Longitudinal wave attenuation in polycrystals with elongated grains: 3D numerical and analytical modeling.

This work develops a second-order approximation (SOA) model and a three-dimensional (3D) finite element (FE) model to calculate scattering-induced attenuation for elastic wave propagation in polycrystals with elongated grains of arbitrary crystal symmetry. The SOA model accounts for some degree of multiple scattering, whereas the 3D FE model includes all scattering possibilities. The SOA model incorporates the accurate geometric two-point correlation function obtained from the FE material systems to enable comparative studies between the two models. Also, the analytical Rayleigh and stochastic asymptotes are presented to provide explicit insights into propagation behaviors. Quantitative agreement is found between the FE and analytical models for all evaluated cases. In particular, the FE simulations support the SOA model prediction that grain shape does not exert influence on attenuation in the Rayleigh regime and its effect emerges as frequency increases to the stochastic regime showing anisotropy in attenuation. This attenuation anisotropy intensifies with the increase in frequency, but it exhibits a complicated behavior as frequency transits into the geometric regime. Wavefield fluctuations captured from the FE simulations are provided to help observe these complex scattering behaviors. The proportionality of attenuation to elastic scattering factors is also quantitatively evaluated.

Finite element evaluation of a simple model for elastic waves in strongly scattering elongated polycrystals.

A simple semi-analytical model for longitudinal scattering-induced attenuation and phase velocity is proposed for strongly scattering cubic polycrystals with statistically elongated grains. It is formulated by iterating the Born approximation of the far-field approximation model and by empirically increasing the coefficient in the quadratic term for the elastic scattering factor. The comparison with the three-dimensional grain-scale finite element calculations shows excellent performance of the semi-analytical model for both attenuation and phase velocity in all studied frequency ranges and especially in the Rayleigh regime in which, for strongly scattering materials, the existing analytical models significantly disagree with the numerical results.

Elastic wave velocity dispersion in polycrystals with elongated grains: Theoretical and numerical analysis.

The phase velocity dispersion of longitudinal waves in polycrystals with elongated grains of arbitrary crystallographic symmetry is studied in all frequency ranges by the theoretical second-order approximation (SOA) and numerical three-dimensional finite element (FE) models. The SOA and FE models are found to be in excellent agreement for three studied polycrystals: cubic Al, Inconel, and a triclinic material system. A simple Born approximation for the velocity, not containing the Cauchy integrals, and the explicit analytical quasi-static velocity limit (Rayleigh asymptote) are derived. As confirmed by the FE simulations, the velocity limit provides an accurate velocity estimate in the low-frequency regime where the phase velocity is nearly constant on frequency; however, it exhibits dependence on the propagation angle. As frequency increases, the phase velocity increases towards the stochastic regime and then, with further frequency increase, behaves differently depending on the propagation direction. It remains nearly constant for the wave propagation in the direction of the smaller ellipsoidal grain radius and decreases in the grain elongation direction. In the Rayleigh and stochastic frequency regimes, the directional velocity change shows proportionalities to the two elastic scattering factors even for the polycrystal with the triclinic grain symmetry.

Maximizing the accuracy of finite element simulation of elastic wave propagation in polycrystals.

Three-dimensional finite element (FE) modelling, with representation of materials at grain scale in realistic sample volumes, is capable of accurately describing elastic wave propagation and scattering within polycrystals. A broader and better future use of this FE method requires several important topics to be fully understood, and this work presents studies addressing this aim. The first topic concerns the determination of effective media parameters, namely, scattering induced attenuation and phase velocity, from measured coherent waves. This work evaluates two determination approaches, through-transmission and fitting, and it is found that these approaches are practically equivalent and can thus be used interchangeably. For the second topic of estimating modelling errors and uncertainties, this work performs thorough analytical and numerical studies to estimate those caused by both FE approximations and statistical considerations. It is demonstrated that the errors and uncertainties can be well suppressed by using a proper combination of modelling parameters. For the last topic of incorporating FE model information into theoretical models, this work presents elaborated investigations and shows that to improve agreement between the FE and theoretical models, the symmetry boundary conditions used in FE models need to be considered in the two-point correlation function, which is required by theoretical models.

