wire and arc additive manufactured metals.

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.

Review of ultrasonic measurement methods for two-phase flow

Two-phase flow is commonly used in many aspects of industrial production, such as the mixed transport of oil and gas in petroleum exploitation and the feeding of coal powder or coal water slurry to coal-fired boilers. In these situations, it is necessary to measure the two-phase flow in real time and then adjust various parameters in order to achieve high efficiency, energy-saving, and safe production. The ultrasonic method is widely used to measure two-phase flow because of its various measurement approaches, wide range of measurable parameters, insignificant effect on the flow field, and its capacity for continuous online measurement. In this Review, the principles, characteristics, application scope, and research examples of different ultrasonic methods used in two-phase flow measurement are summarized, their advantages and disadvantages are compared, and the future development trends are forecast, which will play a positive role in the development of two-phase flow measurement.

Inversion methodology for ultrasonic characterization of polycrystals with clusters of preferentially oriented grains

Titanium alloys are widely used in the aerospace industry. However, due to presence of microtexture, which is characterized by preferred crystallographic orientation clustering of thousands of alpha crystallites, cold dwell fatigue may significantly reduce the part life. To satisfy the practical need for nondestructive microtexture characterization, an inverse ultrasonic methodology is proposed to quantify mean parameters of microtexture regions (MTRs) having ellipsoidal shapes. One limitation of previous model-based ultrasonic inversion methods is required knowledge of elastic constants of the crystallites, which are rarely available for engineering alloys. This study overcomes this constraint by adopting the far field attenuation model, JASA, 137 (5), 2655-2669 (2015), and the backscattering model for ultrasonic wave interaction with microtexture. In the methodology developed, all necessary averaged MTR characteristics are obtained solely from directional ultrasonic measurements (backscattering, attenuation, and velocity) without a prior knowledge of material microstructures or elastic properties of different material phases. The inversion method is illustrated by simulations. To support the inversion methodology, the mean MTR sizes, morphology, and elastic scattering factors are determined from the ultrasonic experiment on a Ti-6242 alloy sample. The inversion results are compared with destructive electron backscatter diffraction (EBSD) analysis from which the MTRs are segmented using a non-contiguous grouping criteria. Good agreement is found.

Transverse-to-transverse diffuse ultrasonic double scattering

Previously, a transverse-to-transverse single scattering model (T-T SSR) was developed for a pulse echo configuration, which may have limitations for strongly scattering materials. In this work, a transverse-to-transverse double scattering model (T-T DSR) is presented to model the transverse ultrasonic backscatter more accurately. First, the Wigner distribution of the transducer beam pattern is extended to a transverse wave. Next, the multiple scattering framework is followed to derive the transverse and longitudinal components of the second-order scattering. Then, a quasi-Monte Carlo (QMC) method is used with Graphics Processing Unit (GPU) acceleration to calculate numerical results of the final expression which contains a five-dimensional integral. The correlation length, the focal length of the transducer, and incident angle are used to investigate differences between the T-T DSR model and the T-T SSR model. Finally, a backscatter experiment is performed on two stainless steel specimens with different grain sizes to determine the respective correlation lengths. The results show that the T-T DSR model has better performance over the T-T SSR model for evaluating the grain size of these relatively strongly-scattering specimens.

Attenuation and Phase Velocity of Elastic Wave in Textured Polycrystals with Ellipsoidal Grains of Arbitrary Crystal Symmetry

