Shape effect of elongated grains on ultrasonic attenuation in polycrystalline materials

Abstract

Longitudinal and transverse wave attenuation coefficients are obtained in a simple integral form for ultrasonic waves in cubic polycrystalline materials with elongated grains. Dependences of attenuation on frequency and grain shape are described in detail. The explicit analytical solutions for ellipsoidal grains in the Rayleigh and stochastic frequency limits are given for a wave propagating in an arbitrary direction relative to ellipsoid axes. The attenuation exhibits classic frequency dependence in those frequency limits. However, the dependence on the grain shape in the stochastic limits is unexpected: it is independent of the cross-section of the ellipsoidal grains and depends only on the grain dimension in the propagation direction. In the Rayleigh region attenuation is proportional to effective volume of the ellipsoidal grain and is independent of its shape. A complex behavior of attenuation on the grain shape/size and frequency is exhibited in the transition region. The results obtained reduce to the classic dependences of attenuation on parameters for polycrystals with equiaxed grains.

Introduction

It is well known that ultrasonic attenuation is determined by different absorption mechanisms and by scattering on material inhomogeneities. In polycrystalline materials the attenuation is dominated by scatter on grain boundaries, due to relative misorientation of the crystallites, and crystalline absorption can be neglected. In such materials attenuation depends strongly on microstructure: elastic anisotropy of grains, their size and shape and material texture [1]. Experimental studies indicate that ultrasonic measurements in the MHz range can provide useful information on material microstructure/microtexture [1] that results from industrial material processing and can be related to their mechanical properties.

To relate measured ultrasonic attenuation to material microstructure robust, quantitative theoretical models are needed and thus development of such models may have significant practical importance. For this reason several fundamental theoretical studies have been devoted to describe scattering-induced attenuation of ultrasonic waves in polycrystals. Lifshits and Parkhomovski [2] have developed a general stochastic anisotropic model which is suitable for a wide frequency range. Hirsekorn [3], [4] has described a scattering model using the Born series which has accounted for multiple scattering and obtained both wave attenuation and velocity. Stanke and Kino [5] extended the Keller approximation to an anisotropic case and by combining it with the Lifshits and Parkhomovski [2] approach have provided a unified theory which is suitable in all frequency ranges. They did not use the Born approximation, as Lifshits and Parkhomovski [2] did, and have derived final results applicable for computation for a cubic equiaxial untextured polycrystalline medium.

Weaver [6] in his work on diffusivity of ultrasound in untextured cubic-symmetry polycrystals has obtained a general solution using the Dyson equation to account for multiple scattering. To simplify the general solution he has employed the Born approximation and has obtained explicit equations for attenuation which are identical to those of the Born approximation of Stanke and Kino [5]. The final results are suitable for the whole frequency range below the geometrical limit. Both the Stanke and Kino [5] and Weaver [6] models considered a cubic equiaxial untextured polycrystalline medium (i.e. uncorrelated crystallographic orientation of crystallites). For textured materials with equiaxial grains Hirsekorn [7] has extended her models [3], [4]. The wave propagation and scattering in polycrystalline materials with texture was considered by Ahmed and Thompson [8], who have modified the unified model [5] and by Turner [9] who has extended approach [6] by the use of the Green’s function for anisotropic materials and has obtained the attenuation model in Born approximation.

All those models are valid for materials with equiaxed grains. Ahmed and Thompson [10] have extended Stanke and Kino’s model [5] to a polycrystalline medium with elongated grains and have obtained longitudinal attenuations by numerically evaluating an integral solution for the Green’s function. They have provided computational results and calculated the attenuation as a function of different microstructural parameters; however, their model is numerical in essence and is difficult for general interpretation of results and analysis.

In this paper, the Weaver [6] model is employed for a polycrystalline medium with elongated grains of ellipsoidal shape of cubic symmetry. As a material system we assume that ellipsoidal grains are aligned in a preferred direction due to prior material thermomechanical processing history; however, crystallographic orientations of different ellipsoidal grains are random and are not correlated and the medium is elastically isotropic macroscopically (the model assumptions are further discussed in Section 2.1). The spatial autocorrelation function introduced by Ahmed and Thompson [10] and also applied to describe backscattering [11], [12] is used in this work to characterize the shapes of ellipsoidal grains. A simple analytical equation for the three-dimensional Fourier transform of this autocorrelation function reported in [12] is utilized to obtain a general solution for attenuation coefficients in a form suitable for effective computation. Explicit expressions are obtained from the general theory for attenuation coefficients in the low frequency (Rayleigh) and stochastic limits. Those solutions are generalizations of the classical Rayleigh and stochastic asymptotes; they allow us to elucidate and obtain new insights on the dependence of the attenuation on different parameters. In particular, in the Rayleigh limit the attenuation is independent of the ellipsoid shape and determined by its volume; however, above the Rayleigh regime the attenuation is dominated by the ellipsoid grain size in the direction of wave propagation and is independent of the ellipsoidal cross-section. This last result is counterintuitive and cannot be discerned from the equiaxed grain model. The shape effect of ellipsoidal grains is further simulated in the entire frequency range below geometrical limit. Also, the relation to the backscattering coefficient in polycrystalline medium with elongated grains is obtained.

Our results are in the Born approximation. As it was noted in [5], [6], this approximation is defined differently from that usually referenced as the Born approximation. Briefly [5], [6], in this approximation the term $(k\overline{d}{)}^2-(k_0\overline{d}{)}^2$ is replaced by $2(k_0\overline{d})[(k\overline{d})-(k_0\overline{d})]$, where k is a wavenumber in polycrystalline medium and k₀ is an “unperturbed” wavenumber and some approximations are made to solve nonlinear dispersion equation regarding k; $\overline{d}$ is an average grain size. As a result the approximation is applicable when the perturbation in wavenumber of the propagating wave is small and is not valid in the geometrical acoustics range. The effect of this approximation has been estimated by Stanke and Kino [5] who showed that both solutions are nearly identical for low-anisotropy crystals such as Al, and have about 12% difference in the Rayleigh regime for high-anisotropy crystals such as iron (this is a relatively small difference when compared with experiment due to variability in grain size and other experimental factors). Thus the results in the Born approximation are valid for most practical applications.