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.

Numerical and analytic modelling of elastodynamic scattering within polycrystalline materials.

The elastodynamic behavior of polycrystalline cubic materials is studied through the fundamental propagation properties, the attenuation and wave speed, of a longitudinal wave. Predictions made by different analytical models are compared to both numerical and experimental results. The numerical model is based on a three-dimensional Finite Element (FE) simulation which provides a full-physics solution to the scattering problem. The three main analytical models include the Far-Field Approximation (FFA), the Self-Consistent Approximation (SCA) to the reference medium, and the herein derived Second Order Approximation (SOA). The classic Stanke and Kino model is also included, which by comparison to the SOA, reveals the importance of the distribution of length-scales described in terms of the two-point correlation function in determining scattering behavior. Further comparison with the FE model demonstrates that the FFA provides a simple but satisfactory approximation, whereas the SOA shows all-around excellent agreement. The experimental wave velocity data evaluated against the SOA and SC reveal a better agreement when the Voigt reference is used in second order models. The use of full-physics numerical simulations has enabled the study of wave behavior in these random media which will be important to inform the ongoing development of analytical models and the understanding of observations.

A multiband approach for accurate numerical simulation of frequency dependent ultrasonic wave propagation in the time domain.

Finite element (FE) simulations are popular for studying propagation and scattering of ultrasonic waves in nondestructive evaluation. For a large number of degrees of freedom, time domain FE simulations are much more efficient than the equivalent frequency domain solution. However, unlike frequency domain simulations, time domain simulations are often poor at representing the speed and the attenuation of waves if the material is strongly damping or highly dispersive. Here, the authors demonstrate efficient and accurate representation of propagated and scattered waves, achieved by combining a set of time domain solutions that are obtained for a set of frequency ranges known as bands, such that, in combination, the authors' multiband solution accurately represents the whole wave spectrum. Consequently, high accuracy is achieved, at minor computational cost, using a modest number of bands. The multiband technique is implemented for ultrasonic wave propagation in highly attenuating polyethylene material, using three frequency bands, and can yield a reduction in empirical acoustic properties fractional error compared with respective time domain simulations, in propagation duration, of a factor of 1.4, and in full-width-half-maximum, of a factor of 10. Last, the accuracy of this approach is further exemplified in a wave scattering simulation.

Ultrasonic attenuation and phase velocity of high-density polyethylene pipe material.

Knowledge of acoustic properties is crucial for ultrasonic or sonic imaging and signal detection in nondestructive evaluation (NDE), medical imaging, and seismology. Accurately and reliably obtaining these is particularly challenging for the NDE of high-density polyethylene (HDPE), such as is used in many water or gas pipes, because the properties vary greatly with frequency, temperature, direction and spatial location. Therefore the work reported here was undertaken in order to establish a basis for such a multiparameter description. The approach is general but the study specifically addresses HDPE and includes measured data values. Applicable to any such multiparameter acoustic properties dataset is a devised regression method that uses a neural network algorithm. This algorithm includes constraints to respect the Kramers-Kronig causality relationship between speed and attenuation of waves in a viscoelastic medium. These constrained acoustic properties are fully described in a multidimensional parameter space to vary with frequency, depth, temperature, and direction. The resulting uncertainties in acoustic properties dependence on the above variables are better than 4% and 2%, respectively, for attenuation and phase velocity and therefore can prevent major defect imaging errors.

Finite-element modelling of elastic wave propagation and scattering within heterogeneous media.