This study extends the second-order attenuation (SOA) model for elastic waves in texture-free inhomogeneous cubic polycrystalline materials with equiaxed grains to textured polycrystals with ellipsoidal grains of arbitrary crystal symmetry. In term of this work, one can predict both the scattering-induced attenuation and phase velocity from Rayleigh region (wavelength >> scatter size) to geometric region (wavelength << scatter size) for an arbitrary incident wave mode (quasi-longitudinal, quasi-transverse fast or quasi-transverse slow mode) in a textured polycrystal and examine the impact of crystallographic texture on attenuation and phase velocity dispersion in the whole frequency range. The predicted attenuation results of this work also agree well with the literature on a textured stainless steel polycrystal. Furthermore, an analytical expression for quasi-static phase velocity at an arbitrary wave propagation direction in a textured polycrystal is derived from the SOA model, which can provide an alternative homogenization method for textured polycrystals based on scattering theory. Computational results using triclinic titanium polycrystals with Gaussian orientation distribution function (ODF) are also presented to demonstrate the texture effect on attenuation and phase velocity behaviors and evaluate the applicability and limitation of an existing analytical model based on the Born approximation for textured polycrystals. Finally, quasi-static phase velocities predicted by this work for a textured polycrystalline copper with generalized spherical harmonics form ODF are compared to available velocity bounds in the literature including Hashin–Shtrikman bounds, and a reasonable agreement is found between this work and the literature.

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.

Universal scaling of transverse wave attenuation in polycrystals

A single mode approximation for the transverse attenuation coefficient in polycrystals is obtained from the far field approximation model (Rokhlin et al., 2015). The model is applicable in all frequency ranges to polycrystals with ellipsoidal grains of triclinic symmetry and is shown to be in favorable agreement with other second order models. In the frame of this model, the transverse wave attenuation coefficient depends on a single elastic scattering factor only (it solely encompasses dependence on the crystallite elastic moduli and the elastic covariance). Therefore in this approximation the attenuation coefficient can be scaled with this factor (normalized), obtaining the universal (master) curve. The admissibility of scaling is supported by the use of the second order model (the type of Stanke-Kino) for a large number of material systems with different grain anisotropies. Within the second order model, the behavior of the scaled transverse wave attenuation coefficient versus frequency is nearly independent of the material system and is a function of the grain geometrical characteristics only. The scaling of the transverse wave attenuation coefficient is tested on the measurements for Ti alloy samples performed in this work and a large set of experimental data obtained for different material systems available in the literature. The results confirm the scaled coefficient independence of the material and good agreement between the data and the universal attenuation curve.

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.

Effect of microstructural elongation on backscattered field: Intensity measurement and multiple scattering estimation with a linear transducer array

The effect of microstructural elongation on ultrasonic backscattered fields was studied. Two methods for determining the elongation direction of macrozones in titanium alloys, using the anisotropic spatial coherence of the backscattered field, are presented. Both methods use a phased array attached on a rotative holder that records the array response matrix at several angles. Two titanium alloys were investigated: TA6V and Ti17. TA6V exhibited a strong macrozone elongation, whereas Ti17 macrozones were found equiaxial. The first method is based on the measurement of backscattered intensity in function of the probe angle relative to the macrozones elongation direction. An angular dependence of backscattered intensity is observed in presence of elongated scatterers, and their elongation direction is collinear with the probe direction corresponding to a minimal intensity. This variability is linked to both piezoelectric shape and the backscattered field spatial properties. The second method is based on the measurement of the relative proportion of single to multiple scattering in a diffusive media, using a simplified version of the single scattering filter developed in Aubry and Derode (2009). It allows the measurement of the level of multiple scattering: both titanium alloys exhibited strong multiple scattering. The elongation direction was determined as the direction of minimal multiple scattering. Furthermore, these results were confirmed by the measurement of the coherent backscattering cone on both samples.

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.

Mode-converted ultrasonic scattering in polycrystals with elongated grains

Elastic wave scattering is used to study polycrystalline media for a wide range of applications. Received signals, which include scattering from the randomly oriented grains comprising the polycrystal, contain information from which useful microstructural parameters may often be inferred. Recently, a mode-converted diffuse ultrasonic scattering model was developed for evaluating the scattered response of a transverse wave from an incident longitudinal wave in a polycrystalline medium containing equiaxed single-phase grains with cubic elastic symmetry. In this article, that theoretical mode-converted scattering model is modified to account for grain elongation within the sample. The model shows the dependence on scattering angle relative to the grain axis orientation. Experimental measurements were performed on a sample of 7475-T7351 aluminum using a pitch-catch transducer configuration. The results show that the mode-converted scattering can be used to determine the dimensions of the elongated grains. The average grain shape determined from the experimental measurements is compared with dimensions extracted from electron backscatter diffraction, an electron imaging technique. The results suggest that mode-converted diffuse ultrasonic scattering has the potential to quantify detailed information about grain microstructure.

