Modeling and simulation of ultrasonic beam skewing in polycrystalline materials
14 May 2018-International Journal of Advances in Engineering Sciences and Applied Mathematics (Springer India)-Vol. 10, Iss: 1, pp 70-78
TL;DR: In this paper, a Voronoi tessellation algorithm is used to represent an equiaxed polycrystalline morphology and numerical simulations are performed on beam skewing in both weak (Aluminum) and strong (Copper) anisotropic media as a function of beam launch angles.
Abstract: Ultrasonic wave propagation through polycrystalline media results in scattering caused by the anisotropy of single grains and randomness in the orientation of the individual grains making up the polycrystal. Scattering leads to variation in phase velocity and beam skewing of elastic waves leading to a loss in energy of the forward propagating wave, significantly affecting the ability to perform material characterization, defect detection and sizing in structural components. The present work addresses the problem of beam skewing of ultrasonic longitudinal waves using FEM-based wave propagation studies in a simulated polycrystal. A well-established Voronoi tessellation algorithm is used to represent an equiaxed polycrystalline morphology. Numerical simulations are performed on beam skewing in both weak (Aluminum) and strong (Copper) anisotropic media as a function of beam launch angles. The effect of a small number of large grains and a large number of small grains on the beam quality is described. The effective refraction in polycrystals is quantified with respect to the corresponding reference isotropic media and the implications for various applications are discussed.
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TL;DR: In this paper, the Voronoi geometry of a polycrystalline object is modeled using tessellations and the orientation of the material's lattice structure is reconstructed using a non-linear least square method.
Abstract: In this paper, we study the inverse problem of recovering the spatially varying material properties of a solid polycrystalline object from ultrasonic travel time measurements taken between pairs of points lying on the domain boundary. We consider a medium of constant density in which the orientation of the material's lattice structure varies in a piecewise constant manner, generating locally anisotropic regions in which the wave speed varies according to the incident wave direction and the material's known slowness curve. This particular problem is inspired by current challenges faced by the ultrasonic non-destructive testing of polycrystalline solids. We model the geometry of the material using Voronoi tessellations and study two simplified inverse problems where we ignore wave refraction. In the first problem, the Voronoi geometry itself and the orientations associated to each region are unknowns. We solve this nonsmooth, nonconvex optimisation problem using a multistart non-linear least squares method. Good reconstructions are achieved, but the method is shown to be sensitive to the addition of noise. The second problem considers the reconstruction of the orientations on a fixed square mesh. This is a smooth optimisation problem but with a much larger number of degrees of freedom. We prove that the orientations can be determined uniquely given enough boundary measurements and provide a numerical method that is more stable with respect to the addition of noise.
5 citations
TL;DR: In this paper , the authors used Laguerre tessellations generated by random sphere packings dividing space into convex polytopes, where cells represent grains in a real polycrystal and act as single scatterers.
Abstract: Ultrasonic testing of polycrystalline media relies heavily on simulation of the expected signals in order to detect and correctly interpret deviations due to defects. Many effects disturb ultrasonic waves propagating in polycrystalline media. One of them is scattering due to the granular microstructure of the polycrystal. The thus arising so-called microstructural noise changes with grain size distribution and testing frequency. Here, a method for simulating this noise is introduced. We geometrically model the granular microstructure to determine its influence on the backscattered ultrasonic signal. To this end, we utilize Laguerre tessellations generated by random sphere packings dividing space into convex polytopes—the cells. The cells represent grains in a real polycrystal. Cells are characterized by their volume and act as single scatterers. We compute scattering coefficients cellwise by the Born approximation. We then combine the Generalized Point Source Superposition technique with the backscattered contributions resulting from the cell structure to compute the backscattered ultrasonic signal. Applying this new methodology, we compute the backscattered signals in a pulse-echo experiment for a coarse grain cubic crystallized Inconel-617 and a fine grain hexagonal crystallized titanium. Fitting random Laguerre tessellations to the observed grain structure allows for simulating within multiple realizations of the proposed model and thus to study the variation of the backscattered signal due to microstructural variation.
References
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Book•
01 Jan 1973
TL;DR: In this article, the authors apply the material developed in the Volume One to various boundary value problems (reflection and refraction at plane surfaces, composite media, waveguides and resonators).
Abstract: This work, part of a two-volume set, applies the material developed in the Volume One to various boundary value problems (reflection and refraction at plane surfaces, composite media, waveguides and resonators). The text also covers topics such as perturbation and variational methods.
5,211 citations
2,121 citations
TL;DR: In this paper, a unified approach to solve for the attenuation and phase velocity variations of elastic waves in single phase, polycrystalline media due to scattering is presented. But the approach is not applicable for any material whose singlecrystal anisotropy is not large, regardless of texture, grain elongation, or multiple scattering.
Abstract: We have developed a unified approach to solve for the attenuation and phase velocity variations of elastic waves in single‐phase, polycrystalline media due to scattering. Our approach is a perturbation method applicable for any material whose single‐crystal anisotropy is not large, regardless of texture, grain elongation, or multiple scattering. It accurately accounts for the anisotropy of the individual grains. It is valid for time‐harmonic waves with all ratios of grain size to wavelength. It uses an autocorrelation function to characterize the geometry of the grains, and thereby avoids coherent artifacts that occur if the grains are assumed to have symmetrical shapes and suggests new methods for characterizing distributions of grains that are irregularly shaped. We have carried out numerical calculations for materials that are untextured and equiaxed, and have cubic‐symmetry grains and an inverse exponential spatial autocorrelation function. These calculations agree with the previous calculations which are valid in the Rayleigh, stochastic, and geometric regions, and show the transitions between these regions. The complex transition between the Rayleigh and stochastic regions for longitudinal waves, and the severe limitations of the stochastic region for grains with fairly large anisotropy are of particular interest.
354 citations
TL;DR: In this article, the diffusivity of ultrasound in an untextured aggregate of cubic crystallites is studied theoretically with a view towards nondestructive characterization of microstructures, and the covariance is found to obey an equation of radiative transfer for which a diffusion limit is taken.
Abstract: T he diffusivity of ultrasound in an untextured aggregate of cubic crystallites is studied theoretically with a view towards nondestructive characterization of microstructures. Multiple scattering formalisms for the mean Green's dyadic and for the covariance of the Green's dyadic (and therefore for the energy density) based upon the method of smoothing are presented. The first-order smoothing approximation used is accurate to leading order in the anisotropy of the constituent crystallites. A further, Born, approximation is invoked which limits the validity of the calculation to frequencies below the geometrical optics regime. Known result for the mean field attenuations are recovered. The covariance is found to obey an equation of radiative transfer for which a diffusion limit is taken. The resulting diffusivity is found to vary inversely with the fourth power of frequency in the Rayleigh, long wavelength, regime and inversely with the logarithm of frequency on the short wavelength, stochastic, asymptote. The results are found to fit the experimental data.
291 citations
TL;DR: In this paper, the elastic properties of copper have been compiled and reviewed, including Young's modulus, the shear modulus and the bulk modulus of copper, and a few theoretical numbers are included.
Abstract: The elastic properties of copper have been compiled and reviewed. Polycrystalline elastic constants included are: Young's modulus, the shear modulus, the bulk modulus, and Poisson's ratio. Single‐crystal constants of second‐, third‐, and fourth‐order are included. Over 200 references to the experimental literature are given. A few theoretical numbers are included. When sufficient data exist, best values are recommended together with their standard errors. Effects on the elastic constants of temperature, pressure, and mechanical (plastic) deformation are included. The Cauchy (central‐force) relationships and the single‐crystal—polycrystal relationship are also discussed.
211 citations