The scattering treated here arises when elastic waves propagate within a heterogeneous medium defined by random spatial fluctuation of its elastic properties. Whereas classical analytical studies are based on lower-order scattering assumptions, numerical methods conversely present no such limitations by inherently incorporating multiple scattering. Until now, studies have typically been limited to two or one dimension, however, owing to computational constraints. This article seizes recent advances to realize a finite-element formulation that solves the three-dimensional elastodynamic scattering problem. The developed methodology enables the fundamental behaviour of scattering in terms of attenuation and dispersion to be studied. In particular, the example of elastic waves propagating within polycrystalline materials is adopted, using Voronoi tessellations to randomly generate representative models. The numerically observed scattering is compared against entirely independent but well-established analytical scattering theory. The quantitative agreement is found to be excellent across previously unvisited scattering regimes; it is believed that this is the first quantitative validation of its kind which provides significant support towards the existence of the transitional scattering regime and facilitates future deployment of numerical methods for these problems.

The symmetry and coupling properties of solutions in general anisotropic multilayer waveguides.

Multilayered plate and shell structures play an important role in many engineering settings where, for instance, coated pipes are commonplace such as in the petrochemical, aerospace, and power generation industries. There are numerous demands, and indeed requirements, on nondestructive evaluation (NDE) to detect defects or to measure material properties using guided waves; to choose the most suitable inspection approach, it is essential to know the properties of the guided wave solutions for any given multilayered system and this requires dispersion curves computed reliably, robustly, and accurately. Here, the circumstances are elucidated, and possible layer combinations, under which guided wave solutions, in multilayered systems composed of generally anisotropic layers in flat and cylindrical geometries, have specific properties of coupling and parity; the partial wave decomposition of the wave field is utilised to unravel the behaviour. A classification into five families is introduced and the authors claim that this is the fundamental way to approach generally anisotropic waveguides. This coupling and parity provides information to be used in the design of more efficient and robust dispersion curve tracing algorithms. A critical benefit is that the analysis enables the separation of solutions into categories for which dispersion curves do not cross; this allows the curves to be calculated simply and without ambiguity.

A generic hybrid model for the simulation of three-dimensional bulk elastodynamics for use in non-destructive evaluation.

A three-dimensional generic hybrid model is developed for the simulation of elastic waves in applications in Non- Destructive Evaluation that efficiently links different solution strategies but, crucially, is independent of the particular schemes employed. This is an important step forward in facilitating rapid and accurate large-scale simulations and this advances the twodimensional generic hybrid methodology recently developed by the authors. The hybrid model provides an efficient and effective tool for creating highly accurate simulations that model the wave propagation and scattering, enabling the interpretation of inspection data; the new methodology is verified against other numerical simulations. Furthermore, its deployment to simulate wave reflection from side-drilled holes, comparing the results with experimental measurements, provides a realistic demonstration as well as further validation.

Modeling guided elastic waves in generally anisotropic media using a spectral collocation method.

Guided waves are now well established for some applications in the non-destructive evaluation of structures and offer potential for deployment in a vast array of other cases. For their development, it is important to have reliable and accurate information about the modes that propagate for particular waveguide structures. Essential information that informs choices of mode transducer, operating frequencies, and interpretation of signals, among other issues, is provided by the dispersion curves of different modes within various combinations of geometries and materials. In this paper a spectral collocation method is successfully used to handle the more complicated and realistic waveguide problems that are required in non-destructive evaluation; many pitfalls and limitations found in root-finding routines based on the partial wave method are overcome by using this approach. The general cases presented cover anisotropic homogeneous perfectly elastic materials in flat and cylindrical geometry. Non-destructive evaluation applications include complex waveguide structures, such as single or multi-layered fiber composites, lined, bonded and buried structures. For this reason, arbitrarily multi-layered systems with both solid and fluid layers are also addressed as well as the implementation of interface models of imperfect boundary conditions between layers.

A Stiffness Reduction Method for efficient absorption of waves at boundaries for use in commercial Finite Element codes.