Stress-dependent ultrasonic scattering in polycrystalline materials

Stress-dependent elastic moduli of polycrystalline materials are used in a statistically based model for the scattering of ultrasonic waves from randomly oriented grains that are members of a stressed polycrystal. The stress is assumed to be homogeneous and can be either residual or generated from external loads. The stress-dependent elastic properties are incorporated into the definition of the differential scattering cross-section, which defines how strongly an incident wave is scattered into various directions. Nine stress-dependent differential scattering cross-sections or scattering coefficients are defined to include all possibilities of incident and scattered waves, which can be either longitudinal or (two) transverse wave types. The evaluation of the scattering coefficients considers polycrystalline aluminum that is uniaxially stressed. An analysis of the influence of incident wave propagation direction, scattering direction, frequency, and grain size on the stress-dependency of the scattering coefficients follows. Scattering coefficients for aluminum indicate that ultrasonic scattering is much more sensitive to a uniaxial stress than ultrasonic phase velocities. By developing the stress-dependent scattering properties of polycrystals, the influence of acoustoelasticity on the amplitudes of waves propagating in stressed polycrystalline materials can be better understood. This work supports the ongoing development of a technique for monitoring and measuring stresses in metallic materials.

Stress-dependent second-order grain statistics of polycrystals

In this article, the second-order statistics of the elastic moduli of randomly oriented grains in a polycrystal are derived for the case when an initial stress is present. The initial stress can be either residual stress or stresses generated from external loading. The initial stress is shown to increase or decrease the variability of the grain's elastic moduli from the average elastic moduli of the polycrystal. This variation in the elastic properties of the individual grains causes acoustic scattering phenomenon in polycrystalline materials to become stress-dependent. The influence of the initial stress on scattering is shown to be greater than the influence on acoustic phase velocities, which defines the acoustoelastic effect. This work helps the development of scattering based tools for the nondestructive analysis of material stresses in polycrystals.

Far-field scattering model for wave propagation in random media

A simple approximate model is developed for ultrasonic wave propagation in a random elastic medium. The model includes second order multiple scattering and is applicable in all frequency ranges including geometric. It is based on the far field approximation of the reference medium Green's function and simplifications of the mass operator in addition to those of the first smooth approximation. In this approximation, the dispersion equation for the perturbed wave number is obtained; its solution yields the dispersive ultrasonic velocity and attenuation coefficients. The approximate solution is general and is suitable for nonequiaxed grains with arbitrary elastic symmetry. For equiaxed cubic grains, the solution is compared with the existing second order models and with the Born approximation. The comparison shows that the obtained solution has smaller error than the Born approximation and shows reasonably well the onset of multiple scattering and the applicability limit of the Born approximation at high frequency. The perturbed wave number in the developed model does not depend explicitly on the crystallite elastic properties even for arbitrary crystallographic symmetry; it depends on two nondimensional scattering elastic parameters and the macroscopic ultrasonic velocity (those are dependent on the crystallite moduli). This provides an advantage for potential schemes for inversion from attenuation to material microstructure.

Contribution of double scattering in diffuse ultrasonic backscatter measurements

Diffuse ultrasonic backscatter measurements are used to describe the effective grain scattering present during high frequency ultrasonic inspections. Accurate modeling of the backscatter is important for both flaw detection and microstructural characterization. Previous models have been derived under the assumption of single scattering for which the ultrasound is assumed to scatter only once in the time between excitation and detection. This assumption has been shown to be valid in many experiments for which the time scales are short or the frequency is sufficiently low. However, there are also many instances (e.g., for strongly scattering materials, unfocused beams, or long propagation paths) for which the single scattering assumption appears to break down. In this article, a model for the double scatter is developed within the previous formalism based on Wigner distribution functions. The final expression allows the effect of double scattering to be estimated for any combination of experimental parameters. The improved proposed model is anticipated to increase the capabilities of ultrasonic microstructural evaluation, especially in terms of probability of detection estimates.