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 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.
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.
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 (FFA) 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 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 quasistatic velocities obtained from the SOA and FE models are in excellent agreement with the static self-consistent velocity.
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 they 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.
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