Commercially available Finite Element packages are being used increasingly for modelling elastic wave propagation problems. Demand for improved capability has resulted in a drive to maximise the efficiency of the solver whilst maintaining a reliable solution. Modelling waves in unbound elastic media to high levels of accuracy presents a challenge for commercial packages, requiring the removal of unwanted reflections from model boundaries. For time domain explicit solvers, Absorbing Layers by Increasing Damping (ALID) have proven successful because they offer flexible application to modellers and, unlike the Perfectly Matched Layers (PMLs) approach, they are readily implemented in most commercial Finite Element software without requiring access to the source code. However, despite good overall performance, this technique requires the spatial model to extend significantly outside the domain of interest. Here, a Stiffness Reduction Method (SRM) has been developed that operates within a significantly reduced spatial domain. The technique is applied by altering the damping and stiffness matrices of the system, inducing decay of any incident wave. Absorbing region variables are expressed as a function of known model constants, helping to apply the technique to generic elastodynamic problems. The SRM has been shown to perform significantly better than ALID, with results confirmed by both numerical and analytical means.

Correction of ultrasonic array images to improve reflector sizing and location in inhomogeneous materials using a ray-tracing model.

The use of ultrasonic arrays has increased dramatically within recent years due to their ability to perform multiple types of inspection and to produce images of the structure through post-processing of received signals. Phased arrays offer many advantages over conventional transducers in the inspection of materials that are inhomogeneous with spatially varying anisotropic properties. In this paper, the arrays are focused on austenitic steel welds as a representative inhomogeneous material. The method of ray-tracing through a previously developed model of an inhomogeneous weld is shown, with particular emphasis on the difficulties presented by material inhomogeneity. The delay laws for the structure are computed and are used to perform synthetic focusing at the post-processing stage of signal data acquired by the array. It is demonstrated for a simulated austenitic weld that by taking material inhomogeneity and anisotropy into account, superior reflector location (and hence, superior sizing) results when compared to cases where these are ignored. The image is thus said to have been corrected. Typical images are produced from both analytical data in the frequency domain and data from finite element simulations in the time domain in a variety of wave modes, including cases with mode conversion and reflections.

Scattering of the fundamental torsional mode at an axial crack in a pipe.

A quantitative study of the interaction of the T(0,1) torsional mode with an axial defect in a pipe is presented. The results are obtained from finite element simulations and experiments. The influence of the crack axial extent, depth, excitation frequency, and pipe circumference on the scattering is examined. It is found that the reflection from a defect consists of a series of the wave pulses with gradually decaying amplitudes. Such behavior is caused by the shear waves diffracting from the crack and then repeatedly interacting with the crack due to circumferential propagation. Time-domain reflection coefficient analysis demonstrates that the trend of the reflection strength for different crack lengths, pipe diameters, and frequencies from a through-thickness crack satisfies a simple normalization. The results show that the reflection coefficient initially increases with the crack length at all frequencies but finally reaches an oscillating regime. Also, at a given frequency and crack length the reflection decreases with the increase in pipe circumference. An additional scattering study of the shear wave SH(0) mode at a part-thickness notch in a plate shows that the reflection coefficient, when plotted against depth of the notch, increases with both frequency and notch depth.

Scattering of the fundamental shear horizontal mode in a plate when incident at a through crack aligned in the propagation direction of the mode.

A study of the scattering of the fundamental guided wave SH(0) at a through-thickness narrow notch directed along the wave's propagation in a plate is presented. The results are obtained from Finite Element simulations and experimental measurements. Good agreement is found between the simulations and the measurements. The results are shown for a range of crack lengths and shapes. The scattered wave field consists of the reflected and diffracted SH(0) mode and also contributions from mode conversions to the S(0) mode. It is found that the coefficient of direct reflection of the SH(0) mode has an undulating nature depending on the length of the crack. This is caused by interference phenomena that are related to the interaction of different surface wave types generated on the crack surfaces and their diffractions at both tips of the crack. It is shown that the dominating part of this reflection is generated by the delayed "Rayleigh type" surface waves reflected from the far tip of the crack.

Scattering of the fundamental shear horizontal guided wave by a part-thickness crack in an isotropic plate.

The interaction of the fundamental shear horizontal (SH0) guided mode with part-thickness cracks in an isotropic plate is studied as an extension within the context and general framework of previous work ["Short range scattering of the fundamental shear horizontal guided wave mode normally incident at a through thickness crack in an isotropic plate," J. Acoust Soc. Am. 122, 1527-1538 (2007); "Angular influence on scattering when the fundamental shear horizontal guided wave mode is incident at a through-thickness crack in an isotropic plate," J. Acoust. Soc. Am. 124, 2021-2030 (2008)] by the authors with through-cracks. The symmetric incidence case where the principal direction of the incident beam bisects the crack face at 90 degrees is studied using finite element simulations validated by experiments and analysis, and conclusions are inferred for general incidence angles using insights obtained with the through-thickness studies. The influence of the crack length and the monitoring distance on the specular reflection is first examined, followed by a study of the angular profile of the reflected field. With each crack length considered, the crack depth and operating frequencies are varied. For all crack depths studied, the trend of the results is identical to that for the corresponding through-thickness case and the values differ only by a frequency dependent scale factor. Theoretical analysis is used to understand the physical basis for such behavior and estimates are suggested for the scale factor--exact for the high-frequency scattering regime and empirical for the medium- and low-frequency regimes.

Torsional waves propagation along a waveguide of arbitrary cross section immersed in a perfect fluid.

Guided torsional waves in a bar with a noncircular cross section have been exploited by previous researchers to measure the density of fluids. However, due to the complexity of the wave behavior in the noncircular cross-sectional shape, the previous theory can only provide an approximate prediction; thus the accuracy of the measurement has been compromised. In this paper, a semianalytical finite element method is developed to model accurately the propagation velocity and leakage of guided waves along an immersed waveguide with arbitrary noncircular cross section. An accurate inverse model is then provided to measure the density of the fluid by measuring the change of the torsional wave speed. Experimental results obtained with a rectangular bar in a range of fluids show very good agreement with the theoretical predictions. Finally, the potentials to use the model for sensor optimization are discussed.

Angular influence on the scattering of fundamental shear horizontal guided waves by a through-thickness crack in an isotropic plate.

The angular influence on the scattering of cylindrical-crested waves of the fundamental shear horizontal (SH0) guided mode by through-thickness cracks in an isotropic plate is studied in the context of array imaging using ultrasonic guided waves. Finite element simulations are used to obtain trends which are subject to analytical study and experimental confirmation. The influence of the incidence angle on reflection behavior is first studied in terms of two complementary cases, that of normal incidence and that of specular reflection at various oblique incidence angles. The normal incidence study suggests that for a given incidence angle, the peak reflection is concentrated around the specular direction, while the oblique incidence studies show that maximum specular reflection occurs in the case of normal incidence. The variation of diffraction with both the angle of incidence and that of monitoring is then taken up and this shows that when the first diffraction from the crack edges can be separated, its angular dependence can be obtained from literature on similar bulk elastic wave scattering problems.

Short range scattering of the fundamental shear horizontal guided wave mode normally incident at a through-thickness crack in an isotropic plate.

Interaction of the fundamental shear horizontal mode with through-thickness cracks in an isotropic plate is studied in the context of low frequency array imaging for ultrasonic guided wave nondestructive evaluation with improved resolution. Circular wave fronts are used and the symmetric case where a line from the wave source bisects the crack face normally is considered. Finite element simulations are employed to obtain trends subject to analytical and experimental validation. The influence of the crack length and of the location of source and measurement positions on the specular reflection from the crack face is first examined. These studies show that low frequency short range scattering is strongly affected by diffraction phenomena, leading to focusing of energy by the crack in the backscatter direction. Study of the diffraction from the crack edges reveals contributions due to a direct diffraction at the edges and multiple reverberations across the crack length. A simple diffraction model is shown to adequately represent cracks up to moderate lengths, providing an easy means of estimating the far field of the waves. The presence of multiple diffraction components is quantitatively established and surface waves on the crack face are identified as equivalent to low frequency symmetric modes of rectangular ridge waveguides.